The present work aims at the determination of thermal buckling loads of various functionally graded material beams with both ends clamped. Thermal loading is applied by applying linear temperature distribution and nonlinear temperature distribution at steady state heat conduction condition, across the beam thickness. Temperature dependences of the material properties, considered in the formulation, make the present problem physically nonlinear. Also, the effect of limit thermal load at which the effective elastic modulus and/or thermal expansion coefficient become theoretically zero is considered. The mathematical formulation is based on Euler-Bernoulli beam theory. An energy based variational principle is employed to derive the governing equations as an eigenvalue problem. The solution of the governing equation is obtained using an iterative method. The validation of the present work is carried out with the available results in the literature and with the results generated by finite element software ANSYS. Four different functionally materials are considered, namely, stainless steel/silicon nitride, stainless steel/alumina, stainless steel/zirconia, and titanium alloy/zirconia. Comparative results are presented to show the effects of variations of volume fraction index, length-thickness ratio, and material constituents on nondimensional thermal buckling loads.
Nonlinear free vibration analysis of clamped isotropic skew plates has been presented. The large amplitude of vibration is imparted statically by subjecting the plate under uniform transverse pressure. The mathematical formulation is based on the variational principle in which the displacement fields are assumed as a combination of orthogonal polynomial or transcendental functions, each satisfying the corresponding boundary conditions of the plate. The large amplitude dynamic problem is addressed by solving the corresponding static problem first and subsequently with the resultant static displacement field, the dynamic problem is formulated. The vibration frequencies are obtained from the solution of a standard eigenvalue problem. Entire computational work is carried out in a normalized square domain obtained through an appropriate domain mapping technique. Results of the reduced problem revealed excellent agreement with other studies and a typical comparison of the actual problem is also carried out successfully. Results are furnished in dimensionless amplitude-frequency plane, in the form of backbone curves, and pictorial representations of some vibration mode shapes are made.
A new formulation is introduced to study the free vibration behavior of a statically loaded beam with geometric nonlinearity. The tangent stiffness of the statically loaded beam is used to investigate the free vibration behavior of the beam about its loaded configuration. The problem is formulated for a linearly tapered beam, and a uniform beam is obtained as a special case. Energy principles based on the variational approach are used to derive the governing equations for the static and dynamic problems. The Ritz method of approximate displacement field is followed to solve the governing equations. The Ritz coefficients are used to derive the tangent stiffness of the loaded beam. Components of the tangent stiffness matrix are derived for a Timoshenko beam with von Kármán-type nonlinearity. Illustrative results are presented for four different classical boundary conditions having in-plane restraint. Results for the first two modes of transverse vibration are presented in the nondimensional deflection-frequency plane. Validation of the work is carried out using finite element software ANSYS. The formulation is new of its kind and can be used for any displacement-based problem following the Ritz method.
Large amplitude forced vibration behavior of thin beams under harmonic excitation is studied, incorporating the effect of geometric nonlinearity. Large amplitude free vibration analysis of the same system is also carried out to obtain the backbone curves. When a very small loading is considered, the system response becomes almost identical to the backbone curve, which confirms the validity of the large amplitude study. The sets of governing equations for both forced and free vibration problems are obtained using Hamilton’s principle, whereas the same for the static problem are obtained by applying the variational form of the energy principle. The solution of the static problem provides system stiffness as a function of deflection and it becomes an input to the corresponding free vibration problem. The results are presented in the nondimensional frequency–amplitude plane in the form of response curves, and the related backbone curves are shown to represent the complete system behavior. Results for different combinations of classical flexural boundary conditions and loading patterns are reported.
In the present work, the non-linear post-buckling load-deflection behavior of tapered functionally graded material beam is studied for different in-plane thermal loadings. Two different thermal loadings are considered. The first one is due to the uniform temperature rise and the second one is due to the steady-state heat conduction across the beam thickness leading to non-uniform temperature rise. The governing equations are derived using the principle of minimum total potential energy employing Timoshenko beam theory. The solution is obtained by approximating the displacement fields following Ritz method. Geometric non-linearity for large post-buckling behavior is considered using von Kármán type non-linear strain-displacement relationship. Stainless steel/silicon nitride functionally graded material beam is considered with temperature-dependent material properties. The validation of the present work is successfully performed using finite element software ANSYS and using the available result in the literature. The post-buckling load-deflection behavior in non-dimensional plane is presented for different taperness parameters and also for different volume fraction indices. Normalized transverse deflection fields are presented showing the shift of the point of maximum deflection for various deflection levels. The results are new of its kind and establish benchmark for studying non-linear thermo-mechanical behavior of tapered functionally graded material beam.
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