Aerospace Structural Mechanics & Plasticity
Ramberg-Osgood Elastic-Plastic Strain & Secant Formula
Calculate Metallic Elastic Strain sigma / E, MMPDS 0.2%-Offset Ramberg-Osgood Plastic Strain 0.002 * (sigma / sigma_02)^n, Total Nonlinear Strain, and Effective Secant Modulus E_sec.
Capture smooth work-hardening yield curves with the 0.2%-offset Ramberg-Osgood law!
Aluminum, titanium, and austenitic stainless alloys lack a sharp upper/lower yield plateau, transitioning smoothly from Hookean elasticity into power-law strain hardening! W. Ramberg and W.R. Osgood (1943, NACA Technical Note 902) and FAA/MMPDS metallic allowables express total uniaxial engineering strain as the sum of Elastic Strain plus 0.2%-Proof Plastic Strain: epsilon_tot = sigma / E + 0.002 * (sigma / sigma_02) ^ n, where E is Young’s Modulus, sigma_02 is the 0.2% offset yield strength, and n is the Ramberg-Osgood hardening exponent! Notice the built-in calibration anchor: whenever applied stress sigma equals sigma_02, the plastic term equals 0.002 * (1)^n = 0.002000 (0.20% permanent offset) for ANY exponent n, while Effective Secant Modulus E_sec = sigma / epsilon_tot governs inelastic column and skin-panel buckling!
Verify on Austenitic Stainless Steel—Applied_Stress sigma = 450 MPa (B2), Youngs_Modulus E = 200 GPa (200,000 MPa, B3 -> Elastic Strain = 450 / 200,000 = 0.002250 in E2), Yield_Proof_Stress sigma_02 = 300 MPa (B4), and Hardening_Exponent n = 4.0 (B5): Plastic Strain equals 0.002 * (450 / 300)^4 = 0.002 * 5.0625 = 0.010125 (E3); Total Strain equals 0.002250 + 0.010125 = 0.012375 (1.2375% in E4); and Effective Secant Modulus drops from 200.00 GPa to (450 / 0.012375) / 1,000 = 36.36 GPa (E5)!
| Ramberg-Osgood Metric (sigma=450 MPa, E=200 GPa, sigma_02=300 MPa, n=4) | Formula | Verified Output |
|---|---|---|
| Hookean Elastic Strain | sigma_MPa / (E_GPa * 1000) | 0.002250 (0.001500 @ 300 MPa) |
| Ramberg-Osgood Plastic Strain | 0.002 * (sigma / sigma_02) ^ n | 0.010125 (0.002000 @ 300 MPa) |
| Total Elastic-Plastic Strain | Elastic_Strain + Plastic_Strain | 0.012375 (0.003500 @ 300 MPa) |
| Effective Secant Modulus E_sec | (sigma_MPa / Total_Strain) / 1000 | 36.36 GPa (85.71 GPa @ 300 MPa) |
MMPDS 0.2%-Offset Ramberg-Osgood Total Strain Formula
Validate sigma > 0, E_GPa > 0, sigma_02 > 0, and strain-hardening exponent n >= 1!
=LET(sg_a, B2, md_e, B3, sg_y, B4, ex_n, B5, IF(OR(sg_a <= 0, md_e <= 0, sg_y <= 0, ex_n < 1), "Invalid Material Law", LET(ep_e, sg_a / (md_e * 1000), ep_p, 0.002 * ((sg_a / sg_y) ^ ex_n), ROUND(ep_e + ep_p, 6))))Proof-Stress Anchor Check (sigma = sigma_02) & Secant Modulus Formula
Set Applied_Stress B2 = 300 MPa (= sigma_02): Plastic Strain E3 hits 0.002 exactly, Total Strain E4 hits 0.0035, and Secant Modulus E5 hits 85.71 GPa!
=IF(ISTEXT(E4), E4, ROUND((B2 / E4) / 1000, 2))Try the example
Use the original values or change them to test your understanding.