What is St. Venant torsion and what assumptions are made?
A theory on how torsion in beams can be dealt with. It comes with assumptions:
Straight beams
Free & unrestrained twist
Exclusive twist around beam axis
Torsion and bending are decoupled
Cross - sectional shape is retained
Cross sections free of warping are assumed
What is the constitutive law for St. Venants torsion?
—> theta´=Mx/GIt
Give shear stress/ strain over a full circular cross section?
They are distributed linearly
What is a relevant learning in a full circular cross section for Leightweight engineering?
The middle of the cross section has a bad utilization (as with regular bending). It is more beneficial to use thin walled cross sections, they have a better utilization.
What assumption is permissable in thin walled closed cross sections?
Constant shear over the thickness - linear variations can be disregarded
Give the shear stress in a thin walled closed cross section
How can be dealt with an arbitrary thin walled closed cross-section in the context of St.Venant torsion?
Issue: Shear stress not constent with respect to the circumference
Solution: Switch to evaluation of shear flow —> shear flow constant with respect to the circumference
Give the shear flow for St. Venant torsion over an arbitrary closed cross section - how is this formula called?
Bredt´s 1st formula
Give the maximum shear stress for a thin walled arbitrary closed cross section in St. Venant torsion?
Where is it to be found?
At the thinnest point on the circumference
How can we calculate the Torsional (Polar) Moment of inertia?
Special case circle:
Why are open (thin walled) cross sections much weaker in regard to torsion than closed (thin walled) cross sefctions?
Instead of the cross section carrying a constant stress, the stress is linearly distributed across the thin cross section —> Utilization very bad.
Give the Ratio of shear stresses in St. Venant torsion in thin walled open and closed cross sections.
By what factor is the max shear stress increased when a cross section is opened?
—> increase by Factor 3*Rm/t
How can generally be dealt with multicellular thin walled cross sections in the context of St. Venant torsion?
Parallel connection of cells with asumed same theta is permissable —> Torsional moment can be split
What is understood as warping torsion?
Warping describes the behaviour of a cross section, where the individual sections will move out of plane under torsion —> The slices move out of their initial plane.
How do the normal stresses due to warping look like, how have the common Torsion hypothesis have to be adjusted to accommodate that?
—> Assumption of the flatness of cross sections is not permissable anymore
—> Flat cross sections can only be assumed in the individual segments of the cross section (e.g. web, flange)
How does warping torsion look like on an I-beam
Torsion can also be described by two Forces acting on the outside —> Flanges will bend accordingly
How do the shear stresses look like for primary and secondary torsion on an I-beam?
Which form will the warping torsion adapt to?
How does the flange behave under warping torsion?
Torsional moment Mx = Mxp + Mxs (Primary + Secondary)
—> Parabolic form
—> No the flange doesn´t warp. A force in the direction of the middle line in the flange will not lead to a torsional moment —> No warping torsion
Give at least 3 cross sections that do not exhibit warping torsion
Give at least 3 cross sections that do exhibit warping torsion
Give the differential equation of warping torsion (for a constant moment load m_x)
- remember constant moment load one differentiate below Moment Mx
How do Primary & secondary torsion behave over a beam, discuss especially in respect to the decaying factor (describe the decaying factor)
For discussion of decaying factor:
The decaying factor (lambda^2 = GIt / EI_omega = Torsional stiffness/ warping stiffness) guides, secondary, primary and warping moment. Thereby a
high decaying factor describes a small warping stiffness —> Small warping moment, sharp rise of secondary moment and a fast drop of primary moment
small decaying factor describes a high warping stiffness —> high warping moment, higher initial secondary moment but small spike, lower Primary moment.
What is described by omega_w and what is it used for?
It describes unit warping with respect to an arbitrary rotation axis D
It can be used to calculate normal stresses due to warping
What is the main issue with warping?
It leads to additional loads where the cross section as a whole is clamped (e.g. clamped edge)
What is omega_D and what is it used for
Omega_D is a unit warping w.r.t to an arbitrary rotation axis D vor a given cross section
It is used to determine displacements u (no need to know the formula):
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