1ME311 Machine DesignW Dornfeld11Sep2014Fairfield UniversitySchool of EngineeringLecture 2: Materials; Stress & Strain; Power TransmissionStress-Strain Curve for Ductile MaterialUsually 0.2% Offset. ε = 0.002ll∆=εSlopeEE===εσεσFractureUltimate TensileYieldProportional LimitElastic Limit2Effects of Hardness; BrittlenessThis slope does NOT change because the Modulus is the same.Hardening by Heat Treating or Cold Working increases the Yield Strength of materials.Terminology:Strength = Material Property;Stress = Applied LoadingStrength, kpsiManufacturing/Metalworking• Metal fabrication, where sheets and bars are bent and formed, obviously depends on going beyond yield into the plastic forming range.• It is common for highly-formed metals to require annealing to reduce their yield strengths for further forming.3Brittle versus DuctileDuctile if %El > 5%Brittle if %El < 5%Ductility is related to the amount of Plastic Deformation (Strain) at fracture.The strain at fracture = % Elongation%100%00×−=lllElfractureHamrock Fig. 3.7How to Crack Sheet Metal• Bend it in a real tight radius. Why?For any wrap angle θ, the circumferential wrap is r θ.The Neutral length is (R+t/2)θ, and the length in Tension is (R+t)θ.The strain is: tDttRttRttRtRtRNeutralNeutralTensionll+=+=+=++−+=−=∆)2()2/(2/)2/()2/()()(θθθTensionCompressionNeutralθRt4How to Crack Sheet Metal• Example: Aluminum DogboneMaterial is 6061-T6 AluminumInner Bend Radius = 0.102”Thickness = 0.101”%33331.0305.0101.0101.0204.0101.02===+=+=∆tRtllFrom MatWeb.com:See Example 3.1, HamrockShould it have cracked? _____What do you think the stress was? _____________Calculate the strain.tRt+=2εPoisson’s Ratioν (nu)What’s nu?εAXIALεLATERAL= -ν ν ν ν εAXIALQuestion: Is there a resulting lateral stress? _______So, could you say there is strain without stress? ______Can you think of another way to get strain without stress? ______Poisson’s ratio is around 0.3 for most metals.Lowest is ≈0.2 for Cast Iron; Highest is ≈0.44 for Lead.Shear modulus (“stiffness in twisting”) isBecause most metals have a ν of about 0.3, this means that for most metals, G ≈ what percent of E? __________)1(2ν+=EGHamrock Eqn. 3.75Hardness Testshttp://www.the-warren.org/ALevelRevision/engineering/materialtesting.htmlCone PyramidSphereUTS vs. Hardness0501001502002503003500 100 200 300 400 500 600 700Brinell HardnessSut (ksi)6Ashby ChartsLook them over.Understand what they are and how they represent differences among materials.My favorite is Fig. 3.19, comparing Modulus of Elasticity and Density.Most Common MetalsBerylliumIncr. Speed of SoundThe natural frequency of a cantilever beam is nearly the same for steel or aluminum, but for Beryllium it is almost 3x higher!What is for Aluminum ______ and for Steel _____? What are the units? _____ρEStresses in Straight Beams• A uniform beam in pure bendingIMc=maxσrMMThe maximum stress iswhere I is the Area Moment of Inertia about the Centroid, andc is the distance from the Neutral Axis, or Centroid.We need to know these properties of the beam cross section to be able to calculate bending stress, one of the most common large stresses on structures.7Hamrock Chapter 4:Area Moment of InertiaDefinitions:• Centroid of an Area• Area Moment of Inertia• Parallel Axis TheoremHamrock Section 4.2This is NOT same as J = Polar Moment of Inertia, or as Mass Moment of Inertia, Im, (which has units Lb.In.Sec2, and sets how fast a torque can rotationally accelerate a device.)22'AdIIdAyIAxdAxxxAxA+===∫∫CENTROIDParallel Axis TheoremIx' = Ix + Ad2 = bh3/12 + bhd2where:• Ix' = moment of inertia about an axis that IS NOT through the center• Ix = moment of inertia about an axis that is through the centroid= bh3/12• d = distance between the x axes• A = area of the cross section = bhNote: the axes must be parallelXX’Y’YOO’dhb8Moment of Inertia Procedure1. Find neutral axis (centroid) of total area. Why? Because we will use I to calculate bending stress about the neutral axis. ( A1 + A2 + A3 ) rtot = A1 r1 + A2 r2 + A3 r3 Solve for rtot .  Cut-outs have negative areas.  Pick a convenient axis to measure r’s from.  If you pick an outer edge, you will always know which direction rtot is! 2. Compute Moment of Inertia about the centroid of each sub-piece. For example: 123bhICG= 644DICGπ= 3. Use Parallel Axis Theorem to translate to about the neutral axis 2AdIICGX+= where “d” is the distance from the sub-piece’s centroid to the total area centroid, and “A” is the area of the sub-piece. 4. Add up the translated moments of inertia of all the pieces, subtracting moments if they are cut-outs. b h A2A1A3r2,3rtotr1Hamrock Section 4.2.3Elementary Load Building BlocksFrom DAWrightat U of Western AustraliaSTRESSAP=σAP=τIMy=σJTr=τHamrock Eqn. 4.22Eqn. 4.46Eqn. 4.329Stresses in Curved BeamsBending Stresses in a curved beam are not linearly distributed across the beam section, but have a hyperbolic distribution that is higher at the inside surface than for a straight beam.Stresses in Curved Beams2) Calculate the centroidal radius, R, based on the section type. (Hamrock § 4.5.3)Rectangular: Circular: 1) Draw a very good picture.• Show the applied Force, F• Show ri, ro, Area3) Compute the neutral radius, rn, based on the section type. (Hamrock § 4.5.3)Rectangular: Circular:2oirrR+=cirrR +=)ln(ioionrrrrr−=222cnrRRr−+=eriroRrnrircbFRLriroCentroidal radius, RNeutral radius, rnCircularRectangular10Stresses in Curved Beams (2)5) Compute the moment about the centroidal radius, R. Here M = F x L, not F x R, because the force is not through the center of curvature.4) Compute the eccentricity, e = R - rn6) Calculate the distances from the neutral axis to the inner and outer surfaces: ci= rn– riand co = ro– rn.7) Calculate the stresses at the inside and outside surfaces:and ,where A = the section areaiiiAerMc=σoooAerMc−=σeriroRrnrircbFRLriroCentroidal radius, RNeutral radius, rniiiAerMc=σoooAerMc−=σStresses in Curved Beams (3)8) Add or subtract any P/A stressesin this section (using superposition).Rectangular: Circular: brrFio)( −=σ2crFπσ=9) As a check, compare the answer to MC / Ifor a straight beam with neutral axis on the centroid and see if it makes sense.eriroRrnrircbFRLriroCentroidal radius, RNeutral radius, rn11Stresses in Curved Beams (4)The