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

Two forms of Completeness Theorem Let Γ be a set of wffs. The following parts are equivalent. If Γ |= A then Γ ` A If Γ is consistent, then Γ is satisfiable

The need for a richer language In P, it is not possible to express assertions about elements of a structure. First Order Logic is a considerably richer logic than propositional logic, but yet enjoys many nice mathematical properties

Syntax Formation Rules for P The The Axiomatic Structure of P Theorems and Derived Rules

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

Previously saw (in Unit 19)that a multi degree-of-freedom system has the same basic form of the governing equation as a single degree-of-freedom system The difference is that it is a matrix equation

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