stress_strain_notes_v2 - Measurements and Instrumentation...

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Class Notes Lecture 7 Review of Stress-Strain Strain Measurement Mark Schulz (some figures are from the Measurements Textbook by Tse and some powerpoint slides are from the Machine Analysis and Design class textbook by Shigley, Mischke, and Budynas) February 12, 2005 Measurements and Instrumentation
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Measurement and Instrumentation Lab Notes for Class Lectures 6 & 7 Chapter Sections: 1 (p. 16-17), 3.4, 5.7-5.8, 10.5 Notes and Example Problems for Strain Measurement Objectives 1. To use strain gages to measure strains 2. To relate measured strains to stresses 3. To relate measured stresses to factors of safety (machine analysis)
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Step-wise Approach to Relate Strain Measurements to Stress and Safety Factors 1. A strain gage measures strain indirectly through the change in resistance of a metallic foil. 2. The gage factor equation (1-6) relates the change in resistance of the gage to the change in strain in one direction. 3. The wheatstone bridge is a circuit that allows more accurate measurements of the change of resistance of the strain gage. 4. Strain can be measured using a strain gage rosette which measures strain in three directions, 0, 45 and 90 degrees. The three measurements will allow the principal strains and shear strain to be computed. 5. Once the strains are obtained, the stress-strain relationships can be used to determine stresses. 6. Mohr’s circle can be used as a graphical method to find the principal stresses (for the plane stress case) and their orientation and the maximum shear stress. 7. The stresses can be used to predict the factor of safety or the fatigue damage of the component or to compare the experimental stresses with design stresses based on an analysis such as a finite element analysis.
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Strain Measurements
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2 ) , , ( CD L A L f R ρ ρ = = ε R dR G f / = Resistance of the strain gage (1-3) Gage Factor (1-6) Strain Gage Properties •A foil strain gage is shown. •Soldering leads to the gage •must be done carefully.
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Strain Gage Configurations General three gages at Three different angles The Rosette strain gage has three gages at angles 0, 45 and 90 degrees A wheatstone bridge is used to increase the sensitivity of the change in resistance of the gage.
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( ) ( ) i o v R R R R R R R v + + + = 0 3 0 2 3 0 3 0 2 3 0 2 0 1 0 4 0 2 0 3 / / R R R R = 3 0 3 3 R R R + = Solve for 3 R For a Balanced Bridge Strain changes resistance of the gage (one quarter bridge active) Solve for strain f G R dR / = ε
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Strain Analysis
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Uniaxial stress but biaxial strain Biaxial stress and usually biaxial strain
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Uniaxial stress-strain is described by Hooke’s law; Poisson’s ratio is the ratio of strains: Can you derive the expression for biaxial stresses in the x,y directions (1,2)? ε σ , , E are the normal stress, young’s modulus and normal strain l, a are the lateral and axial directions
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The general plane stress-strain equations can be developed by considering the Strains in a plane: For biaxial stresses in the directions x,y or 1,2, the strains and stresses are: The strain equations can be used to find the strain at any angle where a strain Gage is placed, relative to a reference axis.
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