The collapse of stone vaulting J. Heyman 3 Banhams Close, Cambridge, CB4 IHX, UK

Transcription

1 The collapse of stone vaulting J. Heyman 3 Banhams Close, Cambridge, CB4 IHX, UK ABSTRACT A study is made of the way in which the masonry vault of a large church might be affected by the collapse (as during a fire) of the covering roof structure. Consideration is also given to the effect of water building up in vaulting pockets. A description of the relevant fundamental properties of masonry is illustrated by the behaviour of the simple voussoir arch. This in turn leads on to a discussion of the normal structural state of a high vault. Estimates are made of the way in which a particular vault of specified dimensions might behave, and the conclusions which are drawn apply only to that vault. Assessments of other structures can be made in the light of knowledge of their own particular dimensions. Nevertheless, it is concluded that the total collapse of a roof may be resisted by a vault of usual thickness and that, if severe damage occurs, it may well be limited to relatively small regions of the vault (adjacent to the north and south windows) and not involve overall collapse of the masonry. Similarly, the completefillingof a vaulting pocket with water (which may be a more severe condition than collapse of the timber roof) may again cause only local damage of the vault. However, the calculations made on the particular vault under study indicate that the forces resulting both from collapse of the high roof and from water pressure are of the same magnitude, or exceed, the basic strength of the vault, so that some damage under these conditions is almost inevitable. INTRODUCTION Vaults may be made from a range of stones, from say "standard" sandstone to relatively weak clunch to lightweight tufa. What they have in common is that they are required normally to carry only their weight, and they do

2 328 Structural Repair and Maintenance of Historical Buildings this at very low ambient stress levels. Typically, the largest compressive stress (which will, of course, depend on particular dimensions in any given case) does not exceed 1 N/mm\ whereas the crushing strength of a medium sandstone may be 30 N/mm^. Whatever the particular dimensions, there is such a large factor of safety on the strength of the material that collapse due to over stressing of the vault itself may be said never to occur. (Local spalling of material may sometimes be seen at pressure points, but these are not signs of overall danger. Indeed, local crushing of material leads to an enlarged area of contact and consequent automatic reduction in stress.) This observation of low stress levels, which applies generally to most elements of masonry construction, leads to a simple approach to the understanding of the mechanics of the masonry building. Individual blocks of stone may be taken as rigid strong building blocks, assembled together to form a stable structure. The stones may be squared and well fitted, or unworked and placed roughly in contact. Mortar may be used tofillinterstices, but this mortar will have decayed with time, and cannot be assumed to add strength to the construction. Thus the stones will be unable to resist any attempt to pull them apart; compressive forces can be transmitted within the masonry structure, but only feeble tensions can be resisted. Stability of the structure is assured, in fact, by the compaction under gravity of the various elements, and these must be assumed to have sufficient friction between their points of contact (or to interlock) so that sliding of one portion on another does not occur. (Occasional examples of slippage can be seen in practice.) Under these conditions, the structure will be satisfactory if it is of the right shape. In essence, the shape of the structure must conform to the pattern of forces it carries, and the greater the conformity, the greater the safety of the structure. This general principle can be made to yield numerical results, as will be seen. THE VOUSSOIR ARCH The general action of masonry may be illustrated by the simplest form of construction, the two-dimensional arch. The arch of fig. l(a) is supposed to be made from identical wedge-shaped voussoirs, assembled without mortar on temporary centering, and fitting exactly between its abutments. When the centering is removed the arch at once thrust horizontally at its abutments. The abutments inevitably give way slightly, and there is now a small geometrical mismatch. The arch must somehow accommodate itself to the increased span, and the way it does this is shown schematically infig.l(b). From the assumed properties of the material, the stones cannot crush and they cannot slip; on the other hand, the construction cannot resist tension. Cracks, in fact, will develop, idealised as hinges in fig. l(b), and it is these hinges which allow an increase in span without loss of overall structural in-

