TITLE

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1 Contents

1 CONTENTS......................................................................................................................2

7 METHODS AND MODELS FOR SSI ANALYSIS........................................................4

7.1 BASIC STEPS FOR SSI ANALYSIS................................................................47.2 DIRECT METHODS (JEREMIC)......................................................................8

7.2.1 Linear and Nonlinear Discrete Methods..................................................87.3 SUB-STRUCTURING METHODS..................................................................15

7.3.1 Principles...............................................................................................157.3.2 Rigid or flexible boundary method........................................................177.3.3 The flexible volume method..................................................................207.3.4 The subtraction method.........................................................................207.3.5 CLASSI: Soil-Structure Interaction - A Linear Continuum

Mechanics Approach.............................................................................227.3.6 Discrete methods...................................................................................287.3.7 Foundation input motion.......................................................................28

7.4 SSI COMPUTATIONAL MODELS.............................................................287.4.1 Introduction...........................................................................................287.4.2 Soil/Rock Linear and Nonlinear Modelling..........................................287.4.3 Structural models, linear and nonlinear: shells, plates, walls, beams,

trusses, solids.........................................................................................347.4.4 Contact Modeling..................................................................................357.4.5 Structures with a base isolation system.................................................387.4.6 Foundation models................................................................................407.4.7 Small Modular Reactors (SMRs)...........................................................427.4.8 Buoyancy Modeling...............................................................................427.4.9 Domain Boundaries...............................................................................437.4.10 Seismic Load Input................................................................................457.4.11 Liquefaction and Cyclic Mobility Modeling.........................................487.4.12 Structure-Soil-Structure Interaction.......................................................497.4.13 Simplified models..................................................................................527.4.14 General guidance on soil structure interaction modelling and analysis 53

7.5 PROBABILISTIC MATERIAL MODELING.............................................547.5.1 Introduction...........................................................................................54

7.6 PROBABILISTIC RESPONSE ANALYSIS................................................557.6.1 Overview................................................................................................557.6.2 Simulations of the SSI Phenomena.......................................................57

BIBLIOGRAPHY.....................................................................................................................65

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7 METHODS AND MODELS FOR SSI ANALYSIS

7.1 BASIC STEPS FOR SSI ANALYSIS

To identify candidate SSI models, model parameters, and analysis procedures, assess:

• Scoping the problem, see what is to be modelled, what is important and then choose numerical tools to do the job (MORE ON THIS) LOOK at a paper by Neb et al, in NED (see email) and use recommendations (reference them) and paraphrase here…

• The purposes of the SSI analysis (design and/or assessment):

– Seismic response of structure for design or evaluation (forces, moments, stresses or

deformations, such as story drift, number of cycles of response)

– Input to the seismic design, qualification, evaluation of subsystems supported in the structure

(in-structure response spectra ISRS; relative displacements, number of cycles)

– Base-mat response for base-mat design

– Soil pressures for embedded wall designs

• The characteristics of the subject ground motion (seismic input motion):

– Amplitude (excitation level) and frequency content (low vs. high frequency)

∗ Low frequency content (2 Hz to 10 Hz) affects structure and subsystem design/capacity; high frequency content (> 20 Hz) only affects operation of mechanical/electrical equipment and components;

– Incoherence of ground motion;

– Are ground motions 3D? Are vertical motions coming from P or S (surface) waves. If from

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S and surface waves, we have full 3D motions. What to do about it (model?) –

Refer to Chapters 3, 4, and 5 for free-field ground motion and seismic input

discussions

• The characteristics of the site:

– Idealized site profile is applicable (Section 5.3.3.1)

∗ Linear or equivalent linear soil material model applicable (visco- elastic model parameters assigned) ∗ Nonlinear (inelastic, elastic-plastic) material model necessary?

– Non-idealized site profile necessary?

– Sensitivity studies to be performed to clarify model requirements for site

characteristics (complex site stratigraphy, inelastic modelling, etc.)?

• The structure characteristics:

– Expected behaviour of structure (linear or nonlinear);

– Based on initial linear model of the structure, perform preliminary seismic response

analyses (response spectrum analyses) to determine stress levels in structure

elements;

– If significant cracking or deformations possible (occur) such that portions of the

structure behave nonlinearly, refine model either approximately introducing cracked

properties or model portions of the structure with nonlinear elements;

– For expected structure behaviour, assign material damping values;

• The foundation characteristics:

– Effective stiffness is rigid due to base-mat stiffness and added stiffness due to

structure being anchored to base-mat, e.g., honey-combed shear walls anchored to

base-mat;

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– Effective stiffness is flexible, e.g. if additional stiffening by the structure is not

enough to claim rigid; or for strip footings;

The end result is to identify the important elements of SSI to be considered in the analysis of the subject structure:

• Seismic input as defined in Chapters 5 and 6;

• Equivalent linear vs. nonlinear (inelastic, elastic-plastic) soil behaviour; equivalent linear –

substructure approach to SSI acceptable;

• Linear, equivalent linear, or nonlinear (inelastic) structure behaviour; equivalent linear

implements approximate stiffness degradation for structures; linear/equivalent linear -

substructure approach to SSI acceptable;

• Foundation to be modeled as behaving rigidly (e.g., first stage of multi-stage analysis) or flexibly;

Select SSI model and analysis procedure. (BE more specific, as per Jim’s suggestion, good practice is comprised to do (list…) verify tools, meet standards (ASME app B or similar, or whatever member states require…

For the SSI model and analysis to be implemented, confirm existence of

• Determine the application domain for all models, elements, etc. (Application domain is

discussed in some detail in section on Verification and Validation, in Chapter 9).this is

coming before V&V in order to determine

• Verification for all algorithms, procedures, elements, etc. (From user side it is actually verify only those that are going to be used for particular project…

• Validation for all (as many as possible) material models, finite elements, modelling procedures, etc.

Before initial results are available, make estimates of what type of behaviour you expect to see (accomplished in steps above and confirmed herein). In (both) cases, if results are similar or not similar to your pre-analysis expectations do the following investigations:

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• investigate alternative parameters, in order to understand sensitivity of results to parameter variations,

• investigate alternative models (with different degree of fidelity, simplifications, etc.), in

order to understand sensitivity of results to (simplifying) modelling assumptions

Modelling sequence should be:

• Linear elastic, model components first then slowly complete the model:

– soil only, (self weight in all three directions, static loads, point, self-weight, eigen

analysis, etc.); dynamic loads (point loads, etc.); free field ground motions (see

chapter 4 and 5)

– components of structural model only (for example containment only, internal

structure only. Aux building, and in general all structures on the nuclear island

(EPR has 9 separate sub-structures), start with a fixed base model for all, etc.), and

then full structural model (just the structure, no soil), static loads (self weight in three

directions to verify model and load paths, point loads to verify model and load paths);

then dynamic loads (point loads, and seismic loads) (Visual verification of results!!!,

visual verification of element shapes

– complete structure and foundation, (apply same load scenarios as above)

– complete structure, foundation, soil system, (apply same load scenarios as above)

• Equivalent linear modeling, and observe changes in response, to determine possible

plastification effects. It is very important to note that it is still an elastic analysis, with

reduced (equivalent) linear stiffness. Reduction in secant stiffness really steams from

plastification, although plastification is not explicitly modeled, hence an idea can be

obtained of possible effects of reduction of stiffness.

One has to be very careful with observing these effects, and focus more on verification of model (for example wave propagation through softer soil, frequencies will be damped, etc...).

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• Nonlinear/inelastic modelling, slowly introduce nonlinearities to test models, convergence

and stability, in all the components as above.

• Investigate sensitivities for both linear elastic and nonlinear/inelastic simulations!

• Develop documentation according to project requirements and procedures on modelling,

choices/assumptions, uncertainties (how are they dealt with) results, etc.

• Peer review (!) independent, with continuous involvement during project

7.2 Direct methods (Jeremic)

7.2.1 Linear and Nonlinear Discrete Methods

Linear and nonlinear mechanics of solids and structures relies on equilibrium of external and internal forces/stresses. Such equilibrium can be expressed as

σij,j = fi − uρ ¨i (6.1)

where σij,j is a small deformation (Cauchy) stress tensor, fi are external (body (fiB) and surface (fi

S) ) forces, ρ is material density and u¨i are accelerations. Inertial forces uρ ¨i follow from d’Alembert’s principle (D’Alembert, 1758).

The above equation forms a basis for both Finite Element Method (FEM) and Finite Difference Method (FDM). Above equation can be pre-multiplied with virtual displacements uδ i and then integrated by parts to obtain the weak form, as further elaborated below in section 6.2.1. This equation can also be directly solved using finite differences, as noted in section 6.2.1.It is important to note that equation 6.1 is usually not satisfied in either FEM or FDM. Rather is is satisfied in an approximate fashion, with a smaller or large deviation, depending on type of FEM or FDM used.

7.2.1.1 Finite Element Method

Equilibrium Equations Development of finite element equations is efficiently done by using principle of virtual displacements. This principle states that the equilibrium of the body requires that for any compatible, small virtual displacements, which satisfy displacement boundary

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conditions imposed onto the body, the total internal virtual work is equal to the total external virtual work.

Finite Element Equations After some manipulations (Zienkiewicz and Taylor, 1991a,b), we can write the finite element equations as:

MPQ u¨¯P + CPQ u¯˙P + KPQ u¯P = FQ P,Q = 1,2,...,(#ofDOFs)N (6.2)

where MPQ is a mass matrix, CPQ is a damping matrix, KPQ is a stiffness matrix and FQ is a force vector. Damping matrix CPQ cannot be directly developed from a formulation for a single phase solid or structure. In other words, viscous damping is a results of interaction of fluid and solid/structure and is not part of this formulation (Argyris and Mlejnek, 1991). Viscous damping can, however, be added through viscoelastic constitutive material models and through Rayleigh damping, or a more general, Caughey damping. Viscous damping can also be added through viscoelastic constitutive material models.

In general Caughey damping is defined as (Semblat, 1997):m−1

C = [M] X aj([M]−1[K])j (6.3)j=0

where the order used depends on number of modes to be considered for damping in the problem. The second order Caughey damping, is also known as a Rayleigh damping, with j = 1 in Equation (6.3).

In reality, damping matrix (more precisely, damping resulting from viscous effects) results from an interaction of soils and/or structures with surrounding fluids (Argyris and Mlejnek, 1991). For porous solid with pore space filled with fluid, a direct derivation of damping matrix is possible (Jeremicet al.,

2008).

Stiffness matrix KPQ can be linear (elastic) or nonlinear, elastic-plastic.

Finite element analysis comprises a discretization of a solid and/or structure into an assemblage of discrete finite elements. Finite elements are connected at nodal points.

It is very important to note that the finite element method is an approximate method. Generalized displacement solutions at nodes are approximate solutions. A number of factors controls the quality of such approximate solutions. For example it can be shown (Zienkiewicz and Taylor, 1991a,b; Hughes, 1987; Argyris and Mlejnek, 1991) that an increase in a number of nodes, finite elements (refinement of discretization) and a reduction of increments (loads steps or time step

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size) will lead to a more accurate solution. However, this refinement in mesh discretization and reduction of step size, will lead to longer run times. A fine balance needs to be achieved between accuracy of the solution and run time. This is where verification procedures (described in some details in section 8.) become essential. Verification procedures provide us magnitudes of errors that we can expect from our finite element (approximate) solutions. Results from verification procedures should thus be used to decide appropriate discretization (in space (mesh) and load/time) to achieve desired accuracy in solution.

Finite Elements There exist different types of finite elements. They can be broadly classified into:

• Solid elements (3D brick, 2D quads etc.)

• Structural elements (truss, beam, plate, shell, etc.)

• Special Elements (contacts, etc.)

Solid finite elements usually feature displacement unknowns in nodes, 3 displacements for 3D elements, and 2 unknowns displacements for 2D elements. The most commonly used 3D solid finite elements are bricks, that can have 8, 20, and 27 nodes. In 3D, tetrahedral elements (4 and 10 nodes) are also popular due to their ability to be meshed into any volume, while solid brick elements sometimes can have problems with meshing. In 2D most common are quads, with 4, 8 or 9 nodes (Zienkiewicz and Taylor, 1991a,b; Bathe, 1996a). Triangular elements are also popular (3, 6 and 10 nodes), due to the same reason, that is triangles can be meshed in any plane shape, unlike quads. Two dimensional finite elements can approximate plane stress, plane strain or axisymmetric continuum. It is important to note that 3 node triangular elements feature constant strain field, and thus lead to discontinuous strains, and possibility of mesh locking.

Solid finite elements are also used to model coupled problems where porous solid (soil skeleton) is coupled with pore fluid (water), as described by Zienkiewicz and Shiomi (1984); Zienkiewicz et al. (1990, 1999). These elements and the underlying formulation will be described in some detail in section 6.4.11.

Structural finite elements use integrated section stress to develop section generalized forces (normal, transversal and moments). Truss elements can have 2 or 3 nodes. Beams usually have 2 nodes, although 3 node beam elements are also used (Bathe, 1996a). Most beam elements are based on a Euler-Bernoulli beam theory, which means that they do not take into account shear deformation, and thus should only be used for slender beams, where the ratio of beam length to (larger) beam cross section dimension is more than 10 (some authors lower this number to 5) (Bathe, 1996a). For beams that are not slender, Timoshenko beam element is recommended (Challamel, 2006), as it explicitly takes into account shear deformation.

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Plate, wall and shell elements are usually quads or triangles. Plate finite elements model plate bending without taking into account forces in the plane of the plane plate. Main unknowns are transversal displacement and two bending (in plane) moments.

In plane forcing and deformation is modeled using wall elements that are very similar to plane stress 2D elements noted above. In plane nodal rotations are usually not taken into account. If possible, it is beneficial to include rotational (drilling) degree of freedom (Bergan and Felippa, 1985), so that wall elements have three degrees of freedom per node (two in plane displacements and out of plane rotation). Shell element is obtained by combining plate bending and wall elements.

Special elements are used for modeling contacts, base isolation and dissipation devices and other special structural and contact mechanics components of an NPP soil-structure system (Wriggers, 2002).

7.2.1.2 Finite Difference Method

Finite different methods (FDM) operate directly on dynamic equilibrium equation 6.1, when it is converted into dynamic equations of motion. The FDM represents differentials in a discrete form. It is best used for elasto-dynamics problems where stiffness remains constant. In addition, it works best for simple geometries (Semblat and Pecker, 2009), as finite difference method requires special treatment boundary conditions, even for straight boundaries that are aligned with coordinate axes.

