Cornell University Library TH 153.B63 Dangerous structures; a handbook for prac 3 1924 004 017 699 Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924004017699 DANGEROUS STRUCTURES. DANGEROUS STRUCTURES a ftanttboolt for J^ractital JMtit. BY GEORGE H. BLAGROVE, AUTHOR OF 'SKOSIHG AND ITS APPLICATION,' ETC. LONDON: B. T. BATSFORD, 52, HIGH HOLBORN. 1892. ^ PREFACE. In the present volume the Author's object has been to deal with some of those awkward problems in building which demand prompt solution, to suggest ready means for getting over difficulties which frequently occur in practice, and to supply data from which efficient and at the same time economical remedies may be designed to counteract evils arising from structural defects. To this end, the results of some practical experience have been summed up and presented to the reader in a concise form, together with so much theoretical demonstration as may not be too abstruse for ready reference. Tables showing the ultimate strength of such structural accessories as lintels, beams, struts, and posts, have been supplied. These tables have been expressly calculated for the purpose, and as they have been carefully checked, it is hoped that they will prove useful and reliable as memo- randa for reference by surveyors, builders, and others, and will be conducive to economy of time. vi Preface. The work comprised in the following pages was first published in 1890, as a series of articles in the Building World, when it met with so much appreciation that it was thought advisable to embody it in its present more permanent form. All the matter in the book has been thoroughly revised for republication, and an index has been provided. The favourable reception accorded to the Author's handbook on ' Shoring and its Application,' * encourages him to hope that the present work will commend itself to the approval of practical men. * London : Crosby Lockwood and Son, 1887. CONTENTS. CHAPTER I. PAGE Foundations . . . . . . . . . . i CHAPTER II. Walls and Piers .. .. .. ..12 CHAPTER III. Roofs .. .. .. .. .. ..21 CHAPTER IV. Arches .. .. .. .. .. ..27 CHAPTER V. Lofty Structures .. .. .. •• 31 CHAPTER VI. Stone Lintels .. .. .. ..40 viii Contents. CHAPTER VII. PAGE Timber Beams •• 44 CHAPTER VIII. Ties •■ 55 CHAPTER IX. Struts .. 58 CHAPTER X. Theory of Shoring . . . . . . . . 69 DANGEROUS STRUCTURES. CHAPTER I. FOUNDATIONS. The surveyor who is called in to inspect a dangerous structure, frequently find himself in a position analogous to that of a medical man in attendance upon a patient who is " in a very bad way." Great issues may depend upon the diagnosis of the professional adviser, and the remedies which he may feel called upon to prescribe. But in one sense, the experiences of the surveyor are capable of a more fundamental application than those of the physician or surgeon ; for not only does it rest with him to mend or mar the ailing structure which he sees before him, but he may find in it examples of what to avoid in other fabrics which may be devised and carried out by him. Constitutional defects in buildings, unlike those of the human subject, can actually be prevented from recurring in future instances. Undoubtedly, defects in old structures are more fre- quently due to faulty foundations than to any other cause. The equal distribution of weights when building upon a site has not been in the past, and is not invariably now, attended to upon scientific principles. To many persons the arrangement of foundations according to the relative compressibility of the soil is almost a new B Dangerous Structures. doctrine. How frequently have we seen one portion of a building carried to a much greater height than the remainder, while yet the same spread of footings and concrete foundations is allowed to prevail throughout ? The results of such experimentalising must often be an unequal subsidence in the foundations, causing danger- ous dislocations in the superstructure. Time-honoured customs, thoughtlessly followed, are sometimes to blame for anomalies in construction which may be attended with disastrous results. For example, admitting what no sane man will deny, namely, that the weights of various parts of a structure should be distributed over proportionate areas of ground, why should the spread of footings be regulated by the thicknesses of walls, instead of by their height and consequent weight ? In reference to footings, it may be questioned, indeed, whether brick ones are of much utiHty when they project more than 4J inches. At a projection of 9 inches, a header ceases to have any tail under the wall, and when it has a tail of anything less than half its length, it seems incredible that the presure of the footing upon the foundation can represent any significant portion of the weight above. Cases have been known in which walls have sunk, and their footings have turned up, clearly proving that the latter were unable to force their way through the soil. Had the bricks in the footings been bad, they would probably have been broken across. Failing this, what could have prevented them from turning, except the adhesive power of the mortar? Clearly, not statical conditions, which are supposed to be independent of " friction and sticktion." When a wall has sunk, and its footings have turned up, it may be asked, what is best to be done ? First, then, let it be shored and needled up as necessary, and if it is to 'be raised and restored to its Foundations. former position, let that be done if feasible. The defec- tive footings should be taken out, a proper excavation should be made, and a solid bed of concrete should be formed beneath the wall, so that its surfj.ce should be level with the bottom of the footings. Finally, the foot- ings should be restored with brickwork in cement, when the concrete is set, and should be well pinned up under the wall. If the weight of the wall is considerable, it may now be asked, over what surface of concrete is it distri- buted ? For reasons stated, it will not be safe to assume that it is evenly distributed over the whole width of the footings, if these are executed in the ordinary way ; but we may assume that it is distributed over a width of \\ inches on each side of the wall. Ordinary concrete, made with six parts of aggregate to one of lias lime, may be safely loaded to the extent of 6 tons per super- ficial foot; and Portland cement concrete, made with seven parts of aggregate to one of cement, will stand from 8 to 10 tons. Of course the strength of concrete increases considerably after it has set for some months. In the case of a heavy structure, such, as a factory chimney, for instance, a better distribution of the weight can be ensured by increasing the depths of the footings, laying them in double or treble courses, or by diminish- ing the projections of their set-offs, so that they shall spread less rapidly than in ordinary cases ; but the best way to equalise the pressure upon the footings is by forming them with through slabs of Yorkshire stone. The practice of laying foo-tings dry and afterwards grout- ing them with liquid cement is, no doubt, a good one. Brick footings should be all headers, and connected with hoop-iron bond, tarred and sanded. The next point to be settled in restoring defective foundations is, bow wide the concrete should spread in B z Dangerous Structures. order to distribute the weight of the structure over d sufficient surface of soil ? This depends upon the nature of the soil. Other things being equal, it may generally be inferred that a bottom at a considerable depth is less liable to be shifty than one of the same nature near the surface. It is made compact, and retained in posi- tion by the pressure of the superincumbent earth. The lateral pressure of the surrounding soil will also fre- quently have considerable influence upon the foundation of a building. If there is a large building sunk to a greater depth in the vicinity, this will help to retain the earth on that side, so that subsidences will be more likely to occur in other places. A building in the south of London, close to the river side, was found to have sunk considerably on the side farthest from the river, the side towards the river having scarcely sunk at all. This was made evident by the directions of the cracks in the walls and arches of the building. The explanation was- that the embankment wall of the river retained the earth on that side, and prevented the foundations from subsiding. If the foundation is of clay, much will de- pend upon whether the stratum runs level or not. It has been noted that the Oxford and London blue clays slip at an inclination of one in ten, one in twelve being considered safe. With regard to the bearing power of different soils, most of the soils in and about London will safely carry 2\ tons per superficial foot; and for ordinary houses they are rarely required to carry mote. The blue till or clay of Glasgow has been stated to be as firm and solid as a rock ; and yet the Townsend chimney, at Port Dundas, which is built upon this s6il, without any bed of concrete, does not weigh more than 3-^ tons per superficial foot upon the foundation. The compact Foundations. 5 coarse gravel or ballast, which overlies the London clay for the space of about two miles on both sides of the Thames, may be loaded to the extent of 4 tons per superficial foot, or more. The chimney of the South Metropolitan Gas Company's works, weighs about .58 ton per foot upon its foundation. Tall chimneys afford a crucial test of the solidity of foundations, because the slightest inequality of subsidence would, with these structures, bring about the most disastrous results. The foundations ,of heavy structures should go, if possible, to greater depths than those of lighter sur- rounding buildings ; but if a building is not very heavy, it is sometimes undesirable to carry its foundations too far down. There may be a thin stratum of clay near the surface, which is capable of bearing the required weight, and beneath this there may be a softer soil, quite unsuit-, able for a foundation. Once pierce the thin stratum of clay, and you may be compelled to go down another 20 feet or 30 feet in order to reach a sufficiently firm bottom. One can tell whether the stratum is thick enough to carry an ordinary building, by sounding it with an iron bar or length of piping, or a wooden pole. The. soil is sounded by bumping the end of the pole or bar, held vertically upon the earth. A dull, heavy thud be- tokens a solid bottom, while a slightly prolonged " ploshy" sound indicates a soft soil beneath. Practice accustoms the ear to judging of these things. If one portion of an old building is found to be sinking more than another, this may often be remedied by putting in a thick bed of concrete under the footings, The concrete, when set, will act as a species of lintelj bridging over the softer portions of the site into which the footings would otherwise sink. This should remind us that a concrete foundation is more or less subject to 6 Dangerous Structures. a transverse strain, and it raises the question, how thick must the concrete be in proportion to its projection be- yond the footings of the wall ? It is obvious that if we place a very thin bed of concrete beneath a very heavy wall, making the concrete project considerably beyond the footings of the wall, the wall may sink with the concrete immediately under it, the concrete at the sides being broken off, and left behind in the downward pro- gress of the structure. If the concrete had been thicker, it would not have broken, because the weight of the wall would have been distributed over