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7x

001583

2024-09-11

Solving Quadratic Equation for Flexural Buckling

Solving Quadratic Equation for Flexural Buckling

\frac{1}{1 - χ \cdot {\bar{λ}}^{2}} \cdot χ \cdot α + χ = 1 | - 1

\frac{1}{1 - χ \cdot {\bar{λ}}^{2}} \cdot χ \cdot α + χ - 1 = 0 | \cdot 1 - χ \cdot {\bar{λ}}^{2}

= > χ^{2} - \frac{2 ϕ}{λ^{2}} \cdot χ + \frac{1}{λ^{2}} = 0

χ_{1, 2} = + \frac{ϕ}{λ^{2}} \pm \sqrt{{(\frac{ϕ}{λ^{2}})}^{2} - \frac{1}{λ^{2}}} = \frac{ϕ}{λ^{2}} \pm \frac{1}{λ^{2}} \cdot \sqrt{ϕ^{2} - λ^{2}} = \frac{1}{λ^{2}} (ϕ \pm \sqrt{ϕ^{2} - λ^{2}})

= \frac{1}{λ^{2}} \cdot \frac{(ϕ \pm \sqrt{ϕ^{2} - λ^{2}}) \cdot (ϕ \pm \sqrt{ϕ^{2} - λ^{2}})}{\cdot (ϕ \pm \sqrt{ϕ^{2} - λ^{2}})} = \frac{1}{λ^{2}} \cdot \frac{ϕ^{2} - (ϕ^{2} - λ^{2})}{(ϕ \pm \sqrt{ϕ^{2} - λ^{2}})}

χ = \frac{1}{ϕ \pm \sqrt{ϕ^{2} - λ^{2}}}

Flexural Buckling

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