d1 PROBLEM 1.2 d2 125 kN B C A Two solid cylindrical rods AB and BC are welded together at B and loaded as shown.
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d1
PROBLEM 1.2
d2
125 kN B
C
A
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.
60 kN 125 kN 0.9 m
1.2 m
SOLUTION (a)
Rod AB: Force:
P
Stress:
AB
Area:
A AB
4
d12
60 103 N 150 106 Pa
d12
4 P A
AB
P AB
d12
(4)(60 103 ) (150 106 )
4P AB
(b)
P
A
d1
22.568 10 3 m
Force:
P
60 103
Stress:
BC
509.30 10 6 m 2
d1
22.6 mm
d2
40.2 mm
Rod BC:
Area:
A BC
d 22
190 103 N
150 106 Pa
4 P A
d 22 4P d 22
4P BC
d2
(2)(125 103 )
(4)( 190 103 ) ( 150 106 )
40.159 10 3 m
1.61277 10 3 m 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. 4
1200 N
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.
A
C
B 1200 N
SOLUTION P A Geometry:
A
4
(d12
P
A
d 22 ) 4A
d 22
d12
d 22
(25 10 3 )2
4P
d12
(4)(1200) (3.80 106 )
222.92 10 6 m 2 d2
14.93 10 3 m
d2
14.93 mm
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. 7
PROBLEM 1.7 0.4 m
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.
C 0.25 m
0.2 m
B E
20 kN D A
SOLUTION Use bar ABC as a free body.
MC
0:
(0.040) FBD FBD
MB
0:
(0.040) FCE
(0.025
0.040)(20 103 )
32.5 103 N
Link BD is in tension. 3
(0.025)(20 10 ) 12.5 103 N
FCE
0 Link CE is in compression.
Net area of one link for tension
(0.008)(0.036
For two parallel links,
320 10 6 m 2
(a)
(b)
BD
FBD Anet
A net
32.5 103 320 10 6
0
0.016)
101.563 106
BD
Area for one link in compression
(0.008)(0.036)
For two parallel links,
576 10 6 m 2
CE
FCE A
12.5 103 576 10 6
160 10 6 m 2
A
21.701 10
6
101.6 MPa
288 10 6 m 2
CE
21.7 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. 9
B
D
PROBLEM 1.11
F
12 ft H
A
C 9 ft
E 9 ft
80 kips
For the Pratt bridge truss and loading shown, determine the average normal stress in member BE, knowing that the crosssectional area of that member is 5.87 in2.
G 9 ft
80 kips
9 ft 80 kips
SOLUTION Use entire truss as free body. MH Ay
0: (9)(80)
(18)(80)
(27)(80)
36 Ay
0
120 kips
Use portion of truss to the left of a section cutting members BD, BE, and CE. Fy BE
0: 120
FBE A
80
12 FBE 15
0
FBE
50 kips
50 kips 5.87 in 2 BE
8.52 ksi
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. 13