Distribution Of Transverse Shear Stresses Research Articles

Prediction of inter-laminar stresses in composite honeycomb sandwich panels under mechanical loading using Variational Asymptotic Method

This work focuses on the formulation of an asymptotically correct theory for symmetric composite honeycomb sandwich plate structures. In these panels, transverse stresses tremendously influence design. The conventional 2-D finite elements cannot predict the thickness-wise distributions of transverse shear or normal stresses and 3-D displacements. Unfortunately, the use of the more accurate three-dimensional finite elements is computationally prohibitive. The development of the present theory is based on the Variational Asymptotic Method (VAM). Its unique features are the identification and utilization of additional small parameters associated with the anisotropy and non-homogeneity of composite sandwich plate structures. These parameters are ratios of smallness of the thickness of both facial layers to that of the core and smallness of 3-D stiffness coefficients of the core to that of the face sheets. Finally, anisotropy in the core and face sheets is addressed by the small parameters within the 3-D stiffness matrices. Numerical results are illustrated for several sample problems. The 3-D responses recovered using VAM-based model are obtained in a much more computationally efficient manner than, and are in agreement with, those of available 3-D elasticity solutions and 3-D FE solutions of MSC NASTRAN.

A Double Thick Beam FEM with Explosion Resistance Dynamics Analysis Based on Spline Wavelets in Underground Structure

Isoparametric B-spline wavelet functions can provide excellent approximation for smooth functions. In this paper, the response of composite laminated plate subjected to explosions is analyzed by using a double thick beam FEM mode based on the spline wavelet function. This mode, which can be considered in the distribution of normal stress and transverse shear stress simultaneously, is very important in the analysis of thick plate or blast-damage mechanism. The mode in analyzing underground structure and earth medium respond has the advantage of high precision and fast constringency and is easy to establish a element displacement function. The comparison between simulative and experimental results shows the effectiveness and precision of this method.

On the simple and mixed first-order theories for plates resting on elastic foundations

This article investigates the bending response of an orthotropic rectangular plate resting on two-parameter elastic foundations. Analytical solutions for deflection and stresses are developed by means of the simple and mixed first-order shear deformation plate theories. The present mixed plate theory accounts for variable transverse shear stress distributions through the thickness and does not require a shear correction factor. The governing equations that include the interaction between the plate and the foundations are obtained. Numerical results are presented to demonstrate the behavior of the system. The results are compared with those obtained in the literature using three-dimensional elasticity theory or higher-order shear deformation plate theory to check the accuracy of the simple and mixed first-order shear deformation theories.

Effects of Boundary Conditions in Laminated Composite Plates Using Higher Order Shear Deformation Theory

This paper extends the applicability of a modified higher order shear deformation theory to accurately determine the in-plane and transverse shear stress distributions in an orthotropic laminated composite plate subjected to different boundary conditions. A simpler, two-dimensional, shear deformable, plate theory accompanied with an appropriate set of through-thickness variations, is used to accurately predict transverse shear stresses. A finite element code was developed based on a higher order shear deformation theory to study the effects of boundary conditions on the behavior of thin-to-thick anisotropic laminated composite plates. The code was verified against three dimensional elasticity results. The study also compared the stresses and deformation results of higher order theory with those obtained using commercial software such as LUSAS, ANSYS and ALGOR. The commercial software are heavily used by designers to design various components/products made of composites. Various combinations of fixed, clamped and simply supported boundary conditions were used to verify a large class of anticipated applications. Results obtained from software are in good agreement for some cases and significantly differ for others. It was found that LUSAS and ANSYS yield better results for transverse deflection and in-plane stresses. But for transverse shear stresses, it is highly dependent on boundary conditions.

