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### Stress-Strain Curve Ramberg-Osgood Equations and Calculator

**Structural Deflections, Stress, Strain Analysis Calculators**

**Stress-Strain Calculator and Plotted Curve Approximation using Ramberg-Osgood Equation:**

The Ramberg–Osgood equation is used to approximate the non linear relationship between stress and strain, the stress–strain curve in materials near their yield points. It is especially applicable to metals that harden with plastic deformation, showing a smooth elastic-plastic transition.

The stress-strain curve may be approximated using the Ramberg-Osgood equation, which calculates the total strain (elastic and plastic) as a function of stress:

Preview example: **Stress-Strain Curve Calculator Results and Plotted Stress-Strain Curve**

ε = σ / E + o_{s} ( σ / σ_{y} )^{1/n}

where σ is the value of stress, E is the elastic modulus of the material, σ_{y} is the tensile yield strength of the material, and n is the strain hardening exponent of the material which can be calculated based on the provided inputs. o_{s} is the % offset of the yield stress which is can be assumed to be 0.2% of 0.002..

Based on the specified material properties, the value of the strain hardening exponent, n, is:

n = log (σ_{2} / σ_{1} ) / log (ε_{2} /ε_{1} ) strain hardening exponent

or

n = log(S_{tu} / S_{ty} ) / log(ε_{f} /ε_{p} )

ε_{p} = .002

Strain Values at Yield and Ultimate Strength Points

The strain value associated with the yield strength is:

ε_{yield} = σ_{y} / E + o_{s}

Note that a plastic strain of o_{s} = 0.002 is assumed. This is consistent with the 0.2% offset method.

The strain value associated with the ultimate strength is:

ε_{ult} = S_{tu} / E + ε_{f}

where e_{f} is the plastic strain at failure and is simply the percent elongation expressed in decimal form:

e_{f} = eL / 100%

References:

- Ramberg, W., & Osgood, W. R. (1943). Description of stress–strain curves by three parameters. Technical Note No. 902, National Advisory Committee For Aeronautics, Washington DC.
- Description of Stress Strain Curves by Three Parameters, Walter Ramburg and William R. Osgood. Techncail Note 902: NASA
- Hollomon, J. R. (1945). "Tensile Deformation". Transaction of AIME. 162: 268–277.
- Gadamchetty, Geethanjali; Pandey, Abhijeet; Gawture, Majnoo (2016-01-05). "On Practical Implementation of the Ramberg-Osgood Model for FE Simulation". SAE International Journal of Materials and Manufacturing. 9 (1): 200–205. doi:10.4271/2015-01-9086. ISSN 1946-3987.
- Hill, H. N. (1944). Determination of stress-strain relations from "offset" yield strength values. National Advisory Committee for Aeronautics. OCLC 647978489.

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