Shells - Basic assumptions and principles

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A shell element

Shell theory is developed under the following assumptions:

  • The shells thickness, t, is much smaller than the shells dimensions in the x-y-plane.
  • All stresses are constant over the shell thickness.
  • The shells material is linear elastic, homgeneous, and isotropic.

A shell introduced to external load wil have a distribution of normal- and shear stresses in the x-y-plane. The shell theory's aim is to establish equations to describe this distribution. To do this, we use three conditions:

  • Equilibrium
    By looking at a small element of the shell, dxdy, we demand that stresses keep the shell element in equilibrium. Elements on the shells boundaries must also have stresses in equilibrium with external forces. The shell can now be comprised of a set of small shell elements, who all satisfy this equilibrium.
  • Kinematic compatibility
    When the shell is deformed, the material still has to be continous. This condition implies that openings or overlap between the small elements are not allowed.
  • Linear elastic material
    For a linear elestic material we have to have a linear relationship between stress and strain. This means that equilibrium and compatibility can be expressed unambiguously by either stress or strain.