Concrete Interaction Diagram - ribbed wall section 2

[staffengw]

I have been trying to come up with an interaction diagram for a wall section with a 24" wide, 2" thick outside face with a rib that projects out from the face 4.875" and is 3.5" wide. I usually do interaction diagrams for rectangular sections and am really concerned that I am getting wrong results with this ribbed section because of the strange shape. See the attached jpg. For the direction where the rib is in tension and the 2" face is in compression the diagram doubles back such that at 160k compression the moment capacity would be insufficient between about 75k*in and 175k*in, but would be OK between about 175k*in and 190k*in.

I followed the procedure in the PCI handbook. Has anyone else seen such a strange diagram? I am trying to find an error in my formulas but have not been able to.

 

[ishvaaag]

Please state your geometry (sketch), effective width, reinforcement (prestressed and reinforced) and concrete strength in order to properly assess the matter. I think I have somewhere interaction diagrams for T sections design buy haven't found them.

 

[staffengw]

I used f'c = 5000 psi, the full 24" width as effective, and 2) #4 bars. Attached is a sketch to clarify further.

 

[ishvaaag]

OK, since a RC section I have already modified one Mathcad worksheet able to solve the strength of this section in compatibility of deformations, and will check the question tomorrow.

Whilst, the plausible explanation is that a level of axial load, the section is able to show equilibrium with different moments (and of course different stresses in the components of the section), something that the way you are proceeding or the algorithm used is able to bring to your attention.

In the compatibility of deformations method of the worksheet, that I may provide later, you can see that starting from different seed values for the strains epsilon1 and epsilon2 that define the position of the trial deformed plane, sometimes the section is able to provide different solutions to the equilibrium (different mathematically satisfactory positions of the deformed plane).

Even if such is the case, by selecting always growing values of the axial loads, by increments, you only will be getting a value of moment for the chart per trial axial load, and only your wit or a visible anomaly in the chart will lead you to review if the point you are wanting to plot is such. These things are cleared by those making the interaction diagrams, either through the algorithm, or as we are doing now, through engineering judgement.

As I have said, these equilibriums are really feasible in compatibility of deformations, and sometimes one can visualize how to come to such state starting from other plausible one.

I will try to document all this tomorrow in the thread.

 

[IDS]

staffengw - There is definitely something wrong with your results. The upper line where compression controls should extend until it intersects the tension controlled line. I have attached the diagram I get using my spreadsheet. Unfortunately it only handles the Australian code and SI units at the meoment, but if you are interested you can download it from:

http://newtonexcelbach.wordpress.co...rete-section-analysis-5-ultimate-limit-state/

There is also some more detail about how it works at

http://newtonexcelbach.wordpress.co...rete-section-analysis-6-ultimate-limit-state/

The Australian code is very similar to ACI in procedure. I note that I get a considerably higher bending moment at 0 axial load than you, probably because the bending capacity includes both bars in tension (I have used a steel yield strength of 60,000 ksi (414 MPa)). The maximum axial capacity is in pretty good agreement.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[staffengw]

I did not consider both bars for the diagram. I neglected the bar when it was on the compression side since there are no ties.

 

[IDS]

It makes very little difference if you remove the top bar, except for very small axial loads.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[ishvaaag]

Well, I post now the interaction diagram based in the unreduced characteristic strengths of both the steel and concrete, and full section of 24 inches total width as effective. It is NOT an interaction diagram for the immediate check of structural members along any code, since the worksheet giving the values in compatibility of deformations is one targeted more to ability to meet some solicitations in accord of the science of construction and mathematics than any code. This said, and for the given section, these capacities (and higher, giving that the average strengths of the steel and concrete will be higher) should be expected of the section of the member.

One likely difference with the two interaction charts of staffengw and IDS is that in this chart, the strain hardening of the steel is taken unto account, what results in some points showing tensile stresses exceeding 60 ksi. This happens mainly in the low range of axial loads.

