RM E Prestressing Basic Part3 Earthquake

RM Bridge Professional Engineering Software for Bridges of all Types

TRAINING PRESTRESSING BASIC - RM - PART 3: DYNAMIC EARTHQUAKE ANALYSIS

RM Bridge V8i

March 2012

RM Bridge

Training Prestressing Basic - RM - Part 3: Dynamic Earthquake Analysis I

Bentley Systems Austria

Contents

1 General ................................................................................................................... 1-1

1.1 Scope .............................................................................................................. 1-1

1.2 Generating a new construction schedule ....................................................... 1-1

2 Definition of Masses .............................................................................................. 2-3

2.1 Definition of load sets for self weight and SDL ............................................ 2-3

2.1.1 Generating the load sets for SW and SDL ................................................. 2-3

2.1.2 Defining the load set for SW ..................................................................... 2-3

2.1.3 Defining the load set for SDL .................................................................... 2-4

2.2 Definition of load case for masses ................................................................. 2-4

2.2.1 Generating a load case for the masses ....................................................... 2-4

2.2.2 Assigning the load sets to the load case ..................................................... 2-5

3 Calculation of Eigenvalues .................................................................................... 3-6

3.1 Generating a stage for the calculation of the eigenvalues ............................. 3-6

3.2 Calculating the eigenvalues ........................................................................... 3-6

4 Preparation of Response Spectrum ........................................................................ 4-8

4.1 Defining a response spectrum diagram .......................................................... 4-8

4.1.1 Generating tables for horizontal and vertical response spectrum ............. 4-8

4.1.2 Defining the tables for horizontal and vertical response spectrum ............ 4-8

4.1.3 Assignment of the tables to variables (formulas) .................................... 4-13

4.2 Response Spectrum defined by formulas ..................................................... 4-13

5 Definition of the Earthquake Load ...................................................................... 5-16

5.1 Defining the earthquake events .................................................................... 5-16

6 Evaluation of the Response Spectrum ................................................................. 6-18

6.1 Generating a stage for the evaluation of the response spectrum .................. 6-18

6.2 Evaluating the response spectrum ................................................................ 6-18

6.2.1 Initializing envelopes for storing the resuts of the evaluation ................. 6-18

6.2.2 Evaluating the response spectra ............................................................... 6-18

7 Superposition of the Seismic Loads ..................................................................... 7-20

7.1 Generating a stage for the superposition of the seismic loads ..................... 7-20

7.2 Superposing the seismic loads ..................................................................... 7-20

RM Bridge

Training Prestressing Basic - RM - Part 3: Dynamic Earthquake Analysis II


8 Result Presentation .............................................................................................. 8-22

8.1 Plotting the response spectra data ................................................................ 8-22

8.2 Plotting of eigenmodes ................................................................................ 8-22

8.2.1 Definition of the DoPlot action ................................................................ 8-22

8.2.2 Definition of the Plot File in the Plot Container ...................................... 8-23

9 Time History Analysis ......................................................................................... 9-25

9.1 Preparation of the tables and variables ........................................................ 9-25

9.2 Definition of the load cases and load sets .................................................... 9-27

9.2.1 Definition of the load sets for the masses ................................................ 9-27

9.2.2 Definition of the load sets for the displacements ..................................... 9-27

9.2.3 Definition of the load cases for the time history calculation ................... 9-28

9.3 Calculation of the Time History Events ...................................................... 9-30

9.4 Definition of Damping ................................................................................. 9-31

RM Bridge

Training Prestressing Basic - RM - Part 3: Dynamic Earthquake Analysis 1-1


1 General

1.1 Scope

In the following the procedure for performing a dynamic earthquake calculation using

the response spectrum analysis shall be explained. All necessary steps including load

case definitions, preparation of response spectrum, and required calculation actions and

superposition in the schedule will be given. The principle input procedure is the follow-

ing:

1.) Definition of masses of all permanent loads (i.e. self weight and superimposed dead load)

2.) Calculation of the eigenvalues

3.) Preparation of the response spectrum

4.) Definition of the earthquake load

5.) Evaluation of response spectrum for earthquake analysis

6.) Superposition of the seismic loads

7.) Result presentation

In addition in chapter 9 there is given a general overview of the necessary input proce-

dure for performing a time history analysis.

1.2 Generating a new construction schedule

To separate schedule actions and results it can be favourable to create a new schedule

variant (it is of course not obligatory), where the dynamic calculation is performed. This

is done as follows:

Create a new

schedule variant

Schedule CS will be skipped deactivate

Name earthquake

Schedule Variants Description response spectrum analysis

Sequence No 1

Top Table

Then change in the main GUI to this construction schedule (drop down menu to the

right of the Recalc button). All input of loads and stages will now refer to this (active) schedule.

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NOTE: If the final superposition file containing the results of the earthquake calculation

is needed and used in the calculation of the DEFAULT schedule (e.g. combination table), the sequence for calculating the different construction schedules has to be

changed in accordance. In the recalculation pad one can then choose Recalc all, which will use this given sequence. All envelope results (*.sup) are stored in the main project

directory of RM and are therefore available for all the construction schedules. Alterna-

tively one can just calculate a particular construction schedule by selecting it in the GUI

first or in the recalculation pad and using Recalc.

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2 Definition of Masses

All permanent loads acting on the structure, i.e. self weight and superimposed dead

loads, have to be defined as masses. These masses must be defined in one single load

case, whereas it is favourable to group them by means of load sets.

