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Fix principal stress docs equations
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CHANGELOG.rst

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@@ -4,7 +4,7 @@ Changelog:
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v2.0.1:
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Fix issue with library module
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- Fix issue with library module
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v2.0.0:
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docs/source/rst/theory.rst

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@@ -871,12 +871,12 @@ The von Mises stress can be determined from the net axial and shear stress as fo
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Principal Stresses
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^^^^^^^^^^^^^^^^^^
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For a cross section subjected to axial force, shear in the :math:`x` and :math`y` axes which are
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For a cross section subjected to axial force, shear in the :math:`x` and :math:`y` axes which are
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perpendicular to the centroidal (:math:`z`) axis, and moments about all three axes, there are no
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axial stresses in the :math:`x` or :math:`y` axes, and so the stress tensor is given by:
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.. math::
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\textbf{\sigma} = \begin{bmatrix} 0 & 0 & \tau_{zx} \\
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\boldsymbol{\sigma} = \begin{bmatrix} 0 & 0 & \tau_{zx} \\
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0 & 0 & \tau_{zy} \\
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\tau_{xz} & \tau_{yz} & \sigma_{zz}
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\end{bmatrix}
@@ -889,7 +889,7 @@ matrix through a coordinate transformation. Since this is the basic eigenvalue p
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principal stresses are then given by:
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.. math::
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\det (\textbf{\sigma} - \lambda \textbf{I}) = 0
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\det (\boldsymbol{\sigma} - \lambda \textbf{I}) = 0
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Of which the characteristic polynomial can then be written:
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@@ -899,9 +899,9 @@ Of which the characteristic polynomial can then be written:
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where the stress invariants :math:`I` are then given by [5]:
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.. math::
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I_1 &= \textnormal(\textbf{\sigma}) = \sigma_{zz} \\
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I_2 &= \frac{1}{2}\left[ (\textnormal(\textbf{\sigma})^2 - \textnormal(\textbf{\sigma}^2)) \right] = -\tau_{zx}^2 - \tau_{yz}^2 \\
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I_3 &= \det(\textbf{\sigma}) = 0
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I_1 &= \textnormal(\boldsymbol{\sigma}) = \sigma_{zz} \\
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I_2 &= \frac{1}{2}\left[ (\textnormal(\boldsymbol{\sigma})^2 - \textnormal(\boldsymbol{\sigma}^2)) \right] = -\tau_{zx}^2 - \tau_{yz}^2 \\
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I_3 &= \det(\boldsymbol{\sigma}) = 0
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and thus, the cubic polynomial reduces to a quadratic, the two roots of which are then the first
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and third principal stresses (with :math:`\sigma_2 = 0`):

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