41 
 
 
CHAPTER 04 
 
4.0  CALCULATION   AND  DISCUSSION  
4.1    0.2% Offset  yield stress in tensile testing  
As per litreature review  (Pg 25) yield stress was calculated for CTD and TMT bars  and 
tabulated in the table 06.  
4.2    0.05% Offset yield stress in three point  bending.    
In accordance with ASTM E855-90 [13] applies to three point bending at smaller strains 
for spring applications,Certain elastic /plastic data may be obtained from load- 
displacement (position) curves  in three point bending including 0.05% offset yield, the 
elastic limit and the elastic modulus in bending. A typical example for calculating 0.05% 
offset yield stress for 12 mm diameter of  CTD  and  TMT  are shown in  fig.15 & 
fig.16.  However,0.05% offset yield in bending measured on many samples shows 
higher yield values and in some cases go beyound  maximum stress in tensile strength.  
 
810 
0.004 
y = 178587x - 90.646 
y = 178587x - 179.94 
-400 
-200 
0 
200 
400 
600 
800 
1000 
1200 
-0.2% 0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4% 1.6% 1.8% 
S
tr
es
s 
 (
N
/m
m
2
) 
Strain %  
Stress vs. Strain  
Stress vs. Strain  
 
Straight line portion  
Fig.15 Stress Vs Strain in three point bending including 0.05% offset yield for CTD bars. 
42 
 
 
 
 
4.3   0.04% Offset yield stress in three point  bending.    
As per the NBRO testing data of tensile test, it was observed that, the ratio of stress at 
elastic limit and 0.2% proof stress is 1.10.  It was shown that,  ratio of 0.04% Offset 
bend stress value and stress at elastic limit in three point bend test also  follows the 
trends in the data measured in the tensile test of NBRO.Therfore,0.04% offset bend 
stress was calculated for all the bars tested.  
Figure 17 & figure 18 show typical examples for calculating 0.04% offset yield in 
bending for CTD and TMT 12mm diameter bars. 
980 
0.0055 
y = 180926x - 13.119 
y = 180926x - 103.58 
-400 
-200 
0 
200 
400 
600 
800 
1000 
1200 
1400 
1600 
-0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 
S
tr
es
s 
 (
N
/m
m
2
) 
Strain %  
Stress vs. Strain  
Stress vs. Strain  
 
Straight line 
portion  
0.05 % Offset  
Fig.16 Stress Vs Strain in three point bending including 0.05% offset yield for TMT 
43 
 
 
 
 
Next to correlate the 0.2% offset yield stress that was measured in the tensile test was 
plotted against the 0.04% offset bend stress measured in bending for 10mm to 32mm 
diameter of CTD and TMT bars are  shown in fig.19 to fig.24. By fitting these data to a 
straight line, a correlation in CTD bars and TMT bars were determined to relate the 
0.2% proof stress in tension and 0.04% Offset yield stress in three point bending. Yield 
780 
0.004 
y = 178587x - 90.646 
y = 178587x - 162.08 
-400 
-200 
0 
200 
400 
600 
800 
1000 
1200 
-0.2% 0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2% 1.4% 1.6% 1.8% 
S
tr
es
s 
 (
N
/m
m
2
) 
Strain %  
Stress vs. Strain  
Stress vs. Strain  
 
Fig.17 Stress Vs Strain in three point bending including 0.04% offset yield for CTD bars. 
960 
0.00525 
y = 180926x - 13.119 
y = 180926x - 85.49 
-400 
-200 
0 
200 
400 
600 
800 
1000 
1200 
1400 
1600 
-0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 
S
tr
es
s 
 (
N
/m
m
2
) 
Strain %  
Stress vs. Strain  
Stress vs. Strain  
 
Straight line 
portion  
0.04 % Offset  
Fig.18 Stress Vs Strain in three point bending including 0.04% offset yield for TMT bars. 
44 
 
stresses are given in table 05.Good correlations were obtained for individual bar 
diameters and shown bellow.    
    
      
 
Fig.19: 0.04% yield stress in bending Vs 0.2% Yield        Fig.20: 0.04% yield stress in bending  
Vs Stress* in tension -10mm dia. Bars.                                0.2% yield Stress* in tension -12mm dia. Bars. 
 
 
  
Fig.21: 0.04% yield stress in bending Vs 0.2% Yield          Fig.22: 0.04% yield stress in bending Vs 0.2%     
             Stress* in tension -16mm dia. Bars.                                      Stress* in tension -20mm dia. Bars. 
 
