STABILITY OF STRUCTURE
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 Pcr.
- 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.
- 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?
- 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.
Post-buckling behavior that is asymmetric about load axis.
- 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