Sources/Anleitungen/FEM-Formulierung für den Euler-Bernoulli-Balken

Diese Seite fasst die wichtigsten Ergebnisse für die FEM-Formulierung zum Euler-Bernoulli-Balken zusammen.

Maxima-Code

Hier finden Sie den Sourcecode für die Herleitung der Elelemt-Matrizen.

/*Maxima Quellcode für die Erzeugung Inhalte dieser Seite */
/* Maxima version 16.04.2*/

declare( "ℓ", alphabetic);

/*Trial-Fucntions*/
phi : [       (xi-1)^2*(2*xi+1),
         ℓ[i]* xi     *(  xi-1)^2,
        -      xi^2   *(2*xi-3), 
         ℓ[i]* xi^2   *(  xi-1)];

/* depict independant and dependant coordinates */
preamble: "set yrange [] reverse";
plot2d(subst([ℓ[i]=1],[0.9,1/2,0.1,1/3].phi),[xi,0,1],
    [x,0,1], [legend, "w(x)"],
    [xlabel, "ξ →"], [ylabel, "← w"],
    [gnuplot_preamble, preamble]);

/* plot trial-functions parameters */
scale: [1,1/ℓ[i],1,1/ℓ[i]];
colours: [blue, red, green, magenta];
legends: ["ϕ1","ϕ2/ℓ[i]","ϕ3","ϕ4/ℓ[i]"];
for j:1 thru 4 do
     (plot2d(scale[j]*phi[j],[xi,0,1], [xlabel,"ξ →"],[ylabel,"ϕ →"], [legend,legends[j]],[color,colours[j]], [style, [lines,3]], same_xy,
              [gnuplot_preamble, "set xtics 1;set ytics 1;set tics font \",11\""]))$	

/* Element-Mass Matrix */
M[i] : funmake('matrix,makelist(makelist(rho*A*integrate(phi[j]*phi[k],xi,0,1)*ℓ[i],j,1,4),k,1,4));
/*       (rho*A*ℓ[i])*matrix([13/35,(11*ℓ[i])/210,9/70,-(13*ℓ[i])/420],
                             [(11*ℓ[i])/210,ℓ[i]^2/105,(13*ℓ[i])/420,-ℓ[i]^2/140],
                             [9/70,(13*ℓ[i])/420,13/35,-(11*ℓ[i])/210],
                             [-(13*ℓ[i])/420,-ℓ[i]^2/140,-(11*ℓ[i])/210,ℓ[i]^2/105])  */
/* Element-Striffness Matrix */
K[i] : funmake('matrix,makelist(makelist(EI*integrate(diff(phi[j],xi,2)/ℓ[i]^2*diff(phi[k],xi,2)/ℓ[i]^2,xi,0,1)*ℓ[i],j,1,4),k,1,4));
/*       (EI/ℓ[i]^3)*matrix([12,6*ℓ[i],-12,6*ℓ[i]],
                            [6*ℓ[i],4*ℓ[i]^2,-6*ℓ[i],2*ℓ[i]^2],
                            [-12,-6*ℓ[i],12,-6*ℓ[i]],
                            [6*ℓ[i],2*ℓ[i]^2,-6*ℓ[i],4*ℓ[i]^2])   */

Trial-Functions

Die Koordinaten eines Finiten Elements sind

${\underline {Q}}(t)=\left({\begin{array}{l}W_{i-1}(t)\\\Phi _{i-1}(t)\\W_{i}(t)\\\Phi _{i}(t)\end{array}}\right)$.

Der Ansatz für die Näherung der Auslenkung w(x,t) ist

$\displaystyle w(x,t)=\sum _{i=1}^{4}Q_{i}(t)\cdot \phi _{i}(x)$

Die Drehwinkel sind hier so gewählt, dass 

$\displaystyle {\frac {d}{dx}}w(x)|_{x=x_{i}}=\Phi _{i}$.

