consent of McGraw–Hill Education.
PROBLEM 11.49
Two forces P can be applied separately or at the same time to a plate that is welded
to a solid circular bar of radius r. Determine the largest compressive stress in the
circular bar, (a) when both forces are applied, (b) when only one of the forces is
applied.
SOLUTION
For a solid section,
24
,,
4
= = =A r I r cr
π
π
Compressive stress
23
4
=−−
=−−
F Mc
AI
FM
rr
σ
ππ
(a) Both forces applied.
2, 0= =F PM
2
2−
=P
r
σπ
(b) One force applied.
,F P M Pr= =
22
4
=−−
r
FP
rr
σππ
2
5−
=P
r
σπ
W8 21×
W8 21×
PROBLEM 11.50
As many as three axial loads, each of magnitude
10 kips,P=
can be
applied to the end of a
W8 21×
rolled–steel shape. Determine the stress at
point A, (a) for the loading shown, (b) if loads are applied at points 1 and 2
only.
SOLUTION
PROBLEM 11.51
A short wooden post supports a 6–kip axial load as shown. Determine the
stress at point A when (a)
0,b=
(b)
1.5in.,b=
(c)
3in.b=
SOLUTION
22 2
44 4
3
(3) 28.27 in
(3) 63.62 in
44
63.62 21.206 in
3
6 kips
Ar
Ir
I
Sc
P M Pb
pp
pp
= = =
= = =
= = =
= =
(a)
00
60.212 ksi
28.27
bM
P
A
s
= =
=−=− =−
212 psi
s
= −
(b)
1.5 in. (6)(1.5) 9 kip inbM= = = ⋅
69
0.637 ksi
28.27 21.206
PM
AS
s
=−− =− − =−
637 psi
s
= −
(c)
3 in. (6)(3) 18 kip in
bM= = = ⋅
6 18 1.061 ksi
28.27 21.206
PM
AS
s
=−− =− − =−
1061 psi
s
= −
PROBLEM 11.52
Knowing that the magnitude of the horizontal force P is 8 kN,
determine the stress at (a) point A, (b) point B.
SOLUTION
2 62
3 3 3 4 94
3
3
(30)(24) 720 mm 720 10 m
45 12 33 mm 0.033 m
11
(30)(24) 34.56 10 mm 34.56 10 m
12 12
1(24 mm) 12 mm 0.012 m 8 10 N
2
(8 10 )(0.033) 264 N m
A
e
I bh
cP
M Pe
−
−
= = = ×
=−= =
== =×=×
= = = = ×
==×=⋅
(a)
36
69
8 10 (264)(0.012) 102.8 10 Pa
720 10 34.56 10
A
P Mc
AI
−−
×
=−− =− − =− ×
××
σ
102.8 MPa
A
σ
= −
(b)
36
69
8 10 (264)(0.012) 80.6 10 Pa
720 10 34.56 10
B
P Mc
AI
σ
−−
×
=−+ =− + = ×
××
80.6 MPa
B
σ
=
consent of McGraw–Hill Education.
PROBLEM 11.53
The vertical portion of the press shown consists of a
rectangular tube of wall thickness t = 10 mm. Knowing that the
press has been tightened on wooden planks being glued
together until P = 20 kN, determine the stress at (a) point A,
(b) point B.
SOLUTION
Rectangular cutout is
60 mm 40 mm.×
3 2 32
3 3 64
64
3
33
(80)(60) (60)(40) 2.4 10 mm 2.4 10 m
11
(60)(80) (40)(60) 1.84 10 mm
12 12
1.84 10 m
40 mm 0.040 m 200 40 240 mm 0.240 m
20 10 N
(20 10 )(0.240) 4.8 10 N m
A
I
ce
P
M Pe
−
−
=−=×=×
=−=×
= ×
= = = += =
= ×
==× =×⋅
(a)
33 6
36
20 10 (4.8 10 )(0.040) 112.7 10 Pa
2.4 10 1.84 10
A
P Mc
AI
σ
−−
××
=+= + = ×
××
112.7 MPa
A
σ
=
(b)
33 6
36
20 10 (4.8 10 )(0.040) 96.0 10 Pa
2.4 10 1.84 10
B
P Mc
AI
σ
−−
××
=− = − =−×
××
96.0 MPa
B
σ
= −
consent of McGraw–Hill Education.
PROBLEM 11.54
Solve Prob. 11.53, assuming that
8 mm.t=
PROBLEM 11.53 The vertical portion of the press shown
consists of a rectangular tube of wall thickness t = 10 mm.
