Transverse Normal Deformation Research Articles

Stress analysis of laminated and sandwich beams subjected to concentrated load by using quasi-two-dimensional theory

This study presents a quasi-two-dimensional higher-order shear deformation theory for stress and displacement analysis of isotropic, laminated, and sandwich beams subjected to concentrated load. The assumed displacement field includes the effect of transverse shear and normal deformations. The condition of zero transverse shear stresses on the upper and lower surface of beams is satisfied, hence the present formulation does not require the shear correction factor generally associated with first-order models. By applying the principle of virtual work, the governing equations and boundary conditions for laminated and sandwich beams are derived. Transverse shear and normal stresses are determined from the stress-equilibrium equations of elasticity theory. The transverse stresses obtained using this approach satisfy the continuity condition at layer interfaces and the stress boundary conditions at the external surfaces. The outcomes of the present theory are compared with those of third-order, and first-order shear deformation models, and the classical beam model. The results highlight the significant deviation in displacements and stresses of laminated and sandwich beams when normal strain is included in the model, as compared to the prediction made by lower-order models.

Evaluation of static and dynamic responses considering thickness stretching effect for layered composite and sandwich arches using exponential shear and normal deformation theory

Present theory investigates the dimensionless vibration frequencies, deformations, and stresses for multilayered and sandwich arches using exponential shear and normal deformation theory (ESNDT). Present studies consider the effect of [1+(z/R)]radius of curvature while selecting the displacement field of arches under the action of uniform load. It includes the effects of transverse shear (γxz≠0) and transverse normal deformation(εz≠0)i.e., effect of thickness stretching. The governing relations are obtained using the Navier's method, and Hamilton's virtual work principle. The proposed layered arches are completely free from shear correction, and its top and bottom surfaces is satisfies zero traction free end conditions. Present study obtained very accurate natural frequencies for layered sandwich arches than other theories because, available literature not considered the thickness stretching effect i.e.,(εz=0). Hence, the present scientific investigation accounts the effect of thickness stretching i.e.,(εz≠0). However, the non-availability of exact elasticity results of bending and dynamic responses for layered sandwich arches. Present theory is obtained vibration frequencies, displacements, and stresses of layered composite and sandwich arches; the results are compared and validates through prior published literature.

Porosity-dependent free vibration and transient responses of functionally graded composite plates employing higher order thickness stretching model

Tailoring the mechanical properties in hybrid materials, like functionally graded materials (FGMs), has become more pragmatic with the recent progress in material manufacturing and design. The porosities in FGMs, which develop during the manufacturing process, may become detrimental to armored plates or bulletproof designs. On the other hand, gradient-porosity distributions may have broad advantages in the design of lightweight and variable-stiffness aircraft components. The present study investigates the free vibration and transient responses of porosity-gradient FGM plates. A higher order theory, which considers the transverse shear and normal deformation, warping of the transverse cross-section, and higher order rotary inertia, is utilized for the first time to model the deformation of the porous FGM plate. The advantages of this model are (i) higher order kinematic terms for membrane and bending deformations due to the coupled membrane-bending behavior of FGMs in addition to the thickness-stretching effects, and (ii) the flexibility of applying external loads at different points along the thickness direction of the FGM plate. An analytical solution technique, popularly known as Navier’s approach, is adopted for the spatial solutions, and Newmark’s average acceleration method is utilized for the temporal solutions. The influence of the geometrical and porosity parameters on the structure’s vibration response is studied. Natural frequencies are affected significantly with uniform porosity distribution, while graded porosities can tweak vibration response in a small range but with better control. In the absence of the elasticity solutions, the present results may be considered the possible benchmark solutions for comparison of the other two-dimensional (2D) kinematic models.

