sm12 89

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PROBLEM (*12.89) The bumper of a car is constructed with a molded plastic material of modulus
of elasticity E = 0.3 x 106 psi. If the car has a weight W = 1.8 kips and is coasting at a speed of v = 2
mph when it strikes a rigid post (Fig. P12.89a), determine:
(a) The maximum deflection of the bumper.
(b) The maximum stress in the bumper.
Assumption: The bumper can be modeled as beam supported on two springs, attached to the rigid frame of
the car (Fig. P12.89b).
Given: I = 750 in.4 c = 3 in. y
σ
= 4 ksi. (bumper); ks = 15 kip/in. (spring)
*SOLUTION
Equilibrium. max
1
2
s
FP=:
2b
ss k
k
δ
δ = or 2
s
b
s
k
k
δ
δ = (1)
Conservation of energy.
2
22 2 2 2
11 1 1
2( )
22 2 24
bss bb
s
k
mv k k k k
δ
δδδ =+ =+
Solving
2
2
b
s
m
vk
kk
δ =
+
(2)
where
32
1.8(10 ) 4.658 .
32.2(12)
w
ml
g
== = ⋅
bins
2(5280 12)
2 35.2
3600
vmph ip
s
×
== =
6
33
48 48(0.3 10 )(750) 50 .
(60)
EI
k kip in
L
×
== =
15 .
s
k kip in=
So
2
3
4.658
35.2 0.251 .
50
10 (50 )
215
bin
δ ==
+×
From Eq. (1),
50 (0.251) 0.418 .
22(15)
sb
s
kin
k
δδ == =
Maximum Deflection
max 0.251 0.418
bs
δδ =+= +
δ
0.669 .in=
Continued on next slide
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Maximum Stress
Maximum force on the beam
max 50(0.251) 12.55
b
Pk kips
δ
== =
Maximum moment is at midspan,
max
max 12.55(60) 188.3 .
44
PL
M
kip in== = ⋅
Hence
max
max 188.3(3) 0.753
750
Mc ksi
I
σ == =
Since max 4,
y
σ
σ <= the solution is valid.
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or
instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright owner is unlawful.