Bending of bilinear quadrilateral shell elements

Formulae are derived for exact inextensional bending solutions for arbitrary quadrilateral shell finite elements of bilinear parametric representation. It is found that the polynomial degree of parametric representation of the rectangular components of displacement requires to be at least cubic in order to describe any inextensional bending modes.     

Fortran computer program for inextensional bending of a doubly curved shell triangular element

A Fortran computer program is described which calculates physical quantities for a class of shell triangular elements undergoing inextensional bending. These elements are in quadratic parametric representation and may have positive, zero or negative Gaussian curvature. The exact inextensional bending solutions for the rectangular displacement components are cubic in the surface co-ordinates and the curvature changes are relatively slowly varying.     

Comparison solutions for assessment of inextensional bending in shells of quadratic and cubic polynomial representation

This paper is intended to complement recent work in facilitating the assessment of inextensional bending in finite element models of thin shells under the linear theory. Details are given of exact solutions for inextensional bending in shells which may have arbitrary depth and Gaussian curvature where the middle surface is explicitly defined by a quadratic or cubic polynomial.It is essential in finite element assessment to have ready access to all the components which constitute the exact solution. For this purpose, a Fortran computer program is described which calculates the displacements, rotations and curvature changes with respect to any chosen orthogonal directions on the shell middle surface.     

Convergence studies of eigenvalue solutions using two finite plate bending elements

The convergence rates of eigenvalue solutions using two finite plate bending elements are studied. The elements considered are the well-known 12 degree of freedom, non-conforming rectangular element and the 16 degree of freedom, conforming rectangular element. Three problems are analysed, a square plate simply supported on two opposite sides with the other two sides clamped, simply supported, or free. Closed form, finite element solutions for these problems are obtained by using shifting E-operators. With few exceptions, eigenvalue solutions found with the non-conforming element converge from below the exact answers at an asymptotic rate of n^?2, where n is the number of elements on a side. However, since the array size needed for such convergence is very large, little can be said about the convergence rates for practical arrays. The conforming element solutions converge from above at an asymptotic rate of n^?4. A comparison of the errors involved in using these two elements shows that the conforming element is far superior to the non-conforming element.     

Powerful FE approaches for Pochhammer's dispersion modelling

Simple and very effective finite element approaches for modelling of the dispersion of torsional, longitudinal and flexural waves propagated in infinite and elastic cylindrical bars (Pochhammer's problem) and hollow cylinders are presented. The approaches allow one to model given problems without using infinite elements and their usage is demonstrated by using dispersion curves, shapes of waves and tables of phase velocities. Comparison of finite element solutions (solved in one task from small‐sized models) with exact solutions shows an excellent agreement. The maximum relative differences among the 20 lowest FEM and analytical phase velocities was less than 0.000015%. The approaches and results hold also for cylinders with finite lengths, simply supported on both ends. Copyright © 2003 John Wiley & Sons, Ltd.     

A simple class of finite elements for plate and shell problems. II: An element for thin shells,with only translational degrees of freedom

This paper is the second of a pair which discuss the development of a class of overlapping hinged bending finite elements, which are suitable for the analysis of thin-shell, plate and beam structures. These elements rely on a simple physically appealing analogy, in which overlapping hinged facets are used to represent bending effects. They are based on quadratic overlapping assumed displacement functions, which results in constant stress/strain representation. Only translational nodal degrees of freedom are necessary, which is a significant advantage over most other currently available beam, plate and shell finite elements which employ translational, rotational and higher-order nodal variables. In paper I the hinged bending element concept has been introduced, and the hinged beam bending (HBB) and hinged plate bending (HPB) elements formulated. In the present paper these concepts are extended to develop a hinged shell bending (HSB) element. The HSB element can be readily combined with the constant strain triangular (CST) plane stress finite element for the modelling of thin-shell structures; and the combined HSB-CST element is tested against a number of 'standard' thin-shell problems. The present paper, like paper I, is conducted entirely in the context of small-displacement elastic behaviour.     

The curved beam/deep arch/finite ring element revisited

In this paper, an attempt is made to understand the errors arising in curved finite elements which undergo both flexural and membrane deformations. It is shown that with elements of finite size (i.e. a practical level of discretization at which reasonably accurate results can be expected), there can be errors of a special nature that arise because the membrane strain fields are not consistently interpolated with terms from the two independent field functions that characterize such a problem. These lead to errors, described here as of the ‘second kind’ and a physical phenomenon called ‘membrane locking’. The findings here emerge from recent research on the effect of reduced integration on shallow curved beam elements and on the use of coupled displacement fields in finite rings. The failures which have occurred in earlier attempts to use independent polynomial displacement fields for curved elements may not have been due to neglect of rigid body motions or failure to achieve constant strain states, but because of locking due to spurious constraints. These emerge in the penalty limits of extreme thinness (an inextensional regime), when exact integration of the energy functional of an element based on low order independent interpolations for the in-plane and normal displacements is used. It seems possible to determine optimal integration rules that will allow the extensional deformation of a curved beam/deep arch/finite ring element to be modelled by independently chosen low order polynomial functions and which will recover the inextensional case in the penalty limit of extreme thinness without spurious locking constraints. The much maligned ‘cubic in w–lincar in u’ curved beam element is now reworked to show its excellent behaviour in all situations. What is emphasized is that the choice of shape functions, or subsequent operations to determine the discretized functionals, must consistently model the physical requirements the problem imposes on the field variables. In this manner, we can restore an old element to respectability and thereby indicate clearly the underlying principles. These are: the importance of ‘field consistency’ so that arch and shell problems can be modelled consistently by independent polynomial displacement fields, and the role that reduced integration or some equivalent construction can play to achieve this.     

