Deflection of thick beams of multimodular materials (original) (raw)

Nonlinear bending of bimodular-material plates

International Journal of Solids and Structures, 1983

The paper presents finite element results for geometrically nonlinear response of fiber-reinforced, single-layer and two-layer cross-ply rectangular plates constructed of materials which have linear elastic properties in tension and compression that are different. A shear deformation theory of layered composite plates, accounting for large rotations (in the von Karman sense) and the bimodular action, is employed to analyze rectangular plates made of two cord-rubber bimodular materials. Numerical results for transverse deflection are presented for simply supported plates under sinusoidally distributed and uniformly distributed transverse loads.

Vibration of bimodular sandwich beams with thick facings: A new theory and experimental results

Journal of Sound and Vibration, 1983

This study deals with both analytical and experimental investigations of three-layer beams with cores of polyurethane foam and facings of unidirectional cord-rubber. Both of these materials are bimodular (i.e., having different behavior in compression as compared to tension). The new theory presented is a shear-flexible laminate version of the well-known Timoshenko beam theory, which, due to the bending-stretching coupling present in the bimodular case, results in a coupled sixth-order system of differential equations. In this theory, a separate derivation is presented for the shear correction factor. Due to the discontinuities in the normal stress distribution and the bimodularity, the shear correction factor is much different than the classical homogeneous material value of 5/6. Theoretical and experimental results are presented for the frequencies of the first three modes of vibration for a pin-ended beam without axial restraint. This work is believed to be the first devoted to vibration of bimodular materials in a sandwich configuration.

A Computationally Attractive Beam Theory Accounting for Transverse Shear and Normal Deformations

A variational higher-order theory has been developed for representing the bending and stretching of linearly elastic orthotropic beams which include the deformations due to transverse shearing and stretching of the transverse normal. The theory assumes a linear distribution for the longitudinal displacement and a parabolic variation of the transverse displacement across the thickness. Independent expansions are also introduced in order to represent the through-thickness displacement gradients by requiring least-square compatibility for the transverse strains and the exact stress boundary conditions at the top/bottom beam surfaces. The theory is shown to be well suited for finite element development by requiring simple C 0-and C'-continuous displacement interpolation fields. Computational utility of the theory is demonstrated by formulating a simple two-node stretching-bending finite element. Both analytic and finite element procedures are applied to a simple bending problem and compared to an exact elasticity solution. It is shown that the inclusion of the transverse normal deformation in the present theory provides an improved displacement, strain and stress prediction capability, particularly for the analysis of thick-section beams. UNCLASS IF IED SECUIQTY CLASSIFICATION O' TWiS PAGE 14thao 0lee F.nre'd) NOMENCLATURE A cross-sectional area of beam Aij inplane rigidities b width of beam's cross-section C O the class of continuous functions possessing discontinuous derivatives at element nodes CI the class of continuous functions that are discontinuous at element nodes Cij elastic stiffness coefficients Dij bending rigidities Ej elastic moduli f consistent load vector G transverse shear rigidity 2h beam thickness IY cross-sectional moment of inertia about y-axis Ke element stiffness matrix L beam span N x , N., Q, force resultants Mx, M, moment resultants q, q+, q applied transverse loads S+, S top and bottom beam surfaces CONTENTS NOMENCLATURE.

Large deflection analysis of beams with variable stiffness

Acta Mechanica, 2003

In this paper, the Analog Equation Method (AEM), a BEM-based method, is employed to the nonlinear analysis of a Bernoulli-Euler beam with variable stiffness undergoing large deflections, under general boundary conditions which maybe nonlinear. As the cross-sectional properties of the beam vary along its axis, the coefficients of the differential equations governing the equilibrium of the beam are variable. The formulation is in terms of the displacements. The governing equations are derived in both deformed and undeformed configuration and the deviations of the two approaches are studied. Using the concept of the analog equation, the two coupled nonlinear differential equations with variable coefficients are replaced by two uncoupled linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under fictitious load distributions. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. Several beams are analyzed under various boundary conditions and loadings to illustrate the merits of the method as well as its applicability, efficiency and accuracy.

Stresses in Curved Beams Made of Heterogeneous Materials

International Journal of Mechanical Systems Engineering, 2015

The main objective of the present paper is a generalization of some classical results for curved beams made of heterogeneous materials. We consider a beam made of nonhomogeneous, isotropic, linearly elastic material. The elastic parameters depend on the cross-sectional coordinates only. Our investigations include the determination of the normal stress (i.e., the generalization of the Grashof formula), the shearing stress and the curvature change. Interestingly, our newly established formulae have the same structure as the classical ones. We conclude with numerical examples which illustrate the applicability of our results.

