FE Review Mechanics Of Materials - Auburn University

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FE ReviewMechanics of Materials

FE Mechanics of Materials ReviewStressFVMNN internal normal force (or P)V internal shear forceM internal momentN P Normal Stress σ A AAverage Shear Stress τ VADouble ShearF/2FF/2V F/2FV F/2τ F2A

FE Mechanics of Materials ReviewStrainNormal Strainε L L L0 δ L0L0L0Units of length/lengthε normal strain L change in length δL0 original lengthL length after deformation (after axial load is applied)Percent Elongation L 100L0Ai A fPercent Reduction in Area AiAi initial cross- sectional areaAf final cross-sectional area 100

FE Mechanics of Materials ReviewStrainShear Strain change in angle , usually expressed in radiansyBγ xyθB'x

FE Mechanics of Materials ReviewStress-Strain Diagram for Normal Stress-Strain

FE Mechanics of Materials Review

FE Mechanics of Materials ReviewHooke's Law (one-dimension)σ Eεσ normal stress, force/length 2E modulus of elasticity, force/length 2 normal strain, length/lengthετ Gγτ shear stress, force/length 2G shear modulus of rigidity, force/length 2 shear strain, radiansγ

FE Mechanics of Materials ReviewEG 2(1 ν )ν Poisson's ratio -(lateral strain)/(longitudinal strain)ε latν ε longε lat δ'rε long δLchange in radius over original radiuschange in length over original length

FE Mechanics of Materials ReviewAxial LoadIf A (cross-sectional area), E (modulus of elasticity), and P (load) are constantin a member (and L is its length):E σ P A ε δ Lδ PLAEChange in lengthIf A, E, or P change from one region to the next:PLδ AEδApply to each section where A,E, & P are constantA / B displacement of pt A relative to pt BδA displacement of pt A relative to fixed end

FE Mechanics of Materials Review-Remember principle of superposition used for indeterminate structures- equilibrium/compatibility

FE Mechanics of Materials ReviewThermal Deformationsδ t α ( T ) L α (T T0 ) Lδtα change in length due to temperature change, units of length coefficient of thermal expansion, units of 1/ T final temperature, degreesT0 initial temperature, degrees

FE Mechanics of Materials ReviewTorsionTorque – a moment that tends to twist a member about its longitudinal axisShear stress, τ , and shear strain, γ , vary linearly from 0 at center to maximum atoutside of shaft

FE Mechanics of Materials ReviewTTrτ Jφ TLJGτ shear stress, force/length 2rT applied torque, force·lengthr distance from center to point of interest in cross-section(maximum is the total radius dimension)J polar moment of inertia (see table at end of STATICSsection in FE review manual), length 4φ angle of twist, radiansL length of shaftG shear modulus of rigidity, force/length 2τ φ z Gγ φ z Gr ( dφ / dz )( dφ / dz ) twist per unit length, or rate of twist

FE Mechanics of Materials ReviewBendingPositive BendingMakes compression in top fibers andtension in bottom fibersNegative BendingMakes tension in top fibers andcompression in bottom fibers

FE Mechanics of Materials ReviewdV q( x )Slope of shear diagram negative of distributed loading value Î dxdMSlope of moment diagram shear value Î Vdx

FE Mechanics of Materials Reviewx2Change in shear between two points neg. of area under V2 V1 [ q( x )]dx distributed loading diagram between those two points Îx1x2Change in moment between two points area underM 2 M1 [V ( x )]dxshear diagram between those two points Îx1

FE Mechanics of Materials ReviewStresses in BeamsMyσ IMcσ max Iεx y ρFromσ normal stress due to bending moment, force/length 2y distance from neutral axis to the longitudinal fiber inquestion, length (y positive above NA, neg below)I moment of inertia of cross-section, length 4c maximum value of y;distance from neutral axis toextreme fiberρ radius of curvature of deflectedaxis of the beamσ Eε E y ρÎ σ MyIand1ρ MEI

