PROBLEM 5.1
w
B
A
L
For the beam and loading shown, (a) draw the shear and bending-moment
diagrams, (b) determine the equations of the shear and bending-moment
curves.
SOLUTION
Reactions:
MB
0:
AL
wL
L
2
0
MA
0:
BL
wL
L
2
0
A
B
wL
2
wL
2
Free body diagram for determining reactions:
Over whole beam,
0
x
L
Place section at x.
Replace distributed load by equivalent concentrated load.
Fy
0:
wL
2
wx
V
0
V
MJ
M
wL
x
2
0:
w
( Lx
2
wx
x
2
M
L
2
x
0
x2 )
M
Maximum bending moment occurs at x
w
w
x( L
2
x)
L
.
2
M max
wL2
8
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681
PROBLEM 5.3
w0
A
B
L
For the beam and loading shown, (a) draw the shear and bendingmoment diagrams, (b) determine the equations of the shear and bendingmoment curves.
SOLUTION
Free body diagram for determining reactions.
Reactions:
Fy
MA
0: RA
0:
MA
w0 L2
3
MA
w0 L
2
0
w0 L
2
RA
w0 L
2
2L
3
0
w0 L2
3
Use portion to left of the section as the free body.
Replace distributed load with equivalent concentrated load.
Fy
0:
w0 L
2
1 w0 x
x
2 L
V
0
w0 L
2
V
MJ
w0 x 2
2L
0:
w0 L2
3
w0 L
( x)
2
1 w0 x
x
2 L
M
w0 L2
3
x
3
M
w0 Lx
2
0
w0 x3
6L
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
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on a website, in whole or part.
683
PROBLEM 5.4
w
B
A
L
For the beam and loading shown, (a) draw the shear and bendingmoment diagrams, (b) determine the equations of the shear and bendingmoment curves.
SOLUTION
Free body diagram for determining reactions.
Reactions:
Fy
0: RA
MA
0:
MA
w0 L2
2
MA
wL
0
RA
wL
(wL)
L
2
0
Use portion to the right of the section as the free body.
Replace distributed load by equivalent concentrated load.
Fy
0: V
w( L
x)
0
V
MJ
0:
M
w( L
x)
L
x
w( L
x)
0
2
M
w
(L
2
x) 2
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.
684
3 kN
A
C
0.3 m
5 kN
2 kN
PROBLEM 5.7
E
B
Draw the shear and bending-moment diagrams for the beam and
loading shown, and determine the maximum absolute value (a) of
the shear, (b) of the bending moment.
2 kN
D
0.3 m
0.3 m
0.4 m
SOLUTION
Origin at A:
Reaction at A:
Fy
0: RA
MA
0: M A
3
2
5
2
0
(3 kN)(0.3 m)
RA
2 kN
(2 kN)(0.6 m)
(5 kN)(0.9 m)
(2 kN)(1.3 m)
MA
0
0.2 kN m
From A to C:
Fy
0:
V
2 kN
M1
0:
0.2 kN m
M
(2 kN)x
0.2
M
0
2x
From C to D:
Fy
0: 2
3
V
0
V
M2
0:
1 kN
0.2 kN m
M
(2 kN)x
0.7
(3 kN)( x
0.3)
M
0
x
From D to E:
Fy
M3
0: V
0:
5
M
M
2
0
V
3 kN
5(0.9
x)
(2)(1.3
1.9
x)
0
3x
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.
687
PROBLEM 5.7 (Continued)
From E to B:
Fy
M4
0: V
0:
2 kN
M
M
2(1.3
2.6
x)
0
2x
(a)
(b) M
V
max
max
3.00 kN
0.800 kN m
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.
688
3 kN
3 kN
PROBLEM 5.11
E
Draw the shear and bending-moment diagrams for the beam and loading
shown, and determine the maximum absolute value (a) of the shear,
(b) of the bending moment.
450 N ? m
A
C
D
300 mm
B
300 mm
200 mm
SOLUTION
MB
0: (700)(3)
450
A
MA
0:
(300)(3)
B
At A:
V
2.55 kN
A to C:
V
2.55 kN
(300)(3)
1000 A
2.55 kN
450
(700)(3)
1000B
M
0
MC
0:
(300)(2.55)
M
V
M
0.45 N m
MD
0:
(500)(2.55)
M
MD
(500)(2.55)
M
0
0:
(200)(3)
450
M
0
3.45 kN
ME
M
M
V
M
1125 N m
At E:
At B:
(200)(3)
675 N m
At D:
V
0
765 N m
At D:
E to B:
0
3.45 kN
At C:
C to E:
0
3.45 kN,
M
0:
(300)(3.45)
0
1035 N m
0
(a)
(b)
M
V
max
max
3.45 kN
1125 N m
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.
694
10 kN
PROBLEM 5.15
100 mm
3 kN/m
C
A
B
1.5 m
1.5 m
200 mm
For the beam and loading shown, determine the maximum
normal stress due to bending on a transverse section at C.
2.2 m
SOLUTION
Using CB as a free body,
MC
0:
M
(2.2)(3 103 )(1.1)
M
0
7.26 103 N m
Section modulus for rectangle:
S
1 2
bh
6
1
(100)(200)2
6
666.7 10
6
666.7 103 mm3
m3
Normal stress:
M
S
7.26 103
666.7 10 6
10.8895 106 Pa
10.89 MPa
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.
698