Nonlinear Behavior of Bonded and Unbonded Two-Way Post-Tensioned Slabs Pre-Strengthened with CFRP Laminates

: In this study, hybrid nonlinear ﬁnite element models (FEM) were developed to examine the ﬂexural performance and the ultimate load capacity of bonded and unbonded two-way reinforced concrete post-tensioned (PT) slabs that were pre-strengthened with external carbon-ﬁber reinforcement polymer (CFRP) laminates. Full 3D simulations, using ANSYS models, have been created for ﬁve different slab samples that were selected from a previously available experimental study. The model results were assessed to enable further numerical analysis. The result calibration included measurements of ﬁrst crack loads, ultimate loads, deﬂections, strains in the extreme ﬁber of concrete, strains in CFRP laminates, and failure modes. The results proved a good correlation between FEM output and experimental ones. Based on this, the inﬂuencing parameters that affect plate stiffness, as well as the bending capacity of PT slabs, were examined by performing a detailed parametric study. The parameters included real-life load simulation, cable-to-CFRP strength contribution, and CFRP laminate location selection. The results demonstrated that strengthening using CFRP laminates have signiﬁcantly increased the ductility index of both bonded and unbonded PT concrete slabs by 62.18% and 59.87%, respectively. In addition, strip strengthening locations near supports are much more effective than in the middle of slabs. Additionally, the CFRP strengthening contribution is very considerable in slabs with low PT ratios.


Introduction
Prestressed concrete has been rapidly used in numerous fields of structural engineering, because of the enormous development in building techniques and the increasing need for long-span members [1].Pre-tensioned and post-tensioned concrete are two types of prestressed concrete, while the post-tension technic has several advantages [2].In pre-tension, the cables are tensioned between two fixed supports then the concrete is cast around the cables, which are cut from the supports once the concrete has reached its strength [3,4].Post-tensioned concrete is used with the formwork in the site, the concrete is cast around corrugated ducts, which are fixed to any required profile [5,6].The cables are usually unstressed in the ducts during the concrete casting.When the concrete has reached its strength, the cables are tensioned.The concrete is compressed during the stressing operation and the prestress is kept after the cables are anchored by bearing the end anchorage plates onto the concrete [7,8].After the cables have been anchored, the corrugated ducts including the cables are often filled with grout.In this method, the cables are bonded to the concrete and are more efficient in controlling cracks and providing ultimate strength [9].In some situations, cables are not grouted for reasons of economy and remain permanently unbonded [10].However, prestressing can be divided into fully prestressing, which focused on the complete removal of tensile stresses in members at normal service loads [10], and partial prestressing, which is a medium solution between fully prestressed concrete and ordinary RC [11].One of the problems that have faced PT applications is that the fully prestressed concrete members experienced low ductility [10,12,13].
On the other hand, because of changes in laws and provisions, as well as changes in the use of buildings, it is highly recommended that old structures be rehabilitated and strengthened [14,15].In the last years of the previous century, real progress in research took place using fiber-reinforced polymers (FRPs) in strengthening applications [16][17][18].FRPs are considered a better alternative than steel plates because of their multiple characteristics such as high tensile capacity, non-corrosive nature, and lightweight [19][20][21].The method of manufacturing pre-cured laminate is about the impregnation of fibers with adhesive, which is then pultruded and cured by the manufacturer.The most important result in manufacturing the pre-cured laminate can create laminates that are stronger and stiffer per unit volume than equivalent wet lay-up laminates [22].Typically, laminates used for concrete strengthening are unidirectional and have all fibers oriented in the longitudinal direction [23][24][25].The bond behavior between FRP and concrete was extremely hard and complicated because of their complex debonding failure mechanisms, and the material and geometrical nonlinearities of FRP-strengthened reinforced concrete slabs.Multiple factors affect the bond behavior between FRP and the concrete surface, such as the concrete grade [26], length of bonding, axial stiffness of the FRP plate, FRP/concrete ratio, stiffness, and strength of adhesive material [27][28][29].Lu et al. [30] studied bond-slip models for FRP sheets/plates bonded to concrete by using the results of 253 pull tests on simple FRP-to-concrete bonded joints.The results concluded that a more accurate model is required.Some researchers have studied strengthening PT concrete elements using FRP.Mohamedien et al. [31] investigated the behavior of hollow core pre-stressed slabs.Nine slabs with dimensions 5.0 m span, 1.2 m wide, and 0.2 m thickness.The slabs were strengthened using CFRP strips and sheets with various arrangements.It is observed that strengthening slabs using CFRP increased the flexural capacity by 40%.Chakrabarti et al. [32] experimentally investigated unbonded PT slabs using slab specimens repaired with different patterns of CFRP.They concluded that using CFRP in repairing slabs reduced the frequency at the high moment region and the crack width and the repaired PT slabs results showed preferable serviceability conditions.As a result of no de-bonding of CFRP sheets, the flexural capacity of the slabs obviously increased.Mohamed et al. [33] studied the flexural behavior of two-way P.T slabs pre-strengthened with external CFRP laminates.The laboratory testing involved four P.T simply supported concrete slabs.They concluded that when compared to control specimens, the P.T concrete slab with bonded tendons and strengthened with CFRP strips showed improvements in ductility, initial stiffness, and deflection of 62.18%, 58.2%, and 37.8%, respectively.
Due to advancements in computing technology and high-end computers' computational power, more research has been done on the behavior of concrete [34,35].In the field of PT slab strengthening applications, some researchers studied the effect of different FRP strengthening methods.Mahmood et al. and Abdulamier et al. [36,37] have presented a novel approach to simulate lab-strengthened slabs with CFRP (Carbon-Fiber-Reinforced-Polymers) laminates using ANSYS.It is concluded that the study gave good agreement with the experimental results.El Mesk and Harajli [38] examined the effect of strengthening unbonded post-tension (PT) one-way slabs using FRP laminates.The results indicated that strengthening using FRP sheets increases the flexural stiffness and capacity of slabs without a considerable reduction in ductility.The behavior of PT slabs strengthened with CFRP laminates has been modeled using 3D finite-element (FEM) models.

