NERA Manual

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Department of Civil Engineering

NERA

A Computer Program for

Nonlinear Earthquake site Response Analysesof Layered Soil Deposits

by

J. P. BARDET and T. TOBITA

April 2001


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Table of Contents

1. INTRODUCTION ......................................................................................................................... 1

2. MODELING SOIL RESPONSE DURING SHEAR CYCLES....................................................... 1

2.1 Viscoelastic Model.................................................................................................................. 1

2.2 Equivalent Linear Model......................................................................................................... 2

2.3 Nonlinear and Hysteretic Model ............................................................................................. 3

2.3.1 Energy dissipated during strain cycles ............................................................................ 5

3. ONE-DIMENSIONAL GROUND RESPONSE ANALYSIS .......................................................... 9

4. FINITE DIFFERENCE FORMULATION OF ONE-DIMENSIONAL SITE RESPONSE

ANALYSIS...................................................................................................................................... 11

4.1 Spatial and time discretizations............................................................................................ 11

4.2 Central difference algorithm ................................................................................................. 12

5. DESCRIPTION OF NERA ......................................................................................................... 14

5.1 System requirement, distribution files and download NERA................................................ 14

5.2 Installing and removing NERA ............................................................................................. 14

5.3 NERA commands................................................................................................................. 15

5.4 NERA worksheets ................................................................................................................ 16

5.4.1 Earthquake data ............................................................................................................17

5.4.2 Soil Profile...................................................................................................................... 18

5.4.3 Material stress-strain damping-strain curves................................................................. 20

5.4.4 Calculation ..................................................................................................................... 21

5.4.5 Output (Acceleration)..................................................................................................... 22

5.4.6 Output (Strain) ............................................................................................................... 23

5.4.7 Output (Ampli)................................................................................................................ 23

5.4.8 Output (Fourier) .............................................................................................................24

5.4.9 Output (Spectra) ............................................................................................................ 25

5.5 Running NERA ..................................................................................................................... 26

6. REFERENCES .......................................................................................................................... 27

7. APPENDIX A: SAMPLE PROBLEM..........................................................................................29

7.1 Definition of problem ............................................................................................................29

7.2 Results.................................................................................................................................. 33

8. APPENDIX B: COMPARISON OF NERA AND EERA RESULTS ............................................39


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1. INTRODUCTION

During past earthquakes, the ground motions on soil sites were found to be generally larger thanthose of nearby rock outcrops (e.g., Seed and Idriss, 1968). One of the first computer programsfor simulating soil site responses was SHAKE (Schnabel et al., 1972). Based on Kanai (1951),

Roesset and Whitman (1969), and Tsai and Housner (1970), SHAKE assumes that the cyclic soilbehavior can be simulated using an equivalent linear model (e.g., Idriss and Seed, 1968; Seedand Idriss, 1970; Kramer, 1996; Sugito, 1995; Idriss and Sun, 1992).

In 1998, the computer program EERA was developed starting from the same basic concepts asSHAKE (Bardet et al., 1998). EERA stands for Equivalent linear Earthquake Response Analysis.EERA implements the well-known concepts of equivalent linear earthquake site responseanalysis taking advantages of FORTRAN 90 and spreadsheet program Excel.

In 2001, the implementation principles used for EERA were applied to NERA, a nonlinear siteresponse analysis program based on the material model developed by Iwan (1967) and Mroz(1967). NERA stands for Nonlinear Earthquake Response Analysis and takes full advantages ofFORTRAN 90 and spreadsheet program Excel. Concepts similar to those in NERA have beenused by Joyner and Chen (1975); Prevost, (1989); and Lee and Finn (1978).

Following the introduction, the second section of this report reviews the material models used formodeling the soil behavior in one-dimensional ground response analysis during earthquakes. Thematerial models include the viscoelastic model, the equivalent linear model and the model of Iwanand Mroz. The third section describes the finite different formulation of one-dimensional groundresponse analysis. The fourth section describes how to use NERA. The appendices contain asample problem and compare NERA and EERA results.

