Users:General FEM Analysis/Elements Reference/BeamCR

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=== Element Type ===
 
=== 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 is a 2 node geometrically 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 kinematics are applied following the "co-rotational" principle: the rigid body movements are separated from the elastic deformations on element level.
−
* The stiffness matrix for the linear calculation is hard-coded, thus not needing any integration. For theory of 2nd order, the normal force in the element leads to modifications in the stiffness matrix.
+
* This beam element has 6 DOFs per node (three translations and three rotations).
 +
* With its implemented mass matrix the BeamCR-element can also be used in dynamic analyses (see examples).
  
 
=== Degrees of Freedom ===
 
=== Degrees of Freedom ===
  
−
For the Beam1 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''.
+
For the BeamCR 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''.
−
 
+
−
=== Theory of 2nd Order ===
+
−
 
+
−
If Theory of 2nd Order should be taken into account, a nonlinear calculation (ANALYSIS STA_GEO_NLIN) has to be executed.
+
  
 
=== Orientation of the local coordinate system ===
 
=== Orientation of the local coordinate system ===
  
−
The Beam1 element uses the following definition for the determination of the local coordinate system (needed for the orientation of IYY and IZZ,...):
+
The BeamCR element uses the following definition for the determination of the local coordinate system (needed for the orientation of IYY and IZZ,...):
 
* the local x-axis is oriented from node 1 to node 2 of the beam
 
* the local x-axis is oriented from node 1 to node 2 of the beam
 
* 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)
 
* 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)
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=== Update of the local coordinate system ===
 
=== Update of the local coordinate system ===
  
−
For updating the local coordinate system, the element base vectors are rotated by a rotation of magnitude dφax of the anti-symmetric rotation increment dφa around the beam axis, followed by the rotation of the whole coordinate system to align the base vector nx with the update beam axis. The complete rotation of the base configuration is represented by a rotation matrix T which is defined by the current element base vectors '''𝑻'=[𝒏𝑥 𝒏𝑦 𝒏𝑧]''.
+
For updating the local coordinate system, the element base vectors are rotated by a rotation of magnitude ''dφax'' of the anti-symmetric rotation increment ''dφa'' around the beam axis, followed by the rotation of the whole coordinate system to align the base vector ''nx'' with the update beam axis. The complete rotation of the base configuration is represented by a rotation matrix ''T'' which is defined by the current element base vectors ''T = [nx ny nz]''.
  
 
=== Orientation of the resultant forces ===
 
=== Orientation of the resultant forces ===
  
−
The BeamCR element has 6 resultant forces in accordance with the degrees of freedom. The resultant forces, ''[ N V1 V2 | MT M1 M2 ]'', are oriented along the local axes on the positive section of the beam. Note that the resultantforces are evaluated in the center of the beam element, so it's the mean value of the two ends. For the orientation, cf. also the orientation sketch:
+
The BeamCR element has 6 resultant forces in accordance with the degrees of freedom. The resultant forces, ''[ N V1 V2 | MT M1 M2 ]'', are oriented along the local axes on the positive section of the beam. Note that the resultant forces are evaluated in the center of the beam element, so it is the mean value of the two ends. For the orientation, cf. also the orientation sketch:
 
[[File:Beam1-Resultant_forces_orientation.png|200px|frame|right|Resultant forces orientation for element BeamCR]]
 
[[File:Beam1-Resultant_forces_orientation.png|200px|frame|right|Resultant forces orientation for element BeamCR]]
  
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|-
 
|-
 
!IT
 
!IT
−
|IT=IYY+IZZ
+
|IYY+IZZ
 
|Definition of the torsional resistance
 
|Definition of the torsional resistance
 
|-
 
|-
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</pre>
 
</pre>
  
−
=== Use of the shear correction factors KY and KZ ===
+
=== Use of the shear sections SHEAR_SECTION_Y and SHEAR_SECTION_Z ===
−
The shear correction factors ''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.
+
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 ===
Line 97: Line 94:
  
 
== Element Loading ==
 
== Element Loading ==
−
For the moment, only nodal forces in the three global directions can be applied (i.e. ''Fx, Fy, Fz'').
+
For now, only nodal forces in the three global directions can be applied (i.e. ''Fx, Fy, Fz'').
  