Research Significance
To date, from the literature survey, it is observed that there were no previous works that have studied the performance of two-way PT slabs pre-strengthened with external CFRP either experimentally or analytically except, lately, the experimental work of [33].The purpose of the early use of CFRP strips in slab fabrication was to identify the corresponding stiffness and capacity of PT concrete slabs having such combinations of bending resistance systems.Therefore, in this paper, the response of two-way post-tensioned reinforced concrete (RC) slabs pre-strengthened with CFRP strips is investigated by carrying out a numerical study using ANSYS software to evaluate the significance and sensitivity of the structural responses.The parameters considered in this investigation were the pre-compression ratio, bonded and unbonded tendons, ductility index, and real load simulations besides the effect of ratio and locations of external CFRP strips on the overall slab capacities.

Experimental Procedure
Experiments were performed by [33], on five half-scale simply supported PT slabs with overall length, width, and depth of 2370, 2370, and 150 mm, respectively, and the slabs were supported with a clear span of 1800 mm on each side, as shown Figure 1a.Both bonded and un-bonded post-tensioned techniques were used in specimens.On the other hand, 1 MPa and 2 MPa were used as the pre-compression ratio in specimens, number of tendons in X-direction and number of tendons in Y-direction.Each specimen was reinforced with four 10 mm steel bars in addition to the pre-stressing steel because this was the minimum amount of bonded reinforcement needed above columns in both the X and Y directions, as shown in Figure 1a.As shown in Table 1, four slabs (UN, US, BN, and BS1) had the same prestressing tendon layout, as shown Figure 1b,c of the unbonded sample UN, US, and the bonded sample BN, BS1, respectively.One had another tendon layout (BS2), which was double the number of tendons, as shown Figure 1d.Two of the tested slabs were tested without strengthening, as shown Figure 1e, as control slabs (one was bonded, BN, and the other was unbonded; UN), whereas the other three were strengthened using CFRP laminates using the same strengthening scheme (two were bonded, BS1 and BS2, and the other one was unbounded, US).According to [33] tendon handling, unbending was done by means of using coating with corrosion-inhibiting grease and encasing in plastic sheathing.On the other hand, the bonded tendons were placed inside a duct that was injected with cementitious grout after applying prestressing.In addition, the slab samples were designed as per international standards [39][40][41][42][43].The concrete mix was intended to have a cube compressive strength of 40 MPa.The mix contains 1024 kg/m 3 coarse aggregates with a nominal maximum size of 12.5 mm, and 447 kg/m 3 fine aggregates.Overall, 560 kg/m 3 CEM I 42.5N Portland cement was used.Sika ® ViscoCrete ® 5930, Sika company, Egypt, is a third-generation super plasticizer for concrete that was added to the mix to ensure workability.One hour before casting, concrete was patched from a ready-mix concrete plant.The same concrete mix was used to cast all the specimens.The cylinders were tested at the same time of testing the specimen in compression and tensile strength.The test-setup of specimens were mounted symmetrically relative to the center of the span with four concentrated point loads and separated by a distance equal to 1/6 of the span length to simulate as accurately as possible the moment diagram generated using the uniformly applied load, as seen in Figure 2. The load was applied statically at a rate of approximately 2 kN/min.All the data was obtained via a framework for data processing.Both strain gauge, LVDT and load cell data were processed within a PC computer and subsequently exported to Excel spreadsheet for analysis.Figure 3 and Table 2 show the load-deflection curves of all tested slabs and all other recorded results [33].

ANSYS Finite Element Model
The FEM models were developed using ANSYS-version 15.0 [44].To save computational time, only one-quarter of each slab was modeled.Direct mesh generation was used to control the model completely; however, it consumed considerably more time than the solid modeling technique.The behaviors of concrete, steel reinforcement, CFRP, and epoxy adhesive were modeled using appropriate element types and constituent models for materials, which were carefully modeled to simulate the experimental trend [33].

A. Concrete
The failure of concrete was defined by a multi-linear stress-strain response along with the William and Warnke model [45].Under uniaxial compression, the behavior of concrete was non-linear plastic obtained from the model of Hognestad [46], as presented in the following equations and shown in Figure 4a,b.
where fc is the compressive stress at any strain εc, f'c and ε 0 are the maximum compressive stress and its corresponding strain, εcu and fu are the ultimate strain and its corresponding stress which are assumed to be 0.003 and 0.85f'c, respectively; and Ec is the initial concrete Young's modulus taken as 4700 f c [47,48].where fc is the compressive stress at any strain εc, f'c and ε0 are the maximum compressive stress and its corresponding strain, εcu and fu are the ultimate strain and its corresponding stress which are assumed to be 0.003 and 0.85f'c, respectively; and Ec is the initial concrete Young's modulus taken as 4700 f' [47,48].Figure 4b shows the curve of concrete tensile stress-strain.In ANSYS, release 14.0 Documentation, 2015 [48], tension softening is the ability of cracked concrete to pick up tensile stresses between cracks.The rapture strength ft of concrete was adopted as 0.62 f' , whereas the open and closed shear crack coefficients were set to 0.20.From ANSYS Elements, SOLID65 was used to model concrete due to its capability to produce concrete crushing as well as tension rupture [44,47,48].

