List of top Strength of Materials Questions asked in GATE CE

Consider the beam ACDEB given in the figure. Which of the following statements is/are correct: 

A hanger is made of two bars of different sizes. Each bar has a square cross-section. The hanger is loaded by three point loads in the mid-vertical plane as shown: the upper bar is $100\, \text{mm}\times100\, \text{mm}$ and the lower bar is $50\, \text{mm}\times50\, \text{mm}$. Ignore self-weight and stress concentration. What is the maximum tensile stress (in N/mm$^2$) anywhere in the hanger? 

The infinitesimal element shown in the figure (not to scale) represents the state of stress at a point in a body. What is the magnitude of the maximum principal stress (in N/mm², in integer) at the point? 

A circular solid shaft of span \( L = 5 \, m \) is fixed at one end and free at the other end. A torque \( T = 100 \, kN.m \) is applied at the free end. The shear modulus and the polar moment of inertia of the section are denoted as \( G \) and \( J \), respectively. The torsional rigidity \( \frac{GJ}{L} \) is \( 50,000 \, kN.m^2/rad \). 
Statement i) The rotation at the free end is \( 0.01 \, rad \). 
Statement ii) The torsional strain energy is \( 1.0 \, kN.m \). 
With reference to the above statements, which of the following is true?

When a simply-supported elastic beam of span $L$ and flexural rigidity $EI$ is loaded with a uniformly distributed load $w$ per unit length, the deflection at the mid-span is \[ \Delta_0=\frac{5}{384}\,\frac{wL^4}{EI}. \] If the load on one half of the span is now removed, the mid-span deflection 

In a two-dimensional stress analysis, the state of stress at a point is shown in the figure. The values of length $PQ$, $QR$, and $RP$ are $4$, $3$, and $5$ units, respectively. The principal stresses are  (round off to one decimal place)

A 2D thin plate (plane stress) has $E=1.0~\text{N/m}^2$ and Poisson's ratio $\mu=0.5$. The displacement field is $u=Cx^2y$, $v=0$ (in m). Distances $x,y$ are in m. The stresses are $\sigma_{xx}=40xy~\text{N/m}^2$ and $\tau_{xy}=\alpha x^2~\text{N/m}^2$. Find $\alpha$ (in $\text{N/m}^4$, integer). 

A 2D thin plate (plane stress) has $E=1.0~\text{N/m}^2$ and Poisson’s ratio $\mu=0.5$. The displacement field is $u=Cx^2y$, $v=0$ (in m). Distances $x,y$ are in m. The stresses are $\sigma_{xx}=40xy~\text{N/m}^2$ and $\tau_{xy}=\alpha x^2~\text{N/m}^2$. Find $\alpha$ (in $\text{N/m}^4$, integer). 

The infinitesimal element shown in the figure (not to scale) represents the state of stress at a point in a body. What is the magnitude of the maximum principal stress (in N/mm², in integer) at the point?