By CHARLES LEAN, M. Inst. C.E.
Having had occasion to get out the stresses in girders of the bowstring form, the author was not satisfied with the common formulae for the diagonal braces, which, owing to the difficulty of apportioning the stresses amongst five members meeting in one point, were to a large extent based on an assumption as to the course taken by the stresses. As far as he could ascertain it, the ordinary method was to assume that one set of diagonals, or those inclined, say, to the right-hand, acted at one time, and those inclined in the opposite direction at another time, and, in making the calculations, the apportionment of the stresses was effected by omitting one set. Calculations made in this way give results which would justify the common method adopted in the construction of bowstring girders, viz., of bracing the verticals and leaving the diagonal unbraced; but an inspection of many existing examples of these bridges during the passing of the live load showed that there was something defective in them. The long unbraced ties vibrated considerably, and evidently got slack during a part of the time that the live load was passing over the bridge.
In order to get some definite formulae for these girders free from any assumed conditions as to the course taken by the stresses, or their apportionment amongst the several members meeting at each joint, the author adopted the following method, which, he believes, has not hitherto been used by engineers:
Let Fig. 1 represent a bowstring girder, the stresses in which it is desired to ascertain under the loads shown on it by the circles, the figures in the small circles representing the dead load per bay, and that in the large circle the total of live and dead load per bay of the main girders. A girder, Fig. 1A, with parallel flanges, verticals, and diagonals, and depth equal to the length of one bay, was drawn with the same loading as the bowstring. The stresses in the flanges were taken out, as shown in the figure, keeping separate those caused by diagonals inclined to the left from those caused by diagonals inclined to the right. The vertical component of the stress in the end bay of the top flange of the bowstring girder, Fig. 1, was, of course, equal to the pressure on the abutment, and the stress in the first bay of the bottom flange and the horizontal component of the stress in the first bay of the top flange was obtained by multiplying this pressure by the length of the bay and dividing by the length of the first vertical.
The horizontal component of the stress in any other bay of the top or bottom flange of the bowstring girder - Fig. 1 - was found by adding together the product of the stress in the parallel flanged girder, caused by diagonals inclining to the right, divided by the depth of the bowstring girder at the left of the bay, and multiplied by the depth of the parallel flanged girder; and the product of the stress caused by diagonals inclining to the left divided by the depth of the bowstring girder at the right of the bay, multiplied by the depth of the parallel flanged girder. Thus the horizontal component of the stress in D=
_ _ | Stress caused by diagonals Length of right Depth of parallel | | leaning to left. vertical. flanged girder. | | | + |_ 15.75 × 1/4.5 × 10 _| _ _ | Stress caused by diagonals Length of ver- Depth of parallel | | leaning to right. tical to left. flanged girder. | | | |_ 24 × 1/8 × 10 _| = 65; and the vertical component = Horizontal component. Length of bay. 65 × 1/10 × (8.0 - 4.5) = 22.75.
In the same way the horizontal and vertical components of the stresses in each of the other bays of the flanges of the bowstring were found; and the stresses in the verticals and diagonals were found by addition, subtraction, and reduction. These calculations are shown on the table, Fig 1B. The result of this is a complete set of stresses in all the members of the bowstring girder - see Fig. 2 - which produce a state of equilibrium at each point. The fact that this state of equilibrium is produced proves conclusively that the rule above described and thus applied, although possibly it may be considered empirical, results in the correct solution of the question, and that the stresses shown are actually those which the girder would have to sustain under the given position of the live load. Figs. 2 to 10 inclusive show stresses arrived at in this manner for every position of the live load. An inspection of these diagrams shows: a. That there is no single instance of compression in a vertical member of the bowstring girder, b. That every one of the diagonals is subjected to compression at some point or other in the passage of the live load over the bridge, c.
