GEOMETRICALLY NON-LINEAR AND ELASTOPLASTIC THREE-DIMENSIONAL SHEAR FLEXIBLE BEAM ELEMENT OF VON-MISES-TYPE HARDENING MATERIAL

A three-dimensional elastoplastic beam element being capable of incorporating large displacement and large rotation is developed and examined. Elastoplastic constitutive equations are applied to the beam element based upon the assumption of small deformational strain leading to a material formulation which is completely objective for the application of stress update procedures. The continuum-type equations of plastic model of J2 mixed hardening are transformed into the beam equations by satisfying beam hypotheses. An effective stress update algorithm is proposed to integrate elastoplastic rate equations by means of the so-called multistep method which is a method of successive control of residuals on yield surfaces. It avoids severe divergence when the displacement increments become large which is usual for the continuation methods. Material tangent stiffness matrix is derived by using consistent elastoplastic modulus resulting from the integration algorithm and is combined with geometric tangent stiffness matrix. Different from other elements, the present element is shear flexible and can satisfy the plasticity condition in a pointwise fashion. A great number of numerical examples are analysed and compared with the literature. The proposed beam element is verified to be not only quite accurate but also very effective for the analyses of pre-buckling and large deflection collapse of spatial framed structures.

[1]  S. Atluri,et al.  Large-deformation, elasto-plastic analysis of frames under nonconservative loading, using explicitly derived tangent stiffnesses based on assumed stresses , 1987 .

[2]  J. Argyris.  An excursion into large rotations , 1982 .

[3]  E. P. Popov,et al.  Accuracy and stability of integration algorithms for elastoplastic constitutive relations , 1985 .

[4]  J. C. Simo,et al.  Consistent tangent operators for rate-independent elastoplasticity☆ , 1985 .

[5]  Jerrold E. Marsden,et al.  On the rotated stress tensor and the material version of the Doyle-Ericksen formula , 1984 .

[6]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[7]  M. Géradin,et al.  Kinematic and Dynamic Analysis of Mechanisms. A Finite Element Approach Based on Euler Parameters , 1986 .

[8]  W. Rheinboldt,et al.  On steplength algorithms for a class of continuation methods siam j numer anal , 1981 .

[9]  Michael Ortiz,et al.  Symmetry-preserving return mapping algorithms and incrementally extremal paths: A unification of concepts , 1989 .

[10]  Daigoro Isobe,et al.  Adaptively shifted integration technique for finite element collapse analysis of framed structures , 1993 .

[11]  E. Reissner,et al.  On One‐Dimensional Large‐Displacement Finite‐Strain Beam Theory , 1973 .

[12]  G.A.O. Davies,et al.  A 3-D, co-rotational, curved and twisted beam element , 1990 .

[13]  Sanjay Govindjee,et al.  Non‐linear B‐stability and symmetry preserving return mapping algorithms for plasticity and viscoplasticity , 1991 .

[14]  K. Bathe,et al.  Large displacement analysis of three‐dimensional beam structures , 1979 .

[15]  E. Dvorkin,et al.  On a non‐linear formulation for curved Timoshenko beam elements considering large displacement/rotation increments , 1988 .

[16]  M. Géradin,et al.  A beam finite element non‐linear theory with finite rotations , 1988 .

[17]  M. Crisfield.  A consistent co-rotational formulation for non-linear, three-dimensional, beam-elements , 1990 .

[18]  W. Rheinboldt.  Numerical analysis of continuation methods for nonlinear structural problems , 1981 .

[19]  John L. Meek,et al.  Geometric and material non-linear behaviour of beam-columns , 1990 .

[20]  U. Hindenlang,et al.  Finite element analysis of two- and three-dimensional elasto-plastic frames—the natural approach☆ , 1982 .

[21]  Matti Ristinmaa,et al.  EXACT INTEGRATION OF CONSTITUTIVE EQUATIONS IN ELASTO - PLASTICITY , 1993 .

[22]  J. D. Downer,et al.  Dynamics of flexible beams for multibody systems: a computational procedure , 1992 .

[23]  Satya N. Atluri,et al.  ELASTO-PLASTIC LARGE DEFORMATION ANALYSIS OF SPACE-FRAMES: A PLASTIC-HINGE AND STRESS-BASED EXPLICIT DERIVATION OF TANGENT STIFFNESSES , 1988 .

[24]  J. C. Simo,et al.  A return mapping algorithm for plane stress elastoplasticity , 1986 .

[25]  Michael Ortiz,et al.  An analysis of a new class of integration algorithms for elastoplastic constitutive relations , 1986 .

[26]  D. Owen,et al.  Studies on generalized midpoint integration in rate-independent plasticity with reference to plane stress J2-flow theory , 1992 .

[27]  K. Spring.  Euler parameters and the use of quaternion algebra in the manipulation of finite rotations: A review , 1986 .

[28]  Alf Samuelsson,et al.  Finite element analysis of elastic–plastic materials displaying mixed hardening , 1979 .

[29]  Michael Ortiz,et al.  A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations , 1985 .

[30]  P. Pinsky,et al.  Numerical integration of rate constitutive equations in finite deformation analysis , 1983 .

[31]  K. Mattiasson,et al.  On the Accuracy and Efficiency of Numerical Algorithms for Geometrically Nonlinear Structural Analysis , 1986 .

[32]  P. M. Naghdi,et al.  On the relationship between the Eulerian and Lagrangian descriptions of finite rigid plasticity , 1988 .

[33]  R. Borst,et al.  Studies in anisotropic plasticity with reference to the Hill criterion , 1990 .

[34]  R. D. Krieg,et al.  Accuracies of Numerical Solution Methods for the Elastic-Perfectly Plastic Model , 1977 .

[35]  J. C. Simo,et al.  A Geometrically-exact rod model incorporating shear and torsion-warping deformation , 1991 .

[36]  C. Rankin,et al.  Finite rotation analysis and consistent linearization using projectors , 1991 .

[37]  Czesław Cichoń,et al.  Large displacements in-plane analysis of elastic-plastic frames , 1984 .

[38]  J. C. Simo,et al.  A three-dimensional finite-strain rod model. Part II: Computational aspects , 1986 .