This section is from the book "Modern Buildings, Their Planning, Construction And Equipment Vol5", by G. A. T. Middleton. Also available from Amazon: Modern Buildings.
Fig. 120 shows the case of a vaulted compartment such as would be occasioned by a semicircular vault passing round a circular building. Abcd is the plan. Aeb is the section of the semicircular vault, which has been made of even thickness throughout, save for the two lowest stones at each corner, the joints radiating from the centre. Develop the line Aod along A1Od1, and by the method of ordinates set up the developed section as shown in Chapter VI (The Geometry Of Masonry). Next find the curves of interpenetration as shown in Chapter VI. Project the joints on the soffit of the smaller vault on to the springing line, as at a, b, c, d, etc.
Using the centre from which the lines AD and BC were struck, draw concentric arcs to pass through the points a, b, c, d, etc., and to cut the line of the groins; and from the points of intersection thus found draw radial lines as shown to cut the line Aod at a11, b11, c11, d11, etc. Along Oa1 develop the distances Oa11, Ob11, Oc11, etc., as shown at Oa1, Ob1, Oc1; and from the points a1, b1, c1, d1, etc., erect perpendiculars to cut the intrados of the larger vault section. The points of intersection thus found will give the position of the joints on the intrados. The joints on the extrados may be found in a similar manner. The vertical joints are put in as shown in Fig. 118, save that in this case they radiate across the vaults on plan and are concentric with the lines AD and BC along the vault. The smaller vault section may be found in a similar manner. The bed and face moulds for the lowest three stones at the corner A and for the keystone are also as shown in Fig. 118. The shapes of these stones are similar to those in Fig. 118, save that the vertical joints radiate.
A more difficult case is presented in Fig. 121, in which an arched opening penetrates a cylindrical vault, having the external face of the wall at an angle with the axis of the vault and also battering vertically.
Let the angle of skew of the face be 10 degrees and the batter of the wall 1 3. Let CB, AD be the lines of contact of the beds of the voussoirs at the springing, and B, J, I, H, G, F, E, A the points at the joints of the stones which form the semicircular arch, the centre being K.
Project lines parallel with the axis K, K2 from C, B, A, and D, and draw LQ the thickness of the wall at the springing line at C. Draw LM at right angles to the axis, and set off MN the difference between the thickness of the wall at L and M. Set NL at 10 degrees off the line NT (drawn parallel to LM), and draw LO at right angles to LN. This will form the line of set back for the projection of the points on the battering face.
From C erect a perpendicular CP to the springing line, and draw Cq1 at a slope of 1 in 3 from the vertical.
Project lines parallel with the springing through all the points of the voussoirs on to Cq1 through CP. The various distances from CP to Cq1, transferred to LO, will give, by means of lines drawn through these points and parallel to LN, the positions of the joints and planes of the several arch stones. Similar lines projected parallel to K, K1, etc., will at their points of contact with the former define the edge of the arch in front. Transfer the points to N1R1 of the section, and from a centre whose horizontal distance from R1 is equal to the radius of the vault describe an arc upwards from R1. This will be the boundary line for the back. Project the various members as before. The points of contact thus found transferred to the plan will give the points for the curve of the arch at the back.
To develop the surfaces of the voussoirs, take, for example, the keystone Hh gG. Draw a horizontal line hh, and from h erect a perpendicular hh1 equal to hh1 on plan.
The four-sided figure hhlg1g on plan gives the true mould of the top of the keystone, as Hh1g2G gives that of the intrados. The figure hhlg1g, reversed as in the projection, with its point h in contact with the horizontal line hh, forms the first plane of development.
Draw a line through h on plan parallel to h1g1, and with this as base line lay off distances on the horizontal, hh equal to the distance of the points of the voussoir from the line through h on plan. The true widths of the surfaces, as at gg1g2G, can be taken from the elevation, the other faces being treated in a similar way.
The developed figure with its sides revolved on the joints Hh1, Gg2, and gg1 (see Plan) will then form the voussoir or keystone.
An isometrical projection of the keystone is also shown, with the enclosing squared block indicated by dotted lines. Another voussoir is shown both developed and in isometrical projection.
It should be borne in mind that in using bevels for the curves of the intrados great care should be exercised to apply them at right angles to the true axial line of the arch.