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.

 

[Pretty Girl]

@IDS

Can you please elaborate a bit, how you did it? I mean, bit more information in simple manner would be useful. Did you set the NA parallel to y axis (90 degrees)? and after that did you divide Mx moment with bh^2 fck? etc? I mean you only take one axis for consideration? Can you please provide the steps of how you did it.

About the diagram source, I just got a screenshot of a random youtube video. Thank you for providing a more clear diagram source.

 

[IDS]

@IDS

Can you please elaborate a bit, how you did it? I mean, bit more information in simple manner would be useful. Did you set the NA parallel to y axis (90 degrees)? and after that did you divide Mx moment with bh^2 fck? etc? I mean you only take one axis for consideration? Can you please provide the steps of how you did it.

About the diagram source, I just got a screenshot of a random youtube video. Thank you for providing a more clear diagram source.

The NA is set parallel to the X axis. Otherwise, exactly as you said

The charts are for uniaxial bending about one axis, so the only other changes from previous calcs were setting the bar centres so that d2/h = 0.15, and setting the bar diameter so that Asfyk/bhfck = 0.2.

One other thing, I set fck = 40 MPa and fyk = 500 (as in the example in the book).

Note that the biaxial calculation in the book uses the simplified method given in the code, i.e. finding Mx and My for uniaxial loading, then combining them using the code formula. They don't do an analysis using a rotated cross section, so you shouldn't expect the charts to work for a section under biaxial loading.

 

[Pretty Girl]

The NA is set parallel to the X axis. Otherwise, exactly as you said

The charts are for uniaxial bending about one axis, so the only other changes from previous calcs were setting the bar centres so that d2/h = 0.15, and setting the bar diameter so that Asfyk/bhfck = 0.2.

One other thing, I set fck = 40 MPa and fyk = 500 (as in the example in the book).

Note that the biaxial calculation in the book uses the simplified method given in the code, i.e. finding Mx and My for uniaxial loading, then combining them using the code formula. They don't do an analysis using a rotated cross section, so you shouldn't expect the charts to work for a section under biaxial loading.

@IDS
Thank you for the clarification.

I'm confused why did you set Asfyk/bhfck = 0.2 when you calculate, and compared your result with Asfyk/bhfck = 0.4 line in the diagram?
So, you used 4 bars and each bar is 25 mm diameter?

Or you used 4 bars with 35 mm diameter (for Asfyk/bhfck = 0.4)?

Further, this is not that useful in biaxial situations as you mentioned. So, is there any other standard charts out there to compare out diagram so we can confirm our diagram is accurate?

may be there are some standard set of other biaxial charts? with varying angle of NA?

When I change it to close to 5 degrees NA angle (if it gets close to zero, it's even worse it seems). It becomes very strange

bh^2 = 62500000
fck = 40
bh^2 fck = 2500000000
bh^2 fck*10^-6 = 2500

Mx/bh^2 fck = Mx/2500
example: 208.76 kNm / 2500 = 0.083


Mx without dividing by bh^2 fck is also shows that exact line pattern at 5 degrees. That means, the way I calculate is wrong somewhere, although I got similar results as yours earlier for Mx and My.

Mx	Mx/bh^2 fck
0.000141971​	5.67883E-08​
0.127429845​	5.09719E-05​
0.508296359​	0.000203319​
1.140465012​	0.000456186​
2.021801273​	0.000808721​
3.150170611​	0.001260068​
4.523438496​	0.001809375​
6.139470397​	0.002455788​
7.996131783​	0.003198453​
10.09128812​	0.004036515​
12.42280489​	0.004969122​
14.94513556​	0.005978054​
17.46744228​	0.006986977​
19.9675143​	0.007987006​
22.44535162​	0.008978141​
24.90095426​	0.009960382​
27.33432219​	0.010933729​
29.74545543​	0.011898182​
32.13435398​	0.012853742​
34.50101784​	0.013800407​
40.413447​	0.016165379​
47.20764146​	0.018883057​
56.56760123​	0.02262704​
66.66532631​	0.026666131​
76.08981669​	0.030435927​
84.91807238​	0.033967229​
93.21609337​	0.037286437​
101.0418797​	0.040416752​
108.4424313​	0.043376973​
115.4597482​	0.046183899​
122.1308304​	0.048852332​
128.4846779​	0.051393871​
132.6042907​	0.053041716​
138.4066689​	0.055362668​
143.9668123​	0.057586725​
147.3557211​	0.058942288​
152.4843951​	0.060993758​
157.4218345​	0.062968734​
162.1800391​	0.064872016​
166.7720091​	0.066708804​
171.2087444​	0.068483498​
175.4992449​	0.070199698​
179.6525108​	0.071861004​
183.676542​	0.073470617​
187.5783385​	0.075031335​
191.3659003​	0.07654636​
195.0432274​	0.078017291​
198.6173198​	0.079446928​
202.0931775​	0.080837271​
205.4758005​	0.08219032​
208.7681889​	0.083507276​

 

[IDS]

@IDS
Thank you for the clarification.

