Ultimate and Yield Plastic Bending Allowables Using Cozzone Stresses

Cozzone Method Equation Derivation

Background

The Cozzone method provides a practical approach for calculating plastic bending allowables based on the material's stress-strain curve. This method accounts for the redistribution of stress across a section as the outer fibers yield and enter the plastic range.

1. Elongation Strain (ε_u')

The elongation strain represents the plastic component of the total ultimate strain:

ε_u' = ε_u - F_tu/E

Where:

• ε_u = total ultimate strain (elastic + plastic)
• F_tu/E = elastic component of strain
• ε_u' = plastic component (elongation)

2. Material Shape Factor in Plastic Range (m)

The parameter m characterizes the shape of the stress-strain curve in the plastic region using the Ramberg-Osgood formulation:

m = log(ε_u' / 0.002) / log(F_tu / F_ty)

This relates the plastic strain at ultimate (ε_u') to the 0.2% offset yield criterion and the ratio of ultimate to yield strength.

3. Ultimate Intercept Stress (f_ou)

The intercept stress is derived by integrating the stress distribution across a beam section in bending, assuming:

• Plane sections remain plane (Bernoulli-Euler assumption)
• Material follows Ramberg-Osgood stress-strain relationship
• Maximum fiber stress equals F_tu

Derivation Steps:

1. The strain distribution across the section is linear: ε = (y/c) × ε_max
2. The stress at any fiber is obtained from the Ramberg-Osgood equation
3. The moment is found by integrating: M = ∫ σ × y × dA
4. The same moment is then assigned to a trapezoidal stress block with f_m at the extreme fiber and an intercept f_o at the neutral axis. For a rectangular section the two moments are equal when f_o = (6/ε_m²) ∫₀^ε_m σ ε dε − 2 f_m; this defines f_o.
5. The trapezoidal block is then applied to any other section through its shape factor k, giving F_b = f_m + f_o × (k - 1).

The intercept stress formula accounts for the nonlinear stress distribution:

f_ou = ((6/ε_u²) × [(1/3) × (F_tu/E)² + ε_u' × ((m+1)/(m+2)) × (F_tu/E) + (m/(2m+1)) × (ε_u')²] - 2) × F_tu

The bracket is the integral ∫ σ ε dε / F_tu evaluated with the strain written as its elastic part σ/E plus its plastic part ε_u' (σ/F_tu)^m. Multiplying out the strain and its differential gives three products:

• (1/3) × (F_tu/E)² - elastic strain × elastic strain increment
• ε_u' × ((m+1)/(m+2)) × (F_tu/E) - the two cross products of the elastic and plastic parts
• (m/(2m+1)) × (ε_u')² - plastic strain × plastic strain increment

Every fiber contributes to all three; the terms are not regions of the section.

4. Yield Intercept Stress (f_oy)

Similarly, the yield intercept stress uses the yield properties:

ε_y = 0.002 + F_ty/E

f_oy = ((6/ε_y²) × [(1/3) × (F_ty/E)² + (ε_y - F_ty/E) × ((m+1)/(m+2)) × (F_ty/E) + (m/(2m+1)) × (ε_y - F_ty/E)²] - 2) × F_ty

5. Section Shape Factor (k)

The shape factor k accounts for different cross-sectional geometries:

• k = 1.000 - Two-noded beam (pure bending, linear stress distribution)
• k = 1.273 - Thin round tube
• k = 1.333 - Hour-glass section
• k = 1.500 - Rectangular section (fully plastic)
• k = 1.698 - Round section (solid circular)
• k = 2.000 - Diamond section

6. Plastic Bending Allowables

The final bending allowables combine the maximum fiber stress with the intercept stress contribution:

F_bu = f_mu + f_ou × (k - 1) (ultimate)

F_by = f_my + f_oy × (k - 1) (yield)

Where:

• f_mu = F_tu (maximum fiber stress at ultimate)
• f_my = F_ty (maximum fiber stress at yield)
• (k - 1) factor adjusts for section geometry

Physical Interpretation

The term (k - 1) represents the benefit gained from stress redistribution as the outer fibers yield. A higher k value indicates greater capacity for plastic stress redistribution, resulting in higher bending strength compared to elastic theory.

Ramberg-Osgood Shape Factors: m vs n

Overview

The Ramberg-Osgood equation describes the stress-strain curve with an elastic term and a power-law plastic term. The exponent is the shape factor. This tool calls its exponent m; the handbooks and the Reference Stress tool call it n. They are the same parameter, fitted to different points of the curve.

The Ramberg-Osgood Equation

Written through the 0.2 percent offset yield stress:

ε = σ/E + 0.002 × (σ/F_ty)ⁿ

The plastic strain is 0.002 at σ = F_ty by construction, and the exponent n sets how fast it grows above yield.

Physical Meaning of n

• Higher n → sharper knee; above yield the stress rises little with strain (closer to elastic-perfectly-plastic)
• Lower n → gradual yielding, a rounded curve with more strain hardening
• n → ∞ → elastic-perfectly-plastic

How This Tool Finds m

Requiring the same curve to pass through the ultimate point (F_tu, ε_u) gives the plastic strain at ultimate ε_u' = ε_u − F_tu/E = 0.002 × (F_tu/F_ty)ⁿ. Solving for the exponent:

m = log(ε_u' / 0.002) / log(F_tu / F_ty)

So m is the Ramberg-Osgood exponent of the curve that passes through both the yield point and the ultimate point. It is not a different quantity and it is not the inverse of n. With the tool's default inputs (E = 10.5 Msi, F_tu = 65 ksi, F_ty = 55 ksi, ε_u = 10 %) it comes out at m = 23.0.

Why m Differs from a Handbook n

Handbook values of n are usually fitted near the knee of the curve, for example from the secant yield stresses F_0.7 and F_0.85 (Hill's method, used in MIL-HDBK-5J). The value of m here is fitted over the whole range from yield to ultimate. A real curve is not an exact power law, so the two fits give different numbers for the same material:

• m matches the curve at ultimate, which is what the plastic bending integral needs, since the extreme fiber is taken to F_tu
• a handbook n matches the curve near yield and is the right choice for buckling and proportional-limit work
• neither is wrong; they describe different parts of the same curve with one parameter

Requirements on the Inputs

• F_tu must exceed F_ty, or the denominator log(F_tu/F_ty) is zero
• ε_u' must exceed 0.002, or the numerator is zero, negative or undefined. The tool reports both conditions instead of calculating.
• ε_u should be the strain at F_tu. The handbook elongation is the strain at fracture; for a material that necks it is larger, which raises m and the intercept stress and leans unconservative.

Use in the Cozzone Method

m enters the intercept stress through the weights (m+1)/(m+2) and m/(2m+1), which come from integrating σ ε dε across the section with the power-law plastic strain. See the Cozzone Method Equation Derivation.