Table of Contents

Bend-twist Coupling Relationships in Beam Elements

This article describes how bend-twist coupling effects can be accounted for in Bladed beam elements. The built-in Bladed model of bend-twist coupling due to shear centre offset from the elastic centre are described. Additionally, the specification of user-defined bend-twist coupling terms is discussed.

Co-incident elastic and shear centres

If the elastic centre and shear centre coincide, the constitutive relationship between strain and internal load for a beam element can be expressed as a diagonal matrix as shown below. Note that this equation is formulated in the local element coordinate system (i.e. it is rotated according to blade structural twist, prebend and sweep).

\[ \begin{equation} \left[\begin{matrix}\bscalar{F_x}\\\bscalar{F_y}\\\bscalar{F_z}\\\bscalar{M_x}\\\bscalar{M_y}\\\bscalar{M_z}\\\end{matrix}\right]=\left[\begin{matrix}\bscalar{EA}&\ &\ &|&\ &\ &\ \\0&{\bscalar{GA}}_y&\ &|&\ &\bmatrix{sym}&\ \\0&0&{\bscalar{GA}}_z&|&\ &\ &\ \\-&-&-&-&-&-&-\\0&0&0&|&{\bscalar{GI}}_x^\ast&\ &\ \\0&0&0&|&0&\bscalar{EI_y}&\ \\0&0&0&|&0&0&\bscalar{{EI}_z}\\\end{matrix}\right]\left[\begin{matrix}\bscalar{\gamma_x}\\\bscalar{\gamma_y}\\\bscalar{\gamma_z}\\\bscalar{\kappa_x}\\\bscalar{\kappa_y}\\\bscalar{\kappa_z}\\\end{matrix}\right]=\ \bmatrix{\bar{\bar{C}}}\left[\begin{matrix}\bscalar{\gamma_x}\\\bscalar{\gamma_y}\\\bscalar{\gamma_z}\\\bscalar{\kappa_x}\\\bscalar{\kappa_y}\\\bscalar{\kappa_z}\\\end{matrix}\right] \end{equation} \]

The 6x6 constitutive matrix is referred to in this document as \(\bmatrix{\bar{\bar{C}}}\), where the double over-bar denotes the local element coordinate system.

It is also noted that this equation may easily be transformed to the local principal elastic coordinate system if the principal elastic axis direction is constant for successive beam elements in the blade. In this case, the principal elastic x-axis equals the element z-axis and the principal elastic y-axis equals the element y-axis in the opposite direction, which implies that the principal elastic z-axis equals the element x-axis. Further details of the relation between the local principal coordinate system and the element coordinate system may be found in the technical document 110052-UKKBR-T-31-B with the title “Orientation of blade local element frame relative to the blade root coordinates”.

Translational offset between elastic and shear centres

It is possible to define a translational offset between the elastic centre and the shear centre within the blade cross-section, as illustrated in Figure 1.

Alt text

Figure 1: Shear centre offset from elastic centre

The translational offset between elastic and shear centres is taken into account using the following calculation, which transforms the shear properties onto the elastic centre.

\[ \begin{equation} \left[\begin{matrix}\bscalar{F_x}\\\bscalar{F_y}\\\bscalar{F_z}\\\bscalar{M_x}\\\bscalar{M_y}\\\bscalar{M_z}\\\end{matrix}\right]=\left[\begin{matrix}\bmatrix{I}&\bmatrix{0}\\\bmatrix{Y_S}^T&\bmatrix{I}\\\end{matrix}\right]\left[\begin{matrix}\bscalar{EA}&\ &\ &|&\ &\ &\ \\0&{\bscalar{GA}}_y&\ &|&\ &\bmatrix{sym}&\ \\0&0&{\bscalar{GA}}_z&|&\ &\ &\ \\-&-&-&-&-&-&-\\0&0&0&|&{\bscalar{GI}}_x^\ast&\ &\ \\0&0&0&|&0&\bscalar{EI_y}&\ \\0&0&0&|&0&0&\bscalar{{EI}_z}\\\end{matrix}\right]\left[\begin{matrix}\bmatrix{I}&\bmatrix{Y_S}\\\bmatrix{0}&\bmatrix{I}\\\end{matrix}\right]\left[\begin{matrix}\bscalar{\gamma_x}\\\bscalar{\gamma_y}\\\bscalar{\gamma_z}\\\bscalar{\kappa_x}\\\bscalar{\kappa_y}\\\bscalar{\kappa_z}\\\end{matrix}\right], \end{equation} \]

