Plane Elasticity Solution Research Articles

Assessment of instability initiation in exposed salt rock roofs

Stability of rocks is the main requirement for the safe operation of mines. For this purpose, certain measures are applied for the protection and support of underground openings, including roof support design and roof arrangement in the most stable rocks. Stability assessment of underground excavations is largely related to their roof stability. Determination of possible instability conditions in mine roofs governs the choice of support system design and parameters of mine excavations. The Upper Kama Potash Salt Deposit represents a stratified layer of solid salt rocks. Roof instability develops as stratification and roof collapse. The Mohr–Coulomb criterion of coherent rocks is currently used to estimate parameters of a possible damage zone in the exposed roof. This criterion allows evaluating shear plane angles in roof rocks and, as a result, finding parameters of the possible collapse zone. The experience of mining operations in the Upper Kama deposit shows different failure conditions as against the Mohr–Coulomb criterion as the stress state is scarcely included in the criterion used. This study is an attempt to assess parameters of rock exposures by solving a Lame problem in terms of a single mine excavation. The analytical results were compared with the parameters obtained from the plane elasticity solutions by the finite element method. Based on the implemented studies, an engineering approach is developed for the assessment of anticipated instability parameters in exposed roofs in horizontal excavations driven in salt rocks.

Unit weight patterns of overlying rocks above commercial-value salt strata in the Upper Kama deposit

Stability of rocks is the main requirement for the safe operation of mines. For this purpose, certain measures are applied for the protection and support of underground openings, including roof support design and roof arrangement in the most stable rocks. Stability assessment of underground excavations is largely related to their roof stability. Determination of possible instability conditions in mine roofs governs the choice of support system design and parameters of mine excavations. The Upper Kama Potash Salt Deposit represents a stratified layer of solid salt rocks. Roof instability develops as stratification and roof collapse. The Mohr–Coulomb criterion of coherent rocks is currently used to estimate parameters of a possible damage zone in the exposed roof. This criterion allows evaluating shear plane angles in roof rocks and, as a result, finding parameters of the possible collapse zone. The experience of mining operations in the Upper Kama deposit shows different failure conditions as against the Mohr–Coulomb criterion as the stress state is scarcely included in the criterion used. This study is an attempt to assess parameters of rock exposures by solving a Lame problem in terms of a single mine excavation. The analytical results were compared with the parameters obtained from the plane elasticity solutions by the finite element method. Based on the implemented studies, an engineering approach is developed for the assessment of anticipated instability parameters in exposed roofs in horizontal excavations driven in salt rocks.

Edge Effect in a Two-layer Orthotropic Strip

The article discusses the edge effect (boundary layer, Saint-Venant effect) of a two-layer strip of orthotropic materials. Two methods are used to solve the problem of plane elasticity (solution in stresses and displacements). Both methods are constructed in such a way that they can describe the edge effect. This allows us to construct a theory for calculating the edge effects of single-layer strips based on which we manage to solve the problem of calculating the edge effect of two-layer strips analytically accurately. In fact, the problem can be summarized by saying that it is calculation of a single-layer strip. The obtained numerical results show that both methods of solution lead to the completely coinciding results.

Wave Propagation Analysis in Beams Using Shear Deformable Beam Theories Considering Second Spectrum

AbstractIn this study, wave propagation in beams is studied using different beam theories like Euler-Bernoulli, Timoshenko and Reddy beam theories. Dispersion curves obtained for these beam theories are compared with the exact plane elasticity solutions. It is obtained that, there are two branches for Reddy beam theory similar to the Timoshenko beam theory. However, one branch is obtained for Euler-Bernoulli beam theory. The effects of in-plane load on Timoshenko and Reddy beam theories are examined and dispersion curves of the Timoshenko and Reddy beams are compared with exact plane elasticity solution. In Timoshenko beam theory, qualitative difference between the two spectrums has been lost with in-plane loads for some wave numbers.

Plane elasticity solutions for beams with fixed ends

The plane stress problem of beams is a typical one in elasticity theory. In this paper a new set of boundary conditions for the fixed end is proposed to improve the accuracy of the plane elasticity solution for beams with fixed end(s). Plane elasticity solutions are then derived for the cantilever beam, propped cantilever beam, and fixed-fixed beam. The new set of boundary conditions is constructed by combining two conventional ones with a parameter. The parameters for different kinds of beams are determined by minimizing the square sum of the longitudinal displacements through the thickness of the fixed end. Comparison with the results obtained by the finite element method (FEM) shows the efficiency of the new type of boundary conditions. When the beam is a deep one, it is found that different boundary conditions yield different errors, and the elasticity solution obtained by the new boundary conditions best approaches the FEM results.

