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Displaying elastic stresses in surfaces


To visualize the elastic stresses in slabs or plates, first click , and then select the chosen partial result from these options:

  • Representation of the stresses in the direction x’ in the upper fibre.
  • Representation of the stresses in the direction z’ in the upper fibre.
  • Representation of the effective stresses in the upper fibre.
  • Representation of the stresses in the direction x’ in the lower fibre.
  • Representation of the stresses in the direction z’ in the lower fibre.
  • Representation of the effective stresses in the lower fibre.
  •  Representation of the first principal normal stress in the upper fibre of the plate.
    = \frac{1}{2} (\sigma_{xx.s} + \sigma_{zz.s} + \sqrt{(\sigma_{xx.s} - \sigma_{zz.s})^{2}+ 4\cdot \sigma_{xz.s}^{2}})
  • Representation of the second principal normal stress in the upper fibre of the plate.
    = \frac{1}{2} (\sigma_{xx.s} + \sigma_{zz.s} + \sqrt{(\sigma_{xx.s} - \sigma_{zz.s})^{2}+ 4\cdot \sigma_{xz.s}^{2}})
  • Representation of the directions of principal normal stresses in the upper fibre of the plate.
  • Representation of the first principal normal stress in the lower fibre of the plate.
    = \frac{1}{2} (\sigma_{xx.i} + \sigma_{zz.i} + \sqrt{(\sigma_{xx.i} - \sigma_{zz.i})^{2}+ 4\cdot \sigma_{xz.i}^{2}})
  • Representation of the second principal normal stress in the lower fibre of the plate.
    = \frac{1}{2} (\sigma_{xx.i} + \sigma_{zz.i} - \sqrt{(\sigma_{xx.i} - \sigma_{zz.i})^{2}+ 4\cdot \sigma_{xz.i}^{2}})
  • Representation of the directions of principal normal stresses in the lower fibre of the plate.

Notes:

  • The sign conventions are listed in this article.
    But the effective stresses are always positive. They are calculated with the use of the plasticity criterion of Huber-Hencky-von-Mises:

        \[\sigma_{eff}= \sqrt{\frac{1}{2}\cdot \left [(\sigma_{x}-\sigma_{y})^{2}+(\sigma_{y}-\sigma_{z})^{2}+(\sigma_{z}-\sigma_{x})^{2} \right ] + 3(\tau_{x}^{2}+\tau_{y}^{2}+\tau_{z}^{2})}\]

In Diamonds, σy = 0, so the formula simplifies to:

    \[\sigma_{eff}= \sqrt{\frac{1}{2}\cdot \left [\sigma_{x}^{2}+(-\sigma_{z})^{2}+(\sigma_{z}-\sigma_{x})^{2} \right ] + 3(\tau_{x}^{2}+\tau_{y}^{2}+\tau_{z}^{2})}\]

In principle, the above formula should be applied for all the points. But in Diamonds, the effective stresses are calculated only for the upper and lower fibers. However, they are overestimated, since it is understood that \tau_{x}=\frac{V_x}{A_x}, \tau_{z}=\frac{V_z}{A_z} and \tau_{y}=\frac{N_{xz}}{A_{xz}} where A_{xz}=min\left( A_x, A_z \right).

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