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Slenderness ratio for column having different cross sections

Slenderness ratio for column having different cross sections

Slenderness ratio for column having different cross sections

(OP)
Hello,

I was researching a way to calculate the effective slenderness ratio for column having different cross sections. The application I need to perform this on is to estimat the axial loading capacity for a deteriorated timber pile in marine environment. The pile diameter is 12" and necks down to 7" then back to 12".
I came across an article which I thought may help (but in Japanese "attached" which I don't understand!).
I was hoping if  somebody advices on an english source on this topic. Any idea on estimating the axial load in this condition or calculation example are also appreciated.

Thanks

RE: Slenderness ratio for column having different cross sections

Timoshenko has an example of this in his Theory of Elastic Stability.  As you can imagine, it's a bit of a bear to solve the DE by hand, but it is out there.  My copy is in my car because I was reading it last night, so I don't have a page number for you.  I'll post the page number tonight, if you need it.

RE: Slenderness ratio for column having different cross sections

Your link does not work for me.  The question of calculating a column with variable cross section is extremely easy, but remember, the cross section will change with additional exposure to marine environment.

BA

RE: Slenderness ratio for column having different cross sections

BA-
Can you tell me what procedure you use to make it extremely easy?

RE: Slenderness ratio for column having different cross sections

It starts on page 113 of the second edition.  The reason I say it's a bear is because you need to write a seperate set of DE's for each section of column.  It's time consuming and there's a lot of places to make simple algebra errors.   

RE: Slenderness ratio for column having different cross sections

StructuralEIT,

The procedure I have used is Newmark's Numerical procedures.  Page 122 of "Theory of Elastic Stability" (second edition) by Timoshenko contains a simple example using this method.  For a more complicated case, you would divide the member into many more sections.

I believe, although I have not done it myself, that the method could be set up on a spread sheet if you had a lot of similar problems to solve.  Happily, these things don't seem to occur too frequently.   

BA

RE: Slenderness ratio for column having different cross sections

i'd've thought the complexity comes in as I is no longer constant but now f(x) which would mess with the DE ... as I is changing as D^4, it'd probably be simplier to replace with a simpler function (A+B*cos(x))

RE: Slenderness ratio for column having different cross sections

I didn't do one by hand, but it looks pretty good.  I don't know I would say it's simple, but definitely easier than solving the DE's from the more traditional method.  I think I'm going to set up a spreadsheet to try this out.  I actually wanted to do something similar for buckling of a thin rod with greatly thickened ends.  It is the opposite of what is efficient, but is what I was faced with and seems to be what the OP is faced with.  The example has a buckling load 2.17 times higher than if it had a constant cross section.  That's a pretty dramatic increase, in my opinion.

RE: Slenderness ratio for column having different cross sections

(OP)
Thanks folks,
I looked at the referenced book but have couple of comments on using it for the application I need:

1) On figure 2-43(b)in page 113, the shown system is opposite to the one I have. For deteriorated pile the thinner section is in between the two thicker ones. So I can not directly use the values at Table 2-10 to obtain the factor "m" since I1/I2 will be greater than 1.

2) It's also a question on if this method could be applied on timber columns. Other factors comes to the play when analyzing the stability of timber column(Fce, Cp, etc.).

I setup a spreadsheet to calculate the stability of a constant section timber column. I think a conservative approach to model the deterioration is to assume that the entire length have this minimized diameter. I realize this may under estimates the actual capacity of the pile but still not sure how to apply the deterioration in the middle for the reasons stated above.
 

RE: Slenderness ratio for column having different cross sections

To solve 4rth degree DE, go to

http://www.seaofsc.org/Alex's%20Corner.htm

and use this spreadsheet
POLYNOM

Great Alex. I have never seen such nice guy. Every time I go to his page, my heart is ponding and my credit card is shivering that now I got to pay but ! ! !
 

RE: Slenderness ratio for column having different cross sections

adfo,

You cannot use the values in Table 2-10 but you can follow the example on page 122, except that your value for EI will be smaller in the central section than the ends.  

If you want more accuracy, use more sections each with the appropriate EI.  Using the Newmark method is very straightforward and easy to carry out.  

BA

RE: Slenderness ratio for column having different cross sections

adfo-
The procedure is the same whether the middle is bigger or smaller, you just change the ratio of I1/I2 as req'd.

RE: Slenderness ratio for column having different cross sections

SEIT,

Table 2-10 considers only I1/I2 ratios less than 1.0 so you can't use the table, but using the transcendental equation on p. 114, a solution is available by using the appropriate value of I1 and I2 for k1^2 and k2^2.

