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First ClassDiscussion is not mandatory, but good to go toMonday, August 23, 201008:57Title: Introductory Solid MechanicsAlso called: mechanics of materials, strength of materials, mechanics of deformable bodiesProfessor JohnsonResearch of materials reacting to electricity and light (solar cells, lasers )Objective: to learn to analyze stress, strain, and displacement in solid materials used in engineeringstructuresTheory course - engineering designCompass will be used for all documentsOutlineI. Introduction to stress, strain, and material behaviorII. Axial loadsIII. TorsionIV. Bending (as in of beams)V. General states of stressVI. BucklingNotes Page 110 point grading scale2 HE X 20% eachFinal exam, 30%Quiz 10%HMWK 20%Quizzes are 5 minutes, basic questions - likely from lecture - will begraded directly by professorExams closed book, closed note, have an equation sheet

Statics Review (methods)Monday, August 23, 201009:17Example: Force analysis of a simple 2-member structureSo force A causes compression along ABThe two force components Cx and Cy together act along the line BCThus forces A and C make AB and BC two-force members - thus these forces must be collinearCould have more easily found the solution by taking a FBD of joint BThought experiment on axial deformation- We have a bar or two force member like BCWhat do we need to know to predict the change in length of the bar?Young's modulus of elasticity, EOriginal length, LCross sectional area, AForce being applied, FNotes Page 2

What do we need to know to predict the change in length of the bar?Young's modulus of elasticity, EOriginal length, LCross sectional area, AForce being applied, FNotes Page 3

Stress4/6 homework problems gradedSolutions posted to allPut a photo on compassHMWK: 1.13, 1.28, 1.32, 1.53, 2.19, 2.26Wednesday, August 25, 201009:01Why is AB thicker?So it is more rigid, and doesn't buckleIt is in compression, only compressed members can buckleQuantity that causes actual material failure is known as stress.Stress: an average distribution of forces over a cross -sectional area ve sign Tensile stress-ve sign Compressive stressEither can cause failureIn case of BC:We assume the distribution of forces is uniform in axial loadingUnits of stressForce per unit area: Nm-2, PaMore typical to use kPa or MPa or GPa in engineeringTo find the maximum stress before it breaks look for"ultimate stress""yield stress"Failure mechanisms are slightly different in tension or compressionOther considerationLoads/stress on joints or bearingsMember BC is said to be in an axial (uniaxial) state of stressThe actual spatial distribution of stress is statically indeterminantA different type of stress: shear stressShear stress: average shear on the planeContrary to axial stress: shear stress is not uniform (can't be uniform) on the cross section nor can it beassumed to beExample: commonly found in bolts or pins or rivetsNotes Page 4

Another kind of stress: bearing stresses: localized stresses at contacting surfaces between pins, bolts,rivets, and the surfaces they connectEX: the lateral area of the bolt rather than the cross sectionCan now completely analyze the stresses in the two-member structureNotes Page 5

Stress on an oblique (not perp.) planeHomework due WednesdayFriday, August 27, 201009:01Angle measured from verticalStress depends on coordinate systemIn 3 dimensions: general loading conditionsRecall:Notes Page 6

We have a set of stress componentsOnly six of the stress components areindependent of one anotherPlanes not lined up with coordinate system need sines and cosines, but there is no reason to do that.Back to 2 dimensionsWhy would we rotate it?Suppose we do not have a simple material - maybe wehave wood rather than steelOr maybe we have an angled joint glued togetherNotes Page 7

Uniaxial DeformationMonday, August 30, 201009:00What does strain have to do with material behaviour?DefinitionsStrainTwo quantities that together allow us to examine the materialStressStrainStrain and stress are linearly relatedNormal strain (axial strain)Also have shear strain laterNoteStrain is dimensionlessTypical numbers1m steel bar, stretched under tension by 0.01 mmStrain: 0.01E-3/1 10 -5, or 10 * 10 -6 (10 microstrain)Typical values are around 10 -4Comparing stress and strain gives us a way to measure the properties of the materialStress-Strain relationship- Can quantify in a lab (ME 330)- Experimental setupStandard tensile test sampleApply an increasing load (know A)Thus compute the stress and measure strainHave a load frameYield stress is yield strengthUltimate stress is ultimate strengthInitial part of curve is very steepAppears linear - have the behaviour we spoke ofMake a line thru 0.002 parallel to initial valueIntersects at yield stressNotes Page 8

Slope of the linear region is E from earlier (Young's modulus)3 important properties- Ultimate stress- Yield Stress- Young's modulusUltimate Stress Yield Stress Young's ModulusSteels250 MPa200 GPaAluminum 40 MPa400 MPa95 MPa70 GPaConcreteN/A25 GPa28 MPaConcrete ultimate stress is for a compression test rather than a tension testIn compression can get a bucklingBuckling effect at failure (also known as barreling)In tension can get neckingSample (before failing) thins out a lotNotes Page 9

Stress and Strain Axial LoadingWednesday, September 01, 201009:06Stress/strain machines must be very strong - more so than materialMaterials behave differently in compressive and tensile loadsStrength is affected by alloying, heat treating and manufacturingbut stiffness is not.If strain disappears when the stress is removed the materialis said to behave elastically.The largest stress for which this occurs is called the elasticlimit.When the strain does not return to zero after the stress isremoved, the material is said to behave plastically.Fatigue: shown on S-N diagramsNotes Page 10

