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CHAPTER 1
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PROBLEM 1.1

Two solid cylindrical rods AB and BC are

welded together at B and loaded as shown.
Knowing that d1  30 mm and d 2  50 mm,
find the average normal stress at the midsection
of (a) rod AB, (b) rod BC.

SOLUTION

(a) Rod AB:

Force: P  60  103 N tension

 
Area: A d12  (30  103 ) 2  706.86  106 m 2
4 4
P 60  103
Normal stress:  AB    84.882  106 Pa  AB  84.9 MPa ⊳
A 706.86  106
(b) Rod BC:

Force: P  60  103  (2)(125  103 )  190  103 N

 
Area: A d 22  (50  103 ) 2  1.96350  10 3 m 2
4 4
P 190  103
Normal stress:  BC    96.766  106 Pa
A 1.96350  103
 BC  96.8 MPa ⊳

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PROBLEM 1.2

Two solid cylindrical rods AB and BC are

welded together at B and loaded as shown.
Knowing that the average normal stress must
not exceed 150 MPa in either rod, determine
the smallest allowable values of the diameters
d1 and d2.

SOLUTION

(a) Rod AB:

Force: P  60  103 N
Stress:  AB  150  106 Pa
 2
Area: A d1
4
P P
 AB   A
A  AB
 P
d12 
4  AB
4P (4)(60  103 )
d12    509.30  106 m 2
 AB  (150  10 )
6

d1  22.568  103 m d1  22.6 mm ⊳

(b) Rod BC:

Force: P  60  103  (2)(125  103 )  190  103 N

Stress:  BC  150  106 Pa
 2
Area: A d2
4
P 4P
 BC  
A  d 22
4P (4)(190  103 )
d 22    1.61277  103 m 2
 BC  (150  106 )
d 2  40.159  103 m d 2  40.2 mm ⊳

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PROBLEM 1.3

Two solid cylindrical rods AB and BC are welded together at B and loaded as
shown. Knowing that P = 10 kips, find the average normal stress at the
midsection of (a) rod AB, (b) rod BC.

SOLUTION
(a) Rod AB:
P  12  10  22 kips
 
A d12  (1.25) 2  1.22718 in 2
4 4
P 22
 AB    17.927 ksi  AB  17.93 ksi ⊳
A 1.22718
(b) Rod BC:
P  10 kips
 
A d 22 
(0.75) 2  0.44179 in 2
4 4
P 10
 AB    22.635 ksi  AB  22.6 ksi ⊳
A 0.44179

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PROBLEM 1.4

Two solid cylindrical rods AB and BC are welded together at B and loaded as
shown. Determine the magnitude of the force P for which the tensile stresses in
rods AB and BC are equal.

SOLUTION
(a) Rod AB:
P  P  12 kips
d2 
A  (1.25 in.) 2
4 4
A  1.22718 in 2
P  12 kips
 AB 
1.22718 in 2
(b) Rod BC:
P P
 
A d2  (0.75 in.)2
4 4
A  0.44179 in 2
P
 BC 
0.44179 in 2
 AB   BC
P  12 kips P
2

1.22718 in 0.44179 in 2
5.3015  0.78539 P P  6.75 kips ⊳

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PROBLEM 1.5

A strain gage located at C on the surface of bone AB indicates that the average normal stress
in the bone is 3.80 MPa when the bone is subjected to two 1200-N forces as shown.
Assuming the cross section of the bone at C to be annular and knowing that its outer diameter
is 25 mm, determine the inner diameter of the bone’s cross section at C.

SOLUTION

P P
   A
A 

Geometry: A (d12  d 22 )
4
4A 4P
d 22  d12   d12 
 
(4)(1200)
d 22  (25  103 )2 
 (3.80  106 )
 222.92  106 m 2
d 2  14.93  103 m d 2  14.93 mm ⊳

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PROBLEM 1.6

Two steel plates are to be held together by means of 16-mm-

diameter high-strength steel bolts fitting snugly inside cylindrical
brass spacers. Knowing that the average normal stress must not
exceed 200 MPa in the bolts and 130 MPa in the spacers,
determine the outer diameter of the spacers that yields the most
economical and safe design.

SOLUTION
At each bolt location the upper plate is pulled down by the tensile force Pb of the bolt. At the same time, the
spacer pushes that plate upward with a compressive force Ps in order to maintain equilibrium.