where

\[ \begin{equation*} \bmatrix{Y_S}=\ \left[\begin{matrix}0&0&0\\\bscalar{{-z}_{cs}}&0&0\\\bscalar{y_{cs}}&0&0\\\end{matrix}\right] \qquad \qquad \bmatrix{I}\ =\ \left[\begin{matrix}1&0&0\\0&1&0\\0&0&1\\\end{matrix}\right] \end{equation*} \]

Entries \(\bscalar{y_{cs}}\) and \(\bscalar{z_{cs}}\) define the position of the shear centre relative to the elastic centre.

Expanding the above expression gives the following constitutive relationship associated with the principal elastic coordinate system. The effect of shear centre offset is to introduce additional coupling between shear strain and torsional moment, and between bending strain and shear force.

\[ \begin{equation} \left[\begin{matrix}\bscalar{F_x}\\\bscalar{F_y}\\\bscalar{F_z}\\\bscalar{M_x}\\\bscalar{M_y}\\\bscalar{M_z}\\\end{matrix}\right]=\left[\begin{matrix}\bscalar{EA}&\ &\ &|&\ &\ &\ \\0&{\bscalar{GA}}_y&\ &|&\ &\bmatrix{sym}&\ \\0&0&{\bscalar{GA}}_z&|&\ &\ &\ \\-&-&-&-&-&-&-\\0&\bscalar{{-z}_{cs}}{\bscalar{GA}}_y&\bscalar{y_{cs}}{\bscalar{GA}}_z&|&{\bscalar{GI}}_x&\ &\ \\0&0&0&|&0&\bscalar{EI_y}&\ \\0&0&0&|&0&0&\bscalar{{EI}_z}\\\end{matrix}\right]\left[\begin{matrix}\bscalar{\gamma_x}\\\bscalar{\gamma_y}\\\bscalar{\gamma_z}\\\bscalar{\kappa_x}\\\bscalar{\kappa_y}\\\bscalar{\kappa_z}\\\end{matrix}\right] \end{equation} \]

where, \({\bscalar{GI}}_x={{\bscalar{GI}}_x}^\ast+\ {\bscalar{GA}}_yz_{cs}^2+\ {\bscalar{GA}}_zy_{cs}^2\) and \({{\bscalar{GI}}_x}^\ast\) is the torsional stiffness defined about the principal shear (torsional) x-axis.

Translation and out-of-plane orientation offset between principal elastic x-axis and principal shear x-axis

In general, the principal elastic x-axis and principal shear x-axis are not parallel, so it can be important to take account of the orientation difference between them. The out-of-plane orientation difference between the principal shear x-axis and the principal elastic x-axis is illustrated by the \(\theta\) terms in Figure 2.

Alt text

Figure 2: Orientation difference between principal shear and principal elastic axes

The combined translational and orientation offset between principal shear and principal elastic axes is taken into account using the following calculation, which transforms the shear properties onto the elastic centre.

\[ \begin{equation} \left[\begin{matrix}\bscalar{F_x}\\\bscalar{F_y}\\\bscalar{F_z}\\\bscalar{M_x}\\\bscalar{M_y}\\\bscalar{M_z}\\\end{matrix}\right]=\left[\begin{matrix}\bmatrix{I}&\bmatrix{0}\\\bmatrix{Y_S}^T&\bmatrix{R_s}^T\\\end{matrix}\right]\left[\begin{matrix}\bscalar{EA}&\ &\ &|&\ &\ &\ \\0&{\bscalar{GA}}_y&\ &|&\ &\bmatrix{sym}&\ \\0&0&{\bscalar{GA}}_z&|&\ &\ &\ \\-&-&-&-&-&-&-\\0&0&0&|&{\bscalar{GI}}_x^\ast&\ &\ \\0&0&0&|&0&\bscalar{EI_y}&\ \\0&0&0&|&0&0&\bscalar{{EI}_z}\\\end{matrix}\right]\left[\begin{matrix}\bmatrix{I}&\bmatrix{Y_S}\\\bmatrix{0}&\bmatrix{R_s}\\\end{matrix}\right]\left[\begin{matrix}\bscalar{\gamma_x}\\\bscalar{\gamma_y}\\\bscalar{\gamma_z}\\\bscalar{\kappa_x}\\\bscalar{\kappa_y}\\\bscalar{\kappa_z}\\\end{matrix}\right], \end{equation} \]

