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. 215
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 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- - vEEE 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
216 Mechanics of Materials §10.1 Wire-wound thick cylinders If the internal and external radii of the cylinder are R,and R2 respectively and it is wound with wire until its external radius becomes R3,the radial and hoop stresses in the wire at any radius r between the radii R2 and R3 are found from: hoop stress r-(+)()} where T is the constant tension stress in the wire. The hoop and radial stresses in the cylinder can then be determined by considering the cylinder to be subjected to an external pressure equal to the value of the radial stress above when r=R2. When an additional internal pressure is applied the final stresses will be the algebraic sum of those resulting from the internal pressure and those resulting from the wire winding. Plastic yielding of thick cylinders For initial yield,the internal pressure P,is given by: R,=录[R好-R1 For yielding to a radius R, P-,[是，-购 and for complete collapse, -,[是] 10.1.Difference in treatment between thin and thick cylinders-basic assumptions The theoretical treatment of thin cylinders assumes that the hoop stress is constant across the thickness of the cylinder wall(Fig.10.1),and also that there is no pressure gradient across the wall.Neither of these assumptions can be used for thick cylinders for which the variation of hoop and radial stresses is shown in Fig.10.2,their values being given by the Lame equations: =M+,片4ndg,=A- B Γ2 Development of the theory for thick cylinders is concerned with sections remote from the
216 Mechanics of Materials $10.1 Wire-wound thick cylinders If the internal and external radii of the cylinder are R, and R, respectively and it is wound with wire until its external radius becomes R,, the radial and hoop stresses in the wire at any radius r between the radii R, and R3 are found from: radial stress = ( -27i-) r2 - R: Tlog, (-) Ri - R: r2 - Rt r2 + R: R; - R: hoop stress = T { 1 - ( - 2r2 )'Oge(r2-Rf)} where T is the constant tension stress in the wire. The hoop and radial stresses in the cylinder can then be determined by considering the cylinder to be subjected to an external pressure equal to the value of the radial stress above when r = R,. When an additional internal pressure is applied the final stresses will be the algebraic sum of those resulting from the internal pressure and those resulting from the wire winding. Plastic yielding of thick cylinders For initial yield, the internal pressure P, is given by: For yielding to a radius R,, and for complete collapse, 10.1. Difference io treatment between thio and thick cylinders - basic assumptions The theoretical treatment of thin cylinders assumes that the hoop stress is constant across the thickness of the cylinder wall (Fig. lO.l), and also that there is no pressure gradient across the wall. Neither of these assumptions can be used for thick cylinders for which the variation of hoop and radial stresses is shown in Fig. 10.2, their values being given by the Lame equations: B B an=A+- and q=A-- r2 r2 Development of the theory for thick cylinders is concerned with sections remote from the
§10.2 Thick Cylinders 217 2t Fig.10.1.Thin cylinder subjected to internal pressure. c,(tensile】 Stress distributions a.(compressive c-p A+B/r2 ,A-B/r2 Fig.10.2.Thick cylinder subjected to internal pressure. ends since distribution of the stresses around the joints makes analysis at the ends particularly complex.For central sections the applied pressure system which is normally applied to thick cylinders is symmetrical,and all points on an annular element of the cylinder wall will be displaced by the same amount,this amount depending on the radius of the element.Consequently there can be no shearing stress set up on transverse planes and stresses on such planes are therefore principal stresses(see page 331).Similarly,since the radial shape of the cylinder is maintained there are no shears on radial or tangential planes,and again stresses on such planes are principal stresses.Thus,consideration of any element in the wall of a thick cylinder involves,in general,consideration of a mutually prependicular,tri-axial, principal stress system,the three stresses being termed radial,hoop (tangential or circumferential)and longitudinal (axial)stresses. 10.2.Development of the Lame theory Consider the thick cylinder shown in Fig.10.3.The stresses acting on an element of unit length at radius r are as shown in Fig.10.4,the radial stress increasing from a,to o,+do,over the element thickness dr (all stresses are assumed tensile), For radial equilibrium of the element: o,+do,)r+dn)d0x1-o,×rd6×1=2oH×dr×1×sin2
