James G. Simmonds

New concepts for linear beam theory with arbitrary geometry and loading

Abstract A new approach is introduced for the analysis and calculation of straight prismatic beams of piecewise constant cross-section under arbitrary loads. This theory can be called “exact” because it determines exact static and kinematic generalized quantities. Moreover, contrary to classical theories, it is not limited to high-aspect ratio (i.e. relatively slender) beams.

Saint-Venant end effects in composite structures☆

Thin-walled structures, such as rifle barrels, rocket casings, helicopter blades and containment vessels, are often constructed of layers of anisotropic, filament or fiber-reinforced materials which must be designed to remain elastic. A proper assessment of end or edge effects in such structures is of fundamental technological importance. The extent to which local stresses, such as those produced by fasteners and at joints, can penetrate girders, beams, plates and shells must be understood by the designer. Thus a distinction must be made between global structural elements (where Strength of Materials or other approximate theories may be used) and local elements which require more detailed (and more costly) analyses based on exact elasticity. Moreover, it must be recognized that it is impossible, in general, to refine global approximate theories (such as various so-called higher-order plate and shell theories) without a simultaneous consideration of local effects. The neglect of end effects is usually justified by appeals to some form of Saint-Venants principle, and years of experience with homogeneous isotropic elastic structures have served to establish this standard procedure. Saint-Venants principle also is the fundamental basis for static mechanical tests of material properties. Thus property measurements are made in a suitable gage section where uniform stress and strain states are induced and local effects due to clamping of the specimen are neglected by an appeal to Saint-Venants principle. Such traditional applications of Saint-Venants principle require major modifications when strongly anisotropic and composite materials are of concern. For such materials, local stress effects persist over distances far greater than are typical for isotropic materials. In this paper, we describe some problems of static and dynamic elasticity where anisotropy induces such extended Saint-Venant end zones. The paper is a review and a comprehensive list of references is given to original work where details of the analysis may be found. The consideration of such extended end zones due to anisotropy is essential in the proper analysis and design of structures using advanced composite materials.

The Nonlinear Thermodynamical Theory of Shells: Descent from 3-Dimensions without Thickness Expansions

A glance at the current engineering literature on shells reveals two strong, interacting trends: the inexorable rise and spread of the finite element method and the pressures of economics that are demanding light, efficient structures for automobiles, air- and spacecraft.

The strain energy density of rubber-like shells☆

Abstract Under the assumption that the state of a shell is described by the change in the first and second fundamental forms of its midsurface from an initially elastic isotropic state, an approximate strain energy density is derived, first using strictly two-dimensional arguments and then by descent from three dimensions assuming incompressibilily. To within errors inherent in shell theory itself, it is shown that the strain energy density is the same as that of a plate of the same material.

Asymptotic analysis of an end-loaded, transversely isotropic, elastic, semi-infinite strip weak in shear

Abstract We use linear elasticity to study a transversely isotropic (or specially orthotropic), semiinfinite slab in plane strain, free of traction on its faces and at infinity and subject to edge loads or displacements that produce stresses and displacements that decay in the axial direction. The governing equations (which are identical to those for a strip in plane stress, free of traction on its long sides and at infinity, and subject to tractions or displacements on its short side) are reduced, in the standard way. to a fourth-order partial differential equation with boundary conditions for a dimensionless Airy stress function ƒ. We study the asymptotic solutions to this equation for four sets of end conditions—traction, mixed (two), displacement—as g3, the ratio of the shear modulus to the geometric mean of the axial and transverse extensional moduli, approaches zero. In all cases, the solutions for ƒ consist of a “wide” boundary layer that decays slowly in the axial direction (over a distance that is long compared to the width of the strip) plus a “narrow” boundary layer that decays rapidly in the axial direction (over a distance that is short compared to the width of the strip). Moreover, we find that the narrow boundary layer has a “sinuous” part that varies rapidly in the transverse direction, but which, to lowest order, does not enter the boundary conditions nor affect the transverse normal stress or the displacements. Because the exact biorthogonality condition for the cigenfunctions associated with ƒ can be replaced by simpler orthogonality conditions in the limit as e→b 0, we are able to obtain, to lowest order, explicit formulae for the coeflicients in the eigenfunction expansions of ƒ for the four different end conditions.

The Eigenvalues for a Self-Equilibrated, Semi-Infinite, Anisotropic Elastic Strip

We have examined the plane stres (or plane strain) problem for an homogeneous anisotropic semi infinite strip, free of tractions on its long sides and subject to self equilibrating end loads

On Application of the Exact Theory of Elastic Beams

A new approach is developed for the analysis and calculation of straight prismatic beams of piecewise constant cross-section under arbitrary loads. The material can be anisotropic and composite; it is only supposed that the beam is x-homogeneous, x being the abscissa. This theory can be called “exact” because it determines exact static and kinematic generalized quantities. Contrary to classical theories, it is not limited to high aspect ratio (i.e. relatively slender beams). The paper is focused on how to use the exact theory of elastic beam for computing 3D stresses. It is shown in particular how to compute the basic operators which depend on the cross-section geometry, the material and the loading which are the basic building blocks of the theory. An example is of an elastic tube with a small thickness submitted to nearly concentrated extremity loads.

Rigorous expunction of poisson's ratio from the reissner-meissner equations☆

Abstract Solutions of the two Reissner-Meissner linear equations for the torsionless, axisymetric deformation of elastically isotropic shells of revolution of constant thickness subject to edge conditions and variable normal pressure are compared with the solutions of a simplified version of these equations obtained by neglecting terms containing Poissons ratio. The relative pointwise differences in the predicted values for the change in the meridional angle and a stress function are shown to be of the same order of magnitude as the inherent errors in classical, first-approximation shell theory. These results do not depend on physical arguments or asymptotic integration techniques, but rather follow from the structure of the Reissner-Meissner equations themselves. The advantage of the simplified equations is that they may be combined into a single comple-valued equation containing no conjugates of the dependent variable.

Extension of Koiter'sL2-error estimate to approximate shell solutions with no strain energy functional

ZusammenfassungEs wird gezeigt, dass sich KoitersL2-Fehlerschätzung für die Lösung linearer elastischer Randwertprobleme, die ein modifiziertes angenähertes Funktional der Verzerrungsenergie enthalten, auf die Lösung von Problemen erweitern lässt, in denen die Spannungs-Dehnungs-Beziehungen nur approximativ von einer Verzerrungsenergiedichte abgeleitet werden können. Die Resultate werden für Untersuchungen bei Schalen mit einer Mittelfläche konstanter Durch-schnittskrümmung und bestehend aus einem Material mit modifizierten Spannungs-Dehnungs-Beziehungen verwendet, um die geltenden Gleichungen zu vereinfachen.

Notes on the nonlinearly elastic boussinessq problem

The nonlinearly elastic Boussinesq problem is to find the deformation produced in a homogeneous, isotropic, elastic half space by a point force normal to the undeformed boundary, using the exact equations of elasticity for an incompressible or compressible material. First we derive the governing equations from the Principle of Stationary Potential Energy and then we examine some of the implications of the conservation laws of elastostatics when applied to the entire half space, assuming that the well-known linear Boussinesq solution is valid at large distances from the point load. Next, we hypothesize asymptotic forms for the solutions near the point load and, finally, we seek solutions for two specific materials: an incompressible, generalized neo-Hookean (power-law) material introduced by Knowles and a compressible Blatz-Ko material. We find that the former, if sufficiently stiffer than the conventional neo-Hookean material, can support a finite deflection under the point load, but that the latter cannot.