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

so that the impact factor can be written as
δmax
2h
n = = 1+ 1+
δ
δ
st
Since the static deflection was calculated previously as δ st = 0.025465 mm, the
impact factor for the 30 mm drop height is
st
n = 2(30 mm)
1 + 1 + 0.025465 mm = 49.551
Ans.
The static results can now be multiplied by the impact factor to give the dynamic
deformation, force, and stress:
δ
σ
max
= nδst
= 49.551(0.025465 mm) = 1.262 mm
Ans.
F = nF = 49.551(1,200 N) = 59,461 N
Ans.
max
max
st
= nσ
= 49.551(6.79061 MPa) = 336 MPa
Ans.
st
ExAmpLE 17.6
The simply supported beam shown in the figure spans 20 ft.
The beam consists of a W14 × 34 shape of A992 steel [E =
29 × 10 6 psi]. Calculate the deflection at B and the largest
bending stress in the beam if the 3,000 lb load is
(a) applied statically.
(b) dropped from a height of h = 5 in.
Plan the Solution
The beam deflection and the maximum normal stress at B due to a static loading are determined
for the 3,000 lb load. Work and energy principles can be used to equate the work
done by the load to the strain energy stored in the beam at the instant of maximum deflection.
From this energy balance, the maximum beam deflection at B can be calculated. The
maximum deformation can then be used to determine the maximum dynamic load, the
corresponding bending stress, and the impact factor n.
SolutioN
(a) load Applied Statically. Applying the load statically means applying the 3,000 lb
load slowly and gradually from a height of h = 0. The W14 × 34 shape used for the beam
has the following section properties:
d = 14.0 in.
I = 340 in. 4
Deflection at B: The deflection formula found in Appendix C for a simply supported
beam with a concentrated load applied anywhere will be used here. For this span, L =
20 ft = 240 in., a = 12 ft = 144 in., and b = 8 ft = 96 in. The 3,000 lb static load produces
a downward deflection at B of
v
st
2 2 2 2
Pab st
(3,000 lb)(144 in.) (96 in.)
= =
6 2 4
3LEI 3(240 in.)(29×
10 lb/in. )(340 in. )
= 0.080757 in. = 0.0808 in.
Ans.
A
W14 × 34
3,000 lb
B
h
12 ft 8 ft
C
x
735