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

Have considered the vibrational behavior of a discrete system. How does one use this for a continuous structure? First need the concept of..... Influence Coefficients

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:

Thus far, we have concentrated on the bending of shell beams. However, in the general case a beam is subjected to axial load. F · bending moments,M · shear forces,S torque(torsional moments)

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

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

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

Thus far have considered separately beam - takes bending loads column -takes axial loads Now combine the two and look at the beam-column (Note: same geometrical restrictions as on others

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

We have looked at basic in-plane loading. Lets now consider a second\building block of types of loading: basic torsion There are 3 basic types of behavior depending on the type of cross-section