# Bending of Beams

 Question 1
The effective stiffness of a cantilever beam of length $L$ and flexural rigidity $EI$ subjected to a transverse tip load $W$ is A $\frac{3EI}{L^3}$ B $\frac{2EI}{L^3}$ C $\frac{L^3}{2EI}$ D $\frac{L^3}{3EI}$
GATE ME 2023   Strength of Materials
Question 1 Explanation:
$K =\frac{W}{\delta}=\frac{3EI}{L^3}$
 Question 2
A cantilever beam with a uniform flexural rigidity $(EI = 200 \times 10^6 N.m^2)$ is loaded with a concentrated force at its free end. The area of the bending moment diagram corresponding to the full length of the beam is $10000 \;N.m^2$. The magnitude of the slope of the beam at its free end is ________micro radian (round off to the nearest integer).
 A 42 B 50 C 65 D 84
GATE ME 2021 SET-2   Strength of Materials
Question 2 Explanation:

Assume:
\begin{aligned} &A=10000 \mathrm{~N}-\mathrm{m}^{2}\\ &\mathrm{El}=200 \times 10^{6} \mathrm{~N}-\mathrm{m}^{2} \end{aligned} As per moment area first theorem.
\begin{aligned} \theta_{\mathrm{B}}-\theta_{A} &=\left(\frac{A}{E I}\right) A B \\ \theta_{\mathrm{B}}-0 &=\frac{10000}{200 \times 10^{6}}=0.5 \times 10^{-4} \mathrm{radian} \\ \theta_{\mathrm{B}} &=50 \mu \text { radians } \end{aligned}

 Question 3
A plane frame PQR (fixed at P and free at R) is shown in the figure. Both members (PQ and QR) have length, L, and flexural rigidity, EI. Neglecting the effect of axial stress and transverse shear, the horizontal deflection at free end, R, is A $\frac{5FL^3}{3EI}$ B $\frac{4FL^3}{3EI}$ C $\frac{2FL^3}{3EI}$ D $\frac{FL^3}{3EI}$
GATE ME 2021 SET-2   Strength of Materials
Question 3 Explanation: \begin{aligned} U&=U_{P Q}+U_{Q P} \\ U&=\frac{M^{2} L}{2 E I}+\int_{0}^{L} \frac{\left(M_{x-x}\right)^{2}(d x)}{2 E I} \\ U&=\frac{(F L)^{2} L}{2 E I}+\int_{0}^{L}\left(\frac{(F x)^{2}(d x)}{2 E I}\right) \\ U&=\frac{F^{2} L^{3}}{2 E I}+\frac{F^{2} L^{3}}{6 E I}=\frac{2 F^{2} L^{3}}{3 E I} \end{aligned}
By Castigliano's theorem:
$\left(\delta_{H}\right)_{R}=\frac{\partial U}{\partial F}=\frac{4 F L^{3}}{3 E I}$
 Question 4
An overhanging beam PQR is subjected to uniformly distributed load 20 kN/m as shown in the figure. The maximum bending stress developed in the beam is ________MPa (round off to one decimal place).
 A 125 B 250 C 325 D 450
GATE ME 2021 SET-1   Strength of Materials
Question 4 Explanation:  \begin{aligned} M_{D}-M_{A} &=\frac{1}{2} \times 15 \times 0.75 \\ M_{D} &=5.625 \mathrm{kN}-\mathrm{m}(\mathrm{s}) \\ \Sigma M_{A}&=60 \times 1.5-R_{B} \times 2=0 \\ R_{B} &=45 \mathrm{kN}(\uparrow) ; \quad R_{A}=15 \mathrm{kN}(\uparrow) \end{aligned}
Location of D:
\begin{aligned} \frac{15}{x} &=\frac{25}{2-x} \\ 6-3 x &=5 x \\ &=\frac{3}{4} m \end{aligned}
\begin{aligned} M_C-M_B &=\frac{1}{2} \times 20 \times 1 \\ M_B&=-10 kN- m \\ M_B &= 10kN - m \;(+1)\\ \text{Max B.M.}&=\text{Larger of }(M_B \text{ and } M_D) \\ M_B&=10kN- m\; (+1) \\ (\sigma _b)_{max} &=\frac{M_{max}}{Z_{N.A.}}\\&=\frac{6 \times 10 \times 10^6}{224 \times 100^2}=250MPa \end{aligned}
 Question 5
A cantilever beam of length, $L$, and flexural rigidity, $EI$, is subjected to an end moment, $M$, as shown in the figure. The deflection of the beam at $x=\frac{L}{2}$ is A $\frac{ML^2}{2EI}$ B $\frac{ML^2}{4EI}$ C $\frac{ML^2}{8EI}$ D $\frac{ML^2}{16EI}$
GATE ME 2021 SET-1   Strength of Materials
Question 5 Explanation: \begin{aligned} Y_{C}-Y_{A}&=\left(\frac{A \bar{X}}{E I}\right)_{A C} \\ Y_{C}-0&=\frac{1}{E I}\left[\frac{-ML}{2} \times \frac{L}{4}\right]\\ Y_{C}&=\frac{ML^{2}}{8 E I} \text { (downward) } \end{aligned}

There are 5 questions to complete.

### 2 thoughts on “Bending of Beams”

1. Question 6 has wrong figure. kindly correct.

2. 