FLEXURE OF BEAMS 



75 



from which the distance of the point of maximum deflection from the left support 

 is found to be 



d('2l-d) 

 ~~3 



and the deflection at this point is 



D = 



3EII 



Problem 78. Find the equation of the elastic curve and the maximum deflec- 

 tion for a simple beam of length Z, bearing a uniform load of w Ib. per unit 

 of length. 



Problem 79. Find the equation of the elastic curve and the maximum deflec- 

 tion for a cantilever of length I, uniformly loaded with a load of w Ib. per unit 

 of length. 



Problem 80. A ( 'arnegie I-beam, No. B 13, is 10 ft. long and bears a load of 

 25 tons at its center. Find the deflection of the point of application of the load. 



NOTE. From the Carnegie handbook, the moment of inertia of the beam about a 

 neutral axis perpendicular to the web is 7=84.9 in. 4 . 



Problem 81. Find the deflection of the beam in the preceding problem at a 

 point 4 ft. from one end. 



67. Limitation to Bernoulli's assumption. In Article 39 it was 

 stated that Bernoulli's assumption formed the basis of the common 



D 



FIG. 68 



theory of flexure. In the case of a prismatic beam subjected to pure 

 bending strain, this assumption is rigorously correct. For if the oppo- 

 site faces of a prism ABCD (Fig. 57) are acted upon by equal bend- 

 ing moments of opposite sign, both faces must, by reason of symmetry, 

 remain plane and take a position such as A'B'C'D' in the figure. 



However, if shearing stress also occurs, Bernoulli's assumption is 



no longer absolutely correct. In Article 55 it was proved that the 



distribution of shear over any cross section limited by parallel sides 



- as the ordinates to a parabola. Consequently, if the beam is 



supposed cut into thin layers by horizontal planes, as represented in 



