STRUCTURElECTURE 3

Kites Team

Equilibrium is the foundation of any science .Each science sees it with different point of view and present it in away which suits the nature of the proposed problem. In the case of studying static structure, equilibrium is essential to model any structural problem, and it means that the structure will be able to sustain the applied external load without movement ,in other words, the external generalized forces equals the support reactions and the structure internal forces equal the external loads.

Equilibrium

Average stress =force /area (vector )

There are two types of stress 1-Axial stress ( axis line is parallel to

surface area )2 -shear stress (axis line is normal to

surface area –Tangential stresses )( friction betweeen two surfaces causses shear stress)

STRESS AND STRAIN

Posion ratio

Shear Bolt

2Shear_stress

4

P

D

2

/ 2Shear_stress

4

P

D

Equilibruim xA P

..... 0.3

X

Y

Y

X

LStrain

Ld

d

The generalized hook’s law

Applying a hydrostatic pressureSO ,the dilatation (reduction in the

volume / original volume ) will beFor the problem to be physically

acceptable the volume must decrease ,so :

HOOK'S LAW

X Y ZX

Y X ZY

Z Y XZ

E E E

E E E

E E E

X Y Z P 3 (1 2 )

X Y Z

Pe

E

0.5

X Y ZX

Y X ZY

Z Y XZ

E E E

E E E

E E E

ZX

Y

VELUMETRIC STRAIN Volume =Length x weight x height

V LWH

(1 2 )

3(1 2 )

V X Y Z

V X Y Z

V LH W LW H WH L

V W H L

V W H L

E

E

'

/

'

/

Block of material with , 0

Find as f ( , , ' .)

( ) ?

( )

( ) _ _ _

( ) ( ) (1 )

( ) , , ( )

(1 ),1

( ) _ 0.3_ _ _

Y X Z

x

z

x

x

z xz x Y

X XX Y Z

X

XX

mat l

a

b E

c compare E with E

aE E E

b EE E

EE

EC Assume a typical val

'_ _ _ _ _ _ tan

ue

the effect of on E is subs tial

/

2

2 2

Block of material confined by rigid die in x directions

a yes ,    and    occur

(b)E

( ) 0.5

( ) 0

( ) 0

____ ( ) 0

(1 ) ( ) 0

, ,1 1

(

x Y

Z

Z

X X Y Z

Y Y X Z

X Y Z

Y Z

Z ZY X

Z Z

c

E

E

E

2

'

/

2)

1

1 2

1

(1 )

(1 )(1 2 )

1: _

2

ZX Y Z

Z Z

Z

Z

E

EE

E AS

6

Block of brass(70cu-30zn)is confined

by die in z directions

0 _ _   60, 100 ,

, ,

101.000 , 0.350

( ) 0

( ) 56

( ) 53.5 10

( )

z x Y

Z x y

Zz Y Z

Z X Y

XX Y Z

YY X Z

if MPa

find

E MPa

E EMPa

xE E

E E

6588 10x

2

2

2

Mg alloy confined-so that_ = =0

=-50MPa,no_friction ,no_yielding

( ) , ?_ 44.7 , 0.291

1( ( ) 0

1( ( ) 0

( ) 0

( ) 0

1 1

X Y

Z

x Y

X X Y Z

X ZY

X Y X Z

X ZX Z

X Z X Z

ZX Z

a E GPa

E

E

4

4 4

'

'

10.291

( 50 ) 20.521 0.291

1( ) ( ( ))

1( 50 0.291( 20.52 2))

44.700

8.51 10

0 0 8.51 10 8.51 10

( ) 58.700 58.7

_

ZY X

Y X

Z Z X Y

V X Y Z

Z

Z

V

Z

Z

similarty

MPa MPa

bE

x MPaMPa

x

x x

c E MPa GPa

E E d

_ _ _

_ _ _x Y

ue to poisson strains

from and

'

Block of Al alloy confined in the z direction , with 

100

70.3 , 0.345

( ) ?, ( ) , ( ) ?

1( ( )) 0

( ) 0.345( 100 100)

69 69 ( )

1( ( ))

x Y

XZ V

X

Z Z X Y

Z X Y

Z

x X Y Z

x

MPa

E GPa

a b E c

E

MPa Compression

E

'4

4

3

1( 10 0.345( 100 69)

70.3000.0005931

100168.6

5.931 10

_ _

2( 5.931 10 ) 0

1.186 10 ( _ _ )

X

X

X

Y X

V X Y Z

V

MPa MPaMPa

MPaE GPa

x

by symmetry

x

x means decreases