Distribution Of Transverse Shear Stresses Research Articles

Investigation of a Cross-Section with a Constant Transverse Shear Stress Distribution Using a Numerical Approach

A structural beam which is subjected to shear forces acting perpendicularly to its longitudinal axis will experience longitudinal and transverse shear stresses. In beams where failure in the transverse direction is plausible, it is desirable to maintain a constant transverse shear stress over the beam cross-section to avoid stress concentrations and to use the least amount of material. A numerical approach to the inverse problem of solving for a beam cross-section with a constant transverse shear stress distribution was investigated in this study using Microsoft Excel’s Solver and Matlab. The efficiency and shape of the developed cross-section were dependent on the number of elements used to discretize the cross-section. As the number of elements approached infinity, the shape of the cross-section became infinitely thin at the top and infinitely wide at the neutral axis, while also approaching an efficiency of 100%. It is therefore determined that this is an ill-posed inverse problem and no such perfect cross-section exists.

Static buckling and vibration analysis of continuously graded ceramic-metal beams using a refined higher order shear deformation theory

Purpose The purpose of this paper is to study the static buckling and free vibration of continuously graded ceramic-metal beams by employing a refined higher-order shear deformation, which is also the primary goal of this paper. Design/methodology/approach The proposed model is able to catch both the microstructural and shear deformation impacts without employing any shear correction factors, due to the realistic distribution of transverse shear stresses. The material properties are supposed to vary across the thickness direction in a graded form and are estimated by a power-law model. The equations of motion and related boundary conditions are extracted using Hamilton’s principle and then resolved by analytical solutions for calculating the critical buckling loads and natural frequencies. Findings The obtained results are checked and compared with those of other theories that exist in the literature. At last, a parametric study is provided to exhibit the influence of different parameters such as the power-law index, beam geometrical parameters, modulus ratio and axial load on the dynamic and buckling characteristics of FG beams. Originality/value Searching in the literature and to the best of the authors’ knowledge, there are limited works that consider the coupled effect between the vibration and the axial load of FG beams based on new four-variable refined beam theory. In comparison with a beam model, the number of unknown variables resulting is only four in the general cases, as against five in the case of other shear deformation theories. The actual model represents a real distribution of transverse shear effects besides a parabolic arrangement of the transverse shear strains over the thickness of the beam, so it is needless to use of any shear correction factors.

Isogeometric static and dynamic analysis of laminated and sandwich composite plates using nonpolynomial shear deformation theory

An effective numerical approach based on the isogeometric analysis (IGA) employing nonpolynomial shear deformation theory (NPSDT) has been proposed and implemented in the present work for the static and dynamic analysis of laminated and sandwich composite plates. The theory assumes the nonlinear distribution of transverse shear stresses, and also satisfy the zero transverse shear deformation at the top and bottom surfaces of the laminates. Using Hamilton’s principle, the governing equation of motion is derived and then discretized based on the IGA technique, which facilitates the use of non-uniform rational B-splines (NURBS) basis functions to easily satisfy the stringent continuity requirement of the NPSDT model (C1-continuity) without any additional variables. The set of governing equations are solved to obtain transient response using Newmark’s time integration scheme. Fourier transformation is carried out on the transient response to obtain the natural frequency. Various numerical examples covering different features of present modeling for laminated and sandwich plates are investigated. The performance of the model has been observed by comparing the evaluated results with different published results available in the open literature which ascertain its precision and range of applicability at a reduced computational cost.

On Static Bending, Elastic Buckling and Free Vibration Analysis of Symmetric Functionally Graded Sandwich Beams

This article presents Navier type closed-form solutions for static bending, elastic buckling and free vibration analysis of symmetric functionally graded (FG) sandwich beams using a hyperbolic shear deformation theory. The beam has FG skins and isotropic core. Material properties of FG skins are varied through the thickness according to the power law distribution. The present theory accounts for a hyperbolic distribution of axial displacement whereas transverse displacement is constant through the thickness i.e effects of thickness stretching are neglected. The present theory gives hyperbolic cosine distribution of transverse shear stress through the thickness of the beam and satisfies zero traction boundary conditions on the top and bottom surfaces of the beam. The equations of the motion are obtained by using the Hamilton’s principle. Closed-form solutions for static, buckling and vibration analysis of simply supported FG sandwich beams are obtained using Navier’s solution technique. The non-dimensional numerical results are obtained for various power law index and skin-core-skin thickness ratios. The present results are compared with previously published results and found in excellent agreement.

