Transverse Shear Stress Effects Research Articles

Finite element analysis of smart composite plate structures coupled with piezoelectric materials: Investigation of static and vibration responses

This research presents a generalized-energy-based finite element modeling of smart composite plates with distributed piezoelectric materials for the static and vibration responses under electrical and mechanical loading. The zigzag kinematics in conjunction with a non-polynomial higher-order shear deformation theory is used to derive the model. The plate theory accounts for the non-linear variations of the transverse shear strains across the thickness of the plates and also accommodates the inter-laminar continuity effects of transverse shear stresses at the interfaces. Eight noded isoparametric serendipity elements are used to discretize the physical domain. The solutions in the time domain are obtained from the discretized system of ordinary differential equations with Newmark’s time integration scheme. Several numerical examples are solved for different types of smart composite plates with various piezoelectric materials, boundary conditions, static and dynamic loadings. The classical negative feedback controller is used for deriving the suppressed and controlled vibration responses of the smart composite plates with distributed actuators and sensors. A comparison of the present FE solutions with the elasticity and other analytical and numerical solutions shows good agreement, thereby ensures the applicability of the present finite element model for studying the coupled electromechanical problems of piezoelectricity.

Nonlinear vibration analysis of graphene sheets resting on Winkler–Pasternak elastic foundation using an atomistic-continuum multiscale model

Based on the higher-order Cauchy–Born (HCB) rule, an atomistic-continuum multiscale model is proposed to address the large-amplitude vibration problem of graphene sheets (GSs) embedded in an elastic medium under various kinds of boundary conditions. By HCB, a linkage is established between the deformation of the atomic structure and macroscopical deformation gradients without any parameter fitting. The elastic foundation is formulated according to the Winkler–Pasternak model which considers both normal pressure and transverse shear stress effects. The weak form of nonlinear governing equations is derived via a variational approach, namely based on the variational differential quadrature (VDQ) method and Hamilton’s principle. In order to solve the obtained equations, a numerical scheme is adopted in which the generalized differential quadrature (GDQ) method together with a numerical Galerkin technique is utilized for discretization in the space domain, and the time-periodic discretization method is used to discretize in the time domain. The effects of the arrangement of atoms, the Winkler and Pasternak coefficients of the elastic foundation, and boundary conditions on the frequency–response curves of GSs are illustrated. It is revealed that the nonlinear effects on the response of GSs with larger size in armchair direction are less important.

Combined effects of transverse shear stresses and nonlinear elastic foundations on the nonlinear dynamic response of heterogeneous orthotropic cylindrical shells

The present work aims to provide a detailed study regarding the nonlinear dynamic response of heterogeneous (HT) orthotropic cylinders resting on the nonlinear elastic foundations. The problem is formulated on the basis of the shear deformation theory (SDT) using von Karman type geometric nonlinearity in the framework of the Donnell's shell theory. Material properties of the cylinders vary continuously in the thickness direction according to the exponential-law distribution. Nonlinear ordinary differential equation is obtained with the use of superposition and Galerkin methods and solved applying the homotopy perturbation method (HPM). The validity of the present method is demonstrated by comparing the present results with existing results in the literature in the limit cases. Numerical simulations are performed to discuss the influences of the nonlinear elastic foundations, vibration amplitude, shear stresses, heterogeneity and cylinder characteristics on the nonlinear frequency parameters.

Nonlinear analysis of concrete shells including effects of normal and transverse shear stresses

The previously developed numerical model of the authors for the analysis of conventional reinforced and prestressed concrete shells under short‐term and long‐term loading was improved by including the effects of transverse shear stresses on the shell failure. The 9‐node degenerated shell element with the layered material model through the thickness of the shell was used. The reinforcement was modelled as a separate layer. To include the effect of transverse shear stresses on the shell failure, the failure criterion for concrete and longitudinal reinforcement was defined by a relation of transverse shear stresses and normal stresses in two mutually perpendicular vertical planes. The total transverse shear bearing capacity of the shell cross‐section is obtained by summing up the concrete and reinforcement contributions. The developed numerical model and appropriate software were verified based on experimental tests.

Evaluation of Weld Root Failure Using Battelle Structural Stress Method

Weld related fatigue failure is one of the most common concerns in welded structures. From the fatigue design point of view, weld toe failure is preferable to weld root failure. Base plate thickness is a controlling parameter for weld toe failure, while weld metal size is a controlling parameter for weld root failure. However, controlling the weld metal size is not easy because the actual weld penetration and weld leg size vary along a weld and from weld to weld. Therefore, analyzing fatigue test data for weld root failure tends to enlarge scatter band due to variability in weld penetration and weld leg size when the nominal weld size is considered. The structural stress based weld fatigue master S-N curve adopted by 2007 ASME section VIII Div. 2 and new API 579/ASME FFS-1 was constructed by incorporating only clearly defined weld toe fatigue data. In this article, a simplified structural stress procedure was developed and a design master S-N curve for weld root failure was established based on the published fatigue test data. Consequently, the mean design master S-N curve for weld root failure is downshifted relative to the mean master S-N curve for weld toe failure, and has a wider scatter band. To be conservative, a crack path along weld throat is recommended for structural stress calculation. Also, the transverse shear stress effects on structural stress calculation can be ignored.

