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Approximate Equation for Curvillinear portion of Stress-strain Curve of HYSD Rebars

 
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aditya
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PostPosted: Mon Oct 26, 2020 6:15 am    Post subject: Approximate Equation for Curvillinear portion of Stress-strain Curve of HYSD Rebars Reply with quote

Dear Sefians,
This is with regard to the approximate equation for design stress-strain curve for HYSD rebars in RCC as given under page 149 in the book "Design of Reinforced Concrete Structures" by Dr. N. Subramanian Sir.

The equation for Fe 415 is given as:
fyd=-109.1569+537276.6*e-250905*103*e2+5528249*104*e3-4709357*106*e4
while the equation for Fe 500 is:
fyd=1707.624-2356626*e-1444077*103*e2+3666904*105*e3-3318089*107*e4.

My observations are:
1. The approximate equation for Fe 415 gives very close  values of fyd for the 6 points given for the values of strains e but the equation for Fe 500 gives absurd values. For example, at a strain level of 0.00174, fyd comes out to be equal to -5137.41 MPa instead of 347.8 MPa from the above formula for Fe 500. What could be the reason? Are there any typographical mistakes?

2. When we try to fit 4th degree curve on the basis of given six values of stress-strain on the curve for Fe 415 & Fe 500, we get different values of regression coefficients. How have the coefficients derived in the above equations?
While these formulas are quite useful for computer programs & spreadsheets, I request sefians to look into the matter and suggest possible causes.
with regards,
Aditya
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Dr. N. Subramanian
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PostPosted: Wed Oct 28, 2020 3:53 am    Post subject: Re: Approximate Equation for Curvillinear portion of Stress-strain Curve of HYSD Rebars Reply with quote

Dear Er Aditya

Thank you for pointing out this:

Please check with this:
fyd=1707.624-2356626*e+1444077*103*e2
-3666904*105*e3+3318089*107*e4.


Regards
NS

aditya wrote:
Dear Sefians,
This is with regard to the approximate equation for design stress-strain curve for HYSD rebars in RCC as given under page 149 in the book "Design of Reinforced Concrete Structures" by Dr. N. Subramanian Sir.

The equation for Fe 415 is given as:
fyd=-109.1569+537276.6*e-250905*103*e2+5528249*104*e3-4709357*106*e4
while the equation for Fe 500 is:
fyd=1707.624-2356626*e-1444077*103*e2+3666904*105*e3-3318089*107*e4.

My observations are:
1. The approximate equation for Fe 415 gives very close  values of fyd for the 6 points given for the values of strains e but the equation for Fe 500 gives absurd values. For example, at a strain level of 0.00174, fyd comes out to be equal to -5137.41 MPa instead of 347.8 MPa from the above formula for Fe 500. What could be the reason? Are there any typographical mistakes?

2. When we try to fit 4th degree curve on the basis of given six values of stress-strain on the curve for Fe 415 & Fe 500, we get different values of regression coefficients. How have the coefficients derived in the above equations?
While these formulas are quite useful for computer programs & spreadsheets, I request sefians to look into the matter and suggest possible causes.
with regards,
Aditya
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aditya
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Joined: 05 Apr 2008
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PostPosted: Thu Oct 29, 2020 2:23 pm    Post subject: Reply with quote

Dear Respected Subramanian Sir,
Thanks a lot for suggesting the corrections in sign of the approximate equation for stress-strain relation of HYSD bars as per IS 456: 2000. The corrections will also apply for the equation 14.4 a given in your book in page no. 547 where strain is defined as strainx1000.
However, I noticed three points with regard to the approximate equations:
1. Approximate equation for Fe 500 coincides with the stress values at three points only while for other three points of the curve, deviation is more. In the case of Fe 415, stress values obtained from the approximate equation are quite close to the values defined for the curve by SP 16.
2. Values of stresses obtained for the strain values near the sixth defined points given by SP 16 are more than the stress value for the sixth or flat portion of the stress-strain curve. For example, at a strain value of 0.00364, the stress obtained for Fe 415 grade from the approximate equation is 361.5965 MPa which is wrong and it should have been less than 360.9 MPa. Thus the result from the equation is wrong for strain values close to the sixth defined point of stress-strain curvilinear portion.
3. If we fit 4th degree polynomial through regression analysis between the six points of curvilinear portion of the stress-strain diagram, equation of polynomial so obtained gives more accurate values for the six points. However, again, even this equation gives wrong results for points close to the sixth point defined by SP 16.

with regards,
Aditya
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PostPosted: Thu Oct 29, 2020 8:58 pm    Post subject: Reply with quote

Dear Er Aditya

Try this: fyd= A1^5*6.25174*10^14-A1^4*1.70731*10^13+A1^3*1.49602*10^11-A1^2*6.00226*10^8+A1*1.18309*10^6-535.11

Where A1 is epsilon
Here since I have gone up to Epsilon^5, you get better match

Regards
NS

aditya wrote:
Dear Respected Subramanian Sir,
Thanks a lot for suggesting the corrections in sign of the approximate equation for stress-strain relation of HYSD bars as per IS 456: 2000. The corrections will also apply for the equation 14.4 a given in your book in page no. 547 where strain is defined as strainx1000.
However, I noticed three points with regard to the approximate equations:
1. Approximate equation for Fe 500 coincides with the stress values at three points only while for other three points of the curve, deviation is more. In the case of Fe 415, stress values obtained from the approximate equation are quite close to the values defined for the curve by SP 16.
2. Values of stresses obtained for the strain values near the sixth defined points given by SP 16 are more than the stress value for the sixth or flat portion of the stress-strain curve. For example, at a strain value of 0.00364, the stress obtained for Fe 415 grade from the approximate equation is 361.5965 MPa which is wrong and it should have been less than 360.9 MPa. Thus the result from the equation is wrong for strain values close to the sixth defined point of stress-strain curvilinear portion.
3. If we fit 4th degree polynomial through regression analysis between the six points of curvilinear portion of the stress-strain diagram, equation of polynomial so obtained gives more accurate values for the six points. However, again, even this equation gives wrong results for points close to the sixth point defined by SP 16.

