A Higher Order Shear Deformation Model for Bending Analysis of Functionally Graded Plates

In this paper, the static response of simply supported functionally graded plates subjected to a transverse uniform load and resting on an elastic foundation is examined by using a new higher order displacement model. The present theory exactly satisfies the stress boundary conditions on the top and bottom surfaces of the plate. No transverse shear correction factors are needed, because a correct representation of the transverse shear strain is given. The material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of material constituents. The foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second parameter is zero. The equilibrium equations of a functionally graded plate are given based on the new higher order shear deformation theory of plates presented. The effects of stiffness and gradient index of the foundation on the mechanical responses of the plates are discussed. It is established that the elastic foundations significantly affect the mechanical behavior of thick functionally graded plates. The numerical results presented in the paper can serve as benchmarks for future analyses of thick functionally graded plates on elastic foundations.     

In-Plane Elastic Stability of Arches under a Central Concentrated Load

This paper is concerned with the in-plane elastic stability of arches with a symmetric cross section and subjected to a central concentrated load. The classical methods of predicting elastic buckling loads consider bifurcation from a prebuckling equilibrium path to an orthogonal buckling path. The prebuckling equilibrium path of an arch involves both axial and transverse deformations and so the arch is subjected to both axial compression and bending in the prebuckling stage. In addition, the prebuckling behavior of an arch may become nonlinear. The bending and nonlinearity are not considered in prebuckling analysis of classical methods. A virtual work formulation is used to establish both the nonlinear equilibrium conditions and the buckling equilibrium equations for shallow arches. Analytical solutions for antisymmetric bifurcation buckling and symmetric snap-through buckling loads of shallow arches subjected to this loading regime are obtained. Approximations for the symmetric buckling load of shallow arches and nonshallow fixed arches and for the antisymmetric buckling load of nonshallow pin-ended arches, and criteria that delineate shallow and nonshallow arches are proposed. Comparisons with finite element results demonstrate that the solutions and approximations are accurate. It is found that the existence of antisymmetric bifurcation buckling loads is not a sufficient condition for antisymmetric bifurcation buckling to take place.     

Biparametric Models for Elastic Wedge Foundation

In this paper, some biparametric models for an elastic foundation are proposed. It is assumed that the foundation has the shape of the wedge. The modeling procedure starts from the linear elasticity equations into which we introduce some simplifying assumptions based on the conceptions of decay functions. The simplified models are described by the ordinary differential equations. Stationary and nonstationary Green’s functions for the foundation considered are obtained by applying the Hankel and Laplace transform methods. An example of the interaction between the rigid plate and the elastic wedge foundation is studied.     

Classic Buckling of Three-Dimensional Multicolumn Systems under Gravity Loads

The elastic stability of three-dimensional (3D) multicolumn systems under gravity loads is analyzed in a condensed manner using the classical Timoshenko stability functions. The characteristic equations corresponding to multicolumn systems with sidesway uninhibited, partially inhibited, and totally inhibited are derived. Using the transcendental equations of the proposed method, the effective length K factor for each column and the total critical axial load of an entire story can be determined directly. The proposed method is applicable to 3D framed structures with rigid, semirigid, and simple connections. It is shown that the elastic stability of framed structures depends on: (1) the axial load pattern on the columns; (2) the variation in size and height among the columns; (3) the plan layout of the columns; (4) the overall floor-torsional sway caused by any asymmetries in the loading pattern, column layout, and column sizes and heights (all of which reduce the flexural-buckling capacity of multicolumn systems); (5) the end restraints of the columns; and (6) the bracings along the two horizontal and rotational directions of the floor plane. The proposed method solves the classical bifurcation stability of 3D frames directly without complex matrix solutions, however, it is limited to frames made up of columns of doubly symmetrical cross section with their principal axes parallel to the global axes. Examples are presented that show the effectiveness of the proposed method and the results compared with those obtained by complex matrix methods.     

Nonexistence of Solution for Horizontally Rigid Half-Space

The Westergaard expressions for stresses and displacements in a half-space of Westergaard material are shown to be invalid because the vertical shear stress does not vanish at the plane boundary of the half-space. The same inadequacy is discovered in the solution for a horizontally rigid cross-anisotropic half-space deduced from Michell’s expressions. A surprising conclusion is drawn, namely, that there is currently no exact elastic solution for the stresses and displacements in a horizontally rigid cross-anisotropic half-space. The Westergaard theory should cease to be regarded as an exact elastic solution for a problem of the theory of elasticity.     