3 Structural Repair and Maintenance of Historical Buildings 329 tegrity. Cracks are not necessarily signs of incipient collapse - they indicate that the structure has responded to some unpredictable shift in the external environment. (Since all arches thrust, and all river banks give way, it is a common sight when passing under a masonry river bridge to see a single crack running along the barrel of the arch near the crown.) The arch of fig. l(b) is, then, stable, and can carry load. Suppose a single point load is imposed as in fig. l(c), and this load is slowly increased so that its magnitude becomes more and more important compared with the dead weight of the arch. It may be imagined that some of the pre-existing cracks, such as those of fig. l(b), might close, and that new cracks might open, but the arch itself will remain stable. However, there is a limit to which the point load can be increased - when four hinges are formed, as in fig. l(c), unrestricted motion of the arch will occur, and it will collapse. The load at which this happens is calculable, and lower values of load can be carried safely. The calculations are made by the use of statics, and equilibrium of the arch can best be understood by reference to the funicular polygon. The funicular polygon for the arch describes geometrically the way in which forces are transmitted from section to section, and gives the shape of the line of thrust. The shape of this compressive funicular polygon is exactly the same as that of a weightless cord in tension. If such a cord is imagined to be hung from two points, and subjected to the same loads that act on the arch, then "as hangs the flexible line, so, but inverted, will stand the rigid arch". This was Robert Hooke's statement of 1675, and fig. 2 shows an eighteenth-century illustration of the principle; the slender catenary arch shown should just stand under its own (uniform) weight. It is of course the real thickness of the arch that confers a margin of safety on an actual construction. The arch of fig. l(a) is stable because the line of thrust is containable within the masonry. The semicircle does not actually conform very well with the catenary, but fig. 3(a) would seem to indicate some margin of safety. Indeed, the margin can be estimated by reference to fig. 3(b), which shows the thinnest possible semicircular arch which will stand under its own weight - the thrust line can only just be contained within the masonry, and collapse is incipient. The increasing point load of fig. l(c) will alter the shape of the thrust line (as such a load would alter the shape of Hooke's hanging cord), and at the limit of fig. l(c) the thrust line just passes through the four hinge points, again reaching the surface of the masonry and indicating incipient collapse. Four such hinges are, however, necessary for collapse; the three-hinge arch is a perfectly satisfactory structural form. Figure l(b) is redrawn in

4 330 Structural Repair and Maintenance of Historical Buildings fig. 4(a); fig. 4(b) shows the corresponding position of the thrust line. If the arch is pointed at the crown, then the thrust line will be unable to "reach" the extrados at that point, and, in theory, the two hinges of figs 4(c) and (d) will form. In practice slight asymmetry will suppress one or other of these crown hinges. THE CROSS VAULT A simple extension of these basic ideas for the arch leads to an understanding of the mechanics of the vault. As afirststep towards this understanding, fig. 5(a) shows the cross section of a uniform barrel vault, drawn roughly to scale (say a vault thickness of 300 mm with a span of 12 m). The vault is supposed to be maintained by external buttresses, with or without flyers spanning the side aisles. As drawn, the vault is too thin to carry its own weight; a thrust line cannot be drawn within the masonry. Fill is therefore shown in fig. 5(a) backing the haunches of the barrel, allowing a pathway for the vault forces in their "escape" from the vault proper. (Thefillshould not necessarily be regarded as an added weight on -the haunches; it could be massless, provided that it had sufficient strength to transmit the forces.) The external buttressing system will give way slightly under the imposed thrusts from the vault, and movements will occur during thefirstfew years of the life of the structure. Movement of the buttresses will virtually cease when the soil under the foundations has consolidated under the heavy pressures. Thus the barrel vault might be expected to show the crack pattern of fig. 5(b), and the crack near the crown would be visible from within the building. Figure 6, from a study of 1934 by Pol Abraham, shows a sketch of typical cracks of this sort near the crown of a slightly pointed quadripartite vault. This is a first kind of chronic defect exhibited in practice by masonry vaults, but other cracks are visible in fig. 6. The vault of fig. 5 is essentially two-dimensional in the sense that the cross-section was supposed to be the same down the length of the church. Figure 7 shows a single bay of a quadripartite vault formed by the intersection of two slightly pointed barrels. In fig. 7(a) an elevation of the vault is shown, looking east down the length of the church; thefill,which serves the same function as before, is placed in the vaulting conoids (cf. the plan of fig. 7(c). The vault is of course supposed to extend for several bays). If now the buttressing of the vault gives way, the portion which runs east/west will crack as before, and the single hinge line near the crown will be seen from within the church. The change in geometry is accommodated by rotation of the three hinges, with a consequent drop of the crown of the vault. There is, however, a severe geometrical constraint on the intersecting barrel which runs north/south. A crack pattern which allows the vault to deform in more or less strain-free monolithic pieces is sketched in figs

