# Structural Analysis

 Question 1
In the frame shown in the figure (not to scale), all four members $(A B, B C, C D$, and $A D)$ have the same length and same constant flexural rigidity. All the joints $A, B, C$, and $D$ are rigid joints. The midpoints of $A B, B C, C D$, and $A D$, are denoted by $E, F, G$, and $H$, respectively. The frame is in unstable equilibrium under the shown forces of magnitude $P$ acting at $E$ and $G$. Which of the following statements is/are TRUE?

 A Shear forces at $\mathrm{H}$ and $\mathrm{F}$ are zero B Horizontal displacement at $\mathrm{H}$ and $\mathrm{F}$ are zero C Vertical displacement at $\mathrm{H}$ and $\mathrm{F}$ are zero D Slopes at E, F, G, and $\mathrm{H}$ are zero
GATE CE 2023 SET-2      Methods of Structural Analysis
Question 1 Explanation:

Due to symmetry horizontal displacement of $\mathrm{H}$ $\& F=$ zero
$\Rightarrow$ option (B) correct
Also due to symmetry slopes at E, F, G, H $=0$ $\Rightarrow$ option (D) is correct
Since $A D$ and $B C$ are subjected to pure bending Hence shear force at $\mathrm{H} \&$ $\mathrm{F}$ are zero
$\Rightarrow$ option (A) correct
Vertical displacement at $\mathrm{H} \& \mathrm{~F} \neq 0$
$\Rightarrow$ option (C) is incorrect.
 Question 2
Muller-Breslau principle is used in analysis of structures for
 A drawing an influence line diagram for any force response in the structure B writing the virtual work expression to get the equilibrium equation C superposing the load effects to get the total force response in the structure D relating the deflection between two points in a member with the curvature diagram inbetween
GATE CE 2023 SET-2      Influence Line Diagram and Rolling Loads
Question 2 Explanation:
Muller Breslass principle is used to draw influence line diagram for determinate and indeterminate structures. It states that influence line for any stress function may be obtained by removing the restraint offered by that function and introducing a directly related generalised unit displacement at that location in the direction of the stress function.

 Question 3
An idealised bridge truss is shown in the figure. The force in Member $U_{2} L_{3}$ is _____ kN (round off to one decimal place).

 A 12.5 B 16.8 C 14.1 D 10.6
GATE CE 2023 SET-1      Trusses
Question 3 Explanation:

$\sum F_{y}=0, \quad 50-20-20-F_{U_{2} L_{3}} \sin 45^{\circ}=0$
$\therefore \quad F_{U_{2} L_{3}}=\frac{10}{\sin 45^{\circ}}=14.14 \mathrm{kN}$
$\therefore$ Rounding off to one decimal.
 Question 4
Consider the pin-jointed truss shown in the figure (not to scale). All members have the same axial rigidity, $\mathrm{AE}$. Members $\mathrm{QR}, \mathrm{RS}$, and $\mathrm{ST}$ have the same length L. Angles OBT, RCT, SDT are all $90^{\circ}$. Angles BQT, CRT, DST are all $30^{\circ}$. The joint $T$ carries a vertical load $P$. The vertical deflection of joint $T$ is $k \frac{P L}{A E}$. What is the value of $k$ ?

 A 1.5 B 4.5 C 3 D 9
GATE CE 2023 SET-1      Trusses
Question 4 Explanation:

Considering joint $T$,

$\sum F_{y}=0, \quad F_{Q T} \sin 60^{\circ}=P$
$\therefore \quad \mathrm{F}_{\mathrm{QT}}=\frac{2 \mathrm{P}}{\sqrt{3}}$
$\sum \mathrm{F}_{\mathrm{X}}=0, \mathrm{~F}_{\mathrm{BT}}=-\mathrm{F}_{\mathrm{QT}} \cos 60^{\circ}=-\frac{P}{\sqrt{3}}$
$\therefore \quad$ Strain energy $(U)=\left(\frac{P^{2} L}{2 A E}\right)_{Q T}+\left(\frac{P^{2} L}{2 A E}\right)_{B T}$
$=\frac{\left(\frac{2 \mathrm{P}}{\sqrt{3}}\right)^{2}(3 \mathrm{~L})}{2 \mathrm{AE}}+\frac{\left(-\frac{\mathrm{P}}{\sqrt{3}}\right)^{2}\left(\frac{3 \mathrm{~L}}{2}\right)}{2 \mathrm{AE}}$
$=\frac{2 P^{2} \times L}{A E}+\frac{P^{2} \times L}{4 A E}$
$\frac{8 P^{2} L+P^{2} L}{4 A E}=\frac{9 P^{2} L}{4 A E}$
$\therefore \quad$ Vertical deflection of joint $T$
$=\frac{\partial \mathrm{U}}{\partial \mathrm{P}}=\frac{18 \mathrm{PL}}{4 \mathrm{AE}}=4.5 \frac{\mathrm{PL}}{\mathrm{AE}}$
$\therefore \quad \mathrm{K}=4.5$
 Question 5
Consider the following three structures:

Structure I: Beam with hinge support at A, roller at $C$, guided roller at $E$, and internal hinges at $B$ and $D$.

Structure II: Pin-jointed truss, with hinge support at $A$, and rollers at $B$ and $D$.

Structure III: Pin-jointed truss, with hinge support at $A$ and roller at $C$.

Which of the following statements is/are TRUE?
 A Structure I is unstable B Structure II is unstable C Structure III is unstable D All three structures are stable
GATE CE 2023 SET-1      Determinacy and Indeterminacy
Question 5 Explanation:
Unstability in beam can be checked

(i) If support reactions are not enough.
(ii) If reactions are concurrent.
(iii) If reactions are parallel.
(iv) If there is mechanism.

where as for truss also along with the above given points the triangular panels are generally stable.
But, a virtual inspection must be conducted to understand the stability

(I)
It is unstable as it shows mechanisms.
Also to understand if we apply a vertical force at slider side, the deflected shape will be as follows.

(II) The frame is internally stable but all the reactions are concurrent and meeting at hinge A, and the frame can rotate about A. Hence, it is unstable.

(III) The frame is unstable because it cannot resist shear in DE and AB since DE and AB are slender members.

There are 5 questions to complete.

### 1 thought on “Structural Analysis”

1. if you also provide gate calculator along with the questions it will be really helpfull for many aspirants