2.1 Definition of load sets for self weight and SDL

2.1.1 Generating the load sets for SW and SDL

Create new load

sets for the self

weight and super-

imposed dead load

Schedule Name LS-SW LS-SDL

Description self weight mass SDL mass

Load Definition

Load Set Defi-

nition

Top table

2.1.2 Defining the load set for SW

Define load set for

the self weight

Schedule Name LS-SW

Loading Uniform

load

Uniform load

Uniform load

Load Definition Type Self weight -

mass

Self weight -

mass

Self weight -

mass

From 101 1201 1301

Load Set Defi-

nition To 135 1204 1304

Step 1 1 1

Bottom table Rx 0 0 0

Ry -1 -1 -1

Rz 0 0 0

Gam

[kN/m3]

0 0 0

The load type for the self-weight mass can be specified as just as mass (only used for dynamic calculation) or load and mass (also used for static calculation).

These defined self weight masses act in the centre of gravity.

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2.1.3 Defining the load set for SDL

Define load set for

superimposed dead

load

Schedule Name LS-SDL

Loading masses masses masses

Load Definition Type

Element

uniform

mass + ec-

centricity

Element

uniform

mass + ec-

centricity

Element

uniform

mass + ec-

centricity

From 101 101 101

Load Set Defi-

nition To 135 135 135

Step 1 1 1

Bottom table g*mx

[kN/m] 35 6.1 6.1

g*my

[kN/m] 35 6.1 6.1

g*mz

[kN/m] 35 6.1 6.1

g*Imx

[kNm] 0 0 0

Ey [m] 0.06 0.45 0.45

Ez [m] 0 +6.3 -6.3

In RM masses are defined as forces (and moments respectively) and internally trans-

formed into masses by dividing them by the gravity acceleration value, which is set to

9.81 m/s2 by default (this can be modified in the Recalculation Pad in the menu Dy-

namic). Since mass is a scalar value the definition of the vectors has to be given for all three

force-components.

The eccentricities are not related to the center of gravity (local element coordinate

system), but to the node (node axis, i.e. connection between start and end node of the

element), that means the internal element eccentricities are added automatically!

2.2 Definition of load case for masses

2.2.1 Generating a load case for the masses

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Create new load

case for the load

sets of self weight

and superimposed

dead load

Schedule Name LC-MASS

Description dead load mass

Load Definition

Load Case Def-

inition

Top table

2.2.2 Assigning the load sets to the load case

Assign load sets to

load case

Schedule Name LC-MASS

Loading Load set input Load set input Load Set LS-SW LS-SDL

Load Definition

Load Case Def-

inition

Bottom table

This load case is needed then as reference for calculating the eigenvalues (see 3.2)

All loads not specified in this manner using the respective load types for masses will

not be considered for the calculation of the mass matrices in the dynamic analysis.

(That means on the other hand that it is possible to include load items, which should

not be taken into account as masses, but only as loads in the static or time history

analysis).

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3 Calculation of Eigenvalues

3.1 Generating a stage for the calculation of the eigenvalues

Create a stage for

the calculation of

the eigenvalues

Schedule Name EIGEN

Description calculation of eigenvalues

Stages

Activation

Top table

Activate all ele-

ments of structure

Schedule Activate From 101

Stages To 1402

Step 1

Activation

Bottom table

3.2 Calculating the eigenvalues

Calculate the ei-

genvalues

Schedule Action Calculation (Dy-

namic) List/plot actions

Type Eigen ListMod

Stages Input-1 30 eigen.mod

Input-2 LC-MASS

Schedule Actions Input-3

Output-1 eigen.mod

Bottom table Output-2 eigen.lst eigen-mod.lst

Descrip-

tion

Calculate eigen

frequencies and

eigenvectors (natural

modes) of structure

Create listfile of

binary modal file

The load case, where the effective masses are defined, has to be given as reference for

the eigenvalue analysis (see 2.2).

The number of eigenvalues (lowest natural frequencies) to be calculated has to be

given. (The tolerance value for determining the accuracy of the calculated eigenval-

ues is specified in the Recalc pad in the menu Dynamic). For receiving 90% of mass participation in vertical direction in this example approximately 30 eigenvalues

were necessary (see comments for the output of mass participation below).

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The eigenvalues and natural modes are stored in a binary modal file, which name has

to be given. A readable output list of this file can be printed using the list/plot action

ListMod The name of the output list with a protocol of all the relevant input and output data

has to be entered (or left to default *). An essential part is the output of the mass participation factors at the end of this list, which should be greater than 90% of the to-

tal mass in the relevant direction of the acceleration of the structure (see Diagram 1).

In some cases the number of calculated eigenmodes is smaller than expected. In such

cases the dimension of the iteration matrix may be increased by additional iteration

vectors (Subspace). This may lead to better iteration results and a higher number of

eigenmodes found.

Note: Eigenvalues and eigenfrequencies (natural frequencies) are calculated on the

un-damped structural system; therefore no damping parameters need to be specified

yet within this action.

After performing this action a number of n load cases is generated and can be accessed

from the load case pool (named name#n), where n is the number of the eigenmode and

the name being taken from Output-1 (e.g. eigen#3). These load cases contain normal-

ized eigenvectors as displacements and may be used for graphic presentation (see 8.2).

Diagram 1: Output of mass participation factors in eigen.lst

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4 Preparation of Response Spectrum

4.1 Defining a response spectrum diagram

The response spectrum is defined in Properties Variables in terms of tables and/or formulas according to the rules of the respective codes.

The response spectrum used in this example is in accordance with Eurocode 8 (Type 1,

Ground type A, 5% damping). It is prepared by means of tables where the particular

values have been evaluated before. Also an alternative will be shown in 4.1.3 illustrat-

ing a direct definition by formulas.

