CUREe-KAJIMA RESEARCH PROJECT

SEISMIC RESPONSE OF UNDERGROUND . STRUCTURES IN SOFT SOILS

CYLINDRICAL SHAFTS IN DRY SAND

NUMERICAL SIMULATIONS OF CENTRIFUGE TESTS

R. F. Scott California Institute of Technology

January 15, 1993

I

Table of Contents

Chapter Page

1. Introduction 1

2. Selected Experimental Results 3

3. Numerical Models 6

4. System Identification 10

5. Comparison and Results 17

5.1 Engineering Model 17

5.1.1 Low Level Test 72B 18

5.1.2 High Level Test 73P 23

5.2 SAP90Model 26

5.2.1 Tests 72B and 73P 27

5.2.2 Test 83 (short tube) 29

5.3 ABAQUS Model 29

5.4 DYSAC2 Model 31

6. Other Models 32

7. Conclusions and Recommendations 32

8. Acknowledgments 34

9. References 35

Tables

Appendices

i

LIST OF FIGURES

Figure No. Title

2.1 Kajima test set up 2.2 Spectra El Centro, Tests 72B, 73P 2.3 Some Kajima test data 2.4 Kajima transfer functions (corrected) 2.5 Project transforms for Kajima tests 2.6 ACC04 vs. ACC12 measured

3.1 Engineering model 3.2 ABAQUS model 3.3 SAP90 model 3.4 DYSAC2 model

4.1 Vertical vs. horizontal acceleration plot for ACC09 vs. ACC01 4.2 Location of center of box rotation

5.1 Uncorrected input acceleration record 72B 5.2 Velocity 72B 5.3 Displacement 72B 5.4 Comparison of EM and ACC12; first 5.12 sees. 5.5 Comparison of EM and ACC12; first 21 sees. 5.6 Comparison of EM and EP5 5.7 Comparison of EM and EP7 5.8 Comparison of EM and ST3 5.9 Comparison of EM and ST5 5.10 Uncorrected input acceleration record 73P 5.11 Velocity 73P 5.12 Displacement 73P 5.13 Comparison of EM and ACC12 5.14 Comparison of EM and EP5 5.15 Comparison of EM and EP7 5.16 Comparison of EM and ST3 5.17 Comparison of EM and ST5 5.18 EM with 3 mass, 10 mass and ACC12 5.19 EM with 3 mass and 10 mass alone 5.20 72B SAP90 vs. ACC12 5 sees. 5.21 72B SAP90 vs. ACC12 21 sees. 5.22 73P SAP90 vs. ACC12 5.23 73P SAP90 vs. EP7 5.24 73P SAP90 vs. ST5 element 107 5.25 73P SAP90 vs. ST5 element 108 5.26 83 SAP90 vs. ACC12

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5.27 ABAQUS at ACC04, node 6 5.28 ABAQUS at ACC12, node 36 5.29 ABAQUS at ACC07, node 406 5.30 ABAQUS at ACC15, node 436 5.31 ABAQUS displacement, node 6 5.32 DYSAC2 at ACC02 5.33 DYSAC2 at ACC04 5.34 DYSAC2 at ACC05 5.35 DYSAC2 at ACC06 5.36 DYSAC2 at ACC07 5.37 DYSAC2 at ACC12 5.38 DYSAC2 at ACC13 5.39 DYSAC2 at ACC14 5.40 DYSAC2 at ACC15

iii

TABLES

1. Kajima centrifuge tests

2. Best-fit engineering model properties

3. Table of properties in SAP90 model in Test 73P

4. Properties employed in ABAQUS model

5. ABAQUS model frequencies

iv

Appendix

1.

2.

3.

4.

APPENDICES

Subject

Toyoura sand properties

Engineering model program and explanation

Tabulation of computer codes

Additional SAP90 results

v

CUREe-Kajima Project

Cylindrical Shaft in Dry Sand

Numerical Simulations of Centrifuge Tests

R. F. Scott California Institute of Technology

1. INTRODUCTION

It had been originally proposed in January 1991 to meet Kajima's requirements for studies

on cylindrical shafts embedded in soft ground that Caltech perform centrifuge tests on an instru­

mented model shaft and follow these with a limited number of numerical evaluations of the test

data. However, when Kajima examined the proposals for the current fiscal year, they decided

that in view of the fact that their own centrifuge was close to operating, they would prefer to

carry out the centrifuge tests themselves, and give a contract to Caltech to provide numerical

simulations of the test data and to evaluate these simulations for correspondence with the actual

cylinder.

The cylinder was to be embedded in dry sand in a box mounted on the centrifuge and

operated at 50 g. The box would be subjected to a horizontal base motion simulating the north­

south component of the El Centro 1940 earthquake. The shaft was to be made of aluminum and

constructed so that the bending properties of the shaft (EI) simulated a full-scale reinforced

concrete shaft 6 meters in diameter. It was to be instrumented with strain gauges, pressure trans­

ducers, and accelerometers; accelerometers were also to be placed in the sand at the base, inside

the soil mass and at the soil surface to record the motions of the model during the input test.

The tests were carried out in the period July to October 1992 and the data were

communicated by Dr. Honda of Kajima at a meeting at Caltech with R. F. Scott and

B. Hushmand. During this period also, work was performed at Caltech on examining numerical

models which would be used to represent the test data when they became available. An

"engineering model" was constructed composed of a number of masses, springs and dashpots to

represent the basic characteristics of the cylinder in sand problem, and a variety of finite element

codes was selected from those available, either on the commercial market or from universities, in

order to simulate the test motion. The finite element models consisted of- ABAQUS, a

commercially available program which has the ability to incorporate nonlinear soil behavior, the

commercial program SAP90, which, in the form employed, has no nonlinear capability but can be

used for plane strain equivalent linearized versions of the test. The third finite element program

used was one named DYSAC2 which has been developed at the University of California at Davis

and is a two-dimensional but fully nonlinear finite element code incorporating a soil constitutive

model of the bounding surface type developed by Dafalias and Herrmann.

Computations have been performed with all of these models, with the engineering model

being used for parametric studies to achieve the best match with the experimental data. The

three-dimensional ABAQUS model presented many numerical difficulties in representing the

dynamic soil behavior; they seem to be attributable to defects in the original program. Even

linear solutions required long computational times. The SAP90 model was used for simulating a

number of tests and proved an economical way of approaching the modeling, while the DYSAC2

representation also involved a great deal of computer time, and it was only possible to study the

model behavior during a limited duration of input motion for both the low and high level earth­

quakes applied to the deep foundation.

In the course of the numerical investigations, it also appeared that the centrifuge tests

carried out by Kajima included a number of problems due to the design of the shaking system

incorporated on the Kajima centrifuge. In that system the application of the shaking force to the

shaking table lies at a level considerably below the center of mass of the system, so that the

movement imparted to the sand box is not a purely translational horizontal input acceleration, as

desired, but involves a pitching motion, so that vertical out-of-phase accelerations are recorded

at both ends of the box at its base as well as some vertical motion at the center of the box,

superimposed on the generally horizontal acceleration developed. This made the experimental

motions very complicated, and made numerical simulations difficult. One consequence of this

was that only the simple engineering model could incorporate a simplified version of this pitching

motion, since the finite element codes in general require the same motion to be input at all base

points at the same time. It was not possible, in the time available, to attempt adaptations of the

finite element systems to account for the pitching. This might be possible given more time.

However, the result was that there were differences between the purely horizontal measured

input motion applied to some of the numerical models and the actual pitching motion imparted to

the sand box in the centrifuge tests as a result of that horizontal motion. Kajima adjusted the

transfer functions obtained from the experimental data to remove the effect of pitching, but the

actual histories of various transducers presented were those obtained in the -tests without

correction.

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2. SELECTED EXPERIMENTAL RESULTS

Many tests were performed by Kajima in their centrifuge on the box full of sand, both

with and without model shafts of two different lengths. The shafts and experimental arrangement

are shown in Figure 2.1; the tests were performed on aluminum cylinders 6 em in diameter and

30 em in length (long cylinder) and 9 em (short cylinder). Both cylinders possessed the same

cross section, with aluminum wall thickness of 2 mm. The cylinders were instrumented with

accelerometers, strain gauges, and pressure transducers, in order to record the deflections of the

cylinder and moments induced in it as a result of the shaking motion. Accelerometers were also

deployed in the sand. In both cylinders the base was closed. The soil employed in the tests was

fine Toyoura sand, with an average grain size of about 0.1 mm. The sand is described in

Appendix 1 of this report, which contains data received from Kajima on the properties of the

material.

The test configuration consisted of a rigid box containing the sand and the cylinder to be

tested along with the other instrumentation. Because the Kajima Corporation was concerned

about side and boundary effects on the sand, they decided to install at each end of the box an

approximately 2 em thick layer of soft silicone rubber, and, at the side walls, they separated the

sand from the wall by a plastic membrane and lubricant, with the idea that the soil would be free

to slide back and forward against the side wall as the test was being conducted. Thus, in effect,

if the cylinder had not been present, the soil would have been subjected to conditions close to

those of plane strain. However, under the circumstances it is not clear that the lubricant would

respond fast enough Jo the input accelerations to which the soil was subjected to give sufficiently

low side shearing stresses. Since it is viscous, the shearing stress, 't, in the lubricant is a function

of the shear strain rate as follows

't = Jl d<l> or Jl dv dt dz

(1)

where Jl is the lubricant viscosity, <1> is shear strain, t time, v velocity and z distance perpendicular

to the shearing direction. In a lubricant layer only a few hundredths of a millimeter in thickness,

the shear strain rate would be high in the tests, and, depending on the lubricant used, this could

give high shearing stresses. More information is needed to evaluate the lubricant effect.

However, the test results as presented do not include tests with sand alone in the box,

and sand with the modified side and end conditions; it is impossible to tell from the actual test

data whether the inclusion of the silicone rubber and the lubricated side walls made a substantial

3

difference to the test results. In the data seen so far, the demonstration of the efficacy of these

measures is only given by means of a computer calculation of the effect of including assumed

properties for these materials. In addition, not enough information has been given on how the

computer was able to incorporate the viscous lubrication at the side wall boundaries. There is

another point: the silicone rubber, which was included apparently because of its softness, has a

Poisson's ratio of almost one half, (that is to say, the material is close to incompressible).

Consequently under the loading conditions, the rubber slab towards the base of the soil layer

would only be subjected essentially to one-dimensional compression; in this case it would be

almost rigid~ and would not function as a soft material at the boundary.

In the engineering model developed at Caltech, no direct simulation of such boundary

conditions was attempted since the model was, in effect, one-dimensional, but, since a parametric

variation study was carried out in an attempt to represent the test results as well as possible, the

effect of the boundary, if there was any, was accounted for by the parameters found to give the

best fit with the test data. As will be seen later, these did indicate a soil stiffness smaller than

would be expected for soil alone. It was possible in the ABAQUS code to include a soft

boundary with the properties of the silicone rubber at each end of the three-dimensional model

test container, and the side lubrication was simulated by including a layer of material next to the

side wall with low modulus. The property of a viscous layer cannot be simulated in these finite

element codes as they are presently constituted.

In summary then, difficulties from the point of view of numerical simulation were caused

by centrifuge test input motion consisting of both horizontal and vertical components associated

with a pitching movement, experienced as a result of the mechanical design of the box.

Additional complications were caused by the attempt to minimize wave reflections from the ends

by using the silicone rubber slabs at the end of the soil specimen as well as the intention to reduce

side friction in the model by including a viscous layer at the side boundaries. In general, since it

is possible to represent the properties of sand in the box in finite element codes we consider it

better to leave the test apparatus full of soil, without changing the boundary conditions, and then

the resulting boundary conditions can be simulated in the numerical model. If test results are

eventually represented satisfactorily by varying the soil properties to give the best fit with experi­

ments, then for prototype circumstances the boundary conditions can be relaxed by extending the

numerical model to represent the behavior of in this case, a shaft embedded in free ground in

which the distant boundary conditions can be included by the use of nonreflecting elements.

4

With these conditions Kajima performed tests on the box with two sizes of shaft

imbedded in sand: the short one of length 1.5 times diameter, and the long one with length 5

times the diameter. Sand was present between the base of the cylinder, and the bottom of the

box. The box was subjected to a variety of input conditions, including sine waves of different

frequencies, (the rocking motion of course was always present) and simulations of the El Centro

1940 earthquake horizontal accelerations (north-south component). The latter simulation was

not entirely successful, as can be seen from the comparison in Figure 2.2 of the spectra from the

real El Centro earthquake and the measured horizontal base motion of the centrifuge bucket

during the tests (both to same prototype scale). A complete list of the Kajima tests carried out is

given in Table 1 and some selected test data as supplied by Kajima are presented in Figure 2.3.

Spectral determinations and transfer functions were also calculated by Kajima from their test

results and some of these are shown in Figure 2.4. The transfer functions H(ro) that are

presented in Figure 2.4 have been corrected for the spectral component induced by pitching

according to the following equation

(2)

where F}, F2 and F3 are Fourier transforms of horizontal acceleration at top center of sand,

center of base (input), and the vertical acceleration at the edge of the base, respectively and K P is

a constant. This was suggested by Kajima in their preliminary report. The test data without the

correction give rise to quite different transfer functions which have been calculated during the

numerical evaluation and are shown in Figure 2.5.

It is seen from the transfer functions that the material in the test container, with or

without the presence of the cylindrical shaft, exhibits peak amplification ratios at approximately

4.5 to 5 Hertz (prototype scale). This is a very high fundamental frequency and would not be

developed by natural soil materials at typical real sites unless the soil were of shallow depth. A

soil, with this frequency, would not respond strongly to a real earthquake input, because there is

not much energy in earthquakes at this frequency. Figure 2.2 reveals that, at this frequency the

El Centro spectrum has an amplitude of about 40% of its peak value. An examination of the

input acceleration spectrum for the Kajima tests as shown in Figure 2.2 indicates that the

spectrum is close to zero above 4 Hertz frequency, and therefore it is concluded that the tests

were carried out with a minimum soil response to the input motion. In fact, the box of sand

behaved almost as a rigid body during at least the low-level tests. As a consequence, the

measured/numerical error minimization exercise, which is described in a later section of this

5

report, is relatively insensitive. In other words, varying the soil properties in a particular model

does not make a great deal of difference in the fidelity of the model response to that obtained in

the centrifuge calculations.

For a meaningful numerical evaluation, the soil response ought to be an important

component of the behavior of the test. This would have been possible, for example, if the tests

had been carried out in a laminar box rather than the rigid box in which ·they were performed.

The rigid box commes the soil to such an extent (and this is assisted by the relatively rigid

cylindrical shaft embedded in the soil) that the soil does not vibrate independently, and therefore

responds directly according to the driving motion of the box. Another way in which the rigid

behavior of the system can be seen is by comparison of the horizontal motion of the

accelerometer attached to the top of the cylindrical shaft (ACC04) with that of the accelerometer

in the soil 30 em away on the midline of the system (ACC12). These are shown in Figure 2.6,

and it can be seen that there is little difference in the two acceleration histories. In other words,

the presence of the shaft does not make much difference to the motion of the soil in the

container. Another way of observing this information is the comparison of the transfer functions

in Figure 2.4 where the transfer function (horizontal acceleration) between the base and the

motion of the shaft top, and that between the base and the soil are almost identical. The situation

would have been different if the input motion had included a substantial amount of energy at

frequencies over the entire range from zero to, say, 10 Hertz which would have included the first

three or four modes of soil vibration.

3. NUMERICAL MODELS

A variety of numerical models was used in the simulation exercises described in this

report. It was decided to construct a simple mass-spring-dashpot model in order to attempt a

simple numerical simulation of what happened in the Kajima tests. There are two reasons for

this. The first is that the early stages of most engineering design analyses of such a shaft would

require the use of a fairly simple representation of the shaft in order to arrive at gross

proportions, dimensions, and amount of reinforcement, from estimates of the bending moment

and soil pressures that would be generated in the shaft by possible design earthquakes. These so­

called "engineering models" are widely used in preliminary estimates of an engineering design.

Secondly, more sophisticated models such as finite element systems or finite difference schemes

are widely used, but, since they are expensive and time-consuming to construct and run for

dynamic simulations, it is usually desirable to use them only in the final stages of design, when

the system parameters are already quite well known. A few runs serve to determine what actual

6

stresses might be like in a more exact simulation. If the fmite element model is constructed for

the purposes of doing a parametric fit to actual test data to represent the material and structural

behavior quite closely, then the amount of time involved in the simulations becomes immense; it

is not practical to use these models in order to explore the fit between experiment and numerical

simulation.

Consequently, the intention was to use the engineering model for an exploration of the

soil constants that would give the best fit between the calculated results and those obtained in the

centrifuge. · When this was complete, those numerical values would be provided to the finite

element codes for simulation of selected tests only. It was hoped in this way to minimize the

amount of effort put into the finite element codes. As an illustration of the reason for doing this,

the ABAQUS code which was originally run on the V AXNMS system at Caltech required

approximately 20 hours of CPU time in order to simulate only a few seconds (prototype scale) of

the input earthquake. The overall running time was usually 2 to 3 days, because the system at

Caltech is a batch system, with a number of users at any one time. The DYSAC2 program had

even longer running times and this led to a decision to run DYSAC2 on a Cray (XMP) computer

in order to cut down the amount of time involved. Even in that case the tests required a running

time of several hours of CPU time on the Cray in order to simulate only a few seconds of

earthquake input at prototype scale.

On the engineering model that was devised, a large amount of time was involved in

performing a best-fit minimization, since it required minimizing the error of fit of the calculated

to the experimental results in an n-dimensional space, where n was 4 to 6 even in the most

minimal model. However, this technique worked out reasonably well as will be seen later when

the results are presented. These results guided the fmite-element formulations.

The engineering model (EM) is represented in Figure 3.1; the number of masses can be

varied at the discretion of the programmer. The program is presented and described in Appendix

2. When the code was being written, it was thought that a system giving several modal

frequencies would be needed in order to represent the test data. It was not until the data were

being analyzed that it was discovered that the first mode required for the centrifuge simulation

needed to be in the range of 4 to 5 Hertz so that all other modes have higher frequencies. But

since the input motion does not include energy at these frequencies, it is not necessary to have a

very complicated model in order to represent what is going on in the centrifuge. Consequently,

most of the fitting efforts, as will be described in the next section, were performed with a model

that only had 3 masses - a subset of the EM shown in Figure 3.1. Any higher frequency

7

components are caused in the centrifuge by the higher frequency vertical vibrations present in the

system; these cannot be properly represented in the engineering model as presently constituted.

Figure 3.2 shows the ABAQUS model which represented the test configuration in three

dimensions, including the presence of the silicone rubber slab at the ends and the lubricated layer

along the side of the box. The ABAQUS model simulated one quarter of the centrifuge space;

the appropriate boundary conditions were applied along lines of symmetry and anti-symmetry as

well as on the external edges of the model. As shown, the model contained 593 elements and

863 nodes.. The SAP90 representation is shown in Figure 3.3 in cross section; it was a plane­

strain model consisting of 216 elements and 250 node points. No attempt was made to simulate

the soft end conditions of the centrifuge test because the material properties were taken from the

engineering model best fit results, which included the behavior of the silicone rubber. The

lubricated side boundaries were also, in effect, included because of the plane-strain conditions.

In Figure 3.4 is shown the plane-strain DYSAC2 model with 94 elements and 143 nodal points.

This test configuration also did not include the boundary details of the centrifuge tests. The long

running time of the DYSAC2 model developed because it was the only one of the three finite

element codes which had the capability of carrying out meaningful nonlinear material behavior.

In the EM it was possible to simulate various pieces of information at different points in

the model. The accelerations, velocities, and displacements of all the masses employed could, of

course, be obtained as output. In addition to these, the forces exerted between the masses and

the rigid rod representing the shaft could be calculated and used as a basis for computing

simulated pressures acting on the shaft. Since these dynamic forces were known between soil

mass and shaft, they could be multiplied by appropriate distances along the shaft in order to give

bending moments. Although the shaft in the simulation is rigid, the EI of the actual model shaft

used by Kajima (or of the relevant prototype) is known and so these calculated bending moments

can be translated into hypothetical strains in the actual existing shaft. The earth pressures and

strains are compared later with the measured responses in the Kajima test results. In particular,

the horizontal acceleration history at the top of the soil column and the acceleration history at the

top of the cylindrical shaft are of interest, and can be used to obtain transfer functions between

soil surface and base and between the shaft top and base for comparison with the centrifuge data.

The horizontal acceleration at the top of the soil column (the top mass) was used in the

parametric study employed to discover which soil properties best fit the Kajima test data.

8

In the finite element models, particularly ABAQUS, there was less difficulty in translating

the results which were obtained from the numerical model to those from the centrifuge test. In

the three-dimensional finite element code it is possible to output the stresses in any element in the

system and therefore the pressures between the shaft and the adjacent soil could be obtained at

the appropriate element. The "shaft" in ABAQUS was actually represented as a cylindrical tube

and consequently the axial stresses (strains) in the wall of that tube correspond to those

measured in the centrifuge model. The accelerations, velocities, and displacements of all node

points can be obtained at points closely corresponding to those at which measurements were

made in the centrifuge tests.

With the two plane strain fmite element models the situation is not so straightforward

since it is not obvious how to translate results from a model including a three-dimensional shaft

to a plane strain model. In the SAP90 case the shaft was represented by a column of elements of

the size and depth of the shaft, and by giving those elements a high modulus so that the material

of the shaft would behave rigidly as it was in the engineering model. The real shaft employed by

Kajima is very close to being rigid and so for the plane strain circumstance this was felt to be a

reasonable approximation. Then the stresses on and in the simulated shaft could be calculated

for comparison with the centrifuge test data. The results from the DYSAC2 model need to be

included here as they are not available yet.

In linear dynamic fmite element models there are two ways of proceeding with the

calculations. The first is direct time step integration in which the forces are applied to each

element in turn at a particular time and the acceleration, velocity, and displacement changes in

the element are calculated at a new time, !::.t later. From these a new set of forces and stresses

are computed, and the incremental calculation continues. For this process to be stable a time

step !::.t has to be determined in advance, depending on the element sizes and properties.

Numerical instabilities arise if the time step selected is too large. On the other hand, if the time

step is very small in order to avoid this numerical problem, the time of computation becomes

excessively large and the cost of a computer run may be large. This proved to be a particularly

difficult problem with the ABAQUS code and many test runs were required with sine-wave input

motion of different frequencies and different values of !::.t to determine the range of !::.t for

stability. This could not be calculated from the manuals supplied with the code. The second

technique in performing linear dynamic computer calculations is to determine the mode

frequencies and shapes, to calculate the effect of the input earthquake on each of the modes, and

to sum these up using modal participation factors in order to give an overall response. If the

computer solution is working correctly and the correct choices have been made in the various

9

parameters, the results obtained from time step and modal superposition calculations should be

very similar, depending on the number of modes (percentage of total mass) used in the

superposition technique. In many of the trials with ABAQUS it was not possible to obtain this

correspondence of the two results and thus, a number of trial runs was done both with modal

superposition and time step integration for comparison. The reason for using modal

superposition is that the calculation time is much less than that required for the time step method

and thus, modal superposition is an economical way to perform the calculations. Modal

superposition was employed in the SAP90 code which was therefore used for most of the earlier

studies exploring the effect of mesh size and time step, but the results were not found to apply to

the ABAQUS code.

4. SYSTEM IDENTIFICATION

When the engineering model had been constructed, and set up in such a way that the

material properties could be readily changed, it was decided to make a test of the method of

system identification in order to see if a best-fit could be accomplished between the EM output

and the recorded output of a Kajima centrifuge test. The first attempt was applied to the low

level ("El Centro") excitation test 72B with the long cylinder (HID= 5.0). In the Kajima tests a

variety of instrumentation included accelerometers at the base, in the soil, on the cylinder, and in

particular, at the soil surface, as well as earth pressure and strain transducers on the shaft. It was

considered that the most sensitive discrimination of the material properties required to fit the

model to the centrifuge test would be given by comparing the model acceleration (ACC12)

output to the acceleration in the centrifuge test at the sand surface 30 mm from the center axis of

the cylindrical tube.

In the EM a variety of variables involving material and model properties can be extracted.

The acceleration, velocity, and displacement of each mass can be selected for output, and the

output can be manipulated so as to give the forces between the center of each mass and the rigid

rod representing the cylinder in the centrifuge tests. Division of these forces by a certain area,

which has to be chosen, will give the earth pressure acting on the cylinder at that level.

Summation of the forces multiplied by distances from the point of action of the force to a

particular point on the cylindrical tube will give the bending moment in the tube at that point, and

the code has been written to calculate this bending moment at any selected level as a function of

time. When the bending moment is divided by the actual EI of the tube, and multiplied by the

tube radius, then the strain in the tube on its surface can be calculated for comparison with the

10

centrifuge test data. These variables, acceleration, earth pressure acting on, and strain in the tube

at a selected level are written to a file resulting from the engineering model calculation.

The acceleration at the soil surface was considered to be represented in the engineering

model by the acceleration of the top mass. The earth pressure and strain at the transducer

locations on the tube were obtained from the EM by selection of the number of masses to be

used. This was done so that the center of one particular mass would occur at the level at which

earth pressure or strain was measured in the tube, and thus the calculated force and moment at

that level corresponded to those at the centrifuge test location. System identification requires the

solution of many numerical calculations on the EM; in each case the base input must be applied,

the engineering model code run, and the acceleration history at the top of the top mass calculated

and filed. These calculations for a number of masses are readily carried out on a PC but the

running time depends on the number of masses (i.e. the complexity of the model) selected. Since

so many were to be run in order to determine the system identification parameters, and the

centrifuge soil model was stiff, it was decided to use only a few modes in the EM for good

system identification. When an optimum set of properties had been established on the low mode

model, then a model with more degrees of freedom was run to see the effect of higher

frequencies, and to calculate earth pressures and bending moments. The engineering model for

system identification purposes was run with three masses only.

The idea of system identification is that, if an analytical or numerical model exists which

can be compared to a real-life test result, or set of results, or conversely to some known·

analytical model whose performance is to be represented, then the material constants, or model

geometrical parameters, can be visualized as axes in a hyperspace. In this multi-dimensional

space, a measure of the error between model and test is represented by a surface, which may

have several local minima, but possesses one absolute minimum. This occurs at that combination

of the variables, (that is, system properties) which best matches the output at the selected

location in the model with the measured value at the corresponding transducer location in the

physical test. It is an important part of system identification to choose a strategy to minimize the

effort required to find the absolute minimum. A variety of techniques has been devised to do this

(ref.) of which the simplest is called the "method of steepest descent".

This can be approximated in a numerical calculation by fixing a value for all of the

variables except one, whose value is then changed systematically while the measure of error is

calculated until a minimum is found at some value of the variable. That particular variable is

fixed at this value, and another variable is then changed until another error minimum is reached.

11

This process is continued for each control variable in tum. After error minima have been found

by variation of all of the parameters in tum, the process starts again since the minimum error

found at the end of the first cycle may not be the actual minimum obtained for the most optimum

set of variables. The nature of the error surface in the hyperspace may be quite complicated with

subsidiary minima or valleys, and a false minimum can be arrived at unless a wide range of

parameters is checked; this was done in the present circumstance. In the calculations the

measure of error is formed by subtracting the calculated result at each time step from the

measured result at the same time, to give a signal difference. The difference is squared and

integrated over the whole duration of the input selected. The final value of the measure of error,

J, is thus derived as given in equation (3) below

(3)

where q m is the measured, and q c the calculated quantity selected, and t 1 if the time at the end of

the calculation. It is clear that, if the calculated result exactly fitted the measured result, the

value of J would be zero over the time interval studied. In point of fact, with any comparison of

an idealized numerical model with real-life data the minimum value of J can only be non-zero.

It is clear that even with three masses which involve three springs and dashpots between

the masses and end wall, another three between the masses and the rigid rod representing the

cylinder, shearing springs in the soil column shear beam, and the damping of the system, a large

number of variables exists in principle to be used in the minimization process. It was necessary

to reduce the number of these in order to give a practical method of arriving at the material

properties. The following approach was adopted: it was decided initially that a power law

variation of soil shear modulus with depth in the form given by Kajima would be employed.

With such a representation, the shear modulus of the soil at the base, and the exponent of the

variation with depth were two of the variables to be selected. Next, the connection of the masses

with the end of the soil container and with the cylinder were related to the Young's modulus, E,

of the soil. This Young's modulus was considered to vary with depth in the same way as the

shear modulus, G, with the same power exponent, and to be related to it by the usual equation

below. The behavior of sand frequently corresponds to a value of Poisson's ratio, v, of about 0.4

and this value was also introduced into the equation

G= E E 2(1+v) 2.8

(4)

12

Thus, no more elastic constants were required to give E as a function of depth. In the system

identification process, no change was made in the Poisson's ratio during the calculations.

Another constant required was the value of the torsional spring constant at the base of the

cylindrical column (rigid rod in the numerical model). This was initially calculated as a

foundation compliance obtained from the rocking of a cylindrical foundation on an elastic half­

space. It turned out later that this value was substantially in error, but it permitted the

calculation process to begin. The other principal constant selected was the damping coefficient

relating the damping matrix to the stiffness matrix. This coefficient is related to damping given in

terms of percentage of critical damping; the relationship was given by a number of numerical

trials and is shown in Table 2. For test 72B, (low level input excitation), the value of this

constant was selected to be quite low as little damping was expected from the soil at the small

strains anticipated.

Thus, the basic number of variables selected for the system identification :Rrocess was

reduced to four: the shear modulus at the base of the soil column, the exponent of the power law

variation with depth, the value of the torsional spring at the base of the cylindrical column, and

the damping coefficient

Clearly there are other variables that play a part in the response of the system and these

require some discussion. In representing the behavior of the soil in one-half of the centrifuge test

container, a question arises as to the proportion of the mass of the soil that supplies input forces

and stresses to the cylindrical column. A decision has to be made regarding the effective cross

sectional area of the soil mass; it is obviously less than the total cross sectional area of the soil in

half the box. Somewhat arbitrarily, the contributing percentage was selected to be 20%. This

yielded prototype scale values of 10ft (61 mm) and 25 ft (152 mm) for the width and breadth

(model scale) of the soil cross section respectively. The rationale employed here was that, since

at prototype scale, the width of the cylinder was about 10 ft, selection of a 10 ft width soil

column would represent the proportion of soil mass reacting on the tube. With a half-length of

test box of 41 ft, and a radius of tube of 5 ft, a length of soil 36ft was left between the tube and

the end wall. Some of the reaction of the soil column would bear on the end wall, and it was

considered therefore that a shear beam of soil about 0.7 of the horizontal length might be

considered to be representative of the vibrating mass of soil. This gave 25 ft for the horizontal

dimension of the column. From the comparison of earth pressures and strains between

engineering model and centrifuge tests, these values seemed to work well, and were not

subsequently modified.

13

Once the number of masses is chosen, the height of each element follows, and, when

multiplied by the cross-sectional area the volume of such an element is given. When multiplied

by the soil density, the mass of the element is obtained. The choice was made to utilize the real

unit weight of the soil as identified in these tests by Kajima. It is, of course, desirable, because it

is a measured property, to use the real soil unit weight in any calculation involving the volume of

soil. The effective cross-sectional area and soil unit weight were not used as variables in the

system identification process, although they could be, and further work could be used to identify

best-fit values. It is also evident that the spring constants representing the soil behavior between

the hypothetical soil column described above and the end of the box presumably should be

different from those indicating the interaction of the soil column with the rigid beam. A variable

was assigned to this in the EM program, and it can be changed; however, it was taken as unity in

all of the system identification calculations and was not selected to be one of the variables in the

minimization process. Again, if work were to be continued, the effect of selecting a different

value for this ~roperty could be investigated. As will be seen later, the quality of fit of the

numerical calculations with the centrifuge tests in both the low level and high level earthquakes

was reasonably good for the "best-fit" models in each case, and therefore, further examination of

these subsidiary variables was not considered necessary at this stage.

The first set of values for the four constants in the system identification model was

selected partly on a basis of engineering judgment, and partly by use of the system transfer

functions or spectra which·were obtained from the measured data. In the latter case, the low

level test (72B) transfer function showed a peak at approximately 4.8 Hertz (prototype scale).

The initial properties of the soil column were selected to give this frequency for the lowest mode

of the system (i.e. for the three mass combination, together with spring constants or soil

properties selected). A small initial value of damping coefficient was chosen for the test 72B; a

higher one for test 73P.

As described earlier, the motion of the box in the Kajima centrifuge tests was not only a

linearly horizontal motion but involved a complicated pitching motion, with both vertical and

horizontal accelerations. In order to attain a reasonable representation of such behavior in the

engineering model, it was necessary to include some form of this pitching behavior in the

calculation algorithm. This was done by first making a plot, as shown in Figure 4.1, of the

vertical acceleration at the end of the box versus the horizontal acceleration at the center as

measured by Kajima. It will be seen in that figure that there is an overall average relation

between the two accelerations, which indicates that the motion of the box might be represented

as a rotary motion about a system center some distance below the surface of the actual box. The

14

distance was·calculated from Figure 4.1 and is shown in Figure 4.2. The assumption was made

that the actual motion of the box could be given by the use of the measured central input

horizontal acceleration applied as a tangential acceleration to a box pivoting about this center.

The motion at different heights above the box base could then be calculated from this

acceleration times a geometrical factor. The consequence of this mechanical model was that the

horizontal accelerations of the box increased with height in the engineering model from the base

to the top in a manner reasonably approximating those which were observed. Vertical

accelerations were not included in the engineering model but could be added to a more

sophisticated representation if desired. Clearly, the radius selected for the pivot point below the

center of the box could also be a variable in the system identification, and a minor study was

made of the effect on the goodness of fit parameter J by varying this amount. It was found that a

value of 1.05 of that given by Figure 4.2 produced a small improvement in fit, but this was not

investigated further. A table of the properties obtained for best-fit of both low level and high

level input motions is given in the next chapter "Comparison and Results".

The same system identification process was followed through for the high level test 73P

after observing that the fundamental frequency of the soil in the transfer function obtained by

Kajima in test 73P was reduced to approximately 4.3 Hertz (all of the transfer functions are

actually more complicated than this, but this is the value for frequency at the center of the

transfer function). Consequently, a new set of elastic properties was given to the model but the

mass properties were retained as in the modeling of test 72B at low level excitation. This

enabled the system identification to be started using a higher damping coefficient than in test 72B

as would be expected with the larger soil strains developed by the high level acceleration input.

When the all of the system identification trials had been completed, new material models

had been established which gave the best-fit of the surface acceleration in the centrifuge tests to

the value obtained for the surface acceleration of the top mass of the numerical engineering

model.

The object of the identification studies was not just to see how well the accelerations

could be matched in the centrifuge tests, but also to examine the earth pressure acting on, and

the strain in the cylinder as a result of the shaking motion. The Kajima tests involved

measurements of earth pressure (see Figure 2.1) and strain at a number of locations down the

cylinder, and it is possible to calculate the corresponding values in the simple engineering model

as well as, of course, in the finite element tests. As described previously, the calculation of earth

pressures in the engineering model require an assumption regarding the area over which the

15

acting forces from the individual masses are applied in order to determine the pressures.

Similarly, identification of the strains in the cylinder need assumptions regarding how the forces

developed between the masses and the cylinder develop strains in the cylinder. The goodness of

these assumptions can be examined by comparing the calculated and measured earth pressures

and strains at different locations on the tubes. In each case two points were selected for

comparis<;>n; the transducers identified as EP5 and EP7, for earth pressure, and the strains at the

locations identified as ST3 and ST5. Plots are given subsequently, comparison of both low and

high input EM calculations with the measured results. These will be discussed in more detail in

the next section.

The best-fit soil properties for both the low level and high level excitation were supplied

to the SAP90 model see how well that model duplicated the test results in spite of the lack of

inclusion of the pitching movement. Some discussion of the results is given in the subsequent

chapter. There was little interaction between the engineering model and either the ABAQUS

finite element test or that performed with the DYSAC2 code because of the difficulties

encountered in making the finite element codes function satisfactorily. A great deal of effort was

put into the ABAQUS code with little or no useful results. Attempts were made to perform tests

both with time step calculations and with modal superposition: the final results presented in this

report were obtained from modal superposition, because of the necessity for reducing the amount

of time required to make the calculations. The DYSAC2 code, which was originally intended for

simulating liquefaction of soils, does not appear to work well, or possibly may be in error, when

dry sand is employed as in the present case. After a certain amount of input motion had been

applied to the model, numerical instabilities were obvious in the large input case. Consequently,

only a portion of the test data are presented here. As mentioned before, the computer CPU time

was excessive in particular, for the high level input case for the DYSAC2 model because of the

way the nonlinearities in the code are implemented in the calculations.

The high level input for the DYSAC2 code was not that supplied by Kajima for the

centrifuge high level tests because the DYSAC2 work started before that input was known and

the calculation times were so long that the computation was not repeated. Consequently, the

high level tests were carried out with DYSAC2 using an input acceleration history which was

taken to be ten times the value of the input acceleration history used for the low level tests

(72B). Since the spectra of the low and high-level tests were similar, the input used still has

some validity. Some results of the calculations using this input are given- subsequently;

obviously, since the input is not that supplied to the Kajima model the acceleration values at the

soil surface cannot be compared directly with that of the centrifuge test results. In the case of

16

the DYSAC2 model, no output was prepared corresponding to earth pressure values or strain in

the cylindrical tube because of the lack of confidence in the model to be run correctly.

The SAP90 model could be employed to obtain stresses acting on the side of the

cylindrical tube, and also, with adjustments, to produce output representing the strain at selected

points on the tube. The earth pressure and strain locations were selected to be the same as those

for the engineering model evaluations and output was prepared from the SAP90 code for the

high level test, for comparison of pressures and strains at these locations. The results are

discussed in- the next section.

The ABAQUS model also, although the tube was properly represented in the

three-dimensional configuration, was not employed to obtain earth pressure against and strain in

the tube because of the lack of confidence in the viability of the results. Data are presented only

for comparison of acceleration values.

