Three-Hinged Arches

A three-hinged arch has hinges at both supports and one internal hinge, commonly at the crown. With a proper geometry this system is statically determinate: all support reactions can be obtained from equilibrium without using member stiffness.

Reactions and horizontal thrust

For an arch whose supports are at the same elevation under vertical loading, the vertical reactions can be obtained in the same way as for the corresponding simply supported beam. The horizontal reactions form the thrust $H$, which is found from the zero-moment condition at the internal hinge.

Crown-hinge condition

If the internal hinge has coordinates $(x_c,y_c)$ relative to a support, the moment of all external actions on either isolated half about that hinge must vanish. The equilibrium equation contains the contribution $H y_c$ and can therefore be solved for the horizontal thrust.

Corresponding-beam relation

For equal support elevations and vertical loading, it is useful to compare the arch with a simply supported beam of the same span. At a section $(x,y)$,

$$M(x)=M_0(x)-H\,y(x),$$

where $M_0(x)$ is the bending moment in the corresponding beam. This relation shows directly how horizontal thrust reduces bending in the arch.

Checks

After calculating the reactions, verify global equilibrium and $M=0$ at every ideal hinge. For a symmetric arch under symmetric vertical loading, the vertical reactions should also be symmetric.