## Description

Base class for all constitutive models of sections of Timoshenko beams.

To be used with ChElementBeamTaperedTimoshenko.

#include <ChBeamSectionTaperedTimoshenko.h>

## Public Member Functions

ChBeamSectionTimoshenkoAdvancedGeneric (const double mAx, const double mTxx, const double mByy, const double mBzz, const double mGAyy, const double mGAzz, const DampingCoefficients mdamping_coeff, const double malpha, const double mCy, const double mCz, const double mSy, const double mSz, const double mmu, const double mJyy, const double mJzz, const double mJyz=0, const double mQy=0, const double mQz=0, const double mMy=0, const double mMz=0)

virtual void SetYshearRigidity (const double mv)
Sets the shear rigidity, for shearing along Y axis, at shear center, usually Ayy*G for uniform elasticity, but for nonuniform elasticity here you can put a value ad-hoc from a preprocessor.

virtual void SetZshearRigidity (const double mv)
Sets the shear rigidity, for shearing along Z axis, at shear center, usually Azz*G for uniform elasticity, but for nonuniform elasticity here you can put a value ad-hoc from a preprocessor.

virtual double GetYshearRigidity () const
Gets the shear rigidity, for shearing along Y axis at shear center, usually Ayy*G, but might be ad hoc.

virtual double GetZshearRigidity () const
Gets the shear rigidity, for shearing along Z axis at shear center, usually Azz*G, but might be ad hoc.

virtual void SetBeamRaleyghDamping (DampingCoefficients mdamping_coeff)
Sets the damping parameters of section. More...

virtual DampingCoefficients GetBeamRaleyghDamping () const
Gets the damping parameters of section.

virtual void SetInertiasPerUnitLength (const double mJyy, const double mJzz, const double mJyz, const double mQy, const double mQz)
Set the Jyy Jzz Jyz components of the sectional inertia per unit length, in centerline reference, measured along centerline main axes. More...

virtual void SetMainInertiasInMassReference (const double Jmyy, const double Jmzz, const double Jmyz, const double mass_phi, const double Qmy, const double Qmz)
Set inertia moments, per unit length, as assumed computed in the Ym Zm "mass reference" frame, ie. More...

virtual void GetMainInertiasInMassReference (double &Jmyy, double &Jmzz, double &Jmyz, double &mass_phi, double &Qmy, double &Qmz)
Get inertia moments, per unit length, as assumed computed in the Ym Zm "mass reference" frame, and the rotation phi of that frame, ie. More...

virtual double GetInertiaJxxPerUnitLengthInMassReference () const
Get inertia moment per unit length Jxx_massref, as assumed computed in the "mass reference" frame, ie. More...

virtual double GetInertiaJyyPerUnitLength () const
Get the Jyy component of the sectional inertia per unit length, in centerline reference, measured along centerline main axes.

virtual double GetInertiaJzzPerUnitLength () const
Get the Jzz component of the sectional inertia per unit length, in centerline reference, measured along centerline main axes.

virtual double GetInertiaJyzPerUnitLength () const
Get the Jyz component of the sectional inertia per unit length, in centerline reference, measured along centerline main axes.

virtual double GetInertiaQyPerUnitLength () const
Get the Qy component of mass moment of area per unit length, in centerline reference, measured along centerline main axes.

virtual double GetInertiaQzPerUnitLength () const
Get the Qz component of mass moment of area per unit length, in centerline reference, measured along centerline main axes.

virtual void SetArtificialFactorForShearDamping (double mv)
Set the artificial correction factor for the structural damping model in the shear deformation, as default it's 1.0 and don't have any correction effect. More...

virtual double GetArtificialFactorForShearDamping () const
Get the artificial correction factor for the structural damping model in the shear deformation.

Public Member Functions inherited from chrono::fea::ChBeamSectionRayleighAdvancedGeneric
ChBeamSectionRayleighAdvancedGeneric (const double mAx, const double mTxx, const double mByy, const double mBzz, const double malpha, const double mCy, const double mCz, const double mSy, const double mSz, const double mmu, const double mJyy, const double mJzz, const double mJyz, const double mMy=0, const double mMz=0)

virtual void SetInertiasPerUnitLength (const double mJyy, const double mJzz, const double mJyz)
Set the Jyy Jzz Jyz components of the sectional inertia per unit length, in centerline reference, measured along centerline main axes. More...

virtual void SetMainInertiasInMassReference (double Jmyy, double Jmzz, double phi)
Set inertia moments, per unit length, as assumed computed in the Ym Zm "mass reference" frame, ie. More...

virtual void GetMainInertiasInMassReference (double &Jmyy, double &Jmzz, double &phi)
Get inertia moments, per unit length, as assumed computed in the Ym Zm "mass reference" frame, and the rotation phi of that frame, ie. More...