Finite Difference Solution Technique The FDM solves dynamic equations of motion directly to obtain displacements or velocities or accelerations, depending on the problem formulation. Within the context of the elasto-dynamic equations, on which FDM is based, elastic-plastic calculations are performed by changes to the stiffness matrix, in each step of the time domain solution.

7.2.1.3 Nonlinear discrete methods

Nonlinear problems can be separated into (Felippa, 1993; Crisfield, 1991, 1997; Bathe, 1996a)

• Geometric nonlinear problems, involving smooth nonlinearities (large deformations, large

strains), and

• Material nonlinear problems, involving rough nonlinearities (elasto-plasticity, damage)

Main interest in modeling of soil structure interaction is with material nonlinear problems. Geometric nonlinear problems are involve large deformations and large strains and are not of much interest here.

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It should be noted that sometimes contact problems where gaping occurs (opening and closing or gaps) are called geometric nonlinear problems. They are not geometric nonlinear problems for cases of interest here, namely, gap opening and closing between foundations. Problems where gap opens and closes are material nonlinear problems where material stiffness (and internal forces) vary between very small values (zeros in most formulations) when the gap is opened, and large forces when the gap is closed.

Material nonlinear problems can be modeled using

• Linear elastic models, where linear elastic stiffness is the initial stiffness or the equivalent

elastic stiffness (Kramer, 1996; Semblat and Pecker, 2009; Lade, 1988; Lade and Kim,

1995).

– Initial stiffness uses highest elastic stiffness of a soil material for modeling. It is

usually used for modeling small amplitude vibrations. These models can be used for

3D modeling.

– Equivalent elastic models use secant stiffness for the average high estimated strain (typically

65% of maximum strain) achieved in a given layer of soil. Eventual modeling is linear elastic,

with stiffness reduced from initial to approximate secant. These models should really be only used for 1D modeling.

Nonlinear 1D models, that comprise variants of hyperbolic models (described in section 3.2), utilize a predefined stress-strain response in 1D (usually shear stress τ versus shear strain γ) to produce stress for a given strain.

There are other nonlinear elastic models also, that define stiffness change as a function of stress and/or strain changes (Janbu, 1963; Duncan and Chang, 1970; Hardin, 1978; Lade and Nelson, 1987; Lade, 1988)

These models can successfully model 1D monotonic behaviour of soil in some cases. However, these models cannot be used in 3D. In addition, special algorithmic measures (tricks) must be used to make these models work with cyclic loads.

Elastic-Plastic material modeling can be quite successfully used for frys frys both monotonic, and cyclic loading conditions (Manzari and Dafalias, 1997; Taiebat and Dafalias, 2008; Papadimitriou et al., 2001; Dafalias et al., 2006; Lade, 1990; Pestana and

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Whittle, 1995). Elastic plastic modeling can also be used for limit analysis (de Borst and Vermeer, 1984).

7.2.1.4 Inelasticity, Elasto-Plasticity

Inelastic, elastic-plastic modeling relies on incremental theory of elasto-plasticity to solve elastic-plastic constitutive equations, with appropriate/chosen material model. Most solutions are strain driven, while there exist techniques to exert stress and mixed control (Bardet and Choucair, 1991). There are two levels of nonlinear/inelastic modeling when elasto-plasticity is employed:

Constitutive level, where nonlinear constitutive equations with appropriate material models are solved for stress and stiffness (tangent or consistent) given strain increment

Global, finite element level, where nonlinear dynamic finite element equations are solved for given dynamic loads and current (elastic-plastic) stiffness (tangent or consistent).

Material Models for Dynamic Modeling At the constitutive level, general 3D strain increments (incremental strain tensor, or in other words, increments in all six independent components of strain, normal (σxx, σyy, and σyy) and shear (σxy, σyz, and σzx)) is driving the nonlinear constitutive solution.

Proper elastic-plastic material models must be chosen to obtain results. Elastic-plastic material models, consist of four main components:

Elasticity, that governs the elastic response, before material yields. Yield function, a function in stress and internal variables (shear strength, friction

angle, back-stress, etc.) space, that separates elastic region from the elastic-plastic region.

Plastic flow directions, that provide directions of plastic strain, once material plastifies. Magnitude of plastic strain is obtained from the solution of constitutive equations.

Hardening/softening rules, that control evolution of yield surface and plastic flow direction, during plastic deformation. There are four main types of hardening/softening rules, that can be combined between each other (for example isotropic and kinematic hardening models can be combined):

– Perfect plastic material behaviour, where yield function and plastic flow directions do not change during plastic deformation. There is no internal variable for this type of hardening/softening.

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– Isotropic hardening/softening material behaviour, where yield function and plastic flow directions change isotropically (proportionally). This type of hardening/softening is only good for monotonic loading and should not be used for cyclic loading. Internal variables are of scalar type, for example friction angle, shear strength, maximum isotropic confinement, etc.

– Kinematic hardening where yield function and plastic flow direction either translate (works well for metals and total stress analysis of undrained, soft clays), or rotate (works well for soils, concrete, rock and other pressure sensitive materials). This type of hardening (usually it is only used for hardening, there is no softening) is good for cyclic loading. Internal variables are the back stress.

– Distortional hardening where yield function and plastic flow direction can have a general change in stress and internal variable space. This type of hardening/softening is the most general case and contains all the previous hardening/softening cases, however it is rarely used, as it requires a large number of tests.

Dynamic modeling, where stresses and strain cyclically change requires models that feature kinematic hardening. In case of pressure sensitive materials, like soil, concrete and rock, rotational kinematic hardening is used. For materials that do not have pressure sensitivity (metals, and saturated clays when modeled with a total stress approach (as opposed to effective stress approach (Jeremicet al., 2008)), translational kinematic hardening is used.

There exist a number of models developed recently that can produce satisfactory modeling of dynamic response of geomaterials (Dafalias and Manzari, 2004; Taiebat and Dafalias, 2008; Dafalias et al., 2006; Mr´oz et al., 1979; Mroz and Norris, 1982; Prevost and Popescu, 1996). Of particular importance is availability of calibration tests, and addressing the issue of uncertainty and sensitivity of material response to changes in parameters.

It is also important to address the issue of spatial variability and uncertainty in material parameters for soils, as the ensuing response can also be quite uncertain. The issue of spatial variability and uncertainty in material modeling will be addressed in more detail in section will be addressed in section 6.5.

Nonlinear Dynamics Solution Techniques On the global, finite element level, finite element equations are solved using time marching algorithms. Most often used are Newmark algorithm (Newmark, 1959) and Hilber-Hughes-Taylor (HHT) α algorithm (Hilber et al., 1977). Other algorithms (Wilson θ, l’Hermite, etc.) also do exist Argyris and Mlejnek (1991); Hughes (1987); Bathe and Wilson (1976), however they are used less frequently. Both Newmark and HHT algorithm allow for numerical damping to be included in order to damp out higher frequencies that are introduced artificially into FEM models by discretization of continua into discrete finite elements.

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Solution to the dynamic equations of motion can be done by either enforcing or not enforcing convergence to equilibrium. Enforcing the equilibrium usually requires use of Newton or quasi Newton methods to satisfy equilibrium within some tolerance. This results in a (much) longer running times, however, provided that the convergence tolerance is small enough, analyst is assured that his/her solution is within proper material response and equilibrium. Solutions without enforced equilibrium are faster, and if they are done using explicit solvers, there is a requirement of small time step, which can then slow down the solution process.

7.3 SUB-STRUCTURING METHODS

7.3.1 Principles

The desire to break the complicated soil-structure problem into more manageable parts have lead to the sub-structure methods. Unlike direct methods of analyses in which all the elements involved in a soil-structure model are gathered in a single model and the dynamic equilibrium equations are solved in one step, sub-structuring methods replace the one-step analysis by a multi-step approach: the global model is replaced by several sub-models of reduced sizes, which are solved independently and successively. To express that the individual models belong to the same global problem, compatibility conditions are enforced between the sub-models: these compatibility conditions simply state that at the shared nodes of two sub-models displacements must be equal and forces must have the same amplitude but opposite signs. The sub-structure approach is schematically depicted in FIG. 7.1.

A

B

Sub-structure 1

Sub-structure 2

FB

FA

FIG. 7.1: Schematic representation of a sub-structure model

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At nodes A and B, which are at the same location in space, the displacement vectors satisfy UA = UB and the force vectors FA = –FB. Note that here the partitioning between the two subsystems is made along the boundary separating the structure and the soil; however, at it will be briefly mentioned below, other possible alternatives. The boundary considered in FIG. 7.1 does not need to be rigid but the size of the problem to handle is significantly reduced when this assumption holds.

Several approaches are proposed to handle the sub-structure methods. They differ essentially by the way in which the interaction between the two sub-structures is handled. One possibility is to define the interface between the two sub-systems along the external boundary of the structure which can be either a plane (2D model) or a surface (3D model); the approach is known as the (rigid or flexible) boundary method and is due to Kausel and Roësset (1974). Other sub-structuring methods have been introduced more recently; they are known as the flexible volume method (Tabatabaie-Raissi, 1982) and the subtraction method (Chin, 1998).

The seismic SSI sub-problems that these three types of substructuring methods are required to solve to obtain the final solution are compared in FIG. 7.2 (SASSI manual). As shown in this figure, the solution for the site response problem is required by all four methods. This is, therefore, common to all methods. The analysis of the structural response problem is also required and involves essentially the same effort for all methods. The necessity and effort required for solving the scattering and impedance problems, however, differ significantly among the different methods. For the rigid boundary and the flexible-boundary methods, two explicit analyses are required separately for solving the scattering and impedance problems. On the other hand, the flexible volume method and the substructure subtraction method, because of the unique substructuring technique, require only one impedance analysis and the scattering analysis is eliminated. Furthermore, the substructuring in the subtraction method often requires a much smaller impedance analysis than the flexible volume method. The latter two methods are those implemented in SASSI.

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FIG. 7.2: Summary of substructuring method

Note also that the domain reduction method (DRM) described in section 6.4.10 can be viewed as a substructure method in which the substructure includes not only the structure but part of the soil. However, reference to substructure methods is usually limited in practice to one of the three methods described above.

Actually, the substructure methods apply only to linear systems. However, the nonlinear behaviour of the soil may be accounted for by using strain compatible properties derived from site response analyses carried out with the viscoelastic equivalent linear model.

7.3.2 Rigid or flexible boundary method

The multi-step approach involved in sub-structuring is based on the so-called superposition theorem established by Kausel and Roësset (1974). It is schematically depicted in Figure 7.3 (Lysmer, 1978) and can be described as follows. Note that FIG 7.3 has been drawn for the special case described below of a 2D problem and a rigid foundation.

The solution to the global SSI problem is obtained by solving successively:

• The scattering problem which aims at determining the kinematic foundation motions, which represent, at each foundation node, the six-components of motion of the massless foundation with its actual stiffness subjected to the incident wave field of the global SSI problem. Note that these motions usually differ from the free field motions at the foundation location due to scattering of the waves by the foundation (see chapter 5). Even if the incident motion is simply composed of vertically propagating body waves, the kinematic foundation

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motions may include a rotational component. When the foundation can be considered infinitely stiff, the foundation input motion reduces to a six-component vector.

• The impedance problem; the impedance function expresses the relationship between a unit harmonic force applied in any direction and at any location along the soil-foundation interface and the induced displacements at any node in any direction along the same interface. In calculating the impedance function the foundation is massless and modelled with its actual rigidity, which is not necessarily infinite. The size of the impedance matrix is kn by kn where n is the number of nodes along the boundary and k the number of degrees of freedom at each node. In the most general case k=6 and the stiffness matrix is full with 6n by 6n non-zero terms. When the foundation can be considered rigid, its kinematics can be described by the displacement of a particular node, for instance the geometric centre of the basemat, and the size of the stiffness matrix is reduced to 6 by 6.

• The structural analysis problem which establishes the response of the original structure connected to a support through the impedance functions and subjected to the kinematic foundation motions.

It can be proved that, provided each of the previous steps are solved rigorously, the solution to third problem (step 3 of the analysis) is strictly equivalent to the solution of the original global SSI problem.

FIG 7.3: 3-step method for soil structure interaction analysis (after Lysmer, 1978)

Obviously, except for very simple geometries (surface foundation on a homogeneous halfspace) continuum solutions are beyond reach for SSI problems. Solutions must rely on numerical techniques like the finite element, finite difference or boundary element methods. These techniques are most efficiently implemented in the frequency domain, which requires that theoretically solutions must be established for each frequency of the Fourier decomposition. This would represent a formidable task and, in practice, only a few frequencies are calculated,

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typically around 50, and interpolation is performed to retrieve the solutions at all frequencies (Tajirian, 1981).

The main steps of this implementation are briefly described below for finite element modelling with reference to Figure 7.3; for more details in the derivation of the following equations, the reader can refer to Kausel and Roësset (1974).

Noting with a superscript ~ the Fourier transform of any quantity, the impedance matrix S()

relates the force vector applied to the foundation to its displacement vector by (solution of step 2, Figure 6.3 (b))

101\* MERGEFORMAT (.)

If is the solution of the scattering problem (Figure 7.3 (a)), it can then be shown that the dynamic equilibrium equation for the structure can be written in the frequency domain

202\* MERGEFORMAT(.)

In the above equation the whole system has been partitioned in degrees of freedom belonging to the structure (indices bb) and degrees of freedom belonging to the structure-foundation interface (indices fb). The damping terms (viscous damping if any) has been omitted for simplicity and the energy dissipation term is represented only by the imaginary part of the complex stiffness terms K (viscoelastic equivalent linear model, see chapter 3).

Equation 02 represents the solution to step 3 (FIG 7.3 (c)). If the foundation is infinitely rigid, the displacement vector can be expressed as a function of the displacement at a particular point through a transformation matrix T:

303\* MERGEFORMAT (.)

and eq.02 then becomes

404\*MERGEFORMAT (.)

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In eq.04, the superscript T represents the transpose of the transformation matrix and the second

line has been left multiplied by TT to render the system symmetric. represents the impedance matrix of the rigid foundation (dimensions at most 6 by 6). Noting that

, eq.04 represents the motion of the structure connected to a support by the

impedance matrix T and subject to a support motion which is the kinematic foundation input motion, solution of the scattering problem.

