Plate-soil elastodynamic coupling using analysis S.F.A. Baretto, H.B. Coda, W.S. Venturini Sao Carlos School of Engineering, University ofsao Paulo, Sao Carlos - SP, Brazil BEM Abstract The aim of this work is to present an elastodynamic coupling of a plate in bending with the three-dimensional soil medium by using the Boundary Element Method formulation. This scheme is adopted in order to better represent some aspects of this kind of structural system, as for instance, the stress concentration. The formulation used to treat the soil media is based on time domain fundamental solution, while the dynamic behavior of the plate in bending is represented making use of the mass matrix approach. 1 Introduction The coupling of any structure with the soil medium is usually idealized by using the Finite Element Method (FEM) and the Boundary Element Method (BEM) formulations, respectively. Thus, one can take the advantages that each method presents to treat each particular sub-domain. For elastostatic cases, the first work reporting such procedure is due to ZIENKIEWICZ et alli.^. Since then, many other works have been presented in the literature improving the technique. However, the finite element formulation presents difficulties to exhibit some particular features of this coupling, for instance stress concentrations along the contact area contour. On the other hand, when one applies the boundary element technique, stress concentrations can be easily represented. In this way, PAIVA & BUTTERFIELD^ have proposed to couple both media, plate and soil, by using only elastostatic BEM formulations. In their work compatibility is enforced only for vertical displacements. An important feature close related to the stress concentration presence is the numerical instability when one is carrying out elastodynamics analysis. For this kind of study, the finite element technique presents less problems because this formulation reduces the stress concentration. On the other hand, when BEM is adopted, its formulation always leads to higher numerical instabilities.
364 Boundary Elements The aim of this work is to propose an elastostatics and elastodynamics coupling for plates in bending and the soil medium, where both sub-domains are treated by BEM. Using this combination one can check out the presence of stress concentrations and their numerical influence on the elastodynamics behavior. For this scheme, the plate is assumed rigid in its plane. The coupling is enforced by using the sub-region technique; the degrees of freedom of the threedimensional (soil) and the bidimensional (plate) media are appropriately combined. An elastostatic example is shown to illustrate the quality of the results obtained by using the proposed approach, comparing them with other numerical solutions. In addition, two elastodynamics problems are analyzed to point out the influence of the plate/soil stiffness ratio on the stress concentration and numerical stability. 2 Three-dimensional formulation For any isotropic body exhibiting a linear elastic behavior, the differential equation which governs its equilibrium, the so-called Navier-Cauchy equation, is given by (C,' -CJ)^,, +C^,^. +6, / p = 6, (for,,; = 1,2,3) (1) where bf represents the body forces, p is the media density, d and 2 are longitudinal and shear wave propagation velocities, respectively. In elastodynamics, it is usual to take a particular case of the Stokes' state [STOKES**] as fundamental solution. The usual way to find the desired fundamental solution is changing the body force term, in eqn(l), by a time dependent concentrated load given by where /(T) is a time-dependent load function, 6(s,q) is the Dirac's delta function and Ski the Kronecker's delta. The time-dependent load function adopted here is the one proposed by CODA & VENTURING, obtained by assuming, (2) /(r)=(#(r)-#(r+af))/af (3) By integrating eqn(l), after replacing the f(i) defined in eqn(3), one obtains the so-called Stokes' fundamental solution, as follows, where,,o/a/ (4)
Boundary Elements 365 4(,.,;,,0) = ^ (^-^- (r'-^ur-- I L ^U.^--l4^--lK^-4 (5) The GrafFi's reciprocity theorem for dynamic problems can be written for any two elastodynamics states, here represented by their displacement field u and w', in the case of quiescent past, as follows, %X9,4^ in which g is used to denote thefieldpoint taken along the boundary. In order to write the basic integral representation of displacements one has only to replace the body force term as in eqn(2) and the function /(T) by the particular time loading given in eqn(3). After replacing these values one has,,f;^^ (7) where Cki(Q,s) is the classical free term of the displacement integral representation that can be easily computed as in HARTMANN^. In order to achieve the final algebraic representations, tractions and displacements have to be approximated by, ^(<2V) = ^V'>/^ (8) in which j stands for the space elements, <t^ is the space approximation function, while v[/ (\ / = 1 is taken) gives the approximation along time. Flat quadratic elements with eight nodes have been adopted for the space discretization. Integrals over singular elements have been computed using a special algorithm where the finite-part concept, the Kutt's quadrature, KUTT^, was adopted for a small region around the load point, while the Gauss quadrature was taken to compute the remaining part of the element together with an appropriate element subdivision, CODA & VENTURING. For non-
