Question 6.11: Determine expressions for the yield moment, the plastic mome......
Determine expressions for the yield moment, the plastic moment, and the shape factor for a beam with rectangular cross section (Fig. 1a).
Because the rectangular cross section has two axes of symmetry, the neutral axis passes through the centroid for all values of applied moment. The stress state corresponding to the yield moment is shown in Fig. 1b, and the stress state for the plastic-moment case is shown in Fig. 1c.
Yield Moment: From Eq. 6.47,
M_Y = \frac{σ_YI} {c} (6.47)
M_Y = \frac{σ_YI} {c} = \frac{σ_Y \left(\frac{bh^3} {12 }\right)} {\left(\frac{h}{2}\right)} (1)
or
M_Y = \frac{1} {6} σ_Ybh^2 (2)
Plastic Moment: From Eq. 6.49,
M_P = \frac{σ_YA}{2} (d_C + d_T) (6.49)
M_P = \frac{σ_YA}{2} (d_C + d_T) = \frac{σ_Y (bh)} {2} \left(\frac{h}{4} + \frac{h} {4}\right) (3)
or
M_P = \frac{1} {4} σ_Ybh² (4)
Shape Factor: From Eq. 6.50,
f = \frac{M_P} {M_Y} = \frac{Z} {S} (6.50)
f = \frac{M_P} {M_Y} = \frac{6} {4} = 1.5 (5)
Thus, for a rectangular beam, the plastic moment is 50% higher than the yield moment.