3. Modeling A#

3.1. Characteristics of modeling#

3D elements (PENTA15 + HEXA20)

Modeling:

1/4 of the cylinder according to the circumference

2 zones:	zone 1 = lower part	\((0\le z\le L/2)\)
	zone 2 = upper part	\((L/2\le Z\le L)\)

Cutting:

20 elements depending on the length

16 elements depending on the circumference

2 elements in the thickness

Coordinates of points \((r,\theta ,z)\)

A2 A’2

H H”

B2 B’2

E2 E’2

H1 H’1

F2 F’2

\(r\)

\(\theta\)

.0.

\(z\)

L/2

\(\mathrm{Ri}\) = inner radius

\(\text{Re}\) = outer radius

The \(\mathrm{A2},H,\mathrm{B2},\mathrm{E2},\mathrm{H2},\mathrm{F2}\) points are in section \(z=L/2\) of zone 1

Points \(A’\mathrm{2,}H’,B’\mathrm{2,}E’\mathrm{2,}H’\mathrm{2,}F’2\) are the opposite sides in zone 2

Boundary conditions:

Support conditions \(w=0\) at the base (section \(z=0.\)) introduced by the keyword LIAISON_OBLIQUE
Symmetry conditions \(v=0.\) on the \(\mathrm{AB}\) side introduced by the LIAISON_OBLIQUE keyword
Symmetry conditions \(u=0.\) on the \(\mathrm{EF}\) side introduced by the LIAISON_OBLIQUE keyword
Identification of the nodes common to the 2 zones (section \(z=L/2\)) by the LIAISON_GROUP keyword.

Charging:

Surface load \(p=q/h=500000N/\mathrm{m2}\), along the axis, or in global coordinate system:

\(\mathrm{Fx}=0.\)

\(\mathrm{Fy}=p/2\)

\(\mathrm{Fz}=p\frac{\sqrt{3}}{2}\)

Node name:

3.2. Characteristics of the mesh#

Number of knots: 4298

Number of meshes and types: 160 HEXA20, 320 PENTA15

3.3. Tested values#

\(U,V,W\) displacement values read from file

Location	Value type	Reference
Point \(G\)	\(U(m)\)	—7.143 x 10—7
	\(V(m)\)
	\(W(m)\)
Point \(H,H’\)	\(U(m)\)	—7.143 x 10—7
Point \(I\)	\(U(m)\)	—7.143 x 10—7
Point \(\mathrm{G1}\)	\(U(m)\)
Points \(\mathrm{H1},H’1\)	\(U(m)\)

Values of \(u,v,{u}_{r}\) movements in local coordinate system calculated from \(U,V,W\)

Location	Value type	Reference
Point \(G\)	\({u}_{r}(m)\)	—7.143 x 10—7
	\(v(m)\)
Point \(H,H’\)	\({u}_{r}(m)\)	—7.143 x 10—7
	\(v(m)\)
Point \(I\)	\({u}_{r}(m)\)	—7.143 x 10—7
	\(v(m)\)
Point \(\mathrm{A2},A’2\) Points \(\mathrm{B2},B’2\)	\(v(m)\)
Point \(\mathrm{G1}\)	\(u(m)\)
	\({u}_{r}(m)\)	—7.143 x 10—7
Points \(\mathrm{H1},H’1\)	\(u(m)\)
	\({u}_{r}(m)\)	—7.143 x 10—7
Point \(\mathrm{I1}\)	\(u(m)\)
	\({u}_{r}(m)\)	—7.143 x 10—7
Points \(\mathrm{E2},E’2\)	\(u(m)\)
Points \(\mathrm{F2},F’2\)	\(u(m)\)
Points \(A,B,G\) \(\mathrm{A2},\mathrm{B2},H\) \(A’\mathrm{2,}B’\mathrm{2,}H’\) \(\mathrm{A3},\mathrm{B3},I\)	\({\sigma }_{\mathrm{YY}}(\mathrm{Pa})\)	1.25 x 105
Points \(A,B,G\) \(\mathrm{A2},\mathrm{B2},H\) \(A’\mathrm{2,}B’\mathrm{2,}H’\) \(\mathrm{A3},\mathrm{B3},I\)	\({\sigma }_{\mathrm{ZZ}}(\mathrm{Pa})\)	3.75 x 105

3.4. notes#

Radial displacement \(\mathrm{ur}\) is obtained with good precision.
The symmetry conditions on face \(\mathrm{AB}\) (\(v=0\) locally, i.e. \(\frac{\sqrt{3}}{2}V–05W=0\)) are verified at the points \(\mathrm{A2},A’\mathrm{2,}G,\mathrm{B2},B’\mathrm{2,}H,H’,I\) considered.

Likewise, the symmetry conditions on face \(\mathrm{EF}\) (\(u=U=0\)) are verified at the envisaged points \(\mathrm{E2},E’\mathrm{2,}\mathrm{F2},F’\mathrm{2,}\mathrm{G1},\mathrm{H1},H’\mathrm{1,}\mathrm{I1}\).

The LIAISON_OBLIQUE keyword is thus validated.

The identification of the nodes common to the 2 zones by the keyword LIAISON_GROUP is also validated: the movements \(U,V,W\) are identical to the points \(A’\mathrm{2,}B’\mathrm{2,}H’,E’\mathrm{2,}F’\mathrm{2,}H’1\) in comparison with the movements to the respective opposite sides \(\mathrm{A2},\mathrm{B2},H,\mathrm{E2},\mathrm{F2},\mathrm{H1}\).