SHORT TERM DEFLECTION OF ORDINARY, PARTIALLY PRESTRESSED AND GFRP BARS REINFORCED CONCRETE BEAMS

The behavior of structural concrete beams is studied under short-term loading. A computer program developed originally by Oukaili to evaluate curvature is modified to evaluate the deflection for flexural structural concrete members. The program deals with actual stress-strain relationships of concrete and steel. The analysis is based on requirements of equilibrium and compatibility of strain in concrete and reinforcement. The proposed model is used in conjunction with the step by step analysis for small loading increments that allows the determination of the history of strain and stress in concrete with prestressing steel or non-prestressing reinforcement only or prestressing and non-prestressing reinforcement together. The evaluation of curvatures for the structural member involves iterations for computing the strains vectors at each analysis step. Newmark's numerical integration is used to evaluate the deflection of the member depending on the curvature values. The stress-strain model that was proposed by (Korpenko et al. 1986) is used and compared with experimental data and other analytical models for each of concrete and steel. The comparison showed good agreement between the model used and the experimental data. This relationship is used in SECTION program and presented in this study. The analytical results for load-deflection diagram are compared with available experimental data. The comparison has shown good agreement.


INTRODUCTION
Deflection is defined according to (ACI 116, 2000) as a movement of a point on a structure or structural element, usually measured as a linear displacement or as succession of displacements transverse to a reference line or axis.There are many empirical and analytical methods for computing deflection of structural concrete members.In this paper an analytical method is modified to compute the deflection of a structural concrete member based on the materials stress-strain relationships, the curvature of the cross-section is determined according to these stress-strain relationships, and then integration is made for the cross-section curvatures along the beam to determine the deflection.It must be mentioned that this analytical method takes into consideration the effect of tension stiffening after cracking of concrete.This analytical model is compared to the available experimental data and hand calculations.

STRESS-STRAIN MODEL Stress-strain model for concrete
In 1986, Korpenko published a descriptive relationship for    diagrams, which became very famous because of its overcoming of major deficient problems as shown in Fig. (1).

Figure (1)
Stress-strain diagram for concrete in compression and tension (Korpenko et al., 1986) It is taken that c  be the stress, c  the relative strain, By solving the second degree formula, two roots are obtained from the formula; the larger root is taken for  (Korpenko et al., 1986) For high strength steel the same equations (5, 6, 7) are used but

Stress-strain model for FRP
The stress-strain diagram for FRP is assumed linearly elastic until failure stage as shown in (Fig. 3).

Comparison of the stress-strain relationships of the materials
The proposed stress-strain model is compared to other models by using available experimental data for concrete (LSC, NSC, HSC) and for steel with its types mild; high strength; and high strength for prestressing as shown in Figures from ( 4) to (12).

MODEL ASSUMPTIONS
The proposed model for calculating the deflection depends on the following assumptions:  Strain in the concrete and reinforcement is proportional to the distance from the neutral axis (plane cross sections remaining plane after bending). The tensile behavior of the FRP reinforcement is linearly elastic until failure stage.(Fig. 3). All stresses in concrete and steel are related to secant modulus of elasticity. Shear and torsion stresses are ignored.
 Perfect bond exists between concrete, steel reinforcement and FRP bars.
 Concrete is divided into a group of small cells with its number related to the required accuracy.Beside that, the longitudinal steel and FRP bars in the section are divided into a group of elements, with their numbers equal to the numbers of steel and FRP elements.Thus the structural element acts as a system of linear elements exposed to compression and axial tension. The beam is considered as simply supported with symmetrical shape along the vertical centerline of the beam.This includes the type of loading and the beam cross section. Newmark's numerical procedure is used to determine the deflection from the section curvatures along the beam. In the case of fully or partially prestressed concrete members, the positive value of deflection indicates downward displacement, and the negative sign indicates that the displacement is upward (i.e. camber). The component of the prestress along the beam axis is assumed constant, but the distance from the prestressing center of gravity to the global axis is taken according to the prestressing steel position in each section.