B. Steel Reinforcement and Plates
Figure 5 shows the behavior of the steel reinforcement that has an elastic-perfectly Figure 4b shows the curve of concrete tensile stress-strain.In ANSYS, release 14.0 Documentation, 2015 [48], tension softening is the ability of cracked concrete to pick up tensile stresses between cracks.The rapture strength f t of concrete was adopted as 0.62 √ f c , whereas the open and closed shear crack coefficients were set to 0.20.From ANSYS El-ements, SOLID65 was used to model concrete due to its capability to produce concrete crushing as well as tension rupture [44,47,48].

B. Steel Reinforcement and Plates
Figure 5 shows the behavior of the steel reinforcement that has an elastic-perfectly plastic non-linear response with a Poisson's ratio of 0.30.The yielding phenomenon defined by the von Mises criterion was employed to define steel performance that was assumed to have a linear elastic material with E and µ of 183 GPa and 0.30, respectively.From ANSYS elements, LINK180 is used to model steel reinforcement [49].The loading and supporting apparatus were modeled using the eight-node solid brick element SOLID185.This element has eight nodes with three degrees of freedom at each node, displacements in the global x, y, and z directions, and among its characteristic's plasticity, stress stiffening, and large deformations, as well as nonlinear properties such as multi-linear modeling.Precautions were taken to the nearby concrete to prevent impulsive local failure due to stress concentration under loading plates.Besides, the crushing capability of nearby concrete elements, SOLID 65, was removed [50].

C. CFRP Laminates and Epoxy Layers
The CFRP laminate material was considered linear elastic isotropic with a Poisson's ratio of 0.35.The behavior of the epoxy adhesive layer with the identical concrete cracking model was defined by a multi-linear elastoplastic diagram without considering the phenomenon of tension softening.The Poisson's ratio was taken as 0.37 for the epoxy adhesive.The schematic stress-strain curves assigned to the two materials are shown in Figures 6 and 7 [49].SOLID185 layered elements that represent in-plane layered thick shells were used for modeling both CFRP laminates and adhesive epoxy layers.It must be noted that the in-plane stiffness in such elements is the average of all individual layer stiffnesses [48,49].

C. CFRP Laminates and Epoxy Layers
The CFRP laminate material was considered linear elastic isotropic with a Poisson's ratio of 0.35.The behavior of the epoxy adhesive layer with the identical concrete cracking model was defined by a multi-linear elastoplastic diagram without considering the phenomenon of tension softening.The Poisson's ratio was taken as 0.37 for the epoxy adhesive.The schematic stress-strain curves assigned to the two materials are shown in Figures 6 and 7 [49].SOLID185 layered elements that represent in-plane layered thick shells were used for modeling both CFRP laminates and adhesive epoxy layers.It must be noted that the in-plane stiffness in such elements is the average of all individual layer stiffnesses [48,49].

D. Epoxy/Concrete and Epoxy/CFRP Interfaces
A continuum damage approach (CDM) was used to analyze epoxy-concrete interface debonding with one of the fracture mechanics modules.Delamination or fracture along an interface between different layers plays a basic role in limiting the ductility and the toughness of the multi-phase materials, such as laminated composite structure.This has motivated much research on the failure of the interfaces in which modeling was done by traditional fracture mechanics methods such as the nodal release technique [51,52].Additionally, the so-called cohesive zone material (CZM) technique, that is used in the current work, is aimed to directly introduce fracture mechanisms by adopting softening relationships between tractions and the separations, which in turn introduce a critical fracture energy that is also the energy required to break apart the interface surfaces.CZM modeling was implemented as given in Ref. [47], in which the discontinuity is technically avoided in Ansys analysis.The CZM model consists of a constitutive relation between the traction T acting on the interface and the corresponding interfacial separation δ (displacement jump across the interface).In ANSYS, the modeling of the CZM traction-separation can be simulated by considering contact elements with zero thickness and interface elements with finite thickness, such as touch components CONTA173, and TARGE170 and interface component INTER205, as given in Figures 8 and 9, respectively [47].

D. Epoxy/Concrete and Epoxy/CFRP Interfaces
A continuum damage approach (CDM) was used to analyze epoxy-concrete interface debonding with one of the fracture mechanics modules.Delamination or fracture along an interface between different layers plays a basic role in limiting the ductility and the toughness of the multi-phase materials, such as laminated composite structure.This has motivated much research on the failure of the interfaces in which modeling was done by traditional fracture mechanics methods such as the nodal release technique [51,52].Additionally, the so-called cohesive zone material (CZM) technique, that is used in the current work, is aimed to directly introduce fracture mechanisms by adopting softening relationships between tractions and the separations, which in turn introduce a critical fracture energy that is also the energy required to break apart the interface surfaces.CZM modeling was implemented as given in Ref. [47], in which the discontinuity is technically avoided in Ansys analysis.The CZM model consists of a constitutive relation between the traction T acting on the interface and the corresponding interfacial separation δ (displacement jump across the interface).In ANSYS, the modeling of the CZM traction-separation can be simulated by considering contact elements with zero thickness and interface elements with finite thickness, such as touch components CONTA173, and TARGE170 and interface component INTER205, as given in Figures 8 and 9, respectively [47].

