Limiting plastic moment is 1.5 times elastic moment

Consider a rectangular section of width $b$ and depth $d$ made of an elastic-perfectly plastic material with yield stress $\sigma_y$.

   Elastic limit        Fully plastic
   +--+                  +------+
   | /|                  |      |   sigma_y
 --|/-|-- NA          ---|------|--- NA
   |\ |                  |      |   sigma_y
   +--+                  +------+

Maximum elastic moment

Stress is linear and just reaches $\sigma_y$ at the extreme fibres:

$$M_y = \sigma_y Z = \sigma_y\cdot\frac{bd^2}{6}$$

Fully plastic moment

Every fibre has reached $\sigma_y$. The compression and tension zones are the two halves of the section, each of area $bd/2$, and each resultant acts at the centroid of its half, $d/4$ from the neutral axis:

$$T = C = \sigma_y\frac{bd}{2}$$

The lever arm between the two forces is $d/2$:

$$M_p = T\times\frac{d}{2} = \sigma_y\frac{bd}{2}\times\frac{d}{2} = \sigma_y\frac{bd^2}{4}$$

(The plastic section modulus is $S = bd^2/4$.)

Ratio

$$\frac{M_p}{M_y} = \frac{\sigma_y\,bd^2/4}{\sigma_y\,bd^2/6} = \frac{6}{4} = 1.5$$

Hence $M_p = 1.5\,M_y$. The shape factor of a rectangular section is 1.5.