The contract for a brick viaduct should clearly define the site and extent of the whole of these sections; the gross amount to be paid, as well as the detail prices at which any additions or extra works are to be paid; an enumeration of the plans, sections, and drawings, which should be signed by the engineer and contractor; and, attached to the contract, the mode in which the work is to be paid for, the time for its completion, and the time the contractor is to uphold the work, together with the penalty for non-completion. It should state in a precise manner any diversions of roads or streams, the nature of the foundations, and whether the contractor is to find cofferdams, centerings, scaffolding, and all other machinery and materials. With respect to the time for completion, it will be best to divide it. If, for example, it is a viaduct of ten to twelve arches of fifty feet span, and forty feet high, in twelve months at the farthest the brick work and masonry of the abutments, piers, and wing walls, should be up to the springing of the arches. The remainder of the work may be divided into two portions of six months each, so that the whole should be complete in two years. We are well aware that such a work could be executed in much less time, but it would be at a corresponding increase of expense.

The contractor should be bound to execute the work to the satisfaction of the engineer, or whoever may be appointed by the parties for whom the viaduct is to be erected, and at all times to keep a sufficient number of men on the work, or the engineer should have power to send the requisite number. This will of course be done at the contractor's expense; and it should be expressly stated, that all materials, scaffolding, and machinery, on the spot, will be used by the engineer, and that he will provide any other materials which may be requisite, in order fully to employ his men. The number of competent foremen to be employed by the contractor should be stated; and the engineer should have power to supersede any of them for incapability or improper conduct, and in default of there being a sufficient number, to employ as many more as should be found requisite, up to the specified number. An account of the foremen and workmen employed, should be furnished to the engineer weekly, each trade being classed under a separate head, under a penalty of five pounds per week.

The contractor should be bound to enclose the whole area before commencing the work, and to comply with all local regulations and the act of parliament, if there is one, under which the work is to be done, so as to prevent all trespass on the adjoining lands, and to maintain the said fence till the works shall be completed; but if any of the adjoining land would be of use to the contractor, it should of course be procured for him, by his undertaking to pay the expenses. He should be bound to deposit any spoil earth taken out for the foundations, wherever the engineer may appoint. Any temporary roads which he may require, should be set out and made entirely at his own expense, he compensating the owners of the land. Before he commences any excavation or work for any of the abutments, piers, or wing-walls, or commences any arch, proper notice should be given to the engineer, under a penalty of not less than twenty pounds.

Should the parties for whom the viaduct is to be built be called upon to pay any damages to the owners of the adjoining property, or to any other person, through the negligence or wilfulness of the contractor's workmen, they should reserve the power of deducting the money so paid from any sums which may be due to the contractor for the performance of the work. No subcontract, except for labour, should be allowed, without the the engineer's consent in writing. Should the contractor become bankrupt, or should he, from any other cause, cease to continue the work, the engineer should have power, after serving him with a written notice, to carry on the work himself, placing the amount of all monies expended by him, in constructing the work, to the contractor's account, and using all the materials and machinery on the ground, which should become the property of the parties for whom the viaduct is built, from the time they are brought upon their ground, till the completion of the works, and should not be removed without their consent. When materials are brought on the ground of an improper sort, or unsound, or of an inferior quality in the opinion of the engineer, the contractor should be bound to remove them forthwith, and to supply their place by others, in default of which the engineer should have power to remove them after two days' notice, without being answerable for either loss or damage in so doing. The expense of the removal, of course, should in all cases fall on the contractor.

Should the engineer discover that any part of the work is unsound, or imperfectly executed, or not performed in strict accordance with the conditions of the specification, plans, sections, or drawings, he should have power to order it to be immediately taken down and executed in a proper manner, without any extra charge whatever, and without any delay ensuing in the completion of the several parts of the work; and if the contractor should refuse or neglect to do so after proper notice, the engineer should have power to cause it to be done, charging the expenses to the contractor, and removing the materials, and then proceeding to erect the work in a proper manner.

The contractor should be bound to make any alterations, additions, or omissions in the work, on being authorized to do so by the engineer; the omissions being set off against quantity against the additions, or by measurement and valuation, as may be agreed on, and the balance accounted for to either party. There should be no alterations whatever allowed, except under the signature of the engineer. The contractor should be furnished with copies of the drawings and specification, but he should be bound to see they are correct, and should also be required himself to set out the whole of the work. Should there be any buildings, trees, or other matters on the site of the works, the property in them should be secured to the parties for whom the viaduct is to be built; the contractor removing them to the nearest public road, giving a week's notice previously to his doing so, and stating the place to which they will be removed. All damage done to the works during their progress, or to the materials or implements thereon, whether arising from accident or the carelessness of the workmen, should be made good at the contractor's expense; and he should be bound, not only to execute such works as are expressly described respectively in either the specification, plans, sections, and drawings, but also such other works as may be necessarily implied, or reasonably inferred to be done; and in case of any dispute arising, the decision of the chief engineer should be considered final.

