Chapter 21

In a three-span bridge continuous girders are lighter than discontinuous ones by about 45% for the dead load and 15% for the live load, if no allowance is made for ambiguity due to uncertainty as to the level of the supports. The cantilever and suspended girder types are as economical and free from uncertainty as to the stresses. In long-span bridges the cantilever system permits erection by building out, which is economical and sometimes necessary. It is, however, unstable unless rigidly fixed at the piers. In the Forth bridge stability is obtained partly by the great excess of dead over live load, partly by the great width of the river piers. The majority of bridges not of great span have girders with parallel booms. This involves the fewest difficulties of workmanship and perhaps permits the closest approximation of actual to theoretical dimensions of the parts. In spans over 200 ft. it is economical to have one horizontal boom and one polygonal (approximately parabolic) boom. The hog-backed girder is a compromise between the two types, avoiding some difficulties of construction near the ends of the girder.

Most braced girders may be considered as built up of two simple forms of truss, the king-post truss (fig. 61, a), or the queen-post truss (fig. 61, b). These may be used in either the upright or the inverted position. A _multiple truss_ consists of a number of simple trusses, e.g. Bollman truss. Some timber bridges consist of queen-post trusses in the upright position, as shown diagrammatically in fig. 62, where the circles indicate points at which the flooring girders transmit load to the main girders. _Compound_ trusses consist of simple trusses used as primary, secondary and tertiary trusses, the secondary supported on the primary, and the tertiary on the secondary. Thus, the Fink truss consists of king-post trusses; the Pratt truss (fig. 63) and the Whipple truss (fig. 64) of queen-post trusses alternately upright and inverted.

A combination bridge is built partly of timber, partly of steel, the compression members being generally of timber and the tension members of steel. On the Pacific coast, where excellent timber is obtainable and steel works are distant, combination bridges are still largely used (Ottewell, _Trans. Am. Soc. C.E._ xxvii. p. 467). The combination bridge at Roseburgh, Oregon, is a cantilever bridge, The shore arms are 147 ft. span, the river arms 105 ft., and the suspended girder 80 ft., the total distance between anchor piers being 584 ft. The floor beams, floor and railing are of timber. The compression members are of timber, except the struts and bottom chord panels next the river piers, which are of steel. The tension members are of iron and the pins of steel. The chord blocks and post shoes are of cast-iron.

33. _Graphic Method of finding the Stresses in Braced Structures._--Fig. 65 shows a common form of bridge truss known as a _Warren girder_, with lines indicating external forces applied to the joints; half the load carried between the two lower joints next the piers on either side is directly carried by the abutments. The sum of the two upward vertical reactions must clearly be equal to the sum of the loads. The lines in the diagram represent the directions of a series of forces which must all be in equilibrium; these lines may, for an object to be explained in the next paragraph, be conveniently named by the letters in the spaces which they separate instead of by the method usually employed in geometry. Thus we shall call the first inclined line on the left hand the line AG, the line representing the first force on the top left-hand joint AB, the first horizontal member at the top left hand the line BH, &c; similarly each point requires at least three letters to denote it; the top first left-hand joint may be called ABHG, being the point where these four spaces meet. In this method of lettering, every enclosed space must be designated by a letter; all external forces must be represented by lines _outside_ the frame, and each space between any two forces must receive a distinctive letter; this method of lettering was first proposed by O. Henrici and R. H. Bow (_Economics of Construction_), and is convenient in applying the theory of reciprocal figures to the computation of stresses on frames.

34. _Reciprocal Figures._--J. Clerk Maxwell gave (_Phil. Mag. 1864_) the following definition of reciprocal figures:--"Two plane figures are reciprocal when they consist of an equal number of lines so that corresponding lines in the two figures are parallel, and corresponding lines which converge to a point in one figure form a closed polygon in the other."

Let a frame (without redundant members), and the external forces which keep it in equilibrium, be represented by a diagram constituting one of these two plane figures, then the lines in the other plane figure or the reciprocal will represent in direction and magnitude the forces between the joints of the frame, and, consequently, the stress on each member, as will now be explained.

