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Research Article: Tawashi & AlAmoudi 68

showed that the most appropriate mathematical Deduction of the Constants for the Proposed
formula for the curve, in this case, is the Rational Stress-Strain Mathematical Formula Model
Function form, Equation 10. which is the general To deduce the constants of the proposed
form: equation for the stress-strain curve. Starting
+ from the initial conditions, which are:
= (10)
1+ + 2 First condition:
: the compressive strength in the concrete. The beginning of the curve we have and
: the strain in the concrete. equal to zero. That is: = 0,

a, b, c, d equation constants. = 0, therefore by substituting the values
The equation shown above and in its general into the equation 12. The result is shown in the

form does not achieve the conditions of the Equation 13:
model we are looking for, in terms of stress and + (0)
relative strain. Figure 8 below shows the 0 = 1+ (0)+ (0) 2 → = 0 (13)
experimental and the equation curves form.
Starting from the performance form of the SCC Second condition:
At the peak of the curve. The value = 1
expressed by the stress-strain curve. The
corresponds to the value of = 1, the
equation was reformulated of the closed model
result is shown in the Equation 14:
in its general form, using the values of relative
stress and strain. So that form of Equation 11. (1)
1 = → = 1 + + (14)
and Equation 12. becomes according to the 1+ (1)+ (1) 2
following: Third condition:
+ ( ) At the peak of the curve. The value = 1,
= 2 (11)
′ 1+ ( )+ ( ) = 1, the curve slope is equal to zero, so:

+ ( ) = 1,( )′ = 0 →d=1 (15)

= 1+ ( )+ ( ) 2 (12) Substituting the value into equation (14).
= : nominal relative stress. Equation 16 has been obtained:
′
= : nominal relative strain. = + 2 (16)
Among the available conditions, the value of

two constants = 0 and d=1 has been found,
and the remaining constants b and c have
ϭ
equation /16/. Thus, the formula model is as
1.0
below in Equation 17:
0.8 ( )
= (17)
1+( −2)( )+( ) 2
0.6
To find the value of the constant in the previous
formula, the analytical process by substituting
0.4 the values of the coordinates = , =
, from the experimental curve and
SCC-HRW-3500-550
0.2 putting them in the previous Equation 17 has
SCC-HRW-3500-500
Rational Function-550 been used. Then obtaining the values of the
ξ=∆L/L
0.0 constant b.There are many solutions equal to
0 0.2 0.4 0.6 0.8 1 1.2 1.4
the number of experimental values throughout
Fig. 8. SCC and Rational Function Equation Stress-Strain the substitution of the coordinate values
Curves chosen from the ascending part of the curve.
There was no noticeable convergence with the

SJSI – 2023: VOLUME 1-1