Internal Forces in Arches

A general section of a planar arch carries three main internal force effects: axial force $N$, shear force $V$ or $Q$, and bending moment $M$. The distinctive feature of an arch is that the local section directions rotate continuously along its curved axis.

Local coordinate system

At a point on the arch axis, define a tangent direction $t$ and a normal direction $n$. If the tangent forms an angle $\varphi$ with the global $x$ axis, the resultant force on an isolated part is resolved along $t$ and $n$ to obtain axial and shear components according to the selected sign convention.

Bending moment

The bending moment at a section follows from moment equilibrium of either part of the arch. For a three-hinged arch with supports at the same elevation and vertical loading, a convenient relation is

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

where $M_0$ is the moment in the corresponding beam, $H$ is horizontal thrust, and $y$ is the arch ordinate.

Axial force and shear

Once the global horizontal and vertical components of the section resultant are known, they are projected onto the tangent and normal. Consequently, the same global force contributes differently to $N$ and $V$ at different locations along the arch.

Structural interpretation

A well-shaped arch carries a large share of its load through axial compression. As the arch axis approaches the funicular shape for a given loading, bending moment and usually bending deformation decrease.

Checks

Bending moment must be zero at ideal hinges. Symmetric geometry and loading should produce the corresponding symmetry or antisymmetry in the internal-force distributions. At every section, the calculated actions must equilibrate the isolated portion of the arch.