IT is well known that a beam heated from the stress free reference

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1 AIAA JOURNAL Vol. 45, No. 3, March 27 Thermal Buckling and Postbuckling of Euler Bernoulli Beams Suorted on Nonlinear Elastic Foundations S.-R. Li Lanzhou University of Technology, 735 Lanzhou, Gansu, Peole s Reublic of China and R. C. Batra Virginia Polytechnic Institute and State University, Blacksburg, Virginia 246 DOI: / Deformations of homogeneous and isotroic inned inned and fixed fixed Euler Bernoulli beams suorted on nonlinear elastic foundations and heated uniformly into the ostbuckling regime have been analyzed numerically. Geometric nonlinearities introduced by finite deflections and curvature of the deformed beams are incororated in the roblem formulation. First buckling due to the uniform temerature rise and buckling mode transitions are investigated analytically by analyzing the linear roblem. Subsequently, the nonlinear boundary-value roblems for ostbuckling of beams are transformed into initial-value roblems and analyzed by the shooting method. For different values of the elastic foundation arameters, ostbuckled configurations are illustrated. Nomenclature A = area of the cross section of the beam E = Young s modulus f = dimensionless deflection of the beam center H = resultant horizontal force I = moment of inertia of the cross section, = dimensionless values of k 1, k 2 mn = stiffness of elastic foundation at which buckling mode transitions k 1, k 2 = linear and cubic stiffness arameters of the elastic foundation l = initial length of the beam M = bending moment m = dimensionless value of M m = dimensionless bending moment at a fixed end N = resultant axial force P H, P V = dimensionless values of H and V q x, q y = distributed mechanical loads on the beam S = dimensionless value of s s = arc length of the deformed centroidal axis T = temerature rise U = dimensionless value of u u = dislacement in x direction V = resultant vertical force W = dimensionless value of w w = dislacement in y direction or beam deflection x, y = rectangular Cartesian coordinate axes = coefficient of thermal exansion = angle between the deformed beam s axis and the x axis = rotational angle of the inned end of the beam = curvature of the deformed beam axis = stretch of the centroidal axis = slenderness ratio of the beam Received 2 Aril 26; revision received 5 October 26; acceted for ublication 6 October 26. Coyright 26 by Romesh C. Batra. Published by the American Institute of Aeronautics and Astronautics, Inc., with ermission. Coies of this aer may be made for ersonal or internal use, on condition that the coier ay the $1 er-coy fee to the Coyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 1923; include the code /7 $1 in corresondence with the CCC. Professor, Deartment of Engineering Mechanics; lisr@lut.cn. Clifton C. Garvin Professor, Deartment of Engineering Science and Mechanics, M/C 219; rbatra@vt.edu. 712 = dimensionless value of x = dimensionless value of T I. Introduction IT is well known that a beam heated from the stress free reference configuration with its edges constrained from moving axially will have an axial comressive stress develoed in it. When this axial comressive stress reaches a critical value, the beam will buckle, and will go into a ostbuckled configuration uon further heating. The ostbuckling deformations of slender comonents, such as robotic arms, otical fibers, satellite tethers, microresonators, and microelectromechanical systems, are of interest for design uroses. Satellite tethers are subjected to large variations in temerature during their life cycle. Microelectromechanical resonators are fabricated as clamed clamed comosite beams, and they sometimes buckle during the manufacturing rocess. These are often used as filters and are imortant for mobile communication systems and signal rocessing alications. Also, the temerature rise caused by the heat roduced by the electric current may be enough to induce buckling or change the buckling mode as discussed below. The analysis of ostbuckling behavior