Nonlinear dynamic co-rotational formulation for membrane elements with in-plane drilling rotational degree of freedom

Abstract

Purpose nonlinear dynamic analysis of triangular and quadrilateral membrane elements with in-plane drilling rotational degree of freedom. Design/methodology/approach The nonlinear analysis is carried out using the updated co-rotational Lagrangian description. In this purpose, in-plane co-rotational formulation that considers the in-plane drilling rotation is developed and presented for triangular and quadrilateral elements, and a tangent stiffness matrix is derived. Furthermore, a simple and effective in-plane mass matrix that takes into account the in-plane rotational inertia, which permit true representation of in-plane vibrational modes is adopted for dynamic analysis, which is carried out using the Newmark direct time integration method. Findings The proposed numerical tests show that the presented elements exhibit very good performances and could return true in-plane rotational vibrational modes. Also, when using a well-chosen co-rotational formulation these elements shows good results for nonlinear static and dynamic analysis. Originality/value Publications that describe geometrical nonlinearity of the in-plane behaviour of membrane element with rotational d.o.f are few, and often they are based on the total Lagrangian formulation or on the rate form. Also these elements, at the author knowledge, have not been extended to the nonlinear dynamic analysis. Thus, an appropriate extension of triangular and quadrilateral membrane elements with drilling rotation to nonlinear dynamic analysis is required.

A new enhanced assumed strain quadrilateral membrane element with drilling degree of freedom and modified shape functions

“…The present quadrilateral elements are designed following the concept of the EAS triangular element presented in . In the Allman type interpolation of Equation , the lateral displacement along each side is taken to be quadratic due corner node rotations, whereas, tangential displacement remain linear.…”

“…Wisniewski and Turska presented the formulation of a four‐node quadrilateral element based on Allman shape functions via the Enhanced Assumed Displacement Gradient (EADG) method. Boutagouga and Djeghaba proposed a triangular element that is equivalent to the element by developed in natural coordinates using four incompatible displacements.…”

Reciprocal mass matrices and a feasible time step estimator for finite elements with Allman's rotations

“…This opens an opportunity for constructing sharper time step estimators using Ostrowski's bound for nonsymmetric matrices, like the product of the reciprocal mass matrix and the stiffness matrix in Equation (14). Computing minimax problem in Equation (19) for continuous the exponent is challenging. Herein, a discrete uniform grid for the exponent is used.…”

“…Therefore, CMM is preferred in implicit analysis. 18,19 Based on this profound research, finite elements with Allman's rotation made its way to academic finite element packages, for example, as 3-node triangle TrPlaneStrRot in the open-source code OOFEM 20 and commercial programs, for example, 4-node tetrahedron (ELFORM = 4) and 8-node hexahedron (ELFORM = 3) in LS-DYNA. The latter two formulations are used in explicit dynamics.…”

“…Moreover, the scaling of the rotary inertia in LMM should be adjusted to the stabilization stiffness of the element. Therefore, CMM is preferred in implicit analysis 18,19 . Based on this profound research, finite elements with Allman's rotation made its way to academic finite element packages, for example, as 3‐node triangle TrPlaneStrRot in the open‐source code OOFEM 20 and commercial programs, for example, 4‐node tetrahedron (ELFORM = 4) and 8‐node hexahedron (ELFORM = 3) in LS‐DYNA.…”