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GT AE 3145 - Shear Center in Thin-Walled Beams Lab

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AE3145 Shear Center Lab (S2k)Slide 1Shear Center in Thin-Walled Beams Lab• Shear flow is developed in beams with thin-walled cross sections– shear flow (qsx): shear force per unit length along cross section• qsx=τsx t• behaves much like a “flow,” especially at junctions in cross section– shear flow acts along tangent (s) direction on cross section• there is a normal component, τnx, but it is very small• e.g., because it must be zero at ±t/2– shear force: qsxds (acting in s direction)• Shear flow arises from presence of shear loads, Vy or Vz– needed to counter unbalanced bending stresses, σx– to determine, must analyze equilibrium in axial (x) direction• Shear center:– resultant of shear flow on section must equal Vy and Vz– moment due to qsx must be equal to moment due to Vy and Vz– shear center: point about which moment due to shear flow is zero– not applying transverse loads through shear center will cause atwisting of the beam about the x axisAE3145 Shear Center Lab (S2k)Slide 2Approach for Lab• Apply transverse loading to tip of a cantilever thin-walled beam– use cross-arm at tip to apply both a lateral force and twisting mom.– measure bending deflection– measure twisting– vary location of load point along cross-arm– repeat for beam rotated 90 deg. about x axis• Data analysis– record deflections using LVDT– plot twisting versus load position on cross-arm– determine location on cross-arm where load produces no twisting• Compare the measured shear center with theoretical location– shear flow calculations used to compute shear center– consider both y axis and z axis loading (rotated 90 deg)AE3145 Shear Center Lab (S2k)Slide 3Review from AE2120 (2751), AE3120• Bending of beams with unsymmetrical cross sections– bending stress depends on Iy, Iz and Iyz– neutral surface is no longer aligned with z or y axes• Shear stresses are computed from axial force equilibrium– shear stress needed to counter changing σx– analysis strictly correct for rectangular sections only• Thin-walled cross sections– thin walls support bending stress just like a solid section (no change)– thin walls support shear stress in tangential direction• transverse shear component is negligable...• because it must vanish at the free surfaces (edges of cross section)– shear flow: τxs t (force/unit length along section)– shear flow must be equivalent to Vy and Vz so it must:• produce same vertical and horizontal force (Vx and Vy )• produce same mumoment about any point in cross section– point about which no moment is developed: SHEAR CENTER• lateral load must be applied through SC to avoid twisting beam• twisting loads will cause section to twist about SC (center of rotation)AE3145 Shear Center Lab (S2k)Slide 4LVDTcross armweightTest ConfigurationLab ApparatusCantilever with thin-walled C sectionCantilever with thin-walled C sectionLVDT measures tipdeflection on cross-armLVDT measures tipdeflection on cross-armSmall weight used to applyload at point on cross-armSmall weight used to applyload at point on cross-armAE3145 Shear Center Lab (S2k)Slide 5Lab Procedure1. Determine the beam material properties from reference material (e.g., referencedtextbooks or MIL Handbook 5 which can be found in the GT Library).2. Find the centroid of the given beam cross-section.3. Determine Iz, Iy, Iyz for the given section.4. Determine the shear flow distribution on the cross-section for a Vy shear load.5. Determine the shear flow distribution on the cross-section for a Vz shear load.6. Determine the shear center for the cross-section.7. Using data from the lab, determine the measured location of the shear center andcompare this with the location determined in step 6 above.AE3145 Shear Center Lab (S2k)Slide 6Beam Cross Section22yyAzzAyzAIzdAIydAIyzdA===00AAzdAydA==Centroidal Axes:Area Moments (of Inertia):1.353in.1.330in.0.420in.0.050in.ZYUse single line approx forcross section (t<<b,h)Use single line approx forcross section (t<<b,h)AE3145 Shear Center Lab (S2k)Slide 7Bending of Beam with Unsymmetrical Cross SectionqZYA12()()yy yz z yz zz yxzz yy yzyI zI M yI zI MII Iσ−+−=−−zxzzyMIσ=−Symmetric cross section, Mz=0:General:But also considerequilibrium ofsegment A1 (seenext slide!)But also considerequilibrium ofsegment A1 (seenext slide!)Acts overcross sectionAE3145 Shear Center Lab (S2k)Slide 8Shear Stresses and Shear Flow110xx sx xAAxdx xF dA q dx dAσσ+éé== +−êêêêëëåòòAxial force equilibrium for element:Complementaryqsx acts on A1 inopposite directionComplementaryqsx acts on A1 inopposite directionsZYXqsxσx+dσxσxA1AE3145 Shear Center Lab (S2k)Slide 9Shear Flow11 1122yzsx yy yz zz yzAA AAyy zz yz yy zz yzVVq I ydAI zdA I zdAI ydAII I II Iææ−−=−+−çççç−−èèResult for qsx:sZYShear flow: qsx(s)AE3145 Shear Center Lab (S2k)Slide 10sZYShear flow: qsx(s)Shear CenterVyezTherefore:Shear center liesdistance ez fromorigin where:M0=VyezTherefore:Shear center liesdistance ez fromorigin where:M0=VyezMoment, M0, atorigin due toshear flow, qsxMoment, MMoment, M00, at, atorigin due toorigin due toshear flow, qshear flow, qsxsxMoment due to Vymust be equal to M0Moment due to VMoment due to Vyymust be equal to Mmust be equal to M00AE3145 Shear Center Lab (S2k)Slide 11Examples of Shear CentersSection Symmetric about y axis:Shear center must lie on y axis(similar argument for z axis symmetry)Section Symmetric about y axis:Shear center must lie on y axis(similar argument for z axis symmetry)Angle Section:Shear center must lie at vertex of legs (regardless oforientation of section)Angle Section:Shear center must lie at vertex of legs (regardless oforientation of section)ZYVyShearCenterlies ony axisZYqsxVyShearCenterqsxAE3145 Shear Center Lab (S2k)Slide 12ZYABShearCenterVyqsxqsxqsxShear Center Must Lie Outside CSum moments from qsx about A:=force in each flange x h/2h/2h/2 eMust equal moment from Vy about A:=Vy x ee must be positivefor qsx as shownso shear centerlies to left ofsectione must be positivefor qsx as shownso shear centerlies to left ofsectionAE3145 Shear Center Lab (S2k)Slide 13Data Acquisition• Use PC data acquisition program to acquire deflection andstrain data and test machine load– Use 2 LVDT displacement gages– Measure vertical displacements at ends of cross arm– Use to determine vertical deflection and cross arm rotation– Use single weight but move to different


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GT AE 3145 - Shear Center in Thin-Walled Beams Lab

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