# Nonlinear Stiffness of Semi-Fixed Dowel Joints in Semi-Integral Bridges
## 5 Stiffness Analysis
### Flexural Stiffness  
Figure 14 shows the curves of loading displacement-flexural stiffness of the tested semi-fixed dowel joints. The flexural stiffness of all the joint specimens exhibited non-linear behavior. They decreased rapidly with the increase of the loading displacement before the loading displacement was 8 mm, and then decreased gently and the curves showed almost straight lines. The flexural stiffness of the joint was greatly affected by the wall thickness of rubber sleeve. The maximum flexural stiffness of JD1, JD2 and JD3 are 334.04 kN\(\cdot\)m/\(\theta\), 450.07 kN\(\cdot\)m/\(\theta\) and 529.79 kN\(\cdot\)m/\(\theta\), respectively.  
With regard to the thickness of the rubber pad, it can be seen from Figure 14 that the flexural stiffness of JD4 was only slightly smaller than that of JD2, with the maximum difference of only 3.7%. Therefore, the very small influence of the thickness of the rubber pad on the flexural stiffness of the joint can be negligible.  
It can be seen from Figure 14 that the flexural stiffness of each joint differs greatly at the loading displacement of 2-4 mm, and each curve in this portion is in straight lines; after that, all curves are close to each other and demonstrate nonlinearity with the loading displacement. Thus, the flexural stiffness of the joint can be expressed by a function of two portions:  
\[k_{M1}=\left[\gamma-\varepsilon\cdot\left(\frac{X}{r_{1}}\right)\right]EI \quad X\leq 4 \tag{9a}\] \[k_{M2}=m\cdot\left(\frac{X}{r_{1}}\right)^{n}\cdot EI\quad 4<X\leq 30 \tag{9b}\]  
where \(k_{M1}\) and \(k_{M2}\) are, respectively, flexural stiffness of the semi-fixed dowel joint in a straight line and a curve (N\(\cdot\)mm/\(\theta\)). \(I\) is the inertia moment of the rubber sleeve (mm\({}^{4}\)). \(\gamma\), \(\varepsilon\), \(m\) and \(n\) are coefficients and dimensionless, and \(\gamma\), \(\varepsilon\) and \(m\) are related to the inner radius \(r_{1}\) and outer radius \(r_{2}\) of the rubber sleeve. \(X\) is the loaded displacement (mm), which is taken as 2 mm when \(X\) is less than 2 mm in Formula (9a), due to the same reason as that for Formula (8).  
Figure 14: Loading displacement–flexural stiffness curve of semi-fixed dowel joint. (**a**) Wall thickness of rubber sleeve. (**b**) Thickness of rubber pad.  
After regression of test data, the following formulas can be obtained:  
\[\gamma =0.14\cdot\left(\frac{r_{1}}{r_{2}-r_{1}}\right)^{2.31}\] \[\varepsilon =0.4\cdot\left(\frac{r_{1}}{r_{2}-r_{1}}\right)^{2.31}\] \[m =0.11\cdot\left(\frac{r_{1}}{r_{2}}\right)^{4.1}\] \[n =-1.6\]  
Substituting the above formulas into Formula (9), we can obtain the expressions to predict the flexural stiffness by Formula (10).  
\[K_{M} = \left\{\begin{array}{ll}\left(\frac{r_{1}}{\delta}\right)^{231} \cdot\left[\left(0.15-0.4\cdot\left(\frac{X}{r_{1}}\right)\right]\right)\cdot EI &X\leq 4\\ 0.11\cdot\left(\frac{r_{1}}{r_{2}}\right)^{41}\cdot\left(\frac{X}{r_{1}}\right) ^{\left(-1.6\right)}\cdot EI&4<X\leq 30\end{array}\right. \tag{10}\]  
where \(X\) is the loaded displacement (mm), taken as 2 mm when \(X\) is less than 2 mm, the maximum loading displacement is 30 mm in the process of test.