Pb  Ps

Fb 4Pb 
For the bolt, b   or Pb   bdb2
Ab  db2 4

Ps 4Ps 
For the spacer, s   or Ps   s (d s2  db2 )
As  (d s  db2 )
2
4

Equating Pb and Ps ,

 
 b db2   s (d s2  db2 )
4 4
 b   200 
ds  1  d  1   (16) d s  25.2 mm ⊳
 s  b  130 

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PROBLEM 1.7

Each of the four vertical links has an 8  36-mm uniform rectangular

cross section and each of the four pins has a 16-mm diameter. Determine
the maximum value of the average normal stress in the links connecting
(a) points B and D, (b) points C and E.

SOLUTION

Use bar ABC as a free body.

M C  0 : (0.040) FBD  (0.025  0.040)(20  103 )  0

FBD  32.5  103 N Link BD is in tension.
M B  0 :  (0.040) FCE  (0.025)(20  103 )  0
FCE  12.5  103 N Link CE is in compression.
Net area of one link for tension  (0.008)(0.036  0.016)  160  106 m 2

For two parallel links, A net  320  106 m 2

FBD 32.5  103
(a)  BD    101.563  106  BD  101.6 MPa ⊳
Anet 320  106
Area for one link in compression  (0.008)(0.036)  288  106 m 2

For two parallel links, A  576  106 m 2

FCE 12.5  103
(b)  CE    21.701  106  CE  21.7 MPa ⊳
A 576  106

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PROBLEM 1.8

Link AC has a uniform rectangular cross section 1 in. thick and 1 in. wide.
8
Determine the normal stress in the central portion of the link.

SOLUTION
Use the plate together with two pulleys as a free body. Note that the cable tension causes at 1200 lb-in.
clockwise couple to act on the body.

M B  0:  (12  4)( FAC cos 30)  (10)( FAC sin 30)  1200 lb  0
1200 lb
FAC    135.500 lb
16 cos 30  10 sin 30

1
Area of link AC: A  1 in. in.  0.125 in 2
8
F 135.50
Stress in link AC:  AC  AC    1084 psi  1.084 ksi ⊳
A 0.125

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PROBLEM 1.9

Knowing that the central portion of the link BD has a uniform cross-
sectional area of 800 mm2, determine the magnitude of the load P for
which the normal stress in that portion of BD is 50 MPa..

SOLUTION
Draw free body diagram of link AC.

FBD   A

 
 50  106 N/m 2 800  106 m 2 
40  103 N

BD   0.56 m 2  1.92 m 2
 2.00 m

Free Body AC: M C  0:

0.56
2.00
 
40  103 1.4  
1.92
2.00
 
40  103 1.4   P  0.7  1.4   0

P  33.1  103 N

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P  33.1 kN ⊳

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PROBLEM 1.10

Link BD consists of a single bar 1 in. wide and

1 in. thick. Knowing that each pin has a 3 -in.
2 8
diameter, determine the maximum value of the
average normal stress in link BD if (a)  = 0,
(b)  = 90.

SOLUTION
Use bar ABC as a free body.

(a)   0.
M A  0: (18 sin 30)(4)  (12 cos 30) FBD  0
FBD  3.4641 kips (tension)
 3  1 
Area for tension loading: A  (b  d )t  1     0.31250 in 2
 8  2 
FBD 3.4641 kips
Stress:      11.09 ksi ⊳
A 0.31250 in 2

(b)   90.
M A  0:  (18 cos 30)(4)  (12 cos 30) FBD  0
FBD  6 kips i.e. compression.
1
Area for compression loading: A  bt  (1)    0.5 in 2
2
F 6 kips
Stress:   BD    12.00 ksi ⊳
A 0.5 in 2

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PROBLEM 1.11

The rigid bar EFG is supported by the truss system shown.

Knowing that the member CG is a solid circular rod of 0.75-
in. diameter, determine the normal stress in CG.

SOLUTION

Using portion EFGCB as free body,

3
Fy  0: FAE  3600  0, FAE  6000 lb
5
Use beam EFG as free body.
3 3
M F  0:   4  FAE   4  FCG  0
5 5
FAE  FCG  6000 lb
Normal stress in member CG
d2
Area: A  0.44179 in 2
4
F 6000
 CG    13580 psi
A 0.44179
 CG  13.58 ksi ⊳

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PROBLEM 1.12

The rigid bar EFG is supported by the truss system shown.

Determine the cross-sectional area of member AE for which
the normal stress in the member is 15 ksi.

