Virtual Principle for Determination Initial Displacements of Reinforced Concrete and Prestressed Concrete (Overtop) Members

Theoretical approach with analytical and numerical procedure for determination initial displacement of a reinforced and prestressed concrete members, simple and cantilever beams, loaded by axial forces and bending moments is proposed. It is based on the principle of minimum potential energy with equality of internal and external forces. The equations for strain internal energy have been derived, including compressed and tensile concrete and reinforcement. The energy equations of the external forces with axial flexural displacement effects have been derived from the assumed sinusoidal curve. The tra-pezoid rule is applied to integrate the segment strain energy. The proposed method uses a non-linear stress-strain curve for the concrete and bilinear elastic-plastic relationship for reinforcement; equilibrium conditions at a sec-tional level to generate the strain energies along the beam. At the end of this article are shown three specific numerical examples with comparative, experimental (two tests) results with the excellent agreement and one calculation result with a great disagreement, by obtaining results of virtual principle method. With this method is avoiding the adoption of an unsure (EJ), as in the case of underestimating or overestimate initial flexural rigidity.


Introduction
The mechanics of continuous environments in dealing with the stress and strain distribution under the influence of external forces start from the assumption that the substance is continuous and therefore deformations are treated as conti-How to cite this paper: Balabušić, M. Some changes occur immediately after a change in stress condition and thus are called initial deformations. The equilibrium process of deformation of elastic bodies under linear interconnection between stress and deformation is the subject of the study of classical elastic theory and falls into reversible processes. A more complete theoretical treatment of the occurrence of increasing material deformation was carried out by the Austrian physicist L. Boltzman, who formulated a theory of subsequent elastic action and laid foundations for a linear theory of flow. The deflection [1] (Branson, D.F. and Shaikh, A.F., 1985) of the prestressed concrete beams is calculated with simple equations by modifying some of the existing methods. The comparison between the experimental and theoretical results shows good agreement. Many reinforced and prestressed concrete bridges throughout the world are either deteriorated or distressed to such a degree that structural strengthening of the bridge or reducing the allowable is necessary to extend the service life of the bridge.
While several methods are available in the literature for evaluation of deflections, this chapter concentrates on the effective moment of inertia method in [2] Building Code Requirements for Reinforced Concrete (ACI 318) and modifications introduced by ACI Committee. These reports include [3] ACI 435.2R, "Deflection of Reinforced Concrete Flexural Members", and [4] ACI 435.1R, "Deflection of Prestressed Concrete Members". The report replaces several reports of this committee (ACI 318) in order to reflect the more recent state of the art in design. The recommendations of current codes show that most of them underestimate or overestimate the initial flexural rigidity.
Simplified method of ACI. According to ACI 318 Building Code [2] instantaneous displacement of reinforced concrete beams, shall be computed with effective moment of inertia J c , given by 3  The European approach suggests calculation deflection based on curvature integration at many sections along the span. Tension stiffening [7] depends mainly on the bond characteristics of the reinforcement that, depend on many factors including the type of reinforcement elastic Modulus, and Poisson's ratio. Ignoring the tension stiffening significantly reduces the accuracy of the deflection prediction to 56%. The CEB-FIP Code 1990 (CEB-FIP 2004) [8] interpolates the effective curvature e κ between the curvature of the gross and the cracked sections g κ and cr κ , respectively. Both the computed and the measured deflections can be found in [9] Abdelrahman (1995). Eurocode 2 (ENV 1992-1-1) [10]. The deflection calculation according to Eurocode 2 is generally determined by double integration of the curve K along the length of the element l, The proposed procedure is not directly applicable to elements that are exposed to significant axial force. Two methods [11] are suggested for calculating the deflection of both reinforced and prestressed concrete flexural members. In the more rigorous approach, the curvature is calculated at a reasonable number of sections along the beam and then the deflection is calculated by numerical integrations. In the approximate method, the deflection is calculated twice assuming the whole member to be in the uncracked and fully cracked conditions. The following equation is used as a function of the uncracked and fully cracked deflections.

