Users:General FEM Analysis/Elements Reference/BeamNL

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== General Description ==
 
== General Description ==
 +
'''This element has never been finalized and is deprecated!'''
 +
  
 
=== Element Type ===
 
=== Element Type ===
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* This beam element is a 2 node non-linear 3D-beam for large rotations and small deformations (Green-Lagrange-strains) taking into account shear deformation (Timoshenko-beam element).
 
* This beam element is a 2 node non-linear 3D-beam for large rotations and small deformations (Green-Lagrange-strains) taking into account shear deformation (Timoshenko-beam element).
 
* This beam element has 6 DOFs per node (three translations and three rotations).
 
* This beam element has 6 DOFs per node (three translations and three rotations).
−
* The stiffness matrix is obtained using a one-point Gaussian quadrature, which results in an under-integration of the shear terms. This should prevent shear-locking.
+
* The stiffness matrix is obtained using a one-point Gaussian quadrature, which results in an under-integration of the shear terms. This prevents shear-locking.
  
 
=== Degrees of Freedom ===
 
=== Degrees of Freedom ===
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=== Orientation of the local coordinate system ===
 
=== Orientation of the local coordinate system ===
−
 
+
[[File:BeamNL_Orientation.png|250px|frame|right|Orientation of the local coordinate system for BeamNL]]
−
The Beam1 element uses the following definition for the determination of the local coordinate system (needed for the orientation of IYY and IZZ,...):
+
The BeamNL element uses the following definition for the determination of the local coordinate system in each node (needed for the orientation of I_22 and I_33,...), based on the idea of sets of members (dt.; ''Stabzug''). These coordinate systems also have the role of a local Frenet-frame that is rotated with the element during a non-linear analysis. The difference between the current and the reference Frenet-frames is used to determine the curvatures of the element:
−
* the local x-axis is oriented from node 1 to node 2 of the beam
+
* the local x-axis ('''x'''<sub>loc</sub>) is defined as the tangent to the element in the considered point. This has important consequences:
−
* the local y-axis lies in the global XY-plane, such that the local z-axis points in the same half-space as the global Z-axis (mathematically spoken: the local z-axis and the global Z-axis result in a positive dot-product)
+
** need for a neighbour-element search (inside one set of members, other elements have to be linked using  [[Users:General FEM Analysis/BCs Reference/Dirichlet|''MPC-COUPLING'']]),
−
* the local z-axis is perpendicular to the other two local axis, following the right-hand-rule for x-y-z
+
** for the definition of kinks, two different sets of members have to be used (else kinks would be smoothed out),
−
* '''exception:''' If the local x-axis (i.e. the beam axis) points in the direction of global Z, the local y-axis points in the direction of the global Z-axis. The local z-axis once again follows the right-hand-rule for x-y-z.
+
** for starting members (i.e. with only 1 adjacent element), the connection of node 1 to node 2 is the local x-axis.
−
 
+
* the local y-axis ('''z'''<sub>loc</sub>) is defined with the help of an orientation point (''ORIENTATION_POINT_BEAMNL'', external to the beam element itself and common for one set of members; to be defined in the element property (see below)): an auxiliary vector ('''vec'''<sub>aux</sub>) pointing from node 1 to this external orientation point is set up. The local axis y is the result of the cross-product of this auxiliary vector and the local x-axis (mathematically spoken: '''y'''<sub>loc</sub>  = '''vec'''<sub>aux</sub> x '''x'''<sub>loc</sub>).
−
In case that a rotation of the local coordinate system is needed (rotated elements, inverse definition of IYY and IZZ,...) an angle ''THETA'' has to be specified. This angle rotates the whole coordinate system around the local x-axis, following the right-thumb rule (i.e. the thumb of the right hand points in the direction of the local x-axis).
+
* the local z-axis ('''z'''<sub>loc</sub>) is perpendicular to the other two local axes, following the right-hand-rule for x-y-z (i.e. once again using the cross-product). This makes z point in the half-space of the orientation point.
 +
* '''addition:''' If the local x-axes at both ends (i.e. nodes) of the element don't point in the same direction (due to the situation of the orientation point), the x-axis at the second node is switched by 180 degrees.
  
 
== Input Parameters ==
 
== Input Parameters ==
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|Definition of the cross-sectional area of the beam
 
|Definition of the cross-sectional area of the beam
 
|-
 
|-
−
!IYY, IZZ
+
!SHEAR_SECTION_Y, SHEAR_SECTION_Z
 
|
 
|
−
|Definition of the moments of inertia
+
|Definition of the shear-cross-sectional area of the beam in y- and z-direction
 
|-
 
|-
−
|colspan="3" style="background:#efefef;"| Optional Parameters
+
!I_22, I_33
 +
|
 +
|Definition of the moments of inertia around the local y- and z-axis
 
|-
 
|-
−
!KY, KZ
+
!I_23
−
|0
+
|
−
|Shear correction factor; if 0 or no value specified, shear is not taken into account (Bernouolli-beam theory)
+
|Definition of the mixed moment of inertia
 
|-
 
|-
−
!THETA
+
!I_T
−
|0
+
|
−
|Angle of rotation of the local coordinate system around the beam axis in degrees
+
|Definition of the torsional resistance I_T
 +
|-
 +
!ORIENTATION_POINT_BEAMNL
 +
|
 +
|External orientation point for the set of beam members.
 
