A three – hinged parabolic arch of span ‘L’ and rise ‘h’ is subjected to a uniformly distributed load of intensity ‘ω’, then the horizontal thrust at the supports is
Concept:
From symmetry, \({V_A} = {V_B} = \frac{{Wl}}{2}\)
\(\sum {M_C} = 0,\)
\({H_A} \times h + W \times \frac{\ell }{2} \times \frac{\ell }{4} = {V_A} \times \frac{\ell }{2} = \frac{{W{\ell ^2}}}{4}\)
⇒ \({H_A} = \left( {\frac{{W{\ell ^2}}}{4} - \frac{{W{\ell ^2}}}{8}} \right) \times \frac{1}{h} = \frac{{W{\ell ^2}}}{{8h}}\)
Important Points
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 |
\(H = \frac{4}{3}\left( {\frac{{wR}}{\pi }} \right)\) |
Two hinged semi-circular arches subjected to uniformly varying load |
\(\rm{H = \frac{2}{3}\times \frac{wR}{\pi}}\) |
Parabolic arch subjected to concentrated load (W) at Crown |
\(H = \frac{{125}}{{128}}\left( {\frac{{WL}}{h}} \right)\) |
Parabolic arch subjected to UDL (w/per unit length) over the entire span |
\(\rm{H = \left( {\frac{{w{L^2}}}{{8h}}} \right)}\) |
Two hinged parabolic arches subjected to uniformly distributed load on the half span |
\(\rm{H = \left( {\frac{{w{L^2}}}{{16h}}} \right)}\) |
Two hinged parabolic arches subjected to varying distributed load on the full span |
\(\rm{H = \left( {\frac{{w{L^2}}}{{16h}}} \right)}\) |
You Should not miss this:
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,
\(H = \left( {\frac{W}{\pi }} \right){\sin ^2}\left( \alpha \right)\;\)