Buckling Mode Interaction in Fixed-End Column with Central Brace

The postbuckling behavior of an elastic fixed-end column with an elastic brace at the center is investigated. Attention is focused on those of brace stiffness near its threshold value at which, under axial load, the column becomes critical with respect to two buckling modes simultaneously. We show that, for the brace stiffness greater than the threshold value, there are precisely two secondary bifurcation points on each primary postbuckling path bifurcating from one of the least two classical buckling loads, and the corresponding secondary postbuckling paths connect all of these secondary bifurcation points in a loop. For the brace stiffness less than the threshold value, no secondary bifurcation occurs. The asymptotic expansions of the primary and secondary postbuckling paths are constructed. The stability analysis indicates that, when the brace stiffness goes beyond its threshold value, the primary postbuckling path with a node in the center becomes unstable from stable by means of the secondary bifurcation (i.e., secondary buckling occurs).     

Forced Vertical Vibration of Circular Plate in Multilayered Poroelastic Medium

This paper considers the vertical vibrations of an elastic circular plate in a multilayered poroelastic half space. The plate is subjected to axisymmetric time–harmonic vertical loading and its response is governed by the classical thin-plate theory. The contact surface between the plate and the multilayered half space is assumed to be smooth and either fully permeable or impermeable. The half space under consideration consists of a number of layers with different thicknesses and material properties and is governed by Biot’s poroelastodynamic theory. The vertical displacement of the plate is represented by an admissible function containing a set of generalized coordinates. Contact stress and pore pressure jump are established in terms of generalized coordinates through the solution of flexibility equations based on the influence functions corresponding to vertical and pore pressure loading. Solutions for generalized coordinates are obtained by establishing the equation of motion of the plate through the application of Lagrange’s equations of motion. Selected numerical results are presented to portray the influence of various parameters on dynamic interaction between an elastic plate and a multilayered poroelastic half space.     

Predicting Shrinkage Stress Field in Concrete Slab on Elastic Subgrade

Concrete is a material that changes volumetrically in response to moisture and temperature variations. Frequently, these volumetric changes are prevented by restraint from the surrounding structure, resulting in the development of tensile stresses. This paper provides a method for computing the stress and displacement fields that develop in response to this restraint by considering the concrete slab as a plate resting on an elastic foundation. The interface between the slab and the foundation is capable of simulating all cases between complete perfect bond and perfect compression∕zero tension bond to permit debonding. In addition, stress relaxation is considered in the concrete to account for the reduction in stress due to creep∕relaxation-related phenomena. For this reason, the stress-strain relationship and equilibrium equations have been considered in the rate or differential form. The history-dependent equilibrium equations are obtained by integrating the differential equations with respect to time. An example is presented to illustrate the favorable correlation that exists between the predicted behavior of the plate model and finite-element modeling.     

Vertical Vibration of a Flexible Plate with Rigid Core on Saturated Ground

In this paper, the vertical vibration of a flexible plate with rigid core resting on a semi-infinite saturated soil is studied analytically. The behavior of the soil is assumed to follow Biot’s poroelastodynamic theory with compressible soil skeleton and pore water, and the response of the time-harmonic excited plate is governed by the classical thin-plate theory. By virtue of the Hankel transform technique, the fundamental solutions of the skeleton displacements, stresses, and pore pressure are derived, and a set of dual integral equations associated with the relaxed boundary and completely drained condition at the soil-foundation contact interface are also developed. These governing integral equations are further reduced to the standard Fredholm integral equations of the second kind and solved by numerical procedures. Comparison with existing solutions for a rigid permeable plate on saturated soil confirms the accuracy of the present solution. Selected numerical results are presented to show the influence of the permeability, the size of the rigid core, and the plate flexibility on the dynamic interaction between the elastic plate with rigid core and the underlying saturated soil.     

Thermal Postbuckling of Preloaded Shear Deformable Laminated Plates

Postbuckling analysis is presented for a simply supported, shear deformable laminated plate subjected to a uniform lateral pressure and thermal loading, and resting on an elastic foundation. The temperature fields considered are associated with a nonuniform tentlike and parabolic distribution over the plate surface. The material properties are assumed to be independent of temperature. The lateral pressure is first converted into an initial deflection, and the initial geometric imperfection of the plate also is taken into account. The formulations are based on Reddy's higher-order shear deformation plate theory and include the plate-foundation interaction and thermal effects. The analysis uses a mixed Galerkin-perturbation technique to determine the thermal postbuckling equilibrium paths. The numerical illustrations concern the thermal postbuckling behavior of preloaded antisymmetric angle-ply laminated plates under a tentlike temperature field and symmetric cross-ply laminated plates under a parabolic temperature field resting on Pasternak-type or softening nonlinear elastic foundations from which the results for Winkler elastic foundations are obtained as a limiting case. The effects played by foundation stiffness, fiber orientation, transverse shear deformation, the plate aspect ratio, thermal load ratio, and initial geometric imperfection as well as initial lateral pressure are studied.     

