Chapter 9 Combined Stresses
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Transcript of Chapter 9 Combined Stresses
![Page 1: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/1.jpg)
Chapter 9Combined Stresses
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9-1 Introduction
• Basic types of loading: axial, torsional and flexural
• Stress formulas:
Axial loading -
Torsional loading -
Flexural loading -
A
Pa
J
T
I
Myf
![Page 3: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/3.jpg)
9-2 Combined Axial & Flexural Loads
f
My
I
a
PA
af
PA
MyI
y
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2
6( )
P Mc MA I bh
3 3 3
2
6 6
20 10 6(0.45 15 10 0.15 20 10 )(0.05)(0.150) (0.05)(0.150)
(2.67 10 ) (20.00 10 ) 22.67 MPa
A
3 3 3
2
6 6
20 10 6(0.45 15 10 0.15 20 10 )
(0.05)(0.150) (0.05)(0.150)
(2.67 10 ) (20.00 10 ) = 17.33 MPa
B
A
B 20
15
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o2000sin(15 )
o2000cos(15 )C D
0 :DM O O3
(6) ( )2000cos(15 ) (4)2000sin(15 )12
264.598 lb.
y
y
C
C
264.598 lb.yC
o2000cos(15 )
0 :xF O2000cos(15 )xD
1931.852 lb.
0 :yF O2000sin(15 ) lb.y yD C
253.04 lb.yD
253.04 lb.yD
1931.852
517.638 lb
A
B264.598 lb.yC
1931.852 lb
517.638 lb
759.12 lb.ft
Section AB:
3(3) ( )1931.852 (1)517.638
12yM C
759.12 lb.ft
![Page 8: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/8.jpg)
1931.852
517.638 lb
A
B264.598 lb.yC
1931.852 lb
517.638 lb
759.12 lb.ft
Normal Stresses
2
6( )
P Mc MA I bh
2
1931.852 6 759.12 12
2 6 2 6A
2920.1 lb/in
2
1931.852 6 759.12 12
2 6 2 6B
2598.1 lb/in
BMD
1012.16 lb.ft
529.20 lb.ft
min 2
1931.852 6 1012.16 12
2 6 2 6
21173.15 lb/in
max 2
1931.852 6 1012.16 12
2 6 2 6
2851.17 lb/in
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2 24 0.0025 0.07854 mA D
4 4 6 464 (0.1) 1.5625 10 mI D
P
P
0.25P
P McA I
0.25 0.050.0025 1.5625
P P
840080 MPa
P
680 1029.92 kN
8400P
![Page 10: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/10.jpg)
4.0 1.5
5.0
122400 [ 25 6 25 3]P
180,000 180,000 360,000 kg.
12
180,000 0.5 180,000 3.0
(1000 15) 15 5
M
90,000 540,000 562,500
112,500 kg-m.
2
6( )
P Mc MA I bh
2
min 2
360,000 6 112,50048,333.33 kg/m
1 9 1 9
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For long slender members or columns, the effect of P- is significant
PA
MyI
y
For stiff members the formula is appropriate
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P P
1in
21
in2
2 21 1( ) in2 4
A
4 4112
1 1( ) in2 192
I
Fig.(a)
Fig.(b)
1 12 4
max,( ) 1 14 192
( )( )28
( ) ( )a
P Mc P PP
A I
max. compressive stress in Fig.(a)
max. compressive stress in Fig.(b)
max,( ) 14
4( )b
P PP
A
max,( )
max,( ) 287 :1
4a
a
PP
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Hw10
allow
B
D1D2
D1=(1+z1) in. D2 = D1(1+z2) in.
I1-1=1000(1+z3) in4 Area=10(1+z4) in2
B =10(1+z5) in. allow=10(1+z6) ksi.
ค่�า z1-z6 ได้�จากเลขประจ�าตั�วนิ�สิ�ตั ด้�งตั�อไปนิ��
46z1z2z3z4z5z6
Fig. P-908
หมายเหตุ� D2 = D1(1+z2) in.