Effect of Thickness Stretching on the Natural Frequencies of Layered Composite Beams

Abstract In this study, a new fifth order shear deformation theory is developed considering the effect of thickness stretching for the free vibration analysis of laminated composite beam. To account the effect of shear in transverse direction and normal de-formations, displacement field uses polynomial shape function in terms of thick-ness co-ordinate expanded up to order five. The virtual work equation is used to have the equations in terms of the unknowns and corresponding boundary conditions. The analytical solution for a laminated composite beam with simple sup-ports is developed using Navier’s solution. To assess the validity of the theory developed, dimension less natural frequencies for laminated composite beam determined by the present model are compared with previously published results.

Modeling of partially delaminated composite plates resting on two-parameter elastic foundation with improved layerwise theory

In this work, we aim to build the mathematical model for delaminated composite plates resting on elastic foundation by means of improved layerwise theory and finite element implementation for free vibration analysis. The proposed model takes into account the transverse normal and shear deformation of the elastic foundation so as to investigate their effect on the dynamic characteristics of delaminated plates. The four-node plate element is used for the finite element formulations with Langrage interpolation functions for in-plane structural unknowns and Hermite cubic interpolation functions for out-of-plane structural unknowns. The delamination is simulated by means of Heaviside unit step functions allowing for the discontinuity of displacement fields. The effects of both longitudinal locations and interfacial locations of delamination are investigated. The results show that when the delamination located closer to the clamped end and at the middle plane, it has much more severe influence on its natural frequencies. The foundation parameters also have significant effect on the natural frequencies and frequency shift due to delamination. Different layup stacking sequences are also considered in this study.

Bending Analysis of Sandwich Plate Under Concentrated Force Using Quasi-Three-Dimensional Theory

This study presents the quasi-three-dimensional, higher-order shear-deformation theory for the stress and displacement analysis of sandwich plate subjected to concentrated force. Using expansion of thickness coordinate in the displacement field, the effects of both transverse shear and normal deformations are included. The theory satisfies the zero shear stress conditions on the top and bottom surfaces of sandwich plate. Hence, the shear correction factor is not required, generally associated with the First order shear deformation theory (FSDT). The boundary-value problem of the theory is derived using the principle of virtual work. Transverse shear and normal stresses are recovered from the stress-equilibrium equations of the theory of elasticity, and the continuity condition is satisfied at the interface between layers and stress boundary condition at the external surfaces. In addition, this study provides the effects of transverse normal strain on flexural response of sandwich plate. Closed-form solutions for simply supported sandwich plate subjected to concentrated force are obtained for various side-to-thickness ratios. The results of the present theory are compared with third-order shear and normal deformation theory, first-order shear-deformation theory, and the classical plate theory. The inclusion of the effect of transverse normal strain in the theory showed significant deviations in displacements and stresses.

Thermoelastic Analysis of Functionally Graded Cylindrical Shells

We perform the analytic investigation of the stress-strain state of a functionally graded cylindrical shell of finite length heated by a two-dimensional temperature field. The properties of the shell material are regarded as analytic functions of the thickness coordinate. In our investigations, we use the equations of the refined theory of shells that takes into account the deformation of transverse shear and the transverse normal deformation. The heat-conduction equation is deduced under the assumption of linear temperature distribution over the thickness of the shell. For the boundary conditions of simply supported shell, the quasistatic uncoupled problem of thermoelasticity is solved by the methods of Fourier and Laplace transforms. Numerical examples are presented and discussed to show that it is important to take into account the influence of inhomogeneity of the properties of materials of the metal–ceramics composites.

