Funicular Arch

A funicular arch is an arch whose axis is shaped to carry a specified loading by axial compression without bending. For that particular load case, the line of thrust coincides with the arch axis and the bending moment is zero along the idealized arch. The same idea is also called a rational arch axis.

What is a funicular arch?

An arch develops bending whenever the internal thrust does not pass through its structural axis. A funicular form removes this eccentricity for a chosen loading. As a result, the idealized arch carries the load through compression rather than a combination of compression and bending.

The word funicular emphasizes that the required geometry depends on the loading. There is no single funicular shape that is optimal for every possible load pattern.

Funicular arch and the thrust line

For a three-hinged arch with supports at the same elevation under vertical loading, the bending moment at a section can be written as

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

where $M_0(x)$ is the bending moment in the corresponding simply supported beam, $H$ is the horizontal thrust, and $y(x)$ is the ordinate of the arch axis above the support line.

For a perfectly funicular axis,

$$M(x)=0,$$

so that

$$y(x)=\frac{M_0(x)}{H}.$$

This result gives a useful interpretation: the funicular arch shape is proportional to the bending-moment diagram of the corresponding beam for the same loading.

In the comparison below, all four systems have the same span $L=12\,\text{m}$ and rise $f=3\,\text{m}$. Only the loading changes. The axis ordinates are calculated from $y(x)=M_0(x)/H$, with the horizontal thrust selected so that each form reaches the specified rise.

Funicular Shapes for Different Load Cases

The funicular axis is a scaled bending-moment diagram of the corresponding simply supported beam: point loads produce straight segments, uniform load produces a parabola, and an added point load creates a slope discontinuity.

A single point load produces a triangular $M_0$ diagram, so the funicular axis degenerates into two straight members. Two point loads produce three straight segments. A vertical load uniform over the horizontal projection produces a parabola. Adding a point load at midspan leaves two parabolic branches but creates a kink at the load point, with a slope jump $\Delta y'=P/H$.

Effect of self-weight. The forms shown above correspond to the specified external loads. In masonry and reinforced-concrete arches, self-weight is often a major part of the total loading, so a strictly funicular axis must be determined for the external load plus self-weight. For an arch of constant thickness, self-weight per unit of horizontal projection is generally not uniform; it increases on steeper portions approximately in proportion to $\sqrt{1+(y')^2}$. The initial parabola therefore requires a small correction. In practice, the geometry is refined iteratively: assume an initial axis, calculate its self-weight, rebuild the thrust line, and repeat.

Funicular arch shape

The ideal shape follows directly from the load distribution. Different loads produce different beam moment diagrams and therefore different funicular curves. A geometry that gives zero bending for one loading will generally develop bending when the loading changes.

Parabolic funicular arch

For a vertical load uniformly distributed over the horizontal projection of a span $L$, the corresponding funicular curve is a parabola. If the rise at midspan is $f$, the axis can be written as

$$y(x)=\frac{4f}{L^2}x(L-x).$$

For a three-hinged arch under this loading, the horizontal thrust is

$H=\frac{qL^2}{8f},$

where $q$ is the uniform vertical load per unit of horizontal length.

Funicular arch vs catenary arch

A funicular arch is not necessarily a catenary. “Funicular” describes the relationship between shape and loading; “catenary” names a particular mathematical curve. An inverted catenary is funicular for the loading associated with a freely hanging uniform cable under its own weight. By contrast, a uniform vertical load specified per unit of horizontal projection produces a parabolic funicular form.

Therefore, statements such as “the ideal arch is always a catenary” are incomplete unless the load distribution is also specified.

Funicular arch vs an ordinary arch

An ordinary arch may have a circular, parabolic, catenary, pointed, or other axis. Its geometric name alone does not tell us whether it is funicular. The arch is funicular only when its axis matches the thrust line for the load case being considered.

If the thrust line departs from the axis, an eccentricity develops and the arch must resist bending in addition to axial force.

Why does the funicular shape depend on loading?

The zero-moment condition contains $M_0(x)$, which is determined by the applied loads. Changing the magnitude, position, or distribution of the load changes $M_0(x)$ and therefore changes the required funicular axis.

This is especially important for real structures. An arch may be nearly funicular under permanent loads but experience bending under asymmetric live load, wind, temperature effects, support movement, or other load cases.

Worked example: horizontal thrust

Consider a three-hinged parabolic arch with span $L=20\,\text{m}$, rise $f=5\,\text{m}$, and a uniformly distributed vertical load $q=10\,\text{kN/m}$ over the horizontal projection.

Using

$$H=\frac{qL^2}{8f},$$

we obtain

$$H=\frac{10\times20^2}{8\times5}=100\,\text{kN}.$$

For this ideal load case, the parabolic axis is funicular, so the bending moment is zero throughout the idealized three-hinged arch while the supports must resist a horizontal thrust of $100\,\text{kN}$.

Engineering significance and limitations

Funicular action can make an arch structurally efficient because it reduces bending and makes greater use of compression. This is particularly advantageous for materials and structural systems that perform well in compression.

However, reducing bending in the arch does not eliminate structural demands. Horizontal thrust must be transferred into abutments, foundations, or ties. In addition, a shape optimized for one load case may not remain funicular under other actions, so practical arch design requires checking all relevant load combinations and stability effects.

Frequently asked questions

What is a funicular arch?

It is an arch whose axis coincides with the thrust line for a specified loading, producing axial compression without bending in the idealized model.

Why is the bending moment zero in a funicular arch?

Because the compressive resultant passes through the arch axis. With no eccentricity between the thrust line and the axis, the load does not create a bending moment about the axis.

Is a funicular arch always a catenary?

No. The funicular curve depends on how the load is distributed. A catenary is one particular funicular form; a uniformly distributed vertical load over the horizontal projection gives a parabola.

What is the ideal shape of an arch?

There is no universal ideal shape independent of loading. For minimum bending, the ideal axis follows the thrust line for the governing load case or is selected to perform acceptably across the relevant combination of load cases.