Knowing that the press has been tightened on wooden planks
being glued together until P = 20 kN, determine the stress at
(a) point A, (b) point B.
SOLUTION
Rectangular cutout is
64 mm 44 mm.×
32
32
3 3 62
64
3
33
(80)(60) (64)(44) 1.984 10 mm
1.984 10 mm
11
(60)(80) (44)(64) 1.59881 10 mm
12 12
1.59881 10 m
40 mm 0.004 200 40 240 mm 0.240 m
20 10 N
(20 10 )(0.240) 4.8 10 N m
A
I
ce
P
M Pe
−
−
=−=×
= ×
=−=×
= ×
= = = += =
= ×
==× =×⋅
(a)
33 6
36
20 10 (4.8 10 )(0.040) 130.2 10 Pa
1.984 10 1.59881 10
AP Mc
AI
σ
−−
××
=+= + = ×
××
130.2 MPa
A
σ
=
(b)
33 6
36
20 10 (4.8 10 )(0.040) 110.0 10 Pa
1.984 10 1.59881 10
BP Mc
AI
σ
−−
××
=−= − =− ×
××
110.0 MPa
B
σ
= −
consent of McGraw–Hill Education.
PROBLEM 11.55
Solve Prob. 11.53, assuming that
8 mm.t=
PROBLEM 11.53 The vertical portion of the press shown
consists of a rectangular tube of wall thickness t = 10 mm.
Knowing that the press has been tightened on wooden planks
being glued together until P = 20 kN, determine the stress at
(a) point A, (b) point B.
SOLUTION
Rectangular cutout is
64 mm 44 mm.×
32
32
3 3 62
64
3
33
(80)(60) (64)(44) 1.984 10 mm
1.984 10 mm
11
(60)(80) (44)(64) 1.59881 10 mm
12 12
1.59881 10 m
40 mm 0.004 200 40 240 mm 0.240 m
20 10 N
(20 10 )(0.240) 4.8 10 N m
A
I
ce
P
M Pe
−
−
=−=×
= ×
=−=×
= ×
= = = += =
= ×
==× =×⋅
(a)
33 6
36
20 10 (4.8 10 )(0.040) 130.2 10 Pa
1.984 10 1.59881 10
AP Mc
AI
σ
−−
××
=+= + = ×
××
130.2 MPa
A
σ
=
(b)
33 6
36
20 10 (4.8 10 )(0.040) 110.0 10 Pa
1.984 10 1.59881 10
BP Mc
AI
σ
−−
××
=−= − =− ×
××
110.0 MPa
B
σ
= −
consent of McGraw–Hill Education.
PROBLEM 11.56
The two forces shown are applied to a rigid plate supported by a steel pipe of 8-in.
outer diameter and 7–in. inner diameter. Determine the value of P for which the
maximum compressive stress in the pipe is 15 ksi.
SOLUTION
4 44
all 15 ksi I (4 in.) (3.5 in.) 83.2 in
44
=− =−=
NA
ππ
s
2 22
(4 in.) (3.5 in.) 11.78 in=−=A
ππ
Max. compressive stress is at point B.
24
12 (5 )(4.0 in.)
11.78 in 83.2 in
15 ksi 1.019 0.085 0.240
13.981 0.325
B
Q Mc P P
AI
PP
P
+
=−− =− −
− =−− −
−=−
s
43.0 kips=P
consent of McGraw–Hill Education.
PROBLEM 11.57
An offset h must be introduced into a solid circular rod of
diameter d. Knowing that the maximum stress after the offset
is introduced must not exceed 5 times the stress in the rod
when it is straight, determine the largest offset that can be
used.
SOLUTION
For centric loading,
c
P
A
σ
=
For eccentric loading,
e
P Phc
AI
σ
= +
Given
5
ec
=
σσ
4
2
5
(4)
41
64
4 2
24
P Phc P
AI A
d
Phc P I
hd
d
I A cA d
π
π
+=
= ∴== =
0.500hd=
PROBLEM 11.58
An offset h must be introduced into a metal tube of 0.75–in. outer
diameter and 0.08–in. wall thickness. Knowing that the maximum
stress after the offset is introduced must not exceed 4 times the
stress in the tube when it is straight, determine the largest offset
that can be used.
SOLUTION
( )
( )
1
22 2 2
1
2
44 4 4
1
34
10.375 in.
2
0.375 0.08 0.295 in.
