Elastic˗Plastic Stress˗Strain Model for Beams

Key Concept: Beams respond to a bending moment by rotation about the neutral axis (also the

centroidal axis). The extension or contraction of "fibers" about the neutral axis is assumed to be

linear. So the bending stress at the neutral axis is zero and increases linearly to a maximum at

the outer fiber of the beam. Once the bending stress reaches the yield stress for the material the bending stress remains constant at the yield stress, σ_y . See the figure below.

In a Nut Shell: The equation relating stress and strain for the linear model is as follows:

σ = E ε

ε = ˗ y / ρ where ρ = EI / M

σ = ˗ My/I

where σ is the bending stress (lb/in², mPa, N/mm²)

E is the modulus of elasticity (psi, N/mm²)

ε is the bending strain (μ in/in, microstrain)

y is the distance from the neutral axis to the fiber (in, mm)

ρ is the radius of curvature of the beam (in, mm)

I is the moment of inertia of the X-section about its neutral axis (in⁴, mm⁴)

M is the bending moment (ft lb, N m)

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