# CSMm 2.2.Elements

Line 1: | Line 1: | ||

[[Category: CSMm 2.Structure]] | [[Category: CSMm 2.Structure]] | ||

− | + | Depending on whether we are working in a 3D problem or in a 2D one, we can distinguix various kinds of elements: | |

− | + | === 3D ELEMENTS === | |

− | ''' | + | *'''SOLID ELEMENT'''. |

+ | |||

+ | Only available if ''Structural Type'' is set to ''Solid'' or ''Generic''. | ||

+ | |||

+ | |||

+ | *'''BEAM ELEMENT''' | ||

This element is based on Euler-Bernoulli formulation. The formulation assumes that a cross section plane orthogonal to the axis of undeformed beam will remain plane and also orthogonal to the axis in deformed configuration. This assumption is valid for thin beams where axial strains (due to the axial forces and also bending moments) are dominant. For short (or thick) beams this formulation is not recommended while it can not reproduce the shear strain of the section. The hypothesis involved in formulation are: | This element is based on Euler-Bernoulli formulation. The formulation assumes that a cross section plane orthogonal to the axis of undeformed beam will remain plane and also orthogonal to the axis in deformed configuration. This assumption is valid for thin beams where axial strains (due to the axial forces and also bending moments) are dominant. For short (or thick) beams this formulation is not recommended while it can not reproduce the shear strain of the section. The hypothesis involved in formulation are: | ||

Line 24: | Line 29: | ||

'''Element info''': | '''Element info''': | ||

− | + | ||

− | + | - '''Input file name''': BeamElement3D2N | |

− | + | ||

− | + | - '''Constitutive Law''': None | |

− | + | ||

− | + | - '''Nonlinearity''': Only linear | |

− | + | ||

+ | - '''Time Schemes''': Backward Euler, Forward Euler | ||

+ | |||

+ | - '''Dofs''': DISPLACEMENT, ROTATION | ||

+ | |||

+ | - '''Properties''': CROSS_AREA,LOCAL_INERTIA, POISSON_RATIO, YOUNG_MODULUS, DENSITY | ||

+ | |||

+ | - '''Elemental Data''': None | ||

[[Category:SolidMechanicsElements]] | [[Category:SolidMechanicsElements]] | ||

− | ''' | + | *'''SHELL ISOTROPIC''' |

+ | |||

+ | |||

+ | === 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 | ||

+ | : <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 | ||

+ | : <math>\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}</math> | ||

+ | 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> | ||

+ | |||

+ | |||

+ | *'''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: | ||

+ | : <math>\varepsilon_{ij} = \begin{bmatrix} | ||

+ | \varepsilon_{11} & \varepsilon_{12} & 0 \\ | ||

+ | \varepsilon_{21} & \varepsilon_{22} & 0 \\ | ||

+ | 0 & 0 & \varepsilon_{33}\end{bmatrix}\,\!</math> | ||

− | ' | + | 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. |

## Revision as of 09:39, 2 October 2013

Depending on whether we are working in a 3D problem or in a 2D one, we can distinguix various kinds of elements:

### 3D ELEMENTS

**SOLID ELEMENT**.

Only available if *Structural Type* is set to *Solid* or *Generic*.

**BEAM ELEMENT**

This element is based on Euler-Bernoulli formulation. The formulation assumes that a cross section plane orthogonal to the axis of undeformed beam will remain plane and also orthogonal to the axis in deformed configuration. This assumption is valid for thin beams where axial strains (due to the axial forces and also bending moments) are dominant. For short (or thick) beams this formulation is not recommended while it can not reproduce the shear strain of the section. The hypothesis involved in formulation are:

Where the *u*,*v*,*w* are the displacements in beams local axis.

This element in Kratos is designed for small strain and in this range produce accurate results. The element does not accept any constitutive law and cannot be used for non linear materials.

**Element info**:

- **Input file name**: BeamElement3D2N

- **Constitutive Law**: None

- **Nonlinearity**: Only linear

- **Time Schemes**: Backward Euler, Forward Euler

- **Dofs**: DISPLACEMENT, ROTATION

- **Properties**: CROSS_AREA,LOCAL_INERTIA, POISSON_RATIO, YOUNG_MODULUS, DENSITY

- **Elemental Data**: None

**SHELL ISOTROPIC**

### 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

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

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

**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:

in which the non-zero 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.