Reference Stress and Proportional Limit

The following analysis calculates the compressive reference stresses at secant Moduli .7 Ec and .7 G. The proportional limits for compression and shear are also calculated.

  • Used to calculate inelastic buckling corrections from nondimensional figures

  • Proportional limit is used to differentiate elastic and inelastic buckling so that corrections can be made

Material Properties Input

Unit System: /

Compression Material Allowables




Shear Material Allowables (Optional)




Reference Stress Derivation
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Reference Stress Derivation

1. Reference Stress at 0.7Ec Secant Modulus (F0.7)

Starting Point: Ramberg-Osgood Equation

The Ramberg-Osgood stress-strain relationship is:

ε = f/E + (3/7) × (F0.7/E) × (f/F0.7)n

Secant Modulus Definition

The secant modulus at any stress level f is defined as:

Esecant = f / ε

Finding F0.7

By definition, F0.7 is the stress where the secant modulus equals 0.7Ec:

Esecant = 0.7Ec

Substituting the Ramberg-Osgood equation at f = F0.7:

ε = F0.7/Ec + (3/7) × (F0.7/Ec) × (F0.7/F0.7)n

ε = F0.7/Ec + (3/7) × (F0.7/Ec)

ε = F0.7/Ec × (1 + 3/7)

ε = F0.7/Ec × (10/7)

Therefore, the secant modulus is:

Esecant = F0.7 / ε = F0.7 / (F0.7/Ec × 10/7) = Ec × (7/10) = 0.7Ec ✓

The 3/7 Factor

The factor 3/7 comes from the secant modulus criterion:

3/7 = 1/0.7 - 1

General Solution for F0.7

For an arbitrary stress f with secant modulus 0.7Ec, solving the Ramberg-Osgood equation gives:

F0.7 = Fcy × (k × Fcy / Ec)1/(n-1)

where k is a plasticity correction factor that depends on the structural element and loading condition.

The Coefficient 214.3

For column buckling with plasticity effects, NACA TN 3781 provides the scaling factor 500. The coefficient in our equation is:

214.3 = (3/7) × 500 ≈ 214.3

This gives the final equation:

F0.7 = Fcy × (214.3 × Fcy / Ec)1/(n-1)

2. Shear Reference Stress at 0.7G Secant Modulus (F0.7s)

Analogous Derivation for Shear

The derivation for shear follows the same principles as the compression case, using the shear Ramberg-Osgood equation:

γ = τ/G + (3/7) × (F0.7s/G) × (τ/F0.7s)ns

where:

  • γ = shear strain
  • τ = shear stress
  • G = shear modulus
  • ns = shear Ramberg-Osgood shape factor

The Coefficient 123.9

For shear buckling with plasticity effects, NACA TN 3781 provides a different plasticity correction factor of 289. The coefficient in the shear equation is:

123.9 = (3/7) × 289 ≈ 123.9

This gives the final equation:

F0.7s = Fsy × (123.9 × Fsy / G)1/(ns-1)

Summary

Both equations follow from the Ramberg-Osgood stress-strain relationship and the definition of secant modulus at 0.7 times the elastic modulus. The numerical coefficients (214.3 and 123.9) incorporate:

  • The factor 3/7 from the 0.7 secant modulus criterion
  • Plasticity correction factors (500 for compression, 289 for shear) from NACA TN 3781


References

Ramberg, W., & Osgood, W. R. (1943). Description of Stress-Strain Curves by Three Parameters. NACA Technical Note 902. National Bureau of Standards, Washington, D.C.

Gerard, G., & Becker, H. (1957). Handbook of Structural Stability, Part I: Buckling of Flat Plates. NACA Technical Note 3781. New York University.

MIL-HDBK-5J (31 January 2003). Department of Defense Handbook: Metallic Materials and Elements for Aerospace Vehicle Structures. U.S. Department of Defense.

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