where

\[ \begin{equation*} \bmatrix{Y_S}=\ \left[\begin{matrix}0&0&0\\\bscalar{{-z}_{cs}}&0&0\\\bscalar{y_{cs}}&0&0\\\end{matrix}\right] \qquad \qquad \bmatrix{R_s}=\frac{1}{\bscalar{L^e}}\ \left[\begin{matrix}\bscalar{L_s^e}&0\ &0\\\bscalar{-\bscalar{\Delta y_{cs}}}&\bscalar{L^e}&0\\\bscalar{-\bscalar{\Delta z_{cs}}}&0&\bscalar{L^e}\\\end{matrix}\right] \end{equation*} \]

Entries \(\bscalar{\Delta y_{cs}}\) and \(\bscalar{\Delta z_{cs}}\) describe the change in position of the shear centre within the beam element, in order to describe the orientation of the principal shear x-axis.

The effect of shear centre translation and orientation offset is to introduce additional coupling between bending and torsional moments, resulting in the following constitutive relationship associated with the principal elastic coordinate system.

\[ \begin{equation} \left[\begin{matrix}\bscalar{F_x}\\\bscalar{F_y}\\\bscalar{F_z}\\\bscalar{M_x}\\\bscalar{M_y}\\\bscalar{M_z}\\\end{matrix}\right]=\ \left[\begin{matrix}\bscalar{EA}&\ &\ &|&\ &\ &\ \\0&{\bscalar{GA}}_y&\ &|&\ &\bmatrix{sym}&\ \\0&0&{\bscalar{GA}}_z&|&\ &\ &\ \\-&-&-&-&-&-&-\\0&\bscalar{{-z}_{cs}}{\bscalar{GA}}_y&\bscalar{y_{cs}}{\bscalar{GA}}_z&|&{\bscalar{GI}}_x&\ &\ \\0&0&0&|&-\frac{\bscalar{\Delta y_{cs}}\bscalar{EI_y}}{\bscalar{L^e}}&\bscalar{EI_y}&\ \\0&0&0&|&-\frac{\bscalar{\Delta z_{cs}}EI_z}{\bscalar{L^e}}&0&\bscalar{{EI}_z}\\\end{matrix}\right] \left[\begin{matrix}\bscalar{\gamma_x}\\\bscalar{\gamma_y}\\\bscalar{\gamma_z}\\\bscalar{\kappa_x}\\\bscalar{\kappa_y}\\\bscalar{\kappa_z}\\\end{matrix}\right], \end{equation} \]

where

\[ \begin{equation} {\bscalar{GI}}_x= \left(\frac{\bscalar{L^e}}{\bscalar{L_s^e}}\right)^2{\bscalar{GI}}_x^\ast+ {\bscalar{GA}}_yz_{cs}^2+ {\bscalar{GA}}_zy_{cs}^2+ \frac{\bscalar{\Delta y}_{cs}^2\bscalar{EI_y}+\bscalar{\Delta z}_{cs}^2\bscalar{{EI}_z}}{{\bscalar{L^e}}^2} \end{equation} \]

and \({{\bscalar{GI}}_x}^\ast\) is the torsional stiffness defined about the principal shear (torsional) x-axis. This transformation results in extra bend-twist off–diagonal coupling terms, as well as a change to the torsional stiffness about the principal elastic x-axis.

