**Publication: Scientific Bulletin. Series, Mathematical Modeling in Civil Engineering***Author: Labdaoui, Kamel*

*Date published: June 1, 2010*

1. INTRODUCTION

Although laboratory tests and on site tests provide relevant information, they are time-consumers and are usually very expensive (materials, instrumentation, equipment, labor, etc.). The development of an analytical modeling tool is therefore a more economic approach. This tool allows us to represent the behavior of reinforced concrete elements at the section level "momentcurvature" or at the element level "force-displacement" and overpass the difficulties of the assessment described above. The modeling tool should accurately describe the behavior of the reinforced concrete elements and structures in service limit state and ultimate limit state.

The program elaborated during this study allowed us to draw the moment-curvature diagram for different characteristic parameters.

Subsequently we propose to establish analytically the moment-curvature relationship of reinforced concrete elements subjected to eccentric compression by studying the reinforced concrete elements in rectangular or T sections. The results are compared with experimental data given by literature. A brief discussion is then made on possible maximum deviations of results, due to the basic assumptions accepted.

The basic assumptions used for the constitutive laws are:

a) The response of the elements at bending can be predicted using the laws of behavior, based on the conditions of equilibrium and compatibility, as defined in the following four basic assumptions:

1 . The stresses in the concrete and reinforcement are deduced from their stress-strain relationships;

2. The equilibrium of the forces developed within the section is permanently observed. So, the sum of internal forces must balance the external efforts applied;

3. The plane sections remain plane (sections parallel to the axis of bending) the concrete strain varies linearly along the height of the section;

4. The strain in the concrete and reinforcement are identical at the same level of the section: changes of strain in the adherent reinforcement are equal to changes of strain in the adjacent concrete.

b) The position of the neutral axis over the length of the element: we will assume that along the length of the element, the neutral axes of cross sections are located at a constant distance from the most compressed fiber. In this way, the calculated curve is the average one.

It is assumed that the rupture of the element occurs when the specific strain of compressed concrete reaches this value: ε^sub bc^ = 3,5 %o or when the specific strain of tense reinforcement reaches the value ε^sub a^ = 1 0 %o.

2. DESCRIPTION OF A SOFTWARE USED FOR AUTOMATIC CALCULATION OF THE MOMENT- CURVATURE RELATIONSHIP

The behavior at the ultimate limit state of the cantilevered walls subjected to symmetrical loads is normally controlled by the zone where the bending moment is maximum, i.e. the based section.

If the diagram of axial strain is linear (hypothesis 3), only two parameters are needed to define each layer (for a position related to the center of gravity y ): curvature F and the depth of neutral axis (c ), which is the distance between the extreme compressed fibers and the point of zero-strain (Fig. 1).

The model fixes the parameters (cet(φ)) and calculates the average axial strain on each layer (hypotheses 3 and 4) and also the stress and strain associated with them (hypothesis 1).

Numerical integration of the section allows us to calculate the axial strain (N) and the moment (M) acting on the section.

At this stage, if the calculated axial strain does not match the proper value, the depth of the neutral axis is changed and the calculation is remade (hypothesis 2). In short, the curvature and depth of the neutral axis are modified in order to balance the efforts on the section (normal effort N ,bending moment M and construct the moment curvature (M - [straight phi] )). Figure 2 shows the iteration used for the discretization done in several layers parallelly oriented to the neutral axis of the section, while Figure 2 describes the approach used.

This double iterative process, over the neutral axis and the curvature is completed in order to construct the curve M - [straight phi] (moment / curvature) of the cracked section.

It also allows us to follow the evolution of different parameters during the increase of the bending moment, as: the strain and stress in the extreme fibers of the concrete and in the reinforcement bars.

-To carry out these operations, which require a large amount of numerical calculations, we developed an algorithm and software written in Visual Basic 6.

-For the analysis we considered an I section, double armed on both ends. This section contains a percentage of reinforcement between 0.2% and 0.7%.

-For the bending moment we considered a dimensionless magnitude of bending moment:

-For the axial force N the axial force intensity was considered:

-For the table size we considered bh / b ratio.

We calculated the relative corresponding curve.

The entry data are:

- The traction stress - Rt and the compression stress of concrete- Rc

- The characteristics of the reinforcement.

- The geometrical data of the section b, h,h^sub 0^, h^sub h^,b^sub h^,µ = A^sub a^ / b^sub h^.h^sub h^, µ' = A'^sub h^ / b^sub h^.h^sub h^,b^sub h^ / b

- Number of points k used for calculating the moments and curves.

- Our software applies to any section T, I or rectangular.

- This software applies also to any mechanical characteristic of the section.

- This program is based on informatics loops; these loops allow us to directly find the correct position of the neutral axis with a maximum error of calculation equal to 0.1%, in order to draw directly the moment/curvature diagram.

PARAMETRIC STUDY REGARDING THE INFLUENCE OF DIFFERENT PARAMETERS ON THE BEHAVIOR LAW "MOMENT- CURVATURE"

To understand the functioning of the model, and particularly the monitoring of plastic strain, the following example is presented in details. The analyzed section is illustrated in Figure 2.5 and the main properties of these materials are summarized in Table 1.

An analysis has been conducted to determine the optimal number of layers in section and the number of calculation points (increments of curvature) to be used for obtaining accurate results while maintaining a reasonable amount of time for calculation. This interpretation, which obviously depends on the type of section, appears to cap for an average number of lOOlayers, and for 100 calculation points.

