Three-dimensional problems in geomechanics

Three-dimensional problems in geomechanics

FINITE ELEMENTS IN ANALYSIS A N D DESIGN ELSEVIER Finite Elements in Analysis and Design 18 (1994) 31 40 Three-dimensional problems in geomechanics...

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FINITE ELEMENTS IN ANALYSIS A N D DESIGN

ELSEVIER

Finite Elements in Analysis and Design 18 (1994) 31 40

Three-dimensional problems in geomechanics I.M. Smith Engineerin9 Department, University of Manchester, UK

Abstract

Using PC hardware it is nowadays perfectlyfeasibleto carry out analyses of foundations, slopes and retaining walls in three dimensions if warranted. Some questions of solution algorithm and of three dimensional element selection are addressed first and then results of some typical analyses presented. The importance of results display is highlighted.

1. Introduction

Analysis of geotechnical problems in three space dimensions has usually been thought of as "too expensive". The reducing cost of hardware has rendered such reservations redundant and what matters now is threefold: the identification of problems in which three-dimensional analysis is really warranted, the choice of suitable elements and finally, and probably crucially, meaningful results display. The current generation of PC hardware is perfectly capable of carrying out nonlinear analyses in three dimensions. Of course software considerations are always important. Much "package" software is useless on restricted memory machines but efficient alternatives exist (Smith and Griffiths [-8] ). When it comes to graphical displays, much commercial software is again useless (typical systems absorb 64 Mb of core) but engineer-designed alternatives can be built using simple FORTRAN primitives and have the added advantage of portability.

2. Algorithm choice The limitations of "conventional" solution strategies for analyses of three-dimensional problems using finite elements are well known. Most (and nearly all commercial) programs still use an elimination strategy for solving the equilibrium equations of solid mechanics. This strategy is limited by the band (front)-width of the assembled equation coefficients and by the fact that, in general, Gaussian elimination algorithms are not easily vectorisable or parallelisable. Further, 0168-874X/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSD! 0 1 6 8 - 8 7 4 X ( 9 4 ) 00011-4

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I.M. Smith / Finite Elements in Analysis and Design 18 (1994) 31 40

out-of-core techniques are not attractive on modern machines and some simple extrapolation calculations will show that no readily-conceivable provision of real memory will be capable of storing the assembled equation coefficients of large, three-dimensional, finite element meshes. The alternative presently available is to consider iterative solution strategies for the equilibrium equations. Here, it pays to use symmetric calculation matrices even although the element "stiffness" matrices may be unsymmetrical, due to non-associated plasticity for example. For such symmetric positive definite coefficient matrices, conjugate gradient techniqt]es, with suitable preconditioning, offer a possibly efficient alternative to direct solution. For very large problems, they offer the only possible route to a solution. Hughes et al. [2] and Smith et al. [-9] discuss effective preconditioning strategies. The examples given later in this paper involve medium-sized three dimensional analyses (up to, say, 10000 degrees of freedom) for which direct solution strategies are still feasible and efficient.

3. Element choice Geotechnical engineering is dominated by "chunky" geometries which have usually been adequately analysed in two dimensions using quadrilaterals, thus alleviating directional dependency in meshes. The natural counterpart of the quadrilateral in 3-d is the hexahedral "brick". If a mesh of such bricks is created from tetrahedra, subdivision into four or five such basic tetrahedra is necessary as shown in Fig. 1. It should be realised that the modes of deformation of which such tetrahedra are capable, assuming the basic 4-node version, are very simplified. Fig. 2 shows these modes for a "corner" tetrahedron from Fig. 1 and a parent unit cube. The numbers in the boxes in Fig. 2 are the eigenvalues of the element stiffness matrix assuming Young's modulus of 1 and Poisson's ratio of 0.3. For the four-node tetrahedron, these eigenvalues are often of the form ~G or /~K where ~ and/3 are constants and G and K are Lam6's parameters. For more complex elements, the eigenvalues are invariably of the form 7G + 6K where 7 and 6 are constants, except for the case of pure volumetric straining.

Fig. 1. Brick dissection into tetrahedra.

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I.M. Smith/Finite Elements in Analysis and Design 18 (1994) 31 40

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A detailed study of the brick element family (Smith and Kidger [10]; Kidger and Smith [4]; Kidger and Smith [5]; Smith and Kidger [-11]) shows that a promising candidate for threedimensional plasticity analyses in geomechanics is the 14-node brick whose eigenmodes are shown in Fig. 3. Again the numbers in the boxes refer to the eigenvalues of the element stiffness matrix for a unit cube with E = 1 and v = 0.3. It can be seen that this element, which is exactly integrated using a 2 x 2 x 2 Gauss rule, is capable of a flexible range of deformation modes (but note that it does not degenerate well into a thin "plate" in the bending mode). The analyses described later in this paper have been done either using this 14-node element or the better known 20-node "serendipity" variety. In the latter case care is needed if "reduced" integration is employed.