0 10.2 Thick Cylinders 217 Fig. 10.1. Thin cylinder subjected to internal pressure. Stress distributions uH=A + B/r2 u,= A-B/r2 Fig. 10.2. Thick cylinder subjected to internal pressure. ends since distribution of the stresses around the joints makes analysis at the ends particularly complex. For central sections the applied pressure system which is normally applied to thick cylinders is symmetrical, and all points on an annular element of the cylinder wall will be displaced by the same amount, this amount depending on the radius of the element. Consequently there can be no shearing stress set up on transverse planes and stresses on such planes are therefore principal stresses (see page 331). Similarly, since the radial shape of the cylinder is maintained there are no shears on radial or tangential planes, and again stresses on such planes are principal stresses. Thus, consideration of any element in the wall of a thick cylinder involves, in general, consideration of a mutually prependicular, tri-axial, principal stress system, the three stresses being termed radial, hoop (tangential or circumferential) and longitudinal (axial) stresses. 10.2. Development of the Lam6 theory Consider the thick cylinder shown in Fig. 10.3. The stresses acting on an element of unit length at radius rare as shown in Fig. 10.4, the radial stress increasing from a, to a, + da, over the element thickness dr (all stresses are assumed tensile), For radial equilibrium of the element: de (a,+da,)(r+dr)de x 1-6, x rd0 x 1 = 2aH x dr x 1 x sin- 2
218 Mechanics of Materials §10.2 +do dr Fig.10.3. a,+do, Unit length de Fig10.4. For small angles: do do sin22 radian Therefore,neglecting second-order small quantities, rdo,+a,dr oHdr dor o,+dr =0H dor or GH-c,=r (10.1) Assuming now that plane sections remain plane,i.e.the longitudinal strain &L is constant across the wall of the cylinder, then 6L=E[aL-vo,-vOH] -E [GL-Y(o,+oH)]=constant It is also assumed that the longitudinal stress aL is constant across the cylinder walls at points remote from the ends. ,+H constant 2A (say) (10.2)
218 Mechanics of Materials 410.2 Fig. 10.3. q +do, length Fig. 10.4. For small angles: . d9 d9 22 sin - - radian Therefore, neglecting second-order small quantities, rda, + a,dr = aHdr .. or do, a, + r- = an dr (10.1) Assuming now that plane sections remain plane, Le. the longitudinal strain .zL is constant across the wall of the cylinder, 1 E 1 E then EL = - [aL - va, - VaH] = - [aL - v(a, + OH)] = constant It is also assumed that the longitudinal stress aL is constant across the cylinder walls at points remote from the ends. .. a, + aH = constant = 2A (say) (10.2)
§10.3 Thick Cylinders 219 Substituting in (10.1)for oH, dor 2A-0,-0,= 'dr Multiplying through by r and rearranging, 2a,7+r2dg-24=0 dr o,2-Ar内=0 d i.e. Therefore,integrating, a,r2-Ar2 constant =-B (say) 6,=A- B (10.3) and from eqn.(10.2) B GH=A+ (10.4) The above equations yield the radial and hoop stresses at any radius r in terms of constants A and B.For any pressure condition there will always be two known conditions of stress (usually radial stress)which enable the constants to be determined and the required stresses evaluated. 10.3.Thick cylinder-internal pressure only Consider now the thick cylinder shown in Fig.10.5 subjected to an internal pressure P,the external pressure being zero. Fig.10.5.Cylinder cross-section. The two known conditions of stress which enable the Lame constants A and B to be determined are: Atr=R1 ,=-P and at r=R2 G,=0 N.B.-The internal pressure is considered as a negative radial stress since it will produce a radial compression(i.e.thinning)of the cylinder walls and the normal stress convention takes compression as negative
$10.3 Thick Cylinders 219 Substituting in (10.1) for o~, dor 2A-ar-ar = r- dr Multiplying through by r and rearranging, dor 2orr + r2 - - 2Ar = 0 dr i.e. Therefore, integrating, .. d -(or? -A?) = 0 dr orrZ - Ar2 = constant = - B (say) and from eqn. (10.2) (10.4) B UH=A+- rz The above equations yield the radial and hoop stresses at any radius r in terms of constants A and B. For any pressure condition there will always be two known conditions of stress (usually radial stress) which enable the constants to be determined and the required stresses evaluated. 10.3. Thick cylinder - internal pressure only Consider now the thick cylinder shown in Fig. 10.5 subjected to an internal pressure P, the external pressure being zero. Fig. 10.5. Cylinder cross-section. The two known conditions of stress which enable the Lame constants A and B to be determined are: At r = R, or= -P and at r = R, or =O N.B. -The internal pressure is considered as a negative radial stress since it will produce a radial compression (i.e. thinning) of the cylinder walls and the normal stress convention takes compression as negative