# Innovative Suspension Structures: The Role of Straight Elements Under Asymmetric Loads
## 1 Introduction
### Calculation of Suspension Structure with Two Straight Elements  
Before discussing the innovative suspension system, we will examine its prototype, which is composed of two stiff elements in bending straight elements (\(EI\
eq 0\)), see Figure 1, SK-1 structure. These elements with bending stiffness allow us to create a flat roof surface. This would not be possible to create if the elements of the suspension structure were absolutely flexible (\(EI=0\)). Such a suspension structure of straight \"rigid\" bars under the influence of asymmetric loads experiences such displacements: the central node (hinge) \"went up\", and the straight \"rigid\" bars incurve (see Figure 2a). The analysis and calculation of this symmetrically loaded straight element suspension structure are presented in [47]. Figure 1: Suspension structure schemes. Analyzing the following case of asymmetric loading, when the suspension structure consists of two straight bars, the temporary asymmetric load s is distributed over half of its span length, i.e., through one of the straight elements \"1-2\" (see Figure 2b). Under the action of loads, the straight tension elements \"1-2\" and \"2-3\" will deflect with displacements \(\Delta f_{1,2}\) and \(\Delta f_{2,3}\), and the central node \"2\" will move not only with a vertical displacement \(\Delta f_{c}\), but also a horizontal displacement \(\Delta h_{2}\). The tensile forces \(H_{1}\) and \(H_{2}\) and corresponding bending moments \(M_{1(x)}\) and \(M_{2(x)}\) will appear in these elements. In the case of asymmetric loading, the calculation of this suspension structure becomes more complicated compared with symmetric loading, because we have one additional unknown--the horizontal displacement of the central node \(\Delta h_{2}\). It is, therefore, necessary to analyze both sides of the structure separately: one subjected to combined \((g+s)\) loads and the other subjected to only the \(g\) load. In this case (in the case of asymmetric loading), there are already two mutually unequal unknowns--the fictitious displacement \(\Delta f_{ific,1}\) and \(\Delta f_{ific,2}\) of both straight elements. It should be noted that the effect of asymmetric load on suspension structures can be estimated by applying the ratio of loads \(\gamma=s/g\). The ratio \(\gamma\) indicates the influence of asymmetric loading on the deformation of the structure. The higher the value of the ratio \(\gamma\), the more dominant the displacements of kinematic origin become. Figure 2: The suspension structures of two straight elements: (**a**) deformed scheme of asymmetric loading; (**b**) calculation scheme of element “1–2” of the suspension structure. According to the value of the load ratio \(\gamma\), it is possible to write the dependency of the fictitious displacements of both straight tension elements from the equilibrium condition in the central node \"2\" (see Figure 2b):  
\[\Delta f_{fc,1}=\Delta f_{fc,2}(1+\gamma). \tag{1}\]  
It can be seen from Formula (1) that the difference in the fictitious displacements of the two tension bars in the loaded part of the structure \(\Delta f_{fc,1}\) and the unloaded part \(\Delta f_{fc,2}\) is increasing when the values of the load ratio \(\gamma\) increasing. The equation of equilibrium of the true tension \"rigid\" element (bar) \"1-2\" can be written as follows (see Figure 2). The actual \"1\" element displacement \(w_{1}(x)\) will be equal to:  
\[w_{1}(x)=\Delta f_{fc,1}\Bigg{[}\frac{8}{(k_{1}l)^{2}}\bigg{(}chk_{1}x+\frac{ (1-chk_{1}l)shk_{1}x}{shk_{1}l}-1\bigg{)}+\frac{4x}{l}-\frac{4x^{2}}{l^{2}} \bigg{]}+\Delta f_{c}\frac{x}{l}, \tag{2}\]  
here \(k_{1}l=\sqrt{\frac{H_{1}l^{2}}{EI\cos^{3}\varphi_{1}}}\)-slederness parameter of element \"1-2\"; \(\Delta f_{c}\)--vertical displacement of the central node (see Figure 2b); \(chk_{1}x=0.5\Big{(}e^{k_{1}x}+e^{-k_{1}x}\big{)}\)--hyperbolic cosine; and \(shk_{1}x=0.5\Big{(}e^{k_{1}x}-e^{-k_{1}x}\big{)}\)--hyperbolic sine.