A MIXED TRIANGULAR ELEMENT FOR THE ELASTO-PLASTIC ANALYSIS OF KIRCHHOFF PLATES

COMPUTATIONAL MECHANICS New Trends and Applications S. Idelsohn, E. Oñate and E. Dvorkin (Eds.) c CIMNE, Barcelona, Spain 1998 A MIXED TRIANGULAR ELEMENT FOR THE ELASTO-PLASTIC ANALYSIS OF KIRCHHOFF PLATES Paola Caracciolo and Emilio Turco Dipartimento di Strutture, Università della Calabria 87030 Rende (Cs), Italy e-mail: emilio@virgo.labmec.unical.it Key words: Mixed finite elements, Elasto-plastic problems, Kirchhoff plates. Abstract. A triangular mixed finite element model for the elasto-plastic analysis of Kirchhoff plates is presented. The proposed discretization is based on rigid in bending elements connected by means of rotational springs between the inter-element sides and constant bending moments on the influence area of each node. The continuity of the normal bending moment through the sides allows a standard Hellinger-Reissner formulation. The advantages are related to a reasonable accuracy and a very simple element algebra. Some numerical results, obtained by using extremal path theory and the arc-length strategy, allow an analysis of the element s performance in the evaluation of the collapse load. 1

1 INTRODUCTION Finite elements based on high order interpolation, in spite of their algebraic complexity and the number of variables bound to the element construction, are suitable for linear elastic problems characterized by solutions with a high degree of continuity. If the analysis is extended in plasticity the discontinuities in the constitutive law require a totally different background. In particular, the localization of the plastic strains, when the load is near to the collapse load, leads to the formation of isolated sliding surfaces. This consideration suggests the use of simple finite elements for which the only way to obtain an accurate solution remains related to a mesh with a high number of elements. Furthermore, since the nonlinear behaviour of the material is strongly influenced by the stress level, compatible elements do not appear to be particularly convenient 1. This reason leads to a preference for mixed finite elements, since when using them, generally, a more accurate reconstruction of the stress field should be expected. This paper proposes a simple triangular mixed finite element developed to analyse elasto-plastic Kirchhoff plates. This element was suggested in Kawai s paper 2 and Argyris and his collaborators paper 3 and characterized by a linear interpolation of the displacement field between the nodes of the triangle making it rigid in bending. Therefore, the flexural deformation is concentrated on the sides of the triangle and measured by the relative rotation between the elements adjacent to the side considered. The further hypothesis of constant bending moments on the influence area of each node and the continuity of the normal bending moment through the sides allows a standard Hellinger-Reissner formulation. The above hypotheses produce an element which has a simple algebra and therefore make it suitable for discretization with many elements. In order to reconstruct the equilibrium path of the structure the load evolution law is discretized into a sequence of finite steps. Starting with a known equilibrium point the extremal path theory developed by Ponter and Martin 4 is used. This theory, in the case of perfect elasto-plasticity, leads to the Haar-Karman holonomic solution 5. Along with these assumptions the arc-length strategy is used to evaluate the next equilibrium point and therefore the algorithm keeps its accuracy even near the limit point which is one of the most important aims of the analysis. In the next sections the triangular mixed elements will be presented and discussed along with the incremental approach particularized for the underlying problem. In order to assess the performance of the proposed mixed element some numerical tests, referring to the elastic perfectly plastic behaviour, are performed and their results are compared with some classic results for plates on regular domains. 2 A SIMPLE TRIANGULAR MIXED FINITE ELEMENT The features of the finite element presented below derive essentially from the Kawai s paper 1 which suggests the use of rigid finite elements connected by means of rotational springs over the contact area of two neighbouring elements. A particular type of Kawai s 2

