chrono::fea::ChElementBeamTaperedTimoshenkoFPM Class Reference

## Description

For composite beams such as wind turbine blades and helicopter rotor blades, the cross-sectional stiffness properties in axial, shear, bending and torsion directions are coupled with each other, hence the fully-populated matrix(FPM) of cross-sectional stiffness properties is used to describe this complex coupling.

The shape functions of classical Timoshenko beam are not applicable for this new Timoshenko beam. In this tapered Timoshenko beam, the shape functions are numerically derived according to the cross-sectional fully-populated stiffness matrix(FPM), and the local mass, stiffness and damping matrices(M K R) are evaluated via Gauss quadrature. When the fully-populated stiffness matrix(FPM) includes only diagonal elements(no coupling in the cross-sectional stiffness matrix, such as a circular section made from steel), the shape functions are reduced to the occasion in classical Timoshenko beam as in ChElementBeamTaperedTimoshenko, then ChElementBeamTaperedTimoshenkoFPM is equal to ChElementBeamTaperedTimoshenko. Note that there are also ChElementCableANCF if no torsional effects are needed, as in cables.

#include <ChElementBeamTaperedTimoshenkoFPM.h>

Inheritance diagram for chrono::fea::ChElementBeamTaperedTimoshenkoFPM:
Collaboration diagram for chrono::fea::ChElementBeamTaperedTimoshenkoFPM:

## Public Types

using ShapeFunctionGroupFPM = std::tuple< ChMatrixNM< double, 6, 12 >, ChMatrixNM< double, 6, 12 > >

## Public Member Functions

void SetTaperedSection (std::shared_ptr< ChBeamSectionTaperedTimoshenkoAdvancedGenericFPM > my_material)
Set the tapered section & material of beam element.

Get the tapered section & material of the element.

void ShapeFunctionsTimoshenkoFPM (ShapeFunctionGroupFPM &NB, double eta)
Computes the shape function matrix 'Nx' and strain-displacement relation matrix 'Bx' at dimensionless abscissa 'eta'. More...

void SetIntegrationPoints (int mv)
Set the order of Gauss quadrature, as default it is four.

int GetIntegrationPoints ()
Get the order of Gauss quadrature.

void ComputeStiffnessMatrix ()
Computes the local (material) stiffness matrix of the element: K = integral( [B]' * [D] * [B] ), Note: the sectional properties at Gauss integration point are linearly interpolated from two ends of tapered beam. More...

void ComputeDampingMatrix ()
Computes the local element damping matrix via Guass Quadrature: R = beta * integral( [B]' * [D] * [B] ), Note: the sectional properties at Gauss integration point are linearly interpolated from two ends of tapered beam. More...

void ComputeConsistentMassMatrix ()
Computes the local element consistent mass matrix via Guass Quadrature: M = integral( [N]' * [D] * [N] ), Note: the sectional properties at Gauss integration point are linearly interpolated from two ends of tapered beam. More...

void ComputeMassMatrix ()
Finally, compute the local mass matrix of element. More...

virtual void EvaluateSectionDisplacement (const double eta, ChVector<> &u_displ, ChVector<> &u_rotaz) override
Gets the xyz displacement of a point on the beam line, and the rotation RxRyRz of section plane, at abscyssa 'eta'. More...

virtual void EvaluateSectionForceTorque (const double eta, ChVector<> &Fforce, ChVector<> &Mtorque) override
Gets the force (traction x, shear y, shear z) and the torque (torsion on x, bending on y, on bending on z) at a section along the beam line, at abscissa 'eta'. More...

virtual void EvaluateSectionStrain (const double eta, ChVector<> &StrainV) override
Gets the axial and bending strain of the ANCF "cable" element.

virtual void EvaluateSectionStrain (const double eta, ChVector<> &StrainV_trans, ChVector<> &StrainV_rot) override
Gets the strains(traction along x, shear along y, along shear z, torsion about x, bending about y, on bending about z) at a section along the beam line, at abscissa 'eta'. More...

