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* '''2D elements'''
 
* '''2D elements'''
 
In Kratos, you can choose between two 2D element types: plain stress and plain strain elements.
 
In Kratos, you can choose between two 2D element types: plain stress and plain strain elements.
In [[continuum mechanics]], a material is said to be under '''plane stress''' if the [[stress (mechanics)|stress vector]] is zero across a particular surface.  When that situation occurs over an entire element of a structure, as is often the case for thin plates, the [[stress analysis]] is considerably simplified, as the stress state can be represented by a [[tensor]] of dimension 2 (representable as a 2 × 2 matrix rather than 3 × 3). <ref name=Meyers/> A related notion, [[plane strain]], is often applicable to very thick members.
 
  
Plane stress typically occurs in thin flat plates that are acted upon only by load forces that are parallel to them.  In certain situations, a gently curved thin plate may also be assumed to have plane stress for the purpose of stress analysis. This is the case, for example, of a thin-walled cylinder filled with a fluid under pressure.  In such cases, stress components perpendicular to the plate are negligible compared to those parallel to it.<ref name=Meyers/>
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PLAIN STRESS
 
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In other situations, however, the bending stress of a thin plate cannot be neglected. One can still simplify the analysis by using a two-dimensional domain, but the plane stress tensor each point must be complemented with bending terms.
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Mathematical definition
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Mathematically, the stress at some point in the material is a plane stress if one of the three [[principal stress]]es (the [[eigenvalues and eigenvectors|eigenvalues]] of the [[Cauchy stress tensor]]) is zero.  That is, there is [[Cartesian coordinate system]] in which the stress tensor has the form 
+
 
+
 
+
______________
+
 
+
 
+
 
+
 
+
_____________________________
+
  
 +
In continuum mechanics, a material is said to be under '''plane stress''' if the stress vector is zero across a particular surface.  When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analysis is considerably simplified, as the stress state can be represented by a tensor of dimension 2 (representable as a 2 × 2 matrix rather than 3 × 3). A related notion, plane strain, is often applicable to very thick members.
  
 +
Plane stress typically occurs in thin flat plates that are acted upon only by load forces that are parallel to them.  In certain situations, a gently curved thin plate may also be assumed to have plane stress for the purpose of stress analysis. This is the case, for example, of a thin-walled cylinder filled with a fluid under pressure.  In such cases, stress components perpendicular to the plate are negligible compared to those parallel to it.
  
 +
In other situations, however, the bending stress of a thin plate cannot be neglected. One can still simplify the analysis by using a two-dimensional domain, but the plane stress tensor each point must be complemented with bending terms.
  
 +
Mathematically, the stress at some point in the material is a plane stress if one of the three principal stresses (the eigenvalues of the Cauchy stress tensor) is zero.  That is, there is Cartesian coordinate system in which the stress tensor has the form 
 +
: <math>\sigma =
 +
\begin{bmatrix}
 +
\sigma_{11} & 0 & 0 \\
 +
0 & \sigma_{22} & 0 \\
 +
0      &    0      & 0
 +
\end{bmatrix}
 +
\equiv
 +
\begin{bmatrix}
 +
\sigma_{x} & 0 & 0 \\
 +
0 & \sigma_{y} & 0 \\
 +
0      &    0      & 0
 +
\end{bmatrix}</math>
 +
 
More generally, if one chooses the first two coordinate axes arbitrarily but perpendicular to the direction of zero stress, the stress tensor will have the form
 
More generally, if one chooses the first two coordinate axes arbitrarily but perpendicular to the direction of zero stress, the stress tensor will have the form
 
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: <math>\sigma =
 
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\begin{bmatrix}
__________________
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\sigma_{11} & \sigma_{12} & 0 \\
 
+
\sigma_{21} & \sigma_{22} & 0 \\
 +
0      &    0      & 0
 +
\end{bmatrix}
 +
\equiv
 +
\begin{bmatrix}
 +
\sigma_{x} & \tau_{xy} & 0 \\
 +
\tau_{yx} & \sigma_{y} & 0 \\
 +
0      &    0      & 0
 +
\end{bmatrix}</math>
 
and can therefore be represented by a 2 × 2 matrix,
 
and can therefore be represented by a 2 × 2 matrix,
 +
: <math>\sigma_{ij} =
 +
\begin{bmatrix}
 +
\sigma_{11} & \sigma_{12} \\
 +
\sigma_{21} & \sigma_{22}
 +
\end{bmatrix}
 +
\equiv
 +
\begin{bmatrix}
 +
\sigma_{x} & \tau_{xy} \\
 +
\tau_{yx} & \sigma_{y}
 +
\end{bmatrix}</math>
  
  
_________________________
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PLAIN STRAIN
  
 
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If one dimension is very large compared to the others, the principal strain in the direction of the longest dimension is constrained and can be assumed as zero, yielding a plane strain condition (Figure 7.2).  In this case, though all principal stresses are non-zero, the principal stress in the direction of the longest dimension can be disregarded for calculations. Thus, allowing a two dimensional analysis of stresses, e.g. a dam analyzed at a cross section loaded by the reservoir.
Constitutive equations
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:''See [[Hooke's law#Plane_stress]]''
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Plane stress in curved surfaces
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In certain cases, the plane stress model can be used in the analysis of gently curved surfaces. For example,
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consider a thin-walled cylinder subjected to an axial compressive load uniformly distributed along its rim, and filled with a pressurized fluid.  The internal pressure will generate a reactive [[hoop stress]] on the wall, a normal tensile stress directed perpendicular to the cylinder axis and tangential to its surface.  The cylinder can be conceptually unrolled and analyzed as a flat thin rectangular plate subjected to tensile load in one direction and compressive load in another other direction, both parallel to the  plate.
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Plane strain
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If one dimension is very large compared to the others, the [[Strain (materials science)|principal strain]] in the direction of the longest dimension is constrained and can be assumed as zero, yielding a plane strain condition (Figure 7.2).  In this case, though all principal stresses are non-zero, the principal stress in the direction of the longest dimension can be disregarded for calculations. Thus, allowing a two dimensional analysis of stresses, e.g. a [[dam]] analyzed at a cross section loaded by the reservoir.
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{{Clear}}
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The corresponding strain tensor is:
 
