ME311 Machine DesignW Dornfeld30Oct2014Fairfield UniversitySchool of EngineeringLecture 8: CylindersThin-Walled Cylinders(You already covered this in Beer & Johnston.)A pressurized cylinder is considered to be Thin-Walled if its wall thickness is less than 2.5% (1/40th) of its inside diameter.Under these conditions:1. We assume the stress distribution is uniform throughout the wall thickness – both in the hoop (circumferential) direction and in the longitudinal (axial) direction.2. We assume that the radial stress is negligible.Then:Hamrock Page 24622θθσσσ===trpAxialandtrpHoopiziThin-Walled Cylinder QuizKnowing all that you do about pressurized cylinders (i.e., that the hoop stress is twice the longitudinal stress), which direction would you predict that these pressurized cylinders will fracture?BAA. Lengthwise due to hoop stress.B. Crosswise due to axial stress.C. On a 45°angle due to shear stress or torque.CStresses in Thick-Walled Cylinders• Thick-Walled cylinders have a wall thickness greater than 1/20thof their average radius.• They are pressurized internally and/or externally.• The principal stresses are circumferential (hoop) σc, radial σr, and longitudinal (axial) σl.Hamrock Section 10.3.2riropopiRσrσlσcCircumferential & Radial Stresses2222222/)()(iooioiooiirrRrrppprprR−−±−=σFor the general case of both internal and external pressure, the circumferential and radial stresses at radius R in the wall are:±−=222221)(RrrrprRoioiiσFor the special case of only internal pressure, po= 0, and the stresses at radius R are:The sign convention is the same.Eqns10.20/10.22Eqns10.23/10.24Where the ± is: + for circumferential, and- for radial stress.riropopiRσrσlσσσσcLongitudinal 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).Un-numbered Equation just below Eqn. 10.5riropopiRσrσlσcStresses 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.±−=222221)(RrrrprRoioiiσNow 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 )Eqns10.23/10.24riropopiRσrσlσσσσcThick-Walled Cylinder with internal pressure of 5300 psi.Stresses for Internal Pressurization( + is hoop, - is radial )±−=222221Rrrrproioiiσ-8-404812160 0.5 1 1.5 2Radius (in.)Stress (KSI)RADIAL STRESSHOOP STRESSHoopRadialStresses vs. Radius - Internal PressureRadial stress is as predicted:• -5300 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.HoopRadialThick-Walled Cylinder with external pressure of 5300 psi.Stresses for External Pressurization±−−=222221Rrrrpriioooσ( + is hoop, - is radial )HoopRadialRADIAL STRESSHOOP STRESS-16-14-12-10-8-6-4-200 0.5 1 1.5Radius (in.)Stress (KSI)Stresses vs. Radius - External PressureRadial stress is as predicted:• 0 at the unpressurized inner surface.• -5300 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.HoopRadialBurst Tubing AnalysisID = 0.395”; OD = 0.505”; p = 16,000psi• What was the hoop stress in the tube?• Analyze it as both thin-wall and thick-wall. Which is it?Stresses vs R for Tube-30,000-20,000-10,000010,00020,00030,00040,00050,00060,00070,00080,0000 0.1 0.2 0.3Radius (in.)Stress (PSI)Circumferential(Hoop)Radial±−=222221RrrrproioiiσRotating RingsStresses (radial & tangential) are similar to those in thick-walled cylinders. The forces come from centrifugal loads on all of the ring particles instead of from the internal pressure.Conditions:1. ro≥ 10 t2. t is constantωωωωrirot++−+++=222222233183rrrrrroioihoopνννρωσsec,38623radianssininLbDensityMass ===ωρwhere−−++=222222283rrrrrroioiradialνρωσEqn 10.35Eqn 10.36Rotating Rings: Effect of Center Bore Radius on StressesStresses for a 2 inch thick steel disk rotating at 5000 RPM.Rotating Ring vs Center Bore Radius (0.125 -> 2")Ri = 2.0Ri = 2.0 1.01.00.50.50.2512,03705,00010,00015,00020,00025,0000 2 4 6 8 10 12Radius (inches)Stress (PSI)Tangential (Hoop) StressRadial StressωωωωrirotAt what radius is the peak radial stress?Remember Differentiation?( )( )90.424)12)(2(220838342223222222222222222222222======−−=−=−=−−+=−−++=oioioioioioioioiradialrrrrrrrrdrdrrrrrrdrdrdrdrrrdrdpeakatrrrrdrdrrrrrrdrddrdνρωνρωσ3212−−−−==rrdrdnxxdxdnnPress FitsIn a press fit, the shaft is compressed and the hub is expanded.Before AfterHubShaftRadial interference, δrHubShaftPress 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.HubCharacteristics of Press Fits1) 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