Bi-axial bending chart - concrete column 29

[Pretty Girl]

This is from "Reinforced concrete design to eurocodes" by prab bhat, page 371 and 372.
It has mentioned the it's for My/ (hb^2) = 2.

But I don't know how to reproduce with that ratio kept constant. So, I tried to create it with making the alpha and beta values at a constant ratio of 0.8 (beta = 0.8 alpha). Then I produced a chart.

Since I didn't reproduce the exact chart in the book, now I have another problem. I have got no reference chart to compare my chart with. Can anyone kindly help me find out if my chart is correct for the column dimensions and data I provided.

Are there any free software/ excel sheet to enter the column details mentioned below and compare it with my chart?

I'm concerned that my chart may not be correct as I don't see the part the book's chart have I have shown in the green rectangle below, when I produce my chart. I understand it cannot be the same chart, but if my chart is correct that "nose" like curve should also be in my chart isn't it.

1. Chart from the book






2. The chart I produced

This is a rectangular column, h = 2000 mm, b = 1000 mm. I maintained "beta/ alpha ratio = 0.8".
4 reinforcement bars, 1 bar each corner. Steel percentage 4% (So, 1% bh area for each bar).
40 mm distance from column surface to the centroid of r/f for each bar.
fck = 30 MPa, fcd = 20 MPa, fyk = 500 MPa, fyd = 434.7 MPa.

 

[IDS]

Did you mean I should range it beyond zero to the right (negative side) like below? (+1500 to -1500 etc?)

No, there has to be at least a corner where the concrete is in compression. I was talking about the direction of the NA. If a line parallel to the X axis is horizontal, then a line sloping down from left to right has a negative slope, but there is no problem with the range of NA positions.

I am saying the interaction diagram for My is inconsistent with your calculations because it shows a capacity of just under 100 kNm for an axial compression of 158 kN, but our calculations find the capacity is less than 50 kNm.

 

[Pretty Girl]

No, there has to be at least a corner where the concrete is in compression. I was talking about the direction of the NA. If a line parallel to the X axis is horizontal, then a line sloping down from left to right has a negative slope, but there is no problem with the range of NA positions.

I am saying the interaction diagram for My is inconsistent with your calculations because it shows a capacity of just under 100 kNm for an axial compression of 158 kN, but our calculations find the capacity is less than 50 kNm.

Thank you for the clarification.
You're right. I found out the fault. Not only My was incorrect, Mx was also incorrect. I have updated the diagrams below, and the full data set was also updated with new data.




Now My is in positive side, but in a strange shape though. Hope it's correct?



Now got, Sydney opera house like structure.

Axial (kN)​	Mx (kN)​	My (kN)​
-853.68878​	-0.5326955​	-0.5333486​
-848.99384​	0.61095898​	0.01591872​
-793.27636​	12.6655118​	5.05326976​
-597.96956​	54.0481724​	21.8794454​
-490.29144​	76.8634545​	30.9568381​
-396.449​	96.427872​	38.33184​
-250.87124​	126.180939​	39.3838435​
-94.15116​	157.538169​	38.469241​
46.73124​	184.727076​	37.9804246​
156.96796​	204.469174​	37.3067869​
279.438​	225.525976​	35.54372​
403.68​	245.482128​	33.404​
524.215​	262.825392​	31.538​
657.007​	274.301592​	31.258​
809.993​	278.573728​	33.094​
955.637​	281.5808​	34.686​
1095.315​	283.208808​	36.079​
1262.013​	276.742752​	34.673​
1421.857​	269.313632​	33.359​
1575.183​	260.929448​	32.183​
1722.968​	251.4532​	31.125​
1854.878​	243.080888​	29.249​
1993.812​	231.102512​	28.378​
2128.78​	217.668072​	27.617​
2255.62​	201.757568​	27.334​
2380.146​	184.447​	27.074​
2491.499​	167.990368​	27.752​
2607.84236​	148.827307​	27.023687​
2713.16956​	130.643191​	25.3534143​
2807.64044​	113.583542​	22.9637032​
2891.4​	97.800848​	20.0786093​
2964.57524​	83.447594​	16.9245005​
3027.27716​	70.6822664​	13.7229892​
3079.60376​	59.6643515​	10.701366​
3121.64104​	50.5573353​	8.08161172​
3153.465​	43.524704​	6.09284178​
3175.14364​	38.7339437​	4.95425458​
3188.115​	36.003​	4.72​
3199.469​	33.648​	4.609​
3210.241​	31.412​	4.504​
3220.474​	29.289​	4.404​
3230.208​	27.269​	4.309​
3239.479​	25.346​	4.218​
3248.317​	23.511​	4.132​
3256.755​	21.761​	4.049​
3264.817​	20.088​	3.97​
3272.53​	18.487​	3.895​
3279.913​	16.955​	3.823​
3286.989​	15.487​	3.754​
3293.776​	14.079​	3.688​
3300.292​	12.727​	3.624​