Material may fail due to fatigue at stress levels significantly below theultimate strength if subjected to many loading cycles.When stress is reduced below the endurance limit, fatigue failures donot occur for any number of cycles.StressNumber of completely reversed cyclesModel material as a springNotes Page 11

More stress-strain relationships, uniaxial deformationFriday, September 03, 201009:13Increase in the yield stress: work hardening (strain hardening, strengthening)Fatigue of materialsFatigue: failure of a material at a stress that is lower than the yield stress due to repeated (cyclical)elastic loadingMust have very large amounts (10 6, 10 9 )Has caused plane crashesCharacterized by S-N curveAmplitude of the cyclical stressx-axis on a log scaleCurve represents the place where you expect it to fail after NloadingsSteel has a fatigue limit - there is a value of stress below which(regardless of times loaded) there is no decrease in its yield stressThis is statistical, not guaranteed (ME 330 has more of this)Number of loadingsDifference between engineering stress and true stress and engineering strain and true strainEngineering stressTrue StressEngineering strainTrue strainForcedivided by initial area The force over the current area Change in length divided by initial length The sum over the current areaHas no decrease during neckingNotes Page 12

Has no decrease during neckingDeformation under uniaxial loadingInternal force can vary in each segmentThen need free body diagramsNotes Page 13

Thermal strain, static indeterminacy, superpositionHomework 3 due next week - problems posted sometime todayWednesday, September 08, 201009:02Another way to increase the length of the barApply a temperature change, get a thermal strainIt is only zero ifthe bar isunconstrainedNo one is applying a force onit - expands freelyConcept of static indeterminacyUntil now, we could always get all of the internal forces by considering a free body diagramNotes Page 14

Many problems have more reaction forces than can be found from statics (a FBD) alone"statically indeterminant"Notes Page 15

Statically Indeterminant and SuperpositionFriday, September 10, 201008:59Statically Indeterminant: statics alone is not sufficient to find all of the reaction forces, so displacementsare also consideredSuperposition Method (Formally)Superposition method: "release" one boundary constraint and replace it with an unknown loadthat gives displacement that is compatible with the original problemSuperimpose the problems to get another oneWorks only with linear mechanics (same as in electriccircuits)Notes Page 16

Non-1-dimensional effectsToday: Poisson effectMonday: Multiaxial loading and generalized Hooke's LawFriday, September 10, 201009:28Lateral contraction characterized by Poisson's Ratio (greeknu)Ratio is a material property - can bemeasured or looked up.Generally positiveTypically ranges from 0 to 0.5.Impossible to be greater than 0.5according to theory. Could benegative though.Negative here - can be made in a lab certain minerals canNegative Poisson materialCurrently have two important material properties for elastic (isotropic) materials:Young's modulusPoisson's RatioAlso have coefficient of thermal expansion if are talking about thermo-elastic materialsMultiaxial loading: pulling or pushing in more than one directionUpon deformation, the new volume of the unit cube should beNotes Page 17

e: dilatationk: bulk modulusBetween pressure and dilitationConsider shear strainShear stresses will deform a cube into an oblique parallelepipedAngle changeAxial case does rectangular parallelepipedLength changeStill only two independent constants - can thus get G and kFor isotropic materialsIsotropic: material that is the same in all directionsRecapping strain as a function of stress2 constants are reduced from 36 by isotropyNotes Page 18

2 constants are reduced from 36 by isotropyWood, bone are not isotropicOffice hours MWF 2:30 to 3:50, 362 MEBHomework 3 has statically indeterminant problems, and Poisson effectNotes Page 19

Stresses near point loads and stress concentrationsHMWK 4 posted later todayOffice hours today are 2:00-2:50 (362B MEB)Wednesday, September 15, 201009:04Physical concept checkConsider a cube of materialRigid walls on each x-side, frictionless contact1. Consider ΔT 0i. What are axial stresses and strains2. Compressive load in z direction3. Tensile load in z directionStresses near point loadsOverall length change in both cases is the sameExact displacements near end are not the sameIn both cases the forces are statically equivalentDisplacements far from the end of the bar are the same in the ideal and actual caseSt. Venant's PrincipleStresses near defects or stress concentrations in materialFar from the loadWhich has higher stress along the centre line? - The one with the whole cut outActually it is much worse than thisExamine stress distribution near holeStress concentration: feature inside material that magnifies the stress relative tosome average or mean caseHow large is the peak stress?Notes Page 20

Stress concentration: feature inside material that magnifies the stress relative tosome average or mean caseHow large is the peak stress?'k' is a property of geometry, but not the materialExample: filleted plate/beamSmall r sharpBig r gradualStress distribution at filletImportant in brittle materials with possibility of defects (glass with bubble/microscopiccrack)Many structural engineering failures due to thisNotes Page 21

Plastic deformationG: shear moduluse: dilatationE: Young's ModulusFriday, September 17, 201009:15Portion of stress-strain behaviour of material that goes beyond the linear elastic region of the stress strain curveTau xy and tau yx are the same because momentsare zeroThis would be an elastic perfectly plastic materialResidual StressUpon unloading in an elastic-plastic material, often have stresses that are built in to the materialBy divine interventionNotes Page 22

Notes Page 23