Orthotropic Bar Research Articles

Timoshenko & Lekhnitskii's puzzle ‒ Rule of swapping for the torsional rigidity of a rectangular bar

Timoshenko presented the torsional rigidity of an isotropic rectangular bar, and Lekhnitskii presented that of an orthotropic rectangular bar. The solutions of Timoshenko and Lekhnitskii (T&L) are functions of the bar's length, width, thickness and shear modulus or moduli. However, the functions of T&L solutions become different from their original ones when the width and thickness are swapped. Swapping the width and thickness definitions does not alter the bar's physical properties, named the “rule of swapping” by the authors. In the last century, no research has shown the T&L solutions to satisfy the rule of swapping, an observation hereinafter referred to as the “Timoshenko & Lekhnitskii Puzzle”. Roughly 90 years later, Tsai et al. re-solved T&L cases using the TSAI technique. The derived solutions are nearly if not completely identical to T&L's numerically and satisfy the rule of swapping automatically. The rule of swapping is a novel issue and has never been mentioned before. Based on the Weierstrass factorization theorem, this study mathematically proves that they are identical for isotropic and orthotropic bars and satisfy the rule of swapping. The result of a torsional pendulum test is analyzed to support the rule.

The torsional rigidity of an orthotropic bar of unidirectional composite laminate part I: Theoretical solution

In 1934, Timoshenko derived and published the torsional rigidity of a rectangular bar of isotropic material by using the membrane analogy. The rigidity depends on the bar’s shear modulus, width and thickness. In 1950, Lekhnitskii followed Timoshenko’s process to derive the torsional rigidity for an orthotropic bar of unidirectional composite laminate, where the rigidity depends on the laminate’s width, the thickness and the shear moduli. In 1990, Tsai, Daniel and Yaniv solved the same case, deriving a quasi-exact solution of torsional rigidity. In the same year, Tsai and Daniel verified their result through multiple experiments. All these rigidities become different when the definitions of thickness and width are swapped. However, they remain identical numerically, lacking mathematic proof for nearly a century. In 2022, Tsai et al. resolved Timoshenko’s case by considering all the conditions and energy minimization criterion. By a completely different approach, the torsional rigidity was derived in a completely different form from that of Timoshenko but numerically identical. Moreover, the solution satisfies the rule of swapping, which makes perfect sense physically. This work applies Tsai’s process to Lekhnitskii’s case. A general solution is derived that satisfies the rule of swapping involving the swapping of shear moduli.

Saint-Venant torsion of cylindrical orthotropic bar

The Saint-Venant torsion of cylindrical orthotropic homogeneous linearly elastic bar is considered. The cross section of the bar is bounded by two straight lines and a curved arc. The geometry of the boundary curve of cross section depends on the shear moduli of the cylindrical orthotropic bar. An analytical method is presented to obtain the Prandtl’s stress functions, warping function, shearing stresses and the torsional rigidity.

Solution of the Saint-Venant torsion of orthotropic bar by the method of boundary collocation

This paper deals with an approximate solution of Saint-Venant torsion problem of orthotropic bar with solid cross section. The boundary collocation is used to get the approximate analytical solution of the torsion problem. Examples illustrate the applications of the presented numerical solution.

Analytical solution for torsion of cylindrical orthotropic annular wedge-shaped bars reinforced by thin shells

This paper deals with the Saint-Venant torsion of elastic, cylindrically orthotropic bar whose cross section is a sector of a circular ring shaped bar. The cylindrically orthotropic homogeneous elastic wedge-shaped bar strengthened by on its curved boundary surfaces by thin isotropic elastic shells. An analytical method is presented to obtain the Prandtl’s stress function, torsion function, torsional rigidity and shearing stresses. A numerical example illustrates the application of the developed analytical method.

Analysis of an orthotropic circular bar weakened by multiple radial cracks under torsional transient loading

In this research, the problem of a circular orthotropic bar with multiple cracks under torsional transient loading is investigated. The solution to the problem weakened by a Volterra-type screw dislocation is first obtained with the aid of the Laplace and finite Fourier sine transform. The bar is considered to be subjected to a torsional transient loading. The solution is obtained for stress fields and displacement in the domain under consideration. Then, the dislocation solution is used to derive a set of Cauchy singular integral equations for analysis of the circular bar containing several cracks. The solution to these equations is used to analyze the torsional rigidity of the bar and also the mode III stress intensity factors at the crack tips. Finally, several examples of a single and multiple cracks are presented.

