Sign of moment in buckling of column

[fonseh]

There would still be moment if the column was not ideally straight or if it was displaced laterally, even for the compression case,

Can you explain further ? Perhaps with diagram ? i still can't imagine it

 

[PhanthomJay]

the deflected shape under tension might be due to eccentricity of the applied load or initial curvature in the column. But in any case, increasing the value of T reduces the deflection, so buckling cannot occur.. It is interesting to note that using
d^2y/dx^2 + Py = 0 versus
d^2y/dx^2 - Py = 0 yields completely different results for the solution (the first equation involves the basic sin function while the second equation involves the hyperbolic sinh (exponential) function). Thus, the signage is very important. The first is the compression case with the Euler buckling solution, and the 2nd I believe is the tension case with no buckling solution . I can only conclude that signage is determined by negative curvature or a negative deflection value.

 

[fonseh]

It doesn't matter in which direction the column is laterally deflected left or right. First you can adopt a convention that curvature is negative when the shape is concave with respect to the beam axis facing the beam, thus, for the pinned pinned case, curvature is negative for both left or right displacements . In answer to your question on moment signage when beam is deflected left or right for the pinned pinned case, when it deflects right, the applied eccentric moment is Py counterclockwise, and the internal moment M must be clockwise, thus
(-Py) + M = 0, or M = Py; and when it is deflected left , then the eccentric moment Py is clockwise and the internal moment M must be counterclockwise, thus (+Py) - M = 0, or M = Py, which is the same result, so again, it does not matter, and in both cases since M is - (EI)(d^2y/dx^2) , the the differential equation becomes
(EI)d^2y/dx^2 + Py = 0.

So , can I conclude that no matter what cicumstances , the moment of the buckling beam should have the moment look like this ?

 

[PhanthomJay]

So , can I conclude that no matter what cicumstances , the moment of the buckling beam should have the moment look like this ?

No. See post #26.