===== Thin walled profile in torsion =====

Please note that the techniques employed rely on the //monoclinic// nature of the material with respect to the cross-sectional plane, i.e. such a plane is required to be a symmetry plane for the material.

Open thin-walled rectangular cross section profile (a longitudinal cut is performed at the lateral wall center line, whose kerf (width) is negligible), 80x40mm at the midsurface,  6mm wall thickness. 
Element size of ~10mm. 
The severed lateral wall may be easily deactivated to reproduce the simplified ladder-frame chassis cross section. 

{{:wikifemfuchde2018:analisi_sezione_a_torsione.png?nolink|}}
{{:wikifemfuchde2018:analisi_sezione_a_torsione.pdf |sorgente ipe}}



At the A cross section, a skew-symmetry material continuity constraint is applied.

The kinematic constraint at the B point (sphere in cylinder joint) is a positioning constraint, along with the $w_A=0$ axial constraint in A. 

A twist per unit length equaling 0.001 radiant/mm is imposed to the profile, i.e. a $\psi_B=0.001 \cdot l$ rotation is imposed at each end, where $l$ is the  $z$ axial coordinate of the endpoints, being z=0 at the skew-symmetry plane.

The reaction moment associated with the constraint will determine the applied torque $T$.

----

{{ :wikifemfuchde2020:profile_in_torsion_a2020_v000.mfd | profile geometry alone}}

{{ :wikifemfuchde2020:profile_in_torsion_a2020_v005.mfd | model at the end of the 2020-04-02 lesson}}

{{ :wikifemfuchde2020:profile_in_torsion_a2020_v008.mfd | model at the end of the 2020-04-06 lesson}}

{{ :wikifemfuchde2020:profile_in_torsion_a2020_v010.mfd |}}

{{ :wikifemfuchde2020:profile_in_torsion_a2020_v011.mfd | model at the end of the 2020-04-09 lesson}}

{{ :wikifemfuchde2020:profile_in_torsion_a2020_v012.mfd | model at the end of the 2020-04-12 lesson}}, and
{{ :wikifemfuchde2020:profile_in_torsion_a2020_v012m.mfd | the same model, with increasingly refined in-cross-section discretizations}}. Please observe (try it!) that no change in results is obtained by further splitting the elements in the axial directions; such a result is related to the fact that stresses are observed to be constant along the same direction.

{{ :wikifemfuchde2020:torsional_stiffness_evaluation_ffcd2020_v000.xls | LibreOffice/MSExcel spreadsheet for the collection and the normalization of the results}}, to be filled as homework.

At the end of the 2020-04-16 lesson: 
{{ :wikifemfuchde2020:profile_in_torsion_a2020_v013.mfd |otw model}},
{{ :wikifemfuchde2020:profile_in_torsion_a2020_v013c.mfd |ctw model}}.


spreadsheet with increased profile length
{{ :wikifemfuchde2020:torsional_stiffness_evaluation_ffcd2020_v001.xls |}}, and the associated 
{{ :wikifemfuchde2020:profile_in_torsion_a2020_v014.mfd |otw model}}
and
{{ :wikifemfuchde2020:profile_in_torsion_a2020_v014c.mfd |ctw model}}.

===== Thin walled profile subject to pure shear =====

Various incremental steps in the definition of the shear loadcase
{{ :wikifemfuchde2020:profile_in_torsion_a2020_v015.mfd |v015}}
{{ :wikifemfuchde2020:profile_in_torsion_a2020_v016.mfd |v016}}
{{ :wikifemfuchde2020:profile_in_torsion_a2020_v017.mfd |v017}}
{{ :wikifemfuchde2020:profile_in_torsion_a2020_v018.mfd |v018}}, and the
{{ :wikifemfuchde2020:torsional_stiffness_evaluation_ffcd2020_v001.xls |updated spreadsheet}}.
{{ :wikifemfuchde2020:profile_in_torsion_a2020_v018.mfd |v018}} is the final model for the response evaluation of the cross section to skew-symmetric internal action components ($M_t$,$Q_x$,$Q_y$).

Please note that, for the application of the presented procedure, the global (O,x) (O,y) axes **must** coincide with the principal axes of (elastic) inertia for the cross section.

===== Thin walled profile in bending =====

For reference/review purposes: {{ :wikifemfuchde2020:last_year_notes_on_bending_and_shear.pdf |}}.
Please read attentively the first paragraph devoted to axial load and bending; if something is not clear, please ask the professor. 
Such a treatise should constitute just a slight extension of what you already encountered in previous //Strength of Materials// courses.

{{ :wikifemfuchde2020:profile_in_bending_a2020_v001.mfd |}}

{{ :wikifemfuchde2020:profile_in_bending_a2020_v002.mfd |}}

{{ :wikifemfuchde2020:profile_in_bending_a2020_v003.mfd |}}

{{ :wikifemfuchde2020:symmprops.ods |}}

{{ :wikifemfuchde2020:symmprops_v003.ods |}}

Solution:
{{ :wikifemfuchde2020:profile_in_bending_a2020_v003a.mfd |FE model with unit curvature loadcases}},
{{ :wikifemfuchde2020:symmprops_v003a.ods |Spreadsheet with the collected results}},
{{ :wikifemfuchde2020:profile_in_bending_a2020_v003a.mfd |FE model re-oriented}} s.t. the global (O,x) (O,y) axes coincide with the principal axes of (elastic) inertia for the cross section, in order to proceed with the pure shear analysis.

Auxiliary {{ :wikifemfuchde2020:eig2x2.wxmx |maxima worksheet}} for literally solving the 2x2 eigenvalue problem.