Reciprocal figures are easily drawn by following definite rules, and afford therefore a simple method of computing the stresses on members of a frame.

The external forces on a frame or bridge in equilibrium under those forces may, by a well-known proposition in statics, be represented by a closed polygon, each side of which is parallel to one force, and represents the force in magnitude as well as in direction. The sides of the polygon may be arranged in any order, provided care is taken so to draw them that in passing round the polygon in one direction this direction may for each side correspond to the direction of the force which it represents.

This polygon of forces may, by a slight extension of the above definition, be called the _reciprocal figure_ of the external forces, if the sides are arranged in the same order as that of the joints on which they act, so that if the joints and forces be numbered 1, 2, 3, 4, &c., passing round the outside of the frame in one direction, and returning at last to joint 1, then in the polygon the side representing the force 2 will be next the side representing the force 1, and will be followed by the side representing the force 3, and so forth. [v.04 p.0555] This polygon falls under the definition of a reciprocal figure given by Clerk Maxwell, if we consider the frame as a point in equilibrium under the external forces.

Fig. 66 shows a frame supported at the two end joints, and loaded at each top joint. The loads and the supporting forces are indicated by arrows. Fig. 67a shows the reciprocal figure or polygon for the external forces on the assumption that the reactions are slightly inclined. The lines in fig. 67 a, lettered in the usual manner, correspond to the forces indicated by arrows in fig. 66, and lettered according to Bow's method. When all the forces are vertical, as will be the case in girders, the polygon of external forces will be reduced to two straight lines, fig. 67 b, superimposed and divided so that the length AX represents the load AX, the length AB the load AB, the length YX the reaction YX, and so forth. The line XZ consists of a series of lengths, as XA, AB ... DZ, representing the loads taken in their order. In subsequent diagrams the two reaction lines will, for the sake of clearness, be drawn as if slightly inclined to the vertical.

If there are no redundant members in the frame there will be only two members abutting at the point of support, for these two members will be sufficient to balance the reaction, whatever its direction may be; we can therefore draw two triangles, each having as one side the reaction YX, and having the two other sides parallel to these two members; each of these triangles will represent a polygon of forces in equilibrium at the point of support. Of these two triangles, shown in fig. 67 c, select that in which the letters X and Y are so placed that (naming the apex of the triangle E) the lines XE and YE are the lines parallel to the two members of the same name in the frame (fig. 66). Then the triangle YXE is the reciprocal figure of the three lines YX, XE, EY in the frame, and represents the three forces in equilibrium at the point YXE of the frame. The direction of YX, being a thrust upwards, shows the direction in which we must go round the triangle YXE to find the direction of the two other forces; doing this we find that the force XE must act down towards the point YXE, and the force EY away from the same point. Putting arrows on the frame diagram to indicate the direction of the forces, we see that the member EY must pull and therefore act as a tie, and that the member XE must push and act as a strut. Passing to the point XEFA we find two known forces, the load XA acting downwards, and a push from the strut XE, which, being in compression, must push at both ends, as indicated by the arrow, fig. 66. The directions and magnitudes of these two forces are already drawn (fig. 67 a) in a fitting position to represent part of the polygon of forces at XEFA; beginning with the upward thrust EX, continuing down XA, and drawing AF parallel to AF in the frame we complete the polygon by drawing EF parallel to EF in the frame. The point F is determined by the intersection of the two lines, one beginning at A, and the other at E. We then have the polygon of forces EXAF, the reciprocal figure of the lines meeting at that point in the frame, and representing the forces at the point EXAF; the direction of the forces on EH and XA being known determines the direction of the forces due to the elastic reaction of the members AF and EF, showing AF to push as a strut, while EF is a tie. We have been guided in the selection of the particular quadrilateral adopted by the rule of arranging the order of the sides so that the same letters indicate corresponding sides in the diagram of the frame and its reciprocal. Continuing the construction of the diagram in the same way, we arrive at fig. 67 d as the complete reciprocal figure of the frame and forces upon it, and we see that each line in the reciprocal figure measures the stress on the corresponding member in the frame, and that the polygon of forces acting at any point, as IJKY, in the frame is represented by a polygon of the same name in the reciprocal figure. The direction of the force in each member is easily ascertained by proceeding in the manner above described. A single known force in a polygon determines the direction of all the others, as these must all correspond with arrows pointing the same way round the polygon. Let the arrows be placed on the frame round each joint, and so as to indicate the direction of each force on that joint; then when two arrows point to one another on the same piece, that piece is a tie; when they point from one another the piece is a strut. It is hardly necessary to say that the forces exerted by the two ends of any one member must be equal and opposite. This method is universally applicable where there are no redundant members. The reciprocal figure for any loaded frame is a complete formula for the stress on every member of a frame of that particular class with loads on given joints.