5. COMPARISON AND RESULTS

5.1 Engineering Model

This section will summarize the results of the comparison of the engineering models with

test data from the Kajima centrifuge studies. All of the simulation tests which were done in order

to validate the engineering model, were carried out at prototype scale with g levels,

displacements, and dimensions appropriate to a model 50 times the size of the Kajima centrifuge

test. Two tests were used for comparison, the first 72B ("low level El Centro input"), and the

second test 73P ("high level El Centro input"). As presented by Kajima.the input acceleration at

accelerometer ACCOl presented some difficulties in the low level tests. (See Figure 5.1). First,

the input acceleration for the first 1.4 seconds does not have an average zero acceleration. For

the engineering model tests, this was changed by moving the baseline of the acceleration so that

the input was, in fact, zero through this period. The second difficulty with the test results is that

the acceleration seems to be somewhat erratic with changes in its mean level at relatively long

period. The consequence of this is that when the acceleration is integrated to give velocity and

displacement, the values obtained are not the usual oscillatory values of typical free surface

records in earthquakes but show evidence of a changing baseline during the excitation motion.

An example of this is shown in Figure 5.2, the velocity record obtained by integration of the

acceleration record of Figure 5.1, (corrected to zero level to 1.4 sec.) and in figure 5.3, the

displacement history. In particular, it can be seen that the displacements go to very substantial

values in prototype scale. Figure 5.3, for example, shows that the displacement in the Kajima

17

model reached approximately 250 inches in 20 seconds of the strong ground motion history. It is

clear that this (5 inches in model scale) is much too great for the shaking table in the centrifuge

to have moved during the excitation. In consequence, it seems likely that the baseline of the

recorded acceleration changed during the input, possibly as a result of electrical discrepancies, to

give a false impression of the displacement history. No correction was made for this in the

calculations since it was not known how the acceleration integration would affect the behavior of

the centrifuge data. One consequence of this is that the displacements at the base and at the top

of the centrifuge model (ACC12) were different and were of course, different from those

experienced in the engineering model. For this reason it was decided to base the fitting process

for obtaining the best model on a comparison of the acceleration histories rather than on a

comparison of, for example, velocity or displacement histories.

It was commented earlier in this report that the spectrum of the input motion supplied for

both the tests, low level 72B, and high level 73P, consisted of very low frequency shaking

compared to the natural frequencies of the container full of sand with the test cylinder in place.

This is illustrated in Figure 2.2 where the El Centro spectrum is shown to the same vertical and

horizontal scales as both spectra of the inputs for tests 72B and 73P. It will be seen that the El

Centro spectrum contains energy over a wide range of frequencies as high as the highest value

shown on the graph, 10 Hertz, whereas both of the shaking histories used by in the tests exhibit 2

peaks at approximately 2 Hertz and 4 Hertz with very little energy elsewhere in the spectrum.

This, of course, has a particular effect on the ground motions observed in the test results. It was

somewhat surprising to find in view of the relatively rigid motion of the soil in the container, that

it was possible to obtain clear best-fit values of the engineering properties in the engineering

model. It had been anticipated before, that the model might be extremely insensitive to the EM

test properties over the range that was normal for soil at these stresses.

5.1.1 Low Level Test 72B

Using the technique described in the previous section of this report, the best-fit

engineering properties were obtained for test 72B and are given in Table 2. Earlier in the report

the correlation between the damping coefficient used in the engineering model and the more

usual percent of critical damping was described. In the present test the comparison for the best­

fit model is shown in Figure 5.4, where the first 5 seconds of the recorded acceleration in the

centrifuge test are compared with the calculated acceleration for the best-fit engineering model.

The comparison can be considered to be quite good except for some high frequency fluctuations

in the test, which are not present on the EM. It is considered that these high frequencies are not

18

generated by higher modes of the test specimen (which are also present in the model) but were

obtained as a result of the higher frequency vertical accelerations inadvertently obtained in the

centrifuge test. These higher frequency accelerations were not input to the EM; only the

horizontal motions as recorded at ACCOl were used, modified by the pitching mechanism.

Figure 5.5 shows the fit over the duration of 21 seconds to give an impression of the behavior

over the longer time scale. It is observed in Figure 5.5 that there is a systematic difference

between the test data and the engineering model at times greater than about 7 seconds with the

EM having an average acceleration higher than shown for ACC12 in test 72B. The reason for

the higher level in the EM at this time is as described earlier, due to the fact that the as-given

ACC01 input acceleration drifts higher as time progresses in the model, and the EM, which

employed the ACC01 input, incorporates this shift. For reasons that are not clear, the test 72B

data at ACC12 do not include this shift, so that, although the shaking motion is similar for times

in excess of 7 seconds, the ACC12 data from the centrifuge test maintains a reasonable zero

value on the average. Why does the ACC12 data have an apparently consistent mean when the

input signal varies?. It is possible that the input accelerometer ACC01 was recorded on

electronic equipment which introduce a drift, and that the real input acceleration did maintain a

mean which is represented in the ACC12 output. Was any processing of the test data employed?

When the engineering properties had been established, the engineering model was run

with an appropriate number of masses to enable the earth pressure to be calculated at the levels

of EP5 and EP7 earth pressure transducers. The forces, arising from both model spring

compression and velocity damping which were calculated at these levels were divided by the

cross sectional diameter of the test cylinder, and by the height of the mass in the engineering

model in the relevant test, to give a pressure. The pressures obtained by calculation are

compared with those from the centrifuge test data in Figures 5.6 and 5.7 for EP5 and EP7

respectively. In the engineering model the values of force were taken to be positive when the

force was compression and negative in extension as is common in geotechnical engineering; the

same convention was adopted for strain subsequently calculated in the tube. It was not known

what convention Kajima used for these two quantities, and so it was necessary first to print out a

comparison between the test data to resolve this.

Another difficulty arose with the Kajima tests m the comparison studies. In the

centrifuge tests carried out with pressure transducers attached to the wall of the cylinder, the

earth pressure measured in the test, of course, reflects the real earth pressure. ·The tube was

installed at 1 g, and, when the sand was filled around the tube it exerted a relatively small

pressure on the earth pressure transducers at this gravitational value. Subsequently, when the

19

centrifuge was brought up to its speed of 50 g, the earth pressure on the tube increased. This

was not represented on the EM since it is a dynamic model only and static pressures are not

included. Consequently, a comparison requires the subtraction of the static values of earth

pressure from the Kajima results in order that only the dynamic component of earth pressure

could be compared with the EM values. This was done by taking the average value of earth

pressure in the 5 second long period used for comparison, and subtracting this from the Kajima

test data for both EP5 and EP7. The result of the comparison of the calculated and test result for

EP5 is shown in Figure 5.6.

It was found when this was done first, that the two results were out-of-phase and the EM

calculations were reversed in sign fit. For EP5 it was surprising to find that the calculated values

of earth pressure were much smaller than the values obtained during the centrifuge test by a

factor of 4 Figure 5.6 calculated values multiplied by -4 to give similar scales. It had been

anticipated that, since, in the engineering model, the force exerted on the cylinder by one of the

masses through the attached spring and dashpot would only be a portion of the actual force on a

cross section through the cylinder and the soil on each side, the earth pressure given by the

engineering model would typically be too high.

In view of the correspondence between other test data to be described later, the question

is raised as to whether the actual recorded data from pressure transducer EP5 during Kajima test

72B was recorded correctly. Is it possible that the gain in the signal conditioning equipment for

EP5 was set incorrectly or misinterpreted? In any case, it can be seen that there is a substantial

correspondence between the EP5 centrifuge data and the adjusted results from the EM as shown

in Figure 5.6.

The question about the EP5 measurements is made clear by reference to Figure 5.7,

which shows the comparison between the pressures measured at the deeper transducer EP7 in

the centrifuge model and EM. The results were again out-of-phase, and so the calculated values

from EM were multiplied by -1 in order to match the Kajima data. In this case, however, no

adjustment to the EM values was required other than their inversion, and it can be seen that the

fit between the calculated and measured results on Figure 5.7 is quite good in this test. In fact, it

is considered to be surprisingly good, both in terms of amplitude and phase correspondence, if it

is remembered that the modes of vibrations of the engineering model at various depths would not

be expected to correspond to those of the soil and tube very precisely in the centrifuge test. And

yet it can be seen that the results, particularly in the area of 2.4 seconds to 5 seconds in Figure

5.7, correspond quite well. Some of the difference prior to that seems to be due to noise in the

20

system during the centrifuge test since the first 1.4 seconds of motion should be relatively quiet

before the strong motion input begins. This may be due to the small values of earth pressure

encountered, and the necessity for using high amplification in the recording system. It is

necessary to explain why the fit for EP7 is good and for EP5 it is not. An additional fact is that

the pressure obtained in EP7, both calculated and measured dynamically, seems to be a

reasonable value for the test conditions. It will be observed in Figure 5.6 for EP5 that the

measured earth pressures are higher than those obtained in test EP7. This could be due, of

course, to the rocking of the cylinder about its base with relative soiVcylinder motion greater at

higher elevations in the sand, but this is counterbalanced by the fact that towards the surface the

modulus of elasticity of the sand gets smaller so that at the actual ground surface the dynamic

earth pressures will be, of course, zero. It is questionable if, at the level of EP5, the measured

dynamic earth pressures could be 50% higher than those observed in the deeper soil location of

EP7.

It is noted that the average level of earth pressure in the centrifuge tests at both locations

EP5 and EP7 remains fairly constant through the entire record. There is no indication of an

overall change in earth pressure resulting as a consequence of the shaking. The situation is

different in test 73P, as discussed later.

The next engineering variable to be discussed is the strain in the tube at the two locations

ST3 and ST5; these data are shown in Figures 5.8 and 5.9 respectively. Strain gauge ST3 is

located higher on the tube than ST5. Once again, the calculated value is multiplied by -1.0 in

order to bring the two signals into phase with one another. Another problem which arose in this

case the reported test strains are given as values in the order of 1 to 2, which is a very large

number in terms of real strain. It is assumed that the factor 10-6 was omitted. The test results,

to compare with the EM calculations, were thus multiplied by 10-6 the usual value associated

with strain (that is to say, strain is usually given in units of micro-strain). The strain in the tube

would also be expected to change as the centrifuge is taken from 1 g to 50 g and then shaken,

and a correction was made for this in the plot of both Figures 5.8 and 5.9 in order to have the

test data start at zero. However, as can be seen in Figure 5.8, the ST3 strain value wanders

considerably in the 5 second duration of the engineering test, decreasing initially and then

gradually rising as the test progresses. This is perhaps a function of the settling of both soil and

cylindrical tube, changing the axial stress on the tube as the shaking progresses, but this effect

would be expected to be visible on the earth pressure at EP5. If the conjecture is correct, the

strain change might be expected to be more significant at the level of ST3 than lower in the tube.

21

Once again, in the 1.4 seconds before the earthquake strong motion begins, it can be seen

that the strain gauge record is quite noisy compared to the calculated value in Figure 5.8. It is

reasonable to expect this noise to be present in the whole record, in spite of which, after about

2.4 seconds on Figure 5.8 the correspondence between the calculated and measured response is

fair.

At the deeper location of strain gauge ST5 in the Kajima test, the changes in the strain

are somewhat smaller perhaps as a result of the more stable position of the tube at that depth.

The results of the comparison between ST5 in the centrifuge test and the EM are shown

in Figure 5.9. A number of things are clearly evident: (the test data are multiplied by 10--6 again,

and the EM calculation are multiplied by -1.0): it can be seen that there are a number of spikes in

the test data at, for example, about 0.3 seconds, 1.0 second, 1.7 seconds, and so on. It is

assumed that these are artifacts of the electronic data acquisition system and probably arise from

the phenomenon known as "bit drop" in the acquisition process. They do not represent sudden

changes in strain at these times but are superimposed upon the general oscillations of the data.

Apart from this unusual noise, it can be seen again that in the first 1.4 seconds there is a fairly

high level of noise in the data compared to the engineering model results. However, after about

2.4 seconds there is a relatively good correspondence (apart from the anomalous spikes) between

the calculated results and the measured results. In particular, the strains are now several times

those recorded at gauge ST3, which would be expected as the bending moment at this depth in

the tube is higher from the accumulation of forces from elements higher in the soil column. Peak

strains at the location of ST5 reach values of approximately 3 x 10--6 in contrast with values of

about 1 x 10--6 at ST3.

In both of these figures that the good correspondence between calculated and measured

strains is surprising in view of the presumably large difference in detailed dynamic behavior

between the EM and centrifuge test material. The correspondence is even more surprising in the

case of strain since the strain at ST5 results from many dynamic loads applied to the tube at

different distances from the location of ST5. H the mode shapes of the model tube in the EM

were different from those of the centrifuge test tube, which might obviously be expected, one

would expect a good deal of out-of-phase motion to be exhibited. The correspondence, may be

due in part to the fact that higher modes do not contribute very much to the behavior of the

system since the input excitation is at a frequency lower than almost all of the -system modal

frequencies (rigid behavior). H the centrifuge test material were, in fact rigid, then some

correspondence between observed and calculated accelerations would still occur, but tube strains

22

would be zero. Consequently, the tube strains are a good test of the validity of the EM. Also,

the relatively good correspondence of the EM with EP7, ST3 and ST5 emphasizes the extreme

lack of correspondence with EP5.

5.1.2 High Level Test 73P

In the high level test 73P, the input acceleration seems to be more under control than in

the low level test to the extent that the drift in the base motion is, although present, not so great

as it was in 72B. The input motion for the first 5 seconds again is shown in Figure 5.10 and the

integrated velocity and displacement records during this period are shown in Figures 5.11 and

5.12 respectively. The only correction that was made to the input data for this test simulation

was again to bring the input prior to the initiation of strong motion to a value of zero in order

that the engineering model would have a stable beginning.

The spectra of the three motions 72B, 73P, and El Centro is as discussed earlier shown in

Figure 2.2 where it can be seen that the spike of test 73P occurs at the peak intensity of motion

in El Centro but, whereas, El Centro has a distribution of energy on both sides of its peak, 73P

has a burst of energy at this frequency with a much smaller peak at approximately 4 Hertz. In

terms of the main peak, the energy in the centrifuge test input is similar to that exhibited by the

El Centro earthquake. Some broadening of the band of the motion in terms of frequency needs

to be performed before further centrifuge tests are contemplated. For the high level test the

fitting process proceeded as described in the previous section, and the final result gave rise to a

comparison between the centrifuge test data for ACC12, and the calculated value in the modified

EM. This is shown in Figure 5.13 where the fit is considered to be reasonably good. Again, high

frequencies are visible in the test data which do not appear in the calculated values, and are

probably again due to the high frequency vertical accelerations which were applied to the test

box. Basically, however, the amplitude and phase of the calculated values at ground surface, and

the spacing between the two functions appear to be relatively satisfactory. The best-fit high-level

properties are shown in Table 2.

When the engineering parameters for best-fit were obtained, the results were then

employed to compare the earth pressures and strains in the cylinder at the same locations as for

test 72B. These values are shown beginning with Figure 5.14, which shows the comparison

between the centrifuge test data for pressure transducer EP5 compared with the calculated

values. In this case, the calculated values were multiplied by -1 as before to bring them into

phase as discussed earlier, but no other adjustment was made to them. The initial offset

23

phenomenon alluded to in the previous discussion on test 72B occurs also in these test data.

However, in order to have the two curves fit on the same plot the EP5 data was established at an

initial value of -0.15 kg/cm2 because the pressure increased throughout the test as a result of the

shaking. This may be attributed to the increase in density of the sand surrounding the soil

cylinder and a change in the lateral earth pressure coefficient as a result of the substantial shaking

intensity imposed. It can be seen in the data that from about 1.6 seconds on, the average value

of the EP5 test results steadily increases through at least the 5 seconds of data shown. It was not

possible to subtract this shaking-induced change from the data, and so it was left in place.

However, it can be seen in Figure 5.14 that again, there is a reasonable correspondence

between the EM dynamic pressures and the test data. From about 2.8 seconds there is a

reasonable correspondence between the two dynamic components. In particular, the amplitudes

are quite similar and any engineering design calculations based on the EM would therefore be

making use of relatively realistic values. It is also interesting to note that the dynamic values of

earth pressure are approximately 10 times those measured and calculated for tests 72B, closely

representing the approximately 10 times increase in the amplitude of input motion. This may be

interpreted to indicate that the soil behavior has not been strongly nonlinear over this range of

input motion. Perhaps the most significant thing with test EP5 in this comparison is that the

calculated values from the engineering model did not have to be multiplied by a factor of 4 to

compare with the output of the earth pressure transducer EP5 in the centrifuge test of test 72B.

This again suggests an error in the EP5 signal in test 72B, and also that the error was removed

before test 73P.

Figure 5.15 represents the earth pressure at gauge EP7 compared with the engineering

model. For the lower earth pressure gauge EP7 there is less evidence of an increase in the lateral

pressure as a result of the shaking intensity than there is at the shallower location EP5 but a

careful inspection of the diagram shows some change is present . The offset on this figure was

left at approximately -0.02 for the initial 1.4 seconds of the centrifuge test data. It can be seen

that the biggest difference between the two calculated results arises in the interval from 2.4 to

approximately 2.8 seconds when the earth pressure, in the centrifuge test, undergoes a very large

pair of spikes compared to the EM dynamic stresses. It is not known why this occurred, in

particular, since the rest of the calculations are in reasonably good correspondence. It is

considered possible that the spike might be a result of some component of the vertical

acceleration applied to the specimen but it is not clear how this would have an effect. It is

remarkable that the double spike is large and yet is clearly a dynamic rather than a soil

24

densification phenomenon since it immediately drops down to the level of the earth pressure data

calculated by the engineering model.

The next comparisons are between the calculated and measured values of strain at the

two locations ST3 and ST5. Once again, it is pointed out that the strains in the EM derive from

a fairly complicated calculation involving a number of different forces acting in the system during

the shaking. In the measurement of strain at ST3, which is compared with EM calculations in

Figure 5.16, it was again necessary to multiply the test data by the factor 10-6 and the EM

calculations by -1 to bring the two into reasonable phase correspondence. Otherwise, no other

adjustment was made to the strain calculations to produce the plot shown as Figure 5.16. The

correspondence appears to be reasonably good in terms of phase and to a lesser extent in

amplitude. On this occasion, it is a little surprising to see that the calculated amplitude is

substantially greater at some time than the measured amplitude. Also, as observed before, there

is a certain amount of noise in the signal and, in fact, it appears that from about 3 seconds to the

end of the record a substantial proportion of the signal measured in the centrifuge could be

attributed to noise which would make the difference between the two results more extreme. It is

not clear why larger strains are not being observed in the centrifuge test at this section of tube

than are recorded in Figure 5.16.

Figure 5.17 shows the comparison of strains between test ST5 and the EM and the

correspondence is better at this location. Again, it is seen that the record is 2 to 3 times higher in

amplitude than the strain at location ST3. For this case, the mean initial value of the strains

could be taken close to zero on the plot for comparison with the EM calculations. Maximum

strains are of the order of 4 to 5 x 10-5 on this plot and this corresponds to a vertical dynamic

stress in the aluminum cylinder at this location of about 30 kg/cm2. The strains are also

approximately a factor of 10 higher than those recorded in test 72B.

One further test was carried out using the high level input earthquake of test 73P. Since

the comparison made to obtain the best-fit model only involved three masses, the question arose:

would more masses in the EM make a substantial difference to the comparison between the

tests? This was not considered particularly likely because, as pointed out earlier, the input

earthquake was at a relatively low frequency compared to the frequencies of the centrifuge

system. However, it was decided to perform an analysis using the EM and the best-fit material

properties using both 3 masses and 10 masses to see if the fit changed; the results are shown in

Figures 5.18 and 5.19. In Figure 5.18 a comparison is given between the measured response at

ACC12 in the centrifuge and the 3 and 10 mass calculations using the same properties and input.

25

It will be seen that there is some slight difference between the two models in terms of the plot

but it is not a particularly significant. The point can be seen more clearly on Figure 5.19 which

compares, for the 73P input, the performance of both the 3 mass and 10 mass engineering

models alone. It can be seen that the response of the 10 mass has slightly higher amplitudes

consistently through the motion but the frequency content does not change significantly in this

section of record.

The relatively close correspondence in the amplitudes of both the earth pressure and

strain signals from the centrifuge tests, and the calculated values of these quantities from the EM

is interesting. It means that the value originally selected fro the thickness of the soil column

( 10 ft prototype or 61 mm at model scale) was appropriate, since division of the acting load from

an element, by the product of this thickness and the element height gives reasonable values of

earth pressure.

5.2 SAP90 Model

Initially, a number of attempts was made using the SAP90 code with the soil properties

assigned to layers in the model according to the formula for the shear modulus of the soil given

for Toyoura sand (see Appendix 1). These tests were trial efforts to see how the SAP90 code in

plane strain would work for this problem. The values of the moduli obtained from the Toyoura

sand equation were too stiff to represent the test behavior correctly and these SAP90 results

were given only in the interim report presented to Kajima on December 4, 1992. However, since

the code was successful in carrying out the calculations fairly rapidly, it was decided to explore

the problem of modeling further with this code using the EM best-fit properties. The values are

given in Table 2. Since they were obtained by fitting the EM to the centrifuge test data the

values of the properties obtained included the presence of the softer silicone layer at the ends of

the model and therefore, the silicone layer was not included in the SAP90 model.

In the SAP90 model a total of 250 nodes and 216 elements was used, (see Figure 3.3)

including a central core section intended to represent the behavior of the essentially rigid cylinder

of the Kajima tests. It was decided not to attempt to model this with structural elements in the

SAP90 code because of the difficulty of determining the properties of vertical structural elements

to represent in plane strain correctly the EI of the cylindrical tube in the three-dimensional model.

Had it been possible to obtain reasonable results from the ABAQUS tests, the comparison of

ABAQUS with SAP90 might have indicated what value to use for such elements. Instead, it was

decided to represent the tube by a column of stiff elements two elements wide down the center of

26

the plane strain tube as shown in Figure 3.3. These elements were arbitrarily assigned a Young's

modulus E of 10,000 kg/cm2 in order to represent the stiff tube. This was arrived at arbitrarily:

some trials showed it to be a reasonable number to represent a material rigid compared to the

soil stiffnesses which were of the order of several hundred kg/cm2 for the deepest soil layers.

5.2.1 Tests 72B and 73P

As was pointed out earlier, it was not possible to operate the SAP mode with any

mechanism representing the pitching motion of the real centrifuge tests since the same input had

to be applied in SAP to all of the boundary nodes. Consequently, the results to be discussed

were obtained from a model in plane strain subjected to a homogeneous horizontal acceleration

history at the base and each end wall. Two tests were performed, one with the low level input of

acceleration represented by test 72B of the Kajima centrifuge tests, and the other with the high

level input, 73P in the centrifuge tests. The results of these simulations will be presented in tum.

Test 72B

In the 72B tests a comparison of the acceleration calculated by SAP90 at the surface

node, 80 of Figure 3.3 which corresponds to the acceleration recorded on accelerometer ACC12

of the Kajima tests was compared with that test acceleration. The results are shown in Figures

5.20 and 5.21 which respectively show the comparison for 5.12 seconds and for the longer

duration of 21 seconds. The comparison may be described as fair under the circumstances of not

including the higher accelerations at the surface caused by pitching in the centrifuge model. .. Since the input motion had been determined before to give only very small earth pressures and

strains in the cylindrical tube (as indicated in the engineering model tests), it was decided not to

make a detailed comparison of the lateral earth pressures and strains for the lower input

acceleration. Consequently, the analysis proceeded to be performed for the input of test 73P.

Test 73P

The model soil properties in that analysis were kept the same as in the engineering model

best-fit for the low level input since the engineering model system identification studies had not

been completed at the time the SAP90 runs were carried out. As a consequence, the model is

slightly too stiff and has too small damping to properly represent the higher level input motions.

Nevertheless, the results have some interest and will be described as follows. First, the usual

comparison of the calculated acceleration at node 80 is compared with the measured acceleration

in the centrifuge test at ACC12 in Figure 5.22. Again, it can be seen that the behaviors are

27

comparable but lack the higher frequencies in the calculated results that exist due to vertical

vibrations in the centrifuge model.

The results for earth pressure have also been compared for the gauge EP7 in the

centrifuge tests which occurs at an elevation in the SAP90 model corresponding to element 107

in the soil, and 108 in the column representing the cylindrical element of the model (see Figure

3.3). It was decided that the horizontal stress component in soil element 107 of the SAP90

model would represent a fair comparison with the EP7 output of the centrifuge tests and the

results are shown in Figure 5.23.

It will be recalled that the results of the comparison of the best-fit engineering model

between the calculated values of horizontal earth pressure and the measured values at EP7 in the

centrifuge test 73P as shown in Figure 5.15 exhibited reasonable agreement with one another,

except for a large double spike which appeared in the centrifuge test records between 2.4

seconds and 2.8 seconds. That large double spike had a value of approximately 3 times the value

of the earth pressure that was calculated in the EM. The rest of the data had amplitudes in which

calculation and the centrifuge test agreed reasonably well. The same effect is present in the

comparison of the centrifuge data with the SAP90 model as shown in Figure 5.23. However, in

this case, the centrifuge test results are all bigger than the SAP90 calculations through the

duration of the record, and can be seen from the figure to be something like 7 to 8 times higher

than the stress measured in the soil element adjacent to the simulated tube. At present, no

reasonable explanation can be given for this large discrepancy. It had been thought that these

high lateral stresses might be better accounted for in the finite element model since it represented

the frequency behavior of the test box more faithfully than the simple engineering model, but this

is not the case.

The test comparison also examines the correspondence between SAP90 calculations and

the behavior in the centrifuge strain gauge at location ST5 in test 73P; these results are shown in

Figures 5.24 and 5.25. Figure 5.24 shows a comparison of the centrifuge data with the results of

using the vertical stress output from the SAP90 model at element 107, the soil element outside

the cylinder at the location of gauge ST5. Figure 5.25 shows the comparison of the centrifuge

result with the vertical stress in element 108 inside the cylindrical tube. It is not clear which of

these is appropriate to use and therefore they are both shown. The conversions from the finite

element stress to strain were made using the Young's modulus E of the appropriate element. In

the case of element 107 which occurred at a depth occupied by layer 5 (see Figure 3.3) the value

of Young's modulus was 478 kg/cm2 according to Table 3, and the value of E used for the

28

cylindrical material was 10,000 kg/cm2. Consequently the vertical stresses at elements 107 and

108 were divided by these values respectively to give the corresponding strains for comparison

with the centrifuge tests. These are shown in Figure 5.24 and 5.25, with the exception that,

once again, the calculated values have been inverted in order to make the phases agree

reasonably well with the test data. It will be seen from the test plot that the measured strains lie

in between the two calculated values; the calculated value based on the soil vertical stress in

element 107 is much higher than the measured strains in the centrifuge test and thar for element

108 in the cylinder is much too small. This problem needs further study to determine how such a

vertical tube can be simulated more appropriately in a plane strain finite element model.

5.2.2 Test 83 (short tube)

It was not possible in the time available to recast the engineering model to represent the

behavior of the short tube in the tests performed on the centrifuge. Consequently, the most

interesting test, 83, in which the short tube was supplied with a high level input "El Centro"

earthquake acceleration was modeled in these test comparisons with the SAP90 code only. The

comparison between the results of the SAP90 code, again using the best-fit low level EM soil

properties, is shown in Figure 5.26 in which the engineering calculations at node 80 (Figure 3.3)

are compared with the test accelerations measured at ACC12. The degree of correspondence is

seen to be similar to that evidenced by SAP90 in the other tests. Actually, it is unlikely that the

results would be much different from this since an examination of Kajima's centrifuge test data

demonstrates that the behavior of the short tube in the soil column is not much different from the

behavior of the long tube in the soil column as far as the effect of the tube on the soil is

concerned. The transfer functions produced by Kajima and demonstrated at the meeting on

December 4, 1992 are virtually identical for the high level tests 83 and 73P.

5.3 ABAQUS Model

The difficulties with the ABAQUS program have been referred to earlier, and, in

consequence, only an abbreviated presentation of the results of these tests will be given here.

The ABAQUS tests employed the soil model with properties given by Kajima on Toyoura sand

as described previously.

The ABAQUS model was, however, supplied with an end layer of boundary elements

with a soft modulus to represent the silicone layer incorporated in the centrifuge model. In order

to account for the lubricated side boundaries in the centrifuge test, the ABAQUS model, which

cannot take lubrication into account, was supplied with a side layer of softer property than the

29

soil of the mass of the model. This softer modulus enabled the side layer to undergo much larger

shearing strains than would be possible if the soil elements had been continued to the wall. The

appropriate soil properties used in ABAQUS are shown in Table 4. The ABAQUS calculations

were carried out, as described above, after several trials and errors, by modal superposition

rather than by time stepping in order to reduce computer time. In the calculation, 15 modes

were used in the model, and the time step duration was 2 x 10-4 seconds with a total duration in

model time of 0.2 seconds of the earthquake; this represents 10 seconds of the earthquake input

at low level (test 72B) at prototype scale. In order to perform this calculation the CPU time on a

V AXNMS system was 9.5 hours with a total running time in excess of 20 hours.

This number of modes used is not generally considered sufficient for a proper evaluation

of the solution but a previous run with 25 modes ran for over 30 hours of CPU time without

terminating. • The modal frequencies obtained from this model to represent test 72B are shown

in Table 5. It will be seen that the first mode has a frequency of 164 Hertz at the model scale, or

approximately 3.3 Hertz prototype scale, which is somewhat lower than the fundamental

frequency observed in the test results. The frequencies of Table 5 were caused to be low by the

presence of the soft silicone layer at the end of the model together with the soft soil boundaries.

The ABAQUS results for comparison with test 72B have been plotted as shown in

Figures 5.27 to 5.31, for an illustration of the ABAQUS output. Unfortunately, it is not easy to

make the ABAQUS code produce separate files of the output data at various nodes and elements

in the model. Consequently, it has not been possible to compare the results from the ABAQUS

tests directly with the centrifuge test data as has been done for the models described so far. The

figures that are given in this connection represent the ABAQUS results only and can be

compared with the relevant 72B test data presented in other figures earlier in this report. The

results are only of value in demonstrating the three-dimensional ABAQUS output, but are not of

interest in any serious comparison of calculation results with centrifuge test output. Figure 5.27

represents the horizontal component of acceleration at node 6 of the ABAQUS finite element

model which can be compared to the performance of the horizontally directed accelerometer

ACC04 at the top of the cylinder wall in test 72B. The accelerometer which has been used for

previous comparisons, ACC12, in this test is represented by comparison with the acceleration

recorded at node 36 in Figure 5.28. In subsequent figures, the accelerations are shown at the

same two horizontal locations but at approximately the half height of the tube. In Figure 5.29,

node 406, horizontal acceleration is given; this lies vertically below node 6 at the soil surface on

• At this time the computer system failed (for other reasons) and the output was lost

30

the cylinder wall (corresponds to ACC07) and in Figure 5.30.8 the horizontal acceleration at

node 436 is exhibited corresponding to the acceleration (ACC15) directly below node 36 on the

soil surface. Finally, the displacement at node 6 at the cylinder wall at the surface is given in

Figure 5.31, and it is seen that, as expected from the discussion previously, this shows a

displacement parabolically increasing with time resulting from the input acceleration record.

The computational times in the ABAQUS code were so great for the linear

three-dimensional program that it was not possible to incorporate any nonlinear model in the

behavior in order to try to describe the response of the centrifuge test 73P, for example.

5.4 DYSAC2 Model

DYSAC2 is a computer code formulated using the bounding surface approach, to study

the problem of liquefaction in a saturated sand in two dimensions. Apparently, it has not

previously been used to simulate a dry sand, and there seems to be some difficulty with this

implementation as seen in this section. The primary reason that the two codes ABAQUS and

DYSAC2 were selected was that both have the capability of including nonlinear hysteretic soil

models. ABAQUS, in addition, can be used to represent three-dimensional circumstances while

DYSAC2 is limited to two dimensions. Although some preliminary studies were carried out to

simulate the behavior in the low level test 72B, the primary emphasis on using this code was to

simulate the nonlinear behavior that would be expected in the high level test 73P. Accordingly,

all of the test data that are represented in this report are those similar to test 73P. As pointed out

earlier, the actual input motion was not that of test 73P, but was the input to test 72B scaled

upward by a factor of 10. The DYSAC2 code does give output in file form which can be

employed for comparison tests with the results of the centrifuge and one or two of these

diagrams is also shown in this report. However, the principal data obtained are given in Figures

5.32 through 5.40, all of which refer to accelerations at various locations identified with the

places where accelerometers were placed in the centrifuge test, that is, accelerometers ACC02

through ACC15. In order to perform these calculations it was necessary to make the run on a

Cray XMP machine in which the results shown in the accompanying figures required a running

time of several hours.

The calculation as shown in all of the accompanying figures indicate a computer analysis

which apparently became unstable at about 6 seconds (prototype scale) into the motion, and

therefore the results from 6 to 10 seconds cannot be employed usefully in comparisons.

However, there may be some value in comparing the test accelerations calculated with those

31

measured from the centrifuge for the time between 2 and 6 seconds, although it is not certain

that the calculations are correct up to that time. In the previous numerical studies described in

this report, the centrifuge output of accelerometer ACC12 at the soil surface has been used for a

comparison basis and this acceleration result for the high level test 73P is given in Figure 5.37. It

can be seen that if the noise in the first 1.8 seconds or so of record, which is typically quiet

during the Kajima 73P test input, is ignored, a reasonable fit through the time duration of 5

seconds is obtained. This would appear to indicate that the program is functioning reasonably

correctly during this time but it is not likely that nonlinear soil behavior played a big part in the

motion of the test specimen.

6. OTHER MODELS

A survey of current numerical models available for the study of the problem of dry and

saturated soil static or dynamic behavior has been carried out, and the results are presented in

Appendix 3 of this report. The Appendix is self-explanatory; it contains a brief description of

each model and the people or agency responsible for producing the code, and where it can be

obtained.

7. CONCLUSIONS AND RECOMMENDATIONS

A great deal of effort has been put into the problem of identifying the Kajima centrifuge

test results with available numerical computer codes. Four test codes were selected for this

purpose, one of which was a simple engineering model (EM) which was designed and coded for

the purpose of this exercise, and which might be used for preliminary engineering design of such

a structure as a cylindrical shaft in soil and its performance during an earthquake. The second

model selected was a commonly available efficient structural engineering code, SAP90, and this

proved to be the most efficient of the complex numerical finite element codes available. The

SAP90 code has the capability of solving only linear plane strain models, although structural

elements can be included. The ABAQUS code was included because of its capability of handling

both three-dimensional problems such as the present one and also nonlinear soil properties

although its capabilities are limited. The last model employed, DYSAC2, was used because it

incorporates a nonlinear constitutive relationship for sand, called "bounding surface" model,

which in the past has been one of the better models to describe cyclic response of sand to applied

loads. However, it suffers from the defect of being only available for plane strain problems at the

present time. All of these computer codes have drawbacks when used to represent real

situations, some of which became obvious during this study.

32

In addition to the defects in the codes, the way in which the tests were carried out on the

centrifuge also posed some difficulties for code representation. It had been originally thought

that the centrifuge tests would consist of the one-dimensional shaking of a three-dimensional box

containing a vertical cylinder. However, the design of the shaking table on the Kajima centrifuge

was such that the model was subjected to both vertical accelerations at a fairly high level

combined with a pitching motion so that the vertical accelerations at the ends of the box were

out-of-phase and of the same magnitude as the horizontal accelerations. The frequency of the

vertical motion input to the box was also considerably higher than the frequency of the horizontal

motion. In the engineering model it was possible, to some extent, to simulate this rocking or

pitching component of motion and it also could be accounted for in the DYSAC2 representation

but, with the time available, it could not be included in the SAP90 and ABAQUS models.

Consequently, the numerical solutions were fundamentally deficient in the representation of the

motion that actually happened.

In addition to this condition, the centrifuge tests, were also carried out with an input

motion that was deficient in higher frequencies. In the tests the relatively stiff soil was so

confined by the small dimensions of the box and the axial cylinder included in it, that it could

only respond at relatively high frequencies. As a consequence, the motion of the box during

shaking included only its first mode of vibration and the higher frequency motions were not

stimulated by the designed input. Indeed, it was only the presence of the silicone rubber at the

ends of the test box, included for reasons of inhibiting reflections, that brought the first mode

response to a low enough frequency to be excited by the input. In addition, the design input

which was referred to as "El Centro" motion was very far from simulating ,at the centrifuge

scale, the real El Centro spectrum. In consequence, although the fmite element codes did not

perform very well, it turned out that the actual behavior of the test system on the centrifuge gave

results in which the numerical models of all kinds were not especially property-sensitive. A

better test would require an input acceleration base motion that was entirely horizontal and

contained a broad band of frequencies that reasonably represent the entire range demonstrated in

the El Centro record.

The instrumentation that was employed was more than adequate, and gave all of the

information that would be required for any modeling or engineering design procedures.

Apart from these limitations, the use of the engineering model in conjunction with a

system identification technique demonstrated that a best-fit of material properties could be

obtained by minimizing the difference between an acceleration measurement in the centrifuge

33

model and a calculated acceleration at the same simulated point in the engineering model.

Besides the acceleration, use of the best-fit properties in the engineering model gave a

surprisingly good fit among strains, and earth pressures as measured in the centrifuge test. It is

concluded that the engineering model, when a rational method is determined of assigning

material properties to it could be used effectively in obtaining preliminary design data for the

construction of civil engineering facilities such as the cylinder in the ground. However, it has to

be pointed out that the match was artificial to the extent that the centrifuge included layers of

soft silicone rubber at the ends of the test box, and a lubrication condition along the lateral

boundaries at which were not included explicitly in the engineering model properties. The

properties were obtained only by making an overall best-fit match. If the centrifuge model were

to be constructed in a bigger box ,or without the silicone rubber inserts at the end, it might not

be easy to represent the behavior of the resulting tests by an engineering model with rational

property determination without a fitting process. This consideration would apply to the

prototype in real-life also. A rational method for selection of the properties is required. Some

consideration of methods of handling extended boundary conditions is also needed.

Among the other models, the SAP90 model provided an efficient way of carrying out

two-dimensional plane strain calculations for such soil vibration problems, if a method can be

obtained (references) to apply the plane strain approach of SAP90 to three-dimensional

problems. The match obtained between the SAP90 model and some of the measurements during

the centrifuge tests, given the absence of pitching input to the SAP90 model, was reasonable.

The ABAQUS code was too difficult to use and caused a great deal of time and effort to be

expended in an attempt to make it perform satisfactorily. It seems apparent the code requires a

great deal of study before being employed in any practical circumstance involving soil. Even if a

solution can be made to work, it is clear that, for dynamic problems, very long running times in,

preferably, a main frame computer are required. The latter comment also applies to DYSAC2,

which in addition, apparently also needs more work before being applied to dry sand soil

conditions. Its performance in this study even warrants doubt about its ability to represent the

dynamic behavior of saturated sand stably.