4. Adaptive mesh refinement A possible limitation to the use of bricks in practice is that, at least for the h-version of adaptivity, suitable adaptive algorithms have not yet been developed. However, solutions obtained using higher order elements are rich in information as shown in Fig. 4. Here, instead of plotting displacement data at the nodes only, as is usual, a coarse mesh has been "regridded" using the

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I.M. Smith~Finite Elements in Analysis and Design 18 (1994) 31 40 ..''-.. p:.

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Fig. 5. Displacements into square excavation. element shape functions. A shear plane is clearly present and it could well be that a form of p-refinement could be effective in this case.

5.

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excavations

Figs. 5 and 6 illustrate the deformations leading to collapse of two 3-d excavations in undrained clay (modelled as an incompressible elastic - Tresca plastic material). In Fig. 5 the excavation is

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I.M. Smith/Finite Elements in Analysis and Desiqn 18 (1994) 31-40

!iili! I Fig. 6. Displacements into slot excavation.

Fig. 7. Eccentrically loaded block foundation.

square in plan whereas in Fig. 6 it is a rectangular slot. In both cases only a symmetrical quarter was analysed. Only nodal displacements have been plotted, illustrating the advantages of regridding as in Fig. 4. The pictures are quite illustrative, but confused by nodal concentrations and the inability of the viewer to distinguish between one plane and another in the third dimension. The question of visualisation is taken up later in the paper.

I.M. Smith/Finite Elements in Analysis and Desion 18 (1994) 31-40

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Fig. 8. Laterally loaded spud can foundation. 6. Rectangular and circular foundations Fig. 7 shows the deformed profile of an eccentrically loaded, buried block foundation which was analysed by Williams and Hicks [13] in competition with several other predictors (none of whom used finite element or similar numerical techniques). While the analyses illustrated a perennial problem in ground engineering-namely the difficulty of knowing accurately the in situ soil conditions sensible predictions of footing displacement, rotation and failure load were computed. Fig. 8 shows a finite element model of an offshore "spud can" foundation for a jack-up drilling platform. Here the problem concerns lateral loading of the foundation after various vertical preloadings to different proportions of the ultimate vertical capacity (also determined by computation). The computed results agreed closely with the well-known Brinch Hansen formulae for a drained, cohesionless, frictional soil. Both of the above examples were computed using the sophisticated constitutive soil model MONOT. Although only analyses of the drained condition are reported above, undrained or partially drained conditions can of course also be treated. 7. Reinforced retaining and nailed soil walls One of the most important developments in geotechnical engineering over the past 30 years has been the reinforcing of soil by metallic and later polymeric inclusions. When a wall is constructed

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I.M. Smith~Finite Elements in Analysis and Design 18 (1994) 31 40

Fig. 9. Displaced profile of extensibly reinforced wall.

from the base upwards incorporating reinforcement we speak of a "reinforced soil" wall whereas when it is built by excavation and placement of reinforcement into the in situ soil we speak of"soil nailing". For reinforced soil walls with stiff (usually steel) inclusions a design methodology has been evolved which is accurate at working loads. The forces in the inclusions balance, at every level, the active forces in the soil between. Thus a stiff monolith of reinforced soil is formed which is actually as effective as a gravity wall since little shear is transferred to the base of the wall near the toe. However, when the reinforcement is very flexible, a complex interaction is set up between reinforcement and soil within the reinforced mass. The reinforcement now does not accept the active thrust from the soil between and passes it on downwards as shear towards the toe of the wall. The presence of reinforcement inhibits formation of a shear plane (unless the reinforcement is more sparse than used in practice) but relatively large deformations result with a concentration of shear across the foundation towards the toe of the wall. The forces in the reinforcement can be quite small, except perhaps adjacent to the wall facing. The mechanism of deformation and load transfer is quite different from that which occurs with stiffly reinforced soil. A picture of the deformed shape of an extensibly reinforced wall is shown in Fig. 9 (see also Smith and Segrestin [-12]).

I.M. Smith~Finite Elements in Analysis and Desiyn 18 (1994) 31 40

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Fig. 10. Tensions, shear and moment in nailed wall.