 

[IDS]

Your new results are still inconsistent with the calculation for 158 kN axial load, and my spreadsheet results.

The screenshots below show interaction diagrams from my spreadsheet (blue line with no markers) overlaid on your graphs (blue line with markers) and the calculated values for 158 kN axial load (red point).

 

[Pretty Girl]

Your new results are still inconsistent with the calculation for 158 kN axial load, and my spreadsheet results.

The screenshots below show interaction diagrams from my spreadsheet (blue line with no markers) overlaid on your graphs (blue line with markers) and the calculated values for 158 kN axial load (red point).
View attachment 5570
View attachment 5571

@IDS
Thank you for the diagram and the comparison provided.
I found out where the fault is. I believe now they are matching.

Full data in the spread sheet were also updated.
https://docs.google.com/spreadsheets/d/1aBSvgExzAAx7aWmUKDzR9JgOKIt75cvZYXzXlfsLUkk/edit?usp=sharing

Axial (kN)​	Mx (kN)​	My (kN)​
-853.68571​	-0.5319302​	-0.5329659​
-846.2325​	1.25556712​	0.33899597​
-676.26098​	37.315033​	15.2642157​
-513.59746​	71.5265442​	29.4495071​
-393.12494​	96.1092476​	39.569718​
-279.9804​	117.91929​	47.386696​
-78.608856​	156.361818​	46.9442891​
90.397696​	185.940978​	46.602345​
251.605256​	211.336917​	46.4387116​
450.155824​	223.008782​	50.7682367​
673.8434​	233.634719​	54.010768​
895.022​	243.600178​	55.558375​
1121.468​	246.596915​	55.467375​
1365.861​	239.796597​	53.008375​
1586.422​	232.155224​	49.982375​
1807.943​	218.736796​	48.155375​
2021.205​	201.611312​	46.557375​
2225.01733​	181.276975​	44.8295598​
2409.45869​	160.614583​	41.7605971​
2563.07604​	142.575744​	38.4490326​
2708.83938​	122.909309​	33.3955363​
2836.20172​	104.305133​	27.8590099​
2940.90305​	86.1760695​	22.5797705​
3027.35037​	69.7629708​	17.5716663​
3097.04468​	55.7226906​	13.0816197​
3150.20498​	44.4640821​	9.47180712​
3187.01728​	36.4029988​	7.09497854​
3207.879​	31.903​	6.28​
3222.582​	28.852​	6.074​
3236.271​	26.011​	5.882​
3249.047​	23.36​	5.703​
3260.999​	20.88​	5.535​
3272.205​	18.555​	5.378​
3282.731​	16.371​	5.23​
3292.637​	14.315​	5.091​
3301.978​	12.377​	4.96​
3308.746​	10.973​	4.667​
3313.628​	9.96​	4.264​
3318.253​	9​	3.883​
3322.641​	8.089​	3.521​
3326.809​	7.224​	3.177​
3330.774​	6.402​	2.85​
3334.551​	5.618​	2.538​
3338.151​	4.871​	2.241​
3341.588​	4.158​	1.958​
3344.872​	3.476​	1.687​
3348.014​	2.825​	1.428​
3351.021​	2.2​	1.18​
3353.904​	1.602​	0.942​
3356.669​	1.029​	0.714​
3359.189​	0.506​	0.506​

 

[IDS]

Looking better from a quick look.

I have to keep that tax man happy at the moment, but I will have a proper look as soon as I can.

