 2004 by W.H.DornfeldPressCylinder:Thick-Walled Cylinders and Press Fits1 2004 by W.H.DornfeldPressCylinder:Stresses in Thick-Walled Cylinders• Thick-Walled cylinders have an average radius less than 20 times the wall thickness.• They are pressurized internally and/or externally.• The principal stresses are circumferential (hoop) σc, radial σr, and longitudinal (axial) σl.riropopiRσrσlσc2 2004 by W.H.DornfeldPressCylinder:Circumferential & Radial Stresses2222222/)(iooioiooiirrRrrppprpr−−±−=σFor the general case of both internal and external pressure, the circumferential and radial stresses at radius R in the wall are:±−=222221RrrrproioiiσFor the special case of only internal pressure, po= 0, and the stresses at radius R are:The sign convention is the same.3Eqns10.20/10.22Eqns10.23/10.24Where the ± is: + for circumferential, and- for radial stress. 2004 by W.H.DornfeldPressCylinder:Longitudinal Stresses222ioiilrrrp−=σThe longitudinal stress is simply given by a Force/Area, where the Force is pitimes the circular inside area πri2 , and the Area is the annular area of the cylinder cross section, π( ro2 - ri2) , or:This is generally only considered for the case of internal pressurization ( po= 0).riropopiRσrσlσc4Un-numbered Equation just below Eqn. 10.8 2004 by W.H.DornfeldPressCylinder:Stresses vs. RadiusFirst, the easy observation: Radial stresses at the inner and outer surfaces are equal to minus the pressurization.• If a surface is unpressurized, the radial stress there is zero.• If a surface is pressurized, the radial stress there = - p, because it is in compression.±−=222221RrrrproioiiσNow let’s look at an internally pressurized cylinder, and how the radial and circumferential stresses vary across the wall thickness at radius R.( + is circumferential, - is radial )5Eqns10.23/10.24 2004 by W.H.DornfeldPressCylinder:Thick-Walled Cylinder with internal pressure of 5330 psi.-8-404812160 0.5 1 1.5 2Radius (in.)Stress (KSI)RADIAL STRESSHOOP STRESSStresses for Internal Pressurization6( + is hoop, - is radial )±−=222221Rrrrproioiiσ 2004 by W.H.DornfeldPressCylinder:Stresses vs. Radius - Internal PressureRadial stress is as predicted:• -5330 psi at the inner, pressurized surface.• 0 at the unpressurized outer surface.Hoop stress is:• Maximum at the inner surface, 13.9 ksi.• Lower, but not zero, at the unpressurized outer surface, 8.5 ksi.• Larger in magnitude than the radial stressLongitudinal stress is (trust me):• 4.3 ksi, considered as a uniform, average stress across the thickness of the wall.Now let’s look at an externally pressurized cylinder.7 2004 by W.H.DornfeldPressCylinder:Thick-Walled Cylinder with external pressure of 5330 psi.RADIAL STRESSHOOP STRESS-16-14-12-10-8-6-4-200 0.5 1 1.5Radius (in.)Stress (KSI)Stresses for External Pressurization±−−=222221Rrrrpriioooσ( + is hoop, - is radial )8 2004 by W.H.DornfeldPressCylinder:Stresses vs. Radius - External PressureRadial stress is as predicted:• 0 at the unpressurized inner surface.• -5330 psi at the outer, pressurized surface.Hoop stress is:• Minimum at the outer surface, -8.9 ksi.• Maximum at the (unpressurized) inner surface, -14.2 ksi.• Larger than the radial stressLongitudinal stress is:• Not usually considered for external pressurization.9 2004 by W.H.DornfeldPressCylinder:Press Fits10In a press fit, the shaft is compressed and the hub is expanded.Before AfterHubShaftRadial interference, δrHubShaft 2004 by W.H.DornfeldPressCylinder:Press FitsPress fits, or interference fits, are similar to pressurized cylinders in that the placement of an oversized shaft in an undersized hub results in a radial pressure at the interface.Hub11 2004 by W.H.DornfeldPressCylinder:Characteristics of Press Fits121) The shaft is compressed and the hub is expanded.2) There are equal and opposite pressures at the mating surfaces.3) The relative amount of compression and expansion depends on the stiffness (elasticity and geometry) of the two pieces.4) The sum of the compression and the expansion equals the interference introduced.5) The critical stress location is usually the inner diameter of the hub, where max tensile hoop stress occurs.HubShaftHubShaft 2004 by W.H.DornfeldPressCylinder:Analysis of Press FitsStart by finding the interface pressure.Rrori()()( )−−−=22222222ioiorrrRrRRrREpδWhere δris the RADIAL interference for hub and shaft of the same material, with modulus of elasticity, E.If the shaft is solid, ri= 0 and−=2212orrRREpδ13Eqn 10.52, rearrangedEqn 10.53, rearranged 2004 by W.H.DornfeldPressCylinder:Analysis of Press FitsIf the shaft and hub are of different materialsOnce we have the pressure, we can use the cylinder equations to compute the hoop stresses at the interface.−−+++−+=iiiioooorRrRERRrRrERprννδ22222222A) The ID of the hub is tensile:B) The OD of the shaft is compressive:2222RrRrpoooc−+=σ2222iicirRrRp−+−=σ= -p if shaft is solid14Eqn 10.51, rearrangedEqn 10.45Eqn 10.49Ei,νiRroriνi,o= PoissonEo,νo 2004 by W.H.DornfeldPressCylinder:Strain Analysis of Press FitsThe press fit has no axial pressure, so σl= 0, and it is a biaxial stress condition.The circumferential strainwhich equals the radial strain (because C = 2πr). Because the radial change δ = R εr, we getthe increase in Inner Radius of the outer member (hub):RroriAnd the decrease in Outer Radius of the inner member (shaft): EErccσνσε −=+−+=oooooRrRrEpRνδ2222−−+−=iiiiirRrREpRνδ222215Eqn 10.50Eqn 10.46 2004 by W.H.DornfeldPressCylinder:Notes on Press FitsAs a check, make sure that The assembly force required will be Fmax= πdLpµwhere p = the interface pressureµ = the coefficient of frictionroiδδδ =+The torque capacity available is T = FR = RπdLpµwhere R = the interference radius, as before.We conveniently know the interface pressure for these equations!Td = 2RLF16 2004 by W.H.DornfeldPressCylinder:Shrink FitsIf heating or cooling a part to achieve a shrink fit, the required radial interference is:∆R = δr= Rα∆Twhere R is the interface radiusα is the coefficient of thermal expansion∆T is the temperature changeTo select an amount of