List of top Structures Questions on Mechanics of Materials asked in GATE AE

Given: The tip deflection and tip slope for a tip-loaded cantilever of length \( L \) are: \[ \delta = \frac{NL^3}{3EI} \quad \text{and} \quad \theta = \frac{NL^2}{2EI}, \] where \( N \) is the tip force and \( EI \) is the flexural rigidity.
A cantilever \( PQ \) of rectangular cross-section is subjected to transverse load, \( F \), at its mid-point. Two cases are considered as shown in the figure. In Case I, the end \( Q \) is free and in Case II, \( Q \) is simply supported.
The ratio of the magnitude of the maximum bending stress at \( P \) in Case I to that in Case II is _________ (rounded off to one decimal place).

A 0.5 m long thin-walled circular shaft of radius 2 cm is to be designed for an axial load of 7.4 kN and a torque of 148 Nm applied at its tip, as shown in the figure.
The allowable stress under uniaxial tension is 100 MPa.
Using the maximum principal stress criterion, the minimum thickness, \( t \), of the shaft so that it does not fail is _________ mm (rounded off to the nearest integer).

A 10 kN axial load is applied eccentrically on a rod of square cross-section (1 cm $\times$ 1 cm) as shown in the figure.
The strains measured by the two strain gages attached to the top and bottom surfaces at a distance of 0.5 m from the tip are \( \varepsilon_1 = 0.0016 \) and \( \varepsilon_2 = 0.0004 \), respectively.
The eccentricity in loading, \( e \), is _________ mm.