Change in geometry of a structure or structural component under compression – resulting in loss of ability to resist loading is defined as *instability. *Instability can lead to catastrophic failure that must be accounted in design. Instability is a strength-related limit state.

**Why did we define instability instead of stability? **

Stability is not easy to define.

- Every structure is in equilibrium – static or dynamic. If it is not in equilibrium, the body will be in motion or a
*mechanism*. - A
*mechanism*cannot resist loads and is of no use to the civil engineer. - Stability qualifies the state of equilibrium of a structure. Whether it is in
*stable*or*unstable*equilibrium. - Structure is in stable equilibrium when small perturbations do not cause large movements like a mechanism. Structure vibrates about it equilibrium position.
- Structure is in unstable equilibrium when small perturbations produce large movements – and the structure never returns to its original equilibrium position.
- Structure is in neutral equilibrium when we cant decide whether it is in stable or unstable equilibrium. Small perturbation cause large movements – but the structure can be brought back to its original equilibrium position with no work.
- Thus, stability talks about the equilibrium state of the structure.
- The definition of stability had nothing to do with a change in the geometry of the structure under compression – seems strange!

**BUCKLING Vs. STABILITY**

- Change in geometry of structure under compression – that results in its ability to resist loads – called
*instability*. - Not true – this is called
*buckling.* *Buckling*is a phenomenon that can occur for structures under compressive loads.- The structure deforms and is in stable equilibrium in state-1.
- As the load increases, the structure suddenly changes to deformation state-2 at some critical load P
_{cr}. - The structure buckles from state-1 to state-2, where state-2 is orthogonal (has nothing to do, or independent) with state-1.

- What has buckling to do with stability?
- The question is – Is the equilibrium in state-2 stable or unstable?
- Usually, state-2 after buckling is either neutral or unstable equilibrium

**TYPES OF INSTABILITY**

Structure subjected to compressive forces can undergo:

- Buckling – bifurcation of equilibrium from deformation state-1 to state-2.
- Bifurcation buckling occurs for columns, beams, and symmetric frames under gravity loads only

- Failure due to instability of equilibrium state-1 due to large deformations or material inelasticity
- Elastic instability occurs for beam-columns, and frames subjected to gravity and lateral loads.
- Inelastic instability can occur for all members and the frame.

**BIFURCATION BUCKLING**

- Member or structure subjected to loads. As the load is increased, it reaches a
*critical*value where:- The deformation changes suddenly from state-1 to state-2.
- And, the 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
- Is the post-buckling path stable or unstable?

**SYMMETRIC BIFURCATION**

- Post-buckling load-deform. paths are
*symmetric*about load axis.- If the load capacity increases after buckling then
__stable__symmetric bifurcation. - If the load capacity decreases after buckling then
__unstable__symmetric bifurcation.

**ASYMMETRIC BIFURCATION**

Post-buckling behavior that is asymmetric about load axis.

**INSTABILITY FAILURE**

- There is no bifurcation of the load-deformation path. The deformation stays in state-1 throughout
- The structure stiffness decreases as the loads are increased. The change is stiffness is due to large deformations and / or material inelasticity.
- The structure stiffness decreases to zero and becomes negative.
- The load capacity is reached when the stiffness becomes zero.
- Neutral equilibrium when stiffness becomes zero and unstable equilibrium when stiffness is negative.
- Structural stability failure – when stiffness becomes negative.

**Failure of Beam-Columns**

**Snap-Through Buckling**

**Shell Buckling failure – very sensitive to imperfections**