Nonlinear Formulations of a Four-Node Quasi-Conforming Shell Element

Abstract

The quasi-conforming technique was introduced in the 1980's to meet the challenge of inter-elements conforming problems and give a unified treatment of both conforming and nonconforming elements. While the linear formulation is well established, the nonlinear formulation based on the quasi-conforming technique that includes geometric and material nonlinearity is presented in this paper. The formulation is derived in the framework of an updated Lagrangian stress resultant, co-rotational approach. The geometric nonlinear formulation provides solutions to buckling and postbuckling behaviour while the material nonlinear formulation considers the spread of plasticity within the element while maintaining an explicit construction of element matrices. Aside from the elasto-plastic constitutive relation, formulations on laminate composites and reinforced concrete are also presented.The formulations of laminate composite and reinforced concrete material are present based on the layer concept, the material properties can vary throughout the thickness and across the surface of a shell element. The various failure criteria for laminate composite are included in the formulation which makes it possible to analyses the progressive failure of fibre and matrix. For the reinforced concrete material, the nonlinearities as a result of tensile cracking, tension stiffening between cracks, the nonlinear response of concrete in compression, and the yielding of the reinforcement are considered. The steel reinforcement is modeled as a bilinear material with strain hardening.

Accurate, Efficient, and Robust Q4-Like Membrane Elements Formulated in Cartesian Coordinates Using the Quasi-Conforming Element Technique

“…in which K is the element stiffness matrix and q is the nodal displacement vector of the element under consideration. A number of the assumed strain elements based on the quasi-conforming element technique have been developed [2,[19][20][21][22][23][24][25]27]. The numerical examples show that the quasiconforming element technique could not only solve the conforming difficulties in plate elements, but also yield very accurate and efficient assumed strain elements.…”

“…The quasi-conforming element technique proposed by Tang and his coworkers [19][20][21][22] is a general assumed strain method to formulate reliable and accurate elements. Although it has been primarily used for the developments of the quasiconforming plate and shell elements that involve with the 1continuity problems, a number of quasi-conforming membrane elements have been presented [2,[21][22][23][24][25]. Chen and Tang [21] presented a quasi-conforming quadrilateral isoparametric membrane element QC6 where two displacement-like internal parameters are used.…”

Assumed stress quasi-conforming formulation for static and free vibration analysis of symmetric laminated plates

“…QC finite element method is a very flexible and efficient theory framework, and its basic idea is the weakening of strain-displacement equations in the element together with that of the equilibrium equations. Many elements have been proposed during past few decades, the readers can refer to the review articles (Lomboy et al , 2009; Hu and Xia, 2012). In the QC technique, the element strain fields are approximated using polynomials and integrated using interpolations.…”

Assumed stress quasi-conforming technique for static and free vibration analysis of Reissner-Mindlin plates

“…In the QC technique, the strain‐displacement functions as well as the equilibrium functions are weakened by integrations, and multiple sets of approximating functions are used for displacement fields on the element boundary and the inner‐element. Many plane elements and plate/shell elements based on the QC technique have been developed during past few decades, and the readers can refer to the review articles . Kim et al presented a 4‐node QC shell element based on assumed strain QC technique for linear and nonlinear analysis of plates and shells.…”

“…Many plane elements and plate/shell elements based on the QC technique have been developed during past few decades, and the readers can refer to the review articles . Kim et al presented a 4‐node QC shell element based on assumed strain QC technique for linear and nonlinear analysis of plates and shells. Hu et al developed a four‐node quadrilateral flat shell element based on assumed displacement QC technique, in which the complete quadratic polynomials were used to discrete membrane and bending displacement fields.…”