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Pusan National University Department of Naval
Architecture and Ocean Engineering
Assignment No 1Appendix C: Equations from Elasticity Theory
Altan AK
201183201
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C.1 Differential Equations of Equilibrium
x; Normal stress y; Normal stress z; Normal stress
xy; Plane shear stress xz; Plane shear stress yz; Plane shear stress (unit of force per unit area)
Xb; Body force Yb; Body force Zb; Body force (unit of force per unit volume)
Figure C-1 Plane differential element Figure C-2 Three dimensional stress element
subjected to stresses
Two dimensional
Summing forces in the x direction is:
Fx = 0 Fx = 0
dxdyXdxdxdy
ydydydx
xbyx
yx
yxxx
x
0
dxdydx
ydxdxdyXdydxdy
xdyF yx
yx
yxbx
x
xx
Summing forces in the y direction is:
Fy = 0 Fy = 0
dxdyYdydydx
xdxdxdy
ybxy
xy
xyy
y
y
0
dydydx
xdYdxdyYdxdxdy
ydxF xy
xy
xYby
y
yy
After cancelling terms and dividing all equations into dxdy it will be:
dx
xy
xy
dy
dy
yx
yx
yx
x
y
dyy
y
y
dxx
y
dxx
xx
xy
Xb
Yb
CZb
x
Xb
y
Yb
z
xz
yx
yz
xy
zy
zx
z
y
x
C
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0
b
yxxx X
yxF
and 0
b
xyy
y Yxy
F
If we take moment from point of C
Mz = 0 Mz = 02222
dydxdy
y
dydx
dxdydx
x
dxdy
yx
yxyx
xy
xyxy
0222222
dydydx
y
dydx
dydx
dxdxdy
x
dxdy
dxdyM
yx
yxyx
xy
xyxyz
After summing and dividing all equations into dxdy it will be:
022
dy
ydx
xM yxyx
xyxyz
if we neglect these two terms it will be
yxxyyxxyzM 0
Three dimensional
Summing forces in the x direction is: Fx = 0
dxdydzXdxdydxdydzz
dxdzdxdzdyy
dydzdydzdxx
Fbzx
zx
zxyx
yx
yxx
x
xx
dxdydzXdxdydzdxdyz
dxdydxdzdydxdzy
dxdzdydzdxdydzx
dydzF bzxzx
zxyx
yx
yxx
x
xx
Summing forces in the y direction is: Fy = 0
dxdydzYdxdydxdydzz
dydzdydzdxx
dxdzdxdzdyy
F bzyzy
zyxy
xy
xyy
y
yy
dxdydzYdxdydzdxdy
z
dxdydydzdxdydz
x
dydzdxdzdydxdz
y
dxdzF bzyzy
zyxy
xy
xyy
y
yy
Summing forces in the y direction is: Fy = 0
dxdydzZdxdzdxdzdyy
dydzdydzdxx
dxdydxdydzz
F byzyz
yzxzxz
xzzz
zy
dxdydzZdxdzdydxdzy
dxdzdydzdxdydzx
dydzdxdydzdxdyz
dxdyF byzyz
yzxzxz
xzzz
zy
After cancelling terms and dividing all equations into dxdydz it will be:
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0
b
zyyxx
x Xzyx
F
, 0
b
zyxyy
y Yzxy
F
, 0
b
yzxzzz Z
yxzF
If we take moment from point of C
Mz = 0
222222
dxdydz
dydxdzdy
y
dydxdz
dydxdz
dxdydzdx
x
dxdydzM xz
yx
yxyxyx
xy
xyxyz
22222
dzdxdydz
z
dzdxdy
dzdxdydz
z
dzdxdy
dxdydzdx
x
zy
zyzyzx
zxzxxz
xz
022
dydxdzdy
y
dydxdz
yz
yzyz
2222222
dxdydz
dydydxdz
y
dydxdz
dydxdz
dxdxdydz
x
dxdydz
dxdydzM xz
yx
yxyx
xy
xyxyz
2222222
dzdxdy
dzdxdy
dzdzdxdy
z
dzdydz
dzdydz
dxdxdydz
x
dxdydz zyzy
zxzxzx
xzxz
02222
dydydxdz
y
dydxdz
dydxdz
dzdzdxdy
z
yz
yzyz
zy
After summing dividing all equations into dxdydz it will be:
0222222
dy
y
dz
z
dz
z
dx
x
dy
y
dx
xM
yz
yz
zy
zyzx
zxxz
xz
yx
yx
xy
xyz
if we neglect these six terms it will be;