Erratum to “Transverse Shear Stress Distribution through Thickness near an Internal Part – Through Elliptical Hole in a stretched Plate” [Composite Structures 92 (2010) 818–825

Erratum to “Transverse Shear Stress Distribution through Thickness near an Internal Part – Through Elliptical Hole in a stretched Plate” [Composite Structures 92 (2010) 818–825

Two new hyperbolic shear displacement models for orthotropic laminated composite plates

Two hyperbolic displacement models, HPSDT1 and HPSDT2, are developed for a bending analysis of orthotropic laminated composite plates. These models take into account the parabolic distribution of transverse shear stresses and satisfy the condition of zero shear stresses on the top and bottom surfaces of the plates. The accuracy of the analysis presented is demonstrated by comparing the results with solutions derived from other higher-order models and with data found in the literature. It is established that the HPSDT1 model is more accurate than some theories of laminates developed previously, and therefore the analysis can be expanded to laminated composite shells.

Static and Free Vibration Analyses of Doubly Curved Composite Sandwich Panels with Soft Core Based on a New Three-Layered Mixed Theory

In this paper, a closed-form solution of static problems and free vibrations of a doubly curved sandwich shell with flexible core based on a newly developed three-layered high-order sandwich panel theory (mixed theory) is presented for the first time. In this theory, displacements and transverse stresses at the interface of the core and face sheets are primary unknowns. Linear distribution of transverse normal and shear stresses through the core thickness is considered in the present model. This model accounts for non-linear distribution of displacements through the core thickness. Hamilton's principle is employed to obtain equations of motion. The formulation includes Donnell's shallow shell theory. In the case of static problems and free vibrations, the results of the present model are validated by a three-dimensional elasticity solution and analytical and numerical results published in the literatures. A parametric study is also included in this paper to investigate the effects of radius of curvature, thickness, and flexibility of the core. New results for thin and thick spherical and cylindrical sandwich shells are established, which can be considered as a basic reference for future investigations.

High-order free vibrations of doubly-curved sandwich panels with flexible core based on a refined three-layered theory

Abstract There are few reports on the free vibration of soft core doubly-curved sandwich shells. Previous studies are largely based on the equivalent single layer theories in which the natural frequencies are grossly overestimated. This study deals with the analytical free vibrations of doubly curved sandwich shell with flexible core based on a refined general-purpose sandwich panel theory. In this theory, equations of motion are formulated based on displacements and transverse stresses at the interfaces of the core. The first order shear deformation theory and assumptions of linear distribution of transverse normal stress and uniform shear stresses over the thickness of core (based on 3D-elasticity solution of weak core) are used in the present theory. In this model, the in-plane displacements take cubic polynomial distributions and the transverse displacement has a quadratic one thorough the core thickness. Hamilton’s principle is used to obtain the equations of motion. The obtained results are validated by the analytical and numerical results published in the literatures. Parametric study is also included to investigate the effects of radius of curvature, thickness and flexibility of core.

Bending of laminated plates with mixed boundary conditions based on higher-order shear deformation theory

