model module¶
The GEMSEO-free version of the model for the beam use case.
- class gemseo_umdo.use_cases.beam_model.core.model.BeamModel(n_y=3, n_z=3)[source]¶
Bases:
objectThe beam model.
We consider an horizontal beam with length \(L\), width \(b\) and height \(h\). This beam is hollow and made of a material with a Young’s modulus \(E\), a Poisson’s ratio \(\nu\) and a thickness \(t\). One of its ends is fixed at \(x=0\) (the “root”) while the other at \(x=L`(the "tip") is free. The :math:`y\)-axis is horizontal and perpendicular to the beam, the \(z\) is vertical and the center of the root is at the origin \((0, 0, 0)\).
A force \(\vec{F}\) of amplitude \(F\) is applied to the beam at \((L, dy, dz)\) with an angle \(\alpha\) w.r.t. \(-\vec{e}_y\) in the xz-plane and an angle \(\beta\) w.r.t. \(-\vec{e}_y\) in the yz-plane, where \(\vec{e}_y\) is the unit vector along the \(y\)-axis.
From these settings, the model computes the weight of the beam \(w=2 \rho L (b + h -2t)\) and several quantities on a regular \(yz\)-grid:
the strain energy vector \(\vec{U}=(U_x,U_y,U_z)\) at the tip,
the normal stress \(\sigma\) at the root,
the torsional stress \(\tau\) at the root,
the displacement \(\delta\) at the tip,
the von Mises stress \(\sigma_{\text{VM}}\) at the root.
The equations are:
Force components - \(F_x=F\sin(\alpha)\) - \(F_y=F\cos(\alpha)\sin(\beta)\) - \(F_z=F\cos(\alpha)\cos(\beta)\)
Inertia - \(I_x=(2tb^2h^2)/(b + h)\) - \(I_y=(bh^3-(b-2t)(h-2t)^3)/12\) - \(I_z=(hb^3-(h-2t)(b-2t)^3)/12\)
Strain energy - :math:`U_x = E^{-1} { frac{ F_x L }{ 2t (b+h-2t) }
zL (F_x dZ - F_z L/2) I_y^{-1} - yL (F_y L/2 - F_x dY) I_z^{-1} }`
\(U_y = E^{-1} \{ (F_y L^3/3 - F_x dY L^2/2)I_z^{-1} - zL \frac{ F_zdY-F_ydZ }{ 2 (1+\nu) I_x } \}\)
\(U_z = E^{-1} \{ (F_z L^3/3 - F_xdZ L^2/2) I_y^{-1} + yL \frac{ F_zdY-F_ydZ }{ 2 (1+\nu) I_x } \}\)
Displacements - \(\delta=\sqrt{U_x^2+U_y^2+U_z^2}\)
Torsional stress - \(\tau_x=(F_zdY-F_ydZ)/(2bht)\) - \(\tau_y= - (0.5|z|(b-t)+(b-t)^2(1-4y^2/(b-t)^2))F_y\text{sign}(z)/(8I_z)\) - \(\tau_z=(0.5|y|(h-t)+(h-t)^2(1-4z^2/(h-t)^2))F_z\text{sign}(y)/(8I_y)\) - \(\tau = \tau_x + \tau_y + \tau_z\)
Stress - \(\sigma = F_x/(2t(b+h-2t)) + y (F_xdY-F_yL)/I_z + z (F_xdZ-F_zL)/I_y\)
von Mises stress - \(\sigma_{\text{VM}} = \sqrt{\sigma^2 + 3\tau^2}\)