A Review of mathematical models for prediction of Stress-strain and moment –curvature behaviour in concrete

In this paper, a mathematical model for predicting the stress –strain and moment curvature relations in concrete is developed. A good number of empirical equations were proposed to represent stress-strain behaviour of conventional concrete. Most of the equations can be used for the ascending portion of the curve only. In 1997 Mansur et al. have adopted Carriera and Chu (1985) model, which was based on the model proposed by Popovics (1973). As such, model proposed by Mansur et al includes both ascending and descending portions of the stress-strain curve for the confined concrete with introduction of two constants for the descending portion of the curve. Several researchers proposed various empirical equations for stress-strain behaviour as briefly reported in the previous chapter. An attempt has been made in this study to develop mathematical models for concrete in unconfined state. These analytical equations can be applied to any concrete with slight modifications. These models are developed to validate the experimental values obtained.


Introduction
Graph obtained by drawing a curve for the values of stresses and strains obtained during testing a material specimen of materials is called a stress -strain curve. By testing cylinders of standard size made with concrete, under uni-axial compression values of stresses and strains are obtained and the stress-strain curves are plotted. Even though the stress strain relation for cement paste and aggregate when tested individually is practically linear, it is observed from the stress-strain plots of concrete that, no portion of the curves is in the form of a straight line. In concrete the rate of increase of stress is less than that of increase in strain because of the formation of micro cracks, between the interfaces of the aggregate and the cement paste. Thus the stress strain curve is not linear. In conventional concrete the value of stress is maximum corresponding to a strain of about 0.002 and further goes on decreasing with the increasing strain, giving a dropping curve till it terminates at ultimate crushing strain.

Analytical Stress-strain equations
Number of empirical equations for stress-strain curve has been proposed for conventional concrete. Early works done by Hognestad and Desayi and Krishnan and proposed a basic model for stress-strain of ordinary concrete. Later Saenz has proposed model duly overcoming the limitation in the model of Desayi and Krishnan. Carriera and Chu [7] provided an extension to the empirical equation proposed by Popovics. Further, Loove improves the early work by Carriera and Chu who proposed a model that can be validate experimental values. Collins et al. also extended the work by Thorenfeldt et al. to examine the relation between compressive stresses at any strain to peak stress. Stressstrain equations proposed by these authors are summarized in the following sections.

Hognestad Model
Stress-strain relation for ordinary concrete upto ascending portion of stress-strain curve defined by  Model is derived from Saenz's original equation is in simple form such that closed-form integration can be evaluated to calculate the stress-block parameters. Due to simplicity in model formulation has encouraged many researchers to use it as general stress-strain model for concrete.

Modified saenz model
Desayi and Krishnan has proposed model for ascending portion of stress-strain curve only. In view of this limitation, Saenz proposed a model considering both ascending and descending portion of stress-strain curve.

Mathematical model for Concrete
In the present investigation only ascending portion of curve is considered. Out of existing, Modified Saenz's model is selected which seem to be valid for ascending portion of stress strain curve. The proposed equation of the curve is in the form of Where Y is the normalized stress ( ) and X is the normalized strain ( ). A, B, C are the constants. Further, equation for non-dimensional stress-strain curve can be written in the following form The constants 1 , 1 , 1 are determined from the boundary conditions of the non-dimensional stressstrain curve. The boundary conditions are as follows.
At the origin, =0 and =0, Slope of the stress-strain curve are evaluated using following equations.
Finally, substituting the above constants, theoretical equation for the stress-strain curve is obtained for concrete. Theoretical stresses are calculated for corresponding strains using developed equations. These theoretical stresses are compared with experimental results of Normal Concrete. Boundary conditions for ascending and descending portions of stress-strain curves are, i. At the origin the ratio of stresses and strains are zero i.e. at origin (Є / Є0) = 0, (σ / σ0 ) = 0 Є0-strain at peak stress, σ0peak stress ii. The strain ratio and stress ratio at the peak of the nondimensional stress-strain curve is unity.
i.e. at (Є / Є0) = 1, (σ / σ0 ) = 1 iii. The slope of non-dimensional stress-strain curve at the peak is At 85% stress ratio the corresponding values of strain ratio is 1.3. I.e at (σ/σ0 ) = 0.85 (Є/Є0) = 1.3 Where σ0corresponds to peak stress and Є0 -corresponds to strain at peak stress σ and Є corresponds to stress and strain values at any other point Boundary conditions i and ii are for determining the constants in the ascending portion of the normalized stress-strain curve and ii, iii and iv are for determining the constants in the descending portion of the curve.
Using the boundary conditions in the non-dimensional stress-strain curves, constants for different SCC mixes are determined and from that the equations are developed.

Stress Block Parameter for Normal Concrete
Stress-block parameters are essential for the analysis and design of structured members. Using these parameters, flexure capacity of beam can be determined.
If assumed stress-strain model is correct, more reliability in strength estimate and deformation behaviour of concrete structural members.   It is in the standard form ∫

Theoretical moments and curvatures
The experimental results of moments have been analyzed by developing procedures for obtaining the complete theoretical moment-curvature diagrams. The models proposed for stress-strain behaviour of concrete mixes are used as the basis for prediction of the analytical behaviour of moment-curvature and in deriving the expressions of the resisting moments and curvatures. For obtaining the complete moment curvature relationship for any cross-section, discrete values of concrete strains (Є) were selected such that even distribution of points on the plot, both before and after maximum was obtained. The procedure used in the computation is given below. i) Calculation of neutral axis depth for given values of concrete strains (Є) ii) Calculation of moment carrying capacities (M) iii) Calculation of theoretical moment curvature values Moment of resistance is given by Mt=CU*Z Substituting fcu=fck Mt= fck bXu(d-XU) Also moment of resistance Mt is given by Mt=ktfckbd 2 Kt=Mt/bd 2 Where kt is moment resistance factor kt= ′Xu/d(1-XU/d) using partial safety factor 1.5 kt= Xu/d(1-XU/d) Theoretical moment is obtained by Mt=ktfckbd 2 The resistance factor Kt for each grade calculated using developed stress-block parameters As mentioned earlier, a final phase of experimental was undertaken to validate the stress block and design parameters which are developed.

Conclusions
After determining the stress-strain behaviour of concrete, empirical equations were developed based on the relevant simplified models proposed by (1) Derived