Robert R. Archer

Axisymmetric stresses in transversely isotropic finite cylinders

Abstract The elastostatic analysis of transversely isotropic finite cylinders with stress-free lateral surfaces is considered using the formulation in terms of a displacement potential. Solution in the form of the eigenfunction expansion where each of the eigen-functions satisfies the lateral boundary conditions is obtained. The end conditions are satisfied by means of the general orthogonality relations for the radial eigenfunctions. The orthogonality technique gives closed-form expressions for the Fourier-Bessel coefficients for certain types of mixed end conditions. For other cases the technique leads to an infinite set of linear equations. The convergence of these equations are examined and solutions are obtained by suitable truncation. Both stress and displacement end conditions are considered and numerical results for two types of transversely isotropic materials are presented.

Method of initial functions for thick shells

Abstract In the present work for circular cylindrical shells, three-dimensional elasticity equations are solved by assuming Taylor series expansions, in the radial direction, for the stresses and displacements. Depending upon the number of terms retained in the expansion, different order shell theories are derived., classical theories (referred to as eighth-order), the shear deformation-transverse normal stress theories (referred to as tenth-order), and higher order theories (referred to as twelfth-order). In each case, by carrying out the symbolic algebra using the digital computer, partial differential equations are derived. The procedure was carried out in detail for the case of a circular cylindrical shell with no loading on the interior surface and a given pressure distribution on the exterior surface. Then, numerical comparisons are made between the current theories and various shell theories, as well as the exact (three dimensional) theory. Thus, using this method with its associated computer programs, one can realize a spectrum of approximate shell theories ranging from the classical thin shell, through all current thick shell theories, right up to the three-dimensional elastic theories.

Solutions of a twelfth order thick plate theory

SummaryA system of equilibrium equations governing a twelfth-order theory for the bending of thick plates is shown to be equivalent to a biharmonic equation together with four Helmholtz equations. These equations are closely related to equations derived by Cheng for an elasticity based thick plate theory. Detailed comparisons between the solutions for the displacements and stresses predicted by the approximate plate theory and an exact theory give some basis for deciding the applicability of the plate theory. As an example of the application of the solution procedure presented here, some earlier results for the decay parameters for the end problem for finite width plates are extended to the present case of twelfth-order plate theory.

Relief of Growth Stresses in Diametrical Planks

The technique of examining the strain relief of Strips cut from diametrical planks has been used extensively in experimental studies aimed at predicting growth Stresses in trees. This paper presents an analysis of the redistribution of growth Stresses associated with the preparation of diametrical planks. The analysis is based on a homogeneous, orthotropic elastic model. The process of stress relief is divided into two stages, one the transverse stress relief which may be treated äs a plane strain problem, and the other the longitudinal stress relief which may be treated äs a plane stress Problem. Ash, a typical hardwood, is chosen äs an example and the theoretical axisymmetric growth stress distribution is adopted. The results show that the relief of Stresses on the sides of the plank does not produce significant strains, which has also been predicted by Gillis using a simplified analysis. The relief of Stresses at the ends of the plank, in addition to causing a uniform strain, results in end effects with large strains which may be responsible for the violent Splitting of diametrical planks äs observed in previous experiments. The contribution of the end displacement, caused by strain relief in the end zones of the plank, to the measured length changes in the plank and strips is examined with the help of the numerical results. These theoretical observations may be of help in devising suitable experimental methods aimed at measuring growth strains using the planks and strips.

On the “End Problem” for Thick Rectangular Plates

Summary The state of stress near the boundary of a thick plate is studied by means of “end solutions” obtained from the shea ˙ deformation plate theory of Reissner. The complex eigenvalues which control the width of the edge zone vary with the thickness. Eigenvalues for simply supported and clamped edges are obtained for a range of thicknesses. The problem of a thick plate loaded over an interval at the center of an otherwise free edge is solved by a superposition of eigenmodes. Of particular interest is the rapid variation of stresses in a very narrow zone near the edges where the adjustment to three prescribed boundary conditions takes place.

Semi-analytical finite element analysis of end problems for orthotropic cylinders

Abstract A semi-analytical finite element method has been developed for the solution of general asymmetric elastostatic end problems for orthotropic cylinders by an eigensolution technique. The finite element equations for the semi-infinite cylinder are derived from the potential energy expression assuming that the solution decays exponentially in the axial direction. The resulting quadratic matrix eigenvalue problem yields the decay parameters of the exponential solutions as eigenvalues. For finite cylinders two sets of eigenfunctions together with zero-eigenvalue solutions, if applicable, are superposed to satisfy the end conditions in a least square sense. Two cases of three dimensional flexure analysis of an orthotropic beam of hollow circular section have been considered for illustration. Accurate results have been obtained using this technique which lends itself to easy programming for a digital computer. Although the method is less versatile than the three dimensional finite element methods, it may prove to be a powerful and economical method for a wide range of elasticity problems.

Relief of Growth Stresses in Planks

The redistribution of growth Stresses associated with the preparation of a general off-diametrical plank is analysed in this paper using a semianalytical finite element method. In this linear elastic analysis the log is assumed to be cylindrically orthotropic which makes an off-diametrical plank inhomogeneously anisotropic. The stress relief process is divided into two stages, one the transverse stress relief which may be treated äs a plane strain problem, and the other the longitudinal stress relief which may be treated äs a plane stress problem. A theoretical axisymmetric growth stress distribution and average hardwood properties are considered in the numerical example. The stress relief causes high strains near the end from the longitudinal stress relief in addition to high bending strains caused by transverse stress relief, the later being present only in off-diametrical planks. The transverse bending present in addition to the above two stages causes small strains. The high strains near the ends may explain the cracks observed in planks. This general theoretical analysis technique may be helpful in devising experimental procedures for measuring growth stress distributions.

Internal Residual Stress Patterns in Tree Stems

Kubler (1959a, b) derived mathematical expressions for the internal stress increments induced by growth stresses in each small growth increment at the periphery of tree stems. His continuum mechanical arguments were based upon the equilibrium of suitably chosen elements, and wood was modeled as an elastic transversely isotropic material. Kubler (1959a, 1986) gives an excellent review of earlier work on this subject. He also pointed out the added protection which these residual stresses provide against damage by wind loads on the tree. The longitudinal stress at the periphery serves as a “pre-stressing” system against excessive compressive stress acting upon the stem opposite to the wind direction, while the tangential compressive stresses at the periphery counteract cracks induced by frost and drying or heating which might cause the tree to dry out.

Higher order theories for thick cylindrical shells

SummaryA variational derivation is used to obtain 10th and 12th order shell theories along with the associated boundary conditions. A computer program produces the coefficients in the reductions of the sets of equilibrium equations to equivalent single 10th and 12th order equations. Exact solutions for closed shells which decay in the axial direction are obtained and compared with exact three dimensional solutions in order to assess the accuracy of the shell theory as an approximation to the elasticity theory.

An Introduction to Growth Stresses

This book is concerned with mechanical stresses and strains which develop in trees as they grow. The term “growth stress” has come to be used (Dinwoodie 1966) when referring to distributions of mechanical stresses and strains which originate in the cambial layers of woody stems of a wide variety of tree species (Jacobs 1945). These stresses must be clearly distinguished from and not confused with other additional stresses which might arise due to other factors; for example, when subsequent moisture changes cause “drying stresses” to develop in wood harvested from trees or when gravitationally induced stresses in standing trees are redistributed by felling.