### Analytic Modeling of Rock-Structure Interaction

#### Jeremy Isenberg

U.S. Bureau of Mines

R-7215-2299

August 1972

A number of recent advances in finite element theory and computer technology are combined into a computer program for analysing structures and cavities in rock. The program applies to general three-dimensional forms, considers non-linear material properties including joints, anisotropic and time-dependent material properties, gravity loading and sequence of construction or excavator. Example problems, demonstrating the ability of the program to reproduce ideal situations having closed-form, analytic solutions are solved.

### Applications of Parallel and Vector Algorithms in Nonlinear Structural Dynamics Using the Finite Element Method

### B.E. Healy, D.A. Pecknold, and R.H. Dodds, Jr.

University of Illinois at Urbana-Champaign

UILU-ENG-92-2011

September 1992

This research is directed toward the numerical analysis of large, three dimensional, nonlinear dynamic problems in structural and solid mechanics. Such problems include those exhibiting large deformations, displacements, or rotations, those requiring finite strain plasticity material models that couple geometric and material nonlinearities, and those demanding detailed geometric modelling.

A finite element code was developed, designed around the 3D isoparametric family of elements, and using a Total Lagrangian formulation and implicit integration of the global equations of motion. The research was conducted using the Alliant FX/8 and Convex C240 supercomputers.

The research focuses on four main areas:

- Development of element computation algorithms that exploit the inherent opportunities for concurrency and vectorization present in the finite element method;
- Comparison of the preconditioned conjugate gradient method to a representative direct solver;
- Investigation of various nonlinear solution algorithms, such as modified Newton-Raphson, secant-Newton, and nonlinear preconditioned conjugate gradient; and,
- Discovery of an accurate, robust finite strain plasticity material model.

### Elasto-Plastic Strain Hardening Mohr-Coulomb Model: Derivation and Implementation into the Finite Element Model using Principal Stress Space

#### Emil Smed Sørensen, Aalborg University

8 June 2012

The purpose of this report is to derive and implement a strain hardening Mohr-Coulomb model based on return mapping in principal stress space by the use of boundary planes. The report aims at modelling strain hardening rock material through a Mohr-Coulomb approximation of the generalized Hoek-Brown criterion. Firstly, the classification of rock materials as well as the generalized Hoek-Brown criterion are presented. Afterwards follows an introduction to the Mohr-Coulomb criterion and the approximations used for the generalized Hoek-Brown criterion. Next, the fundamentals of plasticity and hardening is presented along with the theory behind return mapping in general stress space, including the derivation of the consistent constitutive matrix used in the global FEM equilibrium iterations. Then the advantages of return mapping in principal stress space is outlined. Following is the derivation of a non-associated isotropic strain hardening Mohr-Coulomb model based on the introduced theory. Finally, the derived model is implemented in two examples. The first example tries to model a strip footing while the second example models a tunnel excavation. The obtained results are compared with perfectly plastic solutions utilizing the peak and residual strength of the rock material.

### Evaluation of FEM Engineering Parameters from In situ Tests

#### F.C. Townsend, J. Brian Anderson, and Landy Rahelison

Florida Department of Transportation RPWO-14

December 2001

The purpose of this study was to take a critical look at in situ test methods (SPT, CPT, DMT, and PMT) as a means for developing finite element constitutive model input parameters. The first part of the research examined in situ test derived parameters with laboratory triaxial tests at three sites: Saunder’s Creek, Archer Landfill, and SW Recreation Centre. The triaxial tests on these sands were used to develop baseline input parameters. These parameters were verified by simulating the triaxial tests using two finite element codes. From these comparisons, the following conclusions were drawn:

- FEM simulations of triaxial test stress-strain curves produced excellent results.
- The hardening models (PLAXIS – Hardening Soil and PlasFEM – Sandler Dimaggio) simulated the non-linear behaviour better than the Mohr-Coulomb or Drucker-Prager models.
- In general, E50 triaxial test modulus values agreed with those estimated from DMT and PMT unloading tests, and
- FEM simulations of field PMT curves using triaxial test based parameters were unsuccessful. It was necessary to increase the triaxial E
_{50}values by Ω = 1.3078e^{0.0164pl }R^{2}= 0.8515, where Ω is the triaxial E_{50}modulus multiplier and pl is the PENCEL limit pressure.