366 Boundary Elements with an appropriate element subdivision, CODA & VENTURING For nonsingular integrals, the Gauss quadrature was also adopted with an element subdivision technique. As usual for boundary element formulations, the displacement integral representation, eqn (7), is transformed into an algebraic equation after choosing an appropriate number of load points and carrying out all the space and time integrals. So the classical BEM system of linear equations is achieved, as follows, H»U* = G*P* (9) where 9 ranges from 1 to the integration step number Nf. For any analyzed instant "t" the expression (9) can be written as: HU = GP+A (10) where "A" contains all the history of the movement represented in eqn (9). Eqn (10) has been presented in the standard form appropriate to implement the subregion technique. 3 Plate formulation For a linear elastic thin plate, the dynamic equilibrium is governed by the following differential equation, Z).AAw+ pw - g (11) where w is the transversal displacement, g the transversal load applied on the domain, p the plate density and D the plate rigidity. The fundamental solution employed for both, the elastostatics and elastodynamics cases, is the same one, due to DANSON^. By substituting the domain load g by the Dirac's delta function in eqn (11), in its elastostatic form, written in polar coordinates, one obtains, where r is the distance between the load point q and the field point/?. Applying the Betti's reciprocity theorem, and assuming that the variables employed in the plate formulation are: w and dw/ch (displacement derivative with respect to the outward normal direction) to define the displacement vector, and Vn (shear force) and Mn (bending moment) to define the effort vector, one can write the plate bending displacement integral representation as follows,
Boundary Elements 367! h (13) where ^c/ is the corner reaction and g* is the fundamental load. As it has been done for the soil formulation, in order to solve a plate problem by BEM it is necessary to transform eqn (13) into its algebraic representation. In this case, the plate boundary must to be discretized. For that quadratic approximation elements with linear geometry were adopted. As eqn (13) contains a domain integral with unknown density, domain discretization is needed as well. In this case, a cell similar to the surface element adopted for the 3D formulation was taken, generating therefore the mass matrix M The scheme adopted here to achieve algebraic representations avoids the slope integral equation usually employed for plate bending analysis with BEM Only non singular representation have been selected. Thus, for each node we have taken two collocation points, consequently achieving equal numbers of equations and unknowns. The clear advantage of employing this scheme is avoiding singular integration and keeping representation of the same nature. Schemes similar to that have already proved to give more precise numerical results, as can be seen in VENTURINI & PAIVA^. For the scheme adopted here the collocations are taken along the outward normal boundary direction as showed in figure 1. boundary load points inside points' corner /load point Figure 1. Plate geometry, discretization and collocation points Thus, using the proposed scheme one writes the displacement equation, eqn(13), only for outside and inside collocation points. In the first case, the domain integral containing the fundamental load, the third term of eqn (13) left hand side, becomes zero, while for the second case, it becomes equal to w(q). As there is no singular integral, the algebraic system can be numerically
