The use of virtual work for the formfinding of fabric, shell and gridshell structures

Paper in proceeding, 2018

The use of the virtual work theorem enables one to derive the equations of static equilibrium of fabric, shell and gridshell structures from the compatibility equations linking the rate of deformation of a surface to variations in its velocity. If the structure is treated as a continuum there is no need to consider its micro-structure provided that the grid is fine compared to the overall geometry. Thus we can include fabrics, ribbed shells, corrugated shells and gridshells with a fine grid, such as the Mannheim Multihalle. The equilibrium equations are almost identical to those obtained by assuming that a shell is thin and of uniform thickness, but are more general in their application. Our formulation introduces the concept of geodesic bending moments which are relevant to gridshell structures with continuous laths.

The virtual work theorem is more general than the energy theorems, which it in- cludes as a special case. Hence it can be applied to surfaces which admit some form of potential, including minimal surfaces and hanging fabrics. We can then use the calculus of variations for the minimization of a surface integral to define the form of a structure.

Many existing formfinding techniques can be rewritten in this way, but we concen- trate on surfaces which minimize the surface integral of the mean curvature subject to a constraint on the enclosed volume, producing a surface of constant Gaussian curvature. This naturally leads to the more general study of conjugate stress and curvature directions, and hence to quadrilateral mesh gridshells with flat cladding panels and no bending moments in the structural members under own weight.