Optimal shape of a column with clamped-elastically supported ends positioned on elastic foundation

We determine optimal shape of an elastic column positioned on elastic foundation of Winkler type. The Euler-Bernoulli model of beam is considered. The column is loaded by a compressive force and has one clamped end and the other elastically supported end. In deriving the optimality conditions, the Pontryagin?s principle was used. The optimality conditions for the case of bimodal optimization are derived. Optimal cross-sectional area is obtained from the solution of a non-linear boundary value problem. A first integral (Hamiltonian) is used to monitor accuracy of integration. This system is solved by using standard Math CAD procedure. New numerical results are obtained.

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

By using the Pontryagin's 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.

Bimodal optimization with constraints: Critical value of the constraint and post-critical configurations

By using a method based on Pontryagin?s principle, formulated in [13], and [14] we study optimal shape of an elastic column with constraints on the minimal value of the cross-sectional area. We determine the critical value of the minimal cross-sectional area separating bi from unimodal optimization. Also we study the post-critical shape of optimally shaped rod and find the preferred configuration of the bifurcating solutions from the point of view of minimal total energy.

Optimal Shape of a Rotating Rod With Unsymmetrical Boundary Conditions

Governing equations of a compressed rotating rod with clamped–elastically clamped (hinged with a torsional spring) boundary conditions is derived. It is shown that the multiplicity of an eigenvalue of this system can be at most equal to two. The optimality conditions, via Pontryagin’s maximum principle, are derived in the case of bimodal optimization. When these conditions are used the problem of determining the optimal cross-sectional area function is reduced to the solution of a nonlinear boundary value problem. The problem treated here generalizes our earlier results presented in Atanackovic, 1997, Stability Theory of Elastic Rods, World Scientific, River Edge, NJ. The optimal shape of a rod is determined by numerical integration for several values of parameters.

Bimodal Optimization of Compressed Columns on Elastic Foundations

We consider columns attached to elastic foundations and compressed by axial end loads. Pinned-pinned, clamped-clamped, and pinned-clamped boundary conditions are treated. The columns have rectangular sandwich cross sections with a fixed lightweight core and identical face sheets of variable thickness. For given total volume, we optimize the variation of the thickness along the column so as to maximize the buckling load. In most cases, the optimal design is bimodal (i.e., associated with two buckling modes). The optimal designs depend on the foundation stiffness, and the largest increase in buckling load relative to a column with constant thickness is 22 percent.