E. CFRP Slippage
The bond-slip relationship adopted by [50] was used to simulate FRP-concrete interfacial behavior.Figure 10 shows the relationships that comprised interfacial stress (shear) and related gliding slip between the FRP laminates and concrete faces [48,50].The bond-slip relationship adopted by [50] was used to simulate FRP-concrete interfacial behavior.Figure 10 shows the relationships that comprised interfacial stress (shear) and related gliding slip between the FRP laminates and concrete faces [48,50].

F. Bonded and Unbonded Cable Modeling
LINK180 was used to model the unbonded cables of PT slabs.PT force is transferred to concrete by end anchorages, in which cables are linked to the slab at the end of anchorages.To prevent stress-concentration, steel plates were added at the location of loading.A similar approach was used to prevent local concrete failure as done here for portions of applied concentrated loads (refer to Section B; Steel Reinforcement and Plates).
Simulation of the bonded tendon through the ANSYS program elements library was like the interface system discussed before in contact surfaces for CFRP laminates and epoxy layers.Again, the contact pair manager method was selected to model the contact between the concrete and the tendon [37].Application terms of this method are defining two surfaces: one of them is the target and the other is the contact surface.The part of the concrete face contact cable is considered a rigid face and is modeled as the target face within this model (TARGE170).A contact, sliding, and the deformable surface is defined as the tendon surface, which is modeled by the contact surface (CONTA175).In this system, a zero-friction assumption was selected to model contact between the concrete surfaces and unbonded prestressing tendons.The model could not allow the unbonded tendons and concrete surfaces to penetrate each other, although each of them could displace compared to the other.Furthermore, the unbonded prestressing tendon retains its profile during deformation in both transversal and lateral directions (i.e., y-and z-directions, respectively) to ensure compatibility [50].

FEM Meshing
Figure 11 shows the mesh discretization used in the developed models for all types of slabs.Very fine mesh sizes were selected (12.5 mm) for the whole half-scale model [36,48].

F. Bonded and Unbonded Cable Modeling
LINK180 was used to model the unbonded cables of PT slabs.PT force is transferred to concrete by end anchorages, in which cables are linked to the slab at the end of anchorages.To prevent stress-concentration, steel plates were added at the location of loading.A similar approach was used to prevent local concrete failure as done here for portions of applied concentrated loads (refer to Section B; Steel Reinforcement and Plates).
Simulation of the bonded tendon through the ANSYS program elements library was like the interface system discussed before in contact surfaces for CFRP laminates and epoxy layers.Again, the contact pair manager method was selected to model the contact between the concrete and the tendon [37].Application terms of this method are defining two surfaces: one of them is the target and the other is the contact surface.The part of the concrete face contact cable is considered a rigid face and is modeled as the target face within this model (TARGE170).A contact, sliding, and the deformable surface is defined as the tendon surface, which is modeled by the contact surface (CONTA175).In this system, a zero-friction assumption was selected to model contact between the concrete surfaces and unbonded prestressing tendons.The model could not allow the unbonded tendons and concrete surfaces to penetrate each other, although each of them could displace compared to the other.Furthermore, the unbonded prestressing tendon retains its profile during deformation in both transversal and lateral directions (i.e., y-and z-directions, respectively) to ensure compatibility [50].

FEM Meshing
Figure 11 shows the mesh discretization used in the developed models for all types of slabs.Very fine mesh sizes were selected (12.5 mm) for the whole half-scale model [36,48].

Loading and Boundary Conditions
For symmetry, only half-sized slabs were modeled in which symmetrical constraints were applied at the boundaries of cut lines.Models were limited to simulating experimental works for only one-bay slabs.The four-point loading was applied, as seen in Figure 12.The FEM models were loaded exactly at the same positions located in the experimental work of [33], in which the loading and support patch dimensions were 200 mm × 200 mm × 12.5 mm and 200 mm × 200 mm × 350 mm, respectively.A 12.5 mm thick steel plate was added at the support location to prevent stress-concentrating issues.As stated earlier, Solid185 elements were used to model it, which offered a more balanced distribution of tension over the support field.Moreover, to facilitate the rotation of the slab, a single-line support was mounted under the centerline of the steel plates [53][54][55][56].

Stepped Analysis
A force-displacement control method is utilized for solving non-linear analysis associated with a gradual load increase of 1 kN load increment up to failure.Due to the complexity of the solution that consumes much time and due to the non-linear behavior of all elements, it was difficult to achieve solution convergence for the developed models.Therefore, the force convergence tolerance limit was increased from 0.005 (the default tolerance limit defined in ANSYS) to 0.10 for shortening run-time without any loss in solution accuracy.The Newton-Raphson equilibrium iterations were used to update the global stiffness matrix after the completion of each load increment.

Loading and Boundary Conditions
For symmetry, only half-sized slabs were modeled in which symmetrical constraints were applied at the boundaries of cut lines.Models were limited to simulating experimental works for only one-bay slabs.The four-point loading was applied, as seen in Figure 12.The FEM models were loaded exactly at the same positions located in the experimental work of [33], in which the loading and support patch dimensions were 200 mm × 200 mm × 12.5 mm and 200 mm × 200 mm × 350 mm, respectively.A 12.5 mm thick steel plate was added at the support location to prevent stress-concentrating issues.As stated earlier, Solid185 elements were used to model it, which offered a more balanced distribution of tension over the support field.Moreover, to facilitate the rotation of the slab, a single-line support was mounted under the centerline of the steel plates [53][54][55][56].