under thermal loads of railroad tracks, concrete avements, and ielines either buried underground or resting on ground is needed to fully comrehend their failure. These structural elements are exosed to different environmental conditions and hence to different thermal cycles. Here we consider a roblem in the second category, and simlify it considerably by studying the buckling and ostbuckling resonse of a uniformly heated Euler Bernoulli beam resting on an elastic foundation. Furthermore, the elastic foundation is assumed to exert distributed forces on the beam; these forces may vary either linearly or nonlinearly with both the axial and the transverse dislacements of a oint. Even such a simle model involves nonlinear governing equations and illuminates interesting transitions among buckling modes. The mathematical model alies to a beam/rod wraed in a thick rubberlike sleeve with the action of the sleeve relaced by distributed forces on the beam, and to a microelectromechanical resonator with the action of the electrode relaced by a distributed force on the substrate. Emam [1] has reviewed the literature on vibrations of ostbuckled Euler Bernoulli beams, and has comared analytically redicted interactions among different modes with the exerimental findings; the reader is referred to Emam s dissertation for historical

2 LI AND BATRA 713 develoments in the field. Lestari and Hanagud [2] have studied the nonlinear vibrations of buckled beams with elastic end constraints, and the beam subjected simultaneously to axial and lateral loads. Other works [3 5] have analyzed either analytically or numerically nonlinear ordinary differential equations associated with the ostbuckling behavior of initially straight rods subjected to comressive loads and different boundary conditions. Kreider and Nayfeh [6] have studied exerimentally and theoretically the nonlinear vibrations of a clamed clamed buckled beam. Phungaingam et al. [7] have investigated the ostbuckling resonse of a simle beam under an intermediate follower force acting along the tangent to the deformed axis of the beam. They found that the analytical solution obtained by solving the ellitic integral equations matched well with the numerical solution comuted with the shooting method. Plaut [8] studied the ostbuckling deformations and vibrations of endsuorted elastica ies carrying fluid and also subjected to follower loads. He emloyed the shooting method to solve the nonlinear boundary-value roblem for the equilibrium configuration, and the linear boundary-value roblem for frequencies of the first four modes of vibration. Postbuckling deformations of erfect and geometrically imerfect elastic columns resting on an elastic Winkler foundation have been analyzed by Kounadis et al. [9]. They found that the critical state of a erfect column is a stable symmetric bifurcation oint. Abu-Salih and Elata [1] assumed the beam to be infinite, erfectly bonded to a linear elastic foundation, and subjected to internal comressive stresses. They incororated extensional deformations into the roblem formulation, and found that the wave length of a ostbuckled beam is unaffected by the magnitude of the internal stress. Li and Balachandran [11] have analyzed the buckling and free vibrations of comosite microelectromechanical resonators modeled as a steed comosite Euler Bernoulli beam. They accounted for axial deformations of the beam. It aears that ostbuckling deformations of a thermally loaded Euler Bernoulli beam resting on an elastic foundation have not been analyzed thus far. A difference between the buckling of a beam due to the alied axial force and that due to temerature rise is that in the latter case axial deformations may not be negligible and ought to be considered. Jekot [12] has studied deformations of a thermally ostbuckled beam made of a nonlinear thermoelastic material. However, the geometric nonlinearity introduced by the curvature of the centroidal axis of the deformed beam was not considered, and a simle exression for the axial strain was emloyed. Raju and Rao [13 18] have used the Rayleigh Ritz and the finite