Saint-Venant torsion of orthotropic bars with a circular cross-section containing multiple cracks

In this paper, the problem of the circular orthotropic bars with multiple cracks is investigated based on the Saint-Venant torsion theory. The solution to the problem of an orthotropic bar weakened by a Volterra-type screw dislocation is first obtained by means of the finite Fourier sine transform. In this research, the bar is assumed to be subjected to an axial net torsion when finding the dislocation solution. The closed form solution is then derived for displacement and stress fields in the bar. At the next step, the dislocation solution is employed to derive a set of Cauchy singular integral equations for analysis of the circular bar with curved cracks. The solution to these equations is used to determine the torsional rigidity of the bar and the stress intensity factors of the tips of the cracks. The paper is furnished with several examples of a single crack and multiple cracks.

Implementation of HAM and meshless method for torsion of functionally graded orthotropic bars

Implementation of HAM and meshless method for torsion of functionally graded orthotropic bars

Saint-Venant torsion of orthotropic bars with rectangular cross section weakened by cracks

The solution to problem of a Volterra-type screw dislocation in an orthotropic bar with rectangular cross section is first obtained by means of a finite Fourier cosine transform. The bar is under axial torque which is governed by the Saint-Venant torsion theory. The series solution is then derived for displacement and stress fields in the bar cross section. The dislocation solution is employed to derive a set of Cauchy singular integral equations for the analysis of curved cracks. The solution to these equations is used to determine the torsional rigidity of bar and the stress intensity factors (SIFs) for the tips of the cracks. Several examples of a single straight crack and an arc-crack are solved. Furthermore, the interaction between two cracks is studied. Finally, the stress components around an inclined edge crack tip are used to define the boundary of the plastic region employing von Mises yield criterion.

Deformation of laminated anisotropic bars in the three-dimensional statement 1. Transverse-longitudinal bending and edge compatibility condition

By using the methods of asymptotic splitting, the 3D problem on the transverse-longitudinal bending of elastic anisotropic laminated bars under the action of longitudinal and transverse loads is split into a number of plane boundary-value problems, whose solution allows one to construct asymptotic approximations for all components of the displacement vector and the strain tensor at each point in the bars. A system of ordinary differential equations of transverse-longitudinal bending is obtained for laminated bars with arbitrary anisotropy of its layers. A condition for the edge compatibility of materials in laminated orthotropic bars is deduced.

Torsion of cylindrically orthotropic elastic circular bars with radial inhomogeneity: some exact solutions and end effects

Torsion of elastic circular bars of radially inhomogeneous, cylindrically orthotropic materials is studied with emphasis on the end effects. To examine the conjecture of Saint-Venant’s torsion, we consider torsion of circular bars with one end fixed and the other end free on which tractions that results in a pure torque are prescribed arbitrarily over the free end surface. Exact solutions that satisfy the prescribed boundary conditions point by point over the entire boundary surfaces are derived in a unified manner for cylindrically orthotropic bars with or without radial inhomogeneity and for their counterparts of Saint-Venant’s torsion. Stress diffusion due to the end effect is examined in the light of the exact solutions.

Asymptotic homogenization model for generally orthotropic reinforcing networks in smartcomposite plates

A general three-dimensional micromechanical model pertaining to smart composite layerswith wavy boundaries is applied to the case of thin smart plates reinforced with a networkof generally orthotropic bars that may also exhibit piezoelectric behavior. The method usedfor the development of the structural model is that of asymptotic homogenization, whichreduces the original boundary value problem into a set of three decoupled problems, eachproblem being characterized by two differential equations. These three sets of differentialequations, referred to as ‘unit cell problems’, deal, independently, with the elastic,piezoelectric, and thermal expansion behavior of the network-reinforced smart compositeplates. The solution of the unit cell problems yields expressions for effective elastic,piezoelectric and thermal expansion coefficients which, as a consequence of theiruniversal nature, can be used to study a wide variety of boundary value problemsassociated with a smart structure of a given geometry. The model can be used tocustomize the effective properties of a smart structure by changing some materialor geometric parameters such as the size or nature of the reinforcements. Thedeveloped general methodology is applied to smart network-reinforced compositestructures with generally orthotropic reinforcements and actuators. As particularexamples, spatial rectangular, triangular, and rhombic smart network plates areanalyzed. The general orthotropy of materials is very important from the practicalviewpoint and this orthotropy makes micromechanical modeling significantlymore complex. In the limiting case of isotropic reinforcements and absence ofactuators, the above general orthotropic micromechanical model converges to resultsthat are consistent with those of previous models obtained by either asymptotichomogenization, or stress–strain relationships in the isotropic reinforcements.