Static and free vibration analysis of laminated composite and sandwich spherical shells using a generalized higher-order shell theory

In this article, higher-order closed-form solutions are obtained for static bending and free vibration analysis of laminated composite and sandwich spherical shells using a generalized higher-order shell theory. A theory is independent of the choice of shearing stress function (polynomial/non-polynomial) which eventually results in a theoretical unification of most of the classical and higher-order shear deformation theories. The present theory yields an accurate distribution of transverse shear stresses through the shell thickness; therefore, it does not require problem dependent shear correction factor. Governing equations and associated boundary conditions of the theory are derived by employing Hamilton’s principle. Navier type higher-order closed-form solutions are obtained for simply supported boundary conditions. Displacements, stresses and natural frequencies are presented for laminated composite and sandwich plates as well as shallow and deep spherical shells. The results of parabolic, trigonometric, hyperbolic, and exponential models are compared with each other and previously published results to verify the accuracy and efficiency of the present generalized shell theory.

Stresses in thin-walled composite laminated box-beams with curvilinear fibers: Antisymmetric and symmetric fiber paths

Stress analysis of thin-walled composite laminated box beams having variable stiffness is realized in this study based on an analytical model accounting for flexural-torsional coupling and warping effects. The variable stiffness of the beam is acquired by constructing laminates with curvilinear fibers having certain specific paths. The fiber paths of variable stiffness layers are classified in three groups as antisymmetric, symmetric and asymmetric. A displacement based finite element method is used to solve the analytical model and to calculate the distributions of axial and transverse shear stresses at different locations of the cantilever composite beam subjected to the transverse and the torsional loading at its free end. Numerical results obtained are compared with available results in the literature for specific cases. A detailed investigation is performed to understand the relation between the stress distributions along the cross section of the beam and the shape of curvilinear fibers for antisymmetric and symmetric cases.

Geometrically nonlinear free vibration analysis of laminated composite plates: A finite element assessment of a higher order non-polynomial shear deformation theory

In the current work, a higher order non-polynomial theory based upon nonlinear distribution of transverse shear stresses along the plate thickness is employed to apprehend the geometrically nonlinear free vibration of laminated composite plates. The present theory satisfies the zero traction at the top and bottom of the laminated plate. By considering the Lagrange equation, the governing equations of motion are derived and discretized using finite element procedure. The displacement field function is reflected with seven degrees of freedom. For the analysis, various types of laminates including symmetric, anti-symmetric, cross and angle-ply with various edge supports are considered.

Nonpolynomial Zigzag Theories for Random Static Analysis of Laminated-Composite and Sandwich Plates

The present paper aims to study the random static behavior of laminated composite and sandwich plates using nonpolynomial zigzag theories. The nonpolynomial zigzag theories are proposed earlier by the authors, which satisfy the transverse shear-stress continuity conditions at the layer interfaces including the traction-free boundary conditions on the top and bottom surfaces of the plate. These theories incorporate the realistic nonlinear distribution of transverse shear stresses across the thickness of the plate. A probabilistic procedure is developed using a stochastic finite element method that is in the conjunction of finite element method with a mean-centered first-order perturbation technique, to obtain the second-order statistics of deflections of the laminate. The problem is analyzed considering the material properties of laminated composite and sandwich structures to be random in nature while the other system properties are assumed to be deterministic. The stochastic results in terms of mean as well as standard deviation of the responses have been obtained for various combinations of geometric parameters, stacking sequences, span–thickness ratios, and boundary conditions and compared with those existing in the literature and Monte Carlo simulation approach.