MOMENT-CURVATURE MODEL
A partially prestressed concrete member is taken as a case study for determining the moment-curvature relation of structural concrete members.The concrete is distinguished by the subscript (c), prestressed steel by (ps), non-prestressed steel by (s), and (f) for FRP.
The space that is occupied by the section is assigned by the symbol  .In this space, there are areas occupied by concrete, prestressed steel, nonprestressed steel, and FRP which are symbolized as c A , ps A , A s , and f A respectively, as shown in (Fig. 13).
The external force, which is either normal force (prestressing force) or applied load, leads to change the general form of the element causing strain distribution at individual sections.
The strain energy per unit length is determined by the following formula: where  ‫ا‬  =normal strain and stress, respectively;  ‫ا‬  =shear strain and stress, respectively.
The equation that relates the stress to strain is as follows: ) where E = secant modulus of elasticity of the material, which takes into account the initial stresses and strains, and its calculation is based on    diagrams for the materials as shown in Figs. (14)  and (15).
Eq. ( 10) is substituted in Eq. ( 9).Shear stress and strain which appear in Eq. ( 9) are neglected (assumption No. 5).9) takes the following form: Because a linear strain distribution across the cross-section is assumed, so the third order vector will simulate the strain increment as in Eq. ( 13): ,  = axial strain vector, o  = axial strain, x K = curvature of the member longitudinal axis in OYZ plane, y K = curvature of the member longitudinal axis in OXZ plane, x , y = the distance from the center of gravity of concrete, prestressed, non- prestressed steel, and FRP in the cross-section, to the selected reference coordinates.
By the assumption that a plane cross section before bending remains plane after bending, the strain at any point can be expressed by the following relationship: Considering Eq. ( 13), then Eq. ( 12) becomes as: A matrix C is defined as: C= stiffness matrix relative to the selected reference coordinates.
To identify the loads related to the resultant axial strains, partial differentiation of the strain energy equation is taken as: where N = Longitudinal force; M x =bending moment in OYZ plane; M y = bending moment in OXZ plane; C 1 = general axial stiffness vector; C 2 = general flexural stiffness vector in OYZ plane; C 3 = general flexural stiffness vector in OXZ plane.
The general axial stiffness vector C 1 is determined by the following equation: The flexural stiffness vectors C 2 and C 3 are limited by Eq. ( 19) and Eq. ( 20).
The direct integration of stiffness vectors elements 1 C , 2 C , and 3 C is unknown (not explicit) theoretically because the secant modulus of elasticity depends on the value of the resulted strains, and the strain gradient which is not equal to zero.Accordingly, numerical methods should be used.Therefore, the cross section is covered by a mesh mostly with perpendicular lines.Average value of the stress (strain) is taken in each cell.So the infinite summation of the elements of section stiffness matrix is substituted by a finite summation, that its maximum value is equal to the number of developed mesh cells.
Based on this, the matrix elements take the following form: (26) where: 11 C = axial stiffness, which depends on the loading level and the geometrical properties of the cross section, 12 C =axial-flexural stiffness, which reflects the reciprocal influence of the longitudinal force with the bending moment in the direction of Y-axis, and it depends on the geometrical properties of the cross section, on the resultant stress-strain developments and on the location and direction of selected coordinates axes, 13 C =axial-flexural stiffness, which reflects the reciprocal influence of the longitudinal force with bending moment in the direction of X axis, 22 C =flexural stiffness in the direction of Y axis,

23
C =the stiffness, which reflects the reciprocal influence of bending in the direction of the axes X and Y, and depends on the geometrical properties of the section, on the strain level and on the location of selected coordinates, 33 C =flexural stiffness in the direction of X-axis, k =number of effective cells (strips) in concrete, x , = the distance from the center of gravity of longitudinal FRP bar i to the selected coordinates, j=number of longitudinal FRP bars in the concrete cross-section.
The secant modulus of elasticity E can be expressed as: where E =the initial modulus of elasticity for the material (concrete, steel, or fiber);  = elastic strain factor and it expresses the ratios between the elastic strains to total strains (as given before).
The system (17) can be rewritten in another form to become: where   F = loads vector;   ) ( C =stiffness matrix of the section;    = strains vector. The nonlinear system ( 28) is considered for determining the curvature of any cross-section in the concrete structural members.The elements of Eq. ( 28) change with changing the location of coordinates axis, it means that the axes rotate by an angle with certain value or at the time of moving the axes parallel to the existing axes (Fig. 13).