เพื่��อให้�ห้นิ�าตั�ด้มี�ประสิ�ทธิ�ภาพื่ด้�ในิการร�บห้นิ�วยแรง
![Page 15: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/15.jpg)
Hw11
L1
L2 L3 L4
b
h
L1= (1+z1) in. L2 = (1+z2) in.
L3= (1+z3) in. L4 = (1+z4) in.
b = 0.2(1+z5) in. h = b(1+z6) in.
P = (1+z5) kips. F = (1+z6) kips.
ค่�า z1-z6 ได้�จากเลขประจ�าตั�วนิ�สิ�ตั ด้�งตั�อไปนิ��
46z1z2z3z4z5z6หมายเหตุ� h = b(1+z6) in.
เพื่��อให้�ค่านิมี�ค่วามีล'กไมี�นิ�อยกว�าค่วามีกว�างเสิมีอ
![Page 16: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/16.jpg)
9-3 Kern of Section: Loads Applied off Axes of Symmetry
( )P My Pe a
A I I
Ia
Ae
for b h section
3( /12)2h bh
bh e
6h
e
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That is in designing of masonry or other structures weak in tension, the resultant load should fall in the middle third of the section.
6h
e
The maximum eccentricity to avoid tension
The general case:
( )( ) yx
y x
Pe yPe xPA I I
2 2
( )( )0 yx
y x
Pe yPe xP
A Ar Ar
The position of neutral axis (line of zero stress)
2 20 1 yx
y x
eex y
r r
2
2
x x
y y
I Ar
I Ar
2y
x
ru
e
2x
y
re
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( , )2 2
b h
3 3
( )( )Rectangular section: 0
/12 /12yx
Pe yPe xPbh bh hb
3 3
( )( / 2)( )( / 2)0
/12 /12yx
Pe bPe hP
bh bh hb
1
/ 6 / 6yx
ee
h b
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A compressive load P= 12 kips is applied, as in Fig. 9-8a, at a point 1 in. to the right and 2 in. above the centroid of a rectangular section for which h=10 in. and b=6 in. Compute the stress at each corner and the location of the neutral axis. Illustrate the answers with a sketch similar to Fig. 9-8b.
918
12 kips
2
1
10
6
( )( ) yx
y x
Pe yPe xPA I I
3 3
Rectangular section:
( )( )
/12 /12yx
Pe yPe xPbh bh hb
3 3
12 (12 1) (12 2)0.08 ksi
6 10 6 10 /12 10 6 /12( 5) (3)
A
3 3
12 (12 1) (12 2)0.72 ksi
6 10 6 10 /12
( 5
10 6 /12
) ( 3)B
3 3
12 (12 1) (12 2)0.48 ksi
6 10 6 10 /12 1(5) ( 3)
0 6 /12C
0.32 ksiD
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12 kips
2
1
10
63 3
Position of Neutral Axis:
( )( )0
/12 /12yx
Pe yPe xPbh bh hb
3 3
12 (12 1) (12 2)0
6 10 6 10 /12 10 6 /1( ) (
2)x y
3 21
25 3
x y
on x axis (y=0) 25/ 3 8.33x
on y axis (x=0 3/ 2 1.5) y
N.A.
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921 Calcualte and sketch the kern of a W360 X 122 section.
2 2Position of Neutral Axis: 0 1 yx
y x
eex y
r r
257 363A( , )
2 2
2 2
257 363At corner A: 0 1
63 52 21 3yx
ee
22 63on x-axis ( =0): mm
230.89
57y xe e
22 153on y-axis ( =0): mm
3629 0
31 .x ye e
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9-4 Variation of Stress with Inclination of Element
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Mc
I
Tc
J
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9-5 Stress at A Point
Stress at a point really defines the uniform stress distributed over a differential area.
