344 STEEL HEAD FRAMES AND COAL TIPPLES. CHAP. X. 



if the cable is slack. If a descending cage should stick and then drop, the stress will be equal 

 to the kinetic energy developed and will be very large. The load due to starting a cage suddenly 

 from the bottom of a shaft may be taken as 



K = 2W+R+F (i) 



where K = stress in Ib. at the sheave at the instant of picking up the load; 

 W = gross load in Ib.; 

 R = weight of rope in Ib. ; 



F = friction in Ib., = (W + K)f, where / = coefficient of friction, which may be taken 

 at O.OI to 0.02 for vertical shafts and from 0.02 to 0.04 for inclined shafts with the rope supported 

 on rollers. The working load should not be greater than K plus the stress due to bending, and 

 should not exceed ^ of the ultimate strength of the rope, or f of the ultimate strength for direct pull. 

 For inclined shafts with angle of inclination with horizontal = 0, the stress in the rope due 

 to starting the cage is 



K'.= ( 2 W + R) sine +f(W+ R) cos (2) 



Bending Stresses in Wire Rope. The stresses due to bending will depend upon the diameter 

 of the rope, the make-up of the rope, the angle through which the rope is bent, and the diameter 

 of the sheave. The unit stress due to bending in a round hoisting rope may be obtained from 

 formula (3), the form of which is due to Rankine ("Machinery and Mill Work," p. 533). 



5 = 1, 894,000 -p (3) 



where D = the diameter of the sheaves in inches, and d = the diameter of the rope in inches. 

 The area of the steel in a round hoisting rope is approximately a = 0.4^2, and the total bending 

 stress in a round rope will be 



d 3 

 S b = S-a - 757,600 ^ (4) 



Now the direct breaking strength of a crucible steel round rope is closely 



U = 6o,oood z (5) 



Where bending stress is considered, the safe working load should not exceed I of the ultimate 

 strength, and the safe working load, L, should not exceed 



d 3 

 L = 20,oood 2 757,6oo (6) 



The safe working loads for crucible steel round ropes based on formula (6) are given in Fig. 7.* 

 For plough steel ropes the ultimate strength is U = 7O,oood 2 , and 



L = 26,700^ 757,600 j: (6') 



Mr. William Hewitt in "Wire Rope," published by the Trenton Iron Company, gives the 

 following formula for bending, f 



S> = Ea (7) 



where E = the modulus of elasticity of steel, a = the area of the rope in sq. in., D = the diameter 

 of the sheave in inches, d' the diameter of the individual wires in inches, and C = a constant 



* Redrawn from a diagram prepared by Mr. E. T. Sederholm, Chief Engineer, Allis-Chalmers 

 Company. 



t Also see Engineering News, May 7, 1896. 