|-
 
|-
 
|}
 
|}
 +
  
 
=== Example of a Complete Input Block ===
 
=== Example of a Complete Input Block ===
 
<pre>
 
<pre>
−
EL-PROP 1 : BEAM1
+
EL-PROP 1 : BEAMNL
 
MAT= EL-MAT 1
 
MAT= EL-MAT 1
−
AREA=5.555 IYY=1.222 IZZ=1.333
+
AREA=0.015 SHEAR_SECTION_Y = 0.0075  SHEAR_SECTION_Z = 0.015
−
KY=1.4  KZ=1.5
+
I_22=1.125 I_33=0.03125
−
THETA=90
+
I_23=1.23 I_T=1.15625
 +
ORIENTATION_POINT_BEAMNL = 0.0, 10.0, 0.0
 
</pre>
 
</pre>
  
−
=== Use of the shear correction factors KY and KZ ===
+
 
−
The shear correction factors ''KY'' and ''KZ'' depend on the cross section of the beam. For a rectangle, ''KY'' resp. ''KZ'' are equal to 1.2, which gives a factor of α=1/''Ki''=5/6.
+
=== Use of the shear sections SHEAR_SECTION_Y and SHEAR_SECTION_Z ===
 +
The shear sections ''SHEAR_SECTION_Y'' and ''SHEAR_SECTION_Z'' depend on the cross section of the beam and can be obtained by multiplying the area of the beam ''AREA'' with the shear correction factor α which is dependent on the shape of the sections. For a rectangle, α=5/6.
  
 
=== Use of the rotation parameter THETA ===
 
=== Use of the rotation parameter THETA ===
 
The rotation parameter ''THETA'' and its use is explained in the section concerning the coordinate system above.
 
The rotation parameter ''THETA'' and its use is explained in the section concerning the coordinate system above.
  
−
=== The torsional resistance IT ===
+
=== The mixed moment of inertia I_23 ===
−
The torsional resistance ''IT'' is interpreted as the polar moment ''Ipp'', i.e.: ''IT'' = ''Ipp'' = ''IYY'' + ''IZZ''.
+
The mixed moment of inertia ''I_23'' is equal to 0 if one of the two local axes y and z is an axis of symmetry of the section.
 +
 
 +
=== The external point of orientation ORIENTATION_POINT_BEAMNL ===
 +
The external point of orientation ''ORIENTATION_POINT_BEAMNL'' has to be entered with its coordinates in the global XYZ-frame in order to define the orientation of the local coordinate system of the beam.
  
 
== Element Loading ==
 
== Element Loading ==
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== Tests and Benchmarks ==
 
== Tests and Benchmarks ==
 
=== Static linear analysis ===
 
=== Static linear analysis ===
−
[[File:linBeamTests2.jpg|200px|thumb|right|Static test for element Beam1]]
+
[[File:linBeamTests2.jpg|300px|frame|right|Static linear test for element BeamNL]]
−
For the moment, the element Beam1 has successfully been tested in 3D in all its linear static features, including
+
For the moment, the element BeamNL has successfully been tested in 3D in all its linear static features, including
 
* bending, axial deformation, torsion,
 
* bending, axial deformation, torsion,
 
* shear deformation (separately definable for both local axes),
 
* shear deformation (separately definable for both local axes),
Line 103: Line 118:
 
== Theory ==
 
== Theory ==
  
−
The element implementation mainly follows the implementation of a linear 3D-beam element in  
+
The element implementation mainly follows the implementation of a non-linear 3D-beam element in  
−
FELyX <ref name="FELyX"> https://www.rdb.ethz.ch/projects/project.php?proj_id=8314 </ref>,
+
the PhD-thesis of Roland Sauer <ref name="Sauer1998"> Sauer, R.: Eine einheitliche Finite-Element-Formulierung für Stab- und Schalentragwerke mit endlichen Rotationen, Institut für Baustatik, Universität Karlsruhe, 1998</ref>, and
−
Schwarz <ref name="Schwarz1991">Schwarz, H.: Methode der Finiten Elemente, Teubner, 1991</ref> and
+
some papers of F. Gruttmann et al.
−
Wunderlich <ref name="Wunderlich">Wunderlich, W.: Statik der Stabtragwerke, Teubner, 2004</ref>.
+
  
−
As it is based on the Hermite-form functions, a Bernoulli beam can be modelled by one single element.
+
As it is based on linear ansatz-functions, the BeamNL-element needs sufficiently fine discretization in order to converge to the true solution.
  
 
== References ==
 
== References ==
  
 
<references/>
 
<references/>

Latest revision as of 06:32, 13 January 2017


Contents

General Description

This element has never been finalized and is deprecated!


