Large Thermal Deflections of a Simple Supported Beam with Temperature-Dependent Physical Properties DR. ŞEREF DOĞUŞCAN AKBAŞ Civil Engineer, Şehit Muhtar Mah. Öğüt Sok. No:2/37, 34435 Beyoğlu- Istanbul, Turkey E-mail: serefda@yahoo.com Abstract This paper focuses on large deflections of a simple supported beam subjected to non-uniform thermal rising with temperature dependent physical properties by using the total Lagrangian Timoshenko beam element approximation. It is known that large deflection problems are geometrically nonlinear problems. Also, the material properties (Young s modulus, coefficient of thermal expansion, Poisson ratio) are temperature dependent: That is the coefficients of the governing equations are not constant in this study. This situation suggests the physical nonlinearity of the problem. Hence, the considered problem is both geometrically and physically nonlinear. The considered highly non-linear problem is solved considering full geometric non-linearity by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. In the study, the difference between the geometrically linear and nonlinear analysis of the beam is investigated in detail for temperature rising. The relationships between deflections, thermal configuration, Cauchy stresses of the beams and temperature rising are illustrated in detail in post-buckling case. Also, the differences between temperature dependent and independent physical properties are investigated in detail. Keywords-Large Thermal Deflection, Temperature Dependent Physical Properties, Total Lagrangian Finite Element Model, Temperature Rise I. INTRODUCTION The design of structural elements (beams, plates, shells etc.) in the high thermal environments is very important in engineering applications. In recent years, much more attention has been given to the thermal behaviour of beam structures. Jekot [1] investigated the thermal post-buckling of a beam made of physically nonlinear thermoelastic material is investigated by using the geometric equations in the von-karman strain-displacement approximation. Li [2] examined thermal post-buckling of Rods with Pinned-Fixed Ends using the shooting method. On the basis of exact nonlinear geometric theory of extensible beam and by using a shooting method, computational analysis for thermal post buckling behavior of beams were studied [3] and [4]. Li at al [5] investigated thermal post-buckling responses of an elastic beam, with immovably simply supported ends and subjected to a transversely non-uniformly distributed temperature rising. Li and Zhou [6] studied thermal postbuckling response of an immovably pinned-fixed Timoshenko beam subjected to a static transversely nonuniform temperature by using a shooting method. Li and Song [7] analyzed large thermal deflections for Timoshenko beams subjected to transversely non-uniform temperature rise. Song and Li [8] investiagted both thermal buckling and post-buckling of pinned fixed beams resting on an elastic foundation. Vaz et al. [9] examined a perturbation solution for the initial post-buckling of beams that were supported on an elastic foundation under uniform thermal load. Vaz et al. [10] examined elastic buckling and initial post-buckling behavior of slender beams subjected to uniform heating with temperature-dependent physical properties by using a perturbation solution. Akbaş and Kocatürk [11] investigated post-buckling analysis of a simply supported beam subjected to a uniform thermal loading by using total Lagrangian finite element model of two dimensional continuum for an eight-node quadratic element. Kocatürk and Akbaş [12] studied post-buckling analysis of Timoshenko beams with various boundary conditions subjected to a non-uniform thermal loading by using the total Lagrangian Timoshenko beam element approximation. Yu and Sun [13] investigated large deformation post-buckling of a linear-elastic and hygrothermal beam with axially nonmovable pinned-pinned ends and subjected to a significant increase in swelling by an alternative method. Akbaş and Kocatürk [14] examined post-buckling behavior of Timoshenko beams subjected to uniform temperature rising with temperature dependent physical properties. It is seen from literature that large thermal deflection studies with temperature-dependent physical properties has not been broadly investigated. In this study, the difference between the geometrically linear and nonlinear analysis of the beam is investigated in detail for temperature rising. The related formulations of large thermal deflection analysis of simple supported Timoshenko beam with subjected to a non-uniform thermal loading are obtained by using the total Lagrangian finite element model. In deriving the formulations for large deflection analysis under nonuniform thermal loading and temperature dependent physical properties, the total Lagrangian Timoshenko beam element formulations given by Felippa [15] are used. There is no retstriction on the magnitudes of deflections and 19
