Experimental Stress Analysis & Structural Flight Testing, 0°-45°-90° Foil Rosette Mohr Circle
Rectangular 0-45-90 Strain Rosette & Von Mises Formula
Calculate Maximum & Minimum Principal Strains e1, e2 (microstrain), Biaxial Hooke Maximum Principal Stress sigma_1 (MPa), and Plane-Stress Equivalent Von Mises Stress (MPa).
Bonding only 0° and 90° axial strain gauges on a shaft under pure torsion (+80 / -80 ue) completely misses the +620 ue 45° diagonal shear peak—under-predicting von Mises stress 10-fold!
Whenever principal stress directions on an aircraft wing root, crane boom weldment, or pressure-vessel nozzle are unknown, experimental mechanics bonds a **3-Grid Rectangular `0°–45°–90°` Foil Strain-Gauge Rosette (`epsilon_a, epsilon_b, epsilon_c` in microstrain `ue = 10⁻⁶` m/m)**! Because tensorial **Engineering Shear Strain equals `gamma_{xy} = 2 * epsilon_b - (epsilon_a + epsilon_c)`**, **Mohr’s Strain Circle** fixes **In-Plane Principal Strains `epsilon_{1,2} = 0.5 * (epsilon_a + epsilon_c) +/- 0.5 * sqrt((epsilon_a - epsilon_c)^2 + (2 * epsilon_b - epsilon_a - epsilon_c)^2)`** (`ue`)! Converting to **Plane-Stress Hooke’s Law** (`Elastic Modulus E` GPa, `Poisson’s Ratio nu`) yields **`sigma_1 = ((E * 1,000) / (1 - nu^2)) * (epsilon_1 + nu * epsilon_2) * 10⁻⁶`** (`MPa`), `sigma_2 = ((E * 1,000) / (1 - nu^2)) * (epsilon_2 + nu * epsilon_1)` (`MPa`), and **Distortion-Energy Equivalent Von Mises Stress `sigma_{vM} = sqrt(sigma_1^2 - sigma_1 * sigma_2 + sigma_2^2)`** (`MPa`)!
Verify on a Structural Steel Gusset (`E = 205` GPa, `nu = 0.29`) Under Combined Bending + Shear—`0°`_Grid `epsilon_a` = 425 ue (B2), `45°`_Diagonal_Grid `epsilon_b` = 585 ue (B3), `90°`_Grid `epsilon_c` = -115 ue (B4), Steel_Modulus `E` = 205 GPa (B5), and Poisson_Ratio `nu` = 0.29 (B6): Maximum Principal Strain `epsilon_1` equals **`+662.7` ue** (`56%` higher than the `0°` gauge! E2), Minimum Principal Strain `epsilon_2` equals **`-352.7` ue** (E3), Maximum Principal Tensile Stress `sigma_1` equals **`125.44` MPa** (E4), and Equivalent Von Mises Stress equals **`146.75` MPa** (E5)!
| 0°-45°-90° Rectangular Strain Rosette KPI (E=205 GPa, nu=0.29) | Formula | Verified Output |
|---|---|---|
| Max Principal Strain e1 (ue) | 0.5*(ea+ec) + 0.5*SQRT((ea-ec)^2 + (2*eb-ea-ec)^2) | +662.7 ue (+625.1 ue @ torsion!) |
| Min Principal Strain e2 (ue) | 0.5*(ea+ec) - 0.5*SQRT((ea-ec)^2 + (2*eb-ea-ec)^2) | -352.7 ue (-625.1 ue pure shear) |
| Max Principal Stress s1 (MPa) | ((E_GPa*1000)/(1 - nu^2)) * (e1 + nu*e2) * 1e-6 | 125.44 MPa (99.34 MPa @ 45° helix) |
| Von Mises Equivalent Stress (MPa) | SQRT(s1^2 - s1*s2 + s2^2) | 146.75 MPa (172.07 MPa hidden shear!) |
Mohr’s Circle 0°-45°-90° Rosette Transformation & Plane-Stress Von Mises Formula
Validate `Grid_Strains epsilon_a, epsilon_b, epsilon_c` in `[-5,000, +5,000]` ue (linear elastic foil range), `Elastic_Modulus E` in `[10, 500]` GPa (`~45` Mg, `~70` Al, `~114` Ti, `~205` steel), and `Poisson_Ratio nu` in `[0.05, 0.49]`!
=LET(ea_u, B2, eb_u, B3, ec_u, B4, em_g, B5, nu_p, B6, av_e, 0.5 * (ea_u + ec_u), rd_e, 0.5 * SQRT(((ea_u - ec_u) ^ 2) + (((2 * eb_u) - ea_u - ec_u) ^ 2)), ep_a, av_e + rd_e, ep_b, av_e - rd_e, fc_m, ((em_g * 1000) / (1 - (nu_p ^ 2))) * 0.000001, sp_a, fc_m * (ep_a + nu_p * ep_b), sp_b, fc_m * (ep_b + nu_p * ep_a), IF(OR(ea_u < -5000, ea_u > 5000, eb_u < -5000, eb_u > 5000, ec_u < -5000, ec_u > 5000, em_g < 10, em_g > 500, nu_p < 0.05, nu_p > 0.49), "Invalid Rosette Input", ROUND(SQRT((sp_a ^ 2) - (sp_a * sp_b) + (sp_b ^ 2)), 2)))Pure-Torsion Trap (`e_a = +80` ue, `e_c = -80` ue, `e_b = +620` ue) Diagonal Shear Check
Test a drive tube carrying heavy torque where axial/hoop grids read tiny `B2 = 80` ue and `B4 = -80` ue while the `45°` leg reads `B3 = 620` ue: Principal Strains E2/E3 jump to **`+625.1` / `-625.1` ue** (`pure +/-99.34` MPa tension-compression in E4), driving **Von Mises Stress E5 to `172.07` MPa (`= sqrt(3) * tau`)**—over `10x` higher than a naive `E * epsilon_a = 16.4` MPa single-gauge guess!
=IF(ISTEXT(E5), E5, ROUND(0.5 * (B2 + B4) + 0.5 * SQRT(((B2 - B4) ^ 2) + (((2 * B3) - B2 - B4) ^ 2)), 1))Try the example
Use the original values or change them to test your understanding.