STRESS AND STRAIN. l3 



20,000 



tion as one-thirci. The lougitudinal unit stress = 



2 



^= 10,000 lbs. / sq. in., while the lateral unit stress due to 

 contraction is 1/3 . ro,ooo = — 3333 lbs. / sq. in. The maxi- 

 mum unit shearing stress, neglecting contraction, equals one- 

 half the direct unit stress, or 5000 lbs. / sq. in., and acts at 

 45°. The maximum shearing unit stress when contraction is 

 considered is found to still act at 45°, but its value becomes 

 now one-half the algebraic difference between the longitudinal 

 and lateral unit stresses, or, 1/2 [10,000 — ( — 3333)] = 6666 

 lbs. / sq. in. Thus it is seen that the actual unit shear con- 

 siderably exceeds the value found by the Common Theory. 



Let us take up now the case of a beam under flexure. 

 Here, at the start, it is necessary to distinguish between two 

 distinct phases of stress to which the beam is subjected, 

 namely, local stress and stress due to general bending and 

 shear. The effects of local stress are especially pronounced 

 under the points of application of concentrated loads and 

 over points of support. These effects are superposed upon 

 the general stresses. The results are different with different 

 loadings, and in the present article no attempt will be made 

 to consider tham. Our efforts, therefore, will be confined to 

 the general internal stresses due to bending movement and 

 shear, which is all that can be done if general results are to 

 be obtained. Later, if it be found desirable, the local effects 

 for special cases may be considered, and we shall have only 

 to add these to the general results already found to complete 

 the investigation. 



In the case of a beam subjected to bending moment the 

 upper fibres are in compression and the lower fibres are in 

 tension. According to the Common Theory, no deformation 

 is considered, and the stresses are assumed to vary directly as 

 the distance from the neutral axis. From the principles of 

 statics the horizontal stresses necessary to equilibrate the 

 external forces are calculated. Further, at most points in 