the entire width of the concrete. The general practice is to proportion the thickness of the concrete to the height, and conse- quent weight of the structure. In building factory chimneys, a thickness of i foot 6 inches is adopted for shafts up to 40 feet high, an extra 4 inches of thickness being added for every additional 10 feet of height. But this is a mere arbitrary rule. The true question is, what is the slope or angle at which the weight can be dis- tributed downwards ? We know what it is supposed to be in the case of ordinary footings, where the projection is to the depth as three to four, which gives a slope rather steeper than 60°. I am inclined to think that the slope should be quite as steep as this ; and of course the steeper it is the better. Let us now see how this would apply in practice. Let Fig. i be a section through the footings of a wall which is to have a concrete foundation. We will assume that we know the weight of the wall and all that it carries, and that we also know how many tons per superficial foot the soil upon which we are building will safely bear. This enables us to fix the width of our bed of concrete, which we set off centrally with the wall. The weight is supposed to be distributed first over the width of the footings, and from Foundations. thence over the width of the concrete. But it certainly is not safe to assume that the whole weight is distributed evenly over the width of the footings, and it will be better to leave the footings out of consideration ; there- Fig. I. fore, draw the dotted lines from each face of the wall to the base of the footings, and from thence drav/ the dotted lines sloping outwards at the required angle, as ■shown, and these latter will determine the thickness required for the bed of concrete. If an old wall, which is beginning to sink, has concrete already under its foot- ings, another and wider . bed of concrete may be intro- duced underneath, either throughout the entire length of the wall, or only at places where the soil shows a tendency to subside. If new concrete is to be put in throughout the entire length of the wall, it will have to be done piecemeal, in lengths of from three to four feet at a time ; and it will be best to commence operations near the middle of the wall, which is, of course, greatly supported by the adjacent parts, and proceed outwards 8 Dangerous Structures. towards the quoins or extremities of the building, the whole of which should previously be shored up. Everybody is familiar with the expedient of putting in inverted arches to equalise the pressure upon the founda- tions of a building. An ordinary application of this is in the basement storey of a high house, the piers between the windows pressing only upon a small portion of the foundation, and having to support the whole of the in- termediate brickwork, except a small portion in the base- ment storey. The piers with their load would naturally sink, while those portions of the foundation between them, being only weighted with the brickwork around the basement windows, would scarcely sink at all, the result being the partial crushing of that brickwork, and the breaking of the continuity of the footings. To pre- vent this, inverted arches are introduced between the piers, which have the effect of distributing the pressure of the piers equally upon the foundation. The thrust of each arch is counteracted by the equal thrust of a similar arch opposed to it, until we come to the angle pier of the building. The pressure of the earth against this pier may suffice to counteract the thrust exerted by the arch against it, but by rights the pier itself should be sufficiently massive to overcome the thrust by its own statical resistance. Otherwise, the lower part of the pier may be pushed out of the perpendicular, and the struc- ture may become dangerous. Let us suppose that an angle pier, under such circumstances, is being pushed out of the perpendicular, through the portion of the arch next to it having become partially flattened, so as to thrust the pier out ; what, it may be asked, should be done? First of all, the necessary shoring should be done, the upper part of the angle pier being upheld by needling, so that its lower part can be rebuilt. The Foundations. pier and the defective arch can then be reconstructed ; but how are we to prevent a recurrence of the disaster ? There is no doubt that a sufficiently deep bed of con- crete under the footings would act as a lintel, and do the work that would otherwise be done by the inverted arches. Another Way would be to increase the statical resistance of the angle pier by adding to its weight, just as the mediaeval builders increased the resistance of their buttresses by weighting them with pinnacles. It might be inconvenient as well as expensive to carry up an erection in the nature of a chimney stack above the Fig. 2. ^ pier merely for the purpose of adding to its weight. The best thing to do would be to increase the width of the pier. This, which, probably, need only be done in the basement storey, would not only increase the weight of the pier, but would reduce the span of the arch in pro- portion to its rise, thereby giving the thrust more of an upward direction, as well as shifting the centre of the arch inwards. If it should be impracticable to widen the pier. Fig. 2 shows a suggestion for meeting the difficulty, the original form of the arch being indicated by dotted lo Dangerous Structures < lines. There is yet another expedient for preventing the spread of inverted arches, and that is by the intro- duction of iron, tie-rods, passing right through the brick piers, and being secured to them by means of large iron washers. These rods should be inserted at the springing of the arches ; and they furnish us with a ready means of checking the spread of a dangerous movement which has commenced. Old buildings are found in some parts of London having horizontal balks of timber laid beneath the foot- ings of their walls. These timbers, as might have been anticipated, become rotten through being saturated with wet, and allow the walls above them to subside, in many cases unequally. In such cases, when the mischief has not gone too far, and when the building is worth saving, the whole structure should be shored, and the timber cut out in lengths of two or three feet at a time, and replaced by good concrete. These lengths of concrete should be got in as soon as possible, with intervals be- tween them,, and the concrete should be allowed to set before the intermediate portions are disturbed. As for the amount of timber which can with safety be removed at a time, that will depend upon the condition of the brickwork above, and over how wide a span it is capable of sustaining itself without support. It may be useful, in similar cases, to cut out the timber in lengths of a foot, or eighteen inches, at intervals of two or three feet or more, care- fully to wedge up to the footings with slabs of stone where the portions have been removed, then to remove the intermediate portions of timber, and finally to fill in the excavation with concrete, the slabs of stone being allowed to remain, and becoming imbedded in the con- crete. In all such operations, when a wall has partially sunk, the tendency of concrete to expand in setting Foundations. 1 1 may be of service, in conjunction with shoring and other means to be noticed hereafter, to assist in restoring the wall to its proper position. A mixture of hydraulic and common lime will cause an expansion during the process of slaking, which frequently does not complete itself until after the concrete is laid. Blue lias concrete will, in hot weather, expand as much as one-thirtieth of its bulk, and in frosty weather, about one-fiftieth. Perhaps there will be no harm in protesting here against the practice of throwing in concrete from a height. The practice has been repeatedly condemned before, and yet many builders persist in following it. It ought to be evident that when ingredients of different specific gravi- ties are mixed, the effect of gravitation upon them must be to send the heavier constituents to the bottom ; there- fore, the less concrete is exposed to the effect of gravi- tation before it is set, the less liable is it to become unmixed by the precipitation of the heavier portions of its aggregate. 12 Dangerous Structures. CHAPTER II. WALLS AND PIERS. The conditions which govern the strength respectively of walls and isolated piers are somewhat different. If the brickwork or masonry of a wall does not crack imme- diately under a point where great vertical pressure is applied, it is improbable that any failure will occur lower down. To explain this, let us suppose the common case of a heavily-loaded girder whose end is built into a wall, without any sufficient stone template under it to distribute the pressure over the requisite extent of brickwork. The probability is that the brickwork immediately under the end of the girder will crack, the cracks extending out- wards and downwards, and gradually dying away. The bonding of the brickwork enables the pressure to distri- bute itself over an area of walling which becomes wider and wider as we go further down ; so that if we can only provide for a sufficient distribution of pressure immediately under the girder, we may set our minds at rest regarding the safety of the brickwork down to the footings. Apart from this spreading tendency of the lines of pressure, any quantity of brickwork forming part of a continuous wall is stronger than the same quantity in an isolated pier, because in the former case it is sustained by the lateral pressure of the surrounding work. The importance of this lateral sustaining power is often very great in old walls, which may stand for indefinite periods if left alone, but which often crumble rapidly away when a Walls and Piers. 1 3 few bricks are hacked out, or even when the pointing is tlisturbed. When an isolated pier has been overloaded, the case is altogether different, and, it may be said, much more dangerous than that of a wall. The pressure cannot extend itself over a wider mass of brickwork, but it is augmented, as we go lower down, by the weight of the pier itself, so that cracking or crushing is most likely to occur near the footings. Hence, the work in sight may be perfectly sound, while the part that is failing may be out of view and difficult of access. When a pier is over- loaded, it will probably give warning by sinking. If the sinking takes place equally, that is, without deviation from the vertical, there is reasoii to hope that it may be due only to a subsidence of the foundation, but if unequally, it may be due to a partial crushing of the brickwork. In either case, it will be advisable to relieve the pier of its load. The entire load should be sustained by shoring of sufficient strength, resting upon foundations which will not be disturbed by the necessary excavations. A few light raking shores may be necessary to sustain the pier while the earth around it is being dug out to admit of an inspection of its lower part. Probably the entire pier will have to be rebuilt or replaced by an iron column. A safe limit of 8 tons per superficial foot is often quoted in engineers' tables for the load upon good stock brick- work in mortar, and 10 tons per foot is not too much for the best Staffordshire brickwork in cement. The greatest pressure upon the brickwork in the Townsend chimney at Glasgow is 8^ tons per foot, and that upon the South Metropolitan Gas Company's chimney about "j^ tons. A common stock brick commences cracking under a pressure of 10 tons, and as the area of a brick is little less than one-third of a foot, this does not allow much more than 14 Dangerous Structures. 