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aditya ...
Joined: 05 Apr 2008 Posts: 184

Posted: Mon Oct 26, 2020 6:15 am Post subject: Approximate Equation for Curvillinear portion of Stressstrain Curve of HYSD Rebars 


Dear Sefians,
This is with regard to the approximate equation for design stressstrain 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:
f_{yd}=109.1569+537276.6*e250905*10^{3}*e^{2}+5528249*10^{4}*e^{3}4709357*10^{6}*e^{4}
while the equation for Fe 500 is:
f_{yd}=1707.6242356626*e1444077*10^{3}*e^{2}+3666904*10^{5}*e^{3}3318089*10^{7}*e^{4}.
My observations are:
1. The approximate equation for Fe 415 gives very close values of f_{yd} 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, f_{yd} 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 stressstrain 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 General Sponsor
Joined: 21 Feb 2008 Posts: 5457 Location: Gaithersburg, MD, U.S.A.

Posted: Wed Oct 28, 2020 3:53 am Post subject: Re: Approximate Equation for Curvillinear portion of Stressstrain Curve of HYSD Rebars 


Dear Er Aditya
Thank you for pointing out this:
Please check with this:
_{fyd}=1707.6242356626*e+1444077*10^{3}*e^{2 }^{}3666904*10^{5}*e^{3}+3318089*10^{7}*e^{4}.
Regards
NS
aditya wrote:  Dear Sefians,
This is with regard to the approximate equation for design stressstrain 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:
f_{yd}=109.1569+537276.6*e250905*10^{3}*e^{2}+5528249*10^{4}*e^{3}4709357*10^{6}*e^{4}
while the equation for Fe 500 is:
f_{yd}=1707.6242356626*e1444077*10^{3}*e^{2}+3666904*10^{5}*e^{3}3318089*10^{7}*e^{4}.
My observations are:
1. The approximate equation for Fe 415 gives very close values of f_{yd} 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, f_{yd} 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 stressstrain 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 ...
Joined: 05 Apr 2008 Posts: 184

Posted: Thu Oct 29, 2020 2:23 pm Post subject: 


Dear Respected Subramanian Sir,
Thanks a lot for suggesting the corrections in sign of the approximate equation for stressstrain 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 stressstrain 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 stressstrain curvilinear portion.
3. If we fit 4th degree polynomial through regression analysis between the six points of curvilinear portion of the stressstrain 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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Dr. N. Subramanian General Sponsor
Joined: 21 Feb 2008 Posts: 5457 Location: Gaithersburg, MD, U.S.A.

Posted: Thu Oct 29, 2020 8:58 pm Post subject: 


Dear Er Aditya
Try this: fyd= A1^5*6.25174*10^14A1^4*1.70731*10^13+A1^3*1.49602*10^11A1^2*6.00226*10^8+A1*1.18309*10^6535.11
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 stressstrain 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 stressstrain 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 stressstrain curvilinear portion.
3. If we fit 4th degree polynomial through regression analysis between the six points of curvilinear portion of the stressstrain 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 ...
Joined: 05 Apr 2008 Posts: 184

Posted: Sun Nov 01, 2020 10:23 am Post subject: 


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*f_{y} 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 stressstrain graph.
with best regards and thanks,
Aditya


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abhio ...
Joined: 08 Mar 2010 Posts: 533

Posted: Wed Nov 25, 2020 9:47 am Post subject: 


This gives a good match for Fe 500
8150780000000*e^4+99924733122*e^3465498637.6*e^2+1005216*e443.477


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aditya ...
Joined: 05 Apr 2008 Posts: 184

Posted: Thu Nov 26, 2020 3:55 pm Post subject: 


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 ...
Joined: 08 Mar 2010 Posts: 533

Posted: Sat Nov 28, 2020 2:55 pm Post subject: 


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 recheck?
Regards,
A S Oundhakar


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aditya ...
Joined: 05 Apr 2008 Posts: 184

Posted: Sat Nov 28, 2020 4:17 pm Post subject: 


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 ...
Joined: 08 Mar 2010 Posts: 533

Posted: Sat Nov 28, 2020 5:36 pm Post subject: 


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+09x^{3}  9.05197E+07x^{2} + 3.46850E+05x  2.34055E+01
Regards,
A S Oundhakar


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