Analysis of Lightweight Deflectometer Test Based on In Situ Stress and Strain Response

The lightweight deflectometer (LWD) is gaining acceptance and popularity as an in situ spot-testing device for quality control/quality assurance of earthwork compaction. Little research has been conducted to investigate the stress–strain response within the soil during LWD testing. Similarly, little research has been performed to examine the appropriateness of using homogeneous, isotropic, linear elastic half-space theory to estimate soil modulus (ELWD) from LWD results. With this aim, an array of vertical stress and strain sensors was placed within the soil to measure the stress–strain response during LWD loading. Measured in situ stress values matched well with stresses predicted using homogeneous, isotropic, linear elastic half-space theory. In situ stress data revealed that the contact stress distribution between the soil surface and loading plate is a function of the soil type. Measured in situ strain values did not correspond well with strains predicted using homogeneous, isotropic, linear elastic elasticity. An exponentially increasing modulus function was required to match experimental with theoretical elastic strains. The results indicate that the commonly used form to predict ELWD is inappropriate if the goal is to extract constitutive soil properties. Analysis of strain data suggests the LWD depth of influence (measurement depth) is 0.9–1.1 times the plate diameter.     

Consequences of population models for the dynamics of food chains

A class of bioenergetic ecological models is studied for the dynamics of food chains with a nutrient at the base. A constant influx rate of the nutrient and a constant efflux rate for all trophic levels is assumed. Starting point is a simple model where prey is converted into predator with a fixed efficiency. This model is extended by the introduction of maintenance and energy reserves at all trophic levels, with two state variables for each trophic level, biomass and reserve energy. Then the dynamics of each population are described by two ordinary differential equations. For all models the bifurcation diagram for the bi-trophic food chain is simple. There are three important regions; a region where the predator goes to extinction, a region where there is a stable equilibrium and a region where a stable limit cycle exists. Bifurcation diagrams for tritrophic food chains are more complicated. Flip bifurcation curves mark regions where complex dynamic behaviour (higher periodic limit cycles as well as chaotic attractors) can occur. We show numerically that Shil'nikov homoclinic orbits to saddle-focus equilibria exists. The codimension 1 continuations of these orbits form a 'skeleton' for a cascade of flip and tangent bifurcations. The bifurcation analysis facilitates the study of the consequences of the population model for the dynamic behaviour of a food chain. Although the predicted transient dynamics of a food chain may depend sensitively on the underlying model for the populations, the global picture of the bifurcation diagram for the different models is about the same.     

Postbuckling of Elastic Columns with Second-Mode Imperfection

Initial imperfections of columns are often assumed to have the shape of the first buckling mode. In this technical note, the imperfection has the shape of the second mode. An elastica analysis is performed, and numerical results are obtained for two cases with the use of a shooting method. For the example of a pinned column, bifurcation occurs at a load slightly higher than the critical load for the perfect system. With further increase in load, the deflection changes smoothly from an antisymmetric shape with one node to a shape with no nodes. For the cantilevered-column example, a limit point occurs just beyond the critical load for the perfect system, and the equilibrium shape jumps from one state to another. As the load is increased further, the deflection passes smoothly to the other side of the column and loses its inflection point.     

Two Tangential Forces and a Penny-Shaped Crack: A Complete Solution

The following problem is considered: A penny-shaped crack is located in the plane z = 0 of a transversely isotropic elastic space and interacts with two equal tangential forces, acting in the same direction, which are located arbitrarily, but symmetrically with respect to the plane of the crack. An exact closed-form solution is obtained here. Some of the stresses and displacements in the whole space are expressed in terms of elementary functions. Two methods of solution are considered. The first method considers superposition of an uncracked space subjected to two forces and a space weakened by a crack and subjected to a specially defined loading. The second method reduces the problem to that of a half-space, one force is acting inside this half-space, and its surface is loaded by a pressure, providing zero displacements outside the crack. It is shown that both methods give the same results.     

Nonlinear Analysis of Uniform Pantographic Columns in Compression

The linear analysis of a uniform pantographic deployable column shows that, in bending, its behavior is very similar to that of an equivalent solid column, whereas under axial loading the two columns display distinct differences in their force and deformation distributions. The total change in the height of a particular pantographic unit in the deployable structure consists of two parts, one due to relative rotation of bars in the unit, the other to their bending. To account for configuration changes, the internal forces must satisfy the equilibrium of each unit “after rotation.” The additional pantographic unit deformation due to bending of bars is found to be based on these forces. The set of equilibrium and nonlinear deformation equations is solved iteratively. The “deformation-controlled” approach for solving this system of equations shows the load maximum in the equilibrium paths that corresponds to the snap-through buckling of the top pantographic unit. It is found that the change in the number of units in the column introduces only minor differences in the equilibrium paths as long as the column height and degree of deployment are kept constant. The axial stiffness of the pantographic column is greatly increased and the snap-through buckling considerably postponed if just one additional constraint is introduced, namely the horizontal link between the two nodes at a particular unit interface. The optimal location of the link is found.     

