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.
(Contributed by Walter Hooker)
A problem of some difficulty is encountered when an arch has to be constructed in a wall which is circular on plan, or in a battered wall, or when the axis of the arch runs obliquely to the face of the wall. The following examples will show how to set out such arches, and how to work the various stones for them -
To set out and construct arched openings in circular walls is a problem frequently met with in buildings of the rotunda class, or circular apses of cathedrals, etc.
Fig. 114 shows an opening in a circular wall with an arch surmounting it having the intrados of its outer face semicircular on development.
In this instance the reveals of the opening radiate from the centre, from which the plan of the wall is struck, and as the external arch is semicircular and the soffit at the crown is horizontal, the interior arris of the arch will form an elliptic curve on development, with half its major axis equal to the height of the arch from springing line to the crown, and its minor axis equal to the length of the arc BC on plan.
Fig. 114 shows a plan of the opening with centre line OX, where AB and CD are the reveals.
On any horizontal line YY, with X1 as centre and with radius equal to the length of the arc AX, set up the quadrant A1E. Next set out the points B1, etc., making B1X1 and C1X1 each equal to half the length of the arc BC, and draw the half ellipse B1Ec1.
These two curves A1E and B1E will give the development of the outer and inner line of the stones composing the arch of the soffit. Divide the outer curve A1E into any uneven number of equal parts required to form the voussoirs, and draw lines radiating from the centre X1, which will give the joints. From the points of contact of the joints with the outer arc draw lines parallel to the springing, and cutting the ellipse as shown. These will give the line of the bed joints on the soffit.
Draw GH on plan radiating from the centre O, making AG equal to the required depth of the arch. Make XlGl on the springing line equal to the arc XG on plan, and with the centre X1 and the radius X1G1 describe the arc G1K. This arc is the development of the extrados of the outer face of the arch. Make X1Nl on the springing line equal to half the arc HN on plan, and X1M equal to X1G1. Then with X1N1 as minor axis and X1M as major axis describe the quarter ellipse Mn1 shown dotted on elevation. This quarter ellipse is the developed elevation of the internal extrados. Complete the outline of the bed joints, which will be parallel to the springing line on the extrados and will radiate from the centre X1 on the internal face as indicated by the dotted lines on the right-hand side of the developed elevation. Project the points a, b, c, d, etc., horizontally on to the curve Ec1 at a1, b1, c1, d1, and c1 respectively, and through a1, b1, c1, d1, etc., draw the radial joints.
The arch may be constructed with the extrados and intrados forming concentric circles, as described up to the present. In practice, however, it is almost invariable to step the extrados so as not only to make a sounder job by bonding in the arch stones to the adjacent masonry, but also to save the labour of working twisted faces on the stones next to the arch wing. For this reason a stepped extrados is shown in the figure, the steps being set out upon the developed elevation. It will now be seen that on the left-hand side of the line Xx1 a front developed elevation of the arch has been drawn, while on the right-hand side of Xx1 a development of the internal face is shown.
From f drop the perpendicular fg on to YY, and from the arc XP cut off a part Xf1 equal to X1g. A line drawn from f1, radiating from O, marks the position on plan of the joint f1h1 in the elevation. In the same way all the other bed joints can be found, thus completing the plan. The object of the above drawings is to find the shape of the moulds from which the stones are to be worked. For the sake of example the springer on the left-hand side of the arch will be considered, every operation required to bring it from the rough block to the required form for setting in the arch being explained in order.
For each stone a bed mould and two face moulds are required, one for the internal face and one for the external face. In Fig. 115 the moulds for the springer and the next stone above and the keystone are shown. The bed mould of the springer is simply a replica of that shown upon plan in Fig. 114, each point of the bed mould being similarly lettered to the plan of the springer in Fig. 114. The external face mould is similarly a replica of stone No. 1, as shown on the left-hand side of the developed elevation, and the internal face mould is a replica of the same stone on the right-hand side.
The moulds of stone No. 2 and of the keystone are also replicas of the plans, and the external and internal developed elevations respectively of the stone, as shown in Fig. 114.
