6.3.2Non-idealized Site Profile and Wave Propagation Mechanisms

For site profiles that cannot be idealized with horizontal soil layers, that have non-horizontal topography, and where wave propagation cannot be approximated with vertically propagating P and S waves, other methods need to be used.

One approximate way of dealing with this situation is to increase the variability (standard deviation) of the soil properties of the soil profile for site response analysis and SSI analysis. [J1]

6.3.3Analysis Models andModelling Assumptions

6.3.3.1Perspective

Four basic methods of developing ground motions were introduced in Section 5.2: Empirical Ground Motion Prediction Equations (GMPEs), Point Source Stochastic Simulations (PSSS), Finite-Fault Simulations (FFS), and the Hybrid Empirical Method (HEM).

Currently, the Probabilistic Seismic Hazard Analysis (PSHA) and Deterministic Seismic Hazard Analysis (DSHA) are the most frequently used methods to generate the ground motion at the site at a location, such as top of grade (TOG), an assumed or actual outcrop, or an assumed or actual impedance mis-match. From this location, site response analyses are frequently needed to further define the motion at FIRS or at the boundary of the nonlinear soil island.

In general, GMPEs are developed from large databases of recorded motions. The ground motions comprising these records included the effects of the fault rupture, all wave propagation mechanisms, topographic effects, geological effects, and local site effects at the recording stations. These measurements are acceleration time histories from which spectral accelerations can be calculated and peak ground acceleration (PGA) values can be determined. These large databases may be parsed into smaller databases to permit customization of the GMPEs, e.g., site condition customization based on Vs30 values.

The important point is that all significant elements contributing to the recorded ground motion values as itemized above exist in the recorded motions, but, generally they are not separable. So it is extremely difficult, if not impossible, to determine which portion of the GMRS is due to topographic effects, geological effects, or wave types. One advantage of implementing site response analysis starting from a deep rock or soil outcrop is the possibility to introduce potentially important site specific effects for the purposes of understanding their impact on the seismic input to the soil-structure system.

[J2]The next sub-sections discuss one-dimensional, two-dimensional, and three-dimensional representations of the wave fields and their potential effect on SSI analyses of structures of interest. It is important to recognize that SSI analyses of nuclear installations are three-dimensional, i.e., three-dimensional soil and structure models and three spatial dimensions of earthquake input motion. For calculation purposes, in some instances, the SSI models are analyzed for each spatial direction of input motion separately. However, even in this case, the three dimensional response of the structures is determined through combining the SSI responses from each individual direction of excitation by an appropriate combination rule, e.g., algebraic sum, square-root-of-the-sum-of-the-squares (SRSS), absolute sum, or other rules. [J3]

In general, I recommend we leverage this subsection off of the above introduction I wrote and focus on three-dimensional modelling of free-field motion and soil properties – do not return to one-dimensional modelling – move some of the material that I highlighted below on ELA to Section 6.3.1 (I can do it if you agree).

Development of analysis models for free field ground motions and site response analysis require an analyst to make modeling assumptions. Assumptions must be made about the treatment of 3D or 2D or 1D seismic wave field, treatment of uncertainties, material modeling for soil (see section 7.3), possible spatial variability of seismic motions (see section 4.4).

This section is used to address aspects of modeling assumptions. The idea is not to cover all possible modeling assumptions (simplification), rather to point to and analyze some commonly made modeling assumptions. It is assumed that the analyst will have proper expertise to address all modeling assumptions that are made and that introduce modeling uncertainty (inaccuracies) in final results.

Addressed in this section are issues related to free field modeling assumptions. Firstly, a brief description is given of modeling in 1D and in 3D. Then, addressed is the use of 1D seismic motions assumptions, in light of full 3D seismic motions (Abell et al., 2016). Next, an assumption of adequate propagation of high frequencies through models (finite element mesh size/resolution) is addressed aswell (Watanabe et al., 2016). There are number of other issues that can influence results (for example, nonlinear SSI of NPPs (Orbovi´c et al. (2015)) however they will not be elaborated in much detail here, rather they will be addressed in appendix through select examples.

6.3.3.2One-dimensional (1D)Models.

