ObjectBeamGeometricallyExact2D

A 2D geometrically exact beam finite element, using 2 or 3 nodes of type NodeRigidBody2D. Note that the orientation of the nodes need to follow the cross section orientation in case that includeReferenceRotations=True; e.g., an angle 0 represents the cross section aligned with the \(y\)-axis, while and angle \(\pi/2\) means that the cross section points in negative \(x\)-direction. Pre-curvature can be included with referenceCurvature and axial pre-stress can be considered by using a length different from the reference configuration of the nodes. The localPosition of the beam with length \(L\)=length and height \(h\) ranges in \(X\)-direction in range \([-L/2, L/2]\) and in \(Y\)-direction in range \([-h/2,h/2]\) (which is in fact not needed in the EOM).

Interface

Python names: ObjectBeamGeometricallyExact2D or Beam2D, and VBeam2D for its visualization
Nodes it takes: NodeRigidBody2D, NodePoint2DSlope1
Body markers that can be placed on it: MarkerBodyMass, MarkerBodyPosition, MarkerBodyRigid

Parameters

The parameters of the item; in a dictionary, its type is ‘BeamGeometricallyExact2D’:

name	type	default	description
name	String	‘’	objects’s unique name
nodeNumbers	ArrayNodeIndex	[]	two node numbers for beam element
length	UReal	0.	(symbol: \(L\)) [SI:m] reference length of beam; such that the total volume (e.g. for volume load) gives \(\rho A L\); must be positive
massPerLength	UReal	0.	(symbol: \(\rho A\)) [SI:kg/m] mass per length of beam
crossSectionInertia	UReal	0.	(symbol: \(\rho J\)) [SI:kg m] cross section mass moment of inertia; inertia acting against rotation of cross section
bendingStiffness	UReal	0.	(symbol: \(EI\)) [SI:Nm\(^2\)] bending stiffness of beam; the bending moment is \(m = EI (\kappa - \kappa_0)\), in which \(\kappa\) is the material measure of curvature
axialStiffness	UReal	0.	(symbol: \(EA\)) [SI:N] axial stiffness of beam; the axial force is \(f_{ax} = EA (\varepsilon -\varepsilon_0)\), in which \(\varepsilon\) is the axial strain
shearStiffness	UReal	0.	(symbol: \(GA\)) [SI:N] effective shear stiffness of beam, including stiffness correction
bendingDamping	UReal	0.	(symbol: \(d_{K}\)) [SI:Nm\(^2\)/s] viscous damping of bending deformation; the additional virtual work due to damping is \(\delta W_{\dot \kappa} = \int_0^L \dot \kappa \delta \kappa dx\)
axialDamping	UReal	0.	(symbol: \(d_{\varepsilon}\)) [SI:N/s] viscous damping of axial deformation
shearDamping	UReal	0.	(symbol: \(d_{\gamma}\)) [SI:N/s] viscous damping of shear deformation
referenceCurvature	Real	0.	(symbol: \(\kappa_0\)) [SI:1/m] reference curvature of beam (pre-deformation) of beam
includeReferenceRotations	bool	False	if True, rotation of the cross section at the nodes includes node reference rotations (within referenceCoordinates of NodeRigidBody2D), which are used for the computation of bending strains (this means that a pre-curved beam is stress-free); if False, the reference rotation of the cross section is orthogonal to the reference slope vector. This allows to easily share nodes among several beams with different reference cross section orientation (i.e., only the change of rotation counts).
visualization	VObjectBeamGeometricallyExact2D		parameters for visualization of item

