Users:General FEM Analysis/Analyses Reference/Buckling

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[[File:LinBuckling_det.png | 200px | center | linear estimation of buckling point ]]
 
[[File:LinBuckling_det.png | 200px | center | linear estimation of buckling point ]]
  
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As the linear dependency of '''K'''''geo'' w.r.t. λ is a simplifying assumption, the '''estimated total load carrying factor γ·λ''' is the more exact the closer λ is to 1.
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As the linear dependency of '''K'''''geo'' w.r.t. λ is a simplifying assumption, the '''estimated total load carrying factor γ·λ''' is the more exact the closer λ is to 1. Be aware that in general it is over-estimating the buckling load!
  
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[[Users:General FEM Analysis/Analyses Reference/Static Nonlinear|static nonlinear analysis]]
 
  
  
  
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=== Structure of Equation System ===
 
  
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The linear static analysis solves the problem '''K'''*'''U'''='''F''', with the linear stiffness matrix '''K''', the unknown displacement fields '''U''' and the right hand side vectors '''F'''. This system of equations is solved for the unknown displacements by a linear solver. In general, linear solvers may be separated in iterative and direct solvers. Direct solvers transform the stiffness matrix '''K''' into an upper diagonal matrix. The displacement field follow by a back substitution with the respective right hand side vector. These back substitution requires only small numerical effort which makes direct solvers very attractive if a couple of displacement fields have to be computed.
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=== Defining the eigen problem ===
 +
 
 +
The equation shown above leads to the solution of an eigen problem similar to the one known from
 +
[[Users:General FEM Analysis/Analyses Reference/Eigenfrequency Analysis|eigenfrequency analysis]].
 +
The only difference is that instead of det('''K'''-λ·'''M''')=0 we have to solve det('''K'''''el''+λ·'''K'''''geo'')=0, and so the corresponding eigen problem reads
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[[File:LinBuckling_eigenproblem.png | 300px | center | eigen problem of linear buckling analysis ]]
  
 
== Input Parameters ==
 
== Input Parameters ==
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!Description
 
!Description
 
|-
 
|-
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!SOLVER
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!EIGEN_SOLVER
 
|PC-SOLVER ''int''
 
|PC-SOLVER ''int''
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|Linking to a linear solver (direct or iterative)
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|Linking to an [[Users:General FEM Analysis/Solvers Reference/Eigensolvers | eigen solver]]
 +
|-
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!LINEAR_SOLVER
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|PC-SOLVER ''int''
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|Linking to a [[Users:General FEM Analysis/Solvers Reference/Linear solvers| linear solver]]
 
|-
 
|-
 
!OUTPUT
 
!OUTPUT
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|EL-DOMAIN ''int''
 
|EL-DOMAIN ''int''
 
|Linking to the domain the analysis should work on
 
|Linking to the domain the analysis should work on
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|-
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!NUM_ROOT
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|''int''
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|Number of buckling modes and corresponding buckling load factors to determine
 
|}
 
|}
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 +
 +
It should be mentioned that the two solvers used within this analysis (linear and eigen solver) have to be compatible with respect to the used sparse matrix format (e.g. both using a Skyline formate or both working on a TRILINOS Csr matrix).
  
 
=== Example of a Complete Input Block ===
 
=== Example of a Complete Input Block ===
 
<pre>
 
<pre>
−
PC-ANALYSIS 1: STA_GEO_LIN
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PC-ANALYSIS 1: LINEAR_BUCKLING
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SOLVER = PC-SOLVER 5
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EIGEN_SOLVER = PC-SOLVER 1
 +
LINEAR_SOLVER = PC-SOLVER 2
 
OUTPUT = PC-OUT 1
 
OUTPUT = PC-OUT 1
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COMPCASE = LD-COM 1,2,3
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COMPCASE = LD-COM 1
 
DOMAIN = EL-DOMAIN 1
 
DOMAIN = EL-DOMAIN 1
 +
NUM_ROOT = 3
 
</pre>
 
</pre>
  
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== A Full Example ==
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== Examples ==
  
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The following example describes a simple cantilever problem discretized by [[Users:General FEM Analysis/Elements Reference/Shell8|SHELL8]] elements. The respective input file can be found here:
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=== A Full Example: Buckling of a rod ===
  
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{{PathToBenchmarkExamples}}elements/shell8_quad_lin_canti_I/shell8_canti_2x20_elem_load.dat
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The following example describes a rod buckling problem, where the rod is discretized by ([[Users:General_FEM_Analysis/Elements_Reference/Shell8|SHELL8]])-elements. A built in rod of length 2 is loaded in normal direction by a compressive force of 1. The cross section's bending stiffness is equal to 1750. The buckling load according to Euler's formula computes to
 +
[[File:LinBuckling_euler.png|200px]]
  
