120 



RESISTANCE OF MATERIALS 



This is the same formula as for the sphere, which was to be 

 expected, since the cross section is the same in both cases. 



75. Thick cylinders. Lame's formulas. Consider a thick circular 

 cylinder of external radius a and internal radius 5, subjected to either 



internal or external 

 uniform pressure,or 

 to both simultane- 

 ously, and suppose 

 that a section is cut 

 out of the cylinder by 

 two planes perpen- 

 dicular to the axis 

 at a unit distance 

 FIG. 97 apart (Fig. 97). 



Now consider a 



thin ring of the material anywhere in the given section, of external 

 radius r e and internal radius r { . Then, under the strain, r e will become 



where s e denotes the unit deformation of the fiber, which never 

 exceeds -^-Q-Q-Q for safe working stresses. Similarly, r i will become 



where the unit deformation s. is also very small. Since any safe 

 strain produces no appreciable change in the sectional area of the 

 thin ring here considered, by equating its sectional areas before and 

 after strain we have 



T W - rf-) = * W (1 + O 2 - r? (1 + S< ) 2 ]- 

 Canceling out the common factor TT and reducing, this becomes 



*.* (.' + 2 o = *(.' + 20 ; 



or, since the unit deformations s e and s. are very small, their squares 

 may be neglected in comparison with their first powers, and con- 

 sequently this expression further reduces to 



