9.7 Response of Beams Due to Moving Loads
387
y =
Px
2
(L − x)
2
3E I L
(9.72)
Due to application of P and its movement, the beam is vibrating. In addition to
static load P, there is an inertia force associated with its vertical movement. Therefore,
total force at the point of application of P is
R = P −
ρ
g
d
2 y
dt 2
(9.73)
If the load moves with a constant velocity v on the beam, then
dy
dt
= v
dy
dx
(9.74)
Differentiating both sides of Eq. (9.74) with respect to t, we get
d
2 y
dt 2 = v
d
dt
dy
dx
= v
d
dx
dy
dt
or
d
2 y
dt 2 = v
d
dx
v ·
dy
dx
= v
2 d
2 y
dx 2
(9.75)
Combining Eqs. (9.61) and (9.63), we get
R = P
1 −
v
2
g
d
2 y
dx 2
(9.76)
Replacing P by R given in Eqs. (9.76), (9.72) results in
y = P
1 −
v
2
g
d
2 y
dx 2
x
2
(L − x)
2
3E I L
(9.77)
Equation (9.77) indicates the path of contact of the rolling load with the beam.
Timoshenko has approximated the solution of Eq. (9.77) by assuming v = 0, so that
maximum deflection occurs at the centre and the maximum pressure is given by
R max = P
1 +
v
2
g
P L
3E I
(9.78)
Therefore, the maximum deflection at the centre is
y max = y st
1 +
v
2
g
P L
3E I
(9.79)