### Abstract

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 includes 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 concentrate 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.

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 includes 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 concentrate 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.

Original language | English |
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Pages | 286 – 315 |

Number of pages | 29 |

Publication status | Published - 24 Sep 2018 |

Event | Advances in Architectural Geometry - Chalmers University of Technology, Gothenburg, Sweden Duration: 24 Sep 2018 → 25 Sep 2018 http://www.architecturalgeometry.org/aag18/ |

### Conference

Conference | Advances in Architectural Geometry |
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Abbreviated title | AAG2018 |

Country | Sweden |

City | Gothenburg |

Period | 24/09/18 → 25/09/18 |

Internet address |

### Keywords

- Virtual work
- fabric, shell and gridshell structures
- calculus of variations
- conjugate directions

### Cite this

Williams, C., Shepherd, P., Adiels, E., Ander, M., Hörteborn, E., Olsson, J., ... Sehlström, A. (2018).

*The use of virtual work for the formfinding of fabric, shell and gridshell structures*. 286 – 315. Paper presented at Advances in Architectural Geometry, Gothenburg, Sweden.