Thermal Buckling and Postbuckling of Euler-Bernoulli Beams Supported on Nonlinear Elastic Foundations

Deformations of homogeneous and isotropic pinned–pinned and fixed–fixed Euler–Bernoulli beams supported on nonlinear elastic foundations and heated uniformly into the postbuckling regime have been analyzed numerically. Geometric nonlinearities introduced by finite deflections and curvature of the deformed beams are incorporated in the problem formulation. First buckling due to the uniform temperature rise and buckling mode transitions are investigated analytically by analyzing the linear problem. Subsequently, the nonlinear boundary-value problems for postbuckling of beams are transformed into initial-value problems and analyzed by the shooting method. For different values of the elastic foundation parameters, postbuckled 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 K1, K2 = dimensionless values of k1, k2 K1mn = stiffness of elastic foundation at which buckling mode transitions k1, k2 = linear and cubic stiffness parameters of the elastic foundation l = initial length of the beam M = bending moment m = dimensionless value ofM m0 = dimensionless bending moment at a fixed end N = resultant axial force PH, PV = dimensionless values of H and V qx, qy = distributed mechanical loads on the beam S = dimensionless value of s s = arc length of the deformed centroidal axis T = temperature rise U = dimensionless value of u u = displacement in x direction V = resultant vertical force W = dimensionless value of w w = displacement in y direction or beam deflection x, y = rectangular Cartesian coordinate axes = coefficient of thermal expansion = angle between the deformed beam’s axis and the x axis 0 = rotational angle of the pinned end of the beam = curvature of the deformed beam axis = stretch of the centroidal axis = slenderness ratio of the beam = dimensionless value of x = dimensionless value of T I.