What are general assumptions of the Euler-Bernoulli beam?
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
What are understood as geometric Boundary conditions?
BC that are concerned with deflections and rotations.
What are understood as dynamic Boundary conditions?
BC regarding Forces or Moments
What is the underlying methodology for approximation methods?
An energetic interpretation of the buckling with respect to internal energy stored in the beam and external energy provided.
Give the elastic potential for an Euler Bernoulli Beam in the deformed state
Note - external Energy (-), interpretation: Potential lost.
What is the underlying idea of the Rayleigh quotient?
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:
What are advantages and disadvantages of the Rayleigh quotient?
Relativly easy to implement
Good results for good form functions
Only one form function —> heavily reliant on a good choice
What is the underlying idea of the Ritz method?
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)
What are advantages and disadvantages of the Ritz method?
Multiple shape functions can increase the accuracy
Less reliant on a single choice of shape function
Higher computational power required
What is the basic Ansatz of the Finite Element Method
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)
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