Form-finding and determining geodesic seam lines using the updated weight method for tensile membrane structures with strut and anchorage supports

Abstract

Tensile membrane structures (TMS) consist of the membrane fabric, supporting frame, cables and strut elements. The initial shape of a TMS is not known beforehand and has to be found by a process of ‘form-finding’. The majority of form-finding research has focused on determining the initial shape due to a defined prestress for TMS having only membrane and cable elements, while there has been very little research on TMS supported with struts and anchorage cables. This study extends the updated weight method (UWM) for the form-finding of TMS with struts and anchorage supports. The modified approach provides a robust solution in comparison to constrained strut length approaches. Furthermore, the curved form-found shape is not developable and geodesic lines need to be identified on the final form-found shape to provide seam locations for cutting the fabric. To solve this issue, geodesic lines are included in the UWM as constraints, whereby the geodesic pseudo cable lengths are minimised tangent to the surface. A sequential process is developed to ensure both the equilibrium and the constraint conditions are satisfied. The proposed method is successfully tested on a wide variety of TMS shapes along with the patterning of the cut panels. The study provides an integrated solution for the form-finding and identification of geodesic seam line on TMS having different boundary types.

Appendix A: Gradient vectors

The gradients of the formulated Lagrangian is derived in this section. The gradient $[\nabla _{\textbf{x}} \mathcal {L}_{U}~~\nabla _{\varvec{\lambda }}\mathcal {L}_{U}]=0$ describes the non-linear system of equilibrium equations, derived analytically and is given as input to the optimiser.

$$\begin{aligned} \begin{aligned}&\nabla _\textbf{x}\mathcal {L}_{U} = \sum \limits _{i=1}^{N_m} \sum \limits _{j=1}^{3} W^j_i\nabla _\textbf{x} (L^j_i)^2 +\sum \limits _{i=1}^{N_c} {^cw_i} \nabla _\textbf{x}(^cl_i^2) \\ {}&\qquad +\sum \limits _{i=1}^{N_a} {^aw_i} \nabla _\textbf{x}(^al_i^2) +\sum \limits _{i=1}^{N_s} {^sw_i} \nabla _\textbf{x}(^sl_i^2) \\ {}&\qquad +\sum \limits _{i=1}^{N_g} \lambda _i {\nabla _\textbf{x}} \left( {\nabla _{\textbf{x}_\textbf{q}}} (^gl^2_i) - \left( {\nabla _{\textbf{x}_\textbf{q}}} (^gl^2_i)\cdot \hat{\textbf{N}}_\textbf{q} \right) \hat{\textbf{N}}_\textbf{q}\right) \\&{\small \nabla _{\varvec{\lambda }} \mathcal {L}_{U} = \sum \limits _{i=1}^{N_g} \nabla _{\varvec{\lambda }} \left( \lambda _i\right) \left( {\nabla _{\textbf{x}_\textbf{q}}} (^gl^2_i) - \left( {\nabla _{\textbf{x}_\textbf{q}}} (^gl^2_i)\cdot \hat{\textbf{N}}_\textbf{q} \right) \hat{\textbf{N}}_\textbf{q}\right) } \end{aligned} \end{aligned}$$

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where again $\nabla _{\textbf{x}}$ is the derivative operator with respect to all the free nodal coordinates given as $\nabla _{\textbf{x}} = \left[ \frac{\partial }{\partial \textbf{x}_1}~\frac{\partial }{\partial \textbf{x}_2}~\ldots ~\frac{\partial }{\partial \textbf{x}_{N_n}} \right] ^\textsf{T}$ and $\nabla _{\textbf{x}_m}= \left[ \frac{\partial }{\partial x_m}~\frac{\partial }{\partial y_m}~\frac{\partial }{\partial z_m} \right] ^\textsf{T}$.

1.1 Appendix A.1: Gradient vectors of objective function

Consider the length $l_i$ of any ith line element (boundary edge, anchorage, pseudo-strut and geodesic pseudo cables) connecting nodes r and s. The derivative of the square of cable length with respect to the nodal coordinates m is given as

$$\begin{aligned} \begin{aligned}&l_i(r,s) = \sqrt{(x_r-x_s)^2 + (y_r-y_s)^2 + (z_r-z_s)^2}\\&\nabla _{\textbf{x}_m} (l_i^2)= \left[ \frac{\partial l^2_i}{\partial x_m}~~\frac{\partial l^2_i}{\partial y_m}~~\frac{\partial l^2_i}{\partial z_m} \right] ^\textsf{T} \\&= {\left\{ \begin{array}{ll} 2 \left[ (x_r-x_s)~~(y_r-y_s)~~(z_r-z_s)\right] ^\textsf{T} &{}; {m=r}\\ 2 \left[ (x_s-x_r)~~(y_s-y_r)~~(z_s-z_r)\right] ^\textsf{T} &{}; {m=s} \\ \left[ 0~~0~~0\right] ^\textsf{T} &{}; {m\ne r \cup s } \end{array}\right. } \end{aligned} \end{aligned}$$

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for $m= 1 \ldots N_N$. The derivative $\nabla _{\textbf{x}_m} (L^j_i)^2$ of the pseudo cables length joining membrane nodes r and s can be found by replacing $l_i$ with $L^j_i$. The derivative of the Hessian can be similarly found (Marbaniang et al. 2022).