Something you might like to have a look at. When the concrete is all in compression, Eurocode 2 requires the maximum compression strain to be reduced in accordance with Fig. 6.1.

Have you allowed for that?

 

[Pretty Girl]

Looking better from a quick look.

I have to keep that tax man happy at the moment, but I will have a proper look as soon as I can.

Something you might like to have a look at. When the concrete is all in compression, Eurocode 2 requires the maximum compression strain to be reduced in accordance with Fig. 6.1.

Have you allowed for that?

@IDS
Thank you.
No worries at all, take your time. You gotta keep the tax man happy first. Hope that all goes smoothly for you.

For the time being, I will go through the eurocode figure you mentioned.

 

[Pretty Girl]

@IDS

Thank you for the EC clause you mentioned.
I went through the code,

So, in a case where column is in pure compression (and moment = 0),

1. if we use parabolic shape for the compression block --> we should use 0.0020 as max strain limit (up to C50)
2. if we use simplified (0.8x etc) for the compression block --> we should use 0.00175 as max strain limit

I believe, that's what the code says. Is that correct/wrong?

 

[IDS]

I have updated the plots comparing the results. They are now very close over most of the range, but differences in the zone with maximum moment capacity. It would be interesting to see the detailed calculation for a point where there is a significant difference in results. Also my results are less for high axial loads, which would be due to the reduction in maximum strain where the concrete is all in compression.

 

[Pretty Girl]

I have updated the plots comparing the results. They are now very close over most of the range, but differences in the zone with maximum moment capacity. It would be interesting to see the detailed calculation for a point where there is a significant difference in results. Also my results are less for high axial loads, which would be due to the reduction in maximum strain where the concrete is all in compression.

View attachment 5712
View attachment 5713

@IDS
Thank you for the comparison.

I have another question until I apply the strain limit at the full compression and until I prepare the detailed calculation for a point where there is a significant difference in results.

I noticed in the Eurocode, the parabola is shown as it goes all the way to the bottom surface. So, does the code assumes in that situation the neutral axis is in bottom location? I believe they are showing a uniaxial scenario, but still I need to understand where the neutral axis is at in their diagram.



So, the following parabolic stress block and the neutral axis location is correct or wrong in our biaxial scenario?
I mean I basically want to confirm the fact that the location where parabola ends will be taken as the neutral axis location. It should right, because it's where compression and tension becomes zero, so we are considering it as the NA, right?




Further, the EC2 clause you mentioned,

If we use simplified (0.8x etc) for the compression block --> we should use 0.00175 as max strain limit.
Is this used only to calculate strain gradients of steel r/f?

Since we used, fcd = αcc * fck / γc, we can't plug that 0.00175 limit to this. So, is it only used to calculate steel gradients at a scenario where column is in full compression?

What I meant is, I only see the only place we can plug that in where we used εcu3. We used εcu3 for the steel calculations. So, when the column is in full compression, we can use εc3 (0.00175) instead of εcu3 (0.0035). Do I need to put that 0.00175, somewhere else other than steel calculations?

 

[Pretty Girl]

@IDS

After applying the 0.00175 strain when the column completely in compression, it looks like the following. I believe it's not correct is it?

 

[IDS]

@IDS
Thank you for the comparison.

I have another question until I apply the strain limit at the full compression and until I prepare the detailed calculation for a point where there is a significant difference in results.

I noticed in the Eurocode, the parabola is shown as it goes all the way to the bottom surface. So, does the code assumes in that situation the neutral axis is in bottom location? I believe they are showing a uniaxial scenario, but still I need to understand where the neutral axis is at in their diagram.

View attachment 5716

So, the following parabolic stress block and the neutral axis location is correct or wrong in our biaxial scenario?
I mean I basically want to confirm the fact that the location where parabola ends will be taken as the neutral axis location. It should right, because it's where compression and tension becomes zero, so we are considering it as the NA, right?

View attachment 5718


Further, the EC2 clause you mentioned,

If we use simplified (0.8x etc) for the compression block --> we should use 0.00175 as max strain limit.
Is this used only to calculate strain gradients of steel r/f?