The mode and times of payment for the work, and the nature and extent of any sums which may be retained from the periodical payments, as a security for the due performance of the contract, should be clearly defined, and also the time when the retained money will be paid, which in all cases should be dependent on the engineer being perfectly satisfied with the state of the works. It is usual to make contractors of these works an allowance, when they have large quantity of materials on the ground, as part payment; for instance, when two million or more bricks are on the spot, by per cent. on their value is usually advanced; and, generally speaking, if it is seen that the contractor is proceeding with his work in a spirited manner, he should be encouraged by every reasonable assistance in the power of the parties by whom he is employed.

The schedule of prices for extra and additional works would include every thing which is at all likely to be required; such as fencing, excavations, brickwork, concrete, masonry, specifying the nature and quality of the stone, and the description of the work; such as string courses, coping, ashlar work, impostos, cornices, caps to pilasters, quoins, &c.; the price of wrought and cast-iron work, of culverts, drains, and ditching, specifying the size in each case; of timber, stating the size and description; pruning and levelling at the ends of the viaduct; piling for foundations; the formation of roads; laying railways, distinguishing the price with blocks from those with sleepers; and all other things which it may be necessary to order in the course of the work.

It is usual to take security from the contractors for these works. Many an excellent contractor may not be able to find substantial sureties. In this case, his character from past work will form the best criterion. It is always advantageous to get a zealous, active, and substantial contractor, even at a higher price, for all large works.

The specification should state in a clear manner the site of the viaduct, its intended use, and what work is to be included in the contract, with any other information, so as to enable the contractor clearly to see his way. The level should be stated, the span, height, and form of the arches, whether any are skew, and if so, the angle of their inclination; the radius of the curve, if there is any; the nature and quality of the principal material; and if this should be brick, then whether there are to be stone arch quoins, string courses, string course, plinth, or coping; also, that pilasters, &c., are intended; what is to be the form of the piers, whether they project from the face line of the arch; what description of wing walls, and whether they are built plumb, or battened, or curved, and what arched openings they may have; and if so, whether there are to be vents to them, and also any other internal openings which may be intended; for instance, in railway viaducts, walls will be required in the abutments for the support of each line of rails; but these may be built considerably cheaper, by opening them with arched openings, having inverts underneath. In this case, the wing walls might be built up, and extended just far enough for the required slope of the embankment at each end. The space between the walls, for supporting the four lines of rails, should be arched over with nine-inch brickwork for supporting the earth and lasting of the railway. The spandril walls should be perpendicular, and a wall should be run, if the height requires it, under each rail, extending over the arches, piers, and abutments; the spaces between these walls being arched over in a similar manner to those described above.

The forms and dimensions of every part of the viaduct should be shown in the drawing, which should contain an elevation, plan of foundations, plan of the roadways or waterways, if any; sections through the square and skew arches and piers; the abutments and parapet walls, showing the form of the string course, plinth, and coping; and lastly, a longitudinal section throughout the whole length of the viaduct. The nature of the ground should be carefully studied, before the footings of the piers and abutments are decided on; and if at all doubtful, concrete should be used; but in all cases, the footings should be stepped outwards, carried so low as to rest on an entirely good and solid foundation. Should there be any waterways under any of the viaduct, the arches, the outer course of materials of the piers on each side the water, should be laid in Roman cement, high enough to ensure their safety in this respect. The piers of skew arches may generally be lightened in the interior, and the mode of doing this should be shown in the drawing.

The spring course should if possible be of stone, not less than twenty inches thick on the face, say for an arch of fifty feet span, and rising behind, so as to form the proper skewback of the arch. These should be laid over the tops of the piers and abutments, each stone not being less than three feet six inches in the bed, each course consisting of not less than eight stones in the width of a viaduct of about twenty-eight feet broad; and no stone should be less than three feet six inches in length, dowelled and leaded to the adjoining one. If the viaduct is brick, the upper surface of the stones, at the springing of any skew arches, should be made in steps or notches, for receiving the brick-work of the arch, each step taking five courses of bricks, and being cut at right angles to the brickwork, rising behind at the proper summering for forming the skewback of the arch. The five courses of bricks agreeing with each arch stone, should all fit into a corresponding step in the springing course.

