Chapter 4 (MECH of MAT) 4-4 The copper shaft id subjected to the axial loads shown. Determine the displacement of end A with respect to end D if the diameters of each segment are mm, mm, and mm AB BC CD d d d 20 25 12 . Take GPa cu E 126 . kN kN kN kN kN kN kN x AB AB x BC AB x CD CD F F F F F F F F F 0 40 0 40 0 40 50 0 10 0 30 0 30 / / / N m N . m N . m Pa . m Pa . m Pa . m . m i i BC BC CD CD AB AB DA i i i AB AB BC BC CD CD DA DA FL F L F L F L EA E A E A E A 3 3 3 2 2 2 9 9 9 3 40 10 2 10 10 3 75 30 10 25 126 10 0 02 126 10 0 025 126 10 0 012 4 4 4 3 8484 10 / . mm Ans. DA 3 85
Transcript
Chapter 4 (MECH of MAT) 4-4 The copper shaft id subjected to the axial loads shown. Determine the displacement of end A with respect to end D if the diameters of each segment are
mm, mm, and mmAB BC CDd d d 20 25 12 . Take GPacuE 126 .
kN kN
kN kN kN
kN kN
x AB AB
x BC AB
x CD CD
F F F
F F F
F F F
0 40 0 40
0 40 50 0 10
0 30 0 30
/
/
/
N m N . m N . m
Pa . m Pa . m Pa . m
. m
i i BC BC CD CDAB ABD A
i i i AB AB BC BC CD CD
D A
D A
F L F L F LF L
E A E A E A E A
3 3 3
2 2 29 9 9
3
40 10 2 10 10 3 75 30 10 2 5
126 10 0 02 126 10 0 025 126 10 0 0124 4 4
3 8484 10
/ . mm Ans.D A 3 85
4-13 A spring-supported pipe hanger consists of two springs which are originally unstretched and have a stiffness of k = 60 kN/m, three 304 stainless steel rods, AB and CD which have a diameter of 5 mm, and EF, which has a diameter of 12 mm, and a rigid beam GH. If the pipe and the fluid it carries have a total weight of 4 kN, determine the displacement of the pipe when it is attached to the support.
kN kN
kN kN
kN
y EF EF
y AG CH AG CH
y S AG S AG
F F F T
F F F F F T symmetry
F F F F F C
0 4 0 4
0 4 0 2
0 0 2
The displacement of pipe is from 3 parts 1. The elongation of rod EF 2. The contraction of spring (Consider force SF that produced the contraction in one spring)
3. The elongation of rod AG or CH (Consider force AGF that produced the elongation in one
rod)
N . m N . m N
N/m Pa . m Pa . m
.
pipe EF S AG
S AG AGEF EFpipe
EF EF spring AG AG
pipe
pipe
FL
EA
F F LF L
E A k E A
3 33
32 29 9
4 10 0 75 2 10 0 752 10
60 10193 10 0 012 193 10 0 0054 4
33 867 10 m3
. mm Ans.pipe 33 9
stainless steel GPaE 304 193
4-21 The rigid beam is supported at its ends by two A-36 steel tie rods. The rods have diameters
mmABd 12 and . mmCDd 7 5 . If the
allowable stress for the steel is MPaallow 115 , determine the intensity of the
distributed load w and its length x on the beam so that the beam remains in the horizontal position when it is loaded.
. m
N where : N/m and : m .
N w.
A CD
CD
y AB CD
AB CD
AB
xM F wx
wxF w x
F F F wx
F F wx
wxF wx
2
2
0 2 4 02
14 8
0 0
4 8 here : N/m and : m w x 2
Given; 1. The beam remains in the horizontal position
and
or
Member and will reach allowable stress at the same time
AB CD
CD CDAB ABAB CD AB CD
AB AB CD CD
CDABAB CD
AB CD
FL
AE
F LF LE E L L
A E A E
FF
A A
AB CD
2. MPaallow 115
MPa
MPa. m
. kN
AB allow
AB
AB
AB
AB
F
A
F
A
F
F
2
115
1150 012
413 006
MPa
MPa. m
. kN
CD allow
CD
CD
CD
CD
F
A
F
A
F
F
2
115
1150 0075
45 0805
From (1);
.. N N N/m
.
wxw
x
2 33
2
24 386 105 0805 10
4 8
From (2);
. .. N N
.
.. N . N : m
. m
. N/m . N/m
.
xx
x x
where xx
x
w
3 3 23
2 2
33 3
33
2
24 386 10 24 386 1013 006 10
4 8
24 386 1013 006 10 5 0804 10
1 3483
24 386 1013 414 10
1 3483
. kN/m
. m Ans.
w
x
13 4
1 35
4-36 The A-36 steel pipe has an outer radius of 20 mm and an inner radius of 15 mm. If it is fits snugly between the fixed walls before it is loaded, determine the reaction at the walls when it is subjected to the load shown.
