**Angle of twist is zero on the shaft:**

A shaft 10 m long and 100 mm diameter during its length is fixed at the ends and is subjected to two opposite torques of 8 kN-m and 10 kN-m at a distance of 3 m and 8 m from one end. Discover the fixing torques at the ends. What is the maximum angle of twist and the maximum shear stress in the shaft? At what section the angle of twist is zero on the shaft?

Take E = 80 GPa.

**Solution**

Σ T = 0

⇒ T_{A} - 8 + 10 + T_{D} = 0

∴ T_{A} + T_{D} = - 2

G = 80 GPa = 0.8 × 10^{11} N/m^{2} -------------- (1)

d = 100 mm = 0.1 m

J = (π /32)d ^{4} = (π/32) × (0.1)^{4} = 9.8 × 10^{- 6} m^{4}

R = d /2= 100/2 = 0.05 m , Z_{P} = 1.96 × 10^{- 4} m^{3}

**Figure (a)**

** ****Figure (b) : Line Diagram **

** **

**Portion AB**

θ_{ 1} = (T_{A} . l )/GJ = (T_{A} × 3 )/GJ ---------- (2)

**Portion BC**

θ_{2} = ((T_{A} - 8) × 5 )/ GJ -------- (3)

**Portion CD**

θ_{3 }= ((T_{A} - 8 + 10) × 2)/ GJ = ((T_{A} + 2) × 2)/ GJ ---------- (4)

Angle of twist at D co.r.t A = 0

(T_{A} × 3 )/ G J + ((T_{A} - 8) × 5 )/ GJ + ((T_{A} + 2) × 2)/ GJ = 0

∴ T_{A} = 3.6 kN-m ----------- (5)

From Eqs. (1) and (5),

3.6 + T_{D} = - 2

T_{D} = - 5.6 kN-m = 5.6 kN-m ( )** **

**Angle of Twist**

θ_{1} = ((3.6 × 10^{3} ) × 3) / ((0.8 × 10^{11} ) (9.8 × 10^{- 6} ))

= 1.4 × 10^{- 2} radians

θ_{2} = ((3.6 - 8) × 10^{3} × 5 )/(0.8 × 10^{11} ) (9.8 × 10^{- 6} )= - 2.8 × 10^{- 2} radians

θ_{3} = (3.6 + 2) × 10^{3} × 2 /((0.8 × 10^{11} ) (9.8 × 10^{- 6} ))= 1.4 × 10^{- 2} radians

θ_{AB} = θ_{1 }= 1.4 × 10^{- 2} radians

θ _{AC} = θ_{1} + θ_{2} = 1.4 × 10^{- 2} - 2.8 × 10^{- 2} = - 1.4 × 10^{- 2} radians

θ_{AD } = θ_{AC } + θ_{3} = - 1.4 × 10^{- 2}

**Shear Stress**

+ 1.4 × 10^{- 2} = 0

τ _{max} = (T/ J) . R = T/ Z_{ P}

**Portion AB**

τ_{max }= 3.6 × 10^{3} / (1.96 × 10^{- 4}) = 18.4 × 10^{6} N/m^{2 } = 18.4 N/mm^{2}

**Portion BC**

τ _{max }= (3.6 - 8) × 10^{3}/ (1.96 × 10^{- 4})

= - 22.4 × 10^{6} N/m^{2} = - 22.4 N/mm^{2}

**Portion CD**

τ _{max} = (3.6 + 2) × 10^{3}/1.96 × 10^{- 6 }= 28.6 × 10^{6} N/m^{2} = 28.6 N/mm^{2}

**Zero Angle of Twist**

It occurs between B and C

θ_{1} = 1.4 × 10^{- 2} radians

∴ x = 2.5 m from B.