What does the slenderness ratio of a structural steel column need to be to be in the inelastic buckling range?

What were the slenderness ratios of the central core columns at the collapse initiation sites of the 98th floor in the North Tower and 82nd floor in the South Tower?

I saw this question and jumped into the discussion, posting:

Inelastic buckling occurs for all slenderness ratios under the Euler limit. That's 4.71 * SQRT(E/Fy). Of course extremely stout members won't buckle inelastically, however none of the columns in the upper floors of the WTC were that stout.

Do you have any clue as to what you're talking about? I recommend picking up an AISC Manual of Steel Construction and see exactly how steel is designed these days. We're not in the 1940's, we know how steel fails now. Maybe you should update you knowledge to modern information.

Mr. Szamboti's replied:

How did I know you would come on.This is where Mr. Szamboti shows his lack of knowledge as regards to structural engineering. The slenderness ratio that we are talking about, and what Mr. Szamboti struggles to understand, here is defined as K*L/r.

You used an effective length factor of 1.0 in your letter to Gordon Ross, which is for a pinned connection, when you should have used .5 to .65 for fixed both ends connections for the tower columns. The 1.0 gave you larger slenderness ratios and they still weren't greater than 40. Now you are going to say the tower columns weren't in the short column category and would have been subject to inelastic buckling. The AISC equations you show here and which you used in your paper are conservative for design.

You want to say the tower columns would fail due to buckling. Well how about a test case were an I beam with a slenderness ratio of 20 or lower failed due to inelastic buckling. Do you have any test cases? I have an AISC manual right here. I am familiar with the equations and monograph. You want to go around asking others if they have a clue and you seem to be the one who should be asked that question Mr. Smarty pants.

Where:

k = effective length factor

L = length of the column (in)

r = radius of gyratio of the column (in) - [This is a function of the geometric properties of a column]

For the columns that we are talking about, the variables L and r are very well defined and not argued. The effective length factor 'k' is where he slips up. In the commentary of the AISC Manual of Steel Construction (arguably the Bible of how to design steel in structures) this factor 'k' is defined. The first place is in table C2.2 (shown below).

Table C-C2.2 (click to enlarge)

Under column (a) it defines the theoretical k value of 0.5 and a recommended design value of 0.65. It's fairly easy to see that this is where Mr. Szamboti thinks the factor k is defined, as the tower exterior columns were moment frames, which means that the top and bottom portions of the columns were fixed. Column (a) shows a column element fixed at the top and bottom, so he used it. And he's very wrong. The commentary clearly explains how this table is to be used on the page before the table, "These range from simple idealizations of single columns such as shown in Table C-C2.2 to complex buckling solutions for specific frames and loading conditions". In his rush to prove me wrong, I can only surmise that he went through the commentary to find an answer to his question, and stumbling upon the first table that seemed to show an answer that confirmed his bias, he lept to a conclusion. An incorrect one not supported by the document he was referencing.

The correct way to calculate this effective length factor is with the nomograph chart, shown below. The nomograph table is for frames which can translate horizontally, this contributes significantly to the stability of the frame.

Figure C-C2.4 (click to enlarge)

This table looks nonsensical, but it's fairly simple to use. First Ga and Gb need to be defined, which are simply a comparison of the stiffnesses of the columns to the girders. Ga is the comparative stiffness of the top point of the column and Gb is of the bottom. Then, to get the effective length factor, one merely needs to draw a straight line between these two points (see the figure below with Ga = 1.0 and Gb slightly stiffer).

Example nomograph

In this example, the k factor of a frame that has a column stiffness that is roughly equal to the girder stiffness is about 1.4.

It is very easy to see with this table that the lowest factor k that a column in a moment frame can have is 1.0, rendering Mr. Szamboti's statement that it should be 0.5 or 0.65 completely without merit. The purpose of this isn't to belittle or attack Mr. Szamboti as being an incompetent engineer. I'm sure he is an excellent mechanical engineer, however he is not an expert in structural engineering, far from it. He is most definitely unqualified to "peer-review" papers of a structural engineering focus for anyone.

In this example, the k factor of a frame that has a column stiffness that is roughly equal to the girder stiffness is about 1.4.

It is very easy to see with this table that the lowest factor k that a column in a moment frame can have is 1.0, rendering Mr. Szamboti's statement that it should be 0.5 or 0.65 completely without merit. The purpose of this isn't to belittle or attack Mr. Szamboti as being an incompetent engineer. I'm sure he is an excellent mechanical engineer, however he is not an expert in structural engineering, far from it. He is most definitely unqualified to "peer-review" papers of a structural engineering focus for anyone.