0 yzzyzxxzyxxyzM
If we think about two dimensional we can say yxxy , zxxz , yzzy
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C.2 Strain/Displacement and Compatibility Equations
u; displacement (x) v; displacement (y) w; displacement (z direction)
x; strain y; strain z; strain (Change in length difference/original length)
xy; shear strain xz; shear strain yz; shear strain
Figure C-3 Differential element before and
after deformation (two dimensional) Figure C-4 Differential element before and after
deformation (three dimensional)
Two dimensional
In the x direction line element AB
AB
ABBAx
``
As we know AB = dx and A`B`; we can calculate A`B` from triangle (hipotenus)
22
A`B`= dxx
vdx
x
udx 2
2
2
2
22`` dx
x
vdx
x
u
x
udxBA
If we neglect the value of
2
x
vdue to small strains A`B` is equal to dx
x
udxBA
`
AB
ABBAx
``
x
u
dx
dxdxx
udx
xx
In the y direction line element AD
A`
B
D C
A
B`
C`D`
dx
dy
dxx
udx
d xx
v
dyu
x,u
y,v
x,u
y,v
z,w
A
B
E
C
D
FG
A`
B`
C`
D`
E`
F`
G`
dx
dy
dz
dyv
dy
H
H`
dyw
dxx
udx
dxx
v
dyy
vdy
dyy
u
dzz
v
dxx
w
dzz
wdz
dzz
u
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AD
ADDAy
``
As we know AD = dy and A`D`; we can calculate A`D` from triangle (hipotenus)
22
A`D`= dyy
udy
y
ydy 2
2
2
2
22`` dy
y
udy
y
v
y
vdyDA
If we neglect the value of
2
y
udue to small strains A`D` is equal to dy
y
vdyDA
`
AD
ADDAy
``
y
v
dy
dydyy
vdy
yy
xy= yx is defined to be change in the angle between two lines, AB and AD.
x
v
y
uyxxy
Condition of compatibility
x
v
yxy
u
yxyx
xy
222,
y
v
x
uyx
, yx
xy
xyyx
2
2
2
22
x
v
yyy
u
xyxy
yx
222,
y
v
x
uyx
, yx
yx
xyxy
2
2
2
22
yxxy
Three dimensional
In the x direction line element AB
AB
ABBAx
``
As we know AB = dx and A`B`; we can calculate A`B` from hipotenus
222
A`B`= dxx
wdx
x
vdx
x
udx
22
2
2
2
2
22`` dx
x
wdx
x
vdx
x
u
x
udxBA
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If we neglect the values of
2
x
vand
2
x
wdue to small strains A`B` is equal to dx
x
udxBA
`
AB
ABBAx
``
x
u
dx
dxdxx
u
dxxx
In the y direction line element AD
AD
ADDAy
``
As we know AD = dy and A`D`; we can calculate A`D` from triangle hipotenus
222
A`D`= dyy
wdyy
udyy
ydy
22
2
2
2
2
22`` dy
y
wdy
y
udy
y
v
y
vdyDA
If we neglect the value of
2
y
uand
2
y
wdue to small strains A`D` is equal to dy
y
vdyDA
`
AD
ADDAy
``yv
dy
dydyy
vdy
yy
In the z direction line element AH
AH
AHHAz
``
As we know AH = dz and A`H`; we can calculate A`H` from hipotenus
222
A`H`= dzz
udzz
vdzz
wdz
22
2
2
2
2
22`` dz
z
udz
z
vdz
z
v
z
wdzHA
If we neglect the value of
2
z
vand
2
z
udue to small strains A`H` is equal to dz
z
wdzHA
`
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AH
AHHAz
``
z
w
dz
dzdzz
wdz
zz
xy= yx is defined to be change in the angle between two lines, AB and AD.
xz= zx is defined to be change in the angle between two lines, AB and AH.
yz= zy is defined to be change in the angle between two lines, AD and AH.
