This problem studies the response of two single degree of freedom bridge systems shown in Figure 1 under three loading cases. The problem has two parts. Part A and Part B use the same loading cases but the system is modified. Assume the following three loading cases in both Part A and Part B: (a) Harmonic wind load acting on the bridge deck pw(t) = powsin(ωwt) with amplitude pow and forcing circular frequency ωw. (b) Harmonic displacement base excitation acting at the base of the bridge pier ug(t) = ugosin(ωgt) with amplitude ugo and displacement circular frequency ωg. (c) Rectangular pulse load acting on the bridge deck with amplitude pop and pulse duration td. Part A The system includes part of a bridge deck and a bridge pier shown in Figure 1(a). For each loading case find the symbolic expression of the peak shear force in the bridge pier assuming the following: • The bridge deck is rigid and it has a mass m. • The bridge deck is rigidly connected with the bridge pier (i.e., monolithic connection). • The base of the bridge pier cannot rotate or translate (i.e., fixed base). • The bridge pier is massless. • The bridge pier has height h, square cross section with dimension b, and modulus of elasticity E. • The top of the bridge pier cannot rotate but it can translate in the horizontal direction (cantilever pier-mass system with stiffness: kp = 12EIh3 ). • The system can be approximated as a damped single degree of freedom (SDOF) system with damping ratio ξ = 2%. • ωw ωn = 0.8, ωg ωn =0.5, and td Tn =0.1 where ωn and Tn are the natural circular frequency and the natural period of the SDOF system, respectively. Part B The system includes part of a bridge deck, a bridge pier, and an elastomeric bearing as shown in Figure 1(b). For each loading case find the symbolic expression of the peak shear force in the bridge pier and the symbolic expression of the displacement of the bridge deck relative to the displacement at the top of the bridge pier. The following assumptions are made: • The bridge deck remains rigid and it has a mass m. • The base of the bridge pier cannot rotate or translate. • The bridge pier remains massless. • The bridge pier has the same height h, square cross section with dimension b, and modulus of elasticity E. • The top of the bridge pier can rotate (pin connection) and it can translate in the horizontal direction. (Pier stiffness: kp = 3EIh3 ) • The bridge deck is connected with the bridge pier through an elastomeric bearing with shear stiffness kb. The shear stiffness of the bearing is equal to 10% of the stiffness of the bridge pier in Part B (Bearing stiffness: kb = 0.1kp). The bearing transfers only shear force at the top of the bridge pier. • The system can be approximated as a damped single degree of freedom (SDOF) system with damping ratio ξ = 10% and a spring with the equivalent stiffness of the two springs in series (i.e., Equivalent stiffness of SDOF system assumes that pier stiffness is in series with the stiffness of the bearing).
This problem studies the response of two single degree of freedom bridge systems shown in Figure 1 under three loading cases. The problem has two parts. Part A and Part B use the same loading cases but the system is modified. Assume the following three loading cases in both Part A and Part B: (a) Harmonic wind load acting on the bridge deck pw(t) = powsin(ωwt) with amplitude pow and forcing circular frequency ωw. (b) Harmonic displacement base excitation acting at the base of the bridge pier ug(t) = ugosin(ωgt) with amplitude ugo and displacement circular frequency ωg. (c) Rectangular pulse load acting on the bridge deck with amplitude pop and pulse duration td.
Part A
The system includes part of a bridge deck and a bridge pier shown in Figure 1(a). For each loading case
find the symbolic expression of the peak shear force in the bridge pier assuming the following:
• The bridge deck is rigid and it has a mass m.
• The bridge deck is rigidly connected with the bridge pier (i.e., monolithic connection).
• The base of the bridge pier cannot rotate or translate (i.e., fixed base).
• The bridge pier is massless.
• The bridge pier has height h, square cross section with dimension b, and modulus of elasticity E.
• The top of the bridge pier cannot rotate but it can translate in the horizontal direction (cantilever
pier-mass system with stiffness: kp = 12EIh3 ).
• The system can be approximated as a damped single degree of freedom (SDOF) system with damping
ratio ξ = 2%.
• ωw
ωn = 0.8, ωg
ωn =0.5, and td
Tn =0.1 where ωn and Tn are the natural circular frequency and the natural
period of the SDOF system, respectively.
Part B
The system includes part of a bridge deck, a bridge pier, and an elastomeric bearing as shown in Figure 1(b).
For each loading case find the symbolic expression of the peak shear force in the bridge pier
and the symbolic expression of the displacement of the bridge deck relative to the displacement
at the top of the bridge pier. The following assumptions are made:
• The bridge deck remains rigid and it has a mass m.
• The base of the bridge pier cannot rotate or translate.
• The bridge pier remains massless.
• The bridge pier has the same height h, square cross section with dimension b, and modulus of elasticity
E.
• The top of the bridge pier can rotate (pin connection) and it can translate in the horizontal direction.
(Pier stiffness: kp = 3EIh3 )
• The bridge deck is connected with the bridge pier through an elastomeric bearing with shear stiffness
kb. The shear stiffness of the bearing is equal to 10% of the stiffness of the bridge pier in Part B
(Bearing stiffness: kb = 0.1kp). The bearing transfers only shear force at the top of the bridge pier.
• The system can be approximated as a damped single degree of freedom (SDOF) system with damping
ratio ξ = 10% and a spring with the equivalent stiffness of the two springs in series (i.e., Equivalent
stiffness of SDOF system assumes that pier stiffness is in series with the stiffness of the bearing).
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