Support line

The arc of the Jefferson National Expansion Memorial (Gateway Arch) in St. Louis , Missouri roughly follows a support line.

Representation of the load distribution that makes a semicircular arc the support line. The example shows that a free-standing masonry arch, as shown, would only be feasible with very solid lateral columns and a filigree arch that almost dissolves towards the apex . Almost the entire weight (sum of dead weight and load ) would have to be brought into the area of ​​the two pillars. Any load resting on the central part of the arch will cause the columns to be pushed apart laterally and the arch to collapse.

The support line is the 'line' on which the combined normal forces (compressive forces) resulting from the load run. The term was introduced by Franz Joseph Ritter von Gerstner (1756-1832) in the early 1830s.

In an ideal arch , the normal forces run along the axis of the arch (central pressure). The arch shape of the arch thus corresponds to the support line (support line vault). There are no moments. The support line only ever applies to a certain load. If this load is changed, the support line of an arch can deviate from its actual course. Tensile forces and moments arise.

An arch follows a support line if only compressive stresses are present in its entire cross-section at a given load . Bending , shear and Torsionspannungen however, are not present. Given only with its own weight of loaded sheet, the support line of a following catenoid , at about the arch span distributed uniformly distributed load of a quadratic function ( parabola ). The use of material for an arch running in a support line is minimal; the support line thus represents an optimum .

The principle of the support line can be understood using the example of a chain. If you hang a chain at its ends, the chain deforms downwards: it 'hangs'. This phenomenon is known as a chain line . In a chain, under any load, only tensile forces can occur, as the chain is flexible due to its construction (it cannot absorb any moments) and, under pressure, fails just like a rope. If you mirror the chain line around a horizontal axis, an arc is created in which only compressive forces (no moments) occur under your own weight. It is the support line, see also.

If the arch shape deviates too much from the support line, the bending moments can become so great that the arch fails. In practice, this problem occurs primarily with bridge structures. Vehicle and pedestrian traffic lead to asymmetrical loads that distort the support line. The arch must be reinforced for the bending moments that arise (e.g. by using additional bending beams).

The ideal load curve for a given arch geometry is of interest for structural analysis. In the case of a semicircular arch, this is the function of the perpendicular load:

${\displaystyle q\left(\varphi \right)={\frac {q_{0}}{\sin ^{3}\left(\varphi \right)}}}$

where the perpendicular load is in the middle of the arc, the angle plotted on the semicircle of the arc and the sine of this angle raised to the third power . At the supports of the arch, q becomes infinitely large (see illustration) . The formula shows that a (delicate) semicircular arch, in contrast to a catenoid, can hardly be constructed free-standing. There would have to be a lot of mass on the arch near the supports , while it would be made almost weightless towards the apex . Nevertheless, semicircular arches can be carried out without any problems if they are integrated on the right and left in masonry that is able to absorb the lateral shear forces that arise. ${\displaystyle q_{0}\ }$ ${\displaystyle \varphi }$ ${\displaystyle {\sin ^{3}\left(\varphi \right)}}$

To dimension an arch, the compressive stresses of the material along the support line are determined. According to the stability theory , arch structures must also be examined for buckling , buckling and additional load cases.

The concept of the support line is comparable to that of the stress trajectories .

See also

• Rope statics

literature

• José Luis Moro: Building Construction. From principle to detail. Volume 2: Concept. Springer, Berlin et al. 2009, ISBN 978-3-540-27790-3 , paragraph 4.2, p. 194 f .: The terms of the rope and the support line .
• Wormuth / Schneider (eds.): Construction dictionary online - support line ; Bauwerk publishing house.

Individual evidence

1. ^ Karl-Eugen Kurrer: History of structural engineering. In search of balance . 2nd, greatly expanded edition. Ernst & Sohn, Berlin 2016, ISBN 978-3-433-03134-6 , pp. 231 .
2. ↑ Construction dictionary. Support line. Beuth Verlag GmbH, accessed on January 10, 2017 .