An original four-variable quasi-3D shear deformation theory for the static and free vibration analysis of new type of sandwich plates with both FG face sheets and FGM hard core

This paper presents an original high-order shear and normal deformation theory for the static and free vibration of sandwich plates. The number of unknowns and governing equations of the present theory is reduced, and hence makes it simple to use. Unlike any other theory, the number of unknown functions involved in displacement field is only four, as against five or more in the case of other shear and normal deformation theories. New types of functionally graded materials (FGMs) sandwich plates are considered, namely, both FG face sheets which the properties vary according to power-law function and exponentially graded hard core. The equations of motion for the present problem are derived from Hamilton' principle. For simply-supported boundary conditions, Navier's approach is utilized to solve the motion equations. The accuracy of the present theory is verified by comparing the obtained results with three-dimensional elasticity solutions and other quasi-3D higher-order theories reported in the literature. Other numerical examples are also presented to show the influences of the volume fraction distribution, geometrical parameters and power law index on the bending and free vibration responses of the FGM sandwich plates are studied. It can be concluded that present formulation which takes into account both the transverse shear and normal deformation, predicts the natural frequencies with the same degree of accuracy as that of 3D elasticity solutions and gives a good results of displacements and stress compared with others Quasi-3D theories. It can be also deduced that the central deflection is in direct correlation relation with inhomogeneity parameter and the natural frequency is in inverse relation with this parameter.

Buckling and postbuckling behavior of shell type structures under thermo mechanical loads

The thermo mechanical buckling and post-buckling behavior of layered composite shell type structure are considered with the finite element method under the combination of temperature load and applied mechanical loads. To account for through-thickness shear deformation effects, the thermal elastic, and higher-order shear deformation theory is used in this study. The refined higher order theories, that takes into account the effect of transverse normal deformation, is used to develop discrete finite element models for the thermal buckling analysis of composite laminates. Attention in this study is focused on analyzing the temperature effects on buckling and post-buckling behavior of thin shell structural components. Special attention in this paper is focused on studying of values of the hole in curved panel on thermal buckling behavior and consequently to expend and upgrade previously conducted investigation. Using finite element method, a broader observation of the critical temperature of loss of stability depending on the size of the hole was conducted. The presented numerical results based on higher-order shear deformation theory can be used as versatile and accurate method for buckling and post-buckling analyzes of thin-walled laminated plates under thermo mechanical loads.

A new higher-order theory for the static and dynamic responses of sandwich FG plates

In this study, a static and free vibration analysis of single layer FG and sandwich FG plates is carried out using a fifth order shear and normal deformation theory. The displacement field of the present theory includes the terms considering the effect of transverse shear and normal deformation. Also, the terms of the thickness co-ordinate are expanded upto fifth order to predict the accurate bending behavior of the plates. The equations of motion are derived based on Hamilton’s principle, and further solved using Navier’s solution scheme. The present results of displacement, stresses and natural frequencies in sandwich FG plates are obtained and compared with other higher order theories available in literature to check the validity and efficacy of the theory.

Dynamic response of viscoelastic multiple-core sandwich structures

This work focuses on the free and forced vibrations of sandwich beams and plates hosting an arbitrary number of damping cores. A fractional derivatives Zener-type model is adopted for representing the frequency-dependent viscoelastic behaviour, along with conventional series developments. The structural models are formulated within an established variable kinematics approach, which enables to investigate the role of specific assumptions and to identify the most accurate model at a least number of degrees of freedom. Approximate solutions are found by a computationally efficient Ritz method that allows to take into account any type of boundary conditions. Modal loss factors and damped eigenfrequencies are obtained from a complex eigenvalue problem, for which a modal strain energy approach or a complex eigensolution can be employed. Frequency responses can be computed by a direct approach or by elementary modal projection algorithms. Results are reported for conventional and innovative sandwich configurations. Different modelling and solution strategies are compared and the role of transverse normal deformation of the mechanically weak viscoelastic plies is particularly emphasised.

Transverse shear and normal deformation effects on vibration behaviors of functionally graded micro-beams

A quasi-three dimensional model is proposed for the vibration analysis of functionally graded (FG) micro-beams with general boundary conditions based on the modified strain gradient theory. To consider the effects of transverse shear and normal deformations, a general displacement field is achieved by relaxing the assumption of the constant transverse displacement through the thickness. The conventional beam theories including the classical beam theory, the first-order beam theory, and the higherorder beam theory are regarded as the special cases of this model. The material properties changing gradually along the thickness direction are calculated by the Mori-Tanaka scheme. The energy-based formulation is derived by a variational method integrated with the penalty function method, where the Chebyshev orthogonal polynomials are used as the basis function of the displacement variables. The formulation is validated by some comparative examples, and then the parametric studies are conducted to investigate the effects of transverse shear and normal deformations on vibration behaviors.

