Measurement of Strain using Electrical Resistance Strain Gauges 3B3  Mechanics of Solids

Adam McCreevey
3/15/2013
This is a laboratory to learn how to make measurements using a strain gauge by using different configurations, also to determine experimentally the axial and transverse stress at the surface of the beam and compare them to theoretical calculations
Introduction
If a length of wire is subject to a stress within its elastic limits, the resulting elongation and change of diameter alters the resistance. The resulting principle is used in the resistance strain gauge, which consists of many turns of resistance wire, wound on an insulating former, the strain gauge wire is selected for the minimum temperature coefficient of resistivity. They may be used to measure extremely small displacements, of the order of nanometres. Strain gauges are classified as bonded or nonbonded. The bonded type has a wire mounted on a paper backing and the paper could be pasted to the surface of the body under strain. The unbonded type is simply wire mounted between two supports which move with respect to each other. For the purpose of this experiment, we use the bonded type of strain gauge. In order to convert variations in resistance into recordable electrical signals, the strain gauge is used as one arm of a wheatstone bridge. Each bridge produces a small differential voltage which is amplified, lowpass filtered and converted to a reading in microstrain. Because temperature errors may be encountered, the use of a single gauge element is uncommon. Strain gauges can be configured in different ways to measure different aspects of strain. In this lab, we used 5 strain gauges. Two of the gauges were set up in a linear/ single configuration to measure the strain in one direction. The other three gauges were set up in a rosette configuration. This allows the measurement of the strain locally in all directions. The strain gauges in this configuration are offset from each other by a certain amount of degrees. All of the gauges were bonded to the beam with an adhesive. The amount of deflection was measured in the middle of the beam by using an LDVT.
Theory
The two single/linear gauges are set up such that one is parallel to the beam and one is perpendicular to the beam. The rosette gauges are set up at 55⁰, 100⁰ and 145⁰ to the axis parallel to the length of the beam. The beam is symmetrically supported on either side with over hangs at equal distances from the supports. This arrangement produces a uniform bending moment between the supports and so the upper and lower surfaces of the beam are uniformly strained, with the top surface experiencing tensile strain and the lower surface a compressive strain. The length of the beam was 800mm and the breath and the depth were 32mm and 6mm respectively. The supports were 250mm in from each edge. Here is the shear force diagram and the bending moment diagram for the beam.
For the sake of convenience, we will start x from the first support. So when working out the bending moment, we get this equation: Mx=PxPxa
Mx=PxPx+P(a)
Mx=P(a)
Using this value for the bending moment and the dimensions of the beam we can work out the axial stress of the beam. This is the equation: σaxial=MyI
σTrans= ν×σaxial
Where M is the bending moment, y is the centroid and ν is the Poisson's ratio. To work out the moment of inertia, we use the simple beam theorm: I=bd312
Using these equations we can work out an equation for axial strain. Here it is: εaxial=MyEI
where E is the Young's modulus for the material of the beam. In order to compare the experimental values for deflection and axial strain to theoretical values, we must derive an equation which allows us to calculate the deflection of the beam. After this, we will use the simple beam model to calculate the axial strain in the beam for a given load. We start off with this:
d2Δdx2=mEI
We are looking for a...