![Page 25: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/25.jpg)
• The most general state of stress at a point may be represented by 6 components,
),, :(Note
stresses shearing,,
stresses normal,,
xzzxzyyzyxxy
zxyzxy
zyx
state of stress เม�อแสดงด�วยระบบโคออร�ด�เนตุ (xyz)
xx xy xz x xy xz
yx yy yz yx y yz
zx zy zz zx zy z
σsymmetry
state of stress เม�อแสดงด�วยระบบโคออร�ด�เนตุ (xyz)
xx xy xz x xy xz
yx yy yz yx y yz
zx zy zz zx zy z
σsymmetry
![Page 26: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/26.jpg)
• Plane Stress - state of stress in which two faces of the cubic element are free of stress. For the illustrated example, the state of stress is defined by
.0,, and xy zyzxzyx
• State of plane stress occurs in a thin plate subjected to forces acting in the midplane of the plate.
( , )n n • State of plane stress also occurs on the free surface
of a structural element or machine component, i.e., at any point of the surface not subjected to an external force.
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Two methods to compute the maximum stresses i.e.,
(1) Analytical approach
(2) Using of Mohr’s circle
Plane Stress
x
y
xy
xy x
y yx
yx
x
y
z x
y x
y x
y
z
![Page 28: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/28.jpg)
9-6 Variation of Stress at A Point: Analytical Derivation
A
cosA
sinA
![Page 29: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/29.jpg)
0nF ( sin
( s
)( cos ) ( cocos sin sin
cn o
s
i s
)
)
x xy
yx
yA A
A
A A
0tF ( sin )
( s
sin cos co( s
s
c
in )
os ) c s )
in
( ox xyy
yx
A
A
A A A
![Page 30: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/30.jpg)
0nF ( sin
( s
)( cos ) ( cocos sin sin
cn o
s
i s
)
)
x xy
yx
yA A
A
A A
0tF ( sin )
( s
sin cos co( s
s
c
in )
os ) c s )
in
( ox xyy
yx
A
A
A A A
22cos 2 cossin sinyx xy
2 2sin cosinco sicss os nx yyy xx
2 2sisin 2
No11 cos2
cos ,2
te: , cos sinc 2
,2
n2
osxy yx
cos2 sin 22 2
x y x yxy
sin 2 cos22
x yxy
![Page 31: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/31.jpg)
cos2 sin 22 2
x y x yxy
sin 2 cos22
x yxy
A
cosA
sinA
x y
xy
xy
x y yx
yx
cos2 sin 22 2
x y x yx xy
sin 2 cos22
x yxy xy
cos2 sin 22 2
x y x yy xy
2
2
cos2( ) cos( 2 ) cos2
sin 2( ) sin( 2 ) sin 2
![Page 32: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/32.jpg)
cos2 sin 22 2
x y x yxy
sin 2 cos22
x yxy
A
cosA
sinA
d2 sin 2 2 cos2 0
d 2x y
xy
Find maximum or minimum differentiating Eq.(9-5) w.r.t. and setting the derivative equal to zero
Eq.(9-5)
2tan 2 xy
x y
Eq.(9-6)
Find maximum or minimum differentiating Eq.(9-6) w.r.t. and setting the derivative equal to zero
d2 cos2 2 sin 2 0
d 2x y
xy
tan 2
2x y
sxy
![Page 33: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/33.jpg)
A
cosA
sinA
2tan 2 xy
x y
cos2 sin 22 2
x y x yxy
sin 2 cos22
x yxy
Eq.(9-5)
Eq.(9-6)
At zero shearing stress
0 sin 2 cos22
x yxy
ซึ่'�งเป)นิมี*มีเด้�ยวก�บสิมีการ Eq.(9-7) ด้�งนิ��นิ ค่�า maximum or minimum จะเก�ด้ข'�นิเมี��อ = 0