I'm confused why did you set Asfyk/bhfck = 0.2 when you calculate, and compared your result with Asfyk/bhfck = 0.4 line in the diagram?
So, you used 4 bars and each bar is 25 mm diameter?

Or you used 4 bars with 35 mm diameter (for Asfyk/bhfck = 0.4)?

Further, this is not that useful in biaxial situations as you mentioned. So, is there any other standard charts out there to compare out diagram so we can confirm our diagram is accurate?

may be there are some standard set of other biaxial charts? with varying angle of NA?

When I change it to close to 5 degrees NA angle (if it gets close to zero, it's even worse it seems). It becomes very strange

bh^2 = 62500000
fck = 40
bh^2 fck = 2500000000
bh^2 fck*10^-6 = 2500

Mx/bh^2 fck = Mx/2500
example: 208.76 kNm / 2500 = 0.083


Mx without dividing by bh^2 fck is also shows that exact line pattern at 5 degrees. That means, the way I calculate is wrong somewhere, although I got similar results as yours earlier for Mx and My.


View attachment 6537

View attachment 6544

Mx	Mx/bh^2 fck
0.000141971​	5.67883E-08​
0.127429845​	5.09719E-05​
0.508296359​	0.000203319​
1.140465012​	0.000456186​
2.021801273​	0.000808721​
3.150170611​	0.001260068​
4.523438496​	0.001809375​
6.139470397​	0.002455788​
7.996131783​	0.003198453​
10.09128812​	0.004036515​
12.42280489​	0.004969122​
14.94513556​	0.005978054​
17.46744228​	0.006986977​
19.9675143​	0.007987006​
22.44535162​	0.008978141​
24.90095426​	0.009960382​
27.33432219​	0.010933729​
29.74545543​	0.011898182​
32.13435398​	0.012853742​
34.50101784​	0.013800407​
40.413447​	0.016165379​
47.20764146​	0.018883057​
56.56760123​	0.02262704​
66.66532631​	0.026666131​
76.08981669​	0.030435927​
84.91807238​	0.033967229​
93.21609337​	0.037286437​
101.0418797​	0.040416752​
108.4424313​	0.043376973​
115.4597482​	0.046183899​
122.1308304​	0.048852332​
128.4846779​	0.051393871​
132.6042907​	0.053041716​
138.4066689​	0.055362668​
143.9668123​	0.057586725​
147.3557211​	0.058942288​
152.4843951​	0.060993758​
157.4218345​	0.062968734​
162.1800391​	0.064872016​
166.7720091​	0.066708804​
171.2087444​	0.068483498​
175.4992449​	0.070199698​
179.6525108​	0.071861004​
183.676542​	0.073470617​
187.5783385​	0.075031335​
191.3659003​	0.07654636​
195.0432274​	0.078017291​
198.6173198​	0.079446928​
202.0931775​	0.080837271​
205.4758005​	0.08219032​
208.7681889​	0.083507276​

I'm confused why did you set Asfyk/bhfck = 0.2 when you calculate, and compared your result with Asfyk/bhfck = 0.4 line in the diagram?
So, you used 4 bars and each bar is 25 mm diameter?

Or you used 4 bars with 35 mm diameter (for Asfyk/bhfck = 0.4)?

My mistake. I had 4 bars with 36 mm diameter.

I agree the charts aren't particularly useful for biaxial loading.

I don't know of any biaxial charts, and if there are any they would be likely to be based on the simplified method, rather than analysis with a rotated NA. Note that the simplified method doesn't agree very well with the detailed analysis. See our previous discussion here: https://www.eng-tips.com/viewthread.cfm?qid=508170, and also my blog post: simplified-bi-axial-bending

I'll have a look at my results with different angles and reply separately on that.

 

[Pretty Girl]

My mistake. I had 4 bars with 36 mm diameter.

I agree the charts aren't particularly useful for biaxial loading.

I don't know of any biaxial charts, and if there are any they would be likely to be based on the simplified method, rather than analysis with a rotated NA. Note that the simplified method doesn't agree very well with the detailed analysis. See our previous discussion here: https://www.eng-tips.com/viewthread.cfm?qid=508170, and also my blog post: simplified-bi-axial-bending

I'll have a look at my results with different angles and reply separately on that.

@IDS
Thank you for the response and the links.

