Radioss Coordinate System

Shell and solid (thick shell) elements are introduced for the coordinate systems.

Global system ( $X$ , $Y$ , and $Z$ )
Natural system (isoparametric frame) ( $ξ, η, ζ$ )
Local element coordinate system ( $x$ , $y$ , and $z$ )

( $X$ , $Y$ , and $Z$ ) - Global Cartesian fixed system:

( $ξ, η, ζ$ ) - Natural system (non-normalized coordinate system).

$ξ$ is from middle point of Line 14 to middle point of Line 23.
$η$ is from middle point of Line 12 to middle point of Line 34.
Plane ( $ξ, η$ ) is in the middle surface of shell element and $ζ$ is normal of the middle surface.

(

$x$ ,

$y$ , and

$z$ ) - Local coordinate system (orthogonal, normalized elemental coordinate system):

$z$ is normal of middle surface.
( $x$ and $y$ ) are in the middle surface
$x$ and $y$ are positioned so that they have same angle between $x$ and $ξ$ , $y$ and $η$

The origin of ( $ξ, η, ζ$ ) and ( $x$ , $y$ , and $z$ ) are the same as it is at the intersection point of middle point line.

3-node Shell Element

( $X$ , $Y$ , and $Z$ ) - Global Cartesian fixed system

( $ξ, η, ζ$ ) - Natural system (non-normalized coordinate system).

$ξ$ is from Node 1 to Node 2.
$η$ is from Node 1 to Node 3.
Plane ( $ξ, η$ ) is in the middle surface of shell element and $ζ$ is normal of the middle surface.

(

$x$ ,

$y$ , and

$z$ ) - Local coordinate system (orthogonal, normalized elemental coordinate system).

$z$ is normal of middle surface.
$x$ is from Node 1 to Node 2.
$y$ is orthogonal to $x$ and ( $x$ and $y$ ) are in the middle surface.

The origin of (

$ξ, η, ζ$ ) and (

$x$ ,

$y$ , and

$z$ ) are the same as it is at Node 1.

Solids and Thick Shells (hexa)

Global system ( $X$ , $Y$ , and $Z$ )
Natural system ( $r$ , $s$ , and $t$ )
Local element coordinate system ( $x$ , $y$ , and $z$ )
Material system

( $X$ , $Y$ , and $Z$ ) - Global Cartesian fixed system

(

$r$ ,

$s$ , and

$t$ ) - Natural system (non-normalized coordinate system).

$r$ is from the center of surface (1, 2, 6, and 5) to center of surface (4, 3, 7, and 8)
$s$ is from the center of surface (1, 2, 3, and 4) to center of surface (5, 6, 7, and 8)
$t$ is from the center of surface (1, 4, 8, and 5) to center of surface (2, 3, 7, and 6)

Plane (

$r$ ,

$t$ ) is also in the middle surface (1', 2', 3', and 4').

$r$ is also from middle point of Line 1' and 2' to middle point of Line 3' and 4'.
$t$ is also from middle point of Line 1' and 2' to middle point of Line 3' and 4'
$n$ is normal of middle surface (1', 2', 3', and 4')

( $x$ , $y$ , and $z$ ) - Local coordinate system (orthogonal, normalized elemental coordinate system).

Local coordinate system in middle surface (1', 2', 3', and 4') is the same as the local coordinate system in middle surface (1, 2, 3, and 4) for shell element. r in solid is the same as $ξ$ in shell element.

Tetra Elements

(

$r$ ,

$s$ , and

$t$ ) - Natural system (non-normalized coordinate system).

$r$ is from node 4 to node 1
$s$ is from node 4 to node 2
$t$ is from node 4 to node 3

Material System

For shell element, anisotropic can be defined with property type 9, 10, 11, 16, 17, and 19, using a material system to describe the anisotropic. Vector

$V$ and angle

$ϕ$ requested to define material system. (See /PROP/TYPE9 Format below). Material direction (

$m 1$ and

$m 2$ ) presents the direction of different mechanic characters (Example: E-Modulus, shear Modulus, stress-strain behavior, damage, ...) for anisotropic.

