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. For a rectangular section, this leads to the bending stress equation: 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 three terms in brackets represent:

• (1/3) × (F_tu/E)² - Contribution from elastic region
• ε_u' × ((m+1)/(m+2)) × (F_tu/E) - Transition region contribution
• (m/(2m+1)) × (ε_u')² - Fully plastic region contribution

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 nonlinear stress-strain behavior in the plastic range. There are two common parameterizations using different shape factors:

Original Form (using n)

The classic Ramberg-Osgood equation is:

ε/ε₀ = σ/σ₀ + (σ/σ₀)ⁿ

Where:

• n = strain hardening exponent (material constant)
• σ₀ = reference stress (often F_ty, the 0.2% offset yield stress)
• ε₀ = reference strain (often 0.002 for 0.2% offset)
• n typically ranges from ~5 to 30+ depending on material

Physical Meaning of n:

• Higher n → sharper yield transition, material behaves more like elastic-perfectly-plastic
• Lower n → gradual yielding, more rounded stress-strain curve
• n → ∞ → approaches perfectly elastic-plastic (no strain hardening)

Cozzone Form (using m)

The Cozzone method uses a related parameter m that characterizes the plastic region specifically:

m = log(ε′_u / 0.002) / log(F_tu / F_ty)

Where:

• ε′_u = plastic strain at ultimate (total strain minus elastic strain)
• 0.002 = reference plastic strain (0.2%)
• F_tu/F_ty = ratio of ultimate to yield strength
• m is derived directly from material test data

Relationship Between m and n

The parameters are related but not identical:

• n is a general material constant valid across the entire plastic range
• m is specifically derived for the region between yield and ultimate stress
• Both describe "roundness" of the stress-strain curve in plastic range
• Approximate relationship: m ≈ 1/(n-1) for many materials
• They're inversely related - high n corresponds to low m

Typical Values

Aluminum Alloys:

• n ≈ 15-25
• m ≈ 20-25

Steel:

• n ≈ 5-10 (sharper yield transition)
• m ≈ higher values

Copper:

• n ≈ 20-30
• m ≈ varies widely

Use in Cozzone Method

The parameter m appears in the intercept stress calculations with terms like:

• (m+1)/(m+2) - weight factor for transition region
• m/(2m+1) - weight factor for fully plastic region

These factors come from integrating the stress distribution across the bending section using the Ramberg-Osgood stress-strain relationship.

Key Advantage

The Cozzone method is convenient because m can be directly calculated from standard material properties (F_tu, F_ty, ε_u, E) without needing to fit the entire stress-strain curve!