Branislava N. Novakovic

APPLICATION OF PONTRYAGIN'S PRINCIPLE TO BIMODAL OPTIMIZATION OF NANO RODS

By using the Pontryagins maximum principle, we determine optimal shape of a nonlocal elastic rod clamped at both ends. In the optimization procedure, we imposed restriction on the minimal value of the cross-sectional area. We showed that the optimization may be both unimodal and bimodal depending on the value of the restrictions and the value of characteristic length. Several concrete examples are treated in detail and the increase in buckling capacity is determined.

Rotating Nanorod with Clamped Ends

This paper presents an analytical investigation on the buckling and post-buckling behavior of rotating nanorods subjected to axial compression and clamped at both ends. The nonlinear governing equations are derived based on the classical Euler–Bernoulli theory and Eringens nonlocal elasticity model. The critical load parameters such as angular velocity and compressive axial force are determined for given values of nonlocality parameter. The validity, convergence and accuracy of the solutions are established by comparing them with known classical solutions. The numerical results show that an increase in the nonlocality parameter gives rise to an increase in post-buckling deformation.

OPTIMAL SHAPE OF A HEAVY ELASTIC ROD LOADED WITH A TIP-CONCENTRATED FORCE AGAINST LATERAL BUCKLING

By using Pontryagins maximum principle, we determine the optimal shape of an elastic rod free at one and clamped at the other. The rod is loaded with a concentrated force at the free end and its own weight. The optimality criterion is the volume of the rod guaranteeing lateral stability.

Dynamic Stability of Axially Loaded Nonlocal Rod on Generalized Pasternak Foundation

AbstractIn this paper, the problem of determining the dynamic stability boundary (critical value of the axial force) of an axially loaded nonlocal rod of Eringen’s type is considered. The rod is positioned on a viscoelastic foundation of the Pasternak type. Constitutive equations containing fractional derivatives of real and complex order are used to model the viscoelasticity of the foundation. The influence of various model parameters on the value of critical axial load is examined.

On Post-Critical Behavior of a Beam on an Elastic Foundation

In this paper, we analyze the nonlinear equilibrium equation corresponding to the two-parameter bifurcation problem arising in the stability analysis of an elastic simply supported beam on the Winkler type elastic foundation for the case when bimodal buckling occurs. We perform the bifurcation analysis of the nonlinear problem, by using Lyapunov–Schmidt reduction, thus obtaining the number of the nontrivial solutions to the nonlinear problem and qualitatively characterizing the solution patterns. We also give the formulation of the problem and bifurcation analysis from the total energy viewpoint and determine the energy of each bifurcating solution. We assert that the solution with the smallest energy is the one that will be observed in the post-critical state. For specific choice of parameters, the bifurcating solution in the form of the second buckling mode has the smallest total energy. The numerical results illustrating the theory are also provided.