Static Flexural Analysis of Thick Isotropic Beam Using Hyperbolic Shear Deformation Theory

Flat slabs are highly versatile elements widely used in construction, providing minimum depth, fast construction and allowing flexible column grids. Common practice of design and In the present study, a hyperbolic shear deformation theory is developed for static flexural analysis of thick isotropic beams. The theory assumes a parabolic variation of transverse shear stress across the thickness of the beams. Simply supported thick isotropic beams analysed for the axial displacement, Transverse displacement, Axial bending stress and transverse shear stress. In this theory the hyperbolic sine and cosine function is used in the displacement field to represent the shear deformation effect and satisfy the zero transverse shear stress condition at top and bottom surface of the beams. The Governing differential equation and boundary conditions of the theory are obtained by using Principle of virtual work. The simply supported isotropic beam subjected to varying load is examined using present theory. The numerical results have been computed for various lengths to thickness ratios of the beams and the results obtained are compared with those of Elementary, Timoshenko, Trigonometric and other higher order refined theories and with the available solution in the literature Keywords: Thick beam, shear deformation, isotropic beam, transverse shear stress, static flexure, hyperbolic shear deformation theory, principle of virtual wor The wide spreaduse of shear flexible materials in air craft, automotive, shipbuilding and other industries has stimulated interest in the accurate prediction of structuralbehaviourofbeams. Theories of beams involve basically the reduction of a three dimensional problems of elasticity theory to a one-dimensional problems. Since the thickness dimension is much smaller than the longitudinal dimension, it is possible to approximate the distribution of the displacement, strain and stress components in the thickness dimension. The various methods of development of refined theories based on the reduction of the three dimensional problems of mechanics of elastic bodies are discussed by Gol denveizer [1], Kil chevskiy [2], Donnell [3], Vlasov and Leontev [4], Sayir and Mitropoulos [5]. It is well-known that elementary theory of bending of beam based on Euler-Bernoulli hypothesis that the plane sections which are perpendicular to the neutral layer before bending remain plane and perpendicular to the neutral layer after bending, implying that the transverse shear and transverse normal strains are zero. Thus the theory disregards the effects of the shear deformation. It is also known as classical beam theory. The theory is applicable to slender beams and should not be applied to thick or deep beams. When elementary theory of beam (ETB) is used for the analysis thick beams, deflections are underestimated and natural frequencies and buckling loads are overestimated. This is the consequence of neglecting transverse shear deformations in ETB. Rankine [6], Bresse [7] were the first to include both the rotatory inertia and shear flexibility effects as refined dynamical effects in beam theory. This theory is, however, referred to as the Timoshenko beam theory as mentioned in the literature by Rebello, et al. [8] and based upon kinematics it is known as first-order shear deformation theory (FSDT). Rayleigh [9] included the rotator inertia effect while later the effect of shear stiffness was added by Timoshenko [10]. Timoshenko showed that the effect of shear is much greater than that of rotatory inertia for transverse vibration of prismatic beams. In Timoshenko beam theory transverse shear strain distribution is constant through the beam thickness and therefore requires shear correction factor to correct the strain energy of deformation. Cowper [11] and Murty [13] have given new expressions for this coefficient for different cross-sections of the beam. Stephen and Levinson [15] have introduced a refined theory incorporating shear curvature, transverse direct stress and rotatory inertia effects. The limitations of the elementary theory of bending (ETB) of beams and first order shear deformation theory (FSDT) for beams forced the development of higher order shear deformation theories Soler [16] developed the higher order theory for thick isotropic rectangular elastic beams using Legendre polynomials and Tsai and Soler [17] extended it to orthotropic beams. Effects of shear deformation and transverse normal stress are included. Levinson [18] obtained the higher order beam theory providing the fourth order system of differential equations, satisfying two boundary conditions at each end of the beam. No shear correction factors are required since the theory satisfies the shear stress free surface conditions on the top and bottom of the beam. Krishna Murty [22] formulated a third order beam theory including the transverse shear strain and non classical (nonlinear) axial stress. In this theory the parabolic transverse shear stress distribution across the depth of the beam can be obtained using constitutive relations. Ghugal and Dahake[23] has developed a trigonometric shear deformation theory for flexure of thick or deep beams, taking into account transverse shear deformation effect. The number of variables in the present theory is same as that in the

Study of auxetic beams under bending: A finite element approach

Materials Today: Proceedings, 2020

Materials with a negative Poisson's ratio are termed as auxetic materials. A negative Poisson's ratio implies a lateral strain in accordance to a longitudinal strain under application of load, that is, it expands laterally when under tensile load and contracts under compressive loading. The key feature that imparts the auxetic behavior to a structure is its geometry. This paper initially discusses about the single cell honeycomb and re-entrant configurations. Analytical formulations are used to calculate the Poisson's ratio for various structures and numerical analysis is carried out using the ABAQUS software. Further, various beam configurations such as homogeneous, truss and auxetic re-entrant structures are investigated under four point bending to understand the scope of application of these structures and to observe their behavior when employed in a practically significant entity. The auxetic beam is found to have lowest level of stress in most of the regions. This paper also discusses the reasons why there is a need for a combined analysis of these auxetic structures using Experimental/Numerical and analytical methods.