CHAPTER 10 THICK CYLINDERS Summary The hoop and radial stresses at any point in the wall cross-section of a thick cylinder at radius r are given by the Lame equations: 分 hoop stress aH=A+ B radial stress ,=A- With internal and external pressures P:and P2 and internal and external radii R:and R2 respectively,the longitudinal stress in a cylinder with closed ends is PR-P2R3 0L= (R-R) =Lame constant A Changes in dimensions of the cylinder may then be determined from the following strain formulae: circumferential or hoop strain diametral strain =OH_0, E ,0L E ongn-登- E-V E For compound tubes the resultant hoop stress is the algebraic sum of the hoop stresses resulting from shrinkage and the hoop stresses resulting from internal and external pressures. For force and shrink fits of cylinders made of different materials,the total interference or shrinkage allowance (on radius)is [eH。-eH]r where eand eare the hoop strains existing in the outer and inner cylinders respectively at the common radius r.For cylinders of the same material this equation reduces to Elon-on For a hub or sleeve shrunk on a solid shaft the shaft is subjected to constant hoop and radial stresses,each equal to the pressure set up at the junction.The hub or sleeve is then treated as a thick cylinder subjected to this internal pressure. 215CHAPTER 10 THICK CYLINDERS Summary The hoop and radial stresses at any point in the wall cross-section of a thick cylinder at radius r are given by the Lam6 equations: B hoop stress OH = A + - r2 B radial stress cr, = A - - r2 With internal and external pressures P, and P, and internal and external radii R, and R, respectively, the longitudinal stress in a cylinder with closed ends is P1R: - P2R: aL = = Lame constant A (R: - R:) Changes in dimensions of the cylinder may then be determined from the following strain formulae: circumferential or hoop strain = diametral strain 'JH cr OL =-- v- - v￾EEE OL or OH longitudinal strain = - - v- - v- EEE For compound tubes the resultant hoop stress is the algebraic sum of the hoop stresses resulting from shrinkage and the hoop stresses resulting from internal and external pressures. For force and shrink fits of cylinders made of diferent materials, the total interference or shrinkage allowance (on radius) is CEH, - 'Hi 1 where E", and cH, are the hoop strains existing in the outer and inner cylinders respectively at the common radius r. For cylinders of the same material this equation reduces to For a hub or sleeve shrunk on a solid shaft the shaft is subjected to constant hoop and radial stresses, each equal to the pressure set up at the junction. The hub or sleeve is then treated as a thick cylinder subjected to this internal pressure. 21 5