virtual void ComputeNF (const double U, ChVectorDynamic<> &Qi, double &detJ, const ChVectorDynamic<> &F, ChVectorDynamic<> *state_x, ChVectorDynamic<> *state_w) override
Evaluate N'*F , where N is some type of shape function evaluated at U coordinates of the line, each ranging in -1..+1 F is a load, N'*F is the resulting generalized load Returns also det[J] with J=[dx/du,..], that might be useful in gauss quadrature. More...

virtual void ComputeNF (const double U, const double V, const double W, ChVectorDynamic<> &Qi, double &detJ, const ChVectorDynamic<> &F, ChVectorDynamic<> *state_x, ChVectorDynamic<> *state_w) override
Evaluate N'*F , where N is some type of shape function evaluated at U,V,W coordinates of the volume, each ranging in -1..+1 F is a load, N'*F is the resulting generalized load Returns also det[J] with J=[dx/du,..], that might be useful in gauss quadrature. More...

Public Member Functions inherited from chrono::fea::ChElementBeamTaperedTimoshenko
virtual int GetNnodes () override
Get the number of nodes used by this element.

virtual int GetNdofs () override
Get the number of coordinates in the field used by the referenced nodes. More...

virtual int GetNodeNdofs (int n) override
Get the number of coordinates from the specified node that are used by this element. More...

virtual std::shared_ptr< ChNodeFEAbaseGetNodeN (int n) override
Access the nth node.

virtual void SetNodes (std::shared_ptr< ChNodeFEAxyzrot > nodeA, std::shared_ptr< ChNodeFEAxyzrot > nodeB)

void SetTaperedSection (std::shared_ptr< ChBeamSectionTaperedTimoshenkoAdvancedGeneric > my_material)
Set the tapered section & material of beam element .

Get the tapered section & material of the element.

std::shared_ptr< ChNodeFEAxyzrotGetNodeA ()
Get the first node (beginning)

std::shared_ptr< ChNodeFEAxyzrotGetNodeB ()
Get the second node (ending)

void SetNodeAreferenceRot (ChQuaternion<> mrot)
Set the reference rotation of nodeA respect to the element rotation.

ChQuaternion GetNodeAreferenceRot ()
Get the reference rotation of nodeA respect to the element rotation.

void SetNodeBreferenceRot (ChQuaternion<> mrot)
Set the reference rotation of nodeB respect to the element rotation.

ChQuaternion GetNodeBreferenceRot ()
Get the reference rotation of nodeB respect to the element rotation.

ChQuaternion GetAbsoluteRotation ()
Get the absolute rotation of element in space This is not the same of Rotation() , that expresses the accumulated rotation from starting point.

ChQuaternion GetRefRotation ()
Get the original reference rotation of element in space.

void SetDisableCorotate (bool md)
Set this as true to have the beam behave like a non-corotated beam hence do not update the corotated reference. More...

void SetForceSymmetricStiffness (bool md)
Set this as true to force the tangent stiffness matrix to be inexact, but symmetric. More...

void SetUseGeometricStiffness (bool md)
Set this as false to disable the contribution of geometric stiffness to the total tangent stiffness. More...

void SetUseRc (bool md)
Set this as true to include the transformation matrix due to the different elastic center offsets at two ends of beam element with respect to the centerline reference, in which case the connection line of two elastic centers is not parallel to the one of two centerline references at two ends. More...

void SetUseRs (bool md)
Set this as true to include the transformation matrix due to the different shear center offsets at two ends of beam element with respect to the centerline reference, in which case the connection line of two shear centers is not parallel to the one of two centerline references at two ends. More...

void SetUseSimplifiedCorrectionForInclinedShearAxis (bool md)
Set this as true to use a simplified correction model for the case of inclined shear axis. More...

void ShapeFunctionsTimoshenko (ShapeFunctionGroup &NN, double eta)
Shape functions for Timoshenko beam. More...

virtual void Update () override
Update, called at least at each time step. More...

virtual void UpdateRotation () override
Compute large rotation of element for corotational approach The reference frame of this Euler-Bernoulli beam has X aligned to two nodes and Y parallel to Y of 1st node.

virtual void GetStateBlock (ChVectorDynamic<> &mD) override
Fills the D vector with the current field values at the nodes of the element, with proper ordering. More...