Element Type

  • This beam element is a 2 node non-linear 3D-beam for large rotations and small deformations (Green-Lagrange-strains) taking into account shear deformation (Timoshenko-beam element).
  • This beam element has 6 DOFs per node (three translations and three rotations).
  • The stiffness matrix is obtained using a one-point Gaussian quadrature, which results in an under-integration of the shear terms. This prevents shear-locking.

Degrees of Freedom

For the BeamNL element use the 3 translatoric degrees of freedom DISP_X, DISP_Y, DISP_Z and the 3 rotatoric degrees of freedom ROT_X, ROT_Y, ROT_Z.

Orientation of the local coordinate system

Orientation of the local coordinate system for BeamNL

The BeamNL element uses the following definition for the determination of the local coordinate system in each node (needed for the orientation of I_22 and I_33,...), based on the idea of sets of members (dt.; Stabzug). These coordinate systems also have the role of a local Frenet-frame that is rotated with the element during a non-linear analysis. The difference between the current and the reference Frenet-frames is used to determine the curvatures of the element:

  • the local x-axis (xloc) is defined as the tangent to the element in the considered point. This has important consequences:
    • need for a neighbour-element search (inside one set of members, other elements have to be linked using MPC-COUPLING),
    • for the definition of kinks, two different sets of members have to be used (else kinks would be smoothed out),
    • for starting members (i.e. with only 1 adjacent element), the connection of node 1 to node 2 is the local x-axis.
  • the local y-axis (zloc) is defined with the help of an orientation point (ORIENTATION_POINT_BEAMNL, external to the beam element itself and common for one set of members; to be defined in the element property (see below)): an auxiliary vector (vecaux) pointing from node 1 to this external orientation point is set up. The local axis y is the result of the cross-product of this auxiliary vector and the local x-axis (mathematically spoken: yloc = vecaux x xloc).
  • the local z-axis (zloc) is perpendicular to the other two local axes, following the right-hand-rule for x-y-z (i.e. once again using the cross-product). This makes z point in the half-space of the orientation point.
  • addition: If the local x-axes at both ends (i.e. nodes) of the element don't point in the same direction (due to the situation of the orientation point), the x-axis at the second node is switched by 180 degrees.

Input Parameters

Parameter Description

Compulsory Parameters
Parameter Values, Default(*) Description
MAT EL-MAT int Number for the used Material

e.g. MAT=EL-MAT 1

AREA Definition of the cross-sectional area of the beam
SHEAR_SECTION_Y, SHEAR_SECTION_Z Definition of the shear-cross-sectional area of the beam in y- and z-direction
I_22, I_33 Definition of the moments of inertia around the local y- and z-axis
I_23 Definition of the mixed moment of inertia
I_T Definition of the torsional resistance I_T
ORIENTATION_POINT_BEAMNL External orientation point for the set of beam members.


Example of a Complete Input Block 

EL-PROP 1 : BEAMNL
MAT= EL-MAT 1
AREA=0.015  SHEAR_SECTION_Y = 0.0075   SHEAR_SECTION_Z = 0.015
I_22=1.125  I_33=0.03125
I_23=1.23 I_T=1.15625
ORIENTATION_POINT_BEAMNL = 0.0, 10.0, 0.0


Use of the shear sections SHEAR_SECTION_Y and SHEAR_SECTION_Z

The shear sections SHEAR_SECTION_Y and SHEAR_SECTION_Z depend on the cross section of the beam and can be obtained by multiplying the area of the beam AREA with the shear correction factor α which is dependent on the shape of the sections. For a rectangle, α=5/6.

Use of the rotation parameter THETA

The rotation parameter THETA and its use is explained in the section concerning the coordinate system above.

The mixed moment of inertia I_23

The mixed moment of inertia I_23 is equal to 0 if one of the two local axes y and z is an axis of symmetry of the section.

The external point of orientation ORIENTATION_POINT_BEAMNL

The external point of orientation ORIENTATION_POINT_BEAMNL has to be entered with its coordinates in the global XYZ-frame in order to define the orientation of the local coordinate system of the beam.

Element Loading

For the moment, only nodal forces in the three global directions can be applied (i.e. Fx, Fy, Fz).

Pressure

  • not defined yet

Dead Load

  • not defined yet

Snow Load

  • not defined yet


Tests and Benchmarks

Static linear analysis

Static linear test for element BeamNL

For the moment, the element BeamNL has successfully been tested in 3D in all its linear static features, including

  • bending, axial deformation, torsion,
  • shear deformation (separately definable for both local axes),
  • rotation around the local axis.

As an example, the structure on the right was part of the final tests.


Theory

The element implementation mainly follows the implementation of a non-linear 3D-beam element in the PhD-thesis of Roland Sauer [1], and some papers of F. Gruttmann et al.

As it is based on linear ansatz-functions, the BeamNL-element needs sufficiently fine discretization in order to converge to the true solution.

References

  1. ↑ Sauer, R.: Eine einheitliche Finite-Element-Formulierung für Stab- und Schalentragwerke mit endlichen Rotationen, Institut für Baustatik, Universität Karlsruhe, 1998




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