Vibration response of a functionally graded graphene nanoplatelet reinforced composite beam under two successive moving masses

The vibration response of a functionally graded graphene nanoplatelet reinforced composite (FG-GPLRC) beam under two successive moving masses is investigated in the current study. The weight fraction of graphene nanoplatelets (GPLs) is assumed to vary continuously and smoothly in the thickness direction. The modified Halpin–Tsai micromechanics model is used to evaluate the effective Young’s modulus of the FG-GPLRC beam. A new high order shear deformation theory (HSDT) is proposed for the structural behavior analysis of FG-GPLRC beams. The proposed theory assumes nonlinear distribution of transverse shear stresses and satisfies the traction-free boundary conditions at the bottom and top surfaces of the beam and hence, a shear correction factor is not required. A generalized closed-form solution methodology of the Navier-type is implemented to ensure the validity and efficiency of the present theory for the free vibration and dynamic response of the beam. The Newmark-β method is adopted to obtain the response of the tube in time domain. Two successive moving masses are considered as the moving load acting on the beam. A parametric study is performed to investigate the effects of GPL weight fraction, distribution pattern, and distance between the two masses on the vibration responses of the beam.

Assessment of dynamic instability of laminated composite-sandwich plates

The current work deals with the assessment of dynamic stability behavior of laminated composite and sandwich plates subjected to in-plane static and periodic compressive loads based on a recently developed zigzag theory by the authors. This theory satisfies the traction-free boundary conditions at top and bottom surfaces of the laminate as well as the inter-laminar stress continuity at layer interfaces. Also, it obviates the need of artificial shear correction factor. The theory is based upon shear strain shape function assuming non-linear distribution of transverse shear stresses. An efficient C0 continuous, eight-noded isoparametric element with seven field variables is employed for the dynamic stability analysis of laminated composite and sandwich plates. The boundaries of principal instability domains are obtained following Bolotin's approach and are represented either in the non-dimensional load amplitude-excitation frequency plane or load amplitude-load frequency plane. A series of numerical examples on the dynamic stability analysis of laminated composite and sandwich plates are studied to demonstrate the effects of modular ratio, span to thickness ratio, boundary conditions, thickness ratio, static load factor and various load parameters on the principal instability regions. The predicted results are compared with the available existing results in order to ensure the performance of the proposed model.

A Unified Higher-Order Beam Theory for Free Vibration and Buckling of FGCNT-Reinforced Microbeams Embedded in Elastic Medium Based on Unifying Stress–Strain Gradient Framework

The main object of this research is to formulate the linear free vibration as well as static stability of embedded functionally graded carbon nanotube-reinforced composite microbeams in thermal environment. The nonlocal stress–strain gradient theory in conjunction with the unified higher-order beam theory by considering the temperature dependence of material properties and the initial thermal stresses is used to derive nonclassical governing equations. The eigenvalue problems governing the linear vibration and static stability of microbeams are obtained by using the weak form of partial differential equations and employing Chebyshev–Ritz method. The fast rate of convergence of the method is demonstrated numerically, and its accuracy is verified by comparing the results in the limit cases with existing solutions in the literature. The effects of transverse shear stress distribution along the thickness together with the spring constants of Winkler–Pasternak elastic medium, different distribution patterns of CNTs across the thickness, the temperature dependence of material properties, the temperature rise, boundary conditions, nonlocal stress and strain gradient parameters on the frequency parameters and load-bearing capacity are investigated. Findings show that the effects of Pasternak constant of elastic medium on the natural frequency as well as critical buckling load depend on the boundary conditions. It is also shown that the nonlocal stress and strain gradient parameters have opposite effects on the stiffness. The more effective distribution pattern of CNTs across the thickness which enhances the static stability and vibratory behavior of microbeam is determined as well.

A single variable shear deformable nonlocal theory for transversely loaded micro- and nano-scale rectangular beams

In this paper, a simple single variable shear deformable nonlocal theory for bending of micro- and nano-scale rectangular beams is presented. To incorporate small size effects, the theory uses Eringen’s nonlocal differential constitutive relations. The theory has only one fourth-order governing differential equation involving a single unknown variable. The governing equation and the expressions for the bending moment and shear force of the present theory are strikingly similar to those of nonlocal Euler-Bernoulli Beam Theory (EBT) formulated based on Eringen’s nonlocal elasticity theory. The theory assumes that the axial and lateral displacements have bending and shear components such that the bending components do not contribute towards shear force, and the shear components do not contribute towards bending moment. Also, the chosen displacement functions of the theory give rise to a realistic parabolic transverse shear stress distribution across the beam cross-section. Efficacy of the proposed theory is demonstrated through bending of simply supported, cantilever and clamped-clamped micro- and nano-scale beams of rectangular cross-section. The numerical results obtained by using the present theory are compared with those predicted by other nonlocal first-order and higher-order shear deformation beam theories. The results obtained are quite accurate.

