X 7 40 40 X 2 40 X 3 40 X 4 40 X 5 40 X 6

As this method of exact or true shears is seldom employed, problems illustrating its application will here be omitted.

35. Position of Live Load for Maximum Moments. In order to obtain the maximum moment at any point, the live load must cover the entire bridge. Let the beam of Fig. 34 be considered, and let it be required to obtain the maximum moment at the section a — a. The reaction, as before computed, is: all terms of which are positive. The moment at the section is: The first term of this equation represents the effect of the load on the portion x, and the second term represents the effect of the load on the portion y. The value of M will always be positive. The quantity x varies between 0 and 1. When x = 0, M is equal to 0. When x = 1, the moment is equal to + - 2. For all values of x between 0 and 1, the first term is positive; and the second term being positive in all cases, it is therefore proved that for maximum live-load moments at any point, the entire span should be loaded, as loads on both segments add positive values to the moment value.

36. Warren Truss under Live Load. In order to analyze a truss intelligently, it is necessary to know its physical structure; that is, it must be known what character of stress can be withstood by the different members. The top chords of all trusses are built to take only compression, and the bottom chords are built to take only tension; while some web members of some trusses are built for tension stresses, some for compression stresses, and some for both. The characteristic of the Warren truss is that the web members are built so as to be able to withstand either tension or compression.

Let it be required to determine the live-load stresses in the Warren truss of Fig. 32. Let the live load per square foot of roadway, which is assumed to be 15 feet wide, be 100 pounds. The live panel load is then 100 X 15 X 20 _ 2 = 15 000 pounds, and the live-load reaction under full load is 21 X 15 000 = 37 500 pounds.

As the live load must cover the entire bridge to give maximum moments—and therefore maximum chord stresses, as the chord stress is equal to the moment divided by the height of the truss—a simple method for the determination of live-load chord stresses presents itself. The live load and the dead load being applied at the same points, and being different in intensity, the stresses produced will be proportional to the panel loads. The maximum live-load chord stresses (see Fig. 33) will then be equal to the dead-load chord stresses

multiplied by 15 000 - 6 333 = 2.371, and they are as follows: = — 2.371 X 17 700 = —42 000 U,U, _ — 2.371 X 15 833 = —37 530 f U,U, = —2.371 X 25 333 = —60 050 U,U, _ —2.371 X 28 500 = —67 600 = +2.371 X 7 917 = +18 770 L,L, = +2.371 X 20 583 = +48 800 L,L, = +2.371 X 26 917 = +63 850 The next step in order is to determine the maximum positive shears, and from these write the maximum negative shears. This is done as follows: + Live-Load V — Live-Load V 15 000 V, _ ----- (1 + 2 + 3 + 4+ 5) = +37 500 0 V, — 1a (1 + 2 + 3 + 4) _ +25 000 —2 500 V, = +15000 —7500 _ 15 000 1(1 + 2) _ + 7 500 —15 000 15 V, = = + 2 500 —25 000 + 0 —37510 The stresses produced by the positive shears are called the maximum live-load stresses, and are: cos 0 + 37 500 = 0 .•. = —37 .500 X 1.12 = —42 000 — U,L, cos + 37 500 = 0 .•. U,L, _ +37 500 X 1 12 = +42 000 +L,U, cos ' + 25 000 = 0 .. L,U, = —25 000 X 1.12 = —28 000 — U,L, cos ’ + 25 000 = 0 .•. U,L, = +25 000 X 1.12 = +28 000 + L,U, cos 0 + 15 000 = 0 .. L,U, = —15 000 X 1.12 = —16 800 —U,L,cos U,L, = +15 000 X 1.12 = +16 800The stresses produced by the negative shears are called the minimum live-load stresses, and are: U, cos =0 — U,L, cos ’ + 0 = 0 U,L, = 0 +L,U, cos cb — 2 500 = 0 .. L,U, = +2 500 X 1.12 = +2 800 —U,L, cos sb — 2 500 = 0 = —2 500 x 1.12 = —2 800 +L,U,cos — 7500=0 ..L,U ,= +7500X1.12= +8400 — U,L, cos $ — 7 500 = 0 .-. U,L, = — 7 500 X 1.12 = —8 400 These stresses, together with the dead-load stresses, should now be placed together as a half-diagram, as is done in Fig. 38, the stresses being rounded off to the nearest ten pounds and then ex pressed in thousands of pounds. No minimum live-load stress is given for the chords, as this will evidently be zero in all cases, since no position of the live load will cause a reversal of stress. It will be seen that the stresses produced by the negative shears are of opposite sign from the stress produced by the dead load, and these tend to decrease the dead-load stress by that amount; and in some cases (see U, and L,, Fig. 38) it will be so large as to overcome the dead-load stress and therefore change the total stress from one kind to another. Do not forget, in considering any combination of the above stresses, that the load occurs with either the maximum or the minimum live load, but not with both at the same time.