5 Structural Repair and Maintenance of Historical Buildings 331 7(b) and (c). Cracks have opened in the vault at a distance of a metre or so from the north and south walls (usually containing windows); these cracks are called Sabouret's cracks in Pol Abraham's taxonomy. In addition, cracks will probably open adjacent to the walls, and both wall cracks and Sabouret's cracks are sketched in fig. 6. Thus cracks near the crown are traces of hinge lines in a portion of the vault through which compressive forces are being carried, the forces in fact acting perpendicular to the hinge lines. By contrast the wall cracks and Sabouret's cracks represent complete separation of the masonry, through which a hand may often be passed. No forces can be transmitted across such fissures, and the compressive forces run parallel to the cracks. A wall crack and a Sabouret crack effectively isolate a portion of the north/south barrel, and this portion will then be free to act as a simple arch running east/west, and spanning between adjacent vaulting conoids. In principle similar cracks might develop in a direction at right angles, in the east/west barrel. However, successive bays of the vault buttress each other in the east/west direction, preventing the movement which would lead tofissures.at the west end of a cathedral the westwork is usually sufficiently massive to maintain the last bay of the vault in position, but the east end of the church may be of less robust construction. In this case a wall crack in the main vault will be seen above the east window, again with complete separation of the fabric. Thus the structural action of a vaulting bay may be envisaged as in the sketch offig.8. The edges of the vault may be thought of as two-dimensional arches spanning between the vaulting conoids, and may in many cases have been transformed into such arches, if Sabouret's cracks are visible. Whether visible or not, this is the potentially most severe condition for the vault edges, where the spans are greatest. The central quadripartite portion of the vault in fig. 8 has progressively smaller dimensions as the centre is neared; it is the edges of the vault which are critical. SUPERIMPOSED LOADS ON VAULTING Calculations will be made for a vault of typical dimensions, having bay size 12 m x 5 m and thickness 300 mm. Such a vault has a weight (in sandstone or limestone) of about 5 kn/m\ and it is this self-weight which gives rise to the primary vault forces. Above the vault the great timber trusses, together with their boarding, might weigh 20 kn for each metre run of the nave or choir, and lead sheeting might contribute a further 10 kn. Together, then, these weights are equivalent to about 3 kn/rn^ above the masonry vault, and it is this weight which would be imposed on the vault if the roof should collapse during a fire. If

