# Effects of Flexural Stiffness on Deformation Behaviour of Steel and FRP Stress-Ribbon Bridges
## 2 Design Procedure of the Bridge with Flexural-Stiff Ribbons  
The ribbons are designed to resist substantial axial forces, transferring them to massive foundations. The stress-ribbon bridge shape also has no strict limitations; in some cases, the movement requirements of special vehicles can limit the maximal slope of the bridge deck. However, the stress-ribbon bridges can undergo significant displacements, particularly from concentrated and asymmetrically distributed loads. Thus, the limitations of vertical displacements become the governing design condition [1; 7; 19]. The displacement increment induced by the live load (\(q\)), regarding the dead-load deformed shape, is the design criterion. Figure 1 shows the design scheme of the stress-ribbon bridge, having an initial sag of the ribbons\(f_{0}\). The dead load \(g\) increases the vertical displacement. A parabolic curve approximately describes this deformed shape that is the onset of design calculations. In other words, a live load (\(q\)) induces additional displacement \(u_{q}(x)\) that is the limitation object--various load distribution layouts determine the maximum vertical displacement  
\[u_{q,max}(x)\leq u_{q,lim} \tag{1}\]  
where \(u_{q,lim}\) is the displacement limit. The stress-ribbon systems are sensitive to the load \(q\) distributed over the bridge half span that often induces the maximum kinematic displacement of the bridge. The loading ratio \(\gamma(=q/g)\) and the flexural stiffness of the ribbons control the kinematic displacement magnitude [1; 13]. The following expression determines the functional relationship between the bridge parameters and the vertical displacement [13]:  
\[u_{q}(x)\ =\ \Phi(\gamma,\ x/l,(kl)\ ^{-\ 1},\ \Omega[kl,\ kx])\leq u_{q,lim} \tag{2}\]  
where \(l\) is the bridge span; the coefficient \(k\ =\ \sqrt{H/EI_{y}}\) defines slenderness of the ribbon (\(H\) is the thrust force, \(EI_{y}\) is the flexural stiffness of the ribbon); the operand \(\Omega[kl,\ kx]\) represents a combination of hyperbolic functions of the variables given in the brackets. A mathematical expression of Equation (2) depends on the structural parameters and boundary conditions of the ribbons. [PERSON] and [PERSON] [26] formulated several examples of such equations for the steel ribbon, having three hinges. The proposed approach to the choice of the efficient geometry of the ribbon cross-section, expressed in terms of the flexural stiffness, employs an iterative calculus accounting for ULS (ultimate limit state) and SLS (serviceability limit state) limitations. The following expression defines the required second moment of inertia of the ribbon subjected to an unsymmetrical load distributed over the bridge half span, using the beam analogy and satisfying condition (1):  
\[I_{y}\geq\frac{1}{156}.\frac{q_{eq}^{*}!q_{q}^{4}}{u_{q,lim}\cdot E} \tag{3}\]  
where \(u_{q,lim}\) is the displacement limit; \(q_{eq}^{*}\) is the load uniformly distributed over the span length \(l_{q}\) (in this case, \(l_{q}=l/2\)), i.e., \(q_{eq}^{*}\ =\ q/2\); \(E\) is the elasticity modulus. The ULS limits define the condition for determining the cross-section parameters:  
\[H\leq\frac{A\cdot f_{d}}{\beta} \tag{4}\]  
where \(H\) is the thrust force; \(A\) is the cross-section area; \(f_{d}\) is the design strength of the material; the coefficient \(\beta\) is calculated, as follows:  
\[\beta\ =\ 1\ +\ \frac{e}{2\cdot a_{c}^{2}\cdot h_{c}} \tag{5}\]  
where \(a_{c}\) is the shape coefficient; \(h_{c}\) is the cross-section height. The eccentricity \(e\) can be approximated, using the beam analogy, as follows:  
\[e\approx\frac{48\cdot u_{q,lim}\cdot EI_{y}}{5\cdot l_{q}^{2}\cdot H} \tag{6}\]  
Figure 1: The design scheme of a stress-ribbon bridge. where \(El_{y}\) is the flexural stiffness of the ribbon.