8. ACKNOWLEDGMENTS

A number of people assisted in this investigation. Dr. Behnam Hushmand assembled the

collection of relevant computer programs available for dynamic soil modeling, and also

contributed to the ABAQUS model calculations, which were patiently set up and followed

through by Mr. Li-ping Yan. Mr. M. Halling assembled and performed the SAP90

34

computations. At the University of California, Davis, Mr. Majid Manzari carried out the

DYSAC2 calculations under the supervision of Professor K. Arulanandan. This report was

typed with skill and competence by Ms. D. Okamoto.

9. REFERENCE

Pilkey, W. D., and R. Cohen, "System Identification of Vibrating Structures," The American Society of Mechanical Engineers, Symposium at 1972 Winter Annual Meeting, 1972.

35

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o ~ ~ c::-~:::"':r""' ... ~~ 8-~""":1."' •. -·( ... ~~~M:,.,..v: .... ~_._s_t_J __ ~_c_h_n_a_m_e_~_E_P_o_s_J __

-0.8l 0 . 8 ( Kg/c~}JI""ee73p. 829

-0~~ f ,_ .... (Max=0.62J lch name 1 EP07J

lust. JCUI""ee73p.833 (Max=-37.001

40.~ o.,.. ... ~tw....,-~llll·~ ,,, ....... ,.11161···. ,,,

-40.

lch name 1 ST03J

. ' .... , ..

(ch name 1 STOSJ

2.3 c

........ tlO Cl)

,:::a '--'

Cl) til Cl:l ...c::

0...

curcc72b. Bl7 I curcc72b. B06-1_57V2 8. ~-----~--------~--------.--------.

o. L--------J--------~--------~------~ o. 100. 200. 300. 400.

Frequency (Hz)

180. ~------~--------~------.---------. t::.l:>

90. t::. t::.

~!:> t::.

o.

-90. t::.

t::. -180.

t::. t::.

t::.

t::.

t::. t::.

with shaft:H/0=5 (El Centro Max. 1.22G)

t::.

·------

........ tlO Cl)

,:::a '--' Cl) til Cl:l ...c::

0...

curee73p. Bl7 I curee73p. B06-l_57V2 8. r--------.------~~------~------~

0. 0. 100. 200. 300. 400.

Frequency (Hz)

180. t::. t::. t::. t::.t::.

t::. t::. t::.

t::. t::.

t::. ..1. t::. t::.

90. t::. ~ t::.

t::. t::.

0.

-90.

-180. 1-----..J__---~--=---E-~-__j with shaft:H/0=5.

(EI-C~~if~ lax.9.98G)

2.4

0.. E <(

Kajima: Transfer F'ns Tests 728 & 73P: ACC12/ACC01

20~----------------------------------------------~

18

16

14

12

10

8

6

4

2

s .. .. ------------------------------~--------------------------------------------------J-~-----------------------------------------------------------------" 41 I I :'It :I

I I 1 I I I I I I t r t

-------------------------------------------------------------------------1--!----·-~-----------------------------------------------------------------l I I . ' I I ' I \ I 1

------------------------------------------------------------------- - -~-----~--;---~----------------------------------------------------------------,' \ : :

-----------------------------------------------------------------~

····································;;·······------. . \

' \ ---,----------.

' ' ' I

I ,' ,

' . I

__ , . '. , . ' ' ' '. ' '. . . . ' - --~~----~------------------- --------------------------------------------~·

,. ------;~------!·\························· ' \ / ' ,,

,'', / \ .. , ,'', : \, \ ,~ ,,, \ • • # •

----~~----------------------~-------------,...... ·. ,.., ·--·· \ "·

----------------------~----......... --~!: ______________________________________________________ -

.· ' , ' . ,-· ~.·

0+-----~----~--~----~----~----~----~----~----~--~ 0 1 2 3 4 5 6 7 8 9 10

Frequency, Hz

73P ---------· 728 2.5

Kajima, Test72B Test ACC12 Vs ACC04

30~------------------------------------~

~0 --·-··-··-·---· ·····-··-··- ·-········-··-··-··-··-··--------------·-----·-----------·----··--·-------------------------------··-----·------------

~ 1 0 ----------- -- -- -- -- -- -- -- ---------------------------------------------------------------------------------------------------------------(.) Q)

.5!2 c ·­... . Q) (.)

(.) -1 0 --·--·---- ·- -- -- -- -· . -· --··----··----··-· --·-·---·-··--·-- -------------··--·-··----·-------------------------------------------------<(

-~0 --·----·--·-- ·----··----··-----·-··-··----------··-·--··-··-·---·--·-··-------··----------·--·--··--------------------------------------·---------

-30-+-----r-------..-------.-----....--------l 0 5 10 15 ~0 ~5

Time, sees

- ACC1 ~ ---------· ACC04 2.6

\ \ \ \ \ \ ____ j\ \ __

\ -\ ) , __ ,/

\VLz----:.....,....~_J tl~ \'IR

RJ.GID ~ssLBSS ROD (l:lo\loW cylinder)

(c) 1'1NAL coMl'\Yfl'.D MODEL

DASf\PO'I'

1'0RS10N sPRING

vL\~~~~~~~~~~~ 1-lC \

~I

3.2

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 -..--

I 2 3 4 5 6 7 8 9 10 II 12 13 14 IS 16 17 18 19 20 21 22 23 24 9 19 29 39 49 59 69 79 89 99 109 119 129 139 149 159 169 179 189 199 209 219 229 239 249

25 26 27 28 29 30 31 32 33 34 35 36 8 18 28 38 48 58 68 78 88 98 108 118

49 50 51 52 53 54 55 56 57 58 59 60 1.1) CD 7 \7 27 37 47 57 67 77 87 97 107 117 cO ~ 73 7 4 75 7 6 77 78 79 80 81 82 83 84 ~ 6 16 26 36 46 56 66 76 86 96 106 116 cw; @J 97 98 99 10~ 101 10~ 10: 104 10E 106 10/ 108 ,.... 5 15 25 35 45 55 65 75 85 95 105 115

121 12~ 12: 12~ t25 12~ 121 t28 12s t3e 131 132

-f-

4 14 24 34 44 54 64 74 84 94 104 114

14~ 14~ 141 14~ 145 IS~ IS I 152 15: 15~ 15E 156 3 13 23 33 43 53 63 73 83 93 103 I 13

E 165 (\') 0 2 ®co

17~ 171 17£ 17: 17 ~ 17E 17 6 17/ 17t 175 180 12 22 32 42 52 62 72 82 92 102 112

N II 19: I

19.1 tn t9~ t9/ t9s 195 200 201 20~ 20: 204 11 2t 31. 4·1 Sl 61 71 8t· 91· 101 Ill

37 38 39 40 41 42 43 44 45 46 47 48 128 138 148 158 168 178 188 198 208 218 228 238 248

61 62 63 64 65 66 67 68 69 70 71 72 127 137 147 157 167 177 187 197 207 217 227 237 247

85 86 87 88 89 90 91 92 93 94 95 96 126 136 146 156 166 176 186 196 206 216 226 236 246

109 11~ Ill 112 113 114 115 11~ 11/ 118 119 120 125 135 145 !55 165 175 185 195 205 215 225 235 245

133 134 13f 136 137 138 135 14~ 141 142 14: 144 124 134 144 154 164 174 184 194 204 214 224 234 244

157 ISS ISS 160 161 16~ 16: \6.1 165 16~ 16/ 16S 123 133 143 153 163 173 183 193 203 213 223 233 243

181 18~ 18: 184 185 18~ 18/ 18f 185 19~ 191 19, 122 132 142 152 162 172 182 192 202 212 222 232 242 205 20~ 20/ 208 209 21~ 211 21~ 21: 214 215 21~

121 131 141 151 161 171 181 191 201 211 221 231 241

~,.,____ ___ 11 @ 2 = 22 cm----·+ .. -2

_@_3--+·----- 11 @ 2 = 22 em ----t•l

=Scm

Sl0

UNDEFORMED SHAPE

f...-----------1

OPTIONS JOINT IDS ALL JOINTS ELEMENT IDS WIRE FRAME

I SAP90 3.3

I SHAFT

~ ~

~ ~

~ ~

~ ~

t% ~ ~ ~ ~ ~

~ ~

~ ~

~ ~

~ ~

~ ~

~ ~ ~ ~ ~ ~ '/

~ ~ ~

DYSAC2 MODEL

3.4

Acceleration Horizontal Vs Vertical

0.8~----------------------------------------------~

0.6

0.4

C) 0.2 cJ ~ 0 t ~ -0.2

-0.4

-0.6

-0.8+---~----~--~----~--~----~--~----~--~--~ -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Horiz. Ace., g

4.1

407mm

LOCATION OF VERTICAL ACCELEROMETER ACC 09

(67 ft. PROTOTYPE)

LOCATION OF HORIZONTAL ACCELEROMETER ACC 01

CENTER OF ROTATION

4.2

C\J < (.) Q)

J!}_ E (.)

Q) (.) (.) <(

Kajima: Test 728 Input Accel., ACC01

25~--------------------------------------------~

2() --------------- --- --------------- --- ----------------------------------------------------------------------------------------------------------

1 5 --------------- -- ---- --- ----------------------------------------------------------------------------------------------------------

1()

5

0

-5

-10

-15-+--~-~-~-----;.,---,-----.--~---..----.-----i

0 2 4 6 8 1 0 12 14 16 18 20 Time, sees

5.1

() Q)

~ E ()

Kajima: Test 728 Input Vel., ACC01

8~------------------------------------------------------~

6

4

2

0

Q)

> -2

-4 -----------------------------~----------------------------------------------------------------------------

-6

-8+-----~--~----~----~----~--~----~----~----~--~ 0 2 4 6 8 10

Time, sees 12 14 16 18 20

5.2

Kajima: Test 728 Input Displacement

20

15

10

5 E

0 (.) ------------------------------------------.------------------------------------------------------------------- ----------------------------------

Q) -5 (.)

~ c..

-10 C/)

0 -15

-25 --------------------------------------------- --------------------------------- ------------------------------------------------------------------

-30+---~--~----~--~~~----~--~--~----~~ 0 2 4 6 8 1 0 12 14 16 18 20

Time, sees

5.3

Kajima; Test 728 ACC12: Test vs Eng Model

30~------------------------------------~

20 ------------------------------------------------------------------ --------------------------- ---.--· --------------------- -----------------------

C\J ~ 1 0 --------------·--·-········-·····-····--·--······· . . --······ .. ···-··-·· . ·--------· --------- . ---------· - -----------------------(.) Q)

.5!}_ c

... . Q)

~ -1 () ---------------------------------------------- --------- - - --------- - --------- ------------- ------- - - ------ ----------------------

-20 ---------------------------------------------------------- ---------------------------------------------------------------------------------------

-30+-------r-------r------~------r-------r---------i

0 1 2 3 Time, sees

4

Eng Model -Test; ACC12

5 6

5.4

C/F Test Vs Eng Model ACC12

30~----------------------------------~

20 --------

~ 10 --------(.) a>

JQ c

-30+--------..------~--------r---------..---------l

0 5 1 0 15 20 25 . Time, sec

1- Test 728 - Eng Model I

5.5

Kajima, Test 728 Earth Press, EP5

0.02-r-------------------------,

0.01!:) -------------------------------------------------- ---------- -------------------------------------------------- ---------------------

~ 0.01 E Q 0.00!:> ~ ~ 0 ~

~ -0.005 ~ a.. -0.01

-0.01!:)

-0.02-+--------,..-----r----,-------,------,-------l 0 1 2

728, EP!:>

3 Time, sees

4

EP!:> (Calc)*-4

!:) 6

5.6

Kajima, Test 728 Earth Press, EP7

0.015-r-----------------------,

0.01 -------------------------------------------------- --------------------------------------------------------------------------------------

-0.01 --------------------------------------------------- -------------------------------------------------------------------------------------

-0.015-t-----r----~--~--~--~---i ·. 0 1 2 .

-728, EP7

3 Time, sees

4

-1.0*(EP7 Calc) I

5 6

5.7

Kajima, Test 728 Strain, ST3

1.5~--------------------------,

1 - -------···-------------·-··-------------·--·----- --------- ---------------------------------------··------·· ---------------··-·---

en - 0. 5- ·· · ·------------------------------ ------------- - ---~ -- - ----- ---------- ~ ---- -- ~ ---------- - ---------------------

-~ ~ 0-"' ~ ~-~'/ ~~--- v~- --- ~ -~~u- -~ -~ -~ -rv _IT _________________ _ 'o :; I l II' 1 I \ ~~ .c .~ -0.5- U-- ----- - - -~PJ -rr-/ ----- --- -- -- --• ----- -- ----- -- ---- -- ------------------------ctS ·- ~ !\ ~~-(/) .._. -1- -·· - -·- ------ - ··- -- -·- ... -- ----- . -- ··--------------·-· ----·-··----·-· ----~---------------------------

-1 .5- .................................................................................................................................. .

-2~----~,----~,-------~,----------~,----------~,----------~

0 1 2 3 4 5 6 Time, sees

728, ST3*1.0E-06 - -1.0*(ST3 Calc) 5.8

en -en <0 Q) I

- w E 0

"'0 ~

en c Q)

·ccs E "-- i-~ en ..._..

Kajima, Test 728 Strain, ST5

6~--------------------------------------------------~

5

4

3

~ ----- ---------------- --------------- -----------

1

-1

-~

-3+-----------~--------------~----------~------------~------------~--------~ 0 1 ~ 3

Time, sees 4

7~8, ST5*1.0E-06 -1.0*(ST5 Calc)

5 6

5.9

C\J

Kajima: Test 73P Input Accel., ACC01

200~----------------------------------~

1 !:>0- ----------- - -------------------------------------------------------------------------------------------------------------------------------

1 00- ----------- - -- ----------- ----------------------- --~------------ ----------------------------------------------------------------------

~ !:>0- ---------- -- -- -- --- -- ----------------- -- -- -- --- --- - -- -- -- ------ -- ----- ---------- -- -- -- ---------------------

Q) ~ -!:>0- --------- - -- -- ------- --- ------------------ --- -- ------- --) - -- ------- -------------------------------------------------------<(

-1 00- ------------ -------------------------------------------------------------------------------------------------------------------------------

-1 !:)()- ------------ --------------------------------------------------------------------------------------------------------------------------------

-200-t---------r-------.,.-,------r-, ---------,,r-----------1

() 5 1 () 1 !:) 20 2!:> Time, sees

5.10

() Q)

J!!. E ()

Kajima: Test 73P Input velocity

40~--------------------------------------------~

30

20

10

0

-10

-20~------~------~------~------~----~ 0 5 10 15 20 25

Time,secs 5.11

Kajima: Test 73P Input displacement

350

300

250 E (.)

200 "' ~ c Q)

E 150 Q) (.) ctS c.. 100 en ·-0

50 ----------------------------------------------------------

Q-+-----------~___....~---~----~--·-----------------------------------------------------------------------------------------------------------------

-50-+----~-------,------.-----.,...--------1

0 5 10 15 20 25 Time,secs

5.12

Kajima, Test 73P Best Fit: Soil Ace Camp

300~----------------------------------~

~00 -----·······-········-···-~---·································-··· ··············-········-··············-··-·····-··········--········-····

N ~ 1 00 ············-···········--------------------------- -- ---------- -- --------- - --------------------------------···· ·······-·····-······· (.) Q)

.5!2 c ... .

Q) (.)

(.) -1 00 -------------------------------------------- ····----- . - ------ . ····-··----------------------------------- --- -----········--------< -~00 ···-·····-··-----------·-···········-···-·········------- .. -------·-·····-·····-··--------------------------------------------------------

-3oo~--~--~------,---..,-------r------l

0 1 ~ 3 4 5 6 Time, sees

- ACC12Test- ACC12Calc 5.13

C\J < E Q ~

Kajima, Test 73P Earth Press, EP5

0.25~--------------------,

0.2

0.15

0.1

0.05 0-+---~

-0.05

0 1 --------------·--------------------------- ----------- -- ----------------------------------------------------------------------------------- .

-0.2-+----~----r------r-----r-----.------1

0 1 2

73P,EP5

3 Time, sees

4

EP5 (Calc)* -1

5 6

5.14

Kajima, Test 73P Earth Press, EP?

0.4~--------------------------------------------~

0.~ ······················-···········-······················· . . ····-··-··························-·····-··-·····-··············-··············

().~ ·······--··----------------------------------------------- -- - ---------------------------------------------··-----·------------------··------

(). 1 ··········-·····-··-··-·····················- ··-······· ..... ····················--·········-·······-··-·····-··-···········-······-·-··-·

-().~-+-------.-----.------,.---~-----.---------1

() 1 ~

73P,EP7

~

Time, sees 4

EP7 (Calc)*-1

5 6

5.15

C/) -C/) LO Q) I

~ w E 0

""0 ~

C/)

c Q)

-~ E '- J-+-'

(f) .._..

Kajima, Test 73P Strain, ST3

2~----------------------~~------------------------~

1.5

1

0.5

0

-0.5 ----------------------------------------

-1

-1.5

2 ------------------------------------------------------ -----------------------------------------------------------------------------­-

-2.5-+----~--..,.-------~----r---~--~

0 1 2 3 Time, sees

4 5 6

73P,ST3*1.0E-06 -1.0*(ST3 Calc) 5.16

Kajima, Test 73P Strain, STS

6~------------------------------------~

~ ................................................................... ··································································-····

-~ ·•··••·••••·•······•••••···········••••···········•···· .. . ..................................••......•..••............................

-6~----~----~------~----~----~----~ 0 1 2 3 ~ 5 6

Time, sees

- 73P,ST5*1.0E-06 -1.0*(ST5 Calc) 5.17

<D ()

Kajima, Test 73P ACC12 Camp Test Vs Calc

300~------------------------------------~

~00 ·························-··-····················-· ························································

~ -10 ---- -------- ---------------------- -- ----------------1

-~0 ··········································· .. ······--····-··-··-··-··-··-··-············-·-··-····--··-··

-30~----~----~----~------~----~----~ 0 1

1- ACC1~ Test

~ 3 Time, sees

4 5 6

- ACC1~ 3M Calc ---- ACC1~ 10M Calc j

5.18

C\1 < (.) Q)

J!!. c

Kajima, Test 73P ACC12 3 Mass Vs 10 Mass

250~----------------------------------------~

'" I 200

150 100 50

-50 -100 -150 -200

r··----------------------------------------------------------------------1

--------------------------------------------------~-----------£\ __________________________________________________________ _

I -- 1-----------1 I I I I I 1 I r/ ~---------- ,- :··---------\}L-------------------------------------------------------1 \ I

- L-----------~--------------------------------------------------------------------1 I I

1----------------------------------------------------------------------------------\ I I

-250-+----~--~-...x.----T----.----~----1

0 1 2 3 Time, sees

4

ACC123 Mass ----- ACC12 10 Mass

5 6

5.19

C\J < (.) Q)

~ c ... .

C/F Test Vs SAP90, 1g ACC12

30~--------------------------------------~

~0 ------------------------------------------------------------------------------- ----------------------------------------------------------------

-~0 ---------------------------------------------------------------------- ---------------------------------------------------------------------------

-30-l---.,....-----or---.------,.----r----.-----..----r----.------l 0 0.5 1 1.5 ~ ~.5 3 3.5 4 4.5 5

Time, sec

- Test7~B SAP 90 5.20

... . -

C/F Test Vs SAP90, 1g ACC12

30~--------------------~----------~

~0 --------------- ------------ ---------------------------------------------------------------------------------------------------------------------

Q) (.)

(.) -1 0 ---------- - - -- - - -- ------------------ ----------------- -----------------------------------------------------·---------------------<(

-~0 ------------- -------------------------------------------------------------------------------------------------------------------------------------

-30+----------.---------.----------,---------r------l 0 5 10 15 ~0 25

Time, sec;

- Test728 -SAP 90 5.21

C\J < (.) Q)

.!!!. E (.)

... . -Q) (.) (.) <(

Kajima: Test 73P SAP90 Calc Vs ACC12 Test

800~------------------------------------------~

E>OO --------------------------- ------------------------------------------------------------------------------------------------------------------

400 -----·-·-------------- ....

200 -------------------- - -

0

-200 ----------------·· ---

-400 -----------------------

-E>OO

-800-+----~----r-----r------"T--~--~-----t

0 2 4 E) 8 10 12 14 Time, sees

- ACC12test - SAP90Calc 5.22

Kajima: Test 73P SAP90 EP7 Comparison

0.5-y-----------------------.

0.~ ---------------------------------------------------------------- -------------------------------------------------------------------------------

~ 0.~ ---------------------------------------------------------- -- ---------------------------------------------------------------------------------E ~ ~

... ~ :::J (/) (/)

~ a_

0.~ ---------------------------------------------------------- --- -- -------------------------------------------------------------------------------

0.1 ---------------------------------------~----- ---------- ------- ----------- -----------------------------------------------------------------

-0.1 -······--···········-·-······-··-··--·-···-········-··-· ---····--·--·-··-·········--·----·······-------···············- -·····-·····-·······--

-0.~-+-------.---~---------..---~--__....--;-----l

0 1 ~ ~ ~ 5 6 Time, sees

- 73P,EP7 - SAP90, Calc 5.23

(/) .--..... (/) L() Q) I - w ~

E 0

"'0 ~

(/)

c Q)

·co E ~ i-~

(/) -

0 1

Kajima; Test 73P SAP90 vs Test, Cyl Strain

2

ST5*1.0E-06

3 Time, sees

4

(SAP soil str)*-1

5 6

5.24

en -en LO Q) I

w E 0

""0 ~

en c Q)

·co E ~ F +-" (/) -

Kajima: Test 73P SAP90 vs Test, Cyl Strain

10~--------------------------------~

0 1

ST5*1.0E-06

2 3 Time, sees

(SAP Soil str)*-1

4 5 6

(SAP Cyl Str) * -1 5.25

C\1 < 0 CD en --E 0 .. . -CD 0 0 <(

800

600

400

200

0

-200

-400

-600

-800 0

Kajima: Test 83 SAP90 Calc Ace vs Test

(A ~~. rv"Atf:l.rt:J.tA .. p/)j

JI.Jv vr v "" 1)JJ , v·~

\/ ·~

I I

1 2

_· __ ._J

' ~ i I

A ;

~ \ - ~ " ~.! \

I"

~ v

\.. \

I

3 Time, sees

,, {\

"' \K a r\ ~

1G

~ v

I

4

A 1\ 1\

J ~I

~v v.

1--· SAP90 Cyl Ace - ACC12, Test 83

I

I

5 6

5.26

LINE VARIABLE SCALE FACTOR

! ACC-N(J:1E f. +1 ,t'\H;-O~t

3 .------,.------

-3 ,___ _ ___. __ ____,

c 1 TIME

2

(*10**-1)

5.27

3~------~~----~

LINE VARIABLE SCALE FACTOR

! ACt';-NQm". 3E' +1. t11F~·-O:J

2 1- -

-2 L....--------!.J.._ __ L.___I ____ _

0 1 2

TIME-SEC (*10**-1)

5.28

LINE VARIABLE SCALE FACTOR

1 ACC-NO!'lt~ 406 +1.(lH.;-Vl

2 .-------.-1 --------.,

-2 ._____ ____ __,_! ____ ____,

0 1

TIME-SEC 2

(*10**-1)

5.29

LINE VARIABLE SCALE FACTOR

1 ACG-NOn~-: 436 +1.tl1F;-O)

2 ~--------~------~

-2 L....-..-_____ L__ ___ ____j

0 1 TIME-SEC

2

(*10**-1)

5.30

LINE VARIABLE SCALE FACTOR

DJSP.-NCH.it. 6 +l.('IOF;+O(J

6

5

4

3

>< ..... 2 0 H

"' 'tJ

I n 3:

0

TIME-SEC 2

(*10**-1)

5.31

0.4 !

0.3 ~-····~ Acc2 l

0.2 ,...._ 0.0

0.1 '-'

c .s::

0 -tU ..... ~

0 -0.1 u u <

n

t\~1-~l,llnJJ.t tAJA. .. d~M ~-·· ~nr''"' " VY ~ : w ~ v lfl

. -~ .... :

-0.2 ..........

-0.3 ........

-0.4 0 2 4 6 8 10 12

Time (second)

5.32

0.6

0.4

-bJ) '-'

c 0.2 0

·.;::: ~ 1-o 0 ~ 0 u u <

-0.2

-0.4 0 2 4 6 8 10 12

Time (second)

5.33

0.5

AceS

0.3

-bl) ._.,

c: 0.1 .9 .... ~

'"" 0 0 -0.1 (.) (.)

<

-0.3

-0.5 0 2 4 6 8 10 12

Time (second)

5.34

0.4 :

0.3 ~ .... .J Acc6

0.2 :

······ i···· -b.O '-'

= 0.1 _g -~ ..... ~ 0 0 (.) (.)

< -0.1

·······

~ ~·····~ ~ ~~ I!JuA.,~ ...... ~ ~ !••••

~ ....

~ rvv vv,vv'r y

: \1 w .... ~ ~! ..... :······

:

-0.2 ······················

-0.3 0 2 4 6 8 10 12

Time (second)

5.35

0.4 !

0.3 l ..... l Acc7 ........

0.2 ........ bl) - ~ c:: 0.1

.52 -"' "" (I) 0 0 (.) (.)

< -0.1

I .... ft

diJ.uJ11 "ll!l.. .. M1 ~ j,

'v l r~ ~ ~. j"'''I 1JP" "V' II\"

~ ..... ~ I~ :

~ :

-0.2 , ......

-0.3 0 2 4 6 8 10 12

Time (second)

5.36

1

.......... 0.5 b.O '-'

~

.!:? .... Cl:l .... 0 0 (.) (.) 0 <

-0.5 0 2 4 6 8 10 12

Time (second)

5.37

1

---Acc13

0.5 ..-.. bO ._.,

c= .s: 0 ....

~ ··························--­

'"' 0 0 u u <

-0.5 ·····························i-··-·-···························!································i······················ ········t················ ·········· ·--~---·················------···

l 1 1 i i : : : : :

I I I I I : : : : !

-1

0 2 4 6 8 10 12

Time (second)

5.38

0.8

0.6

0.4 -b.O 0.2 '-'

= .9 0 -~

""' 0 0 -0.2 u u <

-0.4

-0.6

-0.8 0 2 4 6 8 10 12

Time (second)

5.39

1

0.5

,....... 0 bJl .._,

c:: 0

·.;::: -0.5 C<$ .... 0

Q) (.) (.) -1 < 1-·················'·············i-·································o·································i··············+·····+i······+··ll+······+··+++i··+······························--i

-1.5

-2 0 2 4 6 8 10 12

Time (second)

5.40 .,

l Date 1992. 10. 12 1992. 10. 13 1992. 10. 14 1992. 10. 14 1992. 10. 14 1992. 10. 14 1992. 10. 14 Experiment name CUREe No. 72B CUREe No. 73P CUREe No. 75 CUREe No. 76 CUREe No. 77 CUREe No. 78 CUREe No. 79 Centrefugal 50G 50G 50G 50G 50G 50G 50G

acceleration File name CUREE72B CUREE73P CUREE75 CUREE76 CUREE77 CUREE78 CUREE79 Input wave El Centro El Centro Sine 100Hz Sine 150Hz Sine 200Hz . Sine 250Hz Sine 300Hz Acceleration

fraquency 100Hz 150Hz 200Hz 250Hz 300Hz

Acceleration wave 1. 22 G 9. 98 G 4. 53 G 15.47 G 3. 97 G 6. 00 G -5.63 G Shaft H/D=5 H/D=5 H/D=5 H/D=5 H/D=5 H/D=5 H/D=5 h % ~ ~ ~ ~ ~ ~ ~ (em) After 14 IE 1 4 2 8 .

p d % ~ ~ ~ /C /C /C /C (gf/cm 3 ) After 4 1 5 5 5 5 8 e % ~ ~ ~ ~ ~ ~ ~ After 2 4 0 0 0 0 7 D r % ~ ~ ~ ~ ~ ~ ~ (!ll) After 17 ~~ 4 8 0 8 6

TABLE 1(a)

Date 1992.10.21 1992.10.21 1992.10.22 1992. 10. 22 1992.10.22 1992. 10. 2-2 1992.10.22 Experiment name CUREe No.82B CUREe No. 83 CUREe No. 85 CUREe No. 86 CUREe No. 87 CUREe No.88 CUREe No.89 Centrefugal 50G 50G 50G 50G 50G 50G 50G

acceleration File name CUREE82B CUREE83 CUREE85 CUREE86 CUREE87 CUREE88 CUREE89 Input wave El Centro El Centro Sine 100Hz Sine 150Hz Sine 200Hz Sine 250Hz Sine 300Hz Acceleration

fraquency 100Hz 150Hz 200Hz 250Hz 300Hz

Acceleration wave 1. 09 G 9. 83 G 4. 75 G 16. 76 G 4.34 G -6. 78 G 5. 93 ·G Shaft H/D=l. 5 H/D=l. 5 H/D=l. 5 H/D=l. 5 H/D=l. 5 H/D=l. 5 H/D=l. 5 h

~ % ~ ~ % ~ ~ ~ (em) Aftez )4 8 8 3 7 4 3 p d

~ ~ ~ ~ ~ ~ ~ .~ (gf/cm 3 ) Aftei 1 1 3 4 4 5 7 e

~ ~ ~ ~ ~ ~ ~ ~ Aftez 8 4 1 1 1 0 0 Dr

~ ~ ~ ~ %. ~ ~ ~ (%) Aftez 6 5 2 2 2 7 7

TABLE 1 {b)

TABLE2

BEST-FIT ENGINEERING MODEL PROPERTIES (v=0.4)

Depth Test Eat Base Gat Base Torsion Spring Exponent Damping

(EE) (GG) (EF) (B) BETA

psi psi lb-in/rad (% critical)

(kg/cm2 ) (kg/cm2) (kg-m/rad)

72B 9000 3217.5 5.0 x1010 0.5 0.007

634 227 5. 77 X 108 10%

73P 7800 2788.5 7.0 X 1010 1.0 0.035

550 196 8.08 X 108 25%

TABLE3

SAP90 MODEL PROPERTIES

Young's Shear Modulus G

Element Material Density ModulusE

( = E }kgf/cm2) Identification (kg!cm3

) kgflcm2 Poisson's Ratio v 2(l+v)

1 1.6E-06 162 0.4 58

2 1.6E-06 276 0.4 99

3 1.6E-06 357 0.4 128

4 1.6E-06 421 0.4 150

5 1.6E-06 478 0.4 171

6 1.6E-06 527 0.4 188

7 1.6E-06 574 0.4 205

8 1.6E-06 612 0.4 219

9 1.6E-06 643 0.4 230

10 (shaft) l.OE-08 10,000 0.4 3570

TABLE4

PROPERTIES EMPLOYED IN ABAQUS MODEL

Young's Modulus Density Material dynes/cm2 kg/cm2 Poisson's Ratio gm/cm3

"Rigid Box" 2.07 X 1012 2.11 X 106 0.28 7.78

Silicone 2.35 X 106 2.4 0.49 0.8

"Membrane" 6.6 X 103 6.7x1o-3 0.49 0.4

Shaft, Aluminum 7.06 X 1011 7.2x105 0.3 2.8

Soil Layer 1 1.24 X 109 1264 0.4 1.583

Soil Layer 2 1.93 X 109 1967 0.4 1.583

Soil Layer 3 2.37 X 109 2416 0.4 1.583

Soil Layer 4 2.71 X 109 2762 0.4 1.583

Soil Layer 5 3.00 X 109 3058 0.4 1.583

Soil Layer 6 3.25 X 109 3313 0.4 1.583

Soil Layer 7 3.47 X 109 3537 0.4 1.583

Soil Layer 8 3.65 X 109 3721 0.4 1.583

Soil Layer 9 3.80 X 109 3874 0.4 1.583

All materials linearly elastic

TABLES

ABAQUS MODAL FREQUENCIES

Frequency, Hertz

Mode Model Prototype

1 163.72 3.27

2 352.88 7.06

3 434.25 8.69

4 464.21 9.28

5 596.46 11.93

• • •

• • •

• • •

15 888.58 17.77

APPENDIX 1

Toyoura Sand Properties

Physical and mechanical properties of TOYOURA sand

!)Specific gravity Gs=-2.65

2)Bulk density - COQpacted 1.645g/cm 1

-uncompacted 1.335g/cm~

3)Grain size distribution -+ Fig. 1

4)Void ratio(target value) e = 0. 69

5)Friction angle

6)Shear strength on horizontal plane of soil ele~ent 'tt- eq.]-3

7)Iinital shear modulus

8)Dynnmic properti~;

G •l - eq. 4-6

G/Gn -r/rr, h-7/r .. - Fig. 4

100

dfJ

Ol 80 c ·.-l en en 60 10 0.

(1) 40 Ol 10 .j..J c (1) 20 u H (1) ll.

l

.001

I [ I I i !

i I •

I ! I i I I I

I I I ! I, i

!

1-1 I i I I I I •

I i ! i

.01

I I J I I I I i

I 1 I I I I I I I I

I I ·I I I

I I I I

I l ! !

I II J ! I I i : : i

I !

.1 Grain diameter (mm)

Clay Silt . 0.005 0.075 0.425

Fig. 1 Grain SIZe distribution of

.,

I i

I I i I ! I

I I I I I I I

i

i !

1 10

Gravel

2.0 4.75

TOYOURA sand

a11

< 1 I

> a---=::;3.. >\ I -rl .__ ___ ___..

initial stress condition stress condition subject to sheaing

• _ 1 o, -a, ; ~ 1

¢ = s 1 n ( a 1 + a ;) :.t" x J

initial

final(failure)

v a.

a 11 = K o a v

Based on microgranuler theory as well as sophisticated experimental results, o 1 in this test is considered to be al~ost constant throughout entire sheaing stage,

(-+see reference 1)

Fig. 2 Definition of ¢

(Initial and fjnal ~ohr circle for ) £oil element under :3implc sheat· G~uJi.Llun

. '

50.----~----~~~

0. 69 30~--~----~--~

0. 6 0. 7 0. 8 0. 9 VOID RATIO e

Fig. 3 Relationship between e and

1.0

0. 8

0. 6 G/Go\

0.4

0. 2

r,.=,r/Go

(Go from eq. 4-eq. sv G/Go

' f from eq. 1 ..._ eq. 3!

h'

~

0. 5 1. 0 5. 0 10 50 100

r /r,.

Fig.4 Dynamic properties of TOYOURA-~~~~ (G/Go,_r/r,., h,_r/r,.)

50

40 .

30 h·

...•

20 (%)

10 .

0 500

'C r =

where:

2 (

£..!..- K ~2 av • (J \'

(From Fig.2)

a • (_1 + sin~) av =Ku ~1- sin¢;

K u == 0. 52 e

(from experimental results)

Evaluation of 7: r

G n = 900 ( 2. 17 - e ) 2 p o. + 1 + e

where: P'= 1+2Ko av

3

K o = 0. 52 e

(from experimental results)

(4)

(5)

(6)

Evaluation of Initial tangent shear modulus Go

(1)

(2)

(3)

APPENDIX2

ENGINEERING MODEL PROGRAM AND EXPLANATION

Lines Function

30-45 Interpolates earthquake acceleration linearly between input data points, if the time inteiVal is greater than the value of time step H selected (uses parameter AH).

1072 This is the file containing the acceleration base motion for input (in in/sec2 units).

1076-1083 The material properties (at base of column) and dimensions are input here, as well as the number of masses (N) to be used.

1249-1253 Damping matrix is a constant (BETA) times stiffness matrix.

1300-1311 Prints on the printer the values of various parameters calculated in the program, in case it is desired to check the, or use mass and stiffness matrices to obtain modal shakes and frequencies.

3090, 30095 Acceleration equations including effect of pitching motion. If no Pitching, then second term in line 3090 is -EQA1, and line 3095 is +EQAl.

(4045) 4046 Write to output file the calculated values of selected variables at the output time inteiVal (AT). In this section, choice of GOTO statements and apostrophes (') controls which variables are written to the output file.

3 4 5 6 7 8 9

10 11 20 30 32 34 35 37 38 39 40 41 42 43 45 49 50 59 60 61 62 70 98 99 100 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 130 131 132 133 134 135 136

RS8FIN.BAS

'THIS PROGRAM COMPUTES THE DYNAMIC RESPONSE OF A DAMPED MDOF SYSTEM. 'In this form it represents the interaction of a soil shear beam and 'and a rigid adjacent shaft connected to the shear beam by springs. ' It is based on a program in Glen V. Berg's book: ' "Elements of Structural Dynamics", 1989, Prentice Hall, 'modified by K. Rubenacker for a soil column, and rewritten by 'R. F. Scott for imbedded shaft problem, 11/92 - 1/93.

'-------------------------------------------------------------GOSUB 1000 GOSUB 2000 WHILE T < TEND

EQAOLD = EQANEW

cc = 0 WEND

DO WHILE CC < AH CH = CC I AH

LOOP

IF CC = 0 OR CH = INT(CH) THEN INPUT #2, EQANEW EQAX = EQAOLD + CH * (EQANEW - EQAOLDl EQA1 = EQAX cc = cc + 1 GOSUB 5000

'OUTPUT HEADING PRINT #1, " TravelTime="; TRAVT; 11 Tau="; TAU; 11 P/Height= 11

; PONH PRINT #1, CLOSE #1 CLOSE #2 END

'VARIABLE DEFINITIONS:

'INPUT VARIABLES: 'HEIGHT =height of soil column (ft) 'N =# of DOF 'P =amplitude of displacement at base (in) 'T1 =duration of displacement (sec) 'H =time step (sec) 'TEND =ending time (sec) 'GG =shear modulus at base of soil column (psi) 'GAMMAT =unit weight of the saturated soil (pcf) 'B =the power of Z (depth) that G (shear mod.) is proportional to 'BETA =the damping matrix is proportional to the stiffness matrix

by the factor BETA 'X =effective distance in calculation of spring constant, AA (in)

(say, length of box/6 for a start) 'Y =ratio of spring constants, BB/AA 'COLWID =width of soil column considered to be shear beam 'EF =torsional spring constant at base of shaft (pound-in/rad) 'AH =ratio of e/q accel data spacing to time step, H 'AT =ratio of time to time step; number of time steps 'AK =result of MOD calculation: AT MOD RT 'RT =number controlling print time step: 10 for 0.01s; 100 for 0.1s 'CALCULATED AND OUTPUT VARIABLES 'TRAVT =time it takes for stress wave to travel length of soil column 'THETA =rotation of shaft 'BM(Il =bending moment in shaft 'TAU =dimensionless "time" variable

=T1/TRAVT 'PONH =dimensionless "distance" variable

137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 999 1000 1010 1020 1030 1031 1040 1050 1054 1055 1060 1070 1071 1072 1073 1074 1075 1076 1077

=input displacement/column height 'GAMMAB =buoyant unit weight of saturated fLOil ( pcf) 'Wii) =weight of lumped mass #i !pounds)

note: the top mass is 1/2 that of the others 'Z ( i) =depth to middle of "spring" # i ( ft) 'G( i) =shear modulus of soil at depth Z! i) (psi) 'K(i,j) =stiffness matrix 'STIFF!i)=spring constant of "spring" #i (pounds/inch) 'C!i,j) =damping matrix

note: we can input any damping matrix, but for simplicity it has been made proportional to the stiffness matrix.

