L, U, = 0 U,L, I See discussion U L, = 0 following.
If live panel loads were placed at points L and L, the live load shear in c c would be 46.8; and the dead-load shear being + 15, the counter would act, and the stress in would be tensile and equal to the sum of the dead and live panel loads which are at its lower end L,. If points and had live panel loads on them, the resultant shear in c c would be 23.4 + 15.0 = 8.4; the counter would act, and the stress in U,L, would be tensile and equal to the dead panel load which is at L,. There being no live panel load at L the live-load stress in U,L, would be zero under this loading. If a live panel load be placed at L, only, then the shears and the mem bers acting will be as shown in Fig. 51, and V, for dead load = +45.0 the load at U or = 45 10 = +35.0. The V,, for live load = 7.8, and the stress equation U,L, 7.8 = 0, from which U,L, _ 7.8. So this live-load compression stress of 7 800 pounds occurs at the same time as the dead-load tensile stress of 45 000 pounds.
By loading various groups of panel points in succession and determining the resulting live-load stresses in U,L it will be found that under no loading can a negative live-load stress be produced. The minimum live-load stress is therefore zero, and occurs when there is no live load on the bridge.
The stresses should now be placed on an outline diagram similar to that of Fig. 48, and the stresses in corresponding members com pared with those in that figure. This is left for the student.
41. Bowstring and Parabolic Trusses. A bowstring truss is shown in Fig. 13, the full lines representing the main members, which are the members under stress by the dead load. The dotted members represent counters which may be stressed by the action of the live load.
As before mentioned, the stresses in the chords and also in the webbing are quite uniform. When the end supports and the panel points lie on the are of a certain curve, called a parabola, then, under full load, the stresses in all panels of the lower chord are equal; the stress in all verticals is tensile and is equal to the panel load at the lower end; and the stress in all diagonals is zero. Under partial load, the stresses in the webbing are exceedingly small, and the chord stresses remain almost equal.
If it is desired to have a parabolic truss, first decide upon the length of span, the number of panels, and the height at the center. The height of any vertical post is given by the formula: All distances are in feet. Suppose, as an example, that it was desired to determine the heights of the vertical posts in an 8-panel parabolic truss of a height approximately equal to 24 feet. One-half the truss is shown in Fig. 52. At the center, d = 0, and the equation reduces to h = H, which is 24 feet. For d = 20; then, 160' from which, h 22.5 feet.
For h = 24 4 X 40'h = 18.0 feet.
For U,L 4x24x60'h= 10.5 feet.
Inspection of the above results shows that the span or the center height must become quite great before the clearance at will be sufficient to allow the traffic to pass under a portal bracing at this point. For this reason these trusses are usually built as through trusses with bracing on the outside of the truss, which connects to the floor-beams extended.
In the bowstring truss, the panel points of the top chord usually lie on the arc of a parabola which does not pass through the supports. For example, suppose that it was decided to have the span and pan els the same as shown in Fig. 52, but the height at L, was to be' 28 feet, and at 36 feet. By substituting thesevalues in the equation just given, and solving for 1, the place will be deter mined where the para bolic curve cuts the lower chord extended, and the lengths of the vertical posts may be computed as before. Substituting these re sults: which shows that the are cuts the lower chord extended at a point 254.5 _ 2 = 127.25 feet from the center of the span (see Fig. 53).
4 X 36 X 60 U,L, h = 36 254.5' = 28.00 feet, which checks.
The analysis of a bowstring truss will now be given. Both the maximum and minimum stresses will be determined, as reversal of stresses is liable to occur in the intermediate posts. The loading for minimum live-load stresses can be ascertained only by trial, care being taken to compute the dead-load stresses for arrangement of web members caused by that particular live loading.
Let it be required to determine the maximum stresses in the 5-panel 100-foot bowstring truss shown in Fig. 54, remembering that the diagonals take only ten sion. The height of U,L, is 20 feet, and of U2L2 25 feet. The dead panel load is 17 200 pounds, and the live panel load is 50 000 pounds. The full lines show the main members which act under dead-load stress, and the dotted lines show the counters which may act under the action of the live load. One-third of the dead panel load, or 5 730 pounds, is taken as acting at the upper panel points, while the remainder, 11 470 pounds, acts at the lower ones. Articles 27, 28, and 29 should be carefully, reviewed before going further. The shear times the secant method cannot be con veniently employed for the live-load stresses in the members and as the section will cut the member U,U,, and the vertical component of its stress must be reckoned with in the stress equation The method of moments as illustrated in Fig. 27, Article 29, will b used for these members.
The dead-load reaction is 2 X 17.2 = +34.4. The dead-load chord stresses should first be computed.
By resolving the horizontal forces around L it is seen that = (see Fig. 55). Passing the section a a, taking the center of moments, at U and stating the equation of the moments to the left of the section, there results (see Fig. 56) :