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Equilibrium stresses in frameworks via symmetric averaging

Cameron Millar, Bernd Schulze, Louis Theran

Abstract

For a bar-joint framework $(G,p)$, a subgroup $Γ$ of the automorphism group of $G$, and a subgroup of the orthogonal group isomorphic to $Γ$, we introduce a symmetric averaging map which produces a bar-joint framework on $G$ with that symmetry. If the original configuration is ``almost symmetric", then the averaged one will be near the original configuration. With a view on structural engineering applications, we then introduce a hierarchy of definitions of ``localised" and ``non-localised" or ``extensive" self-stresses of frameworks and investigate their behaviour under the symmetric averaging procedure. Finally, we present algorithms for finding non-degenerate symmetric frameworks with many states of self-stress, as well as non-symmetric and symmetric frameworks with extensive self-stresses. The latter uses the symmetric averaging map in combination with symmetric Maxwell-type character counts and a procedure based on the pure condition from algebraic geometry. These algorithms provide new theoretical and computational tools for the design of engineering structures such as gridshell roofs.

Equilibrium stresses in frameworks via symmetric averaging

Abstract

For a bar-joint framework , a subgroup of the automorphism group of , and a subgroup of the orthogonal group isomorphic to , we introduce a symmetric averaging map which produces a bar-joint framework on with that symmetry. If the original configuration is ``almost symmetric", then the averaged one will be near the original configuration. With a view on structural engineering applications, we then introduce a hierarchy of definitions of ``localised" and ``non-localised" or ``extensive" self-stresses of frameworks and investigate their behaviour under the symmetric averaging procedure. Finally, we present algorithms for finding non-degenerate symmetric frameworks with many states of self-stress, as well as non-symmetric and symmetric frameworks with extensive self-stresses. The latter uses the symmetric averaging map in combination with symmetric Maxwell-type character counts and a procedure based on the pure condition from algebraic geometry. These algorithms provide new theoretical and computational tools for the design of engineering structures such as gridshell roofs.

Paper Structure

This paper contains 18 sections, 3 theorems, 47 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Symmetric graphs and frameworks
  3. Background on rigidity theory
  4. Symmetric averaging
  5. Symmetric averaging for the reflection group
  6. Symmetric averaging in general
  7. Properties of the averaging map
  8. Connections to representation-theoretic approaches
  9. Types of self-stress: local vs. extensive
  10. $\Gamma$-localised and $\Gamma$-extensive self-stresses
  11. Self-stress behaviour under the averaging map
  12. General localised and extensive self-stresses
  13. Heuristics for creating self-stressed symmetric frameworks
  14. Finding symmetric realisations with large self-stress dimension
  15. Finding realisations with extensive self-stresses
  16. ...and 3 more sections

Key Result

theorem 1

Let $\Gamma$ be a non-trivial group and let $G$ be a graph whose $(\Gamma,\tau)$-generic realisations as a framework have no non-trivial self-stress. Suppose the (not necessarily symmetric) framework $(G,{\bf p})$ has a strongly $\Gamma$-localised self-stress $\mathbf \omega$ which is supported on a

Figures (12)

  • Figure 1: Hypothetical gridshell design with form diagram (plan/top view) inset. Note that this gridshell is not self-tied and relies upon pin supports along the boundary.
  • Figure 2: The triangular prism graph can be realised as a bar-joint framework in the plane with reflection symmetry $\mathcal{C}_s$ (a,b), half-turn symmetry $\mathcal{C}_2$ (c), dihedral symmetry $\mathcal{C}_{2v}$ of order 4 (d), three-fold rotational symmetry $\mathcal{C}_3$ (e) and dihedral symmetry $\mathcal{C}_{3v}$ of order 6 (f). The symmetry types shown here correspond to the choices of the combinatorial group $\Gamma$ and point group representation $\tau$ that lead to planar realisations. Other choices exist, but the associated drawings have crossings.
  • Figure 3: $(\Gamma,\tau)$-symmetric frameworks in the plane, where $\Gamma$ has order $2$. For (a) and (b), $\Gamma=\{\mathrm{Id}, (1,2)(3,4)(5,6)\}$. However, in (a) the non-trivial element of $\Gamma$ is mapped to a reflection, whereas in (b) it is mapped to the half turn. For (c) and (d), $\Gamma=\{\mathrm{Id}, (1,4)(2,3)(5,6)\}$, where in (c) and (d) the non-trivial element of $\Gamma$ is again mapped to a reflection and the half turn, respectively.
  • Figure 4: Generating a symmetrically averaged framework $(G,{\bf x})$ from a non-symmetric framework $(G,{\bf p})$ for the case of the reflection group $\mathcal{C}_s$ in the plane: $(G,\bar{{\bf p}})$ in (b) is obtained from $(G,{\bf p})$ in (a) by relabelling. $(G,{\bf q})$ in (c) is obtained from $(G,\bar{{\bf p}})$ in (b) by reflecting in the vertical mirror line, so that ${\bf q}=(\gamma \cdot {\bf p})$. Finally, $(G,{\bf x})$ in (d) is the average of (a) and (c) and has mirror symmetry.
  • Figure 5: Generating a symmetrically averaged framework $(G,{\bf x})$ with $\mathcal{C}_3$ symmetry from a non-symmetric framework $(G,{\bf p})$: the framework in (b) is $(G,{\bf q})$, where ${\bf q}=(\gamma \cdot {\bf p})$; the framework in (c) is $(G,{\bf q}')$, where ${\bf q}'=(\gamma^2 \cdot {\bf p})$. Finally, the framework $(G,{\bf x})$ in (d) is the average $(G,A{\bf p})$ of the ones in (a), (b) and (c) and clearly has $\mathcal{C}_3$ symmetry.
  • ...and 7 more figures

Theorems & Definitions (15)

  • definition 1
  • definition 2
  • definition 3
  • theorem 1
  • proof
  • definition 4
  • Proposition 2
  • proof
  • remark 3
  • definition 5
  • ...and 5 more