Once the solution is determined for a set of discrete frequencies and interpolated for the missing frequencies, an inverse Fourier transform provides the time domain solution. It must again be stressed that, as superposition is involved in the frequency domain solution, the substructure methods are limited to linear systems, even though nonlinearities can partially be accounted for in the soil by using strain compatible properties.

7.3.3 The flexible volume method

The flexible volume substructuring method (Tabatabaie-Raissi, 1982) is based on the concept of partitioning the total soil-structure system into three substructure systems. Substructure I consists of the free-field site, substructure II consists of the excavated soil volume, and substructure III consists of the structure, of which the foundation replaces the excavated soil volume. The substructures I, II and III, when combined together, form the original SSI system. The flexible volume method presumes that the free-field site and the excavated soil volume interact both at the boundary of the excavated soil volume and within its body, in addition to interaction between the substructures at the boundary of the foundation of the structure.

7.3.4 The subtraction method

The substructure subtraction method (Chin, 1998) is basically based on the same substructuring concept as the flexible volume method. The subtraction method partitions the total soil structure system into three substructure systems (FIG. 6.4). Substructure I consists of the free-field site, substructure II consists of the excavated soil volume, and substructure III consists of the structure. The substructures I, II and III, when combined together, form the original SSI system. However, the subtraction method recognizes, as opposed to the flexible volume method, that soil structure interaction occurs only at the common boundary of the substructures, that is, at the boundary of the foundation of the structure. This often leads to a smaller impedance analysis than the flexible volume method.

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FIG. 6.4: Substructuring in the subtraction method.

Noting C the matrix K–2M and S the impedance matrix, and identifying the various degrees of freedom with the fllowing subscripts:

b nodes at the boundary of the total system i nodes at the boundary between the ground and the structure w nodes within the excavated soil volume g nodes at the remaining part of the freefield s nodes at the remaining part of the structure

The equation of motion is partitioned as follows:

505\* MERGEFORMAT(.)

21

Where the superscripts refer to the substructures. The complex frequency dependent dynamic stiffness matrix on the left of eq. 05 simply indicates the stated partitioning according to which the stiffness and mass of the excavated soil volume are subtracted from the dynamic stiffness of the free-field site and the structure. As previously the freefield motions, associated with nodes i, are noted with a * superscript; they represent the load vector as in eq.02.

7.3.5 CLASSI: Soil-Structure Interaction - A Linear Continuum Mechanics Approach

7.3.5.1 Overview

Key elements of the CLASSI approach [1] are:

The site is modeled as semi-infinite viscoelastic horizontal layers overlying a semi-infinite viscoelastic halfspace.

1. Complex-valued, frequency-dependent Green’s functions for horizontal and vertical point loads are used in the generation of the foundation input motion (scattering functions) and the foundation impedances. In the standard version of CLASSI, Green’s functions are generated from continuum mechanics principles. The Green’s functions are integrated over the discretized foundation sub-region areas to calculate a resultant set of forces and displacements at sub-region centroids. The foundation impedances are developed by imposing rigid body constraints to the sub-region centroidal displacements corresponding to unit displacements at a defined foundation reference point (FRP). The scattering matrix is developed in a second stage where the free-field ground motions are imposed on the assumed rigid, massless foundation yielding the foundation input motion and, thereby, the scattering matrix. In the Hybrid Method, the advantages of CLASSI and SASSI are combined for generation of scattering functions and foundation impedances.

2. In the same manner as described above, the Green’s functions from SASSI are integrated over the sub-regions creating the unconstrained flexible impedance matrix. Applying the constraints of rigid body motion yields the impedance and scattering matrices from the SASSI flexible impedance matrix for embedded foundations of arbitrary shape. This approach is the subject of this paper.

3. The foundation geometry is discretized for generating the scattering functions and foundation impedances by applying rigid body constraints. The results consist of complex-valued, frequency-dependent scattering functions and impedances. Multiple rigid foundations may be modeled including through soil-coupling.

4. The structures are idealized by simple or very detailed three-dimensional finite element models. The dynamic characteristics of the structure are represented by the fixed-base eigensystem. For the SSI analysis, these dynamic properties are projected onto the

22

foundation, i.e., the fixed-base dynamic characteristics of the structure are represented exactly by its mode shapes, frequencies and modal damping values projected onto the foundation. Multiple structures on a single foundation may be modeled.

5. Material damping is introduced by the use of complex moduli in the soil and modal damping in the structure.

6. The seismic environment may consist of an arbitrary three-dimensional superposition of inclined body and surface waves.

7. The earthquake excitation is defined by time histories of free-field accelerations, called control motion. The control motion is applied at the control point which may be defined on the soil free surface or at a point within the soil column.

8. The Fast Fourier Transform technique is used. 9. The solution of the complete SSI problem is performed in stages: (i) the SSI response of

the foundation is calculated including the effects of the structure (fixed-base modes), foundation (mass), and supporting soil (impedance functions) when subjected to the foundation input motion; and (ii) the dynamic response of the structure degrees of freedom, when subjected to the foundation SSI response, are calculated.

Figure 1 shows schematically the steps in the CLASSI methodology.

7.3.5.2 Important Equations

The SSI problem is solved in the frequency domain, frequency-by-frequency for the frequencies of the Fourier transformed input motion. The equations of motion are written about the foundation reference point (FRP), in terms of the unknown foundation response. Equation 1 shows the equations of motion.

The equations of motion have been re-arranged with the unknown foundation response U(ω) on the left hand side. The first term represents the inertial loading conditions (forces and moments) of the foundation and the structure applied at the foundation reference point (FRP). The foundation loads are a result of the foundation mass being treated as a rigid body, [Mo]. The structure loads are comprised of two parts. The first part is due to the loads induced at the FRP due to the rigid body motion of the foundation (foundation translations and rotations). The second part is the vibrational response of the structure when excited by the foundation motion, i.e., a base excitation comprised of the foundation translations and rotations. The two additional terms represent loads due to the resistance of the supporting soil. The foundation impedances act on the differential motion between the foundation input motion U*(ω) and the resulting foundation response (see Eqn. 2). In all cases, the foundation response U(ω) is that motion of the foundation as a result of SSI, including kinematic and inertial interaction. The vibrational response of the structure is calculated using the fixed-base modes of the structure. The following equations assume M independent structures supported by the foundation.

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M - ω2 [Mo] + ∑ [MB(ω)]i + [Ks(ω)] U(ω)

i=1

= [Ks(ω)] U*(ω) (1) Resistance of the supporting soil:

[Ks(ω)] U(ω) - U*(ω) (2)

[Mo] = the N x N mass matrix of the rigid foundation mat composed of the mass and mass moments of inertia about the FRP; N is the number of foundation degrees of freedom (N ≤ 6).

[MB(ω)]i = the frequency dependent equivalent mass matrix of the i th structure supported on the foundation mat.

[Ks(ω)] = the frequency dependent impedance matrix of the foundation-soil system; the real part of [Ks(ω)] represents the stiffness and inertia of the soil; the imaginary part of [Ks(ω)], when divided by the frequency, corresponds to the radiation and material damping of the soil.

U*(ω) = up to three N x 1 vectors describing the Foundation Input Motion for each of the three directions of excitation of the free-field, i.e., the response of the massless, rigid foundation subjected to the free-field ground motion.

U(ω) = unknown frequency-dependent response of the foundation mat, including all aspects of SSI (kinematic and inertial interaction).

[MB(ω)]i = [MBO]i + [Г]i

T [D(ω)]i [Г]i (3)where

[MBO]i = the 6 x 6 rigid body mass matrix of the ith structure supported on the foundation mat;

[MBO]i = [α]iT [M]i [α]i

[M]i = nodal mass matrix of the ith structure; assume q x q where q is the number of structure dynamic degrees of freedom;

[α]i = structure geometry matrix; a q by 6 rigid body transformation matrix of the ith structure about the FRP; [α]i is calculated as:

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[ α ] i=¿ [ .. . .. .¿ ] [ .. . .. .¿ ] [1 0 00 1 00 0 1

0 z j − y j

−z j 0 x j

y j −x j 0¿]¿

¿

¿¿

(4)

where xj, yj, and zj are the coordinates of node j relative to the the FRP.

The second term on the right hand side of Eqn. 3 represents the dynamic response of the structure when subjected to the base excitation of translations and rotations. The dynamic response characteristics of the structure are represented by its fixed-base modes, frequencies ωi

and mode shapes [Φ]. The modal participation factor matrix for a given mode r with mass normalized mode shape φr can be calculated as:

<Гr> = φrT [M] [α] (5)

Where <Гr> is the modal participation factors for translation and rotations (1 x 6), φrT is the transpose of the rth fixed-base mass normalized mode shape (1 x q), [M] is the structure mass matrix (q x q), and [α] is the rigid body transformation matrix of the structure about the FRP. The matrix [Г] is comprised of the modal participation factors for all fixed-base modes considered in the structure model.

The matrix [D(ω)] is a diagonal matrix comprised of the dynamic amplification factors for each of the fixed-base modes:

[ D ]=[ (ω /ωr )2

(1−ω2

ωr2 )+2iβ r ( ω/ωr ) ]

(6)

Where ωr is the frequency (rad/sec) and βr is the modal damping (fraction of critical) for mode r.

The solution process proceeds by solving for the foundation response U(ω) and then solving the dynamic response problem for the structure subjected to this foundation motion.

The remainder of this paper treats the development of the foundation impedance and scattering matrices by the Hybrid approach. The general formulation follows that of Luco et al. [4].

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7.3.5.3 SSI Parameters By The Hybrid Approach

Foundation Impedance Matrix

The foundation impedance matrix for the rigid, massless foundation, [Ks(ω)], is derived from the SASSI flexible impedance matrix as follows.

For each frequency of interest, assume the unconstrained SASSI flexible impedance matrix is generated. Figure 2 shows the foundation and free-field schematically.

The unconstrained SASSI flexible impedance matrix, [k], relates the displacements, u, occurring at the n degrees of freedom on the foundation to the forces, f, being applied to them. This relationship is only dependent on the properties of the soil and the geometry of the n degrees of freedom.

f = [k]u (7)

For a rigid foundation the impedance functions can be obtained by enforcing the constraint that all displacements, u, are related to each other by rigid-body motion.

u = [a]U (8)

where [a] is a rigid-body transformation matrix, which relates the displacements at each degree of freedom, u, to the six degrees of freedom, U, at the foundation reference point (FRP).

The resultant force vector at the FRP, F, is

F = [a]Tf (9)

= [a]T[k]u

= [a]T[k][a]U

Define [Ks(ω)] = [a]T[k][a] and

F(ω) = [Ks(ω)] U(ω) (10)

[Ks(ω)] is the impedance matrix of the rigid, massless foundation.

Foundation Input Motions (FIMs)

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Let u' be the vector of the unconstrained foundation input motions (FIMs), defined at the n degrees of freedom of the foundation. The unconstrained FIMs are defined by the given free-field motion, U0, i.e., the control motion (amplitude and frequency content at the control point) and the spatial variation of motion over the depth and width of the embedded foundation. At this stage, this assumes the foundation node points are unconstrained by any foundation stiffness. The vector u' is the same as the free-field motion at the node points defining the foundation. If the amplitude of U0 is unity, then u' can be thought of as the unconstrained scattering vector, s'.

Let U* be the unknown FIMs at the FRP, assuming the foundation behaves rigidly. Let u* be the rigid FIMs transformed to the degrees of freedom of the flexible foundation.

u* = [a]U* (11)

where [a] is defined in Eqn. 8. The difference between the displacements of the unconstrained flexible foundation and the assumed rigid

foundation is:

du = u’ - u* (12)

The force pattern required to produce du is df where

df = [k]du (13)

This force pattern must be self-equilibrating and therefore its resultant at the FRP dF must equal zero

dF = [a]T df= [a]T [k]u’ - u* = 0 (14)

Therefore:

[a]T[k][a]U* = [a]T [k]u’

and

U* = [Ks(ω)]-1[a]T [k]u’ (15)

For a unit amplitude free-field motion at frequency ω, u' = s', and U*(ω) is the scattering vector, S(ω).

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7.3.6 Discrete methods

(finite elements and finite difference)

7.3.7 Foundation input motion

(Reference to section 4.5 and 5.1)

7.4 SSI computational models

7.4.1 Introduction

Soil structure interaction computational models are developed with a focus on three components of the problem:

Earthquake input motions, encompassing development of 1D or 2D or 3D motions, and their effective input in the SSI model,

Soil/rock adjacent to structural foundations, with important geological (deep) and site (shallow) conditions near structure, contact zone between foundations and the soil/rock, and

Structure, including structural foundations, embedded walls, and the superstructure

It is advisable to develop models that will provide enough detail and accuracy to be able to address all the important issues. For example, for modeling higher frequencies of earthquake motions, analyst needs to develop finite element mesh that will be capable to propagate those frequencies and to document influence of numerical/mesh induced dissipation/damping of frequencies.

7.4.2 Soil/Rock Linear and Nonlinear Modelling

7.4.2.1 Effective and Total Stress Analysis

Soil and rock adjacent to structural foundations can be either dry or fully (or partially) saturated

(Zienkiewicz et al., 1990; Lu and Likos, 2004).

Dry Soil. In the case of dry soil, without pore fluid pressures, it is appropriate to use models that are only dependent on single phase stress, that is, a stress that is obtained from applying all the loads (static and/or dynamic) without any consideration of pore fluid pressures.

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Partially Saturated Soil. For partially saturated soil, effective stress principle (see equation 6.4 below) must also the include influence of gas (air) present in pore of soils. There are a number of different methods to do that (Zienkiewicz et al., 1999; Lu and Likos, 2004), however computational frameworks that incorporate those methods are not yet well developed. Main approaches to modeling of soil behaviour within a partially saturated zone of soil (a zone where water rises due to capillary effects) are dependent on two main types of partial saturation

Voids of soil fully saturated with fluid mixed with air bubbles, water in pores is fully connected and can move and pressure in the mixture of water and air can propagate, with reduced bulk stiffness of water-air mixture.

