Comparison of Equivalent-Linear Site Response Analysis Software

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Tenth U.S. National Conference on Earthquake Engineering Frontiers of Earthquake Engineering July 21-25, 2014 10NCEE Anchorage, Alaska

COMPARISON OF EQUIVALENT-LINEAR SITE RESPONSE ANALYSIS SOFTWARE

S.J. Lasley1, R.A. Green2 and A. Rodriguez-Marek3

ABSTRACT

The objective of the study presented herein is to compare equivalent-linear site response analysis software. Ground motions recorded at soft soil sites can be drastically different from those recorded at rock sites. Site-specific response analyses provide an important way to predict surface ground motions from bedrock ground motions. Of all methods of site response analysis, the equivalent-linear method is, perhaps, the most popular. It has become an important method of site response analysis since first proposed by Schnabel et al. (1972) and has been shown to give a good approximation for many scenarios. In the years since the algorithm was released, several software programs that implement the equivalent-linear procedure have been written and made available to the engineering community. As part of ongoing research at Virginia Tech, a new equivalent-linear site response program, ShakeVT2, has been written in the Python programming language. In an effort to verify the implementation of the equivalent linear algorithm, extensive comparisons were made between ShakeVT2 and other equivalent-linear codes (i.e., SHAKE91, SHAKEVT, Strata, and DEEPSOIL). It was found that these implementations give similar results when supplied the same inputs. However, the choice of complex shear modulus and effective strain ratio have an impact on the results, especially for motions that contain a lot of energy at high frequencies. Additionally, it has been shown that when using DEEPSOIL, SHAKEVT, or SHAKE91 the discretization of the profile may obscure peaks when plotting the maximum shear strain, shear stress, or acceleration with depth. Finally, SHAKE91 and SHAKEVT suffer from a bug that gives incorrect results when the input motion file has an odd number of columns.

1Graduate Student Researcher, Charles E. Via, Jr. Dept. of Civil Engineering, Virginia Tech, Blacksburg, VA 24060 2 Professor, Charles E. Via, Jr. Dept. of Civil Engineering, Virginia Tech, Blacksburg, VA 24060 3 Associate Professor, Charles E. Via, Jr. Dept. of Civil Engineering, Virginia Tech, Blacksburg, VA 24060

Lasley SJ, Green RA, Rodriguez-Marek A. Comparison of Equivalent-Linear Site Response Analysis Software. Proceedings of the 10th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014.

Comparison of Equivalent-Linear Site Response Analysis Software

S.J. Lasley1, R.A. Green2 and A. Rodriguez-Marek3

ABSTRACT

Introduction

Soft soil profiles can greatly affect the amplitude of incoming earthquake motions. This fact was drastically demonstrated in various earthquakes, including the 1989 Loma Prieta earthquake [1]. Consequently, modern building codes (e.g. [2]) apply a factor to modify motions from rock profiles to those for soil profiles. Moreover, under some conditions, codes require that site- specific response analyses are performed. The equivalent-linear procedure is a popular method of site response analyses because, in part, it is easy to perform compared to other non-linear site response analyses.

Ongoing research at Virginia Tech on liquefaction assessment includes the calculation of dissipated energy from site response analyses of multiple soil profiles and earthquake motions. To aid the research, it was decided to re-implement the equivalent-linear algorithm [3][4] instead of using existing codes (e.g. SHAKE91, SHAKEVT, Strata, DEEPSOIL). This paper gives an

1 Graduate Student Researcher, Charles E. Via, Jr. Dept. of Civil Engineering, Virginia Tech, Blacksburg, VA 24061 2 Professor, Charles E. Via, Jr. Dept. of Civil Engineering, Virginia Tech, Blacksburg, VA 24061 3 Associate Professor, Charles E. Via, Jr. Dept. of Civil Engineering, Virginia Tech, Blacksburg, VA 24061

Equivalent-Linear Site Response Analysis

The 1-D equivalent-linear site response analysis first introduced by Schnabel et al. in 1972 [3][4] is a well-known and well-used method. It is used to estimate the transformation of earthquake motions as they propagate upward through a soil profile. It assumes the vertical propagation of shear waves from a uniform half-space through horizontal layers of a soil profile modeled as visco-elastic material having a constant damping ratio across all frequencies [5]. These calculations are performed in the frequency domain, greatly increasing the speed and numerical stability of the calculations. However, in order to model the non-linear response of soil in the frequency domain, an iterative procedure is required. For each layer in a soil profile, the equivalent-linear procedure approximates the non- linear stress-strain behavior using shear modulus reduction and damping curves. Fig. 1 shows examples of these curves for a range of overburden pressures. An iterative procedure is used to determine the degraded soil properties due to strain induced nonlinearities. For the first iteration of the algorithm, the response of the soil profile is calculated using small-strain values of shear modulus and damping.

