Arches and vaults are characterized by a thrust whose intensity and angle may disturb the stability of the whole. The thrust is the resultant of two forces: the weight of the arch and the horizontal thrust. Thus, the thrust always pushes downwards with an angle which depends on the arch profile and weight. The intensity of the horizontal thrust is generated by the weight of the voussoirs, which rest on each other, and the flatness of the arch. The flatter the arch is, the more intense the horizontal thrust is.

The horizontal thrust (HT) is applied on both springers, but it is also found on top of the arch, as it represents the balance of the second half of the arch. The horizontal thrust can be minimized by the optimization of the arch profile. Nevertheless, there will always be a thrust which can be neutralized by means of buttresses, truss rods or ring beams.

The successive action of the voussoirs on the ones below creates a Line of Thrust (LT). An arch or vault is stable as long as LT remains in the middle third of the arch section.
When LT goes in the inner third of the arch, the latter will tend to burst outwards.
When LT goes in the outer third of the arch, the latter will tend to collapse inwards.

Domes are also characterized by a thrust. As well as arches and vaults, the dome’s thrust is also composed of its weight and the horizontal thrust of the basic arch section. Therefore there is also a line of thrust which corresponds to the arch section.

When a dome is generated by the intersection of two vaults, the forces involved are identical to those of vaults. But when a dome is created by the rotation of an arch around a vertical axis, another force is acting in it: the circular force (CF). Domes generated by the rotation of an arch are built with successive horizontal rings. Each block of this ring behaves like the voussoir of an arch. Therefore, it will create a thrust (in the plan of the ring) against the next blocks.

The circular force in a “circular dome” is acting in a horizontal plan, a ring, and can be assimilated to the thrust which acts downwards in a vertical plan, in the case of arches or vaults. This force explains why it is possible to build circular domes without support. The dome is self supporting at every stage of its construction because the horizontal thrust of one half of the dome is transferred to the other half by the various rings. The force of gravity will obviously transfer vertically the circular force into the line of thrust.

The arch section, which generates the circular dome, rotates around a vertical axis. Therefore, the dome can be assimilated to an infinitesimal number of arches whose thrust radiates from the centre towards the periphery. On the springer level, the combination of all these horizontal thrusts will create a peripheral tension (PT) which will tend to open the wall supporting the dome.

In case of failure of any part of the dome, under an exceptional stress, this net of compressive forces will find another way to act in the dome, and the latter will rarely collapse entirely as long as the supports (walls or columns) are intact.

Example 1:
A heavy central load is applied on top of the arch, or the shape is disproportioned.
The line of thrust passes in the intrados third and will cause failure.

The line of thrust is in the middle third of the arch, but not in the middle third of the pier.
The latter is not wide enough and will collapse.

CATENARY CURVE OF THE LINE OF THRUST

Arches can have various shapes and sizes, but the line of thrust always follows the shape of an inverted catenary curve.

 A catenary is the curve assumed by a freely suspended chain or flexible cable under the action of gravity. The centre line of the links is the line of tensile stress. In an arch, the line of thrust is the line of compressive stress and takes the shape of an inverted catenary because it is the curve of natural load transfer in the masonry.

In an arch which has the shape of a perfect inverted catenary curve, the voussoirs correspond to the links of the chain. As the links of the chain are under tension, the voussoirs of the inverted catenary arch are under compression and LT is centred in the voussoirs. Catenary arches are always the most stable so their thickness can be reduced. Segmental arches are also very stable, as LT is near the centre of the arch.

MODIFICATION OF THE LINE OF THRUST IN A WALL

The arches shown as examples here are considered to be free standing, meaning without masonry above. Adding some load above the arch will modify the line of thrust in the masonry. LT will become a higher catenary, which often does not pass anymore in the arch but in the masonry above it. Thus the principal line of thrust materializes a discharging arch. Therefore the original arch is supporting only its load and the “triangular load” of the wall, situated below the discharging arch. The arch will still have a line of thrust, and this triangular load will slightly increase its intensity. LT will pass more towards the arch extrados and exit nearer the intrados.

Modification of LT in a wall

(Where t is the thickness and S the span)

Instability with too thin an arch: t = S/20

Stability with the proper arch thickness: t = S/5

This relationship explains why the semicircular barrel vaults need to be very thick, as the haunches cannot always be loaded. Note that the thickness of a semicircular arch can be reduced if the haunches are loaded. This will have three effects:

INFLUENCE OF THE MORTAR ON STABILITY

The various methods described hereafter for calculating the stability of arches and vaults don’t take into account the effect of the mortar on the strength of vaulted structures. Calculations are done as if the vaulted structures are built with dry stacked masonry.

The mortar binds the blocks and transmits the compression forces. Note that at the intrados the transmission of forces is directly from block to block: they touch each other. At the extrados the contact is ensured by the mortar which transmits the compression forces.

When vaults and domes are built with the Nubian or Free Spanning techniques, the quality of the mortar is essential to stick the blocks onto each other. But this is needed only while building the structure. Once the structure is completed, the transmission of the forces is also through the mortar, stressed under compression.

Note that cement-sand mortars cannot be used for building vaults and domes without support. Raw earth or stabilised earth mortars are needed to build them this way. Mortars have in general a low tensile strength and they should not be taken into account to give strength to the vaulted structures.

EVALUATION OF THE STABILITY OF DOMES

We saw that domes which are generated by the intersection of two vaults (i. e. groined and cloister domes), have forces identical to those of vaults. Therefore the stability of their arch section can be studied like an arch. These kinds of domes will show a difference from the arches and vaults: they will exert a thrust on four sides and will need a ring beam or abutments to balance it.

When a dome is created by the rotation of an arch around a vertical axis, the circular forces which act in it cannot be yet calculated. Therefore these types of domes require another approach to calculate their stability.

The examples of domes built all over the world through the ages show that domes can have a wider variety of shapes than vaults. For instance, a dome can be conical with any proportions: from a sharp one to a flatter one. But it is obvious that an arch cannot have a triangular section, as a cone is a triangle rotating around a central axis.

Therefore, it appears that if arches or vaults are stable, domes of the same section will necessarily be stable. But the opposite is not necessarily true, as we have seen with the case of the conical dome and the triangular arch.