Useful referenc material for Vlasov/restrained warping torsion theory
P.C.J. Hoogenboom, Vlasov torsion theory
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:
Imposed rotations and imposed torsional moments complementary boundary conditions may be defined as usual.
Maxima worksheet for evaluating the Vlasov characteristic length on the basis of FE simulations
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).
lesson_09_04_2019_models_for_restrained_warping_stiffening.zip