This type of partial saturation can be modeled using fully saturated approaches, given in section 6.4.11 below. It is noted that bulk modulus of fluid-air mixture is (much) lower than that of fluid alone, and to be tested for. Therefore, only methods that assume fluid to be compressible should be used (u − p − U, u − U, see section 6.4.11 for details). In addition, permeability will change from a case of just fluid seeping through the soil, and additional testing for permeability of water-air mixture is warranted. It is also noted, that since this partial saturation is usually found above water table, (capillary rise), hydrostatic pore pressure can be suction.

Voids of soil are full of air, with water covering thin contact zone between particles, creating water menisci, and contributing to the apparent cohesion of cohesionless soil material (think of wet sand at the beach, there is an apparent cohesion, until sand dries up).

This type of partial saturation can be modeled using dry (unsaturated) modeling, where elasticplastic material models used are extended to include additional cohesion that arises from thin water menisci connecting soil particles.

Saturated Soil. In the case of full saturate, effective stress principle (Terzaghi et al., 1996) has to be applied. This is essential as for porous material (soil, rock, and sometimes concrete) mechanical behaviour is controlled by the effective stresses. Effective stress is obtained from total stress acting on material (σij), with reductions due to the pore fluid pressure:

(6.4)

where is effective stress tensor, σij is total stress tensor, δij is Kronecker delta (a diagonal matrix with numbers 1 on a diagonal and numbers 0 on non-diagonal positions, that is δij = 1, when i=j, and δij = 0, when i 6= j), and p is the pore fluid pressure. We use standard mechanics of materials convention that tensile components of stress are positive, and so the pore fluid pressure p is negative when in compression (Zienkiewicz et al., 1999). All the mechanical behaviour of

29

soils and rock is a 0 function of the effective stress σij, which is affected by a full coupling with the pore fluid, through a pore fluid pressure p.

A note on clays. Clay particles (platelets) are so small that their interaction with water is quite different from silt, sand and gravel. Clays feature chemically bonded water layer that surrounds clay platelets. Such water does not move freely and stays connected to clay platelets under working loads.

Usually, clays are modeled as fully saturated soil material. In addition, clays feature very small permeability, so that, while the effective stress principle (from Equation 6.4) applies, pore fluid pressure does not change during fast (earthquake) loading. Hence clays should be analyzed using total stress analysis, where the initial total stress is a stress that is obtained from an effective stress calculation that takes into account hydrostatic pore fluid pressure. In other words, slays are modeled using undrained, total stress analysis, using effective stress (total stress reduced by the pore fluid pressure) for initializing total stress at the beginning of loading.

7.4.2.2 Drained and Undrained Modeling

Depending on the permeability of the soil, on relative rate of loading and seepage, and on boundary conditions (Atkinson, 1993), a decision needs to be made if analysis will be performed using drained or undrained behaviour. Permeability of soil (k) can range from k > 10−2m/s for gravel, 10−2m/s > k > 10−5m/s for sand, 10−5m/s > k > 10−8m/s for silt, to k < 10−8m/s for clay. If we assume a unit hydraulic gradient (reduction of pore fluid pressure/head of 1m over the seepage path length of 1m), then for a dynamic loading of 10 − 30 seconds (earthquake), and for a semi-permeable silt with k = 10−6m/s, water can travel few millimeters. However, pore fluid pressure will propagate (much) faster (further) and will affect mechanical behaviour of soil skeleton. This is due to high bulk modulus of water (Kw = 2.15×105 kN/m2), which results in high speed of pressure waves in saturated soils. Thus a simple rule is that for earthquake loading, for gravel, sand and permeable silt, relative rate of loading and seepage requires use of drained analysis. For clays, and impermeable silt, it might be appropriate to use undrained analysis for such short loading.

Drained Analysis Drained analysis is performed when permeability of soil, rate of loading and seepage, and boundary conditions allow for full movement of pore fluid and pore fluid pressures

during loading event. As noted above, use of the effective stress for the analysis is essential, as is modeling of full coupling of pore fluid pressure with the mechanical behaviour of soil skeleton. This is usually done using theory of mixtures (Green and Naghdi, 1965; Eringen and Ingram, 1965; Ingram and Eringen, 1967; Zienkiewicz and Shiomi, 1984; Zienkiewicz et al., 1999) and will be elaborated upon in some detail in section 6.4.11. During loading events, pore fluid pressures will dynamically change (pore fluid and pore fluid pressures will displace) and will affect the soil skeleton, through effective stress principle. All nonlinear (inelastic) material

30

modeling applies to the effective stresses ( ). Appropriate inelastic material models that are used for modeling of soil (as noted in section 6.4.2) should be used.

Undrained Analysis Undrained analysis is performed when permeability of soil, rate of loading and seepage, and boundary conditions do not allow movement of pore fluid and pore fluid pressures during loading event. This is usually the case for clays and for low permeability silt. There are three main approaches to undrained analysis:

Total stress approach, where there is no generation of excess pore fluid pressure (pore fluid pressure in addition to the hydraulic pressure), and soil is practically impermeable (clays and low permeability silt). In this case hydrostatic pore fluid (water) pressures are calculated prior to analysis, and effective stress is established for the soil. This approach assumes no change in pore fluid pressure. This usually happens for clays and low permeability silt, and due to very low permeability of such soils, a total stress analysis is warranted, using initial stress that is calculate based on an effective stress principle and known hydrostatic pore fluid pressure. Since pore fluid pressure does not affect shear strength (Muir Wood, 1990), for very low permeability soils (impermeable for all practical purposes), it is convenient to perform elastic-plastic analysis using undrained shear strength (cu) within a total strain setup. Since only shear strength is used, and all the change in mean stress is taken by the pore fluid, material models using von Mises yield criteria can be used.

Locally undrained analysis where excess pore fluid pressure (change from hydrostatic pore pressure) can be created. Excess pore fluid pressures can be created, due to compression effects on low permeability soil (usually silt). On the other hand, pore fluid suction can also be created due to dilatancy effects within granular material (silt). Due to very low permeability, pore fluid and pore fluid pressure does not move during loading, and hence, effective stress will change, and will affect constitutive behaviour of soil. Analysis is essentially undrained, however, pore fluid pressure can and will change locally due to compression or dilatancy effects in granular soil. Appropriate inelastic (elastic-plastic) material models that are used for modeling of soil (as noted in section 6.4.2) should be used, while constitutive integration should take into account local undrained effects and convert any change in voids into excess pore fluid pressure change (excess pore pressure).

Very low permeability soils, that can, but to not have to develop excess pore fluid pressure can also be analyzed as fully drained continuum, while using very low, realistic permeability. In this case, although analysis is officially drained analysis, results will be very similar if not the same as for undrained behaviour (one of two approaches above) due to use of very low, realistic permeability. Effective stress analysis is used, with explicit modeling of pore fluid pressure and a potential for pore fluid to displace and pore

31

fluid pressure to move. However, due to very low permeability, and fast application of load (earthquake) no fluid will displace and no pore fluid pressure will propagate.

This approach can be used for both cases noted above (total stress approach and locally undrained approach). While this approach is actually explicitly allowing for modeling of pore fluid movement, results for pore fluid displacement should show no movement. In that sense, this approach is modeling more variables than needed, as some results are known before simulations (there will be no movement of water nor pore fluid pressure). However, this approach can be used to verify modeling using the first two undrained approaches, as it is more general.

It is noted that globally undrained problems, where for example soil is permeable, but boundary conditions prevent water from moving, should be treated as drained problems, while appropriate boundary conditions should prevent water from moving across impermeable boundaries.

7.4.2.3 Linear and Nonlinear Elastic Models

Linear and nonlinear elastic models are used for soil, rock and structural components. Linear elastic model that are used are usually isotropic, and are controlled by two constants, the Young’s modulus E and the Poisson’s ratio ν, or alternatively by the shear modulus G and the bulk modulus K.

Nonlinear elastic models are used mostly in soil mechanics, There are a number of models proposed over years, tend to produce initial stiffness of a soil for given confinement of over-consolidation ratio (OCR) (Janbu, 1963; Duncan and Chang, 1970; Hardin, 1978; Lade and Nelson, 1987; Lade, 1988).

Anisotropic material models are mostly used for modeling of usually anisotropic rock material (Amadei and Goodman, 1982; Amadei, 1983).

Equivalent Linear Elastic Models Equivalent elastic models are linear elastic models where the elastic constants were determined from nonlinear elastic models, for a fixed shear strain value. They are secant stiffness 1D models and usually give relationship between shear stress ( τ = σxz) and shear strain ( ). Determination of secant shear stiffness is done iteratively, by performing 1D wave propagation simulations, and recording average high estimated strain (65% of maximum strain) for each level/depth. Such representative shear strain is then used to determine reduction of stiffness using modulus reduction curves (G/Gmax and the analysis is re-run. Stable secant stiffness values are usually reached after few iterations, typically 5-8. It is important to emphasize that equivalent elastic modeling is still essentially linear elastic modeling, with changed stiffness. More details are available in sections 3.2 and 4.5.

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7.4.2.4 Elastic-Plastic Models

Elastic plastic modeling can be used in 1D, 2D and full 3D. A number of material models have been developed over years for both monotonic and cyclic modeling of materials. Material models for soil (Manzari and Dafalias, 1997; Taiebat and Dafalias, 2008; Papadimitriou et al., 2001; Dafalias et al., 2006; Lade, 1990; Pestana and Whittle, 1995; Prevost and Popescu, 1996; Mroz and Norris, 1982), rock (Lade and Kim, 1995; Hoek et al., 2002; Vorobiev, 2008) have been developed over last many years.

It should be noted that 3D elastic plastic modeling is the most general approach to material modeling of soils and rock. If proper models are used (see section 6.2.1) it is possible to achieve modeling that is done using simplified modeling approaches described above (linear elastic, equivalent linear elastic, modulus reduction curves, etc.). However, calibration of models that can achieve such modeling sophistication requires expertise. The payoff is that important material response effects, that are usually neglected if simplified models are used, can be taken into account and properly modeled. As an example, soil volume change during shearing is a first order effects, however it is not taken into account if modulus reduction curves are used.

7.4.2.5 A Note on Constitutive Level and Global Level Equilibrium. There are two main types of algorithms

for constitutive integrations:

Explicit or Forward Euler, is an algorithm that produces tangent stiffness tensor on the constitutive level. This algorithm does not enforce equilibrium and error in constitutive integrations (drift from the yield surface) is accumulated. This algorithm is simpler and faster than the implicit algorithm (next item) and is implemented and used in most (all) computer programs.

Implicit or Backward Euler) is an algorithm that produces algorithmic (consistent) stiffness tensor (matrix) that can produce very fast convergence (quadratic for Newton scheme) on the global, finite element equilibrium iterations. This algorithm is iterative and does enforce equilibrium (within user specified tolerance). It is usually slower than the explicit algorithm (see above) and implementation can be quite complicated, particularly for elastic plastic material models for soil and concrete (Crisfield, 1987; Jeremicand Sture, 1997; Jeremi´c, 2001).

On the global, finite element level, there are two ways to advance the solutions

Solution advancement without enforcing the equilibrium. In this case, solutions is produced using current tangent stiffness matrix (relying on the tangent stiffness tensor, developed on the constitutive level, as noted above). For each step of loading (static or dynamic) difference between applied loads and internal loads (stresses) is not checked

33

for. This means that error in unbalanced forces is accumulating as computations progress. Usual remedy is to make steps small enough so that error is also reduced. However this reduction in step size (or time step size) can significantly increase computational times. In one specific instance, if lumped mass matrix is used, instead of a consistent mass matrix (which is theoretically more accurate), solution of a large system of equations can be completely circumvented. For particular explicit dynamic computations, only inverse of a diagonal mass matrix is required, which is trivial to obtain.

Solution advancement with enforcement of the equilibrium. In this case, equilibrium is explicitly checked for, and if unbalanced forces are not balanced within certain (user specified) tolerance, an iterative scheme is used until equilibrium is achieved (within tolerance). Alternative method for ending iterations (instead of achieving equilibrium) is to check for iterative displacements and place a low limit below which iterations are not worth while any more and therefor end them. The most commonly used iteration methods are based on Newton iterative scheme Crisfield (1984). This approach is computationally demanding, however it does benefit the solution as it yields (close to) equilibrium solutions. In addition, if consistent (algorithmic) stiffness is used on the constitutive level (see Implicit constitutive algorithm above), a fast convergence (sometimes even close to quadratic) is achieved.

Concluding note for both constitutive and global level solution advancement is that simpler methods (explicit, no equilibrium check) will lead to accumulating error (unbalance stress and force) and will thus render solutions that are not in equilibrium and are possibly quite wrong. This can be remedied by reducing step size (time increment), however computational times are then becoming long On the other hand, methods that enforce equilibrium (within tolerance) are (much) more complicated to develop, implement and execute, yet they enforce equilibrium (again within tolerance).

7.4.3 Structural models, linear and nonlinear: shells, plates, walls, beams, trusses, solids

Linear and nonlinear structural models are not used as much in the industry for modeling and simulation of behaviour of nuclear power plants (NPP). One of the main reasons is that NPPs are required to remain, effectively elastic during earthquakes. Nevertheless modeling of nonlinear effects in structures remains a viable proposition. It is important to note that inelastic behaviour of structural components

(trusses, beams, walls, plates, shells) features a localization of deformation (Rudnicki and Rice, 1975). While localization of deformation is also present in soils, soils are more ductile medium (unless they are very dense) and so inelastic treatment of deformation in soils, with possible localization, is more benign than treatment of localization of deformation in brittle concrete. Significant work has been done in modeling of nonlinear effects in mass concrete and concrete beams, plates, walls and shells (Feenstra, 1993; Feenstra and de Borst, 1995; de Borst and

34

Feenstra, 1990; de Borst, 1987, 1986; de Borst, 1987; de Borst et al., 1993; Bi´cani´c et al., 1993; Kang and Willam, 1996; Rizzi et al., 1996; Menetrey and Willam, 1995; Carol and Willam, 1997; Willam, 1989; Willam and Warnke, 1974; Etse and Willam, 1993; Scott et al., 2004, 2008; Spacone et al., 1996a,b; Scott and Fenves, 2006).

The main issue is still that concrete structural elements still develop plastic hinges (localized deformation zones). Finite element results with localized deformation are known to be mesh dependent (change of mesh will change the result), and as such are hard to verify. Recent work on rectifying this problem (Larsson and Runesson, 1993) shows promises, however these methods are still not widely accepted. One possible, rather successful solution relies on classical developments of Cosserat continua (Cosserat, 1909), where results looked very promising (Dietsche and Willam, 1992), however sophistication required by such analysis and lack of programs makes this approach still very exotic.