30 tons per foot for the ultimate strength. Taking the safe load at one- fourth of that required to produce fracture, it amounts to from 7^ to 8 tons per foot. Doubtless the mortar-joints form an element of weakness, but from experiments upon brick .piers not exceeding twelve diameters in height, the load required to produce fracture has been proved to be 29 tons per superficial foot upon brickwork in mortar, and 30 tons upon brickwork in cement. As regards the strength of Staffordshire bricks tested singly, the common ones are about one-and-a-half times as strong, and the pressed ones about twice as strong, as ordinary stocks. But a great deal of the brick- work in and about London jis far inferior to that built with good stocks and good mortar. Not only are the bricks used of a bad quality, being soft and friable, but their irregular shapes conduce to fracture by concentrat- ing the pressure upon a few points only of their surface. Added to thjs, the bad mortar used with them does not fuliil its purpose in binding the work together, or in equalising the pressure upon the beds, because the greater portion of it, being improperly compounded, has not the property of setting, but becomes and remains a dry crumbling mass. Brickwork of this kind will not safely carry more than 5 tons per superficial foot, and when loaded, as it occasionally is, to the extent of 12 tons, symptoms of danger soon begin to display themselves. As regards the strength of good masonry, we know that the Portland stone piers in St. Paul's Cathedral carry 17 tons per superficial foot; the red sandstone pillar in Elgin Chapter House, rather more; and the coursed rubble piers in St. Peter's, Rome, about 15 tons. From experiments upon stone piers about twelve diameters high, it is found that rubble masonry, flat bedded, in lias lime and sand, begins to crack at 40 tons per foot, while Walls and Piers. similar work with irregularly-shaped stones shows fractujre at only 15 tons. The latter is about half as strong as good brickwork in mortar, and the former possesses about the same strength as the best brickwork in cement. In repairing dangerous structures, the strength of stone templates and bedplates becomes a matter of great importance. It is of the highest importance that a stone template shall lie upon a level bed, and that that bed shall continue to be level. Suppose, for example, that the end of a girder has been built into a wall without a proper template, and that the brickwork beneath it has become cracked or partially crushed. The girder, with its load, must be supported by shoring, the defective brickwork cut out and made good with new work in cement, and a proper template pinned in under the end of the girder. Suppose, now, that the weight is allowed to come upon the template before the cement in the new work has set. The effect will be to compress and squeeze out the cement under the centre of the girder, jamming the bricks closer together under the middle of the tem- plate, so that the latter will lie in a hollow bed and be deflected like a beam. A very slight deflection of this kind will cause the stone to crack from its bed upwards. Of course the weight should be sustained by shoring until the cement is thoroughly set. Another danger to which the stone is exposed arises from the deflection of the girder, which, by concentrating the pressure ujjon the front edge of the stone, may cause it to split or spall at that particular place. It may be urged that, unless the girder is cambered, some deflection, however slight, is sure to take place, and, in view of this, it might be well to give the upper surface of the stone a slight inclination towards the span ; but at any rate, the worst evil may be 1 6 Dangerous Structures. obviated by rounding the front edge of the stone, as shown in Fig. 3, for then, even if the pressure should be concentrated in one place, yet the line of pressure, being normal to the curve of deflection, is sure to fall within the section of the stone. Before putting in a tem- plate, the pressure which is to come upon it should be cal- culated, which may be done by means of Very rough measurements of the walls, floors, and roofing sustained by the girder, reckoning all walls at one cwt. per cubic foot, ordinary floors at 140 lb., and roofing at 50 lb. per square foot, the latter including all allowance for exceptional stress of weather at times. Having ascertained the maximum pressure brought to bear upon one end of the girder, the next point to consider is the superficial extent of its bearing. Some girders are very narrow in the flange in proportion to their depth and consequent load. Suppose the bearing of the girder to be 12 inches, and the width of its bottom flange 6 inches, then it will have to rest upon half a superficial foot of stone. Suppose the entire load upon the girder to be 60 tons, then 30 tons of this will be thrown upon this half superficial foot of surface. In Yorkshire stone, fracture commences under a pressure of 100 tons per foot, and it' would certainly be imprudent to place 30 tons upon a surface where only 50 tons would cause fracture. One-fourth of the fracturing pressure would be a safe limit, and therefore granite, which will resist 300 tons per foot, would be amply strong for the purpose; or if Yorkshire stone were used, the girder should rest imme- Walls and Piers. 1 7 diately upon a cast-iron bed-plate, i foot by i foot 3 inches, which would distribute the pressure over i^ super- ficial foot of stone. The size of the template will, of course, depend upon the number of tons per foot the brickwork below is capable of sustaining. Upon average good brickwork, 2 feet by 3 feet will probably suffice ; but of course . it would be preferable for the bearing of the girder to extend over the whole width of the stone measured from the face. With regard to the depth of a stone template it could be wished, in the interest of sound building, that prevailing customs were less parsimonious. We should like to see the depth of such a stone not less than half its length. This, however, would certainly be considered extravagant, and we must usually content ourselves with a depth of one-third of the length — less than which ought not to be sanctioned. We can only hope that no stone of such dimensions will ever be subjected to a transverse strain. It would appear that a stone about 12 inches square and 4 inches thick, if supported only at its edges, would not stand much more than half a ton upon its centre. The transverse strength varies as the square of the depth, so that a 6-inch stone is more than double as strong as a 4-inch one. With regard to the stability of walls and piers, this depends on a simple application of the theory of the bent lever. In Fig. 4, let A B D E represent a pier which is going to be overturned upon its angle E by a hori- zontal force P. The weight of the pier is conveniently assumed to be concentrated at its centre of gravity C, and in overturning the pier, its weight has to be lifted by leverage. The fulcrum of the lever is at E, The length of the long arm of the lever must of course be measured at right angles to the direction of the power P, so that it is equal to the vertical height of P above the fulgrum C i8 Dangerous Structures. E. The weight of the pier acting vertically, the short arm of the lever is the horizontal distance between the direction of the weight C F and the point E. The moment of stability is the weight multiplied by the distance F E, and the moment of the thrust is the power P, multiplied by its height — the forces and distances being, in both cases, expressed in the same terms. So long as the moment of the power P is less than the moment of stability, the pier will not be overturned, and Fig. 4. if P should cease to act, the pier would return to equi- librium, so that F E would be equal to half D E. Thus we see why the momentof stability of a wall is ordinarily stated to be equal to its weight multiplied by half its thickness, and also we see why, when P acts lower down, it has less effect upon the stability of the pier. When the diagonal A E is vertical, F E is nothing, and the pier has no stability. The , question, at what incHnation from the vertical does a wall or pier become unstable ? Walls and Piers. 19 is therefore unmeaning, for it is obviously possible for a thick wall or massive pier to lean over considerably, and yet be stable, while a wall or pier of attenuated proportions may become unstable at a very slight in- clination. When a wall inclines from the vertical, the first thing to do, in judging whether it is dangerous or not, is to ascertain if all its joints are in a sound condition. If there is no danger of the brickwork or masonry sliding upon its beds, the wall may be safe. The next thing is to ascertain whether the centre of gravity of the wall's section overhangs its base. If it does not, the wall may be safe. Next, we should endeavour, approximately, to ascertain the moments of any thrusts tending to overturn the wall, and the counterpoising effect of cross or end- walls bonded into it, carefully noting, also, whether there is any tendency to draw the bond, such as may be in- dicated by the rupture of wall-paper or plaster. When all these circumstances have been duly considered, the surveyor should be prepared to decide whether the wall must be rebuilt, or whether it will suffice to erect buttresses against it Setting aside the question of thrusts, let us assume the case of a 9-inch boundary wall, 12 feet high, and 20 feet long, the top of which projects 6 inches beyond its base. Its weight is just 9 tons, and, if it were vertical, its moment of stability would be 180 X f = 80 foot-cwts. As it is, the centre of gravity is only ij inch instead of 4J inches within the fulcrum, the moment of stability being only 180 X \ = 22^ foot-cwts." To restore the original stability of the wall, we must increase its moment by 5 7 J, or say 60 in round numbers. This could be done by means of four buttresses, spaced equidistantly, and each having a moment of 15. Suppose each buttress c 2 20 Dangerous Structures. to be carried to the full height of the wall, to be of triangular form in side elevation, and projecting 2 feet at the base, so that its centre of gravity would be at I foot 2 inches measured from its extreme projection inwards; then its weight would have to be 13 cwt., which would necessitate its being i foot 2 inches wide. The amount of brickwork in these buttresses would be just over one-fourth of that contained in the wall. In calculating the stability of a wall which will enable it to resist overturn, we of course assume the existence of an open joint at or near the base. As a matter of fact, any thrust or wind-pressure which would overturn a wall would have to overcome the cohesive strength of the mortar in the lowest joint. ( 21 ) CHAPTER III. ROOFS. When a roof threatens to fail by thrusting out the feet of its rafters, it is convenient to calculate the amount and direction of the thrust, in order that the danger may be arrested in the most economical way. The thrust of one-half of a roof is estimated, as shown in Fig. 5, where A is the apex, and B the foot of the rafter, which rests upon a wall or pier. The latter has to sustain half the weight of the entire roof, and the stresses which come upon it. Draw A C vertically, and C B horizontally. If we assume the weight and the stresses of wind, snow, &c., to be concentrated at the apex, and to act vertically, then let A C represent one-half of the entire