The chart has been built searching with the help of the worksheets points for which equilibrium of moments and forces are met that pertain to the interaction chart. A Pn is tried, then we try to get the true maximum Mn available for the section meeting equilibrium, this in 10 kips steps.

The mathematical procedures, even trying to find a maximum, fail many times (without manual iteration) to find in a single try the Mn to port at the chart. This because there are local maxima that are accepted as solutions. Seeing the chart, the two flattened parts in the bottom part of the curve quite likely will get a more continuous curve by just making some trials more that give a more true (and higher) Mn acceptable as concomitant with the trial Pn.

Since my chart and the one from IDS show what seems to me reasonable continuity, I am as IDS of the opinion of that the actual chart shouldn't be showing the discontinuities shown in the one causing staffengw question.

Of course if you take a true interaction chart for a member like ours and then deviate inwards the safe zone we can still have a safe interaction chart. The algorithms used, or error, may lead to this happen and as long you are satisfactorily convinced that there are no other outwards satisfactory values, one could take it as a true axial-moment interaction chart. As said, the algorithm, or an insufficient number of trials may make this happen.

 

[ishvaaag]

Of course, to build some interaction chart for office use one should temper the math above to what indicated in the code. For example, it is unlikely that anyone would be using to such purpose unreduced strengths of the materials, either from code or as allowance for sustained loads. Neither the full width of the flanges likely would be counted effective, and quite likely as well the maximum axial load would be curtailed to some value.

Just as complementary info, the worksheet also allows perfectly-elastic perfectly-plastic diagrams, but I simply accepted the built-in data for grade 60, that is included. As usual, I didn't count this time any tensile strength of the concrete. The concrete stress-strain diagram is bi-parabolic, 0 to top strength, then decaying somewhat, see attached figure.

 

[staffengw]

I think I found my problem. I was choosing a value of a, then getting the values of c, the compression area (Acomp), the distance to the compression area centroid (y'), the stress in the steel (fs = (0.003/c)*(d-c)), then the phi factor (relating to the c / d ratio), then phi*Pn = phi*Acomp*(0.85*f'c) [neglecting As for any additional compression capacity], then phi*Mn = phi(0.85*f'c*Acomp)+As*fs*(d-cg).
The chart I showed was the comparison between the phi*Pn and phi*Mn from this process. I don't believe this is the correct approach. It does not compare apples to apples.

I changed to choosing an axial load, Pu, then iterating values of c until the sum of compression (0.85*f'c*Acomp + Pu) and tension loads (As*fs) equal zero. I get a result that more closely resembles your results.

Thank you for your input.

 

[IDS]

I have now updated my concrete section analysis spreadsheet to work automatically with any selected units, and to work to the ACI 318 code, as well as the Australian codes. The revised file can be downloaded from:

http://newtonexcelbach.wordpress.com/2012/08/09/uls-analysis-of-concrete-beams-now-with-added-units/

Generating the interaction diagram for the T section to the ACI code (see attached) throws some light on what is happening.

The reduction factor on ultimate moment capacity goes from 0.9 for tension controlled sections to 0.65 for compression controlled sections. It so happens that the depth of the compression zone at the axial load limit for tension control is very close to the bottom of the flange, so the additional axial load when the NA reaches the position for compression control is very small, and Phi.P actually reduces. The result is that instead of getting a gradual transition in the Phi value it jumps straight from 0.9 to 0.65, and there is a jump in the position of the NA.

Looking at the Mu and Phi.Mu lines on the attached graph should clarify wjat is happening.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[ishvaaag]

Just for completeness, from PCA notes to ACI 318-08, the confirmation of that a linear interpolation between 0.65 and 0.9 is permitted for the points in the interaction diagram between the compression controlled zone and tension controlled zone.

PCA notes to 318-08 also has example 6.4 showing the basics of construction of a code-based Axial load-Moment interaction diagram.