LXII. On the Construction of Skew Arches. By [With a Plate.] SKEW bridges have hitherto been comparatively little used; but since railways have been introduced, in which it is highly important to preserve as direct and straight a line as possible, they are very frequently required, as a railway passes through the various districts without the possibility of regarding the angle at which it may cross canals and roads, its course being in great measure controlled by the natural features of the country. Wherever a canal is thus crossed at an angle, we must either divert the canal, so as to bring it at right angles to the railway; or we must build a common square bridge of sufficient span to allow the canal, its course being unaltered, to pass uninterruptedly under it; or we must erect a proper skew bridge. The first of these is often impracticable, as provisions are generally inserted in the Acts of Parliament, for preserving the canal from any alteration in its course; and even if this were not the case, the diversion of a canal causes great expense, and is attended with much inconvenience to its traffic: the second is a most unscientific mode of overcoming the difficulty, and would also involve very serious expense, arising from the necessity of making use of an arch of much larger dimensions than would be required were the proper oblique arch erected in its stead. By referring to Plate III. figs. 1 and 2, this will be apparent: for this diagram I have selected the angle at which the London and Birmingham railway crosses the Grand Junction Canal, being an angle of 30 degrees. It is for the above reasons that oblique arches are now so frequently erected; and a good method of building them is, therefore, of considerable importance. As many practical men with whom I am acquainted have experienced considerable difficulty in the construction of skew bridges, I was led to turn my attention to the subject; and have at length succeeded in rendering the principles of it easy to be understood. All persons are acquainted with the manner in which common square arches are built, where all the courses are square to the face, and parallel both to the direction and surface of the road or river running under it, by which means the thrust or strain is always at right angles to the joints or beds of the * Communicated by the Author. individual stones composing the arch; hence the whole thrust of ordinary arches, which is brought in upon the abutments, is exerted in the direction of the bridge itself, i. e. of the road passing over it. To devise some simple mode of setting out and working the courses of stone in a skew arch, so as to bring in the thrust in the proper direction, was the great object to be obtained. All practical men are aware of the vast difference between having to deal with straight and with twisted lines; and the necessity of introducing twisted lines in the construction of skew bridges will soon be seen. In skew bridges, in order to keep the thrust in the proper direction, it is necessary to place the courses of stones at an angle with the abutment, whereby each stone loses its parallelism with the surface of the road, and is therefore laid on an inclining bed. In a common semicircular arch each course of stones is parallel with the axis of the bridge, and all the beds are wrought so as to point to the axis: the inclination of the stones varies in every course; but although the inclination of the stones varies in every course, both ends of the course have the same inclination, both ends are equally high in the arch, and both ends point to the centre. This is the case in the ordinary bridge; but in a skew bridge, as the courses run obliquely across the arch, one end of the course is necessarily higher up the arch than the other, and therefore would no longer point to the centre; but only make this point to the centre, and we immediately get the twisted form, that is, we make each bed of the courses of stones a true spiral plane. The principle which I have adopted is, to work the stones in the form of a spiral quadrilateral solid, wrapped round a cylinder, or in plainer language the principle of a square threaded screw; hence it becomes quite evident that the transverse sections of all these spiral stones are the same throughout the whole arch. It will be obvious that the beds of the stones should be worked into true spiral planes; but I am not aware that any rule has yet been published that would enable the stones to be wrought at the quarry into the desired form, or of any rule by which the true angle at which the courses cross the axis of the bridge is determined. Fig. 3. is a representation of the courses of the stones, each alternate course being omitted in order to show their form more distinctly; and the course forming the key-stone is carried out so as to show that it really is the thread of a square threaded screw wound round a cylinder, the cylinder being indicated by the two dotted lines. If the threads are cut at right angles to the cylinder, the section would appear as in fig. 4; if cut at right angles to the courses, or as nearly so as the case will admit of, as they are really cut to form the face of the bridge, the section would appear as in fig. 5. In order that these principles may be understood, it is necessary to have a clear idea of the nature of a spiral plane; and perhaps, the best definition of it is, to consider it as being produced by the twofold motion of the radius of a cylinder, 2. e. let a radius revolve upon its axis at an uniform velocity, and at the same time impart to it a progressive motion along the axis itself, and then by apportioning these two motions to the particular case you will obtain any spiral you may desire; hence it is apparent that the outer edge of a spiral plane is produced by a straight line wound round a cylinder everywhere forming the same angle with the axis, while the inner edge actually merges into the axis itself, which of course is a straight line. The question which now naturally suggests itself is how to decide at what angle to place these spiral stones with respect to the axis of the bridge, or in mechanical language, what traverse must we give the screw? In entering upon the investigation of this subject, my first idea was to develop upon a plane surface all the superficies connected with a skew arch. If a semi-cylinder be cut obliquely, the section is a semiellipsis, and if the semi-cylinder be then unfolded, the edge of the developed ellipsis will not be a straight line but a spiral one; and some builders not being aware of this fact, have squared a course from the face of the centring, and having drawn in the remaining courses parallel with this, have taken it for granted that all the courses would be square with the face, which it will be seen is impossible by referring to the development of the intrados, or under surface of the arch, which is the development of the centring itself: they have hereby been led into very serious and perplexing difficulties. Having shown the impossibility of making all the stones square to the face, I will now give the mode of deciding in what direction they should be placed. When the soffit is developed, the edge which formed the face of the arch gives a true spiral line: my first plan was to lay the courses of stone at right angles to a line extending between the two extreme points of the spiral line of the developed soffit (see fig. 6); this line I shall afterwards speak of as the approximate line, as it is the nearest approximation to the line of the face that can be obtained by a straight line. On further consideration I discovered a far more eligible mode of laying out the lines. It is evident from fig. 7, that if spiral planes are considered as composed of spiral lines placed at various distances from the centre of the cylinder, each of these lines will form a different angle with the axis; and therefore, as an arch has always some thickness, that although we have the inner edge of the spiral plane placed at right angles to the thrust, yet every other portion is gradually departing from a right angle, and is, therefore, exerting its force in an improper direction: thus an arch of this description can never exert its thrust in the direction of the bridge, but is endeavouring to push the abutments obliquely. To get the thrust strictly correct, I have supposed the arch to be cut into two rings of equal thickness (see fig. 8); and having considered the external ring as removed, have proceeded to develop the outside surface of the remaining one: this I shall hereafter speak of as the intermediate development, as it is the development of a surface midway between the extrados and soffit or intrados. Upon this intermediate development I place the approximate line, and then draw all the courses square to it; by which means we obtain a line in the centre of each stone exerting its force in the true direction, and thus get rid of the disadvantage of twisted beds to the stones, as in proportion as the one half of this bed exerts its force in an oblique direction on the one hand, the other half acts in the opposite direction, and is therefore always producing a balance of effect, which resolves the various forces into one exerting all its power in the true direction, which is the object to be obtained. Having explained the mode of setting out the beds of the stones, a little may now be said on the situation of the crossjoints: by these will be understood the joints between the various stones constituting a complete course. Where an arch is built of stone throughout, the situation of these joints is of minor importance; but where stone is expensive, it is common to make the faces of the arch only of stone, filling in the intermediate space with brick-work; as in these instances the cross joints form the boundary between stoneand brick-work, it becomes a point of considerable importance. This is the case in the Watford viaduct; each stone here is equal in thickness to five courses of bricks, so that there are five thicknesses of mortar in the brick-work to one in the stone. Mortar always is compressed into a smaller compass when the centring is struck, and the full weight of the arch comes upon it. In consequence of this tendency, that portion of arches constructed of brick-work, always subsides much |