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Chapter 5
5-1 S
y
= 350 MPa.
MSS: σ 1
− σ 3
= S
y
/ n ⇒
1 3
y
S
n
σ σ
DE:
1/2 1/
2 2 2 2 2
A A B B x x y y xy
σ σ σ σ σ σ σ σ σ τ
y
S
n
σ
(a) MSS: σ 1
= 100 MPa, σ 2
= 100 MPa, σ 3
n = = Ans
DE:
2 2 1/
(100 100(100) 100 ) 100 MPa, 3..
σ n Ans
(b) MSS: σ 1
= 100 MPa, σ 2
= 50 MPa, σ 3
n = = Ans
DE:
2 2 1/
(100 100(50) 50 ) 86.6 MPa, 4..
σ n Ans
(c)
2
2
, ( 75) 140, 40 MPa
A B
σ σ
1 2 3
σ = 140, σ = 0, σ = −40 MPa
MSS:
n = = Ans
DE:
1/
2 2
100 3 75 164 MPa, 2..
σ n Ans
(d)
2
2
, ( 50) 11.0, 114.0 MPa
A B
σ σ
1 2 3
σ = 0, σ = −11.0, σ = −114.0 MPa
MSS:
n = = Ans
DE:
2 2 2 1/ 2
σ [( 50) ( 50)( 75) ( 75) 3( 50) ] 109.0 MPa
n = = Ans
(e)
2
, 20 104.7, 15.3 MPa
A B
σ σ
th
1 2 3
σ = 104.7, σ = 15.3, σ =0 MPa
MSS:
n = = Ans
DE:
1/
2 2 2
σ 100 100(20) 20 3 20 98.0 MPa
n = = Ans
______________________________________________________________________________
5-2 S
y
= 350 MPa.
MSS: σ
1
− σ
3
= S
y
/ n ⇒
1 3
y
S
n
σ σ
DE:
1/
2 2 y y
A A B B
S S
n
n
σ σ σ σ σ
σ
(a) MSS:
1 3
100 MPa, 0 3..
σ = σ = ⇒ n = = Ans
DE:
2 2 1/
[100 (100)(100) 100 ]
n = = Ans
(b) MSS:
1 3
100 , 100 MPa 1..
σ = σ = − ⇒ n = = Ans
DE:
1/
2
2
n = = Ans
(c) MSS:
1 3
100 MPa, 0 3..
σ = σ = ⇒ n = = Ans
DE:
1/
2 2
n = = Ans
(d) MSS:
1 3
100, 50 MPa 2..
σ = σ = − ⇒ n = = Ans
DE:
1/
2
2
n = = Ans
( e ) MSS:
1 3
0, 100 MPa 3..
σ = σ = − ⇒ n = = Ans
DE:
1/
2 2
n = = Ans
______________________________________________________________________________
5-3 From Table A-20, S y
= 37.5 kpsi
th
DE:
1/
2 2 2
σ 24 24 24 24 3 15 35.4 kpsi
n = = Ans
______________________________________________________________________________
5-4 From Table A-20, S y
= 47 kpsi.
MSS: σ
1
− σ
3
= S
y
/ n ⇒
1 3
y
S
n
σ σ
DE:
1/
2 2
y y
A A B B
S S
n
n
σ σ σ σ σ
σ
(a) MSS:
1 3
30 kpsi, 0 1..
σ = σ = ⇒ n = = Ans
DE:
2 2 1/
[30 (30)(30) 30 ]
n = = Ans
(b) MSS:
1 3
30 , 30 kpsi 0..
σ = σ = − ⇒ n = = Ans
DE:
1/
2
2
n = = Ans
(c) MSS:
1 3
30 kpsi, 0 1..
σ = σ = ⇒ n = = Ans
DE:
1/
2 2
n = = Ans
(d) MSS:
1 3
0, 30 kpsi 1..
σ = σ = − ⇒ n = = Ans
DE:
1/
2 2
n = = Ans
( e ) MSS:
1 3
10, 50 kpsi 0..
σ = σ = − ⇒ n = = Ans
DE:
1/
2
2
n = = Ans
______________________________________________________________________________
th
5-5 Note: The drawing in this manual may not be to the scale of original drawing. The
measurements were taken from the original drawing.
