Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

Note: Throughout the previous chapters in this book, the symbol v has been used to denote
deflection perpendicular to the longitudinal axis of a beam. In this chapter, velocity
has been introduced as an additional consideration, and the symbol v is also used to
denote velocity. In this example, beam deflections are being considered; therefore, the
symbol v used in this context represents beam deflections.
largest Bending Stress: From Example 17.4, the static beam reaction at A is
A
y
Pb st
=
L
Therefore, the maximum bending moment (which occurs at B) is
M
st
Pb st
A a
L a (3,000 lb)(96 in.)
= y = = (144 in.) = 172,800 lb⋅in.
240 in.
The largest bending stress in the beam is
Mstc
(172,800 lb⋅in.)(14in./2)
σ st = =
= 3,557.65 psi = 3,560 psi Ans.
4
I
340 in.
(b) load Dropped from h = 5 in. The maximum beam deflection when the load is
dropped can be determined from work and energy principles. The external work done
by the 3,000 lb load dropped from height h must equal the strain energy stored by the
beam at its maximum deflection. Recall from Section 17.5 that the strain energy
stored in a flexural member can be expressed in terms of the member deformation by
Equation (17.20):
U =
L
∫ 0
2
M
2EI dx
The total elastic strain energy U for this type of beam and loading was derived in
Example 17.4:
U =
Pab 2 2 2
6LEI
Deflection at B: Equate the internal strain energy of the beam to the work done by gravity
as the 3,000 lb load moves downward:
Since the static deflection at B is
External work = Internal strain enery
2 2 2
Pmaxa b 3LEI
Pst
( h + vmax
) = =
6LEI
2ab v
3LEI
2ab v 2
P ( h v ) 0
2 2 max - st + max =
2 2
2
2Pab
st
vmax
- ( h + vmax
) = 0
3LEI
v
st
Pab st
=
3LEI
the quadratic equation in v max can be rewritten as
2 2
2
v - 2 v ( h + v ) = 0
max
st
max
2
2 2 max
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