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 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:

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