Global optimization of 2D frames with variable cross-section beams

Abstract

 The structural design has been one of the first fields of engineering 
in needing powerful tools for analysis. The methods to asses that the 
structure agrees with any of the design criteria (strength, stability, 
vibrations, etc. ) are usually implemented in numerical applications 
that, even under usual simplifications such as constant cross-section 
or linearization, are computationally very demanding. Nevertheless, 
with the current capacities both of analysis and of manufacturing and 
the use of new materials, it is possible to propose problems like the 
ones shown in this work, where the variation of the dimensions of 
the cross-section of the beams of any 2D frame is determined so that 
its strength to buckle is maximum. Classic solutions exist for isolated 
beams, in some cases even analytical solutions. But for structures 
built of several beams, the problem is more complex and numerical 
solutions are compulsory. The new formulation presented in this 
work can deal with the optimization problem of frames in which the 
geometrical shape of the profi le is found under restrictions such as 
stability (no buckling), elastic behaviour (equivalent von Mises stress 
below the yield stress), limited displacements, etc. With these aim, 
equilibrium equations for each beam are established in its deformed 
configuration, and using the hypothesis of small displacements and 
small deformations (Second Order Theory) a system of differential 
equations of variable coefficients is set. For its solution, numerical 
techniques as quadratic sequential programming are employed.