  
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The problem computes three computation cases (dead load, snow load and pressure load). The [[Users:General FEM Analysis/BCs Reference|boundary conditions]] are visualized by the figure below.
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The corresponding input can be found at:  
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{|
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* ..\examples\benchmark_examples\analyses\linbuckl_shell8_I\cbm_shell8_euler_1.dat
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|[[File:Shell8_canti_org_geom_01.png‎|frame|up|Simple cantilever problem using SHELL8 elements]]
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−
|}
+
  
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The basic goal of each linear static analysis is the computation of the displacement field. For load case 1 this result is depicted in the figure below, whereas the deformation in z-direction is additionally visualized by the color plot.
 
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{|
 
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|[[File:Shell8_canti_def_geo_disp.png‎|frame|up|Displacement of cantilever problem]]
 
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|}
 
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It can be seen that the support region does not show any deformation whereas the tip region deforms by a value of 8.76.
 
  
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Often the stress distribution is visualized by color plots. Shell structures require the specification of the layer on which the stresses are computed. The picture below shows the first principle stress on the top of the cantilever.
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The gallery shows the corresponding buckling modes:
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{|
+
 
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|[[File:Shell8_canti_def_geo_stress1.png|frame|up|First principle stress of cantilever problem (top layer)]]  
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<gallery caption="" widths="250px" heights="300px" perrow="4">
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|}
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File:LinBuckling_mode1.jpg  |  1st buckling mode
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It can be seen that the stresses show maximum values at the support region of the cantilever. At the tip they are nearly zero. In contrast to analytical results the stresses are not exactly zero in numerical models. The reason is the stress computation at the Gauss points. These points are situated inside the elements and not exactly at the tip of the cantilever.
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File:LinBuckling_mode2.jpg  | 2nd buckling mode
 +
File:LinBuckling_mode3.jpg  | 3rd buckling mode
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File:LinBuckling_mode4.jpg  | 4th buckling mode
 +
</gallery>
 +
 
 +
 
 +
The related load factors are computed to:
 +
<pre>
 +
13:22:12    ###########################################
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13:22:12    ### Results of linear buckling analysis ###
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13:22:12    ###########################################
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13:22:12 Number of buckling modes to compute: 4
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13:22:12
 +
13:22:12  Number |   Load Factor    |   Norm of Eigenvector
 +
13:22:12 -------------------------------------------------------
 +
13:22:12    1        1.079266e+03            3.755564e-03
 +
13:22:12    2        9.697325e+03            3.878872e-04
 +
13:22:12    3        2.684653e+04            6.865951e-05
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13:22:12    4        5.234735e+04            2.934980e-05
 +
13:22:12 -------------------------------------------------------
 +
13:22:12
 +
13:22:12 Buckling Analysis finished!
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</pre>
 +
 
 +
 
 +
So even for a load augmentation factor of more than 1000 the analytical result is met very well.
 +
 
 +
If the calculation is done with ([[Users:General_FEM_Analysis/Elements_Reference/Beam1|BEAM1]])-elements, the quality of the results is depending on the numbers of buckling modes and the P/Pcrit ratio. For only one buckling mode the applied load should not be less than 5% of the critical load. The minimum applied load approaches the critical load rapidly by increasing the number of buckling modes. In contrast to this there´s no maximum limit.
 +
 
 +
=== Benchmark examples ===
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* buckling of an Euler-rod discretized with ([[Users:General_FEM_Analysis/Elements_Reference/Shell8|SHELL8]])-elements : ..\examples\benchmark_examples\analyses\linbuckl_shell8_I\cbm_shell8_euler_1.dat
 +
* buckling of an Euler-rod discretized with ([[Users:General_FEM_Analysis/Elements_Reference/Beam1|BEAM1]])-elements : ..\examples\benchmark_examples\elements\beam1_buckling\cbm_beam1_buckling.dat
 +
* buckling of an Euler-rod discretized with ([[Users:General_FEM_Analysis/Elements_Reference/BeamCR|BEAMCR]])-elements : ..\examples\benchmark_examples\elements\BeamCR_LinBuckling\cbm_BeamCR_LinBuckling.dat

Latest revision as of 08:08, 7 December 2016


Contents

General Description

This analysis performs a linear estimation of the buckling load factor. To this purpose the system stiffness K is considered to consist of two parts, the elastic stiffness Kel, which is well known from the field of linear static analysis, and the geometric stiffness part Kgeo which is associated to the current state of stress. The complete stiffness can be determined as K = Kel + Kgeo.

