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

Abstract

The use of the virtual work theorem enables one to derive the equations\ua0of static equilibrium of fabric, shell and gridshell structures from\ua0the compatibility equations linking the rate of deformation of a surface\ua0to variations in its velocity. If the structure is treated as a continuum\ua0there is no need to consider its micro-structure provided that the grid\ua0is fine compared to the overall geometry. Thus we can include fabrics,\ua0ribbed shells, corrugated shells and gridshells with a fine grid, such as\ua0the Mannheim Multihalle. The equilibrium equations are almost identical\ua0to those obtained by assuming that a shell is thin and of uniform thickness,\ua0but are more general in their application. Our formulation introduces\ua0the concept of geodesic bending moments which are relevant to\ua0gridshell structures with continuous laths.The virtual work theorem is more general than the energy theorems,\ua0which it in- cludes as a special case. Hence it can be applied to surfaces\ua0which admit some form of potential, including minimal surfaces\ua0and hanging fabrics. We can then use the calculus of variations for the\ua0minimization of a surface integral to define the form of a structure.Many existing formfinding techniques can be rewritten in this way, but\ua0we concen- trate on surfaces which minimize the surface integral of the\ua0mean curvature subject to a constraint on the enclosed volume, producing\ua0a surface of constant Gaussian curvature. This naturally leads to\ua0the more general study of conjugate stress and curvature directions, and\ua0hence to quadrilateral mesh gridshells with flat cladding panels and no\ua0bending moments in the structural members under own weight