Differenze

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--- wikifemfuchde2019:lez_2019-04-08 [2019/04/08 11:22] – ebertocchi
+++ wikifemfuchde2019:lez_2019-04-08 [2019/04/09 11:32] (versione attuale) – ebertocchi
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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 |}}
+----
+{{ :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
+$$
+=\frac{dT}{dz}=+\frac{dT_\mathrm{dSV}}{dz}+\frac{dT_\mathrm{Vla}}{dz}
+=
+- E C_w \frac{d^4 \psi}{dz^4}
++ G K_t \frac{d^2 \psi}{dz^2}
+$$
+$$
+=- d^2 \frac{d^4 \psi}{dz^4} + \frac{d^2 \psi}{dz^2}
+$$
+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
+{{ :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 |}}