368 Boundary Elements achieved by applying only the Gauss's quadrature with or without element subdivision, according to the required accuracy. Other important aspects to be emphasized are the treatment given to the corners reactions Rci and to the contact reaction and acceleration domain integrals. In order to take into account the corner reactions, one extra displacement equation is written for each node exhibiting geometric discontinuities. For those extra equations, the collocations are taken out of domain as well, following the corner's bisector direction, as shown in figure 1. This procedure generates one more integral equation for each corner. Thus, one has included more equations and more unknowns, corner displacements or reactions, but the final system of equation remains square. Regarding the domain integrals, they are computed numerically, dividing the domain into cells, over which the Gauss's quadrature is applied. These cells are nothing more than flat surface element; in particular, the cell here adopted corresponds to the flat quadrilateral with 8 nodes used to discretize the boundary of the soil domain. After performing the integrals over boundary elements and internal cells, and taking into account the corner values, one can write the global system due to the dynamic equilibrium, making use of the Newmark Method, as follows: (\-20)GP +/3GP + ~ ~ s ~ ~ S-\ J * H\U (14) ~ J ~ _,_, where H is the classical BEM matrix of the plate problem, M is the mass matrix, G is the matrix that contains the subgrade reaction coefficients, P gives the subgrade reactions, U is the displacement vector, At represents the time step and P the integration method parameter. Finally eqn (14) can be easily written, for any instant, in the form given in eqn (9). 4 Plate-soil interaction In this work, the sub-region technique was chosen to establish the coupling between the two media. In order to perform this interaction, one can connect the two regions making use of the equilibrium and displacement compatibility conditions, as follows, lj(v) _ fj(jf) and p(v) + p(j>) =o (15) Finally, one obtains the coupled matricial equation given by, system, giving origin to a single
Boundary Elements 369 //<»> -G<» 0 [0 0 0 (16) where the bar indicates prescribed loads on the interface, e gives the region not belonging to the coupling interface. The first superscript stands for the region of the corresponding interface values and the second indicates the adjacent subregion. The vector F contains all the prescribed value influences. 5 Numerical results For all examples presented in this paper we have adopted the same discretization characterized by: a) for BEM discretization we have taken 16 cells and 16 boundary elements as shown in figure 1; b) for FEM discretization, 96 HCT/CSTfiniteelements. In the first place an elastostatic example with a uniformly distributed load is analyzed and the behavior is compared with the PAIVA & BUTTERFIELD^ and CODA & VENTURING responses. This comparison is made to point out the formulation accuracy. To run this example the following plate geometry was adopted: side length equal to 12m, thickness equal to 0.1/m and the elastic parameters used are: ES = 0.26 x 10^ kn/m^ (soil elastic modulus), Ep = 9.78 x 10^ KN/m^ (plate elastic modulus), vp = vs = 0.3 (Poisson's coefficients). 3. I " " 2 2- PAIVA» CODA * BARRETTO Figure 2. Displacements (w) and subgrade reactions (Sr) along the axis A-B In next example the plate has side length equal to 10m, thickness equal to 0.5m and is subjected to a uniformly distributed load equal to 300 Kgf7nf. First of all, the elastostatic behavior along the direction A-B is shown to emphasize the marked differences in terms of stress concentration near the boundary. After that the elastodynamic analysis is carried out and the transversal displacement at the center of the plate, as well as the subgrade
370 Boundary Elements reaction at the middle of sides, are shown to illustrate the expected differences. For this elastodynamic analysis the time step used was equal to 0.005 s, while the elastic parameters adopted are given by: ES = 7.8 x 10" Kgf/m/s^ ; Ep = 7.8 x 10*0 Kgf/m/s^; vp = vs = 0.3 and the material densities by: p^ = 2500 Kg/nf; p,= 1800Kg/m*.! & 5 ^, A BARRETTO] - CODA I Figure 3. Displacements (w) and subgrade reactions (Sr) along the axis A-B! I -A BARRETTO - CODA,.. -BARRETTO -CODA Figure 4. Displacement (w) at the center Figure 5. Subgrade reaction (Sr) at the middle of the side 6 Conclusions The coupling of boundary element formulations for plate bending and threedimensional solid models has been shown for elastostatics and elastodynamics analyses. The numerical results achieved by using the implemented algorithm were compared with equivalent coupling where finite elements were adopted to represent the plate in bending. The static values shown that the BEM/BEM coupling can capture with accuracy the stress concentration along the boundary, while for BEM/FEM coupling the results are reduced over this region. However, the accuracy exhibited to model stress peak produces instabilities for BEM/BEM elastodynamics analysis. Working with damped values the BEM/FEM coupling is more appropriate to analyze this kind of problem.
