slenderness ratio for column having different cross sections

huangyhg

slenderness ratio for column having different cross sections
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
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.
your link does not work for me.  the question of calculating a column with variable cross section is extremely easy, but re  
ba-
can you tell me what procedure you use to make it extremely easy?
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.
  
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.   
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   
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))
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.
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.

to solve 4rth degree de, go to