Question
Download Solution PDFThere are two hinged semi-circular arches A, B and C of radii 5 m, 7.5 m, and 10 m respectively and each carries a concentrated load W at their crowns. The horizontal thrust at their supports will be in the ratio of
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
For two hinged semi-circular arches:
Horizontal thrust is given by:
\({{\rm{H}}_{\rm{A}}} = {{\rm{H}}_{\rm{B}}} = \frac{{\smallint \frac{{{{\rm{M}}_{\rm{x}}}{\rm{ydx}}}}{{{\rm{E}}{{\rm{I}}_{\rm{c}}}}}}}{{\smallint \frac{{{{\rm{y}}^2}{\rm{dx}}}}{{{\rm{E}}{{\rm{I}}_{\rm{c}}}}}}}\)
Where,
Mx = Moment at A – A’, y = vertical distance,
E = Modulus of elasticity, and
I = Moment of inertia of c/s of the arch.
When W load acts at the crown, then
Horizontal thrust (H) = w/π
∴ H is independent of the radius of the arch.
So, the ratio will be 1: 1: 1.
Note
|
Arches |
Horizontal Thrust |
|
Semicircular arch subjected to concentrated load (W) at Crown |
\(H = \left( {\frac{W}{\pi}} \right)\) |
|
Semicircular arch subjected to UDL (w/per unit length) over the entire span |
\(H = \frac{4}{3}\left( {\frac{{wR}}{\pi }} \right)\) |
|
Parabolic arch subjected to concentrated load (W) at Crown |
\(H = \frac{{25}}{{128}}\left( {\frac{{WL}}{h}} \right)\) |
|
Parabolic arch subjected to UDL (w/per unit length) over the entire span |
\(H = \left( {\frac{{w{L^2}}}{{8h}}} \right)\) |
Important Points:
For a Two Hinged Semicircular arch subjected to concentrated load (W) at any other point which makes an angle α with the horizontal, then the horizontal thrust,
\(H = \left( {\frac{W}{\pi }} \right){\sin ^2}\left( \alpha \right)\;\)
Last updated on May 28, 2025
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