Since we used, fcd = αcc * fck / γc, we can't plug that 0.00175 limit to this. So, is it only used to calculate steel gradients at a scenario where column is in full compression?

What I meant is, I only see the only place we can plug that in where we used εcu3. We used εcu3 for the steel calculations. So, when the column is in full compression, we can use εc3 (0.00175) instead of εcu3 (0.0035). Do I need to put that 0.00175, somewhere else other than steel calculations?

It looks like you have been misled by the Fig. 6.1. The parabolic shape has nothing to do with the stress block, it is just showing the concrete outline for a random non-rectangular section. If you drew stress contours on your left-hand cross section all the lines would be parallel to the Neutral Axis, and the compressive stress region would extend to the NA over the full extent. It is the relationship of stress to strain that is parabolic-rectangular

Regarding the limit on the maximum strain, and the shape of the interaction diagram for high axial load, I will check the Eurocode provisions for axial load limit and get back to you on that one.

 

[Pretty Girl]

It looks like you have been misled by the Fig. 6.1. The parabolic shape has nothing to do with the stress block, it is just showing the concrete outline for a random non-rectangular section. If you drew stress contours on your left-hand cross section all the lines would be parallel to the Neutral Axis, and the compressive stress region would extend to the NA over the full extent. It is the relationship of stress to strain that is parabolic-rectangular

Regarding the limit on the maximum strain, and the shape of the interaction diagram for high axial load, I will check the Eurocode provisions for axial load limit and get back to you on that one.

@IDS

Thank you for the response and for the clarification.
Yes I was mistaken it for a stress block inside a rectangular concrete section.
Now I understand it's just a random section of a concrete element. For a moment I was questioning my self the very basic understanding I gained about the compression block ending at the neutral axis.

 

[Celt83]

@Pretty Girl
While the plots are informative could you instead post your full calculation for say the Mx and My points at P= 2000 N then we can see the values you are getting for strains, stresses, and forces. That may make it a bit easier to help.

 

[Pretty Girl]

@IDS

I noticed, the eurocode clause 6.1 number 4, says the following,
"For cross-sections loaded by the compression force it is necessary to assume the minimum eccentricity, eo = h/30 but not less than 20 mm where h is the depth of the section."
That means, since we apply that minimum eccentricity (in analysis phase even before we develop interaction diagram), we expect our column to never be in pure compression. Isn't that indirectly saying we don't need to use the limits that we should use in pure compression? (like 0.00175 at pure compression), when we develop the interaction diagram? As the pure compression will never occur.

@Celt83

Here's the detailed calculation, at the -289.508168 kN location, where @IDS 's and my diagram does not match.

When column is not in full compression.

fcd = 22.667 N/mm2, h = 500 mm, b = 250 mm, NA position = (190.8975 mm, 150 mm), compression block dimensions = 152.718 mm, 120 mm (as shown below), bar diameter = 25 mm, concrete cover = 30 mm, Distance from top surface to bar centre 42.5 mm, Distance from right surface to bar centre 42.5 mm.



Concrete

Area = 9163.08 mm2
x bar = 40 mm, y bar = 50.906 mm
Distance to centroid of compression block to x axis = 199.094 mm
Distance to centroid of compression block to y axis = 85 mm

axial (Nc) = 22.667 N/mm2 * 9163.08 mm2 = 186926.83 / 1000 = 186.93 kN

Moment x = 186926.83 * 199.094 / 1000 = 37.22 kNm
Moment y = 186926.83 * 85 / 1000 = 15.88 kNm


Steel

Strain
bar 1: 0.001729119
bar 2: -0.005879676 --> limit -0.002173920
bar 3: -0.009729676 --> limit -0.002173920
bar 4: -0.002120881

Stress

bar 1: 345.824
bar 2: -434.784 --> limit -434.783
bar 3: -434.784 --> limit -434.783
bar 4: -424.176

Stress after concrete stress reduction (compression) (N/mm2)

bar 1: 323.157
bar 2: -434.784 --> limit -434.783
bar 3: -434.784 --> limit -434.783
bar 4: -424.176

