BeerMOM_ISM_C09
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BeerMOM_ISM_C09

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CCHHAAPPTTEERR 99 
 
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use. 
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted 
on a website, in whole or part. 
1373 
 
B
A
y
L
P
x
 
PROBLEM 9.1 
For the loading shown, determine (a) the equation of the elastic 
curve for the cantilever beam AB, (b) the deflection at the free end, 
(c) the slope at the free end. 
 
SOLUTION 
 0: ( ) 0JM M P L x\uf053 \uf03d \uf02d \uf02d \uf02d \uf03d 
 ( )M P L x\uf03d \uf02d \uf02d 
 
2
2 ( )
d yEI P L x PL Px
dx
\uf03d \uf02d \uf02d \uf03d \uf02d \uf02b 
 2 1
1
2
dyEI PLx Px C
dx
\uf03d \uf02d \uf02b \uf02b 
 1 10, 0 : 0 0 0 0
\uf0e9 \uf0f9\uf03d \uf03d \uf03d \uf02d \uf02b \uf02b \uf03d\uf0ea \uf0fa\uf0eb \uf0fb
dyx C C
dx
 
 2 3 1 2
1 1
2 6
EIy PLx Px C x C\uf03d \uf02d \uf02b \uf02b \uf02b 
 2 2[ 0, 0] : 0 0 0 0 0\uf03d \uf03d \uf03d \uf02d \uf02b \uf02b \uf02b \uf03dx y C C 
(a) Elastic curve. 
2
(3 )
6
Pxy L x
EI
\uf03d \uf02d \uf02d \uf074 
 (2 )
2
dy Px L x
dx EI
\uf03d \uf02d \uf02d 
(b) at .y x L\uf03d 
2 3
(3 )
6 3B
PL PLy L L
EI EI
\uf03d \uf02d \uf02d \uf03d \uf02d 
3
3B
PLy
EI
\uf03d \uf0af \uf074 
(c) at .dy x L
dx
\uf03d 
2
(2 )
2 2B
dy PL PLL L
dx EI EI
\uf03d \uf02d \uf02d \uf03d \uf02d 
2
2B
PL
EI
\uf071 \uf03d \uf074\uf020
 
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use. 
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted 
on a website, in whole or part. 
1374 
 
BA
y
L
x
M0
 
PROBLEM 9.2 
For the loading shown, determine (a) the equation of the elastic curve 
for the cantilever beam AB, (b) the deflection at the free end, (c) the 
slope at the free end. 
 
SOLUTION 
 0
0
0 : 0KM M M
M M
\uf03d \uf02d \uf02b \uf03d\uf0e5
\uf03d
 
 
2
0
0 1
d yEI M M
dx
dyEI M x C
dx
\uf03d \uf03d
\uf03d \uf02b
 
 0 1 1 0, 0 : 0
\uf0e9 \uf0f9\uf03d \uf03d \uf03d \uf02b \uf03d \uf02d\uf0ea \uf0fa\uf0eb \uf0fb
dyx L M L C C M L
dx
 
 20 1 2
1
2
EIy M x C x C\uf03d \uf02b \uf02b 
 2 2 20 0 2 2 0
1 1[ , 0] 0
2 2
x L y M L M L C C M L\uf03d \uf03d \uf03d \uf02d \uf02b \uf03d 
(a) Elastic curve: 2 20 ( 2 )
2
My x Lx L
EI
\uf03d \uf02d \uf02b \uf074\uf020
 20 ( )
2
My L x
EI
\uf03d \uf02d \uf074\uf020
(b) at 0:y x \uf03d 20 ( 0)
2A
My L
EI
\uf03d \uf02d 
2
0
2A
M Ly
EI
\uf03d \uf0ad \uf074\uf020
(c) at 0:dy x
dx
\uf03d 0 0 0( ) ( 0)dy M M M LL x L
dx EI EI EI
\uf03d \uf02d \uf02d \uf03d \uf02d \uf02d \uf03d \uf02d 
 0\uf071 \uf03dA M LEI \uf074\uf020
 
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use. 
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted 
on a website, in whole or part. 
1375 
 
B
A
y
w
L
x
PROBLEM 9.3 
For the loading shown, determine (a) the equation of the elastic curve 
for the cantilever beam AB, (b) the deflection at the free end, (c) the 
slope at the free end. 
 
