# Theory of Columns and Shear Centre

 Question 1
Consider the cross-section of a beam made up of thin uniform elements having thickness $t(t \lt \lt a)$ shown in the figure. The $(x,y)$ coordinates of the points along the center-line of the cross-section are given in the figure.

The coordinates of the shear center of this cross-section are:
 A x = 0, y = 3a B x = 2a, y = 2a C x = -a, y = 2a D x = -2a, y = a
GATE CE 2022 SET-1   Solid Mechanics
Question 1 Explanation:
Shear centre of section consisting of two intersecting narrow rectangles always lies at the intersection of centrelines of two rectangles.

Coordinate of shear centre (0, 3a).
 Question 2
Consider two axially loaded columns, namely, 1 and 2, made of a linear elastic material with Young's modulus $2 \times 10^{5}$ MPa, square cross- section with side 10 mm, and length 1 m. For Column 1, one end is fixed and the other end is free. For Column 2, one end is fixed and the other end is pinned. Based on the Euler's theory, the ratio (up to one decimal place) of the buckling load of Column 2 to the buckling load of Column 1 is ________
 A 2.5 B 8 C 5.2 D 4
GATE CE 2017 SET-1   Solid Mechanics
Question 2 Explanation:

\begin{aligned} P_{c r 1}&=\frac{\pi^{2} E \cdot I}{(2 l)^{2}}=\frac{\pi^{2} E I}{4 l^{2}} \\ P_{c r 2}&=\frac{\pi^{2} E \cdot I}{\left(\frac{l}{\sqrt{2}}\right)^{2}}=\frac{2 \pi^{2} E I}{l^{2}}\\ \therefore \text{Required ratio }&=\frac{P_{C r 2}}{P_{c r 1}}=\frac{\frac{2 \pi^{2} E I}{l^{2}}}{\frac{\pi^{2} E I}{4 l^{2}}}=8 \end{aligned}

 Question 3
In a system,two connected rigid bars AC and BC are of identical length, L with pin supports at A and B. The bars are interconnected at C by a frictionless hinge. The rotation of the hinge is restrained by a rotational spring of stiffness, k. The system initially assumes a straight line configuration, ACB. Assuming both the bars as weightless, the rotation at supports, A and B, due to a transverse load, P applied at C is
 A $\frac{PL}{4k}$ B $\frac{PL}{2k}$ C $\frac{P}{4k}$ D $\frac{Pk}{4L}$
GATE CE 2015 SET-2   Solid Mechanics
Question 3 Explanation:
Deflection under load $P=\theta L$
Work done by force $P=\frac{1}{2} P(\theta L)$

Strain energy stored in spring
\begin{aligned} &=\frac{1}{2}[k(2 \theta)](2 \theta) \\ \frac{1}{2} P(\theta L) &=\frac{4 k \theta^{2}}{2} \\ \Rightarrow \quad \theta &=\frac{P L}{4 k} \end{aligned}
 Question 4
The possible location of shear centre of the channel section, shown below, is
 A P B Q C R D S
GATE CE 2014 SET-1   Solid Mechanics
Question 4 Explanation:

For no twisting,
\begin{aligned} V \times e &=H \times h\\ \Rightarrow \;\; e&=\frac{Hh}{V}\\ \end{aligned}
Hence, possible location of shear center is P.
 Question 5
Two steel columns P (length L and yield strength $f_{y}$= 250 MPa) and Q (length 2L and yield strength $f_{y}$= 500 MPa) have the same cross-sections and end-conditions. The ratio of buckling load of column P to that of column Q is:
 A 0.5 B 1 C 2 D 4
GATE CE 2013   Solid Mechanics
Question 5 Explanation:
Buckling load,
\begin{aligned} P &=\frac{\pi^{2} E \mid}{\left(l_{e f f}\right)^{2}} \\ \therefore \quad P_{p} &=\frac{\pi^{2} E \mid}{(L)^{2}} ; P_{Q}=\frac{\pi^{2} E \mid}{(2 L)^{2}} \\ \therefore \quad \frac{P_{P}}{P_{a}} &=4 \end{aligned}

There are 5 questions to complete.