Consider a Warren girder (fig. 68), loaded at the top and bottom joints. Fig. 69 b is the polygon of external forces, and 69 c is half the reciprocal figure. The complete reciprocal figure is shown in fig. 69 a.

The method of sections already described is often more convenient than the method of reciprocal figures, and the method of influence lines is also often the readiest way of dealing with braced girders.

35. _Chain Loaded uniformly along a Horizontal Line._--If the lengths of the links be assumed indefinitely short, the chain under given simple distributions of load will take the form of comparatively simple mathematical curves known as catenaries. The true catenary is that assumed by a chain of uniform weight per unit of length, but the form generally adopted for suspension bridges is that assumed by a chain under a weight uniformly distributed relatively to a horizontal line. This curve is a parabola.

Remembering that in this case the centre bending moment [Sigma]wl will be equal to wL²/8, we see that the horizontal tension H at the vertex for a span L (the points of support being at equal heights) is given by the expression

1 . . . H = wL²/8y,

or, calling x the distance from the vertex to the point of support,

H = wx²/2y,

The value of H is equal to the maximum tension on the bottom flange, or compression on the top flange, of a girder of equal span, equally and similarly loaded, and having a depth equal to the dip of the suspension bridge.

Consider any other point F of the curve, fig. 70, at a distance x [v.04 p.0556] from the vertex, the horizontal component of the resultant (tangent to the curve) will be unaltered; the vertical component V will be simply the sum of the loads between O and F, or wx. In the triangle FDC, let FD be tangent to the curve, FC vertical, and DC horizontal; these three sides will necessarily be proportional respectively to the resultant tension along the chain at F, the vertical force V passing through the point D, and the horizontal tension at O; hence

H : V = DC : FC = wx²/2y : wx = x/2 : y,

hence DC is the half of OC, proving the curve to be a parabola.

The value of R, the tension at any point at a distance x from the vertex, is obtained from the equation

R² = H²+V² = w²x^4/4y²+w²x²,

or,

2 . . . R = wx[root](1+x²/4y²).

Let i be the angle between the tangent at any point having the co-ordinates x and y measured from the vertex, then

3 . . . tan i = 2y/x.

Let the length of half the parabolic chain be called s, then

4 . . . s = x+2y²/3x.

The following is the approximate expression for the relation between a change [Delta]s in the length of the half chain and the corresponding change [Delta]y in the dip:--

s+[Delta]s = x+(2/3x) {y²+2y[Delta]y+([Delta]y)²} = x+2y²/3x+4y[Delta]y/3x+2[Delta]y²/3x,

or, neglecting the last term,

5 . . . [Delta]s = 4y[Delta]y/3x,

and

6 . . . [Delta]y = 3x[Delta]s/4y.

From these equations the deflection produced by any given stress on the chains or by a change of temperature can be calculated.

