An Analytical Solution to Buckling of Thick Beams Based on a Cubic Polynomial Shear Deformation Beam Theory (original) (raw)
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Exact Solutions for the Elastic Buckling Problem of Moderately Thick Beams
Revue des composites et des matériaux avancés, 2020
We present the elastic buckling problem of moderately thick and thick beams as a boundary value problem of the classical mathematical theory of elasticity. The study considered homogeneous, isotropic, linear elastic beams. Small deformation assumptions were used together with kinematic, constitutive relations and the differential equations of equilibrium to obtain the governing field equations as a fourth order non-homogeneous ordinary differential equation (ODE) when both axial, compressive and transverse loads are considered, and a fourth order homogeneous ODE when only axial compressive force is considered. Using trial function method, the homogeneous ODE is solved in general for any end support conditions to obtain a general solution for the buckled beam in terms of four unknown constants of integration. The boundary conditions corresponding to the four cases of end support conditions considered were used to obtain the characteristic buckling equations, which were expanded to obtain transcendental equations with an infinite number of roots in each case, thus yielding an infinite number of buckling loads. The least root of the transcendental equations was used to obtain the critical buckling load, which was found to depend on the ratio h/l and the Poisson's ratio, . Critical buckling loads for each end support condition was calculated and tabulated. The results show that for each end support condition, as h/l < 0.02, the critical buckling load coefficient obtained was approximately equal to the critical buckling load coefficient of Euler-Bernoulli beam. As h/l > 0.02, which is the threshold for thin beams, the critical buckling load is found to be much smaller than the critical buckling load obtained from Euler-Bernoulli theory. It is thus concluded that the shear deformable theory is necessary for a more realistic analysis of the critical load buckling capacities of moderately thick, and thick beams for safety in their design.
Correct assumption approach to refined theory for analysis of thick beam
INTERNATIONAL JOURNAL OF ADVANCE RESEARCH, IJOAR .ORG ISSN 2320-9100, 2015
This paper presents correct assumption approach to refined theory for analysis of thick beam. To achieve this aim, the assumption that engineering vertical shear strain is not zero was fully implemented. For deformation involving the classical (elementary) deformation, component and shear deformation, component it was insisted that both are not equal to zero. That is. With these assumptions coupled with the assumption of uniform shear stress across the cross section of the beam, two new approaches, refined beam theories 2 and 3 (RBT2 and RBT3) were developed. Formulas for deflection and axial displacement coefficients were formulated. Also formulas for critical buckling load and fundamental natural frequency were formulated. These formulas were used in computing the values of center deflection, axial displacement, critical buckling load and fundamental natural frequency of a simply supported beam for span-depth (L/t) ratios of 100, 30, 20, 15, 10 and 4. It was discovered that RBT2 and RBT3 are giving exactly the same values. Their values were compared with values from higher order shear deformation theory designated RBT1 and they were very close to each other. The average percentage difference recorded from the comparison is 0.58%. This shows how close the values from RBT2/RBT3 are to values from RBT1.
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
Buckling of thin-walled beams by a refined theory
Journal of Zhejiang University SCIENCE A, 2012
The buckling of thin-walled structures is presented using the 1D finite element based refined beam theory formulation that permits us to obtain N -order expansions for the three displacement fields over the section domain. These higher-order models are obtained in the framework of the Carrera unified formulation (CUF). CUF is a hierarchical formulation in which the refined models are obtained with no need for ad hoc formulations. Beam theories are obtained on the basis of Taylor-type and Lagrange polynomial expansions. Assessments of these theories have been carried out by their applications to studies related to the buckling of various beam structures, like the beams with square cross section, I-section, thin rectangular cross section, and annular beams. The results obtained match very well with those from commercial finite element softwares with a significantly less computational cost. Further, various types of modes like the bending modes, axial modes, torsional modes, and circumferential shell-type modes are observed.
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.
