interactive example

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Stress Matrix Entries:

σ₁₁ =

σ₂₂ =

σ₁₂ =

Force Vectors Scale
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σ = (\begin{matrix} 1 & 1 \\ 1 & 3 \end{matrix})

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Q = (\begin{matrix} 0.92388 & 0.382683 \\ - 0.382683 & 0.92388 \end{matrix})

σ^{'} = Q σ Q^{T} = (\begin{matrix} 0.585786 & 0 \\ 0 & 3.414214 \end{matrix})

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Eigenvalues (Principal Stresses) =

(\begin{matrix} 0.585786 \\ 3.414214 \end{matrix})

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Eigenvalues (Principal Stresses Directions) =

(\begin{matrix} 0.92388 \\ 0.382683 \end{matrix}), (\begin{matrix} - 0.382683 \\ 0.92388 \end{matrix})

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Square aligned with original coordinate system

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Square aligned with eigenvectors coordinate system

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Q_{2} = (\begin{matrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{matrix}), τ_{M A X} = 1.414214

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σ^{″} = Q_{2} Q σ Q^{T} Q_{2}^{T} = (\begin{matrix} 2 & - 1.414214 \\ - 1.414214 & 2 \end{matrix})

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Square aligned with coordinate system oriented at 45 degrees with eigenvectors coordinate system