The developed software allows us to determine the diagram moment / curvature for all values of the characteristic parameters in the accepted assumptions. We have studied the influence of various parameters on the moment-curvature relationship.

* Percentage of tense reinforcement (µ = 0.2% to 0.7%), this percentage refers to the reinforcement section at each extremity of the wall in the total section.

* Table size (in case of I sections, bh / b = 1 to 10).

* Reinforcement quality Fe500.

* The axial force ratio "n": 0.2 <= ? <= 0.5

The concrete has been subject to a force mark FC28 = 25Mpa. The stress to eccentric compression introduced in the calculation has been considered at its normal value.

We have taken into account two types of steel for reinforcement: FE400 and Fe500. Their characteristic curves are shown in Figure 4. The limits of the reinforcement-flowing which has been considered, are the result of an average number of tests.

RESULTS

The results are shown in figure 5 to 8 which show the quantitative results of the influence of parameters taken into account to the moment-curvature relationship for the elements.

i. The increased percentage of tense reinforcement leads to a reduction of plastic strain. This is more obvious in cases of Fe500 reinforcement,

ii. As the percentage of tense reinforcement increases, we find a great difference between the moment corresponding to the reinforcement-flowing and the moment corresponding to the rupture of the reinforcements,

iii. Also, if we use high stress reinforcements like Fe500 there will be developing large plastic deformations, but only if the percentage of reinforcement does not exceed 0.4%.

iv. The ratio of the axial force (n) plays a very important role to stabilize the shape of the curve moment - curvature, the ruptured section obtained by the crushing of concrete if you use a relative ratio of the axial force equal to 0.4 per unit of frame on both ends equal to 0.2%.

v. We have reviewed the charts of moment-bending elements bent with T-section with different sizes of length of the table with frames of type Fe500 and a percentage of capacity ranging between 0.2% and 0.7%).

vi. The developed program can also estimate how or in what way some of the assumptions adopted can be of influence,

vii. For larger values of the axial force, the plastic deformation decreases rapidly during the increased axial force.

3. VALIDATION OF THEORETICAL RESULTS

In order to evaluate the effectiveness of the program developed in the second chapter, we have conducted a comparative study with results obtained in experimental research conducted at the University of Bucharest. The moment-curvature diagrams resulting from our software have been integrated over the whole heights of the walls.

On this basis,we have calculated using the software described in this article , the curvature moments (1/p) corresponding to the reduced moments m which are used then as input to the calculation program (force-displacement).

An examination of Fig 9, 10 shows that in the diagrams "force-displacement" the calculations made correspond to a large extent to those determined experimentally [6]. We can also see that ,systematically, the maximal elastic displacements resulted from calculation are higher than those determined experimentally or, in other words, the elastic stiffness determined analytically is smaller than that determined experimentally. This is due to the failure of taking into account for calculation, the influence of concrete between the cracks on the surface of the deformation element. If we had increased the modulus of average elasticity of reinforcement, a much better correlation regarding the calculated diagram with the experimental one ,in the elastic zone could have been obtained.

N.b.: the force-displacement curves used in this comparison represent the envelope of the forcedisplacement curves under cyclic loading presented by the experimental tests [6]

4. CONCLUSIONS

The software describedin this article allows us to quantify the reserves of stress and to evaluate the plastic deformations. Using this program, we conducted a parametric study to show the influence of various parameters on the relationship which describes the laws of behavior in sections -"moment-curvature" and also at the level of elements -"force-displacement". Given the results, the main conclusions are as follows:

(i) The ratio of the axial load has a significant influence on the deformation mode.

(ii) The maximum displacement ductility decreases while the ratio of axial force increases.

(iii) The increase ratio of the axial force has a negative effect on the degradation of the stress.

(iv) The ratio of the axial force is an important factor in performance criteria such as safety of life and prevention of concrete-crushing.

(v) The increase in percentage of reinforcement leads to higher resistant moments in sections.

(vi) As regards the behavior of slender walls at the level of element -"forcedisplacement ", the parameter "slenderness" plays a very important role to quantify the size of ductility.

(vii) A comparative study between the results obtained using this program and the results obtained experimentally has shown a very good correlation between these two results, and it demonstrates the effectiveness of the assumptions and algorithms used in evaluating the behavior of these elements.

References:

[1]. CRAINIC L. Cercetari asupra comportarli ci calculului structuriilor de beton armât în domeniul plastic, doctorate thesis, UTCB 1974.

[2]. CRAINIC L. Reinforced concrete structures, Bucharest 2003.

[3]. CRAINIC L. Calcul post-élastique des structures, Bucharest 2000.

[4]. KSU_RC. Software "moment-curvature" and "shear force-displacement". http://www.ce.ksu.edu/faculty/esmaeily /KSU_RC.htm.

[5]. PASCU R. Béton armât, Bucharest 2000.

[6]. PAVEL M, SUGIMOTO K, SEKI M, PAVEL C and K. NAGANUMAN. Nonlinear finite element analysis of low reinforced walls.

[7]. Règles parasismiques algériennes RPA99version 2003.

[8]. TALLEB R., ELDJOUZI B . Dimensionnement des voiles en béton arme.

Author affiliation:

Kamel Labdaoui -Eng. (Technical University of Civil Engineering Bucharest), (Reinforced Concrete Department.), e-mail: labdaoui2006@yahoo.fr