The form of arches is so much a matter of taste, that no general rules can be given without assuming some particular one. We shall therefore take a segment of a circle, a very substantial viaduct in that form having been built by Robert Stephenson, Esq. They were fifty feet span, thirty feet eight inches in length, and three feet thick. The thickness of each arch quoins was equal to five courses of bricks; and they were alternately three feet three inches, and two feet six inches in length on the soffit from the face of the arch, and rising in steps to receive the horizontal courses of the spandril walls.

Great care should be taken in giving all the arch quoins a true and perfect bed throughout each stone; and no pinning should be used in any part. The joints of the arch stones should be neatly chamfered rustic, and the faces of them fair tooled, and projecting about an inch from the brick work. The proper summering should be given to the bricks in the arch; and any irregularity of form, arising from imperfection in the laggings, workmanship, centering, or any other cause, should be removed, and immediately corrected by the contractor. The centres should in no case be struck without one week's notice to the engineer. After they are removed, the whole of the brickwork in the soffit of the arches should be carefully and neatly pointed. With respect to keeping the arches dry—a point which it has been found extremely difficult to effect, and without which the brick work soon becomes damaged—we believe that puddling, if carefully and properly done, will always succeed. Drainage by pipes through the piers has been tried, and failed. A mixture of coal tar and lime has been found to succeed, and so has coal tar alone, boiled ten or twelve hours, so as to evaporate the water and ammoniacal liquor. As, in cooling, it sets moderately hard, it must not be laid on in wet weather, and the surface of the brickwork should be clean swept previous to applying it. Leading the arches with about five pound lead is a sure preventive; but the process is expensive. The asphaltic cement is also a certain cure, but is dear compared with coal tar.

The backing should be built up to the required height, with good brickwork, well grouted. The wing walls of a railway viaduct should consist of six walls, built plumb up to the level of the crown of the arches, and extending in the direction of the railway towards the embankment at each end. They should be well bouded into the abutments, and lightened by arched openings. The top of the exterior walls should be brought to a proper level with the upper side of the arches, for receiving the string course. Viaduct and parapet walls. The outer walls for the sized arches we have assumed should be about three feet thick, and constructed with bricks laid in horizontal courses, properly jointed with the arch quoins. The interior walls should be eighteen inches thick:

A string course should be laid along the whole length of each front of the viaduct, projecting uniformly with the face line of the arches. No stone in this string course should be less than three feet six inches in length, neatly tooled on the exterior, with the joints made to fit close for the whole bed of the stone. The parapet walls will have a bolder and handsomer appearance, if they stand on a stone plinth, about eighteen inches high, and eight inches wide, placed upon the string course; the remaining portion of the wall being brickwork, fourteen inches thick; the courses being carefully laid parallel with picked bricks; the exterior faces of the plinth being fair tooled, and the whole of the walls throughout forming a uniform and regular line.

The coping should be of stone, the form and dimensions for which should be shown in the drawing; and no stone should be less than three feet in length, neatly jointed, and the exterior faces fair tooled or chisel-dressed, each stone being dowelled or leaded to the adjoining one. The spaces round the footings of the abutments and wing walls should be firmly rammed up with clay, if it can be procured, to the level of the ground, in courses of one foot in thickness, each layer being properly beaten before a succeeding one is put on. It will sometimes be convenient to make a new channel for water-courses, so as to bring them through a given arch of the viaduct. In this case, the old channel should not in any way be impeded, till the new channel is formed; after which, the excavation which has been taken out to form the new channel, may be used in filling up the old.

Great care should be taken in the selection of materials for the construction of these works. The bricks should be hard, sound, and well burnt; those used in the face of the work being selected of a uniform size and colour. The quarries from which the stone is to be brought, should be named, and care taken that the whole is of a good and hard quality, free from flaws of every description. The mortar should be made of lime to be named in the specification, and mixed in the manner, and in the proportions between the lime and the sand, which is known to be best for the particular kind of lime, and be thoroughly incorporated of the proper consistence. The Roman cement should be recently made, and kept from contact with the atmosphere, or any moisture. An equal quantity of good sand or road dust should be mixed with it; and no more than what is required for immediate use should be mixed, or any that has set be worked up again.