The bending of laminated plates is considered using higher-order transverse shear deformation theory. The principle of virtual work is used to derive a new set of seven governing equations and corresponding boundary conditions. These equations, combined with eighteen relationships between the resultant stress and displacement components, compose a system of first-order partial differential equations that is solved by the generalized differential quadrature method. Numerical results for laminated plates with a variety of mixed boundary conditions are calculated using the proposed method, and good agreement is found with the corresponding solutions obtained using ANSYS. Fiber-reinforced laminated composite materials are widely used in a variety of engineering fields, such as aerospace, civil, marine, mechanical, nuclear, and petrochemical engineering. Such materials are popular for industrial applications due to high strength-to-weight ratios, long fatigue life, good stealth characteristics, and enhanced corrosion resistance. A number of theories describing laminated composite plates exist in the literature. Classical plate theory is based on the Kirchhoff kinematic hypothesis that straight lines normal to the undeformed midsurface remain straight and normal to the middle surface after deformation and undergo no thickness stretching. Neglecting transverse shear effects, this theory produces unacceptable approximations in the analysis of even thin laminated plates and shells. Surveys of various classic shell theories can be found in [Naghdi 1956; Bert and Francis 1974; Bert and Chen 1978]. The development of plate theories with transverse shear effects has improved the accuracy of results considerably. The refined theories are of different orders, based on discretization of the transverse shear effects and the number of terms included in the assumed displacement field. Reissner [1945] was the first to develop a plate theory that included transverse shear deformation for static analysis. Mindlin [1951] then expanded Reissner’s theory for dynamic analysis. Both approaches rely on first-order shear deformation theory (FSDT). Reissner’s theory is stress-based, whereas Mindlin’s is displacement-based. These theories do not satisfy the condition of zero transverse shear stress at the top and bottom surfaces of the plate, and consider a uniform transverse shear stress distribution across the thickness of the plate. Therefore they require the use of a shear correction factor to increase the precision of the results. Later a set of theories, generally known as higher-order shear deformation theories (HSDT), has been developed by a number of researchers. Basset [1890] appears to have been the first researcher to suggest that displacement fields can be expanded in a power series of the thickness coordinate. The higher-order

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.

Transverse shear stress distribution through thickness near an internal part-through elliptical hole in a stretched plate

Abstract A semi-analytical post-processing method, termed the equilibrium/compatibility method here, is used for computation of hitherto unavailable through-thickness variation of transverse shear stresses in the vicinity of the circumferential re-entrant corner line of an internal part-through elliptical cylindrical hole weakening an edge-loaded rectangular plate. A C0-type triangular “composite” plate element, based on the assumptions of transverse inextensibility and piece (“layer”)-wise constant shear-angle theory (LCST), is employed to first compute the inplane stresses and “layer”-wise through-thickness average transverse shear stresses. These serve as the starting point for computation of through-thickness distribution of transverse shear stresses in the vicinity of the circumferential re-entrant corner line of the internal part-through elliptical hole. As in the case of its circular counterpart, the transverse shear stresses computed by the conventional equilibrium method (EM) are, in contrast, in serious error in the presence of the circumferential re-entrant corner line singularity arising out of the internal part-through elliptical hole, and are found to violate the compatibility condition. The computed maximum transverse shear stress can be high enough to cause catastrophic transverse shear fracture in the shape of a cone, of elliptical cross-section starting from the circumferential re-entrant corner line of the internal part-through hole. The results computed by the present analysis are in line with a three-dimensional asymptotic analysis.

Boundary-layer hygrothermal stresses in laminated, composite, circular, cylindrical shell panels

Free-edge effects in laminated, circular, cylindrical shell panels subjected to hygrothermal loading are studied by utilizing displacement-based technical theories. Starting from the most general displacement field of elasticity for long, circular, cylindrical shells, appropriate reduced displacement fields are determined for laminated composite shell panels with cross-ply and antisymmetric angle-ply layups. An equivalent single-layer shell theory is used to analytically determine the constant parameters appearing in the reduced displacement fields. A layerwise shell theory is then employed to analytically determine the local displacement functions and the boundary-layer interlaminar stresses in cross-ply and antisymmetric angle-ply shell panels under hygroscopic and/or thermal changes. Several numerical examples for the distributions of transverse shear and normal stresses in various shell panels under a uniform temperature change are presented and discussed.

Computation of transverse shear stresses in the vicinity of the circumferential re-entrant corner line of an internal part-through hole weakening an edge-loaded plate

