# Group-Theoretic Exploitations of Symmetry in Novel Prestressed Structures

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Group Theory and Its Matrix Representations

- Identity: $\exists E\in \mathit{G}$, for any element ${\mathit{g}}_{i}\in \mathit{G}$, it satisfies ${\mathit{g}}_{i}\cdot E=E\cdot {\mathit{g}}_{i}={\mathit{g}}_{i}$.
- Inverses: ${\mathit{g}}_{i}\in \mathit{G},\exists {\mathit{g}}_{i}^{-1}\in \mathit{G}\iff {\mathit{g}}_{i}\cdot {\mathit{g}}_{i}^{-1}=E$.
- Closure: $\forall {\mathit{g}}_{i},{\mathit{g}}_{j}\in \mathit{G}\Rightarrow {\mathit{g}}_{i}\cdot {\mathit{g}}_{j}\in \mathit{G}$.
- Associativity: $\forall {\mathit{g}}_{i},{\mathit{g}}_{j},{\mathit{g}}_{k}\in \mathit{G}$ $\Rightarrow $ $({\mathit{g}}_{i}\cdot {\mathit{g}}_{j})\cdot {\mathit{g}}_{k}={\mathit{g}}_{i}\cdot ({\mathit{g}}_{j}\cdot {\mathit{g}}_{k})$.

## 3. Stability Analysis on Prestressable Pin-Jointed Structures

#### 3.1. Positive Definiteness of the Tangent Stiffness Matrix for a Prestressed Pin-Jointed Structure

#### 3.2. Necessary Conditions for Structural Stability of Prestressed Pin-Jointed Structures

#### 3.3. Example: C_{2v} Symmetric Cable-Strut Structures

_{c}= 2000 mm, the distance between the struts is 2000 mm, and the minimum distance from the boundary nodes (nodes 5 and 6) to the struts is 1000 mm. The elastic modulus of the cables and that of the struts is ${E}_{t}=1.9\times {10}^{5}$ $\mathrm{MPa}$ and ${E}_{c}=2\times {10}^{5}$ $\mathrm{MPa}$; the cross-sectional area of the cables and that of the struts is ${A}_{t}=500$ ${\mathrm{mm}}^{2}$ and ${A}_{c}=3000$ ${\mathrm{mm}}^{2}$. Both structures have a single mechanism mode ($m=1$) and a self-stress state ($s=1$). Note that the initial prestress of the vertical struts is −100 kN, and prestresses of other members can be uniquely determined by the self-stress state.

_{1}indicates full symmetry and the others indicate lower-order symmetry. The short arrows indicate the motion trend of the free nodes. It turns out the eigenvectors keep consistent with the inherent symmetry properties of specific symmetry subspaces. For instance, the fifth and eighth eigenvectors come from the full symmetry subspace and thus they have full symmetry of C

_{2v}. Similarly, since the first and fourth eigenvectors are from the symmetry subspace A

_{2}, they exhibit rotational symmetry of C

_{2v}. The third and seventh eigenvectors keeps C

_{v}symmetry along the X axis, while the second and the sixth eigenvectors keeps C

_{v}symmetry along the Y axis.

_{2v}symmetry) and an internal mechanism mode with lower-order symmetry (i.e., C

_{v}symmetry). However, it has been verified that the internal mechanism mode cannot be rigidified by prestressing (see Figure 3), as the minimum eigenvalue ${\lambda}_{\mathrm{min}}({\mathit{K}}_{T})=-0.037<0$. Consequently, this structure is unstable, although it satisfies the necessary conditions given by Equation (16). It should be pointed out that the results for this structure are consistent with the reported ones [42].

## 4. Form-Finding Analysis on Tensegrity Structures

#### 4.1. Integral Self-Stress State Obtained from the Block with Full Symmetry

**S**come from the null space of the singular matrix $\mathit{H}$ (i.e., $\mathit{HS}=0$). In the symmetry-adapted coordinate system, the symmetry-adapted equilibrium matrix $\overline{\mathit{H}}$ can be transformed from the original matrix $\mathit{H}$:

#### 4.2. Example: A ${D}_{3}$ Symmetric Tensegrity Structure

## 5. Generalized Eigenvalue Problems of Symmetric Prestressed Structures

#### 5.1. Symmetry-Adapted Frequency Analysis

**M**

**U**in Equation (7), the mass matrix

**M**can be decomposed into similar block-diagonalized forms

#### 5.2. Symmetry-Adapted Buckling Analysis

**V**is an orthogonal transformation matrix for expression in the symmetry-adapted coordinate system. Because each block matrix is independent, the original buckling problem can be solved by solving the subproblems in parallel [52]

#### 5.3. Illustrative Example: A ${C}_{12v}$ Symmetric Cable Dome Structure

_{12v}symmetric cable dome with different prestress levels are analyzed by the proposed method (the symmetry subspaces keep invariant). The prestress levels are respectively 0, 0.25t, 0.5t, 2t and 4t, and the initial prestresses t are determined by the feasible prestress modes [40].