'U(il =displacement of lumped mass #i w.r.t. the base 'S!i) =average strain between lumped masses i and i-1 'V(il =velocity of lumped mass #i w.r.t. the base 'K1,K2,K3,K4= temporary variables used in the Runge Kutta subroutine 'SMAX(i)=maximum strain experienced at DOF #i 'EQA,EQA1 =acceleration of base of soil column 'D ="resistance to movement of mass #i due to damping (pounds) 'Q =force on mass #i due to springs (pounds) 'A(i) =relative acceleration of mass #i w.r.t. the base (in/sec~2)

Springs AA Springs BB I I Ground surface

I I><>I W( 2) I><><I <->U!2l. I I I I I I I I I I I I I <--Shear spring

L! I l I I I I I I I I I I

I I I><>I W( 1) I><><I <->U(1l I<---H(I) I I Wall- I I I I I I I I I I I-Shaft I I v I I I I Z! I l I I Spring I <--Shear spring I I I EF \ I I I I ( ( I ) ) Base (accel @ EQAl)

1//11\\\\\ 1111\\\\\ /1//1//1\\\\\\ 1111\\\\ ll/1\\\/1111

'AA(i) =spring constant effective between soil mass and end of box 'BB(i) =spring constant effective between soil mass and shaft 'L(i) =height from base to center of soil element 'H!i) =height of soil element 'E(i) =Young's modulus of soil at level of center of each element 'BMMAX(I)= maximum bending moment in shaft 'BMBASE =bending moment at base of shaft 'GG(I) =force at each node acting on shaft

'HOUSEKEEPING DEFINT I-J, N F3$ = "####.###" F4$ = "#.####~~~~" F5$ = II ####.#####" GRAV = 386.088 PI = 3.141593

'INPUT THE FILE THAT OUTPUT IS SENT TO (0$) ----------­CLS : INPUT "OUTPUT DEVICE OR FILESPEC: ", 0$ OPEN 0$ FOR OUTPUT AS #1 'INPUT THE EARTHQUAKE SOURCE FILE (E$) INPUT "INPUT DEVICE OR FILESPEC: ", E$ OPEN E$ FOR INPUT AS #2 , 'INPUT DATA AND THEN SEND IT TO THE OUTPUT FILE -----------­INPUT "INPUT: COLWID(ft),X(ft),Y ", COLWID, X, Y INPUT "INPUT: EF(pound-inch/rad),EE(psi) ", EF, EE

1078 1079 1080 1081 1082 B 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1102 1103 1105 1106 1107 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1155 1156 1157 1158 1159 1160 1179 1180 1181 1190 1200

- . -PRINT #1, "COLWID="; COLWID; " X="; X; " Y="; Y PRINT #1, " EF="; EF; "EE="; EE INPUT "INPUT: HEIGHT(ft), N,- RT ", HEIGHT, N, RT INPUT "INPUT: AH, H!sec), TEND!secl ", AH, H, TEND INPUT "INPUT:GG(psi),GAMMAT(pcfl,B(the power of Z for Gl ", GG, GAMMAT,

INPUT "INPUT:BETA( C(i,jl=BETA*K(i,j)) ",BETA PRINT #1, "HEIGHT="; HEIGHT; " N="; N; " RT="; RT PRINT #1, "AH="; AH; " H="; H; " TEND="; TEND PRINT #1, "GG="; GG; " GAMMAT="; GAMMAT; " B="; B; " BETA="; BETA PRINT #1, "FILENAME="; 0$ PRINT #1, 'DIMENSION THE VARIABLE ARRAYS AND MATRICES-----------­DIM C ( N, N) , K ( N, N ) , U ( N ) , V ( N) , A ( N) , AC ( N ) DIM W(Nl, S(N), SMAX(N) DIM STIFF ( N ) , AA ( N ) , BB ( N ) , E ( N ) , G ( N ) , H ( N l , L ( N ) , Z ( N) DIM GG(N), EP(N), SS(N), BM(N), BMMAX(N) DIM OLDU(Nl, OLDV(N), K1(N), K2(N), K3(N), K4(N)

'CALCULATIONS -------------------COLWID = COLWID * 12 TRAVT = (GAMMAT I 1728 I GG I GRAY) A .5 * (HEIGHT* 12) I (1 - B I 2) TAU = T1 I TRAVT PONH = P I (HEIGHT * 12) GAMMAB = GAMMAT - 62.4 FOR I = 1 TO (N- 1)

W(I) = GAMMAT *HEIGHT I N * (COLWID I 12) A 2 NEXT I W(N) = GAMMAT *HEIGHT IN I 2 * (COLWID I 12) A 2 FOR I = 1 TO N II = I - 1

NEXT I

Z(I) = HEIGHT -HEIGHT IN* II -HEIGHT IN I 2 G(I) = GG * (Z(I) I HEIGHT) A B

'HEIGHT TO CENTER OF EACH ELEMENT FROM BASE--------­FOR I = 1 TO (N - 1) II = I - 1

L(I) = HEIGHT IN* II +HEIGHT IN NEXT I L(N) = (HEIGHT* (4 * N- 1)) I (4 * N) 'HEIGHT OF EACH ELEMENT----------------------------­FOR I= 1 TO (N- 1)

H(I) = HEIGHT I N NEXT I H(N) = HEIGHT I N I 2 'CALCULATE YOUNG'S MODULUS AS FUNCTION OF DEPTH----­FOR I = 1 TO N II = I - 1

E(I) = EE * ((HEIGHT- L(I)) I HEIGHT) A B NEXT I 'CALCULATE SPRING CONSTANTS, AA(I) AND BB(I)---------­FOR I = 1 TO N

NEXT I

AA(I) = (E(Il * COLWID * H(I)) I X BB(I) = Y * AA(I)

'CALCULATE DENOMINATOR R J(INCLUDES SPRING CONSTANT OF SHAFTl--­FF = 0 FOR I = 1 TO N

FF = FF + BB(I) * 144 * L(I) ~ 2 NEXT I R = EF + FF

'CREATE STIFFNESS MATRIX FOR I = 1 TO N

FOR J = 1 TO N

1210 1220 1225 1226 1227 1228 1230 1231 1232 1233

K! I, J l = 0 NEXT J

NEXT I FOR I = 1 TO N

STIFF!Il = NEXT I NN = N - 1 FOR I = 1 TO NN

II = I + 1

G(I) I (HEIGHT* 12 IN) * COLWID A 2

K( I, Il = STIFF!Il + STIFF(II) + AA(I) + BB(I) - BB!Il A 2 * (12 * LII) l ~ 2

1238 I R

K!I, II l K(II, I)

= -STIFF!II) - BB(Il * BB(II) * L(I) * L!IIJ * 144 I R = K!I, II) 1239

1240 1241 1242 1243 1244 1245 1246 1247 I R 1248 1249 1250 1251 1252 1253 1254 1255 1256 1260 1270 1280 1290 1292 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1350 1900 1901 1902 2000 2010 2011 2012 2013 2020 2030

NEXT I FOR I = 1 TO N

NEXT I

FOR J = I + 2 TO N

NEXT J

K ( I , J l = - BB ( I ) * K(J, Il = K(I, J)

BB(J) * 144 * L(I) * L(Jl I R

K(N, Nl = STIFF(N) + AA(N) + BB(Nl- ((BB(Nl) ~ 2) * (144 * (L(N)) A 2)

'CREATE DAMPING MATRIX -------------------­FOR I = 1 TO N

FOR J = 1 TO N C(I, Jl = BETA * K(I, J)

NEXT J NEXT I 'THE 11 GOT0 11 STATEMENTS PERMIT SELECTION OF VARIABLES FOR PRINTING ----GOTO 1290 PRINT #1, II FOR I = 1 TO N

T ". '

PRINT #1, II s (II; I; II) II; NEXT I PRINT #1, GOTO 1350

11 W(I)= 11 i W(I); "Z(I)= 11 i ZCI); 11 G(Il= "; G(I); "E(Zl= "; E(Il; "L(I)= "; L(I); "H ( I ) = " ; H ( I ) ; "AA(I)= "; AA(I);

FOR I = 1 TO N LPRINT LPRINT LPRINT LPRINT LPRINT LPRINT LPRINT LPRINT LPRINT LPRINT

"BB ( I ) = II j BB ( I ) j

"STIFF(!)= "; STIFF(!);

NEXT I FOR I = 1 TO N

FOR J =

NEXT I RETURN

NEXT J

1 TO N LPRINT

'LPRINT LPRINT

"K(I,J)= "; "C(I,J)= ";

'STARTER T = 0 F = 0 EQANEW cc = 0 FOR I =

= 0

1 TO N U( I) = 0

K( I, J) i C(I, J);

2035 2036 2040 2041 2042 2044 2050 2060 2065 2070 2080 2900 2901 2902 3000 3004 3010 3030 3040 3050 3060 3070 3080 3085 3089 3090 3095 3100 3110 3900 3901 3902 4000 4010 4014 4015 4019 4020 4030 4040 4041 4042 4043 4044 T--4046 4047 4051 4052 4053 4054 4055 4056 4057 4058 4059 4060 4061 4062 4063 4065 4066 4067 4068 4069 4070

SMAX!Il = 0 S(I) = 0 V(I) = 0 GG( I) = 0 BM (I l = 0 BMMAX(I) = 0

NEXT I GOSUB 3000 'Apostrophe in front bypasses output subroutine----­GOSUB 4000 RETURN

'---------------------------------------------------------------------'ACCELERATION

'EQA1 IS THE BASE ACCELERATION FOR I = 1 TO N

D = 0 Q = 0 FOR J = 1 TO N

NEXT J

D = D + C(I, J) * V(J) Q = Q + K(I, J) * U(Jl

'THIS IS THE DIFFERENTIAL EQUATION THIS PROGRAM SOLVES WITH RUNGEKUTTA: A(I) = -(D + Q) * GRAV / W(I) - .0149812 * (L(I) + 66.75) * EQA1 AC(Il = A(Il + .0149812 * (L(I) + 66.75) * EQA1 NEXT I RETURN

'---------------------------------------------------------------------'OUTPUT

'PRINT #1, USING F3$; T; 'BYPASS PRINTING IF WE WISH GOTO 4043 'PRINT STRAINS FOR I = 1 TO N

PRINT #1, USING F4$; S(I); NEXT I PRINT #1, 'BYPASS PRINTING IF WE WISH 'GOTO 4052 'PRINT DISPLACEMENTS, ROTATION, ACCEL., MOMENT, EARTH PRESSURE, E/Q INPU

WRITE #1, (T- H), SS(N- 3), EP(N- 3) 'WRITE #1, AC(N) 'BYPASS PRINTING IF DESIRED CLS : PRINT "Time = "; T GOTO 4059 'PRINT MAXIMUM STRAIN

FOR I = 1 TO N PRINT #1, USING F5$; SMAX(Il;

NEXT I PRINT #1, GOTO 4063 'PRINT THETA PRINT #1, USING F4$; THETA; PRINT #1, GOTO 4070 'PRINT BENDING MOMENTS

FOR I = 1 TO N

NEXT I PRINT #1, GO'T'O 4100

PRINT #1, USING F4$; BM(I);

4071 4072 4073 4074 4075 4076 4077 4078 4079 4080 4090 4091 4092 4093 4100 4900 4901 4902 5000 5010 5011 5020 5030 5040 5050 5060 5070 5080 5090 5100 5110 5120 5130 5140 5150 5160 5170 5180 5190 5200 5210 5220 5230 5240 5241 5242 5244 5245 5252 5253 5254 5255 5256 5258 5260 5261 5262 5263 5264 5265 5266 5267 5268 5269 5270 !)272

'PRINT MAXIMUM BENDING MOMENT FOR I = 1 TO N

PRINT #1, USING F4$; BMMAX!Il; NEXT I

PRINT #1, 'PRINT BM AT BASE OF SHAFT PRINT #1, USING F4$; BMBASE; PRINT #1, 'PRINT ACCELERATION AT TOP OF SOIL COLUMN PRINT #1, USING F4$; A(N); 'PRINT FORCES AT NODES ACTING ON SHAFT

FOR I = 1 TO N PRINT #1, USING F4$; GG(I);

NEXT I RETURN

'---------------------------------------------------------------------'RUNGE-KUTTA

T = T + H I 2 'RETRIEVE NEXT EARTHQUAKE ACCELERATION FROM E$----­FOR I = 1 TO N

OLDU(Il = UII) OLDV( I) = VII) K1(I) = H * A(I) U(I) = OLDU(Il + H * (OLDVII) I 2 + K1(I) I 8) V(I) = OLDV(I) + K1(I) I 2

NEXT I GOSUB 3000 FOR I = 1 TO N

K2(I) = H * A(I) V(I) = OLDV(I) + K2(I) I 2

NEXT I GOSUB 3000 T = T + H I 2 FOR I = 1 TO N

K3 (I) = H * A (I) U(I) = OLDU(I) + H * (OLDV(I) + K3(I) I 2) V(I) = OLDV(I) + K3(I)

NEXT I GOSUB 3000 FOR I = 1 TO N

K4(I) = H * Ali) U ( I ) = OLDU ( I ) + H * ( OLDV ( I ) + ( K1 ( I ) + K2 ( I ) + K3 (I ) ) I 6 ) V ( I ) = OLDV ( I ) + ( K1 ( I ) + 2 * ( K2 ( I ) + K3 ( I ) ) + K4 ( I ) ) I 6 NEXT I I

'CALCULATE STRAINS FROM DISPLACEMENTS---------­S(1) = U(1) I (HEIGHT IN) I 12 FOR I = 2 TO N

II=I-1 S(I) = (U(I)- U(II)) I (HEIGHT IN) I 12

NEXT I 'PICK OUT MAX STRAIN AT EACH DOF-------------­FOR I = 1 TO N

IF ( s ( I ) ) A 2 > ( SMAX ( I ) ) A 2 THEN SMAX ( I ) = s ( I ) NEXT I 'CALCULATE ROTATION AND ROTATION RATE OF SHAFT THETAR = 0: THETA = 0: THETAS = 0: THETAD = 0 FOR I = 1 TO N

NEXT I

THETAR = THETAR + (BB(I) * 12 * L(I) * U(I)) THETAS= THETAS+ (BB(I) * 12 * L(I) * V(I))

THETA = THETAR I R: THETAD = THETAS I R 'CALCULATE FORCES, EARTH PRESSURES ACTING ON SHAFT------------­FOR T = 1 1'0 N

5273 l - THETAD * 12 5274

GG(Il = BB!Il * lUI I) -THETA* 12 * L!Il) +BETA* BB!Il * (V(I * L(I)) EP(I) = (GG(I) * .07045) I (118.11 * 12 * H(I))

5275 5287 5288 5289 5290 5291 * (V(J)

5292 5293 5294 5296 5297 5298 5299 5300 5310 5320 5330 5335 5340 5350 5400

NEXT I 'CALCULATE BENDING MOMENT, STRAIN IN SHAFT-----------­FOR I = 1 TO N - 1

BM(Nl = 0: BM(I) = 0 FOR J = I + 1 TO N

BM!Il = BM!Il + BB(J) * !(U(J)- THETA* 12 * L(J)) +BETA - THETAD * 12 * L!J))) * !L(J) - L(I)) * 12

SS!I) = !BM!Il * 59.05) I (2.35E+13) NEXT J

BMBASE NEXT I

= THETA * EF + BETA * THETAD * EF

'FIND MAXIMUM BENDING FOR I = 1 TO N

IF ( BM (I) ) A

NEXT I GOSUB 3000

MOMENT IN SHAFT----------

2 > (BMMAX(I)) A 2 THEN BMMAX(I) =

'Apostrophe in front bypasses output subroutine----­AT = T I H AK = AT MOD RT IF AK = 0 THEN 4000 RETURN

BM(I)

APPENDIX 3

Tabulation of Computer Codes

APPENDIX3

SELECTED DYNAMIC NUMERICAL CODES IN GEOMECHANICS

Computer Code ABAQUS(l)

Analysis Method Finite elements linear and nonlinear

Geometrical Dimensions 1-D/2-D/3-D

Selected Material Models Linear/nonlinear elastic, porous elastic, viscoelasticity, cap plasticity, critical state (clay) plasticity, extended Drucker-Prager, modified Drucker-Prager/cap model, rate dependent plasticity (creep and swelling)

Loadings Any type of static and dynamic loading can be applied at any desired location in finite element mesh

Soil-Water Medium Analysis Method Total and effective stress analysis -porous (Total Stress vs. Effective Stress) media stress-strain analysis based on effective

stress principle

Element Types A large number of solid (continuum), structural, slide/contact, and other special purpose elements

Author/Contact Person or Institution Hibbitt, Karlsson and Sorensen, Inc., 1080 Main Street, Pawtucket, RI 02860

ABAQUS

ABAQUS, developed and supported by Hibbitt, Karlsson & Sorensen, Inc. (HKS)*, is designed specifically to serve advanced structural analysis needs. The most challenging of these applications involve either linear analysis of very large linear models, or large models with highly nonlinear response.· ABAQUS is designed to provide efficient simulation for these classes of problem. The program is aimed at production analysis needs, so user aspects, such as ease of use, reliability, flexibility and efficiency have received great attention.

The theoretical formulation is based on the finite element stiffness method, with some "hybrid" (mixed stress-displacement variable) formulations included as necessary. The classes of problem that may be simulated with ABAQUS can be characterized as follows:

1. Geometry Modeling. The models can include structures and continua. One-, two- and three-dimensional continuum models are provided, as well as beams· and shells. The beam and shell elements are based on modern discrete Kirchhoff or shear flexible meth­ods and are very cost effective. Shell elements are provided for heat transfer, as well as for stress analysis: this makes the analysis of shell structures subjected to thermal loads very straightforward. Reinforcing (rebar) can be added to any element for com­posite modeling (reinforced concrete or reinforced rubber components). ABAQUS is a truly modular code: any combination of elements, each with any appropriate material model, can be used in the same model.

2. Kinematics. Except for some special purpose elements, all of the elements in ABAQUS are formulated to provide accurate modeling for arbitrary magnitudes of displace­ments, rotations and strains.

3. Material Modeling. Models are provided for metals, rubber, plastics, composites, con­crete, sand, clay and crushable foam. The material response can be highly nonlinear, and may be dependent on history and direction of straining. Anisotropic material properties are allowed. Very general elastic, elastic-plastic and elastic-viscoplastic models are provided, including the standard design theories for high temperature creep/fatigue evaluation of thin-walled piping components (the "ORNL rules"). An elastic-plastic fracture theory is provided for concrete. A general interface for user specification of material behavior is available.

4. Boundary and Loading Conditions. Boundary conditions can include prescribed kine­matic conditions (single point and multi-point constraints) and prescribed foundation conditions. Loading conditions can include point forces, distributed loads and thermal loading. Follower force effects (for example, pressure, centrifugal and Coriolis forces, fluid drag and buoyancy) are included where appropriate.

• HKS is pleased to acknowledge that the Electric Power Research Institute of Palo Alto, CA., (EPRI) has contributed in part to the development funding for ABAQUS. EPRI member utilities should contact EPRI for information on obtaining ABAQUS.

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ABAQUS has very general capabilities for modeling the interaction between bodies · (contact problems): "gap" elements, interface elements to model possible contact between a deforming structure and a rigid body, and "slide lines" , to model the interaction between two deforming bodies. General friction models are provided for use with these elements. Fully coupled thermal-stress interfaces are provided, where heat and traction may both be transmitted, and where the thermal resistance of the interface may depend on the mechanical separation of the surfaces. Acoustic interface elements are included, to couple structural and acoustic medium models for dynamic and vibration analysis.

5. Analyses. ABAQUS is a general purpose program, and primarily uses implicit inte­gration for time stepping, with automatic choice of the time {or load) increments, as discussed below. It provides static and dynamic, linear and nonlinear stress analysis; transient and steady-state heat transfer analysis; fully coupled seepage flow/ stress­displacement analysis (consolidation) for soils; fully coupled temperature/stress anal­ysis; and fully coupled acoustic medium-structural vibration analysis. Modal extrac­tion is provided for frequency determination or eigenvalue buckling load estimation. Response spectrum, time history response, steady-state response, and response to random loadings may all be computed, based on the natural modes of the model. ABAQUS provides a complete fracture mechanics design evaluation capability, in­cluding "line spring'' elements for modeling part-through cracks in shells, as well as ]-integral calculation (on any geometry) by the domain integral method. FUlly plastic crack solutions may be obtained with deformation theory plasticity models, to support the "engineering fracture mechanics" approach to inelastic fracture evaluation.

ABAQUS includes capabilities for both symmetric and non-symmetric matrices, and automatically uses the non-symmetric matrix scheme when the user's input implies that this is needed.

In nonlinear problems, the challenge is always to provide a convergent solution at minimum cost. This is addressed in ABAQUS by automatic control of time stepping, which is provided for all analysis procedures. The user defines a "step" (a portion of the analysis history, such as a thermal transient, a stage in a manufacturing process, or a dynamic event) and certain tolerance or error measures. ABAQUS then automatically selects the increments to model the step. This approach is highly effective for nonlinear problems, because the model's response may change drastically during an analysis step. Automatic control allows nonlinear problems to be run with confidence without extensive experience with the problem. This capability is a good example of the many features of ABAQUS that make it a truly production oriented tool for advanced analysis applications, and distinguish it from other finite element codes.

In ABAQUS, the analysis procedures can be mixed arbitrarily, so that, for example, a nonlinear static analysis may be followed by nonlinear dynamics (with the final static solution as initial conditions) in the same job; eigenvalue frequency extraction can include initial stress and deflection effects, etc.

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FEATURES

Stress Element Library

• Truss: 2 or 3 node. Hybrid (mixed formulation) versions of these elements are also provided, for systems in which the members are quite stiff but undergo very large rotations.

• Two-dimensional problems: 3- and 6-node triangles; 4- and 8-node quadrilaterals with full or reduced integration. These elements are provided for plane stress, plane strain, generalized plane strain and axisymmetric analysis cases.

• Three-dimensional problems: 4- and 10-node tetrahedra, wedges, 8-, 20- and 27-node bricks.

• Hybrid (mixed) versions of the plane strain, axisymmetric and 3-D solid elements are provided for incompressible cases.

• Beams: 2-node straight or curved or 3-node curved with general, box, rectangular, trapezoidal, pipe, circular, !-section, L-section, hexagonal or arbitrary cross-section. The user may choose numerical integration of the cross-section (to model material nonlinearities) or give a general, linear or nonlinear, cross-section response matrix. The 3-node beams are compatible with the second order shell and solid elements and therefore are frequently used in stiffened shell cases. Hybrid {mixed) versions of the beam elements are available for use in very slender or very stiff cases (almost inexten­sible beams), such as flexible offshore piping and riser systems, or stiff components in elasto-kinematic analysis (such as vehicle suspension system components).

• Pipes: these are 2- or 3-node beam elements that also allow uniform radial expansion of the cross-section, thus allowing modeling of internal pressure effects and, in particular, the influence of hoop stress on the elastic-plastic bending response.

• Shells: 3-node triangular and 4-, 8- or 9-node quadrilateral general layered shells, and 2-or 3-node axisymmetric layered shells are provided. Either numerical integration or a user supplied section stiffness matrix may be used. The shell elements in ABAQUS are true doubly curved shells. Both shear flexible ("thick") and ''thin" shell elements are provided.

• Elbows: deforming pipe section shell elements specifically designed to model ovaliza­tion and warping of elbows and straight pipes in nonlinear problems. These elements have been applied successfully to a wide range of nonlinear high temperature piping problems as well as to offshore pipeline problems, and their performance is extensively documented. They have become a standard design tool for some of these applications.

• Line spring elements to inodel.part-through cracks in shells, with elastic or elastic­plastic material behavior.

Heat Transfer Element Library

• 1-D: 2- or 3-node heat ''link". • 2-D: 3- and 6-node t~angles; 4- and 8-node quadrilaterals, for planar and axisymmetric

cases. • 3-D solids: 4- and 10-node tetrahedra, wedges, and 8- or 20-node bricks.

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• Shells: 2- or 3-node axisymmetric and 4- or 8-node general shells, with piecewise parabolic temperature variation through the thickness.

Acoustic Medium Element Library

• 1-D: 2- or 3-node. • 2-D: 3- or 6-node triangle, 4- or 8-node quadrilaterals, for planar and axisymmetric

cases. • 3-D solids: 4- and 10-node tetrahedra, wedges, and 8- or 20-node bricks.

Coupled Problem Element Library

• ABAQUS includes planar, axisymmetric, shell and 3-D elements which have both dis­placement and linear scalar field {pore pressure or temperature) interpolation. These elements are used for coupled temperature/stress problems, or for effective stress, groundwater flow problems in soil mechanics.

Contact and Interface Element Library

• ABAQUS contains a. very complete set of elements for modeling contact and interface problems for stress analysis, heat transfer analysis, coupled stress-heat transfer cases, and fully coupled acoustic pressure-structural response analysis. These elements in­clude "gap" elements for planar, cylindrical and spherical geometries; interface el­ements for small sliding cases {such a.s Hertz contact problems); interface elements for large sliding of a. deforming body past a. rigid surface; "slide line" elements for general contact between two bodies; heat interface elements, in which the heat trans­fer between the bodies may depend on the separation of the bodies; and acoustic medium-structural interface elements. ·

Special Geometric and Kinematic Modeling Options

• Linear and nonlinear springs and da.shpots. Both the springs and da.shpots may be associated with fixed directions or may be placed between nodes, with their line of action always directed between the nodes a.s they move.

• Diagonal or off-diagonal mass terms. • Multiple coordinate s}rstem input: Cartesian, cylindrical, spherical with any point of

origin. • Constraints: linear and nonlinear default multi-point constraints {MPCs), such a.s

rigid links, rigid beams, and a. shell-solid junction. Linear constraint equations may be defined by data cards; nonlinear MPCs that are not already provided in the ABAQUS constrain library can be defined in a. user subroutine.

• Cartesian, cylindrical or spherical transformation of the degrees of freedom at any node.

• Very general rebar {reinforcing rod) layout may be included in any element type to model materials such as reinforced concrete, or reinforced rubber components such a.s tires.

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• Second order isoparametric elements can all use coincident nodes to model crack tip singularities.

• Wave package for loading of immersed structures (pipes and risers). Linear (Airy) and Stokes 5th order waves provided. Surface penetration is included.

• Anisotropic seabed friction model for on-bottom movement of pipes. • Drag chain model for near bottom bending analysis of offshore pipelines. • Spring/friction/ dash pot combination elements to model the interaction between tubes

and support structures (such as in the study of fiow induced vibration in steam gen­erators).

• FUel rod subassembly interaction model. • Any number of linear or nonlinear user defined element types may be introduced into

a model. Stiffness or mass matrices can be introduced as linear User Element types.

Substructuring

• ABAQUS has a very general substructuring capability. Substructures are kept on a library file, and, once generated, they may be introduced in any analysis model. Substructures may be generated and used in the same job. Any substructure can be used several times in a model with repeated geometry, including rotation with respect to the master version of the substructure. Substructures may be used in nonlinear as well as in linear analyses: this provides an especially effective technique for problems involving contact between elastic bodies, in which the nonlinearity is confined to resolution of the contact problem itself.

Material Definitions

• Temperature dependence of all material definition parameters is allowed. Many ma­terial parameters can also be made to depend on any number of predefined field variables, such as the density of a particular phase in a multi-phase material.

• An orientation option is provided so that a local coordinate system may be defined at each point for material property input and stress/strain component output. This is particularly convenient for laminated composite shell analysis.

• Elasticity: several different definitions of elastic behavior are provided. For linear elas­ticity the elastic moduli, including coefficients of thermal expansion, may be isotropic, orthotropic or anisotropic. Hypoelasticity allows the moduli to be dependent on strain. A voided material model is provided, in which the elastic part of the volume change depends on the logarithm of the pressure stress. For large strain elasticity a hyper­elastic model is included, with a general polynomial strain energy function, for fully incompressible or almost incompressible response. Fully incompressible behavior is allowed through the use of hybrid {mixed displacement and pressure) elements.

• Metal plasticity: isotropic with von Mises yield; anisotropic with Hill's anisotropic hydrostatically independent yield; fiow rule is associated {normal) fiow; hardening .rules are isotropic, kinematic and ORNL theory with perfect plasticity default. The material may be rate independent or rate dependent {visco-plastic).

• Creep: isotropic or anisotropic; time or strain hardening laws; user subroutine for special creep laws. ABAQUS automatically switches from explicit to implicit time

5

stepping when the explicit time step is restricted by numerical stability considerations, thus providing for efficient solution of long time creep problems.

• Volumetric swelling: isotropic or anisotropic volume change with time as a function of field variables; tabular or user subroutine input.

• Critical state plasticity, for clay-like soils-a. generalized Cam-clay model that includes third invariant dependence in the yield definition and a "cap" on the yield surface with respect to pressure stress, with the Cam-clay strain hardening/ softening rule.

• Extended Mohr-Coulomb model, with strain hardening/softening and rate depen­dence, and with non-MSociated flow, for granular material such as sand, and for materials with different yield in tension and compression, such as polymers.

• No tension: provides a failure surface so that the material cannot carry tension. • No compression: provides a failure surface so that the material cannot carry compres­

sion. • Concrete: elastic-plastic-damage theory for concrete, including tension cracking, com­

pression crushing, concrete-rebar interaction (via tension stiffening) and post-crack response using damaged elasticity concepts.

• Permeability: isotropic, orthotropic or fully anisotropic permeability with voids ratio dependence.

• Thermal conductivity: isotropic, orthotropic or fully anisotropic, temperature depen­dent.

• Specific heat: temperature dependent. • Latent heat: an internal energy method is used to ensure accurate prediction of severe

latent heat effects associated with phase changes. • Gap conductance: allows conductivity across an interface to be a function of surface

separation. • Gap radiation: provides cross radiation between closely adjacent bodies. • User material: user subroutine UMAT allows any material model to be implemented.

ABAQUS provides for an arbitrary number of solution dependent state variables to be stored at each material calculation point, as well as for any number of material constants to be read as data, for use in this subroutine. This capability has become very popular with many groups working on advanced material behavior.

Analysis Procedures

The user divides the loading histories into "steps" solely on the basis of input convenience. For nonlinear analysis each step may be subdivided into increments, either by user control or {more usually) under automatic program control. In each nonlinear increment ABAQUS iterates for equilibrium using the full Newton method in most cases.

The initial condition for each step is the state of the model at the end of the previous step. This provides a most convenient method for following complex loading histories, such as manufacturing process analysis.

Within each step a procedure is specified. A vai.lable procedures are:

• Static stress/displacement analysis. ABAQUS offers two approaches for static stress analysis. One is for cases when a prescribed history of loading (such as a temperature

6

transient in a thermal shock) must be followed. The alternative is an arc-length (modi­fied Riks) method, which is provided for unstable static problems such as post-collapse or post-buckling cases. During either type of static analysis the material response may be time independent or time dependent: the user may associate a physical time scale with a static step for rate dependent cases.

• Dynamic stress/displacement analysis for linear problems. ABAQUS offers time his­tory analysis, response spectrum analysis, steady state response analysis, and random response analysis, all based on the natural modes of the system.

• Dynamic stress/displacement analysis for nonlinear problems. For fully nonlinear problems ABAQUS includes direct, implicit, time integration, using the Hilber-Hughes operator (the Newmark method with controllable numerical damping), as well as ex­plicit integration using the central difference method. Automatic time incrementation is used in both cases. For mildly nonlinear cases a projection method is provided, in which the response is developed by using the eigenmodes of the system in its initial configuration as global basis functions to develop the nonlinear solution. This method is very effective for some important nonlinear applications involving locally nonlinear response, such as piping systems with nonlinear restraints.

• Creep and swelling analysis. • Addition or removal of elements from the model. • Transient and steady-state heat transfer analysis. • Natural frequency extraction. Subspace iteration is used for the eigenvalue extraction.

Since this procedure can be invoked at any time in the analysis, pre-load effects can · be included. • Consolidation: steady-state and transient coupled effective stress/groundwater flow

analysis for consolidation problems. • Eigenvalue buckling estimates. Arbitrary pre-load and live load specification is al­

lowed. Boundary conditions may be changed during eigenvalue extraction (for exam­ple, from symmetric to anti-symmetric).

• Sequentially coupled te'mperature and thermal stress analysis (heat transfer analysis followed by stress analysis). During the heat transfer analysis the temperatures are stored at the nodes of the mesh on the ABAQUS results file: one data card directs these to be read into the stress analysis, with ABAQ US choosing automatic mere­mentation to step through the thermal transient. This capability is designed to make thermal shock analysis extremely simple.

• Fully coupled, transient or steady state, temperaturtHiisplacement analysis. • Fully coupled acoustic-structural vibration analysis in the time or frequency domain.

Solution Techniques

• Wavefront solution algorithm. Automatic, internal, wavefront minimization: this means that the user may choose any node and element numberings without invok­ing any penalty in solution time. The solution method is highly tuned, and has been developed to take full advantage of vector processing capabilities in high performance computers. ABAQUS includes both symmetric and non-symmetric solution schemes.

• Elastic re-analysis: based on original stiffness matrix.

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• Geometric nonlinearities: Lagrangian and updated Lagrange formulations for finite strain elastic and elastic-plastic problems. ABAQUS uses complete, consistent formu­lations for finite strain cases.

• For nonlinear equations ABAQUS generally uses the full Newton method: this ap­proach is especially effective for the highly nonlinear cases commonly modeled with the program.

• Constitutive integration: for material models that are written in rate form (such as elastic-plastic models), ABAQUS uses fully implicit integration, to ensure solution stability for the largest possible strain increments. This approach, together with the development of consistent Jacobian contributions, ensures efficiency for large strain problems involving complex material behavior.

Loads

• Nodal: concentrated forces and moments, including follower forces; temperature; field variables; non-zero displacements and rotations or accelerations specified with arbi­trary time variations. Corresponding loadings are provided for thermal and other non-structural models.

• Element: uniform and non-uniform body forces; uniform and non-uniform pressure; hydrostatic pressures; fluid drag; centrifugal load; Coriolis force; elastic foundations; follower force effects (including load stiffness terms) are included where appropriate.

• Load application: any load may have a linear variation over an analysis step or may reference an arbitrary amplitude curve, such as a ground acceleration in a seismic analysis or a complex pressure pulse history.

• Wave package with buoyancy and Morison drag, with Airy or Stokes 5th order wave theory, including free surface penetration, for offshore applications.

Input

• Preprocessor: all input lines are scanned and interpreted for consistency in the pre­processor, which performs extensive input data checking and provides model geometry plots. If no errors are detected in the data, the preprocessor creates the problem data base and passes control to the analysis program.

• Input is provided by keyword cards followed by data cards. Keyword cards are free­format, with parameters specifying options. The data input may be in free or fixed format (the latter for compatibility with external data generation programs).

• Set concept: nodes and elements may be gathered into "sets", each of which is given a name by the user. Sets within sets allowed. This set concept provides a simple, easily understood reference for material, load and restraint definition, output editing, etc. The concept is used throughout ABAQUS and is especially valuable in large, complex models, where it simplifies the data handling during the development of the analysis model.

• Simple mesh generation options: incremental fill along lines or curves, region fill, isoparametric mapping for blocks of nodes.

• Multiple coordinate systems: Cartesian, cylindrical, spherical.

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Output

• Printed output: tabular printed output of stress, strain, displacements and reactions. The user may define which variable appears in each column of each table, thus de­signing the output for specific needs.

• Graphical output: mesh plots; contour plots of results; deformed mesh plots; time history plots, variable versus variable plots.

• External file output: any analysis results may be written selectively to the results file. This file provides the data base for history and variable-variable plotting in the ABAQUS post-processor, and is used as the input to external post-processors. It is the basis of the interfaces to standard commercial post-processing packages.

• Restart: allows segmented solution of problems; protection against unexpected aborts; versatile and extremely easy and convenient to use. The restart file is used as the data base for ABAQUS post-processing for contour and displaced configuration plots and for additional printed or external file output not generated during the analysis job.

• Error messages: ABAQUS gives definitive error messages, and includes explanations and suggestions with each message--the user does not have to consult the manual to interpret the messages.

Graphical Displays

• Mesh plotting: full or partial model plots with arbitrary viewpoint for model checking. Element and/ or node number display is optional.

• Results displays: deformed geometry plots of the full or part of the model; contour plots (contour lines or color shading) of element quantities such as stresses, strains, temperatures, etc.; contours are provided on the faces of 3D solid elements and on layers of beam or shell elements; "moment diagram" type plots for beams. Time history plots and variable-versus-variable plots of any variables. Analysis plots may be obtained as the analysis runs, or by post-processing from the restart file.

User Subroutines

ABAQUS includes provision for the user to add his own subroutines to the library. In­terfaces are provided so that the user can define material models, elements, multi-point constraints, and very general loading conditions. The ABAQUS user subroutine capability has seen extensive use, and the versatility it offers has made the program very popular with advanced development groups.

Documentation

ABAQUS is one of the most thoroughly documented finite element codes, with a three volume manual set:

• User's Manual: a complete description of the elements, material models, procedures, input specifications, etc. This is the basic reference document for using ABAQUS.

• Theory Manual: detailed, precise discussion of all theoretical aspects of the program. Written to be understood by users with an engineering background.

9

• Example Problems Manual: This volume contains two parts. The first part has many worked examples designed to illustrate the approaches and decisions needed to perform meaningful nonlinear analyses. Typical cases are: large motion of an elastic-plastic pipe hitting a rigid wall; inelastic buckling collapse of a thin-walled elbow; explosive loading of an elastic, visco-plastic thin ring; consolidation under a footing; buckling of a composite shell with a hole; deep drawing of a metal sheet. The second part of this manual contains several hundred basic test cases, providing verification of each individual program feature against exact calculations.