Field-consistency and violent stress oscillations in the finite element method

The fact that finite element models can give rise to violent stress oscillations and that there are optimal locations where stresses can be correctly sampled in spite of the presence of these violent stress fluctuations has been known for some time. However, it is less well known that these oscillations arise in a specific class of problems—where there are multiple strainfields arising from one or more field-variables and where one or more of these strain-fields must be constrained in particular physical limits. In this paper, we show that unless the interpolations for these constrained strain-fields are ‘field-consistent’, violent oscillations would set in. These oscillations represent spurious self-equilibrating stress-fields generating spurious energy terms that lead to ‘locking’. The field-consistency interpretation offers a conceptual scheme to delineate these problems and an operational procedure called the functional reconstitution technique allows the errors resulting from field-inconsistency to be anticipated a priori. We demonstrate the power of this approach through an interesting example of a multi-strain-field problem—the inextensional/nearly inextensional deformation of a shear flexible curved beam.     

A finite element weighted residual solution to one-dimensional field problems

The one-dimensional diffusion-convection equation is formulated with the finite element representation employing the Galerkin approach. A linear shape function and two-dimensional triangular and rectangular elements in space and time were used in solving the problem. The results are compared with finite difference solutions as well as the exact solution. As another example, the convective term is set equal to zero and these techniques are applied to the resulting heat equation and similar comparisons are made.     

Exact solutions of axial vibration problems of elastic bars

While exact solutions for linear static analysis of most frame structures can be obtained by the finite element method, it is very difficult to obtain exact solutions for free vibration and harmonic analyses for non‐trivial cases. This paper presents a study on new finite element formulation and algorithms for exact solutions of undamped axial vibration problems of elastic bars. Appropriate shape functions are constructed by using the homogeneous governing equations, and based on the new shape functions, a novel element is formulated. An iterative procedure is proposed for determining both the exact natural frequency values and the corresponding vibration mode shapes. Exact solutions can also be obtained for undamped harmonic response analyses by using the new element, as its stiffness and mass matrices are exact for a specified frequency. Illustrative examples are presented to demonstrate the effectiveness of the proposed element and algorithm. Copyright © 2007 John Wiley & Sons, Ltd.     

Exact stiffness matrix of mono-symmetric composite I-beams with arbitrary lamination

The exact stiffness matrix, based on the simultaneous solution of the ordinary differential equations, for the static analysis of mono-symmetric arbitrarily laminated composite I-beams is presented herein. For this, a general thin-walled composite beam theory with arbitrary lamination including torsional warping is developed by introducing Vlasov’s assumption. The equilibrium equations and force–deformation relations are derived from energy principles. The explicit expressions for displacement parameters are then derived using the displacement state vector consisting of 14 displacement parameters, and the exact stiffness matrix is determined using the force–deformation relations. In addition, the analytical solutions for symmetrically laminated composite beams with various boundary conditions are derived as a special case. Finally, a finite element procedure based on Hermitian interpolation polynomial is developed. To demonstrate the validity and the accuracy of this study, the numerical solutions are presented and compared with the analytical solutions and the finite element results using the Hermitian beam elements and ABAQUS’s shell element.     

An improved 3-D brick Cosserat point element for irregular shaped elements

The original 3-D brick Cosserat point element (CPE) satisfies a nonlinear form of the patch test for general element shapes but the coefficients in the strain energy function for inhomogeneous deformations were motivated by considering solutions for a rectangular parallelepiped. In this paper, a generalized functional form for the strain energy of inhomogeneous deformations is proposed for irregular shaped elements. The coefficients in this strain energy function are determined by matching exact pure bending solutions for three right cylindrical parallelepipeds with their cylindrical axes oriented along three orthogonal base vectors. A generalization is proposed which ensures that this strain energy function remains positive definite for general irregular shaped elements. A number of example problems are considered which show that the performance of this improved CPE is good for irregular shaped elements. Moreover, the improved CPE continues to predict physical results for nonlinear and buckling problems that typically cause lack of convergence with other element formulations due to unphysical hourglassing.     

