Moment-curvature¶

A non-linear analysis of the section: instead of answering "how much steel", it answers how the section behaves as it is loaded, up to failure.

It is what shows the effective stiffness and the ductility — two things that ultimate limit state design does not reveal.

The real stress-strain relationship¶

Design uses the equivalent rectangular block, a convenient simplification for integrating by hand. The non-linear analysis uses the NBR 6118 parabola-rectangle relationship, which is the real curve:

\[ \sigma_c = \begin{cases} 0.85\,f_{cd}\left[1 - \left(1 - \dfrac{\varepsilon_c}{\varepsilon_{c2}}\right)^{n}\right] & 0 \le \varepsilon_c \le \varepsilon_{c2} \\[10pt] 0.85\,f_{cd} & \varepsilon_{c2} < \varepsilon_c \le \varepsilon_{cu} \end{cases} \]

The parabolic branch rises up to \(\varepsilon_{c2}\); from there on the stress stays constant up to the ultimate strain.

The procedure¶

For each curvature value \(\chi\), the section is in equilibrium when the resultant compression in the concrete and the tension in the steel cancel out. The unknown is the depth of the neutral axis.

The program finds it by bisection — scipy.optimize.bisect —, searching between acceptable physical bounds for the position at which equilibrium closes. With the neutral axis known, the corresponding resisting moment comes from integrating the stresses.

Repeating this for a sequence of curvatures — 100 steps by default — builds the \(M\)–\(\chi\) diagram.

Why bisection, and not Newton

Bisection is slower than Newton, but it does not depend on the derivative and does not diverge. The concrete constitutive relationship has a kink at \(\varepsilon_{c2}\), where the parabola meets the plateau — exactly the kind of derivative discontinuity that makes Newton take a wrong step.

With a hundred points per diagram, the speed difference is irrelevant.

What the diagram shows¶

The effective stiffness. The initial slope of the diagram is the stiffness \(EI\) of the uncracked section; after cracking it drops, and it is this reduced stiffness — not that of the gross section — that governs real displacements.

The ductility. The length of the plateau before failure tells you how much rotation the section can withstand. It is what allows (or not) the moment redistribution that the \(\beta\) coefficient in design assumes — see bending.

The failure mode. An under-reinforced section yields the steel before the concrete crushes, and the diagram has a long plateau. An over-reinforced one fails in the concrete, and the diagram ends abruptly — brittle failure, without warning.

Limits of validity¶

Range of application

It is a section analysis, not a member analysis. It does not provide deflections: for that, the curvature would have to be integrated along the span, with the moment diagram.
It does not consider tension stiffening — the contribution of the concrete in tension between cracks. This underestimates the stiffness in the cracked phase.
It does not consider creep or shrinkage, which in service substantially reduce the stiffness over time.
It uses the design values of the strengths. For a diagram that represents the expected behaviour — rather than the design one —, mean values would be needed.
The diagram is monotonic: it does not represent loading and unloading cycles, nor behaviour under repeated actions.