7.4.4 Contact Modeling

In all soil-structure systems, there exist interfaces between structural foundations and the soil or rock beneath. There are two main modes of behaviour of these interfaces, contacts:

Normal contact where foundation and the soil/rock beneath interact in a normal stress mode. This mode of interaction comprises normal compressive stress, however it can also comprise gap opening, as it is assumed that contact zone has zero tensile strength.

Shear contact where foundation and the soil/rock beneath can develop frictional slip.

Contact description provided here is based on recent work by Jeremic(2016) and Jeremicet al. (1989-2016).

Modeling of contact is done using contact finite elements. Simplest contact elements are based on a two node elements, the so called joint elements which were initially developed for modeling of rock joints. Typically normal and tangential stiffness were used to model the pressure and friction at the interface (Wriggers, 2002; Haraldsson and Wriggers, 2000; Desai and Siriwardane, 1984).

The study of two dimensional and axisymmetric benchmark examples have been done by Olukoko et al. (1993) for linear elastic and isotropic contact problems. Study was done considering Coulomb’s law for frictional behaviour at the interface. In many cases the interaction of soil and structure is involved with frictional sliding of the contact surfaces, separation, and re-closure of the surfaces. These cases depend on the loading procedure and frictional parameters. Wriggers (2002) discussed how frictional contact is important for structural foundations under loading, pile foundations, soil anchors, and retaining walls.

Two-dimensional frictional polynomial to segment contact elements are developed by Haraldsson and Wriggers (2000) based on non-associated frictional law and elastic-plastic

35

tangential slip decomposition. Several benchmarks are presented by Konter (2005) in order to verify the the results of the finite element analyses performed on 2D and 3D modelings. In all proposed benchmarks the results were approximated pretty well with a 2D or an axisymmetric solutions. In addition, 3D analyses were performed and the results were compared with the 2D solutions.

7.4.4.1 Contact Modeling Formulation

The formulation for contact is represented by a discretization which establishes constraint equations and contact interface constitutive equations on a purely nodal basis. Such a formulation is called node-tonode contact (Wriggers, 2002). The variables adopted to formulate the model are shown in Figure 6.1: the force (F) and displacement vectors (u).

FIG 7.1: Forces and relative displacements of the element.

Each vector is composed of three terms: the first one acts along the longitudinal direction (nlocal) whereas the other two components lie on the orthogonal plane (mlocal and llocal). The total relative displacement additively decomposed into elastic and plastic components (6.6).

F = [p ; t]T ; u = [v; gs]T (6.5)

el h

el eliT

pl h

pl pliT

el pl (6.6)u = v ; gs ; u = v ; gs ; u = u + u

Elastic behaviour. The elastic behaviour (contact and no slip) is defined by the relation:

KN(p) 0 0

el dF = E · du ; E = 0 KT

0

(6.7)

36

0 0 KT

The normal displacement-normal force relationship, valid for loading and unloading conditions, can be

either

Constant contact stiffness (penalty stiffness, hard contact). One simple rule of thumb in choosing this stiffness is to prescribe contact penalty stiffness K = 1000EA/h, where E is a stiffness modulus of one of the materials adjacent to contact zone (hence there is a possibility that this stiffness might be coming from soil (relatively low stiffness) or from concrete (relatively large stiffness), A is a tributary area for that contact element, and h is a thickness of a contact zone (usually a small number, on the order of few centimeters). It is important to note that above recommended penalty stiffness can vary orders of magnitude, and that numerical experiments need to be performed in order to test contact element performance with chosen stiffness, or

Nonlinear function (soft contact). Functional relationship that works well for concrete – soil contact

is,if ux

< 0(6.8)

,if ux < 0

where kn is a constant and ux is a relative displacement of two contact nodes. Force–displacement equation given in equation 6.8 is a parabola that has a non-zero tangent at ux

= 0. The value of stiffness 0.0001kn is chosen as stiffness at 1/10 of millimeter (0.0001m) of penetration.

The tangential stiffness KT is assumed to be constant. Alternative contact stiffness functions were proposed by Gens et al. (1988, 1990) for a more stiff contact between two rock (or concrete) surfaces.

Plastic model: The plastic model is defined in terms of yield surface and plastic potential surface, as shown in Figure 6.2. Yield surface is a frictional cone (with friction coefficient µ = tanφ), with no cohesion. Plastic potential, that defines plastic flow directions, preserves volume, that is, there is no dilation of compression due to plastic slip.

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FIG. 7.2: Yield surface fs, plastic potential Gs and incremental plastic displacement δup.

7.4.4.2 Geometry description

Figure 6.3 shows geometry of the two node contact element and its main deformations modes. It is

Definition of local axes

Nodes in contact Movement in normal direction

Movement in tangent

direction

FIG. 7.3: Description of contact geometry and displacement responses.

important to note the importance of properly numbering element nodes, consistently with the definition of normal x1. Node I is the first node, node J is the second node and normal goes from node I toward node J. If reversed, elements behaves like a hook.

7.4.5 Structures with a base isolation system

Base isolation system are used to change dynamic characteristics of seismic motions that excite structure and also to dissipate seismic energy before it excites structure. Therefor there are two main types of devices:

Base Isolators (Kelly, 1991a,b; Toopchi-Nezhad et al., 2008; Huang et al., 2010; Vassiliou et al., 2013) are usually made of low damping (energy dissipation) elastomers and are primarily meant to change (reduce) frequencies of motions that are transferred to the structural system. They are not designed nor modeled as energy dissipators.

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J

I

J

I

J

I

contact plane

lz

ly

lxJ

I

z

y

x

systemcoordinateGlobal

NOTE for Alain: what I meant is that base isolators will modify frequency of motions that to into the structure, I changed above item to reflect that.However if it is still not clear, I can add more explanation...

Base Dissipators Kelly and Hodder (1982); Fadi and Constantinou (2010); Kumar et al. (2014) are developed to dissipate seismic energy before it excites the structure. There two main types of such dissipators:

– Elastomers made of high dissipation rubber, and

– Frictional pendulum dissipators

Both isolators and dissipators are usually developed to work in two horizontal dimensions, while motions in vertical direction are not isolated or dissipated. This can create potential problems, and need to be carefully modeled.

Modeling of base isolation and dissipation system is done using two node finite elements of relatively short length.

Base Isolation Systems are modeled using linear or nonlinear elastic elements. Stiffness is provided from either tests on a full sized base isolators, or from material characterization of rubber (and steel plates if used in a sandwich isolator construction). Depending on rubber used, a number of models can be used to develop stiffness of the device (Ogden, 1984; Simo and Miehe, 1992; Simo and Pister, 1984).

Particularly important is to properly account for vertical stiffness as vertical motions can be amplified depending on characteristics of seismic motions, structure and stiffness of the isolators Hijikata et al. (2012); Araki et al. (2009). It is also important to note that assumption of small deformation is used in most cases. In other words, stability of the isolator, for example overturning or rolling is not modeled. It is assumed that elastic stiffness will not suddenly change if isolator becomes unstable (rolls or overturns).

Base Dissipator Systems are modeled using inelastic (nonlinear) two node elements. There are three basic types of dissipator models used:

High damping rubber dissipators Rubber dissipators with lead core Frictional pendulum (double or triple) dissipators

Each one is calibrated using tests done on a full dissipator. It is important to be able to take into account influence of (changing) ambient temperature and increase in temperatures due to energy dissipation (friction) on resulting behaviour. Ambient temperature can have significant variation, depending on geographic location of installed devices and such variation will affects base

39

dissipator system response. In addition, energy dissipation results in heating of devices, and resulting increase in temperature will influences base dissipators response as well.

7.4.6 Foundation models

Foundation modeling can be done using variable level of sophistication. Earlier models assumed rigid foundation slabs. This was dictated by the use of modeling methods that rely on analytic solution, which in turn have to rely on simplifying assumptions in order to be solved. For example, soil and rock beneath and adjacent to foundations was usually assumed to be an elastic half space.

Foundation response plays an important role in overall soil-structure interaction (SSI) response. Major energy dissipation happens in soil and contact zone beneath the foundation. Buoyant forces (pressures) act on foundation if water table is above bottom of the lowest foundation level.

Shallow and Embedded Slab Foundations Foundation slabs and walls are flexible. Their thickness can range from 3 − 5 meters, but they extend for up to 100 meters. Containment and shield buildings are rigidly connected to foundation slabs, and will stiffen it up. In addition, auxiliary buildings, will also stiffen up foundation slabs. However, even with all these stiffening effects, slabs and walls should be modeled using flexible models.

Flexible modeling of foundation slabs is best done using either shell elements (plate bending and in plane wall) or solids.

For shell element models, it is important to bridge over half slab or wall thickness to the adjacent soil. This is important as shell elements are geometrically representing plane in the middle of a solid (slab or wall) with a finite thickness, so connection over half thickness of the slab or wall is needed. It is best if shell elements with drilling degrees of freedom are used (Ibrahimbegovi´c et al., 1990; Militello and Felippa, 1991; Alvin et al., 1992; Felippa and Militello, 1992; Felippa and Alexander, 1992), as they properly take into account all degrees of freedom (three translations, two bending rotations and a drilling rotation).

For solid element models, it is important to use proper number of solids so that they represent properly bending stiffness. For example, a single layer of regular 8 node bricks will over-predict bending stiffness over 2 times (200%). Hence at least 4 layers of 8 node bricks are needed for proper bending stiffness. If 27 node brick elements are used, a single layer is predicting bending stiffness within 4% of analytic solution.

Piles and Shaft Foundations For nuclear facilities built on problematic soils, piles and shafts are usually used for foundation system. Piles can carry loads at the bottom end, and in addition to that, can also carry loads by skin friction. Shafts usually carry majority of load at the bottom end.

Piles (including pile groups) and shafts have been modeled using three main approaches:

40

Analytic approach (Sanchez-Salinero and Roesset, 1982; Sanchez-Salinero et al., 1983; Myionakis and Gazetas, 1999; Law and Lam, 2001; Sastry and Meyerhof, 1999; Abedzadeh and Pak, 2004; Mei Hsiung et al., 2006; Sun, 1994) where a main assumptions is that of a linear elastic behaviour of a pile and the soil represented by a half space, while contact zone is fully connected, and no slip or gap is allowed. More recently there are some analytic solution where mild nonlinear assumptions are introduced (Mei Hsiung, 2003), however models are still far from realistic behaviour. These approaches are very valuable for small vibrations of piles, pile groups and shafts as well as for all modes of deformation where elasto-plasticity will be very mild if it exists at all.

P-Y and T-Z approach where experimentally measured response of piles in lateral direction (P-Y) and vertical direction (T-Z) is used to construct nonlinear springs that are then used to replace soil (Stevens and Audibert, 1979; Brown et al., 1988; Bransby, 1996, 1999; Tower Wang and Reese, 1998; Reese et al., 2000; Georgiadis, 1983). This approach is very popular with practicing engineers. However this approach does make some simplifying assumptions that make its use questionable for use with cases where calibrations of P-Y and T-Z curves do not exist. For example, for layered soils, of for piles in pile groups, this approach can produce problematic results. Moreover, in dynamic applications, dynamics of soils surrounding piles and piles groups is poorly approximated using springs.

Nonlinear finite element models in 3D have been developed recently for treatment of piles, pile groups and shafts, in both dry and liquefiable soils (Brown and Shie, 1990, 1991; Yang and Jeremic, 2003; Yang and Jeremi´c, 2005a,b; McGann et al., 2011; S.S.Rajashree and T.G.Sitharam, 2001; Wakai et al., 1999). In these models, elastic-plastic behaviour of soil is taken into account, as well as inelastic contact zone within pile-soil interface. Layered soils are easily modeled, while proper modeling of contact (see section 6.4.4) resolves both horizontal and vertical shear (slip) behaviour. Moreover, with proper modeling, effects of piles in liquefiable soils can be evaluated as well (Cheng and Jeremi´c, 2009).

Deeply Embedded Foundations In case of Small Modular Reactors, foundations are deeply embedded, and the foundation walls, in addition to the base slab, contribute significantly more to supporting structure for static and dynamic loads. Main issues are related to proper modeling of contact (see more in sections 6.4.4 and 6.4.7), as well inelastic behaviour of soil adjacent to the slab and walls. Of particular importance for deeply embedded foundations is proper modeling of buoyant stresses (forces) as it is likely that ground water table will be above base slab.

Foundation Flexibility and Base Isolator/Dissipator Systems. There are special cases of foundations where base isolators and dissipators are used. In this case there are two layers of foundations slabs, one at the bottom, in contact with soil and one above isolators/dissipators, beneath the actual structure. Those two base slabs are connected with dissipators/isolators. It is

41

important to properly (accurately) model stiffness of both slabs as their relative stiffness will control how effective will isolators and dissipators be during earthquakes.

7.4.7 Small Modular Reactors (SMRs)

Add a link to SMR section in chapter 8 and make a bit of intro here….

Jim: introduce why this has special characteristics and why does in need special treatment than other currently described…

7.4.8 Buoyancy Modeling

For NPP structures for which lowest foundation level is below the water table, there exist a buoyant pressure/forces on foundation base and walls. For static loads, buoyant force B can be calculated using Archimedes principle: ”Any object immersed in water is buoyed up by a force equal to the weight of water displaced by the object”, B = ρwgV where ρw = 999.972 kg/m3 (for salty sea/ocean water values of density are higher ρw = 1020.0 − −1029.0 kg/m3) is the mass density of water (at temperature of +4oC with small changes of less than 1% up to +40oC), g = 9.81 m/s2 is the gravitational acceleration, and V is the volume of displaced fluid (volume of foundation under water table). Buoyant force can be applied as a single force or a small number of resultant forces directed upward around the stiff center of foundation. This is strictly applicable if foundation is rigid, but it can probably work in most cases for static loads.

During dynamic loading, buoyant force (buoyant pressures) can dynamically change, as a results of a dynamic change of pore fluid pressures in soil adjacent to the foundation concrete. This is particularly true for soils that are dense, where shearing will lead increase of inter-granular void space (dilatancy), and reduction in buoyant pressures or for soils that are loose, where shearing will lead to reduction of inter-granular void space (compression), and increase in buoyant pressures.