User-defined bend-twist coupling

The user can directly add extra off-diagonal terms to the constitutive matrix as shown

\[ \begin{equation} \bmatrix{\bar{\bar{C}}}=\ \left[\begin{matrix}\bscalar{EA}&\ &\ &|&\ &\ &\ \\0&{\bscalar{GA}}_y&\ &|&\ &\bmatrix{sym}&\ \\0&0&{\bscalar{GA}}_z&|&\ &\ &\ \\-&-&-&-&-&-&-\\0&0&0&|&{\bscalar{GI}}_x^\ast&\ &\ \\0&0&0&|&\bscalar{C_{xy}}&\bscalar{EI_y}&\ \\0&0&0&|&\bscalar{C_{xz}}&\bscalar{C_{yz}}&\bscalar{{EI}_z}\\\end{matrix}\right] \end{equation} \]

The transformations described in previous sections based on shear centre relative to elastic centre are also applied, resulting in the following relationship by selecting the option “ignore blade shear centre axis orientation transformation” in Additional Items.

\[ \begin{equation} \left[\begin{matrix}\bscalar{F_x}\\\bscalar{F_y}\\\bscalar{F_z}\\\bscalar{M_x}\\\bscalar{M_y}\\\bscalar{M_z}\\\end{matrix}\right]=\left[\begin{matrix}\bmatrix{I}&\bmatrix{0}\\\bmatrix{Y_S}^T&\bmatrix{I}\\\end{matrix}\right]\ \left[\begin{matrix}\bscalar{EA}&\ &\ &|&\ &\ &\ \\0&{\bscalar{GA}}_y&\ &|&\ &\bmatrix{sym}&\ \\0&0&{\bscalar{GA}}_z&|&\ &\ &\ \\-&-&-&-&-&-&-\\0&0&0&|&{\bscalar{GI}}_x^\ast&\ &\ \\0&0&0&|&\bscalar{C_{xy}}&\bscalar{\bscalar{EI_y}}&\ \\0&0&0&|&\bscalar{C_{xz}}&\bscalar{C_{yz}}&\bscalar{{EI}_z}\\\end{matrix}\right]\left[\begin{matrix}\bmatrix{I}&\bmatrix{Y_S}\\\bmatrix{0}&\bmatrix{I}\\\end{matrix}\right]\left[\begin{matrix}\bscalar{\gamma_x}\\\bscalar{\gamma_y}\\\bscalar{\gamma_z}\\\bscalar{\kappa_x}\\\bscalar{\kappa_y}\\\bscalar{\kappa_z}\\\end{matrix}\right] \end{equation} \]

The following transformation is used as default without enabling the option.

\[ \begin{equation} \left[\begin{matrix}\bscalar{F_x}\\\bscalar{F_y}\\\bscalar{F_z}\\\bscalar{M_x}\\\bscalar{M_y}\\\bscalar{M_z}\\\end{matrix}\right]=\left[\begin{matrix}\bmatrix{I}&\bmatrix{0}\\\bmatrix{Y_S}^T&\bmatrix{R_s}^T\\\end{matrix}\right]\ \left[\begin{matrix}\bscalar{EA}&\ &\ &|&\ &\ &\ \\0&{\bscalar{GA}}_y&\ &|&\ &\bmatrix{sym}&\ \\0&0&{\bscalar{GA}}_z&|&\ &\ &\ \\-&-&-&-&-&-&-\\0&0&0&|&{\bscalar{GI}}_x^\ast&\ &\ \\0&0&0&|&\bscalar{C_{xy}}&\bscalar{EI_y}&\ \\0&0&0&|&\bscalar{C_{xz}}&\bscalar{C_{yz}}&\bscalar{{EI}_z}\\\end{matrix}\right]\left[\begin{matrix}\bmatrix{I}&Y_S\\\bmatrix{0}&\bmatrix{R_s}\\\end{matrix}\right]\left[\begin{matrix}\bscalar{\gamma_x}\\\bscalar{\gamma_y}\\\bscalar{\gamma_z}\\\bscalar{\kappa_x}\\\bscalar{\kappa_y}\\\bscalar{\kappa_z}\\\end{matrix}\right] \end{equation} \]

It is noted that the transformation of these equations to the local principal elastic coordinate system as described in Co-incident elastic and shear centres will introduce a change of the sign of the off-diagonal terms that relate to bending about the element y-axis.