Loading and Boundary Conditions
For symmetry, only half-sized slabs were modeled in which symmetrical constraints were applied at the boundaries of cut lines.Models were limited to simulating experimental works for only one-bay slabs.The four-point loading was applied, as seen in Figure 12.The FEM models were loaded exactly at the same positions located in the experimental work of [33], in which the loading and support patch dimensions were 200 mm × 200 mm × 12.5 mm and 200 mm × 200 mm × 350 mm, respectively.A 12.5 mm thick steel plate was added at the support location to prevent stress-concentrating issues.As stated earlier, Solid185 elements were used to model it, which offered a more balanced distribution of tension over the support field.Moreover, to facilitate the rotation of the slab, a single-line support was mounted under the centerline of the steel plates [53][54][55][56].

Stepped Analysis
A force-displacement control method is utilized for solving non-linear analysis associated with a gradual load increase of 1 kN load increment up to failure.Due to the complexity of the solution that consumes much time and due to the non-linear behavior of all elements, it was difficult to achieve solution convergence for the developed models.Therefore, the force convergence tolerance limit was increased from 0.005 (the default tolerance limit defined in ANSYS) to 0.10 for shortening run-time without any loss in solution accuracy.The Newton-Raphson equilibrium iterations were used to update the global stiffness matrix after the completion of each load increment.

Stepped Analysis
A force-displacement control method is utilized for solving non-linear analysis associated with a gradual load increase of 1 kN load increment up to failure.Due to the complexity of the solution that consumes much time and due to the non-linear behavior of all elements, it was difficult to achieve solution convergence for the developed models.Therefore, the force convergence tolerance limit was increased from 0.005 (the default tolerance limit defined in ANSYS) to 0.10 for shortening run-time without any loss in solution accuracy.The Newton-Raphson equilibrium iterations were used to update the global stiffness matrix after the completion of each load increment.

Model Calibration and Validation
Numerical results obtained using FEM analysis for the two unbounded and three bonded slab specimens were compared with the test results produced in the experimental work of [33], for validation.To compare the model conditions with the test setup, a design strip was selected for the critical sagging and hogging areas of the slab in which the lines of zero shear in the x-direction (i.e., about the y -axis) are located.The design strip width was bounded at the column center line as shown in Figure 13, and the following initial values were input.

Model Calibration and Validation
Numerical results obtained using FEM analysis for the two unbounded and three bonded slab specimens were compared with the test results produced in the experimental work of [33], for validation.To compare the model conditions with the test setup, a design strip was selected for the critical sagging and hogging areas of the slab in which the lines of zero shear in the x-direction (i.e., about the y -axis) are located.The design strip width was bounded at the column center line as shown in Figure 13, and the following initial values were input.

(A) Comparison of Camber
Camber deflection due to prestressing force is calculated by the load balancing method as follows: where: P: prestressing force, E: modulus of concrete, e: eccentricity of cable from centerline, I: a gross moment of inertia, L: length of the slab.
A comparison between the cambers obtained from the test with the others obtained from the model is given in Table 3, which provides an acceptable margin.

(A) Comparison of Camber
Camber deflection due to prestressing force is calculated by the load balancing method as follows: where: P: prestressing force, E: modulus of concrete, e: eccentricity of cable from centerline, I: a gross moment of inertia, L: length of the slab.A comparison between the cambers obtained from the test with the others obtained from the model is given in Table 3, which provides an acceptable margin.

(B) Applying of Initial Strain of Tendons
In ANSYS, there is no direct method to impose initial strain on the link element (tendons).To overcome this, initial pre-strains were applied only at the first load step that simulates the measured values given in [33].Considering the compatibility between concrete and carrying tendons, the application of initial strains was imposed directly on concrete.

First Crack Monitoring
According to the experimental loading rate listed in Ref. [33], continuous measuring of loads, deflections, and crack monitoring were recorded during all test stages.The first flexural crack was observed visually at a certain load value then it was later verified from the recorded load-deflection curves at the point at which a remarkable change in the slope at nearly the same load level has occurred.P-∆) curves for the five slabs.Good agreement was obtained between the FEM model and experimentally obtained results [33].Table 4 and Figures 24-26 present the comparison details concerning the ultimate load, cracking load, and its corresponding deflection.The ratio between the experimental and FEM results of the first cracking load varied from 0.7 to 0.9, whereas the results of the ultimate load varied from 0.8 to 1.1 and the ultimate deflections varied from 0.8 to 1.It has to be noted that the cause of results ratio differences would be the differences between the experimental environment and modeling.On the other hand, the strain in the CFRP laminates varied between a minimum of 34% and a maximum of 38% of the specified manufactured rupture strain.Generally, it is ensured numerically that the use of CFRP strengthening laminates has significantly increased the ductility index of both bonded and unbonded PT concrete slabs by 62.18% and 59.87%, respectively.

Load Deflection Behavior
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(B) Applying of Initial Strain of Tendons
In ANSYS, there is no direct method to impose initial strain on the link element (tendons).To overcome this, initial pre-strains were applied only at the first load step that simulates the measured values given in [33].Considering the compatibility between concrete and carrying tendons, the application of initial strains was imposed directly on concrete.

