ANALYSIS OF STRESS IN BEAMS 49 



As another example, consider a beam of length I bearing a uniform load of 

 amount w per unit of length. Then the total load on the beam is wl, and each 



id 

 reaction is Therefore the moment at a point distant x from the left support is 



wl x wx /t 



Jf = -.*-.- = _<|-x). 



From this relation it is evident that M is zero f or x = or J, and attains its maxi- 

 mum value for x = - ; that is to say, the bending moment is zero at either end of 

 the beam and a maximum at the center. 



From the formula M= pS, given in Article 45, it is evident that 

 the maximum value of the stress p occurs where the bending moment 

 M tfl a maximum. Ordinarily tlu> maximum bending moment pro- 

 duces a greater strain than the maximum shear ; therefore the section 

 at which the maximum moment occurs is called the dangerous section, 

 Miii- it is the section at which the material is most severely strained, 

 and consequently tin- nn<- at whirh rupture may be expected to occur. 



In order to find the maximum bending stress in a beam, the formula 

 M = pS is written 



M 



The maximum bending stress is then obtained at once by simply 

 dividing tin- maximum bending moment by the section modulus. 



Problem 57. A rectangular wooden beam 14 ft. long, 4 in. \\idr, and '.' in. deep 

 bean a uniform load of 76 Ib./ft. Find the position and amount of the maximum 

 bending moment. 



Problem 58. Kind the maximum bending stress in the beam in the preceding 

 problem. 



Problem 59. A Cambria I-beam, No. B 83, which weighs 40 Ib./ft., is 16 ft. 

 long and bean a single concentrated load of 6 tons at its center. Find the maxi- 

 mum bending stress in the beam, taking into account the weight of the beam. 



52. Bending moment and shear diagrams. In general, the bending 

 moment and shear vary h'>m point to point along a beam. This 

 variation is shown graphically in the following diagrams for several 

 different systems of loading. 



Simple beam bearing a single concentrated load P at its center 



'5). From symmetry the reactions R l and R t are each equal 

 /' 

 to Let mn be any section of the beam at a distance x from tln i . 



left support, and consider the portion of the beam un the left of this 



