Shape factor, plastic hinge properties, T-beam

Shape factor

The shape factor is $S = \dfrac{M_p}{M_y} = \dfrac{Z_p}{Z_e}$, the ratio of the plastic moment of the section to the yield moment.

Properties of a plastic hinge

1. It forms at a section where the moment reaches $M_p$ and the whole depth of the section has yielded.
2. It rotates at a constant moment $M_p$ (like a hinge that carries a moment) and offers no extra resistance to rotation.
3. It forms at the points of maximum moment: fixed supports, joints, under concentrated loads, and at the section of zero shear under a UDL.
4. Rotation takes place only in the direction of the applied moment; on unloading or reversal the hinge locks and behaves elastically.
5. It extends over a finite length of the member (the plastic hinge length), but is treated as a point in analysis.
6. Each hinge reduces the degree of indeterminacy by one; when enough hinges form (degree + 1), the structure becomes a mechanism and collapses.

Shape factor of the T-section

Assumed dimensions: flange 2000 mm × 150 mm; web 175 mm wide, 950 mm deep below the flange; total depth $D = 150 + 950 = 1100$ mm. (If 950 mm is the overall depth, the web is 800 mm and the shape factor becomes 1.84.)

Area: $A = 2000\times150 + 950\times175 = 300000 + 166250 = 466250$ mm².

1. Elastic neutral axis (distance from the top):

$$\bar y = \frac{300000\times75 + 166250\times(150+475)}{466250} = 271.11\ \text{mm}$$

2. Moment of inertia about the elastic neutral axis:

$$I = \frac{2000\times150^3}{12} + 300000(271.11-75)^2 + \frac{175\times950^3}{12} + 166250(625-271.11)^2 = 4.542\times10^{10}\ \text{mm}^4$$

3. Elastic section modulus (extreme fibre in the web at $1100 - 271.11 = 828.89$ mm):

$$Z_e = \frac{4.542\times10^{10}}{828.89} = 5.480\times10^{7}\ \text{mm}^3$$

4. Plastic neutral axis divides the area into two equal halves, $A/2 = 233125$ mm². The flange area is 300000 mm² $> 233125$, so the axis lies in the flange, at depth

$$y_p = \frac{233125}{2000} = 116.56\ \text{mm below the top}$$

5. Plastic section modulus (moment of each half-area about the plastic axis):

$$Z_p = 2000\times116.56\times\frac{116.56}{2} + 2000\times33.44\times\frac{33.44}{2} + 166250\times\left(33.44 + 475\right) = 9.923\times10^{7}\ \text{mm}^3$$

6. Shape factor:

$$S = \frac{Z_p}{Z_e} = \frac{9.923\times10^{7}}{5.480\times10^{7}} = 1.81$$

Answer: $S \approx 1.81$ (elastic $Z_e = 5.48\times10^7$ mm³, plastic $Z_p = 9.92\times10^7$ mm³, plastic neutral axis in the flange 116.6 mm from the top).