SOLUTION

Using portion EFGCB as free body,

3
Fy  0: FAE  3600  0, FAE  6000 lb
5
Normal stress in member AE = 15 ksi
F
 AE 
A
F 6.00 kips
A 
 AE 15 ksi

A  0.400 in 2 ⊳

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PROBLEM 1.13

An aircraft tow bar is positioned by means of a single

hydraulic cylinder connected by a 25-mm-diameter steel
rod to two identical arm-and-wheel units DEF. The mass
of the entire tow bar is 200 kg, and its center of gravity
is located at G. For the position shown, determine the
normal stress in the rod.

SOLUTION

FREE BODY – ENTIRE TOW BAR:

W  (200 kg)(9.81 m/s 2 )  1962.00 N

M A  0: 850 R  1150(1962.00 N)  0
R  2654.5 N

FREE BODY – BOTH ARM & WHEEL UNITS:

100
tan     8.4270
675
M E  0: ( FCD cos  )(550)  R(500)  0
500
FCD  (2654.5 N)
550 cos 8.4270
 2439.5 N (comp.)
FCD 2439.5 N
 CD   
ACD  (0.0125 m) 2
 4.9697  106 Pa  CD  4.97 MPa ⊳

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PROBLEM 1.14

Two hydraulic cylinders are used to control the position

of the robotic arm ABC. Knowing that the control rods
attached at A and D each have a 20-mm diameter and
happen to be parallel in the position shown, determine the
average normal stress in (a) member AE, (b) member DG.

SOLUTION
Use member ABC as free body.

4
M B  0: (0.150) FAE  (0.600)(800)  0
5
FAE  4  103 N
 
Area of rod in member AE is A d2  (20  10 3 ) 2  314.16  10 6 m 2
4 4
FAE 4  103
Stress in rod AE:  AE   6
 12.7324  106 Pa
A 314.16  10
(a)  AE  12.73 MPa ⊳
Use combined members ABC and BFD as free body.

4  4 
M F  0: (0.150)  FAE   (0.200)  FDG   (1.050  0.350)(800)  0
5  5 
FDG  1500 N

 
Area of rod DG: A d2  (20  10 3 ) 2  314.16  106 m 2
4 4
FDG 1500
Stress in rod DG:  DG    4.7746  106 Pa
A 3.1416  10 6
(b)  DG  4.77 MPa ⊳

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PROBLEM 1.15

Knowing that a force P of magnitude 50 kN is required to punch a hole

of diameter d = 20 mm in an aluminum sheet of thickness t = 5 mm,
determine the average shearing stress in the aluminum at failure.

SOLUTION
Area of failure in plate:
A   dt    0.020 m  0.005 m 
 3.1416  104 m 2
Average shearing stress:
P
 avg 
A
50  103N

3.1416  104 m 2
 avg  159.2 MPa ⊳

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PROBLEM 1.16

Two wooden planks, each 12 in. thick and 9 in.

wide, are joined by the dry mortise joint shown.
Knowing that the wood used shears off along its
grain when the average shearing stress reaches
1.20 ksi, determine the magnitude P of the axial
load that will cause the joint to fail.

SOLUTION

Six areas must be sheared off when the joint fails. Each of these areas has dimensions 5
8
in.  1
2
in., its area
being
5 1 5 2
A   in  0.3125 in 2
8 2 16
At failure, the force carried by each area is

F   A  (1.20 ksi)(0.3125 in 2 )  0.375 kips

Since there are six failure areas,
P  6 F  (6)(0.375) P  2.25 kips ⊳

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PROBLEM 1.17

When the force P reached 1600 lb, the wooden specimen shown failed in
shear along the surface indicated by the dashed line. Determine the
average shearing stress along that surface at the time of failure.

SOLUTION

Area being sheared: A  3 in.  0.6 in.  1.8 in 2

Force: P  1600 lb
P 1600 lb
Shearing stress:     8.8889  102 psi   889 psi ⊳
A 1.8 in 2

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PROBLEM 1.18

A load P is applied to a steel rod supported as shown by an aluminum

plate into which a 12-mm-diameter hole has been drilled. Knowing that
the shearing stress must not exceed 180 MPa in the steel rod and 70 MPa
in the aluminum plate, determine the largest load P that can be applied to
the rod.

SOLUTION
For steel: A1   dt   (0.012 m)(0.010 m)
 376.99  106 m 2
P
1   P  A11  (376.99  106 m 2 )(180  106 Pa)
A
 67.858  103 N
For aluminum: A2   dt   (0.040 m)(0.008 m)  1.00531  103 m 2
P
2   P  A2 2  (1.00531  103 m 2 )(70  106 Pa)  70.372  103 N
A2
Limiting value of P is the smaller value, so P  67.9 kN ⊳

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PROBLEM 1.19

The axial force in the column supporting the timber beam shown is P
 20 kips. Determine the smallest allowable length L of the bearing plate
if the bearing stress in the timber is not to exceed 400 psi.

