# Nonlinear Stiffness of Semi-Fixed Dowel Joints in Semi-Integral Bridges
## 5 Stiffness Analysis
### Radial Stiffness  
The horizontal displacement of the upper concrete block of the specimen is illustrated in Figure 7a. The ratio of the actual horizontal force (\(F_{2}\)) on the joint to the horizontal displacement of the upper part of the specimen is the radial stiffness of the specimen in longitudinal direction. For the specimens in this paper, the friction coefficients in Formulas (4) and (5) were set as \(\mu_{1}=0.3\)[19] and \(\mu_{2}=0.8\)[20], respectively. The radial stiffness and rotational stiffness of the joint are obtained by substituting them into the test results. Figure 13a gives the curves of loading displacement-radial stiffness of specimens JD1, JD2 and JD3 with the wall thickness of the rubber sleeve as the parameter. The radial stiffness of the specimens decreased greatly before the loading displacement of 16 mm, and then gently decreased, and the curves were approximately straight lines. The radial stiffness of the joint increases with the decrease of the wall thickness of rubber sleeve. The maximum radial stiffness of JD1, JD2 and JD3 are 27.7 kN/mm, 12.6 kN/mm and 8.4 kN/mm, respectively. From Figure 13b, it can be found that the radial stiffness of JD2 is slightly smaller than that of JD4, the maximum difference between them is only 6.1%, indicating that the thickness of the rubber pad has little influence on the radial stiffness. Figure 12: Isolator for calculation of force on the joint (unit: mm). Radial deformation of the rubber sleeve in the tube can be obtained by assuming that the sleeve is composed of two independent rubber blocks; one block has a compressive deformation, while the other one has no deformation [21]. The size of the rubber block can be taken as follows: the length is \(l\) (height of the rubber sleeve), the width is half of the average circumference, and the thickness \(\delta\) is the thickness of the rubber dowel sleeve. Based on the force balance of the rubber block, the radial stiffness of the rubber sleeve in the tube is as follows:  
\[k_{r}^{\prime}=\frac{E\cdot A}{\delta}=\frac{E\pi l(r_{1}+r_{2})}{2\delta} \tag{6}\]  
where \(k_{r}^{\prime}\) is the radial stiffness of rubber sleeve in tube (N/mm), \(l\) is the height of the sleeve (mm); \(r_{1}\) and \(r_{2}\) are the inner and outer radii of the rubber sleeve (mm), respectively; \(\delta\) is the wall thickness of the rubber sleeve (mm), and \(\delta=r_{2}-r_{1}\); \(E\) is modulus of elasticity of rubber (N/mm\({}^{2}\)). The lower end of the rubber sleeve in the semi-fixed dowel joint is constrained by the steel dowel, the upper end of the rubber sleeve is free, and the force on the rubber sleeve is not uniform. It can be seen from Figure 13 that the radial stiffness of the joint varies nonlinearly with the loading displacement. The ratio of it to the loading displacement \(X\) and the inner radius of rubber sleeve \(r_{1}\) is logarithmic. It can be calculated by Formula (7). \[k_{r}=k_{r}^{\prime}\cdot[\alpha\cdot\ln(\frac{X}{r_{1}})+\beta] \tag{7}\]  
where \(k_{r}\) is the radial stiffness of the semi-fixed dowel joint (N/mm); \(X\) is the loading displacement (mm); \(\alpha\) and \(\beta\) are coefficients. The coefficients \(\alpha\) and \(\beta\) are undetermined, related to the inner radius \(r_{1}\) and outer radius \(r_{2}\) of the rubber sleeve, and they are dimensionless. After regression of test data, the following formula can be obtained:  
\[\alpha=-0.23\cdot\frac{r_{2}+r_{1}}{r_{2}-r_{1}}\]  
\[\beta=0.7\cdot\frac{r_{2}+r_{1}}{r_{2}-r_{1}}\]  
By substituting \(\alpha\) and \(\beta\) into Formula (7), the calculation of Formula (8) of radial stiffness can be obtained. \[k_{r}=\frac{E\pi l(r_{1}+r_{2})^{2}}{\delta^{2}}\cdot\left[0.35-0.115\ln\left( \frac{X}{r_{1}}\right)\right]\ \ \ \ \ X\leq 30 \tag{8}\]  
where \(X\) is the loaded displacement, which is taken as 2 mm when \(X\) is less than 2 mm, the maximum loading displacement is 30 mm in the process of test. Figure 13: Loading displacement-radial stiffness curve of semi-fixed dowel joint. (**a**) Wall thickness of rubber sleeve.