x
v
y
uyxxy
,
x
w
z
uzxxz
,
y
w
z
vzyyz
Condition of compatibility
xv
yxyu
yxyx
xy
222
,yv
xu
yx
, yx
xy
xyyx
2
2
2
22
x
v
xyy
u
xyxy
yx
222,
y
v
x
uyx
, yx
yx
xyxy
2
2
2
22
yxxy
x
w
zxz
u
zxzx
xz
222,
z
w
x
uzx
, zx
xz
xzzx
2
2
2
22
x
w
xzz
u
xzxz
zx
222,
z
w
x
u
zx
,
zx
zx
xzxz
2
2
2
22
zxxz
y
w
zyz
v
zyzy
yz
222,
z
w
y
vzy
, zy
yz
yzzy
2
2
2
22
y
w
yzz
v
yzyz
zy
222,
z
w
y
vzy
, zy
zy
yzyz
2
2
2
22
zyyz
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C.3 Stress/Strain Relationship
LengthOriginal
LengthinChange=Strain
If we think about x, y and x
total volume is V0=xyz
Final volume is
V= x (1+ x) y (1+ y) z (1+ z)
V=xyz (1+ x+ y+ z+ x y+ y z+ x z+ x y z)
2
therefore we can neglect sum of x y+ y z+ x z+ x y z values. Finally,
V=V0 (1+ x+ y+ z)
0
0
VolumeOriginal
VolumeinChangestrainVolumetric=e
V
VV zyxe
Strain
Stress(constant)elasticityofmodulus=E
StressShear
StressShearPlane
(constant)shearofmodulus=G
When a material is compressed in one direction, it usually tends to expand in the other two directions
perpendicular to the direction of compression. This phenomenon is called the Poisson effect ( )
x
z
x
y
axial
trans
d
d
d
d
d
d
Two dimensional
yxx
yxx
EEE
1 , xyyx
y
yEEE
1
y
x
=
EE
EE1
1
y
x
y
x
y
x
EE
EE
1
1
1
y
x
z
y (1+ y)
x (1+ x)
z (1+ z)
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1
1
1
10
01
10
01
1
1
EE
EE
EE
EE
EE
E
E
E
0
10
1
0
0
1
12
22
22
22
11
1110
01
11
0
10
1
EE
EE
EEE
y
x
z
y
x
y
x
y
x
EE
EE
EE
EE
22
22
1
11
111
1
For normal stress
yxx
E
11
yxy
E
11
For plane shear stress G
GG
xyxy
xy
xy
The relation between modulus of elasticity and shear modulus is
12
EG so, if want to write all
stresses in matrix form x, y, xy
xy
y
x
xy
y
xE
2
100
01
01
11
2
100
01
01
11
ED
[D]; stress/strain or
constitutive matrix
Three dimensional
zyxx
zyxx
EEEE
1,
EEE
zxy
y
,
EEE
zxzz
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z
y
x
=
EEE
EEE
EEE
1
1
1
z
y
x
z
y
x
z
y
x
EEE
EEE
EEE
1
1
1
1
1
1
1
1
100
010
001
100
010
001
1
1
1
EEE
EEE
EEE
EEE
EEE
EEE
EE
EE
E
E
E
E
0
0
00
10
10
1
00
00
00
1
1
1
22
22
EEE
EE
EE
EE
EEE
2
2
2
22
22
2
22
22
1
)(
1
)1(
011
011
1
2100
110
101
0
011
00
101
10
1
222
222
222
222
22
22
21
1
2121
2121
)1(
21
212121
)1(
100
010
001
21
1
2121
011
011
1001
10
101
EEE
EEE
EEE
EEE
EE
EE
z
y
x
z
y
x
z
y
x
z
y
x
EEE
EEE
EEE
EEE
EEE
EEE
)21)(1(
1
)21)(1()21)(1(
21)21)(1(
)1(
)21)(1(
)21)(1()21)(1()21)(1(
)1(
1
1
1
2
1
For normal stress
zyxx
E
1
211
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zyxy
E
1
211
zyxzE
1211
For plane shear stress G
GG
xyxy
xy
xy
GG
yzyz
yz
yz
GG
zxzxzx
zx
The relation between modulus of elasticity and shear modulus is
12
EG so, if want to write all
stresses in matrix form x, y, z, xy, yz, zx,
zx
yz
xy
z
y
x
zx
yz
xy
z
y
x
E
2
210
0
02
210
0
02
21
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
211
[D]; stress/strain or constitutive matrix
2
210
0
02
210
0
02
21
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
211
ED
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