1.2 Appendix A.2: Gradient vectors of geodesic constraint

The derivative of the geodesic constraint ${\nabla ^t_{\textbf{x}_\textbf{q}}} (^gl^2)=0$ with respect to the free nodal co-ordinate m is given as

$$\begin{aligned} \begin{aligned} \sum \limits _{i=1}^{N_g} \left[ {\nabla _{\textbf{x}_m\textbf{x}_\textbf{q}}^2} (^gl^2_i) - \left( {\nabla _{\textbf{x}_m\textbf{x}_\textbf{q}}^2} (^gl^2_i)\cdot \hat{\textbf{N}}_\textbf{q} \right) \hat{\textbf{N}}_\textbf{q} \right. \\ \left. - \left( {\nabla _{\textbf{x}_\textbf{q}}} (^gl^2_i)\cdot {\nabla _{\textbf{x}_m}} \left( \hat{\textbf{N}}_\textbf{q}\right) \right) \hat{\textbf{N}}_\textbf{q} \right. \\ \left. - \left( {\nabla _{\textbf{x}_\textbf{q}}} (^gl^2_i)\cdot \hat{\textbf{N}}_\textbf{q} \right) {\nabla _{\textbf{x}_m}} \left( \hat{\textbf{N}}_\textbf{q}\right) \right] \end{aligned} \end{aligned}$$

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where $\nabla _{\textbf{x}_m\textbf{x}_\textbf{q}}^2$ provides the double derivative of the geodesic pseudo cable constraints with respect to $\textbf{x}_m$ and $\nabla _{\textbf{x}_m\textbf{x}_\textbf{q}}^2 \left( ^gl\right) = 0 ~\forall m\notin \textbf{q}$.

The average normal of a node surrounded by elements (Fig. 8) is given as

$$\begin{aligned} \textbf{N}_\textbf{q} = \sum \limits _{i=1}^{N_b} \dfrac{\hat{\textbf{n}}_i}{A_i}; \quad \forall i \in \textbf{b} \end{aligned}$$

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where $\textbf{b}$ is the set of elements with vertex at a node in $\textbf{q}$ and $N_b$ is the number of such elements (Fig. 8). The weighted normalised normal of each element can be simplified as

$$\begin{aligned} \dfrac{\hat{\textbf{n}}}{A} = \dfrac{2\textbf{n}}{\Vert \textbf{n}\Vert \Vert \textbf{n}\Vert } = \dfrac{2\textbf{n}}{\textbf{n} \cdot \textbf{n}} \end{aligned}$$

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where $A = \Vert \textbf{n} \Vert /2$. The unit average normal is further expressed as

$$\begin{aligned} \hat{\textbf{N}}_\textbf{q} = \dfrac{\sum \nolimits _{i=1}^{N_b} \textbf{n}^w_i}{\sqrt{k}} \end{aligned}$$

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where $\textbf{n}^w_i = \dfrac{2\textbf{n}_i}{\textbf{n}_i \cdot \textbf{n}_i}$ and $k = \sum\nolimits _{i=1}^{N_b} \textbf{n}_i^w \cdot \textbf{n}_i^w$.

The derivative of the average unit normal is given as

$$\begin{aligned} {\nabla _{\textbf{x}_m}} \hat{\textbf{N}}_\textbf{q} = \dfrac{ \sum \nolimits _{i=1}^{N_b}\sqrt{k} \dfrac{\partial \textbf{n}^w_i}{\partial \textbf{x}_m} - \dfrac{1}{2\sqrt{k}}\dfrac{\partial k}{\partial \textbf{x}_m}\textbf{n}^w_i }{k} \end{aligned}$$

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where the derivative of $\hat{\textbf{N}}_\textbf{q}$ is with respect to only the nodes of the surrounding elements belonging to the set $\textbf{b}$. The derivatives $\frac{\partial \textbf{n}^w_i}{\partial \textbf{x}_m}$ and $\frac{\partial k}{\partial \textbf{x}_m}$ are given as

$$\begin{aligned}{} & {} \dfrac{\partial \textbf{n}^w_i}{\partial \textbf{x}_m} =2 \dfrac{(\textbf{n}_i \cdot \textbf{n}_i) \dfrac{\partial \textbf{n}_i}{\partial \textbf{x}_m} - \left( 2 \textbf{n}_i \cdot \dfrac{\partial \textbf{n}_i}{\partial \textbf{x}_m}\right) \textbf{n}_i}{(\textbf{n}_i \cdot \textbf{n}_i)^2} \end{aligned}$$

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$$\begin{aligned}{} & {} \quad \dfrac{\partial k}{\partial \textbf{x}_m} = \sum \limits _{i=1}^{N_b} 2 \textbf{n}_i^w \cdot \dfrac{\partial \textbf{n}_i^w }{\partial \textbf{x}_m} \end{aligned}$$

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The normal $\textbf{n}$ for an element i and it’s derivative, is derived from the cross product of the sides defined by the two vectors $\textbf{v}_1$ and $\textbf{v}_2$ as shown in Fig. 8.

$$\begin{aligned} \textbf{n}_i= & {} \textbf{v}_2 \times \textbf{v}_1 \end{aligned}$$

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$$\begin{aligned}{} & {} \dfrac{\partial \textbf{n}_i}{\partial \textbf{x}_m} = \frac{\partial \textbf{v}_2}{\partial \textbf{x}_m} \times \textbf{v}_1 + \textbf{v}_2 \times \frac{\partial \textbf{v}_1}{\partial \textbf{x}_m} \end{aligned}$$

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