BA

RE: Slenderness ratio for column having different cross sections

(OP)
BAretired,

I don't have Timoshenko's book at the moment until Monday. But, if I'm not wrong, I thought the example was for an axially loaded horizontal beam. If it was, can this method be used for a column similarly. I recall the procedure in that example included calculating the reations at the ends of each segment of the beam (I may be wrong though). I'm not sure if the same procedure can be applied to the columns.   
  

RE: Slenderness ratio for column having different cross sections

Yes, you can use the procedure.  The "calculated reactions" at the end of each segment is a fictitious reaction to get a conjugate shear to use in the equations a pageor two earlier.

RE: Slenderness ratio for column having different cross sections

adfo,

The orientation of the member is not relevant.  An axially loaded member is the same whether it is vertical, horizontal or at any other angle.

 

BA

RE: Slenderness ratio for column having different cross sections

If the 7" is at mid height then I would expect the buckling would not be that much greater than a 7" column. I would be interested to hear if this is not the case.

RE: Slenderness ratio for column having different cross sections

I would expect it to be substantially stiffer with a buckling load maybe twice as much as a 7" column.  But I have not worked it through, so we will see.

BA

RE: Slenderness ratio for column having different cross sections

(OP)
BAretired
Is there any difference if the example on P.120 is used on other end support conditions (i.e. one ende free and the other is fixed), which may be more representative to the column


 

RE: Slenderness ratio for column having different cross sections

adfo,

A column of length L hinged at each end and symmetrical about the midpoint will have the same critical load as a column fixed at the midpoint and free at each end but the two half columns may not buckle in the same direction.

I'm not sure if I answered your question or not.

BA

RE: Slenderness ratio for column having different cross sections

If one end is fixed then the buckling capacity would be substantially larger than a 7" column.

RE: Slenderness ratio for column having different cross sections

csd72,

Not if the other end is free.

BA

RE: Slenderness ratio for column having different cross sections

The top of pile is free.  The base of pile is embedded in soil, so where is the point of fixity?  I think it depends on the flexural rigidity of the pile as well as the modulus of elasticity of the soil.  You could consider it a beam on an elastic foundation, axially loaded.  

It is not a simple problem.

 

BA

RE: Slenderness ratio for column having different cross sections

Is there any reason for not just doing a buckling analysis in a frame analysis program?

 

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
 

RE: Slenderness ratio for column having different cross sections

BAretired,

Think of the cantilever from the fixed end and how much less that will buckle if it was 9" not 7" then translate this into an effective reduction of the buckling length.

It will still be the 7" section in the middle that will be the first thing to fail but this will be significantly stiffer against buckling.

RE: Slenderness ratio for column having different cross sections

csd72,

If we are comparing hinged/hinged to hinged/fixed, then I agree the critical load will be higher in the latter case if L is the same.

If we are comparing hinged/hinged to free/fixed, then the critical load will be lower in the latter case if L is the same.

In the actual structure, it appears that one end is free while the other end is restrained against rotation but not fixed.  If L is the length from top of pile to the soil below the water, the effective length of the pile, assuming a constant EI will be greater than 2L.  

For the tapered shape shown in adfo's latest post, there is insufficient information to determine whether the critical load is controlled by the 12" section at the bottom or the reduced section higher up.

BA

RE: Slenderness ratio for column having different cross sections

Structural EIT,

If you're reading theories on Elastic Stability at home in your spare time, then you're too deep into engineering.

Put down the pencil and go to a baseball game.  Play some Xbox.   

RE: Slenderness ratio for column having different cross sections

BARetired:

A minor point here...  

You said above that "The orientation of the member is not relevant.  An axially loaded member is the same whether it is vertical, horizontal or at any other angle."

If you discount the weight of the member, this is true.  But when you set a compression member on it's side, gravity self-weight loads need to be considered in the buclking model as this will cause lateral instability in the direction of the gravity load, increasing the P delta effect and reducing the ultimate buckling load.

Mike McCann
MMC Engineering

RE: Slenderness ratio for column having different cross sections

Mike,

You are quite right.  I was considering that the only force acting was axial.  The example performed by Timoshenko did not consider any load normal to the member.  It was taken as  horizontal more for the convenience of displaying the calculations than anything else.  

If a normal load is considered to be acting simultaneously with the axial load, the same technique can be used (Newmark's Numerical Procedures) but the moment at each station will be increased by the gravity load moment.

BA

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