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wikifemfuchde2019:lez_2019-04-08

Useful referenc material for Vlasov/restrained warping torsion theory

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.

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

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/lez_2019-04-08.txt · Ultima modifica: 2019/04/09 13:32 da ebertocchi