6 332 Structural Repair and Maintenance of Historical Buildings the imposition were uniform then the vault would not be distressed. The mere increase of a load of 5 kn/m^ to 8 kn/rn^ will have no effect on the geometry of the thrust lines within the masonry, and there will be no danger of hinging collapse. The same conclusion is valid whatever the increase of load (at least, up to a very large value, when crushing of the stone might result), so that a dynamic collapse of the timber roof, leading to larger effective forces, could be accommodated provided still that the loading were uniform. The superimposed loading will, of course, not be uniform. Accordingly, the carrying capacity of the vault under other load distributions must be investigated. In fig. 9, for example, the portion ABC of the vault adjacent to the south wall is critical, and a study may be made of this arch considered in isolation. The calculations can be made reasonably quickly by hand, but the arch was in fact modelled on a computer, and fig. 10 shows a typical result. The most severe loading on the arch occurs when a point load acts at about quarter span, and fig. 10 shows the condition at which the thrust line can only just be contained within the masonry. Figure 11 gives a sketch of the incipient collapse mode. The value of the point load which will just cause this collapse is 7.7 kn for an arch of width 1 m. Now the portion of roof above AB as infig.9 weighs 3 kn/nf and acts on an area 2.5 m x 1 m; if this load of 7.5 kn were to fall near the midpoint of AB, then collapse would probably occur. On the other hand any "balancing" load on portion BC will markedly improve the strength of the arch. A similar calculation may be made for portion DEF of the vault, treating the structure as an arch (of 1 m width) running north/south. The computer-derived sketches are shown in figs. 12 and 13, and the value of point load necessary to cause collapse was calculated as 9.8 kn. In this case the roof above DE in fig. 9 covers an area of 6 m^, so that a weight of potentially 18 kn could fall at the quarter span of this arch. The effect of the weight of water on the back of a vault may be investigated in the same way. Figure 14 shows portion ABC of the vault subjected to the fluid pressure that would arise if the water level were flush with the top of the vault (i.e. the vaulting pockets were completely filled). The vault cannot sustain such pressure, the collapse mechanism being shown in fig. 15, where the haunches are forced inwards with a corresponding rise of the crown. Such collapse will occur before the vault is completely filled. An even more dangerous case arises if the loading is unbalanced. If the quarter EFAB of the vault in fig. 9 isfilledwith water, for example, then the pressure on the portion AB of the arch will not be balanced by a corresponding pressure on portion BC. The computer study indicates,first,that

7 Structural Repair and Maintenance of Historical Buildings 333 the portion ABC of the vault is more critical than DEF, and second, that collapse will occur when the water level is still about 1 m below the crown of the vault. DISCUSSION The analysis which led to figs. 8 and 9 is perhaps conservative. It is, however, the edges of a vault which are critical, and the treatment of these edges as quasi two-dimensional arches gives a good picture of the overall strength of the vault. One example has been used, of a simple quadripartite vault of typical overall dimensions 12 m x 5m and typical thickness 300 mm. Thinner vaults may be encountered; their strengths are in proportion to their unit weights, that is to their thicknesses if the material and overall dimensions are the same. The vault used for calculation was taken to have a level highest surface; slight doming of the vaulting severies will lead to increased strength. The calculations do not encourage optimism about the safety of a vault in the case offire.the calculated collapse loads are of the same magnitudes as the loads liable to be imposed by the fall the high roof, and dynamic effects could well lead to considerable damage. Further, punching phenomena have not been discussed - a heavy fall on a more or less localised area of the masonry will lead to a local hole in the vault. Further, the weight of water building up in the vaulting pockets represents a severe loading case. On the other hand, none of the damage which may occur need be widespread. In fig. 9 the dotted line parallel to ABC represents the position of Sabouret's crack, and isolates the weakest portion of the vault running in this direction. If Sabouret's crack or a wall crack is physically present, then water cannot build up in the vaulting pocket. If Sabouret's crack has not developed (or, more likely, has been repaired as a consequence of good maintenance of the cathedral), then the water pressure will build up slowly and lead to collapse of the arch ABC, but will not necessarily lead to spreading collapse of the whole vaulting system. On the contrary, the main portion of the vault could well remain intact. Similarly, a hole punched through the arch ABC by collapse of the high timber roof would lead to collapse of that arch, but leave the main severies of the vault untouched. A hole punched through a severy nearer the centre of the vault could well remain a stable hole. The masonry vault has a high intrinsic strength, and is well able to resist local damage; it is a fact, however, that the forces engendered by the catastrophes examined here are likely to exceed that basic strength.

8 334 Structural Repair and Maintenance of Historical Buildings Figure 2 Figure 3

9 Structural Repair and Maintenance of Historical Buildings 335 Figure 5 Figure 6

10 336 Structural Repair and Maintenance of Historical Buildings WALL CRACK HINGE LINE SABOURET'S CRACK n II -ii W (c) Figure 7 Figure 8

11 Structural Repair and Maintenance of Historical Buildings 337 Figure 9 Figure 10 Figure 11

12 338 Structural Repair and Maintenance of Historical Buildings Figure 12 Figure 13 Figure 14 Figure 15