7Methods and Models for Ssi Analysis

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1Contents

1Contents

7METHODS AND MODELS FOR SSI ANALYSIS

7.1BASIC STEPS FOR SSI ANALYSIS

7.2Direct methods (Jeremic)

7.2.1Linear and Nonlinear Discrete Methods

7.3SUB-STRUCTURING METHODS

7.3.1Principles

7.3.2Rigid or flexible boundary method

7.3.3The flexible volume method

7.3.4The subtraction method

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

7.3.6Discrete methods

7.3.7Foundation input motion

7.4SSI computational models

7.4.1Introduction

7.4.2Soil/Rock Linear and Nonlinear Modelling

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

7.4.4Contact Modeling

7.4.5Structures with a base isolation system

7.4.6Foundation models

7.4.7Small Modular Reactors (SMRs)

7.4.8Buoyancy Modeling

7.4.9Domain Boundaries

7.4.10Seismic Load Input

7.4.11Liquefaction and Cyclic Mobility Modeling

7.4.12Structure-Soil-Structure Interaction

7.4.13Simplified models

7.4.14General guidance on soil structure interaction modelling and analysis

7.5PROBABILISTIC Material Modeling

7.5.1Introduction

7.6PROBABILISTIC RESPONSE ANALYSIS

7.6.1Overview

7.6.2Simulations of the SSI Phenomena

Bibliography

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7METHODS AND MODELS FOR SSI ANALYSIS

7.1BASIC 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

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;

–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:

•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...).

•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

7.2Direct methods (Jeremic)

7.2.1Linear 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 − ρu¨i(6.1)

whereσij,j is a small deformation (Cauchy) stress tensor, fi are external (body (fiB) and surface (fiS) ) forces, ρ is material density and u¨i are accelerations. Inertial forces ρu¨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 δui 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.

7.2.1.1Finite 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 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 = FQP,Q= 1,2,...,(#ofDOFs)N (6.2)

whereMPQ 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 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 ElementsThere 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.

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).

7.2.1.2Finite 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.

7.2.1.3Nonlinear 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.

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 isclosed.

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

7.2.1.4Inelasticity, 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.