44^ PROBLEMS 



PROBLEM 17. CALCULATION OF THE DEFLECTION OF A STEEL BEAM BY 



GRAPHICS. 



(a) Problem. Given a 12" I @ 31^ Ibs. per foot, span 4o'-o", 

 load 5000 Ibs. applied i6'-o" from the left support. 7 = 215.8 in. 4 . 

 E = 28,000,000. Calculate the maximum deflection due to the load, and 

 the maximum deflection under the load by the graphic method. Scale of 

 beam, i" = 6'-o". Scale of loads, i" = 2ooo Ibs. Pole distance, 

 H = 4.000 Ibs. Scale of areas, i" = 60 sq. ft. Pole distance, H' = 240 

 sq. ft. 



(b) Methods. Construct force polygon (a) and draw bending- 

 moment polygon (b). Divide polygon (b) into segments, and assume 

 that each area acts as a load through its center of gravity. Construct 

 force polygon (c), and draw equilibrium polygon (d). Polygon (d) 

 is a curve which has ordinates proportional to the true deflections. 



(c) Results. The maximum deflection comes between the load 

 and the center of the beam. If the area of the polygon (b) was meas- 

 ured in square inches and the ordinates in (d) measured in inches the 

 deflection would be A = y X H X H' -f- E I. In the problem this result 

 must be multiplied by 1728. The closing lines of polygons (b) and 

 (d) need not be horizontal. The solution given above may be very 

 simply stated as follows : Construct the bending-moment polygon for 

 the given loading on the beam. Load the beam with this bending- 

 moment polygon, and with a force polygon having a pole distance equal 

 to E I, construct an equilibrium polygon ; this polygon will be the elastic 

 curve of the beam. It is not commonly convenient to use a pole dis- 

 tance equal to E I, and a pole distance H is used, where n H equals E I. 

 For a discussion of this subject see Chapter XVa. 



PROBLEM i7a. CALCULATION OF THE DEFLECTION OF A STEEL BEAM 



BY GRAPHICS. 



(a) Problem. Given a 12" I @ 31^ Ibs. per foot span 4o'-o", 

 load 3000 Ibs. applied i6'-o" from the left support, and 3000 Ibs. 

 applied i2'-o" from the right support. 7 = 215.8 in. 4 . = 28,000,- 

 ooo. Calculate the maximum deflection due to the load, and the 

 maximum deflection under the load by the graphic method. Scale of 



