UC-NRLF 
 
 SB Eb ESS 
 
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:-:ESE LIBRARY 
 
 UNIVERSITY OF CALIFORNIA. 
 
 

TREATISE 
 
 B E A I 1ST G 
 
 WITH ITS APPLICATION TO 
 
 BRIDGES 
 
 AND OTHER STRUCTURES OF WOOD OR IRON. 
 
 BY 
 
 ROBERT HENRY BOW, 
 
 CIVIL ENGINEER. 
 
 WITH 156 ILLUSTRATIONS ON STONE. 
 
 OF THE 
 
 UNIVERSITY 
 
 NEW YORK : 
 
 D. VAN NOSTRAND, PUBLISHER, 
 23 MURRAY STREET AND 27 WARREN STREET. 
 
 1874. 
 
PREFACE. 
 
 IT is now several years since the description of bracing 
 chiefly dwelt upon in the following pages suggested itself to 
 the author ; and he was surprised to find that a method of such 
 simplicity and evident excellence should have been employed 
 in only a few unimportant instances, and, in the majority of 
 these, in a mixed or not very evident way."- This naturally 
 led him to investigate its qualifications, and it was his expecta 
 tion to have had opportunities, in the exercise of his profession, 
 of making practical use and exemplification of the results. 
 
 * The first example that is likely to occur to the reader is that of the 
 spandrils of Southwark Bridge ; but the arch, from its construction and 
 depth of material, is quite independent of additional bracing, and the use of 
 the lozenges of the spandrils is merely to connect the arch with the roadway ; 
 thus, the Sunderland Bridge, which is of nearly the same span and of 
 greater rise, and composed of voussoirs of less depth and inferior character 
 for rigidity, is, nevertheless, without spandril-bracing. 
 
 A design somewhat similar to the above, represented by Figure 19, is 
 sometimes used for engine-shafts, etc., and its bracing power is, in such 
 cases, brought into action. 
 
 Another, and, though insignificant, seemingly a very clear application of 
 the method is that employed to brace the transverse pipes between the 
 main chains of the Menai Bridge. 
 
 But the most decided case of its employment, in its simplest form, which 
 the author has met with is that figured in the March part of "The 
 Builder" of the year 1850 (page 100, Vol. VIII.) ; it is the wrought-iron 
 roof over the Strasburg Railway Station at Paris. 
 
 Town s Lattice Bridge depends, no doubt, for its strength upon the method, 
 but presents an example of it in a very compound state, and which might 
 have been adopted without a knowledge of the efficiency of the method in 
 its simplest form. (Figure 22.) 
 
IV PREFACE. 
 
 Such opportunities, however, not having as yet occurred, he 
 adopts the less congenial means of the pen to place his inves 
 tigations in such a position that they may be capable of becom 
 ing useful. In these investigations the simpler geometrical 
 processes have been preferred to those of a more abstruse 
 character. 
 
 The h rst draft of the work was strictly confined to the par 
 ticular kind of bracing above referred to, which may be 
 appropriately named the double-acting or triangular method ; 
 but, upon after-consideration, the plan has been somewhat 
 extended, as it was judged that a few moderate additions 
 would render the work more useful, and better fitted, though 
 very imperfectly, to supply an important desideratum in the 
 library of the engineer. 
 
 For dignifying the little work with the title of "A Treatise 
 on Bracing," it is offered as an explanation that, though de 
 voted almost exclusively to one method, the results obtained 
 may be easily applied to all other varieties, with the exception 
 of plate-bracing. This last kind, indeed, is of so peculiar a 
 nature that a completely different process must be employed 
 to elucidate its action ; probably the results of experiment 
 must be principally depended upon to give us correct ideas of 
 its capabilities. 
 
 As bridges are the structures in which bracing is most exten 
 sively employed, much will be said of them ; but, as the inten 
 tion is merely to explain its application, those parts which 
 either influence or are influenced by the character of the 
 bracing alone come strictly within the scope of this treatise. 
 In a few instances the author may have gone beyond this 
 intention, in the belief that the space so employed would not 
 be deemed by the reader unprofitable. 
 
PREFACE. V 
 
 Iii conclusion, the writer begs to say that in launching 
 his little bark upon the seas of publicity, he is not expectant 
 of its passing triumphantly through the testing billows of criti 
 cism ; these may disclose many deficiencies in its structure 
 with some of which, however, its architect is already acquainted ; 
 but the first vessel built upon a new principle is always, in 
 some degree, experimental, aiding man to construct in the fu 
 ture a perfect one of the same class. Seldom, upon a new 
 subject, was a volume ever published that had not to undergo 
 many changes and enlargements ere it ranked as a standard. 
 
OF THE ^^ 
 
 IVERSITY) 
 OF >/ 
 
 CONTENTS, 
 
 CHAPTER INTRODUCTORY ON BRACING. 
 DEFINITIONS ELEMENTARY BRACED FORMS FAULTY STRUCTURES, 1 
 
 PART FIRST THEORY AND DATA. 
 
 CHAPTER I. 
 
 ON THE DOUBLE-ACTING OR TRIANGULAR METHOD OP BRACING. De 
 velopment Efficiency Advantages, 5 
 
 CHAPTER II. 
 
 APPLICATION OF THE TRIANGULAR METHOD TO VARIOUS STRUCTURES. 
 New View of the Action of a Bracing Division of Structures into 
 Four Classes Table of Symbols Used, 7 
 
 CHAPTER III. 
 
 THE PRESSURES THAT MAY ACT IN STRUCTURES OF CLASS FIRST. Pro 
 cess of Investigation Adopted Examples Formula? for the Maxi 
 mum Strains in the Braces and in the Main Beams, . . 10 
 
 CHAPTER IV. 
 
 THE PRESSURES THAT MAY ACT IN CLASS SECOND. Strains in the 
 Braces, etc. Transition into Class Third Triangular Forms, . 14 
 
 CHAPTER V. 
 
 THE PRESSURES THAT MAY ACT IN CLASS THIRD. Proper Form of 
 Arch Principle of Investigation Examples Means of Facilitat 
 ing its Application General Formulae Table of Formulas for the 
 Strengths of the Braces Pressures in the Arch and Tie A Form 
 having More Uniform Strains Effects from the Line of Pressures 
 not lying in the Arch The Arch with External Bracing, . . 18 
 
Vlll CONTENTS. 
 
 CHAPTER VI. 
 THE PRESSURES THAT MAY ACT m CLASS FOURTH, . . . .26 
 
 CHAPTER VII. 
 
 ON THE INCLINATION OP THE BRACES, ETC. The Proper Degree of In 
 clination of the Braces Poly-systemed Bracings Plate-Bracing, 
 its varieties very numerous Comparison of the Methods, etc., . 27 
 
 PART SECOND CONSTRUCTION. 
 
 CHAPTER I. 
 VARIOUS BRACINGS REQUIRED IN DIFFERENT STRUCTURES, . . .81 
 
 CHAPTER II. 
 
 CONSTRUCTION AND APPLICATION OF THE BRACES. General Remarks 
 Wooden Braces Ditto with Iron Sockets Iron Braces, Cast and 
 Wrought, 35 
 
 CHAPTER III. 
 
 CONSTRUCTION OF OTHER PARTS CONNECTED WITH THE BRACING. 
 Laminated Wooden Arches and Beams Springings of Wooden 
 Arches Iron Structures Springings of Iron Arches, . . .39 
 
 CHAPTER IV. 
 
 SUPPLEMENTARY. The First Class, though Weaker, has many Advan 
 tages over the Others Triangular Tube Bridge Internal generally 
 Superior to External Bracing for an Arch Strength of Wooden 
 Braces The whole Length of a Structure must be Braced Designs 
 Compounded of the First and Third Classes The Strains in the 
 Longitudinals of Class First may be spread more uniformly over 
 them Further observations on the relation between the Arch and 
 the Line of Pressures The Line of Tensions in the Inverted Arch, 41 
 
OF THE 
 
 UNIVERSITY 
 CALIFORNIA- 
 
 TREATISE ON BRACING, 
 
 INTRODUCTORY CHAPTER. 
 
 DEFINITIONS. 
 
 IT is necessary to state here the sense in which several terms 
 will be employed; as it has been found expedient to restrict 
 the meaning in most cases, thus rendering the words so much 
 the more precise. 
 
 When a construction is so framed that, under the action of 
 any arrangement of forces, the angles cannot alter, nor there 
 fore the shape, it is described as a completely traced, or simply 
 a braced^ structure. 
 
 When a fabric is adapted to retain its shape under the action 
 of a force applied in one direction only, it is said to be only 
 partially braced, or braced sufficiently for the circumstances; 
 as in the case of the simple roof, Figure 2. 
 
 When the external figure of the structure is not, of itself, 
 such as to ensure the retention of its form (i.e., when not a tri 
 angle), the addition to effect this is called the tracing, and 
 each piece is named a strut, a tie, or a brace, according to the 
 office it performs. 
 
 When a part is only required and adapted to resist compres 
 sion, it is called a STRUT,* and in the diagrams of this work, 
 when it can conveniently be done, will be represented by two 
 lines, as shown by a, Figure 1. 
 
 When a part is only required and adapted to act against a 
 
 * In speaking of the parts of a structure, if it be desired to distinguish those struts or 
 ties used for bracing it from others, they may be designated as bracing struts or bracing ties. 
 
2 DEFINITIONS. 
 
 force of tension, it is called a tie-beam, tie-piece, tie-bar, or 
 simply a TIE, and will be represented by a single strong line, 
 as b. 
 
 When a part is required and adapted to act both as a strut 
 and a tie, it is called a double-acting piece, complete brace, or 
 simply a BRACE, and will be represented by two lines, as for a 
 strut, with the addition of a central stronger one, as for a tie ; 
 this is shown by c. 
 
 When the parts of a braced figure are so long as to be in 
 danger of bending, and this is prevented by certain additions, 
 these are called, collectively, the trussing, and the beams so 
 stiffened are said to be trussed ; the whole figure so treated, 
 when it does not require bracing, is called a truss. Thus, Fig 
 ure 2 represents a simple roof-truss ; the additional parts, cf,fd, 
 fb, constitute the trussing, and they truss or stiffen the pieces 
 ae, ec, and ac, respectively. 
 
 The term pressure, consistently with the practice of eminent 
 writers, is used to signify statical force, cither of a compressive 
 or tensive character. 
 
 BRACED FORMS. 
 
 As a foundation from which to commence, we assume the 
 following propositions: 
 
 PROP. I. In a triangle, an anirle cannot increase or dimin- 
 
 O O 
 
 ish without the opposite side also increasing or diminishing. 
 
 PROP. II. When the angles of a figure are unchangeable, 
 the shape is unchangeable, and, therefore, the figure is com 
 pletely braced. 
 
 The converse of each of these is also true. 
 
 A triangular structure (Figure 3), having sides that are 
 unchangeable (by I. and II.), is a completely braced form. 
 
 If a quadrilateral figure (Figure 4-) alter its shape (by neg. 
 conv. of II.), the angles alter, and, as the sum of the angles 
 must be equal to four right angles, they cannot all increase or 
 all diminish, therefore (considering the diagonals as third sides 
 
^ orr 6 
 (UNIVERSITY 
 
 Vc, 
 
 ELEMENTARY BRACED FORMS. 
 
 of triangles, by I.) the diagonals must one increase and the 
 other decrease. And hence 
 
 In order that 
 any change of 
 form may take - 
 place in a quad 
 rilatcral figure, 
 
 A diagonal must diminish: 
 A diagonal must increase : 
 
 Any diagonal must either "j 
 increase or diminish- 
 both must change : 
 
 Consequently, all tht f 
 forms are completely 
 braced, as each does noi 
 permit of an effect taking 
 place which would neces 
 sarily result from a 
 change of figure. 
 
 The above forms are the elements of all thoroughly-braced 
 structures. They are of an insulated character. But there are 
 certain modifications of them which depend upon external re 
 sistances; these maybe either partially or completely braced, 
 according as their parts are capable of acting in one or two 
 capacities. The Figures 8-11 represent some of these ; Figure 
 11 is an extreme of Figure 8, one of the legs becoming horizon 
 tal it cannot transmit any vertical pressure to its abutment. 
 
 There are several forms, frequently employed, which are 
 not sufficiently braced, though very considerable stiffness may 
 be produced in them, but only by the use of excessively thick 
 parts. To save time in describing these and their defects, we 
 have given three scries of diagrams in Figures 12, 13, and 14. 
 The first of each, marked A, is a sketch of the objectionable 
 structure ; the second, B, represents, in an exaggerated man 
 ner, the effects resulting from a weight placed eccentrically ; 
 and the remaining diagrams illustrate methods of remedying 
 the defects. 
 
 The last series, Figure 14, requires particular consideration. 
 The first modification of it, C, is too often employed as the 
 bracing of bridges, etc. When the weight is placed eccen 
 trically, it is supported on the principle of Figure 11. It will 
 be readily seen, therefore, that when the bridge is long in pro 
 portion to its depth, and nearly one-half of its length, 1) e, is 
 loaded, a great force will be brought against the point <z, and 
 a very moderate motion of a, or shortening of c a, will -produce 
 
INSUFFICIENT FOKMS. 
 
 a considerable depression of the portion at d. Hence this can 
 not be admitted, as a sufficiently braced form, for any but short 
 spans in conjunction with considerable depth of framing. The 
 second modification, D, on the contrary, is an excellent form ; 
 in it the principle is quite different ; for whereas in the form c 
 an eccentric weight is supported wholly (theoretically speak 
 ing) by one abutment, in this each abutment contributes sup 
 port in a degree inversely proportional to the distance of the 
 weight from it. Of course, any of the completely efficient 
 bracings, Figures 5, f>, and 7, for a four-sided compartment, 
 mav be introduced. The design E is suited to obviate the 
 particular change shown in B, but it is not braced ; other con 
 tortions may take place, from the longitudinals approaching 
 nearer to one another. 
 
|p art tl)c .first. 
 
 THEORY AND DATA. 
 
 CHAPTER I. 
 
 ON THE DOUBLE-ACTING OR TRIANGULAR METHOD OF 
 
 BRACING.* 
 
 To develop this method, let us take the comprehensive case 
 of two parallel beams requiring such bracing between them as 
 shall prevent any change of form in the structure that might 
 result from the action of any forces whatever thereon. 
 
 To satisfy this requisite of permanency of shape, the most 
 efficient methods at present in general use, are, in carpentry, 
 those shown by Figures 15 and 16 ; and, in iron-work, tliose o 
 .Figures 17 and 18. Figure 19 gives the best design for cast- 
 iron beams, etc., when of one piece. 
 
 These are, no doubt, sufficiently strong and effective as brac 
 ings ; but there is, Figure 19 excepted, far more both of mate 
 rial and workmanship used than is absolutely necessary, not 
 only causing needless expense directly, but often also indirect 
 ly, as the excess of weight above what is imperatively required 
 may necessitate the provision of a great deal of additional 
 strength in the structure, simply to bear it ; and this, in the 
 case of bridges, would have the effect of considerably diminish 
 ing the length of span attainable. * 
 
 Let us now consider the method of bracing shown by 
 
 * Tliis method is founded on the simple braced element shown by Fig. 
 7. But the insertion of the chapter is warranted by the further elucidation 
 it affords of this most important subject. 
 
6 DOUBLE-ACTING OK TRIANGULAR BRACING. 
 
 Figure.15. Here, as the forces may vary in their direction, all 
 the diagonals require to be strutted ; but suppose that the ends 
 of one of these diagonal struts (say a. 1) are so jointed that the 
 strut may be enabled to act also as a tie-bar ; such being the case, 
 the other strut (c d) of the parallelogram (add) is no longer 
 required, for when it would have been of use, the modified one 
 (a b), by acting as a tie, is now a sufficient substitute. Giving^ 
 then, to one strut of each parallelogram the power to act either 
 rts strut or tie, and doing away with the remaining one, the 
 braced structure becomes what is seen in Figure 20 or 21, ac 
 cording to the order of procedure. In the figures, the upright 
 pieces are also represented as fitted with the double-acting 
 joints, so that the iron tie-bars or straps, seen in Figure 15, 
 would be superfluous. In Figure 20, the upright pieces are 
 unnecessary in the bracing, as the point a is fixed in position 
 by the braces a 1) and ae doing away, therefore, with the up 
 right pieces of Figure 20, our braced framing becomes what is 
 shown by Figure 22, which is the method of bracing advocated ; 
 and Figure 21 is evidently a variety of it. Figure 22 might 
 have been deduced from Figure 21 by merely altering the in 
 clinations of the braces, producing an arrangement that would, 
 in most situations, be more efficient. 
 
 Figure 22, then, is an effective bracing, capable of resisting 
 a change of form under any disposition of the pressures to which 
 it can be subjected ; and yet, compared with Figure 15 (which 
 is the most generally used design), how simple and light ! The 
 weight of material may be roughly estimated at one-third (the 
 beams braced, of course, not being included); and as to work 
 manship, the saving in that may even be greater, and suscepti 
 ble of much more perfect execution ; and the parts may be made 
 removable for repairs with the greatest ease and safety, which 
 is a matter of primary importance when wood is the material 
 employed ; at the same time, the liability to decay and failure 
 is considerably lessened by the nature of the joint to be used, 
 and by their diminished number. 
 
THE ACTION OF THE BRACING. 
 
 CHAPTER II 
 
 APPLICATION OF THE TRIANGULAR BRACING TO 
 VARIOUS STRUCTURES. 
 
 THE following view of the action of a bracing will often be 
 found useful, in investigating the pressures brought into 
 action : 
 
 In all braced structures, though two longitudinal pieces, at 
 least, are necessary, yet one only of these may be considered as 
 the primary member, all the other parts being looked upon as 
 the means of bracing. Thus, for example, in Figure 23, the 
 beam al> requires bracing; arrange, then, as shown in the fig 
 ure, a series of pairs of braces along it ; each of the triangles 
 a d n, n e <?, etc., is thoroughly braced, and may, therefore, be re 
 presented as solid plates. Now, the application of weights to 
 the beam a 1) will give it the curved appearance represented in 
 Figure 2tt, and this change is necessarily accompanied by the 
 approachment of the apices, d, <?,/, g, of the braced elements to 
 one another, and the dotted lines, which were vertical in Fig 
 ure 23, become here radii of curvature. If we prevent these 
 changes by the insertion of strut-pieces between the apices, as 
 seen in Figures 25, 26, etc., a braced structure will be produc 
 ed. And when the braces are applied beneath, or when the 
 upper beam is considered as the braced member, the intervals 
 between the apices, on the addition of weights, would tend to 
 increase, which must therefore be counteracted by the applica 
 tion of tieSj as seen in Figure 27. 
 
 The action of bracing upon flexible arches, when viewed in 
 this manner, becomes very evident ; and, as this is one of its 
 most important applications, it deserves particular considera 
 tion. In Figure 28, P a Q represents a flexible arch. When 
 a load is placed on the crown, that, of course, is flattened and 
 depressed, and the haunches are raised and further bent ; and 
 
CLASSIFICATION OF STRUCTURES. 
 
 when the weights are placed on the haunches, opposite motions 
 and effects ensue. Now, the arch may be broken in two ways : 
 either from being so unequally loaded that a part of it is rup 
 tured by being too much bent, or, when the arch is so braced 
 as to be retained in its proper form, it is crushed by the 
 pressures transmitted along it ; the load required to destroy the 
 arch in the latter manner is very much greater than is suffi 
 cient in the former ; hence the great utility oi bracing the flexi 
 ble arch. 
 
 In investigating the application of bracing to a great varie 
 ty of structures, the work will be much simplified by arranging 
 them into different classes, though one class may, in reality, be 
 but the extreme of another. Adopting this plan, and commenc 
 ing with the simplest forms 
 
 THE FIRST CLASS that claims our attention is that exempli 
 fied by Figure 29, being two parallel beams with the bracing 
 between them, the whole acting as a girder. This, with the 
 triangular bracing, is an excellent design for bridges, and also 
 for the longitudinal framings of roofs, steamboats, etc., and for 
 scaffolding and other framings. 
 
 THE SECOND CLASS comprehends all structures that act on 
 the girder principle, but which have not the longitudinal beams 
 parallel. The designs shown by Figures 30 to 3 are included 
 in this class. In the first division of these viz., 30-32 though 
 less material may be required than by Figure 29, that advan 
 tage will sometimes be more than counterbalanced by the in 
 creased difficulty of execution. The second division, compre 
 hending Figures 33 and 34, consists of structures not requiring 
 tracing, as that is defined at page 1 the external form, with 
 out addition, being a braced one ; but, when large, they require 
 trussing, and though they do not, therefore, come strictly 
 within the province of this treatise,, yet, as the simplest form 
 of trussing that can be introduced is exactly that of the sim- 
 
SYMBOLS EMPLOYE 
 
 plest bracing, with this difference, that the parts are required to 
 act in only one capacit} r , we may be excused for introducing 
 them. Figure 33 is a design now much used for iron roofs ; 
 the reverse of this, shown by Figure 34, is an excellent design 
 for a rigid suspension bridge. The author has never seen the 
 latter employed with more than three supported points. Of 
 course, to secure rigidity, the sides of tlie triangle must be 
 maintained straiht lines. 
 
 
 THE THIRD CLASS. The structures included in this class 
 are those which derive their strength from a single braced arch. 
 Figures 35 to 42 are examples. These may be divided into 
 three sub-classes -first) those having the bracing arranged 
 along the intrados, but which have not the arches tied, as 35 
 and 36 ; second, those which are braced internally, and have 
 the arches tied, as 37 and 38; and, third, those which have the 
 bracing along the extrados, as 39 to 42. 
 
 THE FOURTH CLASS consists of arrangements of t\vo or 
 more parallel, or nearly parallel, flexible arches, each contribut 
 ing its share of support, and the whole, by the introduction 
 of bracing between them, acting, in a great measure, as a single 
 deep rigid arch Figures 43 and 44. 
 
 TABLE I. 
 
 TABLE OF SYMBOLS USED IN THE FOLLOWING PAGES. 
 
 w = Weight at each loaded point of the span. 
 jS r = Total number of weights or points. 
 W = K w = Total load. 
 s = Distance between the weights or points. 
 S == N" s = Total length or span. 
 
 P = Pressure acting in a given line, a I see Figure 45. 
 Y = The vertical element or component of pressure P, = I c. 
 II == The horizontal element of pressure P, = a c. 
 . = / a I c = The angle at which the given line, a I, is 
 inclined to the vertical, b c. 
 
10 INVESTIGATION^ CLASS I. 
 
 y> = /_ b a c = The angle at which tlic given line, a >, is 
 inclined to the horizontal line, a c. 
 
 T = V sec. 6 -- II sec. g> = ^/H" 4- v f > 
 
 Y = P ~ sec. 6 = P. sin. <p = P cos. = II tan. <p. 
 
 II = P ~ sec. = P sin. = tan. 0. 
 
 CHAPTER III. 
 
 ON THE PRESSURES THAT MAY BE BROUGHT INTO 
 ACTION IN CLASS FIRST. 
 
 THE best process for arriving at the amount of the pressures 
 that may be brought into action in the several parts will be, 
 first, to get the effects produced by each portion of the load 
 separately, and then to find the maximum strains that can be 
 produced by any arrangement of the weights ; for the strains 
 are not all the greatest when the whole length of the structure 
 is loaded, as, by removing the weights from some of the points, 
 the strains in some of the braces may, it will be seen, be in 
 creased. As an example of the method pursued in calculating 
 for the- respite given in the following table, let us investigate 
 the forces arising from one weight = it), placed.at the point 4 of 
 Figure 46. As the support contributed by each pier (from a 
 property of the lever) must be inversely as the distance of the 
 weight from it, the pier Q will support -J-J- w y while the other 
 pier, P, supports the remaining portion of the weight, ^ w. 
 The vertical pressure, \\ w, has therefore to be transmitted, 
 through the bracing, to the pier Q ; the real pressure, P, induc 
 ed in the brace b will therefore be = J| w. sec. / j 5 B, = ] **> 
 sec. 0, and by means of that brace, the vertical element, V= 
 
EXAMPLE OF THE METHOD OF INVESTIGATION. 11 
 
 }-J w, is impressed on the point L This point must bo borne- 
 up by the brace ?/, which is consequently stretched witli the 
 strain of }J iv. sec. 6; this is next transmitted through the 
 brace <?, and so on, till it is finally impressed at Q. The other- 
 portion of the weight, in like manner, causes strains of -? g w. 
 sec. 6, in all the braces from the point 4 to the pier P, of com 
 pression in those which are inclined in the same way as brace 
 8j and of tension in those inclined in the opposite direction.. 
 Proceeding in this manner, for each weight, we obtain the 
 quantities marked against each brace in Figure 47 ; those written 
 on the left side of the brace denote the strains of compression,, 
 and those on the right the strains of extension, which take 
 place in the brace when the bridge is uniformly loaded, each 
 number being the co-efficient of T V sec. 6 ; of course, when two- 
 opposite pressures would act in a brace, the resulting strain, in 
 the brace, is their difference; thus in brace 5 j^ the strain, for" 
 a uniform load, is (+ 25 16). -^ sec. = | sec. .8, which is -f 
 or a compressive strain ; in this way the quantities in column 
 second of Table II. are found. But referring again to the 
 brace 5 j, we see that the tensive forces are produced by the 
 weights 1, 2, 3, and 4 ; consequently, if these be removed, the- 
 full compressive strain of 25 -^ sec. will act in the brace,, 
 which is the maximum strain of compression that can possibly 
 be induced in it by any disposition of the weights ; on the 
 other hand, if all the weights, excepting those at 1, 2, 3, and 4,. 
 be removed, there will be no compression induced in the brace 
 5^ , and therefore the total tensive strain of 61y" 8 sec. will 
 come into play; and this is the maximum of tensive strain that 
 can take place in it ; in this way the third and fifth columns of 
 the table are calculated. 
 
 The maximum strains are those we have to attend to in 
 fixing the strengths of the braces and of their joints ; thus, in, 
 brace w, it is a tensive strain we have chiefly to guard against, 
 whereas the required tensive strength of tho brace I is very 
 moderate, but it must be capable of withstanding four times as* 
 great a compression. 
 
12 
 
 A FOR TIIK MAXIMUM STRAINS. 
 
 TABLE II. 
 
 1. 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 Brace. 
 
 Strains when 
 all the 
 weights arc 
 
 Maximum Com- 
 pressiveStrain 
 by uneven 
 
 Weights re 
 moved to pro 
 duce the Strains 
 
 Maximum Ten 
 sive Strain by 
 uneven Load 
 
 Weights re 
 moved to pro 
 duce the Strain 
 
 
 on. 
 
 Loading. 
 
 in Column 3. 
 
 ing. 
 
 in Column 5. 
 
 e 
 
 + 81 
 
 + 81 
 
 All on. 
 
 Never any. 
 
 
 d 
 
 + 63 +64 
 
 1. 1 
 
 All but 1. 
 
 c 
 
 _|_ 45 -f 49 
 
 1,2. 4 
 
 All but 1,2.! 
 
 b 
 
 4-27 + 86 
 
 1, 2, 3. 
 
 9 
 
 l ,2 ,3,4 ,5,4.i 
 
 a 
 
 4-9 +25 
 
 1, 2, 3, 4. 
 
 16 
 
 l ,2 ,3 ,4 ,5. I 
 
 t 
 
 9 
 
 + 16 
 
 l ,2 ,3 ,4 ,5. 
 
 25 
 
 1,2,3,4. 
 
 u 
 
 27 
 
 + 9 
 
 l ,2 ,3 ,4 ,5,4. 
 
 156 
 
 1,2,3. 
 
 V 
 
 45 
 
 + 4 
 
 All but 2,1. 
 
 49 
 
 1,2. 
 
 w 
 
 63 
 
 + 1 
 
 All but 1. 
 
 64 
 
 1. 
 
 The numbers in the Table are the co-efficients of ~. sec. 0. 
 
 = j?. sec. 0, here, as N = 9. 
 
 From the foregoing and similar data, the following formula 1 
 are deduced, the load being applied above : 
 
 The maximum compression of the last brace (1 Q, Figs. 46 
 and 47) is = i*^- 2 w. sec. ~ w. sec. = ^ sec. 0. 
 
 The maximum tension in in 1, and compression in 2 7/1, = 
 (N-D ? Ror > a 
 
 . It/. OCO <-/. 
 
 The maximum tension in I 2, and compression in 3 Z, = 
 ( ^* W. sec. 6?. 
 
 And putting n to represent the number of the pair of 
 braces as counted from the pier, along the lower beam, the 
 
 (N-n)* 
 general formula will be -^ w. sec. 6. 
 
 When N is even, the formula for the central pair of braces 
 will be (N-~) w. sec. 6 = y w. sec. = y sec. 0. 
 
STRAINS IN THE LONGITUDINAL BEAMS. 13 
 
 Carrying the calculation beyond the centre, we obtain the 
 maxima of the other strains that can act in the braces. 
 
 We see, then, that the extreme braces require to be four 
 times as strong against compression as is necessary for the cen 
 tral ones, when N" is even, arid very nearly four times when !N 
 is odd and large. And starting with the value y sec. 9, at the 
 centre (supposing N even), the tensive strains in the braces 
 parallel to #, Figure .46, gradually diminish as they approach 
 Q, where they vanish, but increase as they approach P, where 
 the last amounts to (N-l) 2 ~ sec. 6 ; and in these braces the 
 compress! ve strains are also = y. sec. 0, at the centre, but gra 
 dually increase as they approach Q, where they amount to y 
 sec. 0, and diminish as they approach P, where the compres 
 sion in the last is only sec. 6. The strains in the braces 
 parallel to s, of course, diminish and increase in the opposite 
 directions. 
 
 N.B. These maxima are obtained on the supposition that 
 the whole value of w may be removed from any point, but, as 
 the weight of the bridge itself is included in W, and cannot be 
 altered in its distribution, the effect of uneven loading, as 
 given by the above formulae, will be rather exaggerated, parti 
 cularly towards the centre of the span. 
 
 The maximum strains of the longitudinal beams take place at 
 the centres of their lengths, where they amount to* ~- = ^."W, 
 1) being the depth of the framing. This amount of strain is 
 compressive in the upper and tensive in the lower beam. The 
 strains diminish towards the ends, and at the last lays they 
 amount to (^ +-^) w tan. 6 = (N"-l) w tan. 9 of compres 
 sion in the upper beam, and to ^ tan. 6 of tension in the 
 lower, 9 being the inclination of the braces with the vertical. 
 
 And if ri denote the number of the bay, counted from either 
 
 S = span or length. 
 
 OP THE 
 
 UNIVERSITY 
 
14 INVESTIGATION OF CLASS II. 
 
 end, we have the following formulas for the strains in the lon 
 gitudinals (the load being on the upper level, and the terminal 
 braces being struts) : 
 
 n (X ri) tan. 0. iv, for the compressions in the upper 
 beam ; and 
 
 < ?l (X n ) y + n \ tan. 0. iv, for the tensile strains in 
 the lower beam. 
 
 When the load is sit the lower level (and the terminal 
 braces are struts), the strains will be as above, except that, for 
 the lower beam, each will be-J-w tan. 6 less. 
 
 NOTE. Turning to Figure 47, we observe that, when the weights aro 
 placed on the upper beam, the maximum strains producible in any two 
 braces which meet at the lower beam have equal vertical elements ; this 
 necessarily results from the lower beam being horizontal ; consequently, in 
 the form of bracing shown by Figure 21, the same equality of the vertical 
 elements must also exist. 
 
 If the load were placed on the lower beam, the foregoing observation 
 would apply to each pair of braces which join at the upper beam. The 
 formulas, etc., which have been given, are calculated on the supposition that 
 the load is applied at the upper beam, and they would therefore require to 
 be slightly modified to suit the case of a suspended load. 
 
 CHAPTER IV. 
 
 ON THE PRESSURES THAT MAY BE BROUGHT INTO ACTION 
 IN CLASS SECOND. 
 
 As a representative of this class, take Figure 49. Here, the 
 distance between the points of support being uniform through 
 out the span, the angle 6 for the braces is variable. 
 
 The maximum compression of the extreme braces, a c and 
 I Z, takes place when all the weights are on, and, as half W is 
 upheld by each, the strain will be = y sec. 0. But the next 
 braces will not be strained with so great a tension as would 
 take place in a bridge of class first with the same depth, c n ; 
 
CLASSES I. AND IT. COMPARED. 15 
 
 being relieved from a portion of their duty of imposing the 
 whole of (N 1) . on the points e and Z, by tlie inclined action 
 of the upper beam. This will be made more evident by the 
 following explanation of Figures 50 and 51 ; in these tho 
 depths at c, and the values of s, are the same, and also, there 
 fore, 6 for the braces. Let a g represent in each figure the 
 upward resistance of the pier, = y, then will a c represent the 
 compression induced in the brace a c ; resolving a c into the 
 directions, b c and c d, of the other brace and of the upper 
 beam, the tension produced in the brace, l> c, is c b in 
 Figure 50, but only = c e in Figure 51. Further, we observe 
 that the compressions in the upper beams of the Figures 50 
 and 51 are a 1} and a e respectively; the tension of the lower 
 beam, beneath c, is the same in each, and g c. 
 
 The strains at the centres of the longitudinal beams in 
 Figure 49 are the same as for a bridge of class first having the 
 
 O O O 
 
 same depth as it at the mid-span. Let us compare the strains at 
 the extremity of the latter, represented by Figure 52, with the 
 same of 49. represented by Figure 51. Make a g the same in 
 each, and let it represent ~, then a c will measure the compres 
 sion in the extreme brace ; that in 52 is the least. The tension 
 of the lower beams is measured by g c in each figure, and is 
 greatest in 51. Resolving a G into the directions e d and b c, cf 
 is the resulting compression in the upper beam in each figure, 
 and it is evidently the least in 52. The resulting tension in 
 the brace c b is represented by c e in each figure ; the compa 
 rative amounts are shown in Figure 51. by c e and c v, and c e 
 diminishes as the an^le a c d increases, vanishing as that be- 
 
 omes = 180. 
 
 The advantages, then, of a design of class second over a 
 similar one of class first having the same depth at the middle 
 may be stated as reducing the strains in all the braces, except 
 the extreme one at each end (which is rather more strained, 
 becoming, in some degree, a continuation of the upper beam) 
 the central ones,, however, are little altered also as reducing 
 
16 TRUSSED TRIANGULAR FORMS. 
 
 the lengths of the braces towards the extremities ; and as 
 rendering the longitudinal beams more uniformly strained 
 throughout their lengths, being equalty so with class first at the 
 centre, but more so than in it at the extremities. In fact, as will 
 afterwards be seen, this class is a step towards some forms of 
 class third ; it becomes identical with them when a c, Figure 
 49, forms a continuation of the curve of the upper beam (sec 
 Figures 37 and 38), or when a pressure is applied at c in the 
 manner of an abutment, in the direction c d, as indicated by the 
 dotted arrow. (See Figure 35.) 
 
 Though the triangular forms, shown by Figures 33 and 34, 
 as stated in chapter second, are not traced, and consequently 
 do not demand our attention, yet, on. account of their utility 
 and the reasons there given, we will introduce them here. 
 
 The weight at point 2, Figure 53, must cause equal strains 
 in 1 a and 3 , as the angles 1 a 2 and 3 a 2 are equal, and 1 a 
 and a c form a straight line ; so that half the weight 2 is trans 
 ferred to point 3, and, along with the weight at 3, is impressed 
 on J, the total weight, ~ c -, at I is distributed, so that ~ is car 
 ried to 1, and ~- to point 4 (see note at end of chapter) ; the 
 total weight accumulated at point 4 is = 3 w, one w being 
 received from each of the points b and d, and this is impressed 
 on c, and thence transmitted, equally, to 1 and 1 . 
 
 Xow, the rule, derived from a property of the lever, that a 
 weight at 2 is upheld by the piers, 1 and 1 , in the proportion 
 of 5 to 1, holds good here ; for though in the first effect of 
 the weight at a only -J- is carried to pier 1, yet at each succeed 
 ing point another fraction will be transmitted to 1, by the rod c 1, 
 until we arrive at the lowest point, e, where the remnant of y, 
 amounting to ! |-, will be equally divided between 1 and 1 , so 
 that *g- will be the only portion of the weight, w, at point 2, 
 which the pier 1 can receive. 
 
 After what has been said, little explanation of the Figures 
 
TBIAKGULAB STBU 
 
 55 and 5G will be required. In these, for the sake of having 
 simpler numbers, the length of the side A C is made a multiple 
 of the central depth h 0. The numbers (co-efficients of w), 
 attached to the parts are calculated on the foregoing principles, 
 and give the real strains or values of P, which take place from 
 a uniform loading. In this form of structure, the maximum 
 strains are produced when all the weights are on. Of course, 
 when, to serve for a roof or other purpose, the design is inverted,, 
 as Figure 57, the strains, with very trifling and evident excep 
 tions,* are all the same in degree, but of opposite character. 
 
 NOTE. In Figure 5i, a weight at 5, = w, = &, is distri 
 buted thus: It is resolved into b c and 1) d; and further 
 resolving b d into a horizontal and a vertical pressure, b x is the 
 latter, which is, therefore, the portion of w, or a I, that the rod 
 1) h receives ; and resolving b c similarly, we get c e, = a x, as 
 the vertical,- which represents the portion of w borne by the rod 
 I g. The triangle b x d is evidently similar to the triangle a x 
 f y . . I x : x d = a x : xf, and alternately, b x : x a= dx: xf.. 
 But triangle d af is similar to g b /i, therefore the lines a x 
 and b a, drawn from corresponding angles, and at right angles 
 with the opposite sides, must divide these sides similarly, there 
 fore d x : x f g a : a h, and consequently b x : x a = g a :. 
 a h. The fact that the weight at I is distributed to h and g in 
 the proportion of g a : a h might have been anticipated from 
 the property of the lever. 
 
 cj a : a h = tan g b a : tan h I a. b d ^ w sec. a b h. 
 And b c ~^ t w sec. a b g ; the resulting horizontal pressures 
 are, of course, the same for each direction, and are d x and b e 
 = b d. sin a b h. 
 
 *Arising from the weights being on the slant sides, instead of being: 
 attached to A B, which would be the true arrangement in Figure 56 inverted, 
 and consequently relieving each perpendicular piece of 1 w. 
 
IS PROPER FORM OF ARCH. 
 
 CHAPTER V. 
 
 ON THE PRESSURES THAT MAY BE BROUGHT INTO 
 ACTION IN CLASS THIRD. 
 
 WE now come to consider the case of a braced flexible urcli. 
 The form given to the arch should, theoretically, be the curve 
 of stability for the fully loaded arch (in the investigations it is 
 assumed to be this, so that, in the internally-braced arch, no 
 action will be induced in the bracing when all the weights are 
 on). This curve, in general, will be found to lie between a 
 catenary and a parabola ; but, as the quantity of the curve 
 .generally used is not great, a circular arc may be adopted, as it 
 very nearly coincides for almost 120 with the theoretical 
 figure. When an arch of a greater number of degrees is re 
 quired, and the structure is of a light character, it is generally 
 .advisable to use the parabolic form. But in a braced arch we 
 may depart considerably from the correct curve without much 
 evil ; some of the braces might require to be made stronger, 
 .and a trifling variation would take place in the strains of some 
 of the other parts. 
 
 When the load is uniformly distributed on the arch there 
 .is no bracing required, and consequently (if the line of pres 
 sures remains in the arch*) there is no action induced in the 
 bracing. When the load is suspended from the arch by means 
 of the braces, of course there are tensive strains produced in 
 them. 
 
 As, in the investigations, the values of 10 and s are constant, 
 and the weights of the arch and other parts are not considered, 
 the curve of equilibrium will be a parabola, or rather will be 
 polygonal, having its angles arranged in a parabola. 
 
 * The curious properties which occasion this restriction will be found 
 mentioned at page 25. 
 
INTKADOSAL BKACIXG. 19 
 
 THE ARCH WITH INTERNAL BKACING. 
 
 The following fundamental principle of investigation is 
 capable of very general application ; it might have been intro 
 duced in Chapter III., but the method there adopted is a 
 simple one deductible from it for that particular case, viz,, 
 where the longitudinal members are all horizontal. From a 
 law of the lever, we know that the vertical elements of the 
 pressures upon the abutments, resulting from a weight, w, 
 placed at the point c of Figure 59, must be inversely as the 
 horizontal distances of c from them ; here, therefore, the pier P 
 will bear .75 10, while the pier Q supports the remaining portion, 
 .25 w. The vertical pressure of the weight w at c must be 
 resolved into the directions c P and c Q, the portions of the 
 bridge, P b c s and Q/"c t, acting as instruments for the trans 
 mission of the respective pressures to the piers P and Q; in 
 acting as such, let us trace the strains produced in the parts. 
 The pressure in c P will be resolved into the directions c b and 
 c s, and the vertical elements, or the values of Y of these two 
 pressures, since they both act downwards, must, added together, 
 equal .75 w ; the pressure in cs is at s resolved into pressures 
 in s P and I s / that in 1) s is a tensive strain, and must, since 
 s P is horizontal, contain the same vertical element as the pres 
 sure in c s. The resultant of the two strains, in c I and I s, 
 conveyed to b, must have the direction l> P, and possess a vertical 
 clement = .75 w , for the resultant of the whole, acting in the 
 line c P, passes through P, and as the pressure 6- P is horizontal 
 and passes through P, the remainiog component pressure, Z> P, 
 must also pass through P, and must contain the whole vertical 
 element. The pressure Z> P, is resolved into I) a and I ?, and 
 I) r is resolved into r P and a r ; the resultant of the compres 
 sion in I) a, and the tension in a r, must have the direction 
 a P, and contain a value of Y .75 w. The final resultant of 
 the pressure in a P, and the horizontal pressures, arising at the 
 points s and ? , must have the direction c P, and its vertical 
 element = .75 ?/?. .Returning to the point <?, the pressure in 
 
20 IXTRADOSAL BRACING. 
 
 c* Q is resolved into the directions c d and c t, and since c d 
 acts upwards, the vertical element of c t must exceed V of c d 
 by .25 w, as that is the downward vertical element of the resul 
 tant to the pier Q ; the pressure in c t is resolved into t Q and 
 d t. The resultant of the two pressures conveyed to d, by the 
 compression in c d and the tension in d t, must have the direc 
 tion d Q, and contain a vertical element = .25 w. The resul 
 tant in d Q is resolved into d e and d u ; but we need not 
 proceed further in tracing the transmission of the pressures to 
 Q ; sufficient has been said to explain the principle. A\ r e see 
 that (when the lower longitudinal member is horizontal) the 
 resultants at the various points of the arch, at either side of the 
 weight, are directed to the pier or spring of the arch at that 
 side ; and all those at one side of the weight have the same 
 vertical element or value of V, which is eventually imposed on 
 the pier at that side. 
 
 We will now give, in considerable detail, the investigation 
 of a simple example having five points or weights, shown by 
 Figure 58. Let a weight be placed at the point #, and repre 
 sented by the line 10 a = 1 ; this will be resolved in the direc 
 tions of the piers P and Q, and the amount of the pressures 
 produced in these directions will be = af and a m = ao ; the 
 vertical elements of these will, as given by the lever rule, be 
 = 0.9 and 0.1 or =fn and n m ; the horizontal element or 
 value of II is the same for each, and = n a. The resultant a & 
 is resolved into a I and a t a t = a g into </ u and rg and so 
 on throughout. 
 
 Table III. gives the results for the various parts, from the 
 action of one eccentric weight at a. 
 
TABLE III. 
 
 Vertical Elemeiits 
 or Values of V. 
 
 Real Stra ns or 
 
 Values of P =- 
 
 V see. 9. 
 
 w a = vertical weight at a 
 
 a f resultant to pier P 
 
 m a = a o = resultant to pier Q 
 
 n a = value of II of each resultant. . 
 
 Pressure in a b (measured by a 1) 
 
 " in a g (measured by a ).... 
 
 gu 
 
 in ff b, tensive 
 
 Resultant b Q 
 
 be , 1 
 
 bh I 
 
 h v 
 
 he ! 
 
 Resultant c Q 
 
 cd | 
 
 a j 
 
 iw ! 
 
 id ! 
 
 Resultant d Q I 
 
 d e. . . 
 
 v 
 
 4* 
 
 je 
 
 Resultant e Q 
 
 Q* 
 
 1.000 
 .900 
 .100 
 
 .300 
 .400 
 
 .400 
 .100 
 .050 
 .150 
 
 .150 
 .100 
 .034 
 .006 
 
 .066 
 .100 
 .075 
 .025 
 
 .025 
 .100 
 
 .100 
 
 1.000 
 
 1.260 
 
 .900 
 
 .895 
 
 .582 
 
 .563 
 
 .580 
 
 .4364 
 
 .333 
 
 .255 
 
 .165 
 
 .125 
 
 .161 
 
 .216 
 
 .173 
 
 .071 
 
 .055 
 
 .0726 
 
 .168 
 
 .1455 
 
 .0273 
 
 .036 
 
 .035 
 
 .140 
 
 .099 
 
 .100 
 
 Horizontal thrust = na = yz = gu-{-hv-\-iic-\-jx-} r Qk. 
 " = 895 = .580 + .125 + .055 + .036 + .099. 
 
 Tables, similar to this, being constructed, one for each weight, 
 it is easy to discover by what arrangement of the weights a 
 maximum strain in any part is produced, and the amount of 
 that maximum. Thus, taking the brace 1) h, we have produced 
 
 in it: 
 
 TABLE IV.* 
 
 By weight a. . 
 b.. 
 
 Real Strains or Values of P. 
 
 .3674 ) 
 
 .2200 - 
 
 .0726 
 
 
 = .660 
 
 = .600 
 
 Values of V. 
 
