Instrumentation Lab. Experiment #7 Report: Strain Measurements 2
- 1. Jordan University of Science and Technology Faculty of Engineering Department of Mechanical Engineering Instrumentation and Dynamic Systems Lab Experiment #7 Strain Measurments 2
- 2. Abstract: Given the measurement of 3 independent strains from the 3 gages in a rectangular rosette, it is possible to calculate the principal strains and their orientation with respect to the rosette gage. It is also readily possible to calculate the state of strain at the gage location with respect to any particular x-y axis system using either the rosette readings or the principal strains and their axis orientation. Introduction: When the mechanical elements subjected to a various stress types, such as tension, compression or shear, Mohr circle is a graphical representation indicates the principal stresses in the element. The principal stresses are dominant stresses, which are responsible of failure in the work-piece. Because the combination of various loading conditions causes the element to fail before reaching the critical point, another concern is considered to the cause of this phenomenon, which is explained by Mohr’s circle, the principal stress which is actually affects the element is different from the individually computed stresses. The principle stress is very important to be computed, in order to guarantee the critical stress which is actually responsible of the earlier failure in the work-piece, so the designer should consider the maximum stress in the work-piece caused due to the applied stresses. In the beams that are subjected to a biaxial strain and stress, it's very important to measure the stress in different directions, in order to calculate the principal stresses, which is used in Mohr circle considerations. Principal stresses are maximum and minimum value, and are computed from the principal strains in the work-piece, which are also maximum and minimum; these are computed from different three strain gages, aligned with definite angles from each other, which are pointing to a specific point, which is called the rosette. Strain rosette is available in several forms according to the angles between the strain gages, such as the DELTA form, and the RECTANGULAR form, and the angles which apart the strain gages are 30º, 45º, 60º, 90º or 120º, which differs according to the required application . Equipment and instrumentations Strain gage: sensitive resistance to the change in length. Digital strain gage indicator: digital device used to compute the change in resistance, and then convert it into a number of strains digitally on the screen.
- 3. Selector: digital device used to make the experiment easier; it enables us to switch between the different strain gages on the work-piece very fast, without the need to rewire the assembled circuit. Work-piece: standard steel work-piece is used, with definite dimensions. Procedure: 1- Assemble the three strain ages to the work-piece, with 45º apart. 2- Calibrate the digital gage, by hanging the work-piece without any load, and then reset the readings of the three strain gages to zero, by revolving the digital screw to reach the zero reading, this is done for all strain gages. 3- Hang the work-piece with the loads as shown in figure 1 with 100g mass. 4- Repeat the load up to 600g, with 100g step, readings should be recorded increasingly and decreasingly to study Hysteresis if existed. 5- Record the strains of each strain gage, and calculate the principal stresses and Poisson’s ratio. Figure 1: The cantilever work-piece dimensions and loading. Results:
