• Structural Engineering and Mechanics
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• v.37 no.2
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• pp.149-162
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• 2011

The aim of this research is a comprehensive review and evaluation of beam theories resting on elastic foundations that used to model mode-I delamination in multidirectional laminated composite by DCB specimen. A compliance based approach is used to calculate critical strain energy release rate (SERR). Two well-known beam theories, i.e. Euler-Bernoulli (EB) and Timoshenko beams (TB), on Winkler and Pasternak elastic foundations (WEF and PEF) are considered. In each case, a closed-form solution is presented for compliance versus crack length, effective material properties and geometrical dimensions. Effective flexural modulus ($E_{fx}$) and out-of-plane extensional stiffness ($E_z$) are used in all models instead of transversely isotropic assumption in composite laminates. Eventually, the analytical solutions are compared with experimental results available in the literature for unidirectional ($[0^{\circ}]_6$) and antisymmetric angle-ply ($[{\pm}30^{\circ}]_5$, and $[{\pm}45^{\circ}]_5$) lay-ups. TB on WEF is a simple model that predicts more accurate results for compliance and SERR in unidirectional laminates in comparison to other models. TB on PEF, in accordance with Williams (1989) assumptions, is too stiff for unidirectional DCB specimens, whereas in angle-ply DCB specimens it gives more reliable results. That it shows the effects of transverse shear deformation and root rotation on SERR value in composite DCB specimens.

• Smart Structures and Systems
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• v.20 no.6
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• pp.709-728
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• 2017

This disquisition proposes a nonlocal strain gradient beam theory for thermo-mechanical dynamic characteristics of embedded smart shear deformable curved piezoelectric nanobeams made of porous electro-elastic functionally graded materials by using an analytical method. Electro-elastic properties of embedded curved porous FG nanobeam are assumed to be temperature-dependent and vary through the thickness direction of beam according to the power-law which is modified to approximate material properties for even distributions of porosities. It is perceived that during manufacturing of functionally graded materials (FGMs) porosities and micro-voids can be occurred inside the material. Since variation of pores along the thickness direction influences the mechanical and physical properties, so in this study thermo-mechanical vibration analysis of curve FG piezoelectric nanobeam by considering the effect of these imperfections is performed. Nonlocal strain gradient elasticity theory is utilized to consider the size effects in which the stress for not only the nonlocal stress field but also the strain gradients stress field. The governing equations and related boundary condition of embedded smart curved porous FG nanobeam subjected to thermal and electric field are derived via the energy method based on Timoshenko beam theory. An analytical Navier solution procedure is utilized to achieve the natural frequencies of porous FG curved piezoelectric nanobeam resting on Winkler and Pasternak foundation. The results for simpler states are confirmed with known data in the literature. The effects of various parameters such as nonlocality parameter, electric voltage, coefficient of porosity, elastic foundation parameters, thermal effect, gradient index, strain gradient, elastic opening angle and slenderness ratio on the natural frequency of embedded curved FG porous piezoelectric nanobeam are successfully discussed. It is concluded that these parameters play important roles on the dynamic behavior of porous FG curved nanobeam. Presented numerical results can serve as benchmarks for future analyses of curve FG nanobeam with porosity phases.

• Steel and Composite Structures
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• v.39 no.5
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• pp.615-630
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• 2021

The main purpose of the present work was to study the dynamic instability of a three-layered, thick composite sandwich beam with the functionally graded (FG) flexible core subjected to an axial compressive follower force. Flutter instability of a sandwich cantilever beam was analyzed using the high-order theory of sandwich beams, for the first time. The governing equations in general for sandwich beams with an FG core were extracted and could be used for all types of sandwich beams with any types of face sheets and cores. A polynomial function is considered for the vertical distribution of the displacement field in the core layer along the thickness, based on the results of the first Frosting's higher order model. The governing partial differential equations and the equations of boundary conditions of the dynamic system are derived using Hamilton's principle. By applying the boundary conditions and numerical solution methods of squares quadrature, the beam flutter phenomenon is studied. In addition, the effects of different geometrical and material parameters on the flutter threshold were investigated. The results showed that the responses of the dynamic instability of the system were influenced by the follower force, the coefficients of FGs and the geometrical parameters like the core thickness. Comparison of the present results with the published results in the literature for the special case confirmed the accuracy of the proposed theory. The results showed that the follower force of the flutter phenomenon threshold for long beams tends to the corresponding results in the Timoshenko beam.

• Steel and Composite Structures
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• v.44 no.4
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• pp.503-517
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• 2022

This paper presents the mechanical buckling of bi-directional functionally graded sandwich beams (BFGSW) with various boundary conditions employing a quasi-3D beam theory, including an integral term in the displacement field, which reduces the number of unknowns and governing equations. The beams are composed of three layers. The core is made from two constituents and varies across the thickness; however, the covering layers of the beams are made of bidirectional functionally graded material (BFGSW) and vary smoothly along the beam length and thickness directions. The power gradation model is considered to estimate the variation of material properties. The used formulation reflects the transverse shear effect and uses only three variables without including the correction factor used in the first shear deformation theory (FSDT) proposed by Timoshenko. The principle of virtual forces is used to obtain stability equations. Moreover, the impacts of the control of the power-law index, layer thickness ratio, length-to-depth ratio, and boundary conditions on buckling response are demonstrated. Our contribution in the present work is applying an analytical solution to investigate the stability behavior of bidirectional FG sandwich beams under various boundary conditions.