FE Mechanics of Materials ReviewS IThencS elastic section modulus of beamMcMσ max ISVQTransverse Shear Stress: τ ItTransverse Shear Flow:VQq IQ y ' A't thickness ofcross-section atpoint of interestt b here

FE Mechanics of Materials ReviewThin-Walled Pressure Vessels (r/t 10)Cylindrical Vesselsσt pr σ1tσ1 hoop stress in circumferential directionprt gage pressure, force/length 2 inner radius wall thicknessprσa σ22t axial stress in longitudinal directionSee FE review manual for thick-walled pressurevessel formulas.

FE Mechanics of Materials Review2-D State of StressStress Transformationσ x σ y σ x σ yσ x' cos 2θ τ xy sin 2θ22σ y' σ x σ y σ x σ y 2τ x' y' σx σ y22cos 2θ τ xy sin 2θsin 2θ τ xy cos 2θPrincipal Stressesσ 1, 2 σx σytan 2θ p 2 τ xy σx σy 2 σx σ y 2 2 2 (τ xy ) No shear stress actson principal planes!

FE Mechanics of Materials ReviewMaximum In-plane Shear Stress σx σ y 2max (τ xy )τ in plane 2 2 σ x σ y / τ xytan 2θ s 2 σ avg σx σy2

FE Mechanics of Materials ReviewMohr's Circle – Stress, 2DCenter: Point C( σ avg σx σy,0)2 τσ, positive to the righttau, positive downward!R (σ x σ avg ) (τ xy )22σ1 σ avg R σ aσ 2 σ avg R σ bτ inmax plane R τA rotation of θ to the x’ axis onthe element will correspond to arotation of 2θ on Mohr’s circle!

FE Mechanics of Materials ReviewBeam Deflections -Fig. 12-2Inflection point is wherethe elastic curve haszero curvature zeromomentεσ My and σ Alsoε EIρy1M ρ EI1ρ radius of curvatureof deflected axis ofthe beam

FE Mechanics of Materials Review2M d2ydy 2 M ( x ) EIρ EI dxdx 21from calculus, for very small curvatures dM ( x ) V dx w( x ) dV ( x ) dxV ( x ) EI w( x ) EId 3ydx3d 4vfor EI constant qdx 4for EI constantDouble integrate moment equation to get deflection; use boundary conditionsfrom supports Î rollers and pins restrict displacement; fixed supports restrictdisplacements and rotations

FE Mechanics of Materials ReviewM ( x ) EId2ydx2 [ M ( x )dx ]dx y EI For each integration the “constant of integration” has to be defined,based on boundary conditions

FE Mechanics of Materials ReviewColumn BucklingPcr π 2 EIA2PcrIAr r Euler Buckling Formula(for ideal column with pinned ends) critical axial loading (maximum axial load that a columncan support just before it buckles) the smallest moment of inertia of the cross-section unbraced column lengthIA radius of gyration, units of lengthI I r2 A AA/rPcrπ 2E σ cr A( A / r )2 slenderness ratio for the column critical buckling stress

FE Mechanics of Materials ReviewEuler’s formula is only valid whenWhenσ cr σ yieldσ cr σ yield., then the section will simply yield.For columns that have end conditions other than pinned-pinned:Pcr π 2 EI(KL )2K the effective length factor (see next page)KL Le the effective lengthσ cr π 2E(KL / r )2KL/r the effective slenderness ratio

FE Mechanics of Materials ReviewEffective LengthFactors

FE Mechanics of Materials Review r T Tr J τ τ shear stress, force/length 2 T applied torque, force·length r distance from center to point of interest in File Size: 2MBPage Count: 29People also search formechanics of materials 10th pdfhibbeler mechanics of materials 10eStatics and mechanics of materialsmechanics of materials summarymechanics of materials tenth editionmechanics of materials 10th edition pdf