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8.4: Effect of Structural Imperfections

  • Page ID
    21521
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    Consider the same discrete strutter as in Section 8.2. This time the rigid rod is not straight but is rotated by the angle \(\theta_o\) before the vertical load is applied. Upon load application the column is subjected to additional rotation \(\theta\), measured from the theoretical vertical direction, Figure (\(\PageIndex{1}\)).

    8.4.1.png
    Figure \(\PageIndex{1}\): The initial inclination angle \(\theta_o\) is a measure of structural imperfection.

    The problem will be solved by means of local equilibrium. The external bending moment at the base is

    \[M_{\text{ext}} = Pl \sin \theta, \text{ for } \theta \geqslant \theta_o \]

    where \(l \sin \theta\) is the arm of the force \(P\). In the case of small angle approximation \(M_{\text{ext}} = Pl \theta\). The internal resisting bending moment is

    \[M_{\text{int}} = K(\theta − \theta_o) \]

    Equating the external and internal bending moments

    \[Pl \theta = K(\theta − \theta_o) \label{9.40}\]

    For a geometrically perfect column \(\theta_o = 0\) and from Equation \ref{9.40}

    \[P = P_c = \frac{K}{l} \]

    Equation \ref{9.40} can be re-written in terms of the normalized compressive force \(P/P_c\)

    \[\frac{P}{P_c} \theta = \theta − \theta_o \]

    Solving this equation for \(\theta\) yields

    \[\theta = \theta_o \frac{1}{1 − \frac{P}{P_c}} \]

    The plot of the above function is shown in Figure (\(\PageIndex{2}\)). The term \(1/(1 − \frac{P}{P_c} )\) is called the magnification factor. It predicts how much the initial imperfections are magnified for a given magnitude of load. When structural imperfections are present, there are no primary and secondary equilibrium paths. There is only one smooth load-deflection curve called the equilibrium path.

    8.4.2.png
    Figure \(\PageIndex{2}\): A family of equilibrium paths for different values of imperfections.

    It is interesting to note that with smaller and smaller initial imperfections, the equilibrium paths are approaching the bifurcation point but never reach it. This type of behavior is common to all imperfect structures.

    As another example of an imperfect structure consider a pin-pin elastic column. The following notation is introduced:

    • \(\bar{w}(x)\) – shape of initial imperfection
    • \(\bar{w}_o\) – amplitude of initial imperfection
    • \(w(x)\) – actual buckled shape measured from the vertical (perfect) position
    • \(w_o\) – central amplitude of the actual deflection

    The internal bending moment is

    \[M_{\text{int}} = EI \Delta\kappa = −EI(w^{\prime \prime} − \bar{w}^{\prime \prime}) \]

    where \(\Delta\kappa\) is the change of curvature from the initial curved (imperfect) column. For a simply supported column, the end (reaction) moments are zero so the external bending moment is

    \[M_{\text{ext}} = P w \]

    8.4.3.png
    Figure \(\PageIndex{3\): A continuous imperfect column and the free body diagram.

    Equating the internal and external bending moments one gets

    \[EIw^{\prime \prime} + P w = EI\bar{w}(x) \label{9.46}\]

    This is a second order, linear inhomogeneous differential equation, where the right hand side is a known shape of initial imperfection. The solution to this equation exists in terms of quadratures, but the integrals are difficult to evaluate for complex shapes of imperfections.

    Let’s consider the simplest case of a sinusoidal shape of imperfections. It can be shown that the solution \(w(x)\) is also of the sinusoidal shape

    \[w(x) = w_o \sin \lambda x \label{9.47a}\]

    \[\bar{w}(x) = \bar{w}_o \sin \lambda x \label{9.47b}\]

    The kinematic boundary conditions are

    \[w(0) = w(l) = 0 \nonumber\]

    which implies that

    \[\sin \lambda l = 0 \quad \rightarrow \quad \lambda l = n \pi \]

    Substituting Equations \ref{9.47a} - \ref{9.47b} into the governing equation \ref{9.46}

    \[−EI \lambda^2 (w_o − \bar{w}_o) \sin \lambda x − P w_o \sin \lambda x = 0 \]

    which is satisfied if

    \[P w_o = EI(w_o − \bar{w}_o) \lambda^2 \label{9.50}\]

    For a perfect column \(\bar{w}_o = 0\), and Equation \ref{9.50} yields

    \[(P_c − EI \lambda^2 )w_o = 0 \\ \text{or }P_c = EI\lambda^2 = \frac{n^2 \pi^2 EI}{l^2} \]

    For an imperfect column

    \[P w_o = P_c(w_o − \bar{w}_o) \]

    or solving for \(w_o\)

    \[w_o = \bar{w}_o \frac{1}{1 − \frac{P}{P_c}} \]

    The form of the magnification factor is identical to the one derived for the district column. The only difference is that a continuous column has infinity buckling mode where \(n = 1\) corresponds to the lowest buckling load. The buckling load corresponding to the second buckling mode is four times larger and so on.


    This page titled 8.4: Effect of Structural Imperfections 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.