load on the roof, and C B will proportionately represent the hori- zontal thrust acting against B. This calculation will do if we have to estimate the effect of a heavy skylight or spirelet erected at the apex of the roof, but in ordinary cases the weight of the roof and the stresses upon it must be considered to be distributed over its entire surface, and for convenience in calculation, they may be assumed to be concentrated at the centre of gravity of its half- section on each side. In Fig. 5 let the point G represent the centre of gravity of this semi-section of the roof. Let the rise A C be 40 feet, the half-span C B 30 feet, and then the length of the rafter A B will be 50 feet. Assuming the roof- trusses to be 10 feet apart, then the superficial extent of roofing carried upon each half-truss 2 2 Dangerous Structures. will be 500 feet. Taking the weight of the roofing, in- cluding the stress of wind and snow as 50 lb. per square foot, we have about 1 1 tons for the total weight to be sustained at B. This weight we assume to be concen- trated at G. The pressure of A B against the opposite rafter, where both meet at the apex, must be horizontal ; therefore draw A D horizontally to represent the direc- tion of this force. The horizontal thrust at B, in the opposite direction, must be equal to this ; that is to say, either may be represented by half C B, if A C repre- sent the weight of n tons. Now draw the vertical through G, from D, to cut C B in E ; and D E, which is equal to A C, represents, in magnitude and position, the weight of 1 1 tons acting upon B, while E B, equal to A D, or to half C B, represents the horizontal thrust against B, which is just equal to A,\ tons. The forces acting upon B are therefore two, viz., the downward force D E, and the outward E B. These two forces, compounded together, produce the diagonal thrust repre- Roofs. 23 sented by joining D B. This line D B, produced beyond B, shows the angle at which a shore should be placed in order to resist the thrust of the roof, supposing that the static resistance of the wall were not to be taken into account. ■ Let us assume that the wall is of stone, 3 feet thick, and that the point B is 18 feet above the ground. We should then have 540 cubic feet of stone to each roof-truss. But the spaces occupied by windows ought to be deducted from this, besides which, if the thrusts are concentrated at the trusses, the intermediate walling can be of little service in resisting it. Probably we should not be safe in assuming that the resistance ex- ceeds that of a pier 5 feet wide at each truss. The weight of such a pier, taken at 140 lb. per cubic foot, would be i6f tons, and its moment of resistance would be i6f X ig = 25^1^ foot-tons. The moment of the thrust E B would be 4|- x 18 = 74 foot-tons. Hence we see that, unless the roof were tied, it would overturn its abutments. It may be asked, how such a structure could be sup- posed to stand at all. But it must be remembered that the allowance made for stress of weather applies only to exceptional periods, and that at ordinary times the moment of the thrust would probably not exceed one- third, or about the same as the moment of resistance. In overturning the wall it would also have to overcome the adhesive . resistance of the mortar, which would probably have a moment of 12 foot-tons. Such a structure might, therefore, stand for years in a sheltered situation. But if the mortar near the bottom of the walls should have become partially loosened, and if a violent gale of wind should suddenly occur, the whole might collapse. 24 Dangerous Structures. Fig. 6. Having ascertained the magnitude and direction of the thrust D B, Fig. 5, and the weight of the pier at B, let us now find the resultant of these two forces. In Fig. 6, let the triangle D E B be redrawn precisely as in Fig. 5, so that D E shall represent the weight of II tons, and E B the thrust of 4-J- tons, the diagonal thrust D B being about equal to i if tons. Draw the vertical F B and make it proportionately represent the weight of the pier, viz., i6f tons. To find the resultant of F B and D B, com- plete parallelogram H F D B, and join H B, which represents the re- quired resultant. I his line, H B, produced below B, shows the direc- tion in which a raking shore should be placed, in order to assist the pier to withstand the thrust of the roof A raking shore or tempyorary but- tress resists by its direction, simply conveying the thrust to the ground, its own weight being practically in- appreciable. A permanent buttress of stone or brick resists by means 01 its weight, as explained in the pre- ' vious chapter (p. 19), but it can also resist by means of its direction, if it be of sufficient projection to allow the line of diagonal thrust to fall within its outline. It would require a buttress of considerable size to neutralise the thrust of the roof in the case we have assumed. The moment of the thrust Roofs. 25 Fig- 7- being 74, and the moment of resistance of the pier only about 25, we should require a buttress having a moment of about 49 foot-tons, so that if its centre of gravity- projected 3 feet beyond the face of the wall it would have to weigh more than 16 tons, or to contain about as much masonry as the pier it is required to support. A buttress 2 feet 6 inches wide, projecting 6 feet, and carried back by set-offs to the top of the wall, would answer the purpose and be enormously extravagant, whereas a buttress projecting 3 feet and weighing 4 tons would contain the line of thrust, while its weight would make the angle of thrust still steeper. Such a buttress is shown in Fig. 7, and as it is assumed to be 2 feet 6 inches thick, its weight will be quite 4 tons. The di- agonal thrust H B (Fig. 6) having been set up in conjunction with a vertical line proportioned to this weight of 4 tons, and the parallelogram having been completed and its diagonal drawn, this latter represents the resultant, which falls at the angle indicated by the dotted line in Fig. 7, lying just within the thickness of the wall. It will be seen that if the buttress is properly bonded to the wall, we may consider both as one mass, so that if the whole should be overturned it would turn upon the outer face or toe of the buttress. First, then, we have the weight of the wall, or rather this portion of it, in- creased from i6|^ tons to 2o| tons, by means of the 4 tons added in the form of a buttress. Then we have the distance from the centre of gravity of the wall to the 26 Dangerous Structures, fulcrum increased from li^ to i^\ feet, so that the moment of resistance of the entire mass is 2of x 4^ = 93^ foot-tons. If the buttress were merely built against the wall with a straight joint, as is sometimes done, it would not be safe to calculate its stability as above, but the weight of the buttress would have to be sufficient to add its own separate stability to that of the wall, as explained in the previous chapter. A buttress is sometimes built against a wall without being bonded to it, because it is feared that the main structure, being much heavier than the buttress, might sink more, thus causing the bond to be broken. In a case such as we have assumed, when a roof has no ties, it is always prudent to intro- duce temporary ones while the structure is undergoing repair. Iron tie-rods offer a ready means of guarding against dangers arising from ill-constructed roofs. The practice of cambering tie-beams still prevails largely amongst carpenters, in spite of all that has been urged against it by Tredgold and others. When a cambered beam settles it has a tendency to thrust out' the walls. If there is danger, the centre of the beam should be temporarily supported from below, while iron tie-rods may be introduced from wall to wall, or buttresses "built outside. A wooden curved rib, which shows signs of splitting, may be strengthened by having a wrought iron plate bolted against its side. Cast iron has been used in such a case with success. It is more rigid than wrought iron, and quite capable of resisting any shearing stress likely to come upon it. CHAPTER IV. ARCHES. The method of ascertaining the direction of the thrust of a roof, as illustrated in Fig. 5, has been applied, in Hurst's edition of Tredgold's " Carpentry," to one-half of a polygonal roof-truss, on the assumption that each half of the truss would behave as one mass, both being, as it were, hinged together at the apex. The same prin- ciple can be applied to ascertaining the thrust of an arch. Thus, in Fig. 8, let A B CD represent one half of an arch and -the mass which it supports, the point G being the centre of gravity of the mass. Draw the vertical through G, as shown, and draw the horizontal line from C, meeting this vertical in E. From D, the springing, 28 Dangerous Structures. draw the horizontal line,- meeting the same vertical in F. Now, let E F, which is equal to the rise of the arch, represent the weight of the mass A B C D ; then E C will proportionately represent the horizontal thrust which each side of the arch exerts against the opposite one, and F D will represent the horizontal thrust of the arch against its abutment. In other words, let w be the weight of the mass, / the horizontal thrust against the abutment, r the rise of the arch, and d the distance of the centre of gravity of the mass from the abutment ; then as r : d : -.w. t, and t = w d In Fig. 8, if we join E D, then E D will represent the amount and direction of the thrust of the arch, which of course follows the tangent of the curve. This method affords us a ready means of calculating the thrust of an arch, and of deciding whether its abut- ments are sufficiently massive for security. The posi- tion of the centre of gravity of the mass can be fixed quite nearly enough by the eye ; indeed, if we generally assume d to be equal to one-fourth of the span, we shall not go far wrong, and shall err upon the safe side. A similar method may be pursued in ascertaining the thrust of a brick or stone vault. In the case of a semicircular vault, d would be equal to about one-seventh of the entire span. These methods are, of course, only approximate, but will be found sufficiently accurate for ordinary practice. A semicircular vault fails usually by opening at the soffit of the crown, and at the extrados, midway between the crown and the springing. These are, therefore, the Arches. 