2. MODELING SOIL RESPONSE DURING SHEAR CYCLES

2.1 Viscoelastic Model

As illustrated in Fig. 1, one of the simplest models for simulating the soil stress-strain responseduring earthquake loading is the viscoelastic Kelvin-Voigt model. The shear stress depends onthe shear strain and its rate as follows:

+=G (1)

where G is shear modulus and the viscosity. In the case of harmonic loadings with a circularfrequency , Eq. 1 becomes:

)()()( ** tGeGeiGet tititi ==+== (2)

where G*is the complex shear modulus; is the amplitude of shear stress; and is the amplitude

of shear strain. After introducing the critical damping ratio so that

= /2G (3)

the complex shear modulus G*becomes:

G G i G i*

( )= + = + 1 2 (4)


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G

G

Figure 1. Schematic representation of viscoelastic Kelvin-Voigt model.

2.2 Equivalent Linear Model

The equivalent linearapproach consists of modifying the Kelvin-Voigt model to account for sometypes of soil nonlinearities. The nonlinear and hysteretic stress-strain behavior of soils isapproximated during cyclic loadings as shown in Fig. 2. The equivalent linear shear modulus, G,

is taken as the secant shear modulus Gs, which depends on the shear strain amplitude . Asshown in Fig. 2a, Gsat the ends of symmetric strain-controlled cycles is:

c

c

sG

= (5)

where cand care the shear stress and strain amplitudes, respectively. The energy dissipatedWdduring a complete loading cycle is equal to the area generated by the stress-strain loop, i.e.:

=c

dWd

(6)

The maximum strain energy stored in the system is:

2

2

1

2

1cccs GW == (7)

The critical damping ratio can be expressed in terms of Wdand Wsas follows:

s

d

W

W

4= (8)

The equivalent linear damping ratio, , is the damping ratio that produces the same energy loss ina single cycle as the hysteresis stress-strain loop of the irreversible soil behavior.


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Strain ()

Stress()

Gmax Gsec

c

c 1

Shear strain (log scale)

Gsec,

Gsec

(a) (b)

Figure 2. Equivalent-linear model: (a) Hysteresis stress-strain curve; and (b) Variation ofsecant shear modulus and damping ratio with shear strain amplitude.

In site response analysis, the material behavior is generally specified as shown in Fig. 2b.

Examples of data for Gs-and -curves can be found in Hardin and Drnevitch (1970), Kramer(1996), Seed and Idriss (1970), Seed et al. (1986), Sun et al. (1988), and Vucetic and Dobry(1991).

2.3 Nonlinear and Hysteretic Model

As illustrated in Fig. 3, Iwan (1967) and Mroz (1967) proposed to model nonlinear stress-straincurves using a series of nmechanical elements, having different stiffness kiand sliding resistanceRi. Herafter, their model is referred to as the IM model. The sliders have increasing resistance(i.e., R1 < R2 <


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k1 k2

R1 RnRn-1Rn-2

kn-1 kn

Figure 3. Schematic representation of stress-strain model used by Iwan (1967) and Mroz

(1967).

Strain

Stress

2R1

2R2

2R1

R2

R1

A

BC

D

E

O

F

Strain

Stress

H1

H2

R1

R2

G/Gmax

O

A

B

C

Figure 4. Backbone curve (left) during loading and hysteretic stress-strain loop (right) of IMmodel during loading-unloading cycle.

As shown in Fig. 4, the stress-strain curve during a loading is referred to a backbone curve. Whenthe loading changes direction (i.e., unloading), the residual stress in slider i decreases; slider i

yields in unloading when its residual stress reaches -Ri , i.e., after the stress decreases -2Ri.Instead of yield stress, it is convenient to introduce the back stress I: slider iyields in loading andunloading when becomes equal to I+ Riand I- Ri, respectively. The IM model asumes thatparameters Ri are constant whereas the back stress I varies during loading processes. Asshown in Fig. 4, the cyclic stress-strain curves is hysteretic, and follows Masing similitude rule(Masing, 1926). Curve CDEF is obtained from curve OABC by a simitude with a factor of 2.