2017
In this paper, single variable beam theories taking into account effect of transverse shear deformation are developed and applied for the bending, buckling and free vibration analysis of thick isotropic beams. The most important feature of the present beam theories is that unlike any other higher order theory, the proposed class of theories contains only one unknown variable and does not require shear correction factor. The displacement field of the present theories is built upon the classical beam theory. The theories account for parabolic distribution of transverse shear stress using constitutive relations, satisfying the traction free conditions at top and bottom surfaces of the beam. Governing differential equation and boundary conditions of these theories are obtained using the principle of virtual work. Results obtained for the displacements, stresses, fundamental frequencies and critical buckling loads of simply supported isotropic solid beams are compared with those obtained...
Trigonometric Shear Deformation Theory for Thick Simply Supported Beams
A trigonometric shear deformation theory for flexure of thick beams, taking into account transverse shear deformation effects, is developed. The number of variables in the present theory is same as that in the first order shear deformation theory. The sinusoidal function is used in displacement field in terms of thickness coordinate to represent the shear deformation effects. The noteworthy feature of this theory is that the transverse shear stresses can be obtained directly from the use of constitutive relations with excellent accuracy, satisfying the shear stress free conditions on the top and bottom surfaces of the beam. Hence, the theory obviates the need of shear correction factor. Governing differential equations and boundary conditions are obtained by using the principle of virtual work. The thick isotropic beams are considered for the numerical studies to demonstrate the efficiency of the theory. It has been shown that the theory is capable of predicting the local effect of stress concentration due to fixity of support. The fixed isotropic beams subjected to parabolic and cosine loads are examined using the present theory. Results obtained are discussed critically with those of other theories.
A direct one-dimensional beam model for the flexural-torsional buckling of thin-walled beams
Journal of Mechanics of Materials and Structures, 2006
In this paper, the direct one-dimensional beam model introduced by one of the authors is refined to take into account nonsymmetrical beam cross-sections. Two different beam axes are considered, and the strain is described with respect to both. Two inner constraints are assumed: a vanishing shearing strain between the cross-section and one of the two axes, and a linear relationship between the warping and twisting of the cross-section. Considering a grade one mechanical theory and nonlinear hyperelastic constitutive relations, the balance of power, and standard localization and static perturbation procedures lead to field equations suitable to describe the flexural-torsional buckling. Some examples are given to determine the critical load for initially compressed beams and to evaluate their post-buckling behavior.
A refined shear deformation theory for flexure of thick beams
Latin American Journal of Solids and Structures, 2011
A Hyperbolic Shear Deformation Theory (HPSDT) taking into account transverse shear deformation effects, is used for the static flexure analysis of thick isotropic beams. The displacement field of the theory contains two variables. The hyperbolic sine function is used in the displacement field in terms of thickness coordinate to represent shear deformation. The transverse shear stress can be obtained directly from the use of constitutive relations, satisfying the shear stress-free boundary conditions at top and bottom of the beam. Hence, the theory obviates the need of shear correction factor. Governing differential equations and boundary conditions of the theory are obtained using the principle of virtual work. General solutions of thick isotropic simply supported, cantilever and fixed beams subjected to uniformly distributed and concentrated loads are obtained. Expressions for transverse displacement of beams are obtained and contribution due to shear deformation to the maximum transverse displacement is investigated. The results of the present theory are compared with those of other refined shear deformation theories of beam to verify the accuracy of the theory.
A New Trigonometric Shear Deformation Theory For Thick Fixed Beam
A new trigonometric shear deformation theory for bending of thick fixed beam, taking into account transverse shear deformation effects, is developed. The number of variables in the present theory is same as that in the first order shear deformation theory. The displacement field used in terms of sinusoidal function in thickness coordinate is replaced with to represent the shear deformation effects. The noteworthy feature of this theory is that the transverse shear stresses can be obtained directly from the use of constitutive relations with excellent accuracy, satisfying the shear stress free conditions on the top and bottom surfaces of the beam. Hence, the theory obviates the need of shear correction factor. Governing differential equations and boundary conditions are obtained by using the principle of virtual work. The thick isotropic beams are considered for the numerical studies to demonstrate the efficiency of the theory. It has been shown that the theory is capable of predicting the local effect of stress concentration due to fixity of support. The fixed isotropic beam subjected to uniformly varying load is examined using the present theory. Results obtained are discussed critically with those of other refined theories.