The second phase of this study was to predict the deformations of a cantilevered sheet pile wall (unloading case), and the deformations of a 2-m diameter shallow footing (loading case). Conventional analyses methods were compared with the FEM using in situ test derived input parameters. Conclusions were:

- Conventional analyses (CWALSHT) under-predicted wall deformations unconservatively, while wall deflections were accurately predicted by using the Hardening Soil Model with input parameters estimated from SPT correlations and “curved matched” PMT values.
- Fundamentally, the stress history of a soil profile, i.e., OCR or preconsolidation pressure, must be known for any settlement prediction either using conventional or finite element methods.
- Of the conventional methods for estimating settlements (CSANDSET), only the SPT based D’Appolonia, and Peck and Bazaraa methods provided reasonable estimates of the observed settlement.
- The conventional DMT method, which correlates OCR values, slightly overestimated measured settlements.
- None of the in situ test derived input parameters (SPT, CPT, DMT, and PMT) coupled with FEM Mohr-Coulomb or Hardening Soil models, accurately predicted the shallow footing settlements

### Evaluation of LS-DYNA Soil Material Model 147 Manual for LS-DYNA Soil Material Model 147

#### J.D. Reid and B.A. Coon, Midwest Roadside Safety Facility (MwRSF)

B.A. Lewis, S.H. Sutherland, and Y.D. Murray, APTEK, Inc.

FHWA-HRT-04-094 and FHWA-HRT-04-095

November 2004

This is a combination of two documents. One report is a user’s manual, the second report is a performance evaluation. The user’s manual, *Manual for LS-DYNA Soil Material Model 147*, thoroughly documents the soil model theory, reviews the model input, and provides example problems for use as a learning tool. The other, *Evaluation of LS-DYNA Soil Material Model 147*, comprises the performance evaluation for the soil model. It documents LS-DYNA parametric studies and correlations with test data performed by a potential end user of the soil model, along with commentary from the developer.

The performance evaluation was a collaboration between the model developer and the model evaluator. Regarding the model performance evaluation, the developer and evaluator were unable to come to a final agreement regarding the model’s performance and accuracy. (The material coefficients for the default soil result in a soil foundation that may be stiffer than desired.) These disagreements are listed and thoroughly discussed in section 9 of the second report. This report will be of interest to research engineers associated with the evaluation and crashworthy performance of roadside safety structures, particularly those engineers responsible for the prediction of the crash response of such structures when using the finite element code LS-DYNA.

### Finite Element Analysis of the Columbia Lock Pile Foundation System

#### Chandrakant S. Desai, Lawrence D. Johnson and Charles M. Hargett

U.S. Army Corps of Engineers Waterways Experiment Station

Technical Report S-74-6

July 1974

The Columbia Lock was designed as a gravity-type structure in which the load is transferred essentially through the foundation piles. Results obtained using Hrennikoff ‘s method did not agree closely with observed field data in terms of the distribution of loads in the piles. The finite element (FE) method was therefore used to predict the behavior of the lock structure. As an approximation and to avoid undue amounts of manpower and computer efforts, the three-dimensional system was idealized as a structurally equivalent two-dimensional plane strain system. The FE method simulated major steps of construction including the in situ stress condition, dewatering, excavation, construction of piles and lock, backfilling, filling the lock with water, and development of uplift pressures. Nonlinear behavior of soils and of interfaces between the lock and surrounding soils and between the piles and the foundation soils was introduced into the analysis. The distribution of load in piles in the FE analysis showed improved agreement with field data in comparison with the agreement shown by Hrennikoff’s method. The FE computations verified the trend shown by the observed field data that the piles on the backfill side carried an increased share of the applied load.

