i6 Wilson, Failnre of certain Cast Steel Dies. 



Let the forces be three pairs of normal pressures/,, P,, p,. 



Let a^ a,., «, be the strains in the cube, and let e,, e,, e, 



be those parts of a^, a,., a, due to p„ p,, p, alone, 



Then 



a, ■= ie, - me,. — me^. 



u,. = {- me, + e,. - me,). 



a, ^ ( - me, - me^ + e^). 



where w is the ratio of lateral to longitudinal extension. 



Since o,. = a, for the tube, it is easily seen that /,.=/, as 



long as the relations between the stresses and strains are 



the same. Now for deformation up to maximum load 



/ = Ce\ 



or e = Af and since there is little or no change in densit}- 



a, + 2 a,. = 0, 



which gives 



p,r + 2// = o, 

 or 



For tension alone a stress of/,, would be required = 6c(" ; 

 hence, the ratio of these is 



A L_ 



within the elastic limit n-Y is in usually taken - 



4 



In copper after yield point n = - nearly, whilst careful 

 density measurements make it reasonable to assume m 

 constant up to maximum load, and equal to the value 

 given by the the extension at this point, 77'c., "604. 



The ratio then becomes .- . — ^— or '8. 

 . I '604 



Thus under the best conditions a reduction of 20 per 



cent, of the pull might occur, but it is probable that 



friction would largely counterbalance this, as it would 



then have to be taken into account. 



