Strain Energy due to Torsion

Q. Strain energy in torsion of a shaft per unit volume is given by (q is maximum shear stress, E-modulus of Elasticity and G is modulus of rigidity)

1. $\frac{q^2}{2G}$
2. $\frac{q^2}{2E}$
3. $\frac{q^2}{4G}$
4. $\frac{q^2}{4E}$

Q. Two shafts of solid circular cross-section are identical except for their diameter ${}^{\prime }{d}_1^{\prime }and{}^{\prime }{d}_2^{\prime }$. They are subjected to the same torque T. The ratio of the strain energies stored $U_1/U_2$ will be

1. ${\left(\frac{d_1}{d_2}\right)}^4$
2. ${\left(\frac{d_1}{d_2}\right)}^2$
3. ${\left(\frac{d_2}{d_1}\right)}^2$
4. ${\left(\frac{d_2}{d_1}\right)}^4$

Q. A circular solid shaft of span L = 5m is fixed at one end and free at the other end. A twisting moment T=100 kN-m is applied at the free end. The torsional rigidity GJ is 50000 kN-$m^2/$ rad.
Following statements are made of this shaft.
I. The maximum rotation is 0.01 rad
II. The torsional strain energy is 1 kN-m

With referance to the above statements, which of the following applies?

1. Both statements are ture
2. Statement I is true but II is a false
3. Statement II is true but I is false
4. Both the statements are false

Q. A Stepped steel shaft is subjected to a clockwise torque of 10 Nm at its free end. Shear modulus of steel is 80 GPa. The strain energy stored in the shaft is

1. 1.73 Nmm
2. 2.52 Nmm
3. 3.46 Nmm
4. 4.12 Nmm

Q. Match List-I (Strain Energy) with List-II (Gradually applied load) and select the correct answer using the codes given below the lists $(\sigma =$ direct moment, E = Modulus of elasticity, G = Modulus of rigidity, I = Area moment of inertia, V= Volume):

List-I
A. $\frac{{\sigma }^{2}V}{2E}$
B. $\frac{{\tau }^{2}V}{2G}$
C. $\frac{{\tau }^{2}V}{4G}$
D. ${\int }_0^L\frac{M_x^{2}dx}{2EI}$

List-II

1. Axial load
2. Bending load
3. Shear load
4. Torsional load

Codes :

$\begin{array}{cccc}A& B& C& D\end{array}$

1. $\begin{array}{cccc}2& 3& 4& 1\end{array}$
2. $\begin{array}{cccc}1& 4& 3& 2\end{array}$
3. $\begin{array}{cccc}2& 4& 3& 1\end{array}$
4. $\begin{array}{cccc}1& 3& 4& 2\end{array}$

Q. Match List-I with List-II and select the correct answer using the codes given below the lists:

List-I
A. Torque-twist relationship for a circular shaft
B. Strain energy of elastic torsion
C. Circumferential shear stress
D. Maximum shearing stress due to combined torsion and direct stress

List-II

1. $\frac{1}{2}\sqrt{{\sigma }^2+A{\lambda }^2}$
2. $\frac{Gr\theta }{L}$
3. $\left(\frac{GJ}{2L}\right){\theta }^2$
4. $\frac{GJ}{L}\theta$

Codes :

$\begin{array}{cccc}A& B& C& D\end{array}$

1. $\begin{array}{cccc}2& 3& 4& 1\end{array}$
2. $\begin{array}{cccc}4& 1& 2& 3\end{array}$
3. $\begin{array}{cccc}2& 1& 4& 3\end{array}$
4. $\begin{array}{cccc}4& 3& 2& 1\end{array}$