2tan 2 xy
x y
1 1
2 2 2 2
sin 2 , cos2
( ) 2 ( )2 2
xy x y
x y x yxy xy
2 2
2 2 2 2
sin 2 , cos2
( ) 2 ( )2 2
xy y x
x y x yxy xy
![Page 34: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/34.jpg)
1 2 2
2
( )2 2
x y x yxy
Maximum or minimum (Principal stresses)
2tan 2 xy
x y
1
2
2
1
Maximum or minimum
tan 22x y
sxy
2 2 1 1max ( )
2 2x y
xy
1
22
1
s
มี*มี และ s ตั�างก�นิ 45O
![Page 35: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/35.jpg)
22000.04 kN/mm 40 MPa, 0, 0
50 100x y xy
P
A
O Ocos2( ) sin 2-40 -( )2 2
40x y x yxy
O O40 0 40 0cos2( ) 0 sin 2(-40 -40 ) 16.5 MPa
2 2
sin 2 cos22
x yxy
O O20 0sin 2( ) 0 c-40 -4os2( ) 9.85 MPa
20
![Page 36: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/36.jpg)
4,000 psi
8,000 psi
6,000 psi
x
y
xy
6,000 psi 4,000 psi
8,000 psi
1 2 2 2 2
2
4000 ( 8000) 4000 ( 8000)( ) ( ) ( 6000)
2 2 2 2x y x y
xy
2 22000 (6000) ( 6000) 10485. 64, psi85.33
![Page 37: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/37.jpg)
O Ocos2 30( ) sin 2( )2 2
30x y x yxy
O O304000 ( 8000) 4000 ( 8000)
cos2( ) ( 6000) sin 2( ) 6,19630 .15 psi2 2
sin 2 cos22
x yxy
O O4000 ( 8000)sin 2( ) ( 6000) cos2( )30 30 2196.15 psi
2
4,000 psi
8,000 psi 6,000 psi
6,196.15 psi
2,196.15 psi
o30
![Page 38: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/38.jpg)
cos2 sin 22 2
x y x yxy
sin 2 cos22
x yxy
Eq.(9-5)
Eq.(9-6)
9-7 Variation of Stress at A Point: Mohr’s Circle
Otto Mohr (1882)
Eq.(a)2 + Eq.(b)2
![Page 39: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/39.jpg)
![Page 40: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/40.jpg)
Rule for Applying Mohr Circle to Combined Stresses
( , )x xy
( , )y xy
x-ax
is
y-ax
is
(0,0)
![Page 41: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/41.jpg)
( , )x xy
( , )y xy
(0,0)
x-ax
is
y-ax
isC
![Page 42: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/42.jpg)
(0,0)
( , )x xy
x-ax
is
y-ax
isC
( , )y xy
n-axis
R
( , )n n
n
n
![Page 43: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/43.jpg)
(0,0)
( , )x xy
( , )y xy
n-axis
R
( , )n n
n
n
x-ax
is
y-ax
isC
![Page 44: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/44.jpg)
( , )x xy
( , )y xy
x-ax
is
y-ax
is
C
( ,0) ( ,0)2
x yC
C
R
2 2( )2
x yxyR
1( ,0)2( ,0)
1
2
C R
C R
max( , )C
max R
1
1
sin 2 or
2tan 2 =
xy
xy
x y
R
o2 12 180 2
![Page 45: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/45.jpg)
( , )x xy
( , )y xy
![Page 46: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/46.jpg)
x-axis
y-axisC
R 1( ,0)2( ,0)
(4000, 6000)
( 8000,6000)
( ,0) ( ,0)2
8000 4000( ,0) ( 2000,0)
2
x yC
C
2 2 2 24000 8000( ) ( ) 6000 6000 2 psi
2 2x y
xyR
1 2, 2000 6000 2 4485.3, 10485.3 psiC R
1
6000sin 2
6000 2xy
R
O1 22.5
12
1
22.5
( 2000,0)
![Page 47: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/47.jpg)
x-axis
y-axis
CR 1( ,0)2( ,0)
(4000, 6000)
