Duke CE 281 - Experimental Systems

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CE 281: Experimental SystemsFall, 2000Lab 2, due October 31, 2000Photo-elastic analysis of a thick ring under diametric compression.AnalysisThe bending stresses in a thick ring can be approximated by1σb(r, θ) =M(θ)(R − r)rA(¯r − R)(1)where M(θ) is the bending moment, r is the distance from the center of curva-ture, A is the cross sectional area of the beam, ¯r is the radius to the centroid,and R is the radius to the neutral axis,R =ARAdA/r. (2)For a ring under diametric compression, derive expressions for the bending mo-ment M (θ) as a function of the angle θ from the loading point. Also deriveexpressions for the shear force V (θ) and the normal force N(θ).If the load is applied along a vertical axis, then the shear force on a horizontalsection passing through the center of the ring is zero. Using the expressionsabove, and those that you derive, determine the stress acting on this horizontalplane as a function of r. Along the inner and outer radii of the ring the shearstress should be zero as well. Again, using your expressions, determine thecircumferential stresses at the inner and outer radii.Knowing the state of stress on these surfaces, and that this is a plane-stressproblem, you can determine the maximum shear stress τmaxalong these surfaces.Plot the maximum shear stress as a function of r along a radial horizontal cut,and the maximum shear stress profile as a function of θ for the inner and outersurfaces.ExperimentWe will calibrate the photo-elastic analysis using a calibration disk of the samematerial as the ring under test. The fringe order at the center of the disk, N,is proportional to the diametric load P , the diameter D, and the phtoelasticsensitivity fσ. fσhas units of [stress/fringe/thickness].fσ=8PπDN(3)Note the dimensions of the calibration disk and the load applied to it.Obtain a digital image of the calibration disk fromhttp://www.duke.edu/ hpgavin/ce281/and determine fσ.1Egor P. Popov, Introduction to mechanics of solids, Prentice Hall, 1968, pp 208–213.1Knowing the relationship between N and τmaxuse the images of the thin ringonly athttp://www.duke.edu/ hpgavin/ce281/and figure 12.22 of Beckwith, to determine the stresses on a horizontal sectionparallel to the cut, and the circumferential stresses along the inner and outerradii. For the image in Beckwith, assume that the material and thickness arethe same as for the calibration disk. Also assume that the ring in the photo-copy distributed in class is full size. Determine the load required to produce thefringe pattern shown.To interpret the images, first find the zero-order fringe (N = 0). Then countfringes moving away from the zero-order fringe. Using a ruler, a protractor, oranother method, tabulate the fringe orders (or stresses) as a function of radiusand circumference for the cases to be analyzed. Compare these stresses withthe analytical prediction of the previous section.Data AnalysisType your data into columns of an ascii file. Plot the data with points and theanalytical prediction with lines.ReportDescribe the objective, analysis, design, testing, and data analysis, and conclu-sions of your experiment in separate sections.Show the derivation of N(θ) and V (θ) in LATEX.Do the analytical and physical models support one another?Labs are to be typeset in LATEX. PostScript figures are to be generated usingGnuplot, Matlab, or another computer plotting package, and included withinthe text or at the end of the lab


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Duke CE 281 - Experimental Systems

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