Abstract A semi-analytical post-processing method, termed equilibrium/compatibility method here, is used for computation of hitherto unavailable through-thickness variation of transverse shear stresses in the vicinity of the circumferential re-entrant corner line of an internal part-through circular cylindrical hole weakening an edge-loaded plate. A C o -type triangular “composite” plate element, based on the assumptions of transverse inextensibility and piece (“layer”)-wise constant shear-angle theory (LCST), is employed to first compute the inplane stresses and “layer”-wise through-thickness average transverse shear stresses, which serve as the starting point for computation of through-thickness distribution of transverse shear stresses in the vicinity of the circumferential re-entrant corner line of the internal part-through hole. The same stresses computed by the conventional equilibrium method (EM) are, in contrast, in serious error in the presence of the circumferential re-entrant corner line singularity arising out of the internal part-through hole, and are found to violate the compatibility condition. The computed maximum transverse shear stress can be high enough to cause catastrophic transverse shear fracture in the shape of a cone, starting from the circumferential re-entrant corner line of the internal part-through hole. The results computed by the present analysis are in line with a three-dimensional asymptotic analysis.

Buckling and free vibration analyses of laminated composite plates by using two new hyperbolic shear-deformation theories

Two new hyperbolic displacement models, HPSDT1 and HPSDT2, are used for the buckling and free vibration analyses of simply supported orthotropic laminated composite plates. The models contain hyperbolic expressions to account for the parabolic distributions of transverse shear stresses and to satisfy the zero shear-stress conditions at the top and bottom surfaces of the plates. The equation of motion for thick laminated rectangular plates subjected to in-plane loads is deduced through the use of Hamilton’s principle. Closed-form solutions are obtained by using the Navier technique, and then the buckling loads and the fundamental frequencies are found by solving eigenvalue problems. The accuracy of the models presented is demonstrated by comparing the results obtained with solutions of other higher-order models given in the literature. It is found that the theories proposed can predict the fundamental frequencies and buckling loads of cross-ply laminated composite plates rather accurately.

Stress analysis of functionally graded plates subjected to thermal and mechanical loadings

A two-dimensional (2D) higher-order deformation theory is presented for the evaluation of displacements and stresses in functionally graded (FG) plates subjected to thermal and mechanical loadings. The modulus of elasticity of plates is assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. By using the method of power series expansion of continuous displacement components, a set of fundamental governing equations which can take into account the effects of both transverse shear and normal stresses is derived through the principle of virtual work. Several sets of truncated Mth order approximate theories are applied to solve the static boundary value problems of a simply supported FG plate. In order to assure the accuracy of the present theory, convergence properties of the displacements and stress components are examined in detail. A comparison of the present results of isotropic plates is also made with previously published results. The transverse stresses have been obtained by integrating the three-dimensional (3D) equations of equilibrium in the thickness direction starting from the top or bottom surface of a plate. The distributions of transverse shear and normal stresses in the thickness direction are obtained accurately by satisfying the surface boundary conditions of a plate. The present numerical results are also verified by satisfying the energy balance of external and internal works are considered to be sufficient with respect to the accuracy of solutions. It is noticed that the present 2D higher-order approximate theories can predict accurately the stresses and displacements of simply supported FG plates subjected to thermal and mechanical loadings.

Applying the grid method to validate a numerical model predicting the transverse shear stress distributions in patched structures

This paper describes the application of a full-field measurement technique to validate a numerical model developed to calculate the stress distribution in metallic structures reinforced with composite patches. The numerical model is based on a finite differences scheme. Tensile tests are performed on circular patches to validate the model. Displacement fields are measured on the patches with the grid method. Strain are derived from these measurements using a suitable technique and stresses are deduced using the constitutive equations of the composite. In most cases, a good agreement is observed between stresses deduced from the measurements and their numerical counterparts.

Buckling and Free Vibration Analysis of Symmetric and Antisymmetric Laminated Composite Plates on an Elastic Foundation

Buckling and free vibration analysis of simply supported symmetric and antisymmetric cross-ply thick composite plates on elastic foundation are examined by a new hyperbolic displacement model in this paper. In this new model, inplane displacements vary as a hyperbolic function across the plate thickness, so account for parabolic distributions of transverse shear stresses and satisfy zero shear stress conditions at the top and bottom surfaces of the plate. In the analysis, the foundation is modeled as two parameter Pasternak type foundation, and Winkler type if the second foundation parameter is zero. The equation of motion for thick laminated rectangular plates resting on elastic foundation and subjected to inplane loads is obtained through Hamilton's principle. The closed form solutions are obtained by using Navier technique, and then buckling loads and fundamental frequencies are found by solving the results of eigenvalue problems. The numerical results obtained through the present analysis for free vibration and buckling of cross-ply laminated plates on elastic foundation are presented, and compared with the ones available in the literature.

