ME 354 MECHANICS OF MATERIALS LABORATORY STRESSES IN STRAIGHT AND CURVED BEAMS
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1 ME 354 MECHNICS OF MTERILS LBORTORY STRESSES IN STRIGHT ND CURVED BEMS OBJECTIVES January 2007 NJS The ojectives of this laoratory exercise are to introduce an experimental stress analysis technique known as "photoelasticity", and to apply this technique to measure the stresses induced in a curved eam. In order to use the method of photoelasticity, we must first measure a material property called the "material fringe value", denoted f s. The f s will e measured for the polymer H-9l1. There are several ways of measuring f s. The technique used for caliration in this la is ased on a straight eam sujected to fourpoint ending. Once f s has een measured, then the stresses induced at several points in a curved eam will e measured using photoelasticity. The experimental measurements will e compared to predicted stress levels otained using analytical solutions as well as numerical solutions from finite element analysis (). EQUIPMENT USED total of two test specimens will e used: one straight eam and one curved eam (made from H-911). four-point loading fixture with load pan and calirated masses (used to load the straight eam) line-loading fixture with load pan and calirated masses (used to load the curved eam) circular polariscope equipped with a monochromatic (green) light source Video camera, B/W monitor and a video printer STRIGHT BEM TESTS EXPERIMENTL PROCEDURES i) Install the straight eam in the four-point loading fixture (see Figure 1a) ii) ttach the load pan (Note: the comined mass of the pan and fixture is ~0.980kg) iii) pply two 10 kg masses, one at a time, to the load pan. iv) Use the video camera/printer to record the fringe pattern oserved with a "dark field" (i.e., the image otained when the polarizer and analyzer axes are crossed) v) Use the video camera/printer to record the fringe pattern oserved with a "light field" (i.e., the image otained when the polarizer and analyzer axes are parallel) CURVED BEM TEST i) Install the curved eam in the line loading fixture (see Figure 1) ii) ttach the load pan (Note: The comined mass of the pan and the fixture is ~0.454 kg) iii) pply one 5-kg mass to the load pan. (Note; Do not apply more than 5kg at any time) iv) Use the video camera/printer to record the fringe pattern oserved with a "dark field" (i.e., the image otained when the polarizer and analyzer axes are crossed) v) Use the video camera/printer to record the fringe pattern oserved with a "light field" (i.e., the image otained when the polarizer and analyzer axes are parallel)
2 NME ME 354 MECHNICS OF MTERILS LBORTORY STRESSES IN STRIGHT ND CURVED BEMS WORKSHEET DTE January 2007 NJS 1) Preliminaries: Specimen Dimensions Complete the following tale (Refer to Figure 1-3). Tale 1: Dimensions and Loading for the Straight and Curved Photoelastic Beams Straight Beam Curved Beam Total pplied Force F s Total pplied Force F c = ( M + M + M g (N) = ( M + M + M g (N) weight fixture pan) weight fixture pan) Outer span L o (mm) Outer radius r o (mm) Inner span L i (mm) Inner radius r i (mm) Height (straight), h (mm) Height h c = r o -r i (mm) Thickness, (mm) Thickness, (mm) Momment of Inertia I=h 3 /12 (mm 4 ) verage radius r ave = r =(r o +r i )/2 (mm) Straight leg length l (mm) Note: the caliration and test force must include the mass of the fixture and pan, and the added mass in kg. Gravitational constant is g = kg m/s 2. r r a) Straight Beam ) Curved Beam Figure 1 Nomenclature for the a) Straight and ) Curved Beams
3 Figure 2 Sketch of the Straight Beam showing Dimensions Figure 3 Sketch of the Curved Beam showing Dimensions
4 2) Determination of the Material Fringe Value f s Interpretation of 2-D photoelastic fringe patterns is ased on the stress-optic law: Nf ( ) = 1 2 where 1 and 2 are the in-plane principle stresses, N is the fringe order, is the thickness of the photoelastic model, and f is the material fringe value. In this la, the fringe patterns oserved in straight eams sujected to pure ending are used to determine f. Recall the stresses induced in a straight eam with a constant cross section are given y (using the x-y-z coordinate system shown in Figure 4): τ = τ = 0 (Out of plane stresses are assumed to e zero (plane stress assumption)) z = xz yz = 0 (normal stresses transverse to the eam are ignored) y My x = (axial stresses are given y the flexure formula ) I VQ τ xy = (shear stresses and given y the shear formula ) I free-ody, shear force, and ending moment diagram for a straight eam loaded in four point flexure loading is shown in Figure 4. Note that this loading arrangement places the central region of the eam in pure ending. That is, over the central region, the shear force is zero (V = 0) and the ending moment is constant (M = F s (L o -L i )/4). Consequently in this inner region, the principle stresses are: My Fs ( Lo Li ) y 1 = x = =, 2 = 0, 3 = 0 I 4I Sustituting these results into the stress optics law and rearranging (slightly), we find: Z = ( f ) N where: Fs ( Lo Li) y Z = 4I Hence f can e determined y plotting the quantity Z versus the fringe order, N and fitting a straight line to this data using a linear regression. If the data ehaves perfectly, then the slope given y the linear regression will equal f and the intercept = 0. Nowadays, linear regression can e performed using scientific hand calculators and computer software (such as EXCEL). Equivalently, y completing Tale 2 you will, in effect, e performing a linear regression of the data otained for the straight eam. Perform the following: a) Enter the required data in columns 1 and 2 of Tale 2 y scaling the recorded images using the dark and light field polariscope settings (also indicate whether a dark or light field was used in column 5) ) Complete all calculations indicated in Tale 2 therey determining f.