( 8000,6000)
o
o
o
30
o
o o
30
cos(15 )
2000 6000 2 cos(15 ) 6196.15 psi
sin(15 ) 6000 2 sin(15 ) 2196.15 psi
C R
R
( 2000,0)
o o30 30( , )
o o120 120( , )
o
o
o
120
o
o o
120
cos(15 )
2000 6000 2 cos(15 ) 10196.15 psi
sin(15 ) 6000 2 sin(15 ) 2196.15 psi
C R
R
30
6196.15
2196.15
10196.15
2196.15
![Page 48: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/48.jpg)
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9-8 Absolute Maximum Shearing Stress
Mohr’s circle: Rotation around z-axis
x1
2
1 2
2zR
zR 12
1
2
![Page 51: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/51.jpg)
1
2
2
2xR
Mohr’s circle: Rotation around x-axis
xR
Mohr’s circle: Rotation around y-axis
1
2yR
yR
![Page 52: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/52.jpg)
1
2
x1
2
1 2
2zR
zR
1
2yR
yR
2
2xR
xR
![Page 53: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/53.jpg)
Mohr’s circles for plane stress
zR
yR
xR
Absolute maximum shearing stress for plane stress is equal to the largest of the following three values
1
2
1 2 1 2, ,2 2 2z z xR R R
![Page 54: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/54.jpg)
Mohr’s circles for general state of stress
zR
yR
xR
1
2
z 3
Absolute maximum shearing stress for general state of stress is equal to the largest of the following three values
1 2 1 3 2 3, ,2 2 2z z xR R R
![Page 55: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/55.jpg)
1 2
2
1 5
201
025 ksi,
2
50 2015 ksi,
2
0 ksi,
2
2
2
2
Maximum in-plane shearing stress =
1 2 50 2015 ksi
2 2
Absolute maximum shearing stress is the largest of
![Page 56: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/56.jpg)
50x
Maximum in-plane shearing stress =
1 2 50 2035 ksi
2 2
Absolute maximum shearing stress is the largest of
1 2
2
1
50 2035 ksi,
5025 ksi,
2 2
2010 ksi,
2
2
2
2
(ksi)
(ksi)
1=-50 2 =20zRyR
xR
Ex.
![Page 57: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/57.jpg)
Hw17 the figure
( ส�าหร�บข้�อน��ให�ค�านวณ ค!า absolute maximum shearing
stress ด�วยโดยกำ�าหนดให� z = 0 )
210( 1) MPaz
110( 1) MPaz
310( 1) MPaz
ค่�า z1-z3 ได้�จากเลขประจ�าตั�วนิ�สิ�ตั ด้�งตั�อไปนิ�� 46xxxz1z2z3
![Page 58: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/58.jpg)
9-9 Application of Mohr’s Circle to Combined Loadings
Combined Loadings (axial, torsional, flexural)
Combined stresses
Mohr’s Circlex-axis
( ),
y-axis
(0, )
12
max
Principal stresses and, Maximum shearing stress
1
2
2
1
1
22
1
s
maxmax
Design Criteria, ,allow allow
![Page 59: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/59.jpg)
Stress Trajectories
1
2
max
Tc
J
12
1
Tc
J
![Page 60: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/60.jpg)
Torsional Failure Modes
• A ductile specimen breaks along a plane of maximum shear
• A brittle specimen breaks along planes perpendicular to 1
• Ductile materials generally fail in shear. Brittle materials are weaker in tension than shear.
