21. UNBRACED ROLLED STEEL BEAMS

In a previous chapter we have examined the behavior of laterally supported beams. Under those conditions, the potential modes of failures were either the formation of a plastic hinge (if the section is compact), or local buckling of the flange or the web (partially compact section). Rarely are the compression flange of beams entirely free of all restraint, and in general there are two types of lateral supports:

1. Continuous lateral support by embedment of the compression flange in a concrete slab.
2. Lateral support at intervals through cross beams, cross frames, ties, or struts. Now that the beam is not laterally supported, we ought to consider a third potential mode of failure, lateral torsional buckling.

This chunk introduces the concept of unbraced rolled steel beams. In essence, when a beam is laterally supported (like being held up by a concrete slab or other supports), it behaves in a predictable manner, having certain failure modes such as the formation of a plastic hinge or local buckling. However, when a beam does not have these lateral supports, it can fail in a different way called lateral torsional buckling. The text specifically mentions that there are generally two types of lateral supports which help prevent such failures, making it essential to understand the nature of the beams’ support systems.

Imagine a tightrope walker (the beam) on a thin wire (the support). If the wire is taut and well anchored, the tightrope walker can move confidently without wobbling (lateral support). But if the wire isn't secured or has too much slack, the walker may lose balance and fall (lateral torsional buckling). Therefore, just like good support is essential for the tightrope walker, proper lateral supports are crucial for beams.

Whereas it is beyond the scope of this course to derive the governing differential equation for flexural torsional buckling, we shall review some related topics in order to understand the AISC equations later on. There are two types of torsion:

1. Saint-Venant’s torsion: or pure torsion (torsion is constant throughout the length) where it is assumed that the cross-sectional plane prior to the application of torsion remains plane, and only rotation occurs.
2. Warping torsion: out of plane effects arise when the flanges are laterally displaced during twisting. Compression flange will bend in one direction laterally while its tension flange will bend in another. In this case part of the torque is resisted by bending and the rest by Saint-Venant’s torsion.

This chunk explains two types of torsion that affect beams. The first, Saint-Venant's torsion, occurs when a beam twists along its length without changing its cross-section's shape. It can be thought of as a uniform twisting motion. The second type, warping torsion, occurs when the beam twists and causes its flanges to bend out of shape, leading to more complex behavior. Understanding these two types of torsion is critical as they influence how beams carry loads and respond to stresses.

Think of twisting a towel (the beam). When you twist it evenly (Saint-Venant's torsion), the towel stays relatively flat; however, if you twist it unevenly, causing one end to deform (warping torsion), you can see how the towel behaves differently, showcasing how each type of torsion impacts the overall structure.

The flexural efficiency of the member increases when X decreases and/or X increases. The governing moments are as follows:

1. For very short beams, the moment is given by the plastic hinge formula.
2. For short inelastic lateral torsional buckling, there is a more complex equation that accounts for multiple factors.
3. For long elastic lateral torsional buckling, the critical moment can be determined with specialized relationships that involve stiffness and geometry.

This part outlines the American Institute of Steel Construction (AISC) equations that help in calculating the moments for unbraced beams. It states that the efficiency of a beam's ability to resist bending increases with certain adjustments to the parameters involved (X values). It also provides equations that determine the moment based on various conditions such as length and the type of buckling occurring. This shows how different scenarios impact the structural stability of the beam.

Consider a gym barbell representing the beam. When the weight is evenly distributed and within limits (efficient X values), the barbell remains stable and easy to lift. However, if the weight is too centralized (representing a failure in structural design), the barbell may bend improperly or even snap. Just like understanding where and how much weight to put on a barbell leads to a safer workout, understanding these equations ensures beams are safely designed for their intended loads.

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