Buckling Deflection Calculation 1

[CivilTanju]

hi,

Does anyone can tell me, how am I supposed to calculate the deflections of compressed elements which have buckling as well. It makes me think that this deflection should be function of X which is ranging from 0 to element length(such as d(x) = X^3+2X^2+....)

Thnx,

 

[corus]

If a structure has buckled then it's safe to assume that the deflection is infinite.

corus

 

[CivilTanju]

corus,

I didn't mean that the structure has reached the plastic limits, just want to know the how to calculate deflections at column elements.

 

[rb1957]

if you have perfect columns, they don't deflect laterally(theoretically) untill they buckle, when as corus posts it is assumed to be infinite (again, theoretically).

if you have an imperfect column (offset loading, intitial out-of-straighness) then the column has a bending moment and the lateral deflection is easily calculated.

 

[Denial]

To illustrate rb1957's point, go back to Structures 202.

Assume column is pin-ended. Assume its out-of-straightness is sinusoidal, so that its unloaded shape is
y = a*sin([π]x/L)
where a is the unloaded midspan lateral out-of-straightness.
Apply axial load P. Let the midspan lateral deflection due to the resulting bending be b. Assume shape is still sinusoidal. Deflected shape of column is thus
y = (a+b)*sin([π]x/L)
and the bending moment is
M = P*(a+b)*sin([π]x/L)
For this shape, the area-moment method tells us that the midspan lateral due to the bending is given by the integral from L/2 to L of (M/(EI))*(L-x) with respect to x, which comes out as
P(a+b)L²/([π]²EI)

This expression must equal b, which gives us an equation to solve for b. Result is
b = a/(P_cr/P - 1)
where P_cr = [π]²EI/L²

Different assumed causes of out-of-straightness, or different column boundary conditions, can be similarly investigated. All you need is a half-way decent assumed shae to get a reasonably accurate answer. Oh, and the maths might not be quite so easy.

 

[UcfSE]

Ustay,

Are you asking for deflection due to initial out-of-straightness or eccentric load, or do you want to know how far a column deflects when it buckles?

The former case has been well explained above. For the latter, you have to understand that Euler's equations that we often use are good only for predicting buckling load and location. You cannot find the actual deflection of a column that has buckled with Euler's theory. To find this you will probably need to turn to the theory of elasticity or general continuum mechanics, or perhaps and good finite element analysis.

 

[prex]

The post buckling deformed shape of a column indeed exists: the lateral deflection will go to infinite only if the load is kept over the critical load, but if it is controlled after the buckling starts, then a definite deformed shape can be observed (it is an experiment easy to conduct using a plastic rule). By the way the assumption of a definite deflection curve is part of the process for calculating the critical load.
For a doubly pinned column the deflection curve should be a sinusoid, as the equation is homogeneous (EJy''+Py=0), but when large deformations are accounted for, of course the equation is more complex. However this deflected shape need to contain an indetermined constant, that will be determined not by the load (that remains at the value P_cr), but by specifying the maximum deflection (or the change in distance of the column ends).

prex

http://www.xcalcs.com

Online tools for structural design

 

[CivilTanju]

thank you all for these valuable posts,

I'll wrote down what I understood from your postings, please correct if I'm wrong with comment.

Denial: The given solution is for elements which are elastic-pinned-ended and has initial out-of-straightness.
So it means that deflection calculations are valid for where 0<P<P_cr. ( which clearly explains rb1957's point, whenever P = P_cr then b = a/0 deflection is infinite).

UcfSE: actually what I was inqirying about was, deflection when the buckling load is reached (my fault, I should have given much more explanation). The book that I'm refering to for buckling calculations says:

-d²x / dz² = M/EI = P*x / EI

d²x / dz² + c²*x = 0
where c² is c² = P/EI

solution of differential equations is :
X = A*sincz + B*coscz; which is equal to the column deflection for Euler calculation.

these solution is contradicting with what u said. Can I use the above function(X = A*sincz + B*coscz) as deflection calculation or not?

 

[prex]

UstayTanju,
the post buckling behaviour of columns is a quite theoretical problem, normally engineers only try to avoid buckling.
Can you specify what is your goal, so that we can follow you?

prex

http://www.xcalcs.com

Online tools for structural design

 

[CivilTanju]

hi prex,

Actually there is not problem that I need to solve for now, this question has been asked by employer when I was at employment interview, it is just an inqury.


Best Regards,

 

[UcfSE]

Ustay,

Your equations are correct, as are your results. Try to solve for your constants A and B. You will then find out what I meant. B will be easy. A will probably prove indeterminate by the method you are using.

 

[Denial]

Ustay,

To say that the result I derived above is valid for 0<P<P_cr would be taking things too far. It should not really be used outside the range of finite-but-small lateral deflections. This is the zone where engineers work, even if the mathematicians among us like to stray further afield.

The derivation (directly or indirectly) uses several assumptions that become less justifiable at larger deflections. These include a sinusoidal initial shape begetting a sinusoidal displaced shape, axial effects being insignificant, and the second derivative d²y/dx² being the same as the curvature. Plus of course the requirement that everything remains linear-elastic.