Post on 25-Apr-2020
1.3 Stress
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0lim
x
x xx A
x x
F dFA dA
0lim
x
y yxy A
x x
F dFA dA
0lim
x
z zxz A
x x
F dFA dA
1.3 Stress
Copyright © 2011 Pearson Education South Asia Pte Ltd
0lim
y
y yy A
y y
F dFA dA
0lim
y
z zyz A
y y
F dFA dA
0lim
y
x xyx A
y y
F dFA dA
1.3 Stress
Copyright © 2011 Pearson Education South Asia Pte Ltd
0lim
z
z zz A
z z
F dFA dA
0lim
z
x xzx A
z z
F dFA dA
0lim
z
y yzy A
z z
F dFA dA
i jdirectionof force
direction of a normal to the plane
Therefore, we can express the state of stress at a point
For 2-D case
* uniaxial stress, biaxial stress, triaxial stress, pure shear
1.3 Stress
Copyright © 2011 Pearson Education South Asia Pte Ltd
[ ]
x xy xz
yx y yz
zx zy z
[ ]
x xy
yx y
-x plane+x plane
Copyright © 2011 Pearson Education South Asia Pte Ltd
' '
' '
' '
00
( ) ( ) 0
,
00
x
x x yx y x
x x yx y x
x x yx y x
z
xy yx
xy yx
Fdydz dydz dxdz dxdz
dydz dxdz
Mdydz dx dxdz dy
dx
τxy
τxzσx
σy’
τy’xτy’z
σx’
τx’z
dz
τx’y
dy
σz
σy
τyx
σx’
σyτyx
τxy
σx
σy’
τx’y
τy’x
dx
dy
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σy
σzτzy
τyz
σz’
τy’z
τz’y
σy’
dy
dz
' '
' '
' '
0
0
( ) ( ) 0
,
00
y
y y zy z x
y y zy z y
y y zy z y
x
yz zy
yz zy
F
dxdz dxdz dxdy dxdy
dxdz dxdy
Mdxdz dy dxdy dz
' ' ', , ,
, , x x y y z z
xy yx yz zy xz zx
Therefore
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1.4 Average Normal Stress
for small area avgF AA
avgdF dA A P
avgPA
dFdA
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z
x
y
oy
zσx
0
0
0, centroid of cross section
y x
y x avg
y avg avg
z x
z avg avg
avg
dM z dA
M z dA z dA
M zdA zA
dM y dA
M y dA ydA
yA
y z