### Finite Element Analysis of Elasto-Plastic Soils

#### W. Allen Marr and John T. Christian

Massachusetts Institute of Technology

NASA Research Report R72-21

June 1972

Prediction of the stresses and displacements in soil masses resulting from changes in load are important in the design and construction of many civil engineering structures. Such predictions require the use of an appropriate constitutive relation which defines the stress-strain behaviour of the soil.

The behaviour of finite element models employing different constitutive relations to describe the stress-strain behaviour of soils is investigated. Three models, which assume small strain theory is applicable, include a non-dilatant, a dilatant and a strain hardening constitutive relation. Two models are formulated using large strain theory and include a hyperbolic and a Tresca elastic perfectly plastic constitutive relation.

These finite element models are used to analyse retaining walls and footings. Excellent solutions are obtained for the failure load of retaining walls in drained frictional material. Attempts to obtain the failure load of footings in drained frictional materials are only moderately successful. Good solutions are obtained for the failure of footings on purely cohesive soil using both the small strain and large strain formulations.

Methods of improving the finite element solutions are investigated. For non-linear problems better solutions can be obtained by using smaller load increment sizes and more iterations per load increment than by increasing the number of elements. Suitable methods of treating tension stresses and stresses which exceed yield criteria are discussed.

### Finite Element Formulation for the Analysis of Interfaces, Nonlinear and Large Displacement Problems in Geotechnical Engineering

#### Adolfo E. Zeevaert

PhD Dissertation

Georgia Institute of Technology

18 September 1980

The purpose of this study is to develop an analytical model that is able to predict the state of stresses and deformations of the soil-fabric system when the system is subjected to external loads. The finite element method is used to obtain the solution of this problem. The material design parameters used by the mathematical model are evaluated with appropriate laboratory tests that are in accordance with the constitutive laws of the analytical model. An important matter to be investigated is: To what extent and what is the mechanism developed by flexible membrane elements embedded in the soil to improve the performance of roadways during construction and of conventional roadways subjected to multiple load applications? The answer requires the knowledge of all design parameters including the geometry of the problem, material properties of all elements of the system, pressure distribution, number of load applications, fabric properties and the interface friction parameters. Environmental loads, drainage, pore pressures and climate will also affect the system. The present thesis presents the analytical solution of the soil-fabric system using the finite element method includes: nonlinear behaviour of soil and fabric materials, the interface behaviour of the soil-fabric system, shear transfer and potential slip at the interface, the membrane action of the fabric material, variation of stress distribution due to large displacements, “no tension” characteristics of the gravel and yielding of the elasto-plastic materials. The mathematical model is formulated for an axisymmetric solid structure with the capability of representing interfaces and fabric materials. The present finite element formulation does not include time effects due to viscosity or consolidation. Strain softening, inertia forces, effect of pore pressures or local effects of the gravel punching into the soft soil are also not included. The normality condition used implies that too high rates of dilation for cohesionless soil under drained conditions are obtained. The plasticity solution used is limited to small strain and small rotation of the elements.

### Finite Elements in Plasticity: Theory and Practice

Even after almost forty years, this is one of the best references on the subject.

#### D.R.J. Owen and E. Hinton

University College of Swansea, Wales

The purpose of this text is to present and demonstrate the use of finite element based methods for the solution of problems involving plasticity. As well as the conventional quasi-static incremental theory of plasticity, attention is given to the slow transient phenomenon of elasto-viscoplastic behaviour and also to dynamic transient problems. It is an attempt to present numerical solution techniques, which have been well tried and tested, for selected important areas of application.