( )
β is a coefficient that takes account of bond properties of the bars: 0.5 for plan bars, 1.0 for high bond bars; 2 β is a coefficient of the duration of the loading or of repeated loading, 1.0 is for short-term loading, and 0.5 is for many cycles of repeated loading; Δ is displacement, Δ I displacement calculated on an uncracked section and Δ II the displacement calculated on the basis of a cracked section According to CEB [11], the modulus of elasticity of concrete, E c , may be estimated from compressive strength c f * , using relationships presented in the design codes.
The present paper has developed an analytical procedure (model) based on energy principles applied very frequently in the past (Pfluger, 1948, Timoshenko andGere, 1961) The most comprehensive theoretical review of the Energy method was given by [12] Bažant and Cedolin (2010). The method proposed in this paper covers the tensile zone of concrete. In most papers the equilibrium conditions and moment-curvature relation may be accurately fulfilled at the mid-span of the Open Journal of Civil Engineering beam only. This paper includes the integration performed in all the segments along the beam span. By this paper, we intend to affirm the variational principle, the principle of the minimum of potential energy, that is, the principle of virtual displacements, for determining the deformation of elements of concrete structures.
The energy potential of the internal stresses and strains along the element is equated with the energy of the external forces for the required deformation. This

Relation between Stresses and Strains in Nonlinear Concrete Theory
When solving some tasks in the theory of reinforced concrete structures, [13] (Ghali A. and Favre R., 1994) such as deformations problems and theories, nonlinear relationships between stresses and concrete strains are also taken into ac- ( ) γ c partial safety coefficient for concrete (γ c = 1.50) and γ c = 1.30 for non-random calculating situation, α = 0.85 coefficient for taking the influence of longacting effects on compressive strength of concrete, f cd = f ck /γ c = β calculating compressive strength of concrete.

Steel for Reinforcing [10]
According to EC 2 for reinforced concrete structures, all types of steel that meet the EN10080 steel standards can be used. In compliance with this, the thermal coefficient α = 10 × 10 −6 C o and the modulus of elasticity E a = 200 kN/mm 2 are taken. Calculation diagram stress-strain of steel for reinforcing is a bilinear diagram ( Figure 1). For steels with a pronounced flow threshold, the best approximation is achieved by an under-angle branch (ideally elastic-plastic), bilinear diagram.

Stress and Strains Analysis of Prestressed Elements [10]
Pre-flow stress and strains (t = 0) govern stresses and forces with the index "o".
So the stress in concrete is In the case of adhesion between concrete and steel, the stress in the fiber at the distance y i from the center of gravity is (Equation (6)), Concrete strains are valid according to (6a) and pursuant to ε is the lower strain.

Steel for Prestressing ([10], EC 2) & [14]
The mechanical properties for prestressing of steel [10] are defined by the basic parameters of steel f p 0.1 k-characteristic value of the conventional stretching limit to which corresponds irreversible dilatation of 0.1%. uk ε : characteristic strain value at maximum stress. f tk : characteristic value at straining Modulus of elasticity is adopted with a value of 200 kN/mm 2 .
In compliance with the above, the bilinear idealized diagram shown in Figure  1 is defined. It is recommended to limit the strain of steel to 10‰, without explaining whether the tot expansion of cables and straining of the section in the tightened zone are considered. The calculation diagram was obtained by applying the partial coefficient for PN (γ = 1.15) for basic actions (Figure 1).
Cables that are strained at a later stage are considered to have high, while pre-straining cables normal ductility.

Virtual Energy Method
External forces acting on the body, execute action during its deformation. We will assume that this action is accumulated as deformation energy (deformation action, internal force action). If it is desired that no part of the defamation action is converted into kinetic energy and is lost in this way, then it is assumed that loads grow immensely slowly starting from zero. In this case, there will be a balance between external and internal forces at all times.
If the deformation of the body is entirely elastic, then the energy accumulated during the loading is recovered. The deformation action of the internal reactions of the body is obtained if in volume element we put together all contributions and then we integrate them by the volume of the body. Potential energy of the structure [12] is defined according to the principle of the minimum potential Open Journal of Civil Engineering energy (Bažant and Cedolin 2010).
The potential energy of the structure is defined by the given expressions (7).
Of all possible internal forces that stand in equilibrium with the given external forces, the real forces are those for which the potential Π i δ has a minimum value.
In which the expression is the change in the free energy (stain internal energy) of the system.
The equalizing the change of the free energy of the system (strain energy) (8a) with (8) potential of the external forces (volume and surface integral), displacement of the structural element is determined.