(0.375 0.295 )
0.168389 in
(0.375 0.295 )
44
9.5835 10 in
cd
c ct
A cc
I cc
ππ
ππ
−
= =
=−= − =
= −= −
=
= −= −
= ×
For centric loading,
cen
P
A
σ
=
For eccentric loading,
ecc
P Phc
AI
σ
= +
ecc cen
3
4 or 4
3 3 (3)(9.5835 10 )
(0.168389)(0.375)
P Phc P
AI A
hc I
h
I A Ac
σσ
−
= +=
×
= = =
0.455 in.h=
PROBLEM 11.50
As many as three axial loads, each of magnitude
10 kips,P=
can be
applied to the end of a
W8 21×
rolled–steel shape. Determine the stress at
point A, (a) for the loading shown, (b) if loads are applied at points 1 and 2
only.
SOLUTION
PROBLEM 11.51
A short wooden post supports a 6–kip axial load as shown. Determine the
stress at point A when (a)
0,b=
(b)
1.5in.,b=
(c)
3in.b=
SOLUTION
22 2
44 4
3
(3) 28.27 in
(3) 63.62 in
44
63.62 21.206 in
3
6 kips
Ar
Ir
I
Sc
P M Pb
pp
pp
= = =
= = =
= = =
= =
(a)
00
60.212 ksi
28.27
bM
P
A
s
= =
=−=− =−
212 psi
s
= −
(b)
1.5 in. (6)(1.5) 9 kip inbM= = = ⋅
69
0.637 ksi
28.27 21.206
PM
AS
s
=−− =− − =−
637 psi
s
= −
(c)
3 in. (6)(3) 18 kip in
bM= = = ⋅
6 18 1.061 ksi
28.27 21.206
PM
AS
s
=−− =− − =−
1061 psi
s
= −
PROBLEM 11.52
Knowing that the magnitude of the horizontal force P is 8 kN,
determine the stress at (a) point A, (b) point B.
SOLUTION
2 62
3 3 3 4 94
3
3
(30)(24) 720 mm 720 10 m
45 12 33 mm 0.033 m
11
(30)(24) 34.56 10 mm 34.56 10 m
12 12
1(24 mm) 12 mm 0.012 m 8 10 N
2
(8 10 )(0.033) 264 N m
A
e
I bh
cP
M Pe
−
−
= = = ×
=−= =
== =×=×
= = = = ×
==×=⋅
(a)
36
69
8 10 (264)(0.012) 102.8 10 Pa
720 10 34.56 10
A
P Mc
AI
−−
×
=−− =− − =− ×
××
σ
102.8 MPa
A
σ
= −
(b)
36
69
8 10 (264)(0.012) 80.6 10 Pa
720 10 34.56 10
B
P Mc
AI
σ
−−
×
=−+ =− + = ×
××
80.6 MPa
B
σ
=
consent of McGraw–Hill Education.
PROBLEM 11.53
The vertical portion of the press shown consists of a
rectangular tube of wall thickness t = 10 mm. Knowing that the
press has been tightened on wooden planks being glued
together until P = 20 kN, determine the stress at (a) point A,
(b) point B.
SOLUTION
Rectangular cutout is
60 mm 40 mm.×
3 2 32
3 3 64
64
3
33
(80)(60) (60)(40) 2.4 10 mm 2.4 10 m
11
(60)(80) (40)(60) 1.84 10 mm
12 12
1.84 10 m
40 mm 0.040 m 200 40 240 mm 0.240 m
20 10 N
(20 10 )(0.240) 4.8 10 N m
A
I
ce
P
M Pe
−
−
=−=×=×
=−=×
= ×
= = = += =
= ×
==× =×⋅
(a)
33 6
36
20 10 (4.8 10 )(0.040) 112.7 10 Pa
2.4 10 1.84 10
A
P Mc
AI
σ
−−
××
=+= + = ×
××
112.7 MPa
A
σ
=
(b)
33 6
36
20 10 (4.8 10 )(0.040) 96.0 10 Pa
2.4 10 1.84 10
B
P Mc
AI
σ
−−
××
=− = − =−×
××
96.0 MPa
B
σ
= −
consent of McGraw–Hill Education.
PROBLEM 11.54
Solve Prob. 11.53, assuming that
8 mm.t=
PROBLEM 11.53 The vertical portion of the press shown
consists of a rectangular tube of wall thickness t = 10 mm.
Knowing that the press has been tightened on wooden planks
being glued together until P = 20 kN, determine the stress at
(a) point A, (b) point B.