### Geotechnical Analysis by the Finite Element Method

#### U.S. Army Corps of Engineers

ETL 1110-2-544

31 July 1995

The objective of this ETL is to provide a basis for understanding what can be learned from finite element analyses, what skills are required for its application, and what resources in terms of time, effort, and cost are involved. The emphasis is on practical applications of the method. Appendix A contains information as to how the FEM can be used in soil- structure interaction, embankment construction, and seepage analysis. Appendix A includes discussions on the details of finite element modelling, case histories, and a section which will help interested engineers find further information on how the FEM can help in the analysis of their problems.

### Improved Methods for Forward and Inverse Solution of the Wave Equation for Piles

#### Don C. Warrington

University of Tennessee at Chattanooga

August 2016

This dissertation discusses the development of an improved method for the static and dynamic analysis of driven piles for both forward and inverse solutions. Wave propagation in piles, which is the result of pile head (or toe) impact and the distributed mass and elasticity of the pile, was analysed in two ways: forward (the hammer is modelled and the pile response and capacity for a certain blow count is estimated) or inverse (the force-time and velocity-or displacement-time history from driving data is used to estimate the pile capacity.) The finite element routine developed was a three dimensional model of the hammer, pile and soil system using the Mohr-Coulomb failure criterion, Newmark’s method for the dynamic solution and a modified Newton method for the static solution. Soil properties were aggregated to simplify data entry and analysis. The three-dimensional model allowed for more accurate modelling of the various parts of the system and phenomena that are not well addressed with current one-dimensional methods, including bending effects in the cap and shaft response of tapered piles. Soil layering was flexible and could either follow the grid generation or be manually input. The forward method could either model the hammer explicitly or use a given force-time history, analysing the pile response. The inverse method used an optimization technique to determine the aggregated soil properties of a given layering scheme. In both cases the static axial capacity of the pile was estimated using the same finite element model as the dynamic method and incrementally loaded. The results were then analysed using accepted load test interpretation criteria. The model was run in test cases against current methods to verify its features, one of which was based on actual field data using current techniques for both data acquisition and analysis, with reasonable correlation of the results. The routine was standalone and did not require additional code to use.

### Introduction to Finite Element Methods

#### Carlos A. Felippa

Department of Aerospace Engineering Sciences and Centre for Aerospace Structures

University of Colorado

Although not a geotechnical presentation per se, this is one of the most straightforward and simplest presentations of the basics of the finite element method anywhere. In current use for an introductory course in finite elements, it is divided into four parts:

- Part I: The Direct Stiffness Method. This part comprises Chapters 1 through 11. It covers major aspects of the Direct Stiffness Method (DSM). This is the most important realization of FEM, and the one implemented in general-purpose commercial finite element codes used by practicing engineers. Following a introductory first chapter, Chapters 2-4 present the fundamental steps of the DSM as a matrix method of structural analysis. A plane truss structure is used as motivating example. This is followed by Chapters 5-10 on programming, element formulation, modelling issues, and techniques for application of boundary conditions. Chapter 11 deals with relatively advanced topics including condensation and global-local analysis. Throughout these chapters the physical interpretation is emphasized for pedagogical convenience, as unifying vision of this “horizontal” framework.
- Part II: Formulation of Finite Elements. This part extends from Chapters 12 through 19. It is more focused than Part I. It covers the development of elements from the more general viewpoint of the variational (energy) formulation. The presentation is inductive, always focusing on specific elements and progressing from the simplest to more complex cases. Thus Chapter 12 rederives the plane truss (bar) element from a variational formulation, while Chapter 13 presents the plane beam element. Chapter 14 introduces the plane stress problem, which serves as a test bed for the derivation of two-dimensional isoparametric elements in Chapter 15 through 18. This part concludes with an overview of requirements for convergence.
- Part III: Computer Implementation. Chapters 20 through 29 deal with the computer implementation of the finite element method. Experience has indicated that students profit from doing computer homework early. This begins with Chapter 5, which contains an Introduction to Mathematica, and continues with homework assignments in Parts I and II. The emphasis changes in Part III to a systematic description of components of FEM programs, and the integration of those components to do problem solving.
- Part IV: Structural Dynamics. This part, which starts at Chapter 30, is under preparation. It is intended as a brief introduction to the use of FEM in structural dynamics and vibration analysis, and is by nature more advanced than the other Parts.