The stress-strain curves of the IM model can be calculated using the algorithm of Table 1. thisalgorithm returns an exact value of stress independently of the strain increment amplitude . Atfirst, the algorithm attempts to calculate the stress increment using the strain increment andmodulus H1. If +1+ R1(loading), then +is accepted; the stress is smaller that the yieldstress of slider 1. If +> 1+ R1, the strain increment was too large, and the stress +exceeded the yield stress of slider 1; the tangential modulus of the stress-strain response was H1

only for the stress increment =i+ Ri - and strain increment /H1. The algorithm is reappliedto slider 2, instead of slider 1, using the remaining strain increment /H1. The algorithm isrepeated for other sliders until +becomes smaller than the yield stress of slider j. Each time,the remaining strain increment referrred to as x in Table 1 becomes smaller. At this time, the


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backstresses of sliders 1 to j-1 are updated. The algorithm of Table 1 works for loading andunloading through the use of variable x, which is set to 1 for loading and 1 for unloading,respectively.

Table 1. Algorithm for stress calculation for given strain increment.

Given , , I,Ri, and Hifor i= 1,,n

x = if > 0 thenx=1 elsex= -1 loading or unloadingFor i= 1 to n

=Hi x trial stress incrementIf | + - i| Ri then inside slider i

+ Go to *

End if

=i+ x Ri - correct + update x x -/ Hi left over strain increment

Next i* If i> nthen i= n avoid n+1

If | - i| < Rior i= nthen i= i-1 strictly inside slider iForj= 1 to i

j= -xRj update jNextj

The nonlinear backbone curve of Fig. 4 can be described in terms a variation of secant shear

modulus Gwith shear strain , especially by ndata points, i.e., Gi-I , i= 1, , n. In this case, thetangential shear modulus Hiis related to the secant modulus Gias follows:

ii

iiiii

GGH

=+

++

1

11i= 2, , n-1 and 0=nH (11)

Assuming that the backstress iis initially equal to zero, Riis:

iii GR = i= 1, , n (12)

Equations 11 and 12 imply that the maximum shear resistance is Rn= Gnn, i.e., is specified bythe last point of the G-curve. When the G/Gmax-are specified, then Eqs. 11 and 12 become:

ii

iiiii

GGGH

=+

++

1

'

1

'

1max i= 2, , n-1 and iii GGR

'

max= i= 1, , n (13)

where max'

/GGG ii = .

2.3.1 Energy dissipated during strain cyclesA shown in Fig. 5, when the stress-strain curve follows Masing similitude rule (Masing, 1926), the

areas Ii and Ji corresponding to an unloading from +I to -I and an reloading from -I to +I,respectively, are four times greater then the area Ai under the stress-strain curve for a loading

from 0 to I. The areasAi, Iiand Jiare defined as:

= i

dAi

0

( ) iii AdI i

i

4==

, and ( ) iii AdJ

i

i

4=+=

(14)


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The dissipated energyid

W during a complete cycle of strain amplitude I, which is the area of the

hysteretic loop, is:

iiiiiiid AJIdddW i

i

i

i

i

i

484 =+=+==

i= 1, , n (15)

Stress

StrainAi

Ii

Ji

Strain

Strain

Stress

Stress

i

i

i

i

i

i

i

i

i

i

i

Figure 5. AreasAi, Ii, and Jiused for calculation of hysteretic loop of IM model duringloading-unloading cycle.

When the stress-strain curve is piecewise linear and generated by ndiscrete points (I, GiI), Aibecomes:

01 =A and ( )( )12

112

1

= += jj

i

j

jjjji GGA i= 2, , n (16)

and Eq. 15 becomes:

2

48 iiid GAW i = i= 1, , n (17)Since the maximum strain energy stored in the system is:

2

2

1iis GWi = (18)

The critical damping ratio iat shear strain ican be expressed:


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1=0 and

== 1

22

42

ii

i

is

d

iG

A

W

Wi

i= 2, ,n (19)

When the shear strain exceeds nthe IM model assumes that the shear stress is equal to the

shear strength Rn. In this case, the secant modulus G and critical damping ratio becomes:

nRG = and ( )

+= 1)(22

n

nnn

R

RAfor > n (20)

For very large shear strain, the secant modulus tends toward zero and the damping ratio tends

toward 2/, i.e.:

0G and

2 when (21)

Equation 19 implies that depends on the shape of the G/Gmax-curve, but is independent ofGmax. The IM model assumes that the hysteretic stress-strain loop follows Masing similitude. Its

material parameters (i.e., Hiand Ri, i= 1,..,n) are computed solely from the data points Gi-I , i= 1,, n,which characterizes the G-

curves. The IM model can be assigned the same G-

curves as

the linear equivalent model. However the damping ratio curves of the IM model are calculatedusing Eq. 19. They can not be defined independently as in the case of the linear equivalent

model. In summary, the IM model and the linear equivalent model can be assigned the same G-curve but in general have different damping ratio curves. Figures 6 and 7 show examples of

calculation of damping ratio from G/Gmax-curves, and comparison to the damping ratio used bylinear equivalent model in the case of clay and sand (Idriss, 1990).

