Transverse Normal Stress Effects Research Articles

Stationary complementary energy high-order theory for the static analysis of beams

A high-order theory is developed for the analysis of beams with general mono-symmetric cross-sections. The theory represents the nonlinear distribution of the longitudinal normal stress across the section depth by a polynomial series expansion up to any order as specified by the analyst. The corresponding shear and transverse normal stresses are obtained by satisfying the 2D infinitesimal stress-flow equilibrium conditions. The resulting statically admissible stress fields are then used in conjunction with the principle of stationary complementary strain energy to formulate the governing compatibility equations and boundary conditions. Closed form solutions are then developed for general loading and boundary conditions. Comparisons with the theory of elasticity and 3D finite element analysis predictions showcase the ability of the present theory to naturally capture shear deformation effects, transverse normal stress effects, nonlinear longitudinal normal stress distributions in deep beams, as well as the effect of support height. Unlike conventional beam solutions that are based on postulated kinematic assumptions, which tend to converge to the displacement response from below, the present theory avoids introducing any kinematic assumptions and is shown to converge to the solution from above. The theory is applicable to beams with doubly symmetric or mono-symmetric cross-sections, with isotropic or orthotropic materials, and subjected to general loading and boundary conditions. The theory is shown to offer advantages compared to other theories when modelling deep beams and/or beams with supports that are offset from the centroidal axis.

Density functional theory calculation of ideal strength of SiC and GaN: Effect of multi-axial stress

First-principles density functional theory calculations are performed to quantitatively evaluate the ideal strength of SiC and GaN crystals. The effect of transverse normal stress on the tensile strength and that of hydrostatic stress on the shear strength are clarified for various crystal structures: 3C, 2H and 4H for SiC, zincblende (Z) and wurtzite (W) for GaN. The effect of polytype SiC structures on the shear strength of some shear systems appears to be significant, while that on the tensile strength is marginal. While the ideal tensile strength generally decreases by transverse (perpendicular) tension, large transverse compression reduces the strength in some structures and orientations ([110] tension in 3C–SiC and Z–GaN; [21¯1¯0] tension in 2H–SiC, 4H–SiC and W–GaN) due to structural phase transition. Although SiC and GaN are basically brittle materials, our investigation suggests that not only cleavage by tension but also sliding due to shear stress take place depending on the multi-axial stress condition.

An RMVT-based third-order shear deformation theory of multilayered functionally graded material plates

Abstract A Reissner mixed variational theorem (RMVT)-based third-order shear deformation theory (TSDT) is developed for the static analysis of simply-supported, multilayered functionally graded material (FGM) plates under mechanical loads. The material properties of the FGM layers are assumed to obey either the exponent-law distributions through the thickness coordinate or the power-law distributions of the volume fractions of the constituents. In this theory, Reddy’s third-order displacement model and the layerwise parabolic function distributions of transverse shear stresses are assumed in the kinematic and kinetic fields, respectively, a priori, where the effect of transverse normal stress is regarded as minor and thus ignored. The continuity conditions of both transverse shear stresses and elastic displacements at the interfaces between adjacent layers are then exactly satisfied in this RMVT-based TSDT. On the basis of RMVT, a set of Euler–Lagrange equations associated with the possible boundary conditions is derived. In conjunction with the method of variable separation and Fourier series expansion, this theory is successfully applied to the static analysis of simply-supported, multilayered FGM plates under mechanical loads. A parametric study of the effects of the material-property gradient index and the span-thickness ratio on the displacement and stress components induced in the plates is undertaken.