Statically admissible finite element solution of the thin plate bending problem with a posteriori error estimation

Two triangular elements of class C⁰ developed on the basis of the principle of complementary work are applied in the static analysis of a thin plate. Some techniques to widen the versatility of the equilibrium approach for the finite element method are presented. Plates of various shapes subjected to diverse types of loading are considered. The results are compared with outcomes obtained by use of the displacement-based finite element method. By use of these two dual types of solutions, the error of the approximate solution is calculated. The lower and upper bounds for the strain energy are found.     

Higher-order finite element modelling of laminated composite plates

P-version finite elements based on higher-order theory are developed for the two-dimensional modelling of general bending and cylindrical bending of thin-to-thick laminated composite plates. In the case of general laminated plate elements, three displacement fields are used. In the special case of cylindrically bent laminated plate elements, two displacement fields are needed. In each case the displacement is expressed as the product of two functions—one in terms of out-of-plane co-ordinates alone and the other in terms of in-plane co-ordinates. The shape functions used to build the displacement fields are based on integral of Legendre' polynomials. The quality and performance of the elements are evaluated in terms of convergence characteristics of displacements and stresses. The predicted response quantities are compared with those available in the published literature based on analytical as well as conventional finite element models.     

On the decomposition of generalized eigenproblems for the free vibration analysis of cyclically symmetric finite element models

Free vibration analysis is a major part of any dynamic analysis. Natural frequencies and related mode shapes may be obtained from free vibration analysis as the solutions of generalized eigenproblems. Although the eigensolutions of large‐scale structures require large computational efforts, these solutions may be achieved simply for symmetric structures. We present an efficient method for the decomposition of generalized eigenproblems related to finite element models with cyclic symmetry (having nodes at the axis of symmetry) into eigensubproblems with significantly smaller dimensions. This decomposition is obtained by block diagonalization of a matrix with a special pattern known as a block circulant, using the concept of the Kronecker product and similarity transformations. The proposed method is applied to three finite element models discretized by triangular and four‐node quadrilateral plate and shell elements, and its efficiency, accuracy and simplicity are evaluated. Copyright © 2009 John Wiley & Sons, Ltd.     

A finite element analysis in polar co-ordinates of the Saint Venant torsion problem

A characteristic feature of the variational functionals for several boundary value problems in polar co-ordinates is the fact that one independent variable occurs explicitly in the denominator. Therefore, the coefficients of the finite element equations for sectors of circular ring shaped elements are not constants but functions of the distance of the elements from the origin of co-ordinates.¹ We name them coefficient functions. In order to show the particular aspects of the calculations in terms of polar co-ordinates we deal here with the solution of the torsion problem by bilinear and bicubic Hermitian interpolation. The finite element equations are arranged according to Schaefer² in the form of block which can easily be transformed into ‘stars’³ or molecules^4,5 similar to those used in finite difference methods. The origin of co-ordinates requires a special consideration, firstly because of the coincidence of several nodes at that point and secondly because of the divergent behaviour of some coefficient functions. It turns out to be advantageous for the numerical calculations to expand the coefficient functions in power series. Besides, the expansions are required to deduce the equations for rectangular elements by limiting processes. The twisting moments and shearing stresses calculated for several cross sections illustrate the numerical suitability of the method. The finite element values are compared partly with exact solutions and partly with experimental results obtained by a moiré method using Prandtl's soap film analogy.⁶ Finally it is shown how the accuracy of the finite element values can be improved by the Richardson extrapolation⁷.     

A simple flat triangular shell element revisited

The 18 degree-of-freedom flat triangular shell element is reformulated by combining the well-known bending triangle with a plane stress triangle incorporating in-plane rotations at each vertex. Both elements are displacement formulated. The plane stress element's displacement interpolation is incomplete and hence convergence to exact solutions is precluded. Comprehensive test results are presented for several types of problem including plane stress, thin shells and folded plates. The results indicate that the element does produce rapidly convergent answers. However these answers are not the correct ones, although they may be acceptable engineering approximations in many applications. Further, the element seems to provide reasonably good results even for relatively coarse element grids.     

Hybrid stress finite element analysis of bending of a plate with a through flaw

A hybrid stress finite element procedure for the solution of bending stress intensity factors of a plate with a through-the-thickness crack is presented. Reissner's sixth-order plate theory including the effects of transverse shear deformation is used. The dominant singular crack tip stress field is embedded in the crack tip singular elements and only regular polynomial functions are assumed in the far field elements. The stress intensity factors can be calculated directly from the crack tip singular stress solution functions. The effects of the plate thickness, the ratio between the crack size and the inplane dimension of the plate, and the singular element size on the stress intensity factor solution are investigated. The effects of the explicit enforcement of traction-free conditions along crack surfaces, which are the natural boundary conditions in the present hybrid stress finite element model, are also investigated. The numerical results of bending of a plate with a straight central crack compare favourably with analytical solutions. It is also found that the explicit enforcement of traction-free conditions along crack surfaces is mandatory to obtain meaningful results for the Mode I type of bending stress intensity factor.