Thermoelastic analysis of laminated composite and sandwich shells considering the effects of transverse shear and normal deformations

In the present study, thermoelastic analysis of laminated composite and sandwich shells (cylindrical/spherical) is presented using fifth-order shear and normal deformation theory. The significant characteristic of the present theory is that it includes the effects of both transverse shear and normal deformations. The mathematical formulation uses the principle of virtual work to derive the variationally consistent governing equations and traction free boundary conditions. To obtain the static solution, these governing equations are solved by employing Navier’s solution technique. The shell is subjected to a mechanical/thermal load sinusoidally distributed over the top surface of the shell. The thermal load linearly varies across the thickness of the shell. The present results are compared with other higher-order models and 3D elasticity solution wherever possible. Thermal stresses presented in this study will act as a benchmark for the future work.

A quasi-3D trigonometric shear deformation theory for wave propagation analysis of FGM sandwich plates with porosities resting on viscoelastic foundation

In the present work, a quasi three-dimensional trigonometric shear deformation plate theory (quasi-3D TSDPT) is proposed for the wave propagation analysis of porous functionally graded material (FGM) sandwich plates resting on a viscoelastic foundation. The present plate theory accounts for the transverse shear and normal deformations by dividing the transverse displacement into bending, shear, and stretching components. Different types of FGM sandwich plates are taken into account. The porosities in the FGM layers of the sandwich structures are described by introducing the porosity volume fraction and the step function. The equations of motion governing the wave propagation behavior of porous FGM sandwich plates are derived by employing Hamilton’s principle. The analytical solutions to the wave dispersion relations are presented. Additionally, the parametric research is conducted to highlight the effects of the wave number, the porosity volume fraction, the viscoelastic foundation, the power-law exponent and the core-to-thickness ratio on the wave propagation. Results manifest that the influences of these parameters are significant on the wave propagation characteristics of porous FGM sandwich plates.

Analysis of Laminated Plates Subjected to Mechanical and Hygrothermal Environmental Loads Using Fifth-Order Shear and Normal Deformation Theory

Bending analysis of laminated composite plates subjected to mechanical and hygrothermal environmental loading is presented in this paper using fifth-order shear and normal deformation theory (FOSNDT). This theory uses polynomial-type shape function in the displacement field up to fifth-order. This theory considers the effect of transverse normal deformations ([Formula: see text]). Simply supported laminated composite and sandwich plates are analyzed using Navier’s solution techniques. The results obtained using this theory are compared with the results available in the literature wherever possible. However, many results are presented for the first time in this paper.

1D Thermal Analysis of Laminated Composite and Sandwich Plates using A New Fifth Order Shear and Normal Deformation Theory

Abstract In the present study a new fifth order shear and normal deformation theory (FOSNDT) which incorporates the effect of transverse shear and normal deformation is presented for the 1D thermal analysis (Cylindrical Bending) of laminated composite and sandwich plates. To account for the effect of transverse shear and normal deformation, the present theory uses polynomial shape function expanded up to fifth-order in terms of the thickness co-ordinate. The variationally consistent governing equations are obtained by using Navier’s solution technique. Numerical results obtained using present FOSNDT are compared with thermoelastic solution available in the literature. It is found that the present theory accurately predicts the thermal response of laminated composite and sandwich plates.

