Describe what Stability is in the context of mechanichs?
Stability refers to the state of the equilibrium.
—> Question to ask: Will a system go back to it´s initial state, after a small deflection: Yes-stable; won´t do anything-indifferent; no, will deflect further - unstable
Give the four Euler buckling cases and their critical loads
What is the basic strategy to solve in Stability theory?
2nd order theory. System is looked at in a state of small deflection
1st order - additional loads don´t show up
3rd order - for larger deflections, were linearity cannot be assumed anymore
What is understood as torsional buckling and which type of cross section shows torsional buckling behaviour?
Torsional buckling describes a reaction in the form of a twist of the beam due to compressive loads. It happens due to a torsional moment appearing in 2nd order evaluation.
Pure torsional buckling behaviour is only found in double symmetric cross sections (shear center = center of gravity)
Give the torsional buckling differential equation
Give the critical buckling load for torsional buckling for an I-beam under compressive load in fork restraints
What is understood as flexural torsional buckling and what is the driving cause?
Flexural torsional buckling is the answer of a system of combined flexural bending and torsion due to a compressive load.
The driving cause are the excentrisities from the shear center. They will (in case load is applied not in the shear center) inflict a torsional moment in case of flexural bending, and they will inflict a flexural bending moment in case of torsional deformation.
Which degrees of freedom are of relevance in flexural torsional buckling?
Flexural bending in y and z
& torsion around x
How many differential equations guide flexural torsional buckling and how are they coupled?
2 for Euler buckling + additional load due to theta
1 for torsional buckling + additional loads due to v and w
—> fully coupled if ez and ey are not equal 0
Which effect will a symmetry have on flexural torsional buckling?
It will decouple the Euler buckling in the direction of the symmetry axis
Which Ansatzes can be used to solve the Differential equation system in flexural torsional buckling for an arbitrary beam in fork restraints?
Why are they beneficial?
—> Beneficial because only even derrivatives in ODE System —> Sin function can be lost…
Give the buckling condition for flexural torsional buckling and discuss symmetries.
—> Cubic equation to solve
Describe lateral buckling
A slender beam that experiences bending around y, in z direction. At some point it will give way/ buckle by bending in y and twisting around x.
How are additional loads imposed on the beam for lateral buckling in 2nd order theory?
Biaxial Bending is introduced by torsion angle theta
Torsion is introduced by bending in y, around z (v, Torsional moment dependent on v´)
Give the differential equations for lateral buckling and how they are coupled?
It is to note, that only the second two equations are coupled. This implies two things:
Regular bending around y, in z (w) is uncoupled and can happen freely, always a stable equilibrium! (makes sense, otherwise, every bending beam would be a stability problem)
torsion and bending around z in y (v) are coupled —> Once the beam buckles and deflects to the side, both buckling modes will happen, flexural buckling to the side and torsion
Give the critical buckling Moment for a beam in fork restraints under edge moments
Don´t skip!
Discuss how the excentricity of the force F influences the stability of the following case
A lower excentricity (< 0, above center of gravity) leads to a high lever arm to introduce torsion.
A higher excentricity (> 0, below center of gravity) leads to a lever counteracting the torsion.
Draw a sketch for flexural torsional buckling, mark important points and give DOF
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