For strong shaking, it also expected that gaps will open between soil and foundation walls and even foundation slab. This will lead to pore water being sucked into the opening gap and pumped back into soil when gap closes. This ”pumping” of water will lead to large, dynamic changes of buoyant pressures.

Different dynamic scenarios, described above, create conditions for dynamic, nonlinear changes in buoyant force.

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Dynamic Buoyant Stress/Force Modeling. Fully coupled finite elements (u-p or u-p-U or u-U, as described in section 6.4.11) are used for modeling saturated soil adjacent to foundation walls and base. Modeling of contact between soil and the foundation concrete needs to take into account effects of pore fluid pressure – buoyant stress within the contact zone, on order to properly model normal stress for frictional contact. This modeling can be done using

coupled contact elements, that explicitly model water displacements and pressures and allows for explicit gap opening, filling of gap with water, slipping (frictional) when the gap is closed, and pumping of water as gap opens and closes. This contact element takes into account the pore water pressure information from saturated soil finite elements, as well as the information about the displacement (movement) of pore water within a gap. It is based on a dry version of the contact element and incorporates effective contact (normal) stress, based in the effective stress principle (see equation 6.4). Potential for modeling of pumping of water during gap opening and closing is important as it might influence dynamic response. Noted pumping of water, can also lead to liquefying of even dense soil, in the zone where gap opening happens (Allmond and Kutter, 2014).

coupled finite elements in contact with impermeable concrete finite elements (solids or shells/plates). In this case, impermeable concrete finite elements create a natural barrier for water flow, thus allowing generation of excess pore fluid pressure in the contact zone during dynamic shaking. This approach precludes formation of gaps and pumping of water.

7.4.9 Domain Boundaries

JIM: motivation, intake structure is not embedded on all sides, therefor it receives pressures from sides that are embedded only, so this is a structure which will be important to model using this approach… Introduce a complexity of the issue, set a context (intake structure it is beleiverd to be a conservative approach however do a sensitivity study and show if it is indeed or not (conservative).

One of the biggest problems in dynamic ESSI in infinite media is related to the modeling of domain boundaries. Because of limited computational resources the computational domain needs to be kept small enough so that it can be analyzed in a reasonable amount of time. By limiting the domain however an artificial boundary is introduced. As an accurate representation of the soil-structure system this boundary has to absorb all outgoing waves and reflect no waves back into the computational domain.

The most commonly used types of domain boundaries are presented in the following:

43

• Fixed or free

By fixing all degrees of freedom on the domain boundaries any radiation of energy away from the structure is made impossible. Waves are fully reflected and resonance frequencies can appear that don’t exist in reality. The same happens if the degrees of freedom on a boundary are left ’free’, as at the surface of the soil.

combination of free and fully fixed boundaries should be chosen only if the entire model is large enough and if material damping of the soil prevents reflected waves to propagate back to the structure.

For cases where compressional and/or shear waves travel very fast, in this case boundaries have to be placed very far, thus significantly increasing the size of models.

• Absorbing Lysmer Boundaries

way to eliminate waves propagating outward from the structure is to use Lysmer boundaries. This method is relatively easy to implement in a finite element code as it consists of simply connecting dash pots to all degrees of freedom of the boundary nodes and fixing them on the other end (Figure 7.5).

Lysmer boundaries are derived for an elastic wave propagation problem in a one-dimensional semi-infinite bar. It can be shown that in this case a dash pot specified appropriately has the same dynamic properties as the bar extending to infinity (Wolf, 1988). The damping coefficient C of the dash pot equals

C = A cρ (6.9)

where A is the section of the bar, ρ is the mass density and c the wave velocity that has to be selected according to the type of wave that has to be absorbed (shear wave velocity cs or

FIG. 7.5: Absorbing boundary consisting of dash pots connected to each degree of freedom of a boundary node

44

compressional wave velocity cp).

In a 3d or 2d model the angle of incidence of a wave reaching a boundary can vary from almost 0 up to nearly 180. The Lysmer boundary is able to absorb completely only those under an angle of incidence of 90. Even with this type of absorbing boundary a large number of reflected waves are still present in the domain. By increasing the size of the computational domain the angles of incidence on the boundary can be brought closer to 90

and the amount of energy reflected can be reduced.

• Infinite elements

• More sophisticated boundaries modeling wave propagation toward infinity (boundary elements)

For a spherical cavity involving only waves propagating in radial direction a closed form solution for radiation toward infinity, analogous to the Lysmer boundary for wave propagation in a prismatic rod, exists (Sections 3.1.2 and 3.1.3 in Wolf (1988)). Since this solution, in contrast to the Lysmer boundary, includes radiation damping it can be thought of as an efficient way of eliminating reflections on a semi-spherical boundary surrounding the computational domain.

More generality in terms of absorption properties and geometry of the boundary are provided by the various boundary element methods (BEM) available in the literature.

7.4.10 Seismic Load Input

MOVE this up and combine with section 7.3 which has different methods for seismic onut (SASSSI, CLASSI and so on…)

A number of methods are used to input seismic motions into finite element model. Most of them are based on simple intuitive approaches, and as such are not based on rational mechanics. Most of those currently still widely used methods cannot properly model all three components of body waves as well as always present surface waves. There exist a method that is based on rational mechanics and can model both body and surface seismic waves input into finite element models with high accuracy. That method is called the Domain Reduction Method (DRM) and was developed fairly recently Bielak et al. (2003); Yoshimura et al. (2003)). It is a modular, two-step dynamic procedure aimed at reducing the large computational domain to a more manageable size. The method was developed with earthquake ground motions in mind, with the main idea to

45

replace the force couples at the fault with their counterpart acting on a continuous surface surrounding local feature of interest. The local feature can be any geologic or manmade object that constitutes a difference from the simplified large domain for which displacements and accelerations are easier to obtain. The DRM is applicable to a much wider range of problems. It is essentially a variant of global–local set of methods and as formulated can be used for any problems where the local feature can be bounded by a continuous surface (that can be closed or not). The local feature in general can represent a soil–foundation–structure system (bridge, building, dam, tunnel...), or it can be a crack in large domain, or some other type of inhomogeneity that is fairly small compared to the size of domain where it is found.

7.4.10.1 The Domain Reduction Method

A large physical domain is to be analyzed for dynamic behaviour. The source of disturbance is a known time history of a force field Pe(t). That source of loading is far away from a local feature which is dynamically excited by Pe(t) (see Figure 7.6).

The system can be quite large, for example earthquake hypocenter can be many kilometers away from the local feature of interest. Similarly, the small local feature in a machine part can be many centimeters away from the source of dynamic loading which influences this local feature. In this sense the term large domain is relative to the size of the local feature and the distance to the dynamic forcing source.

It would be beneficial not to analyze the complete system, as we are only interested in the behaviour of the local feature and its immediate surrounding, and can almost neglect the domain outside of some relatively close boundaries. In order to do this, we need to somehow transfer the loading from the source

FIG. 6.6: Large physical domain with the source of load Pe(t) and the local feature (in this case a soil-structure system.

46

Seismic source

Ω

Ω+

Local feature

Large scale domain

Γ

eu

buiu

)t(eP

to the immediate vicinity of the local feature. For example we can try to reduce the size of the domain to a much smaller model bounded by surface Γ as shown in Figure 6.6. In doing so we must ensure that the dynamic forces Pe(t) are appropriately propagated to the much smaller model boundaries Γ.

It can be shown (Bielak et al., 2003) that the consistent dynamic replacement for the dynamic source forces Pe is a so called effective force, Peff:

eff

Pi 0

Peff = Pbeff = − MbeΩ+u¨e0 − KbeΩ+u0e (6.10)

Peeff MebΩ+u¨0b + KebΩ+u0b

where MbeΩ+ and Meb

Ω+ are off-diagonal components of a mass matrix, connecting boundary (b) and external (e) nodes, Kbe

Ω+ and KebΩ+ are off-diagonal component of a stiffness matrix,

connecting boundary (b) and external (e) nodes, u¨0e and u¨0

b are free field accelerations of external (e) and boundary (b) nodes, respectively and, and u0

b are free field displacements of external (e) nodes, and boundary (b) nodes, respectively and. The effective force Peff consistently replace forces from the seismic source with a set of forces in a single layer of finite elements surrounding the SSI model. The DRM is quite powerful and has a number of features that makes an excellent choice for SSI modeling:

Single Layer of Elements used for Peff. Effective nodal forces Peff involve only the submatrices Mbe, Kbe, Meb, Keb. These matrices vanish everywhere except the single layer of finite elements in domain Ω+ adjacent to Γ. The significance of this is that the only wave-field (displacements and accelerations) needed to determine effective forces Peff is that obtained from the simplified (auxiliary) problem at the nodes that lie on and between boundaries Γ and Γe

Only residual waves outgoing. Solution to the DRM problem produces accurate seismic displacements inside and on the DRM boundary. On the other hand, the solution for the domain outside the DRM layer represents only the residual displacement field. This residual displacement field is measured relative to the reference free field displacements. Residual wave field has low energy when compared to the full seismic wave field, as it is a results of oscillations of the structure only. It is thus fairly easy to be damped out. This means that DRM can very accurately model radiation damping.

This is significant for two more reasons:

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Large models can be reduced in size to encompass just a few layers of elements outside DRM boundary,

Residual unknown field can be monitored and analyzed for information about the dynamic characteristics of the soil structure system

Inside of DRM boundary can be nonlinear/inelastic. This is a very important conclusion, based on a fact that only change of variables was employed in DRM development, and solution does not rely on superposition.

All types of realistic seismic waves are modeled. Since the effective forcing Peff consistently replaces the effects of the seismic source, all appropriate (real) seismic waves are properly (analytically) modeled, including body (SV, SH, P) and surface (Rayleigh, Love, etc...) waves.

A Note on Free Field Input Motions for DRM. Seismic motions (free field) that are used for input into a DRM model need to be consistent. In other words, a free field seismic wave that is used needs to fully satisfy equations of motion. For example, if free field motions are developed using a tool (SHAKE, or EDT or SW4, or fk, &c.) using time step ∆t = 0.01s and then you decide that you want to run your analysis with a time step of ∆t = 0.001s, simple interpolation (10 additional steps for each of the original steps) might create problems. Simple linear interpolation actually might (will) not satisfy wave propagation equations and if used will introduce additional, high frequency motions into the model. It is a very good idea to generate free field motions with the same time step as it will be used in ESSI simulation.

Similar problem might occur if spacial interpolation is done, that is if location of free field model nodes is not very close to the actual DRM nodes used in ESSI model. Spatial interpolation problems are actually a bit less acute, however one still has to pay attention and test the ESSI model for free conditions and only then add the structure(s) on top.

7.4.11 Liquefaction and Cyclic Mobility Modeling

JIM will send examples of liquefaction modelling so that I can motivate this a bit more. Needs to be rewritten to reflect motivation and perhaps shorten already short description…

7.4.11.1 Introduction

Saturated soils should be analyzed using information about pore fluid pressure. Modeling of saturated soils, where pore fluid is able to move during loading (see note on effective and total stress analysis conditions in section 6.4.2) is best done using an effective stress approach.

Effective stress analysis, where stress in soil skeleton and pore fluid pressures are treated separately is appropriate for both drained and undrained (low permeability) cases. This approach is necessary if permeability of soil is such that one can expect movement of pore fluid and pore fluid pressures during dynamic event duration. For example, saturated sandy soils, will feature a

48

fully coupled behaviour of pore fluid and soil skeleton. It is expected that pore fluid and pore fluid pressure will move and, through the effective stress principle, such pore fluid pressure change will affect (significantly) response of soil skeleton. There are three main approaches to modeling fully coupled pore fluid – solid skeleton systems (Zienkiewicz and Shiomi, 1984):

u − p − U, where the main unknowns are displacements of porous solid skeleton (ui), pore fluid pressures (p), and displacements of pore fluid (Ui),

u−U where the main unknowns are displacements of porous solid skeleton (ui), and displacements of pore fluid (Ui),

u−p where the main unknowns are displacements of porous solid skeleton (ui), pore fluid pressures

(p).

Zienkiewicz and Shiomi (1984) notes pros and cons of each approach, and concludes that the u−p−U is the most flexible and accurate for dynamic events where there is a significant rate of change in pore fluid pressures and pore fluid and solid skeleton displacements, as is a case for soil structure interaction problems. In the case of u − U formulation, stumbling block is the high bulk stiffness of the fluid, which can create numerical problems. Those numerical problems are elegantly resolved in the u−p−U formulation through the inclusion of pore fluid pressure into the unknown field (although pore fluid pressures and pore fluid displacements are dependent variables). The u − p formulation is the simplest one, but has issues in treating problems with high rate of change of pore fluid pressure in space and time.

7.4.11.2 Liquefaction Modeling Details and Discussion

Modeling of liquefaction and its effects requires availability of computer programs that model behaviour of fully saturated, fully coupled pore fluid – porous solid materials (soils). Chapter 8 list some of the available programs. Modeling and simulation of liquefaction requires significant test data for soil. Both laboratory and in situ test data is required. Recent publications (Peng et al., 2004; Elgamal et al., 2002, 2009; Arduino and Macari, 1996, 2001; Jeremic et al., 2008; Shahir et al., 2012; Taiebat et al., 2010b,a; Cheng and Jeremic, 2009; Taiebat et al., 2010a) describe various possibilities in modeling of cyclic mobility and liquefaction effects. It is important to properly verify and validate modeling tools for coupled (liquefaction) modeling, as multi–physics modeling of these problems can be difficult and results interpretation requires full confidence in simulation programs (Tasiopoulou et al., 2015a,b).

7.4.12 Structure-Soil-Structure Interaction

(in phase, out phase, distance between structures relative to the surface wave length, etc.)

49

Soil-Structure-Soil-Structure Interaction (SSSSI) sometimes needs to be taken into account, as it might change levels of seismic excitation for adjacent NPPs. There are a number of approaches to model SSSSI.

• Direct Models. The simplest and most accurate approach is to develop a direct model all (two or more) structures on subsurface soil and rock, develop input seismic motions and analyze results. While this approach is the most involved, it is also the most accurate, as it allows for proper modeling of all the structure, foundation and soil/rock geometries and material without making any unnecessary simplifying assumptions.

