COMPUTATIONAL MECHANICS New Trends and Applications S. Idelsohn, E. O˜ nate 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, Universit`a della Calabria 87030 Rende (Cs), Italy e-mail:
[email protected]
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
Paola Caracciolo and Emilio Turco
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 convenient1 . 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 paper2 and Argyris and his collaborators paper3 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 Martin4 is used. This theory, in the case of perfect elasto-plasticity, leads to the Haar-Karman holonomic solution5 . 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 paper1 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
Paola Caracciolo and Emilio Turco
element was proposed by Argyris and his collaborators3 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 paper3
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 ϕi =
1 (wi li − wj lj cos αk − wk lk cos αj ) 2Ae
(1)
where Ae represents the element area and wi , li and αi 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 ϕi , ϕj , ϕk in vector ϕe and nodal displacements wi, wj , wk in vector we , the following relation is obtained ϕ e = D e we
(2)
where De 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)
Paola Caracciolo and Emilio Turco
where D is the global compatibility matrix having nl rows and nn columns, nl and nn 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 De . 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 DT 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 1 2
ΦS =
Z S
¯ T E−1 mdA ¯ m
(5)
¯ collects mx , my and mxy in any points of the part S and where m
1 ν 0 Eh 0 E= ν 1 2 12(1 − ν ) 0 0 2(1 − ν) 3
(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 mx , my and mxy 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 mi = mx cos2 βi + my sin2 βi + 2mxy sin βi
(7)
where βi is the angle between x-axis and the outward normal to the i-side. Similar relations can be obtained for mj and mk . Collecting mi , mj and mk in vector ml rotation matrix R is automatically defined by means of the relation ¯ ml = Rm
(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 1 ¯ T E−1 m ¯ Φe = Ae m 2 4
(9)
Paola Caracciolo and Emilio Turco
or using the normal bending moments on sides of the e-th element 1 Φe = Ae mTl (R−1 )T E−1 RT ml 2
(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 mx , my and mxy as constants on the influence area of each node of the mesh, being the influence area of the i-th node defined as 1X e Ai = Ai (11) 3 and the sum has to be intended as extended on all elements around the node i, Aei 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 1 ¯i+m ¯ j) ml = RTl (m 2
(12)
where Rl is the column of the matrix R related to side l. Now, using the global equivalence relation between the deformation work written using ¯ n and ϕ ¯ n which collect, respectively, the side variables, ml and ϕl , and node variables m mx , my , mxy and ϕx , ϕy , ϕxy on the n-node nl X l=1
ml ϕl =
nn X n=1
¯ Tn ϕn m
(13)
and substituting in it the Eq.(12) nl X 1 l=1
2
¯i+m ¯ j ) ϕl = RTl (m
nn X n=1
¯ Tn ϕn m
(14)
the following relation is obtained once contributions related to each node are assembled nn X n=1
¯ Tn m
X 1
2
RTl ϕl
=
nn X n=1
¯ Tn ϕn m
(15)
The relation immediately above furnishes the definition of ϕn 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 1 π[w, m] = − mT Fm + mT Dw − f T w 2 5
(16)
Paola Caracciolo and Emilio Turco
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 DT m + f = 0
(17) (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 DT 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
Paola Caracciolo and Emilio Turco
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 Martin4 , 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 principle5 which, for the plate bending problem, can be expressed as Φ[m − mE ] =
1Z (m − mE )T F(m − mE )dS = min 2 S
(20)
where m represents the bending moment field in equilibrium with the external loads and which satisfy the yield condition, mE 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 mE 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 7
(21)
Paola Caracciolo and Emilio Turco
where p[λ] = p0 + λˆ p is the load vector described in a linear way by the initial load p0 , ˆ . The term s[u[λ]] represents the structural reaction the λ parameter and a linear part p 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 rj = p − s[uj ] ˜ −1 rj uj+1 = uj + K
(22) (23)
˜ is a symmetric and positive definite matrix chosen in such a way that the iterative where K process converges. ˜ the matrix Kj is defined as In order to precise the features of the iterative operator K s[uj+1 ] − s[uj ] = Kj (uj+1 − uj )
(24)
and substituting Eq.(24) in Eq.(23) the following equation is obtained ˜ −1 )rj rj+1 = (I − Kj K
(25)
So that the iterative process converges, i.e. krj k → 0, the spectral radius of the ˜ −1 has to be less than unity. Further algebraic manipulation leads to the matrix I − Kj K equivalent convergence condition6 ˜ 0 < Kj < 2K
(26)
Indicating with KE the elastic stiffness matrix and choosing the iterative matrix as ˜ = KE K ω
1≤ω