( a ) MSS and DE:
OB
n Ans
OA
( b ) MSS:
OD
n Ans
OC
DE:
OE
n Ans
OC
( c ) MSS:
OG
n Ans
OF
DE:
OH
n Ans
OF
( d ) MSS:
OJ
n Ans
OI
= = = , DE:
OK
n Ans
OI
( e ) MSS:
OM
n Ans
OL
= = = , DE:
ON
n Ans
OL
______________________________________________________________________________
th
5-7 S
y
= 295 MPa, σ
A
= 75 MPa, σ
B
= − 35 MPa,
( a )
1/2 1/
2 2 2
2
y
A A B B
S
n Ans
σ σ σ σ
( b ) Note: The drawing in this manual may not be to the scale of original drawing. The
measurements were taken from the original drawing.
OB
n Ans
OA
______________________________________________________________________________
th
5-8 S
y
= 295 MPa, σ
A
= 30 MPa, σ
B
= − 100 MPa,
( a )
1/2 1/
2 2 2
2
y
A A B B
S
n Ans
σ σ σ σ
( b ) Note: The drawing in this manual may not be to the scale of original drawing. The
measurements were taken from the original drawing.
OB
n Ans
OA
______________________________________________________________________________
th
5-10 S
y
= 295 MPa,
2
2
, 40 3.8, 91.2 MPa
A B
σ σ
( a )
1/2 1/
2 2 2 2
y
A A B B
S
n Ans
σ σ σ σ
( b ) Note: The drawing in this manual may not be to the scale of original drawing. The
measurements were taken from the original drawing.
OB
n Ans
OA
______________________________________________________________________________
th
5-11 S
y
= 295 MPa,
2
, 10 30.9, 80.9 MPa
A B
σ σ
( a )
1/2 1/
2 2 2
2
y
A A B B
S
n Ans
σ σ σ σ
( b ) Note: The drawing in this manual may not be to the scale of original drawing. The
measurements were taken from the original drawing.
OB
n Ans
OA
______________________________________________________________________________
5-12 S
yt
= 60 kpsi, S
yc
= 75 kpsi. Eq. (5-26) for yield is
1
3 1
yt yc
n
S S
σ σ
−
( a )
1 3
σ = 25 kpsi, σ = 0 ⇒
1
n Ans
−
( b )
1 3
σ = 15, σ = −15 kpsi ⇒
1
n Ans
−
th
OH
n Ans
OG
( e )
2
, 15 9, 39 kpsi
A B
σ σ
OJ
n Ans
OI
______________________________________________________________________________
5-14 Since ε f
0.05, and S
yt
≠ S
yc
, the Coulomb-Mohr theory for ductile materials will be
used.
( a ) From Eq. (5-26),
1
1
1 3
yt yc
n Ans
S S
σ σ
−
−
( b ) Plots for Problems 5-14 to 5-18 are found here. Note: The drawing in this manual
may not be to the scale of original drawing. The measurements were taken from the
original drawing.
OB
n Ans
OA
______________________________________________________________________________
5-15 ( a ) From Eq. (5-26),
1
1
1 3
yt yc
n Ans
S S
σ σ
−
−
th
( b ) The plot for this problem is found on the page for Prob. 5-14. Note: The drawing in
this manual may not be to the scale of original drawing. The measurements were taken
from the original drawing.
OD
n Ans
OC
______________________________________________________________________________
2
2
, ( 75) 160, 35 kpsi
A B
σ σ
( a ) From Eq. (5-26),
1
1
1 3
yt yc
n Ans
S S
σ σ
−
−
( b ) The plot for this problem is found on the page for Prob. 5-14. Note: The drawing in
this manual may not be to the scale of original drawing. The measurements were taken
from the original drawing.
OF
n Ans
OE
______________________________________________________________________________
2
2
, 50 47.7, 157.3 kpsi
A B
σ σ
( a ) From Eq. (5-26),
1
1
1 3
yt yc
n Ans
S S
σ σ
−
−
( b ) The plot for this problem is found on the page for Prob. 5-14. Note: The drawing in
this manual may not be to the scale of original drawing. The measurements were taken
from the original drawing.