The corresponding strain tensor is:
  
_______________________________________
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: <math>\varepsilon_{ij} = \begin{bmatrix}
 
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\varepsilon_{11} & \varepsilon_{12} & 0 \\
 
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\varepsilon_{21} & \varepsilon_{22} & 0 \\
 +
0      &    0      & \varepsilon_{33}\end{bmatrix}\,\!</math>
  
in which the non-zero ---------------- term arises from the [[Poisson's ratio|Poisson's effect]]. This strain term can be temporarily removed from the stress analysis to leave only the in-plane terms, effectively reducing the analysis to two dimensions.
+
in which the non-zero <math>\varepsilon_{33}\,\!</math> term arises from the Poisson's effect. This strain term can be temporarily removed from the stress analysis to leave only the in-plane terms, effectively reducing the analysis to two dimensions.
  
  

Latest revision as of 00:34, 24 March 2016

Elements

  • Frame Elements:
    • Euler-Bernoulli beam short explanation
    • Crisfield truss short explanation
  • 2D elements

In Kratos, you can choose between two 2D element types: plain stress and plain strain elements.

PLAIN STRESS

In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular surface. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analysis is considerably simplified, as the stress state can be represented by a tensor of dimension 2 (representable as a 2 × 2 matrix rather than 3 × 3). A related notion, plane strain, is often applicable to very thick members.

Plane stress typically occurs in thin flat plates that are acted upon only by load forces that are parallel to them. In certain situations, a gently curved thin plate may also be assumed to have plane stress for the purpose of stress analysis. This is the case, for example, of a thin-walled cylinder filled with a fluid under pressure. In such cases, stress components perpendicular to the plate are negligible compared to those parallel to it.

In other situations, however, the bending stress of a thin plate cannot be neglected. One can still simplify the analysis by using a two-dimensional domain, but the plane stress tensor each point must be complemented with bending terms.

Mathematically, the stress at some point in the material is a plane stress if one of the three principal stresses (the eigenvalues of the Cauchy stress tensor) is zero. That is, there is Cartesian coordinate system in which the stress tensor has the form

\sigma = 
\begin{bmatrix}
\sigma_{11} & 0 & 0 \\
0 & \sigma_{22} & 0 \\
0      &     0       & 0
\end{bmatrix} 
\equiv 
\begin{bmatrix}
\sigma_{x} & 0 & 0 \\
0 & \sigma_{y} & 0 \\
0      &     0       & 0
\end{bmatrix}

More generally, if one chooses the first two coordinate axes arbitrarily but perpendicular to the direction of zero stress, the stress tensor will have the form

\sigma = 
\begin{bmatrix}
\sigma_{11} & \sigma_{12} & 0 \\
\sigma_{21} & \sigma_{22} & 0 \\
0      &     0       & 0
\end{bmatrix} 
\equiv 
\begin{bmatrix}
\sigma_{x} & \tau_{xy} & 0 \\
\tau_{yx} & \sigma_{y} & 0 \\
0      &     0       & 0
\end{bmatrix}

and can therefore be represented by a 2 × 2 matrix,

\sigma_{ij} = 
\begin{bmatrix}
\sigma_{11} & \sigma_{12} \\
\sigma_{21} & \sigma_{22}
\end{bmatrix} 
\equiv 
\begin{bmatrix}
\sigma_{x} & \tau_{xy} \\
\tau_{yx} & \sigma_{y}
\end{bmatrix}


PLAIN STRAIN

If one dimension is very large compared to the others, the principal strain in the direction of the longest dimension is constrained and can be assumed as zero, yielding a plane strain condition (Figure 7.2). In this case, though all principal stresses are non-zero, the principal stress in the direction of the longest dimension can be disregarded for calculations. Thus, allowing a two dimensional analysis of stresses, e.g. a dam analyzed at a cross section loaded by the reservoir.


The corresponding strain tensor is:

\varepsilon_{ij} = \begin{bmatrix}
\varepsilon_{11} & \varepsilon_{12} & 0 \\
\varepsilon_{21} & \varepsilon_{22} & 0 \\
0      &     0       & \varepsilon_{33}\end{bmatrix}\,\!

in which the non-zero \varepsilon_{33}\,\! term arises from the Poisson's effect. This strain term can be temporarily removed from the stress analysis to leave only the in-plane terms, effectively reducing the analysis to two dimensions.


    • Linear triangular element:


Two-dimensional_Shape_Functions



Element implementation

  • Shell elements:
    • Isotropic shell: (change the name with the usual one!!!!)
    • Ansotropic shell: (change the name with the usual one!!!!)
    • EBST shell: (change the name with the usual one!!!!)
  • Membrane element:
  • Solid elements:
    • Linear tetrahedral element:
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