# Innovative Suspension Structures: The Role of Straight Elements Under Asymmetric Loads
## 1 Introduction
### Analysis of Asymmetrically Loaded Innovative Single-Level Suspension System
#### 2.2.3 Calculation of the Asymmetrically Loaded Suspension System  
The calculation of an asymmetrically loaded suspension system is performed by combining the expressions for the calculation of the upper and lower chords of the left and right sides of the system obtained in Sections 2.2.1 and 2.2.2. The main unknowns of this iterative calculation procedure are the vertical displacement \(\Delta f_{c}=\Delta v_{3}\) and the horizontal displacement \(\Delta l_{3}\) of the central hinge (node \"3\"). Other dependent unknowns are the displacements of nodes \"4\" (\(\Delta f_{4}\)) and \"7\" (\(\Delta f_{7}\)) of the lower chord of the suspension system and the fictitious displacements of the straight elements of the upper chord \(\Delta f_{ific,1}\), \(\Delta f_{ific,2}\) (part loaded with asymmetrical load \(g+s\)) and \(\Delta f_{ific,3}\), \(\Delta f_{ific,4}\) (part loaded with \(g\)), as well as the tensile forces in the upper and lower chord elements: \(H_{1}\), \(H_{2}\), and \(H_{3r}\). An important unknown is the force in the struts \(F_{4}\) and \(F_{7}\).  
The iterative calculation can be divided into a few steps. Having accepted the initial values of the vertical \(\Delta f_{c}=\Delta v_{3}\) and horizontal \(\Delta l_{3}\) displacements of the central hinge (node \"3\"), it is possible to calculate the left and right parts of the suspension system separately. First, the displacements of the nodes of the left \(\Delta f_{4}\), (according to (42)) and right parts \(\Delta f_{7}\) (according to (47) and (49)). Then, the left part fictitious average displacement is calculated \(\Delta f_{ific,mr}\) (see (23) and (24)) and fictitious displacements \(\Delta f_{ific,1}\) and \(\Delta f_{ific,2}\) (see (25) and (26)) and the medium force of their elements \(H_{mr}\) (see (24)) are determined. After that, when \(\Delta f_{ific,1}\) and \(\Delta f_{ific,2}\) are known, the tensile forces \(H_{1r}^{*}\) and \(H_{2r}^{*}\) (see (17) and (18)) and the forces in the strut \"2-4\" are determined (see (28)). Then, the calculation sequence for the right side of the upper chord is analogous to the calculation sequence for the left side. The fictitious displacements of this part are calculated \(\Delta f_{ific,ml}\) according to (29) and \(\Delta f_{ific,3}\) and \(\Delta f_{ific,4}\) according to (32) and (33), and axial forces \(H_{ml}\) and \(F_{5l}\). according to (46) and (48). After that, the equilibrium conditions of the central node are verified as follows:  
\[H_{2r}+H_{3r}=H_{2l}+H_{3l}=\Sigma H=\frac{pL^{2}}{8(f_{0}+\Delta f_{c})}. \tag{50}\]  
If the equilibrium conditions of the central node are satisfied, then the actual displacements and bending moments of the upper chord elements of both parts of the system can be calculated.  
It should be noted that the presented analytical solutions are performed for these reasons. The obtained analytical solutions allow the determination of certain rational parameters of the examined structure without a large number of iterations. Calculating rational parameters using an FEM software would require a large number of convergence iterations.