Rameshchandra P. Shimpi

A Review of Refined Shear Deformation Theories for Isotropic and Anisotropic Laminated Beams

A review of displacement and stress based refined theories for isotropic and anisotropic laminated beams is presented. Various equivalent single layer and layerwise theories for laminated beams are discussed together with their merits and demerits. Exact elasticity solutions for the beam problems are cited, wherever available. Various critical issues, related with beam theories, based on the literature reviewed are presented.

Refined plate theory and its variants

Thedevelopmentofa newree ned platetheoryand its two simplevariantsisgiven. Thetheorieshavestrongcommonality withtheequationsofclassicalplatetheory (CPT).However,unlikeCPT,thesetheoriesassumethatlateral and axial displacements have bending and shear components such that bending components do not contribute toward shearforces and, likewise, shearing components do not contribute toward bending moments. The theory and one of its variants are variationally consistent, whereasthesecond variant isvariationally inconsistent and usesthe relationships between moments, shear forces, and loading. It should be noted that, unlike any other ree ned plate theory, thegoverning equation as well as the expressions for moments and shear forces associated with thisvariant areidentical tothoseassociated withtheCPT,savefortheappearanceofasubscript. Theeffectivenessofthetheory and itsvariantsisdemonstratedthroughanexample. Surprisingly,theanswersobtained by boththevariantsofthe theory, one of which is variationally consistent and the other one is inconsistent, are same. The numerical example studied, therefore, not only brings out the effectiveness of the theories presented, but also, albeit unintentionally, supports the doubts, e rst raised by Levinson, about the so called superiority of variationally consistent methods.

A zigzag model for laminated composite beams

Abstract In the present work, a zigzag model for symmetric laminated beam is developed. This model uses a sine term to represent the non-linear displacement field across the thickness as compared to a third order polynomial term in conventional theories. Transverse shear stress and strain are represented by a cosine term as compared to parabolic term. This model satisfies displacement and transverse shear stress continuity at the interface. Zero transverse shear stress boundary condition at the top and bottom of the beam are also satisfied. The numerical results indicates that the present model predicts very accurate results for displacement and stresses for symmetric cross-ply laminated beam, even for small length to thickness ratio. The results are also compared with a simplified theory of same class.

A new layerwise trigonometric shear deformation theory for two-layered cross-ply beams

A new layerwise trigonometric shear deformation theory for the analysis of two-layered cross-ply laminated beams is presented. The number of primary variables in this theory is even less than that of first-order shear deformation theory, and moreover, it obviates the need for a shear correction factor. The sinusoidal function in terms of thickness coordinate is used in the displacement field to account for shear deformation. The novel feature of the theory is that the transverse shear stress can be obtained directly from the use of constitutive relationships, satisfying the shear-stress-free boundary conditions at top and bottom of the beam and satisfying continuity of shear stress at the interface. The principle of virtual work is used to obtain the governing equations and boundary conditions of the theory. The effectiveness of the theory is demonstrated by applying it to a two-layered cross-ply laminated beam.

A beam finite element based on layerwise trigonometric shear deformation theory

A simple one-dimensional beam finite element, based on layerwise trigonometric shear deformation theory, is presented. The element has two nodes and only three degrees of freedom per node. Yet, it incorporates through the thickness sinusoidal variation of in-plane displacement such that shear-stress free boundary conditions on the top and bottom surfaces of the beam element are satisfied and the shear-stress distribution is realistic in nature. Constitutive relations between shear-stresses and shear-strains are satisfied in all the layers, and, therefore, shear correction factor is not required. Compatibility at the layer interface in respect of in-plane displacement is also satisfied. It is to be noted that the element developed is free from shear locking. The results obtained are accurate and show good convergence. Unlike many other elements, transverse shear-stresses are evaluated directly using constitutive relations. The efficacy of the present element is demonstrated through the examples of static flexure and free vibration.

COMBINING APPROXIMATION CONCEPTS WITH GENETIC ALGORITHM-BASED STRUCTURAL OPTIMIZATION PROCEDURES

This paper presents an approach for combining approximation models with genetic algorithm-based design optimization procedures. An important objective here is to develop an approach which empirically ensures that the GA converges asymptotically to the optima of the original problem using a limited number of exact analysis. It is shown that this problem may be posed as a dynamic optimization problem, wherein the fitness function changes over successive generations. Criteria for selecting the design points where exact analysis should be carried out are proposed based on observations on the steady-state behavior of simple GAs. Guidelines based on trustregion methods are presented for controlling the generation delay before the approximation model is updated. An adaptive selection operator is developed to efficiently navigate through such changing and uncertain fitness landscapes. Results are presented for the optimal design problem of a 10 bar truss structure. It is shown that, using the present approach, the number of exact analysis required to reach the optima of the original problem can be reduced by more than 97 %.

A HIGHER ORDER DISPLACEMENT MODEL FOR THE PLATE ANALYSIS

A higher order displacement model for laminated composite plates is presented. The displacement model has trigonometric terms in addition to thin plate terms and contains six unknowns. The model satisfies shear stress conditions at top and bottom of the plate. Governing equations and boundary conditions are obtained using virtual work principle. The present displacement model gives simple governing equations. The model has been applied to various simply supported rectangular isotropic, orthotropic and layered plates for static and free vibration analysis. This model yields better results as compared to other higher order displacement models.

Layer-by-Layer Analysis of a Simply Supported Thick Flexible Sandwich Beam

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A simple single variable shear deformation theory for a rectangular beam

This paper proposes a simple single variable shear deformation theory for an isotropic beam of rectangular cross-section. The theory involves only one fourth-order governing differential equation. For beam bending problems, the governing equation and the expressions for the bending moment and shear force of the theory are strikingly similar to those of Euler–Bernoulli beam theory. For vibration and buckling problems, the Euler–Bernoulli beam theory governing equation comes out as a special case when terms pertaining to the effects of shear deformation are ignored from the governing equation of present theory. The chosen displacement functions of the theory give rise to a realistic parabolic distribution of transverse shear stress across the beam cross-section. The theory does not require a shear correction factor. Efficacy of the proposed theory is demonstrated through illustrative examples for bending, free vibrations and buckling of isotropic beams of rectangular cross-section. The numerical results obtained are compared with those of exact theory (two-dimensional theory of elasticity) and other first-order and higher-order shear deformation beam theory results. The results obtained are found to be accurate.

On the theory of discrete structural model analysis and design

Abstract Exclusive theory for analysis of Structural Models (comprising of springs, masses, dash pots, etc.) is presented by adapting the electrical network theory. It commences a brief statement of a new Principle of Quasi Work (PQW) , relevant to this theory. Derivations presented here include theorems addressing maximum displacements, relative flexibilities, sensitivity analysis of global flexibilities, inverse problem of load prediction and interpolation of stiffnesses and flexibilities of the Structural Models. Finally a Design Equation capable of providing a starting point which more or less satisfies all the displacement constraints for iterative design employing a pair of estimated starting points for design iterations (within or outside feasible region) is evolved. Simple substantive illustrations are included to demonstrate the potential of these theoretical developments.