element methods to analyze the thermal ostbuckling of uniform and taered columns suorted on elastic foundations. They incororated the geometric nonlinearity similar to that in the von Karman late theory, but ignored the nonlinearity introduced by the curvature of the deformed centroidal axis. For a inned inned beam, Coffin and Bloom [19] accounted for the curvature of the beam deformed due to heating, or the absortion of water, and formulated the roblem for the ostbuckled beam in terms of two couled ellitic integral equations that were solved numerically. For a nonlinear thermal strain-temerature relation, Vaz and Solano [2,21] investigated ostbuckling deformations of inned inned rods and gave a closed-form solution via uncouled ellitic integrals. However, their analysis is valid only for inned inned boundary conditions. Based on the geometric nonlinear theory of extensible beams and with temerature increasing uniformly throughout the beam, Li and Cheng [22] and Li et al. [23] have develoed mathematical models of ostbuckled Euler Bernoulli beams with inned inned, fixed fixed, and inned fixed boundary conditions. The resulting nonlinear boundary-value roblems were solved numerically with the shooting method in conjunction with concets of analytical continuation. The work has been extended [24] to bending and buckling of nonlinear Timoshenko beams subjected to thermomechanical loads. Wu and Zhong [25] studied ostbuckling and imerfection sensitivity of axially comressed beams with clamed clamed and fixed free ends resting on an elastic Winkler foundation. They [25] found that for a linear Winkler foundation and a large range of values of the foundation stiffness arameter, a measure of the beam deformation increases with a decrease in the load during its ostbuckling deformations, and concluded that the ostbuckled aths are unstable. Here we consider geometric nonlinearities, and develo a mathematical model of thermal buckling and ostbuckling of an elastic Euler Bernoulli beam resting on a nonlinear elastic foundation. The load alied by the elastic foundation is assumed to deend uon two comonents of dislacement of a oint on the centroidal axis of the beam. For a satellite tether, the force exerted by the environment could conceivably be modeled by a similar relation. For a microelectromechanical resonator, the force exerted by the iezoelectric layer on the substrate can be aroximated in a similar way. Because deformations of a hygroscoic body due to moisture absortion are similar to those of a thermoelastic body due to temerature rise, the analysis also alies to beams immersed in water with the action of water on the beam aroximated by distributed forces deending uon the dislacements of a oint on the centroidal axis of the beam. The nonlinear boundary-value roblem is numerically solved with the shooting method for ostbuckled configurations of beams and their equilibrium aths. Comuted results are comared with those available in the literature. We delineate effects of arameters of the elastic foundation on the beam s deformations, and give values of the foundation stiffness at which the transition in buckling modes occurs. II. Mathematical Model We consider a uniform elastic beam of initial length l resting on a nonlinear elastic foundation and having its ends constrained from moving axially; a schematic sketch of the roblem studied is shown in Fig. 1. An axially distributed mechanical load q q x ;q y and a uniform slow T deform the beam from its natural state. Assume that E and are indeendent of T, and a lane section initially erendicular to the centroidal axis of the beam remains lane and becomes erendicular to the centroidal axis of the deformed beam. We denote a oint of the centroidal axis of the undeformed beam as C: x; y with x 2 ;l, y, in which x and y are coordinates of a oint in a rectangular Cartesian coordinate system. When the beam is deformed, as shown in Fig. 2, the material oint C moves to the oint C : X; Y x u x ;w x, in which u x and w x are dislacements of the oint C in the x and the y directions, resectively. Here we have resumed that the centroidal axis of the deformed beam is in the xy lane. Thus, where ds du ; dx. y dw cos 1; dx 1 du=dx 2 dw=dx 2 x Beam Elastic foundation sin (1) dx q y q x Fig. 1 (To) A schematic sketch of the roblem studied, and (bottom) a reresentation of forces exerted by the elastic foundation on the beam. (2)