max
Tc
J
1
Tc
J
45o
![Page 61: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/61.jpg)
max
Tc
J
1
Tc
J
Stress Trajectories for Torsion
Stress Trajectories: lines of principal stress direction but of variable stress intensity
![Page 62: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/62.jpg)
Stress Trajectories for Beam
Mohr’s Circle x-axis
( ),
y-axis(0, )
12
max
My
I
VQ
Ib
![Page 63: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/63.jpg)
7 26
(2500 )(0.05)8 10 N/m 80 MPa
1.5625 10McI
6
26
(0.05) 1.6 10 1.6T( ) N/m MPa
3.125 10Tc T TJ
2500 N.mM
100 mm
80 MPa
100 MPa
D
4 46 4(0.1)
1.5625 10 m64 64D
I
4 46 4(0.1)
3.125 10 m32 32D
J
![Page 64: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/64.jpg)
Mohr’s Circle
1.68( )0, T
12
max80 MPa
1.6 MPa
T
1.60,( )T
(40,0)
C40 MPaC
2 21
1.640 40 ( )
TC R
2 2max
1.640 ( )
TR
80 MPa
100 MPa
2 (30)(87.81)P
16,551.8 wattP 87.81 N.mT
2 21.640 ( ) 60 MPa
T
2P f T
![Page 65: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/65.jpg)
4 4
,4 2
r rI J
3
4M
McI r
3
2
Tc TI r
![Page 66: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/66.jpg)
Mohr’s Circle
3 34 2,( )
rTr
M
12
max3
4M
r
3
2T
r
320,( )Tr
(40,0)
C
32 /( )C M r
2 2 2 2max 3 3 3
2 2 2( ) ( )
M TR M T
r r r
2 21 3
2C R M M T
r
![Page 67: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/67.jpg)
10 ksi
If900 12
900 lb-ft 10.8 kips-in1000600 12
600 lb-ft 7.2 kips-in1000
T
M
2 2max 3
2 23 3
2
2 8.2637.2 10.8 ksi
M Tr
r r
2 21 3
2 23 3
2
2 12.8477.2 7.2 10.8 ksi
M M Tr
r r
max 16 ksi
max 10 ksi
16 ksi
0.938 in.r
0.929 in.r
![Page 68: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/68.jpg)
![Page 69: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/69.jpg)
2500 N
1250 N
3750 N
4000 N
2500 N
2875 N
3625 N
1500 N.m
750 N.m
750 N.m
2500 N
1250 N
3750 N
4000 N
2500 N
2875 N
3625 N
1500 N.m
750 N.m
750 N.m
![Page 70: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/70.jpg)
750 N.m 750 N.m
1500 N.m
2500 N
4 m 2 m
750 N.m 750 N.m
4000 N 2500 N
1500 N.m
1 m 2 m 1 m 2 m3625 N 2875 N
3750 N1250 N
2500 N
1250 N
3750 N
4000 N
2500 N
2875 N
3625 N
1500 N.m
750 N.m
750 N.m
BMzD
3625 N.m
2875 N.m
TMD
1500 N.m
750 N.m
BMyD
5000 N.m3750 N.m
1250 N.m
![Page 71: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/71.jpg)
2500 N
1250 N
3750 N
4000 N
2500 N
2875 N
3625 N
1500 N.m
750 N.m
750 N.m
BMzD
3625 N.m
2875 N.m
TMD
1500 N.m
750 N.m
BMyD
5000 N.m3750 N.m
1250 N.m
Cross section of solid shaft
and the resultant moment
zM
yM
2 2| | z yMM M
3834.5 N.m
4725.2 N.m 5000 N.m
|M|A B C D E
A
B
C
D
E
![Page 72: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/72.jpg)
BMzD
3625 N.m
2875 N.m
TMD
1500 N.m
750 N.m
BMyD
5000 N.m3750 N.m
1250 N.m
3834.5 N.m
4725.2 N.m 5000 N.m
|M|A B C D E
2 2max 3
2M T
r
2 21 3
2M M T
r
From Prob. 951 and this problem.
