Define plastic moment capacity of beam

The plastic moment capacity ($M_p$) of a beam is the maximum bending moment that the cross-section can carry when the whole section has yielded, that is, when every fibre on the tension side and on the compression side has reached the yield stress $\sigma_y$.

• Up to the yield moment $M_y = \sigma_y Z$, stress varies linearly and only the extreme fibres yield.
• As the moment increases, yielding spreads inward from the outer fibres.
• At $M_p$ the section is fully plastic. The neutral axis then divides the area into two equal halves (equal-area axis), and the tension and compression forces form a couple:

$$M_p = \sigma_y\,S$$

where $S = A_t\bar y_t + A_c\bar y_c$ is the plastic section modulus, the sum of the first moments of the tension and compression areas about the equal-area axis.

At $M_p$ a plastic hinge forms and the section rotates at constant moment. The ratio $M_p/M_y = S/Z$ is the shape factor: 1.5 for a rectangle, about 1.7 for a circle, and about 1.12 to 1.15 for I-sections.