Transient Hygrothermal Analysis of FG Sandwich Plates Lying on a visco-Pasternak Foundation via a Simple and Accurate Plate Theory

Based on a new four-unknown shear deformation theory, the hygrothermal bending response of functionally graded (FG) sandwich plates lying on visco-Pasternak foundation is investigated in this work. The present sandwich plate is composed of a homogeneous ceramic core covered by two FG layers. The properties of face layers are graded according to a power law distribution in terms of volume fraction. The governing equations are derived from Hamilton principle containing the hygrothermal effects employing the suggested theory. This theory includes only four unknown functions and accounts for quasi-parabolic distribution of transverse shear stress. In addition, it satisfies the free traction conditions at the top and bottom surfaces of the plate. By varying the thickness of the core, several types of FGM sandwich plates are obtained and studied. By comparing the present results with those available in the literature, the present theory is validated. Parametric studies are executed to demonstrate the effects of humidity, temperature rise, foundation coefficients, core thickness, and power law index on the bending of FGM sandwich plates.

Closed-from solutions for thermal buckling analyses of advanced nanoplates according to a hyperbolic four-variable refined theory with small-scale effects

A four-variable plate model is successfully extended here to investigate the thermal buckling analysis of advanced nanoplates. The advanced nanoplate is fabricated from a functionally graded material mixed of ceramic and metal with continuously varying material properties through the nanoplate thickness. Two types of thermal loadings, uniform and nonlinear temperature rises along the nanoplate thickness are taken into consideration. The present model contains four unknown functions as against five or more in other alternative models. The through-the-thickness distributions of transverse shear stresses of the nanoplate are considered to vary parabolically and vanish at upper and lower surfaces. The present model does not require any problem-dependent shear correction factor. Comparison examples are made between results obtained via this model and those via available solutions in the literature.

Post-buckling analysis of shear-deformable composite beams using a novel simple two-unknown beam theory

In this paper, an exact analytical solution is developed for the analysis of the post-buckling non-linear response of simply supported deformable symmetric composite beams. For this, a new theory of higher order shear deformation is used for the analysis of composite beams in post-buckling. Unlike any other shear deformation beam theories, the number of functions unknown in the present theory is only two as the Euler-Bernoulli beam theory, while three unknowns are needed in the case of the other beam theories. The theory presents a parabolic distribution of transverse shear stresses, which satisfies the nullity conditions on both sides of the beam without a shear correction factor. The shear effect has a significant contribution to buckling and post-buckling behaviour. The results of this analysis show that classical and first-order theories underestimate the amplitude of the buckling whereas all the theories considered in this study give results very close to the static response of post-buckling. The numerical results obtained with the novel theory are not only much more accurate than those obtained using the Euler-Bernoulli theory but are almost comparable to those obtained using higher order theories, Accuracy and effectiveness of the current theory.

An efficient higher order non-polynomial Quasi 3-D theory for dynamic responses of laminated composite plates

In this work, a new simple Quasi 3-D theory based on linear variation of transverse displacement along the thickness of plate is proposed for free and forced vibration of laminated composite plates. The theory satisfies both zero transverse shear stress conditions at top and bottom surfaces of the plate and non-linear distribution of transverse shear stresses across the thickness of plate. Therefore, the necessity of shear correction factor can be avoided. The governing equations of motion are obtained by using Lagrange equation. The finite element discretization of the plate is presented by considering an eight noded serendipity element. The forced vibration of the plate under different time dependent loads is also illustrated. The Newmark’s time integration techniques is employed to obtain the forced response of laminated plate. The adaptability of the proposed theory is demonstrated by solving several numerical examples and validated with available results. The effect of various parameters like thickness ratio, aspect ratio, modulus ratio and temperature on fundamental frequency of plate is investigated through several examples. The results shows that, the free vibration regime of the plate is greatly affected by the duration of blast load on plate. The theory gives an excellent agreement with available literature.