OH
n Ans
OG
______________________________________________________________________________
2
2
, ( 75) 180.8, 24.2 kpsi
A B
σ σ
( a ) From Eq. (5-26),
th
BCM: Eq. (5-31 b ),
1
n Ans
−
MM: σ
A
≥ 0 ≥ σ
B
, and | σ
B
/ σ
A
| ≥ 1, Eq. (5-32 b ),
1 1
uc ut A B
uc ut uc
S S
n Ans
S S S
σ σ
− −
( e )
2
2
, ( 15) 5, 35 kpsi
A B
σ σ
BCM: Eq. (5-31 c ),
uc
B
S
n Ans
σ
MM: Eq. (5-32 c ),
uc
B
S
n Ans
σ
______________________________________________________________________________
5-20 Note: The drawing in this manual may not be to the scale of original drawing. The
measurements were taken from the original drawing.
( a ) σ
A
= 25, σ
B
= 15 kpsi
BCM & MM:
OB
n Ans
OA
( b ) σ
A
= 15, σ
B
= − 15 kpsi
BCM:
OC
n Ans
OD
MM:
OE
n Ans
OC
( c )
2
2
A B
σ σ
= 24.14, −4.14 kpsi
BCM:
OG
n Ans
OF
MM:
OH
n Ans
OF
th
( d )
2
2
, ( 15) 17.03, 22.03 kpsi
A B
σ σ
BCM:
OJ
n Ans
OI
MM:
OK
n Ans
OI
( e )
2
2
, ( 15) 5, 35 kpsi
A B
σ σ
BCM and MM:
OM
n Ans
OL
______________________________________________________________________________
5-21 From Table A-24, S ut
= 31 kpsi, S
uc
= 109 kpsi
BCM: Eqs. (5-31), MM: Eqs. (5-32)
( a ) σ
A
= 15, σ
B
= 10 kpsi.
BCM: Eq. (5-31 a ),
ut
A
S
n Ans
σ
MM: Eq. (5-32 a ),
ut
A
S
n Ans
σ
( b), ( c ) The plot is shown below is for Probs. 5-21 to 5-25. Note: The drawing in this
manual may not be to the scale of original drawing. The measurements were taken from
the original drawing.
BCM and MM:
OB
n Ans
OA
______________________________________________________________________________
5-22 S
ut
= 31 kpsi, S uc
= 109 kpsi
th
( a ) BCM: Eq. (5-31 c ),
uc
B
S
n Ans
σ
MM: Eq. (5-32 c ),
uc
B
S
n Ans
σ
( b ), ( c ) The plot is shown in the solution to Prob. 5-21.
BCM and MM:
OJ
n Ans
OI
______________________________________________________________________________
5-25 From Table A-24, S ut
= 31 kpsi, S uc
= 109 kpsi
BCM: Eq. (5-31), MM: Eqs. (5-32)
2
2
, ( 10) 15, 37 kpsi
A B
σ σ
( a ) BCM: Eq. (5-31 b ),
1
n Ans
−
MM: Eq. (5-32 b ),
1 1
uc ut A B
uc ut uc
S S
n Ans
S S S
σ σ
− −
( b ), ( c ) The plot is shown in the solution to Prob. 5-21.
BCM:
OL
n Ans
OK
MM:
OM
n Ans
OK
______________________________________________________________________________
5-26 S
ut
= 36 kpsi, S
uc
= 35 kpsi
BCM: Eq. (5-31),
( a ) σ
A
= 15, σ
B
= − 10 kpsi.
BCM: Eq. (5-31 b ),
1
n Ans
−
th
( b) The plot is shown below is for Probs. 5-26 to 5-30. Note: The drawing in this manual
may not be to the scale of original drawing. The measurements were taken from the
original drawing.
OB
n Ans
OA
______________________________________________________________________________
5-27 S
ut
= 36 kpsi, S uc
= 35 kpsi
BCM: Eq. (5-31),
( a ) σ
A
= 15, σ
B
= − 15 kpsi.
BCM: Eq. (5-31 b ),
1
n Ans
−
( b ) The plot is shown in the solution to Prob. 5-26.
OD
n Ans
OC
______________________________________________________________________________
5-28 S
ut
= 36 kpsi, S
uc
= 35 kpsi
BCM: Eq. (5-31),
( a )
2
2
, ( 8) 16, 4 kpsi
A B
σ σ
BCM: Eq. (5-31 b ),
1
n Ans
−
( b ) The plot is shown in the solution to Prob. 5-26.
