GATE 2022 AE – Question 60

Structures · Strength of materials: stress and strain, stress-strain curves, trusses, bars, beams and shafts, determinate and indeterminate · 2 marks · Numerical answer

A 10 kN axial load acts eccentrically on a square rod of cross section 1 cm ×1 cm. Top and bottom strain gauges 0.5 m from the tip measure $\epsilon_1=0.0016$ and $\epsilon_2=0.0004$. The eccentricity e is ___ mm.

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Correct answer: 1

Explanation

The rod carries an axial load $P$ at eccentricity $e$ from the centroid. The load causes uniform axial strain plus bending strain:

Axial strain: $\epsilon_a=\dfrac{P}{AE}$ (the same at the top and bottom).
Bending strain: $\epsilon_b=\pm\dfrac{Pe\,(h/2)}{EI}=\pm\dfrac{6Pe}{Ebh^2}$ for a rectangular section ($I=bh^3/12$, $A=bh$), positive on one face and negative on the other.

**The two gauge readings:**
$$\epsilon_1=\epsilon_a+\epsilon_b,\qquad\epsilon_2=\epsilon_a-\epsilon_b .$$

**Solve:**
$$\epsilon_a=\frac{\epsilon_1+\epsilon_2}{2}=0.0010,\qquad\epsilon_b=\frac{\epsilon_1-\epsilon_2}{2}=0.0006 .$$

**Eccentricity.** The ratio of bending to axial strain is
$$\frac{\epsilon_b}{\epsilon_a}=\frac{6Pe/(Ebh^2)}{P/(Ebh)}=\frac{6e}{h}=0.6\;\Rightarrow\;e=0.1\,h .$$

With a section depth $h=10$ mm: $e=\mathbf{1\ mm}$.