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Elastoplastic analysis of shells with the triangular element TRIC

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Elastoplastic analysis of shells with the triangular element TRIC
  Computational Methodsfor Shell and Spatial StructuresIASS-IACM 2000M. Papadrakakis, A. Samartin and E. Onate (Eds.)© ISASR-NTUA, Athens, Greece 2000 ELASTOPLASTIC ANALYSIS OF SHELLS WITH THE TRIANGULAR ELEMENT TRIC J. H Argyris 1 , M. Papadrakakis 2 and L. Karapitta 2 (1) Institute for Computer ApplicationsUniversity of Stuttgart, Stuttgart, Germany (2) Institute of Structural Analysis & Seismic Research National Technical University of Athens, Greecee-mail:  Key words:  Natural method, von Mises, equivalent plastic strain, equivalent stress Abstract.  TRIC is a simple but sophisticated 3-node shear-deformable isotropic and composite facet shell element suitable for large-scale linear and nonlinear engineering computations of thin and anisotropic plate and complex shell structures. In the present work an elasto-plastic constitutive model based on the von Mises yield criterion with isotropichardening is incorporated into the element. The characteristic feature of this model is that the non-linear material behaviour is taken into account entirely in the natural system of theelement. This is achieved by transforming quantities such as equivalent plastic strain,equivalent stress, the expression of the yield surface and the components of flow vector fromthe material coordinate system to the natural coordinate system. These transformations lead to simple and elegant expressions for the respective quantities in the natural system. Thisimplementation leads to an efficient and cost effective treatment of the nonlinear analysis of  arbitrary shells including material and geometrical nonlinearities.  J. H. Argyris, M. Papadrakakis, L. Karapitta 2 1. Introduction Finite element analysis of shells has been receiving continuous attention since the early daysof the development of the method. The analysis of shells presented a challenge, since their formulation may become cumbersome and their behavior can be unpredictable with regard togeometry or support conditions. The pure displacement based isoparametric formulationsuffers from various kinds of locking phenomena, which are only partially dealed withreduced or selective integration techniques. On going research efforts have been focus ondevising more elaborate element models that circumvent the deficiencies of the pure-displacement elements. Thus, a number of approaches have been proposed based on mixed or hybrid formulations, incompatible displacement methods, stabilization methods, assumed strain methods and free formulations.An attempt to device a shell element with robustness, accuracy and efficiency has led to thederivation of the TRIC shell element [1,2], a simple but sophisticated triangular, shear- deformable facet shell element suitable for the analysis of thin and moderately thick isotropicas well as composite plate and shell structures. Its formulation is based on the natural modefinite element method  [3], a method introduced by Argyris in the late 1960s that separates the pure deformational modes-also called natural modes-from the rigid body movements of theelement. The natural mode method has substantial computational advantages compared to theconventional isoparametric finite element formulations. The treatment of the elementkinematics eliminates automatically locking phenomena while there is no need to performnumerical integration for the computation of the stiffness matrix which is carried out in closeform. On the other hand, the inclusion of the transverse shear deformations in the formulationof the TRIC shell element based on a first order shear-deformable beam theory is performed in a way that eliminates the shear locking effect in a physical manner.The derivation of TRIC’s stiffness matrix was established upon a rather physical approach based on the observation of the element’s deformational modes and the accumulated experience of Argyris and co-workers obtained from previous shell elements that they havedeveloped using physical lumping procedures. In [1], the formulation of TRIC is presented asan evolution step emerging from three previously elements, namely TRUMP [4], TRUNC [5]and LACOT [6]. Recently in [7], it is proved that the TRIC shell element, despite the fact that it is based on a non-standard finite element formulation, satisfies the individual element testand according to the non-consistent formulation its use guarantees convergence. Furthermore,the element’s robustness and accuracy have proved through a variety of standard benchmark  plate and shell problems where the TRIC element exhibits excellent performance.  