3 714 LI AND BATRA yy, C θ Substituting for N from Eq. (9) into Eq. (8a), and solving the resulting equation for we get Hcos V sin =EA T 1 (1) o Fig. 2 x X C u Y = w l x, X A buckled configuration of a inned inned beam. y o Fig. 3 H M V q x q y θ M + dm d s dx = dx+ du V + dv dy = dw H + dh x Free-body diagram of a differential element. Equation (8b), and equations obtained by substituting from Eqs. (6) and (1) into Eqs. (1) and (3), form seven couled nonlinear differential equations for the seven unknown functions s x, u x, w x, x, H x, V x, and M x defined on the interval ;l. By introducing the following nondimensional quantities ; S; U; W x; s; u; w =l; l A=I 1=2 ; k 1 ;k 2 l 4 =EI (11a) 2 T; P H ;P V l 2 H; V =EI; m lm=ei (11b) and substituting them into Eqs. (1), (3), (8), and (1), we arrive at the following equations in terms of nondimensional variables: ds (12a) A free-body diagram of a differential element of the beam is exhibited in Fig. 3. The equilibrium of forces and of moments acting on a segment, ds, of the deformed beam yield dh dx q x; dv dx q y; dm H sin V cos dx (3) For a beam resting on a nonlinear elastic foundation, the force exerted by the foundation on the beam ooses dislacements of the beam s centroidal axis, and can be exressed as du dw cos 1 (12b) sin (12c) m (13a) q k 1 k 2 2 (4) in which fu; wg T is the dislacement vector of a oint of the centroidal axis, and u cos ; w sin (5) Because u 2 w 2, therefore q x k 1 u k 2 u u 2 w 2 ; q y k 1 w k 2 w u 2 w 2 (6) When the resistance offered by the elastic foundation due to the axial dislacement u can be neglected, then Eq. (6) is simlified to q x ; q y k 1 w k 2 w 3 (7) For an isotroic and homogeneous elastic beam resting on an isotroic and homogeneous elastic foundation, assumtion (4) for the force exerted by the foundation on the beam is reasonable. One could, in rincile, consider different values of stiffness arameters k 1 and k 2 for forces in the x and the y directions, resectively. Constitutive relations for the thermoelastic beam are taken to be N EA 1 T (8a) M EI (8b) dx in which EA and EI are, resectively, the same as the extensional and the bending rigidities of a linear elastic beam. Furthermore, T is assumed to be uniform throughout the beam, and N is erendicular to the cross section. We note that Eqs. (8a) and (8b) are not linear in dislacement gradients. N is related to H and V by N H cos V sin (9) in which dm P H sin P V cos (13b) dp H U U 2 W 2 (14a) dp V W U 2 W 2 (14b) P H cos P V sin = 2 1 (15) is derived from Eqs. (1) and (11b). In terms of nondimensional variables, boundary conditions for inned inned and fixed fixed beams are given below. inned inned: S ; U ; W m (16a) U 1 ; W 1 ; m 1 (16b) fixed fixed: S ; U ; W (17a) U 1 ; W 1 ; 1 (17b)

4 LI AND BATRA 715 Note that in each case S imlies that the length S along the centroidal axis is measured from the left end of the beam. In addition to boundary conditions, the following conditions are imosed to secify a buckled configuration of the beam. inned inned: (18) fixed fixed: m m (19) Thus, for a secified nonvanishing value of or m, we can solve Eqs. (12 19) for S; U; W; ; P H ;P V ;m and the accomanying nondimensional temerature rise. If one is not interested in finding the length of the deformed centroidal axis of the beam, then one can ignore Eq. (12a). III. Solution of the Problem A. Linear Problem for Buckled Shaes and Buckling Mode Transitions The onset of buckling of a uniformly heated beam resting on an elastic foundation is governed by the following linear equation obtained from Eqs. (12 14) by setting 1, sin W, cos 1, and P H, and neglecting nonlinear terms in the unknown functions: W W W (2) Here a rime denotes differentiation with resect to. Equation (2) is the same as that of a beam subjected