Wn If wheel loads are used for moments, the relation that K = m m —I must be +, and that k = ]t' n— (L -l- P) must be—, holds m true when the loads are in correct position for maximum moments. Here m = the number of panels, and n = the panel under considera tion and is to be reckoned from the left end; in fact, all terms have the same value as men tioned in Article 46. A careful review of Articles 44 and 46 should enable the student to follow the example which will now be given.
EXAMPLE. It is required Example. It is required to determine the moments at the points of floor-beam support for a 5-panel through plate girder of 75-foot span. The live loading is Cooper's E 40.
Dead-Load Moments. Through plate-girders, on ac count of the heavy floor system and the fact that the floor sys tem transfers its own weight and that of the live load to the girders as concentrated loads, are about 40 per cent heavier than deck plate-girder bridges of the same span. The weight of the entire span, therefore, is; 1.4 X 75(123.5 + 10 X 75) = 91 700 pounds.
Part of this 91 700 pounds (the weight of the girders themselves) acts as a uniform load; the re mainder (the weight of the floor-beams and stringers) acts as concentrated loads at the points where the floor-beams join the web. Experience has shown that the weight of the floor for a single-track railroad system is about 400 pounds per linear foot. The weight of the stringers and floor-beams for this bridge is therefore 75 X 400 = 30 000 pounds, and 91 700 30 000 = 61 700 pounds acts as a uniform load. This 61 700 pounds is distributed over two girders, and so gives 61 700 _ (2X75) = say, 412 pounds per linear foot of one girder.
The dead load which is concentrated at each panel point is that due to the weight of the steel floor and the weight of ties, rails, and fastenings. It is, for one girder, 15 X (400 + 400) _ 2 - 6 000 pounds.
The dead-load moments are now computed by the methods of Strength of Materials, and are found to be: Mo= D1, = 217, = +4 390 000 pound-inches; = = +6 580 000 pound-inches.
Live-Load Moments. The positions of the wheels for maximum moments are now determined (see Table XII).
The computations for the reactions are best arranged in tabular form. Table XIII gives the values.
The last two values are obtained when the load comes on the bridge from the left. Inspection of the results obtained at points 3 and 4 when the load comes on from the right, shows that they are con siderably smaller than the results obtained at their symmetrical points 1 and 2, and therefore it was not necessary to determine the moments for any points to the right of the center. This is true of all
girder spans, deck or through.
The method of procedure when the girder is a deck plate-girder is the same as that just iilustrated, except that in the computation of the dead-load moments there is no concentration of certain por tions of the dead load, the weight of the girders themselves being a uniform load, as is also the weight of the ties and rails or, if it be a highway bridge, the floor-joists which run transversely. Highway spans are seldom built of deck plate-girders, it being preferable to use the through girders, as then the girders themselves serve as a rail ing and keep the traffic confined to the roadway. The girder span is usually divided into ten equal divisions, the points of division being called the tenth-points. The shears and moments are computed for the center point and those points which lie to the left of the center. After the values are computed, they are laid off as ordinates, with the corresponding tenth-points as abscissae. A curve is then drawn through their upper ends, and the curve of maximum shears or moments is the result. To get the maximum shear or moment at any point other than a tenth-point, the ordinate is scaled at the desired point.
EXAMPLE. Let it be required to determine the maximum Example. Let it be required to determine the maximum moments at the tenth-points of a 100-foot-span deck plate-girder.
Dead-Load Moments. The weight of steel in the span is 100 (123.5 -I- 10 X 100) = 112 350 pounds, and the weight of the track is 400 X 100 = 40 000 pounds, making a total of 152 350 pounds, or 152 350 _ (2 X 100) = say, 762 pounds per linear foot per girder. The dead-load moments are now determined according to the methods of Strength of Materials, and are: M, = 342 800 pound-feet M, ='609 400 " M, _ 799 840 " " bi, = 914 100 " " Ai, = 952 200 " " Live-Load Moments. The determination of the wheel load positions is made by the use of the formuhe h = ( _ L) and - m k = — (L P); only, in this case, n is the number of divisions from the left support to the section, and m is the number of divisions into which the girder is divided.