SOLUTION

Bearing area: Ab  Lw
P P
b  
Ab Lw
P 20  103 lb
L   8.33 in. L  8.33 in. ⊳
 b w (400 psi)(6 in.)

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PROBLEM 1.20

Three wooden planks are fastened together by a series of bolts to form

a column. The diameter of each bolt is 12 mm and the inner diameter
of each washer is 16 mm, which is slightly larger than the diameter of
the holes in the planks. Determine the smallest allowable outer
diameter d of the washers, knowing that the average normal stress in
the bolts is 36 MPa and that the bearing stress between the washers
and the planks must not exceed 8.5 MPa.

SOLUTION

d2  (0.012 m)2

Bolt: ABolt    1.13097  104 m 2
4 4
P
Tensile force in bolt:    P  A
A
 (36  106 Pa)(1.13097  104 m2 )
 4.0715  103 N

Bearing area for washer: Aw 
4
d 2
o  d i2 
P
and Aw 
 BRG
Therefore, equating the two expressions for Aw gives

4
d 2
o 
 di2 
P
 BRG
4P
do2   di2
 BRG
4 (4.0715  103 N)
do2   (0.016 m)2
 (8.5  10 Pa)
6

do2  8.6588  104 m 2

d o  29.426  103 m
do  29.4 mm ⊳

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PROBLEM 1.21

A 40-kN axial load is applied to a short wooden post that is

supported by a concrete footing resting on undisturbed soil.
Determine (a) the maximum bearing stress on the concrete
footing, (b) the size of the footing for which the average bearing
stress in the soil is 145 kPa.

SOLUTION
(a) Bearing stress on concrete footing.
P  40 kN  40  103 N
A  (100)(120)  12  103 mm 2  12  103 m 2
P 40  103
    3.3333  106 Pa 3.33 MPa ⊳
A 12  103
(b) Footing area. P  40  103 N   145 kPa  45  103 Pa
P P 40  103
  A   0.27586 m 2
A  145  103

Since the area is square, A  b 2

b A  0.27586  0.525 m b  525 mm ⊳

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PROBLEM 1.22

The axial load P = 240 kips, supported by a W10  45 column, is

distributed to a concrete foundation by a square base plate as shown.
Determine the size of the base plate for which the average bearing
stress on the concrete is 750 psi.

SOLUTION

P
  or
A
P
A

240  103 lb

750 psi
 320 in 2

Since the plate is square,

A  b2
b 320 in 2

b  17.89 in. ⊳

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PROBLEM 1.23

An axial load P is supported by a short W8  40 column of cross-

sectional area A  11.7 in 2 and is distributed to a concrete foundation
by a square plate as shown. Knowing that the average normal stress in
the column must not exceed 30 ksi and that the bearing stress on the
concrete foundation must not exceed 3.0 ksi, determine the side a of
the plate that will provide the most economical and safe design.

SOLUTION

P
For the column,   or
A
P   A  (30)(11.7)  351 kips

For the a  a plate,   3.0 ksi

P 351
A   117 in 2
 3.0

Since the plate is square, A  a 2

a  A  117 a  10.82 in. ⊳

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PROBLEM 1.24

A 6-mm-diameter pin is used at connection C of the pedal shown.

Knowing that P = 500 N, determine (a) the average shearing
stress in the pin, (b) the nominal bearing stress in the pedal at C,
(c) the nominal bearing stress in each support bracket at C.

SOLUTION
Since BCD is a 3-force member, the reaction at C is directed toward E, the intersection of the lines of act of
the other two forces.

From geometry, CE  3002  1252  325 mm

From the free body diagram of BCD,
125
Fy  0 : CP0 C  2.6P  2.6  500 N   1300 N
325
1 1
C C 2C
(a)  pin  2  2 

d2 d
2
AP
4
2 1300 N 
 pin   23.0  106 Pa
 
2
3
 6  10 m

 pin  23.0 MPa ⊳

(b) b 
C

C

1300   24.1  106 Pa
Ab dt 
6  10 3 9  10 3 
 b  24.1 MPa ⊳
1
(c)
C
b  2 
C

1300   21.7  106 Pa
Ab 2dt 2 6  10 3

9  10 3
 

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 b  21.7 MPa ⊳

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PROBLEM 1.25

Knowing that a force P of magnitude 750 N is applied to the pedal

shown, determine (a) the diameter of the pin at C for which the
average shearing stress in the pin is 40 MPa, (b) the corresponding
bearing stress in the pedal at C, (c) the corresponding bearing stress
in each support bracket at C.

