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GATE 2021 AE – Question 49

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

A beam has deflection $v=\frac{w}{48EI}(2x^4-3Lx^3+L^3x)$. The location of maximum deflection, expressed as x/L, is (two decimals)

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Correct answer: 0.41 to 0.43

Explanation

The deflection is $v=\dfrac{w}{48EI}\left(2x^4-3Lx^3+L^3x\right)$.

**Maximum:** set $\dfrac{dv}{dx}=0$:
$$8x^3-9Lx^2+L^3=0 .$$

With $r=x/L$:
$$8r^3-9r^2+1=0 .$$

**Factor out the root $r=1$** (the support, where the slope condition is trivially met): $(r-1)(8r^2-r-1)=0$.

**Solve the quadratic:**
$$8r^2-r-1=0\;\Rightarrow\;r=\frac{1\pm\sqrt{1+32}}{16}=\frac{1\pm\sqrt{33}}{16}.$$

The positive root inside the span is
$$r=\frac{1+\sqrt{33}}{16}=\frac{1+5.745}{16}=\mathbf{0.4215}.$$