Effect of compressive transverse normal stress on mode II fracture toughness in polymeric composites

Effect of transverse normal stress on mode II fracture toughness of unidirectional fiber reinforced composites was studied experimentally in conjunction with finite element analyses. Mode II fracture tests were conducted on the S2/8552 glass/epoxy composite using off-axis specimens with a through thickness crack. The finite element method was employed to perform stress analyses from which mode II fracture toughness was extracted. In the analysis, crack surface contact friction effect was considered. It was found that the transverse normal compressive stress has significant effect on mode II fracture toughness of the composite. Moreover, the fracture toughness measured using the off-axis specimen was found to be quite different from that evaluated using the conventional end notched flexural (ENF) specimen in three-point bending. It was found that mode II fracture toughness cannot be characterized by the crack tip singular shear stress alone; nonsingular stresses ahead of the crack tip appear to have substantial influence on the apparent mode II fracture toughness of the composite.

Theories and Finite Elements for Multilayered Plates and Shells:A Unified compact formulation with numerical assessment and benchmarking

This work is a sequel of a previous author’s article: “Theories and Finite Elements for Multilayered. Anisotropic, Composite Plates and Shell”, Archive of Computational Methods in Engineering Vol 9, no 2, 2002; in which a literature overview of available modelings for layered flat and curved structures was given. The two following topics, which were not addressed in the previous work, are detailed in this review: 1. derivation of governing equations and finite element matrices for some of the most relevant plate/shell theories; 2. to present an extensive numerical evaluations of available results, along with assessment and benchmarking. The article content has been divided into four parts. An introduction to this review content is given in Part I. A unified description of several modelings based on displacements and transverse stress assumptions ins given in Part II. The order of the expansion in the thickness directions has been taken as a free parameter. Two-dimensional modelings which include Zig-Zag effects, Interlaminar Continuity as well as Layer-Wise (LW), and Equivalent Single Layer (ESL) description have been addressed. Part III quotes governing equations and FE matrices which have been written in a unified manner by making an extensive use of arrays notations. Governing differential equations of double curved shells and finite element matrices of multilayered plates are considered. Principle of Virtual Displacement (PVD) and Reissner’s Mixed Variational Theorem (RMVT), have been employed as statements to drive variationally consistent conditions, e.g.C 0 -Requirements, on the assumed displacements and stransverse stress fields. The number of the nodes in the element has been taken as a free parameter. As a results both differential governing equations and finite element matrices have been written in terms of a few 3×3 fundamental nuclei which have 9 only terms each. A vast and detailed numerical investigation has been given in Part IV. Performances of available theories and finite elements have been compared by building about 40 tables and 16 figures. More than fifty available theories and finite elements have been compared to those developed in the framework of the unified notation discussed in Parts II and III. Closed form solutions and and finite element results related to bending and vibration of plates and shells have been addressed. Zig-zag effects and interlaminar continuity have been evaluated for a number of problems. Different possibilities to get transverse normal stresses have been compared. LW results have been systematically compared to ESL ones. Detailed evaluations of transverse normal stress effects are given. Exhaustive assessment has been conducted in the Tables 28–39 which compare more than 40 models to evaluate local and global response of layered structures. A final Meyer-Piening problem is used to asses two-dimensional modelings vs local effects description.

A simplified approach for incorporating thickness stress in the analysis of sheet metal forming using shell elements

AbstractA simplified scheme for considering the thickness stress of shell elements induced by contact is presented which improves the accuracy of sheet metal forming analysis. The yield function formulated on the basis of plane stress conditions is modified to incorporate the effect of transverse normal stress induced by contact forces acting on shell elements and return mapping routine is used to update in‐plane stresses at each time step. The transverse normal stress distributions in the thickness direction are determined using the analytic solution of the cylindrical tube under the internal pressure. As numerical examples, uni‐axial compression, bi‐axial tension and bending tests are treated. The problem of cylindrical cup drawing is also calculated. Each result is compared with the results obtained by the analysis using ABAQUS continuum elements. Copyright © 2002 John Wiley & Sons, Ltd.