element was proposed by Argyris and his collaborators 3 in the simulation of car crash phenomena. This element is precious in its field of application but the elasto-plastic problem of Kirchhoff plates requires a less simplified element. Following Argyris paper 3 Figure 1: Geometric, kinematic and static variables for the triangular element the plate is subdivided into a suitable number of rigid in bending triangular elements connected with rotational springs on the sides of elements. Owing to the indeformability of elements, the bending deformation is measured by the change of angle between two elements which have a common side. Compatibility equations link the displacements of the element nodes to the change of angle along each side of the triangle. By referring to Figure 1 for the notation, for each element, in the hypothesis of small displacements, the side rotation for the i-th side is given by ϕ i = 1 2A e (w i l i w j l j cos α k w k l k cos α j ) (1) where A e represents the element area and w i, l i and α i are, respectively, the out of plane displacement, the length and the angle relative to the i-node. Two similar equations are obtained for the side j and k, by opportunely exchanging the indices. Therefore collecting side rotations ϕ i, ϕ j, ϕ k in vector ϕ e and nodal displacements w i, w j, w k in vector w e, the following relation is obtained ϕ e = D e w e (2) where D e represents the element compatibility matrix. Now, if vector ϕ collects the change of angle on each side of the mesh and w the out of plane displacements, the compatibility is ensured if ϕ = Dw (3) 3

where D is the global compatibility matrix having n l rows and n n columns, n l and n n are, respectively, the number of sides and the number of nodes of the mesh. This matrix is obtained assembling the contributions deriving from the element compatibility matrix D e. It is important to note that the side rotation contributions derive only from the elements adjacent to the side considered. When the choice of the kinematic variables is made, the static variables can be derived in such a way that they respect the duality of the virtual work theorem. For this reason, a vector f which collects the external nodal force and a bending moment vector m which collects the normal bending moment on the element sides are considered. The link between the static variables can be still derived from the virtual work theorem which gives equilibrium equations D T m = f (4) At this point it remains to specify the constitutive law of this discrete model. This can be done, for example, by defining the complementary elastic energy. Indeed, this energy referred to the surface S of the continuum problem can be written as Φ S = 1 m T E 1 mda (5) 2 S where m collects m x, m y and m xy in any points of the part S and E = Eh 3 12(1 ν 2 ) 1 ν 0 ν 1 0 0 0 2(1 ν) (6) where, following the usual notation, E represents the Young modulus, ν the Poisson ratio and h the plate thickness. Now, two possible ways can be followed introducing some hypotheses on the bending moment distribution. The first way assumes the bending moments m x, m y and m xy constant on the e-th element. This hypothesis define the link between the bending moments on the element sides in a unequivocal way. Indeed, looking at Figure 1 results m i = m x cos 2 β i + m y sin 2 β i +2m xy sin β i (7) where β i is the angle between x-axis and the outward normal to the i-side. Similar relations can be obtained for m j and m k. Collecting m i, m j and m k in vector m l rotation matrix R is automatically defined by means of the relation m l = R m (8) By using the hypothesis on constant bending moments on the element, the complementary elastic energy for the e-th element can be written as Φ e = 1 2 A e m T E 1 m (9) 4

or using the normal bending moments on sides of the e-th element Φ e = 1 2 A em T l (R 1 ) T E 1 R T m l (10) This approach uses as variables the vector of the nodal displacements w and the bending moments on the sides of the mesh collected in vector m. The second way assumes bending moments m x, m y and m xy as constants on the influence area of each node of the mesh, being the influence area of the i-th node defined as A i = 1 3 A e i (11) and the sum has to be intended as extended on all elements around the node i, A e i is the area of each of these elements. Furthermore, the normal bending moment on side l between the i-th and the j-th node of the mesh is defined as the average value m l = 1 2 RT l ( m i + m j ) (12) where R l is the column of the matrix R related to side l. Now, using the global equivalence relation between the deformation work written using thesidevariables,m l and ϕ l, and node variables m n and ϕ n which collect, respectively, m x, m y, m xy and ϕ x, ϕ y, ϕ xy on the n-node n l n n m l ϕ l = m T n ϕ n (13) l=1 n=1 and substituting in it the Eq.(12) n l l=1 ( ) 1 2 RT l ( m i + m j ) ϕ l = n n n=1 m T n ϕ n (14) the following relation is obtained once contributions related to each node are assembled n n ( ( )) 1 m T n n=1 2 RT l ϕ n n l = m T n ϕ n (15) n=1 The relation immediately above furnishes the definition of ϕ n and completes this second approach where the nodal displacements and the bending moments on nodes are assumed as independent variables. The two ways delineated above define the basic ingredient to generate a mixed finite element discretization. Indeed, by using the quantities defined above, the standard form of the Hellinger-Reissner functional can be written as π[w, m] = 1 2 mt Fm + m T Dw f T w (16) 5