5 y x M=F s (L o -L i )/4 - Figure 4 Free Body, Shear Force and Bending Moment Dia grams for Four-Point Flexure Loading Tale 2: Linear Regression of Straight Beam H-911 Data Distance y (mm) Fringe Order* N N 2 Z Fs ( Lo Li) y = N Z 4I Dark/Light Field? ( ) *Define the fringes located at negative y positions as negative fringes The total numer of data points: n = The slope (i.e., the material fringe value) is given y: N Z n NZ f s = = 2 2 N n N ( ) ( ) Note: the handook value for the fringe value for the material used here is around 16 kn/m fringe. The intercept is given y: Z fs N intercept = = n
6 3) Experimentally Measured Stresses in the Curved Beam using Photoelasticity t the free edge of selected location of the curved eam ( B and C in Figure 1), the stress states are uniaxial and the stress-optic law can e used to calculate the normal stress using the relation etween the fringe order at the free edge, the stress optic coefficient for the material, and the specimen thickness. Fill in the tale with the values for the loaded test specimen that are used in the following calculations. Fringe value counted at point, N Fringe value counted at point B, N B Fringe value counted at point C, N C Thickness, (mm) f N t point, the normal stress: = = MPa. f N B t point B, the normal stress: B = = MPa. f NC t point C, the normal stress: C = = MPa. 4) nalytically Determined Stress at Point Using the Straight Beam Relations For the straight part of the eam, a straight eam flexure analysis may e used. The applied ending Moment at point is determined as the applied test force F c = N multiplied y the length of the straight leg l = mm such that M = Fc l = N mm. The moment of inertia is calculated from the height of the eam h c and the thickness of the eam, such 3 hc that I = = mm The distance from the neutral axis to the point is c = h c /2 = mm. straight M c The normal stress at for a straight eam assumption is = = MPa. I (Confirm that the normal stress at should e tensile (positive stress)) 5) nalytically Determined Stresses Using the Curved Beam Relations For the curved part of the eam (in this case points B and C ) The analytical calculation must take into account the initial curvature of the eam. a) t the line in the curve connecting B and C, the radius of the neutral axis for the rectangular cross section can e calculated from the outer radius, r o and the inner radius r i such that: ro ri R = = mm. ln( r / r ) o i The eccentricity, e, can e calculated from the average radius (or centroid) r and the radius of the neutral axis such that e = r - R = mm. The cross sectional area is calculated from the thickness,, and the curve height, h c such that = h c = mm 2. (Note that the distance from the Neutral axis to the point of interest is y = r R).
7 ) t point B, r = r i ; therefore, y B =r i R = mm. The ending moment at point B is determined as the applied test force F c = N multiplied y the length of the straight leg l = mm plus the average radius r =(r o +r i )/2 = mm such that M B = Fc ( r + l) = N mm. curved M B yb The normal stress in the curved eam at B is B = = MPa. e( yb + R) c) t point C, r = r o ; therefore, y C =r o R = mm. The ending moment at point C is the same as at point B such that: M C = M B = N mm. curved M C yc The normal stress in the curved eam at C is C = = MPa. e( y + R) 6) dditional xial Normal Stress Component Because the ending moment at B-C is produced y a transverse force (that is, not a pure ending moment), the total stress at B-C has two components: a tensile axial (in the loading direction) stress and a tensile/compressive ending stress (computed aove). a) The tensile axial stress is calculated from the applied test force F c = N and the cross sectional area = h c = mm 2. axial F The axial stress is = c = MPa. (Make sure that this axial normal stress is tensile (i.e. positive stress) C 7) Comparison of the Total Normal Stress (Bending and xial) at B and C a) t B, the total calculated stress using the curved eam assumption is: total ( curved) axial curved B = + B = MPa. Percent difference etween the actual photoelastically measured stress and the calculated stress is: total curved ( ) B B 100 = %. B ) t B, the numerically determined (from the finite element analysis () solution of Figure 5) normal stress is: B = MPa. Percent difference etween the actual photoelastically measured stress and the numerically B B determined stress is: 100 = %. B c) t C, the total calculated stress using the curved eam assumption is: total( curved) axial curved C = + C = MPa. Percent difference etween the actual photoelastically measured stress and the calculated stress is: total curved ( ) C C 100 = %. C d) t C, the numerically determined (from ) normal stress is: C = MPa. Percent difference etween the actual photoelastically measured stress and the numerically C C determined stress is: 100 = %. C
8 8) Comparison of the Normal Stress (ending) at a) t, the total calculated stress using the straight eam assumption is: straight = MPa. Percent difference etween the actual photoelastically measured stress and the calculated stress is: straight 100 = %. ) t, the numerically determined (from ) normal stress is: = MPa. Percent difference etween the actual photoelastically measured stress and the numerically determined stress is: 100 = %. = 6.25 MPa a) Finite element mesh ) Stress in the x-direction B = MPa C = MPa c) Stress in the y-direction c) simulated fringe pattern Figure 5 Finite Element nalysis Solution to the Curved Beam Bending Prolem
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