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:

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Collaboration diagram for chrono::fea::ChElementBeamTaperedTimoshenkoFPM:

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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.

std::shared_ptr< ChBeamSectionTaperedTimoshenkoAdvancedGenericFPM >	GetTaperedSection ()
	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
	Gets the number of nodes used by this element.

virtual int	GetNdofs () override
	Gets 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 n-th node that are used by this element. More...

virtual std::shared_ptr< ChNodeFEAbase >	GetNodeN (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 .

std::shared_ptr< ChBeamSectionTaperedTimoshenkoAdvancedGeneric >	GetTaperedSection ()
	Get the tapered section & material of the element.

std::shared_ptr< ChNodeFEAxyzrot >	GetNodeA ()
	Get the first node (beginning)

std::shared_ptr< ChNodeFEAxyzrot >	GetNodeB ()
	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	ShapeFunctionsTimoshenko (ShapeFunctionGroup &NN, double eta)
	Shape functions for Timoshenko beam. More...

virtual void	Update () override
	Update: this is 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	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.

virtual int	LoadableGet_ndof_x () override
	Gets the number of DOFs affected by this element (position part)

virtual int	LoadableGet_ndof_w () override
	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
ChKblockGeneric &	Kstiffness ()
	Access the proxy to stiffness, for sparse solver.

virtual void	EleIntLoadResidual_F (ChVectorDynamic<> &R, const double c) override
	(This is a default (a bit unoptimal) book keeping so that in children classes you can avoid implementing this EleIntLoadResidual_F function, unless you need faster code)

virtual void	EleIntLoadResidual_Mv (ChVectorDynamic<> &R, const ChVectorDynamic<> &w, const double c) override
	(This is a default (VERY UNOPTIMAL) book keeping so that in children classes you can avoid implementing this EleIntLoadResidual_Mv function, unless you need faster code.)

virtual void	EleIntLoadResidual_F_gravity (ChVectorDynamic<> &R, const ChVector<> &G_acc, const double c) override
	(This is a default (VERY UNOPTIMAL) book keeping so that in children classes you can avoid implementing this EleIntLoadResidual_F_gravity function, unless you need faster code. More...

virtual void	ComputeMmatrixGlobal (ChMatrixRef M) override
	Returns the global mass matrix. More...

virtual void	InjectKRMmatrices (ChSystemDescriptor &mdescriptor) 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
	Adds 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
	Adds the internal forces, expressed as nodal forces, into the encapsulated ChVariables, in the 'fb' part: qf+=forces*factor (This is a default (a bit unoptimal) book keeping so that in children classes you can avoid implementing this VariablesFbLoadInternalForces function, unless you need faster code)

virtual void	VariablesFbIncrementMq () override
	Adds M*q (internal masses multiplied current 'qb') to Fb, ex. More...

Public Member Functions inherited from chrono::fea::ChElementBase
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
ChMatrix33 &	Rotation ()
	Access the cumulative rotation matrix of the element. More...

Additional Inherited Members
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

std::shared_ptr< ChBeamSectionTaperedTimoshenkoAdvancedGeneric >	tapered_section
	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

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

Member Function Documentation

◆ 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

Description

Public Types

Public Member Functions

Additional Inherited Members

Member Function Documentation

◆ ComputeConsistentMassMatrix()

◆ ComputeDampingMatrix()

◆ ComputeMassMatrix()

◆ ComputeNF() [1/2]

◆ ComputeNF() [2/2]

◆ ComputeStiffnessMatrix()

◆ EvaluateSectionDisplacement()

◆ EvaluateSectionForceTorque()

◆ EvaluateSectionStrain()

◆ ShapeFunctionsTimoshenkoFPM()