36. _Deflection of Girders._-- Let fig. 71 represent a beam bent by external loads. Let the origin O be taken at the lowest point of the bent beam. Then the deviation y = DE of the neutral axis of the bent beam at any point D from the axis OX is given by the relation

d²y M --- = -- , dx² EI

where M is the bending moment and I the amount of inertia of the beam at D, and E is the coefficient of elasticity. It is usually accurate enough in deflection calculations to take for I the moment of inertia at the centre of the beam and to consider it constant for the length of the beam. Then

dy 1 -- = ---[Integral]Mdx dx EI

1 y = ---[Integral][Integral]Mdx². EI

The integration can be performed when M is expressed in terms of x. Thus for a beam supported at the ends and loaded with w per inch length M = w(a²-x²), where a is the half span. Then the deflection at the centre is the value of y for x = a, and is

5 wa^4 [delta] = --- ----. 24 EI

The radius of curvature of the beam at D is given by the relation

R = EI/M.

37. _Graphic Method of finding Deflection._--Divide the span L into any convenient number n of equal parts of length l, so that nl = L; compute the radii of curvature R_1, R_2, R_3 for the several sections. Let measurements along the beam be represented according to any convenient scale, so that calling L_1 and l_1 the lengths to be drawn on paper, we have L = aL_1; now let r_1, r_2, r_3 be a series of radii such that r_1 = R_1/ab, r_2 = R_2/ab, &c., where b is any convenient constant chosen of such magnitude as will allow arcs with the radii, r_1, r_2, &c., to be drawn with the means at the draughtsman's disposal. Draw a curve as shown in fig. 72 with arcs of the length l_1, l_2, l_3, &c., and with the radii r_1, r_2, &c. (note, for a length ½l_1 at each end the radius will be infinite, and the curve must end with a straight line tangent to the last arc), then let v be the measured deflection of this curve from the straight line, and V the actual deflection of the bridge; we have V = av/b, approximately. This method distorts the curve, so that vertical ordinates of the curve are drawn to a scale b times greater than that of the horizontal ordinates. Thus if the horizontal scale be one-tenth of an inch to the foot, a = 120, and a beam 100 ft. in length would be drawn equal to 10 in.; then if the true radius at the centre were 10,000 ft., this radius, if the curve were undistorted, would be on paper 1000 in., but making b = 50 we can draw the curve with a radius of 20 in. The vertical distortion of the curve must not be so great that there is a very sensible difference between the length of the arc and its chord. This can be regulated by altering the value of b. In fig. 72 distortion is carried too far; this figure is merely used as an illustration.

38. _Camber._--In order that a girder may become straight under its working load it should be constructed with a camber or upward convexity equal to the calculated deflection. Owing to the yielding of joints when a beam is first loaded a smaller modulus of elasticity should be taken than for a solid bar. For riveted girders E is about 17,500,000 lb per sq. in. for first loading. W.J.M. Rankine gives the approximate rule

Working deflection = [delta] = l²/10,000h,

where l is the span and h the depth of the beam, the stresses being those usual in bridgework, due to the total dead and live load.

(W. C. U.)

[1] For the ancient bridges in Rome see further ROME: _Archaeology_, and such works as R. Lanciani, _Ruins and Excavations of Ancient Rome_ (Eng. trans., 1897), pp. 16 foll.

BRIDGET, SAINT, more properly BRIGID (c. 452-523), one of the patron saints of Ireland, was born at Faughart in county Louth, her father being a prince of Ulster. Refusing to marry, she chose a life of seclusion, making her cell, the first in Ireland, under a large oak tree, whence the place was called Kil-dara, "the church of the oak." The city of Kildare is supposed to derive its name from St Brigid's cell. The year of her death is generally placed in 523. She was buried at Kildare, but her remains were afterwards translated to Downpatrick, where they were laid beside the bodies of St Patrick and St Columba. Her feast is celebrated on the 1st of February. A large collection of miraculous stories clustered round her name, and her reputation was not confined to Ireland, for, under the name of St Bride, she became a favourite saint in England, and numerous churches were dedicated to her in Scotland.

See the five lives given in the Bollandist _Acta Sanctorum_, Feb. 1, i. 99, 119, 950. Cf. Whitley-Stokes, _Three Middle-Irish Homilies on the Lives of Saint Patrick, Brigit and Columba_ (Calcutta, 1874); Colgan, _Acta SS. Hiberniae_; D. O'Hanlon, _Lives of Irish Saints_, vol. ii.; Knowles, _Life of St Brigid_ (1907); further bibliography in Ulysse Chevalier, _Répertoire des sources hist. Bio.-Bibl._ (2nd ed., Paris, 1905), s.v.