virtual void ComputeInertiaMatrix (ChMatrixNM< double, 6, 6 > &M) override
Compute the 6x6 sectional inertia matrix, as in {x_momentum,w_momentum}=[Mm]{xvel,wvel}.

virtual void ComputeInertiaDampingMatrix (ChMatrixNM< double, 6, 6 > &Ri, const ChVector<> &mW) override
Compute the 6x6 sectional inertia damping matrix [Ri] (gyroscopic matrix damping) More...

virtual void ComputeInertiaStiffnessMatrix (ChMatrixNM< double, 6, 6 > &Ki, const ChVector<> &mWvel, const ChVector<> &mWacc, const ChVector<> &mXacc) override
Compute the 6x6 sectional inertia stiffness matrix [Ki^]. More...

virtual void ComputeQuadraticTerms (ChVector<> &mF, ChVector<> &mT, const ChVector<> &mW) override
Compute the centrifugal term and gyroscopic term. More...

Public Member Functions inherited from chrono::fea::ChBeamSectionEulerAdvancedGeneric
ChBeamSectionEulerAdvancedGeneric (const double mAx, const double mTxx, const double mByy, const double mBzz, const double malpha, const double mCy, const double mCz, const double mSy, const double mSz, const double mmu, const double mJxx, const double mMy=0, const double mMz=0)

virtual void SetAxialRigidity (const double mv)
Sets the axial rigidity, usually A*E for uniform elasticity, but for nonuniform elasticity here you can put a value ad-hoc from a preprocessor.

virtual void SetXtorsionRigidity (const double mv)
Sets the torsion rigidity, for torsion about X axis, at elastic center, usually J*G for uniform elasticity, but for nonuniform elasticity here you can put a value ad-hoc from a preprocessor.

virtual void SetYbendingRigidity (const double mv)
Sets the bending rigidity, for bending about Y axis, at elastic center, usually Iyy*E for uniform elasticity, but for nonuniform elasticity here you can put a value ad-hoc from a preprocessor.

virtual void SetZbendingRigidity (const double mv)
Sets the bending rigidity, for bending about Z axis, at elastic center, usually Izz*E for uniform elasticity, but for nonuniform elasticity here you can put a value ad-hoc from a preprocessor.

virtual void SetSectionRotation (const double mv)
Set the rotation in [rad], abour elastic center, of the Y Z axes for which the YbendingRigidity and ZbendingRigidity values are defined.

virtual void SetCentroidY (const double mv)
Sets the Y position of the elastic center respect to centerline.

virtual void SetCentroidZ (const double mv)
Sets the Z position of the elastic center respect to centerline.

virtual void SetShearCenterY (const double mv)
Sets the Y position of the shear center respect to centerline.

virtual void SetShearCenterZ (const double mv)
Sets the Z position of the shear center respect to centerline.

virtual void SetMassPerUnitLength (const double mv)
Set mass per unit length, ex.SI units [kg/m] For uniform density it would be A*density, but for nonuniform density here you can put a value ad-hoc from a preprocessor.

virtual void SetInertiaJxxPerUnitLength (const double mv)
Set the Jxx component of the inertia per unit length (polar inertia), computed at centerline. More...

virtual void SetInertiaJxxPerUnitLengthInMassReference (const double mv)
Set inertia moment per unit length Jxx_massref, as assumed computed in the "mass reference" frame, ie. More...

virtual double GetInertiaJxxPerUnitLengthInMassReference ()
Get inertia moment per unit length Jxx_massref, as assumed computed in the "mass reference" frame, ie. More...

void SetCenterOfMass (double my, double mz)
"mass reference": set the displacement of the center of mass respect to the section centerline reference.

double GetCenterOfMassY ()

double GetCenterOfMassZ ()

virtual double GetAxialRigidity () const override
Gets the axial rigidity, usually A*E, but might be ad hoc.

virtual double GetXtorsionRigidity () const override
Gets the torsion rigidity, for torsion about X axis at elastic center, usually J*G, but might be ad hoc.

virtual double GetYbendingRigidity () const override
Gets the bending rigidity, for bending about Y axis at elastic center, usually Iyy*E, but might be ad hoc.

virtual double GetZbendingRigidity () const override
Gets the bending rigidity, for bending about Z axis at elastic center, usually Izz*E, but might be ad hoc.

virtual double GetSectionRotation () const override
Set the rotation of the Y Z section axes for which the YbendingRigidity and ZbendingRigidity are defined.

virtual double GetCentroidY () const override
Gets the Y position of the elastic center respect to centerline.

virtual double GetCentroidZ () const override
Gets the Z position of the elastic center respect to centerline.

virtual double GetShearCenterY () const override
Gets the Y position of the shear center respect to centerline.

virtual double GetShearCenterZ () const override
Gets the Z position of the shear center respect to centerline.

virtual double GetMassPerUnitLength () const override
Get mass per unit length, ex.SI units [kg/m].

virtual double GetInertiaJxxPerUnitLength () const override
Get the Jxx component of the inertia per unit length (polar inertia), at centerline.