4.1.1 Generating tables for horizontal and vertical response spectrum

Create a new table

for the horizontal

and vertical re-

sponse spectrum

Properties Name resp_hor_tab resp_vert_tab

Type table table

Variables Description table for horizontal

response spectrum

table for vertical

response spectrum

Top table

Note: Do not use special characters in variable names (formulas and tables) except un-

derline _!

4.1.2 Defining the tables for horizontal and vertical response spectrum

Define the tables

for the horizontal

and vertical re-

sponse spectrum

Properties

Variables

Bottom table

The units used are period (T) [s] for the abscissa (VarA) and the elastic response spec-

trum divided by the design ground acceleration (Se/ag) [-] and (Sve/avg) [-] respectively

for the ordinate (VarB).

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Table for horizontal response spectrum

VarA T [s] VarB Se/ag [-] Interpolation

VarA T [s] VarB Se/ag [-] Interpolation

0.00 1.0000 linear

1.70 0.5882 linear

0.15 2.5000 linear

1.75 0.5714 linear

0.40 2.5000 linear

1.80 0.5556 linear

0.45 2.2222 linear

1.85 0.5405 linear

0.50 2.0000 linear

1.90 0.5263 linear

0.55 1.8182 linear

1.95 0.5128 linear

0.60 1.6667 linear

2.00 0.5000 linear

0.65 1.5385 linear

2.10 0.4535 linear

0.70 1.4286 linear

2.20 0.4132 linear

0.75 1.3333 linear

2.30 0.3781 linear

0.80 1.2500 linear

2.40 0.3472 linear

0.85 1.1765 linear

2.50 0.3200 linear

0.90 1.1111 linear

2.60 0.2959 linear

0.95 1.0526 linear

2.70 0.2743 linear

1.00 1.0000 linear

2.80 0.2551 linear

1.05 0.9524 linear

2.90 0.2378 linear

1.10 0.9091 linear

3.00 0.2222 linear

1.15 0.8696 linear

3.10 0.2081 linear

1.20 0.8333 linear

3.20 0.1953 linear

1.25 0.8000 linear

3.30 0.1837 linear

1.30 0.7692 linear

3.40 0.1730 linear

1.35 0.7407 linear

3.50 0.1633 linear

1.40 0.7143 linear

3.60 0.1543 linear

1.45 0.6897 linear

3.70 0.1461 linear

1.50 0.6667 linear

3.80 0.1385 linear

1.55 0.6452 linear

3.90 0.1315 linear

1.60 0.6250 linear

4.00 0.1250 linear

1.65 0.6061 linear

Diagram 2: Table for the horizontal response spectrum

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Table for vertical response spectrum

VarA T [s] VarB Sve/avg [-] Interpolation

VarA T [s] VarB Sve/avg [-] Interpolation

0.00 1.0000 linear

1.60 0.1758 linear

0.05 3.0000 linear

1.70 0.1557 linear

0.15 3.0000 linear

1.80 0.1389 linear

0.20 2.2500 linear

1.90 0.1247 linear

0.25 1.8000 linear

2.00 0.1125 linear

0.30 1.5000 linear

2.10 0.1020 linear

0.35 1.2857 linear

2.20 0.0930 linear

0.40 1.1250 linear

2.30 0.0851 linear

0.45 1.0000 linear

2.40 0.0781 linear

0.50 0.9000 linear

2.50 0.0720 linear

0.55 0.8182 linear

2.60 0.0666 linear

0.60 0.7500 linear

2.70 0.0617 linear

0.65 0.6923 linear

2.80 0.0574 linear

0.70 0.6429 linear

2.90 0.0535 linear

0.75 0.6000 linear

3.00 0.0500 linear

0.80 0.5625 linear

3.10 0.0468 linear

0.85 0.5294 linear

3.20 0.0439 linear

0.90 0.5000 linear

3.30 0.0413 linear

0.95 0.4737 linear

3.40 0.0389 linear

1.00 0.4500 linear

3.50 0.0367 linear

1.10 0.3719 linear

3.60 0.0347 linear

1.20 0.3125 linear

3.70 0.0329 linear

1.30 0.2663 linear

3.80 0.0312 linear

1.40 0.2296 linear

3.90 0.0296 linear

1.50 0.2000 linear 4.00 0.0281 linear

Diagram 3: Table for the vertical response spectrum

The internal variable for evaluating the response spectrum is the angular velocity () [rad/sec]. Therefore if other values are used for the abscissa in the table of the re-

sponse spectrum diagram (Frequency (Hz), Period (T), or if these are given in terms

of logarithm), a respective transformation has to be performed (see 4.1.3).

The related ordinate value of the ground motion amplitude can either be given as dis-

placement (d), velocity (v) or acceleration (a). (The used value has to be referred to

later when defining the earthquake load see 5.1). The ordinate values of the re-sponse spectrum must be given in the internal units [m] and [s]. When other units (or

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factors of the gravity constant (g)) are used for the motion amplitude of the response

spectrum, a respective transformation has to be performed.

For a detailed description for performing such transformations refer to the RM User

Guide 14.3.2.

The dimensionless ordinate value of the ground motion amplitude used in this example

shall be transformed into acceleration. This can either be done by multiplying it with the

design ground acceleration by defining a respective formula (see explanations above

and also 4.1.3) or within the definition of the earthquake load when defining the respec-

tive directions and intensities of the excitation vectors (as will be performed in this ex-

ample (see 5.1)).

For the practical application it can be helpful to prepare an Excel sheet for the genera-

tion of the values of the response spectrum tables and copy/paste them into a Tcl file,

which then can be imported (added) to the project.

