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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
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
(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```