The main issue to be addressed with this approach is development of seismic motions to be used for input. Possible approach to developing seismic motions is to use incoherent motions with appropriate separation distance. Alternatively, regional seismic wave modeling can be used to develop realistic seismic motions and use those as input through, for example the Domain

• Reduction Method (see section 6.4.10).

• Symmetry and Anti-Symmetry Models. These models are sometimes used in order to reduce complexity and sophistication of the direct model (see recent paper by Roy et al. (2013) for example). However, there are a number of concerns regarding simplifying assumptions that need to be made in order for these models to work. These models have to make an assumption of a vertically propagating shear waves and as such do not take into account input surface waves (Rayleigh, Love, etc). Surface waves will additionally excite NPP for rocking and twisting motions, which will then be transferred to adjacent NPP by means of additional, induced surface waves. If only vertically propagating waves are used for input (as is the case for symmetry and antisymmetry models) energy of input surface waves is neglected. It is noted that depending on the surface wave length and the distance between adjacent structures, a simple analysis can be performed to determine if particular surface waves, emitted/radiated from one structure toward the other one (and in the opposite direction) can influence adjacent structures. It is noted that the wave length can be determined using a classical equation λ = v/f where λ is the length of the (surface) wave, v is the wave speed1 and f is the wave frequency of interest. Table6.1 below gives Rayleigh wave lengths for four different wave frequencies (1,5,10,20 Hz and for three different Rayleigh (very close to shear) wave velocities (300,1000,2500 m/s):

TABLE 6.1: RAYLEIGH WAVE LENGTH AS A FUNCTION OF WAVE SPEED [M/S] AND WAVE FREQUENCY [HZ].

1 For Rayleigh surface waves, a wave speed is just slightly below the shear wave speed (within 10%, depending on elastic

properties of material), so a shear wave speed can be used for making Rayleigh wave length estimates.50

1.0Hz 5.0Hz

10.0Hz

20Hz

300m/s 300m 60m 30m 15m

1000m/s

1000m

200m 100m 50m

2500m/s

2500m

500m

250m 125m

It is apparent that for given separation between NPP buildings, different surface wave (frequencies) will be differently transmitted with different effects. For example, for an NPP building that has a basic linear dimension (length along the main rocking direction) of 100m, the low frequencies surface wave (1Hz) in soft soil (vs ≈ 300m/s) will be able to encompass a complete building within a single wave length, while for the same soil stiffness, the high frequency (20Hz) will produce waves that are too short to efficiently propagate through such NPP structure. On the other hand, for higher rock stiffness (vs ≈ 2500m/s), waves with frequencies up to approximately 5Hz, can easily affect a building with 100m dimension.

Further comments on are in order for making symmetric and antisymmetric assumptions (that is modeling a single building with one boundary having symmetric or antisymmetric boundary condition so as to represent a duplicate model, on the other side of such boundary):

Symmetry: motions of two NPPs are out phase and this is only achievable, if the wave length of surface wave created by one NPP (radiating toward the other NPP) is so large that half wave length encompasses both NPPs. This type of motions (symmetry) is illustrated in figure 6.4.12 below

FIG. 7.7: Symmetric mode of deformation for two NPPs near each other.

Antisymmetry: motions of two NPPs are in phase. This is achievable if distance between two NPPs is perfectly matching wave lengths of the radiated wave from one NPP toward the other one, and if the dimension of NPPs is not affecting radiated waves. This type of SSSSI is illustrated in figure 6.4.12 below

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FIG. 7.8: Anti-symmetric mode of deformation for two NPPs near each other.

Both symmetry and antisymmetry assumptions place very special requirements on wave lengths that are transmitted/radiated and as such do not model general waves (various frequencies) that can be affecting adjacent NPPs.

7.4.13 Simplified models

7.4.13.1 Simplified Models

Simplified models are used for fast prototyping and for parametric studies, as they have relatively low computational demand. It is very important to note that a significant expertise is required from analyst in order to develop appropriate modeling simplifications that retain mechanical behaviour of interests, while simplifying out model components that are not important (for particular analysis).

7.4.13.2 Simplified, Discrete Soil and Structural Models

Using discrete models can provide useful results for some aspects of mechanical behaviour while requiring relatively small computational effort. Discrete, simplified models can be used for analyzing many aspects of SSI behaviour of NPPs, provided that simplifications are made in a consistent way and that thus developed models can be verified for required modeling purposes.

P-Y and T-Z Springs. Simplified models using P-Y and T-Z springs (Bransby, 1999; Allen, 1985; Georgiadis, 1983; Stevens and Audibert, 1979; Bransby, 1996) have been used to model response of piles, pile groups and other types of foundations in elastic-plastic soil. The ideas is based on Winkler foundation springs that are now defined as nonlinear springs. Radial (transversal) behaviour is modeled using P-Y springs, while axial (longitudinal) behaviour is modeled using T-Z spings. It is important to note that calibration of P-Y and T-Z springs is done in full scale tests, for a given pile or foundation stiffness and for a given soil type. In that sense, P-Y and T-Z springs can be understood as post-processing (recording) of the actual response, that is then transferred to nonlinear springs behaviour, through a combination of nonlinear springs, dashpots, sliders and other components that are combined in order to mimic recorded P-Y and T-Z behaviour.

While P-Y and T-Z springs have been successfully used for modeling monotonic loading of single piles in uniform soils (sand and clay), their use for loading in different directions, in

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layered soils and for dynamics applications cannot be fully verified. Particularly problematic is P-Y behaviour in layered soils and for large pile groups Yang and Jeremic (2002, 2003); Yang and Jeremic (2005a).

7.4.13.3 Simplified, Continuum Soil Models

While modeling of SSI is best done using continuum models, there are number of simplifications that can be done as well.

Linear Elastic. One of the most commonly made simplification is to use linear elastic models for modeling of soil. While this might be appropriate for small seismic excitations, it is likely that models with more significant seismic load levels, where plastification will have larger effects, cannot be validated and that these models will miss significant aspects of behaviour that are important for understanding response.

Stiffness Reduction (G/Gmax) and Damping Curve Models. Plastification (not significant) can be taken into account using modulus reduction and damping curves. However, as noted in section 4.5, such results have to be carefully used as this material modeling simplification introduces a number of artifacts. For example amplification factors for 1D models using this approximations are known to be biased (Rathje and Kottke, 2008).

7.4.14 General guidance on soil structure interaction modelling and analysis

NOTE: There is a bit of this in Chapter 2 and then in 7.1 so either movet to 7.,1 or combine somehow

Model Development Model development requires development of a hierarchical set of models with gradual (!) in sophistication. As hierarchy of models are developed, each model set needs to be verified and be capable to (properly, accurately) model phenomena of interest.

Model Verification Model verification is used to verify that mechanical features that are of interest are indeed properly modeled. In other words, model verification is required to prove that results obtained for a given (developed) model are accurately modeling features of interest. For example, if propagation of higher frequency motions is required (analyzed), it is necessary to verify that developed (used) model is capable of propagating waves of certain wave lengths and frequency. Model verification is different than code and solution verification and validation, described in some details in Chapter 8 and also in much more detail in a number of references (Roache, 1998; Oberkampf et al., 2002; Oberkampf, 2003; Oden et al., 2005; Babuˇska and Oden, 2004; Oden et al., 2010a,b; Roy and Oberkampf, 2011). Code and solution verification is necessary (required) to prove that code (program) solves correctly used models. Validation provides evidence that the correct model is solved. Model verification has to be performed for

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each developed model in order to gain confidence that modeling results are usable for design, licensing, etc. Section 4.5 describes in some details procedures for model verification.

7.5 PROBABILISTIC Material Modeling

NOTE: This needs to be moved (at least in part to Chapter \3 where Alain talks about material modeling…

This part is also what has been mentioned/described

7.5.1 Introduction

Uncertainty is present in all phases of modeling and simulation of an earthquake soil structure interaction problem for nuclear facilities. There are two main sources of uncertainty:

• Modeling Uncertainty is introduced when simplifying modelling assumptions are made. In general, only simplifying modelling assumptions that do not introduce significant uncertainties, and inaccuracies in results should be used. However, that is not always the case. Significance of the influence of modeling uncertainties on results can only be assessed if higher sophistication models can be simulated and results compared with lower sophistication models (where modeling uncertainties were introduced).

• Parametric uncertainty is introduced when there are uncertainties in

Material modeling parameters, and

Loads (earthquakes).

Uncertainty in material modelling parameters are a result of measuring errors and transformation errors (Phoon and Kulhawy, 1999a,b; Fenton, 1999; Baecher and Christian, 2003), as well as spatial variability that contributes to uncertainties as averaging of material volume has to be done in order to determine material properties.

Parametric (material modeling) uncertainties can be significant. For example Figure 6.9 shows experimental data for a relationship between the Standard Penetration Test (SPT) and elastic modulus (Ohya et al., 1982). In addition, shown is a mean trend, as well as a histogram of a scatter with respect to the mean. It is important to note that significant uncertainty will be introduced in results if a mean relationship is used alone. For example, if one uses the above relationship, and has a soil with a blow-count of 10, elastic stiffness will be (according to the mean equation/relationship) approximately

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8,000 kPa, while test data shows that values can be as low as 1,000 kPa and as high as 33,000 kPa. Hence, assuming deterministic material property (elastic stiffness in this case), analyst will not model (probable) very soft or very stiff material response. In other words, parametric uncertainty has to be taken into account in order to provide full information about (probabilities of) response.

There are a number of approaches to take probabilistic response into account.

5 10 15 20 25 30 35

SPT N Value Residual (w.r.t Mean) Young’s Modulus (kPa)

FIG. 7.9: Left: Transformation relationship between Standard Penetration Test (SPT) N-value and pressure-meter Young’s modulus, E.

FIG. 7.10.Right: Histogram of the residual with respect to the deterministic transformation equation for Young’s modulus, along with fitted probability density function (PDF). From Sett et al. (2011b).

7.6 PROBABILISTIC RESPONSE ANALYSIS

7.6.1 Overview

In general, probabilistic response analysis is compatible with the definition of a performance goal for design and input requirements for assessments, such as seismic margin assessment (SMA) and seismic probabilistic risk analysis (SPRA). Section 1.3 discusses these needs in the context of performance goals defined probabilistically.

For design, one definition is to calculate seismic demand on structures, systems, and components (SSCs) as 80% non-exceedance probability (NEP) values conditional on the design basis earthquake ground motion (DBE), as specified in ASCE 4. Deterministic response analysis approaches specified in ASCE 4 are developed to approximate the 80% NEP response level based on sensitivity studies and judgment. The preferred approach is to perform probabilistic response analysis and generate the 80% NEP directly.

For BDBE assessments, SPRA or SMA methods are implemented.

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Young’s Modulus, E (kPa)

• For SPRA assessments, the full probability distribution of seismic demand conditional on the ground motion (UHRS or GMRS) is required. The median values (50% NEP) and estimates of the aleatory and epistemic uncertainties (or estimates of composite uncertainty) at the appropriate risk important frequency of occurrence are needed. The preferred method of development is by probabilistic response analyses as described in succeeding sections. An alternative is to rely on generic studies that have been performed to generate approximate values.

• Two approaches to SMA assessments are used, i.e., the conservative deterministic failure margin (CDFM) method and the fragility analysis (FA) method. For the CDFM, the seismic demand is defined as the 80% NEP conditional on the Review Level Earthquake (RLE). The procedures of ASCE 4 are acceptable to do so. For the FA method, the full probability distribution is required. In both cases, the 80% NEP values are needed to apply the screening tables of EPRI NP-6041 (1991), which provide screening values for high capacity SSCs. Most applications of SMA use the CDFM method.

Soil-structure interaction includes the two most significant sources of uncertainty in the overall seismic response analysis process, i.e., the definition of the ground motion and characterization of the in situ soil geometry and its material behaviour. Throughout this document, aleatory and epistemic uncertainties are identified in the site response and SSI models and analyses. Implementing probabilistic response analysis permits the analysts to explicitly take into account some of these uncertainties.

Commentary. In addition to the above described applications of probabilistic analysis methods, to have a rational consistent approach to applying risk-informed techniques to decision-making, requires that the seismic demand on SSCs be quantified probabilistically so that the industry (Licensee and Regulator) knows the likelihood of exceedance of a given loading condition (excitation) applied to individual SSCs and that the likelihood of exceedance when combined with the seismic design and fragility parameters leads to a balanced design.

In today’s world, the framework requires that each SSC be individually designed to meet the code, with nothing in the way of cognizance of how that SSC fits into the overall seismic risk profile, beyond the overall classification of an SSC as either “safety-related” or not. Finally, because the design codes have been developed by different code committees (ASME, ASCE, IEEE, ANS, ACI, etc.) at different times using different ideas, there is little in the way of consistency in the margins that these design codes introduce at or above the design basis, and little in the way of “working together” to produce a rational performance-based design.

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7.6.2 Simulations of the SSI Phenomena

Generally, some type of simulation is performed to calculate probabilistic responses of SSCs. The seismic analysis and design methodology chain is comprised of: (i) ground motion definition (amplitude, frequency content, primary earthquakes contributing to its definition - deaggregation of the seismic hazard, incoherence, location of the motion, site response analysis); (ii) SSI phenomena; (iii) soil properties (stratigraphy, material properties – low strain and strain-dependent; (iv) structure dynamic characteristics (all aspects); and (v) equipment, components, distribution systems dynamic behaviour (all aspects).

An important aspect of the elements of the seismic response process is that all elements are subject to uncertainties.

7.6.2.1 SMACS

Probabilistic methods to analyze or reanalyze SSCs to develop seismic demands for input to SPRA evaluations were pioneered in the late 1970s in the U.S. NRC Seismic Safety Margin Research Program (SSMRP). A family of computer programs called the Seismic Methodology Analysis Chain with Statistics (SMACS) was developed and implemented in the SPRA of the Zion nuclear power plant in the U.S. (Johnson et al., 1981). This method is often called the “SSMRP” method. The method is still used in the development of seismic response of SSCs of interest to the SPRA so-called Seismic Equipment List (SEL) items (Nakaki et al., 2010).

The SMACS methodology is based on analyzing NPP SSCs for simulations of earthquakes defined by acceleration time histories at appropriate locations within the NPP site. Modeling, analysis procedures, and parameter values are treated as best estimate with uncertainty explicitly introduced. For each simulation, a new set of soil, structure, and subsystem properties are selected and analyzed to account for variability in the dynamic properties of the soil/structure/subsystems.

