Users:General FEM Analysis/Analyses Reference/Static Linear

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== General Description ==
 
== General Description ==
  
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The linear static analysis is applied to analyze displacements, strains or stresses of structures. This analysis is valid for linear static mechanical problems without any nonlinearity (geometrically nonlinear, contact, material nonlinear, ...). Static nonlinear problems are can be solved by the [[Users:General FEM Analysis/Analyses Reference/Static Nonlinear|static nonlinear analysis]] whereas transient problems are solved by [[Users:General FEM Analysis/Analysis Reference/Dynamic Analysis|dynamic analysis methods]].  
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The linear static analysis is applied to analyze displacements, strains or stresses of structures. This analysis is valid for linear static mechanical problems without any nonlinearity (geometrically nonlinear, contact, material nonlinear, ...). Static nonlinear problems can be solved by the [[Users:General FEM Analysis/Analyses Reference/Static Nonlinear|static nonlinear analysis]] whereas transient problems are solved by [[Users:General FEM Analysis/Analyses_Reference/Dynamic_Analysis|dynamic analysis methods]].  
  
 
The static linear analysis formulates structural equilibrium on the initial configuration without consideration of deformations. Hence, it is only valid for problems showing small displacements.
 
The static linear analysis formulates structural equilibrium on the initial configuration without consideration of deformations. Hence, it is only valid for problems showing small displacements.
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=== Structure of Equation System ===
 
=== 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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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. This 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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!OUTPUT
 
!OUTPUT
 
|PC-OUT ''int''
 
|PC-OUT ''int''
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|Linking to output objects (specifies the type of output format, e.g. GiD)
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|Linking to [[Users:General FEM Analysis/Data Output|output objects]] (specifies the type of output format, e.g. GiD)
 
|-
 
|-
 
!COMPCASE
 
!COMPCASE
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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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|colspan="3" style="background:#efefef;"| Optional Parameters
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|-
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!ONLY_IBRA_PRE
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| ''int''
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|Get integration domain in JSON format for Isogeometric B-Rep Analysis (IBRA) (1 = true, 0 = false)
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|-
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!SIMULTANEOUS_EIGENVALUE_ANALYSIS
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|''int''
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|Simultaneous eigenvalue analysis (1 = true, 0 = false)
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|-
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!EIGEN_SOLVER
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|PC-SOLVER ''int''
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|Linking to a eigenvalue solver
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|-
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!NUM_EIGEN_MODES
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|''int''
 +
|Number of eigenmodes
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|-
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!BENCHMARK_ELEMENTS
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|''int''
 +
|Ids of Design elements
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|-
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!BENCHMARK_ELEMENTS_U
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|''float''
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|Position u of the benchmark points in the parameter space. If the number of members equals the number of the ids of the design element, only one point per element is printed. If not, all positions for every element is printed
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|-
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!BENCHMARK_ELEMENTS_V
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|''float''
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|Position v of the benchmark points in the parameter space
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|-
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!BENCHMARK_ELEMENTS_W
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|''float''
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|Position w of the benchmark points in the parameter space
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|-
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!BENCHMARK_DISP
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|''int''
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|if displacement at benchmark point should be printed  (1 = true, 0 = false)
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|-
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!BENCHMARK_STRESS
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|''int''
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|if stresses at benchmark point should be printed  (1 = true, 0 = false)
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|-
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!BENCHMARK_OUTPUT_NAME
 +
|''string''
 +
|names of files for each benchmark point (default = benchmark_point). If number of output names and design element ids matches, every element is only printed in the respective benchmark output file. For several names write name1,name2,... without spaces
 
|}
 
|}
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=== Example of a Complete Input Block ===
 
=== Example of a Complete Input Block ===
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</pre>
 
</pre>
  
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== Example ==
 
  
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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 in the SVN repository under
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== Examples ==
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<pre>
+
 
−
carat20/examples/benchmark_examples/elements/shell8_quad_lin_canti_I/shell8_canti_2x20_elem_load.dat
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=== A Full Example: Bending of a cantilever discretized with shell elements ===
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</pre>
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−
{{PathToBenchmarkExamples}}
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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 at
 +
* ..\examples\benchmark_examples\elements\shell8_quad_lin_canti_I\shell8_canti_2x20_elem_load.dat
  
 
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.
 