to an axial force [9,25,26] at the ends. The corresonding boundary conditions are inned inned: W ; W ; at and 1 (21) fixed fixed: W ; W ; at and 1 (22) Assuming that >2, a general solution of Eq. (2) is [9,25] in which W C 1 sin C 2 cos C 1 sin C 2 cos (23) 1=2 =2 =2 2 =2 (24) 1=2 = Pinned Pinned Beams For the inned inned beam, assuming that either C 1, or C 1, and recalling that >, Eq. (23) satisfies boundary conditions (21) rovided that whose solutions are sin sin (25) either m ; or n m; n 1; 2; 3;... (26) Temeratures for different buckling modes, subsequently also called critical temeratures, are given by m m 2 2 m2 2 ; or n n 2 2 n2 2 (27) The condition for the buckling mode transition is m n m n (28) which gives following values of the elastic foundation arameter mn m 2 n 2 4 m n (29) and corresonding values of the load arameter are mn m 2 n 2 2 m n (3) For m 1; 2;...; 6, Fig. 4 shows m = 2 versus = 4 ; each curve is a straight line whose intercet with the vertical axis gives the corresonding temerature at the onset of buckling for. Thus the effect of the linear elastic foundation is to increase the temerature rise required for the onset of buckling. For a fixed buckling mode (i.e., a fixed value of m) the critical temerature deends linearly uon the elastic foundation arameter. The oint of intersection, A mn, of two of these curves gives the value of at which the buckling mode could transition from one to the other. Thus for the linear stiffness arameter and the critical temerature corresonding to the oint mn ; mn fm 2 m 1 4 ; m 2 m g, there can be a transition in the buckling mode. Values of the elastic foundation arameter corresonding to the first three transitions in the buckling modes are 4 4, 64 4, and 144 4, which are the same as those given in Wu and Zhong [25], and Hetenyi [27]. When the temerature of the bar is uniformly raised for a fixed value of, the deformed shaes will follow the ath indicated by the solid curve in Fig. 4 because it requires a temerature lower than that needed to stay on the original dotted straight line. If we assume that C 1 and C 1, then we obtain the following buckled mode shaes: W C 1 sin m m 1; 2; 3;... (31) 5 /π 2 4 A 26 A 45 Buckling temerature A 23 A 34 m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 Fig. 4 A 12 inned-inned beam Critical temerature Elastic foundation arameter /π 4 Deendence of the temerature rise at the onset of buckling uon the foundation stiffness arameter.

5 716 LI AND BATRA The constant C 1 is indeendent of imlying that buckled mode shaes of a inned inned beam do not deend on the elastic foundation arameter. From Fig. 4, we conclude that the critical buckling mode changes from symmetric (i.e., m 1; 3;...) to antisymmetric (i.e., m 2; 4;...) and vice versa in assing through the oints A k k 1 (k 1; 2; 3...) and A 2k 2k 1 (k 1; 2; 3...), resectively. The order of the critical buckling mode increases with an increase in the value of = 1 = 4 = 6 = Fixed Fixed Beams For a fixed fixed beam, we require that the general solution (23) satisfy boundary conditions (22). Recalling that >2,we arrive at the characteristic equation 1 cos cos 2 sin sin (32) which cannot be solved analytically for the load arameter as a function of the stiffness arameter because and deend uon and. We thus emloy the Newton iteration method to find its roots. Figure 5 deicts the lot of the critical temerature as a function of the linear stiffness arameter of the foundation. Points A m1 (m, 1, 2, 3) with coordinates m m 2 m ; m m 2 m (33) corresond to the buckling mode transitions. As for a inned inned beam, for a given value of the linear stiffness of the foundation, they give the lowest temerature at which buckling will ensue. Thus the three lowest values of at which buckling mode transitions occur are 9 4, 64 4, and The critical temerature versus curve and the two lowest values of corresonding to the buckling mode transition coincide with those given in Wu and Zhong [25] for an axially comressed beam. Substitution from Eq. (33) into Eq. (32) gives m m 2 ; m m (34) For different values of the linear elastic