General solutions of plane elasticity of one-dimensional orthorhombic quasicrystals with piezoelectric effect

By introducing four potential functions, the governing equations of plane problems in 1D orthorhombic quasicrystals with piezoelectric effect are composed of four second-order partial differential equations, in which the quasi-harmonic functions are the essential unknowns. The general solution of these equations is further established, and all expressions are expressed in terms of the potential functions. As an application of the general solution, the closed-form solutions are obtained for wedge problems or half-plane problems of 1D orthorhombic piezoelectric quasicrystals.

Solution of a Problem Linear Plane Elasticity with Mixed Boundary Conditions by the Method of Boundary Integrals

A numerical boundary integral scheme is proposed for the solution to the system of eld equations of plane. The stresses are prescribed on one-half of the circle, while the displacements are given. The considered problem with mixed boundary conditions in the circle is replaced by two problems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way, and the problem at this stage is reduced to the solution to a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution and the unknown boundary values of stresses or displacements on proper parts of the boundary. On the basis of the obtained results, it is inferred that a stress component has a singularity at each of the two separation points, thought to be of logarithmic type. The results are discussed and boundary plots are given. We have also calculated the unknown functions in the bulk directly from the given boundary conditions using the boundary collocation method. The obtained results in the bulk are discussed and three-dimensional plots are given. A tentative form for the singular solution is proposed and the corresponding singular stresses and displacements are plotted in the bulk. The form of the singular tangential stress is seen to be compatible with the boundary values obtained earlier. The efficiency of the used numerical schemes is discussed.

Plane elasticity solution for a half-space weakened by a circular hole and loaded by a concentrated force

AbstractWe present a complete analytical solution for the plane elasticity problem of a concentrated force acting on a half-space weakened by a circular hole. The solution is based on the algorithm proposed by G.B. Jeffery in 1920 for elasticity problems, using bipolar coordinates to characterize the geometry. We found the biharmonic stress function describing stresses and strains in the half-space and the associated biharmonic function that allows determining the displacement field. Both functions are given in the form of Fourier series with quite compact coefficients. Particular attention is paid to the displacements at the plane boundary of the half-space. It is discussed how these displacements can be used in an inverse procedure for the identification of a circular hole diameter and position.

An XFEM crack-tip enrichment for a crack terminating at a bi-material interface

Crack-tip enrichment functions are determined for a crack terminating at a bi-material interface. The stress and displacement components at the crack-tip are expressed considering plane elasticity solution based on Airy functions. The crack-tip characteristic exponents and the enrichment functions are determined using the problem boundary conditions. Single real, double real or complex characteristic exponents are found depending on the crack-angle and the relative mechanical properties of the considered bi-material. Crack-tip enrichment functions are determined for cases where the characteristic exponent is real and complex. To validate the efficiency of the enrichment functions at the crack-tip singularity, the eXtended Finite Element Method (XFEM) is used to solve a mode-I crack terminating at the bi-material interface. Thanks to two Level-set functions, the crack and the bi-material interface are located and the nodes belonging to elements cut by these discontinuities are enriched. The Stress Intensity Factors (SIF) are calculated for a crack perpendicular and terminating at the bi-material interface using the body force method (BFM). The implemented method was tested and good agreement is found when compared to other investigations in the literature.

The effect of transverse normal strain in contact of an orthotropic beam pressed against a circular surface

The contact problem of a straight orthotropic beam pressed onto a rigid circular surface is considered using beam theories that account for transverse shear and transverse normal deformations. The circular nature of the rigid surface emphasizes the difference between Euler Bernoulli theory behavior, where point loads develop at the edge of contact, and the higher order theories that predict non-singular pressure distributions. While Timoshenko beam theory is the simplest theory that addresses this behavior, the prediction of a maximum value of pressure at the edge of contact contradicts the elasticity theory result that contact pressure must drop to zero. Transverse normal strain is therefore introduced, both to study this fundamental discrepancy and to include an important effect in many contact problems. To investigate this effect, higher order beam theories that account for both constant and linear transverse normal strain through the beam thickness are derived using the principle of virtual work. The resulting orthotropic beam theories depend on the bending stiffness (EI), shear stiffness (GA), axial stiffness (EA1) and transverse normal stiffness (EA2), which are independent stiffness parameters that can differ by orders of magnitude. The above mentioned contact problem is then solved analytically for these theories, along with the Timoshenko beam model which assumes zero transverse normal strain. The results for different orthotropic materials show that inclusion of transverse normal deformation has a significant effect on the contact pressure solution. Furthermore, the solution using higher order beam theories encompasses the two extremes of a Hertz-like contact pressure when the half contact length is smaller than the thickness of the beam, and the Timoshenko beam theory case when the half contact length is much larger than the thickness. Concerning the behavior of the pressure at the edge of contact, adherence to the boundary conditions required by the principle of virtual work, shows that while the pressure does tend to zero, it does not become zero unless artificially enforced. In this regard the solution for the case of linear strain is better than that for constant strain. All beam solutions are validated with plane elasticity solutions obtained using the commercial finite element software ABAQUS.