• HKS also maintains a Quality Assurance Plan, which is designed to meet the stan­dards of the U.S. Nuclear Regulatory Commission. This Plan is made available as a controlled document.

Hardware Compatibility

ABAQUS is written in Fortran, and versions are maintained and supported on all standard engineering computers. The code is supplied as fully single precision or fully double pre­cision, depending on the computer. ABAQUS supports all standard graphics and plotting devices.

Problem Size and Program Performance

ABAQUS has no built-in limits on problem size. Smaller problems run entirely in main memory, with spill to secondary storage occurring automatically as the problem size in­creases. ABAQUS performs efficiently on a wide range of computers, and is particularly effective for large problems running on advanced computer architectures. HKS provides timing data for a set of benchmark calculations done on various computer systems, as a basic comparison of performance of the code on those systems.

Maintenance and Support

ABAQUS is supplied with full maintenance and support services. Customers automatically receive each latest version as it is released, and have "hotline" service for assistance with the code. Status reports, listing known deficiencies and their resolution, are provided to all customers on a regular, frequent schedule.

Installation

In most cases HKS or HKS' local agerit performs the initial installation of the program at a customer's site. Self-installation is offered for workstation systems. Installation service includes check-out and verification of the code so that the installation meets the most stringent Q/ A requirements; interfacing to local graphics/plotting systems; and the pre­sentation of a training seminar on the usage of the code. Based on the initial installation, a customer file is maintained for the purpose of supplying subsequent releases with all local modifications and interfaces included, so that they may be installed easily by customer personnel.

10

Training

HKS offers training seminars at the customer's site as well as regularly scheduled training classes in HKS' and HKS's agents' offices. The basic on-site training seminar may be from one to five days in duration, at the customer's option, and includes lectures and workshops in which the code is exercised on the customer's computer. The standard three day introductory seminar covers basic usage in the first day, followed by study of nonlinear applications, including large displacement, inelastic, dynamic, and thermal shock examples. Workshops are run to provide as much "hands on" experience with the code as possible.

Advanced seminars cover specific topics. Those taught on site focus on topics of par­ticular interest to the customer, based on the customer's prior specification. The advanced seminars offered in HKS' or HKS' agents' offices cover such items as inelastic constitutive modeling, large strain elasticity, metal forming, and fracture mechanics.

User Benefits

• A single code with easy-to-use input allows access to a wide library of linear and nonlinear analysis capabilities and thus minimizes personnel training and retraining costs.

• Full support and maintenance service assures the user of effective software utilization, and of the availability of expertise in cases of advanced analysis.

• Simple keyword, free-format input. • Set definition for easy cross-reference. • Consistent data checking. • Efficient wavefront solution. • User subroutines for flexibility in modeling and analysis of more advanced applications. • Independent material and element libraries-any material model can be used with

any element. No limit on the number of different materials or elements in a model. • Mixed analyses in a single run. • Automatic increment choice (time stepping) in statics, dynamics, creep, transient

heat transfer, coupled temperature/stress, coupled seepage/stress, etc .. ensures high reliability even in the most difficult nonlinear applications.

• Large displacement/large rotation, finite strain analysis. • Proven, modem, element library: reduced integration elements, with "hourglass con-

trol" as needed, for efficiency and stress accuracy. • Multiple options for display of mesh and results. • Versatile restart. • Selective output control, with concise, tabular printed output of user selected variables,

and an external file for storing results that are required for post-processing. • Compatible heat transfer and stress analysis, sequential or fully coupled. • Proven effectiveness: ABAQUS has a large, worldwide, customer base and is routinely

and heavily used in a broad spectrum of applications. The very high volume of usage ensures reliability.

• ABAQUS and its associated support services are designed .to make complex, linear and nonlinear analysis as simple and reliable as present numerical inethods allow.

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ABAQUS /Explicit

ABAQUS/Explicit, developed and supported by Hibbitt, Karlsson & Sorensen, Inc. {HKS) is a transient dynamics program ,designed specifically to serve advanced, nonlinear continuum and structural analysis needs. ABAQUS/Explicit has the standard ABAQUS user interface and is highly efficient on vector/parallel computers, as well as on Unix workstations. The program is aimed at production analysis needs, so user aspects, such as ease of use, reliability, flexibility and efficiency have received great attention.

The most challenging nonlinear dynamic applications involve large, complex models with multiple, arbitrary contact conditions. The contact algorithms in ABAQUS/Explicit are efficient, robust, and very easy to invoke. Contact is defined by specifying surfaces which may interact. Contact with a rigid body is accomplished simply by specifying that one of the contacting surfaces is rigid.

ABAQUS/Explicit is designed to run effectively on a wide range of vector/parallel computers. The code is highly vectorized, and uses a compact, efficient data management system.

Geometry Modeling. The models can include structures and continua. One-, two­and three-dimensional continuum models are provided, as well as beams, membranes and shells. ABAQUS/Explicit is a truly modular code: any combination of elements, each with any appropriate material model, can be used in the same model.

Kinematics. All of the elements in ABAQUS/Explicit are formulated to provide ac­curate modeling for arbitrary magnitudes of displacements, rotations and strains.

Material Modeling. Elasticity and metal plasticity models are provided, including rate dependence and adiabatic heat generation. Equations of state can be included, and a general interface for user specification of material behavior is available.

Boundary and Loading Conditions. Boundary conditions can include prescribed kine­matic conditions {single point and multi-point constraints) and specification of dis­placement, velocity and acceleration histories. Loading conditions include point forces and distributed loads such as pressure, body forces, centrifugal and gravity loads.

ABAQUS/Explicit has very general capabilities for modeling contact problems, with or without friction. The definitions of surfaces which may come into contact may be changed upon restart of the analysis. Surfaces may be specified as being bonded: this provides a simple mesh refinement technique in both two and three dimensions.

1

FEATURES

Element Library

• Truss: a 2 node truss/rod element is provided. • Two-dimensional problems: 3-node triangles and 4-node quadrilaterals are provided

for plane stress, plane strain and axisymmetric analysis. • Three-dimensional problems: 4-node tetrahedra, 6-node prisms and 8-node solid ele­

ments are available. • Beams: 2-node beams in a plane and in space are included, with a library of cross­

section definitions. • Membranes: 3-node triangular and 4-node quadrilateral membrane elements are in­

cluded. • Shells: 3-node triangular, 4-node quadrilateral general shells and 2-node axisymmetric

shells are provided. The shell section may be homogeneous or layered (for laminated composite analysis).

Additional Geometric and Kinematic Modeling Options

• Linear and nonlinear springs and dashpots. • Mass and rotary inertia elements. • Constraints: linear and nonlinear default multi-point constraints (MPCs), such as

rigid links, rigid beams and a shell-solid junction.

Material Definitions

• Temperature dependence of all material definition parameters is allowed. • Elasticity: isotropic linear elasticity is provided. • Metal plasticity: an isotropic hardening, von Mises, associated flow, rate dependent

material is provided. Adiabatic heat generation can be included. Ductile failure criterion can be included.

• Equations of state: the JWL equation of state and the linear Hugoniot equation of state are included.

• User material: user subroutine VUMAT allows user material models to be implemented. The interface passes data in vector blocks, so that the user may take advantage of vectorization when coding VUMAT.

Loads

• Nodal: concentrated forces and moments, including follower forces; non-zero displace­ments, velocities and accelerations specified with arbitrary time variations.

• Element: body forces, pressure, centrifugal and gravity loads are provided. • Load application: loads may be constant or may vary during an analysis by reference

to an arbitrary amplitude curve.

2

Input

• Preprocessor: all input lines are scanned and interpreted for consistency in the pre­processor, which performs extensive input data checking and provides model geometry plots. If no errors are detected in the data, the preprocessor creates the problem data base and passes control to the analysis program.

• Input is provided by keyword cards followed by data cards. Keyword cards are free­format, with parameters specifying options. The data input may be in free or fixed format.

• Set concept: nodes and elements may be gathered into "sets," each of which is given a name by the user. Sets may be nested. This set concept provides a simple, easily understood reference for material, load and restraint definition, output editing, etc. The concept is especially valuable in large, complex models, where it simplifies the data handling during the development of the analysis model.

• Simple mesh generation options: incremental fill along lines or curves, region fill, isoparametric mapping for blocks of nodes.

• Multiple coordinate systems: Cartesian, cylindrical, spherical.

Output

• Graphical output: mesh plots; contour plots of results; deformed mesh plots; time history plots, variable versus variable plots.

• External file output: analysis results may be written selectively to the results file which is used as the input to external post-processors. It is the basis of the interfaces to standard commercial post-processing packages.

• Restart: allows segmented solution of problems; protection against unexpected aborts; versatile and extremely easy and convenient to use. The restart file is used as the data base for post-processing using ABAQUS /Post, and for additional printed or external file output not generated during the analysis job.

• Error messages: ABAQUS gives definitive error messages, and includes explanations and suggestions with each message-the user does not have to consult the manual to interpret the messages.

Graphical Displays

• Mesh plotting: full or partial model plots with arbitrary viewpoint for model checking. Element and/ or node number display is optional. _

• Results displays: deformed geometry plots of the full or part of the model; contour plots (contour lines or color shading} of element quantities such as stresses, strains, temperatures, etc.; contours are provided on the faces of 3D solid elements and on layers of beam or shell elements; "moment diagram" type plots for beams. Time history plots and variable-versus-variable plots of any variables. Analysis plots are obtained by post-processing from the restart file.

3

User Subroutines

ABAQUS/Explicit includes provision for the user to add material models to the library via user subroutine VUMAT.

Documentation

The ABAQUS/Explicit manuals are quite compact. They consist of: • User's Manual: a description of the elements, material models, procedures, input · specifications, etc. This is the basic reference document for ABAQUS/Explicit. It

also contains a short tutorial guide to using the program. • Example Problems Manual: This volume contains worked examples designed to il­

lustrate the approaches and decisions needed to perform meaningful explicit dynamic calculations.

/

Hardware Compatibility

ABAQUS/Explicit is written in Fortran, and versions are maintained and supported on all standard engineering computers. ABAQUS/Explicit uses the standard ABAQUS plotting capabilities which support all standard graphics and plotting devices.

Problem Size and Program Performance

ABAQUS/Explicit has no built-in limits on problem size. Smaller problems run entirely in main memory, with spill to secondary storage occurring automatically as the problem size increases. ABAQUS/Explicit performs efficiently on a wide range of computers, and is particularly effective for large problems running on vector /parallel computer architectures. IlKS provides timing data. for a. set of benchmark calculations done on various computer systems, as a. basic comparison of performance of the code on those systems.

Maintenance and Support

ABAQUS/Explicit is supplied with full maintenance and support services. Customers automatically receive each latest version as it is released, and have "hotline" service for assistance with the code. Status reports, listing known deficiencies and their resolution, are provided to all customers on a. regular, frequent schedule.

Installation

In most cases HKS or HKS's local agent performs the initial installation of the program a.t a. customer's site. Self-installation is offered for workstation systems. Installation service includes check-out and verification of the code, and the presentation of a. training seminar on the usage of the code.

4

Training

HKS offers training seminars at the customer's site as well as regularly scheduled training cla.sses in. HKS's and HKS's agents' offices. The basic on-site training seminar may be from one to five days in duration, at the customer's option, and includes lectures and workshops in which the code is exercised on the customer's computer. The standard three day introductory seminar covers basic usage in the first day, followed by study of nonlinear applications, including large displacement, inelastic, impact examples. Workshops are run to provide as much "hands on" experience with the code as possible.

Advanced seminars cover specific topics. Those taught on site focus on topics of par­ticular interest to the customer, based on the customer's prior specification. The advanced seminars offered in HKS' or HKS' agents' offices cover such items as inelastic constitutive modeling, metal forming, and crashworthiness calculations.

User Benefits

• ABAQUS/Explicit is fully compatible with the standard ABAQUS program, thus minimizing personnel training and retraining costs.

• FUll support and maintenance service assures the user of effective software utilization, and of the availability of expertise in cases of advanced analysis.

• Simple keyword, free-format input. • Set definition for easy cross-reference. • Consistent data checking. • User subroutines for flexibility in modeling and analysis of more advanced applications. • Independent material and element libraries-any material model can be used with

any ele~ent. No limit on the number of different materials or elements in a model. • Automatic time incrementation ensures high reliability even in the most difficult non­

linear applications. • Large displacement/large rotation, finite strain analysis. • Proven, modern, element library: reduced integration elements, with "hourglass con-

trol" for efficiency and stress accuracy. • Multiple options for display of mesh and results. • Versatile restart. • Selective output control, with concise, tabular printed output of user selected variables,

and an external file for storing results that are required for post-processing. • ABAQUS/Explicit and its associated support services are designed to make complex,

nonlinear dynamic analysis as simple and reliable as present numerical methods allow.

5

APPENDIX 3 (continued)

Computer Code BALL

Analysis Method Distinct element- explicit time marching formulation

Geometrical Dimensions 2-D ~

Selected Material Models Granular material modeled with distinct elements

Loadings Gravity and static loading and steady state or transient dynamic loading applied at side walls or base

Soil-Water Medium Analysis Method Dry granular soil only (Total Stress vs. Effective Stress)

Element Types Discrete 2-D ball shape elements

Author/Contact Person or Institution Cundall, P. A., 1978 "BALL- A Program to Model Granular Media Using the Distinct Element Method", Technical Note, Advanced Technology Group, Dames & Moore, London. Dr. Wolfgang H. Roth, Dames & Moore 911 Wilshire Boulevard, Suite 700, Los Angeles, CA 90017, Tel: (213) 683-0471

1'.

II

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.... "':.i.u"·~.t~.....,;.:.:.;.·.:....::t·~.:.·aw-,~'2·~-t~ ............. -. .... ~W"-~.-~I:.i~--~~ ......... ~~.:...:......c,w.;.·.,..-... ..;.....,,.., . .- .... ~·c:r· · · •..• , .. "~~ ............. ~-.,.__ .. -.

.......... .......

.

DAMES & MOORE

. ..• ~ ---:----·.

BALL - A Program to Hodel Granular 1-tedia using the Distinc= Element Method.

by

Peter A. Cundall

Advanced Technology Group London Task

Job

TN-LN-13

2633-93 10369-ool-60 April 20, 1978

•.

..... ..._,• ....

1

·----;--··

!ABSTRACT .

A computer program has been developed that 'can model the behaviour of arbitrary assemblies­of discs contained with~n boundaries that can be mcve~ ~n any given fashion. Tf~ friction and stiffness at particle contacts may be prescribed~ and there is no Umit on the displacements aZ­lo~ed. Comprehensive plotting routines are incorporated in the program~ including stress/ strain plots~ force vector plots, veZocity zilots­and energy dissipation plots.

' . The present study is the first stage in a two-year project to develop constitutive Zaws for granular material by perforrrrr,.'ng nwnericaZ tests on assemblies of particles.

This Technical Note gives details of the program, its assumptions ~ad use~ and presents­some example runs •

•. '

• I i I I I I I ( ,)

I I

1.

2.

3.

4.

5 •.

6.

7.

8.

ii

TABLE OF CONTENTS

INTRODUCTION

LINKED-LIST SCHEME

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

Recapitulation of Original Procedure for Blocks ·changes to Updating and Contact Storage in BALL Changes to Mapping Scheme Overall Storage Allocation Use of Half-Words and Full-Words Ball Data Arrays Wall ~ata Arrays Box Array Contact Data Arrays Box Entries Empty Lists

REBOXING AND CONTACT DETECTION ALGORITill1

3.1 Integer Boundaries 3.2· Efficiency 3.3 Tolerance

AOTOl1ATIC PARTICLE GENERATION

HATHEMATICAL TREATMENT

. USE OF PROGRAM

6.1 6.2 6.3 6.4 6.5 6.6

Choice of Coordinates Files Time-step· Damping Speed of Loading Plots , 1

EXAMPLE RUN3

7.1 Summary of Test Sequence 7.2 Initial Assembly (Stages l to 3) 7.3 Simple Shear Test (Stages 4 to 6)

.7.4 Biaxial Test (Stages 3A, 3B)

CONCLUSIONS •.

l

3

3 3 5 5 6 7 8 s 9

10 10

11

11 11 12

14

16

22

22 22 23 24 26 27

29

29 30 33 34

36

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APPENDIX I - Commands

I-1 Introduction I-2 The first Input Line ~-3 Other Commands

iii

APPENDIX II - Subroutine Guide to program BALL

··- ·---:--····

APPENDIX III - Meaning of Fortran Variables and Program Listing

APPENDIX IV - Post-Processor Program for producing stress/strain plots

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63 63 64

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1. INTRODUCTION

The work described in thiz Technical Note was funded jointl~· by Dames

& Moore and the National Science Foundation. A two-year grant was awa~ded by

the NSF to the University of Minnesota, with Dames & Moore as subcontractor.

The co-principal investigators are Otto Strack of the University of l1innesot-J.

and Peter Cundall of the Advanced Technology Group of Dames & Moore in London.

Dames & Moore's contribution to the study has been_directly, via Task Number

2633-93 and indirectly, via reduced billing-rates on Job 10369-ool-60.

The Techr.ical Note describes the first stage of the \olork in which

the comput~r program BALL was developed and used.in some initial test ~,s.

The program,has been set up both on the London computer of Dames & Moore {PDP 11/45)

and on the Interdata 8/32 {bought on NSF funds) at the University of Mifu,esota •.

Validation tests on the program have been described elsewhere.*

The object of the research ultimately is to develop constitutive laws

for soil based on numerical tests performed on assercblies of discrete particles.

These particles can be simulated in the computer using the distinct element method.

Stress or displacement boundary conditions may be applied to the assemblies, and

friction; cohesion and stiffness may be specified for particle contacts. Using

appropriate graphical displays, the mechanisms occuring within a granular material

during a test may be appreciated, and used to confirm or invalidate existing

theories, or form the basis for new theories.

The mechanical principles embodied in program BALL are almost identical **

to those described in a report by Cundall {1974) for interaction of angular blocks.

It is assumed that the reader is familiar with that report. BALL models circular

discs rather than blocks, but the contacts are still assumed to be at points.

The same explicit time-marching formulation is used.

I ;

Howev~r, the scheme that keeps track-of contacts is somewhat different

from the 1974 report, although the system of "boxes" is retained. It may be

recalled that the box area is a grid that covers the portion of space enclosing

the particles. Each particle maps into one or more boxes. ~fuen it is nec­

essary to determine those particles that are neighbours to a given particle,

only the local boxes need be searched, rather than the .,1hole problem area. The

resulting contacts are ::t'ored in a linked list memory structure and not in se-

................

- t:.-

quentin.l arrays. 'l'his type of storage allows very fast re-allocation of

memory when contacts are created or deleted.·

The program BALL treats two types of physical entity: "......-alls" and

"balls" (also called "particles" in this note). Balls can interact both Hith

other nulls and walls; the full equation o£ motion is solved for balls. t·lalls

can only interact with balls, and not with other walls; the equation of motion

is not solved for l-ralls, which means that the forces acting on a wall do not

influence its motion directly.

and are constant •

The wall velocities are specified by the .user,

* Cundall, P.A. & o. Strack (1978) paper submitted to GeotecP~ique.

**" CUNDALL, P.A., 1974 "Rational design of tunnel supports: a computer model for rock mass behaviour using interactive graphics for the input and output of geometrical data," Technical Repo.t;t HRD-2-74, Missouri River Division, U.S. Army Corps of Engineers.

•.

APPENDIX 3 (continued)

Computer Code DSAGE-2.1

Analysis Method Finite difference

Geometrical Dimensions 2-D plane strain

Selected Material Models Incremental nonlinear elastic models combined with Mohr-Coulomb plasticity law with a nonassociated flow rule, without a yield cap, and with an empirical pore pressure generator

Loadings Gravity loading and earthquake base motion

Soil-Water Medium Analysis Method Effective stress model with an empirical pore (Total Stress vs. Effective Stress) pressure generator

Element Types 2-D plane strain continuum elements

Author/Contact Person or Institution Cundall, P. A., 1976, "Explicit Finite Difference Methods in Geomechanics," 2nd Conference on Numerical Methods in Geomechanics. Version 2.1 developed in 1985 by Dames & Moore. Dr. Wolfgang H. Roth, Dames & Moore, 911 Wilshire Boulevard, Suite 700, Los Angeles, CA 90017, Tel: (213) 683-0471

••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• APPEND :IX

NON-LINEAR EFFECTIVE -STRESS ANALYSIS WITH DSAGE - 2.1

*****************************************************************

LIMITATIONS OF STRESS-ORIENTED ANALYSIS METHODS

Conventional ~ethods of evaluating the earthquake performance of earth structures utilize stress• (rather than deformation) oriented methods of analysis. Induced dynamic stresses are related to available soil resistance (dynamic strength), and conclusions are drawn from this relationship regarding the overall stability of the structure. The most commonly used computer programs based on the stress-oriented approach include the equivalent-linear codes of SHAKE, QUAD4, FLUSH, and SUPERFLUSH, developed over the past 20 years at the University of California, Berkeley (UCB). These codes have significant li~itations associated with their linear-elastic ~aterial laws. They are referred to as 11 equivalent-linear 11

,

because they allow an approximation of the nonlinear material behavior by interactively adjusting elastic and da~ping parameters according to induced strain levels. Later versions of these codes also atte~pt to allow for softening by shaking-induced pore pressures. Nevertheless, because of the elastic material model, cumulative permanent deformations cannot be computed. The generated transient deformations are completely recovered as soon as the dynamic loading stops, and permanent deformations have to be estimated by other means.

Several methods have been developed over the years in an attempt to somehow interpret the stress-oriented results from equivalent­linear analyses in terms of permanent deformations:

Strain Potentials. The method of expressing the induced dynamic stresses in terms of "strain potentials 11 has evolved in the last 15 years. The strain potential of a given element in a numerical model is the maximum axial strain that would be reached in a cyclic triaxial test subjected to a field-equivalent cyclic loading history. A good deal of engineering judqment is necessary to come to a conclusion about the overall deformation behavior of an earth structure based on this type of data.

Compatible Strain Potentials. This method attempts to evaluate dynamically induced deformations by making the strain potentials of individual elements compatible with the deformation constraints of the continuum. The dynamic-analysis results are post-processed by performing static finite element analyses with modified material properties and/or body forces governed by the calculated strain

A-1

potentials (Lee, 1974; Serff et al., 1976). This method, although somewhat difficult to backup theoretically, is assumed to give believable results where small deformations are involved.

Newmark Analysis. The Newmark (1965) analysis is probably one of the more logical and straightforward methods of estimating cumulative dynamic deformations. It consists of double-integrating acceleration peaks which exceed the so-called iield acceleration of a potential slide body. The Newmark method has been shown to give realistic results where distinct failure planes are likely to occur, such as in cohesive materials. However, for problems where wide-spread shear distortion and/or cyclic strength deterioration due to pore pressure build-up is expected, this method is not applicable.

NONLINEAR EFFECTIVE-STRESS ANALYSES

There are two categories of nonlinear methods of analysis, elasticity-based formulations, which directly model the stress­strain curve, and models which are based on plasticity theory. When soil is highly stressed it flows plastically and its behavior is different from that at low stress levels where elastic theory is adequate. The "flow rule" provided by plasticity theory simulates the soil's behavior close to failure more realistically than a formulation that simply modifies elastic parameters incrementally.

An example of the elasticity-based category is the 2-0 program TARA-3 (Finn et al, 1986). In contrast to Finn's more practice­oriented approach, the category of plasticity-based models usually involves rather complex constitutive laws, often ill understood by the practitioner. The latter models require input data not readily available to the engineer, and have "a mind of their own" when it comes to following other than laboratory-prescribed stress paths, such as generated by real earthquakes. Perhaps the most widely known programs of the complex-plasticity type are DYNAFLOW (Prevost, 1981) and DIANA-J (Kawai, 1985).

THE DSAGE-2.1 PROGRAM

The program DSAGE-2 .1 utilized for this investigation, combines the practice-oriented simplicity of the incremental elastic models with the advantage plasticity theory has to offer in terms of realistic soil behavior. This program employs the simple Mohr-coulomb plasticity law, which only requires "standard" soil mechanics parameters, such as friction angle, cohesion and elastic moduli. In combination with a "robust" empirical pore pressure generator, OSAGE-2.1 handles the most irregular stress paths without computational difficulties or erratic behavior.

BASIC COMPUTATION SCHEME

The OSAGE program was developed for Dames & Moore based on original

A-2

work by Cundall (1976), with the. 2 .1-version developed in 1985 specifically for the analysis of Pleasant Valley dam (Roth et al, 1991). OSAGE's operating mode is based on a time-marching dynamic scheme, depicted in Fiqure A-1. The physical continuum to be analyzed is divided into a mesh of discrete elements (zones connected to each other by grid points). The finite difference equations that govern the motions of the continuum are applied to lumped masses at the grid points. The calculations are performed in time increments small enough to preclude interference between adjacent grid points during any one computational step.

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rigure A-Z. Dynamic Analysis with DSAGE. The incremental, time-marching computation scheme is performed in very small time steps to preclude interference between adjacent grid points.

The incremental displacements produced by the grid accelerations in every time step are converted into strains. stress increments are then derived from these strains using any given explicit material law. When stresses have been calculated for all elements, the complete mesh is scanned again, and the remaining imbalance forces are converted into accelerations. This scheme is repeated until equilibrium is reached between the internal stresses and the external forces.

One of OSAGE's major advantages is its extreme simplicity when it comes to coding of material-law subroutines. In contrast to conventional (implicit) finite element programs, where the constitutive relations affect the overall stiffness matrix, the stress-strain relationship for OSAGE is formulated for one isolated element at a time.

A-3

CHOICE OF MATERIAL LAW

There is no universal material law in existence today which perfectly simulates the mechanical behavior of soil under dynamic loading. The choice of the material law depends on the purpose of the analysis and the material and loading characteristics which mostly affect the analysis results. From a practical point of view, the material law should also be simple and easy to apply.

DSAGE-2 .1 utilizes the elastoplastic Mohr-coulomb material law with an empirical strength-degradation scheme which is based on laboratory-measured pore pressures. The necessary input data for the Mohr-coulomb model are bulk and shear modulus, and the shear strength. described by friction angle and cohesion. The material behaves in a linear elastic manner when the induced shear stress is less than the strength, but it produces nonrecoverable shear strains through plastic flow when the strength limit is reached. The schematic stress-strain diagrams in Figure A-2 show the basic difference between a linear-elastic material law and the nonlinear law utilized in DSAGE-2.1.

en Loading

en en en Q) Q) .... .... - -en en ... ... n:s n:s Q) Q) .r; .r:;. en en

Unloading

Shear Strain Shear Strain

Permanent Deformation

LINEAR ELASTIC ELASTOPLASTIC FJgure A-2. Linear-ElastJc and ElastoplastJc Xaterlal Law•· The elastl~ lav result• in ~omplete re~overy of deformatlons at the end of shaking, whereas the elastoplastj~ law •~cumulates permanent deformation••

The credibility of the material law with respect to its prediction of shaking-induced permanent deformations, was established through verification with centrifuge model tests. While these tests were carried out with model embankments of dry sand (without pore pressures), they did verify the material law in terms of effective

A-4

..

(grain-to-grain contact) stresses which govern the soil's shear strength. Hence, as long as the pore pressure generation scheme (described in a later section) produces realistic effective stresses, the shear-strength-governed permanent deformations of the numerical model can be expected to be realistic.

VERIFICATION OF MATERIAL LAW BY CENTRIFU~E TESTING

Because of the dependence of the mechanical properties of soil on ambient stress conditions (i.e., gravity), centrifuge modeling has evolved as an important tool for physical testing of geomechanical models. · Centrifuge models are tested under a centrifugal acceleration field, typically 50 to 150 times the earth's acceleration (g). Under these conditions, the "weight" of the model is increased by a factor of 50 to 150, and the ambient stresses in a model embankment of only several inches in height are boosted to the prototype conditions of a dam with realistic dimensions.

If the centrifugal acceleration field is H times gravity, then the ratio of linear prototype dimensjons to those o~ the centrifuge model is H, the

2ratio of area is H and volume is I(. Forces in the

prototype are H times those in the model, so that stresses remain unchanged.· Deformation in the prototype isH times larger than in the model, but strains are the same. For dynamic problems, time in the prototype is H times the model time. This means that velocities are unchanged, but accelerations and frequencies of a modeled earthquake need to be H times larger than for the prototype.

While the testing of "static" geotechnical problems in the centrifuge has become almost routine, dynamic testing, involving forced shaking of earth models in the centrifuge, has lagged far behind. A few years ago, a servo-hydraulic centrifuge shaking apparatus was developed by Dames & Moore and Caltech for an 8-foot diameter centrifuge, and a series of shaking tests with model embankments was performed (Roth, et al., 1986). The results of these model tests were used to verify the elastoplastic material law utilized in DSAGE-2.1.

SETUP OF CENTRIFUGE TESTING

Fiqure A-3 show~! a cross section of Cal tech's centrifuge, and Fiqure A-4 presents a top view of a cut-away section of the earthquake simulator "in ·flight". The model embankments, consisting of dry, slightly compacted fine sand, were subjected to a constant centrifugal acceleration field of 50g, and then shaken with an "earthquake" with an equivalent prototype peak acceleration of 0.6g, and 20 seconds of prototype duration.

A-5

0'UATIIIG POSIT I Oil

Figure A-3. Cross Section of Cal tech •s Centrifuge. By sp:inning the model embankment of 50g centrifugal acceleration, the l111lb;i.ent soil stresses were boosted to represent those of a 50-times larger prototype.

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Figure A-4. 'I'op View of Centrifuge Shaker. To simulate an earthquake with 0. 6g of prototype peak acceleration, acceleration peaks of JOg had eo be generated by th;i.s shaker.

A-6

The model embankments were instrumented with micro-accelerometers connected to a digital data acquisition system with direct core memory, capable of performing 100,000 measurements per second. In addition, a 3,000-frames-per-second movie camera, mounted on the vertical axis of the centrifuge, recorded the displacement patterns of the embankment (via the mirror shown in Figure A-3) through a glass wall of the test box. Small silver-coated, light reflecting nylon balls, embedded within the embankment, moved with the model with respect to a fixed reference grid on the glass wall. The light reflection from these balls was easily detectable on the individual movie frames.

Displacement records of several points in the embankment cross section were obtained from enlargements of individual frames of the high-speed movie taken during centrifuge testing.

COMPARISON WITH NUMERICAL ANALYSIS

Figure A-5 shows the discrete-element model representing the prototype dimensions of the centrifuge test embankment. This model was subjected to the prototype~equivalent of the base acceleration generated by the centrifuge shaker.

SLUMP f

7igvre A-5. Discrete-Element lfesh of Centrifuge ltodel Embanla:aent. A comparison ol measured and computed shaking-induced deformations wa• used to verify the elastoplastic material law.

A-7

The measured and computed vertical deformation (slump) of the model embankment versus time is plotted in Figure A-6. The initial sudden slump at the beginning of the plot was attributed to compaction of the loose sand in response to the very first acceleration spike. This type of volume change is not addressed by .the utilized elastoplastic law. However, because this initial slump was unique to the model setup, it was considered irrelevant for the purpose of this verification. Only the subsequent portion of gradually accumulated deformation in response to shaking-induced cyclic shear stresses was compared with the analytical prediction. Based on this comparison it was concluded that the chosen elastoplastic material law would be appropriate for the prediction of shaking-induced deformations of an earth embankment.

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ri..gure A-6. lleasured and Computed Deformati..ons (i..n Prototype D.i.mensions). The centrifuge model suffered initial sudden compaction, followed by gradual slumping due to cyclic shear strains. The latter deformation component was satisfactorily predicted by the numerical model.

A-8

SHAKING-INDUCED STRENGTH REDUCTION

The shear strength of a granular soil is governed by the frictional resistance created by grain-to-grain contact forces in the soil skeleton. cyclic loading induced by earthquake shakinq can destroy this skeleton, causing the grains to re-arrange into a denser structure. If the soil is dry, this results in compaction without much effect on the shear strength. However, if the soil is saturated, the densific~tion of the soil mass is prevented by the ~elatively incompressible pore water between the grains. Hence, the collapse of the soil structure causes a load transfer from grain-to-grain contact stresses (effective stresses) to the pore water, thereby generating excess pore pressures. This reduction in effective stresses causes ·a decrease in shear strength which, ultimately, may lead to "liquefaction" of the qranular soil.

Pore-Pressure Generation Scheme

The utilized pore-pressure generation scheme essentially resembles an incremental form of the well-established dynamic analysis procedures developed in the last 20 years by Seed and others. Through real-time coupling of this scheme with the dynamic analysis, effective stresses are continuously updated and, as pore pressures gradually increase, a state of liquefaction is approached in frictional materials. As the available shear strength of the soil decreases, increments of permanent deformations are accumulated in the form of crest settlements and/or distributed slumping of the analyzed earth structure.

Fiqure A-7 presents a flow chart of the computational scheme implemented in DSAGE-2 .1. After defining the static (gravity) state of stress, the static-stress dependent parameters affecting the dynamic strength are stored for each element for subsequent use in the dynamic phase. Pore pressures are driven by dynamic shear stresses which are continuously tracked for each element during the analysis (where appropriate, this scheme could also be driven by shear strains). For each element of the model mesh the "Cumulative Damage" (D) is updated every time a shear cycle is completed. The amplitude of a qiven shear cycle is characterized by "Ntt , " the number of cycles which (at that amplitude) would induce development of 100 percent pore-pressure ratio. The cumulative damage is computed as

A-9

UPOA fl .-oRl 'RESSURES.

MAftAIAL f'AOI'EfiTI£S

--..,

rigure A-7. Deformation-oriented Dynamic Analysis. Ambient- stress-dependent parameters vhich govern dynamic strength and deformation are stored for use in the dynamic-analysi• phase. Pore pressures are updated tvice per •tress cycle, according to •Accumulated Damage•.

A soil subjected to random-amplitude shear cycles will reach a pore-pressure ratio (r) of unity when 0•1. For example, 20 cycles of relatively low-amplitude shearinq characterized by Nuq•40 would result in partial liquefaction (0=20*1/40•0.5). Addinq 10 more cycles corresponding to Nliq=20 would then induce full liquefaction (0•0.5+10*1/20=1); etc. After every cycle, the current value of "D" is related to the pore-pressure ratio "r" by an empirical function.

The input data for the pore-pressure generating scheme are the cyclic strength and the pore-pressure function of the soil. The cyclic-strength relationship of "Nuq" vs. shear-stress amplitude for a qiven soil is expressed by a series of curves determined from cyclic-loading tests in the laboratory, or from SPT or CPT in-situ testing (e.q., Seed and DeAlba, 1986). The empirical pore-pressure function of "r" vs. "D" may be obtained from cyclic-loading tests with pore-pressure measurements or, in lieu of such data, from published results for soils of different types and densities (e.g., Seed et al, 1975, and later works).

A-10

Most recent test results on Monterey #0 sand published by Wang and Kavazanj ian ( 1989} suggest that the shape of the r-vs. -D function depends on the uniformness and sequence of the applied load cycles, in addition to soil type and density.

Notwi thstapding the above, the exact shape of the pore pressure function actually turns out to have little effect on the analysis results with regard to overall shaking-induced deformations of an embankment. Differences in the rate of pore pressure generation tend to be smoothed out by the shear-stress driven model. An accelerated buildup at the beginning of shaking, for example, leads to a slow-down later on, because of the inability of the softened soil to transfer large shear stresses; and vice-versa. This relative insensitivity leads to the adoption in DSAGE-2.1 of the most simple linear function (r-D}, which fits well within the range of curve shapes suggested in the literature.

A-ll

REFERENCES (Appendix)

Cundall, P.A., 1976, "Explicit Finite Geomechanics," 2nd Conf. on Num. Blacksburg, Virginia.

Difference Methods in Methods in Geomech.,

Finn, W.O. Liam, M. Yogendhakumar, N. Yoshida, and H. Yoshida, 1986, "TARA-3: A Program for Nonlinear Static and Dynamic Effective Stress Analysis," Soil Dynamics Group, University of British Columbia, Vancouver, B.C.

Kawai, T. , 19 8 5, "Summary Report on the Development of the Computer Program, DIANA-Dynamic Interaction Approach and Non-Linear Analysis", Science University of Tokyo.

Lee, K.L., 1974, "Seismic Permanent Deformations In Earth Dams," Report UCLA-ENG-7497.

Newmark, N.M., 1965, "Effects of Earthquake on Dams and Embankments," Geotechnique, Vol. 15, No. 2, P. 139.

Prevost, J. H. , 1981, "DYNAFLOW: A Nonlinear Transient Finite Element Analysis Program," Princeton University, Department of Civil Engineering, Princeton, N.J.

Roth, W.H., Bureau, G., Brodt, G., 1991, "Pleasant Valley Dam: An Approach to Quantifying the Effect of Foundation Liquefaction," to be presented at the 17th International Congress on Large Dams, Vienna, June.

Roth, W.H., Scott, R.F., Cundall, P.A., 1986, "Nonlinear Dynamics Analysis of a Centrifuge Model Embankment," 3rd u.s. National Conference on Earthquake Engineering, Aug. 24-28, Charleston, South Carolina, ,Proc., vol. I, pp. 505-516.

Seed, H.B., Idriss, I.M., Makdisi, F., and Banerjee, N., 1975, "Representation of Irregular Stress Time Histories by Equivalent Uniform Stress Series in Liquefaction Analyses: Report:• EERC 75-29, u.c. Berkeley.

Seed, H.B. and DeAlba, P., 1986, "Use of SPT and CPT Tests for Evaluating the Liquefaction Resistance of Sands," ASCE Geotechnical Special Publication.

Serff, N., et al., 1976, "Earthquake-Induced Deformations of Earth Dams," Report EERC-76, u.c. Berkeley.

Wang, J.N., and Kavazanjian, E., 1989, "Pore Pressure Development during Non-Uniform Cyclic Loading," Japanese Society of Soil Mech. and Found. Eng., vol. 29, no. 2.

