Approximation Methods

• Cross-Sections remain planar

• Normal Hypothesis (all cross-sections remain perpendicular to neutral line)

• Thickness h doesn´t change

• No stresses in the z - direction —> Plane state of stress

• Small deformations

• Linear Elastic behaviour

BC that are concerned with deflections and rotations.

BC regarding Forces or Moments

An energetic interpretation of the buckling with respect to internal energy stored in the beam and external energy provided.

Note - external Energy (-), interpretation: Potential lost.

Displacement w is approximated with W, a form function * constant factor

Approximation is inserted into full potential pi. pi is differentiated with respect to c and called for to be zero

—> This is the case at exactly that deformation (C describes the “height” of the form function and therby the displacement), where the state is indifferent —> tipping point from stable to unstable.

Non trivial solution:

• Relativly easy to implement

• Good results for good form functions

• Only one form function —> heavily reliant on a good choice

It advances on the Rayleigh Quotient that approximates deflection with an (arbitrary) test function by utilizing a series expansion as an approximation

Again it is called for an indifferent state in regard to the Coefficients

Constituting an eigenvalue problem (allows for different buckling modes)

• Multiple shape functions can increase the accuracy

• Less reliant on a single choice of shape function

• Higher computational power required

As with the Rayleigh Quotient and the subsequent Ritz Method, the deformation is approximated by shape functions.

Therby the Object is discretized into elements. For every element node and deformation of interest (e.g. deflection and rotation) a shape function is introduced)

Then for every Element a Potential equation is set up by the means of the Ritz method.

During Assembly, a larger set of equation is build up. Therby the equations are coupled via the continuity requirements of the object (One node should have the same deflection and rotation in the neighbouring elements)