foundation stiffness arameter we have lotted in Fig. 6 the deformed shae of the beam at the initiation of buckling in one of the first three buckling modes. As also ointed out by Wu and Zhong [25], buckling modes of a fixed fixed beam deend uon the arameter, and with an increase in the value of, the buckling mode changes from symmetric to antisymmetric, back to symmetric, and so on. However, buckling modes of the inned inned beam do not deend on the value of the linear elastic foundation stiffness arameter. /π 2 Buckling temerature fixed-fixed beam A 11 : (9π 4,1π 2 ) A 1 : (, 4π 2 ) A 21 : (64π 4,2π 2 ) A 31 : (225π 4,34π 2 ) Elastic foundation arameter /π 4 Fig. 5 Deendence of the temerature rise at the onset of buckling uon the linear foundation stiffness arameter.. a) = 1, 2, 4, 6, = 1 = 3 = 4 = 5 = b) = 1, 2,, = 8 = 1 = 12 = 14 = 16 = c) = 8, 1, 12,, 2 Fig. 6 For secified values of ( ), deformed shaes of a fixed fixed beam at the onset of buckling. B. Nonlinear Problem for Postbuckled Shaes It is difficult to solve analytically the nonlinear couled boundaryvalue roblem defined by Eqs. (12 19), therefore we find its aroximate solution numerically by the shooting method that relaces the two-oint boundary-value roblem by a sequence of initial-value roblems. That is, values of unknown functions at the initial oint and unknown arameters are estimated to start comutations [26], and these estimates are modified until secified boundary conditions at the terminal oint are satisfied; the transformation of the boundary-value roblem into an initial-value roblem for use in the shooting method is described in the Aendix. For a inned inned beam, is secified, and for a fixed fixed beam m is secified. Then P H, P V, and are varied until Eq. (16b) or Eq. (17b) is satisfied. The Runge Kutta method is used to integrate Eqs. (12 19) and the Newton Rahson iteration method is used to comute the varying arameters. By gradually increasing

6 LI AND BATRA 717 the arameter (or m ), finding the corresonding equilibrium configuration, and using the continuation method, we determine ostbuckled configurations of the beam as a continuous function of the temerature rise. By assigning the arameter (or m ) a very small value in the comutation, we can obtain the critical buckling load and the corresonding mode shae. In comuting numerical results resented below we have set 1. Figure 7 exhibits the ostbuckled configurations obtained by starting from the first (symmetric) and the second (antisymmetric) buckled modes of the inned inned beam for different values of the nondimensional temerature rise. For each case the deflection of a oint continues to increase monotonically with an increase in the temerature. For a fixed fixed beam, Fig. 8 deicts the first three ostbuckled configurations comuted by starting from the buckling modes of Fig. 6. For a rescribed value of between 4 4 and 9 4 that corresond to transition in the buckling modes, equilibrium aths for the two beams in terms of lots of f W :5 versus are illustrated in Fig. 9. They are qualitatively similar in the sense that for a fixed value of f, an increase in requires a higher value of the temerature rise. To exlore the influence of the nonlinear foundation stiffness arameter on the ostbuckling resonse, we set 2. Figure 1 shows the variation of the central deflection arameter f W :5 with the temerature rise for five values of the stiffness arameter. It is clear from these lots that effects of the nonlinear foundation stiffness arameter are noticeable only for large ostbuckling deformations of the beam. For the fixed fixed beam and for different values of the stiffness arameter, the variation of the bending moment m 1 with the temerature rise is exhibited in Fig. 11. It shows that the end moment m 1 decreases with an increase in. In contrast to the symmetric buckling of beams without an elastic foundation [22], the end transverse force P V 1 will be roduced when the buckled beam is.12 +U() a) = = = = = = = 6, - b) = 2, = = = = = = 3, = = = = = = = =1 4, = = = a) = U() = 6, - +U() = = = = = b) = 6 Fig. 7 For secified values of