Slenderness ratio for column having different cross sections

[Awre]

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

 

[Lion06]

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.

 

[BAretired]

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

 

[Lion06]

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

 

[Awre]

StructuralEIT - Thanks, can you tell me the page number to research it.
BAretired - The file is pdf and has japanese characters. I'am reattaching it anyway.

 

[Lion06]

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.

 

[BAretired]

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

 

[rb1957]

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))

 

[Lion06]

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.

 

[Awre]

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.

 

[dgkhan]

To solve 4rth degree DE, go to

http://www.seaofsc.org/Alex's Corner.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 ! ! !

 

[BAretired]

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

 

[Lion06]

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

 

[BAretired]

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

 

[Awre]

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.

 

[Lion06]

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.

 

[BAretired]

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

 

[csd72]

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.

 

[BAretired]

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

 

[Awre]

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