 
=== Pressure ===
 
=== Pressure ===
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=== Snow Load ===
 
=== Snow Load ===
 
* not defined yet
 
* not defined yet
−
 
  
 
== Tests and Benchmarks ==
 
== Tests and Benchmarks ==
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=== Static non-linear analysis ===
 
=== Static non-linear analysis ===
−
Test herefore is a large displacement response of a cantilever 45-degree bend subjected to a concentrated end load <ref name="Statik  Ergänzung - Skript"> Bletzinger, K.-U.: Statik  Ergänzung - Skript </ref> p. 980
+
Test herefore is a large displacement response of a cantilever 45-degree bend subjected to a concentrated end load <ref name="Bathe"> Bathe, K.-J.: Large Displacement Analysis Of Three-Dimensional Beam Structures, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 14,961-986 (1979) </ref> p. 980
−
The structure is modeled by eight beam_CR-elements. One single load is applied at the free end of the cantilever. The non-dimensional tip deflection for P=7R²/EI=583.33 is w/R=0.53.
+
The structure is modeled by eight BeamCR-elements. One single load is applied at the free end of the cantilever. The non-dimensional tip deflection for P=R²/EI=583.33 is w/R=0.150.
−
 
+
[[File:45degree_bend_non-linear.jpg|400px|frame|right|Static test for element BeamCR]]
−
 
+
  
−
[[File:45degree_bend_non-linear.png|400px|frame|right|Static test for element Beam1]]
+
=== Benchmark examples ===
 +
* 45°-bend example for non-linear static analysis with large displacement and torsion: ..\examples\benchmark_examples\elements\beam_CR_nln_static\cbm_Beam_CR_Bathe_45_circular_bend.dat
 +
* spatial frame including local orientations THETA (see depicted above) in a linear computation under tip load: ..\examples\benchmark_examples\elements\beam_CR_lin_static\cbm_beamCR_stalin_timoshenko.dat
 +
* buckling analysis of a single-span column (Euler 2): ..\examples\benchmark_examples\elements\BeamCR_LinBuckling\cbm_BeamCR_LinBuckling.dat
 +
* non-linear dynamic analysis of a cantilever that has been pre-bent:  ..\examples\benchmark_examples\analyses\dynamic_nonlinear_BeamCR_cantilever_5ele\cbm_dynamic_nonlinear_BeamCR_cantilever_5ele.dat
  
 
== Theory ==
 
== Theory ==
  
−
The element implementation mainly follows the implementation of a non-linear co-rotational 3D-beam element in Krenk <ref name="Krenk"> https://www.rdb.ethz.ch/projects/project.php?proj_id=8314 </ref>.
+
The element implementation mainly follows the implementation of a non-linear co-rotational 3D-beam element in Krenk <ref name="Krenk"> Krenk, S.: Non-linear Modeling and
 +
Analysis of Solids and Structures, Cambridge, 2009 </ref>.
  
−
As it is based on linear ansatz-functions, the BeamCR-element needs sufficiently fine discretization in order to converge to the true solution.
+
As it is based on linear ansatz-functions, the BeamCR-element needs sufficiently fine discretization in order to converge to the true solution. As the calculation of the antimetric rotation increment is based on a small-angle approximation, the BeamCR-element needs sufficiently fine discretization of load steps.
  
 
== References ==
 
== References ==
  
 
<references/>
 
<references/>

Latest revision as of 07:42, 13 January 2017


Contents

General Description

Element Type

  • This beam element is a 2 node geometrically non-linear 3D-beam for large rotations and small deformations (Green-Lagrange-strains) taking into account shear deformation (Timoshenko-beam element).
  • The kinematics are applied following the "co-rotational" principle: the rigid body movements are separated from the elastic deformations on element level.
  • This beam element has 6 DOFs per node (three translations and three rotations).
  • With its implemented mass matrix the BeamCR-element can also be used in dynamic analyses (see examples).