The brickwork should be laid either in English or Flemish band, as may be ordered; and in no case should any joint of mortar exceed a quarter of an inch in thickness; no broken bricks should be allowed to be used, either internally or externally, unless absolutely necessary as closers; nor should any difference be made in the exterior and interior of the work. The whole of the brickwork should be grouted at every course, and the exterior neatly pointed. All the stonework should be truly bedded, and the vertical joints cut square back for the whole bed of the stone; no pinning of any kind being allowed.

After the piers have all been brought up to the level of the springing of the arches, the contractor should fix the centering, and complete the brickwork and masonry of the three arches, and their backing next adjoining to an abutment; the wing walls and spandril walls being also carried up to the same height as the backing. When these three arches have all been completed, and properly backed, the centering may be removed from the arch nearest the abutment, and fixed in the proper place for the completion of the fourth arch. As soon as the fourth arch is completed, and backed as above, and not before, the centering may be moved from the second arch from the abutment, and fixed in the proper situation for completing the fifth arch; and so on with the rest, the centering never being removed from the two last arches which have been turned, until a third, or the one in advance of these two, has been finished. The centering used should be subject to the approval of the engineer.

The string course, plinth, parapet walls, and coping, should not be put on, until the whole of the centres have been withdrawn; and in no case should the centering be struck without orders from the engineer. If the foundations require cofferdams, the contractor should be bound to find them at his own expense. Great care should be taken that the viaduct is properly placed, so that, in the case of a railway, the outside line of rails, when laid in their uniform and proper position, shall be equidistant from, and parallel with the faces of the parapet walls; and in case the line in that particular part where the viaduct is, should be curved, this will require the most vigilant attention; also, that before the string course is laid on, every part of the brickwork is properly level, so that the eye, in running along the string course, shall not be able to detect any deviation from the given line. We have known more than one case in which a noble and handsome structure has been entirely defaced, to a professional eye, through the inattention or the incapacity of overlookers who have been entrusted with this important part of the work.

After the expense of an embankment equals that of a viaduct, there will almost always be a gain in building the latter, if an increased height is required; because the addition to the piers and abutments for this extra height, will cost much less than the additional earthwork at the base of the embankment. Let us take a case in point. In 250 yards of railway, forty-five feet high, there are required three bridges. A viaduct for this 250 yards, with fifty feet arches, would cost L17,000; each bridge would cost L1400, and the remaining embankment sloping 2:1, would cost L6150; at 1s. per yard; total, L10,650. Hence the viaduct is much the dearest method. If the height was seventy-five feet, the bridges may be taken at L2000 each, and the remaining embankment at L15,250; total, L21,250, besides the extra land; while the viaduct, having only 216 cubic yards of brickwork, at L1 for each yard additional height, after full allowance for extending the wing walls, &c., would only cost L19,000. We may conclude, therefore, that viaducts are cheapest only at very great heights. Much of course depends on the length of lead, whether building stone can be got from an adjacent cutting, &c. The cost may however be lessened, by a judicious use of wood and iron. Several wooden viaducts have lately been constructed for railways; and in America they are largely used. A novel plan has been adopted in that country by Mr Town. (See Stephenson's Civil Engineering of North America.)

In the preceding articles in this work, the subject of arches has been so fully discussed, that nothing remains except to give a guide to practical men, how to construct the arch of equilibrium, and to show the variation which it undergoes under different dimensions. Supposing, then, the roadway on the viaduct to be horizontal, (fig. 1, Plate CCCCC) let $K = y$, $AQ = h$, $CI = w$, $DK = a$, and $KQ = c$, then

$$\log \frac{w + \sqrt{w^2 - a^2}}{a} = \log \frac{c + \sqrt{c^2 - a^2}}{a}$$

where the denominator has only to be computed once, and the whole formula is very easy. In the following table, if the number under \( xh \) is multiplied by any half space having given values of \( a \) and \( c \), it gives points in the curve or lines of \( y \); and if more points are wanted, the column logs the logarithm of the denominator in the formula; here the numerator only will require computing.

