Stability of Structures

Stability of structures

General concepts of stability

Collapse: In a structure, we can have different types of collapse:

• Material level: It can be distinguished on the basis of the behaviour of the structure.
  • Brittle behaviour ⇒ local failure
  • Ductile behaviour ⇒ global failure

In many cases, the response is given by a time-independent behaviour. But failure can happen also with damage accumulation induced by time-variable actions.

Geometrical effects can also be responsible for the collapse of a structure. In fact, there can be failure conditions with stress and/or damage conditions far from reaching limit conditions.

Consider the case: a beam with an imperfection that causes an initial deformation ε_o. Applying a compressive force P, the presence of imperfection has an effect of destabilization and induces a bending moment that cannot be resisted ⇒ Unstable condition. Conversely, in the case of tensile action, we have a stabilizing moment tending to straight geometry ⇒ Stable configuration.

Based on experience, in many situations, when axial compressions reach a critical value, any small disturbance can induce unstable configurations ⇒ Stability problems, usually treated with dynamic formulations, but in many cases, they can be treated also by an equivalent static approach.

The classical theory of structure is generally based on the small displacements hypothesis, which stands upon two independent assumptions:

1. It is possible to rely on the (linearized) small strain definition.
2. Displacements are small enough to allow the equilibrium to be imposed in the reference (undeformed) configuration.

Simple statics examples:

• _{Rigid system ➔ Lagrangian coordinate}
  1. We try to place equilibrium in the undeformed configuration. In this way, where P*d can state write moment equilibrium.
  2. We can go through large displacements and rotations description, and we have angle, the transverse force POR. We get the exact solution to the problem, writing the equilibrium in the deformed configuration, and solving it.

Introducing the assumption of small displacements is small enough in a way that we can enter into the 1^st order approximation, the non-linear operators will still depend on the Lagrangian coordinate, 1^st order approximation of Taylor expansion.

We can compare in a graph the results:

Example: Axial force only

Trivial equation

OBS: approximation

Example: Geometric imperfections

Differentiation point possible bifurcation paths. We note in both cases a divergent behaviour for p = 1. It can be interpreted as a complete loss of stiffness of the system corresponding to a bifurcation point in the perfect problem.

Example: Transversal force only

1. Undeformed configuration: kθ = kFe ⇒ αp = θ
2. Deformed configuration: kθ = αPlcosθ ⇒ αpe =_Pθ_Eθ = E_π/₂cosθθ = _π/₂ if φ -> ∞

By combining the results we can say kθ = Fecosθ + Pe₂θ R̄ = 1. If p > 0 ⇒ R̄ > By introducing the compressive force, we are decreasing the stiffness.

Stability

An equilibrium configuration is defined stable if and only if small perturbations produce small oscillations around the undisturbed configuration. Conversely, if any perturbations entered in the system can produce a divergent behaviour further response which is larger than the initial perturbation, the system is called unstable.

We can investigate stability using a dynamic approach - we have to introduce inertia force due to mass in the equilibrium equation. Note: we can decide to use a distributed mass or a lumped mass, at this stage is completely arbitrary. For θ θ (PL - k) θ + l² θ ●● = 0* inertia force is ⊥ to beam axis Rθ = PLemθ + ⨍ẍdx = 0 θ̇̈ + α² θ = 0* If α² > 0 ⇒ p < 1, λ = ±iα θ(t) = e^αt · Asin(ωt) + Bcos(ωt)+ I.C. ➢ θ(0) = θ₀ B = θ₀ θ̇(0) = θ̇₀ A = θ̇₀/ω ⇒ θ(t) = θ₀ sin(ωt)+θ̇₀/ω cos(ωt).