9.7 Response of Beams Due to Moving Loads
391
9.9 Deduce the equation of motion for forced flexural vibrations of a beam
subjected to an applied force. Consider the material of the beam as
viscoelastic, for which the stress–strain curve is
σ n = E
σ x + c
∂ε x
∂t
9.10 For a uniform cantilever beam, determine the maximum deflection at the free
end and maximum bending stress at the clamped end due to half-sine pulse.
9.11 A simply supported beam is subjected to a vertical motion of right support,
of amount
y(L , t) = y o sin ωt
If at t = 0, the velocity and deflection at every point of the beam is zero, what
will be the deflection equation for any time thereafter?
9.12 A vehicle having small mass exerts a pressure P on the deck beam. The
magnitude of P remains constant as it crosses the span with uniform velocity
v. Derive the equation of the deflection curve of the beam.
9.13 A simply supported bridge of span 20 m, EI = 8 GN m
2 and ρ A = 20,000
kg/m is subjected to a load P sin 2π f t, which moves with a velocity 20 m/s.
If f = 5 Hz, what is the largest amplitude of vibration in the bridge?
9.14 Determine the expressions for the displacement and bending moment of a
uniform simply supported beam to a step function force p 0 at a distance from
the left end.
Prob. 9.14
9.15 Determine the response of a cantilever beam when its base is given a motion
y b (t) normal to the beam axis as shown in the figure.
Prob. 9.15
391
9.9 Deduce the equation of motion for forced flexural vibrations of a beam
subjected to an applied force. Consider the material of the beam as
viscoelastic, for which the stress–strain curve is
σ n = E
σ x + c
∂ε x
∂t
9.10 For a uniform cantilever beam, determine the maximum deflection at the free
end and maximum bending stress at the clamped end due to half-sine pulse.
9.11 A simply supported beam is subjected to a vertical motion of right support,
of amount
y(L , t) = y o sin ωt
If at t = 0, the velocity and deflection at every point of the beam is zero, what
will be the deflection equation for any time thereafter?
9.12 A vehicle having small mass exerts a pressure P on the deck beam. The
magnitude of P remains constant as it crosses the span with uniform velocity
v. Derive the equation of the deflection curve of the beam.
9.13 A simply supported bridge of span 20 m, EI = 8 GN m
2 and ρ A = 20,000
kg/m is subjected to a load P sin 2π f t, which moves with a velocity 20 m/s.
If f = 5 Hz, what is the largest amplitude of vibration in the bridge?
9.14 Determine the expressions for the displacement and bending moment of a
uniform simply supported beam to a step function force p 0 at a distance from
the left end.
Prob. 9.14
9.15 Determine the response of a cantilever beam when its base is given a motion
y b (t) normal to the beam axis as shown in the figure.
Prob. 9.15