# Sign of moment in buckling of column

PhanthomJay
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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.

CivilSigma and fonseh
I believe the paradox is resolved when you consider that the internal moment is , when written in terms of the curvature - stiffness relationship where the absolute value of M = EI(d^2y/dx^2), that actually M= - EI(d^2y/dx^2), because the curve is concave with a negative curvature.
why this is not stated in the book ? I have Hibbler and Beer books with me , but it's not stated in it

PhanthomJay
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It is not stated in any source I can find. They all seem to throw the minus sign in there without explanation. I am still unclear why. Further , when you apply a tensile load P instead of a compressive load P, the direction of the internal moment changes, but the result is still the same, M = Py, and you get the same differential equation for the buckling solution, although the column will never buckle under tension load, because the column veil self restore to straight under increasing load, so I am still stuck here. I do remember a lecture on this in college about 50 years ago, but my notes have long since disappeared,

fonseh
It is not stated in any source I can find. They all seem to throw the minus sign in there without explanation. I am still unclear why. Further , when you apply a tensile load P instead of a compressive load P, the direction of the internal moment changes, but the result is still the same, M = Py, and you get the same differential equation for the buckling solution, although the column will never buckle under tension load, because the column veil self restore to straight under increasing load, so I am still stuck here. I do remember a lecture on this in college about 50 years ago, but my notes have long since disappeared,
Ya， i agreed that the beam doesn't buckle under tension load , IF the beam doesnt buckle , why there is moment Py ?

PhanthomJay
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Ya， i agreed that the beam doesn't buckle under tension load , IF the beam doesnt buckle , why there is moment Py ?
There would still be moment if the column was not ideally straight or if it was displaced laterally, even for the compression case, with lateral displacement or a not straight column, the collimn would not buckle if the applied load was less than the critical load.

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 cant imagine it

PhanthomJay
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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 ?

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PhanthomJay