Equations for iteration in one-dimensional ground

Department of Civil Engineering

NERA program can be found and downloaded from the following website. https://sites.google.com/site/tt60898/home/software

NERA A Computer Program for Nonlinear Earthquake site Response Analyses of 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 than those of nearby rock outcrops (e.g., Seed and Idriss, 1968). One of the first computer programs for 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 soil behavior can be simulated using an equivalent linear model (e.g., Idriss and Seed, 1968; Seed and Idriss, 1970; Kramer, 1996; Sugito, 1995; Idriss and Sun, 1992). In 1998, the computer program EERA was developed starting from the same basic concepts as SHAKE (Bardet et al., 1998). EERA stands for Equivalent linear Earthquake Response Analysis. EERA implements the well-known concepts of equivalent linear earthquake site response analysis taking advantages of FORTRAN 90 and spreadsheet program Excel. In 2001, the implementation principles used for EERA were applied to NERA, a nonlinear site response 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 of FORTRAN 90 and spreadsheet program Excel. Concepts similar to those in NERA have been used 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 for modeling the soil behavior in one-dimensional ground response analysis during earthquakes. The material models include the viscoelastic model, the equivalent linear model and the model of Iwan and Mroz. The third section describes the finite different formulation of one-dimensional ground response analysis. The fourth section describes how to use NERA. The appendices contain a sample 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 response during earthquake loading is the viscoelastic Kelvin-Voigt model. The shear stress τ depends on the shear strain γ and its rate γ as follows:

τ = G γ + η γ

(1)

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

τ (t ) = Σe iωt = (G + iωη )Γe iωt = G * Γe iωt = G *γ (t )

(2)

*

where G is the complex shear modulus; Σ is the amplitude of shear stress; and Γ is the amplitude of shear strain. After introducing the critical damping ratio ξ so that

ξ = ωη/2G

(3) *

the complex shear modulus G becomes:

G * = G + iωη = G (1 + 2iξ)

(4)

-1-

γγ

τ

Gγ

G

ηγ

η Figure 1.

Schematic representation of viscoelastic Kelvin-Voigt model.

2.2 Equivalent Linear Model The equivalent linear approach consists of modifying the Kelvin-Voigt model to account for some types of soil nonlinearities. The nonlinear and hysteretic stress-strain behavior of soils is approximated 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 γ. As shown in Fig. 2a, Gs at the ends of symmetric strain-controlled cycles is:

Gs =

τc γc

(5)

where τc and γc are the shear stress and strain amplitudes, respectively. The energy dissipated Wd during a complete loading cycle is equal to the area generated by the stress-strain loop, i.e.:

Wd = ∫ τ dγ

(6)

τc

The maximum strain energy stored in the system is:

1 1 Ws = τ cγ c = Gγ c2 2 2

(7)

The critical damping ratio ξ can be expressed in terms of Wd and Ws as follows:

ξ=

Wd 4πWs

(8)

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

-2-

Stress (τ)

Gsec Gmax

Gsec

τc

γc

Gsec , ξ

1

Strain (γ)

ξ Shear strain (log scale) Figure 2.

(a) (b) Equivalent-linear model: (a) Hysteresis stress-strain curve; and (b) Variation of secant 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-γ and ξ-γ 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-strain curves using a series of n mechanical elements, having different stiffness ki and sliding resistance Ri. Herafter, their model is referred to as the IM model. The sliders have increasing resistance (i.e., R1 < R2 < … < Rn). Initially the residual stresses in all sliders are equal to zero. During a monotonic loading, slider i yields when the shear stress τ reaches Ri. After having yielded, slider i retains a positive residual stress equal to Ri. As shown in Fig. 4, the stress-strain curve generated by the IM model for two sliders (i.e, n = 2) is piecewise linear, whereas the corresponding slope and tangential modulus H varies in steps. In the case of an IM model with n sliders, the stress increment dτ and strain increment dγ are related through:

dτ =H dγ

(9)

where the tangential modulus H is:

 H1 = k1  H = k −1 + k −1 −1 1 2  2  H = −1 −1 −1 −1  H n −1 = k1 + k2 + ... + kn −1  H n = k1−1 + k2−1 + ... + kn−−11 + kn−1  0

(

(

if 0 ≤ τ < R1 if R1 ≤ τ < R2

)

(

)

)

−1

if Rn − 2 ≤ τ < Rn −1 if Rn −1 ≤ τ < Rn if τ = Rn

-3-

(10)

k1

k2

kn-1

kn

R1

Rn-2

Rn-1

τ Rn

γ Figure 3.

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

B

C

Stress

C R2

A H1

G/Gmax

R2

A

R1

R1

H2 Strain

2R2

Stress

O

2R1 O D

2R1

F

E Strain

Figure 4.

Backbone curve (left) during loading and hysteretic stress-strain loop (right) of IM model during loading-unloading cycle.