where F is the flexibility matrix. It has to be pointed out that in the immediately above functional vectors and matrices have a different meaning for the two finite elements above-mentioned. Indeed, by referring in particular to the hypothesis of constant bending moments on each node, vector w collects the nodal displacements and vector m the constant bending moments on each node. Consequently, flexibility matrix F can be obtained by assembling the contributions of each node using Eq.(5) referred to the node influence area. Vector Dw represents the dual quantity to m and it can be obtained using Eq.(15) and Eq.(3). Once the stationarity condition is imposed compatibility equations and equilibrium equations are derived Fm + Dw = 0 (17) D T m + f = 0 (18) In order to solve the immediately above algebraic problem, the bending moment variables contained in vector m can be condensed by generating the system of equations D T FDw = f (19) which solved furnish the nodal displacement vector w and successively the bending moment vector m. The two elements delineated before, corresponding respectively to the hypothesis of moment constants on elements and on the influence area of each node, appear somewhat similar. However, the second way appears to be richer, since the bending moment on each side is obtained as an average between the nodal values and therefore represents a intermediate level between a constant and a linear function. Furthermore, since each node is independent of the other nodes and therefore the flexibility matrix F presents a diagonal block structure which, as will be made clear in the next Section, is useful in the solution of the elasto-plastic step of the incremental analysis. Finally, it has to be pointed out that the proposed mixed element respects the necessary condition for the stiffness matrix to have a sufficient rank. Indeed, this condition requires that the number of independent stress parameters be greater than or equal to the number of nodal displacement diminished by the number of rigid-body degrees of freedom. This condition is usually respected for both the finite elements presented above. 3 ELASTO-PLASTIC INCREMENTAL ANALYSIS The finite element analysis of elasto-plastic bodies subjected to monotonous loading can be performed following two ways. The first one derives essentially from the extension of the limit analysis which point to an accurate evaluation of the collapse load by completely ignoring the behaviour of the body before the collapse. The second approach follows an incremental iterative procedure which numerically simulates the elasto-plastic response to successive load increments. This approach appears 6

more attractive since it reconstructs the complete equilibrium path giving information not only on the limit point but also on the displacements and deformations reached. Furthermore, it is easily feasible for the most common computer. The incremental iterative approach requires that the loading path has to be discretized into a sequence of finite load increments. In each step, defined by the initial state and the load increment, a time interpolation law must be assumed, explicitely or implicitly, for the state variables: stresses and/or strains. This interpolation for the state variables is required since the elasto-plastic behaviour is path dependent. Naturally, various choices of the interpolation law produce different algorithms which give more or less accurate results. No interpolation law can be considered absolutely the best. Therefore, it could be useful to choose the interpolation law on the basis of computational convenience. A suggestive way, which is also convenient from the computational point of view, to define the loading path was suggested by Ponter and Martin 4, which proposed to follow the so-called extremal paths, which fulfil both the maximum strain work and the minimum complementary work in the loading step. For material which follows the elasto-perfectly plastic behaviour the solution on the extremal path is given by the Haar-Karman principle 5 which, for the plate bending problem, can be expressed as Φ[m m E ]= 1 (m m E ) T F(m m E )ds = min (20) 2 S where m represents the bending moment field in equilibrium with the external loads and which satisfy the yield condition, m E is the elastic solution which corresponds to the load increment and to the assigned initial state. By solving Equation (20) the elasto-plastic solution at the end of the incremental step can be obtained. Generally, the solution of this problem is very complicated, essentially because it is difficult to satisfy a priori all the equilibrium equations. However, this is not a complicated problem when finite elements are used. Indeed, in this case the Haar-Karman problem has to be solved in the particular case of nodal displacements assigned. This makes the calculation of m E straightforward by using the compatibility equations and the elastic constitutive law. Furthermore, being the displacement field assigned, no equilibrium equation has to be satisfied and the elasto-plastic bending moment m is bound to remain in the yield surface. These reasons lead, remembering the assumptions of moments constants on the influence area of each node, to solving the Haar-Karman problem for each node separately and making its solution simple. In other words, for each node the geometric problem of the contact between the Von Mises cylinder and the elastic complementary energy function has to be solved. Having defined the constitutive law in the incremental step it remains to satisfy the nonlinear equilibrium equations which can be written in general as p[λ] s[u[λ]] = 0 (21) 7