It is necessary to do 3-d analysis of this problem because of the stress flow around the discrete strips of reinforcement. If a 2-d equivalent is sought, the reinforcement appears as infinite "plates" which inhibit stress flow. Alternatives of a "smeared" system in 2-d have also been tried and discarded. This problem is far from simple to analyse using pseudo-limit equilibrium analyses (Juran et al. [3]; discussion by Leshchinsky [-6] ). When soil is excavated and inclusions placed to form a "nailed" system, currently used inclusions are relatively stiff and the only design question arising is that of density and length of reinforcement. Analytically the problem is more difficult because of the unknown in situ stresses before excavation begins, and the more uncertain bond between the inserted nails and the soil. With these provisos, typical results for nail tension, shear stress and bending moment are shown in Fig. 10 for two walls with different densities of nailing. For the same vertical nail spacing of 1 m, horizontal spacings of 1 m and 2 m have been analysed. In the latter case the wall is close to failure at an excavation depth of 4 m when large bending moments (6 times those in the stable wall) are recorded in the vicinity of a potential rupture plane. Again this problem is not easy to analyse by any simplified limit equilibrium technique. For more details see Smith [-7] and Ho and Smith [-1]).

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1.M. Smith/Finite Elements in Analysis and Design 18 (1994) 31-40

8. Visualisation It will be clear f r o m the p r e c e d i n g t h a t visualisation of results is one of the m o s t i m p o r t a n t , if not the m o s t i m p o r t a n t feature of t h r e e - d i m e n s i o n a l analyses. Typical c o m m e r c i a l graphics o p e r a t i n g systems such as AVS need substantial core storage a n d c o u l d not be used at present on PCs. O t h e r systems will o n l y r u n on specific m a n u f a c t u r e r s ' h a r d w a r e . In the a u t h o r ' s l a b o r a t o r y , a n i m a t e d displays are p r o d u c e d o n P C h a r d w a r e using p r o g r a m s written in F O R T R A N and using only a few very simple graphics primitives. This seems to be a fruitful field for future graphics d e v e l o p m e n t s .

References [1] D.K.H. Ho and I.M. Smith, Modelling of soil nailing construction by 3-D finite element analysis, Proc. Int. Conll Retaining Structures, Cambridge, 1992. [-2] T.J.R. Hughes, R.M. Ferencz and J.O. Hallquist, "Large scale vectorized implicit calculations in solid mechanics on a Cray X-MP/48 utilizing EBE preconditioned conjugate gradients", Comput Methods Appl. Mech. Eng. 61, pp. 215 248, 1987. [3] 1. Juran, M.I. Halis and K. Farrag, "Strain compatibility analyses for geosynthetics reinforced soil walls", ASCE, J. Geotech. Eng. 116 (2), pp. 312 329, 1990. [-4] D.J. Kidger and I.M. Smith, "Eigenvalues of element stiffness matrices, Part I: 2-d plane elements", Engineering Computations, 9, 1992a. [5] D.J. Kidger and I.M. Smith, "Eigenvalues of element stiffness matrices, Part II: 3-d solid elements", Engineering Computations, 9, 1992b. ]-6] D. Leshchinsky, "Discussion of Juran et al." ASCE, J. Geotech. Eng. 118 (5), pp. 816-824, 1992. [7] I.M. Smith (1992). Some results of computations regarding reinforced soil, International Conjbrence: Geotechnics and Computers, Paris. [8] I.M. Smith and D.V. Griffiths, Programming the Finite Element Method, Wiley, Chichester, 2nd edn., 1988. ]-9] I.M. Smith S.W. Wong, I. Gladwell and B. Gilvary, "PCG methods in transient FE analysis, Part I: First order problems", Int. J. Numer Methods Eng. 28 (7), pp. 1557-1566, 1989. [10] I.M. Smith, and D.J. Kidger, "Properties of the 20-Node Brick", Int. J. Numer. Analytical Methods Geomech. 15, pp. 871 891, 1991. [11] I.M. Smith, and D.J. Kidger, "Elastoplastic analysis using the 14-Node brick element family", Int. J. Numer Methods Eng. 35, 1992. [-12] I.M. Smith, and P. Segrestin, Inextensible reinforcements versus extensible ties (FEM comparative analyses of reinforced or stabilized earth structures). Proc. Int. Symp. Earth Reinforcement Practice, Kyushu, 1992. [-13] I. Williams and M.A. Hicks, "Finite element prediction of a footing subjected to inclined loading", Geotechnik, 15, pp. 66 72, 1992.