The transverse impact characteristics of a rubber pipe expansion joint are studied. A pair of joints assembled end to end with an inserted middle mass is tested on a drop shock testing machine. Based on the test results, an equivalent fixed-fixed beam model with polynomial stiffness and damping is applied to predict the transverse impact response and identify the nonlinear impact parameters. The least square residual between the computed and test results is defined to drive the identification optimization. The response surface methodology in combination with the generalized reduced gradient method is used to search the best matching coefficients. Final results show that the equivalent bending stiffness of the tested rubber expansion joint gradually decreases with the transverse deformation and is greatly influenced by its internal working pressure.