Have dealt, thus far, with perfect columns, loading eccentricities, and beam-columns. There are, however, many more issues in buckling/(static) structural instability, most of which will try to touch on a) Buckling versus Fracture Have looked at columns that are long enough such that they buckle

Thus far have considered only static response. However, things also move, this includes structures Can actually identify three \categories\ of response A.(Quasi)-Static [quasi because the load must first be applied

Now consider the case of compressive loads and the instability they can cause. Consider only static instabilities (static loading as opposed to dynamic loading [ e.g., flutter) From Unified, defined instability via a system becomes unstable when a negative stiffness overcomes

For a number of cross-sections we cannot find stress functions. However, we can resort to an analogy introduced by Prandtl(1903) Consider a membrane under pressure p, Membrane\. structure whose thickness is small compared to surface

Earlier looked at Simple Beam Theory in which one considers a beam in the x-z plane with the beam along the x-direction and the load in the z-direction Figure 14.1 Representation of Simple Beam Now look at a more general case Loading can be in any direction

Before we look specifically at thin-walled sections, let us consider the general case (i.e, thick-Walled) Hollow thick-walled sections Figure 12.1 Representation of a general thick-walled cross-section 中=c2 on one boundary φ=c1 on one boundary This has more than one boundary(multiply-connected do=0 on each boundary

Return to the simplest system the single spring-mass This is a one degree-of-freedom system with the governing equation

The logical extension of discrete mass systems is one of an infinite number of masses. In the limit, this is a continuous system. Take the generalized beam-column as a generic representation:

Summarizing what we've looked at in elasticity, we have: 15 equations in 15 unknowns 3 equilibrium -6 strains -6 strain-displacement -3 displacements

As we've previously noted, we may often want to describe a structure in various axis systems. This involves... Transformations (Axis, Deflection, Stress, Strain, Elasticity Tensors) e.g., loading axes material principal axis Figure 7.1 Unidirectional Composite with Fibers at an Angle