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

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

1 Crore+ students have signed up on EduRev. Have you?

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)

The document Eccentrically Loaded Columns(Secant Formula) - Columns and Struts, Strength of Materials - Notes | Study Strength of Material Notes - Agricultural Engg - Agricultural Engineering is a part of the Agricultural Engineering Course Strength of Material Notes - Agricultural Engg.
All you need of Agricultural Engineering at this link: Agricultural Engineering
34 docs
Use Code STAYHOME200 and get INR 200 additional OFF
Use Coupon Code
34 docs
Download as PDF

Download free EduRev App

Track your progress, build streaks, highlight & save important lessons and more!

Related Searches

pdf

,

Objective type Questions

,

Important questions

,

Eccentrically Loaded Columns(Secant Formula) - Columns and Struts

,

Free

,

study material

,

Extra Questions

,

Eccentrically Loaded Columns(Secant Formula) - Columns and Struts

,

Eccentrically Loaded Columns(Secant Formula) - Columns and Struts

,

ppt

,

Exam

,

shortcuts and tricks

,

Strength of Materials - Notes | Study Strength of Material Notes - Agricultural Engg - Agricultural Engineering

,

video lectures

,

Previous Year Questions with Solutions

,

MCQs

,

Summary

,

Strength of Materials - Notes | Study Strength of Material Notes - Agricultural Engg - Agricultural Engineering

,

Viva Questions

,

Semester Notes

,

past year papers

,

Sample Paper

,

Strength of Materials - Notes | Study Strength of Material Notes - Agricultural Engg - Agricultural Engineering

,

mock tests for examination

,

practice quizzes

;