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PROBLEM 11.1
Knowing that the couple shown acts in a vertical plane, determine
the stress at (a) point A, (b) point B.
SOLUTION
3
1
PROBLEM 11.2
Knowing that the couple shown acts in a vertical plane, determine
the stress at (a) point A, (b) point B.
SOLUTION
3
1
PROBLEM 11.3
Using an allowable stress of 155 MPa, determine the largest bending
moment M that can be applied to the wide-flange beam shown. Neglect the
effect of fillets.
SOLUTION
Moment of inertia about x-axis:
32
1
64
3 64
2
64
31
1(200)(12) (200)(12)(104)
12
25.9872 10 mm
1(8)(196) 5.0197 10 mm
12
25.9872 10 mm
= +
= ×
= = ×
= = ×
I
I
II
6 4 64
123
6
66 3
56.944 10 mm 56.944 10 m
1
with (220) 110 mm 0.110 m
2
with 155 10 Pa
(56.944 10 )(155 10 ) 80.2 10 N m
0.110
−
−
=++= × = ×
= = = =
= = ×
××
= =×⋅
x
III I
Mc c
I
I
Mc
M
σ
σσ
80.2 kN m= ⋅
x
M
PROBLEM 11.4
Solve Prob. 11.3, assuming that the wide-flange beam is bent about the y
axis by a couple of moment My.
PROBLEM 11.3. Using an allowable stress of 155 MPa, determine the
largest bending moment M that can be applied to the wide-flange beam
shown. Neglect the effect of fillets.
SOLUTION
Moment of inertia about y axis:
3 64
1
3 34
2
64
31
1(12)(200) 8 10 mm
12
1(196)(8) 8.3627 10 mm
12
8 10 mm
= = ×
= = ×
= = ×
I
I
II
6 4 64
123
6
66 3
16.0084 10 mm 16.0084 10 m
1
with (200) 100 mm 0.100 m
2
with 155 10 Pa
(16.0084 10 )(155 10 ) 24.8 10 N m
0.100
−
−
=++= × = ×
= = = =
= = ×
××
= =×⋅
y
y
III I
Mc c
I
I
Mc
M
σ
σσ
24.8 kN m= ⋅
y
M
PROBLEM 11.5
A nylon spacing bar has the cross section shown. Knowing that the
allowable stress for the grade of nylon used is 24 MPa, determine the
largest couple Mz that can be applied to the bar.
SOLUTION
34
1
π
PROBLEM 11.6
Using an allowable stress of 16 ksi, determine the largest couple that can be
applied to each pipe.
SOLUTION
(a)
( )
4 4 4 4 34
3
(0.6 0.5 ) 52.7 10 in
44
0.6 in.
(16)(52.7 10 )
:0.6
oi
I rr
c
Mc I
M
Ic
ππ
σ
σ
−
−
= −= − = ×
=
×
= = =
1.405 kip in.= ⋅M
(b)
4 4 34
3
(0.7 0.5 ) 139.49 10 in
4
0.7 in.
(16)(139.49 10 )
:0.7
I
c
Mc I
M
Ic
π
σ
σ
−
−
= −= ×
=
×
= = =
3.19 kip in.= ⋅M
PROBLEM 11.7
Two
W4 13×
rolled sections are welded together as shown. Knowing that for the steel
alloy used
58 ksi
U
s
=
and using a factor of safety of 3.0, determine the largest couple
that can be applied when the assembly is bent about the z axis.
SOLUTION
Properties of W4 × 13 rolled section.
PROBLEM 11.8
Two
W4 13×
rolled sections are welded together as shown. Knowing that for the steel
alloy used
58 ksi
U
s
=
and using a factor of safety of 3.0, determine the largest couple
that can be applied when the assembly is bent about the z axis.
SOLUTION
PROBLEM 11.9
Two vertical forces are applied to a beam of the cross section shown. Determine
the maximum tensile and compressive stresses in portion BC of the beam.
SOLUTION
A
0
y
0
Ay
18
5
90
18
1
18
Σ
36
108
0108 3 in.