Bar area
= 490.87 mm2

Axial force (N/mm2)

bar 1: 158629.49
bar 2: -213423.61
bar 3: -213423.61
bar 4: -208216.91

Total steel axial = -476434.63 /1000 = -476.435 kN

Steel moments

Mx (Nmm)
bar 1: 32915618.15
bar 2: 44285398.25
bar 3: 44285398.25
bar 4: 43205008.22

Total steel Mx = 78281406.44 / 1000 = 78.281 kNm

My (Nmm)
bar 1: 13086932.52
bar 2: -17607447.5
bar 3: 17607447.5
bar 4: 17177894.84

Total steel My = 30264827.35 / 1000 = 30.265 kNm

	Concrete​	Steel​	Total​
Axial​	186.926832​	-476.435​	-289.508168​	kN​
Mx​	37.21601069​	78.281​	115.4970107​	kNm​
My​	15.88878072​	30.265​	46.15378072​	kNm​

NA angle = atan(190.8975/150) = 51.84

And fixed some error for the charts as well.

 

[slickdeals]

I haven't read all the threads you've started - but what's your motivation behind doing a spreadsheet for something that plenty of commercial software are available?

 

[IDS]

@IDS

I noticed, the eurocode clause 6.1 number 4, says the following,
"For cross-sections loaded by the compression force it is necessary to assume the minimum eccentricity, eo = h/30 but not less than 20 mm where h is the depth of the section."
That means, since we apply that minimum eccentricity (in analysis phase even before we develop interaction diagram), we expect our column to never be in pure compression. Isn't that indirectly saying we don't need to use the limits that we should use in pure compression? (like 0.00175 at pure compression), when we develop the interaction diagram? As the pure compression will never occur.

@Celt83

Here's the detailed calculation, at the -289.508168 kN location, where @IDS 's and my diagram does not match.

When column is not in full compression.

fcd = 22.667 N/mm2, h = 500 mm, b = 250 mm, NA position = (190.8975 mm, 150 mm), compression block dimensions = 152.718 mm, 120 mm (as shown below), bar diameter = 25 mm, concrete cover = 30 mm, Distance from top surface to bar centre 42.5 mm, Distance from right surface to bar centre 42.5 mm.

View attachment 5846

Concrete

Area = 9163.08 mm2
x bar = 40 mm, y bar = 50.906 mm
Distance to centroid of compression block to x axis = 199.094 mm
Distance to centroid of compression block to y axis = 85 mm

axial (Nc) = 22.667 N/mm2 * 9163.08 mm2 = 186926.83 / 1000 = 186.93 kN

Moment x = 186926.83 * 199.094 / 1000 = 37.22 kNm
Moment y = 186926.83 * 85 / 1000 = 15.88 kNm


Steel

Strain
bar 1: 0.001729119
bar 2: -0.005879676 --> limit -0.002173920
bar 3: -0.009729676 --> limit -0.002173920
bar 4: -0.002120881

Stress

bar 1: 345.824
bar 2: -434.784 --> limit -434.783
bar 3: -434.784 --> limit -434.783
bar 4: -424.176

Stress after concrete stress reduction (compression) (N/mm2)

bar 1: 323.157
bar 2: -434.784 --> limit -434.783
bar 3: -434.784 --> limit -434.783
bar 4: -424.176

Bar area
= 490.87 mm2

Axial force (N/mm2)

bar 1: 158629.49
bar 2: -213423.61
bar 3: -213423.61
bar 4: -208216.91

Total steel axial = -476434.63 /1000 = -476.435 kN

Steel moments

Mx (Nmm)
bar 1: 32915618.15
bar 2: 44285398.25
bar 3: 44285398.25
bar 4: 43205008.22

Total steel Mx = 78281406.44 / 1000 = 78.281 kNm

My (Nmm)
bar 1: 13086932.52
bar 2: -17607447.5
bar 3: 17607447.5
bar 4: 17177894.84

Total steel My = 30264827.35 / 1000 = 30.265 kNm

	Concrete​	Steel​	Total​
Axial​	186.926832​	-476.435​	-289.508168​	kN​
Mx​	37.21601069​	78.281​	115.4970107​	kNm​
My​	15.88878072​	30.265​	46.15378072​	kNm​

NA angle = atan(190.8975/150) = 51.84

And fixed some error for the charts as well.
View attachment 5841

View attachment 5843

I agree with the results of your calculations above but note the result includes the 0.9 reduction factor for the triangular section, but this wasn't shown in the calculation.