and, ostbuckled configurations of a inned inned beam for two values of. - c) = 1 4 Fig. 8 For secified values of the temerature rise, and, ostbuckled configurations of a fixed fixed beam for three values of. suorted on an elastic foundation. The variations of the transverse force P V 1 with temerature for different values of are resented in Fig. 12 for both inned inned and fixed fixed beams. For large values of (>6 for the inned inned beam, and >125 for the fixed fixed beam), the end transverse force P V 1 increases with an increase in the foundation stiffness arameter. IV. Remarks A challenging task is to determine exerimentally the stiffness of an elastic foundation. One could use the following inverse method to do so. By comaring the deformed shae of the beam for different values of temerature rise with those comuted by the resent method, one can find the stiffness of the elastic foundation. The challenging task then is to ensure that this value of stiffness will result in the correct temeratures at which buckling mode shaes transition

7 718 LI AND BATRA = 2 f = a) Pinned-inned beam, = 2 = 1 = 3 = 4 = 5 = 1. = 1 = f f a) Pinned-inned beam = 2 = 1 = 2 = 4 = 6 f = 1 = 5 = 1 = 15 = b) Fixed-fixed beam, K2 Fig. 9 Variation of the dimensionless deflection f W :5 with for mode 1 ostbuckling deformations of beam. from one to another. Li et al. [28] used a similar rocedure to identify the buckled shae of a microresonator. For the resent roblem, results comuted by taking 1 in Eq. (8b) differ from those resented here by less than.1%, suggesting thereby that the consideration of the stretching of the midsurface of the beam in Eq. (8b) does not have much influence on the ostbuckled shaes comuted here. However, for a microelectromechanical beam, Batra et al. [29] used the von Karman aroximation to consider stretching of the beam s midsurface and found that it significantly affects the ull-in voltage. Tiersten [3] has recently revisited the Euler Bernoulli beam theory and delineated various aroximations made to simlify the roblem. Several investigators [5,9,31] have set 1 in Eq. (8b) while studying ostbuckling deformations of beams. V. Conclusions We have studied the buckling and the ostbuckling deformations of uniformly heated inned inned and fixed fixed Euler Bernoulli beams suorted on nonlinear elastic foundations. The geometric nonlinearity introduced by the curvature of the deformed beam, and the constraint force of the elastic foundation in both longitudinal and transverse directions, are incororated in the roblem formulation. Buckling modes and transitions among them are comuted by solving analytically the linear roblem. The deendence of the initiation of the first few buckling modes uon the linear foundation stiffness arameter is lotted from which values of the Winkler foundation arameter corresonding to the transition in the buckling mode are determined. The nonlinear boundary-value roblem for ostbuckling deformations is solved by the shooting method by first transforming it to an initial-value roblem. For different values of the elastic foundation arameter, equilibrium aths and configurations derived from the first buckling mode are illustrated. As exected, the nonlinear foundation stiffness arameter does not influence the b) Fixed-fixed beam Fig. 1 Dimensionless deflection f W :5 vs for beams deformed beyond the critical buckling temerature. buckling temerature, and has a small effect on the ostbuckling deformations as comared with the effect of the linear foundation stiffness arameter. The roblem studied here is directly related to the comressed elastic column. The analysis resented here alies to buckling and ostbuckling deformations of a hygroscoic beam suorted on an elastic foundation, and exosed to moisture. Aendix: Shooting Method for the Nonlinear Problem We briefly describe the rocedure used to numerically solve the nonlinear two-oint boundary-value roblem defined by Eqs. (12 19). We write it as m(1) dy H ; Y ; < <1 (A1a) = 1 = 3 = Fig. 11 Deendence of the bending moment m 1 uon for beams deformed beyond the critical buckling temerature.