29 points at which a vault may be strengthened by cramp- ing, and the cramps should be inside at the crown, and outside at 45° above the springing. A horizontal tie-rod inserted at between four-sevenths and five-sevenths of the height from the springing to the crown may save a vault from failure. Ties lower than this may be useful, but rarely above it. An excellent way to strengthen a vaulted roof is to carry up the side walls to about 45° above the springing, and fill up the haunches with good concrete. The concrete should be filled in slowly, say to a height of about 2 feet each day, otherwise it may injure the vault by expanding. A semicircular arch, having a solid mass above it, which finishes at a level, like that shown in Fig. 8, is sub- ject to very simple conditions. According to Rankine, the horizontal thrust of one-half of such an arch is approximately equal to the weight supported between the Fig. 9. crown and that part of the soffit embraced by an angle of 45" ; that is to say, the thrust is nearly equal to the weight of the portion shaded in Fig. 9. But we must not forget that in most cases an arch does not have to sustain the weight of the brickwork or masonry above it for 30 Dangerous Structures. any considerable height, a large portion of the weight being transferred to the piers by a species of corbelling which necessarily results from the bonding of the wall. Therefore arches, like lintels, support little more than the triangular mass of brickwork which is immediately above them. Semicircular or eHptical arches have a tendency, when failing, to sink at the crown and rise at the haunches, while pointed arches usually tend to rise at the crown and to fall inwards, or become flattened at the haunches. Hence the importance, when shoring up an arch, of supporting its crown, if it belongs to the former class, and of strutting out the haunches, if it belongs to the latter class. ( 31 ) CHAPTER V. LOFTY STRUCTURES. Structures of considerable height in proportion to the superficial extent of their foundations offer some diffi- cult and delicate problems for solution when they show a tendency to lapse into a dangerous condition. A devia- tion from the vertical which, in the case of a house, may be of little consequence, may involve absolute destruc- tion in the case of a tall factory shaft. The effect of wind becomes a highly important consideration in such cases. In factory shafts, and even some stone spires, a visible deflection takes place under strong wind, which proves that there is a certain amount of flexibility about such materials as brick and stone. It has been stated that the Townsend chimney, at Glasgow, oscillates to the extent of 3 feet, that is to say, 18 inches on each side of the vertical. The chimney is 454 feet high, and the oscillations have been observed from a distance through a telescope, having a vertical thread or wire inside it. The amount of oscillation may be measured with tolerable accuracy in such cases by counting the number of bricks, or half bricks, which mark the varia- tion of any point from the vertical. If the oscillations take place equally on both sides of the vertical, the shaft may be considered safe, for it will maintain a vertical position in calm weather. A little oscillation of this kind is considered to be a good sign, the chimney behaving on the same principle as reeds, which bend to Dangerous Structures. the wind but do not break. If there were no oscillation at all, we should conclude that the wind-force must be expended somewhere, and that it would probably take effect at the weakest point, so that a sudden bend might occur some time, under exceptional stress of weather. If the oscillations occur in one direction only, or more in one than in the opposite direction, the chimney is certainly in a dangerous condition. If plumbed in calm weather, it will probably be found to deviate from the vertical ; and this deviation may be expected to increase, unless tneasures are taken for remedying it. In observing the oscillation of a chimney, it is advisable to notice whether it assumes a slight curve in departing from the vertical, or whether the greater portion remains straight, the shaft bending, so to speak, at one point. The question is an important one, and it requires some careful obser- vation to arrive at the truth. A curved deflection shows that the resistance is equally distributed, and the shaft structurally sound ; whereas, bending at one point only is a sure indication of weakness at that point. Some attention should be given to the leverage exerted by the wind upon the opposite side of a structure against which it blows. If we consider the wind-force to. be directed against a single point in the height of a shaft, we can then estimate its effect at any given point lower down on the opposite side. But the wind-force is distri- buted over the entire portion of the chimney exposed to its action, and this is equivalent to its being concen- trated at the centre of gravity of the surface acted upon. In the case of a parallel shaft, this point would be half way up the height, and in the case of a tapering shaft, this point would be a little lower. To find the height of the centre of gravity of the tapering surface, measure up two-thirds of the height of the shaft, and Lofty Structures. 33 divide the upper third into two parts, so that the upper part shall be to the lower as the diameter of the shaft at the base is to the diameter at the top. The point where this division is made is the centre of gravity required. Suppose a shaft to be 100 feet high, the diameter at the base being 10 feet, and that at the top 5 feet 10 inches. The diameters are as 12 to 7, and one-third of the height, or 33 feet 4 inches has to be divided into 1 9 parts, of which 7 must be added on to the lower 33 feet 4 inches, so that we have 12 feet 3 inches «,dded to 33 feet 4 inches — equal to 45 feet 7 inches — for the height of the centre of gravity above the base. The leverage with which the wind force acts upon any joint in the structure is represented by the height of this centre of gravity above such joint. The greatest wind-force acting against a vertical struc- ture in this country is not likely to exceed 40 lb. per square foot of surface, although as much as 55 lb. has been registered on rare occasions and in exposed situa- tions. Assuming the wind to act at 40 lb. per foot against a shaft of the above-named dimensions and square upon plan, the total wind-force expended against one face of the shaft would be a little over 10 tons. Acting upon a joint just 7 inches above the base, this force would have a moment of 450 foot-tons. Dividing this by the diameter at this point, we have 45 tons for the pressure brought to bear upon this joint by the wind, in addition to the weight already pressing upon it. At 25 feet above the base, the moment would be 200 foot-tons, and as the diameter at that point would be 7 feet 6 inches, the actual additional pressure caused by the wind would be 26f tons. By such calcu- lations we are enabled to ascertain the precise extent of the danger to which a lofty shaft may be exposed 34 Dangerous Structures. when any portion of its brickwork shows signs of being defective. If a shaft be circular or polygonal upon plan, the centre of gravity of its exposed surface may be ascer- tained by the method explained, only it is usual to reckon that the wind-force against a circular shaft is equal to half that against a square one of the same dia- meter. In the case of an octagonal shaft it is about two- thirds. As the amount of taper assumed in the fore- going calculations agrees with the proportions sanctioned for the construction of factory shafts under the Metro- politan Regulations, it may be useful to remember that the centre of gravity of the exposed surface in a shaft of that proportion is just -^^ of the height above the base. A shaft which inclines from the vertical may be restored to a safe position by the method adopted more than thirty years ago in the case of the Townsend chimney, which had been deflected by the wind as much as 7 feet 9 inches from the vertical, losing several feet in height, which were regained when it was straightened. The deflection commenced at about 100 feet above the base, and seems to have followed a curve. The remedy was to make a series of horizontal saw-cuts, at twelve suc- cessive heights, on the side opposite to the inclination, the shaft gradually returning to the vertical by its own weight. A shaft at Pitchcombe, Gloucestershire, being 132 feet high, was deflected 3 feet 10 inches from the vertical, about sixteen years ago. The shaft was octa- gonal, and the method adopted in straightening it was to cut out one course of bricks from five of the sides, and substitute a thinner course, the intervening spaces being filled with iron wedges. The work occupied about three weeks, and when all was ready, the wedges were gradu- ally withdrawn, and the shaft returned to the upright Lofty Structures. 35 by its own weight. In such operations it is important to be quite sure how much to take out in order to efifect the desired result ; for a slight error will make a con- siderable difference. For instance, in the Pitchcombe shaft, it was calculated that \ inch taken out from the height on the side opposite to the inclination, at 40 feet above the base, would make a difference of 7 inches in the deflection at the -top. The safest plan in such a case would be to measure the total height on the convex and concave sides of the deflection, respectively, when, of course, the difference between the two heights would be the amount to be taken out on the convex side. If such measurements cannot be taken, calculations are rather apt to be misleading. It might be supposed that when a shaft has bent at one point only, the inclination of the joint, at that point above the horizon, would be equal to the angle of deflection of the axis of the shaft. But this does not necessarily follow in practice. In fact, it is difficult to fix, with any accuracy, the point upon which a shaft will pivot in returning to the vertical. It is safer to rely upon experiment. The use of wedges, which can be gradually withdrawn, is decidedly to be recommended ; and when the wedges have been withdrawn sufficiently far, the work between them can be pinned up to the requisite height with tiles and cement, and the wedges may be entirely withdrawn when the cement is set, after which the spaces occupied by the wedges can be filled up with bricks or tiles and cement. One cause which conduces to the decay of all brick structures is the gradual crumbling of the mortar in the joints. This crumbling process may be postponed by pointing the work before it has been allowed to go too far, and there is no doubt that an external rendering of Portland cement is a great protection, especially to lofty structures such as factory shafts, D 3 36 Dangerous Structures. A shaft may incline from the vertical through a partial sinking of its foundations. When this has occurred to a sufficient extent to cause danger, the case is serious, as the shaft will probably have to be rebuilt. It should not be forgotten that the pressure of the wind against the shaft is transmitted to its foundation on the side oppo- site to that exposed to the wind. It may be asked, what amount of deflection is necessary to constitute danger ? We should say almost any deflection which is perceptible, it being manifest that the walls of an ordi- nary dwelling may safely lose stability to an extent that would be highly dangerous in a lofty shaft. The batter of 2\ inches to lo feet of height, and the half-brick set- offs usually made inside at every 20 feet in shafts, tend to throw the centre of gravity of each wall inwards, and when there is any tendency to fall over towards one side, that tendency is assisted by the opposite wall. The batter of a shaft is very shght, and its effect in diminishing the pressure of the wind is not worth con- sidering. With a spire, however, the case is different. The wind force, acting horizontally, must be resolved in a direction at right angles to the slope of the spire, and again resolved in a horizontal direction, in order to esti- mate its effect in tending to overturn the spire. To understand this, let a = the inclination of the slope from the vertical ; P = the original force of the wind, acting horizontally ; / = its resolved force, at right- angles to the slope ; and n = its final horizontal effect Then P is to /, as the secant of a is to the radius ; and / is to n, as the radius to the cosine of a. Hence, we need not find/ in order to find n because sec. a : cos. a:: P : «, and P. cos. a n = sec. a Lofty Structures. 