| \( a \) | \( b \) | \( w \) | \( xh \) | Log. | |-------|-------|-------|--------|-----| | 2 | 20 | 3 | -3217 | 1-299943 | | | | 4 | -44 | | | | | 5 | -52334 | | | | | 6 | -592 | | | | | 10 | -766 | | | | | 15 | -903 | | | 5 | 10 | 6 | -47064 | -574286 | | | | 7 | -65573 | | | | | 8 | -79175 | | | | | 9 | -90213 | | | 5 | 15 | 6 | -35304 | -76552 | | | | 7 | -49192 | | | | | 8 | -59397 | | | | | 10 | -75019 | | | 5 | 20 | 6 | -30163 | -89614 | | | | 7 | -4201 | | | | | 10 | -63823 | | | | | 15 | -85424 | | | 5 | 23 | 6 | -27149 | -995635| | | | 7 | -37779 | | | | | 8 | -45668 | | | | | 10 | -57678 | | | | | 15 | -768876| | | 5 | 25 | 6 | -2815 | -995591| | | | 7 | -37824 | | | | | 10 | -57683 | | | | | 15 | -76891 | | | | | 20 | -90121 | | | 20 | 29 | 21 | -34394 | -39794 | | | | 22 | -4841 | | | | | 23 | -59117 | | | | | 26 | -82253 | |

Fig. 2 shows this arch with \( a = 2 \), \( b = 20 \), and with \( h \) varying from twenty to seventy feet. These give the result of increase or decrease in that element. Taking \( h \) and \( c \) each \( = 20 \), and making \( a = 5 \), we see in fig. 3, the effect of increasing the road on the arch, keeping the lower curve height of the road the same, having fifteen feet height, and five feet road; and in the upper curve, where \( h = 20 \), \( a = 5 \), and \( c = 23 \), we see the effect of increasing the road to five feet, and retaining the same height of arch, eighteen feet. Fig. 5 is with \( h = 9 \), \( a = 20 \), and \( c = 23 \). This would be the curve for one of the bridges under the London and Birmingham railway, near the Bull Inn, at the Birmingham end of the line. Fig. 9 shows the effect of lowering the crown; the heights to the road being 10, 15, 20, and 25 respectively, while in each case \( h = 20 \), and \( a = 5 \).

These arches of equilibrium are more adapted for brick than stone, owing to the difficulty of making the arch-stones in the latter adobe properly. In elliptic arches, however, this is not always done; they are made to radiate to one core, sometimes from ignorance, but oftener because, when properly done, each arch-stone requires a mould for one side; in the curve of equilibrium it will generally be best to assume a centre to the nearest regular figure; otherwise let \( KD \) (fig. 1) \( = a \), \( AQ = t \), \( DQ = c \), \( DP = x \), and \( CP = y \), and \( Q^2 = \frac{t}{a} \); then

\[ \text{Hy log. } \frac{a + c + \sqrt{2ac + c^2}}{a} \]

the radius of curvature \( R \) is \( R = \frac{2ax + x^2 + Q^2}{Q^2 + a + x} \); and this is a minimum when \( x = \sqrt{\frac{Q - a^2}{2}} - a \). The subtangent is equal to \( \sqrt{2a + x^2} \). Hy log. \( \frac{a + x + \sqrt{2ax + x^2}}{a} \).

From any point C in the arch, draw the tangent \( eb \) (fig. 1), then the pressure on the arch-stone at \( Cg \) is as the secant of \( bcp \) directly and radius inversely. Hence, taking unity as radius, if the depth of the arch-stone at the crown \( (a \cdot \sec. bcp)^2 \) is multiplied into the sec. \( bcp \), it gives the length of the arch-stone at any point proportioned to its pressure.

The areas being proportional to the weights set off \( K' \), the centre of gravity, draw \( KN \) perpendicular to \( EN \), join \( KB \), then the pressure in the directions \( KN \) and \( NB \), is as these lines, the former to a lever \( EN \), and the latter to a lever \( EF \). Put the area of the arch \( = a \), \( KN = b \), \( EN = d \), \( BN = f \), \( FG = g \), and \( EF = x \), then \( \frac{b}{f} da \cdot x \) is the effort to overturn the pier, and \( xa \) the resistance; and \( \frac{bd}{f} - xa \), or \( \frac{bd - fx}{f} \cdot a \) is the difference. The height of the pier is \( zg \), and its resistance on the lever \( \frac{1}{2}x \) is \( zg \cdot \frac{1}{2}x \), or \( \frac{x^2g}{2} \); hence for an equilibrium these must be equal, or

\[ \frac{bd - fx}{f} \cdot a = \frac{x^2g}{2}, \quad \text{whence } x = \sqrt{\frac{2bdag + 4af}{g^2f}} - \frac{2a}{g}. \]

The centre of gravity may be found near enough, by drawing equidistant lines parallel to \( KH \) (fig. 1), taking the centre of gravity of each part, and tracing a curve through the points' lines parallel to \( KQ \), which treated similarly, will give another curve, and when this cuts, the first is the centre of gravity. The area may be found by a corresponding process.