BEAM OF RECTANGULAR SECTION. 



Art. (')'(. 



69. Bending Stresses and Moments of Resistance for a 

 Beam of Rectangular Cross Section. The moment is the same 

 at any section between the loads, of the beam in Fig. 65, because 

 the forces on either side of such section form a couple (30\ 

 The bending moment is, therefore Pa, and M K must equal Pa, 

 that is, M = Pa = M K . 



M n is easily found for a rectangular cross section as shown 

 l** in Fig. 69. If the maximum al- 



lowed stress in the extreme fibers 

 t be 1000 Ibs. per sq. in., the 

 * width of the section 10", and its 

 depth 18", then the average unit 

 stress in tension and compression 

 is 500 Ibs. per sq. in. ; these 

 stresses act over an area of 10X9 = 90 sq. ins.; hence the total 

 tension below and the total compression above the neutral axis 

 equals 72 = 90X500 = 45000 Ibs. It is evident that the neutral 

 axis must be in the middle of the section in order that this con- 

 dition of equilibrium may hold. 



These resultants of 45000 Ibs. each act at the centers of grav- 

 ity of the stress diagrams; and since these are triangles (for con- 

 stant width), they act at two thirds of the altitudes from the 

 vertexes, or at 6" from the neutral axis. The lever arm of the 

 couple is then 12" and its moment is 45000X12 = 540000 in. Ibs. 



than 



If a = 4. 5 ft., P should not be greater 

 10000 Ibs. 



In the same manner a general formula may be deduced. 

 The average unit stress is V^, ; it acts over an area l /M' m > VsJ*h 

 is one force of the couple, and its lever arm is | h; its moment 

 is $ sJ)h 2 = M n . 



This value of 3/ R will bo again gotten from the general for- 

 mula for M , applicable to any cross section, which will now be 

 deduced. 



70. Moment of Resistance for a Beam of Any Cross 

 Section. Referring to Fig. 67,?; denotes tlio vortical distance, from 

 the neutral axis, of any element of area, and .s the correspond- 

 ing unit stress. Having decided upon the maximum unit stress 

 (working stress) to be allowed in the extreme fibers s 1 or s z 