37 The accompanying diagram, Fig. lo, illustrates this, the angle A C B representing the inclination of the slope from the vertical, and the triangle C D B being similar to C B A, C B being the hypothenuse of C D B, as C A is of C B A. Then, if C A be taken to represent the original wind-force, C B is proportioned to its effect at right angles to the slope, and C D is proportioned to its final effect. The general outline of any spire being triangular, the centre of gravity of any face of its surface is at one- third of the total height above the base. Let us assume that the wind blows with a pressure of 40 lb. per foot against one face of a square spire, 12 feet in diameter, and 30 feet high. Here 1:he measurement on the slope is just 30^ feet, the vertical height 30 feet, and the third proportional required, about 29 J feet. The final effect of the wind is therefore about 38-7 lb. per square foot of surface upon a vertical triangle, whose height is 30 feet and its base 12 feet. The superficial extent is 180 square feet, and the wind-force tending to overturn the spire is 3^ tons, which we must consider to be concen- trated at the centre of gravity of the exposed surface, 38 Dangerous Structures. namely, at lo feet above the base. The moment of this force is, therefore, 32 foot-tons, and it is capable of overturning a wooden spire weighing 5^ tons. If the spire were of stone, however, 6 inches thick, it could not weigh much less than 18 tons. With a circular or octagonal spire, the same rule will apply as in the case of shafts, namely, that the wind- pressure upon a circular spire is half, and upon an octa- gonal spire, two-thirds, of what it would be upon a square one of the same diameter. The thrust of a spire against the walls of the tower supporting it may be calculated on the same principle Fig. II. as the thrust of an ordinary roof. Whatever the form of the spire, provided its sides be straight, the thrust increases at a regular rate towards the base, so that if it Lofty Structures. 39 be sufficiently tied, or otherwise held together there, there need be little fear of its giving way higher up. The thrust at the base of a spire is considerably reduced by tying it at some point above. This will be under- stood from Fig. 11, which shows the half-sections of two spires. The right-hand half-section shows one side of a spire which has not been tied. The weight of any triangular strip of spire is, therefore, assumed to be con- centrated at its centre of gravity, namely, at one-third of the height, and the moment of the thrust is found by multiplying this weight by the horizontal distance of the centre of gravity of the section from the abutment. The direction of the thrust is also shown by a dotted slanting line. The section on the left side shows part of a spire which has been tied at a few feet above the abutment. This has the effect, as regards thrust, of concentrating the weight at the point where the tie occurs. Conse- quently the moment of the thrust is reduced in propor- tion to the proximity of this point to the abutment, while the direction of the thrust is made very much steeper, as is shown by the slanting dotted line. 40 Dangerous Structures. CHAPTER VI. STONE LINTELS. Stone lintels which have become cracked so far as to be dangerous can generally be safely and expeditiously restored. In such cases, the first question that suggests itself to us is, whether there is a relieving arch over the lintel, and if so, whether the arch derives its support independently of the lintel.^ If the span of the arch extends as far as the extremities of the lintel, the latter can be removed without in any way disturbing the arch. In such a case, supposing the arch and the work above it to be in a sound condition, we may firgt strut up the opening, then remove the brick or stone spandril filling from under the arch, and then remove the lintel. After that, we can put in a new lintel, make good the jambs upon which it rests, with brick or stone in cement, and lastly, restore the spandril filling. Of course, if the arch and work above it are in a dangerous state, the whole must be pulled down from above and restored from below. If there is no relieving arch, one may be in- serted before the lintel is removed. To do this, let the outline of the arch first be marked on with chalk. The work is then cut ofi", and the arch built in, bit by bit, after which the lintel is restored. Arches of considerable span may thus be put in without the aid of timber centering, the existing work below being afterwards removed, according to the exigencies of the case. Sometimes a relieving arch Stone Lintels. 41 cannot be put in, because the abutments are inadequate to resist its thrust. In such a case, it may be necessary to substitute an iron beam for a stone lintel. Sometimes the rather extravagant device is adopted of employing a stone lintel sufficiently deep to allow of a portion being cut out at the top, so as to form a skewback at each end to receive the arch. The lintel is thus made to serve as a tie to its own relieving arch. A stone lintel is ordinarily capable of withstanding all the tensile strain which it is supposed to receive under the circumstances, and which it would receive if the whole depth of the stone were subjected to tension at once. But the strain of the arch is thrown principally upon the upper part of the stone, which may have the effect of causing it to crack at the springings of the arch. . We cannot, there- fore, regard this method of construction as a safe one to adopt. Sometimes it is convenient to put in a pair of stone springers, and to connect these by means of an iron tie-rod, strong enough to resist the thrust of the arch. The tie should be galvanised to prevent rust. Before restoring a stone lintel we should, of course, feel certain that the new one is not likely to become cracked in the same manner as the original one. Such considerations may induce us to put in a stone of better quality or greater depth, or to substitute a beam of iron or of cement concrete containing a core of rolled _L or I iron imbedded in it. In calculating the strength of stone or other lintels, it is convenient to use the well-known formula applicable to beams : — s where b = the breadth and d the depth, in inches ; s = 42 Dangerous Structures. the span in feet ; c = z. coefficient of strength, dependent upon the nature of the material ; and W - the breaking weight. From the comparatively small transverse strength of stone, it is convenient to express W in cwts. From the results of experiments, quoted in Gwilt's " Encyclopedia of Architecture," we obtain the following values of c, the load being assumed to be evenly dis- tributed over the span, as it usually is in practice : Brown Portland stone, -535 ; white Portland stone, -4 ; burnt clay ballast concrete, -4 ; coke breeze concrete, •312; pit-gravel ballast concrete, -285; Box Ground Bath stone, -129 ; and Corsham Down Bath stone, •096. In the concrete lintels, it is assumed that the ingredients are in the proportion of one part of cement to four of aggregate, the sectional area of the iron core being about one-eleventh of that of the lintel. In arriving at the foregoing coefficients of strength, a considerable margin of safety has been allowed. Ac- cording to Gauthey, a French authority, the value of c would be -697 for hard limestones, and ■616 for soft limestones ; but the coefficients previously given are to be preferred, because experiments are usually made with selected materials, whereas materials ordinarily in use differ in quaUty. The following table shows the breaking weights of Portland stone lintels, the load being in each case distributed. The breath of the lintel is assumed to be 4J inches, which is the most ordinary dimension. Since the strength varies directly as the breadth, it will be easy to calculate the strength of similar lintels of greater or less breadth. It will be observed that the strength of the white Port- land lintels agrees with that of the burnt-clay ballast con- crete ones with an iron core of one-eleventh of the sectional area. The core may be enlarged for additional strength. Stone Lintels. Strength of Portland Stone Lintels. Span ill feet. Depth in inches. Breaking Weight in cwts. Brown Portland. White Portland. I 4 38 28.8 I* 6 S7-7 43-2 2 6 43-3 32-4 2* 8 61.6 46 3 9 65 48.6 1 31 12 3 I22i 4i ,. 4 67i Si .. 4 loof 7 .. 4 i63i 4i .. 4i 76 Si .. Si I38f 8 „ 4 2i3i 5 .. 3 63 6 ., 3 90 8 „ 6 3191 S .. 4 83i 6 „ 4 120 8 „ 8 • 426i Span 7 feet. S by 3 S4 6 by 3 77* 8 by 6 274i 5 .. 4 7if 6 „ 4 I02| 8 .. 8 36SI S .. S 89i 6 „ 6 iS4i 9 .. 3 i73i Si .. 3 64! 7 >. 3 los 9 „ 4i 26of Si ., 4 86i 7 ,. 4 140 9 ,, 6 347 Si ., Si 97i 8 „ 4 182J 9 ,. 9 520I Span 8 feet. 5 by 4 62i 8 by 6 240 10 by 6 37S S ,. S 78* 8 „ 8 320 10 , , 10 62s Si .. 4 7Si 9 ,. 4i Z27f , 4 302i Si .. Si 96i 9 >. 6 3°3l . Si 416 6 „ 4 90 9 .. 9 4SSI . 7 S291 6 „ 6 I3S 10 „ 4 250 . 8 605 8 „ 4 i6o 10 „ S 3i2i , 11 8311 Timber Beams. 49 Strength of Fir Beams. — continued. Scantling in inches. Breaking Weight Scantling in inches. Breaking Weight Scantling in inches. Breaking Weight Span 9 feet. S* by 4 67 9 by 6 270 II by 7 47oi Si .. Si 8SI 9 ,. 9 405 II . 8 S38 6 „ 4 8o 10 ,, 4 222i II , 11 739i 6 „ 6 I20 10 „ 5 277I 12 , 4 320 8 „ 4 I42J 10 ,, 6 333i 12 , 6 480 8 „ 6 2I3I 10 „ 10 SSSi 12 , 8 640 8 „ 8 284J II .. 4 269 12 , 9 720 9 ,, 4i 202J II „ Si 369I 12 , 12 960 Span i< 3 feetj Si by 4 6oi 9 by 6 243 11 b] ' 7 403 Si » Si 77i 9 ,. 9 364i II > 8 484 6 „ 4 72 10 „ 4 200 II , II 64s 6 „ 6 108 10 ., S 250 12 , 4 288 8 „ 4 128 10 „ 6 300 12 , 6 432 8 „ 6 192 10 „ 10 Soo 12 , 8 S76 8 „ 8 256 II ., 4 242 12 , 9 648 9 .. 4i l82i II „ Si 322I 12 , 12 864 50 Dangerous Structures, Strength of Fir Beams — continued. Scantling in inches. Breaking Weight in cwt. Scantling in inches. Breaking Weight in cwt. Scantling in inches. Breaking Weight in cwt. Span II feet. 6 by 6 98i 10 by 10 454I 12 by 9 S88i 8 „ 6 174* II „ SJ 293 12 „ 12 7851 8 „ 8 232I II „ 7 36&f 13 ., 5 3841 9 „ 6 221 II ., 8 440 13 .. 6. 461 9 .. 9 331J II „ II 586 13 .. 7 5371 10 „ 5 2271 12 „ 6 392! 13 .. 8 6i4f lo „ 6 272! 12 „ 8 523! 13 „ 13 998f Span 12 feet. 6 by 6 90 n by 7 3351 13 by 6 422| 8 „ 6 160 II , 8 403I 13 „ 7 492| 8 „ 8 2i3i II , II S37i 13 „ 8 S63i 9 .. 6 202i 12 , 6 360 13 >. 13 9154 9 >^ 9 303* 12 , 8 480 14 „ 6 490 10 ,, 6 250 12 , 9 540 14 ., 7 S7ii 10 ,, 10 4i6i 12 , 12 720 14 .. 9 735 II „ Si 2681 13 . S 352 14 » 14 1 143 J Timber Beams. 51 Strength of Fir '^Y.kM.^-^continued. Scantling in inches. Breaking Weight Scantling in inches. Breaking Weight Scantling in inches., Breaking Weight in cwt. Span 13 feet. 6 by 6 83 II by 7 310 13 by 7 4SS 8 „ 6 1471 II „ 8 354* 13 .. 8 520 8 „ 8 197 II .. II 496 13 ,> 13 84s 9 „ 6 i86i 12 ., 6 332i 14 „ 6 4S2i 9 ., 9 28oi 12 „ 8 443 14 5. 7 S27f 10 „ 6 230I 12 .. 9 498i 14 .. 9 678I 10 „ 10 384 12 ,. 12 664J 14 ,. 14 loSSi II ., Si 248 13 ,, 6 390 Span 14 feet. 8 by 6 137 II by 8 328J 13 by 7 422i 8 „ 8 182I II 5, II 452 13 „ 8 482§ 9 .. 6 173* 12 ,, 6 3o8i 13 >. 13 784 9 „ 9 26oi 12 ,. 8 411J 14 ,. 6 420 10 „ 6 213* 12 ., 9 462! 14 ,. 7 490 10 „ 10 356 12 „ 12 617 14 „ 9 630 II ., 7 287! 13 „ 6 362 14 .. 14 980 E 2 52 Dangerous Structures. Strength of Fir Beams — continued. Scantling in inches. Breaking Weight Scantling in inches. Breaking Weight in cwt. Scantling in inches. Breaking Weight in cwt. Span 15 feet. 8 by 6 128 II by 8 322i 13 by 7 394i 8 , 8 i7oi II „ II 430 13 ,. 8 45°^ 9 , 6 162 12 „ 6 288 13 =. 13 732I 9 . 9 243 12 „ 8 384 14 „ 6 392 10 , 6 200 12 „ 9 432 14 „ 7 457J 10 , 10 333i 12 „ 12 576 14 ,. 9 588 II . 7 268* 13 „ 6 338 14 ,. 14 9141- Span l6 feet. 8 by 8 160 12 by 9 40s 14 by 14 8S7§ 9 ,, 9 2271 12 „ 12 S40 IS .. nh S27i 10 „ 10 312* 13 » 8 422I 15 .. 9 632I II „ 8 302J 13 .. 13 6861 IS „ 10 703i II „ II 4iSf 14 .. 