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 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. 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 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. overview and comparison of the existing implementations of the equivalent linear algorithm. 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