RUPTURE STRESSES IN BEAMS AND CRANE HOOKS. 217 



The very approximate nature of the results obtained by this theory of combined 

 tension and bending moment is dealt with in a paper * by Mr E. S. Andrews, B.Sc., 

 and Professor Karl Pearson, F.R.S., the magnitude of the error being there proved 

 to depend upon the radius of curvature to which the hook is bent. They too arrive 

 at the conclusion that the stress at any point in any fixed section under a given load 

 is only a function of the distance of that point from the centroidal line, the maximum 

 stress being obtained at the maximum distance from that line. Since the stress, then, 

 depends directly on the load and this function, it follows that there will be a true 

 neutral axis. An expression for its distance from the centroidal line is obtained, and 

 this is a constant for a given section. The stresses, however, do not vary directly as 

 the distance from this neutral line. 



If the section, which is that of a 15 -ton hook, is as shown in fig. 2a, with the 

 centroid at N, then the diagram AaiNi&iB of fig. 3a represents the diagram of 

 stresses to some scale, as got by Andrews' theory, right up to the true elastic limit 

 on the tension side, and N x is a fixed point. 



If it were possible for the material to behave elastically right up to. the point of 

 fracture, then our ultimate stress diagram in accordance with the two theories would 

 be as in figs. 3 and 3a, but of greater magnitude, and would be represented by 

 AaiNi&xB in figs. 4 and 4a. 



But the extreme stresses Aa x on the tensile side, as calculated from (2a) and by 

 Andrews' formula and as shown on these diagrams, are much greater than the 

 tensile strength of the material. Let AT and BC represent the fracture value of 

 the material in direct tension and compression respectively. (While the fracture 

 value in direct tension is easy to determine, that of a ductile material in direct com- 

 pression cannot be determined absolutely. The term is here used to mean that value 

 of the stress at which the strain exceeds a certain fraction, having regard to initial 

 section and length of test piece.) Through T and C draw lines parallel to AB to 

 intercept aibi in T x and Ci. Then the areas TaiT x and C&iC 1} where the stresses 

 would exceed the fracture value, cannot exist. Note the area C&iCi is a minus 

 quantity in fig. 4a. 



But the resisting moment of the internal stresses just before rupture must be 

 equal to the external bending moment, and therefore areas within the stress limits, 

 the summation of whose moments is equal to the summation of the moments of 

 these excess areas, must be found to replace them. These can be represented by 

 NiTiN 2 and NiCiC 2 N 2 , possibly involving another alteration of the neutral axis to N 2 . 



The conditions involved in this redistribution of stress areas are (a) that the 

 summation of the moments of the complete resisting stress areas CC 2 N 2 and TT 2 N 2 

 should be equal to the moment of the external load W about an axis through N 2 ; 

 and (b), that the algebraic sum of the tensional and compressive stress areas should be 

 equal to the external load W. 



* Drapers' Go. Research Memoirs, Technical Series I., London, 1904. 