SOLUTION 
 
2
0: ( ) 0
2
1
2
J
xM wx M
M wx
\uf053 \uf03d \uf02b \uf03d
\uf03d \uf02d
 
 
2
2
2
3
1
1
2
1
6
d yEI M wx
dx
dyEI wx C
dx
\uf03d \uf03d \uf02d
\uf03d \uf02d \uf02b
 
 3 31 1
1 1, 0 : 0
6 6
\uf0e9 \uf0f9\uf03d \uf03d \uf03d \uf02d \uf02b \uf03d\uf0ea \uf0fa\uf0eb \uf0fb
dyx L wL C C wL
dx
 
 3 31 1
6 6
dyEI wx wL
dx
\uf03d \uf02d \uf02b 
 4 3 2
1 1
24 6
EIy wx wL x C\uf03d \uf02d \uf02b \uf02b 
 4 4 2
1 1[ , 0] 0 0
24 6
x L y wL wL C\uf03d \uf03d \uf03d \uf02d \uf02b \uf02b \uf03d 
 4 42
1 1 3
24 6 24
C wL wL\uf0e6 \uf0f6\uf03d \uf02d \uf03d \uf02d\uf0e7 \uf0f7\uf0e8 \uf0f8 
(a) Elastic curve. 4 3 4( 4 3 )
24
wy x L x L
EI
\uf03d \uf02d \uf02d \uf02b \uf074 
(b) at 0.y x \uf03d 
4 43
24 8A
wL wLy
EI EI
\uf03d \uf02d \uf03d \uf02d 
4
8A
wLy
EI
\uf03d \uf0af \uf074\uf020
(c) at 0.dy x
dx
\uf03d 
3
6A
dy wL
dx EI
\uf03d 
3
6A
wL
EI
\uf071 \uf03d \uf074 
 
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use. 
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted 
on a website, in whole or part. 
1376 
 
w0
x
B
A
y
L 
PROBLEM 9.4 
For the loading shown, determine (a) the equation of the elastic curve 
for the cantilever beam AB, (b) the deflection at the free end, (c) the 
slope at the free end. 
 
SOLUTION 
Use Free body AJ. 
 
 
0:\uf053 \uf03dJM
2
0 0
2 3
\uf02b \uf0d7 \uf03dw x xM
L
 
 [ , 0]
, 0
\uf03d \uf03d
\uf0e9 \uf0f9\uf03d \uf03d\uf0ea \uf0fa\uf0eb \uf0fb
x L y
dyx L
dx
 
3
01
6
\uf03d \uf02d w xM
L
 
 
2 3
01
6
\uf03d \uf02dd y w x
dx L
 
 
4
0
1
1
24
\uf03d \uf02d \uf02bdy w xEI C
dx L
 
 
5
0
1 2
1
120
\uf03d \uf02d \uf02b \uf02bw xEIy C x C
L
 
 , 0 :dyx L
dx
\uf0e9 \uf0f9\uf03d \uf03d\uf0ea \uf0fa\uf0eb \uf0fb
3
0 1
1 0
24
w L C\uf02d \uf02b \uf03d 
 31 0
1
24
\uf03dC w L 
 [ , 0]\uf03d \uf03dx L y 4 40 0 21 1 0120 24\uf03d \uf02d \uf02b \uf02b \uf03dEIy w L w L C 
 42 0
1
30
C w L\uf03d 
(a) Elastic curve. 5 4 50 ( 5 4 )
120
\uf03d \uf02d \uf02d \uf02bwy x L x L
EIL
 \uf074 
 4 40 ( )
24
\uf03d \uf02d \uf02bdy w x L
dx EIL
 
(b) @ 0:\uf03dy x 
4
0
30
\uf03d \uf02dA w Ly EI 
4
0
30
\uf03d \uf0afA w Ly EI \u25c4 
(c) @ 0:\uf03ddy x
dx
 
3
0
24
\uf03d
A
dy w L
dx EI
 
3
0
24
\uf071 \uf03dA w LEI \uf074 
 
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use. 
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted 
on a website, in whole or part. 
1377 
 
y
A
w
B
C
x
P 5 23
wa
a2a 
PROBLEM 9.5 
For the cantilever beam and loading shown, determine (a) the equation of 
the elastic curve for portion AB of the beam, (b) the deflection at B, (c) the 
slope at B. 
 