- 4. Table 1: Strain Measurements at the Rosset Mass (gram) Є1 (µm/m) Є2 (µm/m) Є3 (µm/m) Єp Єq 100 75 100 20 110 -35 200 140 190 20 205 -45 300 200 275 27 297 -70 400 260 362 25 391 -106 500 325 450 30 487 -132 600 300 545 24 569 -245 500 315 450 25 485 -145 400 260 375 40 400 -100 300 200 305 50 320 -70 200 140 220 40 229 -49 100 75 130 45 132 -12 Sample of Calculation: Calculating the principal Stresses (400 grams): Є 𝒑 = Є 𝟏 + Є 𝟑 𝟐 + √ (Є 𝟏 − Є 𝟐) 𝟐 𝟐 + (Є 𝟐 − Є 𝟑) 𝟐 𝟐 = 𝟐𝟓+𝟐𝟔𝟎 𝟐 + √ ( 𝟐𝟔𝟎−𝟑𝟔𝟐) 𝟐 𝟐 + ( 𝟑𝟔𝟐−𝟐𝟓) 𝟐 𝟐 = 𝟑𝟗𝟏 µm/m Є 𝒒 = Є 𝟏 + Є 𝟑 𝟐 + √ (Є 𝟏 − Є 𝟐) 𝟐 𝟐 + (Є 𝟐 − Є 𝟑) 𝟐 𝟐 = 𝟐𝟓+𝟐𝟔𝟎 𝟐 − √ ( 𝟐𝟔𝟎−𝟑𝟔𝟐) 𝟐 𝟐 + ( 𝟑𝟔𝟐−𝟐𝟓) 𝟐 𝟐 = −𝟏𝟎𝟔 µm/m
- 5. Calculating Poisson’s ratio (γ): Table 2: Poisson’s Ratio (γ= - σq/ σp) Єp(µm/m) Єq (µm/m) γ 110 -35 0.3206 205 -45 0.2207 297 -70 0.2349 391 -106 0.2720 487 -132 0.2716 569 -245 0.4307 485 -145 0.2994 400 -100 0.2508 320 -70 0.2188 229 -49 0.2149 132 -12 0.0881 The Average Poisson’s ratio is: γavg = 0.2566 Sample of Calculation (400 grams): γ = 106/391 = 0.272 Principal Stresses: Load Єp Єq σp σL 100 110 -35 1148 801 200 205 -45 2135 1601 300 297 -70 3086 2402 0 100 200 300 400 500 600 0 100 200 300 400 500 600 700 Strain(µm/m) Load (grams) Figure 2:Strain over a range of loads (Hysterisis)
- 6. 400 391 -106 4071 3202 500 487 -132 5069 4003 600 569 -245 5919 4803 500 485 -145 5047 4003 400 400 -100 4165 3202 300 320 -70 3328 2402 200 229 -49 2385 1601 100 132 -12 1369 801 Sample of Calculation(400 gram): σp = 𝟏𝟎. 𝟒𝒆𝟔 ∗ 𝟑𝟗𝟏+𝟎.𝟐𝟕𝟐∗−𝟏𝟎𝟔 𝟏−𝟎.𝟐𝟕𝟐 𝟐 = 𝟒𝟎𝟕𝟏 𝒑𝒔𝒊 σL = 𝟔 ∗ 𝟎𝟎𝟐𝟐𝟎𝟒𝒑𝒐𝒖𝒏𝒅 𝒈𝒓𝒂𝒎 ∗ 𝟏𝟎𝟎 𝒈𝒓𝒂𝒎 ∗ 𝟏𝟎.𝟐𝟑𝒊𝒏 𝟏𝒊𝒏∗𝟎.𝟎𝟏𝟑 𝟐 𝒊𝒏 𝟐 = 𝟑𝟐𝟎𝟐 𝒑𝒔𝒊 The Rosset gage calibration:
- 7. Sample of calculation (at 400 grams): 𝑐𝑎𝑙𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛 constant = 𝜎𝐿 𝜎 𝑃 = 3202 4071 = 0.787 Average value = 0.768 Constructing Mohr’s Circle, at M =500 grams (See Figure 4): 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 Stress(psi) Theoretical Stress (psi) Figure 3: CalibrationCurve for the Rosset Gage Calibrated curve Measured Stress
- 8. Discussionof results: The results show that we can calculate the principal stresses and strains by only measuring the strains at 3 different angles by a rosset gage. Table 1 shows the normal strains and the shear strain at the location of the rosset. These values were used to calculate the principal strains according to the equations, and then convert these strains to there equivalent stresses depending on the Modulus of elasticity of the material (which is 210 Gpa for mild steel, as in this case). Then the Mohr’s circle was constructed at m=500 grams to easily visualize the stress state, and the stress state is shown by the line crossing the circle. 325, 100 148, 0 0, -100 0 -240 -200 -160 -120 -80 -40 0 40 80 120 160 200 240 -80 -40 0 40 80 120 160 200 240 280 320 360 400 ShearStrain(µm/m) Normal Strain (µm/m) Figure 4: Mohr's Circle at the Rosset (500 grams)
- 9. The major sources of errors are due to the zero offset of the gage indicator device, and this is clearly shown in the readings for gage number 3, as the reading should be zero for all loading conditions. Conclusions: Rosset strain gages are used to identify the stress state at a current location on a loaded specimen or part by identifying the strain values at three different oriented axes. Mohr’s circle is a very helpful illustration tool, that makes the visualization of the stress state easier. The strain gage indicator was a source of error due to the zero offset.