Modulus for sand (Seed & Idriss 1970) - Upper Range and damping for sand (Idriss 1990) - (about LRng from SI 1970)

Strain (%) G/Gmax Strain (%) Damping (%) Area Ai

Calculated

Damping (%)

0.0001 1 0.0001 0.24 0.000000 0.00

0.0003 1 0.0003 0.42 0.000000 0.00

0.001 0.99 0.001 0.8 0.000000 0.00

0.003 0.96 0.003 1.4 0.000004 0.61

0.01 0.85 0.01 2.8 0.000044 2.53

0.03 0.64 0.03 5.1 0.000321 7.340.1 0.37 0.1 9.8 0.002288 15.08

0.3 0.18 0.3 15.5 0.011388 25.84

1 0.08 1 21 0.058288 29.11

3 0.05 3 25 0.288288 17.91

10 0.035 10 28 2.038288 10.49

100 0.0035 33.538288 58.34

0

0.2

0.4

0.6

0.8

1

0.0001 0.001 0.01 0.1 1 10 100

Shear Strain (%)

G/G

max

0

10

20

30

40

50

60

70

DampingRatio(%)

Shear Modulus

Damping Ratio

Calculated DampingRatio

Figure 6. First example of calculation of damping ratio from a G/Gmax-curve, and

comparison to a damping ratio used by linear equivalent model.


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Figure 7. Second example of calculation of damping ratio from a G/Gmax-curve, andcomparison to a damping ratio used by linear equivalent model.

Compared to the linear equivalent model, the IM model has no damping ratio at small strain, andits damping ratio may temporarily decrease for some strain range due to the relative variation ofAiand strain energy Wswith shear strain amplitude. As derived in Eq. 21, the damping ratio

increases again and tends toward 2/for large shear strain. Using Eq. 20, the first derivative of w.r.t.is:

2

4

n

nnn

R

AR

d

d = for > n (22)

which is always positive because Rnnis always larger thanAn. Equation 22 therefore explains

the re-increase of damping ratio for large strain. The damping ratio always increases with shearstrain once the material has failed at constant shear strength.

The IM model can simulate rigid-perfectly plastic material assuming that H1and n= 1, whichleads to the following dissipated energy Wd, maximum strain energy Wsand damping ratio forcycles of strain amplitude :

14RWd = , 12

1RWs = and

2

4==

s

d

W

W (23)

Modulus for clay (Seed and Sun, 1989) upper range and damping for clay (Idriss 1990)

Strain (%) G/Gmax Strain (%) Damping (%) Area Ai

Calculated

Damping (%)

0.0001 1 0.0001 0.24 0.000000 0.00

0.0003 1 0.0003 0.42 0.000000 0.00

0.001 1 0.001 0.8 0.000000 0.00

0.003 0.981 0.003 1.4 0.000004 0.34

0.01 0.941 0.01 2.8 0.000048 0.84

0.03 0.847 0.03 5.1 0.000396 2.460.1 0.656 0.1 9.8 0.003581 5.85

0.3 0.438 0.3 15.5 0.023281 11.53

1 0.238 1 21 0.152571 17.96

3 0.144 3.16 25 0.822571 17.15

10 0.11 10 28 6.184571 7.92

100 0.011 105.184571 58.09

0

0.2

0.4

0.6

0.8

1

0.0001 0.001 0.01 0.1 1 10 100

Shear Strain (%)

G/Gmax

0

10

20

30

40

50

60

70

DampingRatio

(%)

Shear Modulus

Damping Ratio

Calculated DampingRatio


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Equation 22 gives the upper bound of the damping ratio for the IM model, as the rigid perfectlyplastic model has the largest hysteretic loop.The IM model can also simulate elastic-perfectly plastic material by selecting n= 1, H1= Gmaxand

R1= max. When the response is elastic, = 0. The G-and damping curves become:

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