Recently, research and practical application of free-form shells are very active owing to development of computer software tools as well as progress of technology of construction and material. Design problem of free-form shells can be naturally formulated as an optimization problem considering mechanical performances. Parametric surfaces such as Bézier surfaces are effectively used for generating smooth and complex surfaces of continuum and latticed shells. However, to design a practically acceptable design, non-structural performances such as cost of construction should be taken into account. In this study, a shape optimization approach is presented for free-form shells using ruled surface. Latticed shells consisting of ruled surface have high constructability, because members on generating lines require no torsion.
Boundary shape of the latticed shell is defined using a pair of Bézier curves to reduce the number of variables. The points with the same parameter value of the two curves are connected by a line to model a ruled surface. The locations of control points of Bézier curves are chosen as design variables.
A latticed shell modeled by ruled surface consisting of two parabolic boundary curves is selected as the initial shape for optimization. Since a simple ruled surface does not have high degree of freedom for shape representation, two ruled surfaces are connected to model a roof with a ridge line. The members of latticed shell is modeled using beam elements, and the four corners are fixed. The vertical self-weight and live loads as well as horizontal loads representing seismic loads are considered in the process of optimization.
First, we minimize the strain energy under volume constraint to obtain a stiff structure. It is confirmed that the strain energy due to vertical and horizontal loads is drastically reduced through shape optimization. Constructability is maintained, because the stiff beams along the generating lines can be manufactured without torsion.
Although a stiff structure is obtained through strain energy minimization, the local stress may not be reduced.
Therefore, we solve the additional optimization problem, which minimizes the material volume under stiffness and stress constraint. The parameter defining the cross-sectional property of each member is considered as design variables of this optimization problem, which is carried out after shape optimization. By solving this problem, the member stresses are reduced through the process of minimizing the material volume, while maintaining the stiffness.