Degrees of Freedom

For the BeamCR 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

The BeamCR element uses the following definition for the determination of the local coordinate system (needed for the orientation of IYY and IZZ,...):

  • the local x-axis is oriented from node 1 to node 2 of the beam
  • 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)
  • the local z-axis is perpendicular to the other two local axis, following the right-hand-rule for x-y-z
  • 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 Y-axis. The local z-axis once again follows the right-hand-rule for x-y-z.

These coordinate systems have the role of a local Frenet-frame that is rotated with the element during a non-linear analysis.

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).

Update of the local coordinate system

For updating the local coordinate system, the element base vectors are rotated by a rotation of magnitude dφax of the anti-symmetric rotation increment dφa around the beam axis, followed by the rotation of the whole coordinate system to align the base vector nx with the update beam axis. The complete rotation of the base configuration is represented by a rotation matrix T which is defined by the current element base vectors T = [nx ny nz].

Orientation of the resultant forces

The BeamCR element has 6 resultant forces in accordance with the degrees of freedom. The resultant forces, [ N V1 V2 | MT M1 M2 ], are oriented along the local axes on the positive section of the beam. Note that the resultant forces are evaluated in the center of the beam element, so it is the mean value of the two ends. For the orientation, cf. also the orientation sketch:

Resultant forces orientation for element BeamCR

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
IYY, IZZ Definition of the moments of inertia
Optional Parameters
IT IYY+IZZ Definition of the torsional resistance
SHEAR_SECTION_Y, SHEAR_SECTION_Z Definition of the shear-corss-sectional area of the beam in y- and z-direction
THETA 0 Angle of rotation of the local coordinate system around the beam axis in degrees

Example of a Complete Input Block 

EL-PROP 1 : BEAM_CR
MAT= EL-MAT 1
AREA=0.015  IYY=1.125  IZZ=0.03125
SHEAR_SECTION_Y=0.0075   SHEAR_SECTION_Z=0.015
THETA=90

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 torsional resistance IT

The torsional resistance IT is interpreted as the polar moment Ipp, i.e.: IT = Ipp = IYY + IZZ.

Element Loading

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

Pressure

  • not defined yet

Dead Load

  • implemented

Snow Load

  • not defined yet

Tests and Benchmarks

Static linear analysis

Static test for element BeamCR

For the moment, the element BeamCR 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. The displacement of the end-point is (1.432, 0.336, 0.716), which leads to a total displacement of 1.636.

Static non-linear analysis

Test herefore is a large displacement response of a cantilever 45-degree bend subjected to a concentrated end load [1] p. 980 The structure is modeled by eight BeamCR-elements. One single load is applied at the free end of the cantilever. The non-dimensional tip deflection for P=R²/EI=583.33 is w/R=0.150.

Static test for element BeamCR

Benchmark examples

  • 45°-bend example for non-linear static analysis with large displacement and torsion: ..\examples\benchmark_examples\elements\beam_CR_nln_static\cbm_Beam_CR_Bathe_45_circular_bend.dat
  • spatial frame including local orientations THETA (see depicted above) in a linear computation under tip load: ..\examples\benchmark_examples\elements\beam_CR_lin_static\cbm_beamCR_stalin_timoshenko.dat
  • buckling analysis of a single-span column (Euler 2): ..\examples\benchmark_examples\elements\BeamCR_LinBuckling\cbm_BeamCR_LinBuckling.dat
  • non-linear dynamic analysis of a cantilever that has been pre-bent: ..\examples\benchmark_examples\analyses\dynamic_nonlinear_BeamCR_cantilever_5ele\cbm_dynamic_nonlinear_BeamCR_cantilever_5ele.dat

Theory

The element implementation mainly follows the implementation of a non-linear co-rotational 3D-beam element in Krenk [2].

As it is based on linear ansatz-functions, the BeamCR-element needs sufficiently fine discretization in order to converge to the true solution. As the calculation of the antimetric rotation increment is based on a small-angle approximation, the BeamCR-element needs sufficiently fine discretization of load steps.

References

  1. ↑ Bathe, K.-J.: Large Displacement Analysis Of Three-Dimensional Beam Structures, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 14,961-986 (1979)
  2. ↑ Krenk, S.: Non-linear Modeling and Analysis of Solids and Structures, Cambridge, 2009




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