Einführung & System of rigid bars

We are in search of that point in the system (mostly a force) where the system shifts from a stable equilibrium (the System will always strive back to its initial state) to an unstable equilibrium (the system will not go back to it´s inital state, rather it is experience a change/ deformation to an energetically more favorable state)

Second order theory - We look at an equilibrium (of forces) in the deformed state.

—> This will give us a hint whether or not the system will go back to the initial state or not

A System made out of idealy stiff bars (—> this means no energy is stored in the bar)

However Energy may be stored in springs that are part of the system.

(Systems of rigid bars without springs only make little sense)

• INTERNAL The Energy stored in the system, usually in the form of 1/2 * stiffness * coordinate^2

• EXTERNAL The potential lost by the load mostly in the form of -F * coordinate

We call for an equilibrium of the system energy —> We call for the first derrivative being 0 (delta pi/ delta coordinate = 0)

After determaning the equilibrium, we look at the second derrivative to decided whether or not the equilibrium is stable

• + —> minimum —> stable

• 0 —> indifferent

• - —> maximum —> unstable (every other position provides a lower system energy —> System will strive away from the current equilibrium)

Lineraization is mostly ok, because the point of interest is the critical load, behaviour after that is not really of interest.

An Imperfection is understood as a stress free initial deformation from the undeformed initial state.

There is no critical load any more at which the state of equilibrium changes from stable to unstable. The system will deform with the load, however deflections might not be linear.

We will be able observe large deflections.

It will act like an initial deflection --> drastically reducing the load carrying abilities of the system.

The Dischinger Factor connects the first order and second order deflection in the case of a load horizontal to the critical load. (it is usually abbreviated with alpha)

This is necessary because in first order theory the critical load will not have an influence on the deflection, only second order theory reveals the deflection that is added by the critical force.

They are good to understand the basic concepts of stability, imperfections and ‘horizontal loads‘

A system that experiences unstabilities with a rising load, however will fall into a second state of stability.

We derrive the determinant of the Hessian matrix of the energy function. Then we use the known Analysis criteria (> 0 minimum —> stable)

The equilibrium works by describing a force/ moment or Energy equilibrium in second order state. To solve the trivial solution of the displacements being zero has to be avoided. Therefore it is solved for the determinant of the dependency matrix equling zero.