116 



LINES OF PRINCIPAL STRESS. 



Art. 77. 



becomes s' = s s and equation (27) gives < =45 and 135. 

 which agrees with the conclusions in Art. 17. At the neutral 

 axis the principal stresses are equal, of equal intensity with the 

 shearing stresses, and act on planes making angles of 45 with 

 the neutral plane. 



From equation (26), for maximum or minimum s' a 

 ds/ 



d<f> 



. = _ 8t cos 2 <f> 2 s s sin 2 = 



(d) 



tan 24= - (29) 



This value is the negative cotangent of the angle determined 

 by equation (27), therefore the value of 2 <> for maximum or 

 minimum s B f differs from that for maximum or minimum $/ by 

 90, and the values of < differ by 45. This fact has already 

 been brought out for points on the neutral axis. 



By comparing equations (d) and (25), it is apparent that 

 */ does not become zero when S B ' is a maximum or minimum; 

 its value is s t ' = % 5 t ; this is true in the case of Fig. 5, Art. 12, 

 when the inclination of AB is 45, as may readily be proven. 

 By substitution in equation (26) from equation (29), 



. , 



Equation (30) shows that the maximum and minimum 

 values of the shearing stresses are numerically equal, which 

 agrees with the conclusions of Art. 17. 



77. Lines of Principal Stress. Knowing the intensity of 

 the shearing stress (s s ) and the bending stress (s t or s c ) at any 

 point in a vertical section of a beam, the inclinations of the 

 planes on which the principal stresses act may be gotten from 

 equation (27) ; these will also be the inclinations of the lines 

 of principal stress, because they are at right angles to each other., 



Curves having these 

 inclinations for every 

 point through which 

 they pass are lines of 

 principal maximum 

 and principal mini 

 muni stress. Fig. 8V 

 shows approximately 

 the location of such 



Fig. 87. lines in a beam, sup- 



ported at its ends and carrying a load at the center. 



