530 STRUCTURAL MECHANICS. CHAP. XVI. 



The shearing stresses are not uniformly distributed and for a rectangular beam / = %V/A, 

 while in a circular beam/,, = $V/A. 



Resisting Moment. The bending moment at any section is resisted by the moment of the 

 tensile and compressive stresses which act as a couple with an arm equal to the distance between 

 the centroids of the tensile and compressive stresses. The moment of this internal couple is 

 called the resisting moment. If / = the unit stress at any extreme fiber on the surface of the 

 beam due to bending moment, c = distance from that fiber to the neutral axis, and M = the 

 bending moment, or the resisting moment, then 



,, // , M-c 



M = J , or f = ~Y~ ' 



where 7 = the moment of inertia of the cross section of the beam. 



Moment of Inertia. The moment of inertia of any area about any axis is equal to the sum 

 of the products obtained by multiplying each differential area, dA, by z 2 , the square of the distance 

 of each elementary area from the axis, 7 = ~Lz 2 -dA. The moment of inertia of any section is a 

 minimum when the axis passes through the center of gravity of the cross section. 



Section Modulus. In designing beams it is convenient to use the ratio S = I/c, so that 

 M = f'S, or f = M/S. The ratio 5 is known as the section modulus. 



Tables of Moments of Inertia and Section Modulus. Values of moment of inertia, 7, and 

 section modulus, S, for different sections are given on pages 548 to 551, inclusive. Values of 

 moment of inertia and section modulus of structural shapes are given in Part II. 



Deflection of Beams. In a simple beam carrying vertical loads the upper fibers are shortened 

 and the lower fibers are lengthened, while the fibers on the neutral axis are not changed in length 

 but the neutral axis assumed the form of a curve. The differential equation of the elastic curve 

 of a horizontal beam carrying vertical loads will be 



*y - JL i\ 



dy? E-I' 



Substituting proper values of E, I and M, integrating twice and giving proper values to the 

 constants of integration, the values y, or the deflection may be calculated for any point in the 

 beam. The equation of the elastic curve of beams of various types are given on pages 531 to 

 547, inclusive. 



The maximum bending moments and shears in beams due to moving concentrated loads are 

 given on page 542. 



The moments and shears in continuous beams are given on page 543, page 544 and page 545. 



Formulas for stresses in reinforced concrete beams are given on page 546, and stresses in 

 columns, safe working stresses, and safe loads on slabs are given on page 547. 



