wikifemfuchde2019:lez_2019-04-08
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wikifemfuchde2019:lez_2019-04-08 [2019/04/08 11:55] – ebertocchi | wikifemfuchde2019:lez_2019-04-08 [2019/04/09 11:32] (versione attuale) – ebertocchi | ||
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+ | ---- | ||
+ | Useful referenc material for Vlasov/ | ||
+ | {{: | ||
+ | |||
+ | {{ : | ||
+ | |||
+ | |||
+ | ---- | ||
+ | |||
+ | {{ : | ||
+ | |||
+ | 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 $T_\mathrm{Vla}$, | ||
+ | $$ | ||
+ | T_\mathrm{dSV}=G K_t \frac{d \psi}{dz}. | ||
+ | $$ | ||
+ | |||
+ | |||
+ | 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, | ||
+ | |||
+ | $$ | ||
+ | 0=\frac{dT}{dz}=+\frac{dT_\mathrm{dSV}}{dz}+\frac{dT_\mathrm{Vla}}{dz} | ||
+ | = | ||
+ | - E C_w \frac{d^4 \psi}{dz^4} | ||
+ | + G K_t \frac{d^2 \psi}{dz^2} | ||
+ | $$ | ||
+ | $$ | ||
+ | 0=- d^2 \frac{d^4 \psi}{dz^4} + \frac{d^2 \psi}{dz^2} | ||
+ | $$ | ||
+ | |||
+ | 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 // | ||
+ | |||
+ | $$ | ||
+ | B=M_{xx} \cdot h | ||
+ | $$ | ||
+ | |||
+ | 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: $\frac{d^2 \psi}{dz^2}=0$, | ||
+ | * no warping: $\frac{d \psi}{dz}=0$, | ||
+ | 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/ | ||
+ | |||
+ | {{ : |