where p[λ] =p 0 + λˆp is the load vector described in a linear way by the initial load p 0, the λ parameter and a linear part ˆp. The term s[u[λ]] represents the structural reaction depending, clearly, on the nodal displacements vector and so on the λ parameter. The immediately above nonlinear equation, written for the generic step, can be tackled by means of a residual iterative scheme as r j = p s[u j ] (22) u j+1 = u j + K 1 r j (23) where K is a symmetric and positive definite matrix chosen in such a way that the iterative process converges. In order to precise the features of the iterative operator K the matrix K j is defined as s[u j+1 ] s[u j ]=K j (u j+1 u j ) (24) and substituting Eq.(24) in Eq.(23) the following equation is obtained r j+1 =(I K j K 1 )r j (25) So that the iterative process converges, i.e. r j 0, the spectral radius of the matrix I K K 1 j has to be less than unity. Further algebraic manipulation leads to the equivalent convergence condition 6 0 < K j < 2 K (26) Indicating with K E the elastic stiffness matrix and choosing the iterative matrix as K = K E 1 ω<2 (27) ω inequalities (26) are satisfied apart from the collapse zone where K j 0 and therefore a slow convergence has to be expected. This gap can be efficiently eliminated using the arc-length strategy proposed by Riks 7 for nonlinear elastic structures. Indeed, the loss of convergence of the numerical scheme derives from the parameter chosen to describe the equilibrium path, i.e. λ, sincethecurve is not analytical in this parameter. The main idea of the arc-length strategy consists in the choice of another parameter, the arc-length η, to describe the equilibrium curve: u = u[η] (28) λ = λ[η] (29) From this choice, the equilibrium equation (21) has to be completed with a constraint equation in order to be able to solve the problem. Different arc-length schemes are derived from different constraint equations. A simple and efficient way consists of choosing 8

the orthogonality between the extrapolation and the correction as constraint equation 6, deriving in this manner the following iteration scheme λ j+1 = λ j ω ut K 1 E r j (30) λ + ω u T û u j+1 = u j + ω ( K 1 E r j +(λ j+1 λj)û ) (31) where r j = p 0 + λ j ˆp s[u j ] (32) is the equilibrium error obtained by using λ j and u j and û = K 1 E ˆp. The quantities λ and u are the increments of the load parameter and displacements in the last step. The behaviour of the above described iterative process can be improved by choosing the parameter ω and the arc-length on the base of the previous steps as exposed in the Casciaro and Cascini paper 6. The mixed element exposed in Section 2 can be easily coded in the arc-length iteration scheme. Indeed, at the j-step the displacements vectors w j is assigned and the use of the compatibility equations provides the side rotations ϕ j = Dw j (33) and by means of the elastic constitutive law the elastic solution can be evaluated m Ej = m 0 + F 1 (ϕ j ϕ 0 ) (34) Successively, by using the Haar-Karman principle, the elasto-plastic solution m j can be found for each node. Finally, the structural reaction s can be derived from the equilibrium equations s[w j ]=D T m j (35) and the stiffness matrix of the mixed model can be used as iteration matrix K E = D T F 1 D (36) 4 NUMERICAL RESULTS In order to analyse the performance of the triangular mixed element presented in Section 2 a series of numerical tests, relative to simple form plates, was executed. Although, plates defined on complex domains could be analysed, these simple schemes are chosen since they allow a large comparison with results reported in technical literature. In Table 1 the normalized collapse load for a square plate varying the boundary conditions and the applied load are reported. In particular, the results for a simply supported and a clamped plate are reported and as load condition a distributed load q and a concentrated load P in the middle of the plate are considered. This Table reports the normalized 9

