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Chapter 2.8 Arches, Vaults and Domes

Figure 2.8.5 Thrust Line Across Voussoirs

Compression structures are ubiquitous as traditional bridges, cathedrals and viaducts. They have been adopted in modern forms using steel, concrete and timber. Traditional forms of vault and arch are reviewed, leading from a simple bridge with a single arch to multiple arches and to a simplified analysis of a cathedral buttressed section. Finally, the reader is invited to try out the methods of simple analysis to size a steel arch structure (based on a real stadium roof). By way of this example step by step conceptual design short cuts are followed to illustrate the way that a practising engineer might conceive a design without the use of a computer.

When the load is in the middle third the blocks have a compressive stress across their width. At the collapse point the thrust line moves outside the middle third. The stones pull apart at the outer edge. Since it is masonry no tension may form. At a critical point the stones may crush causing the bridge to collapse.
Arch compression (top tubes): see figures 2.8.25 to 2.8.27

Approximately 60% of the roof width is carried by the arch and 40% by the columns to the back of the stand (the beam is similar to a short cantilever with backspan – see figure 2.3.7 right hand support (3.125/(3.125+1.875).

Line load supported by arch = 2 (roof load) x 20 x 0.60 = 24 kN/m

Reference figure 2.8.6 (wL²/8H):

Horizontal Reaction = 24kN/m x 120²/8/15 (H) = 2880kN

Vertical Reaction = 24kN/m x 120 (L)/2 = 1440kN Note the arch is a low angle at the support, the vertical reaction is expected to be less than the horizontal reaction – just by looking at the angle of inclination. A scale sketch is of use to appreciate this.

Thrust in the arch = √ (2880² + 1440²) = 3220kN (i.e. angle of inclination of reactions Ø = cos -1 (2880/3220) = 26˚ – this looks correct by observation of the scale elevation of the arch.

A scale sketch helps to check visually the proportion of calculated reactions and inclination.

Figure 2.8.25 Stadium Elevation
The remainder of this chapter will demonstrate that a stunning stadium roof be designed with the approximate methods described hitherto. In fact this particular roof which may be attributed to Stephen Morley as structural engineer will have been conceptually designed using very similar methods described herein.