General plane elasticity solution for non-homogeneous materials

In this study, we consider the general plane elasticity problem for non-homogeneous materials subjected to mechanical and thermal loads. The general equations are derived in x–y coordinate system. It is shown that, if the non-homogeneity varies exponentially, then the governing equations are reduced to partial differential equations with constant coefficients. To illustrate the validity and usefulness of the formulation, problems involving both mechanical and thermal loads with the non-homogeneity varying in both directions are considered. As examples, infinite non-homogeneous beams subjected to axial load and bending moment are considered and the solutions obtained in closed form. Then an example illustrating the effect of temperature change is presented. Again a closed form solution is obtained for the stress distribution. Also for limiting values of the non-homogeneity parameter, extensive asymptotic analyses are presented. In addition two problems where the Young’s Modulus or the coefficient of thermal expansion coefficient varies in both directions are presented. From the examples solved, it is found that non-homogeneity affects the stress distribution significantly, and leads to counterintuitive results with serious implication to the study of fracture problems in functionally graded materials.

Structural dynamic analysis using hybrid and mixed finite element models

This paper presents and discusses a hybrid-mixed stress finite element model for the dynamic analysis of structures, assuming a physically and geometrically linear behavior. In this model, both the stress and the displacement fields are approximated in the domain of each element. The displacements along the static boundary are also independently approximated. Orthonormal Legendre polynomials are used as approximation functions. The use of these functions enables the use of analytical closed form solutions for the computation of all structural operators and leads to the development of very effective p-refinement procedures. The linear dynamic analysis is performed using time integration procedures. For this purpose, the classical Newmark, Wilson-θ and α-HHT methods are implemented and tested. A recent and promising alternative approach, based on the definition of a mixed approximation in the time domain is also implemented and assessed. The model being discussed is applied to the solution of plane elasticity and Reissner–Mindlin plate bending problems. To validate the model, to illustrate its potential and to assess its accuracy and efficiency, several numerical examples are discussed and comparisons are made with analytical solutions and solutions provided by other numerical techniques.

Contact solutions for a circular orthotropic beam accounting for transverse normal strain

A higher order orthotropic circular beam theory, which accounts for radial and circumferential normal strain, as well as deformations due to transverse shearing and bending, is developed using the principle of virtual work. As such, the governing differential equations are expressed in terms of the four independent stiffness quantities, EA1, EA2, GA and EI. Special cases that can be obtained by appropriate limits include a Timoshenko beam (GA, EI) and an Euler beam (EI) with or without axial extension (EA2). Using these equations the frictionless contact problem of a compressed ring is solved analytically and the significance of transverse normal strain is studied by comparing with solutions from the more elementary beam theories. Furthermore, the limitations of these theories are studied through comparison with plane elasticity solutions obtained using finite elements. The effect of the radial stiffness (EA1) on the contact pressure is quantified and shown to be important for many practical problems, such as a tire that includes the effect of a tread.

The appropriate edge conditions for two-dimensional quasicrystal semi-infinite strips with mixed edge-data

For two-dimensional quasicrystal semi-infinite strips with mixed edge-data, the Betti–Rayleigh reciprocal theorem and the general solution of plane elasticity of quasicrystals are applied in a novel way to obtain the appropriate edge conditions accurate to all order. By introducing two definitions for the decaying and regular states, the necessary conditions deduced form the reciprocal theorem, for the edge-data to induce only a decaying elastostatic are directly translated into the appropriate edge conditions for the existence of a rapidly decaying solution of the strips. Once a suitable regular state, which fulfills load-free boundary conditions, is constructed for the relevant edge-data, the translation is immediate. However, this is not the situation for general edge-data. For the case of transverse bending and in-plane extension of the strips, these decaying state conditions are obtained explicitly for the first time when the mixed edge-data are imposed on the strip edge. Besides, for a degenerated form, an analytical solution of the decaying state is formulated to verify validity of these edge conditions.