For the SSI portion of the seismic analysis chain, the substructure approach is used and the basis for the SMACS SSI analysis is the CLASSI (Continuum Linear Analysis for Soil Structure Interaction) family of computer programs (Wong and Luco, 1980). CLASSI implements the substructure approach to SSI (see Sec. 6.4).

The basic steps of the SMACS methodology are:

• Seismic Input

• The seismic input is defined by sets of earthquake ground motion acceleration time histories at a location in the soil profile defined in the PSHA or the site

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response analyses (Chapter 5). Three spatial components of motion, two horizontal and the vertical, comprise a set.

• The number of earthquake simulations to be performed for the probabilistic analyses is defined. Thirty simulations generally provide a good representation of the mean and variability of the input motion and the structure response. Although recent studies have suggested that a greater number of simulations improves the definition of the probability distributions of the response quantities of interest.

• Often, it is advantageous to use the deaggregated seismic hazard as the basis for the definition of the ground motion acceleration time histories as discussed in Chapter 5.

• Development of the median soil/rock properties and variability.

• Chapter 3 describes site investigations to develop the low strain soil profiles – stratigraphy and other soil properties. Chapter 3, also, describes laboratory testing of soil samples the results of which form the basis for defining the strain-dependent soil material properties. For the idealized case (semi-infinite horizontal layers overlying a half-space), Chapter 5 describes site response procedures to develop an ensemble of probabilistically defined equivalent linear viscoelastic soil profiles – individual profiles from the analyses and their derived median-centered values and their variability. These viscoelastic properties are defined by equivalent linear shear moduli and material damping - and Poisson’s ratios.

• The end result of the process is used to define the best estimate and variability of the soil profiles to be sampled in the SMACS analyses.

• Structure model development.

The structure models are general finite element models. The input to the SMACS SSI analyses is the model geometry, mass matrix, and fixed-base eigen-system. Adequate detail is included in the model to permit the generation of structure seismic forces for structure fragility evaluation and in-structure response spectra (ISRS) for fragility evaluation of supported equipment and subsystems.

The structure model is developed to be median-centered for a reference excitation corresponding to an important seismic hazard level for the SPRA or SMA. Stiffness properties are adjusted to account for anticipated cracking in reinforced concrete elements and modal damping is selected compatible with this anticipated stress level.

• SSI parameters for SMACS: 58

• Foundation input motion. The term “foundation input motion” refers to the result of kinematic interaction of the foundation with the free-field ground motion. The foundation input motion differs from the free-field ground motion in all cases, except for surface foundations subjected to vertically incident waves. The motions differ for two reasons. First, the free-field motion varies with soil depth. Second, the soil-foundation interface scatters waves because points on the foundation are constrained to move according to its geometry and stiffness. The foundation input motion is related to the free-field ground motion by means of a transformation defined by a scattering matrix.

• Foundation impedances. Foundation impedances describe the force-displacement characteristics of the soil/rock. They depend on the soil/rock configuration and material behaviour, the frequency of the excitation, and the geometry of the foundation. In general, for a linear elastic or viscoelastic material and a uniform or horizontally stratified soil deposit, each element of the impedance matrix is complex-valued and frequency dependent. For a rigid foundation, the impedance matrix is 6 x 6, which relates a resultant set of forces and moments to the six rigid body degrees-of-freedom.

• The standard CLASSI methodology is based on continuum mechanics principles and is most applicable to structural systems supported by surface foundations assumed to behave rigidly. Hybrid approaches exist to calculate the scattering matrices and impedances for embedded foundations with the computational efficiency of CLASSI. Median-centered foundation impedances and scattering functions are calculated using the embedment geometry and the median soil profile.

• Statistical sampling and Latin Hypercube experimental design.

Input to this step in the SMACS analyses are the number of simulations (N), the variability of soil/rock stiffness and damping, and the variability of the structure’s dynamic behaviour (structure frequencies and damping). As presented above, the median-centered properties of the soil and structure are derived in initial pre-SMACS activities. The probability distributions of the soil stiffness, soil material damping, structure frequency, and structure damping are represented by scale factors with median values of 1.0 and associated coefficients of variation. Stratified sampling is used to sample each of the probability distributions for the defined parameters. Latin Hypercube experimental design is used to create the combinations of samples for the simulations. These sets of N combinations of parameters when coupled with the ground motion time histories provide the complete probabilistic input to the SMACS analyses.

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• SMACS analyses.

Using the free-field time histories, median-centered structural model, median-centered SSI parameters, and experimental design obtained above, perform N SSI analyses. For each simulation, time histories of seismic responses are calculated for quantities of interest: (i) internal forces, moments, and stresses for structure elements from which peak values are derived; (ii) acceleration time histories at in-structure locations, which provide input for subsystem analyses, i.e., equipment, components, distribution systems, in the form of time histories, peak values, and in-structure response spectra (ISRS).

On completion of the SMACS analysis of N simulations, the N values of a quantity of interest, e.g., peak values of an internal force, ISRS at subsystem support locations, are processed to derive their median value, i.e., 50% non-exceedance probability (NEP), and other NEP values, e.g., the 84% NEP. This then permits fitting probability distributions to these responses for combination with probability distributions of capacity thereby creating families of fragility functions for risk quantification. The specific requirements of the SPRA, guide the identification of these probabilistically-defined in-structure response quantities based on structure and equipment fragility needs.

• Example results

Nakaki et al. (2010) present the SMACS probabilistic seismic response analyses of selected Muhleberg Nuclear Power Plant structures for the purposes of the SPRA.

• For the Muhleberg site, the ensemble of X-direction seismic input motions displayed as response spectra (5% damping) is presented in Figure 6.5-1. The input motion variability is apparent.

• For the Muhleberg NPP Reactor Building, a representative ensemble of ISRS for the X-direction at Elevation 8m (approximately 20m above the basmat) is shown in Figure 6.5-2.

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FIG 7.5-1 Mühleberg Nuclear Power Plant – Ensemble of free-field ground motion response spectra – X-direction, 5% damping

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FIG. 7.5-2 Mühleberg Nuclear Power Plant – Ensemble of in-structure response spectra (ISRS) – X-direction, 5% damping – median and 84% NEP

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7.6.2.2 Monte Carlo

Monte Carlo approach is used to estimate probabilistic site response, when both input motions (rock motions at the bottom) and the material properties are uncertain. For a simplified approach, using equivalent linear (EqL) approach (strain compatible soil properties with (viscous) damping, a large number of combinations (statistically significant) of equivalent linear (elastic) stiffness for each soil layer are analyzed in a deterministic way. In addition, input loading can also be developed into large number (statistically significant) rock motions. Large number of result surface motions, spectra, etc. can then be used to develop statistics (mean, mode, variance, sensitivity, etc.) of site response. Methodology is fairly simple as it utilizes already existing EqL site response modeling, repeated large number of times. There lies a problem, actually, as for proper (stable) statistics, a very large number of simulations need to be performed, which makes this method very computationally intensive.

While Monte Carlo method can sometimes be applied to a 1D EqL site response analysis, any use for 2D or 3D analysis (even linear elastic) creates an insurmountable number of (now more involved, not 1D any more) of simulations that cannot be performed in reasonable time even on large national supercomputers. Problems becomes even more overwhelming if instead of linear elastic (equivalent linear) material models, elastic-plastic models are used, as they feature more independent (or somewhat dependent) material parameters that need to be varied using Monte Carlo approach.

7.6.2.3 Random Vibration Theory

Random Vibration Theory (RVT) is used for evaluating probabilistic site response. Instead of performing a (statistically significant) large number of deterministic simulations of site response (all still in 1D), as described above, RVT approach can be used Rathje and Kottke (2008). RVT uses Fourier Amplitude Spectrum (FAS) of rock motions to develop FAS of surface motions. Developed FAS of surface motions can them be used to develop peak ground acceleration and spectral acceleration at the surface. However, time histories cannot be developed, as phase angles are missing.

7.6.2.4 Stochastic Finite Element Method

Instead of using Monte Carlo repetitive computations (with high computational cost), uncertainties in material parameters (left hand side (LHS) and the loads (right hand side, RHS), can be directly taken into account using stochastic finite element method (SFEM). There are a number of different approaches to SFEM computations. Perturbation approach was popular in early formulations of SFEM (Kleiber and Hien, 1992; Der Kiureghian and Ke, 1988; Mellah et al., 2000; Gutierrez and De Borst, 1999), while spectral stochastic finite element method (SSFEM) proved to be more versatile (Ghanem and Spanos, 1991; Keese and Matthies, 2002;

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Xiu and Karniadakis, 2003; Debusschere et al., 2003). Review of various issues with different SFEM approaches were recently provided by Matthies et al. (1997); Stefanou (2009); Deb et al. (2001); Babuska and Chatzipantelidis (2002); Ghanem (1999); Soize and Ghanem (2009). Most previous approaches pertained to linear elastic uncertain material. Recently, (Sett et al., 2011a) developed Stochastic Elastic Plastic Finite Element Method (SEPFEM), that can be used for modeling of seismic wave propagation through inelastic (elastic-plastic) stochastic material (soil).

Both SFEM and the SEPFEM rely on discretization of finite element equations in both stochastic and spatial dimensions. In particular, (a) Karhunen–Lo`eve (KL) expansion (Karhunen, 1947; Lo`eve, 1948; Ghanem and Spanos, 1991) is used to discretize the input material properties random field into a number of independent basic random variables, (b) Polynomial chaos (PC) expansion (Wiener, 1938; Ghanem and Spanos, 1991) is used to discretize degrees of freedom into stochastic space(s), and (c) (classical) shape functions (Zienkiewicz and Taylor, 2000; Bathe, 1996b; Ghanem and Spanos, 1991) are used to discretize the spatial components of displacements into the (above) polynomial chaos expansion.

In addition to the above, SEPFEM relies on Probabilistic Elasto Plasticity Jeremic et al. (2007); Sett et al. (2007); Jeremic and Sett (2009); Sett and Jeremic (2010); Sett et al. (2011b,a); Karapiperis et al. (2016) to solve for probabilistic elastic-plastic problem and supply probability distributions of stress and stiffness to finite element computations.

Both SFEM and SEPFEM provide very accurate results in terms of full probability density functions (PDFs) of main unknowns (Degrees of Freedom, DoFs) and stress (forces). Of particular importance is the very accurate calculation of full PDF, which supplies accurate tails of PDF, so that Cumulative Distribution Functions (CDFs, or fragilities) can be accurately obtained. However, while SFEM and SEPFEM are (can be) extremely powerful, and can provide very useful, full probabilistic results (generalized displacements, stress/forces), it requires significant expertise form the analyst. In addition, significant site characterization data is needed in order for uncertain (stochastic) characterization of material properties.

If such data is not available, one can of source resort to (non-site specific) data available in literature (Baecher and Christian, 2003; Phoon and Kulhawy, 1999a,b). However, use of non-site specific data significantly increases uncertainties (tails of material properties distributions become very ”thick”) as data is now obtained from a number of different, non-local sites, and is averaged, a process which usually increases variability.

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References for 6 (Pecker)

[1] Kausel, E., and Roësset, J.M. (1974). Soil Structure Interaction Problems for Nuclear Containment Structures, Electric Power and the Civil Engineer, Proceedings of the ASCE Power Division Conference held in Boulder, Colorado.

[2] Tabatabaie-Raissi, M. (1982). The Flexible Volume Method for Dynamic Soil-Structure Interaction Analysis, Ph.D. Dissertation, University of California, Berkeley.

[3] Chin, C.C. (1998). Substructure Subtraction Method and Dynamic Analysis of Pile Foundations, Ph.D. Dissertation, University of California, Berkeley.

[4] Lysmer, J. (1978). Analytical procedures in soil dynamics, ASCE Specialty conference on Earthquake Engineering and Soil Dynamics, vol III, 1267-1316, Pasadena, California.

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[5] Tajirian, F. (1981). Impedance Matrices and Interpolation Techniques for 3-D Interaction Analysis by the Flexible Volume Method, Ph.D. Dissertation, University of California, Berkeley.

[6] Ostadan, F. (2006). SASSI 2000, A System for Analysis of Soil-Structure Interaction, Theoretical manual

References for 6.3 (Johnson)

1. Wong, H.L., and Luco, J.E., “Soil-Structure Interaction: A Linear Continuum Mechanics Approach (CLASSI),” Report CE, Department of Civil Engineering, University of Southern California, Los Angeles, 1970.

2. Johnson, J.J., ”Soil Structure Interaction,” Earthquake Engineering Handbook, Chap. 10, W-F Chen, C. Scawthorn, eds., CRC Press, New York, NY, 2003.

3. Ostadan, F., “SASSI2000, A System for Analysis of Soil-Structure Interaction,” Version 1, 1999.

4. Luco, J.E. et al., “Dynamic Response of Three-Dimensional Rigid Embedded Foundations,” Final Research Report, NSF Grant – NSF ENV 76-22632, University of California, San Diego, 1978.

References for 6.5 (Johnson)

ELECTRIC POWER RESEARCH INSTITUTE, “A Methodology for Assessment of Nuclear Power Plant Seismic Margin,” Report EPRI-NP-6041-SL, Rev. 1, Palo Alto, California, 1991.

Johnson, J. J., Goudreau, G. L., Bumpus, S. E., and Maslenikov, O. R. (1981). “Seismic Safety Margins Research Program (SSMRP) Phase I Final Report—SMACS—Seismic Methodology Analysis Chain With Statistics (Project VII).” UCRL-53021, Lawrence Livermore National Laboratory, Livermore, CA (also published as U.S. Nuclear Regulatory Commission NUREG/CR-2015, Vol. 9).

Nakaki, D. K., Hashimoto, P. S., Johnson, J. J., Bayraktarli, Y., and Zuchuat, O. (2010). “Probabilistic seismic soil structure interaction analysis of the Muhleberg nuclear power plant

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reactor and SUSAN buildings.” ASME Pressure Vessel and Piping 2010 Conference, Bellevue, WA, July.

Wong, H.L., and Luco, J.E., 1980, “Soil-Structure Interaction: A Linear Continuum Mechanics Approach (CLASSI),” Report CE79-03, Dept. of Civil Engineering, University of Southern California, Los Angeles.

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