Wednesday, July 13, 2011

How To Design Rib Vaulting

The following procedure is reproduced from "Modern Practical Masonry", 1929 by Edmund George Warland.

Setting out rib and panel vaulting.

The chief items to be considered when setting out this type of vaulting are: 1) the arrangement of the curves; 2) the setting up of the elevation of the several ribs so that the face moulds for the working of the voussoirs may be obtained, and the bevels for working the boss stones; 3) the determination of the bed moulds for the springers.

First decide upon the general form of the vault, and plot the centre line diagram of the ribs in plan.  Add the thickness for the various ribs, making the diagonal and transverse ribs wider than the intermediate and ridge ribs.

Draw in the plan of the boss stones at the intersection of the ribs as shown, and decide upon the position for the springing points of the ribs on the nosing line, as at E on the plan of the diagonal rib.

Now determine the elevation of the ribs, commencing with the diagonal rib, this rib having the largest span.  In the example given, the diagonal rib has been made semicircular in elevation, so that the centre for striking the curve of the rib is at the centre of the vault in point O.  Draw a line from the striking centre perpendicular to the centre line of the rib in the plan, and draw the curve from point E.

Where this curve cuts the perpendicular line from point O, determines the height of the vault, if the ridges are horizontal, as in the example given.  Next proceed to erect elevations for the other ribs.  To do this, it is necessary to decide upon the form required for the curve of the ribs, owing to the ribs being the same height, but of unequal span.

Setting Out of Gothic Vault

These curves may be composed of a combination of arcs, or obtained by ordinates, thereby producing elliptical curves.  In the example, the curve of the ribs is struck from two centres, the radius of the lower curve being common to all the ribs.

Determine in plan the highest point at which the ribs are separated, as at G, and project this point up to the point G’ in the elevation of the diagonal rib, on the soffit line of the in-filling.  Through G’ draw a normal joint line cutting the nosing of the diagonal rib in point K.  Produce the joint line through G’ to the curve representing the extrados of the rib in point L.  This determines the top height for the horizontal bed line of the springers.

Now draw lines from the centre of each boss stone perpendicular to the centre line of the rib in plan, and mark on these perpendicular lines the vertical height of the vault, this height being taken from the centre height of the diagonal rib.  Produce the centre line of the ribs in plan, and mark off on these lines, from the springing points of the rib nosings, a distance equal to the radius of the diagonal rib, thus determining the centres for striking the lower curve of the ribs in elevation.  Now draw concentric curves representing the soffit line of the in-filling, and measure the vertical height of point G’ from the springing line, and transfer this height to the elevation of the other ribs, cutting them in points M and N.  Through these points draw the normal joint lines cutting the nosing lines in points J and H, and place in the top horizontal bed line in each case.  This determines the elevation of the springers.  A diagram showing the method for determining the curvature of the ribs is given in the figure.

Draw the centre line plan of the ribs.  Determine the point D, which is the point where the ribs separate.  This point in the setting out is taken on the soffit line of the in-filling.  With O as centre, swing all the ribs into the vertical plane, which in the diagram is parallel to the plan of the transverse rib.  From point C and radius C A, draw the elevation curve of the diagonal rib, to cut a perpendicular line drawn from C in point C x.  This determines the height of the vault.  Now draw a horizontal line from C x, representing the elevation of the horizontal ridge rib.

Erect perpendicular lines from the points E F G H, where the other ribs are rotated into the line O C, to cut the elevation line of the ridge rib in points E’ F’ G’ H’.  Draw a normal line from point D through point C, producing it indefinitely beyond C.  Now draw lines bisecting the points D E’, D F’, D G’, and D H’, to cut the normal line D C, produced in points C1 C2 C3 C4.  These are the centres for drawing the upper portion of the ribs from point D.

Now project the plan of the boss stones up to elevation, and place in the normal joints in each elevation, as shown in figure, thus determining the bevel for working the normal joints.  The top surfaces of the boss stones are usually arranged to be horizontal in order to facilitate working.  The bed mould for boss stone B is shown in figure.

The ridge ribs being horizontal, radiating or key joints should be arranged so as to prevent the stones from falling out of position.  Under these circumstances a raking section mould would be required for marking on the radiating joint surfaces of the ridge stones.

Sketch of Springers Showing Beds and Jointing

The moulds for working the springing stones should now be determined.  These are shown divided into horizontal courses, so that, with the exception of the normal joints of the top springer, between points K and L, raking section moulds will be required for marking on the horizontal bed surfaces.  To obtain the outline for these moulds, starting with the first springer, as in figure, cut a true section of each rib, and apply these to their corresponding nosing lines on the springing, as at point E, adjusting each mould to the centre line of each rib.  Where these sections intersect each other determines the outline for the bottom bed mould at springing level.
Now mark the position of the horizontal joint lines in each elevation, and project to the plan the points where these bed lines cut the lines of the ribs, as at Q and R, thus obtaining their outline in plan as shown.  These bed moulds are shown projected out from the drawing in figure.  The top springing course would be rather large if worked in one stone, therefore it would be best to place a vertical normal joint through point G in plan, as suggested in the drawing, thus making the top springing course complete in three stones.

The top normal joints should be worked to a bevel applied from the top horizontal surface, the correct angle for setting the bevel being taken from the elevation of the ribs.  As all the ribs are struck with the same radius up to point K, and the height of this point from the springing is the same in each case, the bevel which is shown applied on the normal joint of the wall rib, in figure, will be the same for the other normal joints at this level.

A detailed section of the diagonal rib is given in figure.  This shows the rebate for the support of the infilling, which is usually laid in thin courses.  A slight camber should be arranged on each course, thereby forming a series of flat arches, so that when the joints of these stones are filled with mortar from the back surface it is impossible for the stones to drop or fall out.

The bed lines of the in-filling courses can be arranged inclined or at right angles to the main ribs in plan.  The material for the in-filling should be as light as possible.

Diagrams showing method for Determining Curvature of Ribs.
A diagram showing the method of obtaining the curves for lierne ribs is given in figure.  Set out the centre line in plan of the ribs and erect an elevation of the diagonal rib, thereby obtaining the height of the vault.  Next determine the height of point D, where the ribs separate, and erect the elevation of the intermediate rib by drawing the lower curve of the rib up to the height D’ with the same radius as the diagonal rib, the centre for the upper portion of the curve being determined as already explained.

From point E on this rib, which is the junction of the lierne ribs, in plan erect a perpendicular to cut the curve line of the intermediate rib in point E’.  From E’ draw a horizontal line to cut the perpendicular line from O in point O’.  Mark off on the line E’ O’ any number of points, as at 1’ 2’, and project these points to the plan of the intermediate rib in points 1 and 2.  From these points draw lines parallel to the ridge rib Ox Ox, cutting the plan of the lierne ribs, and mark on these lines the lengths of the ordinates O’, O”, 2’, 2”, 1’,1”, as shown in points 1 3, 2 3, O 3.  A fair curve drawn through these points determines the curve for the lierne ribs.  A Sketch of the springer stones showing the horizontal beds and jointing for the top springer is given in figure.