I have updated your latest interaction diagrams with my My results, showing excellent agreement up to the balance load, but significant differences for high axial loads (see below). The horizontal steps in the results at 2000 kN axial load don't look right, especially for Mx. Do you know what is causing that?

Regarding the application of the strain reduction when the NA is outside the concrete section, I will get back to you on that but I want I want to check my spreadsheets are handling this correctly first.

 

[IDS]

I haven't read all the threads you've started - but what's your motivation behind doing a spreadsheet for something that plenty of commercial software are available?

I can't speak for Pretty Girl, but in my opinion setting up spreadsheets for standard calculations is the best way to understand in detail code requirements, and also to explore differences between different codes, as illustrated by this thread.

 

[Pretty Girl]

I haven't read all the threads you've started - but what's your motivation behind doing a spreadsheet for something that plenty of commercial software are available?

Because, for me it's more intuitive to use a spreadsheet to learn the core concept and understand what actually happens in the concrete element. It's not about getting some results, or diagrams through couple of key presses/clicks with a software. Of course I want to produce an accurate, code compliant diagram, but also I need to learn the core concept behind it as well. I've learnt a lot through this thread. Most importantly it made my brain to re-think and re-wire. So, spreadsheets are the best for this purpose as I see.

Kindly share any useful software for this purpose, if you have got any better suggestions. I'm open to try them out.

 

[Pretty Girl]

I agree with the results of your calculations above but note the result includes the 0.9 reduction factor for the triangular section, but this wasn't shown in the calculation.

I have updated your latest interaction diagrams with my My results, showing excellent agreement up to the balance load, but significant differences for high axial loads (see below). The horizontal steps in the results at 2000 kN axial load don't look right, especially for Mx. Do you know what is causing that?

Regarding the application of the strain reduction when the NA is outside the concrete section, I will get back to you on that but I want I want to check my spreadsheets are handling this correctly first.

View attachment 5936

@IDS

Thank you for the diagram comparison and the comments.

In the previous post, the 0.9 reduction factor was applied to the the calculation "axial (Nc) = 22.667 N/mm2 * 9163.08 mm2 = 186926.83 / 1000 = 186.93 kN", but I mistakenly have not typed it. But the answer has already accounted for the 0.9 reduction factor. So it should have been "axial (Nc) = 0.9 * 22.667 N/mm2 * 9163.08 mm2 = 186926.83 / 1000 = 186.93 kN".

I will go through it and will try to see what causes the reduction above the 2000 kN.

However, I noticed, if I didn't apply the 0.00175 strain at pure compression (the Clause 6.1), it aligns more with your diagram. Only the points above 2700 kN, is largely incorrect, I believe.
Did you apply the reduction at pure compression in your chart?
Pls check if it's even required as we're already accounting for eccentricity even before the development of diagram (at analysis phase), I believe we don't even need to apply the reduction factor for pure compression at this phase, as we're not even expecting the column to be in pure compression as per the Eurocode.

 

[IDS]

A couple of points regarding the reduction in maximum strain:

The reduction is not just for pure compression. With increasing axial load and reducing bending moment, the full strain at the compression face is maintained until the NA reaches the bottom of the section. From that point the strain at (approximately) mid-height is maintained constant, at 0.00175 or 0.002, and the strain at the compression face reduces.

On the other hand, under pure compression the stress distribution is exactly the same for the parabolic-rectangular and the uniform stress block. They both have uniform strain and stress across the full section, so it makes no sense to me to require a reduced strain for the uniform stress block. My suggested procedure is:

1) Find the section capacity with the NA at the base of the section and the full compressive strain at the top face.
2) Find the axial load for uniform stress across the full section with a strain of 0.002 (or as required for higher strength concrete)
3) Use linear interpolation to find the maximum axial load with the specified minimum load eccentricity, and section capacity for any intermediate axial loads.

Also note that for uniform compression the 0.9 reduction factor for triangular sections is not required, because again the parabolic-rectangular and the full uniform stress block have exactly the same stress distribution in this case.