# Stability of Structure

Any kind of alteration in the geometry of a structure or change in form of structural component under compression –leading to the loss of capacity to resist loading is known as *instability. *Instability could result to catastrophic breakdown which must be seriously considered in design.

## Why do we define instability instead of stability?

Stability is a very comprehensive and often an ambiguous term to define.

- All structure are in the state of equilibrium – be it dynamic or static. In case it is not in equilibrium, the subject would be in motion.
- A
*mechanism*will not avoid the loads and hence is of no utilization the civil engineer. - Stability validates the equilibrium state of a structure. Irrespective of if it is in
*stable*or*unstable*equilibrium. - The structure is in stable equilibrium state when minute perturbations don’t result in big movements like a mechanism. Structure quiver about its equilibrium position.
- The structure is said to be in the state of unstable equilibrium when minute perturbations lead to big movements – and the structure never retains its initial equilibrium position.
- The structure is in unbiased equilibrium when we cannot ascertain if it is in stable or unstable equilibrium state. Small perturbation lead to big movements – yet the structure is possible to be brought back to its initial equilibrium position with no action.
- Hence, stability revolves around the equilibrium state of the structure.
- The meaning of stability has nothing to do with a alteration in the geometry of the configuration under compression.

## Stability vs Buckling

- Alteration in the geometry of configuration when exposed to compression- that leads to capability to resist the loads is understood as instability which is not true. It is buckling. Buckling refers to the phenomenon which can happen for configurations under the massive compressive loads.
- The structure configuration deforms and is in the state equilibrium in state -1.
- As the weight augments, the structure abruptly alters to deformation state -2 at critical load P
_{cr}.

- Structures buckle from the state -1 to state -2, in which case state -2 is orthogonal which is independent with state -1.

**Relation between buckling and stability**

Is the equilibrium state achieved in state-2 stable or unstable? Ans: Normally, state-2 after buckling is impartial or unstable equilibrium.

## Varieties of instability

Structure exposed to the compressive forces could endure:

- Buckling – bifurcation of equilibrium from deformation state-1 to state-2.
- Bifurcation buckling happens for columns, beams, and symmetric frames under gravity loads.
- Failure because of instability of equilibrium state-1 because of big deformations or material inelasticity
- Elastic instability happens for beam-columns, and frames exposed to gravity and lateral loads.
- Inelastic instability could hapen for all members and the frame.

**Bifurcation Buckling**

- Member or structure exposed to loads. As the load is augmented, it achieves a
*critical*value where: - The deformation alteration abruptly from state-1 to state-2.
- Equilibrium load-deformation path bifurcates.
- Critical buckling load when the load-deformation path bifurcates
- Primary load-deformation path before buckling
- Secondary load-deformation path post buckling
- Post-buckling path is either stable or unstable?

**Symmetric Bifurcation**

- In case the load capacity amplifies after buckling then the condition is known as stable symmetric bifurcation.
- In case the load capacity diminishes after buckling then it is known as unstable symmetric bifurcation.

**Asymmetric bifurcation**

Post buckling nature that is not symmetric about load axis.

**Instability failure**

- There is not any bifurcation of the load-deformation path. The deformation stays put in state-1 throughout
- The structure rigidity diminishes as the loads are amplified. The alteration in stiffness is because of big deformations and / or substance inelasticity.
- The structure’s rigidity diminishes to zero and becomes negative.
- The load capacity is acquired when the rigidity becomes zero.
- Neutral equilibrium when rigidity becomes nil and unstable equilibrium when rigidity is in negative state.
- Structural stability failure – when stiffness becomes negative.