void GetField_dt (ChVectorDynamic<> &mD_dt)
Fills the Ddt vector with the current time derivatives of field values at the nodes of the element, with proper ordering. More...

void GetField_dtdt (ChVectorDynamic<> &mD_dtdt)
Fills the Ddtdt vector with the current time derivatives of field values at the nodes of the element, with proper ordering. More...

void ComputeStiffnessMatrix ()
Computes the local (material) stiffness matrix of the element: K = integral( [B]' * [D] * [B] ), Note: in this 'basic' implementation, constant section and constant material are assumed, so the explicit result of quadrature is used. More...

void ComputeDampingMatrix ()
Computes the local (material) damping matrix of the element: R = beta * integral( [B]' * [D] * [B] ), Note: in this 'basic' implementation, constant section and constant material are assumed, so the explicit result of quadrature is used. More...

void ComputeGeometricStiffnessMatrix ()
Computes the local geometric stiffness Kg of the element. More...

void ComputeKiRimatricesLocal (bool inertial_damping, bool inertial_stiffness)
Compute the inertia stiffness matrix [Ki^] and inertial damping matrix [Ri^] which are due to the gyroscopic effect.

virtual void ComputeKRMmatricesGlobal (ChMatrixRef H, double Kfactor, double Rfactor=0, double Mfactor=0) override
Sets H as the global stiffness matrix K, scaled by Kfactor. More...

virtual void GetKRMmatricesLocal (ChMatrixRef H, double Kmfactor, double Kgfactor, double Rmfactor, double Mfactor)
Gets the material mass, material stiffness, material damping and geometric stiffness matrices in local basis. More...

virtual void ComputeInternalForces (ChVectorDynamic<> &Fi) override
Computes the internal forces (e.g. More...

virtual void ComputeInternalForces (ChVectorDynamic<> &Fi, bool Mfactor, bool Kfactor, bool Rfactor, bool Gfactor)
Computes the inertial forces, damping forces, centrifugal forces and gyroscopic moments, then you could consider them as applied external forces, if you want to do the static solve when including nodal velocites and accelarations. More...

virtual void ComputeGravityForces (ChVectorDynamic<> &Fg, const ChVector<> &G_acc) override
Compute gravity forces, grouped in the Fg vector, one node after the other. More...

virtual void EvaluateSectionFrame (const double eta, ChVector<> &point, ChQuaternion<> &rot) override
Gets the absolute xyz position of a point on the beam line, and the absolute rotation of section plane, at abscissa 'eta'. More...

virtual void EvaluateElementStrainEnergy (ChVector<> &StrainEnergyV_trans, ChVector<> &StrainEnergyV_rot)
Gets the elastic strain energy(traction along x, shear along y, along shear z, torsion about x, bending about y, on bending about z) in the element.

virtual void EvaluateElementDampingEnergy (ChVector<> &DampingEnergyV_trans, ChVector<> &DampingEnergyV_rot)
Gets the damping dissipated energy(traction along x, shear along y, along shear z, torsion about x, bending about y, on bending about z) in the element.

Gets the number of DOFs affected by this element (position part)

Gets the number of DOFs affected by this element (speed part)

virtual void LoadableGetStateBlock_x (int block_offset, ChState &mD) override
Gets all the DOFs packed in a single vector (position part)

virtual void LoadableGetStateBlock_w (int block_offset, ChStateDelta &mD) override
Gets all the DOFs packed in a single vector (speed part)

virtual void LoadableStateIncrement (const unsigned int off_x, ChState &x_new, const ChState &x, const unsigned int off_v, const ChStateDelta &Dv) override
Increment all DOFs using a delta.

virtual int Get_field_ncoords () override
Number of coordinates in the interpolated field, ex=3 for a tetrahedron finite element or a cable, = 1 for a thermal problem, etc.

virtual int GetSubBlocks () override
Get the number of DOFs sub-blocks.

virtual unsigned int GetSubBlockOffset (int nblock) override
Get the offset of the specified sub-block of DOFs in global vector.

virtual unsigned int GetSubBlockSize (int nblock) override
Get the size of the specified sub-block of DOFs in global vector.

virtual bool IsSubBlockActive (int nblock) const override
Check if the specified sub-block of DOFs is active.

virtual void LoadableGetVariables (std::vector< ChVariables * > &mvars) override
Get the pointers to the contained ChVariables, appending to the mvars vector.

virtual double GetDensity () override
This is needed so that it can be accessed by ChLoaderVolumeGravity.