IASS-IACM 2000, Chania-Crete, Greece 3The features of the element proposed in the past incorporate large displacements, largerotations and small strains. The blend of the natural mode method and a path followingstrategy based on arc-length method has shown many advantages over classical formulations,i.e.: analytical and elegant expressions for all elemental matrices; a series of vector and matrix multiplications that can be easily optimised for maximum speed; accurate location of  birfucation; limit and displacement points; computational efficiency and economy [2].In the present study, the geometrically nonlinear formulation of the TRIC shell element isextended to include physical nonlinearities as well. The analysis of shell structures exhibitingmaterial and geometrical nonlinearities has reserved considerable attention over the pastyears [8-11]. In this work a layered elasto-plastic constitutive model based on the von Misesyield criterion, the associated flow rule and isotropic hardening is adopted. The mainadvantage of this formulation is that the elasto-plastic stiffness matrix is formed on thenatural coordinate system and can be expressed analytical for each layer. Then, the totalnatural tangent stiffness matrix is computed by adding together the tangent stiffness matrix of each layer. Numerical examples exhibiting highly nonlinear behavior, are presented in this work, inorder to test and verify the proposed elasto-plastic large displacement formulation of TRICshell element. Efficiency and accuracy of the proposed element is demonstrated by a set of numerical examples taken from the literature. 2. The TRIC shell element 2.1 Kinematics of the element For the multilayered composite triangular shell element the following coordinate systemsshown in Fig. 1 are adopted. The natural coordinate system which has the three axes parallelto the sides of the triangle. The local elemental coordinate system, placed at the triangle’scentroid, and the global Cartesian coordinate system where global equilibrium refers to.Finally, for each ply of the triangle, a material coordinate system 1, 2, 3 is defined with axis 1 being parallel to the direction of the fibers. The use of these different coordinate systemsmakes TRIC suitable element in modeling a multilayer anisotropic shell structure that candegenerate, as special case, to a sandwich or a single – layer configuration.Fig. 2 depicts the three total natural axial strains which are measured parallel to the edges of the triangle and replace the Cartesian strains in the natural mode formulation. These strains t γ  are measured directly parallel to the triangle’s sides, while by definition straining of one  J. H. Argyris, M. Papadrakakis, L. Karapitta 4side leaves all other triangular sides unstrained. Similarly, the total natural transverse shear strains s γ  are defined for each one of the triangle’s edges: = γβ ttt α t   (1) γγγ= γβα s   (2)Fig. 3 depicts the total natural transverse shear strain for side  α  of the triangle. As shown,transverse shearing of one side leaves all other side angles orthogonal.The total natural axial strains t γ  are related to the three in-plane Cartesian strains   according to the expression =⇔= ′′′′′′′′′′′′ y'x'y'y'x'x'x  x  x  2x  2x  x  x  2x  2x α x α x α 2x α 2t  t  t α tt  2  cs2sc cs2sc cs2sc  'B   (3)where  , α ,i,xi,sins  , α ,i,xi,cosc xixi =′==′= ′′ (4)and  x  ,,x  ,,x α ,  ′′′  are the angles that the triangle’s edges  α ,    and     form with the localx ΄  axis, respectively. The total transverse shear strains s γ  are related to the two out-of-planetransverse shear strains s γ′  via =⇔= z'y'z'x'  x'  x' α x'  x'  x' α x' α 'sss  sssccc  T   (5)The corresponding natural stresses c σ  to the total natural axial strains t γ  are grouped in thevector   IASS-IACM 2000, Chania-Crete, Greece 5 = c  c  c α c σσσσ  (6)while the corresponding natural transverse shear stresses are = s  s  s α s σσσσ  (7)The constitutive relations between the natural stresses and the total natural strains areestablished by initiating the following sequence of coordinate system transformationsMaterial system  →  Local system  →  Natural systemWith simple geometric transformations and by contemplating the invariance of the strainenergy density in the different coordinate systems, one can easily reach to an expression for the constitutive matrix in the natural coordinate system for both axial and transversedeformations r str sctr sc   σσ ⋅⋅= (8)valid for each layer r. Matrix ct κ  defines the constitutive matrices of axial and symmetrical bending while matrix s χ  corresponds to antisymmetrical bending and transverse shear modes. Additional information for the derivation of the natural constitutive matrix can befound in [1]. 2.2 Natural modes and generalized forces and moments The multilayered triangular shell element TRIC has 6 Cartesian degrees of freedom per node.Its natural stiffness is only based on deformations and not on associated rigid body motions.The element has 18 degrees of freedom but the actual number of straining modes is 12:18 Cartesian d.o.f. – 6 rigid body d.o.f. = 12 straining modesThe element TRIC includes 6 rigid body and 12 straining modes which are illustrated in Figs.4 and 5 and grouped in the vector  =′ (12x1) N(6x1)0(18x1)e ρρρ  (9)
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