Renamed parameters, still taken with a DeprecationWarning: physicsLength (deprecated since 1.12.258, removed in 2031): use length; physicsMassPerLength (deprecated since 1.12.258, removed in 2031): use massPerLength; physicsCrossSectionInertia (deprecated since 1.12.258, removed in 2031): use crossSectionInertia; physicsBendingStiffness (deprecated since 1.12.258, removed in 2031): use bendingStiffness; physicsAxialStiffness (deprecated since 1.12.258, removed in 2031): use axialStiffness; physicsShearStiffness (deprecated since 1.12.258, removed in 2031): use shearStiffness; physicsBendingDamping (deprecated since 1.12.258, removed in 2031): use bendingDamping; physicsAxialDamping (deprecated since 1.12.258, removed in 2031): use axialDamping; physicsShearDamping (deprecated since 1.12.258, removed in 2031): use shearDamping; physicsReferenceCurvature (deprecated since 1.12.258, removed in 2031): use referenceCurvature.

Visualization parameters

The parameters of VObjectBeamGeometricallyExact2D, given as visualization:

name	type	size	default	description
show	Bool		True	set true, if item is shown in visualization and false if it is not shown
drawHeight	float		0.	if beam is drawn with rectangular shape, this is the drawing height
color	Float4	4	[-1.,-1.,-1.,-1.]	RGBA color of the object; if R==-1, use default color

Drawing

Drawn as all finite elements; a band of height drawHeight along the axis, of bodies.beams.axialTiling segments, or a line if drawHeight is 0; the bodies.beams.drawVertical... settings draw a value across it.

Settings beyond those of all finite elements: bodies.beams.axialTiling, bodies.beams.crossSectionFilled, bodies.beams.drawVertical, bodies.beams.drawVerticalColor, bodies.beams.drawVerticalFactor, bodies.beams.drawVerticalLines, bodies.beams.drawVerticalOffset, bodies.beams.drawVerticalValues, general.rendererPrecision.

Output variables

Available as OutputVariableType in sensors, Get...Output() and other functions:

output variable	symbol	description
Position		global position vector of local axis (X) and cross section (Y) position
Displacement		global displacement vector of local axis (X) and cross section (Y) position
Velocity		global velocity vector of local axis (X) and cross section (Y) position
Rotation		3D Tait-Bryan rotation components, containing rotation around \(Z\)-axis only
StrainLocal		6 (local) strain components, containing only axial (\(XX\), index 0) and shear strain (\(XY\), index 5); evaluated at beam axis ONLY
CurvatureLocal		3D vector of (local) curvature, only \(Z\) component is non-zero
ForceLocal		3D vector of (local) section normal force, containing axial (X) and shear force (Y)
TorqueLocal		3D vector of (local) torques, containing only bending moment (Z)
KineticEnergy	\(T = \frac{1}{2} \dot\qv\tp \Mm\, \dot\qv\)	kinetic energy from the mass matrix of the current state and the velocities of the nodes; current configuration only; localPosition must be \([0,0,0]\)
PotentialEnergy	\(V = \frac{1}{2}\int_0^L EA\,\gamma_1^2 + GA\,\gamma_2^2 + EI\,\theta_x^2\, dx\)	elastic energy of the axial, shear and bending strains, integrated with the rule of the elastic forces; localPosition must be \([0,0,0]\)

Detailed description

A shear deformable beam after Simo and Vu-Quoc (1986): the positions and the rotations of the cross section are interpolated independently.

Nodes and coordinates

Two nodes (linear element) or three nodes (quadratic element) of type NodeRigidBody2D, each with the position \([r_{x},\, r_{y}]\) of the axis and the rotation \(\theta\) of the cross section; 6 or 9 coordinates. The local axial coordinate is \(x \in [-L/2,\, L/2]\).

Kinematics and interpolation

Position and rotation are interpolated with Lagrange shape functions - linear, \(S_0 = (L/2 - x)/L\), \(S_1 = (L/2 + x)/L\), or quadratic in \(\xi = 2x/L\) -

\[ \rv(x) = \sum_i S_i(x)\, \rv_i , \quad \theta(x) = \sum_i S_i(x)\, \theta_i . \]

With includeReferenceRotations = False (default) the rotation of the cross section is measured from the direction of the reference axis, \(\mathrm{atan2}(r'_{y,\mathrm{ref}},\, r'_{x,\mathrm{ref}})\); with True the reference rotations of the nodes are used.