First Crack Monitoring
According to the experimental loading rate listed in Ref. [33], continuous measuring of loads, deflections, and crack monitoring were recorded during all test stages.The first flexural crack was observed visually at a certain load value then it was later verified from the recorded load-deflection curves at the point at which a remarkable change in the slope at nearly the same load level has occurred.P-Δ) curves for the five slabs.Good agreement was obtained between the FEM model and experimentally obtained results [33].Table 4 and Figures 24-26 present the comparison details concerning the ultimate load, cracking load, and its corresponding deflection.The ratio between the experimental and FEM results of the first cracking load varied from 0.7 to 0.9, whereas the results of the ultimate load varied from 0.8 to 1.1 and the ultimate deflections varied from 0.8 to 1.It has to be noted that the cause of results ratio differences would be the differences between the experimental environment and modeling.On the other hand, the strain in the CFRP laminates varied between a minimum of 34% and a maximum of 38% of the specified manufactured rupture strain.Generally, it is ensured numerically that the use of CFRP strengthening laminates has significantly increased the ductility index of both bonded and unbonded PT concrete slabs by 62.18% and 59.87%, respectively.Further analysis reveals that the achieved ultimate loads from experimental results, EXP, are somewhat less than those obtained from FEM models.This is because of ignorance of the radial stresses exerted by the tendon on the concrete cover in the developed FEM models.The FEM-EXP difference between the cracking loads and ultimate deflections lies within the usual tolerance and could be attributed to the modeling deviations from real situations.Regarding deflection behavior, Figures 19-23 show that the FEM models of slabs had nearly the same trend as the tested specimens except at the end parts; this remark can be attributed to the technical problem that had occurred during the slab test in the laboratory that caused the test to be stopped [33].However, the differences of pre-yield stiffness are in the accepted range.

Concrete and CFRP Strain Behavior
At different loading levels, there is a good correlation between FEM results and experimental concrete and CFRP strains.Further analysis reveals that the achieved ultimate loads from experimental results, EXP, are somewhat less than those obtained from FEM models.This is because of ignorance of the radial stresses exerted by the tendon on the concrete cover in the developed FEM models.The FEM-EXP difference between the cracking loads and ultimate deflections lies within the usual tolerance and could be attributed to the modeling deviations from real situations.Regarding deflection behavior, Figures 19-23 show that the FEM models of slabs had nearly the same trend as the tested specimens except at the end parts; this remark can be attributed to the technical problem that had occurred during the slab test in the laboratory that caused the test to be stopped [33].However, the differences of pre-yield stiffness are in the accepted range.

Concrete and CFRP Strain Behavior
At different loading levels, there is a good correlation between FEM results and experimental concrete and CFRP strains.Further analysis reveals that the achieved ultimate loads from experimental results, EXP, are somewhat less than those obtained from FEM models.This is because of ignorance of the radial stresses exerted by the tendon on the concrete cover in the developed FEM models.The FEM-EXP difference between the cracking loads and ultimate deflections lies within the usual tolerance and could be attributed to the modeling deviations from real situations.Regarding deflection behavior, Figures 19-23 show that the FEM models of slabs had nearly the same trend as the tested specimens except at the end parts; this remark can be attributed to the technical problem that had occurred during the slab test in the laboratory that caused the test to be stopped [33].However, the differences of pre-yield stiffness are in the accepted range.

Concrete and CFRP Strain Behavior
At different loading levels, there is a good correlation between FEM results and experimental concrete and CFRP strains.

Failure Modes
The failure stage of tested slabs can be expected to be either of the following: a. Concrete crushing; when the strain of concrete reaches 0.002, b.Tendon cut; due to reaching its ultimate capacity (not occurring here), c.CFRP debonding; concrete cover or CFRP laminate separation at the surface plane.
The energy criteria or the stress-intensity-factor criterion are usually used in fracture analysis.The critical value of the magnitude of the stress and deformation fields characterizes the fracture toughness when the stress-intensity-factor criterion is applied.The von Mises failure criterion and the William and Warnke model are used by the multilinear isotropic material to define concrete failure.The research done by Kachlakev et al. [47] served as a foundation for calculating the shear transfer coefficients for open and closed cracks.When the shear transfer coefficient for the open crack fell below 0.2 and the coefficient for the open and close cracks were both set to 0.2, convergence issues appeared.The modulus of rupture was used as the basis for the uniaxial cracking stress.In this approach, the uniaxial crushing stress was derived from the uniaxial compressive strength.As recommended by earlier studies [47], it was entered as -1 to disable the concrete element's capacity for crushing.When the crushing capability was activated, convergence issues kept occurring.

Failure Modes
The failure stage of tested slabs can be expected to be either of the following: a. Concrete crushing; when the strain of concrete reaches 0.002, b.Tendon cut; due to reaching its ultimate capacity (not occurring here), c.CFRP debonding; concrete cover or CFRP laminate separation at the surface plane.
The energy criteria or the stress-intensity-factor criterion are usually used in fracture analysis.The critical value of the magnitude of the stress and deformation fields characterizes the fracture toughness when the stress-intensity-factor criterion is applied.The von Mises failure criterion and the William and Warnke model are used by the multilinear isotropic material to define concrete failure.The research done by Kachlakev et al. [47] served as a foundation for calculating the shear transfer coefficients for open and closed cracks.When the shear transfer coefficient for the open crack fell below 0.2 and the coefficient for the open and close cracks were both set to 0.2, convergence issues appeared.The modulus of rupture was used as the basis for the uniaxial cracking stress.In this approach, the uniaxial crushing stress was derived from the uniaxial compressive strength.As recommended by earlier studies [47], it was entered as -1 to disable the concrete element's capacity for crushing.When the crushing capability was activated, convergence issues kept occurring.