Postbuckling of Elastic Beams Considering Higher Order Strain Terms

Nonlinear relations between the beam displacement and generalized strain measures, which have basic effects on postbuckling behavior of elastic beams, are presented. The complex coupling phenomena associated with the higher order strain terms is reviewed for the special case of planar and rectilinear pinned-pinned beams. Special consideration was made for the physical assumptions used in the various column-beam models. A natural hierarchy results yielding that all the higher order terms can, for a specific beam formulation, be steadily obtained by dissimilar polynomial approximations of the generalized strains. The asymptotic expansions method and the minimum energy criterion are used to perform analytical calculation of the postbifurcation equilibrium path at the neighborhood of a bifurcation point when only a unique buckling mode is assumed to occur. As a result, postbuckling branches are easily obtained even when accounting for both beam centerline extensional deformation and shear strain. They show that the critical load is scarcely affected by the higher order strain terms unlike the postbuckling paths which are found to be very sensitive to them.     

Strain-induced shape changes of cubic precipitates in a cubic matrix with positive anisotropy

This paper considers the evolution of equilibrium shape during the coarsening of a system of cubic precipitates in a cubic matrix with positive anisotropy (Δ = c₁₁ − c₁₂ − 2c₄₄ > 0). The system is assumed to have homogeneous elastic constants and isotropic surface tension. The low-energy shapes include the sphere, and the octahedron, tetrahedron and plate with {111} faces. Minimization of the sum of the elastic and surface energies shows that the sphere is preferred at arbitrarily small sizes, but ordinarily transforms into a tetrahedron, and finally into a plate as size increases. When Δ > 0.79c₁₁ the octahedron is preferred for a small range of sizes between sphere and tetrahedron. Analytic expressions are given for the equilibrium shape transitions. The results are compared to experimental observations of shape changes during coarsening in (Mg, Y)-ZrO₂. The experimentally observed absence of octahedral shaped precipitates and the size at which tetrahedral shaped precipitates become stable in this system are consistent with the theoretical predictions.     

Escape of carbon from ferrite plates in austenite

A newly developed computer program for the simulation of diffusional transformations has been applied to study the escape of carbon from a plate of ferrite assuming that the plate initially formed by a partitionless reaction from an FeC austenite. Thereafter the ferrite-austenite interface was assumed to be immobile and local equilibrium was assumed for carbon but not for iron. The process first follows a parabolic rate law and is there controlled by the rate of diffusion in ferrite. Later stages are not parabolic and are controlled by the diffusivity in austenite. Its concentration dependence was taken into account. It was found that the rate could be estimated analytically using the maximum value rather than the average value.     

The elastic rod model for DNA and its application to the tertiary structure of DNA minicircles in mononucleosomes

Explicit solutions to the equations of equilibrium in the theory of the elastic rod model for DNA are employed to develop a procedure for finding the configuration that minimizes the elastic energy of a minicircle in a mononucleosome with specified values of the minicircle size N in base pairs, the extent w of wrapping of DNA about the histone core particle, the helical repeat h(0)b of the bound DNA, and the linking number Lk of the minicircle. The procedure permits a determination of the set Y(N, w, h(0)b) of integral values of Lk for which the minimum energy configuration does not involve self-contact, and graphs of writhe versus w are presented for such values of Lk. For the range of N of interest here, 330 < N < 370, the set Y(N, w, h(0)b) is of primary importance: when Lk is not in Y(N, w, h(0)b), the configurations compatible with Lk have elastic energies high enough to preclude the occurrence of an observable concentration of topoisomer Lk in an equilibrium distribution of topoisomers. Equilibrium distributions of Lk, calculated by setting differences in the free energy of the extranucleosomal loop equal to differences in equilibrium elastic energy, are found to be very close to Gaussian when computed under the assumption that w is fixed, but far from Gaussian when it is assumed that w fluctuates between two values. The theoretical results given suggest a method by which one may calculate DNA-histone binding energies from measured equilibrium distributions of Lk.     

Cracks and screw dislocation arrays in anisotropie bimaterial plates

This article examines the equilibrium configuration of dislocation arrays in anisotropic bimaterial plates. Both components in the plate are assumed to be orthotropic with respect to plate axes. Forces on dislocations due to external loadings are found in closed form.Image forces due to phase boundary and free surfaces are also taken into consideration.Discussions on the behavior of screw dislocation arrays can be easily extended to the behavior of antiplane shear cracks. The crack opening displacements for several biomaterial systems have been calculated. In these systems, the crack opening displacements are found to be insensitive to the variation of plate thickness. However, they are strongly affected by the relative magnitude of elastic constants of the component phases. This is demonstrated in several diagrams. Formerly Graduate Student, Department of Mechanical and Aerospace Engineering, University of Delaware