134 



DEFLECTION OF A BEAM OR GIRDER. 



Art. 8G. 



which is the same as was deduced from the elastic line equation, 

 Art. 83. 



86. Deflection of a Beam or Girder Having a Variable 

 Cross Section. 1 When the cross section of a beam varies con- 

 tinuously, 7 may be expressed in terms of x in equation (39). 

 The equation of the elastic line may then be deduced as in the 

 cases above, if the integrations are possible. 



In a girder with flange plates, the cross section changes 

 abruptly at the ends of these plates. The sections of abrupt 

 change are similar to sections at which concentrated loads are 

 applied. Dealing with the case shown in Fig. 90 in a somewhat 

 different way from that used in Art. 85, the elastic line for the 

 part AB is gotten at once by treating this part as a beam carry- 

 ing a uniform load of length AB, and a concentrated load at its 

 end B, equal to the sum of all the other loads. The slope at 7? 

 being calculated, the deflection at D, due to the bending of AB, 

 may be obtained. The parts BC and CD would be treated in a 

 similar manner. If there were an abrupt change of section at E, 

 the segment BC would be divided into two parts, because 7 

 changes at E and is a factor in the deflection formula.- The slope 

 and deflection at E would be found from the general formulas 



for -p and y, for a beam of length BC, carrying a uniform load 



and a load at its end C. See Figs. 94 and 100. 



A similar procedure applies to a beam supported at its ends, 

 if the direction of the tangent to the elastic line at some point 

 is known ; any part of the beam may be considered like a canti- 

 lever beam, the reaction and loads acting in opposite directions. 



To find the maximum deflection of a beam supported at its 

 ends, the graphic method is the simplest and safest except in 

 special cases. In this method the moment diagram is divided 

 into a number of strips, at whose centers of gravity imaginary 

 forces, proportional to the areas of the strips, are supposed to 

 act on the beam. Drawing ;> strin.ir polygon (37) for these loads, 

 the ordinates will represent deflections if the pole distance is 

 taken equal to El, as is shown by the following comparison 

 between the elastic line and the moment (M'). 



*,/ 



=Sf. 



= w. 



M 



dx 2 



^ue Trans. Am. Soc. C. E., Vol. 51, page 1. 



