# Relation to Equivalent-Linear Methods

The “equivalent-linear” method is common in geotechnical earthquake engineering for modeling wave transmission in layered sites and dynamic soil-structure interaction. Because this method is widely used and the fully nonlinear method embodied in FLAC3D is still not, it is worth pointing out some of the differences between these two methods.

In the equivalent-linear method (Seed and Idriss 1969), a linear analysis is performed, with some initial values assumed for damping ratio and shear modulus in the various regions of the model. The maximum cyclic shear strain is recorded for each element and used to determine new values for damping and modulus, by reference to laboratory-derived curves that relate damping ratio and secant modulus to amplitude of cycling shear strain. Some empirical scaling factor is usually used when relating laboratory strains to model strains. The new values of damping ratio and shear modulus are then used in a new numerical analysis of the model. The whole process is repeated several times, until there are no further changes in properties. At this point, it is said that “strain-compatible” values of damping and modulus have been found, and the simulation using these values is representative of the response of the real site.

In contrast, only one run is done with a fully nonlinear method (apart from parameter studies, which are done with both methods), since nonlinearity in the stress-strain law is followed directly by each element as the solution marches on in time. Provided that an appropriate nonlinear law is used, the dependence of damping and apparent modulus on strain level are automatically modeled.

Both methods have their strengths and weaknesses. The equivalent-linear method takes drastic liberties with physics, but is user-friendly and accepts laboratory results from cyclic tests directly. The fully nonlinear method correctly represents the physics, but demands more user involvement and needs a comprehensive stress-strain model in order to reproduce some of the more subtle dynamic phenomena. Important characteristics of the two methods are examined in the next two subsections.

FLAC3D contains an optional form of damping (called hysteretic damping) that incorporates strain-dependent damping ratio and secant modulus functions, allowing direct comparisons between the equivalent-linear method and the fully nonlinear method. This form of damping is described in Mechanical Damping and Material Response.

There is a comparison between FLAC3D and SHAKE (a one-dimensional equivalent-linear program — Schnabel, Lysmer and Seed 1972) in a Layered Linear-Elastic Soil Deposit and in a Layered Nonlinear-Elastic Soil Deposit.

## Characteristics of Equivalent-Linear Method

The equivalent-linear method exhibits the following characteristics:

1. The method uses linear properties for each element that remain constant throughout the history of shaking, and are estimated from the mean level of dynamic motion. During quiet periods in the excitation history, elements will be overdamped and too soft; during strong shaking, elements will be underdamped and too stiff. However, there is a spatial variation in properties that corresponds to different levels of motion at different locations.

2. The interference and mixing phenomena that occur between different frequency components in a nonlinear material are missing from an equivalent-linear analysis.

3. The method does not directly provide information on irreversible displacements and the permanent changes that accompany liquefaction, because only oscillatory motion is modeled. These effects may be estimated empirically, however.

4. It is commonly accepted that, during plastic flow, the strain-increment tensor is related to some function of the stress tensor, giving rise to the “flow rule” in plasticity theory. However, elasticity theory (as used by the equivalent-linear method) relates the strain tensor (not increments) to the stress tensor. Plastic yielding, therefore, is modeled somewhat inappropriately.

5. The material constitutive model is built into the method; it consists of a stress-strain curve in the shape of an ellipse (see Cundall 1976). Although this pre-choice relieves the user of the need to make any decisions, the flexibility to substitute alternative shapes is removed. However, the effects of a different shape to the curve are partially allowed for by the iteration procedure used in the method. It should be pointed out that a frequency-independent hysteresis curve in the form of an ellipse is physically impossible, since the continuous change in slope prior to reversal implies preknowledge (and rate information is not available to the model because the model is defined as being rate-independent).

6. In the case where both shear and compressional waves are propagated through a site, the equivalent-linear method typically treats these motions independently. Therefore, no interaction is allowed between the two components of motion.

7. Equivalent linear methods cannot be formulated in terms of effective stresses, to allow the generation and dissipation of pore pressures during and following earthquake shaking.

## Characteristics of Fully Nonlinear Method

The following characteristics of the fully nonlinear method should be compared to the corresponding points listed in Characteristics of Equivalent-Linear Method.

1. The method follows any prescribed nonlinear constitutive relation. If a hysteretic-type model is used and no extra damping is specified, then the damping and tangent modulus are appropriate to the level of excitation at each point in time and space, since these parameters are embodied in the constitutive model. By default, if Rayleigh or local damping is used, the associated damping coefficients remain constant throughout shaking and throughout the grid. Spatially varying damping can also be specified. Consult Mechanical Damping and Material Response for more details on damping.

