Three-dimensional Elasticity Formulation Research Articles

DETERMINATION OF CONTACT STRESSES IN THE RAILS P50, WHICH ARE OPERATED IN THE METRO

Purpose. In the research one should do: 1) to determine a 3-d stress-strain state of the rail head in contact with the rolling wheels; 2) to analyze different forms of contact interaction; 3) to obtain the data necessary to calculate the durability of railway track rails. Methodology. The basis for calculating the 3-d stress-strain state is the finite element method. The basis for calculating the volume of the stress-strain state is the finite element method. The problem was solved in the elastic 3-D conditions. Real geometrical bodies parameters were used during the solving. Findings. The calculation of the 3-d stress-strain state of the rail head in contact with the rolling wheels for various cases of the contact surfaces geometry is performed. The results of calculation are presented in the graphic and tabular form. The comparison of different options contact conditions is performed. The results are analyzed and conclusions about the optimality conditions of contact interaction are made. Originality. The results of the calculation showed that within the criterion of minimizing the contact stresses in the rails P50 for the conditions specific to the direct contact route section, the use of rolling wheels with a profile according to the drawings of CJSC «MINETEK» is not the most rational decision. The most rational in this case, among the considered is the laying of rails in track with gradient 1:20 and the use of the wheel with the rolling surface profile of 1:10 conicity. The lack of rail gradient eliminates the benefits of the wheel running surface with 1:10 conicity, and a case of contact interaction is the least rational. Practical value. The results of analysis of the contact interaction of the rail head with a rolling stock wheel in a three-dimensional elastic formulation for different conditions of contact interaction were obtained. These data can be used to optimize the conditions of contact interaction and scientific substantiation of the causes of defects of the contact fatigue origin in the railway railhead. The presented models can be upgraded, including the residual stresses in the rails, hardening of the surface layer, and the presence of initial defects as a result of imperfections in the manufacturing process and others in the calculations.

Analytical beam theory for the in-plane deformation of a composite strip with in-plane curvature

The variational-asymptotic-method (VAM) provides a mathematically rigorous way to reduce a three-dimensional elasticity formulation to a one-dimensional beam theory without ad hoc assumptions. In this work, the VAM is employed to develop a beam theory to analyze the in-plane deformation of a laminated strip-beam with initial in-plane curvature. The cross-sectional stiffness constants and recovery relations for stress and strain are presented as analytical expressions. For the case of zero initial curvature, consistency of the expressions with those of plate theory is demonstrated. For strip-beams with initial curvature in the in-plane direction, results obtained show explicit dependence on the curvature. Results are verified by comparison with those obtained from VABS, the accuracy and consistency of which with three-dimensional finite elements has been reported in several published works. In addition to the internal consistency check this work provides and its utility in helping to validate VABS (which is based on the principles of VAM), it is hoped that the results obtained herein, since they are all analytical expressions, will help researchers and engineers validate the effect of initial curvature in their beam theories, whether existing or new.

Three-dimensional temperature dependent free vibration analysis of functionally graded material curved panels resting on two-parameter elastic foundation using a hybrid semi-analytic, differential quadrature method

In this study, free vibration analysis of initially stressed thick simply supported functionally graded curved panel resting on two-parameter elastic foundation (Pasternak model), subjected in thermal environment is studied using the three-dimensional elasticity formulation. The material properties are temperature dependent and the temperature is assumed to have uniform and non-uniform distributions through the thickness direction of the curved panel. In order to discretize the governing equations, the differential quadrature method in the thickness direction and the trigonometric functions in longitudinal and tangential directions in conjunction of the three-dimensional form of the Hamilton’s principle are used. The convergence of the method is demonstrated and to validate the results, comparisons are made with the available solutions for both isotropic and functionally graded material (FGM) curved panels. By examining the results of thick FGM curved panels for various geometrical parameters and temperature distribution models with the inclusion of supporting elastic foundation, the influence of these parameters and in particular, those due to functionally graded material (FGM) parameters are studied.

Modelling of rotors with three-dimensional solid finite elements

A shaft is modelled using three-dimensional solid finite elements. The shear-deformation and rotary inertia effects are automatically included through the three-dimensional elasticity formulation. The formulation allows warping of plane cross-sections and takes care of gyroscopic effect. Unlike a beam element model, the present model allows the actual rotor geometry to be modelled. Shafts with complicated geometry can be modelled provided that the shaft cross-section has two axes of symmetry with equal or unequal second moment of areas. The acceleration of a point on the shaft is determined in inertial and rotating frames. It is found that the finite element formulation becomes much simpler in a rotating frame of reference that rotates about the centre-line of the bearings with an angular velocity equal to the shafts spin speed. The finite element formulation in the above frame is ideally suited to non-circular shafts with solid or hollow, prismatic or tapered sections and continuous or abrupt change in cross-sections. The shaft and the disc can be modelled using the same types of element and this makes it possible to take into account the flexibility of the disc. The formulation also allows edge cracks to be modelled. A two-dimensional model of shaft disc systems executing synchronous whirl on isotropic bearings is presented. The application of the two-dimensional formulation is limited but it reduces the number of degrees of freedom. The three-dimensional solid and two-dimensional plane stress finite element models are extensively validated using standard available results.

