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Built-up / Reinforced - Stepped Column 1

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RFreund

RFreund

Structural
Aug 14, 2010
1,885
Thanks to BAretired's Thread on Newmark's Numerical Methods (found here: thread507-267603 ) I have the following question regarding a stepped column (prior to this I probably wouldn't have considered a stepped column)

If you have an existing column that needs to be strengthened and instead of making it a full length built up section you only want to reinforce a section of it. Would the following procedure be acceptable (see attached)?

Basically:
1. Determine Fe by Newmarks method.
2. Determine equivalent L/r.
3. Use appropriate Flexural buckling equation.

Is there a design guide similar to this situation or a better method that exists? I put some preliminary sizes and dimensions together for reference on the attached sheet. Also what other reasonable considerations could be made?

One other question came to mind when going through Newmarks procedure: Can it be used for combined axial and flexural forces (parallel and normal to the beam-column)?

EIT
 
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EIT
 
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RFreund,

For an axially loaded column reinforced in the mid-section, you are finished when you have found Pcr. Steps 2 and 3 are not necessary because the buckling load has been determined whether the column is short, intermediate or slender.

In "Theory of Elastic Stability" by Timoshenko and Gere, your problem is solved on p. 122 using the Newmark method which is described in the two preceding pages.

Table 2-10 on p. 115 of the same reference tabulates values of 'm' for various I1/I2 ratios and various a/l ratios where a is the length of the centrally placed reinforced section. The value of m is used in the expression:

Pcr = mEI2/l[sup2]/sup]


BA
 
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Sorry, that last equation should read:

Pcr = mEI2/l2

One other question came to mind when going through Newmarks procedure: Can it be used for combined axial and flexural forces (parallel and normal to the beam-column)?
Click to expand...

Yes. The procedure is the same except that moment from the normal load must be added to moment from the axial load times the trial eccentricity.

BA
 
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Thanks BA your help is always appreciated.
I did recently purchase Theory of Elasticity (have only started reading it) and have seen the example your referring to. However I have yet to thoroughly read through it.

I'm slightly confused however as I thought that in completing the numerical method that you have found the elastic buckling load/stress. However in order to determine the critical buckling stress accounting for inelastic buckling then you imply steps 2 and 3?

EIT
 
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I believe you are correct. I may be the one who is slightly confused. For Intermediate columns, you would need to use a reduced modulus which is a function of E and tangent modulus Et. For short columns, the yield stress is the limiting value.

I'm a little rusty on the subject of inelastic buckling as I haven't used it recently. Perhaps someone else can step in and clarify the procedure to be used.

BA
 
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Ok thats what I thought. Really this all came about because an existing column needed to be reinforced and normally I would treat it as a built-up section. However I wondered if instead of reinforcing the entire column if you could just reinforce part of it. Maybe this procedure is better used for slender columns.

EIT
 
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I believe that reinforcing the central part of a column is more cost effective than reinforcing the entire column, but the theory gets messy for the inelastic case.

Timoshenko and Gere devote all of Chapter 3 to "Inelastic Buckling of Bars". On p. 184, for bars of variable cross section with variable Et, "We shall always be on the safe side if in such cases we use formulas derived for elastic conditions and substitute in them for E the tangent modulus Et calculated for the cross section with the maximum compressive stress."

Proceding in that way with modified E, the Newmark method could be used for the case of inelastic buckling and the result would be conservative.

BA
 
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