Governing equations and general solutions of plane elasticity of cubic quasicrystals

By introducing a displacement function, huge numbers of complicated equations of plane elasticity of cubic quasicrystals are simplified to an eighth-order partial differential governing equation, and then general solutions are established through an operator method. By taking a decomposition and superposition technique, the general solutions are further simplified in four cases in terms of four second-order displacement functions.

New general solutions of plane elasticity of one‐dimensional quasicrystals

AbstractA theory of general solutions of plane problems is developed for the coupled equations of plane elasticity of one‐dimensional quasicrystals. By introducing a displacement function, very large numbers of complicated equations of the in‐plane problem are simplified to a sixth‐order partial differential governing equation, and then three general solutions are presented through an operator method. Since the displacement function is required to satisfy the sixth‐order governing equation, it is still difficult to obtain rigorous analytic solutions from the higher‐order equation directly and is not applicable in most cases. Therefore, a decomposition and superposition procedure is employed to replace the higher‐order displacement function with three second‐order displacement functions, and accordingly the general solutions are simplified in terms of these functions. In consideration of different cases of three characteristic roots, the general solutions involve three cases, but all are in simple forms that are conveniently applied. Furthermore, it is worth noting that the general solutions obtained here are complete in x3‐convex domains. To illustrate the application of the general solutions obtained, the closed form solutions are obtained for wedge problems subjected to point phonon force and phason force at the apex. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Bending analysis of a functionally graded piezoelectric cantilever beam

A new analysis based on Airy stress function method is presented for a functionally graded piezoelectric material cantilever beam. Assuming that the mechanical and electric properties of the material have the same variations along the thickness direction, a two-dimensional plane elasticity solution is obtained for the coupling electroelastic fields of the beam under different loadings. This solution will be useful in analyzing FGPM beam with arbitrary variations of material properties. The influences of the functionally graded material properties on the structural response of the beam subjected to different loads are also studied through numerical examples.

Trefftz Finite Element Method and Its Applications

This paper presents an overview of the Trefftz finite element and its application in various engineering problems. Basic concepts of the Trefftz method are discussed, such as T-complete functions, special purpose elements, modified variational functionals, rank conditions, intraelement fields, and frame fields. The hybrid-Trefftz finite element formulation and numerical solutions of potential flow problems, plane elasticity, linear thin and thick plate bending, transient heat conduction, and geometrically nonlinear plate bending are described. Formulations for all cases are derived by means of a modified variational functional and T-complete solutions. In the case of geometrically nonlinear plate bending, exact solutions of the Lamé-Navier equations are used for the in-plane intraelement displacement field, and an incremental form of the basic equations is adopted. Generation of elemental stiffness equations from the modified variational principle is also discussed. Some typical numerical results are presented to show the application of the finite element approach. Finally, a brief summary of the approach is provided and future trends in this field are identified. There are 151 references cited in this revised article.

The final governing equation and fundamental solution of plane elasticity of icosahedral quasicrystals

The potential function theory for plane elasticity of three-dimensional icosahedral quasicrystals has been developed, huge numbers of complicated equations involving the elasticity are reduced to a single partial differential equation, fundamental solution of the equation is expressed in six analytic functions of complex variable.

Flexural vibration of rectangular plates subjected to sinusoidally distributed compressive loading on two opposite sides

Natural frequencies and mode shapes of rectangular plates subjected to sinusoidally distributed in-plane compressive loading on two opposite edges were considered in this work. Although vibration analysis of rectangular plates was investigated by numerous authors, flexural vibration of plates with nonlinearly distributed in-plane loading received much less attention. However, certain problems involving thermal stresses may have such nonlinear loading conditions in the plane of the plate. The analysis procedure for the title problem involves first finding a plane elasticity solution for the in-plane problem satisfying all boundary conditions. Then using this in- plane solution, flexural vibration analysis has to be carried out. In a related work investigated recently by the present authors, an in-plane elasticity solution was developed and was compared with existing analytical solutions in the literature and with the finite element method. One aspect of this study was the stress diffusion (i.e., reduction of stress) phenomenon along the length of the plate as well as the presence of all three in-plane stress components. Using this in-plane solution, vibration analysis of rectangular plates was carried out for various plate aspect ratios and the results were compared with corresponding values from finite element analysis.