Dynamic Analysis of Multi-Cracked RC Beams Strengthened with FRP Plates Using the State Space Method

Using the state space method, a novel analytical method is proposed to investigate the dynamic behavior of multi-cracked reinforced concrete (RC) beams strengthened with fiber-reinforced polymer (FRP) plates, while taking damping effects into account. The flexural cracks in RC beams are modeled as rotational springs, with their stiffness dependent on the crack depths. A numerically stable technique and a frequency-scanning strategy are employed to solve the homogeneous state equations associated with mode function vectors and the frequency equation, respectively. This allows for the determination of natural frequencies and corresponding modal shapes of FRP-strengthened multi-cracked RC beams under generalized boundary conditions. The orthogonality relation of vibration modes is established based on the state space formulae and the concept of symplectic inner product. Analytical solutions for the dynamic responses of FRP-strengthened multi-cracked RC beams subjected to arbitrary dynamic loads are derived using the orthogonality relation and the mode superposition method. Numerical examples are provided to predict the dynamic responses of strengthened cracked beams under a dynamic uniformly distributed step load and a concentrated triangular impulse load. The effectiveness of the proposed analytical method is validated through comparisons with finite element simulations and experimental results. Furthermore, parametric studies are performed to analyze the effects of damping ratio, crack depth, crack position and the number of cracks on the dynamic responses of these strengthened cracked beams. The results demonstrate that the strengthening strategy using externally bonded FRP plates is effective, while the effects of cracks and the first-order damping ratio on the dynamic responses are significant and cannot be ignored. Additionally, the proposed method can efficiently and accurately analyze the dynamic behavior of FRP-strengthened RC beams with arbitrary distributions of cracks and dynamic loads.