As shown in Fig. 4, the stress-strain curve during a loading is referred to a backbone curve. When the 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 τ decreases -2 Ri. Instead of yield stress, it is convenient to introduce the back stress αI: slider i yields in loading and unloading when τ becomes equal to αI + Ri and αI - Ri, respectively. The IM model asumes that parameters Ri are constant whereas the back stress αI varies during loading processes. As shown 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. this algorithm returns an exact value of stress τ independently of the strain increment amplitude ∆γ. At first, the algorithm attempts to calculate the stress increment ∆τ using the strain increment ∆γ and modulus H1. If τ +∆τ ≤ α1 + R1 (loading), then τ +∆τ is accepted; the stress is smaller that the yield stress of slider 1. If τ +∆τ > α1 + R1, the strain increment ∆γ was too large, and the stress τ +∆τ exceeded the yield stress of slider 1; the tangential modulus of the stress-strain response was H1 only for the stress increment ∆τ = αi + Ri - τ and strain increment ∆τ/H1. The algorithm is reapplied to slider 2, instead of slider 1, using the remaining strain increment ∆γ − ∆τ/H1. The algorithm is repeated for other sliders until τ +∆τ 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 -4-

backstresses of sliders 1 to j-1 are updated. The algorithm of Table 1 works for loading and unloading 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 τ , ∆γ, αI, Ri, and Hi for i = 1,…,n ∆x = ∆γ if ∆γ > 0 then x =1 else x = -1 loading or unloading For i = 1 to n ∆τ =Hi ∆x ∆τ trial stress increment If |τ + ∆τ - αi| ≤ Ri then τ inside slider i τ ← τ + ∆τ Go to * End if ∆τ = αi + x Ri - τ correct ∆τ τ ← τ + ∆τ update τ ∆x ← ∆x - ∆τ / Hi left over strain increment Next i If i > n then i = n avoid n+1 If |τ - αi| < Ri or i = n then i = i -1 τ strictly inside slider i For j = 1 to i αj = τ - x Rj update αj Next j

*

The nonlinear backbone curve of Fig. 4 can be described in terms a variation of secant shear modulus G with shear strain γ, especially by n data points, i.e., Gi-γI , i = 1, …, n. In this case, the tangential shear modulus Hi is related to the secant modulus Gi as follows:

Hi =

Gi +1γ i +1 − Gi γ i i = 2, …, n-1 and H n = 0 γ i +1 − γ i

(11)

Assuming that the backstress αi is initially equal to zero, Ri is:

Ri = Giγ i i = 1, …, n

(12)

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

H i = Gmax where

Gi'+1γ i +1 − Gi'γ i ' i = 2, …, n-1 and Ri = Gmax Giγ i i = 1, …, n γ i +1 − γ i

(13)

Gi' = Gi / Gmax .

2.3.1 Energy dissipated during strain cycles A 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 +γI to -γI and an reloading from -γI to +γI, respectively, are four times greater then the area Ai under the stress-strain curve for a loading from 0 to γI. The areas Ai, Ii and Ji are defined as: γi

Ai = ∫ τ dγ 0

Ii = ∫

−γ i

γi

(τ i − τ )dγ

= −4 Ai , and J i = ∫

γi

−γ i

-5-

(τ i + τ )dγ

= 4 Ai

(14)

The dissipated energy

Wd i during a complete cycle of strain amplitude γI, which is the area of the

hysteretic loop, is: −γ i

γi

Wd i = ∫ τ dγ = ∫ τ dγ + ∫ τ dγ = − I i + J i − 4τ i γ i = 8 Ai − 4τ i γ i i = 1, …, n γi

−γ i

Stress

γi

τi

Ai Stress

−γi

(15)

Strain γi

τi

Ii

Strain

−γi γi

Stress

−τi τi

Strain

−γi γi

Ji −τi Figure 5.

Areas Ai, Ii, and Ji used for calculation of hysteretic loop of IM model during loading-unloading cycle.

When the stress-strain curve is piecewise linear and generated by n discrete points (γI, GiγI), Ai becomes:

A1 = 0 and Ai =

1 i ∑ (G jγ j + G j −1γ j −1 )(γ j − γ j −1 ) i = 2, …, n 2 j =2

(16)

and Eq. 15 becomes:

Wd i = 8 Ai − 4Giγ i2 i = 1, …, n

(17)

Since the maximum strain energy stored in the system is:

1 Wsi = Giγ i2 2

(18)

The critical damping ratio ξi at shear strain γi can be expressed:

-6-

ξ1 =0 and ξ i =

Wd i 4πWs i

=

 2  2 Ai  − 1 i = 2, …,n 2 π  Giγ i 

(19)

When the shear strain γ exceeds γn the 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 ξ becomes:

G=

Rn 2  2(An + Rn (γ − γ n ) )  − 1 for γ > γn and ξ =  γ π Rnγ 

(20)