Analytical solution using fifth order shear and normal deformation theory for FG plates resting on elastic foundation subjected to hygro-thermo-mechanical loading

In the present study, functionally graded (FG) plates resting on two parameter Winkler-Pasternak elastic foundation subjected to non-linear hygro-thermo-mechanical load is investigated for the bending analysis using a new fifth-order shear and normal deformation. The theory mainly contributes the effects of transverse shear and normal deformations i.e. thickness stretching. This theory involves polynomial shape functions expanded up to fifth-order in terms of the thickness coordinate to account for the effects of transverse shear and normal deformations. Navier’s solution technique is used to obtain analytical solutions for simply supported FG plates and obtained non-dimensional results are compared with published results available in literature.

Analytical solutions for the hygro-thermo-mechanical bending of FG beams using a new fifth order shear and normal deformation theory

A new analytical solution is presented for functionally graded (FG) beams to investigate the bending behaviour under hygro-thermo-mechanical loading using a new fifth order shear and normal deformation theory (FOSNDT). The material properties of the FG beam are varied along the thickness direction according to the power law index. In the present theory, a polynomial shape function is expanded up to fifth-order in terms of thickness coordinate to consider the effects of transverse shear and normal deformations. The present theory is free from the shear correction factor. Using the Navier’s solution technique the closed-form solution is obtained for simply supported FG beams. All the results are presented in non-dimensional form and validated it by developing the classical beam theory (CBT), first order shear deformation theory (FSDT by Mindlin) and third order shear deformation theory (TSDT by Reddy) considering the hygro-thermo-mechanical loading effects which is mostly missing in the literature. It is noticed that the presented FOSNDT is very simple and accurate to predict the bending behaviour of FG beams under linear and non-linear hygro-thermo-mechanical loadings.

A study on the structural behaviour of functionally graded porous plates on elastic foundation using a new quasi-3D model: Bending and free vibration analysis

This work investigates a new type of quasi-3D hyperbolic shear deformation theory is proposed in this study to discuss the statics and free vibration of functionally graded porous plates resting on elastic foundations. Material properties of porous FG plate are defined by rule of the mixture with an additional term of porosity in the through-thickness direction. By including indeterminate integral variables, the number of unknowns and governing equations of the present theory is reduced, and therefore, it is easy to use. The present approach to plate theory takes into account both transverse shear and normal deformations and satisfies the boundary conditions of zero tensile stress on the plate surfaces. The equations of motion are derived from the Hamilton principle. Analytical solutions are obtained for a simply supported plate. Contrary to any other theory, the number of unknown functions involved in the displacement field is only five, as compared to six or more in the case of other shear and normal deformation theories. A comparison with the corresponding results is made to verify the accuracy and efficiency of the present theory. The influences of the porosity parameter, power-law index, aspect ratio, thickness ratio and the foundation parameters on bending and vibration of porous FG plate.

Elastostatic assessment of several mixed/displacement-based laminated plate theories, differently accounting for transverse normal deformability

Several multilayered physically-based plate theories are derived under different limiting assumptions on displacement, strain and stress fields, either in displacement-based or mixed Hu-Washizu and Helinger-Reissner form, and assuming different layerwise functions. Their features are reminiscent to those of theories published in the literature, or are entirely new. The present study aims to evaluate how different forms of description of the transverse normal deformation and stress affect accuracy. At the same time, the purpose is also to see if a much broader degree of generalization of what characterizing currently available physically-based zig-zag theories can be achieved through a redefinition of coefficients obtained by imposing the fulfillment of physical constraints, namely interfacial stress compatibility and local equilibrium equations across the thickness through use of symbolic calculus tool. Besides calculating exactly quantities, this tool enables users to choose representation form and zig-zag functions as desired, keeping fixed the d.o.f. to five. Challenging benchmarks with strong layerwise effects are considered, for which an accurate description of the transverse normal deformation effect is important. They include distributed/localized step loading and different boundary conditions. Effects of constraint stresses at supports are accounted for. Numerical results show that whenever the whole set of physical constraints is enforced across the thickness to redefine coefficients, the choice of the representation form and of zig-zag functions is immaterial and more importantly, accurate results are obtained with few variables and a low expansion order of solutions. Otherwise results are sensitive to the choices made.