Public Member Functions inherited from chrono::fea::ChElementBeam
double GetMass ()
The full mass of the beam, (with const. section, density, etc.)

double GetRestLength ()
The rest length of the bar.

void SetRestLength (double ml)
Set the rest length of the bar (usually this should be automatically done when SetupInitial is called on beams element, given the current state, but one might need to override this, ex for precompressed beams etc).

Public Member Functions inherited from chrono::fea::ChElementGeneric
ChKblockGenericKstiffness ()
Access the proxy to stiffness, for sparse solver.

virtual void EleIntLoadResidual_F (ChVectorDynamic<> &R, const double c) override
Add the internal forces (pasted at global nodes offsets) into a global vector R, multiplied by a scaling factor c, as R += forces * c This default implementation is SLIGHTLY INEFFICIENT.

virtual void EleIntLoadResidual_Mv (ChVectorDynamic<> &R, const ChVectorDynamic<> &w, const double c) override
Add the product of element mass M by a vector w (pasted at global nodes offsets) into a global vector R, multiplied by a scaling factor c, as R += M * w * c This default implementation is VERY INEFFICIENT.

virtual void EleIntLoadResidual_F_gravity (ChVectorDynamic<> &R, const ChVector<> &G_acc, const double c) override
Add the contribution of gravity loads, multiplied by a scaling factor c, as: R += M * g * c This default implementation is VERY INEFFICIENT. More...

virtual void ComputeMmatrixGlobal (ChMatrixRef M) override
Calculate the mass matrix, expressed in global reference. More...

virtual void InjectKRMmatrices (ChSystemDescriptor &descriptor) override
Tell to a system descriptor that there are item(s) of type ChKblock in this object (for further passing it to a solver)

virtual void KRMmatricesLoad (double Kfactor, double Rfactor, double Mfactor) override
Add the current stiffness K and damping R and mass M matrices in encapsulated ChKblock item(s), if any. More...

virtual void VariablesFbLoadInternalForces (double factor=1.) override
Add the internal forces, expressed as nodal forces, into the encapsulated ChVariables.

virtual void VariablesFbIncrementMq () override
Add M*q (internal masses multiplied current 'qb').

Public Member Functions inherited from chrono::fea::ChElementBase
virtual int GetNdofs_active ()
Get the actual number of active degrees of freedom. More...

virtual int GetNodeNdofs_active (int n)
Get the actual number of active coordinates from the specified node that are used by this element. More...

virtual void ComputeNodalMass ()
Compute element's nodal masses.

virtual void EleDoIntegration ()
This is optionally implemented if there is some internal state that requires integration.

Public Member Functions inherited from chrono::ChLoadableUVW
virtual bool IsTetrahedronIntegrationNeeded ()
If true, use quadrature over u,v,w in [0..1] range as tetrahedron volumetric coords (with z=1-u-v-w) otherwise use default quadrature over u,v,w in [-1..+1] as box isoparametric coords.

virtual bool IsTrianglePrismIntegrationNeeded ()
If true, use quadrature over u,v in [0..1] range as triangle natural coords (with z=1-u-v), and use linear quadrature over w in [-1..+1], otherwise use default quadrature over u,v,w in [-1..+1] as box isoparametric coords.

Public Member Functions inherited from chrono::fea::ChElementCorotational
ChMatrix33Rotation ()
Access the cumulative rotation matrix of the element. More...

Protected Member Functions inherited from chrono::fea::ChElementBeamTaperedTimoshenko
void ComputeTransformMatrix ()
compute the transformation matrix due to offset and rotation of axes.

void ComputeTransformMatrixAtPoint (ChMatrixDynamic<> &mT, const double eta)
compute the transformation matrix due to offset and rotation of axes, at dimensionless abscissa eta.

