## 3 Description of the Solution Sets for Bar Structures
### Frames  
Let us take into the consideration a frame, which is composed of \(e\) bars and whose nodal deflections are described with \(s\) degrees of freedom. According to [18; 19], the linear system for frames made of thin bars is given by  
\[(\mathbf{B}^{\intercal}\mathbf{E}\mathbf{B}+2(^{*}\mathbf{B})^{\intercal} \mathbf{D}^{*}\mathbf{B}+(^{*}\mathbf{B})^{\intercal}\mathbf{D}\mathbf{B}^{*}+ (\mathbf{B}^{*})^{\intercal}\mathbf{D}^{*}\mathbf{B}+2(\mathbf{B}^{*})^{ \intercal}\mathbf{D}\mathbf{B}^{*})\,\mathbf{q}=\mathbf{Q}. \tag{22}\]  
while for frames made of moderately thick bars by  
\[\left(\mathbf{B}^{\intercal}\mathbf{E}\mathbf{B}+(^{*}\mathbf{B})^{\intercal} \mathbf{D}^{(1)*}\mathbf{B}+(^{*}\mathbf{B})^{\intercal}\mathbf{D}^{(2)} \mathbf{B}^{*}+(\mathbf{B}^{*})^{\intercal}\mathbf{D}^{(2)*}\mathbf{B}+( \mathbf{B}^{*})^{\intercal}\mathbf{D}^{(1)}\mathbf{B}^{*}\right)\mathbf{q}= \mathbf{Q}, \tag{23}\]  
Figure 1: Diagram of searching for a solution set in the form of inequalities, assuming truss structure. in which \(\mathbf{E}=\text{diag}(E_{K})\in\mathbb{R}^{e\times e}\), \(\mathbf{D}=\text{diag}(D_{K})\in\mathbb{R}^{e\times e}\), \(\mathbf{D}^{(1)}=\text{diag}(D_{K}^{(1)})\in\mathbb{R}^{e\times e}\) and \(\mathbf{D}^{(2)}=\text{diag}(D_{K}^{(2)})\in\mathbb{R}^{e\times e}\) (\(K=1,\ldots,e\)) are constitutive matrices, \(\mathbf{B}\in\mathbb{R}^{e\times s}\), \({}^{*}\mathbf{B}\in\mathbb{R}^{e\times s}\) and \(\mathbf{B}^{*}\in\mathbb{R}^{e\times s}\) are geometrical matrices, \(\mathbf{q}\in\mathbb{R}^{s}\) is a vector of unknown degrees of freedom and \(\mathbf{Q}\in\mathbb{R}^{s}\) is a load vector. Identically like in trusses \(E_{K}=Y_{K}A_{K}\cdot L_{K}^{-1}\), while \(D_{K}=2Y_{K}I_{K}\cdot L_{K}^{-1}\), \(D_{K}^{(1)}=2Y_{K}I_{K}\mu_{K}(1+3\varrho_{K})\cdot(3L_{K})^{-1}\), \(D_{K}^{(2)}=Y_{K}I_{K}\mu_{K}(1-6\varrho_{K})\cdot(3L_{K})^{-1}\), in which \(\varrho_{K}=Y_{K}I_{K}\cdot(H_{K}I_{K}^{2})^{-1}\), \(\mu_{K}=6\cdot(12\varrho_{K}+1)^{-1}\). Components of matrices \(\mathbf{B}\), \({}^{*}\mathbf{B}\), \(\mathbf{B}^{*}\) are defined by the frame geometry and its construction method is described in [18, 19]. Let the bars be characterized by uncertain Young's modulus \(Y_{K}\), and the imprecision \(Y_{K}\) be described by interval bounds \(Y_{K}\in[Y_{K}]\), collected on the diagonal of parameter matrix \([\mathbf{p}]\). It is then possible to decompose the matrices \([\mathbf{E}]\), \([\mathbf{D}]\), \(\left[\mathbf{D}^{(1)}\right]\), \(\left[\mathbf{D}^{(2)}\right]\) to the parts related and unrelated to the uncertain parameters, in form \([\mathbf{E}]=[\mathbf{p  
for the frame made of thin bars and  
\[ \begin{split}&\mathcal{V}(\mathbf{q})=\mathbf{Q}-\sum_{i=1}^{e} \left(\widehat{E}_{ii}\langle\mathbf{B}^{(i)},\mathbf{q}\rangle\mathbf{B}^{(i)} +\widehat{D}_{ii}^{(1)}\left(\langle(^{*}\mathbf{B})^{(i)},\mathbf{q}\rangle(^ {*}\mathbf{B})^{(i)}+\langle(\mathbf{B}^{*})^{(i)},\mathbf{q}\rangle(\mathbf{ B}^{*})^{(i)}\right)+\\ &+\widehat{D}_{ii}^{(2)}\left(\langle(^{*}\mathbf{B})^{(i)}, \mathbf{q}\rangle(\mathbf{B}^{*})^{(i)}+\langle(\mathbf{B}^{*})^{(i)},\mathbf{ q}\rangle(^{*}\mathbf{B})^{(i)}\right)\right)*[p_{i}]\end{split}  \tag{30}\]  
for the frame made of moderately thick bars. Obtaining the description of the solution set in the form of inequalities is possible with the use of (16). For thin bars the scalar products (14) are in the form  
\[ \begin{split}&\mathcal{C}^{(i)}(\mathbf{u})=\widehat{E}_{ii} \langle\mathbf{B}^{(i)},\mathbf{q}\rangle\langle\mathbf{B}^{(i)},\mathbf{u} \rangle+\widehat{D}_{ii}\langle(^{*}\mathbf{B})^{(i)},\mathbf{q}\rangle\langle 2(^{*} \mathbf{B})^{(i)}+(\mathbf{B}^{*})^{(i)},\mathbf{u}\rangle+\\ &+\widehat{D}_{ii}\langle(\mathbf{B}^{*})^{(i)},\mathbf{q}\rangle \langle 2(\mathbf{B}^{*})^{(i)}+(^{*}\mathbf{B})^{(i)},\mathbf{u}\rangle, \end{split}  \tag{31}\]  
while for the moderately thick bars  
\[