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Useful referenc material for Vlasov/restrained warping torsion theory

{{:wikitelaio2016:b16_chap7.pdf|P.C.J. Hoogenboom, Vlasov torsion theory}}

{{ :wikifemfuchde2019:warping_torsion.pdf |}}


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{{ :wikifemfuchde2019:i_section_vlasov_torsion_blade_model.pdf |}}

Various formulas

x+ flange bending
$$
V=\frac{dM_x}{dz}
$$

$$
T=h V=h\frac{dM_x}{dz}
$$

since $\frac{M_x}{\overline{EJ}_{xx}}=\frac{1}{\rho_x}=-\frac{d^2 v}{dz^2}$, we substitute in the equation above $M_x=-\overline{EJ}_{xx}\frac{d^2 v}{dz^2}$, thus obtaining

$$
T=-h \overline{EJ}_{xx}\frac{d^3 v}{dz^3}
$$

The $v$ transverse displacement may be determined based on the local twist angle $\psi$ as 

$$
v=\frac{h}{2} \psi
$$

and a torque to (third derivative of) twist angle may be finally determined as
$$
T=-\frac{h^2}{2}\overline{EJ}_{xx}\frac{d^3 \psi}{dz^3}=-\overline{EC_w}\frac{d^3 \psi}{dz^3}
$$
where the $\overline{EC_w}$ cross-sectional constant for warping has been defined for the I beam as 
$$
\overline{EC_w}=I\frac{h^2}{2}.
$$

Such $T$ torsional moment, which is transmitted based on the flange shear load under restrained warping condition, will be referred to in the following as $T_\mathrm{Vla}$, as opposed to its counterpart according to the de St. Venant torsion theory, i.e.
$$
T_\mathrm{dSV}=G K_t \frac{d \psi}{dz}.
$$
 

Characteristic length of the cross section with respect to the Vlasov (restrained warping) torsion theory.
$$
d=\sqrt{\frac{EC_w}{G K_t}}
$$

The cross-sectional constant for warping may then be evaluated as
$$
E C_w = d^2 G K_t
$$
where $G K_t$ is the torsional stiffness for the cross-section (material properties included) according to the free-warp, de St. Venant torsion theory.

Since the overall torsional moment is constant along the beam in the absence of distributed torsional actions, and it consists in the sums of the two $T_\mathrm{Vla}$ and $T_\mathrm{dSV}$ contributes, we have

$$
0=\frac{dT}{dz}=+\frac{dT_\mathrm{dSV}}{dz}+\frac{dT_\mathrm{Vla}}{dz}
=  
- E C_w \frac{d^4 \psi}{dz^4}
+ G K_t \frac{d^2 \psi}{dz^2}
$$
$$
0=- d^2 \frac{d^4 \psi}{dz^4} + \frac{d^2 \psi}{dz^2}
$$

which is a 4th-order differential equation in the $\psi$ unknown function, whose solutions take the general form

$\psi(z)=C_1 \sinh{\frac{z}{d}} + C_2 \cosh{\frac{z}{d}}+C_3 \frac{z}{d} +C_4$

In the theory of restrained torsion warping, an auxiliary, higher order resultant moment quantity named //bimoment// is introduced, that for the pedagogical I-section example is related to the flange bending moment by the identity

$$
B=M_{xx} \cdot h
$$

In general, we have

$$
B=-EC_w \frac{d^2 \psi}{dz^2};
$$

axial stresses along the cross section linearly scale with the bimoment quantity, if the material behaves elastically.

Warping related boundary conditions may be stated as follows:
  * free warping: $\frac{d^2 \psi}{dz^2}=0$, i.e. absence of bimoment, $B=0$;
  * no warping: $\frac{d \psi}{dz}=0$, i.e. absence of de St. Venant transmitted moment, $T_\mathrm{dSV}=0$;
Imposed rotations and imposed torsional moments complementary boundary conditions may be defined as usual.

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Maxima worksheet for evaluating the Vlasov characteristic length on the basis of FE simulations

{{ :wikifemfuchde2019:vlasov_torsion_cw_da_fem_nr_2019_bis.wxmx |}}

MSC.Marc/Mentat models used for evaluating the torsional stiffness; the //"shell element drilling mode factor"// has been raised from the 0.0001 default value to a unit value, to correct the bending/twisting behaviour of the plate elements at the profile angular rounding ($\tau_{12}$ at extremal layers was expected to be continuous along the wall).

{{ :wikifemfuchde2019:lesson_09_04_2019_models_for_restrained_warping_stiffening.zip |}}