Compatibility;
where and
. m . m
C A
B A C B
BC BCAB ABAB BC AB BC
AB AB BC BC
A B
A B
FL
EA
F LF LE E A A
E A E A
F F
F F
0
0
0
0 3 0 7 0
72
3
Insert (2) into (1);
kN
. kN
kN . kN . kN
B B
B
A
F F
F
F
716
34 8
16 4 8 11 2
. kN
. kN Ans.A
A
F
F
11 2
4 8
x AB A AB A
x C BC BC C
F F F F F
F F F F F
0 0
0 0
Equilibrium;
kN
kN
x A B
A B
F F F
F F
0 16 0
16 1
4-45 The distributed loading is supported by the three suspender bars. AB and EF are made from aluminum and CD is made from steel. If each bar has a cross-sectional area of 450 mm2, determine the maximum intensity w of the distributed loading so that an allowable stress of MPaallow st 180 in the steel and
MPaallow al 94 in the aluminum is not
exceeded. GPa, GPast alE E 200 70 .
Compatibility; the system is symmetry, so same displacement at A, C, and E
where and
GPa GPa
al st
al al st stal st al st
al al st st
al st
st al
FL
EA
F L F LL L A A
E A E A
F F
F F
70 20020
27
Insert (2) into (1);
.
.
al al
al
st al
F F w
F w w
F F w w w
202 3
721
0 617653420 20 21 30
1 76477 7 34 17
Because and st al st alF F A A 3 , the stress in steel is 3 times of in aluminum while allow allowst al 2 ,
therefore the system is controlled by steel. Case 1; Assume steel failed
MPa
Pa m
. N/m
allowst st
st
st
F
A
w
w
16
6 2
31
180
3017 180 10
450 10
45 9 10
Case 2; Assume aluminum failed
MPa
Pa m
. N/m
allowal al
al
al
F
A
w
w
26
6 2
32
94
2134 94 10
450 10
68 486 10
Because w w1 2 ; so the system is controlled by steel member and the maximum intensity w is 45.9
kN/m Ans.
Equilibrium;
. m . m
C E A
E A al
y al st
al st
M F F
F F F
F F F w
F F w
0 1 5 1 5 0
0 2 3 0
2 3 1
4-112 The rigid link is supported by a pin at A and two A-36 steel wires, each having an unstretched length of 300 mm and cross-sectional area of 7.8 mm2. Determine the force developed in the wires when the link supports the vertical load of 1.75 kN.
Equilibrium;
. kN mm mm mm
. . kN
A B C
B C
M F F
F F
0 1 75 150 100 225 0
2 25 2 625 1
Compatibility; From similar triangle
mm mm
.
. where and
.
C B
C B
C C B BC B C C B B
C C B B
C B
FL
AE
F L F LL L A E A E
A E A E
F F
225 100
2 25
2 25
2 25 2
Insert (2) into (1);
. . . kN
. kN
. . . kN . kN
B B
B
C B
F F
F
F F
2 25 2 25 2 625
0 43299
2 25 2 25 0 43299 0 97423
. kN
. kN Ans.B
C
F
F
0 433
0 974
4-84 The rigid block has a weight of 400 kN and is to be supported by posts A and B, which are made of A-36 steel, and the post C, which is made of C83400 red brass. If all the posts have the same original length before there are loaded, determine the average normal stress developed in each post when post C is heated so that its temperature is increased by 10 oC. Each post has a cross-sectional area of 5000 mm2.
Compatibility; The system is symmetry, the displacement of all bars are same
o o
where and
/ C C m Pa m Pa
. N
.
A B
st st B Bbr B st B st B
st st B brass
st B
B st
B st
FL
AE
F L F LTL L L A A
A E A E
F F
F F
F F
6
6 2 9 6 2 918 10 10
5000 10 200 10 5000 10 101 10
0 505 90900
90900 0 505 2
Insert (2) into (1);
N . N
. N
N . N . . N . N
st st
st
B st
F F
F
F F
3
3
3 3
2 90900 0 505 400 10
123 39 10
90900 0 505 90900 0 505 123 39 10 153 21 10
Determine the stress in each post;
. N. MPa
mm
. N. MPa
mm
stA C
BB
F
A
F
A
F
A
3
2
3
2
123 39 1024 678
5000
153 21 1030 642
5000
. MPa and . MPa Ans.A C B 24 7 30 6
Equilibrium;
m m
kN
kN
G C A
A C st
y st B
st B
M F F
F F F
F F F
F F
0 1 1 0
0 2 400 0
2 400 1
4-114 The 2014-T6 aluminum rod has a diameter of 12 mm and is lightly attached to the rigid supports at A and B when T1 =25oC. If the temperature becomes T2 = -20oC, and an axial force of P = 80 N is applied to the rigid collar as shown, determine the reactions at A and B.
5-12 The solid shaft is fixed to the support at C and subjected to the torsional loading shown. Determine the shear stress at points A and B and sketch the shear stress on volume elements located at these points.