Public Member Functions inherited from chrono::fea::ChBeamSectionEuler
virtual void ComputeInertialForce (ChVector<> &mFi, ChVector<> &mTi, const ChVector<> &mWvel, const ChVector<> &mWacc, const ChVector<> &mXacc)
Compute the total inertial forces (per unit length). More...

void SetArtificialJyyJzzFactor (double mf)
The Euler beam model has no rotational inertia per each section, assuming mass is concentrated on the centerline. More...

double GetArtificialJyyJzzFactor ()

virtual void SetBeamRaleyghDampingAlpha (double malpha)
Set the "alpha" Rayleigh damping ratio,
the mass-proportional structural damping in: R = alpha*M + beta*K

double GetBeamRaleyghDampingAlpha ()

virtual void SetBeamRaleyghDampingBeta (double mbeta)
Set the "beta" Rayleigh damping ratio, the stiffness-proportional structural damping in: R = alpha*M + beta*K

double GetBeamRaleyghDampingBeta ()

virtual void SetBeamRaleyghDamping (double mbeta, double malpha=0)
Set both beta and alpha coefficients in Rayleigh damping model: R = alpha*M + beta*K. More...

Public Member Functions inherited from chrono::fea::ChBeamSection
void SetDrawShape (std::shared_ptr< ChBeamSectionShape > mshape)
Set the graphical representation for this section. More...

std::shared_ptr< ChBeamSectionShapeGetDrawShape () const
Get the drawing shape of this section (i.e.a 2D profile used for drawing 3D tesselation and visualization) By default a thin square section, use SetDrawShape() to change it.

void SetDrawThickness (double thickness_y, double thickness_z)
Shortcut: adds a ChBeamSectionShapeRectangular for visualization as a centered rectangular beam, and sets its width/height. More...

Shortcut: adds a ChBeamSectionShapeCircular for visualization as a centered circular beam, and sets its radius. More...

void SetCircular (bool ic)
OBSOLETE only for backward compability

## Protected Attributes

double GAyy
shear regidity along yy, evaluated in the elastic center and elastic principal axis. GAyy=ky*G*A

double GAzz
shear regidity along zz, evaluated in the elastic center and elastic principal axis. GAzz=kz*G*A

double Qy
Mass moment of area, defined as : $$Q_{y} = \int_\Omega \rho z d\Omega$$, $$Q_{z} = \int_\Omega \rho y d\Omega$$,. More...

double Qz
mass moment of area along zz, in centerline reference, measured along centerline main axes

DampingCoefficients rdamping_coeff
Damping parameters. More...

double artificial_factor_for_shear_damping = 1.0
An artificial correction factor for the structural damping model in the shear deformation, as default it's 1.0 and don't have any correction effect. More...

double Jzz

double Jyy

double Jyz

double Ax

double Txx

double Byy

double Bzz

double alpha

double Cy

double Cz

double Sy

double Sz

double mu

double Jxx

double My

double Mz

Protected Attributes inherited from chrono::fea::ChBeamSectionEuler
double rdamping_beta

double rdamping_alpha

double JzzJyy_factor

Public Attributes inherited from chrono::fea::ChBeamSectionEuler
bool compute_inertia_damping_matrix = true
Flag that turns on/off the computation of the [Ri] 'gyroscopic' inertial damping matrix. More...

bool compute_inertia_stiffness_matrix = true
Flag that turns on/off the computation of the [Ki] inertial stiffness matrix. More...

bool compute_Ri_Ki_by_num_diff = false
Flag for computing the Ri and Ki matrices via numerical differentiation even if an analytical expression is provided. More...

## Constructor & Destructor Documentation

 chrono::fea::ChBeamSectionTimoshenkoAdvancedGeneric::ChBeamSectionTimoshenkoAdvancedGeneric ( const double mAx, const double mTxx, const double mByy, const double mBzz, const double mGAyy, const double mGAzz, const DampingCoefficients mdamping_coeff, const double malpha, const double mCy, const double mCz, const double mSy, const double mSz, const double mmu, const double mJyy, const double mJzz, const double mJyz = 0, const double mQy = 0, const double mQz = 0, const double mMy = 0, const double mMz = 0 )
inline
Parameters
 mAx axial rigidity mTxx torsion rigidity mByy bending regidity about yy mBzz bending rigidity about zz mGAyy shear rigidity along yy mGAzz shear rigidity along zz malpha damping coefficients section rotation about elastic center [rad] mCy elastic center y displacement respect to centerline mCz elastic center z displacement respect to centerline mSy shear center y displacement respect to centerline mSz shear center z displacement respect to centerline mmu mass per unit length mJyy inertia Jyy per unit lenght, in centerline reference, measured along centerline main axes mJzz inertia Jzz per unit lenght, in centerline reference, measured along centerline main axes mJyz inertia Jyz per unit lenght, in centerline reference, measured along centerline main axes const mQy inertia Qy per unit lenght, in centerline reference, measured along centerline main axes mQz inertia Qz per unit lenght, in centerline reference, measured along centerline main axes mMy mass center y displacement respect to centerline mMz mass center z displacement respect to centerline