Asymptotic homogenization modeling of thin composite network structures

Asymptotic homogenization models for composite plates reinforced with orthotropic bars are developed and the effective elastic coefficients are obtained. The original problem for the regularly non-homogeneous composite structure reduces to a system of two simpler types of problem, called “unit cell” problems. It is precisely these unit cell problems that enable the determination of the aforementioned coefficients. These effective coefficients are universal in nature and can be used to study a wide variety of boundary value problems associated with a composite structure of a given geometry. The derived model is applied to a number of practical cases involving composite plates reinforced with different networks of orthotropic bars. It is shown that the model can be used to tailor the effective properties of a given composite structure to meet the requirements of a particular application by changing some material or geometric parameters. In the limiting case of isotropic reinforcements, the results are shown to converge to those of previous models obtained by means of asymptotic homogenization or stress–strain relationships in the reinforcements.

The Pairing of Energy Momentum Tensor and Eigentransformation Rate in a Cylindrically Orthotropic Elastic Circular Tube or Bar

ABSTRACTThe pairing of a chemical potential and its associated concentration rate in the thermodynamic identity is well known. The existence of an experimentally determinable molar volume as a function of molar concentrations is also widely used in chemical engineering. What is perhaps less known and infrequently used is the fact that a spatially nonuniform molar volume leads to a field of geometrically incompatible eigenstrain, or eigentransformation in finite deformation. This incompatibility forces the material environment to deform, and the result is a strain energy trapped inside the material body. The change of this energy with respect to the eigentransformation is a generalized configurational stress, which, in the limit as the eigentransformation tends to the identity transformation, tends to the classical energy momentum tensor of Eshelby [1], or the so-called configurational stress. It is shown that the generalized configurational stress is an integral part of the chemical potentials that are responsible for atomic diffusion. This cycle of cause and effect is demonstrated in an axially symmetric setting where the material configuration is taken to be cylindrically orthotropic.Symmetry can be used in many occasions to bare the simple meaning of a complex mathematical expression hiding under the disguise of tensors and index notation. We used it to elucidate the “missing” term in surface chemical potential in 1996 [2] and are now applying it to identify the chemical potential in the bulk. Both Professor Thomas C. T. Ting and I are civil engineering graduates of Tai-Da, the world-renowned National Taiwan University, but did not know each other until we joined UIC. It has been a wonderful friendship of many stimulating anisotropic discussions and numerous delicious potluck dinners. Happy birthday, Tom!

A numerical solution of torsional stress wave propagation in layered orthotropic bar of rectangular cross-section

A numerical scheme based on the method of numerical integration along bicharacteristics is developed for the numerical analysis of torsional wave propagation in layered orthotropic media. The dynamic behavior of a layered orthotropic bar of rectangular cross-section due to impact torque is studied. Moreover, the stability of the present method is investigated.

Reliable solution of an elasto–plastic torsion problem

AbstractA variational inequality with uncertain coefficients, which are given in certain bounded intervals, is considered. The inequality corresponds to a torsion problem of an elasto–plastic orthotropic bar, when employing the Haar–Kármán principle. Two maximization problems with respect to the admissible coefficients are formulated. The solvability of continuous and approximate problems is proven and a convergence analysis is presented. Copyright © 2001 John Wiley & Sons, Ltd.

RELIABLE SOLUTION OF A TORSION PROBLEM IN HENCKY PLASTICITY WITH UNCERTAIN YIELD FUNCTION

The worst scenario method is applied to the torsion problem of a homogeneous orthotropic elasto-plastic bar with uncertain yield function. According to the Haar–Kármán principle, the problem leads to a minimization of the complementary energy over a convex set of admissible stresses. The latter set is determined by a yield function with uncertain coefficients. We formulate two maximization problems with respect to the set of admissible coefficients, prove the solvability of the corresponding continuous and approximate problems, and discuss the convergence of some sequences of finite-element approximations.

Torsion of orthotropic bars with L‐shaped or cruciform cross‐section

AbstractFor an orthotropic torsion bar with L‐shaped or cruciform cross‐section, the studied torsion problem can be reduced to a boundary value problem of elliptic partial differential equation. The studied region is separated into several rectangular sub‐regions, and the series solution is suggested to solve the problem for the individual sub‐region. By using the continuation condition for the functions on the neighbouring sub‐regions, the investigated solution is obtainable. Finally, numerical results for the torsion rigidities of bars are given to demonstrate the influence of the degree of orthotropy. Copyright © 2001 John Wiley & Sons, Ltd.

Torsional stress wave propagation in orthotropic bar of rectangular cross-section

In this paper, a torsional wave propagation problem of anisotropic medium is analyzed by using the method of numerical integration along bicharacteristics. The dynamic behavior of an orthotropic bar of rectangular cross-section due to an impact torque on the end surface of the bar is studied and the stability of the present method is investigated.

Bending of a thin orthotropic bar with an elliptic cut

To study the stressed state of an infinite orthotropic bar with an elliptic slit we use the solutions obtained by integrating the three-dimensional equations of the theory of elasticity. We give a comparison of the results of numerical computations with those obtained in the applied theory. One table. Bibliography: 4 titles.