Reference Stress and Proportional Limit

Reference Stress Derivation

1. Reference Stress at 0.7E_c Secant Modulus (F_0.7)

Starting Point: Ramberg-Osgood Equation

The Ramberg-Osgood stress-strain relationship is:

ε = f/E + (3/7) × (F_0.7/E) × (f/F_0.7)ⁿ

Secant Modulus Definition

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

E_secant = f / ε

Finding F_0.7

By definition, F_0.7 is the stress where the secant modulus equals 0.7E_c:

E_secant = 0.7E_c

Substituting the Ramberg-Osgood equation at f = F_0.7:

ε = F_0.7/E_c + (3/7) × (F_0.7/E_c) × (F_0.7/F_0.7)ⁿ

ε = F_0.7/E_c + (3/7) × (F_0.7/E_c)

ε = F_0.7/E_c × (1 + 3/7)

ε = F_0.7/E_c × (10/7)

Therefore, the secant modulus is:

E_secant = F_0.7 / ε = F_0.7 / (F_0.7/E_c × 10/7) = E_c × (7/10) = 0.7E_c ✓

The 3/7 Factor

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

3/7 = 1/0.7 - 1

Solving for F_0.7 from F_cy and n

The handbook gives F_cy, not F_0.7. The two are linked by the definition of the yield stress: F_cy is the stress at a plastic strain of 0.2 percent (the 0.002 offset). Setting f = F_cy in the plastic term of the Ramberg-Osgood equation:

0.002 = (3/7) × (F_0.7/E_c) × (F_cy/F_0.7)ⁿ

F_0.7^n-1 = [3 / (7 × 0.002)] × F_cyⁿ / E_c

F_0.7 = F_cy × (k × F_cy / E_c)^1/(n-1)

The Coefficient 214.3

The constant k is nothing more than the 3/7 secant factor divided by the yield offset:

k = 3 / (7 × 0.002) = (3/7) × 500 = 214.3

(500 is 1/0.002.) It carries no empirical content and does not depend on the structural element; the plasticity reduction factors of NACA TN 3781 are applied later, in the buckling tools that use F_0.7 and n.

This gives the final equation:

F_0.7 = F_cy × (214.3 × F_cy / E_c)^1/(n-1)

2. Shear Reference Stress at 0.7G Secant Modulus (F_0.7s)

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) × (F_0.7s/G) × (τ/F_0.7s)^n_s

where:

• γ = shear strain
• τ = shear stress
• G = shear modulus
• F_sy = shear yield stress
• n_s = shear Ramberg-Osgood shape factor

Shear Inputs: F_sy and n_s

The shear section of the form takes the shear modulus G and the tensile ultimate, tensile yield and shear ultimate strengths (F_tu, F_ty, F_su), all of which the material dropdown fills from the handbook. The handbook gives neither a shear yield stress nor a shear shape factor, so the tool derives both.

Shear yield stress. The shear yield-to-ultimate ratio is taken equal to the tensile yield-to-ultimate ratio:

F_sy = F_su × (F_ty / F_tu)

Shear shape factor. If the shear stress-strain curve is the von Mises image of the compression curve, the stresses and plastic strains map as

τ = σ / √3     and     γ_p = √3 × ε_p

Applying that map to the plastic term of the compression curve, (σ/F_0.7)ⁿ, gives (√3 τ / F_0.7)ⁿ = (τ / F_0.7s)ⁿ, with F_0.7s = F_0.7/√3. The map rescales the stress axis and the strain axis but leaves the exponent unchanged, so

n_s = n

The tool uses this value when the shear shape factor is left blank. A value typed in the n_s field, for example from a measured shear stress-strain curve, overrides it.

The Coefficient 123.7

The same algebra applies in shear once the offset that defines F_sy is fixed. Under the von Mises map above, the 0.2 percent normal-strain offset becomes a shear-strain offset of 0.002 × √3 = 0.00346, so

k_s = 3 / (7 × 0.002 × √3) = (3/7) × 288.7 ≈ 123.7

Older references round 288.7 to 289 and use 123.9; the tool uses the exact 123.7. The difference is 0.15 percent in k_s before the 1/(n_s-1) power, far below the precision of the inputs.

This gives the final equation:

F_0.7s = F_sy × (123.7 × F_sy / G)^1/(n_s-1)

Summary

Both equations follow from the Ramberg-Osgood stress-strain relationship, the definition of the secant modulus at 0.7 times the elastic modulus, and the offset that defines the yield stress. The numerical coefficients (214.3 and 123.7) combine:

• The factor 3/7 from the 0.7 secant modulus criterion
• The reciprocal of the yield offset: 1/0.002 = 500 in compression, and 1/(0.002 × √3) = 288.7 in shear, the von Mises image of the same offset

The reference stress F_0.7 and the nondimensional buckling curves indexed by F_0.7 and n are due to Cozzone and Melcon (1946); the three-parameter curve is Ramberg and Osgood's (NACA TN 902); the shape factor n can be found from two offset yield strengths by Hill's method (NACA TN 927); the plasticity reduction factors that use these quantities are tabulated by Gerard and Becker (NACA TN 3781).