Eccentrically Loaded Columns(Secant Formula) - Columns and Struts, Strength of Materials - Notes | Study Strength of Material Notes - Agricultural Engg - Agricultural Engineering

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)