Failure Modes
The failure stage of tested slabs can be expected to be either of the following: a. Concrete crushing; when the strain of concrete reaches 0.002, b.
Tendon cut; due to reaching its ultimate capacity (not occurring here), c.
CFRP debonding; concrete cover or CFRP laminate separation at the surface plane.
The energy criteria or the stress-intensity-factor criterion are usually used in fracture analysis.The critical value of the magnitude of the stress and deformation fields characterizes the fracture toughness when the stress-intensity-factor criterion is applied.The von Mises failure criterion and the William and Warnke model are used by the multilinear isotropic material to define concrete failure.The research done by Kachlakev et al. [47] served as a foundation for calculating the shear transfer coefficients for open and closed cracks.When the shear transfer coefficient for the open crack fell below 0.2 and the coefficient for the open and close cracks were both set to 0.2, convergence issues appeared.The modulus of rupture was used as the basis for the uniaxial cracking stress.In this approach, the uniaxial crushing stress was derived from the uniaxial compressive strength.As recommended by earlier studies [47], it was entered as -1 to disable the concrete element's capacity for crushing.When the crushing capability was activated, convergence issues kept occurring.
Basically, the ANSYS program records concrete cracks and crushing at each applied load step.A circle outline in the plane of the crack represents cracking.When the Solid65 element achieves the ultimate strain of 0.002, according to the stress-strain curve of com-pressed concrete, a crushing limit is immediately applied at that element.An octahedron outline represents crushing.Each integration point can crack into up to three different planes.The first, second, and third cracks at an integration point are shown with a red circle outline, green circle outline, and blue circle outline, respectively, as shown in Figure 30 for bonded and unbonded slab failure development.It was observed that failure modes match exactly what has occurred in experiments [33].Finally, it is observed that the rigidities of concrete slabs after cracking in the finite element analysis are much lower than those of the test.Basically, the ANSYS program records concrete cracks and crushing at each applied load step.A circle outline in the plane of the crack represents cracking.When the Solid65 element achieves the ultimate strain of 0.002, according to the stress-strain curve of compressed concrete, a crushing limit is immediately applied at that element.An octahedron outline represents crushing.Each integration point can crack into up to three different planes.The first, second, and third cracks at an integration point are shown with a red circle outline, green circle outline, and blue circle outline, respectively, as shown in Figure 30 for bonded and unbonded slab failure development.It was observed that failure modes match exactly what has occurred in experiments [33].Finally, it is observed that the rigidities of concrete slabs after cracking in the finite element analysis are much lower than those of the test.

Parametric Study
As the FEM simulation closely matches the experimental results, the developed analytical model was used to examine the effect of changing some main parameters on the flexural performance of bonded PT slabs strengthened with CFRP laminates.Three parameters were studied: real loading simulation schemes, using different arrangements of CFRP laminates, and PT/CFRP strength contribution.

Real Load Simulation
All tested slabs in [33] were equipped with four equivalent concentrated point loads applied symmetrically relative to the center of the span and separated by a distance equal to 1/6 of the span length to simulate the actual distributed load in real cases.Similarly, the corresponding FEM models were given the same conditions of loading.To represent the effect of real load simulation as a uniformly distributed load, the equivalent values of concentrated loads are put directly as uniform loads.Figure 31 displays the simulation of the real delivery load; meanwhile, Figure 32 shows the comparison between both cases' results, which reflect the sensitivity of this simulation.

Parametric Study
As the FEM simulation closely matches the experimental results, the developed analytical model was used to examine the effect of changing some main parameters on the flexural performance of bonded PT slabs strengthened with CFRP laminates.Three parameters were studied: real loading simulation schemes, using different arrangements of CFRP laminates, and PT/CFRP strength contribution.

Real Load Simulation
All tested slabs in [33] were equipped with four equivalent concentrated point loads applied symmetrically relative to the center of the span and separated by a distance equal to 1/6 of the span length to simulate the actual distributed load in real cases.Similarly, the corresponding FEM models were given the same conditions of loading.To represent the effect of real load simulation as a uniformly distributed load, the equivalent values of concentrated loads are put directly as uniform loads.Figure 31 displays the simulation of the real delivery load; meanwhile, Figure 32 shows the comparison between both cases' results, which reflect the sensitivity of this simulation.

Different Arrangements of CFRP Laminates
To investigate the effect of the optimum CFRP laminate position selection, two different schemes of distribution were investigated.The first scheme of laminate attachment location was selected to be adjacent to columns only (column laminate), and the other was at the middle span only (field laminate), as shown in Figures 33 and 34.From the results, it was observed that the column laminate had a significant impact, such as delaying the first crack load and producing less deflection, as can be seen from Figures 35 and 36.

Different Arrangements of CFRP Laminates
To investigate the effect of the optimum CFRP laminate position selection, two different schemes of distribution were investigated.The first scheme of laminate attachment location was selected to be adjacent to columns only (column laminate), and the other was at the middle span only (field laminate), as shown in Figures 33 and 34.From the results, it was observed that the column laminate had a significant impact, such as delaying the first crack load and producing less deflection, as can be seen from Figures 35 and 36.

Different Arrangements of CFRP Laminates
To investigate the effect of the optimum CFRP laminate position selection, two different schemes of distribution were investigated.The first scheme of laminate attachment location was selected to be adjacent to columns only (column laminate), and the other was at the middle span only (field laminate), as shown in Figures 33 and 34.From the results, it was observed that the column laminate had a significant impact, such as delaying the first crack load and producing less deflection, as can be seen from Figures 35 and 36.

PT to CFRP Strength Contribution
As shown in Figures 37-39, different ratios of PT cables (tendons) to CFRP laminates were selected to investigate the strength contribution and how efficient is the use of CFRP strengthening in different cases.From the graphs, it is concluded that the strengthening of CFRP laminates is more significant in cases of low PT reinforcement than that in high PT reinforcement.

PT to CFRP Strength Contribution
As shown in Figures 37-39, different ratios of PT cables (tendons) to CFRP laminates were selected to investigate the strength contribution and how efficient is the use of CFRP strengthening in different cases.From the graphs, it is concluded that the strengthening of CFRP laminates is more significant in cases of low PT reinforcement than that in high PT reinforcement.