Protected Attributes inherited from chrono::fea::ChElementBeamTaperedTimoshenko
std::vector< std::shared_ptr< ChNodeFEAxyzrot > > nodes

Tapered section & material of beam element.

ChMatrixDynamic Km
local material stiffness matrix

ChMatrixDynamic Kg
local geometric stiffness matrix NORMALIZED by P

ChMatrixDynamic M
local material mass matrix. More...

ChMatrixDynamic Rm
local material damping matrix

ChMatrixDynamic Ri
local inertial-damping (gyroscopic damping) matrix

ChMatrixDynamic Ki
local inertial-stiffness matrix

ChQuaternion q_refrotA

ChQuaternion q_refrotB

ChQuaternion q_element_abs_rot

ChQuaternion q_element_ref_rot

bool disable_corotate

bool force_symmetric_stiffness

bool use_geometric_stiffness
whether include geometric stiffness matrix

bool use_Rc
whether use the transformation matrix for elastic axis orientation

bool use_Rs
whether use the transformation matrix for shear axis orientation

bool use_simplified_correction_for_inclined_shear_axis = false
whether use the simplified correction model for shear axis orientation, it's false as default.

ChMatrixDynamic T
transformation matrix for entire beam element

ChMatrixDynamic Rc
transformation matrix for elastic axis orientation

ChMatrixDynamic Rs
transformation matrix for shear axis orientation

Protected Attributes inherited from chrono::fea::ChElementBeam
double mass

double length

Protected Attributes inherited from chrono::fea::ChElementGeneric
ChKblockGeneric Kmatr

Protected Attributes inherited from chrono::fea::ChElementCorotational
ChMatrix33 A

## ◆ ComputeConsistentMassMatrix()

 void chrono::fea::ChElementBeamTaperedTimoshenkoFPM::ComputeConsistentMassMatrix ( )

Computes the local element consistent mass matrix via Guass Quadrature: M = integral( [N]' * [D] * [N] ), Note: the sectional properties at Gauss integration point are linearly interpolated from two ends of tapered beam.

Also, this local element consistent mass matrix is constant, computed only at the beginning for performance reasons; if you later change some material property, call this or InitialSetup().

## ◆ ComputeDampingMatrix()

 void chrono::fea::ChElementBeamTaperedTimoshenkoFPM::ComputeDampingMatrix ( )

Computes the local element damping matrix via Guass Quadrature: R = beta * integral( [B]' * [D] * [B] ), Note: the sectional properties at Gauss integration point are linearly interpolated from two ends of tapered beam.

Only the stiffness term(beta) is used for this implemented Rayleigh damping model. Also, this local material damping matrix is constant, computed only at the beginning for performance reasons; if you later change some material property, call this or InitialSetup().

## ◆ ComputeMassMatrix()

 void chrono::fea::ChElementBeamTaperedTimoshenkoFPM::ComputeMassMatrix ( )

Finally, compute the local mass matrix of element.

It could be in lumped or consistent format, depending on the parameter 'use_lumped_mass_matrix' in its sectional settings.

## ◆ ComputeNF() [1/2]

 void chrono::fea::ChElementBeamTaperedTimoshenkoFPM::ComputeNF ( const double U, ChVectorDynamic<> & Qi, double & detJ, const ChVectorDynamic<> & F, ChVectorDynamic<> * state_x, ChVectorDynamic<> * state_w )
overridevirtual

Evaluate N'*F , where N is some type of shape function evaluated at U coordinates of the line, each ranging in -1..+1 F is a load, N'*F is the resulting generalized load Returns also det[J] with J=[dx/du,..], that might be useful in gauss quadrature.

Parameters
 U parametric coordinate in line Qi Return result of Q = N'*F here detJ Return det[J] here F Input F vector, size is =n. field coords. state_x if != 0, update state (pos. part) to this, then evaluate Q state_w if != 0, update state (speed part) to this, then evaluate Q

Reimplemented from chrono::fea::ChElementBeamTaperedTimoshenko.

