STRESSES IN BRIDGE TRUSSES. 501 



f.xti-rnal force acting from the outside toward the cut section, and UiLt X 70.7 + RI X 60 

 - W X 80 = o. Now Ri - 6 tons and W - 3 tons, and U\Lt X 70.7 120 ft. -tons, and 



> U\Li = 1.70 tons. The other dead load stresses are calculated as shown. 



Live Load Stresses. The live load chord stresses are equal to the dead load chord stresses 

 multiplied by 8/3. The maximum stress in U\L\ will occur with loads at Li, Lt, and L\', while 

 the maximum stress in counter U*L\ will occur with a load at L\ only. The maximum tension 

 in /'_/.- will occur with all the live loads on the bridge, while the maximum compression will 

 occur when there is a maximum stress in the counter UtLt, loads at Lt and L/. The details 

 of the solution are shown in the problem. 



(c) Results. The stress in the counter UtLt and the chords UtUt and LtLt may be 

 calculated by the method of coefficients, and will be the same as for a truss with parallel chords 

 having a depth of 25' o". The maximum stress in UtLt will occur with loads Lt and L/ on the 

 bridge, when the left reaction equals 2 X 3-P/5 = f-P. The stress in UtLt = $P-sec0 

 = 6.15 tons. 



PROBLEM 8. MAXIMUM AND MINIMUM STRESSES IN A THROUGH WARREN TRUSS BY 



GRAPHIC MOMENTS. 



(o) Problem. Given a through Warren truss, span 140' o", panel length 20' o", depth 

 20' o", dead load 800 Ib. per lineal foot per truss, live load 1,200 Ib. per lineal foot per truss. 

 Calculate the maximum and minimum stresses by graphic moments. Scale of truss, i" = 20' o". 

 Scale of loads, i" = 50,000 Ib. 



(6) Methods. Chord Stresses. Calculate the center ordinate of the parabola = w- L*/8d 

 = 98,000 Ib., and lay it off at 5 to the prescribed scale. Now lay off the vertical line 1-5 at the 

 left and right abutments. Make 1-2 = 2-3 = 3-4 = 2 (4-5). Draw the inclined lines 1-5, 

 2-5, 3-5, 4-5, 5-5. The intersections of these lines with verticals let drop from the lower chord 

 points are points in the stress parabola for the upper chord stresses. The stresses in the lower 

 chords are the arithmetical means of the stresses in the upper chords on each side. By changing 

 the scale the live load stresses may be scaled directly from the diagram. 



Web Stresses. At the distance of a panel to the left of the left abutment lay off the vertical 

 line 1-8 equal to one-half the total live load on the truss, to the prescribed scale, equal 1,200 X 70 

 = 84,000 Ibs. Now divide the line 1-8 into as many equal parts as there are panels in the truss, 

 and mark the points of division 2, 3, 4, etc. Connect these points of division with the panel 

 point 7, the first panel point to the left of the right abutment. Drop verticals from the panel 

 points of the lower chord of the truss to the line 1-8, and the intersections of like numbered lines 

 will give points on the curve of maximum live load shears. 



To construct the dead load shear diagram, lay off $W, downward to the prescribed scale 

 under the left abutment, and reduce the shear under each load to the right by W, until the dead 

 load shear is i>W at the right abutment. The dead load shear diagram is then constructed as 

 shown. 



Maximum and Minimum Web Stresses. The maximum shear in any panel is then the ordinate 

 to the right of the panel point on the left end of the panel, and the stresses in the web members 

 are calculated by drawing lines parallel to the corresponding member as shown. Positive stresses 

 are measured downwards from the live load shear curve, and negative stresses are measured 

 upwards from the live load shear curve. 



(c) Results. This method is an excellent one for illustrating the effect of the different 

 systems of loads, but consumes too much time to be of practical use. It should be noted that 

 the maximum ordinate to the chord parabola is not a chord stress in a Warren truss with an 

 odd number of panels. 



PROBLEM 9. MAXIMUM AND MINIMUM STRESSES IN A PETIT TRUSS BY ALGEBRAIC 



MOMENTS. 



(o) Problem. Given a Petit truss, span 350' o", panel length 25' o", depth at the hip 

 50' o", depth at center 58' o", dead load 0.9 tons per lineal foot per truss, live load 1.4 tons per 

 lineal foot per truss. Calculate the maximum and minimum stresses due to dead and live loads 

 by algebraic moments. Scale of truss, i" = 40' o". Scale of lever arms, any convenient scale. 



(b) Methods. Construct a truss diagram carefully to scale as shown. Construct one- 

 half the truss to scale on a large piece of paper and calculate the lever arms as shown, and check 

 by scaling from the diagram. The methods of calculation will be shown by two examples: 



i. Stresses in Tie 6-7. Dead Load Stress. Pass a section cutting members 7~X, 6-7, and 

 6-F, and cutting away the truss to the right. The center of moments will be at A, the inter- 

 section of chords 7~X and 6-F. Now assume the stress in 6-7 as an external force acting from 

 the outside toward the cut section. Then for equilibrium 6-7 X 477-O -f- RI X 575 $W 



37 