Diagram 4: (Partial) Tcl file for horizontal response spectrum

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Diagram 5: Graphical presentation of horizontal and vertical response spectrum

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4.1.3 Assignment of the tables to variables (formulas)

In order to enable the correct evaluation of the data of a table the used value for its ab-

scissa (abscissa of the response spectrum) has to be defined, i.e. the table needs to be

related to the value of its abscissa. This can be done in the variable definition (as shown

in the following) or alternatively within the definition of the earthquake load when se-

lecting the table (see 5.1).

In the following the defined tables of the horizontal and vertical response spectrum are

assigned to formulas to enable the evaluation of their data for the response spectrum

analysis.

Create two varia-

bles of the type

formula for assign-

ing the two tables

of the response

spectrum

Properties Name resp_hor resp_vert

Type formula formula

Variables Expression resp_hor_tab(2*pi/om

ega)

resp_vert_tab(2*pi/om

ega)

Description formula for horizontal

response spectrum

formula for vertical

response spectrum

Top table

resp_hor and resp_vert are the variables (formulas) that describe the response spec-

trum using the values of the tables resp_hor_tab and resp_vert_tab, which have been

defined as functions of the period T. Since the internal variable for the evaluation of

the response spectrum is omega, a respective transformation has to be performed

within the assignment in terms of F = f(T) = f(2*pi/omega):

resp_hor = resp_hor_tab(2*pi/omega) and

resp_vert = resp_vert_tab(2*pi/omega).

Referring to 4.1.2 it would be possible to perform the transformation into acceleration

here by including a factor for the design ground acceleration (ag) and (avg) respectively

in the expressions above (e.g. resp_hor = ag*resp_hor_tab(2*pi/omega)). In this case

the normalized vectors have to be used within the definition of the earthquake load in-

stead of referring to the design ground acceleration there (see 5.1).

4.2 Response Spectrum defined by formulas

An alternative to setting up a table with a series of particular evaluated values is to di-

rectly define the formulas that describe the response spectrum. The definition for the

horizontal response spectrum is illustrated below. The particular parameters are defined

by variables and thus easily can be adjusted by the user. In that way also other types of

the response spectrum can be defined by changing the respective values.

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Name

EC

Denota-

tion

Description Definition (default values)

S S soil factor 1 [-]

xi viscous damping ratio in percent 5 [%]

eta damping correction factor max(SQR(10/(5+xi)),0.55) [-]

TB TB limit of the constant spectral acceleration branch (1) 0.15 [s]

TC TC limit of the constant spectral acceleration branch (2) 0.4 [s]

TD TD value of beginning of constant displacement response range 2.0 [s]

Se Se/ag function for elastic response spectrum in terms of Se/ag (T) S*Se_tab(2*pi/omega) [-]

Se_tab table of resp. spectr. in terms of Se/ag/S (T) VarA VarB

0 1+TabA/TB*(eta*2.5-1)

TB eta*2.5

TC eta*2.5*(TC/TabA)

TD eta*2.5*(TC*TD/TabA^2)

4 eta*2.5*(TC*TD/4^2)

Diagram 6: Definition of the horizontal response spectrum by formulas

The definition of the vertical response spectrum is performed analogously. Both sets of

definitions can be viewed in the corresponding example (see Properties Variables; Group EC8_RESP_form).

The advantage of this method is the general definition that can easily be adapted for all

types of response spectra and the exact evaluation of the particular values of the re-

sponse spectrum (while the accuracy of the values derived from a table relies on the

(linear) interpolation of the defined values).

REMARK:

The formulas of the elastic response spectra and design spectra according to Euro-

code 8 can be generated automatically within RM Bridge by using the respective

options in the menu Extras > Loading and Stages > Response Spectra:

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Diagram 7: Generating the design spectrum in RM Bridge

See online-help (F1) within the input window for more details.

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5 Definition of the Earthquake Load

The earthquake load is defined in Schedule Load Definition Earthquake Load.

5.1 Defining the earthquake events

The upper table contains the basic parameters of the particular earthquake events.

Define a seismic

event for each of

the three directions

Schedule Number 1 2 3

Modal-File eigen.mod eigen.mod eigen.mod

Load Definition Rule CQC CQC CQC

Duration [s] 60 60 60

Earthquake

Load Description

earthquake in

x-dir (longi-

tudinal)

earthquake in

y-dir (verti-

cal)

earthquake in

z-dir (trans-

versal)

Top table

Number of the earthquake event for storage in the database

Name of the modal file containing the results of the eigenvalue analysis (see 3)

Combination rule for superposing the particular contributions of the different natural

modes (see RM User Guide 14.3.1 for the different available rules in RM)

Duration [s] of the seismic event (influencing the results only in combinations with

rules using duration dependent correlation factors (DSC, CQC, CQCX))

The lower table contains the related ground motion parameters and the assigned re-

sponse spectrum.

Define a seismic

event for each of


Schedule

Type of re-

sponse spec-

trum graph

a a a

Vec-Vx 1.5

Load Definition Vec-Vy 1.35

Vec-Vz 1.5

Earthquake

Load Damp-Fact 0.05 0.05 0.05

Var-

Name(Graph) resp_hor resp_vert resp_hor

Bottom table

Type of the specified ground motion in the response spectrum (displacement (d), ve-

locity (v) or acceleration (a)) (see 4.1.2)

Vector of the ground motion, multiplied by the respective design ground acceleration

to transform the dimensionless ground motion amplitude of the defined response

spectrum into an acceleration value (see 4.1.2). The design ground accelerations in

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this example are assumed with ag = 1.5 m/s2 and avg = 0.9 ag = 1.35 m/s

2 respec-

tively.