### A Large Strain Plasticity Model for Implicit Finite Element Analyses

#### R.H. Dodds, Jr. and B.E. Healy, University of Illinois Champaign-Urbana

UILU-ENG-91-2001

January 1991

The theoretical basis and numerical implementation of a plasticity model suitable for finite strains and rotations are described. The constitutive equations governing J_{2} flow theory are formulated using strains-stresses and their rates defined on the unrotated frame of reference. Unlike models based on the classical Jaumann (or corotational) stress rate, the present model predicts physically acceptable responses for homogeneous deformations of exceedingly large magnitude. The associated numerical algorithms accommodate the large strain increments that arise in finite-element formulations employing an implicit solution of the global equilibrium equations. The resulting computational framework divorces the finite rotation effects on strain-stress rates from integration of the rates to update the material response over a load (time) step. Consequently, all of the numerical refinements developed previously for small-strain plasticity (radial return with subincrementation, plane stress modifications, kinematic hardening, consistent tangent operators) are utilized without modification. Details of the numerical algorithms are provided including the necessary transformation matrices and additional techniques required for finite deformations in plane stress. Several numerical examples are presented to illustrate the realistic responses predicted by the model and the robustness of the numerical procedures.

### Review of Finite Element Procedures for Earth Retaining Structures

#### Robert M. Ebeling

U.S. Army Corps of Engineers

Miscellaneous Paper ITL-90-5

December 1990

This miscellaneous paper presents a review of previous work in which the finite element method was used to analyse the soil-structure interaction of earth retaining structures such as U-frame locks, gravity walls, and basement walls. This method of analysis results in the computation of stresses and displacements for both the structure and the soil backfill. Applications of the procedure have shown the importance of modelling the actual construction process as closely as possible and the use of a nonlinear stress-strain soil model. Additional requirements include modelling the interface between the soil backfill and the wall using interface elements.

This paper also includes two recent applications of the finite element method for the analysis of earth retaining structures which are loaded so heavily that a gap develops along the interface between the base of the structure and its foundation. The results are compared to those computed using the conventional force equilibrium method of

analysis.

### Three-Dimensional Finite Element Program to Predict the Behaviour of Soils and Substructure Components

#### Richard Long and Peter Lai, Florida Department of Transportation

Michael McVay, Zafar Ahmad, Girish Bhanushali, Brian Basterrachea & Sayed Hashimi, University of Florida

State Project No.: 99700-3583-119

May 2001

The University of Florida’s Geotechnical Group has been developing a one, two and three dimensional finite element code for modeling general Geotechnical problems for the past eight years (1992-2000). Of interest is the ability to model the construction process, either embankment construction, or excavation, as well as the application of surface loads (tractions). Since many soils in Florida are saturated, it was important that the program be capable of modeling both the solid phase and the fluid phase under both static and dynamic loading. The outgrowth of this effort is the PlasFEM code. Due to the significant amount of information required for input (i.e. nodal coordinates, element connectivity, etc.) as well as voluminous output, a preprocessor (PlasGEN) and postprocessor (PlasPLOT) have also been developed. This contract was for the development of the preprocessor (PlasGEN). However, this report describes the user input, features and capabilities of PlasGEN, PlasFEM, and PlasPLOT as well. The computer system requirements for this suite of programs are a Pentium III computer with at least a 500 Mz CPU, and 128 Mb of RAM. A readme file is provided with the installation CD. Chapter 2 describes the capabilities of PlasFEM, Chapter 3 gives PlasGEN features, and Chapter 4 presents PlasPLOT’s capabilities. It is strongly suggested that the user work along with the user manual when reading Chapters 3 and 4. The example problems for these chapters are also provided on the installation CD. A line by line input guide to PlasFEM is also provided in Appendix A.