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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|}
 
|}
 
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.
 
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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 +
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=== Benchmark examples ===
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* Cantilever under different surface loads, discretized [[Users:General FEM Analysis/Elements Reference/Shell8|SHELL8]]-elements ..\examples\benchmark_examples\elements\shell8_quad_lin_canti_I\shell8_canti_2x20_elem_load.dat
 +
* Truss under tension discretized by [[Users:General FEM Analysis/Elements Reference/SolidHexa1|SOLIDHEXA1]]-elements: ..\examples\benchmark_examples\analyses\stalin_cantilever_hexa_I\cbm_biegebalken.txt

Latest revision as of 15:23, 7 November 2018


Contents

General Description

The linear static analysis is applied to analyze displacements, strains or stresses of structures. This analysis is valid for linear static mechanical problems without any nonlinearity (geometrically nonlinear, contact, material nonlinear, ...). Static nonlinear problems can be solved by the static nonlinear analysis whereas transient problems are solved by dynamic analysis methods.

The static linear analysis formulates structural equilibrium on the initial configuration without consideration of deformations. Hence, it is only valid for problems showing small displacements.

Structure of Equation System

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. This back substitution requires only small numerical effort which makes direct solvers very attractive if a couple of displacement fields have to be computed.


Input Parameters

Parameter Description

Compulsory Parameters
Parameter Values, Default(*) Description
SOLVER PC-SOLVER int Linking to a linear solver (direct or iterative)
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
Optional Parameters
ONLY_IBRA_PRE int Get integration domain in JSON format for Isogeometric B-Rep Analysis (IBRA) (1 = true, 0 = false)
SIMULTANEOUS_EIGENVALUE_ANALYSIS int Simultaneous eigenvalue analysis (1 = true, 0 = false)
EIGEN_SOLVER PC-SOLVER int Linking to a eigenvalue solver
NUM_EIGEN_MODES int Number of eigenmodes
BENCHMARK_ELEMENTS int Ids of Design elements
BENCHMARK_ELEMENTS_U float Position u of the benchmark points in the parameter space. If the number of members equals the number of the ids of the design element, only one point per element is printed. If not, all positions for every element is printed
BENCHMARK_ELEMENTS_V float Position v of the benchmark points in the parameter space
BENCHMARK_ELEMENTS_W float Position w of the benchmark points in the parameter space
BENCHMARK_DISP int if displacement at benchmark point should be printed (1 = true, 0 = false)
BENCHMARK_STRESS int if stresses at benchmark point should be printed (1 = true, 0 = false)
BENCHMARK_OUTPUT_NAME string names of files for each benchmark point (default = benchmark_point). If number of output names and design element ids matches, every element is only printed in the respective benchmark output file. For several names write name1,name2,... without spaces

Example of a Complete Input Block 

PC-ANALYSIS 1: STA_GEO_LIN
SOLVER = PC-SOLVER 5
OUTPUT = PC-OUT 1
COMPCASE = LD-COM 1,2,3
DOMAIN = EL-DOMAIN 1


Examples

A Full Example: Bending of a cantilever discretized with shell elements

The following example describes a simple cantilever problem discretized by SHELL8-elements. The respective input file can be found at

  • ..\examples\benchmark_examples\elements\shell8_quad_lin_canti_I\shell8_canti_2x20_elem_load.dat

The problem computes three computation cases (dead load, snow load and pressure load). The boundary conditions are visualized by the figure below.

Simple cantilever problem using SHELL8 elements

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.

Displacement of cantilever problem

It can be seen that the support region does not show any deformation whereas the tip region deforms by a value of 8.76.

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.

First principle stress of cantilever problem (top layer)

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.


Benchmark examples

  • Cantilever under different surface loads, discretized SHELL8-elements ..\examples\benchmark_examples\elements\shell8_quad_lin_canti_I\shell8_canti_2x20_elem_load.dat
  • Truss under tension discretized by SOLIDHEXA1-elements: ..\examples\benchmark_examples\analyses\stalin_cantilever_hexa_I\cbm_biegebalken.txt




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