Referring to the comments given in 4.1.3 the normalized components (value of 1) of

the vectors have to be given here, if the design ground acceleration is already in-

cluded in the values of the ground motion amplitude of the response spectrum.

Damping value only used for the calculation of the correlation factors with combina-

tion rules DSC, CQC and CQCX (note that the damping ratio for the response spec-

trum is already incorporated into the table!).

Name of the variable or table representing the respective response spectrum (see

4.1.2). The particular response spectra must be given in terms of the same type.

The selected variable must already be defined as a function of the value of the abscis-

sa of the response spectrum in the variable definition (as done in this example). If a

table is selected it must be defined as a function of the value of the abscissa of the re-

sponse spectrum within the input here.

If the response spectra are given by the formulas as illustrated in 4.2, the respective

input for Var-Name(Graph) would be the variables Se and Sve respectively. The input

is performed in the schedule variant formula in the corresponding example.

So the possible inputs for the horizontal loads would be:

Var-Name(Graph) Remark

resp_hor resp_hor = resp_hor_tab(2*pi/omega) as defined in the Variable menu

resp_hor_tab(2*pi/omega) The reference value of the defined table has to be given here

Se Se = S*Se_tab(2*pi/omega) as defined in the Variable menu

S*Se_tab(2*pi/omega) The reference value of the table is given and the multipl. with soil factor is performed

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6 Evaluation of the Response Spectrum

6.1 Generating a stage for the evaluation of the response spectrum

Create a stage for

the evaluation of

the response spec-

trum

Schedule Name RESP

Description response spectrum evaluation

Stages

Schedule Ac-

tions

Top table

6.2 Evaluating the response spectrum

6.2.1 Initializing envelopes for storing the resuts of the evaluation

First a superposition file (envelope) has to be created for the storage of the results of the

evaluation of the response spectrum for all three directions.

Initialize envelopes

for the storage of

the results of the

response spectrum

evaluation

Schedule Action

LC/Envelope

actions

LC/Envelope

actions

LC/Envelope

actions

Type SupInit SupInit SupInit

Stages Input-1

Input-2

Schedule Actions Input-3 - - -

Output-1 resp-x.sup resp-y.sup resp-z.sup

Bottom table Output-2 - - -

Descrip-

tion

envelope for

storing results

of x-dir (lon-

gitudinal)

envelope for

storing results

of y-dir (ver-

tical)

envelope for

storing results

of z-dir

(transversal)

6.2.2 Evaluating the response spectra

The evaluation of a response spectrum is performed with the dynamic calculation action

RespS.

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Evaluate the re-

sponse spectra for


Schedule Action

Calcula-tion (Dynam-

ic)


ic)


ic)

Type RespS RespS RespS

Stages Input-1 1 2 3

Input-2 ALL ALL ALL


Output-1 resp-x.sup resp-y.sup resp-z.sup

Bottom table Output-2 * * *

Descrip-

tion

evaluate

response

spectrum in

x-dir (longi-

tudinal)

evaluate

response

spectrum in

x-dir (verti-

cal)

evaluate

response

spectrum in

x-dir (trans-

versal)

Number of the seismic event for evaluating the structural response referring to the

given number in the earthquake load definition

Selection of elements to be considered for the calculation (ALL or ACTIVE)

Name of the envelope where the results shall be stored

Name of the output list for the data of the response calculation (default resp0001.lst

for seismic event 1) NOTE: In this example constant damping is assumed (no input in optional Input-3).

For how to apply modal damping with sinlge response spectrum, tables of response

spectra and weighted element damping see separately available example.

The results of this action are stored in the given superposition file and are extreme

forces and displacements. As the superposition rules are statistic, only leading values

may be obtained. With the use of a special algorithm, called TDV-Superposition method (set in the Recalc-option), it is possible to obtain affiliated results in the superposition file.

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7 Superposition of the Seismic Loads

7.1 Generating a stage for the superposition of the seismic loads

Create a stage for

the superposition

of the seismic

loads

Schedule Name SUP

Description superposition of the seismic loads

Stages

Schedule Ac-

tions

Top table

7.2 Superposing the seismic loads

The particular results of the three basic directions are now combined according to the

30%-rule. An envelope is initialized for the maximum results of each direction (resp-

max-x.sup, resp-max-y.sup, resp-max-z.sup), where the particular results of the respec-

tive directions are added up with 100% (factor 1.0) and the other two with 30% (factor

0.3). Then one final envelope (resp-max.sup) is initialized for evaluating the maximum

results out of these three envelopes.

Initialize envelopes

for the superposi-

tion of the maxi-

mum results for

each direction and

superpose results

according to 30%-

rule

Schedule Action

LC/Envelope

actions

LC/Envelope

actions

LC/Envelope

actions

Type SupInit SupAddSup SupAddSup

Stages Input-1 resp-x.sup resp-max-

x.sup

resp-max-

x.sup

Input-2 1.0 resp-y.sup resp-z.sup

Schedule Actions Input-3 - 0.3,0.3 0.3,0.3

Output-1 resp-max-

x.sup


Descrip-

tion

Initialize

envelope for

max results in

x-dir and add

up 100% of

x-direction

Add up 30%

of y-direction

Add up 30%

of z-direction

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Action LC/Envelope actions LC/Envelope actions LC/Envelope actions Type SupInit SupAddSup SupAddSup