An analytical solution for bending and vibration responses of functionally graded beams with porosities

This work presents a static and free vibration analysis of functionally graded metal–ceramic (FG) beams with considering porosities that may possibly occur inside the functionally graded materials (FGMs) during their fabrication. A new displacement field containing integrals is proposed which involves only three variables. Based on the suggested theory, the equations of motion are derived from Hamilton\'s principle. This theory involves only three unknown functions and accounts for parabolic distribution of transverse shear stress. In addition, the transverse shear stresses are vanished at the top and bottom surfaces of the beam. The Navier solution technique is adopted to derive analytical solutions for simply supported beams. The accuracy and effectiveness of proposed model are verified by comparison with previous research. A detailed numerical study is carried out to examine the influence of the deflections, stresses and natural frequencies on the bending and free vibration responses of functionally graded beams.

Variable kinematic shell elements for composite laminates accounting for hygrothermal effects

ABSTRACTThis article presents advanced shell models for the steady-state hygrothermal analysis of composite laminates. The Carrera Unified Formulation is used to derive refined models that include both layer-wise and equivalent single layer models. The governing equations are derived from the principle of virtual displacement taking into account thermal and hygroscopic effects. The geometrical relations for the exact cylindrical geometry are here considered. Through-the-thickness variations of temperature and moisture concentration are calculated by solving the Fourier equation and the Fick law, respectively. The mixed interpolation of tensorial component method is applied to a nine-node shell element to contrast the membrane and shear locking phenomena. Simply supported cross-ply cylindrical shells with antisymmetrical lamination subjected to bisinusoidal thermal/hygroscopic loads are analyzed considering various thickness/curvature ratios. Results obtained with assumed linear and calculated temperature/hygroscopic profiles are presented. Variable kinematics are compared regarding both accuracy and computational costs. The results show that all the kinematics can approximate the transverse shear stress distribution through the thickness with satisfactory accuracy when sufficient expansion terms are adopted. In some cases, miscellaneous expansions can lead to significant reductions in computational costs. The results presented here can be used as benchmark solutions for future works.

Hygrothermal analysis of multilayered composite plates by variable kinematic finite elements

ABSTRACTAdvanced plate models with variable kinematics for steady state hygrothermal analysis of composite laminates are proposed. The refined models discussed include both layer-wise (LW) and equivalent single layer (ESL) models, and the Carrera Unified Formulation (CUF) is used. The mixed interpolation of tensorial component (MITC) method is applied to a nine-node element to contrast the shear locking phenomena. The governing equations are derived from the principle of virtual displacement (PVD) taking into account elastic mechanical, thermal and hygroscopic effects. Through-the-thickness variations of temperature and moisture concentration are calculated by solving the Fourier equation and the Fick law, respectively. Cross-ply plates with symmetrical lamination and simply supported edges subjected to bisinusoidal thermal/hygroscopic loads are analyzed considering various thickness ratios. Results obtained with assumed linear and calculated temperature/hygroscopic profiles are compared. Variable kinematics with a variety of thickness functions are compared regarding both accuracy and computational costs. The results show that all the kinematics proposed can approximate the transverse shear stress distribution through the thickness with satisfactory accuracy when sufficient expansion terms are adopted. In some cases, miscellaneous expansions can lead to significant reductions in computational costs. The results presented here can be used as benchmark solutions for future works.

Bending and free vibration response of sandwich laminate under hygrothermal load using improved zigzag theory

In this work, the effects of variations in temperature and moisture concentrations on the static and free vibration response of sandwich laminated plates with low-density core have been studied. A higher order zigzag laminate theory is used which satisfies the continuity in transverse shear stress at the layer interfaces and zero transverse shear stress condition at the top and bottom surfaces of the laminate . The displacement field in this theory suitably allows for the quadratic distribution of transverse shear stress across the thickness and transverse compressibility effect of the core. An effective finite element formulation is carried out by employing a nine-node C0isoparametric element for the above plate model. Numerical examples of composite and sandwich laminates with different material properties, effect of temperature and moisture variation, aspect ratios, boundary conditions, number of layers and ply orientations are considered for the analysis. Efficiency of the present plate model in predicting various responses subjected to hygrothermal loading is verified by comparing with the available published results.