The analysis refers to the load factor γ defined by the user in the input deck. Starting from this load factor a linear dependency of the geometric stiffness with respect to the load augmentation factor λ is assumed, and so the critical load augmentation factor where buckling occurs can be estimated by demanding singularity of the complete stiffness (linearized w.r.t. λ):

linear estimation of buckling point

As the linear dependency of Kgeo w.r.t. λ is a simplifying assumption, the estimated total load carrying factor γ·λ is the more exact the closer λ is to 1. Be aware that in general it is over-estimating the buckling load!



Defining the eigen problem

The equation shown above leads to the solution of an eigen problem similar to the one known from eigenfrequency analysis. The only difference is that instead of det(K-λ·M)=0 we have to solve det(Kel+λ·Kgeo)=0, and so the corresponding eigen problem reads

eigen problem of linear buckling analysis

Input Parameters

Parameter Description

Compulsory Parameters
Parameter Values, Default(*) Description
EIGEN_SOLVER PC-SOLVER int Linking to an eigen solver
LINEAR_SOLVER PC-SOLVER int Linking to a linear solver
OUTPUT PC-OUT int Linking to output objects (specifies the type of output format, e.g. GiD)
COMPCASE LD-COM int Linking to computation case objects which specify the boundary conditions (loading and supports)
DOMAIN EL-DOMAIN int Linking to the domain the analysis should work on
NUM_ROOT int Number of buckling modes and corresponding buckling load factors to determine


It should be mentioned that the two solvers used within this analysis (linear and eigen solver) have to be compatible with respect to the used sparse matrix format (e.g. both using a Skyline formate or both working on a TRILINOS Csr matrix).

Example of a Complete Input Block 

PC-ANALYSIS 1: LINEAR_BUCKLING
EIGEN_SOLVER = PC-SOLVER 1
LINEAR_SOLVER = PC-SOLVER 2
OUTPUT = PC-OUT 1
COMPCASE = LD-COM 1
DOMAIN = EL-DOMAIN 1
NUM_ROOT = 3

Examples

A Full Example: Buckling of a rod

The following example describes a rod buckling problem, where the rod is discretized by (SHELL8)-elements. A built in rod of length 2 is loaded in normal direction by a compressive force of 1. The cross section's bending stiffness is equal to 1750. The buckling load according to Euler's formula computes to LinBuckling euler.png


The corresponding input can be found at:

  • ..\examples\benchmark_examples\analyses\linbuckl_shell8_I\cbm_shell8_euler_1.dat


The gallery shows the corresponding buckling modes:

  • 1st buckling mode

  • 2nd buckling mode

  • 3rd buckling mode

  • 4th buckling mode


The related load factors are computed to: 

13:22:12     ###########################################
13:22:12     ### Results of linear buckling analysis ###
13:22:12     ###########################################
13:22:12 Number of buckling modes to compute: 4
13:22:12
13:22:12   Number |   Load Factor    |    Norm of Eigenvector
13:22:12 -------------------------------------------------------
13:22:12     1        1.079266e+03            3.755564e-03
13:22:12     2        9.697325e+03            3.878872e-04
13:22:12     3        2.684653e+04            6.865951e-05
13:22:12     4        5.234735e+04            2.934980e-05
13:22:12 -------------------------------------------------------
13:22:12
13:22:12 Buckling Analysis finished!


So even for a load augmentation factor of more than 1000 the analytical result is met very well.

If the calculation is done with (BEAM1)-elements, the quality of the results is depending on the numbers of buckling modes and the P/Pcrit ratio. For only one buckling mode the applied load should not be less than 5% of the critical load. The minimum applied load approaches the critical load rapidly by increasing the number of buckling modes. In contrast to this there´s no maximum limit.

Benchmark examples

  • buckling of an Euler-rod discretized with (SHELL8)-elements : ..\examples\benchmark_examples\analyses\linbuckl_shell8_I\cbm_shell8_euler_1.dat
  • buckling of an Euler-rod discretized with (BEAM1)-elements : ..\examples\benchmark_examples\elements\beam1_buckling\cbm_beam1_buckling.dat
  • buckling of an Euler-rod discretized with (BEAMCR)-elements : ..\examples\benchmark_examples\elements\BeamCR_LinBuckling\cbm_BeamCR_LinBuckling.dat




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