APPENDIX 3 (continued)

Computer Code DESRA-2

Analysis Method Lumped mass-nonlinear spring system analysis (1-D fmite elements)

Geometrical Dimensions 1-D layered soil

Selected Material Models Incrementally nonlinear hysteretic model (hyperbolic shear stress-strain law) with Masing behavior (incrementally elastic approach)

Loadings Gravity loading and base earthquake motion

Soil-Water Medium Analysis Method Total stress model or effective stress model (Total Stress vs. Effective Stress) with empirical constitutive laws for pore-water

pressure generation derived from 1-D compression tests and cyclic simple shear tests/also models pore water dissipation

Element Types 1-D lumped mass elements

Author/Contact Person or Institution Lee, M. K. W., and Liam Finn, "DESRA-2," University of British Columbia, June 23, 1978

THE UNIVERSITY OF BRITISH COLUMBIA

FACULTY OF APPLIED SCIENCE

DESRA 2

DYNAMIC EFFECTIVE STRESS RESPONSE ANALYSIS OF SOIL

DEPOSITS WITH ENERGY TRANSMITTING BOUNDARY

INCLUDING ASSESSMENT OF LIQUEFACTION POTENTIAL

by

Michael K.W. Lee

&

W.O. Liam Finn

Sponsored by Fugro Inc., Long Beach, California

June 23, 1978

1.

2.

3.

4.

5.

6.

TABLE OF CONTENTS

PROGRAM IDENTIFICATION AND ABSTRACT

DESCRIPTION OF INPUT CARDS

SUBROUTINES OF D E S R A - 2

SAMPLE PROBLEM

D E S R A - 2 SOURCE LISTING

THEORY

6.1 Method of Analysis: Rigid Boundary

6.2 Energy-Transmitting Boundary

PAGE

1

2

8

10

34

51

51

57

1

1. PROGRAM IDENTIFICATION AND ABSTRACT

1.1

1.2

1.3

1.4

Program Name

Program Title

Date

Authors

DESRA-2

Dynamic ~ffective ~tress ~esponse ~alysis

of Soil Deposits with Energy Transmitting

Boundary including Assessment of Liquefaction

Potential

June 23, 1978

Michael K.W. Lee and W.D. Liarn Finn

Faculty of Applied Science

University of British Columbia

Vancouver, B.C.

V6T lWS

1.5 Computer Requirements

The computer program is written in Fortran IV and has

been developed and test run through the use of an IBM 370/168

computer. 46,000 words of core storage on this computer is

required to run the program.

1.6 Abstract

A method of effective stress analysis is developed for

the restricted but practically important case of a horizontally

layered saturated sand deposit shaken by horizontal shear waves

propagating vertically upwards. The method is based on a set of

constitutive laws which take into account important factors that

are known at present to affect the response of saturated sands to

earthquake loading including the generation and dissipation of

pore water pressures. Effect of finite rigidity at the base 0~

the deposit is approximated by using a dashpot model to simulate

the energy transmitting boundary.

2

2. DESCRIPTION OF INPUT CARDS

In general, three main types of problems can be solved by

this computer program, namely,

a) Dynamic Response Only in this case, only the dynamic res-

ponse of a horizontal soil deposit is calculated; the effect of

pore water pressure is not included in the analysis.

b) Dynamic Response Including the Effect of Pore Water Pressure

Generated as a Result of Cyclic Loading in this analysis

the pore water pressure calculated for each layer is assumed to

be confined within the layer and no redistribution nor dissipa­

tion is allowed.

c) Dynamic Response Including Generation, Redistribution and Dissi-

pation of Pore Water Pressure this includes calculation as

for (a) and (b) and, in addition, the consolidation-dissipation

equation is applied to the pore water pressure values for each

time step.

The type of analysis to be carried out by the computer is controlled

by specifying the value for NPTYPE as described in Card 2.2. A maxi­

mum of 20 layers can be analysed and the layers are numbered 1 to 20

from the surface down. Key input variables are explained in the order

of input cards as follows:-

2.1 Title Card

Cols. 1-80 TITLE

2.2 Analysis Control Card

Cols. 1-4 NPTYPE

(20A4)

Eighty characters to describe the

problem.

(20I4)

Analysis control number

= r, dynamic response only;

APPENDIX 3 (continued)

Computer Code DYNARD

Analysis Method Explicit fmite difference for linear and nonlinear static and dynamic problems

Geometrical Dimensions Two-dimensional (2-D)

Selected Material Models Nonlinear soil model - a two-dimensional bounding surface soil model similar to those of Cundall (1979) and Dafalias and Herrmann (1982), (see detailed description in Appendix for references)

Loadings Gravity loading and earthquake motion applied at energy absorbing base

Soil-Water Medium Analysis Method Total stress approach with degradable (Total Stress vs. Effective Stress) undrained soil models - excess pore pressures

are computed from empirical relations

Element Types 2-D plane strain continuum elements

Author/Contact Person or Institution Woodward-Clyde Consultants,.Dr. Mohsen Beikae, 2020 E. First Street, Suite 400 Santa Ana, CA 92705, Tel: (714) 835-6886

1.0 DYNARD COMPUTER PROGRAM

The computer program DYNARD, which was developed by Woodward-Clyde Consultants

has been used to perform two-dimensional, nonlinear, dynamic deformation analyses for

earth structures using an explicit finite difference method. The program analyzes the

deformation and response of earth structures to simultaneous effect of gravity and seismic

shaking using the total stress approach and degradable undrained soil models. A selected

earth structure is discretized into homogenous, isotropic elements and nodal points in a way

similar to the finite element method. Each element is characterized by attributes that are

relatively easily obtainable: its geometry, total unit weight, maximum shear modulus,

undrained shear strength, a variation of shear modulus with shear strain, and the pre­

earthquake bulk modulus through Poisson's ratio. The input motion is specified at the

energy absorbing base of the discretized system. The program has been applied to many

seismically induced deformation evaluations involving case histories, analyses, and design

of earth structures.

DYNARD uses the Lagrangian formulation of the momentum equations, representing

Newton's second law of motion. The Lagrangian formulation inherently takes into account

the mass conservation law and allows elements with fixed masses to translate, rotate,

compress, expand, and distort in space. The equations of motion to be solved at each nodal,

point are replaced by corresponding finite difference expressions using Gauss' divergence

theorem. As summarized below, the numerical evaluation of the finite difference equations

for a garendiscretized system has two main calculation components: element calculations and·

nodal point calculations:

Element Calculations:

For all the elements, assuming that all the nodal velocities and displacements art! KJI\ .. wn.

a) The strain increments are computed for each element from the known

velocities and displacements of the nodal points surrounding that element.

b) The stress increments are computed for each element from the strain

increment using a soil model, which relates strain increments to stress

23092GIYMTE.XT Ol-21-93(08:44aml/23092/KAJ

increments.

Nodal Calculations:

For all the nodal points (after the stress increments are added to the previous stresses to

obtain new stresses for each element).

a) The out-of-balance forces are computed for each node from the known

stresses in the elements surrounding each node.

b) The acceleration of each nodal point is computed from these forces by

Newton's second law and integrated twice to give new nodal velocities and

displacements, updating the previous nodal velocities and displacements.

The above calculational loop can be initiated by assuming initial values of nodal velocities

and displacements, which are usually zero. No iterations are involved in the calculations

discussed above. In performing elemental calculations above, the velocities and

displacements of all nodal points can be assumed to be fixed while the stresses are being

computed as explicit functions of nodal velocities and displacements. Similarly, in

performing nodal calculations above, all stresses can be assumed to be fixed while the

velocities and displacements are being calculated as explicit functions of elemental stresses.

This manner of performing finite difference calculations using explicit functions is call

explicit.

DYNARD allows for a compliant base, which models the ability of the material underlying

the discretized system to absorb seismic energies hitting the base boundary using a linear

elastic halfspace. The properties of the halfspace are defined by its unit weight and shear

wave velocity. The compliant base prevents the numerical and unrealistic trapping of

seismic energies within the discretized system.

2.0 DYNARD ANALYSIS PROCESS

The DYNARD deformation analysis usually consists of three parts: 1) turn-on gravity

analysis to obtain initial stresses in an earth structure before an earthquake motion is applied,

23092G/YMTEXT 01-21-93(08:44am)/23092/KAJ

2) dynamic analysis to obtain deformations and response of the earth structure due to gravity

and the earthquake motion, and 3) post-earthquake analysis to evaluate the deformation of

the earth structure following the earthquake shaking and under the gravity load alone. Each

of these parts are discussed below.

2.1 Turn-on Gravity Analysis

In the turn-on gravity analysis the program calculates the initial static stresses in all the

elements due to gravity load. This is done by fixing the base of the earth structure as a rigid

base and applying gravity body force. The vertical lateral boundaries are set on vertical

rollers to simulate the static free field soil columns. The gravity load is applied by slowly

turning on the gravity body force from zero to its full value and then maintaining it for the

entire duration of DYNARD analysis. For this part of the analysis a high damping ratio is

specified for each nodal points, and the analysis is continued until all the vibrations in the

earth structure induced by the perturbation of introducing the gravity are damped out. At

the end of the turn-on gravity analysis the calculated stresses (horizontal, vertical, and shear

stresses) would satisfy the horizontal and vertical force equilibrium conditions. In addition,

the strain compatibility conditions in the earth structure are also satisfied.

2.2 Dynamic Analysis

Before starting the dynamic analysis, the specified high damping ratio, used for turn-on

gravity analysis, is switched to a very low value. The fixed base of the earth structure is·

changed to a compliant base; the vertical rollers for the lateral boundaries are replaced by

horizontal rollers; and horizontal forces calculated by DYNARD are imposed on the lateral

boundaries to simulate the free field earth columns. These forces at the lateral boundaries

are in equilibrium with the static horizontal and shear stresses that exist in those elements

adjacent to tite !ateral boundaries.

The whole earth structure under the full gravity load is then excited by an input motion

specified at the compliant base. At this stage the dynamic stresses induced by the base

excitation are added to the static stresses, and, as a result, the earth structure may translate,

rotate, compress, expand, and distort in space.

23092GIYMTE.XT Ol-21-93(08:44am)/230921KAJ

2.3 Post-Earthquake Analysis

In the post-earthquake analysis, the input motion is stopped, but gravity load is maintained

on the earth structure to evaluate deformation that may be induced by readjustment of ·

stresses and strains developed during the earthquake shaking. The gravity force is

maintained until a final static equilibrium is achieved. If the earth structure keeps deforming

and the final static equilibrium is not achieved in a reasonable time after the cessation of

shaking, a failure of the earth structure may be considered a likely scenario for the system

depending on the deformation pattern.

3.0 NONLINEAR SOIL MODEL

The soil reflects the undrained degrading cyclic behavior of saturated soils and the total

stress behavior of non-saturated soils. However, the soil properties are assigned based on

the initial effective stresses, and the model is capable of calculating cyclically induced excess

pore water pressures and associated degradation of soil moduli and shear strengths. The soil

parameters needed in the model are relatively simple, and, therefore, easily assignable even

based on limited subsurface information.

The cyclic and nonlinear effects of soils are incorporated in the analyses by a two­

dimensional bounding surface soil model similar to those of Cundall (1979) and Dafalias and

Herman (1982). Details of some aspects of the computation process used in the soil model

are presented by Cundall (1979). In this model the following three basic considerations are·

incorporated: 1) a yield surface, 2) a flow rule, and 3) a hardening rule.

3.1 Yield Surface

A yield surface is specified for sheat kads. However, it is not specified for the compression

loads. The Von Mises criterion is used for the yield surface in shear. The criterion states

that plastic flow occurs when the maximum shear stress equals to the allowable shear

strength. This criterion is considered adequate for undrained strength analyses.

230920/YMTEXT OI-2I-93(08:44aml/23092/KAJ

3.2 Flow Rule

For the flow rule, it is assumed that a total strain increment is comported of linear elastic

and plastic strain increments. The elastic stress increment is in the same direction as the

elastic strain increment. Therefore, for a given strain increment and an elastic modulus,

both the magnitude and direction of the stress increment can be calculated.

However, the direction of plastic stress increment in general should not be the same as that

of the strain increment. Furthermore, the magnitude of plastic stress increment is also

unknown. Following plasticity theory, the direction of the plastic stress increment in the soil

model is provided by the normal to the plastic potential function, assumed to be the same as

that of the yield surface, where the stress state make contact with it. The plastic stress

increment is assumed to be proportional to the projection of a plastic strain increment on the

unit normal to the plastic potential function. Such a flow rule is referred to as "associative."

The next step is to determine the amplitude of the plastic stress increment.

3.3 Hardening Rule

The hardening rule is described by the shape of a backbone curve. A backbone curve is

defined as a basic monotonic stress-strain curve spanning from the zero strain to a large

failure strain. The backbone curve is developed based on a given variation of shear modulus

with shear strain. The small strain shear modulus is based on maximum shear modulus, and

the failure level shear moduli are based on undrained shear strength. The variation of shear·

modulus with shear strain can be selected in almost any manner. For example, it can be

selected to be similar to that of the sand by Seed and Idriss (1970) or to that of clay by

Vucetic and Dobry (1991), depending on the material. The hardening rule is used to

calculate the slope of stress-strain curve at any point along the backbone curve. On the basis

of the difference between the slope and the small strain inaximum shear modulus, the

magnitude of the plastic stress increment is assigned.

3.4 Cyclic Behavior

The soil model is assumed to "know" at what stress level it will fail and adjust its

instantaneous modulus accordingly. The backbone curve is scaled and shifted, depending

230920/YMTE.XT Ol-21-93(08:44am)/23092/KAJ

on the direction and distance from the current stress point to a point on the failure envelope

along the direction of the strain increment to produce cyclic stress-strain curves. The strain

increment, computed from the equations of motion, are used with the basic backbone curve

to calculate the direction and magnitude of stress increment, consistent with the flow rule and

the hardening rule discussed above.

As part of the computation, the scale of the basic backbone curve is changed whenever the

incremental strain vector changes direction. The incremental shear modulus reverts to the

initial value when the incremental strain vector rotates through an angle of 180 degrees in

the plane of incremental deviator strain. At the other extreme of no rotation, nothing is

changed. A linear interpolation is used between these two extreme conditions.

3.5 Hysteretic and Nonhysteretic Dumpings

Soil exhibits material damping under cyclic loading conditions. The amount of damping in

general increases with increasing cyclic strain and is proportional to the inner area of the

hysteresis loop (Seed and ldriss 1970). The soil model described above generates hysteretic

loops under cyclic loading conditions. These loops provide hysteretic damping, which

increases with increasing cyclic strain.

However, for small cyclic strain, say less than about 0.001 percent, the amount of hysteretic

damping generated by the model becomes essentially zero. To account for some damping,

usually about 2 to 4 percent, in this cyclic strain range observed in cyclic laboratory tests·

on soils (Seed and Idriss, 1970), viscous damping can be specified and is included in the

model through the equations of motion. The viscous damping is specified to provide the

desired level of hysteretic-like damping at the fundamental frequency of the system at very

low strain.

3.6 Degradation

When cyclic loading is applied to soils under certain conditions, they degrade in both

stiffness and shear strength. The degradation of the shear strength can be based on (1)

amount of excess pore water pressure generated during cyclic loading ("cyclic degradation"),

and (2) the amount of seismically or monotonically induced large shear strain ("large strain

230920/YMTEXT 01-21-93(0~:44amli23092/KAJ

degradation"). The soil model is capable of progressively reflecting cyclic degradation of

stiffness and shear strength as a function of cyclically induced excess pore water pressures

generated during dynamic analyses.

3. 7 Tension Cracks

The soil model incorporates a tension failure mechanism to disallow development of any

tensile stress beyond a specified tensile strength. For this purpose the minor principal stress

is routinely calculated and checked against the assigned tensile strength for all the elements

during the analysis. As long as the minor principal stress for each element is within the

tensile strength, no action is taken. However, once the minor principal stress exceeds the

tensile strength, the direction of principal stresses are calculated and fixed along with the

total vertical stress for the element. Based on the known direction of the minor principal

stress, a value of the minor principal stress (now set equal to the tensile strength), and total

vertical stress; horizontal and shear stresses are calculated for the element. The newly

calculated state of stress in the element is such that the major principal stress is in

compression and the minor principal stress is equal to the tensile strength of the soil. These

modified stresses are now included in the computation, and the element is considered as

"cracked." However, once stresses go back into utensile zone, they can be considered

"healed."

3.8 Bulk Modulus

The initial bulk modulus of the model is specified through Poisson's ratio for each element

and kept constant throughout the analysis.

4.0 EXAMPLE APPLICATIONS

DYNARD computer program has been applied to many seismically induced deformation

evaluations involving case histories, analyses, and design of earth structures. In the

following sections selected results from the following three DYNARD applications are

presented: Lexington Dam in northern California during the 1989 Lorna Prieta earthquake,

Upper San Fernando Dam in the southern California during the 1971 San Fernando

earthquake, and a tailing dam design.

230920/YMTEXT OJ ·2 I -93(08:44am)/230921KAJ

4.1 Lexington Dam Case History (1989 Lorna Prieta Earthquake)

Lexington dam, completed in 1953, is located on the Santa Cruz mountains of California,

about 15 miles north of Santa Cruz. The dam is 207 feet high and is a zoned earthfill with

upstream and downstream slopes of 5.5:1 (horizontal:vertical) and 3:1, respectively. The

crest of the dam at elevation 667 feet is 40 feet wide and about 810 feet long. The dam is

founded on Franciscan formation, which is composed chiefly of interbedded sandstone and

shale. The dam was built of densely compacted local materials. The embankment zone

consists of a thick central impervious core bounded by more pervious shells, with an internal

drain zone located between the core and the downstream shell.

The dam was instrumented as part of the California Strong Motion Instrumentation Program.

The strong motion instrumentation at the dam site consisted of 3 accelerographs. One set

is located at a rock outcrop at the left abutment, another set, on the left crest and the third

set, on the right crest. At each of the three locations the accelerographs were oriented in

three orhogonal directions: transverse (normal to the dam axis), longitudinal (along the dam

axis), and vertical. The instruments have recorded motions during several earthquakes

including the October 17, 1989 Lorna Prieta earthquake with a moment magnitude of 7.

During the Lorna Prieta earthquake, peak accelerations in the transverse direction of 0.39g

and 0.45g were recorded at the left and right crest of the dam respectively, and 0.45g at the

rock formation of the left abutment. The dam was about 13 miles from the source of this

earthquake.

The seismic analysis of the dam was performed using DYNARD. An analysis section

representing a maximum section of the dam was considered for the analysis. This section

is located next tot he right crest accelerograph. The material properties were specified based

on available geotechnical data. The motion recorded at the left abutment in the direction

transverse tu the axis of the dam was used as an input motion with a maximum acceleration

of 0.45g in the analysis.

Figure I compares the recorded and computed results at the right crest of the dam in terms

of response spectra at a damping of 5 percent and horizontal acceleration time histories.

While the computed results show somewhat higher response in the high frequency range, the

recorded and computed results compare favorably. If you reduce the peak acceleration of

230920/YMTE.XT Ol-21-93(08:44am)/23092/KAJ

the input motion from 0.45g (which, being from the left abutment, may reflect a topographic

effect) to a lower value, the comparison will improve. Figure 2 shows a comparison of

recorded and computed horizontal and vertical displacements of the crest. Although the

amounts of seismically induced displacements are relatively small, the comparison appears

to be reasonable.

4.2 Upper San Fernando Dam Case History (1971 San Fernando Earthquake)

The upper San Fernando dam, which was located northwest of Los Angeles, was an 80 foot­

high hydraulic fill dam with a reservoir of about 1850 acre-feet. Many details of its

construction, its damage due to the 1971 San Fernando earthquake, and its seismic analysis

are presented in a report by Seed et al (1973). The San Fernando earthquake had a surface

wave magnitude of 6.6 and a focal depth of about 8 miles. The dam was about 8-112 miles

from the source of this earthquake, and the peak acceleration at the dam site was considered

to have been no greater than 0.55 to 0.60g (Scott, 1972).

The earthquake apparently created severe longitudinal cracks running almost the full length

of the dam on the upstream slope slightly below the pre-earthquake reservoir level. The

crest of the dam reportedly moved downstream about 5 feet and settled vertically about three

feet. At the downstream toe of the dam a two feet high pressure ridge was observed. These

major observations of the dam following the 1971 earthquake are summarized for a major

section of the dam in Figure 3.

The seismic analysis of the dam was performed using DYNARD (Moriwaki, Beikae, and

ldriss, 1987). A modified Pacoima accelerogram (Boore, 1973) scaled to a peak acceleration

of 0.6g was used as an input accelerogram at the base. The results of the computed

deformation at the end of shaking are summarized in Figure 3 in terms of a deformed finite

difference mesh (using a magnihcatio,; factor of 2). The computed deformations correspond

in a reasonable way with the observed deformation pattern of the dam.

4.3 A Tailing Dam Design

DYNARD was used in design of a tailing dam in a seismically active region of the United

States. The analysis section presented herein has a maximum height of 255 feet, a vertical

23092GNMfEXT 01·21·93(08:44aml/23092/KAJ

upstream slope, a 3:1 (horizontal:vertical) downstream slope, and a crest width of 100 feet.

The seismic considerations and in particular seismically induced shear strength reduction in

the foundation soils were the major design issue. The analysis accelerogram corresponded

to a magnitude 7-112 event with a peak horizontal acceleration of 0.39g. This accelerogram

was assigned at the compliant base of the model. The results of the computed deformation

at the end of shaking are summarized in Figure 4 in terms of post-seismic displacement

vectors and a deformed mesh (using a magnification factor of 5). The results indicate that

the entire embankment shell translated horizontally as a unit by a large amount. The results

shown in Figure 4 and many other results for other trial sections were used as the controlling

factor in developing the design section of this tailing dam.

5.0 REFERENCES

Boore, D. (1973), "The Effect of Simple Topography of Seismic Waves: Implications for

the Accelerations Recorded at Pacoi rna Dam, San Fernando Valley, California,"

Bulletin of the Seismological Society of America, Vol. 63, 1603-1609.

Cundall, P.A. (1979), "The 'Failure Seeking Model' for Cyclic Behavior in Soil-An Initial

Formulation for Two Dimensions," Technical Note PLAN-I, Peter Cundall

Associates, July.

Dafalias, Y.F. and Hermann, L.R. (1982), "Bounding Surface Formulation of Soil

Plasticity," Soil Mechanics - Transient and Cyclic Loads, G. Pande and O.C. ·

Zienkieicz, Eds., John Wiley & Sons, Inc., London, U.K., pp. 253-282.

Makdisi, F.I., and Seed, H.B. (1978), "Simplified Procedure for Estimating Dam and

Embankment Earthquake-Induced Deformations," ASCE, Journal of Geotechnical

Engineering, Vol. 104, No. GT7, pp. 749-867.

Moriwaki, Y., Beikae, M., and Idriss, I.M. (1988), "Nonlinear Seismic Analysis of the

Upper San Fernando Dam Under the 1971 San Fernando Earthquake," Proceedings

of Niufh?? World Conference on Earthquake Engineering, Tokyo, Japan, pp. VIII-

237 - VIII-241.

230920/YMTEXT 01-21-93(08:44am)/23092/KAJ

Scott, R.F. (1972), "The Calculation of Horizontal Accelerations from Seismoscope

Records," Paper presented at a Seismological Society of America Conference in

Hawaii.

Seed, H.B., and Idriss, I.M. (1970), "Soil Moduli and Damping Factors for Dynamic

Response Analyses," Report No. EERC 70-10, University of California, Berkeley,

December.

Seed, H. B., Lee, K.L., Idriss, I.M., and Makdisi, F. (1973), "Analysis of the Slides in the

San Fernando Dams During the Earthquake of February 9, 197?," University of

California, Berkeley, Rep. No. EERC 73-2.

Vucetic, M., and Dobry, R. (1991), "Effect of Soil Plasticity on Cyclic Responses," ASCE,

Journal of Geotechnical Engineering, Vol. 117, No. GTI.

23092GfYMTE.XT Ol-21-93(08:44am)/23092/KAJ

1.0 -bD --CJ 0.5 CJ < - 0.0 ~ -a t: 0 N -0.5 ·-s.. 0

::c -1.0

1.0 -bD --CJ 0.5 CJ < - 0.0 ~ -a t: 0 N -0.5 ·-s.. 0

::c -1.0

5% Damping 5.0 --------------------------------------.

-~ -4.0 t: 0 ·--a ~ s.. 3.0 Q) -Q)

CJ CJ < 2.0 -~ s.. -a CJ cv 1.0 c..

C/)

0.0 10 -I

0 2

0 2

-Recorded at risht crest - Computed at nght crest

I II I II I II 10 _, 1 10 Period (sec)

Recorded at right crest

4 6 8 10 12 14 16 18

Computed at right crest

4 6 8 10 12 14 16 18 Time (sec)

20

20

FIGURE 1 - COMPARISON BETWEEN RECORDED AND COMPUTED RESPONSE SPECTRA AND HORIZONTAL TIME HISTORIES UNDER THE 1987 LOMA PRIETA EARTHQUAKE

Woodward-Clyde Consultants

Obseved Deformations Mter 1987 Loma Prieta earthquake

Computed Deformations 0 100 200 SCALE (ft)

FIGURE 2- OBSERVED AND COMPUTED DEFORMATIONS OF LEXINGTON DAM UNDER THE 1987 LOMA PRIETA EARTHQUAKE

Woodward-Clyde Consultants

3ft

Observed Deformations After 1971 San Fernando Earthquake

2ft pressure ridge

J ----

0 50 100 0.3 ft SCALE (ft)

0.5 ft pressure. ridge ~ , 1 I '/ / / / I / I f ' \ \ \ \ \~

..,.,.,...--, _/ / I /// / / / / ~ I 1 1 I I I I I I I ( / / I I \ _\ \ \ \ ~ )

~' 1 .t 1 ' / 1 / L J 1 ' r~ -- I I J -~

Computed Deformations

Note: Magnification factor for defonned mesh is 2.

FIGURE 3 - OBSERVED AND COMPUTED DEFORMATIONS OF UPPER SAN FERNANDO DAM UNDER THE 1971 SAN FERNANDO EARTHQUAKE

W oodward-Ciyde Consultants

Initial Upstream Slope Location

~

DISPLACEMENT VECTORS

I h-

"" t<:t}

34 ft

Average Shell Displacement

t:W

-DEFORMED MESH

Note : Magnification factor for displacement vectors and deformed mesh is 5.

~

0 200 400 SCALE (ltl

FIGURE 4 - POST-SEISMIC DISPLACEMENT VECTORS AND DEFORMED MESH FOR A TAILING DAM

Woodward-Clyde Consu Ita nts

APPENDIX 3 (continued)

Computer Code DYNAFLOW

Analysis Method Finite element analysis program for static and transient response of linear and nonlinear problems

Geometrical Dimensions Two and three-dimensional (2-D and 3-D)

Selected Material Models Linear elastic, nonlinear hyperelastic, Newtonian fluid, Von Mises elasto (-visco)-plastic, Drucker-Prager elasto ( -visco)-plastic, Matsuoka (Mohr-Coulomb) elasto (-visco)-plastic, a family of multi-yield elasto ( -visco )-plastic surfaces

Loadings Arbitrary static and dynamic loading (force and initial displacements) of the mesh-earthquake acceleration can be applied at base of mesh

Soil-Water Medium Analysis Method Uses coupled field equations for saturated (Total Stress vs. Effective Stress) porous media (effective stress analysis for a

two-phase soiVpore fluid medium) -pore pressure generation and dissipation capabilities

Element Types 1-D element, 2-D plane with axisymmetric options, 3-D brick element, various contact and slide-line elements, structural elements (truss, beam, plate, shell, membrane), 2-D and 3-D boundary and link elements

Author/Contact Person or Institution J.-H. Prevost, Department of Civil Engineering, Princeton University, Princeton, NJ 08544, Tel: (609) 258-5424

December 8, 1992

DESCRIPTION OF COMPUTER CODE liDlfNA.IFIL<CDW

Jean H. Prevost

Department of Civil Engineering and Operations Research School of Engineering and Applied Science

Princeton University Princeton, NJ 08544

Phone: (609) 258-5424

DYNAFLOW is a finite element analysis program for the static and transient response of linear and nonlinear two- and three-dimensional systems. In particular, it offers transient analysis capabilities for both parabolic and hyperbolic initial value problems in solid, structural and fluid mechanics. There are no restrictions on the number of elements, the number of load cases, the number of load-time functions, and the number or bandwidth of the equations. Despite large. system capacity, no loss of efficiency is encountered in solving small problems. In both static and transient analyses, an implicit­explicit predictor-(multi)corrector scheme is used. The nonlinear implicit solution algorithms available include: successive substitutions, Newton-Raphson, modified Newton and quasi-Newton (BFGS and Broyden updates) iterations, with selective line search options. Some features which are available in the program include:

Vectorized coding designed to fully exploit the architecture of vector and/or parallel machines. Selective element reordering options applicable to unstructured as well as structured meshes in order to allow parallel and/or vector processing of elemental arrays in blocks. Selective specification of high- and low-speed storage allocations options. Direct symmetric and non-symmetric matrix column equation solvers (in-core and out-of-core Crout profile solvers). Symmetric frontal solver (in-core and/or out-of-core). Iterative matrix equation solvers: preconditioned conjugate gradients with diagonal and/or element-by-element Crout or Gauss-Seidel preconditioning. Iterative "memoryless" quasi-Newton/conjugate gradient solution procedures. Eigenvalue/vector solution solvers including determinant search, subspace iterations and various Lanczos algorithms. Equation numbering optimization option to reduce bandwidth and column heights of stiffness matrix. Capabilities to slave nodes to share the same equation number for any specified degree of freedom. Selective specification of element-by-element implicit, explicit or implicit­explicit options. Reduced/selective element-by-element integration options. Coupled field equation capabilities for treatment of thermoelastic and saturated porous media. Arbitrary Euler-Lagrange description options for fluid and/or fluid structure(­soil) interaction problems. Generalized convective, radiative and enclosure radiative boundary condition options for heat transfer and/or thermoelastic analysis. Prescribed nodal and/or surface forces options. Prescribed nodal displacement, velocity or acceleration options.

December 8, 1992

IIDlfNAIFlLCCD'Wooo DESCRIPTION (Cont'd)

Prescribed arbitrary load-time functions. Eanhquake acceleration time history generation capability, for eanhquake motions compatible with prescribed acceleration response spectra. Wave transmitting boundaries. Isoparametric data generation schemes. Birth/death options to model addition (birth) or removal (death) of elements (materials) in the physical system. Capability to perform constitutive experiments along prescribed stress and/or strain paths on selected material elements within the finite element mesh. Complete restart capabilities with options to selectively change input data. Free input format mode organized into data blocks by means of corresponding macro commands. Graphics post-processing capabilities including mesh plots (2D/3D), displacement/velocity vector plots, contour plots of selected nodal/field quantities, flownets, time history line plots of selected nodal/field quantities. (Fully interactive color postprocessors (2D/3D), with animation options, are available for SGI IRIS graphics workstations.)

The element and material model libraries are modularized and may be easily expanded without alteration to the main code.

The element library contains a one-dimensional element, a two-dimensional plane element with axisymmetric options, and a three-dimensional brick element. A contact element, a slide-line element with either perfect friction or frictionless conditions, a slide­line element with Coulomb friction, a truss element, a beam element, a plate element, a shell element, a membrane element, a boundary element and a link element are also available for two- and 1three-dimensional analysis.

The material library contains a linear elastic model, a nonlinear hyperelastic model, a linear/nonlinear thermoelastic model, a linear/nonlinear heat conduction model, a Newtonian fluid model, a Von Mises elasto( -visco)-plastic model, a Drucker-Prager elasto(-visco)-plastic model, a Matsuoka (Mohr-Coulomb) elasto(-visco)-plastic model, a multi-mechanism (Ishihara's) elasto-plastic model, a family of multi-yield elasto( -visco)­plastic models developed by the author and a viscoelastic Mises model.

The program is written in standard FORTRAN IV/F77.

COPYRIGHT (C) PRINCETON UNIVERSITY 1983.

TinS PROGRAM IS PROPRIETARY TO PRINCETON UNIVERSITY, PRINCETON, NEW JERSEY. IT MAY ONLY BE USED AS AUTI!ORIZED

IN A LICENCE AGREEMENT CONTROLLING SUCH USE.

APPENDIX 3 (continued)

Computer Code DYSAC2(2)

Analysis Method Finite element code for static and transient linear and nonlinear problems

Geometrical Dimensions Two-dimensional (2-D) plane strain

Selected Material Models Isotropic linear elastic, nonlinear bounding surface (Dafalias and Herrmann) elasto-plastic models for cohesive and cohesionless soils

Loadings Gravity loading and horizontal dynamic base motion

Soil-Water Medium Analysis Method Uses equations modeling coupled behavior of (Total Stress vs. Effective Stress) soil and pore fluid (effective stress analysis of

a two-phase soil/pore fluid medium)- pore pressure generation and dissipation capability

Element Types 2-D plane strain continuum elements

Author/Contact Person or Institution Program was developed at U.C. Davis by Professors L. R. Herrmann andY. F. Dafalias and K. A. Arulanandan and several graduate students. Professor K. A. Arulanandan, Civil Engineering Department, University of California, Davis, Davis, CA 95616, Tel: (916) 752-0895

COMPUTER CODE DYSAC2

( Dynamic Soil Analysis Code for 2 - dimensional problems)

DYSAC2 is a finite element computer program for the dynamic analysis of two dimensional

geotechnical engineering structures. DYSAC2 is based on the rigorous mathematical

formulation of the coupled dynamic behavior of soil skeleton and pore fluid. This

computer code can be used to predict the behavior of dams, embankments, levees and other

geotechnical engineering structures, including the prediction of pore pressure generation

and dissipation at various locations, during dynamic loading events such as eanhquakes.

Linear and nonlinear material behavior can be modeled in DYSAC2. Nonlinear material

behavior is modeled using bounding surface elastoplastic effective stress models.

DYSAC2 has isoparametric mesh generation scheme and graphics post-processing

capabilities such as mesh plots, contour plots of selected nodal/field quantities and time

history line plots of selected nodal/field quantities. Predictions made by DYSAC2 have

been verified using centrifuge model test results.

User's Manual For DYSAC2

A Dynamic Soil Analysis Code for 2-Dimensional Problems

K.K. Muraleethar'an

K.D. Mish

K. Arulanandan

and

C. Yogachandran

Department of Civil Engineering

University of California, Davis

June 1991

/ I ~

/

1.0 GENERAL

The computer program DYSAC2 has been written to perform dyna­

mic analysis of two-dimensional geotechnical engineering struc­

tures under plane strain conditions. DYSAC2 essentially gives the

finite element solution of the fully .. coupled dynamic governing

equations of a saturated porous media (a two phase media)

(Muraleetharan, 1990) . In DYSAC2, stress-strain behavior of

soils can be described by isotropic linear elastic model and

bounding surface elastoplasticity models. Horizontal base motion

type of loading is the only dynamic loading option currently

available in DYSAC2.

DYSAC2 is written in FORTRAN 77 and is quite modular in

nature. Variable names are carefully selected to • reflect the

quantities they are representing. These programming steps facili­

tate easy incorporation of other constitutive models as well as

other dynamic loading scenarios. A finite element mesh generation

scheme is included in DYSAC2 and quite a number ·of output files

are created for easy post-processing.

2.0 INPUT TO THE DYSAC2 COMPUTER PROGRAM

All input to DYSAC2 (with the exception of alphanumeric

strings) is list-directed (i.e., format free). It is recommended

that the user gives specific values for all the required input

quantities, since default values for only few quantities are used

if no specific values are given.

1

Names of all the input and output files are read from a file

named FILE·DAT. Modify the subroutine FILES of the program DYSAC2

to specify exact location of the file FILE· OAT.

following files are read from the file FILE•DAT:

1. Input data file

2. Control data file

3. Initial stress data file

4. Base motion data file

s. output data file

6. Output file for displaced mesh date

7. output file for selected nodal displacements

e. output file for selected nodal accelerations

Names of the

9. output file for selected element excess pore pressures

10. output file for selected element stresses and strains

11. output file for element stresses and strains . •

12. Output file for information regarding convergence of the iteration scheme

13. output file to write debug information (empty for normal execution)

In describing the input data required by the DYSAC2 computer

program, the following convention is used: fields within a record

are listed in order, with each field described by: variable name,

variable type in parentheses (where A = alphanumeric, I = integer,

R = real), and a short description. Explanatory notes are given

at the end of each record, whenever they are deemed necessary. In

general, a record is a line of FORTRAN input.

2

APPENDIX 3 (continued)

Computer Code DYSLAND

Analysis Method Finite element analysis program for static and dynamic impact loading (developed particularly for simulating impact penetration problems in soil mechanics)

Geometrical Dimensions 2-D plane strain with axisymmetric option

Selected Material Models Linear elastic, nonlinear elasto-plastic and visco-plastic constitutive model based on multiple nested yield surface theory including strain hardening and softening effects (isotropic/kinematic). Multiple yield surface theory is extended to account for volumetric plasticity - a large deformation/rotation formulation of the constitutive equations is included

Loadings Gravity and static loading and impact transient (or other dynamic loadings) applied at desired locations of the finite element mesh

Soil-Water Medium Analysis Method Dry soils only (Total Stress vs. Effective Stress)

Element Types Triangular (three-node) or quadrilateral (four-node) elements for plane strain, plain stress, or torsionless axisymmetric analysis (plane stress element is valid only for elastic analysis)

Author/Contact Person or Institution Dr. Said Salah-Mars, Woodward Clyde Consultants, 500 12th Street, Suite 100, Oakland, CA 94607-4014, Tel: (510) 874-3051

A MULTIPLE YIELD SURFACE PLASTICITY MODEL FOR

RESPONSE OF DRY SOIL TO IMP ACT LOADING

A DISSERTATION SUBMITIED TO THE DEPARTMENT OF CIVIL ENGINEERING

AND THE COMMIT lEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY

IN PARTIAL FULFILLivffiNT OF THE REQUIREJ.\1ENTS FOR THE DEGREE OF

DOCfOR OF PHILOSOPHY

By

Said Salah-mars December 1988

ABSTRACT

The objective of this work is to develop a dynamic nonlinear finite element

model to simulate impact penetration problems in soil mechanics. Model

development was oriented toward modelling the Standard Penetration Test. An

elasto-plastic constitutive model based on multiple nested yield surface theory is

implemented in a global finite element program. The multiple yield surface model is

developed such that it can account for strain hardening and strain softening. The

evolution of the yield surfaces may follow both kinematic and isotropic

hardening/softening rules depending on the soil under investigation. Pure deviatoric

plasticity and pressure dependency of the yield surfaces (Drucker & Prager type)

are also accommodated in the model. Multiple yield surface theory is extended to

account for volumetric plasticity through the use of multiple caps that expand (or

yield) in an isotropic fashion. A linear viscous term is introduced to account for the

dependence of soil behavior on the rate of loading.

A large deformation formulation of the constitutive equations is developed.

