What is even stress

Level state of tension


engl: plane stress condition Category: Level 2 mechanics

A plane state of stress in a plane element is present when a 3-dimensional component has stresses in two spatial directions, but no stresses occur in the third spatial direction.

For example, the condition of the plane state of tension is present

  • on free surfaces and edges of structures,
  • with flat ribs (as sketched on the right as ribbing of a mast foot, or leaf roots of trees),
  • ...

These structures are flat, thin-walled, they absorb loads in the plane of the rib. The plane stress state is given by the fact that the loads in the plane lead to strains in this plane and also correspondingly to stresses. In the direction normal to the plane of the rib, elongation occurs due to the transverse contraction, but there are no stresses normal to the plane because there is no constraint or retention. The material is free to expand normal to the plane of the rib. So there is basically a three-dimensional state of strain given a plane stress state.

The plane state of stress is in an infinitely thin sheet. In practice, the thicker the sheet, the less applicable this condition.


By assuming that there is a plane stress state, a 3-dimensional component can be simulated with a 2-dimensional model. The model then covers the area of ​​the section. During the discretization planar elements are selected and set to "plane stress". The plane stress state is taken into account in the element shape functions. There are plane displacements and plane tensions, but 3-dimensional expansions. This results in a significant reduction in the modeling and calculation effort.

The model then represents a section of the component that is "1" long in the 3rd dimension (i.e. 1 [m] if you calculate with the unit of length [m], or 1 [mm] if you calculate with the unit of length [mm ] calculate). This should be noted for loads.

Tooth of a spur gear

In the picture on the right you can see a spur gear of a gearbox. The teeth are straight (straight teeth). When the teeth on the opposite side engage, the force that is transmitted between the teeth acts (theoretically) along a line on the tooth and essentially creates a bending stress and a shear stress in the tooth. In the middle and at the end of the tooth there are special conditions: in the middle (almost) a level state of stretch and at the end a level state of tension.

The flat state of tension at the end results from the fact that the side surface is exposed at the end of the tooth, ie "nothing" is adjacent and therefore no tension can be present normal to the surface. So there can only be stresses in the plane of the surface. This is a level state of tension.

The conditions at the end of the tooth can be simulated with an FEM model that contains planar elements. The behavior of these elements must then be set to "plane stress".

Rib of the mast foot

In a simulation of the rib of the mast foot, planar elements (disc elements in the plane) can be used if the rib in the figure above - and only the rib in itself - is to be calculated. Then only the behavior in the x-y plane oriented to the rib surface is examined.

If, on the other hand, the neighboring components - i.e. the mast foot itself and the other ribs - are to be included in the model, a 3-dimensional component is available for which shell elements should be used.

Other terms

Other special conditions that allow such a reduction to a 2-dimensional model are the plane strain state and axial symmetry.