## ◆ GetInertiaJxxPerUnitLengthInMassReference()

 virtual double chrono::fea::ChBeamSectionTimoshenkoAdvancedGeneric::GetInertiaJxxPerUnitLengthInMassReference ( ) const
inlinevirtual

Get inertia moment per unit length Jxx_massref, as assumed computed in the "mass reference" frame, ie.

centered at the center of mass NOTE: To be safe, it is recommended to use GetMainInertiasInMassReference() instead, because Qmy,Qmz are ignored, although they are exactly zero in theory.

## ◆ GetMainInertiasInMassReference()

 void chrono::fea::ChBeamSectionTimoshenkoAdvancedGeneric::GetMainInertiasInMassReference ( double & Jmyy, double & Jmzz, double & Jmyz, double & mass_phi, double & Qmy, double & Qmz )
virtual

Get inertia moments, per unit length, as assumed computed in the Ym Zm "mass reference" frame, and the rotation phi of that frame, ie.

inertias centered at the center of mass and rotated by phi angle to match the main axes of inertia: $$Jm_{yy} = \int_\Omega \rho z_{m}^2 d\Omega$$, $$Jm_{zz} = \int_\Omega \rho y_{m}^2 d\Omega$$. Assuming the center of mass is already set.

## ◆ SetArtificialFactorForShearDamping()

 virtual void chrono::fea::ChBeamSectionTimoshenkoAdvancedGeneric::SetArtificialFactorForShearDamping ( double mv )
inlinevirtual

Set the artificial correction factor for the structural damping model in the shear deformation, as default it's 1.0 and don't have any correction effect.

But when the numerical divergence occured, you could try to set this value be larger than 1.0 to improve the numerical stability in long-time simulation.

## ◆ SetBeamRaleyghDamping()

 virtual void chrono::fea::ChBeamSectionTimoshenkoAdvancedGeneric::SetBeamRaleyghDamping ( DampingCoefficients mdamping_coeff )
inlinevirtual

Sets the damping parameters of section.

You have a chance to assign different coefficients for axial, bending and torsion directions. This would be helpful for those anisotropic material, such as wind turbine blade.

## ◆ SetInertiasPerUnitLength()

 void chrono::fea::ChBeamSectionTimoshenkoAdvancedGeneric::SetInertiasPerUnitLength ( const double mJyy, const double mJzz, const double mJyz, const double mQy, const double mQz )
virtual

Set the Jyy Jzz Jyz components of the sectional inertia per unit length, in centerline reference, measured along centerline main axes.

These are defined as: $$J_{yy} = \int_\Omega \rho z^2 d\Omega$$, also Jyy = Mm(4,4) $$J_{zz} = \int_\Omega \rho y^2 d\Omega$$, also Jzz = Mm(5,5) $$J_{yz} = \int_\Omega \rho y z d\Omega$$, also Jyz = -Mm(4,5) = -Mm(5,4) It is not needed to enter also Jxx because Jxx=(Jzz+Jyy) by the polar theorem.

## ◆ SetMainInertiasInMassReference()

 void chrono::fea::ChBeamSectionTimoshenkoAdvancedGeneric::SetMainInertiasInMassReference ( const double Jmyy, const double Jmzz, const double Jmyz, const double mass_phi, const double Qmy, const double Qmz )
virtual

Set inertia moments, per unit length, as assumed computed in the Ym Zm "mass reference" frame, ie.

centered at the center of mass and rotated by phi angle to match the main axes of inertia: $$Jm_{yy} = \int_\Omega \rho z_{m}^2 d\Omega$$, $$Jm_{zz} = \int_\Omega \rho y_{m}^2 d\Omega$$. Assuming the center of mass is already set.

## ◆ artificial_factor_for_shear_damping

protected

An artificial correction factor for the structural damping model in the shear deformation, as default it's 1.0 and don't have any correction effect.

But when the numerical divergence occured, you could try to set this value be larger than 1.0 to improve the numerical stability in long-time simulation.

## ◆ Qy

protected

Mass moment of area, defined as : $$Q_{y} = \int_\Omega \rho z d\Omega$$, $$Q_{z} = \int_\Omega \rho y d\Omega$$,.

mass moment of area along yy, in centerline reference, measured along centerline main axes