FLEXURE OF BEAMS 



73 



Problem 75. Find the equation of the elastic curve and the deflection at the 

 center of a simple beam of length J, bearing a single concentrated load P at its 

 center. 



Solution. The elastic curve in this case consists of two branches, AB and BC 



(Fig- 

 Consider the portion of the beam on the left of any section win, distant x from 



p 

 the left support. Then M = R& = x, and consequently the differential 



equation of the branch AB of the elastic curve is 



Integrating twice, 



and 



dx* 



- 





At J5, x = - and = 0, since the tangent at B is horizontal. Substituting these 



values in the first integral, C\ = At A, x = and y = ; hence C a = 0. Con- 



16 

 aequently, the equation <>f the left half of the elastic curve is 



The deflection D at the center is the value of y for x = - ; hence 



48 El 



Problem 76. Find the 

 equation of the elastic 

 curve and the maximum 

 deflection for a cuitil* \T 

 of length J, bearing a 

 gle concentrated load P at 

 the end. 



Problem 77. Find the 

 equation of thrrlastic curve 

 ami the maximum deflec- 

 tion for a simple beam of 

 length /, bearing a single concentrated load P at a distance d from the left support. 



Solution. The elastic curve in this case consists of two branches, AB and BC 



- P(l-d)x 



(Fig. 56). For a point in AB distant x from the left support, M = 

 Therefore 



I 



Integrating twice, 



P<l-d)x 



I 