Note: * Upper Yield stress for thermal treated bars (TMT) 
y = 5.3228x - 1950.2 
R² = 0.9887 
0 
200 
400 
600 
800 
1000 
1200 
450 500 550 600 0
.0
4
%
 y
ie
ld
 s
tr
e
ss
 in
 b
e
n
d
in
g 
0.2% yield stress* in tension N/mm2 
y = 5.1819x - 1865 
R² = 0.9254 
0 
200 
400 
600 
800 
1000 
1200 
500 520 540 560 580 
0
.0
4
%
 Y
ie
ld
 S
tr
e
ss
 in
  b
e
n
d
in
g 
(N
/m
m
2
) 
 
 
0.2% yield stress *in tension (N/mm2) 
 
y = 4.7226x - 1631.3 
R² = 0.7779 
0 
100 
200 
300 
400 
500 
600 
700 
800 
900 
1000 
500 520 540 560 
0
.0
4
%
 y
ie
ld
 s
tr
e
ss
 in
 b
e
n
d
in
g 
(N
/m
m
2
) 
 
0.2% yield stress* in tension 
(N/mm2)  
y = 4.261x - 1376.2 
R² = 0.7156 
0 
200 
400 
600 
800 
1000 
1200 
480 500 520 540 560 
0
.0
4
%
 y
ie
ld
 s
tr
e
ss
 in
 b
e
n
d
in
g 
(N
/m
m
2
) 
0.2% yield stress* in tension (N/mm2) 
45 
 
 
 
 
 
Fig.23: 0.04% yield stress in bending Vs 0.2% Yield          Fig.24: 0.04% yield stress in bending Vs 0.2%             
Stress* in tension -25mm dia. Bars.                                     Yield Stress* in tension -32mm dia. Bars. 
 
Note: * Upper Yield stress for thermal treated bars (TMT) 
The above relations may vary as per the test methods and equipments used. Having this 
in mind following calculation was carried out for the linear equations obtained for 
different bar diameters. 
Table 3 : Relationship between gradient (m) and intercept  (c) drive from the 
                    equations  obtained  
                     from σb Vs σt          
                      (Where,  σb - 0.04% yield stress in bending and σt - 0.2% yield stress in  
                   tensile) 
  
 
      Nominal  Bar Span Span/Mandrel  m C 
  Diameter(mm) (mm)                     Diameter(S)     
  10 200 8.0 5.322 -1950.2 
  12 240 9.6 5.182 -1865.0 
  16 320 12.8 4.723 -1631.3 
  20 400 16.0 4.261 -1376.2 
  25 500 20.0 4.114 -1361.2 
  32 640 25.6 3.252 -886.2 
   
y = 4.1135x - 1361.2 
R² = 0.9376 0 
200 
400 
600 
800 
1000 
1200 
500 520 540 560 580 0
.0
4
%
 y
ie
ld
 s
tr
e
ss
 in
 b
e
n
d
in
g 
(N
/m
m
2
) 
0.2% yield stress* in 
tension(N/mm2) 
y = 3.2522x - 886.2 
R² = 0.8834 
0 
200 
400 
600 
800 
1000 
1200 
500 550 600 
0
.0
4
%
 y
ie
ld
 s
tr
e
ss
 in
 b
e
n
d
in
g 
(N
/m
m
2
) 
0.2% yield stress* in tension(N/mm2) 
46 
 
 
Figure 25: Intercept © Vs gradient (m) 0f σb & σt 
The linear relation was observed and    c = m1 m + c1 Where, c= m σt - σb 
Therefore,- σb+ m σt =  m1 m + c1 ,        m σt =  -508.12m + 762.44 + σb 
 m σt  =  σb  - 508.12 m + 762.44    
The above graph shows a liner relationship between σb & σt if m is constant. But, 'm' 
may depend on many factors such as span, mandrel diameter and steel bar diameter. 
The relationship observed for m Vs bar diameter is as follows 
 
Figure 26: m Vs sample diameter 
y = -508.12x + 762.44 
R² = 0.9974 
-2500.0 
-2000.0 
-1500.0 
-1000.0 
-500.0 
0.0 
0.000 1.000 2.000 3.000 4.000 5.000 6.000 
C
 