For very large shear strain, the secant modulus tends toward zero and the damping ratio tends toward 2/π, i.e.:

G → 0 and ξ →

2 when γ → ∞ π

(21)

Equation 19 implies that ξ depends on the shape of the G/Gmax-γ curve, but is independent of Gmax. The IM model assumes that the hysteretic stress-strain loop follows Masing similitude. Its material parameters (i.e., Hi and Ri, i = 1,..,n) are computed solely from the data points Gi-γI , i = 1, …, n, which characterizes the G-γ curves. The IM model can be assigned the same G-γ curves as the linear equivalent model. However the damping ratio curves of the IM model are calculated using 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-γ curve but in general have different damping ratio curves. Figures 6 and 7 show examples of calculation of damping ratio from G/Gmax-γ curves, and comparison to the damping ratio used by linear 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) Calculated Damping (%) Area Ai G/Gmax Strain (%) Damping (%) Strain (%) 1 1 0.99 0.96 0.85 0.64 0.37 0.18 0.08 0.05 0.035 0.0035

0.0001 0.0003 0.001 0.003 0.01 0.03 0.1 0.3 1 3 10

0.24 0.42 0.8 1.4 2.8 5.1 9.8 15.5 21 25 28

0.000000 0.000000 0.000000 0.000004 0.000044 0.000321 0.002288 0.011388 0.058288 0.288288 2.038288 33.538288

1

70 60

0.8

50

G/Gmax

Shear Modulus 0.6

40

Damping Ratio 0.4

30

Calculated Damping Ratio

20

0.2 0 0.0001

0.00 0.00 0.00 0.61 2.53 7.34 15.08 25.84 29.11 17.91 10.49 58.34

Damping Ratio (%)

0.0001 0.0003 0.001 0.003 0.01 0.03 0.1 0.3 1 3 10 100

10

0.001

0.01

0.1

1

10

0 100

Shear Strain (%)

Figure 6.

First example of calculation of damping ratio from a G/Gmax-γ curve, and comparison to a damping ratio used by linear equivalent model. -7-

Modulus for clay (Seed and Sun, 1989) upper range and damping for clay (Idriss 1990) G/Gmax 1 1 1 0.981 0.941 0.847 0.656 0.438 0.238 0.144 0.11 0.011

Strain (%) 0.0001 0.0003 0.001 0.003 0.01 0.03 0.1 0.3 1 3.16 10

Damping (%) 0.24 0.42 0.8 1.4 2.8 5.1 9.8 15.5 21 25 28

Calculated Damping (%) Area Ai 0.000000 0.00 0.000000 0.00 0.000000 0.00 0.000004 0.34 0.000048 0.84 0.000396 2.46 0.003581 5.85 0.023281 11.53 0.152571 17.96 0.822571 17.15 6.184571 7.92 105.184571 58.09

70

1

60

0.8

G/Gmax

50 0.6

Shear Modulus 40 Damping Ratio

0.4

30

Calculated Damping Ratio

20

0.2 0 0.0001

Damping Ratio (%)

Strain (%) 0.0001 0.0003 0.001 0.003 0.01 0.03 0.1 0.3 1 3 10 100

10

0.001

0.01

0.1

1

10

0 100

Shear Strain (%)

Figure 7.

Second example of calculation of damping ratio from a G/Gmax-γ curve, and comparison 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, and its damping ratio may temporarily decrease for some strain range due to the relative variation of Ai and strain energy Ws with shear strain amplitude. As derived in Eq. 21, the damping ratio increases again and tends toward 2/π for large shear strain. Using Eq. 20, the first derivative of ξ w.r.t. γ is:

dξ 4 Rnγ n − An for γ > γn = dγ π Rnγ 2

(22)

which is always positive because Rnγn is always larger than An. Equation 22 therefore explains the re-increase of damping ratio for large strain. The damping ratio always increases with shear strain once the material has failed at constant shear strength. The IM model can simulate rigid-perfectly plastic material assuming that H1 →∞ and n = 1, which leads to the following dissipated energy Wd, maximum strain energy Ws and damping ratio ξ for cycles of strain amplitude γ:

1 Wd = 4γR1 , Ws = γR1 and 2

ξ=

Wd 2 = 4πWs π -8-

(23)

Equation 22 gives the upper bound of the damping ratio for the IM model, as the rigid perfectly plastic model has the largest hysteretic loop. The IM model can also simulate elastic-perfectly plastic material by selecting n = 1, H1 = Gmax and R1 = τmax. When the response is elastic, ξ = 0. The G-γ and damping curves become:

G Gmax

 1  =  R1  H1γ

when γ < R1 / H1 when γ ≥ R1 / H1

and

0    R ξ = 2 1 − 1 π  H1γ 

when γ < R1 / H1  (24)  when γ ≥ R1 / H1 

The damping ratio ξ is initially zero when γ