MET 87 nstrumentation and utomatic Controls Lecture Sensors July 6-9, 00 Stress and Strain Measurement Safe Load Level monitoring Force (indirect measurement by measuring strain of a flexural element Pressure (by measuring strain in a flexible diaphragm Temperature (by measuring thermal expansion of a material Strain Gauge (SG Basic Device o Bonded metal foil a thin foil of metal (Constantan o grid pattern o Thin plastic backing material (polymide o Large metallic pads for soldering o Total resistance of the SG is given by where L ρ ρ foil metal resistivity (Ω-m L total length of the grid line (m the grid line cross-sectional area (m Figure. Typical Strain Gauge
Cantilever Beam Mounted with a SG SG mounted on a cantilever beam of uniform and symmetric cross section. push-down normal force (stress P is added at the unsupported end, as shown in Figure. P δ L x SG mounted on a cantilever beam of uniform and symmetric cross section Figure s the object is deformed, the SG foil is deformed, causing its electric resistance to change P L For the strength materials, the deflection at the unsupported end is δ, where P E normal force, L length of beam, moment of inertia, and E modulus of elasticity The force applied to a solid object resulting the deformation of the object. The effect of applied force is called Stress and the resulting deformation is called Strain. The Tensile Stress-Strain o tensile force applied to an object cause the object to elongate or pull apart o The stress (unit of pressure or tensile is defined as F Tensile Stress (N/m in S unit or lb/in in the English unit where F applied force in N cross-sectional area of the object in m o The resultant strain is defined as the fractional change in length: l Tensile Strain ε l
where l change in length in m ( or inch l original length in m (or inch The beam develops a tensile stress on the top surface, and total surface stress σ at location is where P x ( t / σ E ε t beam thickness, moment of inertia E modulus of elasticity or Young s modulus is given by Stress F / E Strain l / l For aluminum beam E 6.89 x 0 0 N/m Therefore we can find P in terms of other terms: P σ E ε x t x t P L Then substituting the applied stress P into deflection δ equation: δ E P L δ E E ε L x t E ε L x t Solve for strain ε ε x t δ L Gauge Factor (GF Electric Properties o ttached the SG to an object by a suitable adhesive o s the object is deformed, the SG foil is deformed, causing its electric resistance to change GF is given by / GF ε where change in resistance due to stress (Ω original resistance of strain gauge (Ω ε strain (in/in, m/m, cm/cm, etc
Example 9. f a 0 Ω strain gauge with a GF.0 is used to measure a strain of 00micro ε or 00με, how much does the resistance of the gauge change from the unloaded state to the loaded state? nswer: From the equation: GF / ε We can solve it for : GF ε ε GF (0*00*0-6 *.0 0*0.000* 0.0Ω Measuring with a Wheatstone Bridge Wheatstone bridge a circuit for measure small change in resistance four-resistor network with a DC voltage source For the static balanced mode, as shown in Figure. o and are precision resistors; precision potentiometer; the strain gauge o djust until voltage between and B is zero volt: B B 0 or B B o Since we will use a High mpedance olt Meter to measure the voltage B, so the meter will draw any current. This condition makes and. ex ex Substitute and into, we obtain the following expression:
So we know and accurately, and note the adjusted value for calculating unknown (strain resistance as : Strain gage ex B olt Meter High mpedance oltmeter Static Balanced Bridge Circuit Figure
Dynamic Unbalanced Bridge Circuit The circuit is shown in Figure Figure The put voltage can be expressed as The input voltage ex can be express as ( ( ex Solving for in terms of ex ( ex When the bridge is balanced: 0 and has a known value When change value, as the strain gauge is loaded, the is related to earrange the equation to give ( ( ex ex
eferences Strain Gages, ccessories and nstruments, http://www.omega.com/toc_asp/subsectionsc.asp?subsectione0&bookpressure Strength of Materials, http://en.wikipedia.org/wiki/strength_of_materials Strain Gauge Measurement Tutorial, National nstruments pplication Note 078, http://www.eidactics.com/downloads/efs-methods/n_strain_gauge_tutorial.pdf Strain osette for Strain Measurement, efunda Engineering fundamentals, http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/strain_gage_rosette.cf m Electrical esistance Strain Gage Circuits, http://www.ae.gatech.edu/people/jcraig/classes/ae5/lab/strain-gages.pdf Stress, Elasticity and Deformation, http://arch.umd.edu/tech/tech_/lectures/stress_elasticity_and_deformation.pdf Stress and Strain Formulas, Engineering Edge, http://www.engineersedge.com/strength_of_materials.htm Beam Diagram and Calculator nput, efunda Engineering fundamentals, http://www.efunda.com/formulae/solid_mechanics/beams/casestudy_display.cfm?casec antilever_uniformload Structure Beam Bending Equation/ Calculation Cantilevered Beam with One Load pplied at End, http://www.engineersedge.com/beam_bending/beam_bending9.htm