36
Y= =
consent of McGraw-Hill Education.
PROBLEM 11.10
Two vertical forces are applied to a beam of the cross section shown. Determine
the maximum tensile and compressive stresses in portion BC of the beam.
SOLUTION
22 2
11
2
22
4 (4)(25)
(25) 981.7 mm 10.610 mm
22 3 3
25
(50)(25) 1250 mm 12.5 mm
22
r
Ar y
h
A bh y
ππ
ππ
= = = = = =
= = = =−=− =−
11 2 2
12
(981.7)(10.610) (1250)( 12.5) 2.334 mm
981.7 1250
Ay Ay
yAA
+ +−
= = = −
++
1
2 4 2 4 2 64
1 11 11
11
2 3 2 34
1 1 11
3 3 34
2
22
(25) (981.7)(10.610) 42.886 10 mm
88
10.610 ( 2.334) 12.944 mm
42.866 10 (981.7)(12.944) 207.35 10 mm
11
(50)(25) 65.104 10 mm
12 12
12.5
x
I I Ay r Ay
dyy
I I Ad
I bh
d yy
ππ
=−=−= − = ×
= − = −− =
=+ = ×+ = ×
= = = ×
= −=−
2 3 2 34
2 2 22
3 4 94
12
top
bot
( 2.334) 10.166 mm
65.104 10 (1250)(10.166) 194.288 10 mm
401.16 10 mm 401.16 10 m
25 2.334 27.334 mm 0.027334 m
25 2.334 22.666 mm 0.022666 m
I I Ad
II I
y
y
−
−− =
=+ = ×+ = ×
=+= × = ×
=+= =
=−+ =− =−
33
0: (4 10 )(300 10 ) 1200 N mM Pa M Pa −
−= ==× × = ⋅
top 6
top 9
(1200)(0.027334) 81.76 10 Pa
401.16 10
My
I
σ
−
−
= =− =−×
×
top
81.8 MPa
σ
= −
6
bot
bot 9
(1200)( 0.022666) 67.80 10 Pa
401.16 10
My
I
σ
−
−−
==−=×
×
bot
67.8 MPa
σ
=
PROBLEM 11.2
Knowing that the couple shown acts in a vertical plane, determine
the stress at (a) point A, (b) point B.
SOLUTION
3
1
PROBLEM 11.3
Using an allowable stress of 155 MPa, determine the largest bending
moment M that can be applied to the wide-flange beam shown. Neglect the
effect of fillets.
SOLUTION
Moment of inertia about x-axis:
32
1
64
3 64
2
64
31
1(200)(12) (200)(12)(104)
12
25.9872 10 mm
1(8)(196) 5.0197 10 mm
12
25.9872 10 mm
= +
= ×
= = ×
= = ×
I
I
II
6 4 64
123
6
66 3
56.944 10 mm 56.944 10 m
1
with (220) 110 mm 0.110 m
2
with 155 10 Pa
(56.944 10 )(155 10 ) 80.2 10 N m
0.110
−
−
=++= × = ×
= = = =
= = ×
××
= =×⋅
x
III I
Mc c
I
I
Mc
M
σ
σσ
80.2 kN m= ⋅
x
M
PROBLEM 11.4
Solve Prob. 11.3, assuming that the wide-flange beam is bent about the y
axis by a couple of moment My.
PROBLEM 11.3. Using an allowable stress of 155 MPa, determine the
largest bending moment M that can be applied to the wide-flange beam
shown. Neglect the effect of fillets.
SOLUTION
Moment of inertia about y axis:
3 64
1
3 34
2
64
31
1(12)(200) 8 10 mm
12
1(196)(8) 8.3627 10 mm
12
8 10 mm
= = ×
= = ×
= = ×
I
I
II
6 4 64
123
6
66 3
16.0084 10 mm 16.0084 10 m
1
with (200) 100 mm 0.100 m
2
with 155 10 Pa
(16.0084 10 )(155 10 ) 24.8 10 N m
0.100
−
−
=++= × = ×
= = = =
= = ×
××
= =×⋅
y
y
III I
Mc c
I
I
Mc
M
σ
σσ
24.8 kN m= ⋅
y
M
PROBLEM 11.5
A nylon spacing bar has the cross section shown. Knowing that the
allowable stress for the grade of nylon used is 24 MPa, determine the
largest couple Mz that can be applied to the bar.