rotations in contradistinction to von-karman strain displacement relations of the beam. The difference between is investigated in detail. The relationships between deflections, Cauchy stresses, large deflection configuration of the beams and temperature rising are illustrated in detail. Also, the difference between the geometrically linear and nonlinear analysis of the beam is investigated in detail for temperature rising. II. THEORY AND FORMULATIONS a simple supported beam of length L, width b, height h as shown in Figure 1. Figure. 1. Simple Supported beam subjected to temperature rising and cross-section. In this study, the Total Langragian Timoshenko beam element is used and the related formulations are developed for temperature dependent physical properties by using the formulations given by Akbaş and Kocatürk [14] which was developed for thermal loading by using the formulations given by Felippa [15]. Interested reader can find the related formulations in Akbaş and Kocatürk [14] and Felippa [15]. The second Piola-Kirchhoff stresses with a temperature rise can be expressed by inclusion of the temperature term as follows In Equation (9),,,, and indicate the coefficients of temperature T and are unique to the constituent materials. In this study, the unit of the temperature is Kelvin (K), the unit of the Young s modulus E is Pascal (Pa) and the unit of the thermal expansion coefficient is 1/K. The beams considered in numerical examples are made of Zirconia. The coefficients of temperature T for Zirconia are listed in Table 1 (from Reddy and Chin [17]). TABLE I. The coefficients of temperature T for Zirconia (from Reddy and Chin [17]) The Material Properties Thermal expansion 12.766 coefficient α X (1/K) 10-6 0-1.491 1.0006-6.778 10-6 10-5 10-11 Young s modulus E (Pa) Poisson s ratio ν 0.2882 0 Coefficientof thermal conductivity k (W/ mk) (3) 244.27 10 9 0-1.371 1.214-3.681 10-6 10-6 10-10 1.7000 0 1.133 10-4 0 0 1.276 6.648 10-4 10-8 0 Using constitutive equations (1), axial force N, shear force V and bending moment M can be obtained as (4) (5) (6) (7) (8) where, are initial stresses, E is Young s modulus and G is the shear modulus, is coefficient of thermal expansion in the X direction and, where is installation temperature and is the temperature rise. The physical properties of the material (Young s modulus, coefficient of thermal expansion, Poisson ratio) are dependent on temperature T. Variation of the temperature along the beam height can be expressed as, where and are the temperature rise of the top and the bottom surfaces of the beam. The effective material properties of the beam, P, i.e., Young s modulus E, coefficient of thermal expansion, coefficient of thermal conductivity K, Poisson s ratio ν and shear modulus G are function of temperature T (see Touloukian [16]) as follows; (1) (2) where A, e, and are the cross section area, second moment of inertia axial strain, shear strain and curvature of the beam, respectively. The details of these expressions can be found in Akbaş and Kocatürk [14] and Felippa [15]., (9) For the solution of the total Lagrangian formulations of Timoshenko beam element, small-step incremental approaches from known solutions with Newton-Raphson iteration method are used in which the solution for n+1th load increment and i th iteration is obtained in the following form: (10) where is the system stiffness matrix corresponding to a tangent direction at the i th iteration, is the solution increment vector at the i th iteration and n+1 th load increment, is the system residual vector at the i th iteration and n+1 th load increment. This iteration procedure is continued until the difference between two 20
successive solution vectors is less than a selected tolerance criterion in Euclidean norm given by A series of successive approximations gives where (11) (12) (13) Timoshenko plane beam element. After that, the solution process outlined in the previous section is used for obtaining the related solutions for the total Lagrangian finite element model of the Timoshenko plane beam element. The beams considered in numerical examples are made of Zirconia. The coefficients of temperature T for Zirconia are listed in Table 1. In the numerical integrations, five-point Gauss integration rule is used. Unless otherwise stated, it is assumed that the width of the beam is b=0.1m, height of the beam is h=0.1m, length of the beam is L=3m and in the numerical results. In numerical calculations, the number of finite elements taken as n=120. In figure 2, transversal displacements of the midpoint v(l/2) versus temperature rising are shown for the geometrically linear and the non-linear case for. The residual vector follows; for a finite element is as (14) where f is the vector of external forces and p is the vector of internal forces given by Felippa [15]. The element tangent stiffness matrix for the total Lagrangian Timoshenko plane beam element is as follows which is given by Akbaş and Kocatürk [14] and Felippa [15] (15) where is the geometric stiffness matrix, and is the material stiffness matrix given as follows by Akbaş and Kocatürk [14] and Felippa [15] (16) The interested reader can find the explicit forms of the expressions in Eq. (16) in Akbaş and Kocatürk [14] and Felippa [15]. After integration of Eq. (16), can be expressed as follows; (17) where is the axial stiffness matrix, is the bending stiffness matrix, is the shearing stiffness matrix and explicit forms of these expressions are given by Felippa [15]. III. NUMERICAL RESULTS In the numerical examples, the linear and the