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9.2: Deflection Behavior for Beam with Compressive Axial Loads and Transverse Loads

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    21526
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    9.2.1.png
    Figure \(\PageIndex{1}\): Simply supported beam with intermediate transverse load.

    Consider a simply supported beam with a fixed load \(f\) applied at the middle as shown in Figure (\(\PageIndex{1}\)). Additionally, the beam is subjected to a compressive axial load \(P\). The total potential energy for this mechanical system is

    \[\left. \prod \right._{\text{total}} = \int_{0}^{L} \frac{1}{2} EI(v^{\prime\prime})^2 dx − \int_{0}^{L} \frac{1}{2} (v^{\prime})^2 dx − fv \left(\frac{L}{2}\right) \]

    If \(f = 0\), we are looking at a classical buckling problem; viz., the beam remains straight until a critical load is reached after which the beam bends suddenly. The critical load for the configuration shown is \(P_{cr} = \pi^2 EI/L^2\). Let’s investigate the behavior for \(f \neq 0\).

    The stationary points of the potential energy still give the solutions \(v(x)\) which satisfy equilibrium. Let’s compute an approximate solution using the form

    \[ v(x) \approx C \sin \left(\pi \frac{x}{L}\right) \]

    The derivatives of this function are

    \[ v^{\prime}(x) = C \frac{\pi}{L} \cos \left(\pi \frac{x}{L}\right) \nonumber \]

    \[v^{\prime\prime}(x) = - C \left(\frac{\pi}{L}\right)^2 \sin \left(\pi \frac{x}{L}\right) \nonumber\]

    Inserting these into the potential energy yields

    \[\left. \prod \right._{\text{total}} = \int_{0}^{L} \frac{1}{2} EI \left(\frac{\pi}{L}\right)^4 C^2 \sin^2 \left(\pi \frac{x}{L}\right) dx \\ − P \int_{0}^{L} \frac{1}{2} \left(\frac{\pi}{L}\right)^2 C^2 \cos^2 \left(\pi \frac{x}{L}\right) dx - fC \sin \left(\pi \frac{L/2}{L}\right) \\ = \int_{0}^{L} \frac{1}{2} EI \left(\frac{\pi}{L}\right)^4 C^2 \left[ \frac{1}{2} - \frac{1}{2} \cos \left(\frac{2 \pi x}{L}\right) \right] dx \\ − P \int_{0}^{L} \frac{1}{2} \left(\frac{\pi}{L}\right)^2 C^2 \left[ \frac{1}{2} + \frac{1}{2} \cos \left(\frac{2 \pi x}{L}\right) \right] dx - fC \\ = \frac{1}{4} EI \left(\frac{\pi}{L}\right)^4 C^2L − P \frac{1}{4} \left(\frac{\pi}{L}\right)^2 C^2L - fC \nonumber\]

    The stationary condition yields

    \[0 = \frac{d \prod_{\text{total}}}{dC} = \frac{1}{2} EI \left(\frac{\pi}{L}\right)^4 CL − P \frac{1}{2} \left(\frac{\pi}{L}\right)^2 CL - f = C \left[ \frac{1}{2} EI \left(\frac{\pi}{L}\right)^4 L − P \frac{1}{2} \left(\frac{\pi}{L}\right)^2 L \right] − f = 0 \]

    and thus

    \[C = \frac{f}{\frac{EI \pi^4}{2L^3} − P\frac{\pi^2}{2L}} \\ = \frac{f2L/\pi^2}{\frac{EI\pi^2}{L^2} − P} \\ = \frac{2L}{\pi^2}\frac{f}{P_{cr} − P}\]

    The approximate solution has the form

    \[v(x) \approx \frac{2L}{\pi^2}\frac{f}{P_{cr} − P} \sin \left( \pi \frac{x}{L}\right)\]

    The central deflection \(w_o = v(x = \frac{l}{2}) \) is

    \[w_o = \frac{fl^3}{48.7EI}\frac{1}{1 − \frac{P}{P_c}} \label{10.31}\]

    For zero axial load, Equation \ref{10.31} predicts a linear relation between the lateral point load and deflection \(w_o\). The approximate coefficient \(\frac{\pi^4}{2} \cong 48.7\) is very close to the exact value 48 for the pin-pin column loaded by the point force \(f\). The linear relation holds also for any constant value of \(P/P_c\). A much more interesting picture is obtained by fixing the lateral load and changing the axial load. Equation \ref{10.31} can be written as

    \[w_o = \frac{\eta}{1 − \frac{P}{P_c}}, \text{ where } \eta = \frac{fl^3}{48.7EI} \]

    which is plotted in Figure (9.3.1). Note that the positive force is in compression while the negative in tension. Application of the lateral force deflects the beam by the amount \(\eta\). Then, on application of the in-plane compressive load, the beam-column behaves as an imperfect column. By reversing the sign of the in-plane load from compression into tension, the central deflection becomes smaller and vanishes with \(P/P_c \rightarrow \infty\). This is fully consistent with our everyday experience that by tightening the rope/cable, its deflection is reduced.


    This page titled 9.2: Deflection Behavior for Beam with Compressive Axial Loads and Transverse Loads is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Tomasz Wierzbicki (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.