SOLUTION
34
1
π
PROBLEM 11.6
Using an allowable stress of 16 ksi, determine the largest couple that can be
applied to each pipe.
SOLUTION
(a)
( )
4 4 4 4 34
3
(0.6 0.5 ) 52.7 10 in
44
0.6 in.
(16)(52.7 10 )
:0.6
oi
I rr
c
Mc I
M
Ic
ππ
σ
σ
−
−
= −= − = ×
=
×
= = =
1.405 kip in.= ⋅M
(b)
4 4 34
3
(0.7 0.5 ) 139.49 10 in
4
0.7 in.
(16)(139.49 10 )
:0.7
I
c
Mc I
M
Ic
π
σ
σ
−
−
= −= ×
=
×
= = =
3.19 kip in.= ⋅M
PROBLEM 11.7
Two
W4 13×
rolled sections are welded together as shown. Knowing that for the steel
alloy used
58 ksi
U
s
=
and using a factor of safety of 3.0, determine the largest couple
that can be applied when the assembly is bent about the z axis.
SOLUTION
Properties of W4 × 13 rolled section.
PROBLEM 11.8
Two
W4 13×
rolled sections are welded together as shown. Knowing that for the steel
alloy used
58 ksi
U
s
=
and using a factor of safety of 3.0, determine the largest couple
that can be applied when the assembly is bent about the z axis.
SOLUTION
PROBLEM 11.9
Two vertical forces are applied to a beam of the cross section shown. Determine
the maximum tensile and compressive stresses in portion BC of the beam.
SOLUTION
A
0
y
0
Ay
18
5
90
18
1
18
Σ
36
108
0108 3 in.
36
Y= =
consent of McGraw-Hill Education.
PROBLEM 11.10
Two vertical forces are applied to a beam of the cross section shown. Determine
the maximum tensile and compressive stresses in portion BC of the beam.
SOLUTION
22 2
11
2
22
4 (4)(25)
(25) 981.7 mm 10.610 mm
22 3 3
25
(50)(25) 1250 mm 12.5 mm
22
r
Ar y
h
A bh y
ππ
ππ
= = = = = =
= = = =−=− =−
11 2 2
12
(981.7)(10.610) (1250)( 12.5) 2.334 mm
981.7 1250
Ay Ay
yAA
+ +−
= = = −
++
1
2 4 2 4 2 64
1 11 11
11
2 3 2 34
1 1 11
3 3 34
2
22
(25) (981.7)(10.610) 42.886 10 mm
88
10.610 ( 2.334) 12.944 mm
42.866 10 (981.7)(12.944) 207.35 10 mm
11
(50)(25) 65.104 10 mm
12 12
12.5
x
I I Ay r Ay
dyy
I I Ad
I bh
d yy
ππ
=−=−= − = ×
= − = −− =
=+ = ×+ = ×
= = = ×
= −=−
2 3 2 34
2 2 22
3 4 94
12
top
bot
( 2.334) 10.166 mm
65.104 10 (1250)(10.166) 194.288 10 mm
401.16 10 mm 401.16 10 m
25 2.334 27.334 mm 0.027334 m
25 2.334 22.666 mm 0.022666 m
I I Ad
II I
y
y
−
−− =
=+ = ×+ = ×
=+= × = ×
=+= =
=−+ =− =−
33
0: (4 10 )(300 10 ) 1200 N mM Pa M Pa −
−= ==× × = ⋅
top 6
top 9
(1200)(0.027334) 81.76 10 Pa
401.16 10
My
I
σ
−
−
= =− =−×
×
top
81.8 MPa
σ
= −
6
bot
bot 9
(1200)( 0.022666) 67.80 10 Pa
401.16 10
My
I
σ
−
−−
==−=×
×
bot
67.8 MPa
σ
=