Nodal and independent Coordinates

Trialfunctions

Die einzelnen Trial-Functions sind:

linker Rand (Knoten "i-1")	rechter Rand (Knoten "i")
Trial-Function for Wi-1	Trial-Function for Wi
$\phi _{1}(\xi )=(\xi -1)^{2}\cdot (2\;\xi +1)$	$\phi _{3}(\xi )=\xi ^{2}\cdot (3-2\;\xi )$
Trial-Function für Φi-1	Trial-Function für Φi
$\phi _{2}(\xi )=\ell _{i}\cdot \xi \cdot (\xi -1)^{2}$	$\phi _{4}(\xi )=\ell _{i}\cdot \xi ^{2}\cdot (\xi -1)$

Faltungsintegrale

... für die symmetrische Massenmatrix. Tabelliert sind die Werte der Integrale

$\displaystyle {\frac {m_{ij}}{\varrho A}}=\int _{0}^{\ell }\phi _{i}\cdot \phi _{j}\;dx$

	ϕ1	ϕ2	ϕ3	ϕ4
ϕ1	$\displaystyle {\frac {13}{35}}\;\ell _{i}$	$\displaystyle {\frac {11}{210}}\;\ell _{i}^{2}$	$\displaystyle {\frac {9}{70}}\;\ell _{i}$	$\displaystyle -{\frac {13}{420}}\;\ell _{i}^{2}$
ϕ2		$\displaystyle {\frac {1}{105}}\;\ell _{i}^{3}$	$\displaystyle {\frac {13}{420}}\;\ell _{i}^{2}$	$\displaystyle -{\frac {1}{140}}\;\ell _{i}^{4}$
ϕ3			$\displaystyle {\frac {13}{35}}\;\ell _{i}$	$\displaystyle {\frac {11}{210}}\;\ell _{i}^{2}$
ϕ4	symm.			$\displaystyle {\frac {1}{105}}\;\ell _{i}^{3}$

.. für die symmetrische Steifigkeitsmatrix. Tabelliert sind die Werte der Integrale

$\displaystyle {\frac {k_{ij}}{EI}}=\int _{0}^{\ell }\phi _{i}''\cdot \phi _{j}''\;dx$

	ϕ1	ϕ2	ϕ3	ϕ4
ϕ1	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
ϕ2		${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
ϕ3			${\displaystyle }$	${\displaystyle }$
ϕ4	symm.			${\displaystyle }$

---

	$\Phi _{1}$	$\Phi _{2}$	$\Phi _{3}$	$\Phi _{4}$
$\Phi _{1}$	$\displaystyle {\frac {13}{35}}\;\ell _{i}$	$\displaystyle {\frac {11}{210}}\;\ell _{i}^{2}$	$\displaystyle {\frac {9}{70}}\;\ell _{i}$	$\displaystyle -{\frac {13}{420}}\;\ell _{i}^{2}$
$\Phi _{2}$		$\displaystyle {\frac {1}{105}}\;\ell _{i}^{3}$	$\displaystyle {\frac {13}{420}}\;\ell _{i}^{2}$	$\displaystyle -{\frac {1}{140}}\;\ell _{i}^{4}$
$\Phi _{3}$			$\displaystyle {\frac {13}{35}}\;\ell _{i}$	$\displaystyle {\frac {11}{210}}\;\ell _{i}^{2}$
$\Phi _{4}$	symm.			$\displaystyle {\frac {1}{105}}\;\ell _{i}^{3}$

.. für die symmetrische Steifigkeitsmatrix. Tabelliert sind die Werte der Integrale

$\displaystyle {\frac {k_{ij}}{EI}}=\int _{0}^{\ell }\phi _{i}''\cdot \phi _{j}''\;dx$

	$\Phi ''_{1}$	$\Phi ''_{2}$	$\Phi ''_{3}$	$\Phi ''_{4}$
$\Phi ''_{1}$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\Phi ''_{2}$		${\displaystyle }$	${\displaystyle }$	${\displaystyle }$
$\Phi ''_{3}$			${\displaystyle }$	${\displaystyle }$
$\Phi ''_{4}$	symm.			${\displaystyle }$

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