Evaluation of Strain Vector at Different Loading Cycles:
The solution of the nonlinear system (Eq.( 28)) with the third order, which is specified to determine the curvature of any cross-section in the structural concrete members, depends on the iterative methods of numerical analysis.There are several numerical methods for converting the nonlinear Eq. ( 28) to linear.One of these methods is called the direct iterative technique (Cook, 1981), with its iteration diagram for solving Eq. ( 28), takes the following form as shown in Fig. (16).

Figure (16)
Iterative technique for solving the nonlinear equation (Cook, 1981) In this method, full load is applied once at the first iteration and then a new stiffness matrix has to be formed in each iteration.Then the full system of Eq. ( 28 To solve Eq. ( 29), a correction to the stiffness matrix should be done at each iteration cycle with fixing the vector of external applied loads.

NEWMARK NUMERICAL INTEGRATION
In the case of a symmetrical simply supported beam, the following Eqs.( 30) can be used by considering the following symbols:  referring to the curvature EI M / ,  referring to the equivalent concentrated load (or curvature in the real beam), ) (i slope referring to the shear (or average slope in the real beam between the points i and 1  i ,  referring to deflection, subscript i referring to section number, c referring to section number in the center of the beam, and x  refers to segment length which is equal to the distance between two concentrated loads.
For a simply supported beam, there are no known values of  and slope in both ends; therefore these values can be substituted by zero. where , where for 2 nd degree parabolic curvature ( EI M / ).
Since the beam is symmetric the values of deflection for other half of the beam is determined according to the following formula: where p =1,…c-1, n = number of sections along the beam.

Ordinary reinforced concrete beams
Beam SB1-2 is tested by Thandamoorthy (1999).This beam had 150*300 mm section, 3m span length, the ultimate compressive strength for concrete (  GB1 control beam is tested by Alsayed, et al., (2002).The typical beam is 150*200 mm in cross-section.It is reinforced with three 10  mm steel bars for tension with an effective depth of 152 mm and one 6  mm for compression with depth of 46 mm and with s E equal to 200000     GFRP Benmokrane et al. (1996) built a series of 3300 mm long doubly reinforced concrete beams.
Series 1 is taken for comparison.The beam tested had two  19.1 mm GFRP bars in tension and two  6 mm GFRP bars in compression.The beam section was 200*300 mm.It had a clear span of 3000 mm and was loaded by two concentrated loads each at 500 mm from midspan.The ultimate

CONCLUSIONS
The following conclusions can be stated:  The analytical stress-strain model used in the SECTION program which was developed by Korpenko et al., (1986) gave very good agreement in comparison with the experimental data and other analytical models for each of concrete, mild steel, high strength steel, and prestressing steel. SECTION program presented in this study has shown to be capable of predicting the full history for the structural concrete beams under short-term loading. The SECTION program has the ability of dealing with structural beams with different values of partial prestressing ratio (PPR). The SECTION program has shown to be capable of predicting the deflection state for each of ordinary reinforced concrete, partially prestressed concrete, and bar fiber reinforced concrete beams. In the pre-crack stages, there is very good agreement between the theoretical results and the experimental data of deflection for all the three types of beams used. In the loading stages corresponding to 20% of the failure load, the average discrepancy of the analytical deflections values with respect to experimental values reached 18.3% for ordinary reinforced concrete, 4.9% for partially prestressed concrete beams, and 24.92 % for beams reinforced with fiber bars.One problem that seems to be common to all deflection prediction methods is that of predicting the cracking moment.This is primarily due to the difficulty in accurately predicting the modulus of rupture r f (standards tend to require the use of relatively low conservative values). In the loading stages corresponding to 50% from the failure load, the average discrepancy of the analytical deflections values with respect to experimental values reached 6.83% for ordinary reinforced concrete, 27.4% for partially prestressed concrete beams, and 8.73% for beams reinforced with fiber bars. In the loading stages corresponding to 70% of the failure load, the average discrepancy of the analytical deflections values with respect to experimental values reached 8.63% for ordinary reinforced concrete, 30.75% for partially prestressed concrete beams, and 7.45% for beams reinforced with fiber bars.