9 S5«i IS ,. IS ioS4i Span 17 feet. 8 by 8 iSof 12 by 9 3811 14 by 14 807 9 „ 9 2I4i 12 „ 12 So8i 15 .. 9 596 10 „ 10 294i 13 „ 8 396 IS .. IS 993i II „ 8 284! 13 .. 13 646 16 ,, 12 903i II „ II 39ii 14 » 9 S17 16 „ 16 I204i Timber Beams. 53 Strength of Fir Beams — continued. Scantling in inches. Breaking Weight Scantling in inches. Breaking Weight in cwt. Scantling in inches. Breaking Weight Span 1 8 feet. 8 by 8 I42i 12 by 9 360 14 by 14 762A 9 .. 9 202J 12 „ 12 480 15 .. 9 562! 10 ,, lO 277I 13 „ 8 375* 15 ,. 15 •937* II „ 8 269 13 ,. 13 6ioi 16 „ 12 853i II „ II 369I 14 >> 9 490 16 „ 16 1137I Span 19 feet. 9 by 9 1911 12 by 12 454i 15 by 9 S32| 10 „ 10 263i 13 ,, 8 356 IS ,. IS 888 II „ 8 254I 13 .. 13 578 16 „ 12 8o8i II „ II 339i 14 >, 9 464, 16 „ 16 1078 12 „ 9 341 14 .. 14 722 18 „ 18 iS34f Span 20 feet. 10 by 10 250 13 by 8 338 15 by- 15 843I II „ 8 242 13 ,. 13 S49i 16 „ 12 768 II „ II 322* 14 ,> 9 441 16 „ 16 1024 12 „ 9 324 14 ,. 14 686 18 „ 12 972 12 „ 12 432 IS .. 9 S06* 18 „ 18 1458 54 Dangerous Structures, Strength of Fir '^y.ku.%— continued. Scantling in inches. Breaking Weight Scantling in inches. Breaking Weight Scantling in inches. Breaking Weight Span 25 feet 13 by 13 439i 15 by 9 40s 16 by 16 8i9i 14 .. 9 352i IS ,, IS 67s 18 „ 12 777i 14 „ 14 548f i6 „ 12 6l4i 18 „ 18 u66| According to most authorities, the safe load upon timber beams should not exceed \ of the breaking weight for stationary loads, or \ for moving loads. It should be borne in mind that in every inhabited building there is a certain amount of vibration caused by frequent movements, and that most loads are consequently, to some extent, moving ones. By adopting \ of the breaking-weight as a constant coefficient of safety, and estimating the load proportionately in excess when there is likely to be much movement, we can arrive at the same result as we should by varying the coefficient of safety. Thus, for example, it is customary to calculate the load upon the floor of a public room at ij cwt. per superficial foot, including the weight of the floor itself. This is a greater load than would ever be put upon the floor of a room used for purposes of entertainment ; but it is by no means an excessive allowance to make for any room where dancing takes place. ( 55 ) CHAPTER VIII. TIES. Although it is true that a tie can only be strained in the direction of its length, yet the force producing that strain may act in another direction, and may be greater or less than the strain upon the tie. In the diagram (Fig. 14) let A C represent a vertical wall-face, C B a hinged flap such as is used outside a warehouse loop, Fig. 14. H D \ ' ~ ~ ~* ^- ■ — — ^ _ — ^ \. N- 1 1^ ^ 1 >. X \ t •V N. \i -vG and A B a chain supporting the flap, which is hinged at C ; and let the dotted triangle A D B be similar and equal to A C B. Suppose a load to be placed at B, and let D B represent the amount of the load. , Then D B or A C represents the stress which would be produced upon a chain D B, supposing that the load were suspended 56 Dangerous Stniciures. from D, instead of resting upon the platform or flap, and A B represents the stress upon the chain A B. If the load were placed at C we should assume it to be sup- ported by tlie wall, and there would be, theoretically, no stress upon the chain. If the load were placed somewhere between C and B, the stress upon the chain would be something less tliati A B. Let the load be shifted to E. Draw E F parallel to A B, and cutting A C in F. Then F E represents tlie stress upon the chain. We will now suppose the flap to be extended to G, beyond the point where the chain is fixed. If the load be placed at G the stress upon the chain will be greater than A B. From G draw G H parallel to A B, and cutting C A, produced, in H. Then H G represents the stress upon tlie chain when the load, represented by D B, is placed at G. By this means the stresses upon diagonal ties in any direction may be ascertained, when we know the magnitudes and directions of the forces acting upon them. The metliod is applicable to iron ties by which chimney-stacks are secured to lower structures, to prevent them from being overturned by the wind, or to ties which are similarly employed for securing adver- tisement boards, or those metal frames in which the names of business firms formed with huge letters are suspended against the sky. Structures of the latter description offer little resistance to the wind. The length of a tie does not affect its resistance, except in so far as the additional weight of metal adds to tlie strain. Good wrought-iron ties ought to have an ultimate tensile strength of twenty tons per inch of sectional area, but in many cases we cannot safely reckon upon more than sixteen tons. In old ties tlie Ties. 57 sectional area available for strength is often virtually- reduced by corrosion, especially at the fixings, where the portions rusted may be out of sight. Thus, a \ inch tie may be reduced to f inch in some places. In view of the tendency to lose strength by corrosion, the stress upon external ties ought not to exceed one-tenth of the ultimate strength, while it may be one-eighth or one- seventh in protected situations. The following are the sectional areas, in inches and decimals, of circular ties of the usual dimensions : ;iinch, ■ 049; f inch, -iio; \ inch, -196; f inch, -306; f inch, -441; finch, -600; i inch, -785; i^ inch, -993; 1:1^ inch, 1-224; and It inch, 1-764. These figures may be multiplied by the number of tons per inch which it is thought safe to allow in any special case. 58 Dangerous Structures. CHAPTER IX. STRUTS. The conditions affecting diagonal struts are similar to those affecting ties. If a strut be used to resist a force acting in any given direction, the stress upon the strut will be proportioned to its length. Suppose it be required to strut up the ceiling of a room which is 10 feet high in the clear, the surplus pressure to be sustained by the strut being 10 cwt., a strut 10 feet long, placed ver- tically between the floor and ceiling, will be subjected to a stress of 10 cwt. ; but if the strut be 11 feet long, and placed obliquely, as it must of necessity be, it will be subjected to a stress of 11 cwt. If two raking struts, at different slopes, are applied to resist a pressure at one point, the strut which is more nearly vertical will be subjected to the greater stress, and may, in certain circumstances, be called upon to sustain the whole pressure. In Fig. 15 let A C and B C represent two raking struts, which sustain a vertical pressure at C. The vertical line C D represents this pressure, and the parallelogram of forces being completed in F and E, the distances C E and C F respectively represent the stresses upon the struts. Draw H E and F G parallel to A B, and then C H represents the proportion of the load sustained by the strut C B, and C G the proportion by the strut C A. By such means the stresses upon any number of struts may be determined, and, when these are known, we have only Struts. 59 to take care that the struts themselves are of sufficient strength, for the purposes to which they are appKed. There are two modes of fracture to which a post or strut is liable. It may be crushed, or it may be bent until it breaks across the middle. In the majority of cases, however, a post is partly crushed and partly broken across. It is probable that no material is abso- lutely homogeneous, and that however carefully the pressure may be applied, there is always a tendency for the strut to bend in one direction more than another. When a strut is of such squat proportions that it can be crushed without appreciable bending, we may consider that its strength is proportioned to its sectional area, which, in a square timber, is equivalent to the square of its diameter. When a strut bends beneath a load and afterwards' breaks, it is broken by leverage. The leverage is proportioned to the deflection, which varies directly as the length and as the elasticity, and inversely as the diameter of the strut. The elasticity being proportioned to the length, it follows that the leverage of fracture 6o Dangerous Structures. varies directly as the square of the length and inversely as the diameter ; or, if / = the length and d the diameter the leverage varies as -^. The resistance, like that of a square beam, varies as (P, so that the leverage is to the resistance as — to d^, or as P to . . . consequently the flying shore has to resist the thrust of the opposite house oaly. But when both are party walls it will be best to allow sufficient strength in the shore to resist the thrust of both the houses together," &c. When a wall is to be sustained by means of a flying shore, iiis obvious that the smallest amount of pressure requisite could be applied at the top. If placed too near the top, however, the resistance of the shore may have the effect of c^splacing the masonry at that point, while the portion below continues to move forward. There must therefore be a sufficient load of masonry above to resist lateral displacement. With a wall of uniform thickness, and without reckoning the resistance arising from the adhesive power of the mortar, calculations show that a flying shore might be placed very near indeed to the top of a wall — much nearer than, it is ever placed in practice. This has reference to walls alone in which the thrust is proportioned to the weight Considered as part of a building, however, a wall may be acted upon by thrusts which far exceed what would arise from its own weight, and which would make it dangerous to place a shore near the top. * London : B. T. Batsford, 1882. 72 D.afigerous Structures. Before inquiring into the extents of thrusts that may arise from an entire building leaning over from the per- , pendicular, it may be ■ stated that when a wall which is not bulged shows a disposition to turn over, hinging, upon an oj>en joint near the base, the best point for applying resistance is at the centre of oscillation. Hence it is advantageous to place a flying shore at about two^ thirds of the height above the fulcrum, the chances of a. bulge occurring above or below this point being reduced to a minimum. To ascertain the thrust that may arise from a building; which is leaning over, we must approximately estimate the weight of the building. A few main measurements will enable this to be done with sufficient accuracy for the purpose. The weight of the walls may be taken at I cwt. per cubic foot ; the floors with their loads may be taken at i|- cwt. per square foot ; and the surface of the roofing may be taken at 40 lb. per square foot, which will include the weight of all timbers with an allowance for wind force. The centre of gravity of the building being assumed to be upon its vertical centre line, the force necessary to overturn or sustain it may be cal- culated upyon the same principle as when dealing with a wall or pier. For example, if a tower 40 feet high and 10 feet iri, diameter weighs 100. tons, its moment of stability will be 300 X S = 500 foot-tons ; and at a height of 30 feet it will reiquire a force of ^-§-§ = 1 6f tons to push it over or to prevent it from falKng, assuming that the force is applied horizontally. When several horizontal struts, or flying shores, are placed against a wall at different levels to prevent it from moving over, those which are placed lower should be stronger in the proportion that the thrust increases at Theory of Shoring. 