_{12v}symmetric cable dome [40]. It can be noticed that this symmetric dome structure has many repeated eigenvalues and equivalent eigenvectors, such as modes 1–2, modes 3–4, and modes 5–6. In fact, this phenomenon is ubiquitous for most symmetric structures. This is because the roots computed from the symmetry spaces for multi-dimensional irreducible representations are identical, and the generalized eigenvalues for these symmetry subspaces are exactly the same. In addition, each vibration shape obtained from lower-order symmetry subspace ${\mathit{U}}^{(i-h)}$ does not maintain its full symmetry (i.e., ${C}_{12v}$). Then, it may be reduced to a low-order symmetry. Because the seventh vibration shape is obtained from the symmetry subspace ${\Gamma}^{(2-1)}={A}_{2}$, it has rotational symmetry (i.e., ${C}_{12}$). In other words, the symmetry of all the vibration modes can be predicted from symmetry subspaces in advance [16,29], without numerical computing.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Two-dimensional ${C}_{2v}$ symmetric cable-structures (

**a**) with horizontal cables; (

**b**) with cross cables.

**Figure 2.**Eigenvalues extracted from different symmetry subspaces for of the cable-strut structure shown in Figure 1a.

**Figure 3.**Static and kinematic indeterminacy of the cable-strut shown in Figure 1b: (

**a**) C

_{2v}symmetric self-stress state; (

**b**) C

_{v}symmetric mechanism mode.

**Figure 4.**A simple tensegrity structure with ${D}_{3}$ symmetry [48]: (

**a**) three two-fold rotations; (

**b**) integral prestress mode with full symmetry.

**Figure 5.**A ${C}_{12v}$ symmetric cable dome structures [40]: (

**a**) twelve rotations and twelve reflections indicated in the plan view; (

**b**) 3D geometric configuration.

**Figure 6.**Distribution patterns of nonzero entries of stiffness matrices of the ${C}_{12v}$ symmetric cable dome: (

**a**) original tangent stiffness matrix ${\mathit{K}}_{T}$; (

**b**) block-diagonalized stiffness matrix $\overline{{\mathit{K}}_{T}}$.

**Figure 7.**First 100 frequencies of the C

_{12v}symmetric cable dome with different prestress levels [40].

**Table 1.**Different blocks ${\overline{\mathit{K}}}_{T}^{(i)}$ of symmetry-adapted tangent stiffness matrix $(\times {E}_{c}{A}_{c}/{L}_{c})$.

$\mathrm{Irreducible}\mathrm{Representation}{\mathbf{\Gamma}}^{(\mathit{i})}$ | ${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{B}}_{1}$ | ${\mathit{B}}_{2}$ |
---|---|---|---|---|

Figure 1a | $\left[\begin{array}{cc}2.1121& 0.1118\\ 0.1118& 0.4288\end{array}\right]$ | $\left[\begin{array}{cc}0.1125& 0.1118\\ 0.1118& 0.1121\end{array}\right]$ | $\left[\begin{array}{cc}2.1125& 0.1118\\ 0.1118& 0.1121\end{array}\right]$ | $\left[\begin{array}{cc}0.1121& 0.1118\\ 0.1118& 0.4285\end{array}\right]$ |

Figure 1b | $\left[\begin{array}{cc}2.2241& 0\\ 0& 0.2241\end{array}\right]$ | $\left[\begin{array}{cc}0.2241& 0\\ 0& 0.2238\end{array}\right]$ | $\left[\begin{array}{cc}2.1120& 0.1119\\ 0.1119& 0.1120\end{array}\right]$ | $\left[\begin{array}{cc}0.1120& 0.1119\\ 0.1119& 0.1117\end{array}\right]$ |

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**MDPI and ACS Style**

Chen, Y.; Feng, J.
Group-Theoretic Exploitations of Symmetry in Novel Prestressed Structures. *Symmetry* **2018**, *10*, 229.
https://doi.org/10.3390/sym10060229

**AMA Style**

Chen Y, Feng J.
Group-Theoretic Exploitations of Symmetry in Novel Prestressed Structures. *Symmetry*. 2018; 10(6):229.
https://doi.org/10.3390/sym10060229

**Chicago/Turabian Style**

Chen, Yao, and Jian Feng.
2018. "Group-Theoretic Exploitations of Symmetry in Novel Prestressed Structures" *Symmetry* 10, no. 6: 229.
https://doi.org/10.3390/sym10060229