70 MPa
120 MPa
Mohr’s Circle
x-axis
( ), y-axis
(0, )
12
max3
4 M
r
3
2Tr
2 2max 3
24725.2 1500 1000 mm
r
70 MPa
35.6 mmr
37.2 mmr
2 21 3
24725.2 4725.2 1500 1000
r
120 MPa
2 2max 3
25000 750 1000 mm
r
70 MPa
At section D
35.8 mmr
37.7 mmr
2 21 3
25000 5000 750 1000
r
120 MPa
37.7 mm≥r
At section C
![Page 73: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/73.jpg)
state of stress on the element on the surface of vessel
![Page 74: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/74.jpg)
1
2
67.5
67.5
R
R
Absolute maximum shearing stress
1 2
1
2
| |50 MPa
2| | 67.5
50 MPa2 2
| | 67.550 MPa
2 2
R
R
R
50 MPa
32.5 MPaR
2
2 2
2x y
xyR
2 2 2 222.5 32.5xyR 2 2 232.5 22.5 550xy
23.45 MPaxy
23.45 MPaTc
J
4 4
(455 mm)23.45 MPa
920 90032
T
301.8 kN.mT
![Page 75: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/75.jpg)
20 mm
120 mm
36 420 120
=2.88 10 mm12
I
A
20 mm
40 mm
4 3
(20 40) 40
=3.2 10 mm
Q
N.A.
250 mm
40 kNP
30 kNV
7500 kN.mmM
6
40 7500 20
20 120 2.88 10 68.75 MPa
P My
A I
4
6
30 3.2 1016.67 MPa
2.88 10 20VQI b
![Page 76: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/76.jpg)
250 mm
40 kNP
30 kNV
7500 kN.mmM
6
40 7500 20
20 120 2.88 10 68.75 MPa
P Mc
A I
4
6
30 3.2 1016.67 MPa
2.88 10 20VQI b
2 2
2 2
( )2
68.75( ) 16.67 38.20 MPa
2
x yxyR
1 2, 34.375 38.20
72.578, 3.825 MPa
C R
O
16.67sin 2
38.20
12.94
xy
R
12.94
72.58
72.58 3.83
3.83
( ,0) ( ,0)2
68.75 0( ,0) (34.375,0)
2
x yC
C
y-ax
is
Mohr’s Circle at point A
,1(6 68.75 .67)
12
max
160,( .67)
(34.375,0)C
x-ax
is
2
![Page 77: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/77.jpg)
20 mm
120 mm
36 420 120
=2.88 10 mm12
I
B
20 mm
40 mm
4 3
(20 40) 40
=3.2 10 mm
Q
N.A.
300 mm
40 kNP
30 kNV
9000 kN.mmM
6
40 9000 ( 20)
20 120 2.88 10 45.83 MPa
P My
A I
4
6
30 3.2 1016.67 MPa
2.88 10 20VQI b
![Page 78: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/78.jpg)
2 2
2 2
( )2
45.83( ) 16.67 28.34 MPa
2
x yxyR
300 mm
40 kNP
30 kNV
9000 kN.mmM
45.83 MPa
16.67 MPa
O16.67sin 2 2 36.03
28.34xy
R
Mohr’s Circle at point B
45.83,( )16.67
x-axis
160,( .67)C
y-axis
o60 o36.06
22.9( )15,0
48.81, 11 1( ).5
( ,0) ( ,0)2
45.83 0( ,0) ( 22.915,0)
2
x yC
C
0
o
3028.34sin (23.97 ) 11.51 MPa
0
o o
30
o
cos(60 36.03 )
22.915 28.34cos(23.97 )
48.81 MPa
C R
45.83 MPa
16.67 MPa
48.81 MPa
11.51 MPa
![Page 79: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/79.jpg)
Hw18
1L
2L
3L
4L
D
1.2D
1.2D
L1= 4(1+z1) in. L2 = 4(1+z2) in.
L3= 4(1+z3) in. L4 = 4(1+z4) in.
D = 4(1+z5) in.
ค่�า z1-z5 ได้�จากเลขประจ�าตั�วนิ�สิ�ตั ด้�งตั�อไปนิ�� 46xz1z2z3z4z5
![Page 80: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/80.jpg)
Hw19
L= 0.4(1+z1) m. P = 4(1+z2) kN
H= 40(1+z3) mm. W = 40(1+z4) mm
ค่�า z1-z4 ได้�จากเลขประจ�าตั�วนิ�สิ�ตั ด้�งตั�อไปนิ�� 46xxz1z2z3z4
Also find the maximum shearing stress at point A. Show your results on a complete sketch of a differential element.
