Discrete vs continuum structure; stability

Discrete and continuum structures

• Discrete structure: made of separate, clearly identifiable members joined at joints (nodes), so it can be analysed member by member with a finite number of unknowns. Examples: trusses, beams and frames.
• Continuum structure: the material is continuous and cannot be divided into members; stresses vary continuously through the body. Examples: dams, slabs, shells, retaining walls, a deep beam. They are analysed by elasticity theory or finite elements.

Stability

A structure is stable if it can carry any system of loads without rigid-body motion or collapse, and the supports prevent movement of the whole body. It is unstable if the reactions are fewer than needed, or if they are arranged so that they cannot resist some loading:

• Fewer than 3 independent reactions in 2D (e.g. a beam on two rollers).
• All reactions parallel (two rollers on horizontal ground cannot resist horizontal load).
• All reactions concurrent at one point (the moment about that point cannot be resisted).
• Internal arrangement forming a mechanism (e.g. a four-bar panel of a truss without a diagonal).

Determinacy and indeterminacy

• Beams and frames (2D): degree of indeterminacy D = r − (3 + c), where r is the number of reaction components and c the number of extra equations from internal hinges (each hinge gives one). D = 0 determinate, D > 0 indeterminate, D < 0 unstable.
• Trusses: externally, D_e = r − 3. Internally, D_i = m − (2j − 3). Total: m + r − 2j (m members, j joints, r reactions).

Statically determinate structures can be solved by equilibrium alone and do not develop stresses from support settlement or temperature change; indeterminate structures need compatibility conditions but are stiffer and have lower moments.