A three-hinged symmetrical arch is loaded as shown in the figure below. Which one of the following is the magnitude of the correct horizontal thrust?

Concept:

A three-hinged arch is a statically determinate structure and its reactions are calculated using equilibrium equations only. It is mounted on two hinged support and contains one hinge anywhere in the arch rib, generally at the crown which makes the structure determinate.

Let the reactions be as shown below.

∵ ΣF_Y = 0

⇒ R_A + R_B = 2P

∵ ΣF_X = 0

⇒ H_A = H_B

∵ ΣM_A = 0

⇒ 2P × 5 - R_B × 8 = 0

\( \Rightarrow {{\rm{R}}_{\rm{B}}} = \frac{{10{\rm{P}}}}{8}{\rm{and\;}}{{\rm{R}}_{\rm{A}}} = \frac{{3{\rm{P}}}}{4}\)

Again,

∵ ΣM_C = 0 (Because it is an internal hinge)

⇒ R_A × 4 - H_A × 1.5 = 0

\( \Rightarrow {{\rm{H}}_{\rm{A}}} = \frac{{3{\rm{P}}}}{{1.5}}\)

⇒H_A = 2P and H_B = 2P