gemseo_umdo / use_cases / beam_model / core

# 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: object

The 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 - $$\sigma_{\text{VM}} = \sqrt{\sigma^2 + 3\tau^2}$$

Parameters:
• n_y (int) –

The number of discretization points in the y-direction.

By default it is set to 3.

• n_z (int) –

The number of discretization points in the z-direction.

By default it is set to 3.