# Arches

 Question 1
A semi-circular bar of radius R m, in a vertical plane, is fixed at the end G, as shown in the figure. A horizontal load of magnitude P kN is applied at the end H. The magnitude of the axial force, shear force, and bending moment at point Q for $\theta =45 ^{\circ}$, respectively, are A $\frac{P}{\sqrt{2}}kN,\frac{P}{\sqrt{2}}kN \text{ and } \frac{PR}{\sqrt{2}}kNm$ B $\frac{P}{\sqrt{2}}kN,\frac{P}{\sqrt{2}}kN \text{ and } 0 \; kNm$ C $0 \; kN,\frac{P}{\sqrt{2}}kN \text{ and } \frac{PR}{\sqrt{2}}kNm$ D $\frac{P}{\sqrt{2}}kN,0 \; kN \text{ and } \frac{PR}{\sqrt{2}}kNm$
GATE CE 2022 SET-1   Structural Analysis
Question 1 Explanation: \begin{aligned} (F_Q)&=P\sin \theta =\frac{P}{\sqrt{2}}&(\; at\; \theta =45^{\circ})\\ (S_Q)&=P\cos \theta =\frac{P}{\sqrt{2}}&(\; at\; \theta =45^{\circ})\\ M_Q&=PR \sin \theta =\frac{PR}{\sqrt{2}}&( \; at\; \theta =45^{\circ}) \end{aligned}
 Question 2
A perfectly flexible and inextensible cable is shown in the figure (not to scale). The external loads at F and G are acting vertically. The magnitude of tension in the cable segment FG (in kN, round off to two decimal places) is _______
 A 4.68 B 3.78 C 5.64 D 8.25
GATE CE 2021 SET-2   Structural Analysis
Question 2 Explanation: \begin{aligned} \Sigma M_F=0\\ \left ( 10.667+\frac{H}{6} \right )2-H \times 3=0\\ H=8\; kN\\ V_E=10.667+\frac{8}{6}\\ V_E=12\; kN\\ V_H=10\; kN \end{aligned}

Consider left side of section (1)-(1)

\begin{aligned} \Sigma H=0\\ T \cos \theta =H\\ T \cos \theta =8\\ T \sin \theta =2\\ \therefore \;\; T^2 \cos ^2\theta + T^2 \sin ^2 \theta =8^2+2^2\\ T=8.246\; kN \end{aligned}

Tension in segment GF is 8.246 kN.

 Question 3
A planar elastic structure is subjected to uniformly distributed load, as shown in the figure Neglecting self-weight, the maximum bending moment generated in the structure (in kNm, round off to the nearest integer), is ___________ .
 A 12 B 192 C 96 D 48
GATE CE 2020 SET-1   Structural Analysis
Question 3 Explanation: $V_A=V_B=\frac{wL}{2}=\frac{12 \times 8}{2}=48 kN$
As horizontal thrust is zero so it behaves like a beam (curved beam)
$M_{max}=\frac{wL^2}{8} \; (\text{ At crown})$
$=\frac{12 \times 8^2}{8}=96KNm$
 Question 4
The figure shows a two-hinged parabolic arch of span L subjected to a uniformly distributed load of intensity q per unit length The maximum bending moment in the arch is equal to
 A $\frac{qL^{2}}{8}$ B $\frac{qL^{2}}{12}$ C Zero D $\frac{qL^{2}}{10}$
GATE CE 2017 SET-1   Structural Analysis
Question 4 Explanation:
If a two hinged or three hinged parabolic arch is subjected to UDL throughout its length, bending moment is zero everywhere.
 Question 5
A three hinged parabolic arch having a span of 20m and a rise of 5m carries a point load of 10 kN at quarter span from the left end as shown in the figure. The resultant reaction at the left support and its inclination with the horizontal are respectively A 9.01 kN and 56.31$^{\circ}$ B 9.01 kN and 33.69$^{\circ}$ C 7.50 kN and 56.31$^{\circ}$ D 2.50 kN and 33.69$^{\circ}$
GATE CE 2010   Structural Analysis
Question 5 Explanation: Let the vertical reactions at left and right support be $V_{L}$ and $V_{R}$ upwards respectively.
Taking moments about right support, we get,
\begin{aligned} V_{L} \times 20-10 \times 15&=0 \\ \Rightarrow \quad V_{L}&=7.5 \mathrm{kN} \\ \therefore \quad V_{R}&=10-7.5=2.5 \mathrm{kN} \end{aligned}
Let the horizontal reaction at left support be H from left to right.
Taking moments about the crown from left, we get,
\begin{aligned} 7.5 \times 10-10 \times 5-H \times 5 &=0 \\ \Rightarrow \quad H&=5 \mathrm{kN} \end{aligned}
Resultant reaction,
\begin{aligned} R &=\sqrt{H^{2}+V_{L}^{2}} \\ &=\sqrt{5^{2}+(7.5)^{2}} \\ &=9.01 \mathrm{kN} \end{aligned}
Let the resultant reaction at the left support makes an angle $\theta$ with the horizontal.
\begin{aligned} & \tan \theta=\frac{7.5}{5} \\ \Rightarrow & \theta=\tan ^{-1}(1.5) \\ \Rightarrow & \theta=56.31^{\circ} \end{aligned} There are 5 questions to complete.