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Unformatted text preview: MENG 302L Lab 5: Principal Strains and Stresses Page 1 of 10 6/28/10 Introduction 1 : The purpose of this experiment is to measure the strains along three different axes surrounding a point on a cantilever beam, calculate the principal strains and then the principal stresses from these strains, and compare the result with the stress calculated from the flexure formula for such a beam. Strains from two orthogonal axes will then be used to create a Mohr’s Circle plot of the tensile and shear stresses. In a general biaxial stress or strain field, three strains along different axes at the same point must be measured to determine the principal strains and stresses with strain gages. While the stress field on the surface of a symmetrically loaded cantilever beam is uniaxial (except near the clamped end, and near the loading point), the stress at any point nevertheless varies with angle around that point. The strain field (which, in this case, is biaxial because of the Poisson strain) varies similarly. The accompanying sketch (Figure 1) shows a polar plot of the normal stress and strain at a point in a uniaxial stress field. The three axes along which strains are to be measured can be arbitrarily oriented about the point of interest. For computational convenience, however, it is preferable to space the measurement axes apart by submultiples of π , such as π /3 (60°) or π /4 (45°). An integral array of strain gages intended for simultaneous multiple strain measurements about a point is known as a “rosette”. Three gage strain rosettes are commercially available in two principal forms corresponding to the above angles. These are known as the “delta” or equiangular rosette and the 45° rectangular rosette, respectively. The two rosette configurations are shown in Figure 2. The delta rosette is sonamed because arrangement of the strainsensitive elements in the form of an equilateral triangle is equivalent to the configuration shown, as is an arrangement of two gages symmetrically disposed 60° either side of a third gage. 1 This lab is based on E103 PRINCIPAL STRAINS AND STRESSES – FLEXURE, ©Vishay Measurements Group, Inc., 1982, printed March 2002. Portions of this text were taken verbatim from that document. Figure 1: Uniaxial Stress Field Strains Figure 2: Rosette Configurations MENG 302L Lab 5: Principal Strains and Stresses Page 2 of 10 6/28/10 For the delta rosette, the principal strains can be calculated from the three measured strains with the following relationship: g Gu¡ ¢ g £ ¤g ¥ ¤g ¦ ¦ § U¥ ¦ ¨©g £ ª g ¥ « ¥ ¬ ©g ¥ ª g ¦ « ¥ ¬ ©g ¦ ª g £ « ¥ (1) The corresponding relationship for the rectangular rosette is: g Gu¡ ¢ g £ ¤g ¦ ¥ § £ U¥ ¨©g £ ª g ¥ « ¥ ¬ ©g ¥ ª g ¦ « ¥ (2) where: ϵ p,q = algebraically maximum and minimum principal strains, respectively, in/in (m/m) ϵ 1 , ϵ 2 , ϵ 3 ¢ strains measred alon® correspondin® axes of rosette elementsu in/in (m/m) The algebraically maximum and minimum principal strains correspond to the plus and minus...
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