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SUMMARYNeural network (NN) based constitutive models can capture non-linear material behaviour. These models are versatile and have the capacity to continuously learn as additional material response data becomes available. NN constitutive models are increasingly used within the finite element (FE) method for the solution of boundary value problems. NN constitutive models, unlike commonly used plasticity models, do not require special integration procedures for implementation in FE analysis. NN constitutive model formulation does not use a material stiffness matrix concept in contrast to the elasto-plastic matrix central to conventional plasticity based models. This paper addresses numerical implementation issues related to the use of NN constitutive models in FE analysis. A consistent material stiffness matrix is derived for the NN constitutive model that leads to efficient convergence of the FE Newton iterations. The proposed stiffness matrix is general and valid regardless of the material behaviour represented by the NN constitutive model. Two examples demonstrate the performance of the proposed NN constitutive model implementation.

The United States Geological Survey national seismic hazard maps have historically been produced for a reference site condition of VS30 = 760 m/s. For other site conditions, site factors are used, which heretofore have been developed using ground motion data and simulations for shallow earthquakes in active tectonic regions. Research results from the Next Generation Attenuation–East (NGA-East) project, as well as previous and contemporaneous related research, demonstrate different levels of site amplification in central and eastern North America (CENA) as compared to active regions. We provide recommendations for modeling of ergodic site amplification in CENA based primarily on research results from the literature. The recommended model has three additive terms in natural logarithmic units. Two describe linear site amplification: an empirically constrained VS30-scaling term relative to a 760 m/s reference and a simulation-based term to adjust site amplification from the 760 m/s reference to the CENA reference of VS = 3000 m/s. The third term is a nonlinear model that is described in a companion document. All median model components are accompanied by epistemic uncertainty models.

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Site amplification models for Central and Eastern North America are developed from simulation results presented in a companion paper. Linear and nonlinear amplification functions are developed for response spectral (RS) accelerations and smoothed Fourier amplitude spectra (FAS). Linear RS model components include ground motion scaling with 30 m time-averaged shear wave velocity ( VS30 scaling) and the effects of site period and sediment depth. These models are modular and can be used with or without period or depth terms. Including these terms, especially site period, is desirable and improves model estimation. Modularity also allows linear and nonlinear amplification terms to be developed and combined with linear amplification models without bias. Nonlinear RS models reduce linear amplification as VS30 decreases and the intensity of rock outcrop motions increases. Linear FAS models are tabulated amplification values as functions of VS30 and depth; nonlinear FAS models are analogous to those for the RS. A linear model for correcting a VS30 = 760 m/s rock condition to VS = 3,000 m/s is produced.

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