C D 



ANALYSIS OF STRESS IN BEAMS 69 



, . e-Msina 



e- OK sin a = - ; 



p<F 



whence 

 (36) 



If, in the handbooks issued by iron and steel companies, the 

 inertia ellipse and core section were drawn on each cross section 

 tabulated, the calculation of the 

 maximum bending stress by for- 

 mula (36) would be extremely 

 simple, requiring merely the 

 measurement of the distance OK. 



Problem 74. Calculate the maxi- 

 mum bending stress in Problem 68 by P. 68 

 means of the core section. 



Solution. The loading is as represented in Kii;. 63, in which the portion BC 

 is subjected to pure bending strain. From Problem 68, M = 20,000 ft. Ib. and 

 F= 48 in. 2 . From the diagram of the core section drawn to scale, OK is found 

 to measure .9 in. Therefore, from formula (36), p = o*>65 Ib./mA 



64. Stress trajectories. In Article 27 the principal stresses at any 

 point in a body were denned as the maximum and minimum normal 

 stresses at this point Lines which everywhere have the direction 

 of the principal stresses are called stress trajectories. 



In order to determine the stress trajectories, a number of cross 

 sections of the body are taken, and the shear and normal stress cal- 

 culated for a number of points in cadi section. The directions which 

 the principal stresses at these points make with the axis of the body 

 can then be found by formula (6), Article 26, as explained in Prob- 

 lem 24. The stress trajectories are thus determined as the envelopes 

 of these tangent-. 



Since the principal stresses at any point are always at right angles, 

 the stress trajectories constitute a family of orthogonal curves. 



65. Materials which do not conform to Hooke's law. The preced- 

 inu' articles of this chapter are based on Hooke's law, and consequently 

 the results are applicable only to materials which conform to this 

 law, such as steel, wrought iron, and wood. Other materials, such as 



iron, M'>ne, brick, cement, and concrete, are so lacking in homo- 

 geneity that their physical properties are very uncertain, differing not 