Strains

With the tangent \(\tv = [\cos\theta,\, \sin\theta]\tp\) and the normal \(\nv = [-\sin\theta,\, \cos\theta]\tp\) of the cross section,

\[ \gamma_1 = \tv\tp \rv' - 1 , \quad \gamma_2 = \nv\tp \rv' , \quad \kappa = \theta' - \kappa_0 , \]

the axial strain, the shear strain and the curvature, with the reference curvature \(\kappa_0\) = referenceCurvature. The virtual work of the elastic forces is

\[ \delta W_e = \int_{-L/2}^{L/2} \left( EA\,\gamma_1\,\delta\gamma_1 + GA\,\gamma_2\,\delta\gamma_2 + EI\,\kappa\,\delta\kappa \right) dx , \]

with the damping terms \(d_\varepsilon \dot\gamma_1\), \(d_\gamma \dot\gamma_2\) and \(d_K \dot\kappa\) added in the same way.

Mass matrix and integration

The mass matrix is constant: \(\rho A\) for the positions and \(\rho J\) for the rotations, integrated with Gauss rules of order 3 (linear) or 5 (quadratic). The elastic forces are integrated with reduced integration against shear locking: one point for the linear element, the rule of order 3 for the quadratic element.

Marker interfaces

The position and rotation Jacobians follow from the interpolation above; a marker at a local position \((x,\, y)\) moves with the cross section.

Limitations

Planar; the output variables StrainLocal and ForceLocal give \(\gamma_1\) and \(\gamma_2\) and the forces \(EA\gamma_1\), \(GA\gamma_2\).

Mini example

#a cantilever of four planar geometrically exact beam elements on rigid body nodes, loaded at the tip
L = 1; nElements = 4; EI = 100; GA = 1e4; F = -0.1
mGround = mbs.AddMarker(MarkerNodeCoordinate(nodeNumber=nGround, coordinate=0))
n0 = mbs.AddNode(NodeRigidBody2D(referenceCoordinates=[0,0,0]))
for i in range(3): #clamped: x, y, rotation
    mbs.AddObject(ObjectConnectorCoordinate(markerNumbers=[mGround,
                  mbs.AddMarker(MarkerNodeCoordinate(nodeNumber=n0, coordinate=i))]))
for k in range(nElements):
    n1 = mbs.AddNode(NodeRigidBody2D(referenceCoordinates=[L*(k+1)/nElements,0,0]))
    mbs.AddObject(ObjectBeamGeometricallyExact2D(nodeNumbers=[n0,n1], length=L/nElements,
                  massPerLength=1, crossSectionInertia=0.01, bendingStiffness=EI,
                  axialStiffness=1e5, shearStiffness=GA))
    n0 = n1
mbs.AddLoad(LoadForceVector(markerNumber=mbs.AddMarker(MarkerNodePosition(nodeNumber=n1)), loadVector=[0,F,0]))

mbs.Assemble()
mbs.SolveStatic()

#Timoshenko beam: F*L^3/(3*EI) + F*L/GA = -0.3433e-3, approached with more elements
exu.sys['testResult'] = mbs.GetNodeOutput(n1, exu.OutputVariableType.Displacement)[1]*1000

Relevant Examples (Ex) and TestModels (TM) with weblink to github: pendulumGeomExactBeam2D.py (Ex), ANCFBeamEigTest.py (TM), energiesFlexibleBodiesTest.py (TM), flexiblePendulumBeamComparison.py (TM), geometricallyExactBeam2DquadraticTest.py (TM), geometricallyExactBeam2Dtest.py (TM), gridGeomExactBeam2D.py (TM), LShapeGeomExactBeam2D.py (TM)