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# Eccentrically Loaded Columns(Secant Formula) - Columns and Struts, Strength of Materials Agricultural Engineering Notes | EduRev

## Agricultural Engineering : Eccentrically Loaded Columns(Secant Formula) - Columns and Struts, Strength of Materials Agricultural Engineering Notes | EduRev

The document Eccentrically Loaded Columns(Secant Formula) - Columns and Struts, Strength of Materials Agricultural Engineering Notes | EduRev is a part of the Agricultural Engineering Course Strength of Material Notes - Agricultural Engg.
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Eccentrically Loaded Columns – Secant Formula

Consider an eccentrically loaded slender column as shown in Figure 23.1a. Fig. 23.1.

The equation of elastic line becomes,

${{{d^2}y} \over {d{x^2}}}=-{{{M_x}} \over {EI}}$                                     (23.1)

$\Rightarrow {{{d^2}y} \over {d{x^2}}} + {P \over {EI}}\left( {e + y} \right) = 0$     (23.2)

$\Rightarrow {{{d^2}y} \over {d{x^2}}} + {k^2}y=-{k^2}e$                                          (23.3)

where, ${k^2} = {P / {EI}}$

The general solution of equation (23.3) is,

$y=A\cos kx + B\sin kx - e$                           (23.4)

Constants A and B are determined from the boundary conditions as follows,

$y(x = 0)=0 \Rightarrow A=e$

$y(x = l)=0 \Rightarrow B={{1 - \cos kl} \over {\sin kl}}e$

Substituting A and B in equation (23.4) yields,

$y=\left( {\cos kx + {{1 - \cos kl} \over {\sin kl}}\sin kx - 1} \right)e$                               (23.5)

Lateral deflection at mid-height (y = l/2),

${\left. y \right|_{x = l/2}}=\left( {\cos {{kl} \over 2} + {{1 - \cos kl} \over {\sin kl}}\sin {{kl} \over 2} - 1} \right)e$                          (23.6)

$\Rightarrow {\left. y \right|_{x = l/2}}=\left( {\sec {{kl} \over 2} - 1} \right)e$                                                (23.7)

Writing equation (23.7) in terms of Euler load ${P_E}={\pi ^2}EI/{l^2}$  ,

${\left. y \right|_{x = l/2}}=\left[ {\sec \left( {{\pi\over 2}\sqrt {{P \over{{P_E}}}} } \right) - 1} \right]e$                                    (23.8)

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