 -f .150 
 -f .450 
 
 .334 
 - .200 
 
 .066 
 
 + .600. 
 .600 
 
 * These values are taken from tables similar to Table III., but it would 
 be useless to give them here. The sign -f- indicates a compressive, and a 
 tensive strain. 
 
22 FOKMUL-E FOR THE MAXIMUM 
 
 From Table IV. it is evident that when the weights a and 
 b alone are on, there is a maximum compression produced in 
 the brace o h ; and when these are removed, and the other 
 weights, c, d, and c, are placed on the arch, a maximum strain 
 of tension takes place of exactly the same amount. When all 
 the weights are on (agreeably with a rule previously given, since 
 the contour of the arch is such as to render it equilibrated 
 under the whole load), the compressive and tensive strains 
 neutralize each other, and no effect is produced in the brace. 
 
 The maximum compression for any I race is equal to the 
 maximum tension for the same ; hence, calculating the com 
 pression is sufficient, as the result is the measure of the neces 
 sary strength to be given to the brace to resist both compression 
 and extension. 
 
 At any point of the arch (<?, Figure 59) we can easily find, by 
 the lever rule, the value of V for each resultant from that 
 point to one of the piers (Q), caused by each of the weights (c, 
 6, a) situated at and to the other side of the point (it is .25, 
 .15, and .05); now, the greatest compound resultant that can 
 act in the same line, will evidently have the value of V in it 
 equal to the sum of the values of V above mentioned (thus, .25 
 _|_ .15 _[_ .05 = .45), and its amount will be V multiplied by 
 the secant of its angle with the vertical (P = .45. sec. 0). 
 Having thus the compound resultant, in value and direction, 
 all we have to do, in order to find the maximum strain (= 1.097) 
 in the first brace (ct) from the point, is to resolve it into the 
 directions of the chord of the arch (cd) and of the brace. But 
 the following formula? offer very ready means of calculating the 
 necessary strengths of the braces, as they give at once the 
 value of V for the maximum strain in each. 
 
 The formula for the end pairs of braces (as a r, r b, and./ z y 
 z i, of Figure 59) is g. (N-l)= ^. w == V. 
 
 The general formula, putting n to denote the number of the 
 pair as counted from the pier, is 
 
STRAINS IN THE BRACES, ETC. 23 
 
 The second column of the following table gives the formula 
 for each of the first six pairs of any bridge. The last two col 
 umns contain the results for the case of Figure 59, wherein 
 N" == 10. 
 
 The values of Y for the braces are not altered by a change 
 of the rise or span of the arch, while N and w remain the same ; 
 hence having calculated the vertical elements for the given 
 value of N (as is done in Figures 60 and 61), it only remains to 
 multiply each by the particular value of w. sec. 0, in order to 
 obtain the strains, or values of P. 
 
 TABLE Y. 
 
 Value of n 
 or No. of 
 the Pair of 
 Braces. 
 
 Equations for the 
 Value of V, a maximum 
 in the Braces. 
 
 Values of V 
 whenN 10, 
 as in Fig. 59. 
 
 Values of P = V. sec. 6 
 or Real Strains for Fiff. 
 59, wherein the Hise to 
 Span = 1 : 4.75. 
 
 1. 
 
 L^H). W 
 
 2N 
 
 0.45 W 
 
 (ar = 0.702 w 
 (r b = 0.496 w 
 
 2. 
 
 2 (N-2) 
 ~2N~" 
 
 0.80 w 
 
 ( b s = 0.882 w 
 ( s c 0.836 w 
 
 3. 
 
 3 (N-3) 
 
 2N 
 
 1.05 w 
 
 \ct = 1.097 ic 
 ( t d = 1.085 w 
 
 4 - 
 
 4 (N-4) w 
 2N 
 
 1.20 ID 
 
 (du = 1.240 w 
 \ u e = 1.235 w 
 
 5. 
 
 5 < N - 5) w 
 ~2~N~ 
 
 1.25 w 
 
 (e v = 1.286 w 
 \v f= 1.286 w 
 
 6. 
 
 n. 
 
 6 (N-6) w 
 
 1.20 ID 
 
 (fw = 1.235 w 
 \ w g xr 1.240 w 
 
 ( * i 
 
 2N 
 n <X-n) w 
 
 2 N 
 
 Each weight produces compression throughout the arch ; 
 therefore the compression will be a maximum when all the 
 weights are on. The vertical element of the compression at 
 the springing or foot will evidently be the half-load ; and for 
 any point, it will be half the load above the level of that point.. 
 
24 ADVANTAGES OF FIGURE 49. 
 
 Of course, the real compression at any point in the arch is = 
 the vertical element multiplied by the secant of 6 there. 
 
 It is scarcely necessary to mention the well-known fact that 
 the horizontal element is, for an equilibrating load, uniform 
 throughout;* and that the horizontal thrust for the wholly 
 loaded arch is the measure of the maximum compression of the 
 arch at the crown. 
 
 The strain in the lower beam, when it acts as a tie to the 
 arch, is greatest when all the weights are on ; the tension is 
 then uniform throughout the length, and equal to the horizontal 
 thrust of the arch, or to the compression in the arch at the 
 crown. 
 
 When the lower beam does not act as a tie f to the arch, 
 but simply as an adjunct to the bracing, it undergoes, principally, 
 compressive strains, and these are rendered a maximum for one 
 half-length of the beam, when the weights above that half are 
 removed. Under such a loading, the compressive pressures in 
 the lower beam increase as they approach the pier at the un 
 loaded side, where the compression may amount to about one- 
 half the value of the horizontal thrust for the wholly loaded 
 arch. (See XOTE.) 
 
 SCHOLIUM. Reviewing what has been said, we observe that 
 in class first the strains in the braces increase towards the 
 extremities, and that there, for the same brace, the tensive and 
 compressive maxima are very different in value ; whereas in the 
 
 -When the load is not an equilibrating one, the braces come into 
 action, and transmit parts of the horizontal pressures to the piers; but if a 
 vertical section be made at any part of the span through the arch, bracing, 
 etc., then the sum of the horizontal elements, in the various parts of the sec 
 tion, is a constant quantity for any one loading. From this property, the 
 line, which in the subsequent pages is called the line of pressures, has been 
 named, by Mr. W. H. Barlow, the line of equal horizontal tkrutt ; it is the 
 resultant of all the pressures acting in the structure, with the exception, 
 when the arch is a tied one, of the strains occasioned, by the horizontal 
 thrust of the arch, in the tie-beam. 
 
 \ A tied arch of this form is evidently the extreme of the change of claps 
 first (Figure 4G) into Figure 40 of class second. It might, indeed, without inr 
 propriety, have been included as a variety of the latter class. 
 
EXTRADOSAL BRACING. 2/5 
 
 form treated of above (Figures 59, etc.), the central braces are 
 those most strained, and every brace has its compressive and 
 tensive maxima equal. Now, there is an intermediate form 
 between these two, and its parts will undergo strains of a 
 medium character; such is Figure 49 of the second class, and 
 it is possible so to modify that form as to secure a very equable 
 distribution of the strains, both in the braces and in the longi 
 tudinal members. 
 
 NOTE. In the form Figure 59, with the lower beam not 
 acting as a tie when all the loading is on, the line of 
 pressures, of course, corresponds with the line of the arch. 
 But when the load is only placed on a space at the crown, the 
 line of pressures resulting is shown by the dotted line in Figure 
 62. Where the line rises above the arch, the lower beam, im 
 mediately beneath that part, acts negatively or as a tie ; and 
 when the line falls within the arch, the lower beam, at that 
 part of the span, acts positively, or as a strut, conveying part 
 of the compression to the abutment. These observations upon 
 the relation between the position of the line of pressures, with 
 regard to the arch and the nature of the pressures induced in 
 the lower beam, apply universally. Figures 63 and 64 give an 
 idea of the positions the line may assume when half the length 
 of the arch is loaded. 
 
 THE ARCH WITH EXTERNAL BRACING. 
 
 In this variety (exemplified by Figure 65), the line of pres 
 sures for the fully-loaded arch does not necessarily coincide 
 with the arch; for it has been demonstrated by Mr. "W. II. 
 Barlow (in an excellent and valuable paper read at the I.C.E., 
 July, 1847, a report of which is given in the Civil Engineer 
 and Architect s Journal, Vol.X.,p. 211) that the real curve of 
 action is that which produces the least horizontal thrust- 
 namely, that curve (parabola here) which rises to the neighbor 
 hood of the extrados, which is here the upper beam, as shown 
 by the dotted line."" 
 * The curve or line of pressures may extend beyond the structure when 
 
26 AKCIIES WJTII INTERMEDIA!, BRACING. 
 
 Nevertheless, it will be proper to give sufficient stability to 
 the abutments, and strength to the arch and bracing, to with 
 stand the pressures produced when the central line of action 
 coincides with the arch, as such a coincidence would take place 
 if the upper part, <?, should become injured or materially short 
 ened. If the part c become slightly shortened (from the great 
 compression), the crown of the line of pressures will descend 
 proportionately. 
 
 Unlike the foregoing case of Figure 37, of internal brac 
 ing, the values of V for the braces, when 1ST is given, are not 
 constant. 
 
 CHAPTER VI. 
 
 OX THE PRESSURES THAT MAY BE BROUGHT INTO 
 ACTION IN CLASS FOURTH. 
 
 THE introduction of bracing between the parallel, or nearly 
 parallel, flexible arches of this class renders the whole struc 
 ture in its action a single rigid arch. The dotted line in 
 Figure 6G shows, as stated in last chapter, the line of pressures 
 (W being all on) as long as the structure remains perfectly 
 without change ; but the centre of the upper arch and the ends 
 of the lower one, from their compressibility, will become 
 somewhat shortened, and from these changes the line of pres 
 sures will sink at the crown, and not descend so far at the 
 springings, becoming, in fact, more nearly identical with the 
 midway line; it approaches this as the pressures become more 
 nearly equal to the ultimate strength of the compound arch ; 
 and if one arch be stronger than the other, the line will lie 
 nearer to it, as it recedes from an arch in proportion to the 
 compressibleness thereof. 
 
 that ia braced ; but if the parabola rise above the upper beam, the notion 
 will partake, in a proportionate degree, of the nature of that of the girder 
 the arch beneath the part being then in a state of tension. 
 
AKCIIES WITH INTEKMKDIAL JJKACING. 27 
 
 Such arc the deductions in theory; but to ensure these, we 
 mast have a structure far more perfectly framed than is 
 practicable. Let us see what the shortcomings of practice may 
 entail. 
 
 If, at the time of the erection of the structure, or from 
 after-causes, such as decay or other damage, the abutments, at 
 the different footings, be not all applied with the same accu 
 racy, disturbing causes will be introduced ; and when thefaulti- 
 ness of an abutment is so great that the arch there is relieved 
 from pressure, it becomes only a part of the bracing of the 
 other arch, through the spring of which the line of pressures 
 must necessarily then pass. From this it is evidently proper 
 that one arch should be made sufficiently strong to carry the 
 whole load, unaided by the other as such. 
 
 CHAPTER VII. 
 
 WE will now investigate the effect of inclination, o^tlie value 
 of 0, of the braces, on economy. Let Figures 67 and 68 repre 
 sent two structures, having the same span and depth of framing, 
 and loaded to the same amount. Referring back to Chapter 
 ILL, we find that the strains in. and therefore the necessary 
 strengths of, the braces are proportionate to W. sec. ; and as 
 W is here constant, the requisite strengths are proportionate to 
 sec. 6. The length of a brace in each figure, since the depths 
 are alike, may be taken to represent sec. 6. Let the values of 
 sec. be equal to & and Z> . 
 