BRIDGET, BRIGITTA, BIRGITTA, OF SWEDEN, SAINT (c. 1302-1373), the most celebrated saint of the northern kingdoms, was the daughter of Birger Persson, governor and _lagman_ (provincial judge) of Uppland, and one of the richest landowners of the country. In 1316 she was married to Ulf Gudmarson, lord of Nericia, to whom she bore eight children, one of whom was [v.04 p.0557] afterwards honoured as St Catherine of Sweden. Bridget's saintly and charitable life soon made her known far and wide; she gained, too, great religious influence over her husband, with whom (1341-1343) she went on pilgrimage to St James of Compostella. In 1344, shortly after their return, Ulf died in the Cistercian monastery of Alvastra in East Gothland, and Bridget now devoted herself wholly to religion. As a child she had already believed herself to have visions; these now became more frequent, and her records of these "revelations," which were translated into Latin by Matthias, canon of Linköping, and by her confessor, Peter, prior of Alvastra, obtained a great vogue during the middle ages. It was about this time that she founded the order of St Saviour, or Bridgittines (_q.v._), of which the principal house, at Vadstena, was richly endowed by King Magnus II. and his queen. About 1350 she went to Rome, partly to obtain from the pope the authorization of the new order, partly in pursuance of her self-imposed mission to elevate the moral tone of the age. It was not till 1370 that Pope Urban V. confirmed the rule of her order; but meanwhile Bridget had made herself universally beloved in Rome by her kindness and good works. Save for occasional pilgrimages, including one to Jerusalem in 1373, she remained in Rome till her death on the 23rd of July 1373. She was canonized in 1391 by Pope Boniface IX., and her feast is celebrated on the 9th of October.

BIBLIOGRAPHY.--Cf. the Bollandist _Acta Sanctorum_, Oct. 8, iv. 368-560; the _Vita Sanctae Brigittae_, edited by C. Annerstedt in _Scriptores rerum Suedicarum medii aevi_, iii. 185-244 (Upsala, 1871). The best modern work on the subject is by the comtesse Catherine de Flavigny, entitled _Sainte Brigitte de Suède, sa vie, ses révélations et son oeuvre_ (Paris, 1892), which contains an exhaustive bibliography. The Revelations are contained in the critical edition of St Bridget's works published by the Swedish Historical Society and edited by G.E. Klemming (Stockholm, 1857-1884, II vols.). For full bibliography (to 1904) see Ulysse Chevalier, _Répertoire des sources hist. Bio.-Bibl._, _s.v._ "Brigitte."

BRIDGETON, a city, port of entry, and the county-seat of Cumberland county, New Jersey, U.S.A., in the south part of the state, on Cohansey creek, 38 m. S. of Philadelphia. Pop. (1890) 11,424; (1900) 13,913, of whom 653 were foreign-born and 701 were negroes; (1905) 13,624; (1910) 14,209. It is served by the West Jersey & Sea Shore and the Central of New Jersey railways, by electric railways connecting with adjacent towns, and by Delaware river steamboats on Cohansey creek, which is navigable to this point. It is an attractive residential city, has a park of 650 acres and a fine public library, and is the seat of West Jersey academy and of Ivy Hall, a school for girls. It is an important market town and distributing centre for a rich agricultural region; among its manufactures are glass (the product, chiefly glass bottles, being valued in 1905 at $1,252,795--42.3% of the value of all the city's factory products--and Bridgeton ranking eighth among the cities of the United States in this industry), machinery, clothing, and canned fruits and vegetables; it also has dyeing and finishing works. Though Bridgeton is a port of entry, its foreign commerce is relatively unimportant. The first settlement in what is now Bridgeton was made toward the close of the 18th century. A pioneer iron-works was established here in 1814. The city of Bridgeton, formed by the union of the township of Bridgeton and the township of Cohansey (incorporated in 1845 and 1848 respectively), was chartered in 1864.