Structural Properties of the Blade

NeutralAxisCoordinateSystem

The NeutralAxisCoordinateSystem defines the rotation and locations used to create the underlying finite element model of the blade. The NeutralAxisCoordinateSystem should be positioned and oriented at the point where the section experiences zero axial strain due to applied bending moments. Additionally, the axial stiffness and bend-twist coupling terms are defined at this neutral axis location. Finally, the associated property group, StructuralProperties, is applied to the finite element model, resulting in a representation of the structural blade.

Figure 1: Illustration of the NeutralAxisCoordinateSystem.

ShearAxis

The shear centre is the point on the section where if a force were applied, it would result in zero torsion. The shear centre has no orientation associated with it. It is a single locus, rather than a full coordinate system. The TorsionalStiffness in StructuralProperties are defined with respect to the shear centre and the \(x_{cs}\), \(y_{cs}\) values in Equation \(\eqref{eq:beamconstitutiverelationship}\) are calculated using the location of the ShearAxis centre.

Figure 2: Illustration of the ShearAxis.

StructuralProperties

The inputs for the property group, StructuralProperties, all relate to the cross-sectional 6x6 stiffness matrix used in the Bladed beam model as shown in Table 1. The constitutive relationship for the full beam model can be expressed as:

\[ \begin{equation} \left[\begin{matrix}\bscalar{F_x}\\\bscalar{F_y}\\\bscalar{F_z}\\\bscalar{M_x}\\\bscalar{M_y}\\\bscalar{M_z}\\\end{matrix}\right]= \left[\begin{matrix} \bscalar{GA}_x & & & | & & & \\ 0 & \bscalar{GA}_y & & | & & \bmatrix{sym} & \\ 0 & 0 & \bscalar{EA} & | & & & \\ - & - & - & - & - & - & - \\ 0 & 0 & 0 & | & \bscalar{EI_x} & & \\ 0 & 0 & 0 & | & \bscalar{C_{xy}} & \bscalar{EI_y} & \\ - \bscalar{GA}_x y_{cs} & \bscalar{GA}_y x_{cs} & 0 & | & \bscalar{C_{xz}} & \bscalar{C_{yz}} & \bscalar{GI}_z \\ \end{matrix}\right] \left[\begin{matrix}\bscalar{\gamma_x}\\\bscalar{\gamma_y}\\\bscalar{\gamma_z}\\\bscalar{\kappa_x}\\\bscalar{\kappa_y}\\\bscalar{\kappa_z}\\\end{matrix}\right] \label{eq:beamconstitutiverelationship} \end{equation} \]

where \(\bscalar{GI}_z = \bscalar{GI}_z^\ast + \bscalar{GA}_x \cdot y_{cs}^2 + \bscalar{GA}_y \cdot x_{cs}^2\), and \(x_{cs}\) and \(y_{cs}\) are the offsets between the NeutralAxisCoordinateSystem and ShearAxis centre along the NeutralAxisCoordinateSystem axes.

The constitutive relationship is related to the NeutralAxisCoordinateSystem location and orientation. It can include user-defined bend-twist and bending-bending coupling terms (read more in Bend-twist Coupling Relationships in Beam Elements).

Table 1 provides an overview over the user inputs for defining the cross-sectional stiffness properties in Bladed by relating them to Equation \(\eqref{eq:beamconstitutiverelationship}\).

Table 1: Overview over structural stiffness properties in Bladed

Variable identifier	Property	Description	Unit
\(EI_x\)	BendingStiffnessAboutX	Principal bending stiffness about the neutral x-axis	\(Nm^2\)
\(EI_y\)	BendingStiffnessAboutY	Principal bending stiffness about the neutral y-axis	\(Nm^2\)
\(EA\)	AxialStiffness	Axial stiffness of cross-section	\(N\)
\(GI_z^\ast\)	TorsionalStiffness	Torsional stiffness about the shear z-axis	\(Nm^2\)
\(GA_x\)	ShearStiffnessInX	Principal shear stiffness along the neutral x-axis	\(N\)
\(GA_y\)	ShearStiffnessInY	Principal shear stiffness along the neutral y-axis	\(N\)
\(C_{xy}\)	BendingXYCoupling	Bending-Bending coupling stiffness along neutral x- and y-axes	\(Nm^2\)
\(C_{xz}\)	TorsionBendingXCoupling	Torsion-Bending coupling stiffness along neutral x-axes	\(Nm^2\)
\(C_{yz}\)	TorsionBendingYCoupling	Torsion-Bending coupling stiffness along neutral y-axis	\(Nm^2\)

Example of a blade section with all the properties defined:

"StructuralProperties": {
    "BendingStiffnessAboutX": 71901150000.0,
    "BendingStiffnessAboutY": 50453410000.0,
    "AxialStiffness": 27330360000.0,
    "TorsionalStiffness": 25594570000.0,
    "ShearStiffnesses": {
        "ShearStiffnessInX": 3918299000.0,
        "ShearStiffnessInY": 2492965000.0
    },
    "CouplingTerms": {
        "BendingXYCoupling": 1252955600.0,
        "TorsionBendingXCoupling": 3216255000.0,
        "TorsionBendingYCoupling": 3216255000.0
    }
}

Note on the Orientation of the NeutralAxisCoordinateSystem

The orientation of the underlying element nodes follows the direction of the elements as illustrated in Figure 3, which depicts a prebend blade. In this figure, the element frames (shown in orange) may have a different orientation compared to the NeutralAxisCoordinateSystem (highlighted in yellow), particularly noticeable in section 3. This discrepancy can be problematic because the StructuralProperties are applied directly to the element frames, not the NeutralAxisCoordinateSystem. The RotationAboutReferenceZ defined in the NeutralAxisCoordinateSystem is directly applied as the relative rotation about the element node ends.

As a result, it is advisable to define the ReferenceCoordinateSystem as closely as possible to the NeutralAxisCoordinateSystem, or at the very least, carefully consider the implications of any orientation differences.

Figure 3: Illustration of a prebend blade with a ReferenceCoordinateSystem, NeutralAxisCoordinateSystem and the element coordinate system.