2. Using a nonlinear material law, interference and mixing of different frequency components occur naturally.

3. Irreversible displacements and other permanent changes are modeled automatically.

4. A proper plasticity formulation is used in all the built-in models, whereby plastic strain increments are related to stresses.

5. The effects of using different constitutive models may be studied easily.

6. Both shear and compressional waves are propagated together in a single simulation, and the material responds to the combined effect of both components. For strong motion, the coupling effect can be very important. For example, normal stress may be reduced dynamically, thus causing the shearing strength to be reduced in a frictional material.

7. The formulation for the nonlinear method can be written in terms of effective stresses. Consequently, the generation and dissipation of pore pressures during and following shaking can be modeled.

Although the method follows any stress-strain relation in a realistic way, it turns out that the results are quite sensitive to seemingly small details in the assumed constitutive model (see Cundall 1976 and Dames and Moore and SAI 1978). The various nonlinear models built into FLAC3D are intended primarily for use in quasi-static loading, or in dynamic situations where the response is mainly monotonic (e.g., extensive plastic flow caused by seismic excitation). A good model for dynamic soil/structure interaction would capture the hysteresis curves and energy-absorbing characteristics of real soil. In particular, energy should be absorbed from each component of a complex waveform composed of many component frequencies. (In many models, high frequencies remain undamped in the presence of a low frequency.) It is possible to add additional damping into the existing FLAC3D constitutive models in order to simulate the inelastic cyclic behavior. This procedure is described in Integration of Damping Schemes and Nonlinear Material Models for Geo-materials.

A comprehensive model for dynamic soil behavior may not yet exist. A review of current models is provided in Mechanical Damping and Material Response. Also, the user is free to experiment with candidate models by writing a model in C++ and loading it as a DLL (dynamic link library) file. (See Writing New Constitutive Models.)

It is possible to simulate cyclic laboratory tests on the new model and derive modulus and damping curves that may be compared with those from a real target material. The model parameters may then be adjusted until the two sets of curves match. This approach is discussed in Hysteretic Damping Formulation, Implementation and Calibration. Even standard elastic/plastic models (e.g., Mohr-Coulomb) can produce such curves. An example is shown in Integration of Damping Schemes and Nonlinear Material Models for Geo-materials.

## Applications of the Fully Nonlinear Method in Dynamic Analysis

The standard practice for dynamic analysis of earth structures, and especially analyses dealing with liquefaction, is based primarily upon the equivalent-linear method. The nonlinear numerical method has not been applied as often in practical design. However, as more emphasis is placed on making reliable prediction of permanent deformations and liquefaction-induced damage of earth structures, practical applications with nonlinear numerical codes have increased. Byrne et al. (2006) provide an overview of the different methods used for liquefaction assessment and discuss the benefit of the nonlinear numerical method over the equivalent-linear method for different practical applications.

There are several publications describing applications of nonlinear numerical models for analysis and design of earth structures subjected to seismic loading. [1] Many of the publications describing nonlinear numerical models pertain to back-analyses of geotechnical case histories that recorded large permanent ground deformations and failures of earth dams. These studies revisit analyses previously performed with equivalent-linear models. The response of the Upper and Lower San Fernando Dams to the 1971 San Fernando earthquake is one of the most commonly cited case histories. See Beaty and Byrne (2001) for a review of the observed response of both dams and an assessment of the key parameters affecting the response. An important observation from this case history is that although the characteristics of the dams were similar, the earthquake-induced responses were quite different. While the Upper San Fernando Dam experienced large lateral displacements of approximately 2 meters, a flow slide occurred at the upstream face and crest of the Lower San Fernando Dam some 20 to 30 seconds after the earthquake, nearly resulting in a catastrophic failure. Beaty and Byrne (2000) describe nonlinear numerical analyses of both dams using two-dimensional FLAC, incorporating a liquefaction constitutive model based on a total stress procedure. The analyses directly consider the triggering of liquefaction and post-liquefaction response of the dam material. Beaty and Byrne (2000) conclude that the total stress approach is a logical extension of the equivalent-linear method because it incorporates both liquefaction triggering and residual strength charts in the approach. The approach calculates progressive liquefaction-induced ground deformations that compare reasonably well with observed response, especially for the Upper San Fernando Dam. However, excess pore pressures are not computed directly in the total stress approach, and Beaty and Byrne (2000) state that an effective stress analysis is warranted to investigate the response of the Lower San Fernando Dam properly.