A 2D equivalent single-layer formulation for the effect of transverse shear on laminated plates with curvilinear fibres

The Hellinger–Reissner mixed variational principle in conjunction with Lagrange multipliers is used to model transverse shear stresses, in bending of variable stiffness laminated plates, using a reduced two-dimensional formulation. The effect of transverse shear stresses on the bending deflection of variable angle tow (VAT) laminates is assessed. The novel formulation features multiple shear correction factors that are functions of the bending rigidity terms Dij, their first and second derivatives and the Timoshenko shear factor χ. The new set of governing equations are solved, in their strong form, using the Differential Quadrature Method (DQM) and the accuracy and robustness of the solution technique verified using a 2D “thick” shell and 3D high-fidelity finite element model. The derived theory is superior to the “thick” 2D shell model in capturing transverse shear effects and shows good accuracy compared to the full 3D solution for thicknesses within the range of practical engineering laminates. The derived equations degenerate to Classical Laminate Analysis for very thin configurations but discrepancies as large as 43% are observed for span-to-thickness ratios of 10:1. Finally, the specific VAT panels under investigation are affected more by transverse shear deformation than a corresponding homogeneous quasi-isotropic laminate.

Electric potential approximations for an eight node plate finite element

The aim of this work is to develop a computational tool for multilayered piezoelectric plates: a low cost tool, simple to use and very efficient for both convergence velocity and accuracy, without any classical numerical pathologies. In the field of finite elements, two approaches were previously used for the mechanical part, taking into account the transverse shear stress effects and using only five unknown generalized displacements: C 0 finite element approximation based on first-order shear deformation theories (FSDT) [Polit O, Touratier M, Lory P. A new eight-node quadrilateral shear-bending plate finite element. Int J Numer Meth Eng 1994;37:387–411] and C 1 finite element approximations using a high order shear deformation theory (HSDT) [Polit O, Touratier M. High order triangular sandwich plate finite element for linear and nonlinear analyses. Comput Meth Appl Mech Eng 2000;185:305–24]. In this article, we present the piezoelectric extension of the FSDT eight node plate finite element. The electric potential is approximated using the layerwise approach and an evaluation is proposed in order to assess the best compromise between minimum number of degrees of freedom and maximum efficiency. On one side, two kinds of finite element approximations for the electric potential with respect to the thickness coordinate are presented: a linear variation and a quadratic variation in each layer. On the other side, the in-plane variation can be quadratic or constant on the elementary domain at each interface layer. The use of a constant value reduces the number of unknown electric potentials. Furthermore, at the post-processing level, the transverse shear stresses are deduced using the equilibrium equations. Numerous tests are presented in order to evaluate the capability of these electric potential approximations to give accurate results with respect to piezoelasticity or finite element reference solutions. Finally, an adaptative composite plate is evaluated using the best compromise finite element.

Stress Singularity of Edge Delamination in Angle-Ply and Cross-Ply Laminates

By utilizing the general solutions derived for the plies with arbitrary fiber orientations under uniform axial strain (Huang and Chen, 1994), the explicit solutions of the edge-delamination stress singularities for the angle-ply and cross-ply laminates are obtained. The dominant edge-delamination stress singularities for the angle-ply laminates are found to be a real constant, −1/2, and a pair of complex conjugates, −1/2±i/2πln{(b+b2−a2)/a}. For the cross-ply laminates, the significant effect of transverse shear stresses of the laminate is considered and the dominant edge-delamination stress singularities are shown as −1/2 and −1/2±i/2πln{(c2+c22−4c1c3)/2c1}. a, b, cl, c2, and c3 are the corresponding combined complex constants. In addition, two elementary forms of edge-delamination stress singularity, say, r−1/2 and rδr(lnr)n(δr=n−3/2,n=1,2...) exist for both the angle-ply and cross-ply laminates. Excellent correlations between the present results and available solutions show the validity of the approach. The deficiencies of the solutions available in the literature are compensated. New results for other angle-ply and cross-ply laminates are also provided.

On the shear deformation coefficient in beam theory

The effect of transverse shear stresses on the deflection of straight prismatic beams with arbitrarily shaped cross sections is reconsidered. A method for calculating shear deformation coefficients is developed by combining the engineering beam theory with an elasticity solution for a cantilever beam under a tip load. These coefficients can be found by a geometric approach or an energy method. The latter gives a symmetric tensor of shear deformation coefficients and is more accurate. A new general beam stiffness matrix which accounts for both bending and shear deflection is derived. Numerical examples are presented to show the influence of the Poisson's ratio and the choice of coordinate axes on the shear deformation coefficients.

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.

The effect of transverse shear stresses on the yield surface for thin shells

Several approximate yield surfaces for a thin shell have been examined in the general case where transverse shear stresses are not negligible. It is found that if a simple modification is made to existing approximations, the resulting bounds on the limit load are nearly identical to those obtained from a simple shell theory where transverse shear is neglected. A new expression accurate to plus or minus 2 per cent is also derived.