Figure 1. Shear modulus and damping degradation curves [6].

Using the initial shear modulus and damping values, a shear strain time history response is calculated, wherein the soil properties are held constant from the beginning to the end of the earthquake motion. From this time history response of each layer, a representative shear strain, , is chosen. This representative strain is used to determine the degraded values of shear modulus and damping that are used in the next iteration of the analysis (see Fig. 1), and the process repeats until modulus and damping values reasonably converge to the specified degradation curves. Historically, various methods and values have been used to calculate the effective shear strain (). An effective strain value of 65% of the maximum shear strain is most commonly used today.

Implementation of the Equivalent-Linear Algorithm

Several sources were consulted ([3][4][5][7][8][9][10][11][12]) in order to code the core of the equivalent-linear algorithm. As mentioned, the equivalent-linear algorithm requires iteration until the assumed shear modulus and damping from the beginning of the iteration agree with the shear modulus and damping from the calculated effective strain. Both Kramer [7] and the EERA manual [8] give the equation for the time-domain shear strain within a layer for a harmonic motion. However, the frequency-domain equation for shear strain in a layer for non-harmonic motions is not explicitly defined in these references. To clarify, the equation for strain in the frequency domain is given by: ∗ ∗ ∗ ,= () exp −() exp− (1) where is the depth from the top of layer j; is an array of discrete angular frequencies defined by the Fourier transform and of length ; () and () are arrays of amplitudes of the ∗ up-going and down-going harmonic shear waves of layer j; is the complex number; and is the complex wave number of the layer (/∗). Similarly, the frequency-domain shear stress in a layer is given by:

∗ ∗ ∗ ∗ ,= () exp −() exp− (2)

∗ where is the complex shear modulus of the layer. Regarding Eq. 2, some software may use , the shear modulus of the layer, instead of ∗, the complex shear modulus. When this is done, the stress-strain curve (in time-domain) becomes a straight line with a slope of instead of a series of hysteresis loops. Using G in lieu of G* is inconsequential for computing acceleration time histories and response spectra, but use of G* is essential for computing dissipated energy, which is was the motivation for this study.

Comparison of Existing Software

SHAKE91

The original SHAKE program, written by Per B. Schnabel et al. 1972 [13], was the first well- known equivalent-linear code. It has undergone several updates, and SHAKE91 [5] is a direct descendant of the original. SHAKE91 is the de facto standard equivalent-linear code, and several subsequent codes either directly use its code or are based on it. SHAKEVT [14] added the option to output dissipated energies for each layer and increased the maximum length of the input motion, among other modifications. SHAKE2000 is a pre- and post-processor for SHAKE91 [15]. WESHAKE is a modification and expansion of the SHAKE program [16]. Although the EduSHAKE and ProSHAKE programs are a rewrite of the SHAKE code, they are based on SHAKE, as well [17].

SHAKE91 continues to be a standard because it is such a simple program. Its input and output are simple text files and its executable is a slight 500 kB. Where EduSHAKE and ProSHAKE require an XP emulator to run in Windows 7 (or newer versions), SHAKE91’s Fortran executable still runs on most platforms without problems. However, SHAKE91 lacks a graphical user interface, a batch mode, and a catalog of degradation curves. It also has a few limitations that additionally hamper its utility. For example, the recommended limit to the number of points in the input motion is 3800. This was fine when most motions were discretized at intervals of 0.01 seconds. However, many motions are now discretized at 0.005 second intervals, and thus, SHAKE91 will not run with these motions. Additionally, a maximum of 13 distinct shear modulus and damping degradation curves can be used, limiting the discretization of the soil profile to some extent.

Additionally, SHAKE91 has an implementation bug: if the input motion has an odd number of columns, the motion is interpreted incorrectly by the program [18]. For example, if the input motion has 5 columns (the standard PEER database format), every sixth data point of the ground motion is interpreted as a zero by SHAKE91. This causes a frequency shift in the motion and cuts off the tail of the motion. This is caused by the way SHAKE91 stores the input motion (line 171, MAIN.FOR). In an apparent effort to save memory, SHAKE91 splits the real input motion and stores it in the real and imaginary parts of a variable. For odd-columned input motion formats, the last data point of the column goes into the real part and zero is assumed for the imaginary part.