 The requisite sections * of the braces may be assumed to be 
 proportional with the strains, and therefore also with sec. 6, or 
 
 * Of course this is not strictly correct with regard to compressive strains 
 (and therefore will be a very little less than deduced). But while the 
 number of systems remains the same, the disparity will not be important. 
 And in the case of an increase in the number of systems, the more signifi 
 cant error that might take place will be moderated in practice by the braces 
 being riveted, bolted, or trenailed together at the crossings. 
 
INCLINATION OF TIIK BKACES. 
 
 TABLE VI. 
 
 as b : I . The total lengths of the bracings are evidently as 
 16 1 : 8 I . 
 
 Now, the total weights of the bracings are to one another 
 in the same proportion as the total lengths multiplied by the 
 respective sections, or as 16 1> x o : 8 I x o = 16 I* : 8 I * 
 weight of bracing of Figure 67 : weight of bracing of Figure 
 68. Or, as a general rule, the weight of a bracing is propor 
 tional with 2 N". (sec. #) 2 , but X = 1^577 which is proportion 
 ate to afcSTi* Substituting this for X, we have the weight of 
 the "bracings proportionate to ^nf > which is a minimum when 
 
 e = 45. 
 
 Though the minimum of weight of bracing is arrived at 
 when is made 45, yet we may vary this angle consider- 
 ablv without much increase of the weight, as will be seen from 
 Table VI. 
 
 But when, with 6 = 45, the distances be 
 tween the supported points are very much too 
 great, the difficulty is not to be overcome by 
 lessening the value of #, but by means of the 
 intermediate props shown in Figure 69, or by 
 using two or more systems or ranges of brac 
 ings, as shown in- Figures 70, 86, and 87. We 
 need only make this remark, at present, upon 
 the theory of systems : that as the value of?/? 
 is reduced as much as the number of systems 
 is increased, the total weight of the bracing 
 (supposing the section to represent the 
 strength of a brace) remains the same as fora 
 single system; but the quantity of workman 
 ship will be much greater. 
 
 PLATE BRACING. 
 
 But it would be difficult to define the limits of this descrip 
 tion of bracing. In one direction it includes the simple beam ; 
 
 Values 
 of 9. 
 
 Values of 
 (8y. 
 
 tan.~0. 
 
 10 
 
 202.40 
 
 15 
 
 200.00 
 
 20 
 
 155.60 
 
 25 
 
 130.55 
 
 30 
 
 115.50 
 
 35 
 
 106.40 
 
 40 
 
 101.55 
 
 45 
 
 100.00 
 
 50 
 
 101.55 
 
 55 
 
 106.40 
 
 CO 
 
 115.50 
 
 G5 
 
 130.55 
 
 70 
 
 155.60 
 
 75 
 
 200.00 
 
 80 
 
 292.40 
 
PLATE 
 
 in another, the cylindrical tube, and even hollow spherical forms 
 cannot be excepted as varieties of it. Among tlfe more obvious 
 cases of its employment, we may mention the feather on a cast 
 ing, the sides of a box, the hull of a ship, and the floor, walls, 
 and roof of the majestic tube of Stephenson and his coadjutors. 
 It is the bracing employed in nature, and that in all its varieties. 
 In the plate and shell form, we see it in the beak of a bird, in 
 the bones of the cranium, and in animals with an external 
 skeleton, as the crab, etc. ; as a feather strengthening a plate, 
 we view it, most conspicuously, in the keel of the breast-bone 
 of the falcon tribe, and in the spine of the scapula of the land 
 mammalia ; and in the form of the tube, we witness it uniVer- 
 sally adapted to the wants of the animal and vegetable king 
 doms the bones of the extremities, the quills of the wing, 
 the humbler grasses, and the lofty bamboo, may be cited as 
 familiar examples. We shall, however, confine ourselves to 
 the consideration of it in its simplest form, and endeavor 
 to compare it with the brace or linear method. 
 
 We have observed above that, to derive the plate from 
 the linear bracing, the lateral width must shrink into the thick 
 ness of the resulting plate ; consequently, it has comparatively 
 little lateral stiffness, and would readily warp m pucker wher 
 ever the compressive pressures were considerable ; fortunately, 
 however, this can be remedied with a moderate addition of 
 material in the form of featherings, and of easy application in 
 the case of the material (malleable iron) which would generally 
 be employed for this kind of bracing. 
 
 At those parts where the tensive strains considerably pre 
 dominate, we may expect to find the plate stronger than the 
 linear form, from the following considerations : Let Figure 71 
 represent a bracing of numerous series, and let a tensive strain 
 act in a I. Now the strain is only borne by the parts lying in 
 the direction of a 1) ; but when the systems become infinite and 
 the whole solid, there are, as it were, two plates one resulting 
 from the braces lying in one direction, and the other from those 
 in the opposite one ; and each of these plates serves to bear the 
 
30 PLATE BRACING. 
 
 strain in a 1. And, further, the direction of a strain is r.ot 
 confined to a particular angle. 
 
 The comparison, then, may be enunciated thus : " The plate 
 bracing possesses much greater strength against tensive strains, 
 but less against compressions, than the linear. " And, as a 
 corollary, we must infer that, generally, the plate form is better 
 suited for shallow than for deep bracings. 
 
 The general distribution of material in the plate bracing 
 must be subject to the laws regulating it in the other methods ; 
 thus, in the tirsf class, the plate must be made more substantial 
 towards the extremities ; and in Figure 37 of the third class, the 
 central part must receive the greater share of material. Figures 
 72 and 73 give an idea of the strains in class first ; the lines 
 show the general action of the pressures, and the degree of 
 shade produced is an index to the strength required. But 
 these diagrams are, of course, made much simpler than would 
 really occur, and they are only adapted to illustrate the action 
 of a uniform loading. In Figure 72, at the pier P, the perpen 
 dicular lines indicate that the pressures impressed at a by the 
 lines of suspension, shown in Figure 73, are supported from 
 beneath ; but at pier Q these are supposed to be received by a 
 higher part of the pier from the edge &, as is done by means of 
 the ball-rollers in the Conway and Britannia tubes. 
 
|p art the 0CCOU&. 
 
 CONSTRUCTION". 
 
 CHAPTER I. 
 
 THE VARIOUS BRACINGS REQUIRED IN DIFFERENT 
 STRUCTURES. 
 
 BESIDES the longitudinal vertical bracings, upon which 
 depends its strength to support the loading, a bridge requires 
 other bracings, and to the consideration of these we now proceed. 
 
 Let Figure 74 represent the skeleton of a bridge, composed 
 of two vertical frames. That these may act with effect, they 
 must be retained in straight lines, but there is the pressure ot 
 the wind, etc., tending to bend them laterally ; this, therefore, 
 must be sedulously guarded against ; let us add, then, a series 
 of horizontal braces arranged between the two upper beams, 
 as in Figure 75. The upper beams are now r rendered perma 
 nent in their position, and a roadway may be carried along 
 them ; the lower beams will be less steady, they may be 
 moved by a lateral blast, but, as they undergo a tensive strain, 
 they are in a state of stable equilibrium, and consequently re 
 turn to their places. This wriggling motion would, however, 
 very soon injure the structure, by loosening the joints of the 
 vertical bracings to the upper beams ; such a design is, there 
 fore, inadmissible for a permanent structure ; for such, the 
 lower beams must be retained perpendicularly beneath the 
 upper ones, which may be effected either by the introduction 
 of bracing between them, similar to that used for the upper 
 ones, or by the application of oblique transverse braces, as 
 shown in Figures 76 and 77. The first of these methods will 
 
METHODS OF HORIZONTAL BRACING. 
 
 be sufficient, without addition, for bridges having moderate 
 spans compared with their widths ; while the second may be 
 trusted to in any case, but, of course, in it the horizontal brac 
 ing employed in the upper part will require to be of double 
 strength, as it must resist the whole pressure of the wind. 
 The common practice is to employ both of these methods in 
 the same structure, and in long spans, if it do not interfere 
 with the roadway. The practice is to be commended. Figure 
 77. 
 
 The principal recommendation of the first method, when 
 used alone, is that it permits of the roadway being placed at 
 the level of the lower beams, which is of the utmost importance 
 in situations where the bridge spans over roads or navigable 
 waters, and sufficient height could not otherwise be given for 
 the passage of carriages or shipping beneath. Further, it re 
 quires little expense for parapets, and offers great facilities 
 lor the addition of a roof, or even of a second roadway, 
 along the upper beams. But particular attention must be paid 
 to carrying up the abutments, so that they may offer suffi 
 cient resistance to the outsides of the upper beams. 
 
 In the design, Figure 38, when the rise of the arch is very 
 small and the span moderate, and when the arches have a 
 considerable width of section (which may be conveniently 
 given to them, if built up with deals, or constructed as tubes 
 with wrought iron, as Figures 80 and 144-), then a single series 
 of horizontal braces along the lower part may be sufficient, 
 and the roadway may be on that level. (This also applies to 
 class first when the depth of the framing is small.) But when 
 the structure does not comply with the above conditions, it will 
 be necessary to have a series of braces between the arches also ; 
 for, as they are in a highly compressed state, a very moderate 
 deviation from a horizontally straight line might bring about 
 their rupture. But horizontal bracing being introduced be 
 tween the arches, the road cannot be carried along the chord 
 level ; it must be raised to the level of the crown of the arch,* 
 
 * Unless some new arrangement be made in the design, as in the follow- 
 
PLATE-BRACIXG. 33 
 
 and the design then becomes that of Figure 37, and the road 
 way will also require bracing. 
 
 Most of the other structures of class third will be secure 
 with two lines of horizontal bracings, and of these Figures 
 35 and 42 admit the roadway at the lower level. Bat when 
 Figures 36 and 41 are of considerable span, it will be prudent 
 to bestow three series on them. 
 
 In structures belonging to class fourth, each arch will 
 require to be braced horizontally, and, generally, the roadway 
 also. 
 
 PLATE-BRACING. 
 
 The forms of bridges to which the plate-bracing is more 
 especially adapted are the tubular and the segrnental. 
 Figures 29, 35, 38, etc. 
 
 In the iirst, there is this important source of economy, 
 that each side of the tube is useful in two capacities iirst, as 
 the bracing against pressures acting in its plane ; and, second, 
 as a longitudinal member against pressures acting at right 
 angles to its face. In the form, Figure 35, the strains in the 
 plate are more uniform throughout. 
 
 In the tubular form, let us consider, for an instant, the 
 action under a load, confining ourselves, for the present, to 
 
 ing structures. Figures 78 and 79, which are examples of the use of plate- 
 bracing, may be used when the rise is not great. 
 
 Figure 81 is an arrangement suitable for linear bracings, and there is less 
 waste in it than would at first sight appear. The inside arches are kept 
 perpendicularly over their chords by the transverse bracing, which occupies 
 the central compartment throughout the whole length of the biidge ; and 
 cross connecting pieces or braces, throughout as great a length at the crowns 
 as can be permitted, retain the outer arches parallel with the inner ones. 
 In this design the rise must necessarily be considerable. 
 