Two-dimensional sublamination theory for analysis of functionally graded plates

Presented is a two-dimensional sublamination theory for analysis of plates made of functionally graded materials. The sublamination plate theory extends a layer-wise higher-order shear-deformation theory for laminated plates by considering material properties being continuous functions of the thickness coordinate and using a new concept of sublamination to increase the degrees of freedom. The theory accommodates free shear stress conditions on the bonding surfaces, accounts for non-uniform deformation-dependent distributions of transverse shear stresses through the thickness, can be used for evaluating boundary restraint effects, and can be used for analyzing thin and thick plates with any boundary conditions. A sublamination plate element based on this theory is developed and validated for static and dynamic analysis. The degrees of freedom of the element is adaptable. For an element away from boundaries, it's degrees of freedom can be reduced at the elemental level without loss of accuracy. Analytical shear warping functions are presented. Moreover, modal analysis of functionally graded plates with different boundary conditions is performed to show the capability and accuracy of the theory.

Influence of the Wetted Surface Curvature on the Laminar Liquid Film Thickness

Liquid films flowing down vertical surfaces under the influence of gravity are encountered in many types of heat and mass transfer equipment consisting of vertical tubes and vertical plane surfaces. Therefore, a complete understanding of liquid film flow on vertical surfaces from the entrance to the exit should be of interest industrially and academically. The main reason for that is the influence exerted by the surface cross curvature and the film itself on heat transfer characteristics on the surface, transverse shear stress distribution, and correspondingly the thickness of the liquid film. This paper presents the evaluation of the surface cross curvature on the hydromechanical parameters of a laminar liquid film. The cross curvature of a wetted surface and the film alter the hydromechanical parameters of the liquid film. Simultaneously, the intensity of heat exchange between the wetted surface and liquid film is altered. Calculations evaluating velocity distribution across water and transformer oil film and correspondingly the thickness of the film were performed.

A conceptual model for meander initiation in bedload‐dominated streams

AbstractA simple analytic model is presented relating local sediment transport capacity to variance in the transverse shear stress distribution in a stream channel. The model is used to develop a physically based conceptual model for the initiation of meandering in straight, bedload‐dominated streams as a result of a feedback mechanism. The feedback maximizes the cross‐sectional shear stress variance and – in order to achieve stability – ultimately minimizes the energy slope at repeated locations along the channel, subject to steady‐state mass flux and the stability of the channel boundary. These locations develop into pools in a fully developed meandering channel; they represent attractor states wherein sediment continuity is satisfied using the least possible energy expenditure per unit length of channel. However, since the cross‐sectional geometry of a pool (and the adjacent bar) is asymmetric, these attractor states are only conditionally stable, requiring strong, curvature‐induced secondary circulation to maintain their asymmetry. Between two successive pools, a stream occupies a metastable, higher energy state (corresponding to a riffle) that requires greater energy expenditure per unit length of channel to transport the same volume of sediment. The model we present links processes at the scale of a channel width to adjustments of the channel sinuosity and slope at the scale of a channel reach. We argue that the reach‐scale extremal hypotheses employed by rational regime models are mathematical formalisms that permit a one‐dimensional theory to describe the three‐dimensional dynamics producing stream morphology. Our model is consistent with the results from stream table experiments, with respect to both the rate of development of meandering and the characteristics of the equilibrium channel morphology. Copyright © 2006 John Wiley & Sons, Ltd.