Equations From Elasticity Theory

7/31/2019 Equations From Elasticity Theory

1/12

Pusan National University Department of Naval

Architecture and Ocean Engineering

Assignment No 1Appendix C: Equations from Elasticity Theory

Altan AK

201183201


2/12

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


3/12

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:


4/12

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


5/12

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


6/12

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


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


7/12

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


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


2

z

vand

2

z

udue to small strains A`H` is equal to dz

z

wdzHA

`


8/12

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


9/12

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)


10/12

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


11/12

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


12/12

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

Equations From Elasticity Theory

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