SOLUTION
Since BCD is a 3-force member, the reaction at C is directed toward E, the intersection of the lines of action
of the other two forces.

From geometry, CE  3002  1252  325 mm

From the free body diagram of BCD,
125
Fy  0 : CP0 C  2.6 P  2.6  750 N   1950 N
325
1 1
C C 2C
(a)  pin  2  2 
AP  2 d2
d
4

2C 2 1950 N 
d    5.57  103 m
 pin 
 40  10 Pa 6

d  5.57 mm ⊳

(b) b 
C

C

1950   38.9  106 Pa
Ab dt 
5.57  103 9  10 3  
 b  38.9 MPa ⊳
1
(c)
C
b  2 
C

1950   35.0  106 Pa
Ab 2dt 2 5.57  10 3
5  10 3
 

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 b  35.0 MPa ⊳

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PROBLEM 1.26

The hydraulic cylinder CF, which partially controls the position of rod
DE, has been locked in the position shown. Member BD is 15 mm
thick and is connected at C to the vertical rod by a 9-mm-diameter
bolt. Knowing that P  2 kN and   75, determine (a) the average
shearing stress in the bolt, (b) the bearing stress at C in member BD.

SOLUTION

Free Body: Member BD.

40 9
M c  0: FAB (100 cos 20)  FAB (100 sin 20)
41 4
(2 kN) cos 75(175sin 20)  (2 kN)sin 75(175cos 20)  0
100
FAB (40 cos 20  9sin 20)  (2 kN)(175) sin(75  20)
41
FAB  4.1424 kN
9
Fx  0: C x  (4.1424 kN)  (2 kN) cos 75  0
41
Cx  0.39167 kN
40
Fy  0: C y  (4.1424 kN)  (2 kN) sin 75  0
41
C y  5.9732 kN
C  5.9860 kN 86.2°
C 5.9860  10 N 3
(a)  ave    94.1  106 Pa  94.1 MPa ⊳
A  (0.0045 m) 2
C 5.9860  103 N
(b) b    44.3  106 Pa  44.3 MPa ⊳
td (0.015 m)(0.009 m)

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PROBLEM 1.27

For the assembly and loading of Prob. 1.7, determine (a) the average
shearing stress in the pin at B, (b) the average bearing stress at B in
member BD, (c) the average bearing stress at B in member ABC,
knowing that this member has a 10  50-mm uniform rectangular cross
section.

PROBLEM 1.7 Each of the four vertical links has an 8  36-mm

uniform rectangular cross section and each of the four pins has a 16-mm
diameter. Determine the maximum value of the average normal stress in
the links connecting (a) points B and D, (b) points C and E.

SOLUTION

Use bar ABC as a free body.

M C  0 : (0.040) FBD  (0.025  0.040)(20  103 )  0

FBD  32.5  103 N

FBD
(a) Shear pin at B.   for double shear
2A
 2 
where A d  (0.016) 2  201.06  106 m 2
4 4
32.5  103
  6
 80.822  106 Pa   80.8 MPa ⊳
(2)(201.06  10 )

(b) Bearing: link BD. A  dt  (0.016)(0.008)  128  106 m2

1 FBD (0.5)(32.5  103 )
b  2
  126.95  106 Pa  b  127.0 MPa ⊳
A 128  106
(c) Bearing in ABC at B. A  dt  (0.016)(0.010)  160  106 m2
FBD 32.5  103
b    203.12  106 Pa  b  203 MPa ⊳
A 160  106

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PROBLEM 1.28

Two identical linkage-and-hydraulic-cylinder systems control the

position of the forks of a fork-lift truck. The load supported by the one
system shown is 1500 lb. Knowing that the thickness of member BD is
5
8
in., determine (a) the average shearing stress in the 12 -in.-diameter
pin at B, (b) the bearing stress at B in member BD.

SOLUTION

Use one fork as a free body.

M B  0: 24E  (20)(1500)  0
E  1250 lb
Fx  0: E  Bx  0
Bx   E
Bx  1250 lb
Fy  0: By  1500  0 By  1500 lb

B Bx2  By2  12502  15002  1952.56 lb

(a) Shearing stress in pin at B.

2
  1
Apin  2
d pin     0.196350 in
2
4 42
B 1952.56
    9.94  103 psi   9.94 ksi ⊳
Apin 0.196350
(b) Bearing stress at B.
B 1952.56
    6.25  103 psi   6.25 ksi ⊳
dt  12  85
 

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McGraw-Hill Education.