Boundary Elements 371 7 References 1. Barretto, S. F. A. Elastostatic and elastodynamic analysis of plates, using the Boundary Elements Method, with soil-structure interaction. Ph. D. Thesis, University of Sao Paulo, Sao Carlos, 1995. 2. Brebbia, C. A. The boundary element method for engineers. London Pentech Press, 1978. 3. Coda, H. B. ; Venturini, W. S. Alternative boundary element formulation for elastodynamics. In: International Conference on Boundary Elements in Engineering, 12., Sapporo, Japan, 24-27 Sept., 1990. Proceedings. Southampton, CML; Berlin, Springer-Verlag, v. 1, p. 517-534, 1990. 4. Coda, H. B. Three-dimensional transient analysis of structures by a BEM- FEM 1993. combination. Ph. D. Thesis, University of Sao Paulo Sao Carlos 5. Coda, H. B. ; Venturini, W. S. Three-dimensional transient BEM analysis. Computers & Structures, v. 56, n. 5, p. 751-768, 1995. 6. Coda, H. B ; Venturini, W S. Numerical evaluation of flat singular boundary elements in elastostatics and elastodynamics, Boundary Elements communications, v. 6, p. 06-10, 1995. 7. Coda, H B ; Venturini, W. S. Non-singular time-stepping BEM for transient elastodynamic analysis. Engineering Analysis with Boundary Elements, v. 15, p. 11-18, 1995. 8. Coda, H.B. ; Venturini, W.S. Time domain BEM/FEM approach applied to elastodynamic analysis. Conference of Italian Group of Computational Mechanics, Padova, 1996 (to appear). 9. Danson, D. J Analysis of plate bending problems by direct boundary element method. Southampton. M. Sc. Dissertation - University of Southampton, 1979. lo.hartmann, F. Computing the C-matrix on non-smooth boundary points. In: Brebbia, C. A, ed. New developments in boundary element methods Southampton, CMP Publ, 1980. ll.kutt, H. R. WISK 178: quadrature formulae for finite-part integrals. Pretoria, National Research Institute for Mathematical Sciences. CSIR Special Report, 1975. 12.Paiva, J. B. ; Butterfield, R. Numerical analysis of plate-soil interaction. In: Papadrakakis, M. et Topping, B H V., eds. Advances in computational mechanics. (Proc. 2nd International Conference on Computational Structures Technology, Athens, Greece, 30th Aug. - 1st Sept.). Edinburgh, Civil-Comp Press, p. 275-281, 1994. 13.Patterson, C. ; Sheik, M. A. ; ScholGeld, R. P. On fae a^acof/om q/vac indirect discrete method for three dimensional design problems. In: 1st. Boundary Element Technology Conference, Adelaide, Australia. Proceedings. p. 211-223, 1985. 14. Stokes, G. G On the dynamical theory of diffraction. Transactions of the Cambridge Philosophical Society, v. 9, p. 1, 1849.
372 Boundary Elements 15.Venturing W.S. ; Paiva, J.B. Boundary element for plate bending analysis. Engineering Analysis with Boundary Elements, v. 11, n. 1, p. 1-8, 1993. 16.Warburton, G. B 77?2 dynamical behavior of structures. New York, Pergamon Press, 1964. IT.Zienkiewicz, O.C. ; Kelly, D.W. ; Bettess, P. The coupling of the finite element method and boundary solution procedures. Int. J. Num. Meth. Eng., v.ll, p. 355-375, 1977.