Figure 2: Mesh sequence used in the test problems collapse load (m u is the ultimate bending moment) obtained using meshes generated by an algorithm which subdivide each triangle in four triangles of the same form following the scheme reported in Figure 2. In particular, meshes having 41, 145 and 545 nodes are used. In the same Table are reported the results published by other authors using analytical calculation, lower and upper bounds, and numerical methods 8,9,10,11,12,13. The results reported in this Table suggest that the proposed finite element give quite accurate results. Figure 3 reports the normalized bending moments m x /m u and m y /m u on the middle line of a simply supported square plate in the collapse point. As can be seen from this Figure the curves obtained with the proposed model are very close to the curves reported in Casciaro and Di Carlo paper 12 that can be considered an accurate model as shown in Table 1. Finally, in Figure 4 are reported the equilibrium path and the evolution of the Von Mises function for a simply supported square plate loaded uniformly. The equilibrium curve is described by the load multiplier λ and the normalized middle point displacement w/w E where the index E refers to the elastic limit point. 5 CONCLUDING REMARKS The triangular mixed element proposed in this paper appears interesting when it is used in elasto-plastic analysis of plate bending problems. Indeed, elasto-plastic problems alway require discretization with many elements and therefore finite elements based on very simple algebra are advantageous if they are compared to more complicate finite elements. This advantage is raised when the analysis process requires a large number of operations on each element such as in the incremental iterative approach to the elastoplastic problems. The incremental approach to the elasto-plastic analysis becomes still more convenient when it is implemented by using the extremal path theory and the arclength strategy to reconstruct the loading path as far as the collapse load. 10

Clamped Simply supported Nodes ql 2 /m u P/m u ql 2 /m u 41 42.51 7.822 25.43 145 43.47 7.401 25.18 545 43.89 7.420 25.06 Hodge and Belytschko 8 42.86..49.25 7.120..7.831 24.86..26.54 Ranaweera and Leckie 9 41.04..44.26 23.88..25.06 Lubliner 10..52.01 23.81..27.71 Ang and Lopez 11 45.62 24.74 Casciaro and Di Carlo 12 43.63 7.438 25.03 Owen and Hinton 13 25.00 exact 7.257 Table 1: Collapse load for square plates The present work belongs to an initial stage of the investigation of the proposed model. Further extensions regard the study of plates defined on complex domains and the development of shell elements both in static and dynamic field. Acknowledgments The authors wish to thank Prof. Raffaele Casciaro for his suggestions and comments. 11

Figure 3: Bending moments for a square clamped plate on the A-B line 12

Figure 4: Equilibrium path and evolution of the Von Mises function 13

REFERENCES 1. R. Casciaro and L. Cascini, A Mixed Formulation and Mixed Finite Elements for Limit Analysis, Int. j. numer. methods eng., 18, 211-243, 1982. 2. T. Kawai, New Discrete Structural Models and Generalization of the Method of Limit Analysis, Finite Elements in Nonlinear Mechanics, P. G. Bergan et al. eds, Tapir publishers, 885-906, 1977. 3. J. Argyris, H. Balmer and I. S. Doltsinis, Some Thoughts on Shell Modelling for Crash Analysis, Computer methods in applied mechanics and engineering, 71, 341-365, 1988. 4. A. R. S.Ponter and G. B. Martin, Some extremal properties and energy theorems for inelastic materials and their relationship to the deformation theory of plasticity, Int. j. mech. phys. solids, 20/5, 281-300, 1972. 5. A. Haar and T. von Karman, Zur Theorie der Spannungzustande in Plastischen und Sandartingen Medien, Gottinger Nachrichten, math-phys. K1 1909. 6. R. Casciaro and L. Cascini, Limit analysis by incremental-iterative procedure, IU- TAM Conf. on Deformation and Failure of Granular Materials, Delft, 1982. 7. E. Riks, An Incremental Approach to the Solution of Snapping and Buckling Problems, Int. j. solids struct., 15, 524-551, 1979. 8. P. G. Jr. Hodge and T. Belyschko, Numerical Methods for the Limit Analysis of Plates, J. Applied Mechanics, 796-802, Dec. 1968. 9. M. P. Ranaweera and F. A. Leckie, Bound Methods in Limit Analysis, in Finite Element Techniques in Structural Mechanics, H. Tottenham and C. A. Brebbia eds. Proceedings of a seminar at the University of Southampton, Apr. 1970. 10. J. Lubliner, Plasticity Theory, Macmillan, New York 1990. 11. A. H. Ang and L. A. Lopez, Discrete Model Analysis of Elasto-Plastic Plates, Proc. ASCE, Eng. Mech. Div., 94, 271-293, 1968. 12. R. Casciaro e A. Di Carlo, Formulazione dell Analisi Limite delle Piastre come Problema di Minimax, Giornale del Genio Civile, 5, May 1971. 13. D. R. J. Owen and E. Hinton, Finite Element in Plasticity: Theory and Practice, Pineridge Press Limited, 1980. 14