Question 14.3.2: Sketch the moment-curvature curve for a rectangular section.
Sketch the moment-curvature curve for a rectangular section.
Refer again to Fig. 14.3-3 in Example 14.3-1. For M not exceeding the yield moment M_{y}, the curvature is:
\qquad \kappa=\frac{M}{EI} (14.3-7)
which follows directly from the well-known relations \sigma /y=M/I=E \ \kappa.
For M > My, the flexural stiffness of the section is that of the elastic core only;
Eqn 14.3-7 is therefore modified as
\qquad \kappa=\frac{M_{e}}{EI_{e}} (14.3-8)
where Me and Ie are respectively the resistance moment and the second moment of
area of the elastic core. Referring to Fig. 14.3-3,
\qquad I_{e}=bh^{3}\alpha ^{3}/12 and M_{e} = \alpha ^{2}\sigma _{y}bh^{2}/6 as derived in Example 14.3-1.
Substituting into Eqn 14.3-8
\qquad \kappa =\frac{1}{\alpha }\left(\frac{2\sigma _{y}}{Eh} \right) =\frac{1}{\alpha }\kappa _{y} (14.3-9)
where κy is the value of κ for α equal to unity and is hence the curvature for M = My.
From Eqn 14.3-6
\qquad M=\left(1-\alpha ^2/3\right) M_p (14.3-10)
Eliminating α from Eqns 14.3-9 and 10, we have
\qquad \frac{\kappa }{\kappa _{y}}=\frac{1}{\sqrt{3\left(1-M/M_{p}\right) } } (14.3-11)
Eqn 14.3-11 is plotted in Fig. 14. 3-4, which also shows an approximate curve for a typical British niversal beam section or American wide-flange section.
I In Fig. 14.3-4, the ordinate values 1/1.15 and 1/1.5 are respectively the reciprocals of the shape factors for the I section and the rectangular section; thus M p/1.15 = My for an I section, and M p/1.5 = My for a rectangular section. The moment-curvature relations are, as expected, linear up to M = My , after which curvatures increase rapidly with increase in moments and becomes very large as M approaches Mp.