In working stone No. 1 a roughly square block is taken and the top and bottom beds are worked parallel to each other. The bed mould is then scribed upon both beds, as shown at T, Fig. 115, where represent the block from which the stone is cut, with the bed mould scribed upon its upper surface. Next, the stone is cut along the face Pp and Ka, care being taken to keep PQ vertical. The convex and concave faces are next worked, being boned across from the top and bottom beds, the stone then presenting the appearance of U, Fig. 115. The face moulds are then bent against and scribed on to their respective faces and the joint fKam is boned across. It should be noted that there will be a slight twist on this joint. A chisel draft is cut on both faces along the lines Ak and C1a1, and the stone is completed by boning the soffit across.
The same method of procedure is carried out for the remaining voussoirs, the keystone being a little different, as the soffit and the outer and inner face only would be curved. The soffit would be in this instance of very flat camber and slightly twisted. In actual practice the joints would be slightly spiral.
Oblique and Battered Arches - To find the beds, faces, and moulds of the stones in an oblique arch, one face of which has a batter of 10 degrees off the vertical.
Let Abcd (Fig. 116) be the plan of an oblique opening in a plane wall battered on the external face, Adef and Bcgh being the plans of the bed joints of the springers.
Let A1Jb1 be the elevation of the arch as projected on to a plane parallel to the face of the wall. This elevation is made semicircular.
Divide the arc A1Jb1 into 7 (or any other odd number) equal parts, and draw radiating lines from the centre X through the points thus found, as a, b, c, d, e, and f. These lines will show the joints of the voussoirs of the arch. Complete the elevation. The extrados may be stepped as explained in Chapter VII (Arches - Plane)., or it may be a semicircle as shown.
At H1K draw H1K perpendicularly to F1H1.
From H1 draw H1 L, making an angle of 10 degrees with H1K to indicate the batter of one face of the arch.
From the points a, b, c, etc., let fall perpendiculars on to and beyond AB on plan. From the points a, b, c, etc., draw lines parallel to A1B1 intersecting H1K, and with H1 as centre describe arcs from these intersections cutting the battering line H1L at 1, 2, and 3. From 1, 2, and 3 draw horizontal lines cutting H1K at 11 21, and 31. Then a1a2 and f1f2 on plan are each made equal to 111; b1b2 and exe2 are made equal to 221; c1c2 and d1d2 are made equal to 331.
A curve drawn through a2, b2, c2, etc., will give the plan of the intrados. The points a4, b4, c4, etc., are found in a similar manner, and a line drawn through them will give the extradosal line (or plan of the extrados).
To obtain the rectangular section through the arch, let MN be drawn at right angles to AD produced. Project the points of the voussoirs both of the intrados and extrados to meet MN and parallel to AD. Mark off the heights of the joints - shown above A1B1 in elevation - above MN, as at a5, b5, c5, etc., and connect as before. The intrados of the arch will then be defined by drawing a curve through the points a5, b5, c6, etc., as in the illustration. The points a6, b6, c6, etc., are found in a similar manner, and the elevation is completed as shown. To project the faces of any voussoirs - Take, for example, the keystone. Draw any convenient straight line yz at right angles to AD on plan through the stone required, and with this as a base take the distances above and below, and mark the same above and below any convenient straight line
YZ; the distances of the faces along the line of YZ being transferred from the upper elevation as h1, g1, c4, c2, d2, d4, and h1.
These points being joined and the curved surfaces transferred will give the bed moulds and joints. The faces are found by rotating the lines on a base such as the soffit mould, until they form the true shape of the stone as shown in the projection.
If the development is set out on the above lines on a piece of cardboard and rotated as above described an accurate model of the stone should be the result.
An isometrical view of the apex stone is also shown with the projected faces in true juxtaposition, enclosed in its corresponding block of stone.
A somewhat similar though rather more complex example of an oblique arch in a battered wall is shown in Fig. 121, where the arch is shown penetrating a semicircular vault.