Static inconsistencies in certain axiomatic higher-order shear deformation theories for beams, plates and shells

AbstractStatic inconsistencies that arise when modelling the flexural behaviour of beams, plates and shells with clamped boundary conditions using a certain class of axiomatic, higher-order shear deformation theory are discussed. The inconsistencies pertain to displacement-based theories that enforce conditions of vanishing shear strain at the top and bottom surfaces a priori. First it is shown that the essential boundary condition of vanishing Kirchhoff rotation perpendicular to an edge (w,x=0 or w,y=0) is physically inaccurate, as the rotation at a clamped edge may in fact be non-zero due to the presence of transverse shear rotation. As a result, the shear force derived from constitutive equations erroneously vanishes at a clamped edge. In effect, this boundary condition overconstrains the structure leading to underpredictions in transverse bending deflection and overpredictions of axial stresses compared to high-fidelity 3D finite element solutions for thick and highly orthotropic plates. Generalised higher-order theories written in the form of a power series, as in Carrera’s Unified Formulation, do not produce this inconsistency. It is shown that the condition of vanishing shear tractions at the top and bottom surfaces need not be applied a priori, as the transverse shear strains inherently vanish if the order of the theory is sufficient to capture all higher-order effects. Finally, the transverse deflection of the generalised higher-order theories is expanded in a power series of a non-dimensional parameter and used to derive a material and geometry dependent shear correction factor that provides more accurate solutions of bending deflection than the classical value of 5/6

Buckling and vibration analysis of laminated composite plate/shell structures via a smoothed quadrilateral flat shell element with in-plane rotations

This paper presents buckling and free vibration analysis of composite plate/shell structures of various shapes, modulus ratios, span-to-thickness ratios, boundary conditions and lay-up sequences via a novel smoothed quadrilateral flat element. The element is developed by incorporating a strain smoothing technique into a flat shell approach. As a result, the evaluation of membrane, bending and geometric stiffness matrices are based on integration along the boundary of smoothing elements, which leads to accurate numerical solutions even with badly-shaped elements. Numerical examples and comparison with other existing solutions show that the present element is efficient, accurate and free of locking

Integrated force method versus displacement method for finite element analysis

A novel formulation termed the integrated force method (IFM) has been developed in recent years for analyzing structures. In this method all the internal forces are taken as independent variables, and the system equilibrium equations (EE's) are integrated with the global compatibility conditions (CC's) to form the governing set of equations. In IFM the CC's are obtained from the strain formulation of St. Venant, and no choices of redundant load systems have to be made, in constrast to the standard force method (SFM). This property of IFM allows the generation of the governing equation to be automated straightforwardly, as it is in the popular stiffness method (SM). In this report IFM and SM are compared relative to the structure of their respective equations, their conditioning, required solution methods, overall computational requirements, and convergence properties as these factors influence the accuracy of the results. Overall, this new version of the force method produces more accurate results than the stiffness method for comparable computational cost

A triangular thin shell finite element: Linear analysis

The formulation of the linear stiffness matrix for a doubly-curved triangular thin shell element, using a modified potential energy principle, is described. The strain energy component of the potential energy is expressed in terms of displacements and displacement gradients by use of consistent Koiter strain-displacement equations. The element inplane and normal displacement fields are approximated by complete cubic polynomials. The interelement displacement admissibility conditions are met in the global representation by imposition of constraint conditions on the interelement boundaries; the constraints represent the modification of the potential energy. Errors due to the nonzero strains under rigid body motion are shown to be of small importance for practical grid refinements through performance of extensive comparison analyses

A higher-order theory for geometrically nonlinear analysis of composite laminates

A third-order shear deformation theory of laminated composite plates and shells is developed, the Navier solutions are derived, and its finite element models are developed. The theory allows parabolic description of the transverse shear stresses, and therefore the shear correction factors of the usual shear deformation theory are not required in the present theory. The theory also accounts for the von Karman nonlinear strains. Closed-form solutions of the theory for rectangular cross-ply and angle-ply plates and cross-ply shells are developed. The finite element model is based on independent approximations of the displacements and bending moments (i.e., mixed finite element model), and therefore, only C sup o -approximation is required. The finite element model is used to analyze cross-ply and angle-ply laminated plates and shells for bending and natural vibration. Many of the numerical results presented here should serve as references for future investigations. Three major conclusions resulted from the research: First, for thick laminates, shear deformation theories predict deflections, stresses and vibration frequencies significantly different from those predicted by classical theories. Second, even for thin laminates, shear deformation effects are significant in dynamic and geometrically nonlinear analyses. Third, the present third-order theory is more accurate compared to the classical and firt-order theories in predicting static and dynamic response of laminated plates and shells made of high-modulus composite materials

Flutter of thermally stressed plated at hypersonic speeds Interim report for period ending 30 Jun. 1969

Eigenvalues and finite difference theory for flutter of thermally stressed plates at hypersonic speed

Stability analysis of three-dimensional thick rectangular plate using direct variational energy method

This study investigated the elastic static stability analysis of homogeneous and isotropic thick rectangular plates with twelve boundary conditions and carrying uniformly distributed uniaxial compressive load using the direct variational method. In the analysis, a thick plate energy expression was developed from the three-dimensional (3-D) constitutive relations and kinematic deformation; thereafter the compatibility equations used to resolve the rotations and deflection relationship were obtained. Likewise, the governing equations were derived by minimizing the equation for the potential energy with respect to deflection. The governing equation is solved to obtain an exact deflection function which is produced by the trigonometric and polynomial displacement shape function. The degree of rotation was obtained from the equation of compatibility which when equated to the deflection function and put into the potential energy equation formulas for the analysis were obtained after differentiating the outcome with respect to the deflection coefficients. The result obtained shows that the non-dimensional values of critical buckling load decrease as the length-width ratio increases (square plate being the highest value), this continues until failure occurs. This implies that an increase in plate width increases the probability of failure in a plate. Hence, it can be deduced that as the in-plane load on the plate increase and approaches the critical buckling, the failure in a plate structure is abound to occur. Meanwhile, the values of critical buckling load increase as the span-thickness ratio increases for all aspect ratios. This means that, as the span-thickness ratio increases an increase in the thickness increases the safety in the plate. It also indicates that the capacity of the plate to resist buckling decreases as the span-depth ratio increases. To establish the credibility of the present study, classical plate theory (CPT), refined plate theory (RPT) and exact solution models from different studies were employed to validate the results. The present works critical buckling load varied with those of CPT and RPT with 7.70% signifying the coarseness of the classical and refined plate theories. This amount of difference cannot be overlooked. The average total percentage differences between the exact 3-D study (Moslemi et al., 2016), and the present model using polynomial and trigonometric displacement functions is less than 1.0%. These differences being so small and negligible indicates that the present model using trigonometric and polynomial produces an exact solution. Thus, confirming the efficacy and reliability of the model for the 3-D stability analysis of rectangular plates