m 
y = 6.6784e-0.022x 
R² = 0.9738 
0 
1 
2 
3 
4 
5 
6 
0 10 20 30 40 
m
 
Sample Diameter(mm) 
m Vs Sample Diameter 
47 
 
The above graph was plotted ‘m’ vs sample diameter. Since the exponential equation is 
like y=a{exp(bx)} , we can assume ‘b’ to be a value with inverse length units so that part 
inside exp function is dimensionless. Note that in graphs of σ-t vs σ-b the gradient is a 
dimensionless factor which is the stress gain while intercept is the threshold stress. In 
graph of m vs diameter, ‘a’ is a dimensionless quantity while ‘b’ is an inverse length 
parameter. 
It can be observed that for all diameters tested, there is a definite linear relationship 
between the bending yield strength and tensile yield strength. In addition there is a linear 
relationship between the stress gain and the threshold stress values for all diameters 
tested as shown by the Stress gain vs threshold stress graph. This shows that the 
relationship between bending stress and tensile yield stress is definite and predictable for 
any diameter within the range of tests conducted. It can also be seen from the graph of 
Stress gain vs sample diameter that there is a definite and predictable relationship 
between the diameter of the sample and the stress gain, which is exponential in nature. 
This concludes that knowing the sample size and bending yield stress, it is possible to 
predict the tensile yield stress. 
The accuracy of this prediction was determined by using t- analysis for the observed 
0.2% proof yield stress and for the calculated yield stress by using the equation obtained 
for the different bar diameters. 
 
 
 
 
 
 
 
 
48 
 
Table 4: Summary of Statistical Calculation   
Bar. Diameter σ(Square root of 
variance between 0.2% 
proof yield stress and 
calculated yield stress 
using the equations for 
different bar diameters) 
T-Calculated T-Table 
95% 
10 
12 
16 
20 
25 
32 
32.72 
15.31 
12.10 
12.10 
14.74 
12.10 
0.01 
0.75 
0.77 
0.77 
1.09 
0.77 
 
 
1.76 
 
 
 
 
 
Analysis shows that the all the calculated t-values are below the table value. Therefore, 
it can be stated that the tensile yield stress calculated from bending yield stress are 
accurate within 95% confidence level. 
 
4.4 Rib geometry 
The degree of cold twisting of CTD steel bars may have an influence on the tensile and 
bending properties of the bar due to case hardening. Assuming that the degree of 
twisting will be reflected by the pitch of the longitudinal rib, the relationship between 
the pitch to bar diameter ratio and the 0.04% 0ffset bend stress was examined from the 
test results given in table 5. 
It is seen from figures 27 and 28 that there was no significant linear or exponential 
correlation between the 0.04% bend offset stress and the pitch to bar diameter ratio 
because the correlation coefficient R obtained in both cases were rather low (0.3&0.29). 
49 
 
Thus it may be concluded that the pitch to bar diameter ratio has no significant influence 
on the 0.04% offset bend stress. 
 
Table 5: 0.04% Offset bend stress value Vs Pitch/Diameter ratio 
Nominal 
Diameter(mm) 
Pitch(mm) 
Pitch/Diameter 
Ratio 
0.04% offset bend 
stress (N/mm
2
) 
10 
12 
16 
20 
25 
32 
 
110 
120 
203 
217 
235 
345 
 
 
11 
10 
13 
11 
09 
11 
 
665 
804 
790 
748 
845 
825 
 
 
 
 
Figure 27:  0.04% Off set bend stress Vs Pitch/Bar Diameter Ratio – Linear Relationship 
 
       
y = -18.172x + 975.67 
R² = 0.0959 
0 
100 
200 
300 
400 
500 
600 
700 
800 
900 
0 5 10 15 
0
.0
4
%
o
ff
se
t 
b
e
n
d
 s
tr
e
ss
(N
/m
m
2
) 
Pitch/Diameter Ratio 
0.04% offset bend stressVs Pitch/Diameter Ratio 
50 
 
 
Figure 28: .0.04% offset bend stress Vs Pitch/Bar Diameter Ratio- Exponential Relationship 
 
 
 
 
 
 
 
 
 
 
 
 
 
y = 995.05e-0.023x 
R² = 0.0854 
0 
100 
200 
300 
400 
500 
600 
700 
800 
900 
0 5 10 15 
0
.0
4
%
o
ff
se
t 
b
e
n
d
 s
tr
e
ss
(N
/m
m
2
) 
Pitch/Diameter Ratio 
0.04% offset bend stressVs Pitch/ Bar Diameter Ratio 
51 
 
                                                    CHAPTER 05 
5.0 Conclusion 
Three point bend testing and tensile testing were performed on Cold Twisted reinforced 
bars and Thermo Mechanical Treated bars of 10mm diameter to 32mm diameter. The 
following were concluded. 
1. Using the experimental data obtained for reinforcement steel bars subjected to 
tensile testing  shown, that the 0.2% yield (proof yield) stress can be measured 
from 0.04% offset yield stress measured from three point bend load displacement 
(Position) curves. 
2. Result of three point bend testing and tensile testing show different relationships 
for different bar diameters, which are exponentially related. 
3. Extent of work hardening measured by means of pitch value of rib geometry test 
does not show a clear relationship with 0.2% proof stress in tensile testing for 
different bar diameters. 
Although the three points bend test results, show good relations with tensile yield stress 
measured, ductility and durability tests are also very important for compliance of the test 
result in accordance to the design guild lines.   
 
 
 
 
 
 
 
52 
 
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