# MiguelAngel

(Difference between revisions)
 Revision as of 10:40, 17 July 2013 (view source)Maceli (Talk | contribs)← Older edit Revision as of 14:56, 18 July 2013 (view source)Maceli (Talk | contribs) Newer edit → Line 6: Line 6: * '''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. + + PLAIN STRESS 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). A related notion, [[plane strain]], is often applicable to very thick members. 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). A related notion, [[plane strain]], is often applicable to very thick members. Line 68: Line 70: − Plane strain + PLAIN STRAIN + [[File:Plane strain.svg|600px|right|thumb|Figure 7.2 Plane strain state in a continuum.]] [[File:Plane strain.svg|600px|right|thumb|Figure 7.2 Plane strain state in a continuum.]] {{main|Infinitesimal strain theory}} {{main|Infinitesimal strain theory}}

## Revision as of 14:56, 18 July 2013

#### 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). <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/>

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.

Mathematical definition

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

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

For example, consider a rectangular block of material measuring 10, 40 and 5 cm along the x, y, and z, that is being stretched in the x direction and compressed in the y direction, by pairs of opposite forces with magnitudes 10 N and 20 N, respectively, uniformly distributed over the corresponding faces. The stress tensor inside the block will be

$\sigma = \begin{bmatrix} 500\mathrm{ Pa} & 0 & 0 \\ 0 & -4000\mathrm{ Pa} & 0 \\ 0 & 0 & 0 \end{bmatrix}$

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,

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

See Hooke's law#Plane_stress

PLAIN STRAIN

File:Plane strain.svg
Figure 7.2 Plane strain state in a continuum.

Template:Main 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. Template:Clear

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:

• 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: