X 7 = 40 + 40 X 2 + 40 X 3 + 40 X 4 + 40 X 5 + 40 X 6, and can still be simplified by writing: which is the form customarily used, the panel length being taken as a unit of measurement. The other shears are now easily computed in a similar manner: (1 + 2+3+4+5)=+H5.71 In computing the maximum negative shears, sometimes called the minimum shears, the reaction is not the same as the shear, as there are loads to the left of the section, and these must be sub tracted. The loadings are: For maximum V in 1st panel, load no points.
" "2d " " L, " " "3d " " L, and L,.
" " V " 4th " " L L and " V " 5th " " L. L L and L,.
" " V " 6th " " L L L,, and L.
" " V " 7th " " L,, L2, L3, L,, L,, and L,.
It is evident that the maximum V, is equal to zero, there being no loads on the span. The maximum negative shear in the second panel is equal to the reaction produced by loading the panel point L and the load at L,. Thus, X6 load at L, = 40 7 (6) _ 40 = 5.71 The other shears are next computed as follows: -V, = 40 + 5) - 2 X 40 = -17.14 - V, = 40 (6 + 5 + 4) - 3 X 40 = -34.28 - V, = (6 + 5 + 4 + 3) - 4 X 40 - -57.14 - -5 X 40 = -85.71 - (6+5+4+3+2+1)-6 X40= -120.00 The maximum positive and the maximum negative live-load shears should now be written side by side, and inspected, in order to observe any existing relations which might help to lessen the labor of future computations. The values are given in thousands of pounds below: It is at once seen that the negative shears are numerically equal in value to the positive ones, but that they occur in order. This simplifies the labor required in the derivation of the negative shears; for, after computing the maximum positive shears, these may be written in reverse order, and the negative sign prefixed; the result will be the maximum negative shears.
The above method for maximum live-load shears is called the conventional method. It is the one that is almost universally used, and its use will be continued throughout this text.
Exact Method. On account of the fact that the floor stringers or joists transfer the loads to the panel points, it would be impossible to have a full panel live load at one panel point and no load at the panel point 4head or behind. In order to have a full panel load at one point, the stringers in the panels on both sides of the point must be full-loaded, and this would give a load at the panel point ahead, provided the bridge was fully loaded up to and not beyond the panel point ahead, equal in value to one-half of a full panel load (see Fig.
36). The uniform live load, in order to produce full panel loads at L, and will also produce one-half a panel load at L,.
By the methods of differential calculus, it can be proved that the true maximum positive live-load shear occurs in a panel when the live load extends from the panel point to the right into that panel an amount (see Fig. 37) equal to in which, n = Number of the panel point to the left of the panel under considera tion, counting from the right; m = Total number of panels in the bridge; p = Panel length.
Let the truss of Fig. 35 be considered, the live load being 2 000 pounds per linear foot of truss, and let it be required to determine the true maximum positive live-load shear in the 5th panel from the right end.
7 4 1 X 20 = 13.333 feet.
There will now be (4 X 20 + 13.333) X 2 000 = 186 666 pounds on the truss; and the left reaction will be{186666 X (4 X 20+13.333) 2} =140 = 62 200 pounds. From this must be subtracted the amount of the load on the 13.333 feet, which is transferred to the point This is equal to the reaction of a beam of a span equal to the panel length, loaded for a distance of 13.333 feet from the right support with a uniform load of 2 000 pounds per linear foot. This The + V as computed by the conventional method, was + 57 140, making a difference of 3 730 pounds between the two. If the true shears were computed and compared with the conventional shears, it would be found that the V, would be the same, and that the remainder of the conventional shears would be greater than the corresponding true shears. The difference between any two corre sponding shears would increase from the left to the right end; that is, the difference between the conventional and exact shears would be greatest in the panel To get the maximum negative shear in any panel, load from the left support and out into the panel under consideration an amount p x, and proceed in a manner similar to that above described.