That the maximum horizontal component of the stresses in each of the diagonals is a constant quantity, not only for tension and compression, but for all the diagonals. The diagrams also show the following facts, which are, however, recognized in the common formulae: d. The maximum stress in any vertical is equal to the sum of the amounts of the live and dead loads per bay of the girder. e. The maximum horizontal component of the stresses in any bay of the top flange is the same for each bay, and is equal to the maximum stress in the bottom flange. Having taken out the stresses in several forms of bowstring girders, differing from each other in the proportion of depth to span, the number of bays in the girder, and the amounts and ratios of the live and dead loads, similar results were invariably found, and a consideration of the various sets of calculations resulted in the following empirical rule for the stresses in the diagonals: "The horizontal component of the greatest stress in any diagonal, which will be both compressive and tensile, and is the same for every diagonal brace in the girder, is equal to the amount of the live load per bay multiplied by the span of the girder, and divided by sixteen times the depth of girder at center." The following formulae will give all the stresses in the bowstring girder, without the necessity of any diagrams, or basing any calculations on the assumed action of any of the members of the girders:
Let S = span of girder. D = depth at center. B = length of one bay. N = number of bays. L = length of any bay of top flange. l = length of any diagonal. w = dead load per bay of girder. w¹= live load per bay of girder. W = total load per bay of girder = w + w¹. Then: S/B = N. Bottom Flange. WNS/8D = maximum stress throughout. (1) Top Flange.--In any bay the maximum stress = + WNS/8D × L/B = + WLN²/8D (2) Verticals.--The maximum stress = -W. (3) Diagonals.--The maximum stress is ± w¹lS/16DB = ± w¹lN/16D (4)
These results show that the method generally adopted in the construction of bowstring girders is erroneous; and one consequence of the method is the observed looseness and rattling of the long embraced ties referred to at the commencement of the article during the passage of the live load; the fact being that they have at such times to sustain a compressive stress, which slightly buckles them, and sets them vibrating when they recover their original position.
Another necessity of the common method of construction is the use of an unnecessary quantity of metal in the diagonals; for, by leaving them unbraced, the set of diagonals which does act is subjected to exactly twice the stress which would be caused in it if the bridge was properly constructed. A comparison of the results of a set of calculations on the common plan with those given in this paper, shows at once that this is the case; for the ordinary system of calculation the stresses, in addition to showing compression in the verticals, gives exactly twice the amount of tension in the diagonals which they should have.
FIG. 1B. _______________________________________________________________________________ | Top Flange Stresses. | Stresses in Diagonals. Hor. Ver. | | C= 31.5 × 10/4.5 = +70.00 = 31.50 |a = 70 -65 =+5.00 = 2.25 | 15.75 × 10/4.5 = 35 |b = " " =-5.00 = 4.00 \ | D > +65.00 = 22.75 |c = 65 -58.33-5 =+1.67 = 1.33 / | 24 × 10/8 = 30 |d = " " " =-1.67 = 1.75 \ | E > +58.33 = 14.58 |e = 58.33-55.83-1.67 =+ .83 = .88 / | 29.75 × 10/10.5 = 28.33 |f = " " " =- .83 = 1.01 \ | F > +55.83 = 8.37 |g = 55.83-54.50- .83 =+ .50 = .59 / | 33 × 10/12 = 27.5 |h = " " " =- .50 = .61 \ | G > +54.50 = 2.72 |i = 54.50-53.67- .50 =+ .33 = .43 / | 33.75 × 10/12.5 = 27 |j = " " " =- .33 = .41 \ | H > +53.67 = 2.68 |k = 53.67-53.09- .33 =+ .24 = .28 / | 32 × 10/12 = 26.67 |l = " " " =- .24 = .24 \ | I > +53.09 = 7.97 |m = 53.09-52.67- .24 =+ .18 = .20 / | 27.75 × 10/10.5 = 26.42 |n = " " " =+ .18 = .16 \ | J > +52.67 = 13.17 |o = 52.67-52.36- .18 =+ .13 = .11 / | 21 × 10/8 = 26.25 |p = " " " =- .13 = .06 \ | K > +52.36 = 18.33 | / | 11.75 × 10/4.5 = 26.11 | | L= 23.5 × 10/4.5 = +52.22 = 23.50 | ____________________________________________|___________________________________ | Bottom Flange Stresses. | Stresses in Verticals. | Hor. | Ver. M same as C = 70.00 | r = 15 - 4 = - 11.00 N " D = 65.00 | s = 5 + 2.25 - 1.75 = - 5.50 O " E = 58.33 | t = 5 + 1.33 - 1.01 = - 5.32 P " F = 55.83 | u = 5 + .88 - .61 = - 5.27 Q " G = 54.50 | v = 5 + .59 - .41 = - 5.18 R " H = 53.67 | w = 5 + .43 - .24 = - 5.19 S " I = 53.09 | x = 5 + .28 - .16 = - 5.12 T " J = 52.67 | y = 5 + .20 - .06 = - 5.14 U " K = 52.36 | z = 5 + .11 = - 5.11 V " L = 52.22 | ____________________________________________|____________________________________
- The Engineer.