Paul Seide

An Improved Approximate Theory for the Bending of Laminated Plates

Summary The theory of bending of laminated plates is extended to include shearing deformation in each layer. The shear strain is constant in each layer of the laminate but differs from layer to layer. Transverse displacements are assumed to be constant through the entire plate thickness. Equations of equilibrium and constitutive relations for each layer are developed. The final equations of the theory are expressed compactly in terms of the transverse displacement and the tangential displacements at the top and bottom of each layer. Thus, for a plate with N layers there are 2N – 3 equations and associated boundary conditions to be solved. The theory is applied to the problem of cylindrical bending of a three-layer laminated plate for which an exact three-dimensional elasticity solution is known. Agreement between exact and approximate results for displacement and stresses is good.

An approximate semi-analytical method for prediction of interlaminar shear stresses in an arbitrarily laminated thick plate

Abstract An approximate semi-analytical method for determination of interlaminar shear stress distribution through the thickness of an arbitrarily laminated thick plate has been presented. The method is based on the assumptions of transverse inextensibility and layerwise constant shear angle theory (LCST) and utilizes an assumed quadratic displacement potential energy based finite element method (FEM). Centroid of the triangular surface has been proved, from a rigorous methematical point of view (Aubin-Nitsche theory), to be the point of exceptional accuracy for the interlaminar shear stresses. Numerical results indicate close agreement with the available three-dimensional elasticity theory solutions. A comparison between the present theory and that due to an assumed stress hybrid FEM suggests that the (normal) traction-free-edge condition is not satisfied in the latter approach. Furthermore, the present paper is the first to present the results for interlaminar shear stresses in a two-layer thick square plate of balanced unsymmetric angle-ply construction. A comparison with the recently proposed Equilibrium Method (EM) indicates the superiority of the present method, because the latter assures faster convergence as well as simultaneous vanishing of the transverse shear stresses on both the exposed surfaces of the laminate. Superiority of the present method over the EM, in the case of a symmetric laminate, is limited to faster convergence alone. It has also been demonstrated that the combination of the present method and the reduced (quadratic order) numerical integration scheme yields convergence of the interlaminar shear stresses almost as rapidly as that of the nodal displacements, in the case of a thin plate.

An approximate method for prediction of transverse shear stresses in a laminated shell

Abstract A method has been presented wherein the surface-parallel stresses in a laminated shell are first computed using standard finite element formulation and then approximate transverse shear stress variation through the shell thickness is obtained utilizing the first two (stress) equations of equilibrium and divergence theorem. Numerical results have been presented for both homogeneous isotropic and laminated anisotropic cylindrical shells using the Cartesian-like Riemann coordinate approximation and compared to the corresponding analytical solutions.

Large deflections of rectangular membranes under uniform pressure

Abstract The Foppl large deflection equations for laterally loaded membranes are solved for uniform load and for edges which are fixed normal to the edge but are free to move parallel to the edge. Both the deflection function and the Airy stress function are expanded in Fourier series. The resulting coupled non-linear cubic equations for the deflection function coefficients are truncated and solved by means of an iterative procedure. Results for the center normal deflection and the stress resultants at selected points are calculated with the use of 100 or more equations and are found to differ significantly from the previously accepted approximate results.

Triangular element for analysis of a stretched plate weakened by a part-through hole

Abstract A C 0 -type triangular element formulation in orthogonal curvilinear coordinates has been developed, based on assumptions of transverse inextensibility and layerwise constant shear angle for analysis of a stretched homogeneous plate weakened by a part-through hole. The element stiffness matrix and consistent load vector have been derived using assumed quadratic displacement (in curvilinear coordinate plane) potential energy approach. Numerical results obtained using the straight-sided triangular element version indicate the presence of transverse shear deformation in a stretched homogeneous plate weakened by a symmetrically located concentric hole

Triangular element for analysis of perforated plates under inplane and transverse loads

Abstract A C0-type triangular element formulation in orthogonal curvilinear co-ordinates has been developed, based on assumptions of transverse inextensibility and constant shear angle through thickness for analysis of perforated plates subjected to inplane and transverse loads. The assumed quadratic displacement potential energy approach is utilized in obtaining an element stiffness matrix and consistent load vector, which are numerically integrated. Numerical results have been obtained using a straight-sided triangular version, which behaves like a subparametric element, for stretching and bending analyses of perforated plates.

Snap-Through of Initially Buckled Beams Under Uniform Random Pressure*

Abstract : Aircraft structural components such as engine air intake ducting and rear fuselage and empennage structures which are located in the vicinity of jet engine exhausts experience combined heating and random dynamic excitation which result from the acoustic or pseudoacoustic noise emitted by the jet efflux.

Large deflections of prestressed simply supported rectangular plates under uniform pressure

Abstract The von Karman large deflection equations for laterally loaded rectangular plates are extended to include uniform prestresses parallel to the edges and are solved for uniform load and for edges which are simply supported against movement normal to the plane of the plate and which are either held or free to move as a rigid body in the plane of the plate. Calculated values of center deflection and maximum stress parameters are given as functions of the load parameter for plates of various aspect ratios.

Postbuckling behavior of circular rings with two or four concentrated loads

Abstract Exact solutions are obtained for the deformations of circular rings which have buckled under two or four equal and equally spaced normal or centrally directed concentrated loads. The ring is assumed to be inextensible and the local moment to be proportional to the local change of curvature. The results indicate that the ring with four loads is neutrally stable at its bifurcation load while the ring with two loads is unstable.

Some implications of strain energy expressions for the Love-Kirchhoff non-linear theory of plates and shells

Abstract The exact non-linear theory of bending of shells under the Love-Kirchhoff hypothesis is discussed from the point of view of three-dimensional elasticity theory. The assumption of various forms of the strain energy function for the shell is shown to imply corresponding expressions for the stress distribution throughout the shell wall. Exact equilibrium and constitutive relations for the study of the elastic stability of shells are derived. The equations are specialized and fully discussed for flat plates, for which a correction to the classical equations is obtained.