Input-1 resp-y.sup resp-max-y.sup resp-max-y.sup

Input-2 1.0 resp-x.sup resp-z.sup

Input-3 - 0.3,0.3 0.3,0.3

Output-1 resp-max-y.sup

Output-2 - - -

Descrip-

tion

Initialize envelope for

max results in y-dir and

add up 100% of y-

direction

Add up 30% of x-

direction

Add up 30% of z-

direction

Action LC/Envelope actions LC/Envelope actions LC/Envelope actions Type SupInit SupAddSup SupAddSup

Input-1 resp-z.sup resp-max-z.sup resp-max-z.sup

Input-2 1.0 resp-x.sup resp-y.sup

Input-3 - 0.3,0.3 0.3,0.3

Output-1 resp-max-z.sup

Output-2 - - -

Descrip-

tion

Initialize envelope for

max results in z-dir and

add up 100% of z-

direction

Add up 30% of x-

direction

Add up 30% of y-

direction

Initialize envelope

for the maximum

results of the three

envelopes and

superpose

Schedule Action

LC/Envelope

actions

LC/Envelope

actions

LC/Envelope

actions

Type SupInit SupOrSup SupOrSup

Stages Input-1 resp-max-

x.sup resp-max.sup resp-max.sup

Input-2 resp-max-

y.sup

resp-max-

z.sup

Schedule Actions Input-3 -

Output-1 resp-max.sup


Descrip-

tion

Initialize

envelope for

max results

and add up

first envelope

Superpose

second enve-

lope with OR

Superpose

third enve-

lope with OR

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8 Result Presentation

8.1 Plotting the response spectra data

The tables and diagrams of the response spectra defined in Properties Variables can be plotted either directly in this menu by clicking on the blue info button in the top table

or within the schedule actions using the List/plot action PlVar and selecting the respec-

tive table.

8.2 Plotting of eigenmodes

The load cases deriving from the eigenvalue calculation (e.g. eigen#1, eigen#2, eigen#3

etc., see 3.2) containing the normalized eigenvectors as displacements can be used for

graphic presentation.

The corresponding RM training to this document contains a sample using Plot Contain-

er with variables and special settings within the DoPlot action to allow for plotting mul-

tiple load cases (i.e. all eigenmodes) with just one command. Also special internal vari-

ables can be used for printing the values of the particular eigenmodes.

A brief principle description shall be given in the following:

8.2.1 Definition of the DoPlot action

Diagram 8: Definition of the DoPlot action

The names of the prepared plot container and plot file have to be given as usual.

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The plot file contains a variable EigLC for the load case to be plotted (see 8.2.2). The

load cases to be plotted are eigen#1, eigen#2 up to eigen#30. Therefore the variable is

defined to be the load case eigen#, i.e. the consecutive number is again de-

fined by another variable (given in angle brackets), which is defined in a second in-

put: num = {1 30} to define the range of values that it should take.

Note that also the output file name needs to be changed for each generated plot file,

therefore also here a variable is used (e.g. names of the load cases, i.e. eigen#1, eig-

en#2 etc.).

See also the input help when pressing F1 within the input window of the action.

8.2.2 Definition of the Plot File in the Plot Container

A plot file can be set up from the scratch or easier can automatically be generated by

making use of the Macro function (option Eigenform) and be adapted afterwards.

The specific input shall be explained below:

Diagram 9: Definition of the load case to be plotted

The load case is not given directly, but referred to by the variable EigLC.

To print the particular values of the eigenmodes a text field can be added and defined by

variables:

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Diagram 10: Definition of the text field for plotting the values of the eigenmodes

E.g. to plot the values of the angular velocity or frequency of the particular eigenmode

load cases, the internal variables _OMEGA and _FREQU can be used.

In order to retrieve the values of each load case it has to be referred to by its variable

name using the following syntax:

or

.

Also press F1 for a description and a list of all available internal variables within the

general window of the plot definition.

RM Bridge



9 Time History Analysis

The following input procedures can be followed when changing to the schedule variant

time_history in the corresponding example.

9.1 Preparation of the tables and variables

The data sets describing the time history events are set up in terms of tables and varia-

bles as described in chapter 4 of this document.

Three different real events with a probability of exceedance of 2% in 50 years are exam-

ined in this example. The respective tables are stored in the group Time_Hist with the

names T1_acc, T2_acc and T3_acc.

The amplitude (ordinate) is given in terms of acceleration values (m/s2). RM Bridge

provides an option for automatically converting a table using acceleration or velocity

values into a table using displacements values (see diagram below).

Diagram 11: Conversion of tables from acceleration to displacements

Using this option the three tables are converted into (additional) tables with the names

T1_dis, T2_dis and T3_dis (the names have to be given within the input window).

In the same way as shown in chapter 4.1.3 the tables have to be assigned to respective

variables in dependency of the time (t).

In regard to this one has to take care that the internal variable for the time (t) possibly is

not equal to zero at the time the time history calculation is performed (e.g. in case creep

calculation has been performed in advance). Therefore a new variable for the time his-

tory calculation tint is set up in terms of

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tint = t tstart,

where

t is the internal variable for the time (global time axis in the schedule) and

tstart the internal variable for the time at the point of the global time axis where the ac-

tion is started.

That means in that way at the begin of the action of the time history calculation tint = 0

(since t = tstart) and at the end of the time interval t of the action tint = t (since t = tstart + t).

