21.3.2. Governing Moments

1. L < L : "very short" Plastic hinge M = M = Z F (21.5) n p x y

In cases where the span length (L) is less than the plastic length (L_p), we refer to it as a 'very short' plastic hinge. This implies that the moment capacity (M) at the hinge is equal to the plastic moment ( M_p) which can be expressed through the section modulus (Z) and the yield strength (F_y) of the material. The equation tells us how much moment the section can resist without undergoing plastic deformation.

Imagine a metal rod that can flex a little like a bending tree branch. If you push down on the end of the rod, it bends, but if you push too hard, it bends permanently, similar to a tree branch breaking. The 'very short' plastic hinge shows us the point right before it breaks permanently.

2. L < L < L : "short" inelastic lateral torsional buckling M = C M (M M ) b (cid:0) p M (21.6) n b p p r p

In cases where the length of the beam is between the plastic length (L_p) and the lateral torsional buckling length (L_r), we deal with short beams that might buckle under lateral torsional stresses. This formula shows how the moment (M) can be expressed in terms of coefficients that depend on the specific geometry and loading. The 'C' factor adjusts the moment to account for imperfections in the buckling behavior.

Consider a soda can. When it's pressed uniformly, it resists the force well. But if it's bent just slightly, the moment can cause a different kind of failure along the sides, just like a beam buckling when loads exceed its capacity.

C = 1.75 + 1.05 1 + 0.3 L / L (21.7) b p where M_1 is the smaller and M_2 is the larger end moment in the unbraced segment.

This equation helps us calculate the coefficient 'C' used in the moment formula for short inelastic lateral torsional buckling. By plugging in the ratios of the lengths of the unbraced segment, we can understand how the varying moments (M_1 and M_2) affect the relationship and strength of the beam under load.

Think of balancing multiple groceries on a long stick. If you only put light items on one end and heavy items on the other, the stick will bend towards the heavy side, just as different moments affect the configuration of a beam under load.

3. L < L "long" elastic lateral torsional buckling, and the critical moment is the same as in Eq.21.2 M = C C I + EI GJ C M (21.9) cr b w y y b r p

For beams with lengths longer than the lateral torsional buckling length (L_r), we enter the regime of elastic lateral torsional buckling. In this case, the critical moment is calculated differently and factors in both the moment of inertia (I) and the axial rigidity (E and GJ). Here, the beam’s behavior under load changes and can lead to significant buckling effects; hence we need to consider these variables in our equations.

Imagine a long metal ruler being pushed from the ends. When you push gently, it bends slightly without damage (elastic behavior). But if you push too hard, it bends considerably, just like a beam when it experiences elastic lateral torsional buckling when stressed beyond its limits.

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