Membrane Finite Element Research Articles

Formulation of a membrane finite element with drilling degrees of freedom

A 4-noded membrane element with drilling degrees of freedom is developed. A functional for the linear problem, in which the drilling rotations are introduced as independent variables, is obtained from the functional for the corresponding nonlinear problem, initially derived by Atluri. A careful examination of the Euler equations shows that the Allman-type element fails to pass a one-element patch test with a minimum number of constraints. A 4-noded element, which passes the one-element patch test without constraining the drilling rotations, is proposed here with the use of the separate kinematics variables of displacement and rotation. Detailed numerical studies show the excellent performance of the element.

A novel membrane finite element with an enhanced displacement interpolation

A new quadrilateral membrane finite element is presented in this work. Besides translational, the element possesses rotational (drilling) degrees of freedom. When combined with a plate bending element, the element also provides a versatile tool for analysis of shells. Both displacement based and mixed approaches are investigated. The finite element interpolation field is enriched by a set of incompatible modes. The element exhibits superior accuracy to currently available membrane elements.

A robust quadrilateral membrane finite element with drilling degrees of freedom

AbstractA quadrilateral membrane finite element with drilling degrees of freedom is derived from variational principles employing an independent rotation field. Both displacement based and mixed approaches are investigated. The element exhibits excellent accuracy characteristics. When combined with a plate bending element, the element provides an efficient tool for linear analysis of shells.

Reinforced Concrete Membrane Element Formulations

Finite‐element formulations are presented for the analysis of reinforced concrete membrane structures. Cracked reinforced concrete is treated as an orthotropic material based on a smeared, rotating crack model. Secant‐stiffness moduli are defined for concrete and reinforcement, and these are used in the development of linear displacement rectangular and triangular membrane finite elements. Procedures are discussed by which these elements can then be incorporated into a nonlinear analysis algorithm. Extensions to the formulations are also described that permit the consideration of prestrain effects in the component materials such as prestressing of the reinforcement, shrinkage of the concrete, or thermal expansion. The constitutive relations currently utilized are those of the modified compression field theory, although the element formulations are sufficiently generic to easily accommodate other constitutive models. A numerical example is provided to illustrate the simplicity of the calculation procedure ...

The impact modeling of structures using solid, shell, and membrane finite elements

A comparison of modeling results and efficiency is made for a 4-node, pressure corrected, plane strain quadrilateral element and the 4-node, constant-stress bending quadrilateral of Belytschko, Lin, and Tsay for modeling the transient dynamic response of a complex buried structure. A detailed derivation of the 4-node, constant-stress bending quadrilateral and associated hourglass control is presented.

Assessing the effect of shape distortion in Q9 membrane elements

The Q9 membrane finite element is unable to experience arbitrary change of uniaxial curvature without the occurence of artificial membrane stains. The error which accumulates due to this inability can be significant when a large change of curvature is experienced by this element. This paper investigates the effect of element shape distortion on the accumulation of error in the Q9 membrane element, and presents two methods for evaluating shape distortion effects in problems with large changes of curvature: a series of ‘patch tests’, and single element tests.

A Study of Two-Dimensional Plane Elasticity Finite Elements for Optimal Design

ABSTRACT Design of structures using a variety of two-dimensional finite elements is considered in this paper. An efficient technique of computing first-order derivatives of pointwise stress constraints for simple and higher-order two-dimensional (membrane) finite elements is presented. Computational aspects of design sensitivity vector calculation, using a semi-analytical method versus traditional methods are presented. Implementation of a fully-stressed design approach to find a suitable initial estimate leads to increased computational efficiency. These aspects of the design procedure are illustrated through analysis of numerical examples. Experience indicates that a suitable mix of low and higher-order elements yields the most efficient and accurate design model.

Free vibrations of a tire as a toroidal membrane

A tire is modeled as a toroidal membrane under internal pressure and mounted on a rim, to investigate its free vibration characteristics using a 12 d.o.f. rectangular membrane finite element. Such a modeling is valid if the tire is assumed to be incapable of supporting any weight in the absence of internal pressure. To verify the formulations of the membrane finite element, a flat rectangular membrane subject to in-plane loads and a circular cylindrical membrane under internal pressure are first analyzed. Analytical solutions for these cases are also derived. The analytical and numerical results are in good agreement. A toroidal membrane under internal pressure, assumed to model a low pressure tire, is studied next. Both the analytical derivation and the finite element solutions are presented. For the analytical solution the equations of motion yield a complicated differential equation for which an approximate solution is obtained by assuming that the parallel circle radius is constant as in the case of a bycycle wheel. The finite element solution successfully predicts the symmetrical and the twisting modes of vibration documented by other researchers, and is also in good agreement with the analytical results. The present formulations are useful to obtain a good first approximation of the free vibration response of a tire.

Buckled plate vibrations and large amplitude vibrations using high-order triangular elements

A high-order triangular membrane finite element is combined with a fully conforming triangular plate bending element to solve the geometrically nonlinear problems of plates where the membrane and flexural behaviors are coupled and the effect of the inplane boundary conditions is as significant as the flexural boundary conditions. Each of the three orthogonal displacement components is represented by a two-dimensional polynomial of the same quintic order with no bias against one another giving the element a total of 54 degrees of freedom. The nonlinear stiffness matrices are formulated and a Newton-Raphson iteration procedure is used. Examples include the analyses of plane stresses of a parabolically loaded square plate, large deflections of a square plate under lateral pressure, postbuckling of a square plate, linear free vibration of a buckled rectangular plate, large amplitude free vibration of a square plate with and without inplane stresses. Various flexural and inplane boundary conditions are considered. Results are compared with those obtained by alternative finite element methods, analytical approximate methods, and an experiment. Physical interpretations of the results and explanations of the discrepancies among various solutions are provided. The results indicate that the present development is capable of accurately solving a wide variety of geometrically nonlinear plate problems.

A nonlinear viscoelastic membrane model applied to the human cervix

An exponentially elastic and viscous membrane model is developed and used to represent the human cervix during the first stage of labor in the vertex presentation. An axisymmetric orthotropic membrane finite-element is derived and evaluated. A procedure for solving for large displacement of structures confined to curved loci is derived and extended to include nonlinearly elastic and linear viscous material properties. The cervix is modeled, and the results are compared quantitatively with a previous simpler model of the cervix and qualitatively with published clinical observations. The model suggests a resolution to the controversy over the time course of cervical dilatation.