Dawson et al. (2001) present a back-analysis of the Lower San Fernando Dam based upon an effective stress analysis with FLAC and a semi-empirical constitutive model. The constitutive model is described as a “decoupled” effective stress model, because it generates pore pressure directly in response to the number of shear stress cycles required to trigger liquefaction. Pore pressures are generated incrementally in relation to the cyclic strength of the material as defined by a cyclic strength curve. The same modeling approach is also applied to a back-analysis of the Upper San Fernando Dam as described in Inel et al. (1993).

The delayed failure observed in the Mochikoshi tailing dam failure (which occurred in 1978 in Izu-Ohshim-Kinkai, Japan, as a result of a magnitude M7 earthquake followed by a magnitude M5.8 aftershock) is similar to the failure observed in the Upper San Fernando Dam. Two dams failed: Dam No. 1 failed during the main shaking, and Dam No. 2 failed approximately 24 hours after the main shock. Byrne and Seid-Karbasi (2003) suggest that the delayed failure of Dam No. 2 may be related to the low permeability silt layers contained within the sands of the tailings dam. These layers could impede vertical drainage of excess pore pressures and greatly reduce stability because they cause a water bubble to develop beneath the layers. Byrne and Seid-Karbasi performed coupled, nonlinear effective stress analyses to evaluate the excess pore pressures and deformations that develop during the earthquake and help assess the suggested failure mode.

Back-analyses of full-scale case histories are subject to many uncertainties with respect to material behavior and input motions, making it difficult to verify nonlinear numerical analyses. Confidence in the accuracy of the nonlinear seismic deformation analysis is primarily subject to the uncertainty related to the understanding of liquefaction. Mitchell (2008) lists four difficulties that contribute to this uncertainty:

Difficulties in the constitutive modeling of liquefiable soils, in estimating the extent of liquefaction, in determining the time at which liquefaction is triggered during shaking and in estimating the post-liquefaction residual strength

Centrifuge model tests are commonly used to attempt to address these difficulties and permit verification of nonlinear numerical models. The VELACS (Verification of Liquefaction Analysis by Centrifuge Studies) project (Arulmoli et al. 1992) is one example that has provided experimental data for use in the verification of nonlinear liquefaction analysis. Comparisons are typically made in terms of excess pore pressure, acceleration, and displacement time histories. Publications by Inel et al. (1993), Byrne et al. (2003), Andrianopoulos et al. (2006b), and Kutter et al. (2008) describe different constitutive models that have been tested in FLAC by comparison to results from centrifuge tests.

Nonlinear numerical analyses are presently being applied to provide seismic vulnerability assessments and evaluate remedial measures for dam rehabilitation projects. The application of the decoupled effective stress model to assess liquefaction potential of the Pleasant Valley Dam in California is described by Roth et al. (1991). Deformation analyses using this constitutive model helped determine a safe operating level for the reservoir and supported the renewal of Pleasant Valley Dam’s operating license for the lower pool level. Seismic retrofitting of the Success Dam in Southern California is being guided by a combination of deformation analysis methods, ranging from simplified procedures based on the equivalent-linear method and limit equilibrium analyses to decoupled and fully coupled effective-stress analyses with FLAC. Perlea et al. (2008) provide an overview of the analyses and remediation design. Salah-Mars et al. (2008) report the use of nonlinear deformation analyses with FLAC as part of a probabilistic seismic-hazard analysis to estimate the seismic hazard of the Sacramento-San Joaquin Delta levees in California.

In addition to seismic analyses for earthfill dams and levees, nonlinear numerical models have been used to assess the seismic stability of concrete gravity dams (e.g., Bureau et al. 2005), concrete water reservoirs (e.g., Roth et al. 2008), mechanically stabilized earth (MSE) walls (e.g., Lindquist 2008), and bridge foundations (e.g., Yegian et al. 2008). Several other applications of the fully nonlinear method can also be found in the proceedings edited by Zeng et al. 2008.

So far, the majority of nonlinear numerical models have been two-dimensional. Three-dimensional analyses have not been as common, primarily because of the long runtimes required for dynamic simulations in three dimensions. With the recent advancements in computer processing speed and parallel processing, three-dimensional simulations are now becoming more common. Mayoral et al. (2008) describe the FLAC3D nonlinear dynamic analyses of a cellular-raft foundation in soft clay. Rayhani and Naggar (2008) report the calibration of a FLAC3D model with dynamic centrifuge experiments simulating the behavior of rectangular ten-story buildings on soft clay. Unutmaz and Cetin (2008) assess the liquefaction triggering potential of soils under structures via FLAC3D simulations.