 Figures 82, 83, and 84 show varieties of another arrangement ; it is a 
 compound of the first and third classes. The arch has the roadway sus 
 pended from it by iron rods, and, in 84, it is stiffened by separate braced 
 frames, placed one on either side of it ; bolts are passed through the whole 
 wherever the arch crosses a part of the frames ; the rise is made sufficient 
 for the roadway to pass beneath the horizontal bracing between, the upper 
 beams. ^ 
 
 UNIVERSITY 
 
34 PLATE-BKACING. 
 
 the consideration of tlie bottom plate; this is stretched, but 
 it will evidently be more so immediately below the vertical 
 bracings, and rupture thereof would commence by the tear 
 ing of the edges. The same quantity of material collected 
 into tie-beams, and attached to the bottoms of the vertical 
 bracings, would offer a greater resistance; but then, on the 
 other hand, an additional horizontal bracing would become 
 necessary. Similar reasoning applies, and with greater force, 
 to the top plate ; and on the determination of which of these 
 arrangements is the most economical depends, in a great 
 measure, the question of the respective merits, for this case, 
 of the plate and linear bracings when constructed of wrought 
 iron ; for other materials the plate form of bracing is not 
 suited.* 
 
 The want of uniformity between the strains at the edges 
 and those at the middle line of the plate, when it acts as a 
 longitudinal member, may be obviated by giving it a slight 
 convexity outwards, as in Figure 85. This may be done to 
 the top and bottom plates, as, though it weakens them as 
 bracings, sufficient strength, as such, will remain to counter 
 act any lateral disturbing influences. But this must be 
 looked upon rather as a refinement in theory than a practical 
 improvement. 
 
 In the form of Figure 38, the plate-bracing may be modified 
 so as to give lateral stability to the arch ; this is shown more 
 particularly in the design Figure 78, wherein the plate is 
 double. The deep featherings in Figure 70 are designed to 
 effect the same object. 
 
 The featherings alluded to at page may be arranged 
 on two plans : first, in perpendicular pieces, as in the Britannia 
 tubes; and, second, radiating from each pier, as do the lines of 
 
 -Cast-iron may, indeed, be used for plate-bracing, and, since that form 
 of bracing demands comparatively little tensive strength in the material, 
 probably with good effect, at least where heaviness would not bo a defect. 
 Considerable skill must, however, be exercised when the bracing is com 
 posed of many parts. 
 
THE BRACES. 35 
 
 compression in Figure 72 ; in this last arrangement they 
 require to be of larger section to be equally effective, but, 
 besides stiffening the plate, they carry a portion of the com 
 pressions to the piers. Perhaps, from various considerations, 
 a combination of the two plans would be the best, the feather 
 ings being applied in the one manner to the inside, and in 
 the other way to the outside of the plate. 
 
 CHAPTER II. 
 
 OX THE CONSTRUCTION AND APPLICATION OF THE BRACES. 
 
 the bracing is poiy-systemed, the braces may be 
 arranged on different plans; thus: All the braces of the 
 same system, as shown by Figure 86, may be in the same plane, 
 and the systems so near that the braces touch, notch, or 
 halve into one another at the crossings. Or, as seen in 
 Figure 87, all the braces lying in the same direction may be 
 in the same plane, and those inclined the contrary way in an 
 adjacent plane."" 
 
 In designing the joints of the braces, the following observa 
 tion on the action of the bracing is of such great importance 
 that, though indicated in the previous part, we again place it 
 before the reader. That part of a pressure, in a brace, which 
 is resolvable at right angles to the longitudinal member is, 
 when that member is straight, wholly transferred to the other 
 brace; and when the longitudinal beam is curved, the part 
 transferred is sUghtly diminished or increased, according as 
 the curvature is concave or convex towards the braces, when 
 the longitudinal is compressed; and when it is stretched, 
 the reverse is the case. Arfd the only pressure resulting in 
 the longitudinal is in the direction of its length. 
 
 *" Or, us a variety of this second arrangement, suitable to a very broad 
 longitudinal member, there may be several layers of parallel braces, inclined 
 alternately in opposite direction?. 
 
WOODKN BRACKS. 
 
 WOODEN BRACES. 
 
 1. Those requiring little or no iron. Braces of this 
 charactcr would generally be used in those situations only 
 where iron-work might be difficult to procure. The simplest 
 kind of jointing for them is that by means of trenails, as in 
 Figure 88, or by dovetail notching, as in Figures 89 and 90. 
 These methods are very suitable when the braces are thin 
 or plank-shaped, as in Town s Lattice Bridge. But when n 
 brace is liable to undergo great compression, its section 
 should be nearly square or circular. For braces of a stouter 
 character, the Figures 92 to 95 illustrate methods of jointing 
 adapted to a longitudinal member constructed in a similar 
 way to that shown by Figure 95. Modifications of these 
 joints might be easily constructed, capable of correcting the 
 length of the brace ; but such a property is unnecessary, a.<* 
 there is no difficulty in the way of preparing the braces with 
 the requisite accuracy. 
 
 2. Those jointed with iron. The most appropriate way of 
 applying wrougktiron is in the form of straps, of which several 
 varieties are shown by Figures 9G to 101. But generally the 
 method illustrated by the succeeding Figures 102 to 117, ot 
 jointing the braces by means of CAST-IRON SOCKETS, is recom 
 mended by the author. These sockets are of two kinds; tin* 
 h rst, including Figures 102 to 109, requires the end of thr 
 brace to be of a dovetail shape ; the second, shown by 
 Figures 110 to 117, to be square-headed. The first is best 
 suited to braces formed of a wood that is easily split, the second 
 for those of a tougher material. 
 
 The disadvantages attending the braces with the dovetail 
 joints are, that they require stranger sockets, in order to resist 
 their wedge-action ; and that they might be sensibly lengthened 
 by the compressibility of the dovetails allowing the ends to 
 draw a little, unless they be wedged very tightly in, which 
 would increase the former evil. 
 
 The only remark, since the subject is so fully illustrated by 
 
1511 ACE SOCKETS. 37 
 
 figures, that it is necessary to make upon the construction of 
 the square-headed braces is this : If the joint were formed as in 
 Figure 116, there would be great danger of the piece a being 
 wrenched off by any slight wriggling motion that might take 
 place in the brace before the structure was perfected ; it is 
 therefore constructed, as shown by Figure 117, so that there is 
 no angle at &, which, by acting as the tooth of a crowbar, could 
 split off the piece a ; but the slant of If c must be slight, other 
 wise there would be the wedge-action objected to above. 
 
 Figures 109, 114, and 115 are forms of double sockets for 
 Hie case, mentioned in the beginning of the chapter, of the 
 braces of the same system being arranged in two planes; see 
 Figure 87. And Figures 102 to 106, and 110 to 112, arc forms 
 of single sockets, which may sometimes be found useful in 
 t ransverse and other bracings. The other forms are for the case 
 of the braces of one system being all in the same plane, as 
 seen in 86. 
 
 The braces are retained in the sockets by short screw-bolts, 
 either wUh or without covers. 
 
 The head of the brace and the inside of the socket should 
 bo coated with some preservative substance, such as a heated 
 mixture of tar and pitch. 
 
 One great advantage attending the use of the socket-joint, 
 and also participated in by most of the previous forms, is 
 that if a brace becomes injured by decay or accident, it can 
 readily be removed and replaced by a sound one. 
 
 The designs given are calculated for braces of moderate 
 dimensions. 
 
 IKOX BRACKS. 
 
 To braces of iron the forms that may be given are very 
 numerous ; but if we exclude those which do not admit of 
 being individually removed, the number will be very consid 
 erably reduced ; and the best forms of sockets, such as Figures 
 118 and 119, are liable to this objection. 
 
38 IRON BRACKS. 
 
 1. Cast-Iron braces may have various forms of section, 
 either feathered or tubular ; but those of the latter kind are not 
 removable. The design 124 is an excellent one ; it possesses 
 erery desirable property, especially that of the pair of braces 
 interchanging their pressures without the intervention of the 
 longitudinal, the advantages of which were pointed at in the 
 second paragraph of this chapter. But it cannot, without 
 modification, be applied to a longitudinal member of the tubu 
 lar class, as the bolts must not be inserted from the inside of 
 the tube, for, should one break, it could not be easily replaced. 
 
 When tubular forms of the longitudinals are used, whether 
 of cast or wrought iron, they must be furnished with sufficient 
 leatherings, single or double, to which the bracings are to be 
 attached ; the leatherings may be placed either longitudinally 
 or transversely. 
 
 Figures 125 and 126 show modifications of Figure 124 
 
 o o 
 
 when it is connected with a tube. But when braces are ap 
 plied to the sides of a longitudinal feather, the designs shown 
 by Figures 132 and 134, which are suitable for the arrange 
 ment shown in Figure 87 of a poly-systemed bracing, possess 
 considerable advantages. 
 
 2. Wrought iron braces may be of round or rectangular 
 section, or rolled with a section possessing greater strength 
 against compression, or they maybe constructed of thin plates, 
 strengthened by angle iron, or by being formed into pipes. 
 Braces with solid ends may be jointed, as shown in Figures 
 131 to 134, for cast ones; but those with thin and broad ends 
 will be most effectively joined to the main beams by being 
 riveted to a projecting feather thereon (Figures 135 and 13(5). 
 
 On the subject of the respective merits of cast and wrought 
 iron as the material of the bracing, the only general remark we 
 shall make is that for very long braces the former, and for very 
 short ones the latter, kind of iron is to be preferred.* 
 
 i * Wrought iron may, however, be very properly employed for groat 
 lengths, if formed into tube?, or otherwise braced, especially when light 
 ness is a consideration . 
 
WOODEN BEAMS AND AKCIIES. 9 
 
 CHAPTER III. 
 
 ON THE CONSTRUCTION OF OTHER PARTS OF BRIDGES. 
 
 As the contents of this chapter are in a great measure 
 supererogatory, the meagreness of the account of such im 
 portant subjects needs no apology. 
 
 WOODEN STRUCTURES. 
 
 The longitudinal members. "When these are too long or 
 too much curved to be formed of one piece, the best construc 
 tion is to build them up with planks, using comparatively thin 
 ones when the part is to act as a tie or is an arch of moderate 
 radius, and thisker planks or deals when the part is straight 
 and only subject to compression. In the latter case, care must 
 be taken to bring the butting ends into forcible contact ; in 
 deed, it might even be advisable to place in the joints thin 
 wedges of hard wood or iron, which could, when needful, be 
 tightened up to the proper degree. But in arches formed ot 
 thin planks, if care be taken in the construction, the wedges 
 may be dispensed with ; the purpose will be effected, in 
 a great measure, by the great friction, and this may be further 
 increased by coating the planks with a hot mixture of pitch 
 and tar, or by placing in all the joints a layer of strong brown 
 paper dipped in boiling tar (as used by Messrs. J. & B. Green 
 in their laminated timber arches); but the most important 
 benefit arising from this treatment is its preservative effect by 
 the exclusion of moisture.* The friction here mentioned is 
 also of great importance from its stiffening effect, rendering the 
 arch, when, sufficient, equivalent to one of whole timber ; it 
 primarily depends upon the compression induced by the screw- 
 bolts, straps, etc., that act upon the arch in the direction of its 
 radius ; these, therefore, should be numerous and well tight 
 ened up. When brace-sockets are employed, the bolts connect- 
 
 * Or, the planks being dressed with the plane, white-lead paint may be 
 used, as in building large masts. 
 
40 THK SPRINGING OF THE ARCH. 
 
 ing them with the arch contribute to this end, and, of course, 
 effect a corresponding saving; these bolts, with the addition of 
 screw-bolts, such as shown in Figures 138 to 140, would con 
 stitute a good arrangement. Trenailing forms a cheap addi 
 tional means of stiffening the arch, but renders the removal of 
 a plank, in the event of its becoming decayed, an almost im 
 possible operation. 
 
 Figures 141 and 142 show an American method of drawing 
 the planks together; by the use of these dovetail-gibs and 
 wedges and trenails, a stout arch may be formed without the 
 aid of iron. 
 
 For the case of Figure 38, the arch should have a broad 
 section, and then it might be expedient to use more than one 
 system of braces ; a section of such an arrangement as is meant 
 is shown by Figure 144 (see also foot-note, page 35). 
 