NOTATIONS:
The following symbols are used in this paper: c e 1 , c e 2 =Factors expressing the type of the concrete. =Concrete elastic modulus factor that expresses the ratio of elastic strains to the total strains.o  =Factor depending on stress level in the material.s  = Steel elastic modulus factor that express the ratio of elastic strains to the total strains. =Stress.


Concrete behavior in compression and in tension and steel behavior are assumed to follow Korpenko's model Fig. (1) and Fig. (2).

Figure ( 13 )
Figure (13) Cross-section of the structural element Figure (15) Variation of secant modulus of elasticity for concrete distance from the center of gravity of the concrete strip i to the selected coordinates, m=the number of non-prestressed longitudinal steel bars in the cross section, si A = the cross-sectional area of non-prestressed longitudinal steel bar i , of gravity of non-prtestressed longitudinal steel bar i to the selected coordinates; n =number of prestressed longitudinal steel bars in the cross-section; ) is solved in each iteration to evaluate the strain vector   .Initially, a value of     strains vector in previous and current iteration cycles at one loading level, respectively.
with an effective depth of 261 mm and 8 2 bars in compression with depth of 37 mm and with s E equal to 200000 MPa .The beam has been loaded in two points.Loading points have been located at 500 mm on either side of midspan.Momentcurvature and load-deflection diagrams of this beam are shown in Figs.(17 and 18).

Figure ( 17 )
Figure (17) Theoretical moment-curvature diagram Figure (18) Load-deflection diagram for Thandamoorthy beam(Thandamoorthy, 1999) simply supported over a span of 2050 mm and loaded by two concentrated loads, each placed at 100 mm from the beam centerline.The ultimate compressive strength for concrete is ( -curvature and load-deflection diagrams of this beam are shown in Figs.(19 and 20).

Figure ( 25 )
Figure (25) Theoretical moment-curvature diagram Figure (26) Load-deflection diagram for Benmokrane et al. beam (Benmokrane et al., 1996) Beam F3 tested by Pecce et al., (1998) has 500*185 mm in cross section, reinforced with seven  12.7 mm GFRP bars in tension at a depth of 145 mm and two  12.7 mm GFRP bars in compression at a depth of 40 mm.The clear span of the beam is 3400 mm, and it is loaded by two concentrated loads at 1200 mm from the simple supports.The ultimate compressive strength for concrete is ( MPa f c 30 '  ).The theoretical moment-curvature and load-deflection diagrams are shown in Figs.(27 and 28).

Figure ( 27 )
Figure (27) Theoretical moment-curvature diagram Figure (28) Load-deflection diagram for Pecce et al. beam (Pecceet al., 1998) Factors expressing the type of the steel.E = secant modulus of elasticity of the material.

cE=
Modulus of elasticity of concrete.

sE=
Modulus of elasticity of reinforcing steel.

K
=Curvature about x-axis.y K = Curvature about y-axis.LSC =Low Strength Concrete.xM =Bending moment in OXZ level.


=FRP ultimate strain.s  =Strain in steel. c
level of elastic concrete stress.fu  =FRP ultimate stress.s  =Stress in steel.s  ~= Steel stress level.
level of elastic steel stress. = Space occupied by the section.REFERENCES:  ACI Committee 116, "Cement and Concrete Terminology", American Concrete Institute, March 1, 2000.