STRUCTURE

KITES TEAM

LECTURE 4

1-If it has a uniform cross section 2-loaded at its ends 3-Rod is homogenous ( E constant )

X

U dX

U XE

PXU

EA

2

L

LL

UofB

UofA L

( )

( ) ( )

du P x

dx E x A x

Stress and strain - Axial loading

It is not always possible to determine the forces in the members of structure by applying only the principles of statics , because it is based on the assumption of undeformable rigid structure.

By considering engineering structures as deformable and analyzing the deformation in their various members.

It will be possible for us to compute forces that are statically indeterminate.

Indetermine within the framwork of statics You will consider the deformations of a

structural member such as a rod , bar or plate under axial loading

Normal strain is the deformation of the member per unit length or stress per modulus of elasticity

للجسم الليبتحدث elongationاالستطالهdeflection

اتحركتها اللي ال Bالمسافة rod(ACB)مثالفي

_

_ _ int

BBA AC

Pstress

AP L

strainAE L

PLelongation L

EAdeflection of po B

A3-mm-thick hollow polystyrene cylinder (E=3Gpa) and arigid circular plate (only part of which is shown )are used to support a 250-mm long steel rod AB(E=200GPa) of 6-mm diameter, if a3.2KN load P is applied atB Determine (a)The elongation of rod AB(b)the deflection of point B(c) the average normal stress in the rod AB

3

9 6

4

3

9 2 2 6

5

4

3.2

(3.2)(1000)(250)(10 )36

(200)(10 )( )(10 )4

1.4147(10 )

( 3.2)(1000)(30)(10 )1

(3)(10 )( )(50 44 )(10 )4

7.224(10 )

2.137(10 )

AB

AB ABBA

AB AB

BA

Ac AcAc

Ac Ac

Bc

B BC AC

ABA

P KN

P L

E A

m

P L

E A

m

m

P

6

3.2(1000)113.17

36( )(10 )4

B

AB

MPaA

Two solid cylindrical rods are joined at B and loaded as shown .Rod AB is made of steel (E=200GPa)determine (a) the total deformation of the composite rod ABC(b)the deflection of point B

3

29 6

5

3

9 2 6

5

30

(40 30) 10

( 30)(1000)(250)(10 )

30(200)(10 )( )(10 )

4

5.305(10 )

(10)(1000)(300)(10 )1

(105)(10 )( )(50 )(10 )4

1.4551(10 )

3

AB

BC

A AB BC

AB ABBA

AB AB

BA

Bc BcBc

Bc Bc

Bc

A

P KN

P KN

P L

E A

m

P L

E A

m

5

5

.85(10 )

1.4551(10 )B BC

m

m

Both portions of the rod ABC are made of an aluminum for which (E=70GPa).Knowing the magnitude of P is 4 KNdetermine (a) the value of Q so that the deflection at Ais zero (b) the corresponding deflection of B

29 6

5

9 2 6

6

6

4

(4 )

(40)(1000)(0.4)

20(70)(10 )( )(10 )

4

7.2756(10 )

(4 )(1000)(0.5)1

(70)(10 )( )(60 )(10 )4

2.5263(4 )(10 )

0

2.5263(4 )(10 ) 7

AB

BC

AB ABBA

AB AB

BA

Bc BcBc

Bc Bc

Bc

A BC AB

P KN

P Q KN

P L

E A

m

P L Q

E A

Q m

Q

5

5

.2756(10 ) 0

32.799

7.2756(10 )B BC

Q KN

m

The rod ABC is made of an aluminum for which (E=70GPa) ,Knowing that p=6KN and Q=42KN determine the deflection of (a) point A (b) point B

6 5

5 5 5

_ _ _

_ 6

42

2.5263(6 42)(10 ) 9.0946(10 )

69.0946(10 ) 7.2756(10 )( ) 1.8187(10 )

4

B

A

from the last example

if p KN

Q KN

m

m

The length of the 2-mm-diameter steel wire CD has been adjusted so that with no load applied , agap of 1.5mm exists between the end Bof the rigid beam ACB and aconstant point E .Knowing that E=200GPa) determine where a 20.Kg block should be placed on the beam in order to cause contact between B and E

29 6

4

(0.08) 20(9.8)( ) 0

2450

(2450 )(0.25)

2(200)(10 )( )(1

_ 20 _ _ _ _

_ tan ( ) _ _ in

0 )4

9.7480(10 )

_ _ _ _ _

t

A DC

DC

CD CDCD

CD CD

CD

let the Kg Block be pla

M F

F N

P L

E A

m

from geometry of the deflected struct

ced at

a dis ce a from the po A

4

0.32 0.08

0.08

48.74 (10 )

_ 1.5

0.308

B C

B

B

ure

m

for mm

m

Each of the links AB and CD is made of steel (E=200GPa ) and has auniform rectangular cross section of 6x24mm. Determine the longest load which can be suspended from point Eif the deflection of E is not to exceed 0.25mm

3

9 6

8

3

9

(375 250) (250) 0

2.5 ( )

0

1.5 ( )

2.5 (200)10

(200)(10 )(6)(24)(10 )

1.736 (10 ) ( )

1.5 (200)(10 )

(200)(10 )(6)(

B DC

DC

Y DC BA

BA

CD CDCD

CD CD

CD

BA BABA

BA BA

M P F

F P Tension

F F F P

F P Tension

F L P

E A

P m Downward

P L P

E A

6

8

24)(10 )

1.0416 (10 )( )BA P UPwnward

8 8

8

8 3

_ _ _ _ _

250 375 375

250 250

(2.5)( 1.736 )(10 ) (1.5)(1.0416 )(10 )

2.7776 (10 )

_ max. 0.25

2.7776 (10 ) 0.25(10 )

.max. 9.57

E C B

E

E

E

from geometry of the deflected structure

P P

P m

for deflection m

P

P K

N