wikifemfuchde2019:lez_2019-03-26
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wikifemfuchde2019:lez_2019-03-26 [2019/03/27 09:49] – ebertocchi | wikifemfuchde2019:lez_2019-03-26 [2019/03/27 10:38] (versione attuale) – ebertocchi | ||
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+ | Curiosità: {{ : | ||
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+ | {{ : | ||
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+ | ---- | ||
+ | **Prandtl stress function** $\phi$ | ||
+ | |||
+ | shear stresses: $\tau_{zx}=\frac{\partial \phi}{\partial y}$ , $\tau_{yz}=-\frac{\partial \phi}{\partial x}$ | ||
+ | |||
+ | shear stresses are hence defined as the 90° clockwise rotated $\phi$ | ||
+ | |||
+ | |||
+ | equilibrium equation $\frac{\partial \tau_{zx}}{\partial x} + \frac{\partial \tau_{yz}}{\partial y}=0$ satisfied due to Schwarz. | ||
+ | |||
+ | Compatibility relationship, | ||
+ | |||
+ | Here, $\nabla^2\phi=\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2}$. | ||
+ | |||
+ | Origin of the compatibility equations: the twist rate defines the derivatives | ||
+ | $\frac{\partial u}{\partial z}=-\beta y$ and | ||
+ | $\frac{\partial v}{\partial z}=+\beta x$; the two terms | ||
+ | $\frac{\partial w}{\partial x}$ and $\frac{\partial w}{\partial y}$ complete the $\gamma_{zx}$ and the $\gamma_{yz}$ shear strain component definition by introducing the $w$ warping displacement. | ||
+ | Those shear strains are related to the shear stresses by the constitutive equations $G\gamma_{zx}=\tau_{zx}$ and $G\gamma_{yz}=\tau_{yz}$. | ||
+ | |||
+ | Let's consider a closed path along the section consisting of four segments defined as | ||
+ | {{tablelayout? | ||
+ | ^ from: ^ to: ^ $\frac{d x}{d s}=\frac{d s}{d x}$ ^ $\frac{d y}{d s}=\frac{d s}{d y}$ ^ | ||
+ | | $x , | ||
+ | | $x+dx, | ||
+ | | $x+dx, | ||
+ | | $x , | ||
+ | |||
+ | Then, the circuital integral along such a loop | ||
+ | $$ | ||
+ | \oint \frac{\partial w}{\partial s} ds = 0 | ||
+ | $$ | ||
+ | must be zero to enforce a single valued $w$ displacement at the $x,y$ point, thus avoiding a [[https:// | ||
+ | |||
+ | {{: | ||
+ | |||
+ | Then we have | ||
+ | $$ | ||
+ | 0= | ||
+ | \oint \left( | ||
+ | \frac{\partial w}{\partial x}\frac{d x}{d s}+ | ||
+ | \frac{\partial w}{\partial y}\frac{d y}{d s} | ||
+ | \right) ds, | ||
+ | $$ | ||
+ | whose warping slope components may be expressed based on the twist rate and the shear stresses, thus obtaining | ||
+ | $$ | ||
+ | 0= | ||
+ | \oint \left( | ||
+ | \left(\frac{\tau_{zx}}{G}+\beta y\right)\frac{d x}{d s}+ | ||
+ | \left(\frac{\tau_{yz}}{G}-\beta x\right)\frac{d y}{d s} | ||
+ | \right) ds. | ||
+ | $$ | ||
+ | By accumulating the contributions on the four segments that constitute the loop we obtain | ||
+ | $$ | ||
+ | \left(-\frac{1}{G}\frac{\partial\tau_{zx}}{\partial y}dy-\beta dy\right)dx | ||
+ | + | ||
+ | \left(+\frac{1}{G}\frac{\partial\tau_{yz}}{\partial x}dx-\beta dx\right)dy | ||
+ | =0 | ||
+ | $$ | ||
+ | and | ||
+ | $$ | ||
+ | +\frac{\partial\tau_{zx}}{\partial y} | ||
+ | -\frac{\partial\tau_{yz}}{\partial x} | ||
+ | =-2 G \beta. | ||
+ | $$ | ||
+ | By substituting the shear stress component definition based on $\phi$, the compatibility equation is finally obtained. | ||
+ | |||
+ | boundary conditions: $\phi=c_i$, where $c_i$ is a constant on the $i$-th boundary; such condition, along with the way the $\tau_{zx}, | ||
+ | |||
+ | Torque: $T=2\iint_A \phi dA$ |