Diagram 12: Definition of the variable tint

The definitions of the tables and variables are summarized below:

Name Type Expression Description

T1_acc Table - time history set1 in terms of acceleration



T1_dis Table - time history set1 in terms of displacements



var_T1_dis Variable T1_dis(tint) variable for the time history set T1



tint Variable t-tstart time to be used for time history analysis

t

tint = t = 40 tint = 0

tint = t tstart

t = tstart + t = 120 t = tstart = 80

t = 40

80

t2 = 30 t1 = 50

STG 1 STG 2 STG 3 STG 4

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9.2 Definition of the load cases and load sets

A load case to be used for the time history analysis needs to contain the masses of the

structure and the displacements of the time history event described by the variable. In

this example for each of the three particular events two load cases are set up in regard to

the two horizontal directions (x and z); each load case consists of one load set for the

self-weight, one for the superimposed dead loads and one for the displacements in the

particular direction.

9.2.1 Definition of the load sets for the masses

The definition of the load sets for the masses of the self-weight and superimposed dead

loads are defined as shown in chapter 2.1.

The load sets for the masses just needs to be defined once and can be used for all the six

load cases.

9.2.2 Definition of the load sets for the displacements

For each direction of each time history event one load set defining the element end dis-

placements of the soil springs in the particular direction is generated as shown below.

Create new load

sets for the time

history events

Schedule Name LS-T1_dis_x LS-T1_dis_z

Description displacements

event T1 for x-dir

displacements

event T1 for z-dir

Load Definition

Load Set Defi-

nition

Top table

Name LS-T2_dis_x LS-T2_dis_z LS-T3_dis_x LS-T3_dis_z

Description displacements

event T2 for x-dir

displacements

event T2 for z-dir

displacements

event T3 for x-dir

displacements

event T3 for z-dir

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Define load sets

for the for the time

history events for

each direction

Schedule Name LS-T1_dis_x LS-T1_dis_z

Loading Actions on Element End

Actions on Element End

Load Definition Type Element end

displacements

Element end

displacements

From 1100 1100

Load Set Defi-

nition To 1400 1400

Step 100 100

Bottom table Vx -1 0

Vy 0 0

Vz 0 -1

Rx 0 0

Ry 0 0

Rz 0 0

Global / Local Global Global Begin / End End End

Name LS-T2_dis_x LS-T2_dis_z LS-T3_dis_x LS-T3_dis_z

Loading Actions on Element End




Type Element end

displacements

Element end

displacements

Element end

displacements

Element end

displacements

From 1100 1100 1100 1100

To 1400 1400 1400 1400

Step 100 100 100 100

Vx -1 0 -1 0

Vy 0 0 0 0

Vz 0 -1 0 -1

Rx 0 0 0 0

Ry 0 0 0 0

Rz 0 0 0 0

Global / Local Global Global Global Global Begin / End End End End End

9.2.3 Definition of the load cases for the time history calculation

The prepared load sets are now assigned to the respective load cases to be used for the

time history calculation; each load case consists of the load set for the self-weight, the

load set of the SDL and the respective load set of the displacements. As can be seen in

the input below the load set for the displacements gets its constant factor set to zero, but

the variable factor is defined by the variable describing the time dependent loading of

the time history event which has to be entered in the respective input field.

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Create new load

cases for the time

history calculation

for each direction

Schedule Name LC-T1-x LC-T1-z

Description load case for event

T1 for x-dir

load case for event

T1 for z-dir

Load Definition

Load Case Def-

inition

Top table

Name LC-T2-x LC-T2-z LC-T3-x LC-T3-z

Description load case for event

T2 for x-dir

load case for event

T2 for z-dir

load case for event

T3 for x-dir

load case for event

T3 for z-dir

Assign load sets to

the load cases

Schedule Name LC-T1-x

Loading Load set

input

Load set input

Load set input

Load Definition Load Set LS-SW LS-SDL LS-

T1_dis_x

Const-Fac 1 1 0

Load Case Def-

inition Var-Fac 0 0 var_T1_dis

Bottom table

Name LC-T1-z LC-T2-x

Loading Load set

input

Load set input

Load set input

Load set input

Load set input

Load set input

Load Set LS-SW LS-SDL LS-

T1_dis_z LS-SW LS-SDL

LS-

T2_dis_x

Const-Fac 1 1 0 1 1 0

Var-Fac 0 0 var_T1_dis 0 0 var_T2_dis

Name LC-T2-z LC-T3-x

Loading Load set

input

Load set input

Load set input

Load set input

Load set input

Load set input


T2_dis_z LS-SW LS-SDL

LS-

T3_dis_x

Const-Fac 1 1 0 1 1 0

Var-Fac 0 0 var_T2_dis 0 0 var_T3_dis

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Name LC-T3-z

Loading Load set

input

Load set input

Load set input


T3_dis_z

Const-Fac 1 1 0

Var-Fac 0 0 var_T3_dis

9.3 Calculation of the Time History Events

The defined load cases describing the time dependent loadings are now evaluated using

the schedule action Tint. The maximum and minimum results are stored in envelopes,

which have to be initialized (SupInit) in advance.

Perform a time

history calculation

for the respective

time dependent

load cases

Schedule Action Calculation

(Dynamic)

Calculation (Dynamic)

Type TInt TInt

Stages Input-1

(Load Case) LC-T1-x LC-T1-z

Input-2

(Delta-t) 40.94 40.94


(RM-Set) T-x-Mz T-z-My

Output-1 T1-x.sup T1-z.sup

Bottom table Output-2 tint_T1-x-Mz tint_T1-z-My

Description

Perform time

history calcula-

tion for load

case LC-T1-x

Perform time

history calcula-

tion for load

case LC-T1-z

Action Calculation

(Dynamic)




Type TInt TInt TInt TInt

Input-1

(Load Case) LC-T2-x LC-T2-z LC-T3-x LC-T3-z

Input-2

(Delta-t) 40.94 40.94 40.94 40.94

Input-3

(RM-Set) T-x-Mz T-z-My T-x-Mz T-z-My

Output-1 T2-x.sup T2-z.sup T3-x.sup T3-z.sup

Output-2 tint_T2-x-Mz tint_T2-z-My tint_T3-x-Mz tint_T3-z-My

Description

Perform time

history calcula-

tion for load

case LC-T2-x

Perform time

history calcula-

tion for load

case LC-T2-z

Perform time

history calcula-

tion for load

case LC-T3-x

Perform time

history calcula-

tion for load

case LC-T3-z

The load case to be evaluated (Input-1) and the time period to be considered (Input-2)

have to be given. The results are stored in the given envelope (Output-1). Optionally an

RM-Set of the type Time integration (TINT) can be given (Input-3), within which

RM Bridge



specific result components at particular element positions for graphical output can be

defined. These results are as well printed in a list file (Output-2).