## ◆ ComputeNF() [2/2]

 void chrono::fea::ChElementBeamTaperedTimoshenkoFPM::ComputeNF ( const double U, const double V, const double W, ChVectorDynamic<> & Qi, double & detJ, const ChVectorDynamic<> & F, ChVectorDynamic<> * state_x, ChVectorDynamic<> * state_w )
overridevirtual

Evaluate N'*F , where N is some type of shape function evaluated at U,V,W coordinates of the volume, each ranging in -1..+1 F is a load, N'*F is the resulting generalized load Returns also det[J] with J=[dx/du,..], that might be useful in gauss quadrature.

Parameters
 U parametric coordinate in volume V parametric coordinate in volume W parametric coordinate in volume Qi Return result of N'*F here, maybe with offset block_offset detJ Return det[J] here F Input F vector, size is = n.field coords. state_x if != 0, update state (pos. part) to this, then evaluate Q state_w if != 0, update state (speed part) to this, then evaluate Q

Reimplemented from chrono::fea::ChElementBeamTaperedTimoshenko.

## ◆ ComputeStiffnessMatrix()

 void chrono::fea::ChElementBeamTaperedTimoshenkoFPM::ComputeStiffnessMatrix ( )

Computes the local (material) stiffness matrix of the element: K = integral( [B]' * [D] * [B] ), Note: the sectional properties at Gauss integration point are linearly interpolated from two ends of tapered beam.

Also, this local material stiffness matrix is constant, computed only at the beginning for performance reasons; if you later change some material property, call this or InitialSetup().

## ◆ EvaluateSectionDisplacement()

 void chrono::fea::ChElementBeamTaperedTimoshenkoFPM::EvaluateSectionDisplacement ( const double eta, ChVector<> & u_displ, ChVector<> & u_rotaz )
overridevirtual

Gets the xyz displacement of a point on the beam line, and the rotation RxRyRz of section plane, at abscyssa 'eta'.

Note, eta=-1 at node1, eta=+1 at node2. Results are not corotated.

Reimplemented from chrono::fea::ChElementBeamTaperedTimoshenko.

## ◆ EvaluateSectionForceTorque()

 void chrono::fea::ChElementBeamTaperedTimoshenkoFPM::EvaluateSectionForceTorque ( const double eta, ChVector<> & Fforce, ChVector<> & Mtorque )
overridevirtual

Gets the force (traction x, shear y, shear z) and the torque (torsion on x, bending on y, on bending on z) at a section along the beam line, at abscissa 'eta'.

Note, eta=-1 at node1, eta=+1 at node2. Results are not corotated, and are expressed in the reference system of beam.

Reimplemented from chrono::fea::ChElementBeamTaperedTimoshenko.

## ◆ EvaluateSectionStrain()

 void chrono::fea::ChElementBeamTaperedTimoshenkoFPM::EvaluateSectionStrain ( const double eta, ChVector<> & StrainV_trans, ChVector<> & StrainV_rot )
overridevirtual

Gets the strains(traction along x, shear along y, along shear z, torsion about x, bending about y, on bending about z) at a section along the beam line, at abscissa 'eta'.

It's evaluated at the elastic center. Note, eta=-1 at node1, eta=+1 at node2. Results are not corotated, and are expressed in the reference system of beam.

Reimplemented from chrono::fea::ChElementBeamTaperedTimoshenko.

## ◆ ShapeFunctionsTimoshenkoFPM()

 void chrono::fea::ChElementBeamTaperedTimoshenkoFPM::ShapeFunctionsTimoshenkoFPM ( ShapeFunctionGroupFPM & NB, double eta )

Computes the shape function matrix 'Nx' and strain-displacement relation matrix 'Bx' at dimensionless abscissa 'eta'.

Note, eta=-1 at node1, eta=+1 at node2.

Parameters
 NB shape function matrix 'Nx' and strain-displacement relation matrix 'Bx' are stored here. eta abscissa 'eta'. eta=-1 at node1, eta=+1 at node2.

The documentation for this class was generated from the following files:
• /builds/uwsbel/chrono/src/chrono/fea/ChElementBeamTaperedTimoshenkoFPM.h
• /builds/uwsbel/chrono/src/chrono/fea/ChElementBeamTaperedTimoshenkoFPM.cpp