The large deformation/rotation formulation is based on the J aumann rate of stress.

The rotation generated by the spin or vorticity is used to transform the constitutive

equations such that the small deformation formulation is also valid for the finite

deformation case. From the global view point, the large deformation capability is

implemented with reference to an updated spatial configuration known as the

Eulerian representation, which produces an additional stiffness component in the

governing equations referred to as the initial stress stiffness matrix.

The constitutive equations are implemented into a fmite element program

capable of representing the dynamic behavior of viscous material. To efficiently

integrate the evolution equations of the constitutive model, a new concept of virtual

surfaces is developed. The virtual surfaces help locate precisely the active yield

surface so that the trial stress may be updated in accordance with the radial return

mapping algorithm after an intermediate stress update is completed. Global system

nonlinearity is accommodated through an explicit-predictor implicit-multi-corrector

algorithm. Material nonlinearity is approached iterativelly via either a consistently

v

Abstract

updated tangent operator (known as the Newton-Raphson method), or the simpler

initial tangent operator method. The initial tangent operator method is characterized

by a lower convergence rate, but no update is required.

Program performance is tested against several standard problems for which

closed form solutions existed. The program is then applied to study the behavior of

the Standard Penetration Test (SPT) when driven into dry Sacramento River Sand.

Laboratory test results for this sand at relative densities of 40% and 90% are used in

the analysis. For each density, three confining pressures are analyzed for

penetration resistance. The finite element adopted to analyze the SPT and the

surrounding soil is a torsionless axisymmetric four node isoparametric

quadrilateral. The loading is a longitudinal shock wave applied along the axis of the

drill rod. A direct time integration scheme is used based on a mesh partitioning

technique. The high frequency material (sampler and rods) is treated via an implicit

unconditionally stable Newmark scheme to take advantage of a less stringent time

step increment. The soil mass is treated via an explicit conditionally stable scheme,

since this medium (soil) is characterized by much lower frequency content,

resulting in less required storage as opposed to an implicit formulation.

The parametric study of the S.P.T. leads to two practical results. First, a

correlation between blow count and relative density is obtained as a function of

confining pressure. This correlation compares well with results from controlled

experiments. Second, a correction chart to account for the rod length effect on the

SPT penetration resistance is developed.

vi

APPENDIX 3 (continued)

Computer Code LUSH,FLUSH,ALUSH,TLUSH,QUAD4 computer programs for 2-D and 3-D dynamic analysis of soil-structure interaction and earth dams

Analysis Method- Finite elements- except for QUAD4 the programs solve dynamic problems in frequency domain rather than time domain. QUAD4 is only used for dynamic response of earth structures

Geometrical Dimensions 2-D and 3-D

Selected Material Models Linear elastic, multiple nonlinear soil properties for equivalent linear analysis

Loadings Static gravity and nodal forces or displacements, dynamic base motion (earthquake loading only)

Soil-Water Medium Analysis Method Dry condition only (Total Stress vs. Effective Stress)

Element Types Solid elements (2-D and 3-D) and linear bending elements (beam elements), transmitting (energy absorbing) boundary elements, 2-D elements: plane strain and stress and axisymmetric elements

Author/Contact Person or Institution U.C. Berkeley Reports: (1) EERC 75-30 (November 1975), (2) UCB/EERC-81/14 (September 1981), and (3) EERC 73-16 (July 1973)

(1): FLUSH "A Computer Program for Approximate 3-D Analysis of Soil-Structure Interaction Problems," by J. Lysmer, T. Udak:a, C. F. Tsai, and H. B. Seed

(2): TLUSH "A Computer Program for the Three-Dimensional Dynamic Analysis of Earth Dams," by T. Kagawa, L. H. Mejia, H. B. Seed, and J. Lysmer

(3): QUAD-4 "A Computer Program for Evaluating the Seismic Response of Soil Structrures by Variable Damping Finite Element Procedures," by I.M. Idriss, J. Lysmer, R. Hwang, and H.B. Seed.

EARTHQUAKE ENGINEERING RESEARCH CENTER

FLUSH A COMPUTER PROGRAM FOR APPROXIMATE 3-D ANALYSIS

OF SOIL - STRUCTURE INTERACTION PROBLEMS

by

John Lysmer

Takekazu Udaka

Chan-Feng Tsai

H. Bolton Seed

Report No. EERC 75-30

November 1975

College of Engineering

University of California

Berkeley, California

rABLE OF CONTENTS

1. INTRODUCTION

2. SOIL-STRUCTURE INTERACTION ANALYSIS

Basic Requirements

Two-dimensional Finite Element Analysis

Three-dimensional Effects

Effects of Building-Building Interaction

Proposed Computational Method

3. COMPUTATIONAL MODEL

General

Identification of Nodes and Elements

Boundary Conditions

Mass Distribution

Stiffness and Damping

Free Field Motions

4. NUMERICAL PROCEDURE

The Method of Complex Response

The Frequency Domain

Interpolation

The Equivalent Linear Method

Effective Shear Strain Amplitudes

The Equation Solver

Baseline Correction

Summary of Numerical Procedure

ii

1

1

1

3

5

11

11

16

16

16

19

20

20

21

23

23

24

25

27

29

32

32

34

5. PROGRAMMING DETAILS

MODEl - The Initation Mode

MODE2 - The Extraction Mode

MODE3 - The Iteration Mode

Subprograms

Tapes

Punched Output

Core Requirements

Time Estimates

Auxiliary Programs

6. COMMENTS ON INPUT A..~D OUTPUT

Frequency Content

Control Motions

Solid Elements and Soil Layers

Void Elements

Beam Elements

Transmitting Boundaries

Fixed Boundary Conditions

Symmetric Models

Material Curves

Iterations

Output Accelerations

Response Spectra

Amplification Functions

Bending Moments

7. LISTING OF MAIN PROGRAM

8. EXAMPLE PROBLEM

General discussion

Input Data Cards

9. ACKNOWLEDGEMENTS

10. REFERENCES

iii

36

36

36

37

38

49

52

53

54

55

56

56

56

57

58

58

60

61

61

63

63

65

65

66

66

67

75

75

78

81

82

iv

APPENDIX A - Program Abstract and Availability A-1

APPENDIX B - "Comparison of Plane Strain and Axisymmetric B-1 Soil-Structure Interaction Analyses," by E. Berger, J. Lysmer, and H. B. Seed.

APPENDIX C - "A Simplified Three-Dimensional Soil-Structure Interaction Study," by R. N. Hwang, J. Lysmer, and E. Berger.

C-1

s i

1

1. INTRODUCTION

The computer program FLUSH is a further development of the complex

response finite element program LUSH (Lysmer et al, 1974). The new program

is considerably faster than LUSH, thus the name FLUSH = Fast ~£~.!!., and it

includes a large number of new features such as transmitting boundaries,

beam elements, an approximate 3-D ability, deconvolution within the

program, out-of-core equation solver, new input/output features, etc.,

all of which make the program more efficient and versatile.

2. SOIL-STRUCTURE INTERACTION ANALYSIS

Basic Requirements

Analyses of soil-structure interaction effects during earthquakes for

nuclear power plant structures are usually made by one of two methods-­

either by means of a complete interaction analysis involving consideration

of ~he variation of motions in the structure and the adjacent soil, or by

an inertial analysis in which the motions in the adjacent soil are assumed

to be the same at all points above foundation depth. For surface structures,.

the distribution of free field motions in the underlying soils has no in­

fluence on the structural response and thus, provided the analyses are made

in accordance with good practice, good results may be obtained using either

method of approach. For embedded structures, however, consideration of the

variation of ground motions with depth is essential if adequate evaluations

of soil and structural response are to be obtained without undue conserva­

tism. At the present time, analyses including these effects have only been

developed using the finite element method of approach, although other

computational methods might also be used for this purpose. Not only does

an idealized complete interaction analysis using finite element methods offer

a greater prospect of improved accuracy for analysis of embedded structures

on theoretical grounds but recent observations of the response of the

Humboldt Bay Nuclear Power Station to strong shaking induced by the Ferndale,

California earthquake of June, 1975 show that this method of approach provides

response evaluations which are in excellent accord with those observed under

field conditions.

2

This does not mean that all finite element analyses of soil-structure

interaction provide adequate evaluations of response. Like all analyses,

they can be performed with different degrees of approximation or sophisti­

cation. The basic requirements for a good analytical procedure may be

summarized as follows:

1. The analysis should consider the variation of soil characteris­

tics with depth.

2. The analysis should consider the non-linear and energy-absorbing

characteristics of the soils.

3. For embedded structures, the analysis should consider the varia­

tion of ground motions with depth.

4. The analysis should be capable of taking into account the three­

dimensional nature of the problem.

5. The analysis should be capable of considering the effects of

adjacent structures on each other.

It is not always necessary to meet all of these requirements--for

example, where a simple structure is involved accurate evaluations of the

motions at the base of a structure can be obtained using a two-dimensional

analytical model--but, in general, all of the requirements listed above

should be taken into account.

One of the primary arguments against the use of finite element methods

of analysis is their high cost. This of course depends on the efficiency of

the computer program used but it is true that in the recent past, analyses

of this type have been substantially more costly than analyses using the

inertial interaction approach in conjunction with half space theories and

although savings may be realized by virtue of the lower degree of conservatism

involved in the finite element approach, it is clearly desirable to reduce

the analytical costs to the fullest extent possible.

Accordingly, it is the purpose of this chapter to review the current

level of accomplishment which may be achieved using finite element tech­

niques for the performance of complete interaction analyses and to describe

an efficient procedure which meets all desirable requirements without in­

curring excessive costs for design and analysis.

1

?.-1

APPENDIX A

FLUSH A COMPUTER PROGRAM FOR SEISMIC SOIL-STRUCTURE INTERACTION ANALYSIS

The Geotechnical Engineering Group at U. c. Berkrley is pleased to

announce the release of the computer program FLUSH on January 15, 1976.

This program which is a further development of an earlier finite element

code LUSH provides a complete tool for seismic soil-structure interaction

analysis by the complex response method. FLUSH includes the following

features:

1. Plane strain quadrilateral elements for modeling of soils

and structures.

2. Beam elements for modeling of structures.

3. Multiple nonlinear soil properties for equivalent linear

analysis. This allows for different damping in each element.

4. An approximate 3-D ability which makes it possible to per­

form meaningful structure - soil - structure, interaction

analyses at essentially the same cost as a 2-D analysis.

5. Transmitting boundaries which greatly reduce the number of

elements required.

6. A new out-of-core equation solver which essentially eliminates

core-size problems.

7. Internal deconvolution. This feature eliminates the need to

perform on independent site response analysis for determina­

tion of the rigid base motions.

8. Convenient tape handling features which provide restart ability

and a permanent record for later recovery of details of the

solution.

9. Printed, plotted or punched output time histories of accelera­

tion and bending moments.

10. Computation of maximum shear forces in beam elements.

11. Printed or ?Unched acceleration and velocity response spectra.

12. Plotting of Fourier amplification functions.

13. Several new features which improve the efficiency, utility

A-2

and clarity of the program. (RMS method for strain computations,

new interpolation scheme, etc.)

Program Availability

The FLUSH documentation and CDC or UNIVAC source deck (approximately

9000 cards) are available from

Professor John Lysmer 440 Davis Hall University of California Berkeley, California 94720 (415) 642-1262

The cost including delivery by surface mail is $500 payable by check

to: "The Regents of the University of California". Airmail, magnetic tapes,

~~'· or revisions for other computers can be provided by special arrangements.

On or about May 1, 1976, the program and any auxiliary programs which

are completed at that time will be available from the National Information

Service on Earthquake Engineering at:

NISEE/Computer Applications Davis Hall University of California Berkeley, California 94720 (415) 642-5113

In the interim period minor modifications of the program may be

expected. All purchasers of the program from Professor Lysmer will be kept

informed of these modifications and other news related to the program.

Users are encouraged to report any bugs or suggested improvements.

• \ f'\ c.J.N·· J c

{ Ct-1 •l ,- d-,1.. ~ ~I oJ 1"-,J... •t I, .0 ~ 0--{·-~.. 'tv" C. { &-r· l r;..J_

" 7 1\ f2 · A , 1...~ •

-c. l.a-:. L 1 L 4---. J .\ '~" v-:-r ~·/ a_,.;l )

I V\ v ~ (_,./ <' J

EARTHQUAKE ENGINEERING RESEARCH CENTER

TLUSH: A COMPUTER PROGRA~1 FOR THE

THREE-DIMENSIONAL DYNAMIC ANALYSIS OF EARTH DAI~S

by

Takaaki Kagawa

Lelio H. Mejia

H. Bolton Seed

John Lysrner

Report No. UCB/EERC-81/14

September 1981

A report on research sponsored by the National Science Foundation

College of Engineering

University of California

Berkeley, California

ucej-/lfc_ <g 1-J./.{

ki4 1 9 8' I

U04759

CAL TECH EARTHQUAKE ENG. RES. LIBRARY

MAR 1 2 1982

RECEIVED

TABLE OF CONTENTS

1. INTRODUCTION

2. ANALYTICAL PROCEDURE

2.1 Equation of Motion

2.2 The Method of Complex. Response

2.3 Interpolation in the Frequency Domain

2.4 The Equivalent Linear Method

2.5 Effective Shear Strain

3. PROGRAM DESCRIPTION

3.1 Program Structure

3.2 Description of Routines

3. 3 Tape Usage \,

3.4 Error Messages

3. 5 Core Herrory

4. COMMENTS ON INPUT

4.1 Mesh Size Requirements

4.2 Identification of Nodes and Elements

4.3 Element !1atrices

4.4 Material Curves

4.5 Shear Strain Computation

4.6 Frequency Controls

4.7 Interpolation

Page No.

1

3

3

5

8

9

11

13

13

16

24

26

28

29

29

30

31

34

34

36

36

Page No.

4. COMMENTS ON INPUT (Contd.)

4.8 Printer Plots 38

4.9 Punched Output 39

4.10 Execution Time 39

5. LISTING OF MAIN PROGRAM 41

6. EXAMPLE PROBLEM 48

6.1 Problem Description 48

6.2 Input Data Cards 52

6.3 Computer Output 55

7. ACKNOWLEDGEMENTS 97

8. REFERENCES 98

1. INTRODUCTION

Significant progress has been made over the past two decades in the

developrrent of analytical procedures for evaluating the response and

stability of earth dams subjected to seismic loads. Current methods of

stability analysis involve procedures such as that proposed by Seed et al.,

(1973) which consists of the following steps:

1. Determination of the initial stresses existing throughout the

dam and the foundation before the earthquake.

2. Determination of the characteristics of the earthquake motions

that are likely to affect the dam.

3. Computation of the response of the embankment and foundation

to the selected earthquake motions.

4. Determination in the laboratory or by means of empirical cor­

relations of the response to the induced dynamic stresses of

representative samples of the embankment and foundation

materials.

5. Evaluation of the overall deformations and stability of the

embankment dam.

Due to the fact that the finite element method can easily handle

georretrical irregularities, complex material behavior and arbitrary boun­

dary conditions it is perhaps the most flexible tool currently available

to perform the dynamic response analysis of an earth dam.

Limitations of computer speed and storage capacity have restricted

until recently the use of the finite element method to two-dimensional

problems. Although many earth dams fall within "this category, there are

also many cases in which the assumption of plane strain behavior gives

2

only approximate results and therefore a full three-dimensional analysis

is warranted. Thus, the availability of a numerical procedure for the

dynart".ic analysis of earth dams in t.'I-J.ree-dimensions seems desirable.

Two-dimensional finite element techniaues which use the complex

response method and therefore permit variations in modulus and damping in

different elements of a soil structure, were developed by Lysrner et al.,

( 19 74,19 75) • The·se procedures were extended to three dimensions, with a

constraint on the possible deformations of the finite element model, by

Kagawa (1977). The present version of the computer program TLUSH con-

stitutes a further development of these procedures (Mejia, 1981) and

incorporates additional fe·atures a:nong which are the following:

1) Complete freedom for the selection of the direction of the earthquake

motions, 2) Complete freedom in the deformational modes of the ~cdel,

3) A new interpolation scheme, 4) A nodal point and element data genera-

tion routine, 5) More efficient element stiffness generation routines and l 6) A more efficient program structure that has lower memory requirements.

The program TLUSH can take into acco~~t the strong nonlinear effects

characteristic of soil masses subjected to strong earthquake motions.

This is ac'I-J.ieved by a combination of the equivalent linear method (Seed

and Idriss, 1969) and the complex response method. Typical relationships

between stiffness, damping and effective shear strains for sand and clay

are provided within the program. Special options that permit creation

of a permanent record of both input and basic information on the complete

solution, and recovery of this information for iteration and output

purposes are available within the program.

EARTHQUAKE ENGINEERING RESEARCH CENTER

QUAD - 4

A COMPUTER PROGRAM FOR EVALUATING

THE SEISMIC RESPONSE OF SOIL STRUCTURES

BY VARIABLE DAMPING FINITE ELEMENT PROCEDURES

(

by

I. M. Idriss

J. Lysmer

R. Hwang

H. B. Seed

A computer program distributed by NISEE/Computer Applications

Report No. EERC 73-16

July 1973

College of Engineering University of California

Berkeley, California

QUAD-4

A COMPUTER PROGRAM FOR EVALUATING

THE SEISMIC RESPONSE OF SOIL STRUCTURES

BY VARIABLE DAMPING FINITE ELEMENT PROCEDURES

by

I. M. Idriss, 1 J. Lysmer, 2 R. Hwang3 and H. B. Seed

4

Introduction

The finite element method of analysis has been shown to be a power-

ful tool for the solution of various problems in continuum mechanics.

Among its many uses, it has been applied extensively for the evaluation of

the seismic response of a variety of soil deposits and earth structures

(e.g. Clough and Chopra, 1966; Idriss and Seed, 1966, Finn and Khanna,

1966; Wilson, 1968; Valera, 1968; Dibaj and·Penzien, 1969; Dezfulian and

Seed, 1969; Seed et al., 1969; Seed et al., 1970). Recent studies of the

response of small scale prototypes of clay banks on a shaking table

(Kovacs et al., 1971) indicated that the finite element method can provide

response values in reasonable agreement with measured values.

Although the analytical formulations permit the use of different

stiffness characteristics in each element, all applications to date for

evaluating the response during earthquakes have utilized a constant

damping ratio for the entire finite element representation. The use of

1 . Associate, Woodward-Lundgren & Assocs., Oakland, Calif., and Assist. Research Engineer, University of California, Berkeley.

2 Associate Professor of Civil Engineering, University of California, Berkeley.

3 Director of Research & Development, Harding-Lawson & Assocs., formerly grad-uate student, Dept. of Civil Engrg., University of California, Berkeley.

4 Professor of Civil Engineering, University of California, Berkeley.

(e.g. Clough, 1965; Clough and Chopra, 1966; Idriss and Seed, 1966;

Wilson, 1968; Desai and Abel, 1972).

In earthquake response evaluations, ·the following set of equations

are solved:

[M]{u} + [C]iu} + (K]{u} = tR(t)} (1)

in which [M] = mass matrix for the assemblage of elements

[C] = damping matrix for the assemblage of elements,

[K] = stiffness matrix for the assemblage of elements,

{u} = nodal displacements vector (dots denote

differentiation with respect to time), and

{R(t)} =earthquake load vector.

A detailed description for the formulation of [M], [K] and {R(t)} is

available elsewhere

In previous studies, Eq. 1 has been solved by either one of the

following procedures:

a. Modal Superposit~on (e.g. Clough and Chopra, 1966; Idriss and

Seed, 1966): The nodal displacements are expressed in terms of the normal

coordinates and mode shapes by

{uJ = [¢]{X} (2a)

where [¢] are the mode shapes of the system and {X} are the normal coordi-

nates. The mode shapes and frequencies are determined from a solution of

the eigenvalue problem for the undamped free vibration equations of the

system (i.e. for (C] = 0 and {R(t)} m 0 in Eq. 1):

(:lb)

2

modal superposition in the solution (e.g. Clough and Chopra, 1966;

Idriss and Seed, 1966; Finn and Khanna, 1966; Dezfulian and Seed, 1969)

has required the use of the same damping value for all elements. In

cases where a direct integration procedure has been used (e.g. Wilson,

1968; Valera, 1968; Dibaj and Penzien, 1969), Rayleigh damping was

utilized for the entire finite element representation and thus the same

value of damping was assigned to all elements.

In fact, however, damping in soils is strain dependent, and the

damping value to be used in each element should be based on the strain

developed in that element. Furthermore, with irregular boundary

conditions and variations in material properties, these strains may vary

considerably. Thus, the use of a constant damping value, even though it

may be the weighted average of the damping values for each element

(Idriss et al., 1969), can lead to inaccurate results for some conditions.

Accordingly, an analytical procedure that permits the use of a

different damping ratio for each individual element has been formulated.

This procedure allows the incorporation of both stiffness and damping

values, that are strain-dependent, for each.element. The formulation of

this variable damping finite element procedure, its use in response

evaluation and a listing of the computer program used in these evaluations

are presented in this report.

Analytical Procedure

The finite element method is a numerical procedure by means of which

the actual continuum is represented by an assemblage of elements inter­

connected at a finite number of nodal points. Details of the formulation

of the general method are available in several recent publications

3

Each column, {¢n} of the matrix [¢] represencs the mode shape of the nth

mode of vibration whose natural circular frequency is w • The normal n

coordinates for each mode n, are evaluated from a solution of the normal

equations:

X + 2.\ w X + w 2 X n n n n n n

T .., { cbn} [M] {R( t)}

M n

(2c)

in wh~ch ,\ = damping ratio n

M = {¢n}T[M]{¢n} and T denotes the transpose of the vector. n

Thus, the damping matrix [C] is not used directly in this procedure;

it is actually replaced by:

(2d)

b. Direct Integration (e.g. Dibaj and Penzien, 1969; Wilson, 1968;

Valera, 1968): The equations of motion (Eq. 1) are solved directly as a

set of simultaneous equations. Such a solution would then require that the

damping matrix, [C], be known. The most commonly utilized relationship for

expressing this material characteristic is the one originally proposed by

Rayleigh (1945), viz:

[C] = A(M] + B[K] (3a)

in which A and B are constants. It can readily be shown that the damping

ratio, ;... , n

of the nth mode is related to these constants by the expression:

~­n

A Bwn =--+--2w 2

n ( 3b)

Both of these procedures (the modal superposition and the direct inte-

gration), therefore, utilize the same damping ratio for the entire system.

4

Variable Damoing Solution

In a variable damping solution, a damping submatrix must be formulated

for each individual element and then all element submatrices added, in the

appropriate way, to obtain the damping matrix for the entire assemblage of

elements. The formulation of such a submatrix is proposed herein. It is

also based on utilizing the Rayleigh damping expression, but instead of

using Eq. 3a for the entire system, the following relationship is used for

each element, q:

[ c] = a [m] + 6 [k] q q q q q (4)

in which [c] 1 [m] and [k] are the damping, mass and stiffness submatrices q q q

respectively for element q, and a and 6 are parameters that are functions q q

of the damping value and stiffness characteristics of element q. The para-

meters aq and 6q are given by:

a • A • w q q 1 (Sa)

(Sb)

The value of A 1 which represents the damping ratio for element q, is chosen q

based on the strain developed in the element. The parameter w1

is equal to

the fundamental frequency of the system and is obtained from the solution of

Eq. 2b for n a 1.

The damping matrix for the entire assemblage of elements is obtained

by appropriate addition of the damping submatrices of all the elements in

the assemblage. Thus, if c (q) represents the (ij)th term of the damping ij

submatrix [c) (Eq. 4) of a typical element (q) 1 the (IJ)th term of the q

damping matrix of the entire system is given by

(6)

It should be noted that c1J ~ 0 only it I is equal to J or I is adjacent

to J,

The resulting damping matrix (C) is symmetric and usually sparsely

populated.

Response Evaluation

The response of the finite element system to an earthquake loading

can then be evaluated by the solution of Eq. 1. The mass and stiffness

matrices and the earthquake load vector can be readily formulated by

methods discussed elsewhere. The damping matrix is obtained using the

procedure outlined in the previous section.

The equations of motion (Eq- 1) are most readily solved by a direct

numerical method such as the step-by-step method (Wilson and Clough, 1962).

If a linear variation of acceleration is assumed over the time increment

of integration, ~t, then the unknown response values at the nodal points

at time, t, can be expressed in terms of the known values at time, t-~t,

as follows:

{u} = t [K]-t{R}

t (7a)

[K] = [K] + 6[M]/~t 2 + 3[C]/t.t (7b)

{ RJ '"' t {R} +

t {AIT[M] +

J t {B}T[C] t (7c)

{A} 6 {u} t. 6 {"' + z{ii} ~ (7d) .. - +- u t t. t 2 t- t t.t 1 t-t.t t- t

{B} 3 { u} t. + 2lu} ~ + ~t {ii} (7e) ,._ t t.t t- t t- t 2 t-t.t

{u} J {u} iB} (7f) .. - -t 6t t . t

{u:· 6 r ' {A} (7g) = '.UJ -t ~tl t

The stresses and strains developed in each element can then be readily

compu:ed us1ng the values of tu;t.

It should be noted thaL unless an appropriate value of ~t is used in

the solution of Eqs. i, instability may result- The stability of the

solution is usually dependent on the value of ~t used in the integration

and on the material properties and division of the finite element mesh.

A method that p~ovides stability in the solution (Wilson et al, 1973) was

recently introduced and has been incorporated in the computer program for

solving Eqs. 7.

The solution procedure outlined in Eqs. 7 requires the use of the

modified stiffness matrix, [K], and the damping matrix, [C], in every

integration step. Therefore, both of these matrices-must be available in

core throughout the response evaluation. Normally, the order of these

matrices would be 2N by 2N, where N is the number of free nodal points.

Because these mat~ices are symmetric and usually sparsely populated, it is

more ~onvenient to place each in a banded matrix where only the diagonal

terms and the non-zero components on one side of the diagonal are retained.

The order of each matrix then reduces-to 2N by L, where Lis the band width

and is equal to twice the largest difference between any two adjacent nodal

points plus two, With appropriate layout and numbering of nodal points,

the band width can be kept to a minimum. The use of banded matrices not

only reduces storage requirements, but also reduces considerably the

computer time needed to solve Eq. 7a.

A computer program (QUAD-4) has been written to carry out the required

operat~ons of this procedure and to evaluate the response of any soil

deposit or earth structure dur~ng a given seismic excitation. The program

has been written for elements in plane-strain; triangular and quadrilateral

7

elements can be used in representJ.ng the continuum. A listing of this

program is presented in Appendix A.

The solution proceeds by assigning modulus and damping values to

each element. Because these values are strain-dependent, they would not

be known at the start of the analysis and an iteration procedure is

required. Thus, at the outset, values of shear moduli and damping are

estimated and the analysis is performed. Using the computed values of

average strain developed in each element, new values of modulus and

damping are determined from appropriate data relating these values to

strain (e.g. Seed and ldriss, 1970; Hardin and Drnevich, 1972). Proceeding

in this way, a solution is obtained incorporating modulus and damping

values, for each element, which are compatible with the average strain

developed.

Sample Problem

To illustrate the use of this program in site response evaluation,

the 100-ft layer of dense sand shown in Fig. 1 has been analyzed. The

properties of the sand were considered to be as follows:

Total unit weight = 125 pcf

(K2

) = 65 max

K = 0. 5 0

The parameter (K ) relates the maximum shear modulus, G , and 2 max max

effective mean pressure at any depth, y, below the surface as follows:

G = 1000(K2

) 0,1/2

( 8) max max m

(1 + 2K ) where a' 0 I

= a m 3 v

8

APPENDIX 3 (continued)

Computer Code LIN OS

Analysis Method Finite elements for static and transient loading analysis of linear and nonlinear problems

Geometrical Dimensions 2-D and 3-D

Selected Material Models Linear isotropic elastic and visco-elastic models; Von Mises, Drucker-Prager, and Mohr-Coulomb elasto-plastic models; critical state bounding surface, and multiple yield surface plasticity models; hypoelastic large-deformation and Nelson-Baron variable moduli models

Loadings Static and dynamic loads can be applied at any node of the mesh, gravity loading, nodal displacements, velocities, and accelerations (e.g., earthquake loading at base)

Soil-Water Medium Analysis Method Coupled field equations for saturated porous (Total Stress vs. Effective Stress) media (effective stress analysis with pore-

pressure generation and dissipation capabilities)

Element Types 2-D plane strain and axisymmetric elements; 3-D brick element; 2-D/3-D structural elements (truss, beam, plate, shell, and membrane); 2-D/3-D boundary element, spring element, unconfined/confined seepage element, contact frictional element, slide-line and interface elements

Author/Contact Person or Institution Professor J. P. Bardet, Civil Engineering Department, University of Southern California, Los Angeles, CA 90089-2531, Tel: (213) 740-0608

ATTACHMENT A

The finite element program LIN OS has been developed to analyze the linear and nonlinear

tw<r to three-dimensional problems that are encountered in geomechanics and geotechnical

engineering, especially those problems difficult, or even impossible, to solve by using the

commercial finite element packages of structural mechanics, such as Msc/Nastran, Adina,

Abaqus, etc. LIN OS is capable of solving problems combining solid, structural and fluid

mechanics in the static and transient domains, which include quasistatic boundary value • problems (settlement and ultimate loads), parabolic initial values problems (consolidation and

diffusion problems) and hyperbolic initial value problems (wave propagation and dynamic

resonance problems).

• Examples of geotechnical problems solved with LIN OS are: Dynamic responses of artificial islands in the Beaufort Sea, including the build-up and dissipation of pore water pressure and liquefaction of their saturated sand cores. Rockbursts of underground excavations. Site-response analysis during earthquakes, e.g., three-dimensional response of the Marina District of San Francisco during the Lorna Prietta earthquake. Twcr and three-dimensional consolidation of non-linear and irreversible materials. Determination of ultimate failure loads of Arctic caissons. Seismic responses of landfills.

• The main features of LIN OS are: Non-linear solvers: Newton-Raphson, modified Newton-Raphson and quasi-Newton with optional line search. Implicit-explicit Newmark time integration schemes for transient analyses. Symmetric and non-symmetric matrix equation solvers. Eigenvalue/vector solvers for dynamic and bifurcation problems. B-Bar strain projection rulers for incompressibility constraint. Solution of coupled field equations for saturated porous media. Prescribed boundary conditions may combine nodal displacement, velocity or acceleration. · lsoparametric data generation schemes. Optimization of equation numbering for reduction of bandwidth of stiffness matrix. Capabilities to slave degrees of freedom. Batch or interactive execution modes. Complete restart capabilities. Graphic capabilities by separate post-processors including two- and three-dimensional mesh plots, vector field plots, contour plots of selected nodal/field quantities. Interface with pre- and post-processor PATRAN. Interactive, free format and graphical input with on-line help.

1

• The element library includes the following elements: Two-dimensional element with plane strain/axisynunetric options. Three-dimensional brick element. Truss element (2D/3D). Beam element (2D/3D). Plate/shell element (2D/3D). Membrane element (2D/3D). Boundary element (2D/3D). Spring element (2D/3D). Unconfined/confmed seepage element (2D/3D). Contact frictional element (2D/3D). Slide-line element (2D/3D). Interface element (2D/3D).

• The material library, which applies to two- and three-dimensional solid elements, includes the following constitutive models:

Linear isotropic elastic model. Linear viscoelastic model. von Mises elastoplastic model. Drucker-Prager elastoplastic model. Mohr-Coulomb elastoplastic model. Critical state plasticity models. Bounding surface plasticity models. Multiple yield surface plasticity models. Hypoelastic large-deformation model. Nelson-Baron variable moduli model.

2

APPENDIX 3 (continued)

Computer Code SAP90(3)

Analysis Method Finite elements for static and dynamic loading analysis of linear problems

Geometrical Dimensions 2-D and 3-D

Selected Material Models Linear elastic

Loadings Gravity, thermal and prestress conditions, nodal force and displacements, dynamic loading in the form of a base acceleration response spectrum, or time varying loads and base accelerations

Soil-Water Medium Analysis Method Only dry condition (Total Stress vs. Effective Stress)

Element Types Spring type boundary element, 3-D frame element, prismatic or non-prismatic elements, 3-D SHELL element, 2-D ASOLID element, and 3-D SOLID element, and 2-D frame, truss, membrane, plate, axisymmetric, and plane strain elements

Author/Contact Person or Institution Professor Edward L. Wilson, Department of Civil Engineering, 781 Davis Hall, University of California, Berkeley, CA 94720 or Computers and Structures, Inc., 1995 University Avenue, Berkeley, CA 94704, Tel: (415) 845-2177.

I.

A. The "SAP" Series of Programs

Over the past two decades the SAP series of computer programs (see References II ,2,3]), operating on mainframe computers, have established a worldwide reputation in the areas of struc­tural engineering and structural mechanics.

', These programs represent the research work conducted at the University of California, Berkeley, by Professor Edward L. Wilson over the past 25 years.

The name "SAP" was coined in 1970 with the release. of the first SAP program.

In the years that followed, further research and development in the area of finite element formulation and numerical solution techniques resulted in the release of a series of SAP programs in the form of SOLIDSAP, SAP 3 and finally SAP IV.

Since they were first introduced, the SAP series of programs have been used by hundreds of engineering firms internation-

' ally, and numerous firms have spent millions of dollars in creating modified versions of the programs to meet specific needs.

______ ->_t_\l-::1\J .Jll Ul.llllill J-\llillYIIIII Ullr;;:;J:> IVJc.liiUc.ll

Many major commercially available structural analysis pro­grams are based upon the element formulations and numerical methods that were originally developed for SAP.

The program has acquired the status of being the most reputable and widely used computer program in the field of structural analysis.

n. The SAP80 and SAP90 Programs

The SAP IV computer program was released almost twenty years ago and it represented the state of the art at that time.

Since the release of SAP IV, major advances have occurred in the fields of numerical analysis, structural mechanics and com­puter technology. TI1ese advances led to the release of SAP80, the first structural analysis program for microcomputers, over n decade ngo, and more recently to the release of Si\P90.

Si\P90 represents new technology and was written by the author of the original SAP series of programs. The program is not a modification or an adaptation of SAP IV. The element fonnulations, equation solvers and eigensolvers are all new.

SAP90 represents the current state of the art; it is the technology of today. The program will remain under a constant state of development in the years to come to retain this status.

The program development is being conducted in the ANSI Fortran-77 subset environment, which guarantees portability of the software from the level of the small personal computers to the large mainframe super computers. SAP90 has been de­signed to nm equally well on personal, mini or mainframe com pulers.

-~

,·-" ·' '

'

\

\

Introduction 1-.) ----~-

This version of the program is designed to be used on a MS-DOS based computer system. On computers with 640K of memory and a 30MB hard disk, the problem-size capacity is about 4,000 joints or 8,000 equations. With a larger hard disk and with versions of the programs utilizing extended memory beyond 640K, very large problems can be solved. All numerical operations are executed in full 64-bit double precision.

The program has static analysis and dynamic analysis options. These options may be activated together in the same run. Load combinations may include results from the static and dynamic analyses.

All data is input in list-directed free format. Generation options are available for convenience. Undefonned and deformed shape plotting capabilities exist for data verification of the model geometry and for studying the structural behavior of the system.

The program is built around a blocked out-of-core active col­umn equation solver with an automatic profile minimization algorithm. The out-of-core eigensolution procedure uses an accelerated subspace iteration algorithm.

The finite element library consists of four elements, namely, a three-dimensional FRAME element, prismatic or non-pris­matic, a three-dimensional SHELL element, a two-dimensional ASOLID element and a three.-dimensional SOLID element. The two-dimensional frame, truss, membrane, plate bending, axisymmetric and plane strain elements are all available as subsets of these elements. All necessary geometric and loading options associated with the elements have been incorporated. A boundary element in the fonn of spring supports is also included.

There is no restriction on mixing or combining element types within a particular model.

Loading options allow for gravity, thennal and prestress con­ditions in addition to the usual nodal loading with specified forces or displacements. Dynamic loading can be in the form of a base acceleration response spectrum, or time varying loads and base accelerations.

C. 1'he "SAP" Warning

The effective application of a computer program for the analy­sis of practical situations involves a considerable amount of experience. The most difficult phase of the analysis is assem­bling an appropriate model which captures the major charac­teristics of the behavior of the structure. No computer program can replace the engineering judgment of an experienced engi­neer. It is well said that an incapable engineer cannot do with a ton of computer output what a good engineer can do on the back of an envelope. Correct output interpretation is just as important as the preparation of a good structural model. Verification of unexpected results needs a good understanding of the basic assumptions and mechanics of the program. Equilibrium checks are necessary not only to check the computer output but to understand basic structural behavior.

-I )

/ - ...... ~ \

\

\

Introduction 1-5

Back in 1970 the original SAP publication carried the following statement:

"The slang name SAP was selected to remind the user that this program, like all computer programs, lacks intelligence. It is the responsibility of the engineer to idealize the structure correctly and assume responsibil­ity for the results."

The name SAP has been retained for this program for exactly the same reason.

APPENDIX 3 (continued)

Computer Code NONSAP

Analysis Method Finite elements

Geometrical Dimensions 2-D and 3-D

Selected Material Models Linear and nonlinear elastic, Mooney-Rivlin material, elastic-plastic, Von Mises or Drucker-Prager yield conditions, variable tangent moduli model, curve description model (with tension cut-off)

Loadings Gravity, dynamic base input acceleration, nodal forces and displacements

Soil-Water Medium Analysis Method Dry condition only (Total Stress vs. Effective Stress) .

Element Types 3-D truss, 2-D plane stress and plane strain, 2-D axisymmetrical shell or solid element, 3-D solid element, 3-D thick shell

Author/Contact Person or Institution Bathe, K. J., E. L. Wilson, and R. H. !ding, "NONSAP, a Structural Analysis Program for Static and Dynamic Response of Nonlinear Systems," College of Engineering, University of California Berkeley, February 197 4 (Report No. UC SESM 74-3)

Structures and Materials Research Department of Civil Engineering

Division of Structural Engineering and Structural Mechanics

Report No. UCSESM 74-3

NONSAP

A STRUCTURAL ANALYSIS PROGRAM FOR STATIC AND

DYNAMIC RESPONSE OF NONLINEAR SYSTEMS

by

Klaus-JUrgen Bathe

Edward L. Wilson

Robert H. Idi ng

Structural Engineering Laboratory University of California

Berkeley, California

February 1974

ABSTRACT

The current version of the computer program NONSAP for linear and

nonlinear, static and dynamic finite element analysis is described.