DEEPSOIL

DEEPSOIL was first developed in 1998 under the direction of Youssef M.A. Hashash at the University of Illinois at Urbana-Champaign [12]. The program has several features in addition to those of SHAKE91: it has a graphical user interface, it contains a catalog of shear modulus and damping degradation curves, it can output data and plots to Microsoft Excel format, and it has a batch mode for handling multiple input motions. Additionally, both time domain non-linear and frequency domain equivalent-linear analyses are supported; only the frequency domain equivalent-linear analyses were evaluated herein. One minor issue was encountered: the output shear stress time history is calculated using shear modulus (i.e., G) instead of the complex shear modulus (i.e., G*); thus, it cannot be used to calculate dissipated energy without additional post- processing.

Strata

Strata was written by Albert Kottke and Ellen M. Rathje at the University of Texas at Austin [10]. It has a graphical user interface, an extensive and customizable catalog of shear modulus and damping degradation curves, an option to use Random Vibration Theory (RVT), and options to randomize the profile properties. It also has an extensive set of output options (including dissipated energy), outputs to universal comma-separated value (.csv) format, and multiple input motion batch support. The source code of Strata is completely open and can be compiled on virtual any desktop operating system. (Windows binaries are available precompiled.) In contrast with other implementations of the equivalent-linear algorithm, Strata has an option to auto- discretize the layers of a soil profile, splitting each soil type into smaller sub-layers for analysis. For the comparisons in this paper, auto-discretization of the layers was turned off. Strata is currently under development, but the most recent version (399) is stable enough for most analyses.

ShakeVT2

ShakeVT2 was written in the Python language. Python is an object-oriented, interpreted language that excels at readability and portability of the code [19]. It was chosen for use in this research because it is easy to learn, free to use and distribute, and a large number of scientific libraries are readily available to extend the base language. At the time of this writing, a graphical user interface is not available for ShakeVT2. However, ShakeVT2 well performs the functions for which it was designed: calculating dissipated energy and the number of equivalent cycles for multiple profiles and motions using the low cycle fatigue implementation of the Palmgren-Miner hypothesis outlined in Green and Terri (2005) [20]. Because ShakeVT2 is written in a high-level language (compared to Strata’s C++ or SHAKE91’s Fortran), the core equivalent-linear algorithm of ShakeVT2 is much slower than those of Strata, DEEPSOIL, and SHAKE91. In most cases, however, the difference in the speed of the core algorithm is not an issue.

Comparison of Computation Results

In order to adequately compare the performance of the above-mentioned implementations of the equivalent-linear algorithm, a comparison of their results is presented herein. The Treasure Island E-W ground motion (TRI090) from the 1989 Loma Prieta earthquake is compared with the Yerba Buena Island E-W motion (YBI090) applied at the base of the Treasure Island soil profile (after [21]). Table 1 summarizes the profile.

Figure 2 shows the Fourier amplitude spectrum of the accelerations at the top of the profile after convergence of the equivalent-linear algorithms. The various algorithms show agreement within the accuracy of floating-point numbers for this profile and motion. Although not shown in this figure, discrepancies between SHAKE91/VT and the other software exist at amplitudes below about 10e-4 g/s. This is due to SHAKE’s lack of precision.

Table 1. Treasure Island soil profile used in the analysis (after [21]). Unit Weight Shear Wave Material Thickness (m) (kN/m3) Velocity (m/s) Sand (Fill) 2.9 19.7 270 Sand (Fill) 5.5 19.7 142.4 Sand (Fill) 5 19.7 175 Young Bay Mud 8.1 16.6 179 Young Bay Mud 8.2 16.6 179 Sand 5.2 20.4 320 Sand 6.6 20.4 320 Old Bay Mud 17.6 19.5 270 Old Bay Mud 17.6 19.5 270 Old Bay Mud 12.5 19.5 390 Weathered Rock 11.9 21 660 Bedrock - 22 1072

In this and the subsequent plots, SHAKE91 and SHAKEVT are plotted as the same line since they gave identical results for motions with less than 4096 data points. And, because the Yerba Buena Island motion had more than 4096 points, SHAKE91 could not be used and it is assumed that SHAKE91 and SHAKEVT would yield the same results. There are several different normalization formats commonly used for Fourier amplitude spectrum (FAS). The FAS format used by SHAKE91/VT is different from that used by Strata and DEEPSOIL. As a result, to compare the FAS computed by the different codes, the SHAKE91/VT FAS was normalized using Eq. 3:

= =0.5 Δ (3) where NFFT is the number of points used in the FFT, and Δt is the time step of the input motion.

Figure 2. Comparison of the calculated Fourier amplitude spectra. Note that Strata smooths the FAS but that functionality was removed to aid in comparison.

