sm12 12 15

________________________________________________________________________
PROBLEMS (12.12 through *12.15) A beam is supported and loaded as shown in Figs. P12.12
through P12.15. Determine the strain energy of the beam caused by bending.
SOLUTION (12.12)
A
B
x
2
1
2
x
M
wx=−
L
x
w
22
22
0
11 1
()
22 2 40
L5
M
dx w L
Uwxdx
EI EI EI
==− =
∫∫
SOLUTION (12.13)
Segment AC
22 23
2
00
22 6
aa
AB
M
dx P P a
Uxdx
EI EI EI
== =
∫∫
Segment CD
22 23
3
02
a
CD Pa Pa
Udx
EI EI
==
∫
=
By symmetry: UU.
AC BD
Total strain energy :
323 2
4
263
Pa Pa Pa
3
EI EI EI
=+=
U
x
P
a
B
a MAC=Px
MCD=Pa
D
2a
A
P
P
a
P
C
P
M
Continued on next slide
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SOLUTION (12.14)
22
1()
222
wL wx
M
xwLx=−= −x
22
22
00
1()
224
LL
Mw
Udx Lxx
EI EI
== −
∫∫
dx
222 3 4
0(2 )
8
L
wLx Lx x dx
EI
=−+
∫
After integration
223 4 5
0
2
83 45
L
wLx Lx x
UEI
=−
+25
1
240 wL EI
=
SOLUTION (*12.15)
B
L
A
2
wL
2
wL
w
x
a
B
u
aP
L
A
P
C
L
x
22
0
()
26
a
BC Pu du P a
UEI EI
−
==
∫3
222
0
()
26
L
AB
aPx PaL
L
Udx
EI EI
−
==
∫
Total strain energy
23 22 22
()
666
Pa PaL Pa
UL a
EI EI EI
=+ = +
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.