The solution capabilities, the numerical techniques used, the finite

element library, the logical construction of the program and storage

allocations are discussed. The user's manual of the program is given.

Some sample solutions are included, which are standard data cases

available with the program.

i

ABSTRACT . • . . .

TABLE OF CONTENTS

TABLE OF CONTENTS

. . . . . . . . . ' . . . . . . . . . .

- PART A -

DESCRIPTION OF NONSAP

i

;;

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 1

2. THE INCREMENTAL EQUILIBRIUM EQUATIONS OF STRUCTURAL SYSTEMS. 6

2.1 Element to Structure Matrices and Force Vectors ... ~ 7

2.2 Boundary Conditions . . . . . . . ...

3. PROGRAM ORGANIZATION . . . . . . . .

3.1 Nodal Point Input Data and Degrees of Freedom

3.2 Calculation of External. Load Vectors

3.3 Read-in of Element Data ....... .

3.4 Formation of Constant Structure Matrices

3.5 The Compacted Storage Scheme

3.6 Equation Solution ......•

4. THE ELEMENT LIBRARY . . . . . . . .

4.1 Truss Element.

4.2 Plane Stress and Plane Strain Element .

4.3 Axisymmetric Shell or Solid Element ..

4.4 Three-Dimensional Solid or Thick Shell Element

ii

12

14

19

22

25

26

26

29

32

33

37

37

37

5.

6.

7.

8.

9.

10.

11.

TABLE OF CONTENTS (Continued)

THE MATERIAL MODELS . . . . . . . . . 5.1 · Truss Element Material Models . . . . . 5.2 Two-Dimensional Element Material Models

5.3 Three-Dimensional Element Material Models

EIGENSYSTEM SOLUTION

STEP-BY-STEP SOLUTION

7.1 Linear Static Analysis .

7.2 Linear Dynamic Analysis

7.3 Nonlinear Static Analysis

7.4 Nonlinear Dynamic Analysis .

ANALYSIS RESTART

DATA CHECK RUN . . . . . . INSTALLATION OF NONSAP

CONCLUDING REMARKS . . .

- PART B -

SAMPLE ANALYSES

. . . .

Page

. . . 38

. . . . . 38

38

40

41

43

. . . . 43

43

45

. . . . 47

. . . . . . 50

50

51

. . . . 53

1. Static and Frequency Analysis of a Tower Cable . . . . . 56

2. Large Displacement and Large Strain Static Analysis of a Rubber Sheet . . . . • . . . . . . . . . . . . . . . . . 60

3. Elastic-Plastic Static Analysis of a Thick-Walled Cylinder 63 ·

4. Static Large Displacement Analysis of a Spherical Shell 67

5. Static and Dynamic Analysis of a Simply Supported Plate 69

REFERENCES . . . . . • . . . . . . . . . . . • 72

iii

TABLE OF CONTENTS (Continued)

- PART C -

APPENDICES

APPENDIX - DATA INPUT TO NONSAP .

I. HEADING CARD .....

II. MASTER CONTROL CARDS .

III. NODAL POINT DATA ..

IV. APPLIED LOADS DATA

V. RAYLEIGH DAMPING SPECIFICATION .

VI. CONCENTRATED NODAL MASSES

VII. ·CONCENTRATED NODAL DAMPERS ..

VIII. INITIAL CONDITIONS

IX. TRUSS ELEMENTS ..

. • . . I . 1

1.1

. I I. 1

I I I. 1

. . . IV. 1

. . . . v. 1

o VI.1

VII. 1

.VIII.1

• I X. 1

X. TWO-DIMENSIONAL CONTINUUM ELEMENTS . o X.1

XI. THREE-DIMENSIONAL SOLID OR THICK SHELL ELEMENTS o • XI.1

XII. FREQUENCIES SOLUTION DATA o • • • • • • • • • • • XII o 1

APPENDIX A~ CONTROL CARDS AND DECK SET-UP FOR ANALYSIS RESTART A.1

APPENDIX B - IMPLEMENTATION OF USER-SUPPLIED NONLINEAR MATERIAL MODELS . . . . . . . . . . . . . . . o • o B. 1

iv

- PART A -

DESCRIPTION OF NONSAP

1. INTRODUCTION

The endeavor to perform nonlinear analyses has steadily

increased in recent years [1], [13], [17], [23]. The safety of a structure

may be increased and the cost reduced if a nonlinear analysis can

be carried out. Primarily, nonlinear analyses of complex structures

have become possible through the use of electronic digital

computers operating on discrete representations of the actual

structure. A very effective discretization procedure has proven

to be the finite element method [25]. Based on this method, various

large-scale general purpose computer programs with nonlinear capabi­

lities are now in use [11].

The development of a nonlinear finite element analysis program

is a formidable challenge. The proper formulation of the nonlinear

problem and its idealization to a representative finite element

system demands a modern background in structural mechanics. For

the solution of the equilibrium equations in space and time, stable

and efficient numerical techniques need be employed. The efficiency

of a nonlinear program depends largely on ?Ptimum usage of computer

hardware and software where, specifically, the appropriate allocation

of high- and low-speed storage is important.

The earliest attempts to obtain nonlinear analysis programs

essentially involved simple modifications of estabished programs for

linear analysis, much in the same way as the linear structural

theory was modified to account for some nonlinearities. However,

to analyze systems with large geometrical and material nonlinearities,

the program should be designed specifically for the required

iteration process and not be merely an extension of a linear

analysis program. Naturally, a linear analysis program should be

flexible and easy to modify or extend; however, this applies even

more to a nonlinear analysis program. In particular, it should be

realized that a great deal of research is still required and

currently pursued in the nonlinear static and dynamic analysis of

complex structures. Therefore, unless the general nonlinear

analysis code is easy to modify, it may be obsolete within a few

years after completion.

The nonlinear analysis program NONSAP presented in this report

is not an extension of the linear analysis program SAP [6 ], but

rather a completely new development [ 2]. Program NONSAP is designed

with two primary objectives. The first aim is the efficient

solution of a variety of practical nonlinear problems with the

current capabilities of nonlinear analysis procedures and computer

equipment. The second objective is to have a program which can be

used effectively in the various research areas pertaining to non­

linear analysis. Because of continuous improvements in nonlinear

analysis procedures, both objectives are attained simultaneously

by th~ development of an efficient, modular, and easily modifiable

general analysis code. The program is designed for a general

incremental solution of nonlinear problems, but naturally can

also be used for linear analysis.

The structural systems to be analyzed may be composed of

combinations of a number of different finite elements. The program

presently contains the following element types:

(a} three-dimensional truss element

(b) two-dimensional plane stress and plane strain element

?

d

(c) two-dimensional axisymmetric shell or solid element

(d) three-dimensional solid element

(e) three-dimensional thick shell element

The nonlinearities may be due to large displacements, large

strains, and nonlinear material behavior. The material descrip­

tions presently available are:

for the truss elements

(a) linear elastic

(b) nonlinear elastic

for the two-dimensional elements

(a) isotropic linear elastic

(b) orthotropic linear elastic

(c) Mooney-Rivlin material

(d) elastic-plastic materials, von Mises or Drucker-Prager

yield conditions

(e) variable tangent moduli model

(f) curve description model (with tension cut-off)

for the three-dimensional elements

(a) isotropic linear elastic

(b) curve description model

Program NONSAP is an in-core solver. The capacity of the

program is essentially determined by the total number of degrees

3

of freedom in the system. However, all structure matrices are

stored in compacted form, i.e. only nonzero e 1 ements are processed,

resulting in maximum system capacity and solution efficiency.

The system response is calculated using an incremental

solution of the equations of equilibrium with the Wilson e or

Newmark time integration scheme. Before the time integration is

carried out, the constant structure matrices, namely the linear

effective stiffness matrix, the linear stiffness, mass and damping

matrices, whichever applicable, and the load vectors are assembled

and stored on low-speed storage. During the step-by-step solution

the linear effective stiffness matrix is updated for the non­

linearities in the system. Therefore, only the nonlinearities are

dealt with in the time integration and no efficiency is lost in

linear analysis.

The incremental solution scheme used corresponds to a modified

Newton iteration. To increase the solution efficiency, the user

can specify an interval of time steps in which a new effective

stiffness matrix is to be formed and an interval in which equili­

brium iterations are to be carried out.

There is practically no high-speed storage limit on the total

number of finite elements used. To obtain maximum program capa­

city, the finite elements are processed in blocks according to

their type and whether they are linear or nonlinear elements. In

the solution low-speed storage is used to store all information

pertaining to each block of finite elements, which, in the case

of nonlinear elements, is updated during the time integration.

The purpose in this part of the report is to present briefly

the general program organization, the current element library

A

and the numerical techniques used. The different options available

for static and dynamic analyses are described. In the presentation

emphasis is directed to the practical aspects of the program. For

detailed information on the formulation of the continuum mechanics

equations of motion, the finite element discretization, and the

material modelsused, reference is made to [5].

5

APPENDIX 3 (continued)

Computer Code TARA-2

Analysis Method Finite element

Geometrical Dimensions 2-D plain strain

Selected Material Models Incrementally nonlinear hysteretic model (hyperbolic shear stress-strain law) with Masing behavior (incrementally elastic approach)

Loadings Gravity loading and base earthquake motion

Soil-Water Medium Analysis Method Total stress model or effective stress model (Total Stress vs. Effective Stress) with empirical constitutive laws for pore-water

pressure generation derived from 1-D compression tests and cyclic simple shear tests/also models pore water dissipation

Element Types 2-D continuum elements- beam element

Author/Contact Person or Institution TARA-2 has been developed and improved over many years at the Universities of British Columbia and Nevada, Reno by Professor W. D. Liam Finn, Civil Engineering Department, University of British Columbia Vancouver, B.C. V6T 1Z4, Canada, Tel: (604) 822-4938 and Professor Raj Siddharthan, Department of Civil Engineering, University of Nevada, Reno, NV 89557, Tel: (702) 784-1411

reasons behind the selection of these five configurations are that, 1) since the

development of the TARA-2 methodology has been based on the characterization of

sandy soils, we would like to concentrate on such problems and 2) TARA-2 has been

used in cases similar to these test models and has been ~ound to yield a good

comparison between the computed and the measured centrifuge responses.

2.0 BRIEF DESCRIPTION OF TARA-2

The methodology and application of TARA-2 and its more recent versions have

been reported extensively in the literature. In four recent major conferences,

i.e., Earthquake Engineering and Soil Dynamics II in Utah in 1988; the Symposium

on Seismic Design of World Port 2020 in Los Angeles in 1990; Recent Advances

in Geotechnical Earthquake Engineering and Soil Dynamics in St. Louis in 1991;

and the Symposium on Safety and Rehabilitation of Tailings Dams in Sydney,

Australia in 1990, the state-of-the-art presentation of soil liquefaction and

its effects was given by Professor Liam Finn. In these reports, the applica-

bility of the TARA program to a variety of problems under liquefying soil

conditions has been documented. The problems reported include a buried heavy

structure simulating a nuclear power plant, dams (e.g., the Sardis Dam in

Mississippi), and a rigid surface foundation. Therefore, only a brief

description of the methodology along with some past applications are presented

below.

This method is based on the finite element method, and solutions to the

dynamic equilibrium equations are obtained in the time domain. It is basically

an extension of the method of nonlinear dynamic effective stress analysis

developed by Finn ct al. (1977) for level ground conditions. The soil response

is modeled by combining the effects of shear and normal stresses. In shear, the

soil is treated exactly as in the level ground analysis where it is considered

as a nonlinear hysteretic material exhibiting Masing behavior during unloading 5

and reloading. The shear stress-strain behavior is characterized by a tangent

shear modulus which depends on the shear strain, the state of effective stress,

and the previous loading history. The shear model has been described in detail

by Finn et al. (1977) and has been verified in both laboratory tests (Finn, 1981;

Hushmond et al., 1987) and, to a limited extent, by field data (Finn et al.,

1982).

While extending the one-dimensional method to two dimensions, an additional

material parameter is necessary. The tangent bulk modulus or Poisson's ratio

can be selected for this purpose. Soil behavior in relation to changes in mean

normal effective stresses may be taken to be nonlinear and effective stress

dependent, but essentially elastic, compared to shear response.

An effective stress response analysis requires a porewater pressure

generation model. Siddharthan (1984) extended the Martin-Finn-Seed model to

include the effects of initial static shear stresses. The porewater pressure

is computed in two steps . First, the "apparent" plastic volume change is

evaluated from the shear strain history of an element. The constants (volume

change constants) required to compute this are estimated from the drained

behavior of samples in a simple shear device. The second step is to estimate

the rebound modulus and multiply it by the increment in volume change to

determine the increment in porewater pressure. The constants that define the

rebound modulus are referred to as rebound modulus constants. The procedure to

be adopted to obtain these constants from the tests to be performed under the

VELACS projecc will be discussed subsequently in Section 2.2.

The computed porewater pressures are used to evaluate the current effective

stresses which, in turn, are used to modify the soil properties which depend upon

effective stress. The porewater pressure dissipation was not considered in the

initial version of the program.

6

Siddharthan (1984) reported the details of a validation study that was

carried out using a series of seismic tests on centrifuge models. These tests

on embankments carrying surface loads were conducted at the Cambridge University

Geotechnical Centrifuge. Two selected acceleration and porewater pressure

response histories reported by Finn and Siddharthan (1984) are shown in Figs.

1 and 2. The recorded acceleration response and the response computed by the

program TARA-2 show remarkable agreement. There is high frequency noise present

in the recorded acceleration response (Fig. la). This noise is considered to

come from the walls and top of the model container and not to be propagated as

shear waves from the base.

The recorded porewater pressure response at any time during the dynamic

loading has two basic components: instantaneous and residual (Fig. 2). The

instantaneous component is due to the elastic coupling of soil and water. This

has a one to one relationship with the instantaneous state of stress. The

residual component is independent of the instantaneous state of stress, and it

occurs due to plastic volume change. It should be noted that the soil behavior

is affected by only this residual component; and, therefore, only the residual

component is computed by the program. The agreement between the computed and

the measured residual porewater pressure is ver; good.

2.1 RECENT MODIFICATIONS TO TARA-2

The original version of the program (TARA-2) has undergone a number of

modifications at the University of British Columbia (UBC) and at the University

of Nevada, Reno. TI1ese important changes are as follows.

(1) A drainage model is included to study the dissipation of porewater pressure

after the cessation of the earthquake excitation.

(2) A triggering criterion has been incorporated to assign steady state

strength to liquefied elements.

7

-.: ...

.. ci ...

c.e Ll l.O 4.0 S..G t.L TJu Jn Sees

Fig.1.a-Recorded acce!erationof ACC 1244 in Test 1 .

1:~~~·\1\fVVtNW ~ c:•

-~~.--~~~--~-r~.~~-..--r-~.~~~--r-~~-. :.= Lt u 'lC u s.c u i.e

T.ll''l: .lr. St:;

Fig.l b-Computed acceleration of ACC ~244 in Test 1 (with and without slip ele:nents) .

...... n a..., ~. _,.., ....

--- Recorded Computed 'With

------and Without Slip Elements

a.o 1.0 '2.0 3.0 4.0 S.O t.O 1.0 8.0

Tlne 1n Sees Fig.l., -Recorded and computed porewater pressure of PPT 2342 in Test 2.

8

(3) A procedure has been adopted to compute progressive deformation after

liquefaction in elements.

(4) The finite element mesh dimensions are continually updated since the

deformations can be substantial in cases where liquefaction is extensive.

In the version available at the University of Nevada, Reno the drainage model

and beam elements have been included. However, the other additions have not been

incorporated since they are important only for cases involving substantial

liquefaction.

2.2 SELECTION OF MATERIAL PROPERTIES FOR TARA

Before the computation of the dynamic response, a static analysis is

performed with TARA to estimate the in situ static stresses. The procedures

adopted are similar to those outlined by Duncan and Chang (1970), in which a

hyperbolic relationship is used. Layer by layer construction is also simulated.

The procedures to obtain the material parameters required to do this evaluation

can be easily estimated from static monotonic drained triaxial test results

(Duncan et al., 1980).

The material properties that are required for the computation of the

dynamic response and how they will be evaluated are presented below.

(a) Estimation of Gmax

Maximum shear modulus - 1000 (K2)max<u~)~

Here (K2)max will be estimated from resonant column test results.

(b) Damping at Low Strain Level

This will also be estimated from resonant column tests.

(c) Estimation of Volume Change Characteristics

From the TARA-2 methodology described in Section 2.0, it is clear that the

volume change characteristics of the sand used in the centrifuge test are

required. The proposed tests cannot be used to directly obtain these constants.

9

In practice, we have used the volume change constants that have been obtained

for Ottawa sand of the same relative density. A data base for Ottawa sand and

of some sandy mine tailings is available (Bhatia, 1980; Bhatia et al., 1988).

(d) Estimation of Rebound Modulus Constants

The rebound modulus, Er, is computed using

a' (1-m) v

Er - -------------------------------(K2m){a{,o(n-m)}

The constants K2, m, and n are obtained from oedometer tests and from the

results of cyclic simple shear and cyclic triaxial tests. The cyclic tests will

also give porewater pressure generation rates and the liquefaction potential of

the soil. These rates and liquefaction strength will be compared with those

given by the porewater pressure model used by the program for a single element

subjected to uniform cyclic shear stresses. If necessary, the volume change

constants will also be slightly modified to obtain a match between the values

(porewater pressure generation rates and liquefaction strength) given by the

porewater pressure model and those given by the laboratory cyclic tests.

(e) Other Material Properties

The material properties, such as permeability (K), unit weight, etc. , will

be obtained from the results of other standard testing programs to be given at

a later date.

There is a complication when liquefaction occurs in the test models to be

studied. Recent studies by Scott (1986), Hushmond et al. (1987), and Arulanandan

et al. (1989) have shown that extreme care should be taken in selecting values

for permeability of the soil near liquefaction (low effective stresses). If

liquefaction occurs, then drastic structural changes in the soil fabric occur

and the effective permeability of the liquified soil may be substantially higher

than the initial permeability (Hushmond et al., 1987). Scott (1986) pointed out

10

that the liquefied elements undergo a combined process of solidification and

consolidation; and, thus, characterization of the liquefied soil is very

difficult. If the participants of this validation program are given the

centrifuge results obtained with the model shown in Fig. 0 or Fig. 1 in the RFP,

these results can be used to calibrate the numerical model predictions during

the excitation and also in the dissipation phase. As first try in the absence

of such data, the procedures suggested by Arulanandan et al. (1989) may be used

to model porewater dissipation involving liquefied soil.

2.3 PAST PREDICTIONS USING TARA/DESRA

A number of studies have been reported in which TARA/DESRA predictions have

been compared with those measured in the centrifuge. It should be noted that

the results given by TARA for one-dimensional problems are identical to those

given by its predecessor, DESRA-2. A brief description of the centrifuge test

models used in these studies are presented below.

(a) Level Deposit Response in Centrifuge

Reference: Hushmond et al. (1987)

Cal Tech centrifuge results were independently compared by Hushmond (1988)

with those given by DESRA-2. The soil deposit studied is similar to that in Test

Model No. 1 given in the RFP.

(b) Level Sand Deposit with Surface Load

References: Siddharthan and Norris (1988, 1990a)

These centrifuge test results were reported by Whitman and Lambe (1982),

and the results were compared with those given by TARA-2. The test model

configuration used in the centrifuge test is very similar to that in Test Model

No. 12.

(c) Sand Island Response with Surface Loads

References: Siddharthan (1984), Finn and Siddharthan (1984)

11

Cambridge centrifuge test results for two different surface load levels

(simulated by thin metal strips of different thickness) and for two levels of

excitation (~ax - O.llg and ~ax - 0.17g) were compared with those given by

TARA-2. This centrifuge model configuration is also similar to that in Test

Model No. 12 given in the RFP.

(d) Sand Island Response with Embedded Structure

Reference: Finn (1988)

Cambridge centrifuge test results obtained with one level of excitation

<~ax- 0 .13g) were compared with those given by TARA-3. This test configuration

is also similar to that in Test Model No. 12 given in the RFP.

(e) Flexible Retaining Wall Behavior

Reference: Siddharthan and Maragakis (1989a)

The behavior of a flexible retaining wall supporting dry soil has been

studied. Though Test Model No. 10 given in the RFP is similar to the case

reported, there are additional factors, such as anchor and saturated soil

conditions, that are present in the Test Model.

(f) Rigid Retaining Wall Behavior

References: Siddharthan and Norris (1989b, 1990b)

The behavior of a rigid retaining wall supporting dry soil has been

presented. Factors, such as an increase in the lateral stress caused by

grainslip, have also been included. The computed results were compared with

those measured at the Cambridge centrifuge facility. Though Test Model No. 11

given in the RFP is similar to the one used in the study, additional factors,

such as surface loads from a 2mm lead sheet and saturated soil conditions, are

present in the test model.

3.0 TEST MODEL SELECTION AND COST BREAKDOWN

It is quite clear from the preceding text that the TARA-2 program can

12

an analytical procedure's methodology and assumptions may be valid, the

practitioner may not be using it correctly (e.g., material property selection).

In the case of the EPRI validation study, for example, there were a number of

predictors who used the well-known program FLUSH and arrived at different results

for the same problem. This means that the predictions can be user dependent;

and, therefore, necessary steps should be taken to avoid the wrong use of an

analytical procedure. The effective way to handle that is to let the developers

of the analytical procedure undertake the validation procedures.

The investigator has used the program TARA-2 in the past for validation

studies using data obtained from centrifuge model tests and is, therefore,

qualified to undertake this task. There are other methods that are founded on

much more sophisticated constitutive models based on plasticity. Some of these

models can simulate the undrained cyclic behavior of soils while retaining some

of the convenient features of classical plasticity theory. In comparison, the

procedures used by TARA are simpler and, thus, computer efficient and yet have

the capability of accounting for a number of important factors that affect

saturated soil behavior. This approach is being used by the Waterways

Experimental Station in dynamic dam stability analyses.

5.0 REFERENCES

1. Arulanandan, K. and Muraleetharan, K.K., "Level Ground Soil-Liquefaction Analysis Using In Situ Properties: II," Journal o£ Geotechnical Engineering, Vol. 114, No. 7, ASCE, July 1988, pp. 771-790.

2. Bhatia, S .K., "The Verification of Relationships for Effective Stress Method to Evaluate Liquefaction Potential of Saturated Sand," Ph.D. Thesis Department of Civil Engineering, University of British Columbia, Vancouver, 1980.

3. Bhatia, S.K. and Nanthikesan, S., "The Development of Constitutive Relationship for Seismic Pore Pressure," Soil Dynamics and Liquefaction -Developments in Geotechnical Engineering, No. 42, edited by A.S. Cakmak, Elsevier Science Publishing Co., Inc., 1988, pp. 19-30.

4. Duncan, J.M. and Chang, C.Y., "Nonlinear Analysis of Stress and Strain in

15

Soils," Journal of the Soil 11echanics and Foundation Engineering, Vol. 96, No. SM5, ASCE, Sept. 1970, pp. 1629-1653.

5. Duncan, J .M., Byrne, P.M., Wong, K.S., and Mabry, P., "Strength, Stress­Strain and Bulk Modulus Parameters for Finite Element Analysis of Stresses and Movements in Soil Masses," Report No. UCB/GT/80-01, Department of Civil Engineering, University of California, Berkeley, AugUst 1980.

6. EPRifNRC/TPC Lotung SSI Workshop, Palo Alto, California, December 9-11, 1987.

7. Finn, W.D.L, Lee, K.W., and Martin, G.R., "An Effective Stress Model for Liquefaction," Journal of Geotechnical Engineering Division, ASCE, GT6, 1977, Vol. 103, 517-533.

8. Finn, W.D.L., "Liquefaction Potential Development Since 1976," Proceedings, International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, St. Louis, Missouri, 1981, pp. 655-681.

9. Finn, W.D.L., Iai, S., and Ishihara, K., "Performance of Artificial Offshore Islands Under Wave and Earthquake Loading," Offshore Technology Conference, Vol. 1, 1982, pp. 661-672.

10. Finn, W.D.L. and Siddharthan, R., "Seismic Response of Caisson-Retained and Tanker Islands," Proceedings, 8th World Conference on Earthquake Engineering, San Francisco, August 1985, pp. 751-757.

11. Finn, W.D.L., "Dynamic Analysis in Geotechnical Engineering," State-of-the­Art Report, Earthquake Engineering and Soil Dynamics II, Utah, June 1988, pp. 523-591.

12. Hushmond, B., Crouse, C.B., Martin, G.R., and Scott, R.F., "Site Response and Liquefaction Studies Involving the Centrifuge," Structures· and Stochastic 11ethods, (3rd International Conference on Soil Dynamics and Earthquake Engineering), Elsevier/Computational Mechanics Publications, No. 45, 1987, pp. 3-24.

13. Scott, R.F., "Solidification and Consolidation of a Liquefied Sand Column," Soils and Foundations, Vol. 26, No. 14, December 1986, pp. 23-31.

14. Siddharthan, R., "A Two-Dimensional Nonlinear Static and Dynamic Response Analysis of Soil Structures," Ph.D. Thesis, University of British Columbia, May 1984.

15. Siddharthan, R. and Norris, G.M., "Performance of Foundations Resting on Saturated Sands," Proceedings, Earthquake Engineering and Soil Dynamics II Conference, Geotechnical Special Publication No. 20, ASCE, June.l988, pp. 508-522.

16. Siddharthan, R., and Maragakis, E.M., "Performance of Flexible Retaining Walls Supporting Dry Cohesionless Soils to Cyclic Loads," International Journal for Numerical and Analytical 11ethods in Geomechanics, Vol. 13, June 1989, pp. 309-326.

16

17. Siddharthan, R., Norris, G.M., and Maragakis, E.A., "Deformation Response of Rigid Retaining Walls to Seismic Excitation," Proceedings, 4th International Conference on Soil Dynamics and Earthquake Engineering, Mexico City, October 1989b, pp. 315-330.

18. Siddharthan, R. and Norris, G.M., "Residual Porewater Pressure and Structural Response," International Journal of Soil Djrnamics and Earthquake Engineering, Vol. 9, No. 5, September 1990a, pp. 265-271.

19. Siddharthan, R. and Norris, G.M., "On the Seismic Displacement Response of Rigid Walls," accepted for publication in Soils and Foundations (to appear in July 1991), June 1990b.

17

APPENDIX 3 (continued)

Computer Code TARA-3

Analysis Method Finite element

Geometrical Dimensions 2-D plain strain

Selected Material Models Incrementally nonlinear hysteretic model (hyperbolic shear stress-strain law) with Masing behavior (incrementally elastic approach)

Loadings Gravity loading and base earthquake motion

Soil-Water Medium Analysis Method Total stress model or effective stress model (Total Stress/Effective Stress) with empirical constitutive laws for pore-water

pressure generation derived from 1-D compression tests and cyclic simple shear tests/also models pore water dissipation

Element Types 2-D continuum elements, structural elements, slip or contact elements

Author/Contact Person or Institution TARA-3 has been developed at the University of British Columbia by Professor W. D. Liam Finn and his students, Civil Engineering Department, University of British Columbia, Vancouver, B.C. V6T 1Z4, Canada, Tel: (604) 822-4938

1

1.0 PROGRAM IDENTIFICATION

1.1 Program Name

1.2 Program Title

1.3 Date

1.4 Authors

Two-dimensional Non-linear Static and··!:

Dynamic Response Analysis 11-11 Hr--- ~~~ 'j ~ceo,.,~ IJ-tl~ ~o-tbtl"'"' ~~

Nanmi s~r i9&r +' :YuN LC{ ft+. .J

R. S iddharthan and W. D. Liam .Finn·.

Faculty of Applied Science

The University of British Columbia:.i

Vancouver, B.C., Canada,

V6T lWS

1.5 Computer Requirements and Storage

The computer program, written in Fortran IV, was deve-

loped and tested on an AMDAHL 470 V/6 computer. .Dimension

statements and the amount of storage required to run a given

problem depend on factors such as number of elements, number

of nodes, number of types of materials, etc. The program has

dynamic storage facility; all variable arrays are assigned

from a large array, the.dimension of which depends on the

problem being analysed. Problems with diff.~rent storage re-

quirements can be accommodated by just chan~ing the dimensions

of the large array.

2.0 GENERAL DESCRIPTION OF THE PROGRAM

TARA~, a two-dimensional finite element program, has options

for solving either static or dynamic problems.. The dynamic loading

is limi:ed to earthquake loading. The static and dynamic .anal;yaa$

can be performed in either effective or total stress modes. Non-

linear stress-strain behaviour of soil was modelled by using ari.::.

incrementally elastic ~pproach. Tangent shear modulus and tangent

2

bulk modulus were taken as the two "elastic" parameters. Material

response in shear is described by a hyperbolic shear stress-strain

law with Masing behaviour; response to changes in mean normal stress

is assumed to be non-linear, elastic and pressure dependent. For

static analysis, gravity may be·switched on at once for entire

structure or the construction sequence can be modelled by layer

analysis.

In static analysis, the effect of dilation is also taken into

account using the dilation angle. Slip or contact·elements also

are included to model the interface between soil and structural

elements. The slip element properties were assumed to be elastic-

perfect plastic, with failure at the interface given by ·the Mohr-

Coulomb failure criterion. The stress-strain conditions determined

by the static analysis give the initial conditions for the dynamic

analysis.

When the dynamic analysis is performed in the total stress

mode, the shear strength, T · , initial shear moduli, G , and max max

tangent bulk moduli, Bt, values are kept constant throughout the

dynamic analysis. The above three parameters can be either computed·

from the static stress conditions or can be input directly, if known.

The tangent shear moduli, Gt, will be modified for the corresponding

shear strains developed during analysis. ·

In the effective stress mode, residual porewater pressures are

calculated using a modification of the model proposed by Finn et al

(1977). The parameters, G , and max T are modified for the max

effects of residual porewater pressure. In all dynamic analyses,

accelerations, velocit.ies and dynamic and residual displacements are

3

also computed. The program can be run in any consistent set of

units.

:::V ,3 ~s-e-rt A I 3.0 DESCRIPTION OF INPUT CARDS

3.1 READ

where

TITLE

3.2 READ

where

ATM

GACC

GAMAW

THETA

HWATER(l)*

COMPRE(2)

TITLE (20A4) 1 Card

• Title of the problem using maximum of

80 characters.

ATM,GACC,GAMAW,THETA,HWATER,COMPRE,NLAY,ICHANG,

NPREX,NCPLE (6Fl0.4,4I5) 1 Card

• Atmospheric pressure.

• Acceleration due to gravity.

• Unit weight of water.

c 0.0, for static analysis. For dynamic analy­

sis, a choice of 3 methods is available

for numerical integration of the equa­

tion of motion. The desired method is

selected by the following ~alues of

THETA;

• 1.0, for linear acceleration method;

• -1.0, for constant average acceleration

method;

• 1.4, for Wilson's e method.

• Height of watertable during layer construc­

tion.

• Compaction pressure used on the layer

during construction;

• 0.0, imP,lies normally consolidated state.

*Numbers in parentheses refer to clarifying statements in Section 4.0, ·Explanatory notes.

z

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 L y

I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 9 19 29 39 49 59 69 79 89 99 109 119 129 139 149 159 169 179 189 199 209 219 229 239 249

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 S8 8 18 28 38 48 58 68 78 88 98 108 118 128 138 148 158 168 178 188 198 208 218 228 238 248

49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 UNDEFORMED 7 17 27 37 47 57 67 77 87 97 107 117 127 137 147 157 167 177 187 197 207 217 227 237 247 SHAPE 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 6 16 26 36 46 56 66 76 86 96 106 116 126 136 146 156 166 176 186 196 206 216 226 236 246

97 98 99 10~ 101 10~ 10~ 104 10E 10t 10/ 108 109 11~ 111 112 113 11~ 11E 116 117 118 119 120 5 15 25 35 45 55 65 75 85 95 105 115 125 135 145 155 165 175 185 195 205 215 225 235 245

121 122 123 124 12E 12c 12/ 128 12S 130 131 132 133 134 135 136 137 13E 13S 140 141 142 14J 144 OPTIONS 4 14 24 34 44 54 64 74 84 94 104 114 124 134 144 154 164 174 184 194 204 214 224 234 244 JOINT IDS

ISJ 15E 16E 16/ 14E 146 147 14E 14S 150 lSI 152 154 156 157 15E ISS 160 161 16l 16~ 164 IM 16E ALL JOINTS 3 13 23 33 43 53 63 73 83 93 103 113 123 133 143 153 163 173 183 193 203 213 223 233 243 16S 170 171 17l 17~ 174 17E 176 177 17E 175 180 181 18l 18~ 184 18S 18t 18/ 18E 18S 190 191 19~ ELEMENT IDS 2 12 22 32 42 52 62 72 82 92 102 112 122 132 142 152 162 172 182 192 202 212 222 232 242 WIRE FRAME 19: 194 19E 19t 19/ 19E 195 200 20 I 202 203 204 205 20t 20/ 208 209 21~ 21 I 21l 21~ 21L 21E 21t I 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201 211 221 231 241

KAJIMA TEST 72 8, 73 P, 83 UNDEFORMED MESH

I SAP90 I

z

L y

I I I I I I I I I I I 10 10 I I I I I I I I I I I

2 2 2 2 2 2 2 2 2 2 2 10 10 2 2 2 2 2 2 2 2 2 2 2 S8

3 3 3 3 3 3 3 3 3 3 3 10 10 3 3 3 3 3 3 3 3 3 3 3 UNDEFORMEO SHAPE

4 4 4 4 4 4 4 4 4 4 4 10 10 4 4 4 4 4 4 4 4 4 4 4

5 5 5 5 5 5 5 5 5 5 5 10 10 5 5 5 5 5 5 5 5 5 5 5

6 6 6 6 6 6 6 6 6 6 6 10 10 6 6 6 6 6 6 6 6 6 6 6 OPTIONS ALL JOINTS

7 7 7 7 7 7 7 7 7 7 7 10 10 7 7 7 7 7 7 7 7 7 7 7 PROPERTY IDS 8 8 8 8 8 8 8 8 8 8 8 10 10 8 8 8 8 8 8 8 8 8 8 8 WIRE FRAME

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

KAJIMA TEST 72 8, 73 P PROPERTY IDENTIFICATION NUMBERS

I I SAP90

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I I I I I I I i I I I 10 10 I I I I I I I I I I I

2 2 2 2 2 2 2 2 2 2 2 10 10 2 2 2 2 2 2 2 2 2 2 2 Sl0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 UNOEFORMED

SHAPE 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

5 5 5 5 5 5 s s s s s s s s s s s s s s s s s s

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 OPTIONS

7 7 7 7 7 7 7 7 7 7 7 7 7 ALL JOINTS

7 7 7 7 7 7 7 7 7 7 7 PROPERTY IDS 8 8 8 8 8 8 8 8 8 8 8 8 B B B B B B B B B B B B WIRE FRAME

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

KAJIMA TEST 83 PROPERTY IDENTIFICATION NUMBERS

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-KAJIMA, TEST 83

'ACCELERATION (cm/sec2) -

ACC 1 TEST -

.1 I I I I I I I I I ,-

30 40 50 60 70 80 90

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l....L

100

St0

TIME HISTORY TRACE

JOINT t TYPE AA DIRN YT FACTOR

0 I t 000E +0 t

ENVELOPES MIN -0.9S9SE+04

AT 0.04720 MAX

0.964SE+04 AT 0.05660

SAP90

350 T I M E X 10 2 f-' I I I I I I

f-

280

1--

210

r-

140

f-

70

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!jl ~ fl 1'''' r !Tijjill'rl 1 '11 1~ r'iiuW'~ U W '"'" -

-70

-

-140

-

-210

f-

-280

-

-350 -, I I I I I I

12 24 36 48 60 72

I I I I ,_

-

-

-

-

-

-

-

-

-KAJIMA, TEST 83 ACCELERATION (cm/sec2 ) -ACC 1 TEST -_L 1 I I I I I ,-

84 96 108

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D

120

Sl0 TIME HISTORY TRACE

JOINT l TYPE AA DIRN YT FACTOR

0 I 1000E +01

ENVELOPES MIN -0.9595E+04

AT 0.04720 MAX

0.9645E+04 AT 0.05660

SAP90

T I M E X 10 J S10 350

f-' I I I I I I I I I I '- TIME I- - HISTORY 280

TRACE 1- -

210,

- (\ - JOINT 90 140 ~ C"J TYPE AA

I

A~ (\

~ -CSl DIRN YT 1- ~

70 I >< FACTOR

~A~ 1/V I 0 I 1000E +01 - :z:

0 ....... /'1 A__M C) v1 vv

V1 t----1 -I---70 '-'~ ENVELOPES v \ v (_)

- - :z: MIN -140 :::=J

v v l....L -0.2420E+0S I- -

-210 AT 0.04840 MAX - . -KAJIMA, TEST 83 0.3262E+0S -280 ACCELERATION (cm/sec2} -

ACC 12 CALCULATED AT 0.05620 I- -

-350 b_ I I I I I I I I I I I I I ,-

0 10 20 30 40 50 60 70 80 90 100 I SAP90 I

350 f-' I I I I

f-

280

f-

210

f-

140

f-

70

1\1\ '-

!lAA ~' ~ {\

-70

v v "v ~ i

f-

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-140

1-

-210

f-

-280

-350 i\=" If-

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10 20 30

T I M E X 10 J I I I I I I '·

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A -

v

(\ -A

I \ A ~ -

II

A /\

(\_~ v

1\ 1\ I v ~ \A

I

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-KAJIMA, TEST 83 ACCELERATION (cm/sec2) -ACC 12 TEST -

I I I I I I I J J LLLL

40 50 60 70 80 90

C"J I CSl

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t--1

1--

c.._)

:z::

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LL

100

s 1 1

TIME HISTORY TRACE

JOINT TYPE DIRN FACTOR

1 AA YT

0 I 1000E +01

ENVELOPES MIN -0.3262E+05

AT 0.05080 MAX

0.2498E+05 AT 0.05680

SAP90

T I M E X l0 2 s l l 350

TIME 280 HISTORY

TRACE 210

JOINT l 140 TYPE AA

DIRN YT 70 >< FACTOR

:z:: 0 I l000E +0l 0 D

t---i

-70 t--

ENVELOPES c..__)

:z:: MIN -140 =::)

LL -0.3262E+05 -210 AT 0.05080

MAX -280

KAJIMA, TEST 83 0.2498E+05 ACCELERATION (cm/sec2 ) ACC 12 TEST AT 0.05680

. -350 0 12 24 36 48 60 72 84 96 108 120 SAP90