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2.7: Equilibrium in the Theory of Moderately Large Deflections of Beams

  • Page ID
    21697
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    In Chapter 1, it was shown that finite rotation of the beam element introduced the additional term \(\frac{1}{2}\theta^2\) in the expression for the axial strain. Let’s see if consideration of finite slope would require modification of the equation of the equilibrium.

    In Figure (2.5.3) the beam element is shown in the theory of small deflections (infinitesimal rotation) and moderately large deflections (finite rotation).

    2.7.1.png
    Figure \(\PageIndex{1}\): In finite rotation the axial force contributes to the total shear force.

    The so-called effective shear force \(V^*\) is a sum of the cross-sectional shear \(V\) and projection of the axial force into the vertical direction. Thus,

    \[V^* = V + N \frac{dw}{dx} \label{3.74}\]

    Note that this result is valid as long as \(\cos \theta \approx 1\) and \(\sin \theta \approx \tan \theta \approx \theta\). Will the derivation of the force equilibrium change? The answer is no.

    2.7.2.png
    Figure \(\PageIndex{2}\): Direction of forces to equilibrate the infinitesimal beam element.

    To ensure vertical equilibrium

    \[(V^* + dV^* ) − V^* + qdx = 0\]

    or

    \[\frac{dV^*}{dx} + q = 0 \label{3.76}\]

    where \(V^*\) is defined by Equation \ref{3.74}. Equilibrium of the horizontal (axial) forces stays the same as before since \(\cos \theta \approx 1\).

    \[\frac{dN}{dx} = 0 \label{3.77}\]

    Eliminating \(V^*\) between Equation \ref{3.74} and Equation \ref{3.76} gives

    \[\frac{dV}{dx} + \frac{d}{dx} \left(N\frac{dw}{dx}\right) + q = \frac{dV}{dx} + \frac{dN}{dx} \frac{dw}{dx} + N \frac{d^2w}{dx^2} + q = 0\]

    The second term vanishes on account of Equation \ref{3.77}. The modified force equilibrium equation becomes

    \[\frac{dV}{dx} + N \frac{d^2w}{dx^2} + q = 0 \label{3.79}\]

    The new nonlinear term vanishes if (i) the axial force is zero or (ii) for small deflections and rotation. The moment equilibrium equation, Equation (2.6.15) is not affected by moderately large rotations. Together with Equation \ref{3.79} we arrive at the governing equation of the theory of moderately large deflection of beams

    \[\frac{d^2M}{dx^2} + N \frac{d^2w}{dx^2} + q = 0 \label{3.80}\]

    On closing this section, two important remarks should be made. All equations of equilibrium for infinitesimal deformations of 3-D bodies and small deflections of beams involved only static quantities and their gradients \((M, V, N)\). In the theory of moderately large deflections there is coupling between static and kinematic quantities through the second nonlinear terms.

    Secondly, Equation \ref{3.80} includes leading in the in-plane direction (through \(N\)) and out-of-plane direction through \(q\). Therefore, it is after referred as the equation describing a beam/columns.


    This page titled 2.7: Equilibrium in the Theory of Moderately Large Deflections of Beams 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.