The maximum shear strain profiles compute by the various codes differed (see Figure 3). By requesting the maximum shear strain at 1 meter depth intervals, ShakeVT2 shows that the peak value of shear strain occurs at approximately 15 meters. In comparison, Strata also outputs the maximum shear strains at relatively small depth increments but the output values are not as high as those of ShakeVT2. However, this disparity relates to how Strata outputs its maximum shear strain profile and is not due to differences in the algorithms used to compute the maximum shear strains at a given depth. Strata calculates the maximum shear strains at the center of each layer and interpolates between these depths in plotting the shear strain profile. However, the shear strain time histories computed by ShakeVT2 and Strata at a given depth are identical. SHAKE91/VT and DEEPSOIL also output the maximum shear strains at only at one depth per layer, at the top and center of each layer, respectively. As a result, the soil profile would need to be discretized by more/thinner layers in order to improve the accuracy of the computed strain profiles.

Figure 3. Maximum shear strain profile for the YBI090 motion.

Comparing the Effects of the Method to Calculate Response Spectrum

Several methods are available to calculate the response spectra from an acceleration time history. DEEPSOIL [10] allows the user to select from one of three methods to compute response spectra: ‘Frequency Domain’, ‘Duhamel Integral’, or ‘Newmark’. These respectively refer to a method that uses the FAS to calculate the response spectra, Nigam and Jennings’ piecewise- exact method [22], and the Newmark β method. Using the same soil profile and motion, the 5% damping response spectra at the surface of the profile was computed for each option. Fig. 4 shows that for this profile and ground motion, the choice of method to calculate response spectra makes little difference.

Figure 4. Comparison of methods to compute the response spectrum.

Comparing the Effects of the Complex Shear Modulus

Historically, several different equations have been used to calculate the complex shear modulus, ∗. SHAKE91/VT (lines 421-422 of B1.FOR) calculates the complex shear modulus of a layer using Eq. 4: ∗ =1−2 +21− (4) where is the shear modulus (/) of the layer and is the viscous damping ratio of the layer (as a decimal). Strata [9] uses the complex shear modulus calculated from Eq. 5. Both DEEPSOIL and ShakeVT2 allow the user to choose Eqs. 4, 5, or 6, but default to Eq. 6. All equations give similar results when damping is small. ∗ =(1− +2) (5)

∗ =(1+2) (6)

Fig. 5 illustrates how the choice of complex shear modulus affects the FAS at the surface and the maximum acceleration profile. The FASs are very similar below a frequency of about 0.3 Hz, but the values diverge above that frequency. This translates into a difference in maximum acceleration at several depths in this soil profile.

Comparison of Effective Strain Ratios

All of the software mentioned in this paper allow the user to choose a ratio of effective shear strain to the maximum shear strain. The default value in Strata, DEEPSOIL, and ShakeVT2 is 0.65. The SHAKE91 manual [5] suggests using values between 0.4 and 0.75 “depending on the input motion and which magnitude earthquake it is intended to represent.” Using ShakeVT2 and values of 0.5, 0.65, and 0.8 for the effective strain ratio, the response of the soil profile was calculated. Fig. 6 shows the resulting Fourier amplitude spectra and maximum acceleration profiles. Once again, the Fourier amplitude spectra diverge most at high frequencies and the maximum acceleration profiles are significantly different at several depths. This is not a surprise because effective strain ratio directly affects the stiffness and damping of a profile, and, thus, its capacity to amplify the incoming ground motion.

Conclusions

The equivalent-linear procedure, since its introduction with SHAKE in 1972, continues to be a useful method to estimate site response. Five implementations of the equivalent-linear algorithm have been compared for a single profile and ground motion. It has been shown that these implementations give similar results when supplied the same inputs. However, the choice of complex shear modulus and effective strain ratio have an impact on the results, especially for motions that contain a lot of energy at high frequencies. Additionally, it has been shown that when using DEEPSOIL, SHAKEVT, or SHAKE91 the discretization of the profile may obscure peaks when plotting the maximum shear strain with depth. Finally, SHAKE91 and SHAKEVT suffer from an implementation bug that gives incorrect results when the input motion file has an odd number of columns.

Figure 5. Differences in the Fourier amplitude spectrum and the max acceleration profile caused by using different equations for shear modulus.

Figure 6. Fourier amplitude spectra and maximum acceleration profiles as affected by the effective shear strain ratio.

Acknowledgments

This research is partially funded by National Science Foundation (NSF) grants CMMI-1030564 and CMMI-1306261. This support is gratefully acknowledged. However, any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

References

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