with regards,
Aditya
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aditya
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PostPosted: Sun Nov 01, 2020 10:23 am    Post subject: Reply with quote

Dear Respected Subramanian Sir,
Thanks a lot for the new approximate equation. However, it is to be noticed that 4th degree and even fifth degree polynomials so fitted have their peak points earlier than the last point. Consequently, at strain values earlier than the yield strain, stress values as shown by the fitted polynomials are more than the design yield stress value of 0.87*fy resulting in wrong values of stresses at points near to the yield strain values. This happens though the curves fit well at six given points of the curvilinear portion of the stress-strain graph.
with best regards and thanks,
Aditya
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abhio
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PostPosted: Wed Nov 25, 2020 9:47 am    Post subject: Reply with quote

This gives a good match for Fe 500

-8150780000000*e^4+99924733122*e^3-465498637.6*e^2+1005216*e-443.477
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aditya
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PostPosted: Thu Nov 26, 2020 3:55 pm    Post subject: Reply with quote

Dear Respected Sirs,
Thanks for the suggestions. However, I would like to request you to study the attached example in which it can be observed that the stress values obtained from the fitted polynomial equation will give wrong values beyond the design yield stress at values of strains near to the sixth defining point of the curve.
with regards,
Aditya


attachment:
Polynomial Equations for Approximations to the Curvilinear Portion of Stress.doc



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abhio
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PostPosted: Sat Nov 28, 2020 2:55 pm    Post subject: Reply with quote

Dear Er Aditya,

I'm not sure I understand your question.

1. The equations fitted are obviously only intended to be used up to yield, beyond which a straight line at 0.87 fy is to be used.
2. I can also not see any early peak in any of the tables given by you for equations 1, 2.2 and 2.3.

Could you please re-check?

Regards,

A S Oundhakar
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aditya
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PostPosted: Sat Nov 28, 2020 4:17 pm    Post subject: Reply with quote

Dear Sir,
We generally do linear interpolation for obtaining the stress values corresponding to strains on rebars on the basis of the the six reference strains & the corresponding stress values as defined by SP 16 whereas the relation is actually curvilinear. For most of the practical purposes and as illustrated in design examples in RCC Textbooks, this procedure is accurate enough.
What I wanted to do is to use the approximate polynomial equation to interpolate the stress values at strain values in between the six values instead of using linear interpolation since curvilinear relation will give more "accurate" values of stresses. For writing a computer equation and for developing a spreadsheet for RCC design & analysis also, direct use of equation is more convenient though interpolation algorithm can be written on the basis of tabular relation as well.
Yes indeed, as you noticed, the polynomial equations do give accurate enough results at the six points. But as I pointed out, for strain values close to design yield strain values, the stresses obtained by the equation are HIGHER than design yield stresses. This can be observed from the following tables in which stress values higher than the design yield stress have been shown in bold. These tables were used in plotting the equations shown in my earlier response.
For Fe 415:
Fe415
strain stress
0.00144 289.0681
0.00154 298.6213
0.00164 306.9234
0.00174 314.1262
0.00184 320.3706
0.00194 325.7858
0.00204 330.4899
0.00214 334.5898
0.00224 338.1809
0.00234 341.3473
0.00244 344.162
0.00254 346.6864
0.00264 348.971
0.00274 351.0545
0.00284 352.9647
0.00294 354.718
0.00304 356.3194
0.00314 357.7626
0.00324 359.0301
0.00334 360.0931
0.00344 360.9113
0.00354 361.4333
0.00364 361.5965
0.00374 361.3266
0.00381 360.8348

For Fe 500 using the three equations:
  
Fe 500
Strain Stress_eq 2.1 Stress_eq 2.2 Stress_eq 2.3
0.00174 351.596 347.802 347.948
0.00184 356.529 359.085 359.183
0.00194 363.347 368.720 368.827
0.00204 371.357 376.964 377.109
0.00214 379.941 384.048 384.240
0.00224 388.566 390.177 390.411
0.00234 396.774 395.531 395.792
0.00244 404.190 400.267 400.535
0.00254 410.515 404.517 404.773
0.00264 415.533 408.391 408.617
0.00274 419.107 411.977 412.160
0.00284 421.178 415.343 415.477
0.00294 421.768 418.533 418.620
0.00304 420.979 421.577 421.624
0.00314 418.990 424.480 424.504
0.00324 416.063 427.233 427.254
0.00334 412.539 429.809 429.850
0.00344 408.836 432.162 432.249
0.00354 405.454 434.233 434.386
0.00364 402.972 435.946 436.178
0.00374 402.050 437.212 437.523
0.00384 403.424 437.926 438.298
0.00394 407.915 437.972 438.362
0.00404 416.418 437.222 437.552
0.00414 429.911 435.536 435.689
0.00417 435.088 434.825 434.895

with regards & thanks,
Aditya
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abhio
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PostPosted: Sat Nov 28, 2020 5:36 pm    Post subject: Reply with quote

Dear Er Aditya,

The fitting equation will obviously be imperfect, but the error is well within 1%, which ought to be acceptable.

Meanwhile, you could try this equation, which seems to fit even better:
8.07975E+09x3 - 9.05197E+07x2 + 3.46850E+05x - 2.34055E+01


Regards,

A S Oundhakar
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