EG-268 Experimental Studies 2019/2020 EG-268: AN EXPERIMENT TO MEASURE STRAIN AND CALCULATE STRESSES IN BEAMS SUBJECT TO BENDING AND BARS SUBJECT TO TENSION 1. Introduction Stress, simply defined as the

internal force per unit area is the cause of strains and deformations which can lead to multiple engineering failures. Therefore, a Design Engineer, when choosing appropriate materials and dimension for components, will need to calculate the stresses in advance within the scope of their design work to ensure appropriate material selection and dimensioning of the components. Two common types of failure modes include bending and tension. These experiments will compare the stresses produced for rectangular cross-section bending beams of three different materials as well as for bars under tension, also in three different materials. The materials being assessed are aluminium, copper and brass. It is often of interest to compare the stresses subsequently encountered in operation to the design forecasts. This experimental facility enables the stress to be derived from experimental data for a range of loads. These values can then be compared with theoretical values obtained using conventional formulae. Stress cannot generally be measured directly. The material strain however is more easily measured experimentally and is directly related to the material stress by known stress/strain relationships. The component stresses can thus be determined by way of strain measurement. 2. Experimental Equipment The experimental set-up is shown in Figures 1 and 2. The bending beam in Figure 1 is loaded by means of a dead-weight system via a cantilever arm. The beam is rigidly supported at one end can be tested in bending or torsion, but for this experiment we will only consider pure bending. Figure 2 shows the set up for the tension bar experiment which is loaded directly below the bar and fixed to the top of the frame. 1 EG-268 Experimental Studies 2019/2020 000 Figure 1. Bending Beam experimental set-up 250 mm (0) Table 2. Material Properties Bending beam Figure 2. Tension bar experimental set-up symbol L H b 2.1. Bending Beam and Tension Bar The beam used for the bending experiment is a solid rectangular cross-section steel cantilever beam clamped at one end. The dimensions are given in Table 1. The bar used for the tension test on the other hand has a cross sectional area of 10 x 2 mm² (20 mm²). Both the bending beams and tension bars are tested in three different materials: brass, copper and aluminium. Their material properties are summarised in Table 2, with the beams and bars shown in Figure 3. Table 1. Bending Beam Dimensions Dimensions (mm) Bending length to loading point Beam height Beam Width Tensile bar Value 250 4.75 19.75 2 EG-268 Experimental Studies 2019/2020 Material Properties Symbol Modulus of Elasticity (N/mm²) Poisson's ratio Strain Factor a. LOGO! E μ k b. Materials Copper 123000 0.33 2.050580 998 Brass 88000 0.33 2.050077 294 Aluminium Copper Brass Figure 3. (a) Tension Bars (b) Bending Beams Aluminium 69000 0.33 2.0505809 98 2.2. Strain Gauge Technique An important branch of experimental stress analysis is based on the principle of strain measurement. The use of the strain-gauge technique enables strain to be measured at the surface of the component. As the maximum stress is generally found at the surface, this does not represent a restriction. The most commonly used strain gauges are metallic strain gauges. They function on the principle that mechanical strain causes a change in the electrical resistance of the metal strips or wire. The gauges are bonded to the surface of the beam thus causing a change in resistance (and hence a change in voltage over the gauge) as the beam deforms. See Figure 4. 3 EG-268 Experimental Studies 2019/2020 Insensitive to lateral force Tension increases resistance Figure 4. Strain gauge principal At any point in a material, both normal strains and shear strains generally exist and in order to completely evaluate the stresses at a point, knowledge of both normal and shear strains is required. For determination of the total bending stresses, the bending beam is fitted with four strain gauges (two on the compression side and two on the tension side). The strain gauges are arranged in a bridge circuit shown schematically in Figure 5. This arrangement leads to the summation of all changes in resistance and a high level of sensitivity. Bending moment R1 R2 R3 R₁ R4 UE JUE R2 Measured resistance R4 R3 Stress distrbution Figure 5. Schematic of strain gauge configuration on test beam The strain-gauge element (full-bridge circuit) is attached in the proximity of the clamping point. The measuring amplifier shown in Figure 6 supplies the bridge feed voltage and displays the output voltage in mV/V. 4 EG-268 Experimental Studies 2019/2020 0.050 DSP PAR FIA F2 RST mV/V E= Figure 6. Digital 4-position LED measuring amplifier 3. Analysis of experimental data 3.1. Bending Beam The output signal, UA, of the measuring bridge is referenced to the feed voltage, UE (Note that the output is given in mV). Therefore UA/UE in the strain equation relates to the reading being generated by the strain gauge in mV/V which will need to be converted into V/V. The sensitivity k of the strain gauge enables the strain of the bending beam &, to be calculated for the full bridge as shown in Equation 1. ε = un FL 100 Dehnungsmessstreifen-Lehrsystem MAMBLIKE Strain Gauge Training System UA KUE Eingang / Input 1 4 UA 2(1+μ) k UB E Equation (1) The corresponding stresses can then be obtained using Hooke's Law. 3.2. Tension Bar The strain of the tension bar ɛ is similarly calculated using the output value from the strain sensor converted into UA/UE, along with the sensitivity k of the strain gauge and the poisons ratio u as shown in Equation 2. Equation (2) As with the bending beam, the corresponding stresses can then be obtained using Hooke's Law. 4. Theoretical analysis using simple bending theory 4.1. Bending beam 5