SOLUTION 
 
Using ABC as a free body, 
 20: 2 0
3
4 4
3 3
y A
A
F R wa wa
R wa wa
\uf053 \uf03d \uf02b \uf02d \uf03d
\uf03d \uf02d \uf03d \uf0af
 
 20: (2 )( ) (3 ) 0
3A A
M M wa a wa a\uf0e6 \uf0f6\uf053 \uf03d \uf02d \uf02b \uf02d \uf03d\uf0e7 \uf0f7\uf0e8 \uf0f8 
 0AM \uf03d 
Using AJ as a free body, 
 
2
40: ( ) ( ) 0
3 2
1 4
2 3
J
xM M wa x wx
M wx wax
\uf0e6 \uf0f6 \uf0e6 \uf0f6\uf053 \uf03d \uf02b \uf02d \uf03d\uf0e7 \uf0f7 \uf0e7 \uf0f7\uf0e8 \uf0f8 \uf0e8 \uf0f8
\uf03d \uf02d
 
 
2
2
2
3 2
1
1 4
2 3
1 2
6 3
d yEI wx wax
dx
dyEI wx wax C
dx
\uf03d \uf02d
\uf03d \uf02d \uf02b
 
 1 10, 0 : 0 0 0 0
dyx C C
dx
\uf0e9 \uf0f9\uf03d \uf03d \uf03d \uf02d \uf02b \uf05c \uf03d\uf0ea \uf0fa\uf0eb \uf0fb 
 
4 3
2
1 2
24 9
EIy wx wax C\uf03d \uf02d \uf02b 
 2 2[ 0, 0]: 0 0 0 0x y C C\uf03d \uf03d \uf03d \uf02d \uf02b \uf05c \uf03d 
(a) Elastic curve over AB. 
4 3(3 16 )
72
wy x ax
EI
\uf03d \uf02d \uf074\uf020
\uf020 \uf020 3 2( 4 )
6
dy w x ax
dx EI
\uf03d \uf02d 
(b) at 2 .y x a\uf03d 
410
9B
way
EI
\uf03d \uf02d 
410
9B
way
EI
\uf03d \uf0af \uf074 
(c) at 2 .dy x a
dx
\uf03d 
34
3B
dy wa
dx EI
\uf0e6 \uf0f6 \uf03d \uf02d\uf0e7 \uf0f7\uf0e8 \uf0f8 
34
3B
wa
EI
\uf071 \uf03d \uf074\uf020
 
FBD ABC: 
FBD AJ: 
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use. 
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted 
on a website, in whole or part. 
1378 
 
y
A
w
B
L a
C x
MC 5
wL2
6
 
PROBLEM 9.6 
For the cantilever beam and loading shown, determine (a) the equation of 
the elastic curve for portion AB of the beam, (b) the deflection at B, (c) the 
slope at B. 
 
SOLUTION 
Using ABC as a free body, 
 
 
0:\uf053 \uf03dyF 0\uf02d \uf03d \uf03dA AR wL R wL 
 [ 0, 0]
0, 0
x y
dyx
dx
\uf03d \uf03d
\uf0e9 \uf0f9\uf03d \uf03d\uf0ea \uf0fa\uf0eb \uf0fb
 
 
0:\uf053 \uf03dAM
2
2
( ) 0
2 6
1
3
\uf0e6 \uf0f6\uf02d \uf02d \uf02b \uf03d\uf0e7 \uf0f7\uf0e8 \uf0f8
\uf03d \uf02d
A
A
L wLM wL
M wL
 
 Using AJ as a free body (portion AB only), 
 
 
0:\uf03dJM ( ) 02
\uf0e6 \uf0f6\uf02b \uf02d \uf02d \uf03d\uf0e7 \uf0f7\uf0e8 \uf0f8 A A
xM wx R x M 
 21
2
\uf03d \uf02d \uf02b \uf02bA AM wx R x M 
 2 21 1
2 3
\uf03d \uf02d \uf02b \uf02dwx wLx wL 
 
2
2 2
2
1 1
2 3
\uf03d \uf02d \uf02b \uf02dd yEI wx wLx wL
dx
 
 3 2 1
1 1 1
6 2 3
\uf03d \uf02d \uf02b \uf02d \uf02bdyEI wx wLx wLx C
dx
 
 1 10, 0 : 0 0 0 0 0
dyx C C
dx
\uf0e9 \uf0f9\uf03d \uf03d \uf02d \uf02b \uf02d \uf02b \uf03d \uf03d\uf0ea \uf0fa\uf0eb \uf0fb 
 4 3 2 2
1 1 1
24 6 6
\uf03d \uf02d \uf02b \uf02d \uf02byEI wx wLx wLx C 
 2 2[ 0, 0]: 0 0 0 0 0\uf03d \uf03d \uf02d \uf02b \uf02d \uf02b \uf03d \uf03dx y C C 
(a) Elastic curve over AB. 4 3 2 2( 4 4 )
24
\uf03d \uf02d \uf02b