Three-dimensional vibration analysis of a cantilevered parallelepiped: Exact and approximate solutions

The paper investigates the hypothetical assumption of neglecting transverse normal stress in vibration analysis for cantilevered thick plates and rectangular parallelepiped. The analysis solves the three-dimensional elasticity energy functional including, as well as excluding, transverse normal stress and obtains free vibration solutions for a cantilevered parallelepiped. Although it is widely accepted, the omission of transverse normal stress is well justified in Kirchhoff–Love thin-plate theory and higher-order thick-plate models; the transverse normal stress effects and thickness extent to which the thick-plate models apply as the thickness increases are practically unknown. The inconsistency of assuming constant transverse normal displacement through thickness for thick-plate models is also addressed. The paper concludes that for a rectangular parallelepiped with thickness exceeding a certain limit, there is considerable discrepancy if transverse normal stress is neglected.

Fundamental solution and boundary element analysis of thick plates on Winkler foundation

This paper introduces for the first time a full derivation of a fundamental solution and boundary integral equations for thick Reissner plates resting on a Winkler elastic foundation, where the effect of transverse normal stresses resulting from the foundation reaction on the plate surface has been considered. The fundamental solution is derived by means of Fourier and Hankel integral transforms in terms of complex Bessel functions. Reduction of domain integral loading terms is provided for cases with uniformly distributed loadings, and concentrated shear forces. Case studies, with known analytical solutions, were analyzed using the developed theory, and it has been shown that the given formulations provide a very accurate solution for a wide range of plate thickness.

A nonlinear dynamical theory of non-classical plates

In this paper, a new nonlinear formulation of plates, including shear and rotatory inertia and transverse normal stress effects, is developed by means of general assumptions, of which the von Karman-type formulation and some thick plate theories are special cases. To keep the formulation fairly general, the problem addressed in this paper simultaneously includes: the effects of shear deformation according to the geometric deformation similarity of the cross-section, the rotatory inertia, and the transverse normal stress. The three-dimensional compatible equations are applied to derive the basic equations. Numerical results are given for linear and non-linear analysis of plates.

Green's function method solution of the Reissner's plate problem

Green's functions can play a significant role in the development of numerical procedures based on the BEM. For that reason the construction of Green's functions and matrices is of great practical importance. In this study, the extension of the Green's function formalism is introduced to Reissner's plate theory, which accounts for the effect of transverse normal stress and transverse shear deformation. The development of Green's matrix for the governing boundary value problem is described based on the separation of variables. The explicit expressions for the entries of Green's matrix are derived. The algorithm is developed for computation of the stress-strain for a rectangular plate. A validation example is presented, illustrating an equal level of accuracy attained for both displacement and stress.

FLUTTER OF GEOMETRICALLY-IMPERFECT SHEAR-DEFORMABLE LAMINATED FLAT PANELS USING NON-LINEAR AERODYNAMICS

The dynamic instability resulting from a high-supersonic flow over a simply supported laminated flat panel subjected to uniform in-plane edge compression is studied. The structural model incorporates geometrical non-linearities, transverse shear deformation, and transverse normal stress effects, and satisfies the traction-free condition on both faces of the panel. In-plane edge restraints and small initial geometric imperfections are also considered. Aerodynamic loads based on the third-order piston theory are used and the panel flutter equations are derived via Galerkin's method. The arclength continuation method is used to determine the static equilibrium state whose dynamic stability behavior is subsequently examined. The effects of transverse shear flexibility, aerodynamic non-linearities, initial imperfections and in-plane plate theory generally overpredicts the critical flow speed and compressive load, and a shear deformation theory is required when considering panels that are flexible in transverse shear. When aerodynamic non-linearities are included, multiple flutter speeds may exist. The nature of the flutter boundary for perfect panels is determined by the method of?!multiple scales, and it is seen that the presence of aerodynamic non-linearities could result in the hard flutter phenomenon. Results indicate that non-linear aerodynamics is important for panels that are not sufficiently thin (i.e., panels characterized by a high flutter Mach number).

