Dynamic response of bridge affected by periodic roughness with a long spatial wavelength

Abstract

An object prestressed concrete bridge with a 37.47 m span that vibrated greatly upon a truck running on it was investigated through examination and monitoring to clarify the cause of the vibrations. The results of running examinations showed that the bridge vibrated greatly only when the test truck ran in the 2nd slow lane, and the bridge resonated with the truck suspension’s spring vibration that was caused by the road roughness with a certain spatial wavelength in that lane. Then, monitoring was conducted in a 2.5 year period before and after the repair of the periodic roughness to detect the abnormal stress due to large dynamic amplification and to confirm the effect of reduction and validation of the repair under the movement of ordinary trucks. The monitoring results showed that the diurnal maximums of the dynamic stress after repair decreased by 40 %, and the diurnal maximum dynamic increment factor DIF-1 after repair decreased to approximately one-third. Moreover, dynamic response analysis was performed to investigate the relation among the bridge dynamic amplification and various periodic roughnesses with long spatial wavelengths at the expansion joint. The simulation results showed that a periodic roughness with a spatial wavelength of 10 m affected the dynamic amplification of the bridge when the truck model runs over it at a speed of 80 km/h.

Appendix

The following steps are carried out for solving the bridge-truck interaction system:

• 1. Form mass M, damping C, and stiffness K matrices of the bridge and truck models.

• 2. Initialize displacement, velocity, and acceleration vectors of the bridge and truck.

  

  (A1)

• 3. Select time step size Δt and integration parameters in the Newmark β method α, β, a₀–a_7.

  $$\alpha \geqslant 0.5, \, \beta \geqslant 0.25{(0.5 + \alpha )^2}.$$

  (A2)
  $${{\text{a}}_0} = \frac{1}{{\beta \Delta {t^2}}},\,{{\text{a}}_1} = \frac{\alpha }{\beta \Delta t},{{\text{a}}_2} = \frac{1}{\beta \Delta t},{{\text{a}}_3} = \frac{1}{2\beta } - 1,{{\text{a}}_4} = \frac{\alpha }{\beta } - 1,{{\text{a}}_5} = \frac{\Delta t}{2}(\frac{\alpha }{\beta } - 2),{{\text{a}}_6} = \Delta t(1 - \alpha ),{{\text{a}}_7} = \alpha \Delta t.$$

  (A3)

• 4. Form the effective stiffness matrix ${\hat K_b}$ and ${\hat K_v}$ for the bridge and truck, respectively.

  $${\hat K_b} = {K_b} + {{\text{a}}_0}{M_b} + {{\text{a}}_1}{C_b}.$$

  (A4)
  $${\hat K_v} = {K_v} + {{\text{a}}_0}{M_v} + {{\text{a}}_1}{C_v}.$$

  (A5)

• 5. Check the current position of the truck at time t. If the contact point of the truck wheel lies outside the bridge, each reaction force of the truck $^{t + \varDelta t}{f_v}$ is calculated using the road roughness of the approach road at the contact point. Furthermore, if the contact point of the truck wheel lies on the bridge, each reaction force of the truck $^{t + \varDelta t}{f_v}$ is calculated using the displacement and road roughness of the bridge at the contact point.

  

  (A8)

• 6. Form the force vector of the bridge $^{t + \varDelta t}{f_b}$ using the reaction force of the truck if the truck acts on the bridge.

• 7. Solve the displacement of the bridge ${}^{t + \varDelta t}{u_b}$.

  

  (A9)

• 8. Using the calculated new displacement ${}^{t + \varDelta t}{u_b}$, iterate steps 5–7 until the convergent value ${}^{t + \varDelta t}{u_b}$ is obtained.

• 9. Calculate the acceleration and velocity of the bridge and truck at time t + Δt.

  

  (A10)
  

  (A11)
  

  (A12)
  

  (A13)

• 10. Go to step 5 by setting the next time t = t + Δt.

  

  (A14)
  

  (A15)