Cylinder Under Rotational Force

The aim of this test case is to validate the following functions:

• centrifugal force

The simulation results of SimScale were compared to the numerical results presented in [HPLA100]. The mesh used in (A) and (C) was created with the automatic-tetrahedralization-tool on the SimScale platform. The mesh used in (B) und (D) was created locally.

Import validation project into workspace

Geometry

	A	B	C	D
x [m]	0.0195	0.0205	0.0205	0.0195
y [m]	0	0	0	0
z [m]	0.01	0.01	0	0

To obtain the solid body the face ABCD is rotated 45° around the z-axis. Because of the symmetry of the cylinder only one quarter was modelled.

Analysis type and Domain

Tool Type : Code_Aster

Analysis Type : Static

Mesh and Element types :

Case	Mesh type	Number of nodes	Element type
(A)	quadratic tetrahedral	2893	3D isoparametric
(B)	quadratic hexahedral	2583	3D isoparametric
(C)	quadratic tetrahedral	17460	3D isoparametric
(D)	quadratic hexahedral	9285	3D isoparametric

Simulation Setup

Material:

• isotropic: E = 200 GPa, ν
  $\nu$

   

  = 0.3, ρ

  $\rho$

   

  = 8000 kg/m^3

Constraints:

• Face AA’BB’ and face CC’DD’ zero z-displacement
• Symmetry boundary condition on face ABCD and A’B’C’D’

Loads:

• Centrifugal force with a rotational speed ω
  $\omega$

   

  = 1 rad/s around the z-axis applied on the whole body

Reference Solution

u(r)=−(1+ν)(1−2ν)(1−ν)Eρω2r38+Ar+Br(1)

$$\begin{array}{}\text{(1)}& u(r)=\frac{-(1+\nu )(1-2\nu )}{(1-\nu )E}\rho {\omega }^2\frac{r^3}{8}+Ar+\frac{B}{r}\end{array}$$

 

σzz(r)=−ν(1−ν)ρω2r22+2νE(1+ν)(1−2ν)A(2)

$$\begin{array}{}\text{(2)}& {\sigma }_{zz}(r)=\frac{-\nu }{(1-\nu )}\rho {\omega }^2\frac{r^2}{2}+\frac{2\nu E}{(1+\nu )(1-2\nu )}A\end{array}$$

 

A=(3−2ν)(1+ν)(1−2ν)4(1−ν)Eρω2R2(1−x2)=7.13588⋅10−12(3)

$$\begin{array}{}\text{(3)}& A=\frac{(3-2\nu )(1+\nu )(1-2\nu )}{4(1-\nu )E}\rho {\omega }^2{R}^2(1-x^2)=7.13588\cdot {10}^{-12}\end{array}$$

 

B=(3−2ν)(1+ν)8(1−ν)Eρω2R4(1−x2)2=3.561258⋅10−15m2(4)

$$\begin{array}{}\text{(4)}& B=\frac{(3-2\nu )(1+\nu )}{8(1-\nu )E}\rho {\omega }^2{R}^4(1-x^2{)}^2=3.561258\cdot {10}^{-15}{m}^2\end{array}$$

 

x=h2R=0.001m2⋅0.02m=0.025(5)

$$\begin{array}{}\text{(5)}& x=\frac{h}{2R}=\frac{0.001m}{2\cdot 0.02m}=0.025\end{array}$$

 

All stated equations used to solve the problem are derived in [HPLA100]. The parameters h

$h$

 and R

$R$

 in (5)

$\text{(5)}$

 are the thickness of the cross section and the radius of the middle surface of the cylinder respectively.

Results

Important

The values for the comparison of the displacement ur

$u_r$

 and the stresses σzz

${\sigma }_{zz}$

 are averaged over on edge and an area respectively

Comparison of the displacement ur

$u_r$

 and the stresses σzz

${\sigma }_{zz}$

 of the inner and the outter face obtained with SimScale and the results derived from [HPLA100].

Comparison of the displacement ur

$u_r$

 

 and the stresses σzz

${\sigma }_{zz}$

 

 

Case	Quantity	[HPLA100]	SimScale	Error
(A)	ur(r=0.0195m) $u_r(r=0.0195m)$ [m]	2.9424E-013	2.83491E-013	3.653%
(A)	ur(r=0.0205m) $u_r(r=0.0205m)$ [m]	2.8801E-013	2.8972E-013	-0.594%
(A)	σzz(r=0.0195m) ${\sigma }_{zz}(r=0.0195m)$ [N/m^2]	0.99488	0.992259	0.263%
(A)	σzz(r=0.0205m) ${\sigma }_{zz}(r=0.0205m)$ [N/m^2]	0.92631	0.929711	-0.367%
(B)	ur(r=0.0195m) $u_r(r=0.0195m)$ [m]	2.9424E-013	2.94195E-013	0.015%
(B)	ur(r=0.0205m) $u_r(r=0.0205m)$ [m]	2.8801E-013	2.87966E-013	0.015%
(B)	σzz(r=0.0195m) ${\sigma }_{zz}(r=0.0195m)$ [N/m^2]	0.99488	1.00893	-1.412%
(B)	σzz(r=0.0205m) ${\sigma }_{zz}(r=0.0205m)$ [N/m^2]	0.92631	0.914645	1.259%
(C)	ur(r=0.0195m) $u_r(r=0.0195m)$ [m]	2.9424E-013	2.94234E-013	0.002%
(C)	ur(r=0.0205m) $u_r(r=0.0205m)$ [m]	2.8801E-013	2.88003E-013	0.002%
(C)	σzz(r=0.0195m) ${\sigma }_{zz}(r=0.0195m)$ [N/m^2]	0.99488	0.995217	-0.034%
(C)	σzz(r=0.0205m) ${\sigma }_{zz}(r=0.0205m)$ [N/m^2]	0.92631	0.926616	-0.033%
(D)	ur(r=0.0195m) $u_r(r=0.0195m)$ [m]	2.9424E-013	2.94237E-013	0.001%
(D)	ur(r=0.0205m) $u_r(r=0.0205m)$ [m]	2.8801E-013	2.88007E-013	0.001%
(D)	σzz(r=0.0195m) ${\sigma }_{zz}(r=0.0195m)$ [N/m^2]	0.99488	0.994995	-0.012%
(D)	σzz(r=0.0205m) ${\sigma }_{zz}(r=0.0205m)$ [N/m^2]	0.92631	0.926415	-0.011%

References

[HPLA100]

(1, 2, 3, 4) HPLA100 – Cylindre creux thermoélastique pesanten rotation uniforme