0004 - Columns with P-Delta Effects

1 min read • 159 words

Nonlinear geometry in frames.

$$P_{e L} = \frac{\pi^2 E I}{L^2} \qquad\text{ and }\qquad \lambda = \frac{\pi}{2} \sqrt{\frac{\alpha P_r}{P_{e L}}}$$

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$$u_y = H L^3/(3 E I) + H L/(\kappa G A)$$ $$u = \frac{H L^3}{3 E I}\left[\frac{3(\tan 2\lambda - 2\lambda)}{(2\lambda)^3}\right]$$ $$M = H L\left[\frac{\tan 2\lambda}{2\lambda}\right]$$

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$$u = \frac{H L^3}{12 E I}\left[\frac{3(\tan \lambda - \lambda)}{\lambda^3}\right]$$ $$M = \frac{H L}{2}\left[\frac{\tan \lambda}{\lambda}\right]$$

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$$u = \frac{5 w L^4}{384 E I}\left[\frac{12\left(2 \sec \lambda - \lambda^2 - 2\right)}{5 \lambda^4}\right]$$ $$M = \frac{w L^2}{8}\left[\frac{2(\sec \lambda - 1)}{\lambda^2}\right]$$

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$$v = \frac{w L^4}{384 E I}\left[\frac{12(2 - 2 \cos \lambda - \lambda \sin \lambda)}{\lambda^3 \sin \lambda}\right]$$ $$M = \frac{w L^2}{12}\left[\frac{3(\tan \lambda - \lambda)}{\lambda^2 \tan \lambda}\right]$$

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$$u = \frac{w L^4}{192 E I}\left\{\frac{6}{\lambda^4}\left[(2 \sec \lambda - 2\lambda - 2) - \frac{(\tan \lambda - \lambda)(\sec \lambda - 1)}{\left(\frac{1}{2\lambda} - \frac{1}{\tan 2\lambda}\right)}\right]\right\}$$ $$M = \frac{w L^2}{8}\left[\frac{2(\tan \lambda - \lambda)}{\lambda^2\left(\frac{1}{2\lambda} - \frac{1}{\tan 2\lambda}\right)}\right]$$

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References  

• R.C.Kaehler, D.W.White, Y.D.Kim, “Frame Design Using Web-Tapered Members”, AISC 2011