. m N.m . m. Pa
. m . m
. m N.m . m. Pa
. m . m
A
B
T
J
T
T
2 6
4 4
1 6
4 4
0 035 500 0 0357 4241 10
0 035 0 0352 2
0 02 800 0 026 7878 10
0 035 0 0352 2
. MPa
. MPa Ans.A
B
4 45
6 79
N.m800
N.m300
CT
N.m800
N.m800
N.m300
T1
T2
N.m
N.m
xM T
T
1
1
0 800 0
800
x
N.m N.m
N.m
xM T
T
2
2
0 800 300 0
500
5-13 A steel tube having an outer diameter of 62.5 mm is used to transmit 3 kW when turning at 27 rev/min. Determine the inner diameter d of the tube to the nearest multiples of 5 mm if the allowable shear stress is MPa.allow 70
W
N.m rev/min rad/rev min/ sec
P T
PT
33 10 10000
27 2 1 60 3
max
. m
Pa. m
. m
allow
allow
Tc J
Tc
J
d
d
6
4 4
3
10000 0 06253 2
70 100 0625
32
56 8345 10
mm Ans.d 60
5-56 The motor delivers 32 kW to the 304 stainless steel solid shaft while it rotates at 20 Hz. The shaft has a diameter of 37.5 m and is supported on smooth bearings at A and B, which allow free rotation of the shaft. The gears C and D fixed to the shaft remove 20 kW and 12 kW, respectively. Determine the absolute maximum stress in the shaft and the angle of twist of gear C with respect to gear D.
WN.m
Hz
WN.m
Hz
WN.m
Hz
C
D
P T
PT
PT
PT
3
3
3
32 10 800
2 20
20 10 500
2 20
12 10 300
2 20
Determine the internal load in each section;
N.m
x AC
AC
M T T
T T
0 0
800
N.m N.m
x CD C
CD C
M T T T
T T T
0 0
800 500 300
x BD
BD
M T
T
0 0
0
Determine the shear stress in each section;
. m N.m
. Pa. m
. m N.m
. Pa. m
ACAC
CDCD
BDBD
Tc
J
T c
J
T c
J
T c
J
6
4
6
4
800 0 03752
24 593 100 0375
32300 0 0375
29 2225 10
0 037532
0
Angle of twist;
o
N.m . m. rad .
Pa . m
CD CDCD
CD CD
TL
GJ
T L
G J
49
3000 2
0 0013116 0 07515275 10 0 0375
32
max
o
. MPa
. CCW Ans.CD
24 6
0 0752
5-66 The device serves as a compact torsion spring. It is made of A-36 steel and consists of a solid inner shaft CB which is surrounded by and attached to a tube AB using a rigid ring at B. The ring at A can also be assumed rigid and is fixed from rotating. If the allowable shear stress for the material is MPaallow 84 and
the angle of twist at C is limited to oallow 3 ,
determine the maximum torque T that can be applied at the end C.
Case 1; Consider MPaallow 84 in tube AB
max
. m MPa
. m . m
. mPa
. m . m
. N.m . N.m
AB allow
tube
case
Tc J
T
T
T T
4 4
6
4 4
1
0 02584
0 025 0 018752
0 02584 10
0 025 0 018752
1409 34 1409 34
Case 1; Consider MPaallow 84 in shaft BC
max
. m MPa
. m
. m Pa
. m
. N.m . N.m
BC allow
shaft
case
Tc J
T
T
T T
4
6
4
2
0 012584
0 01252
0 012584 10
0 01252
257 71 257 71
Case 3; Consider oallow 3
o
o
and
. m . m radian
Pa . m . m Pa . m
. N.m
c allow c B A C B
shaft shafttube tubeallow
tube tube shaft shaft
TL GJ
T LT L
G J G J
T T
T
4 4 49 9
0 3 0 6 3
18075 10 0 025 0 01875 75 10 0 01252 2
240 02 . N.mcaseT 3 240 02
Because ,case case caseT T T3 2 1, therefore the system is controlled by allow and max N.mT 240 Ans.
x tube tube
x shaft shaft
M T T T T
M T T T T
0 0
0 0
5-77 The shaft is made of L2 tool steel, has a diameter of 40 mm, and is fixed at its ends A and B. If it is subjected to the couple, determine the maximum shear stress in regions AC and CB.
Equilibrium;
N.m
N.m
x A B
A B
x AC A AC A
x BC B BC B
M T T
T T
M T T T T
M T T T T
0 200 0
200 1
0 0
0 0
Compatibility;
where
. m . m
B A
C A B C
AC AC BC BCAC AC BC BC
AC AC BC BC
A B
A B
TL GJ
T L T LG J G J
G J G J
T T
T T
0
0
0
0 4 0 6 0
32
2
Insert (2) into (1);
N.m
N.m
N.m N.m
B B
B
A B
T T
T
T T
3200
280
3 380 120
2 2
Shear stress in each section;
N.m . m. Pa
. m
N.m . m. Pa
. m
ACAC
BCBC
Tc J
T c
J
T c
J
6
4
6
4
120 0 04 29 5493 10
0 0432
80 0 04 26 3662 10
0 0432
. MPa
. MPa Ans.AC
BC
9 55
6 37
5-79 The shaft is made from a solid steel section AB and a tubular portion made of steel and having a brass core. If it is fixed to a rigid support at A, and a torque of T = 50 N.m is applied to it at C, determine the angle of twist that occurs at C and compute the maximum shear stress and maximum shear strain in the brass and steel. Take GPa, GPast brG G 80 40 .