The particular envelopes of the evaluated load cases are then superposed exclusively

(SupOr) in one final envelope T-final.sup which thus contains the maximum and mini-

mum results deriving from the three events.

9.4 Definition of Damping

Differently from the response spectrum analysis where the damping is incorporated by a

constant value, in the time history analysis the Rayleigh damping is dependent on the

frequency. Design codes usually define a damping as a percentage of the critical damp-

ing not dependent on frequency and the Raleigh coefficients are usually not known.

Therefore RM Bridge provides a function to approximate this presumption with Ray-

leigh damping as accurate as possible by recalculating the coefficients from damping

ratios given for two relevant frequencies. In order to do that the user has to determine

the natural frequencies and to assign the given damping ratio to the two most important

ones. The assignment of these values is done in the Recalc pad in the submenu Dy-namic. When clicking on one of the buttons next to the Rayleigh coefficients Alpha or Beta a menu opens where two pairs of values (w1, xi1 and w2, xi2) describing the first

and second frequency value (w1, w2) with the corresponding predefined damping ratio

(xi1, xi2) have to be given. After confirming with Ok the Rayleigh coefficients Alpha

and Beta are evaluated (note that the input of the factors w1, w2, xi, xi2 is not stored).

In this example the natural modes 1 and 6 have been assumed to be the two relevant

frequencies derived from the list eigen.lst:

MASS PARTICIPATION FACTORS [%]

MODE phi*M*phi X Y Z SUM-X SUM-Y SUM-Z HERTZ

-------------------------------------------------------------------------

1 0.4550E+04 89.10 0.00 3.21 89.10 0.00 3.21 0.849

2 0.2055E+04 2.40 0.00 72.05 91.50 0.00 75.26 1.581

3 0.1016E+04 0.01 5.21 0.04 91.50 5.21 75.30 2.929

4 0.2175E+04 1.14 0.01 0.05 92.65 5.22 75.36 3.568

5 0.1288E+04 0.24 0.01 0.01 92.89 5.23 75.36 5.112

6 0.1349E+04 0.00 58.91 0.01 92.89 64.14 75.37 5.921

7 0.2050E+04 0.44 0.02 7.41 93.33 64.16 82.79 6.683

8 0.1079E+04 0.10 0.00 0.05 93.43 64.16 82.83 9.120

9 0.6110E+04 0.11 0.04 3.61 93.54 64.20 86.44 9.449

10 0.1840E+04 0.17 0.00 0.01 93.72 64.21 86.46 10.958

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Therefore the factors for evaluating the coefficients of the Rayleigh damping for a pre-

defined damping of 5% are approximated as follows:

Relevant Modes Frequency f [Hz] w (in terms of angular velocity [rad/s]) xi (damping)

1 0.85 w1 = 5 xi1 = 0.05

6 5.9 w2 = 35 xi2 = 0.05

Thus the evaluated Rayleigh coefficients used for the calculation are:

Alpha = 0.4375

Beta = 0.0025

1 General1.1 Scope1.2 Generating a new construction schedule

2 Definition of Masses2.1 Definition of load sets for self weight and SDL2.1.1 Generating the load sets for SW and SDL2.1.2 Defining the load set for SW2.1.3 Defining the load set for SDL

2.2 Definition of load case for masses2.2.1 Generating a load case for the masses2.2.2 Assigning the load sets to the load case

3 Calculation of Eigenvalues3.1 Generating a stage for the calculation of the eigenvalues3.2 Calculating the eigenvalues

4 Preparation of Response Spectrum4.1 Defining a response spectrum diagram4.1.1 Generating tables for horizontal and vertical response spectrum4.1.2 Defining the tables for horizontal and vertical response spectrum4.1.3 Assignment of the tables to variables (formulas)

4.2 Response Spectrum defined by formulas

5 Definition of the Earthquake Load5.1 Defining the earthquake events

6 Evaluation of the Response Spectrum6.1 Generating a stage for the evaluation of the response spectrum6.2 Evaluating the response spectrum6.2.1 Initializing envelopes for storing the resuts of the evaluation6.2.2 Evaluating the response spectra

7 Superposition of the Seismic Loads7.1 Generating a stage for the superposition of the seismic loads7.2 Superposing the seismic loads

8 Result Presentation8.1 Plotting the response spectra data8.2 Plotting of eigenmodes8.2.1 Definition of the DoPlot action8.2.2 Definition of the Plot File in the Plot Container

9 Time History Analysis9.1 Preparation of the tables and variables9.2 Definition of the load cases and load sets9.2.1 Definition of the load sets for the masses9.2.2 Definition of the load sets for the displacements9.2.3 Definition of the load cases for the time history calculation

9.3 Calculation of the Time History Events9.4 Definition of Damping

Documents

RM E Prestressing Basic Part3 Earthquake