# Calculation of Trusses System in MATLAB--Multibody
## 2 Example of Analytical Design Based on In-Plane Deformation Conditions  
The system in Figure 1 is composed of five bars. It is externally and internally statically \(1\times\) indeterminate. When one degree of freedom looseness is suitably removed at node A or D, which binds the system to the world frame, the system does not turn into a mechanism. Similarly, after removing any of the bars 1 to 5, the structure remains unchanged. This means that the given system is externally and internally \(1\times\) statically indeterminate. Thus, we partially relax the system and express the unknown reactions in terms of the deformation conditions. The example mainly involves comparing theoretical results with simulations in MATLAB--Multibody and then conducting verification in SOLIDWORKS. The structure of the trusses system loaded by the force at node B after release from the bonds at node(s) A, D, and release of the bar between nodes AB is displayed in Figure 2.  
Figure 1: Structure of the trusses system loaded by the force at node B. For the energy change of the whole system under the \(N_{\rm{AB}}\) force [13] (p. 204), [18] (p. 160) in the bar, the following applies:  
\[\frac{\partial U}{\partial N_{AB}}=0 \tag{1}\]  
To determine the forces in individual nodes or gussets, we use the method of joint [19] (p. 70), [20; 21]. By successively releasing the nodes, see Figures 3-6, we denote the forces acting on the bars via the index of the nodes where the bar is attached. Node A:  
In general, the following applies to the equilibrium of forces at a node:  
\[\sum\mathbf{F}=0 \tag{2}\]  
By decomposing into the individual axes of the coordinate system, the following applies:  
\[\sum F_{x}=0;\;F_{Ax}+\frac{F_{AC}}{\sqrt{2}}-N_{AB}=0 \tag{3}\]  
\[\sum F_{y}=0;\;F_{Ay}+\frac{F_{AC}}{\sqrt{2}}=0 \tag{4}\]  
Node B:  
Figure 3: Reactions after release from the bonds at node A and release of bar AB, AC. Figure 2: Structure of the trusses system loaded by the force at node B after release from the bonds at node A, D, and release of the bar between nodes AB. \[\sum F_{x}=0;\ N_{AB}-F_{Bx}-\frac{F_{BD}}{\sqrt{2}}=0 \tag{5}\]  
\[\sum F_{y}=0;\ F_{BC}-F_{By}+\frac{F_{BD}}{\sqrt{2}}=0 \tag{6}\]  
Node C:  
Figure 1:0  
Figure 1:1  
Node D:  
Figure 1:2  
Figure 1:3 Reactions after release from the bonds at node B and release of bar AB, BD, BC. Figure 1:4 Reactions after release from the bonds at node C and release of bar AC, BC, CD. Figure 1:5 Reactions after release from the bonds at node D and release of bar BD, CD. Figure 1:6  
Consider the system of given Equations (5)-(8), where the variables without reactions at nodes A and D are searched for:  
Figure 1:7  
Figure 1:8  
Figure 1:9  
\[\frac{\partial U}{\partial N_{AB}}=0 \tag{1}\]0  
And its transcription into matrix entry for automated computer calculation:  
\[ \begin{bmatrix}1&0&0&-\frac{\sqrt{2}}{2}&0\\ 0&0&1&\frac{\sqrt{2}}{2}&0\\ 0&-\frac{\sqrt{2}}{2}&0&0&-1\\ 0&-\frac{\sqrt{2}}{2}&-1&0&0\end{bmatrix} \cdot \begin{bmatrix}N_{\text{AB}}\\ F_{\text{AC}}\\ F_{\text{BC}}\\ F_{\text{BD}}\\ F_{\text{CD}}\end{bmatrix} = \begin{bmatrix}F_{\text{Bx}}\\ F_{\text{By}}\\ 0\\ 0\end{bmatrix}  \tag{15}\]  
From system (15), the unknown forces in the bars expressed in terms of the known variables \(F_{\text{Bx}}\), \(F_{\text{By}}\), and \(N_{\text{AB}}\) are as follows:  
\[F_{\text{AC}}=-\sqrt{2}\cdot\left(F_{\text{Bx}}-N_{\text{AB}}+F_{\text{By}}\right) \tag{16}\]  
\[F_{\text{BC}}=F_{\text{Bx}}-N_{\text{AB}}+F_{\text{By}} \tag{17}\]  
\[F_{\text{BD}}=\sqrt{2}\cdot\left(N_{\text{AB}}-F_{\text{Bx}}\right) \tag{18}\]  
\[F_{\text{CD}}=F_{\text{Bx}}-N_{\text{AB}}+F_{\text{By}} \tag{19}\]  
Based on the deformation condition (1) regarding the energy change in the entire system under the \(N_{\text{AB}}\) force in the bar, the following applies:  
\[\sum\frac{F_{i}\cdot l_{i}}{E_{i}\cdot S_{i}}\cdot\frac{\partial F_{i}}{ \partial N_{AB}}=0 \tag{20}\]  
The cross-sections \(S_{i}\) and Young's modulus \(E_{i}\) are the same in the present case.