On the equations of an eighth-order theory for nonhomogeneous transversely isotropic plates

We obtain the equations of a two-dimensional eighth-order theory for the transverse bending of transversely isotropic plates, through use of a variational equation for displacements and transverse stresses. We show that, for sufficiently thin plates, the solution involves an equation of the fourth order for an interior solution contribution, and two equations of the second order for edge zone solution contributions. One of these accounts for transverse shear stress effects, in nearly the same manner as this effect occurs in the sixth-order theory, and the other accounts for transverse normal stress effects which are neglected in the sixth-order theory.

Cylindrical bending of thick plates

Sixth and twelfth order plate theory solutions for cylindrical bending plate problems, efficiently obtained by a preliminary reduction of the twelfth order theory for general cylindrical bending to a boundary value problem for a fourth order ODE, demonstrate the adequacy of these theories for the interior of thick plates with a thickness-to-span ratio λ up to ½ (at least). In contrast. Kirchhoff's classical thin plate theory is not adequate for values of the same ratio greater than 1/10. Comparison with the corresponding interior solutions of 3-D elastostatics shows that the twelfth order theory captures the (numerically small) first order correction term in the thicknessto-span parameter ε not possible by lower order plate theories. While it is a qualitatively important feature, this additional term associated with the effect of transverse normal stress does not significantly improve the plate theory approximation quantitatively for Isotropic plates.

Free vibration and buckling of unsymmetric cross-ply laminated plates using a refined theory

The free vibration and buckling of unsymmetric cross-ply plates, with a variety of boundary conditions, are analyzed by using a refined shear deformation theory used in reference [18]. This theory incorporates tranverse shear deformation and transverse normal stress effects, higher order effects are accounted for and the static conditions on the external bounding planes of the plate are satisfied. The results obtained are compared with their classical and first-order transverse shear deformation counterparts, and with results available in the literature. Excellent agreement is found with simply supported edge-condition results in the literature. Further evidence is provided that classical laminated plate theory over-predicts both the eigenfrequencies and the buckling loads.

Vibrations of Orthotropic, Thick, Cylindrical Shells

An approximate theory of vibrations of orthotropic, thick, cylindrical shells, in which the effect of transverse normal stress is retained, is deduced from the three-dimensional theory of elasticity. The present theory contains, in addition to the usual membrane and bending terms, the influence of rotatory inertia and transverse-shear deformation. As an application, the propagation of free harmonic waves in an infinite cylindrical shell is studied. Numerical results are given for relatively thin and thick shells of isotropic and nonisotropic materials.

On a Nonlinear Problem in Axisymmetric Bending of Circular Thick Plates : 2nd Report, Circular Plates on Rigid Foundation under Uniform Load and Edge Moment

This paper deals with a nonlinear problem of a circular thick plate pressed by a uniform load against an absolutely rigid, smooth, horizontal foundation and sustaining a moment uniformly distributed along the edge of the plate by using E. Reissner's theory. The results obtained here contain the classical plate theory predictions in its limiting case. In order to clarify the significance of the effect of transverse normal stress in addition to the effect of shear deformation, the stress distribution in the plate, the force field on the foundation and the relation between the load and moment are numerically obtained and compared with the results of the classical plate theory.

Axially Symmetric Motions of Thick Cylindrical Shells

Abstract An approximate theory of axially symmetric motions of thick, elastic, cylindrical shells, in which the effect of transverse normal stress is retained, is deduced from the three-dimensional theory of elasticity. The present theory contains, in addition to the usual membrane and bending terms, also the influence of rotatory inertia and transverse shear deformation. Thus it may be specialized to a variety of shell, plate, and solid-cylinder equations. The propagation of free harmonic waves in an infinite shell is studied on the basis of the present theory and the three-dimensional theory of elasticity. Excellent agreement is obtained for the phase velocity of the lowest mode of motion for a wide range of the parameters involved.