One dimensional (1D) wave propagation of seismic motions is a commonly made assumption in free ground motion and site response modeling. Modeling seismic waves in a 1D settings stems from an assumption of ideal horizontal soil and rock layers. Snell’s law (Aki and Richards, 2002) can then be used to show that refraction of any incident ways will produce (almost) vertical seismic waves at the surface. Using Snell’s law will not produce exactly vertical waves, waves will be at least few degrees off vertical, however, for a large number of horizontal layers, with stiffness of layers increasing with depth, waves will be off vertical less by than 10°. However, seismic waves will rarely (never really) become fully 1D. Rather, seismic wave will feature particle motions in a horizontal plane, in two directions. There will not exist a single plane in which these particle motions will be restricted. In addition, even if it is assumed that seismic waves propagating from great depth, are almost vertical at the surface, incoherence of waves at the surface at close distances, really questions the vertical wave propagation assumption. Nonetheless, much of the free and site response analysis is done in 1D.[J4]

It is worth reviewing 1D site response procedures as they are currently done in professional practice and in research. Currently most frequently used procedures for analyzing 1D wave propagation are based on the so called Equivalent Linear Analysis (ELA). Such procedures require input data in the form of shear wave velocity profile (1D), density of soil, stiffness reduction (G/Gmax) and damping (D) curves for each layer. The ELA procedures are in reality linear elastic computations where linear (so called strain compatible) elastic constants are set to secant values of stiffness for (a ratio of) highest achieved strain value for given seismic wave input. This means that for different seismic input, different material (elastic stiffness) parameters need to be calibrated from G/Gmax and damping curves. In addition, damping is really modeled as viscous damping, whereas most damping in soil is frictional (material) damping (Ostadanet al., 2004, Argyris and Mlejnek, 1991). Due to secant choice of (linear elastic) stiffness for soil layers, the ELA procedures are also known to bias estimates of site amplification (Rathje and Kottke, 2008). This bias becomes more significant with stronger seismic motions. In addition, with the ELA procedures, there is no soil volume change modeling, which is quite important to response of constrained soil systems (diPrisco et al., 2012), and can lead to not modelling properly (missing completely!) amplifications of high or low frequency motions[J5]

Instead of Equivalent Linear Analysis approach, a 1D nonlinear approach is also possible. Here, 1D nonlinear material models are used (described in section 3.2). Nonlinear 1D models overcome biased estimates of site amplification, as they model stiffness in a more accurate wave. While modeling with nonlinear models is better than with ELA procedures, it is noted, that volume change information is still missing, and thus amplifications of high or low frequency motions is again missed.[J6]

3DModels.

In reality, seismic motions are always three dimensional, featuring body and surface waves (see more in section 4.2). However, development of input, free field motions for a 3D analysis is not an easy task. Recent large scale, regional models (Bao et al., 1998; Bielak et al., 1999; Taborda and Bielak, 2011; Cui et al., 2009; Bielak et al., 2010, 2000; Restrepo and Bielak, 2014; Bao et al., 1996; Xu et al., 2002; Taborda and Bielak, 2013; Dreger et al., 2015; Rodgers et al., 2008; Aagaard et al., 2010; Pitarka et al., 2013, 2015, 2016) have shown great promise in developing (very) realistic free field ground motions in 3D. What is necessary for these models to be successfully used is the detailed knowledge about the deep and shallow geology as well as a local site conditions (nonlinear soil properties in 3D). Often this data is not available, however, when this data is available, excellent modelling of 3D SSI can be performed, with possible reduction of demand due to nonlinear effects and due to use of more realistic motions. In addition, it should be noted, that due to computational requirements, large scale regional models are usually restricted to lower frequencies (below 3Hz)[J7] while there are current projects (US-DOE) that will extend simulations to 10Hz, for very large regions (200km x 150km x 4km). Another problem is that seismic source, fault slip models currently cannot produce high frequency motions, and stochastic high frequency motions need to be introduced.

In addition, when the data is available, a better understanding of dynamic response of an NPP can be developed. Developed nonlinear, 3D response will not suffer from numerous modeling uncertainties (1D vs. 3D motions, elastic vs. nonlinear/inelastic soil in 3D, soil volumetric response during shearing, influence of pore fluid, etc.). Recent US-NRC, CNSC and US-DOE projects have developed and are developing a number of nonlinear, 3D earthquake soil structure interaction procedures, that rely on full 3D seismic wave fields (free field and site response) and it is anticipated that this trend will only accelerate as benefits of (a more) accurate modeling become understood.

An alternative analytic procedure is the so called Exponential Window Method (EWM) that can be used for lightly damped systems (Kausel (2016)). The method is quite efficient and accurate.