I updated with 36 mm dia r/f and 57 mm cover to come up with d'/h = 0.15. For the 5 degrees NA to x axis, I get the following. I went through the calculation. It also seems ok. It's possible to get inaccurate when it's very close to zero.

Even with detailed analysis more precise, the fact that my charts being totally different at this 5 degrees NA, compared to simplified standard method is very strange.

Please provide some comparisons and insights so I can rectify this.

N​	N/bh fck​	Mx​	Mx/bh^2 fck​
-1770.2204​	-0.3540441​	0.00014275​	5.7102E-08​
-1769.707​	-0.3539414​	0.12813093​	5.1252E-05​
-1768.1652​	-0.353633​	0.51108496​	0.00020443​
-1765.5953​	-0.3531191​	1.14670395​	0.00045868​
-1761.9976​	-0.3523995​	2.03282973​	0.00081313​
-1757.3719​	-0.3514744​	3.16730416​	0.00126692​
-1751.7184​	-0.3503437​	4.5479691​	0.00181919​
-1745.0368​	-0.3490074​	6.17266639​	0.00246907​
-1737.3274​	-0.3474655​	8.03923788​	0.0032157​
-1728.59​	-0.345718​	10.1455254​	0.00405821​
-1718.8248​	-0.343765​	12.4893709​	0.00499575​
-1708.2064​	-0.3416413​	15.024965​	0.00600999​
-1697.4989​	-0.3394998​	17.5604118​	0.00702416​
-1686.7913​	-0.3373583​	20.0733779​	0.00802935​
-1676.0838​	-0.3352168​	22.5638633​	0.00902555​
-1665.3762​	-0.3330752​	25.0318679​	0.01001275​
-1654.6687​	-0.3309337​	27.4773918​	0.01099096​
-1643.9611​	-0.3287922​	29.900435​	0.01196017​
-1633.2536​	-0.3266507​	32.3009975​	0.0129204​
-1622.546​	-0.3245092​	34.6790793​	0.01387163​
-1563.9425​	-0.3127885​	45.4166803​	0.01816667​
-1500.5129​	-0.3001026​	56.9758007​	0.02279032​
-1401.2844​	-0.2802569​	74.7774403​	0.02991098​
-1298.3598​	-0.259672​	93.2035992​	0.03728144​
-1203.1203​	-0.2406241​	110.262277​	0.04410491​
-1114.6417​	-0.2229283​	126.115475​	0.05044619​
-1032.1471​	-0.2064294​	140.899192​	0.05635968​
-954.9696​	-0.1909939​	154.729428​	0.06189177​
-882.53904​	-0.1765078​	167.707183​	0.06708287​
-814.36649​	-0.1628733​	179.916457​	0.07196658​
-750.02394​	-0.1500048​	191.433251​	0.0765733​
-689.14239​	-0.1378285​	202.322564​	0.08092903​
-654.46884​	-0.1308938​	208.602397​	0.08344096​
-599.57428​	-0.1199149​	218.397749​	0.0873591​
-547.27873​	-0.1094557​	227.716619​	0.09108665​
-520.43118​	-0.1040862​	232.56001​	0.093024​
-472.68963​	-0.0945379​	241.036919​	0.09641477​
-426.95008​	-0.08539​	249.140348​	0.09965614​
-383.05552​	-0.0766111​	256.899296​	0.10275972​
-340.86297​	-0.0681726​	264.337764​	0.10573511​
-300.24342​	-0.0600487​	271.47875​	0.1085915​
-261.08287​	-0.0522166​	278.341256​	0.1113365​
-223.27832​	-0.0446557​	284.944281​	0.11397771​
-186.73376​	-0.0373468​	291.303826​	0.11652153​
-151.36221​	-0.0302724​	297.43689​	0.11897476​
-117.08866​	-0.0234177​	303.354473​	0.12134179​
-83.839108​	-0.0167678​	309.069575​	0.12362783​
-51.549556​	-0.0103099​	314.595197​	0.12583808​
-20.157004​	-0.0040314​	319.941338​	0.12797654​
10.388548​	0.00207771​	325.116998​	0.1300468​
40.1421​	0.00802842​	330.131177​	0.13205247​

 

[IDS]

@PG Can you confirm what bar diameter and cover you are using for these latest results. Also concrete dimensions and steel and concrete strengths.

 

[Pretty Girl]

@PG Can you confirm what bar diameter and cover you are using for these latest results. Also concrete dimensions and steel and concrete strengths.

@IDS
I updated mine with your parameters. 36 mm dia r/f, steel strength 500 N/mm2, eff cover (from outer suface to r/f centre) = 57 mm, fck = 40 N/mm2. Column dimensions 250 (b) * 500 (h), NA angle = 5 degrees.