SOLUTION
Rectangular cutout is
64 mm 44 mm.×
32
32
3 3 62
64
3
33
(80)(60) (64)(44) 1.984 10 mm
1.984 10 mm
11
(60)(80) (44)(64) 1.59881 10 mm
12 12
1.59881 10 m
40 mm 0.004 200 40 240 mm 0.240 m
20 10 N
(20 10 )(0.240) 4.8 10 N m
A
I
ce
P
M Pe
−
−
=−=×
= ×
=−=×
= ×
= = = += =
= ×
==× =×⋅
(a)
33 6
36
20 10 (4.8 10 )(0.040) 130.2 10 Pa
1.984 10 1.59881 10
AP Mc
AI
σ
−−
××
=+= + = ×
××
130.2 MPa
A
σ
=
(b)
33 6
36
20 10 (4.8 10 )(0.040) 110.0 10 Pa
1.984 10 1.59881 10
BP Mc
AI
σ
−−
××
=−= − =− ×
××
110.0 MPa
B
σ
= −
consent of McGraw–Hill Education.
PROBLEM 11.55
Solve Prob. 11.53, assuming that
8 mm.t=
PROBLEM 11.53 The vertical portion of the press shown
consists of a rectangular tube of wall thickness t = 10 mm.
Knowing that the press has been tightened on wooden planks
being glued together until P = 20 kN, determine the stress at
(a) point A, (b) point B.
SOLUTION
Rectangular cutout is
64 mm 44 mm.×
32
32
3 3 62
64
3
33
(80)(60) (64)(44) 1.984 10 mm
1.984 10 mm
11
(60)(80) (44)(64) 1.59881 10 mm
12 12
1.59881 10 m
40 mm 0.004 200 40 240 mm 0.240 m
20 10 N
(20 10 )(0.240) 4.8 10 N m
A
I
ce
P
M Pe
−
−
=−=×
= ×
=−=×
= ×
= = = += =
= ×
==× =×⋅
(a)
33 6
36
20 10 (4.8 10 )(0.040) 130.2 10 Pa
1.984 10 1.59881 10
AP Mc
AI
σ
−−
××
=+= + = ×
××
130.2 MPa
A
σ
=
(b)
33 6
36
20 10 (4.8 10 )(0.040) 110.0 10 Pa
1.984 10 1.59881 10
BP Mc
AI
σ
−−
××
=−= − =− ×
××
110.0 MPa
B
σ
= −
consent of McGraw–Hill Education.
PROBLEM 11.56
The two forces shown are applied to a rigid plate supported by a steel pipe of 8-in.
outer diameter and 7–in. inner diameter. Determine the value of P for which the
maximum compressive stress in the pipe is 15 ksi.
SOLUTION
4 44
all 15 ksi I (4 in.) (3.5 in.) 83.2 in
44
=− =−=
NA
ππ
s
2 22
(4 in.) (3.5 in.) 11.78 in=−=A
ππ
Max. compressive stress is at point B.
24
12 (5 )(4.0 in.)
11.78 in 83.2 in
15 ksi 1.019 0.085 0.240
13.981 0.325
B
Q Mc P P
AI
PP
P
+
=−− =− −
− =−− −
−=−
s
43.0 kips=P
consent of McGraw–Hill Education.
PROBLEM 11.57
An offset h must be introduced into a solid circular rod of
diameter d. Knowing that the maximum stress after the offset
is introduced must not exceed 5 times the stress in the rod
when it is straight, determine the largest offset that can be
used.
SOLUTION
For centric loading,
c
P
A
σ
=
For eccentric loading,
e
P Phc
AI
σ
= +
Given
5
ec
=
σσ
4
2
5
(4)
41
64
4 2
24
P Phc P
AI A
d
Phc P I
hd
d
I A cA d
π
π
+=
= ∴== =
0.500hd=
PROBLEM 11.58
An offset h must be introduced into a metal tube of 0.75–in. outer
diameter and 0.08–in. wall thickness. Knowing that the maximum
stress after the offset is introduced must not exceed 4 times the
stress in the tube when it is straight, determine the largest offset
that can be used.
SOLUTION
( )
( )
1
22 2 2
1
2
44 4 4
1
34
10.375 in.
2
0.375 0.08 0.295 in.
(0.375 0.295 )
0.168389 in
(0.375 0.295 )
44
9.5835 10 in
cd
c ct
A cc
I cc
ππ
ππ
−
= =
=−= − =
= −= −
=
= −= −
= ×
For centric loading,
cen
P
A
σ
=
For eccentric loading,
ecc
P Phc
AI
σ
= +
ecc cen
3
4 or 4
3 3 (3)(9.5835 10 )
(0.168389)(0.375)
P Phc P
AI A
hc I
h
I A Ac
σσ
−
= +=
×
= = =
0.455 in.h=