# Assessment of Modal Characteristics of Steel-Concrete Composite Grider Bridge with V-Shaped Piers
## 4 Finite Element Model Optimization  
The two-unit rod system model established by the conventional method cannot accurately simulate the shear joint action because the concrete deck slab and the steel main beam units are directly solidified, and the solid model would make the calculation too complicated. Here, the equivalent shear stiffness is used to optimize the finite element model to simulate the action of the shear connectors.  
The stiffness parameter of a single shear nail can be calculated from Equation (1) according to the code [20] for welded nail connectors:  
\[\mathrm{k}_{\mathrm{ss}}=13.0d_{\mathrm{ss}}\sqrt{E_{c}f_{ck}} \tag{1}\]  
Among them:  
\(\mathrm{k}_{\mathrm{ss}}\) indicates the shear stiffness of the peg (N/mm);  
\(d_{\mathrm{ss}}\) indicates the diameter of the peg (mm);  
\(E_{c}\) indicates the concrete elastic modulus (MPa);  
\(f_{ck}\) indicates the standard value of compressive strength of concrete (MPa).  
The vertical stiffness of the shear nail can be viewed as a whole as an axially tensioned bar; its vertical deformation can be regarded as a vertical lift-off displacement. The stress is proportional to the strain when it is in the elastic phase, and the calculation formula is shown in Equation (2), which can simplify its vertical stiffness to 10 times the lateral stiffness [21].  
\[k_{v}=\frac{E_{\mathrm{sf}}A_{\mathrm{sf}}}{l} \tag{2}\]  
Among them:  
\(k_{v}\) indicates the vertical stiffness of the peg (N/mm);  
\(E_{\mathrm{sf}}\) indicates the modulus of elasticity of the peg (MPa);  
\(A_{\mathrm{sf}}\) indicates the cross-sectional area of the spigot bar (mm\({}^{2}\));  
\(l\) indicates the length of the spigot rod (mm).  
For the equivalent shear bearing capacity of the shear nail group, the formula for the equivalent stiffness of the shear nail group is proposed by calculating the strain energy generated by the deformation of the shear nail and concrete [22]. When the shear nail group is in the elastic deformation, the calculation formula is  
\[\mathrm{K}=\mathrm{n}\beta.\frac{(b-d)^{2}}{h}G_{c} \tag{3}\]  
Figure 6: (**a**) Schematic diagram of combined section force transmission; (**b**) schematic diagram of double unit section force transmission.  
Among them;  
n indicates the number of transverse rows of the shear nail group;  
\(\beta\) is a constant related to the material properties of the shear nail;  
\(b\) indicates the total directional spacing along the shear transfer release direction;  
\(d\) indicates the diameter of the shear nail (mm);  
\(h\) denotes the height of the shear nail (mm);  
\(G_{c}\) indicates the shear modulus of concrete (MPa).  
From Equation (3), it can be seen that the equivalent stiffness of the shear nail group when the shear nail is in the elastic phase is only related to the shear nail arrangement form, the shear nail structure form, and the concrete shear modulus of elasticity.  
When the shear nail enters the plastic state, the equivalent shear stiffness of the shear nail group is  
\[\mathrm{K}=\mathrm{n}\frac{\pi d}{4}G_{p} \tag{4}\]  
where \(G_{p}\) denotes the hardening modulus of the steel after it enters the plastic state (MPa).  
The shear connectors are simulated by the equivalent stiffness of the shear nail group and the calculated equivalent stiffness to elastically connect the concrete deck slab units to the steel girder units.