The Strain Energy of the RC Beam Subjected to Bending [15]
In accordance with the expression (9), the strain energy of compressed concrete along the cross-section is derived from the constitutive relation diagram, calculated by [15] Balabusic and Folic (2015) for concrete and steel ( Figure 2).
Strain c ε of compressed concrete exposed to bending is obtained from the equilibrium condition (see Equation (19, 19a, 19b). The strain energy at the ith section of compressed concrete after numerical integration along entire length of the beam by trapezoidal rule is finally given as,

Strain Energy Transmitted through Tensile and Compressive Reinforcement
The strain energy of tensile and compressive reinforcement [15] is given by The following notations have been used in expressions (8) The strain so ε stated in Equation (12) represents the maximum strain of the steel in central section of simple beam. The strain energy transmitted by tensile reinforcement [15] per section is: Per section is: Per volume integral The strain energy that is transmitted by compressive reinforcement is:

The Strain Energy of the Prestressed Concrete (PSC) Beams [15]
The integration is performed directly per volume because the strain energy is or the cantilever beam

Strain Energy of Tensile Concrete of the RC Beam Subjected to Bending
The strain energy of tensile concrete per cross-section is derived in accordance with the constitutive relations Equation (8)

Equilibrium Conditions [10] EC 2
The strains in the sections have been defined by adopting the tensile reinforcement strain ε s and the N(ε c ) force from the equilibrium conditions (Equation (19), later). From the proposed tensile reinforcement strain ε s and responsible force N(ε c ), it is necessary to define the compressive and tensile concrete strains (ε b , ε ct ) of the beam's cross-sections. (Figure 2) The dimensionless values (Equation (17) The value e 0 stated in Equation (18)   Equation (19) and (19a) serves to determine the strains of compressed and tensile concrete , c ct ε ε , and neutral axis for the adopted strain of tensile reinforcement ε s as an independent variable.

Energy Potential of External Forces
According to [12] Bazant Z.P. and Cedolin L. the value of the integral is ( Figure   3), Energy of external forces is Uniform load expressed as a function of bending moment in cross-section, Therefore after two partial integrations, for boundary conditions In order to determine the deformation (displacement) of a structural element (Simple and Cantilever Beams), it is necessary to adopt a deformation line, where v is usually the maximum displacement, then the energy of external forces can be expressed (Figure 3).
For CANTILEVER BEAM applies: Whereby are uniform load, P iy concentrated forces, M iz concentrated moments at element (rod). N pk : prestressing force in cross-section.
M pk : moment from the prestressing force in the element. v: displacement of the deflection of a structural element.
EXAMPLE NUMBER 1, D. Jeftić. 1979. [17], Prestressed beam, Comparison experimental results with virtual principle method In Figure 4, the cross-section and static diagram of the prestressed beam girder (bracket) are shown, which was examined by Prof. D, Jevtić [17].
The beam was prestressed on the track, and the release was performed 8 days after the concrete was embedded. At the period of 40 days, the beam has been loaded with its own weight g = 0.295 kN/mL and force P = 2.07 kN at the free end of the console (cantilever), which amounted to 42% of fracture force P u . Concrete strength after 28 days f ck = 55.0 MPa. The force of the previous stress is N k = 93.60 kN.
The cross-sectional area of the concrete is A c = 118 cm 2 , moment of inertia ideals. It is a cross section J ci = 1656 cm 4 .
The task of this numerical example is to determine the initial displacement of the end of the prestressed beam by the energy method and compare it with the experimental results.  In Table 1 Figure 6).
The first yield displacement, Δy, corresponds to the intersection of the tangents to the load-displacement.
The load-defection curves can be classified into three distinct zones; the first zone is the initial part of the curve up to the cracking point, the post-cracking zone, continued up to the yielding point, and the post-yield zone, up the failure. The task of this example is to determine displacement in the middle of the reinforced concrete simple beam when steel strain ε s is at the cracking point and on the post-yield zone, up the failure. It will be compared and checked with experimental results of this paper [15] (Figure 7). a.1) The cracking point for steel strain ε s = 3.60‰ and on the post-yield zone, up to failure ε s = 14.40‰, ε c1 = 2.0‰.   In accordance with (15) These results given by Principle of Virtual Load correspond with experimental results taken from the original article [14] (Figure 7).

Conclusions
The main objective of this paper is to present a theoretical and analytical model that determines the elastic and inelastic structural response, strains and strength with the finally initial displacement of the simple and cantilever RC & PCT beams. The proposed method defines clearly the initial displacements of reinforced and prestressed beams with calculation strains and strain energy along with the member. The effective flexural rigidity proposed in this paper covers the tensile zone of concrete, which produces a more realistic estimation of curve section and element deformation. The presented method with calculation strain and stresses along the beam including axial forces and bending moments, avoid calculation the member stiffness with defining approximate value the modulus of elasticity. At first and third example is noticed significantly agreement, experimental and calculated results by virtual principle method, on PSC and RC beams. Also calculated results of PSC girder as simple beam have a great disagreement, with results obtained by virtual principle method. The cause is in stiffness EJ which is given as constant, but in fact, it is a complex function smaller than adopted flexural rigidity and displacements calculated by the present virtual method have a bigger value.