The ULS Design Functions spreadsheet (last updated here) has had another update (2.09) to fix a problem when bars in different layers had different E values. The new version assigns the correct E and strain at yield to each layer.

The new version can be downloaded from:

ULS Design Functions.zip

Results for circular or cylindrical sections designed to Eurocode2 can be checked using the calculator at a new site, EurocodeApplied, which has detailed examples of analysis and use of various Eurocodes. Results for a circular section with an axial load of 2000 kN are shown below (click on images for larger view):

The results include detailed numerical output, optionally an interaction diagram (Moment capacity plotted against axial loads), and a detailed description of the analysis procedure, with reference to the relevant code clauses.

The spreadsheet results for the same section and loading to Eurocode 2 are:

There are small differences in the results because:

- The on-line calculation uses the parabolic-linear stress block, whereas the spreadsheet uses the rectangular stress block.
- The on-line calculation uses a much greater number of layers to represent the circular section than the spreadsheet.

The Circu function in the RC Design Functions spreadsheet has an option to use the parabolic-linear stress block, and uses an analysis of the actual circular shape, rather than subdividing into trapezoidal layers, and gives near exact agreement with the on-line results. An example will be provided in a few days.

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Hi Doug – thanks for sharing your work as always.

I was just wondering your conceptual thoughts on cross-section asymmetries. Particularly, where cross-section asymmetry (either due to concrete shape, reinforcement quantity etc.) leads to a ‘skew’ in the interaction diagram, with apparent moments associated either Crush or Max Tension capacities.

I am thinking that (depending on the structure configuration) this skew may actually manifest or may not – e.g. depending on whether the section is loaded by an action with a determinate eccentricity (like a single bridge pier loaded from beams above), or whether it forms part of an indeterminate frame (like a pile group, where any minor eccentricity would be distributed).

Is this something you have looked at before? I am yet to see this treated well in any textbooks.

Cheers

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Hi James,

There’s actually always asymmetry in the cross-section in a cracked section unless the section is all in tension or all in compression, but the effect of that is included in the moment calculation.

For a circular section (assuming uniform reinforcement) the all tension and uniform compression cases will always have zero moment, but for a rectangular section with non-symmetrical reinforcement, or a non-symmetrical section the extreme load conditions will have an associated bending moment, since the moment is calculated with respect to the uncracked concrete centroid, ignoring reinforcement.

Does that answer your question?

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Hi Doug,

Sorry, maybe I didn’t explain the question very well.

Yes, I’m talking about non-symmetric sections under significant compression or tension loads, which as you note would have an associated bending moment relative to the uncracked concrete centroid.

My question was about the comparison of design actions (e.g. out of an analysis package, or hand calculations) against the interaction diagram. Is it always appropriate to take that uncracked concrete centroid as the basis point for generating the interaction diagram, or would the plastic centroid (for significant compression loads) be a more appropriate point, at least in some cases?

For interest, I checked what RAPT does for a column interaction diagram and it seems to evaluate moments relative to the plastic centroid (so at crush load there is no moment).

Cheers

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you can evaluate moments at any point in the cross section as long as you transform your analysis results to that point/axis orientation.

The geometric centroid is convenient because most modeling programs have you modeling bar elements using the elements centroidal axis, so the analysis results are reported about the cross section centroid.

The use of the plastic center is convenient for the compression side of the interaction diagram because summing moments about that point yields 0 moment at maximum compression, the same isn’t always true for the tension side of the interaction curve depending on how the plastic center was computed.

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DonB’s reply is pretty much what I was going to say. Since analyses are normally done with columns treated as lines positioned on the concrete centroid, it seems to me that it makes sense to use that point as the reference for bending moments due to axial loads. A column with very high axial loads would normally have symmetrical reinforcement anyway, but if it doesn’t it seems to me that the analysis should allow for that.

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Doug I know you’ve done a good amount of research on this topic. They indicate that they split the compression block into layers of rectangles and use the compressive stress computed at the centroid of the rectangle for computation of the overall compression force per slice, is the inherent error in doing it this way significant or is there enough factor of safety throughout that the error doesn’t significantly impact the overall design.

Overall I would guess if using the parabolic stress block the error is likely in the same range the rectangular stress block so probably limited impact on design strength by maybe some impact if you need to get into curvature checks.

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They use a very large number of segments, so the error introduced by that approximation is very small, and much less than the difference between the parabolic and rectangular stress block. In the ULS design functions spreadsheet I use trapezoidal elements with a rectangular stress block, but I will shortly be posting an update to my RC Design Functions which has a CircU function which has an option to use the parabolic-linear stress block with circular section properties, rather than dividing into multiple layers. That gives very close agreement with the on-line calculator.

The rectangular stress block isn’t too bad with circular sections with low strength concrete, but with higher strength concrete, as the Eurocode curve becomes more triangular with a very small (or zero) rectangular part, the difference becomes more significant.

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