I am going to analyze a rectangular beam under bending moment in AceFEM. To prescribe the bending I would like to prescribe the rotation angle of the section at the point $z=L_z$ (see the AceFEM file below). In addition, I would like to constrain this section to remain plane after bending. Theoretically, I know how to do it but in AceFEM I need help.
$\textbf{In theory}$:
Supposing that the prescribed rotation angle of the section at $z=L_z$ is $\theta_s$, the following multi-point constraint imposes the plane section remain plane hypothesis on this section, i.e.,
$\tan(\theta_s)=\frac{z_0-z_i}{x_0-x_i}$, $\qquad$ for each node on this section.
where $x_0$ and $z_0$ are the bottom nodes of this section.
The above constraint can be easily imposed by writing a quasi-potential that makes $\tan(\theta_s)-\frac{z_0-z_i}{x_0-x_i}$ equal to zero. In other words,
$\Pi = \int_s \lambda (\tan(\theta_s)-\frac{z_0-z_i}{x_0-x_i}) \rm{d}s$
where $\lambda$ is a Lagrangian multiplier that makes the above equation zero at each node of the section.
$\textbf{In AceFEM}$:
I have made a simple example in AceFEM of a rectangular beam. The beam is clamped at the section located at $z=0$. While the rotation should be prescribed the section located at $z=L_z$.
<< AceFEM`
{Lx, Ly, Lz} = {5, 5, 50};
{Nx, Ny, Nz} = {10, 10, 100};
points = {{0, 0, 0}, {Lx, 0, 0}, {Lx, Ly, 0}, {0, Ly, 0}, {0, 0,
Lz}, {Lx, 0, Lz}, {Lx, Ly, Lz}, {0, Ly, Lz}};
setup1[] := (
SMTInputData[];
SMTAddDomain[{"A","OL:SED3H2DFLEH2Hooke", {"E *" -> 35000, "ν *" -> 0.1}}];
SMTAddMesh[Hexahedron[points], "A", "H2", {Nx, Ny, Nz}];
SMTAddEssentialBoundary[
Polygon[ {{0, 0, 0}, {Lx, 0, 0}, {Lx, Ly, 0}, {0, Ly, 0}}, "D"],
1 -> 0, 2 -> 0, 3 -> 0]; SMTAnalysis["Output" -> "Example.out"];
)
setup1[];
Now I am stuck at this point. Because I don't know how to impose the above constraints discussed. Of course, it's not mandatory for me to use the method discussed and any other idea is welcomed.
Thanks in advance