8 LI AND BATRA = 5 = 1 = 3 = 2 We consider the following initial-value roblem corresonding to the boundary-value roblem (A1): dz H ; Z ; > (A2a) P V (1) in which a) Pinned-inned beam P V (1) = 4 = 6 = b) Fixed-fixed beam Fig. 12 Deendence of the P V 1 uon the nondimensional temerature rise for beams deformed beyond the critical buckling mode. B Y b B 1 Y 1 b 1 Y fy 1 ;y 2 ;y 3 ;y 4 ;y 5 ;y 6 ;y 7 ;y 8 g T fs; U; W; ; m; P H ;P V ; g T (A1b) (A1c) H f ; cos y 4 1; sin y 4 ; y 7 = ;F 1 ;F 2 ;F 3 ; g T h i F 1 y 5 sin y 4 y 6 cos y 4 ; F 2 y 2 y 2 2 y2 3 h i F 3 y 3 y 2 2 y2 3 y 5 cos y 4 y 6 sin y 4 y 8 = 2 1; b 1 f; ; g T B b f; ; ; ; g T 1 2 B for inned-inned beam 1 2 b f; ; ; ;m g T 3 1 B for fixed-fixed beam 1 Z I D (A2b) in which Z fz 1 ;z 2 ; :::; z 8 g T, and D fd 1 ;d 2 ;d 3 g T is an unknown vector related to initial values of functions z 6, z 7, and z 8 ; I D f; ; ; ;m ;d 1 ;d 2 ;d 3 g T with m for a inned inned beam, and for a fixed fixed beam. A solution of the initial-value roblem (A2) may be symbolically written as Z ; ;m ; D I D Z H ; Z (A3) For a given nonzero value of,orm (one of them must be zero), we find that solution of (A3) which satisfies boundary condition (A1c), that is, B 1 Z 1; ;m ; D b 1 (A4) Obviously, if D D is a root of Eq. (A4), then a solution of the boundary value roblem (A1) is Y Z ; ;m ; D (A5) We emloy the Runge Kutta method to integrate the system (A2) of ordinary differential equations, and the Newton Rahson iteration method to search for a root D of Eq. (A5) to find a numerical solution of the boundary-value roblem (A1). Acknowledgments S. R. L. gratefully acknowledges the artial suort received for this work from the Natural Science Foundation of China (147239) and the China Scholarshi Council, and R. C. B. from the Office of Naval Research (Grants No. N and No. N ). Oinions exressed in this aer are those of the authors and not of funding agencies. The authors are deely indebted to an anonymous reviewer for constructive suggestions that heled imrove the work. References [1] Emam, S. A., A Theoretical and Exerimental Study of Nonlinear Dynamics of Buckled Beams, Ph.D. Dissertation, Virginia Polytechnic Institute and State University, Blacksburg, VA, 22. [2] Lestari, W., and Hanagud, S., Nonlinear Vibrations of Buckled Beams: Some Exact Solutions, International Journal of Non-Linear Mechanics, Vol. 38, No. 26, 21, [3] Stemle, T., Extensional Beam Columns: An Exact Theory, International Journal of Non-Linear Mechanics, Vol. 25, No. 6, 199, [4] Wang, C. Y., Postbuckling of a Clamed-Simly Suorted Elastica, International Journal of Non-Linear Mechanics, Vol. 32, No. 6, 1997, [5] Filiich, C. P., and Rosales, M. B., A Further Study on the Postbuckling of Extensible Elastic Rods, International Journal of Non-Linear Mechanics, Vol. 35, No. 6, 2, [6] Kreider, W., and Nayfeh, A. H., Exerimental Investigation of Single- Mode Resonses in a Fixed Fixed Buckled Beam, Nonlinear Dynamics, Vol. 15, No. 3, 1998, [7] Phungaingam, B., Chucheesakul, S., and Wang, C. M., Postbuckling of Beam Subjected to Intermediate Follower Force, Journal of Engineering Mechanics, Vol. 132, No. 1, 26, [8] Plaut, R. H., Postbuckling and Vibration of End-Suorted Elastica Pies Conveying Fluid and Columns Under Follower Loads, Journal of Sound and Vibration, Vol. 289, No. 1 2, 26, [9] Kounadis, A. N., Mallis, J., and Sbarounis, A., Postbuckling Analysis of Columns Resting on an Elastic Foundation, Archive of Alied Mechanics, Vol. 75, No. 6 7, 26, [1] Abu-Salih, S., and Elata, D., Analytical Postbuckling Solution of a Prestressed Infinite Beam Bonded to a Linear Elastic Foundation,

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