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This paper is concerned with the analysis of framed structures with inextensible and rigid members, i.e., members without axial and bending strains. Rigid and inextensible members may be useful in educational software because they capture the essence of the structural behavior with a reduced the number of variables. In addition, they allow a comparison with results obtained by hand calculation using classical structural analysis methods. There are three main approaches to constraint handling: transformation, penalty function and Lagrange multiplier methods. The transformation methods do not allow the determination of axial forces in inextensible member or bending moments in rigid members. On the other hand, the penalty function method allows the computation of internal forces in constrained members and its implementation is simple. However, as the penalty factor increases, the stiffness matrix becomes increasingly ill-conditioned, which may lead to large solution errors. This paper presents a methodology for considering structural member deformation constraints using Lagrange multipliers. It consists of adding strain constraints into the total potential energy minimization, leading to a quadratic programming problem. In addition, this approach is very suitable for computational implementation because it does not affect the generic characteristic of a matrix structural analysis. The solution gives rise to one Lagrange multiplier per constraint, which is essential for computing member internal forces. However, there are situations in which inextensible and rigid member constrains may be redundant, which prevents the determination of dependent Lagrange multipliers. Although not implemented, a special treatment is indicated for the solution of this problem. |