 THE SPRINGING of a tied arch is a part that demands great at 
 tention. The most perfect arrangement is that which distri 
 butes the thrust of the arch equally over the section of the tie. 
 T fence the method of Figure 146 is perhaps the best for an 
 arch and tie of the nature shown ; each of the iron steps is fixed 
 to its plank of the tie by numerous screw-nails. Figures 148 
 and 149 are methods for the case of a laminated arch with a 
 tie beam of a broad and thin section. 
 
 IRON STRUCTURES. 
 
 The lower beam should be constructed of wrought iron ; the 
 upper one may be of either kind, and, generally speaking, the 
 tubular will be the best form that can be given to it, for which 
 form wrought iron, as a material, is perhaps the more prefer 
 able; but is the more difficult of the two to manage when the 
 member is of the arched form, as the whole must be put to 
 gether on the ground and thence raised with powerful tackle, 
 or the riveting must be performed, with great inconvenience, 
 on the centring. 
 
 The springing of a tied wrought-iron arch may be formed, 
 as shown in Figure 150, by surrounding the end with strong 
 
OF THE 
 
 UNIVERSITY 
 
 41 
 
 THE FIRST C L 
 
 angle-iron, which is then riveted to the flat end of the tic; or 
 by inserting the foot of the arch into a cast-iron socket, as is 
 done with the wooden one in Figure 148. A cast-iron arch 
 may have its terminal pieces formed, as in Figure 151, with 
 solid expanded soles, bolted down to the tie. 
 
 For untied arches a common cast-iron socket, with its bear 
 ing against the abutment expanded and normal to the arch, 
 will generally be sufficient. 
 
 CHAPTER IV. 
 
 SUPPLEMENTARY. 
 
 IT is not here intended to institute a comparison of the 
 merits of the various designs, as that would demand a very 
 large space, depending so much, as the excellence of a structure 
 does, upon its adaptation to the particular combination of the 
 requirements and circumstances of situation in which it may 
 be assumed to be placed ; such an extended enquiry would be 
 more appropriate to a treatise on bridges. But we shall reca 
 pitulate a little, and add anything that may suggest itself as 
 closely connected with the subject of the work, but which came 
 not conveniently under any of the foregoing heads. 
 
 Though the first class may be surpassed in strength by those 
 wherein the arch is used, it possesses many advantages in other 
 respects ; foremost of these is its simplicity, offering easy con 
 struction ; there is great uniformity of parts, which, especially 
 when cast iron is employed, may be a source of economy; it 
 admits of a roadway at either level, without interfering with the 
 horizontal bracings, of which only two series are necessary; it 
 exerts no horizontal thrust ; and, as practised by Mr. Stephen- 
 
42 INTRADOSAL KRACIXG PREFERABLE. 
 
 son, with the tubular bridges, it may be put together at some 
 convenient spot, and thence removed in a nearly complete state 
 to its permanent site ; or, if erected there, only a very simple 
 and inexpensive scaffolding will be necessary. 
 
 The form, of which Figure 152 is a transverse section, may 
 be considered as a variety of the first class; in it the main 
 braces require to be longer ; the horizontal braces are also 
 necessarily longer, but this renders them more powerful ; its 
 advantages are, that by concentrating the upper beams into 
 one, their joint strength against compression will be in 
 creased ; that one series of horizontal braces is sufficient ; 
 and that it may be roofed in at very little expense. 
 
 Speaking generally, the internally-braced arches are su 
 perior in strength to those with external bracing ; the ad 
 vantage attends the deepening of the bracing towards the 
 mid-span. 
 
 When a structure of class fourth has the arches parallel, 
 it may possess the recommendation, when cast iron is em 
 ployed for the bracing, of uniformity of the castings. 
 
 The method of connecting the braces to the main beams 
 by means of trenailing may appear very rude, but it is ca 
 pable of producing considerable strength ; and, as it may in 
 some situations be a convenient one, the reader is referred, 
 if unacquainted with better data, to a table of the strengths 
 of trenails, given in the last edition of the " Encyclopaedia Bri- 
 tannica," art. " Shipbuilding/ 
 
 We are not aware of any satisfactory experiments having 
 been made to determine that strength of tinfbcr which arises 
 from the resistance to severance by the sliding longitudi 
 nally of a part of the material, upon which primarily depends 
 the strength of the socketed wooden braces against a tensive 
 strain. This strength or resistance to severance will be in 
 creased in the dovetail form by the lateral compression, but 
 it may be lessened in the square-headed variety, if care be 
 not taken to obviate any splitting action that is, any force 
 acting so as to wrench the fibres separate. 
 
THE WHOLE LENGTH MUST BE BRACED. 4o 
 
 The following observations are made in order to show the 
 applicability of the formula to the forms of bracing seen in 
 Figures 15, 1G or 17, and 18 : In Figure 15, if a weight be 
 placed at c, the portion of it which must be imposed on the 
 pier to the left side will not influence cb or la, as 5 a cannot 
 act the part of a tie. Jt is therefore conveyed by cd, da, etc. ; 
 and through these parts would, in like manner, pass the 
 strains arising from all the weights placed to the right of c, 
 which w r ould ultimately arrive at the pier to the left side. 
 When all such weights are alone on, the strains in cd and da 
 are maxima. A similar course would be pursued by the strains 
 in the bracing of Figure 21. The effect of the portion of the 
 weight at e, which is to be supported by the pier to the right, 
 must flow 7 through the strut, and not through cl>. Thus it is 
 readily seen by what loading the strain in a part is rendered a 
 maximum, and also the formula by which its amount will be 
 given, after substituting the proper values for the symbols. 
 
 The bracing must be continued throughout the length of the 
 structure. This will be clearly seen by inspecting Figures 15IJ 
 and 154, which show changes that may take place when only a 
 part is braced. 
 
 An arch may be employed without having the bracing ar 
 ranged along its intrados or extrados ; it may be stiffened by 
 being connected with a braced structure of class first. Figures 
 82, 83, 84, 155, and 156 are illustrative of this compound ar 
 rangement. In the arch, the radius of curvature may be made 
 to diminish more rapidly in approaching the crown than if the 
 curve were a parabola, as it. is principally required to aid the 
 class-first portion at the mid-span. The maximum strains in 
 the parts of the class-first portion will be very much reduced 
 by the addition of the arch. Figure 156 is a good design for 
 very large structures, as it permits of the roadway being at the 
 lower level, and as then the horizontal bracing between the 
 upper beams would be sufficiently high ; the arch and each of 
 the straight longitudinals will require horizontal bracing. 
 
 In class first a saving of half the scantling at the centre of 
 
44 CHANGE FROM ARCH TO GIRDEK ACTION. 
 
 the main beams might be effected, if a pressure of exactly J. 
 were made to act tensively at eacli end of the upper beam, and 
 eompressively at the ends of the lower one; for then one-halt 
 of the load would be borne up on the principle of the girder 
 supported at both ends, while the other half would be upheld 
 on the principle of a projecting beam fixed at one end and 
 unsupported at the other. The strains in the longitudinals 
 would be subject to a peculiar arrangement. Their maxima 
 would occur at the mid-span and at the extremities, and would 
 bo = ^, or half what would otherwise take place at the 
 centres, and the strains at the extremities would be tensive 
 >hould that at the centre be compressive, and vice versa ; and 
 at the points midway between the mid-span and the extremi 
 ties, where the strains change their character, the longitudinals 
 would of course be free from all action. Advantage is taken, 
 t<> some extent, of this compound action in the Britannia 
 Bridge, by the method pursued in uniting the tubes over the 
 central pier or tower. 
 
 The following observations are an extension of a principle 
 employed in the latter part of Chapter V., Part First. Let 
 Figure G4| represent a portion of a voussoired arch ; now, it 
 the line of pressures, from not being capable of conforming t< 
 the curve of the arch, rise above it at #, as shown by the dot 
 ted line, the joint a b will open at b. But let us suppose the 
 joint at I to become solid, or that it is made good there by 
 means of bolting, the line may then rise above a or any joint 
 sufficiently united at its lower point ; and might also sink be 
 low a part of the arch, were the upper points of the joints 
 tied. This is the transition condition between an arch and a 
 girder action ; it is that in which a cast-iron (or other continu 
 ous rigid) arch may be placed by an uneven loading too severe 
 for its depth of section. When strengthened by bracing, the 
 joints cannot open, as the voussoirs are then prevented chang 
 ing their proper radial directions. 
 
 The strains in the several parts of Figure 31 will be exact 
 ly the same in amount, but of opposite character, with those in 
 
INVERTED LINE OF PRESSURES. 45 
 
 the corresponding parts of Figure 38. And generally, what 
 ever lias been said with regard to the comportment of the line 
 of pressures in the braced erect arch applies, in a reversed 
 state, to the curve or line of tensions in the braced inverted 
 arch, or rigid suspension bridge. 
 
 And here the writer would almost be induced to speculate 
 upon the possible extent to which bracing might be employed, 
 imagining structures wherein each brace would offer in itself 
 no contemptible example of braced construction ; but he re 
 strains himself, deeming the opportunity objectionable for such 
 sugestions. 
 
 / /OF THE \ 
 
 UNIVERSITY)* 
 ^y 
 
A P P E N D I X . 
 
 x the printer had made considerable progress, and 
 consequently when it would have been very inconvenient to 
 notice it at the proper places, I observed, at page 391 of the 
 Civil Engineer and Architects 1 Journal for December, 
 1850, an account of the failure of a girder bridge in which 
 the triangular method of bracing was employed. I was not 
 previously aware of a patent having been granted for any 
 structure employing the method, however imperfectly. 
 
 The instance of failure of the structure of which I speak 
 offers an opportunity of employing the results obtained in 
 Chapter III., Part First. I will, however, merely outline the 
 application, as it is a very simple matter : The braces, on ap 
 proaching the piers, ought to increase in strength, as seen in 
 Figure 47. In that figure, 1 m will represent the fractured 
 brace; it had to undergo a very great tensive strain, but was 
 made of cast iron of no greater section than the central braces ; 
 it and we cannot wonder at it broke. The great virtual 
 loading impressed on the point 2 by the brace I 2 being thus 
 left to be upheld by the transverse strength of the upper longi 
 tudinal part, that part became fractured near the said point. 
 The remaining fracture probably resulted from the direct 
 action of the load on the portion left projecting. It is not 
 stated that any further breakage took place ; if none did, the 
 circumstance may be explained by the peculiar construction of 
 the "girder." The lower longitudinal part, which was com 
 posed of wrought-iron chain-work, though it could not give 
 aid in supporting the point m, while it continued quite hori 
 zontal, offered, after the failure, sufficient to uphold the wreck ; 
 
48 APPKNDIX. 
 
 and the terminal braces of what remained as a " girder 
 proved sufficient for the diminished load and span. 
 
 The writer of the article suggests that the accident would 
 have been averted, if the lower longitudinal part had been of 
 cast iron, and all cast in one piece. The longitudinals require 
 very great strength at the mid-span; consequently, if of cast 
 iron, the lower one would demand a very large section there 
 to resist the great tensive strain. But let it not be supposed 
 that I approve of the mixture of cast and wrought iron as 
 managed here. The danger of the combination lies Jn this: 
 that the elasticity of the wrought-iron tie may be so great as 
 to allow the points (say^ and &, Figure 47) of the braced ele 
 ments to separate further than the elasticity of the upper cast- 
 iron longitudinal can permit without fracture (at point 4). 
 If, however, the upper longitudinal member be jointed at each 
 of the points, 2, 3, 4, 5, etc., the latter danger will be provided 
 against. 
 
: 
 
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