LP
H
W
![Page 81: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/81.jpg)
![Page 82: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/82.jpg)
2(1 )
EG
![Page 83: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/83.jpg)
http://www.kyowa-ei.co.jp/english/products.htm
![Page 84: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/84.jpg)
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![Page 88: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/88.jpg)
Strain and deformation of line element
0, 0, 0x y xy 0, 0, 0x y xy 0, 0, 0x y xy
O
( )IIA
Aydy
O
A
( )IIIAxydy
Oxdx
( )IAA
O
A
A
0, 0, 0x y xy
O
A
dx
dy
ds
![Page 89: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/89.jpg)
cos2 sin 22 2
x y x yxy
sin 2 cos22
x yxy
Eq.(9-5)
Eq.(9-6)
A
cosA
sinA
![Page 90: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/90.jpg)
![Page 91: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/91.jpg)
12
6300 2 10 radR
6300 10 rad2xy
6800 10 radx
6200 10 rady
6500 10 radC
(800,300)
(200, 300)
![Page 92: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/92.jpg)
![Page 93: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/93.jpg)
If we use the stress-strain relation directly the same answer can be obtained
![Page 94: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/94.jpg)
![Page 95: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/95.jpg)
![Page 96: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/96.jpg)
![Page 97: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/97.jpg)
![Page 98: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/98.jpg)
![Page 99: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/99.jpg)
จงพื่�สิ+จนิ, สิมีการ (9-19) (9-20) ด้�วยภาษาของตั�วเองHw20a
Hw20b
a= 100(1+z1) b= -100(1+z2)
c= 100(1+z3)
ค่�า z1-z3 ได้�จากเลขประจ�าตั�วนิ�สิ�ตั ด้�งตั�อไปนิ�� 46xxxz1z2z3
Hw21
![Page 100: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/100.jpg)
ปร�มาณทาง Physics สามารถแทนด�วย Tensor
Order 0 = zero order Tensor (Scalar) – Magnitude (มีวล, ค่วามีห้นิาแนิ�นิ)
Order 1 = first order Tensor (Vector) – Magnitude, Direction (ค่วามีเร.ว, แรง)
Order 2 = second order Tensor – Magnitudes, Directions (stress, strain)
… Higher order ….
ปร�มาณทาง Physics ไม!เปลี่�ยนแปลี่งไปตุามระบบโคออร�ด�เนตุท�ใช้�ในกำารว�ด
mass
length
2 kg.= ?? lb.mass temperature
5 in. = 12.7 cm.length O O50 C = 122 Ftemperature
![Page 101: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/101.jpg)
1
1
0
1 0.5 0.2
0.5 3 1
0.2 1 4
σ
x
y
z
xy
z
0.6
0.8
1
2 2manitude 1 1 2 2 2 2manitude 0.6 0.8 1 2
P
P
ปร�มาณทาง Physics ไม!เปลี่�ยนแปลี่งไปตุามระบบโคออร�ด�เนตุท�ใช้�ในกำารว�ด
แรง ย�งคงม�ข้นาดแลี่ะท�ศทางเท!าเด�ม ไม!ว!าจะแสดง component ข้องเวคเตุอร�ด�วยระบบโคออร�ด�เนตุอ�น
P
สถานะข้องหน!วยแรง (state of stress) ย�งคงม�ค�ณสมบ�ตุ�เหม�อนเด�ม ไม!ว!าจะแสดงด�วยระบบโคออร�ด�เนตุอ�น
![Page 102: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/102.jpg)
O
A
A
0, 0, 0x y xy
O
B
A
0, 0, 0x y xy
![Page 103: Chapter 9 Combined Stresses](https://reader033.fdocuments.net/reader033/viewer/2022050616/56813564550346895d9cc9d2/html5/thumbnails/103.jpg)