73' lower levels ; and when there is a tendency to bulging in the wall, it is prudent to make each strut sufficiently- strong to withstand the entire thrust independently. When the joints are sound, however, and there is no tendency to bulging, we may economise by providing struts of sufficient strength only to resist that share of the thrust that will fall to each. Thus in the case of the tower assumed above, strutting might be placed at heights of 10, 20, and 30 feet from the ground respec- tively. The entire thrust at each of those points, taken separately, would be 50 tons at 10 feet, 25 tons at 20 feet, and i6f tons at 30 feet. The strutting may then only be required to take one-third- of the thrust at each level, namely, i6§ tons at 10 feet, 8^ tons at 20 feet, and 5f tons at 30 feet high. We will now examine the conditions affecting raking shores as applied to resisting horizontal thrusts. When a wall, which is moving over has to be shored from the ground, the shore should be so placed as to resist the horizontal thrust without heaving up the masonry abo-\'e. In the accompanying diagram, let D C represent the face of a wall, which is about to be pushed over with a thrust F B, so that it will turn upon the point C. Let A B represent a raking shore, put up for the purpose of sustaining the wall. If the point B be fixed, and the wall move forward, the point A will tend to slide outwards in the direction C A, with a thrijst equal tp F B. Draw F H, parallel to B A, cutting B C in H, and draw H G, parallel to C A, cutting B A in G. Then H G is equal to the thrust F B, and B G is pro- portioned to the compressive strahi upon the shore B A. If the point A be fixed, and the point B tree, then, when the wall moves forward, B will tend to slide up the wall with a thrust proportioned to H B. When the head of 74 Dangerous Structures. the shore B is fixed by means of a needle or stud, it is necessary that the weight D B shall be at least equal to the upward thrust H B. Let D B represent the weight of this upper portion of the wall. Draw E D parallel and equal to F B, and E F parallel and equal to D B, and join E B. Then E B represents the resultant Fig. i6. of these two forces, D B the weight, and F B the thrust Now since it is necessary that H B should not exceed D B, it follows that the slope of the shore must not be steeper than E B, otherwise it will heave up the upper portion of the wall. Theory of Shoring. 75 I If we cannot conveniently get sufficient weight above B, and cannot place the foot of the shore further out from the wall, we must continue the wall-piece some distance below B, and secure it there with an additional needle or stud. The weight of the shore takes effect in the following manner. One-half the weight acts vertically at A, and exerts a horizontal thrust in the direction C A, which thrust bears the same proportion to the half weight as C A does to B C. The other half of the weight acts ver- tically down B C, and exerts a proportionate thrust in the direction B F. The effect is to neutralise a portion of the thrust F B, and consequently to reduce the weight D B, which is required to produce equilibrium. The ratio of F B to D B therefore remains unaltered. It is possible that cases might occur where it would be convenient to increase the weight of shoring in order to counteract opposing thrusts. It is evident that a weight suspended from B would produce a thrust in the direction B F, which would bear the same proportion to the weight as B C bears to C A. The condition of a house front will often necessitate placing the head of a shore near the top of the wall, while the narrowness of the street footway will prevent the foot of the shore from being placed as far out as is desirable. Under such circumstances, many shores are, in practice, so placed that the top raker could not, as a single strut, exert its full power in sustaining the wall without forcing up the brickwork above, and thereby defeating the object of shoring. The necessity for more than one raker in a single system or battery of shores is then apparent, quite apart from any tendency to bulging that may exist at any point in the height of the wall. The best points at which to fix the heads of the rakers 76 Dangerous Structures. will differ in different cases. The heads of the rakers should be ordinarily at the floor levels, where they will prevent the joists from pushing out the wall, and the joists in their turn will prevent the shores from pushing the wall in. When the joists run crosswise, however, the circumstances differ, and more dependence is placed upon the rigidity of the wall-piece, and its effect in dis- tributing the resistance of the shore, and neutralising its tendency to push in the wall. 'W^hen the latter is at all likely to occur, the need for internal strutting from wall to wall becomes apparent. The head of a top-raker should be placed no higher than will permit a hne in continuation of its upper surface to fall within the section of the roof. We will now examine the mechanical conditions which indicate the most advantageous positions for the heads of the separate rakers in a system of shores, apart from the question of floor-thrusts or the resistance afforded by party walls. Let us assume that a system of raking shores has to be applied to a portion of a structure 30 feet high above the street, tending to lean forward with a thrust equivalent to one ton at the top. We will assume that the curb of the footway is only 5 feet away from the wall, that the shoring must not encroach upon the roadway, and that the head of the top raker must be placed no lower than S feet from the wall, i. e. 25 feet above the street level. The moment of the thrust at the top of the wall is 30 foot-tons, and its effect at 25 feet high is therefore ff = li ton = 24 cwt. We want to know whether there will be a sufficient load of masonry above the shore to resist its vertical thrust. This will depend upon the condition of the wall, and whether or not a narrow strip of masonry is liable to be pushed upwards, breaking its Theory of Shoring. T] connection with the masonry on each side of it — a highly dangerous contingency. We will assume that the vertical resistance of the wall is equal to 5 cwt. for every vertical foot of height, bearing in mind that it would not be safe to allow so much as this in respect of a parapet above the roof. We have now a horizontal thrust of 24 cwt. and a vertical pressure of 25 cwt. impinging against the head of our shore. It is evident that the resultant of these two forces inclines at an angle very little steeper than 45°, and as the shore is very much steeper than this, it cannot resist the horizontal thrust without overcoming the vertical pressure, and so heaving up the masonry above. This shore must only be calle"d upon to resist a portion of the thrust whose resultant coincides with the slope of the shore ; the remainder of the thrust must be resisted by another shore placed lower down. The shore, the footway, and the wall together form a right-angled triangle, of which the base is to the per- pendicular as I to 5. The forces resisted by the shore must be in the same proportion, if their resultant is to coincide with its slope ; that is to say, the horizontal thrust must be to the vertical pressure as i to 5, and since the vertical pressure is 25 cwt., we have 5 cwt. for the thrust resisted by the shore, leaving another 19 cwt. unresisted. The moment of this surplus thrust is 19 X 25 = 475 foot-cwts. We have now to find the height at which the second shore may most economically be placed. We will assume that the two shores are to have their feet bound together, as is ordinarily the case. The foot of the second shore will, therefore, be 5 feet from the base of the wall, and its head will be at a height not yet determined, which we will call x, assuming it to be 78 Dangerous Structures. expressed in feet. It will have to resist a horizontal thrust which we will call t, and will be kept down by a vertical pressure /, due to the weight of the masonry below the upper shore. The values of / and / are to be expressed in cwts. In order, then, that the slope of this lower shore may coincide with the resultant of / and /, the following proportion must be maintained : — X t As 5 : a: : : / : /, and — = /. But * ^ is the moment of /, 475 which is of course equal to 475, so that/ = — = 95 cwt. ; and as the vertical weight is assumed at 5 cwt. per foot, we have ^-g = 19 feet for the distance of the head of the lower shore from that of the upper one, the height X being = 6 feet. Subject to the above conditions, this is the most economical position for the lower shore. It remains, in concluding these fragmentary notes upon a subject which would admit of more exhaustive treatment, to remind the reader of the very simple means by which the required scantlings of the shores can be ascertained, when the extent of the thrusts acting upon them is known. The resultant of i and /, for in- stance, in the case assumed above is = v' (^' + f^) ', 3.nei—1Z; 76 — 78 Tie-beams, cambering, condemned, 26 ' Ties, 55—57 ,, circular, sectional areas of, 57 ,, various stresses on, 55, 56 Timber beams, 44, 54 ,, ,, calculated breaking weights, 47 — 54 ,, ,, safe loads on, 54 ,, under footings, how to remove, 10 Townsend shaft, Glasgow, 4, 13, 31, 34 Transverse strength of stone, 1 7 ,, timber, 46, 47 Walls and piers, 12—20 „ „ stability of, 17, 20, 23, 25, 26 Wedges to straighten shafts, 35 Weight of buildings, how to calculate, 72 ,, ,, roofing, 22, 72 ,, ,, shoring, how it acts, 75 Wind-force, leverage of, 32 ,, on roofs, 22; against lofty structures, 32, 33; against inclined surfaces, 36 LONDON : PRINTED BY WILLIAM CLOWES AND SONS, LIMITED, STAMFORD STREET AND CHARING CROSS. I j.\.^<^jL jLK^^ij^ t'l^ ui\.i\.ci jor the use of Archi- tects, Surveyors, Students, &c. Published by B. T. BATSF0RD,52, High Holborn, London. 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Illustrated by 89 Diagrams, crown 8vo, cloth. Contents : — I. Loads on Supports — II. Shearing Stresses in Can- tilevers and Beams — III. Stresses in Flanges — IV. Rectangular Beams — v. Method of Designing a Wrought-Iron Girder — VI. Columns and Struts — VII. Framed Cantilevers — VIII. Framed Beams and Girders— IX. Room Dead Load only— X. Roofs, Wind Pressure and Combined Loads. Price 4/. Nett 3/4 Approved by the Science & Art Department. Building Construction and Drawing (Elementary Stage, Part i.) ,- specially adapted for the use of Students in Science and Technical Schools. Compiled by Charles F. Mitchell (Lecturer on Building Con- struction and Technical Carpentry and Joinery). Revised by the Technical Teachers of the Polytechnic Institute, Regent Street. With 550 Illustrations, fully dimensioned. Second Edition, revised. Crown 8vo, cloth. Price 3/- Nett 2/6 The illustrations are executed in the best possible manner, the typography is excellent, and the book has been in every way most carefully prepared, and will doubtless be acknow- ledged as the best and cheapest Text Book on the subject. Dry Rot in Timber; by W. H. Bidlake, M.A., A.R.I.B.A., &c., numerous diagrams on two plates, 8vo, cloth . . . . Price 1/6 Nett i/. Hints to Househunters and Householders ; by Ernest Turner, F.R.I. B.A., &c., &c., treating of Situation and Aspect, Construction, Water Supply, Drains, Ventilation and Drainage, Warmingand Ventilation, Lessors, Lessees, and their liabilities, &c., &c. Crown 8vo, cloth gilt, illustrations (Published for the National Health Society) . . . . Price 2/6. Nett 2/- Hot- water Heating on the Low Pressure System; comprising some of the Principles involved, an Explanation of the Apparatus and its Parts, by F. A. Fawkes, F.R.H.S., author of "Horticultural Buildings,' illustrated with 32 woodcuts, Bvo, boards .. .. .. .. Price 1/