Sign of moment in buckling of column

  • Thread starter fonseh
  • Start date
  • #26
PhanthomJay
Science Advisor
Homework Helper
Gold Member
7,167
507
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.
 
  • Like
Likes CivilSigma and fonseh
  • #27
529
2
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
 
  • #28
PhanthomJay
Science Advisor
Homework Helper
Gold Member
7,167
507
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,
 
  • #29
529
2
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 ?
 
  • #30
PhanthomJay
Science Advisor
Homework Helper
Gold Member
7,167
507
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.
 
  • #31
529
2
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
 
  • #32
PhanthomJay
Science Advisor
Homework Helper
Gold Member
7,167
507
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.
 
  • #33
529
2
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 ?
 

Attachments

  • 493.png
    493.png
    22.2 KB · Views: 308
  • #34
PhanthomJay
Science Advisor
Homework Helper
Gold Member
7,167
507
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.
 

Related Threads on Sign of moment in buckling of column

Replies
0
Views
162
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
24
Views
3K
Replies
3
Views
602
  • Last Post
2
Replies
47
Views
23K
  • Last Post
Replies
1
Views
3K
Replies
12
Views
9K
  • Last Post
Replies
8
Views
1K
Replies
2
Views
2K
Replies
1
Views
2K
Top