3Dversus1DSeismicModels

We start by pointing out one of the biggest simplifying assumptions made, is that of a presence and use of 1D seismic waves. As pointed out in section 4.2.1 above, worldwide records do not show evidence of 1D seismic waves. It must be noted, that an assumption of neglecting full 3D seismic wave field and replacing it with a 1D wave field can sometimes be appropriate. However, such assumption should be carefully made, taking into account possible intended and unintended consequences. For example full 3D motions can be approximated using 3 x 1D motions for lower freqneucies, where wave lengths are much longer (than object (NPP) dimension. For example is the wave length of the surface (Rayleigh) wave is more than 12 times longer than the longest horizontal dimension of an NPP, motions from such Rayleigh wave can be approximated using 3 x 1D modelling. For shorter wave lengths (higher frequency motions) such approximation might procure results that overestimate influence of high frequency motions if 1D vertical waves are used (from 3 x 1D modelling) as shown in recent paper by Abell et al. (2017).

A brief discussion on 1D, 3 x 1D and 3D seismic wave modelling and effects on SSI is provided below[J8]:

  • 1D modelling of seismic waves is possible if material modelling for soil is linear or equivalent linear elastic. In this case, 1D motions from different directions (horizontal) can be combined, as superposition principle applies for linear elastic systems (soil in this case). Modeling of vertical motions using 1D approach is abit different as an analysis needs to be performed to decide if the vertical wave is a compressional wave (primary, P wave) or if vertical motions are a consequence of vertical components of surface waves. More on those options is provided below in 3x1D modelling option.
  • 3 x 1D modelling of seismic waves is possible, similarly to the above case, if soil material is linear or equivalent linear elastic. SAs noted above, superposition principle can be applied and motions from each direction can be superimposed to obtained 3D motions at the surface. Since most of the time vertical motions are a results (consequence) of Rayleigh surface waves, it is important to analyse vertical motions and decide if modelling as 1D is appropriate. To this end, a wave length of surface wave plays an important role. If the Rayleigh surface wave length (which features both horizontal and vertical components) is longer than 12 times the dimension of the object (NPP), than object rotations, due to differential vertical displacements at object ends, are indeed fairly small and object does move up and down as if excited with a vertical wave. This is shown in Figure 6.x_wave below, as the upper case. On the other hand, if the wave is long less than 12 object dimensions, then vertical motions are gradually replaced by object rotations, while vertical motions are reduced. Case in the lower left corner of Figure 6.x-wave shows a limiting case where seismic wave is 4 times longer than object dimension, which results in minimal vertical motions of the object, and maximum rotations, due to differential motions of object ends. For shorter surface waves, as shown in Figure 6.x_wave, lower right case, waves might not even be exciting any significant dynamic behaviour of the object (except local deformation) as their wave lengths are shorter than twice object length.
  • 3D modelling, when done properly will capture all the body and surface wave effects for SSI analysis of NPPs.

Figure 6.x_wave. Three different cases of surface wave wave length. Upper case is where the surface wave length is 12 or more times longer than the object (NPP) dimension). Lower left case is where the surface wave length is only four times longer than the wave length, and lower right case is where the surface wave length is only two times longer than the object length.

An example is developed in the appendix, that illustrates differences between use 3D seismic motions and 1D seismic motions. Example is based on a recent paper by Abell et al, (2017).

[J1]From Philippe Renault.

[J2]New by JJJ.

[J3]New by JJJ.

[J4]This paragraph is similar to the paragraph in Section 5.1 just before Fig. 5-1. These should be combined and edited to eliminate overlap of ideas.

[J5]Part of this paragraph can be added to Section 6.3.1 discussion of SRA in detail adding bullets to the derivation thereby emphasizing that ELA properties are developed simultaneously with the FIRS, which are needed for the SSI analyses.

[J6]This paragraph seems to say that one can perfrom 1D nonlinear analysis, but the “high or low frequency motions are again missed” – it seems to me that if this is the case, we should simplify this paragraph and issue warning about high and low frequency responses in the free-field.

[J7]Here says 3 Hz – other location Chapter 5 says 5 Hz.

[J8]Boris - recall our meeting in Alamo, in current SSI analyses, 1D is only performed for site response analysis not SSI analyses. In some methodologies (e.g., SASSI), three SSI analyses, one for each of the three spatial directions of excitation, are performed for a 3-D soil-structure model. The resulting responses combined by post-processing of the SSI analyses' results.