nonlinear static deflections of the beams are calculated and presented in figures for temperature dependent and independent physical properties for different thermal loads. To this end, with the use of the usual assembly process, the system tangent stiffness matrix and the system residual vector are obtained by using the element stiffness matrixes and element residual vectors for the total Lagrangian 21 Figure 2. Temperature rising - transversal displacements curves for temperature dependent and independent physical properties, a) Linear case, b) Non-linear case. It is seen from figure 2, with increase in temperature, the difference between the displacements for the temperature dependent and independent physical properties increases. figure 2 shows that the transversal displacements for the temperature-dependent physical properties are greater than those for the temperature-independent physical properties. This situation may be explained as follows: In the temperature-dependent physical properties, with the temperature increase, the intermolecular distances of the material increase and intermolecular forces decrease. As a result, the strength of the material decreases. Hence, the beam becomes more flexible in the case of the temperature-dependent physical properties. In figure 3, transversal displacements of the midpoint v(l/2) versus temperature rising are presented for
for. Figure 4. The effect of temperature dependent and independent physical properties on the deflected shape of the beam for,. a) Linear case b) Non-linear case. Figure 3. Temperature rising - transversal displacements curves for geometrically linear case and geometrically non-linear case, a) temperature independent physical properties case, b) temperature dependent physical properties case. It is observed from figure 3 that there is a significant difference between the geometrically linear case and nonlinear case. Increase in temperature causes increase in difference between the displacement values of the linear and the nonlinear solutions. Also it is seen figure 3 that Although the curve of geometrically linear solution for temperature independent physical properties case is linear, the curve of geometrically linear solution for temperature dependent physical properties case is nonlinear. This situation may be explained as follows: In the temperaturedependent physical properties, the material properties (Young s modulus, coefficient of thermal expansion, Poisson ratio) are temperature dependent: That is the coefficients of the governing equations are not constant This situation suggests the physical nonlinearity of the problem. Hence, the solution of geometrically linear case for temperature dependent physical properties is nonlinear. Figure 4, 5 and 6 display the effect of temperature dependent and independent physical properties on the thermal deflected shape of the beam for,, respectively for for the geometrically linear and the non-linear case. Figure 5. The effect of temperature dependent and independent physical properties on the deflected shape of the beam for,. a) Linear case b) Non-linear case. 22
Figure 6. The effect of temperature dependent and independent physical properties on the deflected shape of the beam for,. a) Linear case b) Non-linear case. It is seen from figures 4, 5 and 6 that, there is a significant difference between the deformed configurations of the beam for temperature dependent and temperature independent physical properties. It is expected that the deflections of the beam for the temperature-dependent physical properties are greater than those for the temperature-independent physical properties. Also it is seen from figures 4, 5 and 6 that, with increase in temperature, the difference between the geometrically linear and the non-linear case increases significantly. In figure 7, Cauchy normal stresses at midpoint of the beam (X=1,5m and Y=-0.05m) versus temperature rising are presented for temperature dependent and independent physical properties. Figure 7. Cauchy normal stress midpoint of the beam (X=1,5m and Y=-0.05m) versus temperature rising temperature rising for. a) Linear case b) Non-linear case. It is seen from figures 9, with increase in temperature, the difference between the Cauchy normal stresses for the increases. IV. CONCLUSIONS Large thermal deflections of a simple supported beam subjected to non-uniform thermal rising is studied for temperature dependent physical properties by using the total Lagrangian Timoshenko beam element approximation. The considered highly non-linear problem is solved considering full geometric non-linearity by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. The relationships between deflections, thermal configuration, Cauchy stresses of the beams and temperature rising are illustrated. Also, The difference between the geometrically linear and nonlinear analysis of the beam is investigated. It is observed from the investigations that there are significant differences of the analysis results for the in the post-buckling case. Hence, the temperaturedependent physical properties must be taken into account for safe design of beams and for obtaining more realistic results. Otherwise an important error is inevitable. 23
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