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`V`_X		`V`_Y		`V`_Z		$ϕ$

Use vector $V$ and angle $ϕ$ material direction 1 ( $m 1$ ) can be defined (along normal n project vector $V$ to middle surface and get vector $V'$ . Rotate angle $ϕ$ of vector $V'$ then get material direction $m 1$ . Material direction $m 1$ is normally the fiber direction. For composite, a different ply could be defined with one vector $V$ and different $ϕ$ .

Second material axis

$m 2$ is perpendicular to

$m 1$ (except for /PROP/TYPE16, angle between

$m 1$ and

$m 2$ could be defined with α).

$n$ is normal of shell middle surface

Figure 5.

In /PROP/TYPE11, I_orth can determine the relative orientation of the material system.

I_orth=0 (default): The orthotropic direction follows the local co-rotational reference. The angle between $x$ and $m 1$ is constant during the simulation. Internal force is computed in local frame and then rotated to the global system. This formulation is more accurate, if a large rotation occurs.
I_orth=1: The orthotropic direction is attached to the local isoparametric frame. The angle between $ξ$ and $m 1$ is updated during the simulation. It is updated in a way that projection of vector $m 1$ to $ξ$ and $η$ is always constant during the simulation. Pure shear could not well descripted with this method, but traction could well be described. So, this method usually is used to defined fiber direction in airbag.

Figure 6.

For brick and thick shell elements, use the same process to determine the material direction and orthotropic direction (I_orth), like shell elements. In /PROP/TYPE6 (SOL_ORTH) use the option I_P to determine the reference plane.

I_P=0: Use Skew_ID
I_P=1: Plane ( $r$ , $s$ ) + angle $ϕ$
I_P=2: Plane ( $s$ , $t$ ) + angle $ϕ$
I_P=3: Plane ( $t$ , $r$ ) + angle $ϕ$
I_P=11: Plane ( $r$ , $s$ ) + orthogonal projection of references vector $V$ on plane ( $r$ , $s$ )
I_P=12: Plane ( $s$ , $t$ ) + orthogonal projection of references vector $V$ on plane ( $s$ , $t$ )
I_P=13: Plane ( $t$ , $r$ ) + orthogonal projection of references vector $V$ on plane ( $t$ , $r$ )

Definition is the same for any I_solid and I_frame parameters
In the simplest case, material directions $m 1$ , $m 2$ and $m 3$ directly with skew (I_P=0) are recommended
For I_P > 0 the isoparametric, non-orthogonal system $r$ , $s$ , and $t$ , is used to determine material directions.
- First material axis $m 1$ is determined according to I_P.
- For example, for I_P=1
- The first material axis $m 1$ and $m 2$ is orthogonal and rotated by angle $ϕ$ in the ( $r'$ and $s'$ ) plane.
- The third material axis $m 3$ is normal of $m 1$ and $m 2$ plane (vector product of $m 1$ and $m 2$ ).
  
  Figure 7.
- The ( $r'$ , $s'$ , and $t'$ ) system is orthogonal and it is generated from non-orthogonal isoparametric system ( $r$ , $s$ , and $t$ ).
  
  Figure 8. Orthogonalization of the Isoparametric System

Depending on I_solid and I_frame parameters, three definitions of systems are used in Radioss for hexa elements (8-noded bricks) using /PROP/TYPE6 (SOl_ORTH).

Global System Definition

Definition 1: Solids, I_solid=1, 2, 17 + I_frame=0, 1 (default)
Global system is used, no element system (non-co-rotational formulation) available.

Element System Definition

Definition 2: Solids, I_solid=1, 2, 17 + I_frame=2
Element system (with I_frame=2 co-rotational formulation) is used.
Definition 3: Solids, I_solid=14 or 24
I_frame parameter has no effect. Element system is used, and co-rotational formulation defined already.

Note: If the co-rotational formulation is used, the orthotropic frame (defined with I_orth) keeps the same orientation with respect to the local (co-rotating) frame and is therefore also co-rotating.