Notes 6 "Torsion"

Torsion

• Introduction:

- In the preceding 2 chapters we studied the stress and strain subjected to axial loads .in this chapter we will study the torsion :analyze stress and strain of circular cross section subjected to twisting torques .

•Notes :

- Twist …….rotation………shafts

- Loads…….axial…………..rod

• To deal with torsion there are some conditions:

1.Line  remains line .

2.The circular shape remains in his shape .

3.γ & φ are very small  .

• Notes:

- Φ decreases towards the fixed support

- γ decreases towards the center of the shaft

γ - shear strain

Φ - angle of twist 

Δ - arc

Before we complete  there are some important notes we should take care of them:

• At the center of the shaft the shear = zero

• At the surface of the shaft:

  1.Shear ………………max

  2.γ  ……………………max

• The is a linear relation between γ  and the distance far away the center (p) 

 

J : is the tensional const. Or polar second moment.

G : is the modulus of rigidity.

• Some important relations(τ and t)

α :  is the angel per unit length


• Note : if the Shaft is Hollow :

 

& C2 > C1

•If we have a shaft supports from its 2 edges:

•We free one of supports and solve it as a general ex

• In this type of examples we free one of the supports and we solve it then we use a torque  that opposite to the first torque to achieve the equilibrium .
 

• Example (1):

Solution :
- Free the right support

- Put the opposite torque

- Then complete it by equal these angels

• Example (2):

- Φ2= the angel w.r.t the support at b
- Φ1=the angel w.r.t the shaft A
 
- the is a relative angel between SHAFTS (A&B)…..ώ
- Φ2 R2 = θR2 
- relative angel  ώ=φ2*((R2/R1))
- θ=ώ+φ1
 
N.B:
Lama nin2el angles or torques men tirs l turs tany lazem ni5alibalna men el ratio between R1&R2.

Notes 6 "Slides"