On the buckling and collapse of moderately thick composite cylinders under hydrostatic pressure

Abstract This paper presents a summary of the work carried out at Washington University in recent years on the buckling and associated non-linear response and collapse of moderately thick composite cylindrical shells. Ring elements in conjunction with a three-dimensional elasticity formulation are employed in the analysis. The buckling and postbuckling imperfection sensitivity in individual modes is studied first. The problem of interaction between local and overall instabilities is then investigated in detail. Imperfection sensitivity of typical ring-stiffened shells is established by using a simple and effective approach that combines the asymptotic procedure and the amplitude modulation technique. The influence of dynamic application of the hydrostatic pressure is investigated with the simplified model. The results obtained are compared with those produced by a two-dimensional program package which includes full-fledged non-linear analysis with ring elements, and commercial programs wherever possible. The study has thrown light on several issues regarding the modeling and behavioral aspects of thick composite shells which are summarized at the conclusion of the paper.

Three-dimensional analysis of thick composite plates with multiple layers

Abstract A straightforward three-dimensional (3-D) analysis of thick laminated structures composed of many layers (in practical situations there may be dozens, even hundreds of them) is a computationally expensive task. Indeed, if a three-dimensional elasticity formulation is used, each layer has to be treated as a distinct 3-D anisotropic body with the continuity of displacements and transverse stresses imposed between the layers. If adopting such a formulation, a finite element discretization of the multilayered structure would require, besides a large number of the in-plane elements, at least one element per layer in the through-thickness direction. Thus, a huge amount of degrees of freedom, proportional to the number of layers in the laminate, will be present in the computational model. An analogous situation is faced when using any other numerical method. In order to develop accurate and, at the same time, computationally inexpensive 3-D analysis of multilayered structures, the following approximate models are proposed and applied in the framework of the earlier developed 3-D Variational Deficient Approximation Function Approach: • • Globally Averaged Stiffness (GAS) model, where the actual step-wise variation of stiffnesses of the laminate are substituted by the “effective” stiffnesses of an “equivalent” single layer, calculated through the combined stiffness/compliance averaging procedure. • • Sub-Laminate Averaged Stiffness (SLAS) model, where the groups of adjacent layers in the laminate are represented as “sub-laminates” with their stiffnesses calculated through the combined stiffness/compliance averaging procedure analogous to GAS. • • Polynomial Stiffness Approximation (PSA) model, where the actual step-wise stiffness variation is approximated by Bernstein polynomials using the method of least squares. Implementation of the above models and their accuracy is studied on the examples of cross-ply and angle-ply composite rectangular plates with symmetric and non-symmetric ply lay-ups. Accuracy of the results is evaluated by comparison with the Discrete Layer Stiffness (DLS) model, where the actual step-wise stiffness variation in the thickness direction is used.

Three-Dimensional Elasticity Solution for the Buckling of Transversely Isotropic Rods: The Euler Load Revisited

The bifurcation of equilibrium of a compressed transversely isotropic bar is investigated by using a three-dimensional elasticity formulation. In this manner, an assessment of the thickness effects can be accurately performed. For isotropic rods of circular cross-section, the bifurcation value of the compressive force turns out to coincide with the Euler critical load for values of the length-over-radius ratio approximately greater than 15. The elasticity approach predicts always a lower (than the Euler value) critical load for isotropic bodies; the two examples of transversely isotropic bodies considered show also a lower critical load in comparison with the Euler value based on the axial modulus, and the reduction is larger than the one corresponding to isotropic rods with the same length over radius ratio. However, for the isotropic material, both Timoshenko’s formulas for transverse shear correction are conservative; i.e., they predict a lower critical load than the elasticity solution. For a generally transversely isotropic material only the first Timoshenko shear correction formula proved to be a conservative estimate in all cases considered. However, in all cases considered, the second estimate is always closer to the elasticity solution than the first one and therefore, a more precise estimate of the transverse shear effects. Furthermore, by performing a series expansion of the terms of the resulting characteristic equation from the elasticity formulation for the isotropic case, the Euler load is proven to be the solution in the first approximation; consideration of the second approximation gives a direct expression for the correction to the Euler load, therefore defining a new, revised, yet simple formula for column buckling. Finally, the examination of a rod with different end conditions, namely a pinned-pinned rod, shows that the thickness effects depend also on the end fixity.

Calculating stress intensity coefficients in a lock joint in a gas turbine disk in a three-dimensional elastic formulation. Part 2

Methods are compared for calculating stress intensity coefficients in a three-dimensional elastic formulation. A method is proposed that uses approximation formulas that incorporate the crack geometry with stresses referred to a uniform distribution as acting at points on the crack front. A finite-element treatment gives the stress intensity distributions on the path of the supposed crack growth, which can be used to evaluate the effects from the geometry, loading conditions, and reinforcement of the actual structure. It also gives the decisive factor in calculating the stress intensity coefficients for real working conditions for a dovetail lock joint.

A Surface Crack in Shells Under Mixed-Mode Loading Conditions

The problem of a shallow shell containing a surface crack and subjected to general loading conditions is considered. It is shown that, as in the three-dimensional elasticity formulation, the mode I state can be separated whereas modes II and III remain coupled. A line spring model is developed to formulate the part-through crack problem under mixed-mode conditions. A shallow shell of arbitrary curvature having a part-through crack located on the outer or the inner surface of the shell is then considered. Reissner’s transverse shear theory is used to formulate the problem by assuming that the shell is subjected to all five moment and stress resultants. The uncoupled antisymmetric problem is solved for cylindrical and toroidal shells having a surface crack in various orientations and the primary and the secondary stress intensity factors are given. The results show that, unlike the through crack problems, in surface cracks the effect of shell curvature on the stress intensity factors is relatively insignificant.