Conclusions
A hybrid nonlinear finite elements (FEM) model was developed to examine the flexural performance and the ultimate load capacity of bonded and unbonded two-way reinforced concrete post-tensioned (PT) slabs that were pre-strengthened with external carbon-fiber reinforcement polymer (CFRP) laminates.From this study, the following conclusions can be listed:

•
The FEM simulation showed great agreement with experimental results.The ratio between the experimental and FEM results of the first cracking load varied from 0.7 to 0.9, whereas the results of the ultimate load varied from 0.8 to 1.1 and the ultimate deflections varied from 0.8 to 1.

•
At different loading levels, there is a good correlation between FEM results and experimental results for CFRP strains either for the final debonding stage or for the locations of slippage.

•
From both FEM and experiments, it was concluded that the failure modes of the tested slabs were concrete crushing, CFRP rupture, and/or CFRP debonding.

•
Performance of the fully prestressed post-tensioned two-way slabs pre-strengthened with CFRP laminates have better performance in bonded rather than unbonded slabs.

•
Strengthening using CFRP materials is extremely effective in increasing the ductility index of both bonded and unbonded PT concrete slabs.Ductility increased by 62.18% and 59.87%, respectively, when compared with the control samples.

•
Real load simulation rather than test-equivalent criteria is considered a major and sensitive factor that has to be carefully represented in either experimental or numerical simulations to reflect reliable results.

•
In selecting an optimized scheme for strengthening PT slabs, it is noted that using laminates adjacent to columns is more efficient than in other slab locations.

•
CFRP strength contribution to PT cabling is very considerable in slabs with low PT reinforcement ratios, whereas it is not effective in slabs with larger PT ratios.

•
Finally, the results that were obtained using a numerical FEM model with the ANSYS program demonstrated good agreement with experimental results for illustrated slabs.
Further research is recommended to apply CFRP strengthening in existing PT slabs and study the behavior in that case.Additionally, as the slippage or debonding of CFRP laminates was the major outcome in this study, future solutions should be considered to prevent such failure modes to achieve the most efficiency of CFRP strengthening.

Buildings 2023 ,
13, x FOR PEER REVIEW 4 of 26 (a) Specimen dimensions and reinforcement.(b) UN-bonded tendon slab with 1 MPa, of the sample UN and US.(c) Bonded tendon slab with 1 MPa, of the sample BN and BS1.(d) Bonded tendon slab with 2 MPa, of the sample BS2.Buildings 2023, 13, x FOR PEER REVIEW 5 of 26 (e) Slabs strengthen with CFRP strips.

Figure 8 .
Figure 8.The geometry of interfacial contact pair elements.Figure 8.The geometry of interfacial contact pair elements.

Figure 8 .
Figure 8.The geometry of interfacial contact pair elements.Figure 8.The geometry of interfacial contact pair elements.

Figure 8 .
Figure 8.The geometry of interfacial contact pair elements.

Figure 10 .
Figure 10.Relationships between the local shear stress and the associated slip.

Figure 10 .
Figure 10.Relationships between the local shear stress and the associated slip.

Figure 12 .
Figure 12.Loading and support locations.

Figure 12 .
Figure 12.Loading and support locations.
(a) Mesh of the developed FEM (b) Mesh of the CFRP strips

Figure 12 .
Figure 12.Loading and support locations.

Figure 13 .
Figure 13.Design strip for model calibration.

Figure 13 .
Figure 13.Design strip for model calibration.

Figures 14 -
show the FEM deflection contour and Figures 19-23 compare the numerical and experimental (P-∆) curves for the five slabs.Good agreement was obtained between the FEM model and experimentally obtained results[33].Table4and Figures24-26present the comparison details concerning the ultimate load, cracking load, and its corresponding deflection.The ratio between the experimental and FEM results of the first cracking load varied from 0.7 to 0.9, whereas the results of the ultimate load varied from 0.8 to 1.1 and the ultimate deflections varied from 0.8 to 1.It has to be noted that the cause of results ratio differences would be the differences between the experimental environment and modeling.On the other hand, the strain in the CFRP laminates varied between a minimum of 34% and a maximum of 38% of the specified manufactured rupture strain.Generally, it is ensured numerically that the use of CFRP strengthening laminates has significantly increased the ductility index of both bonded and unbonded PT concrete slabs by 62.18% and 59.87%, respectively.

Figures 14 -
show the FEM deflection contour and Figures 19-23 compare the numerical and experimental (P-Δ) curves for the five slabs.Good agreement was obtained between the FEM model and experimentally obtained results[33].Table4and Figures 24-26 present the comparison details concerning the ultimate load, cracking load, and its corresponding deflection.The ratio between the experimental and FEM results of the first cracking load varied from 0.7 to 0.9, whereas the results of the ultimate load varied from 0.8 to 1.1 and the ultimate deflections varied from 0.8 to 1.It has to be noted that the cause of results ratio differences would be the differences between the experimental environment and modeling.On the other hand, the strain in the CFRP laminates varied between a minimum of 34% and a maximum of 38% of the specified manufactured rupture strain.Generally, it is ensured numerically that the use of CFRP strengthening laminates has significantly increased the ductility index of both bonded and unbonded PT concrete slabs by 62.18% and 59.87%, respectively.

Figure 24 .
Figure 24.Ultimate load comparison between FEM and EXP.

Figure 24 .
Figure 24.Ultimate load comparison between FEM and EXP.

Figure 26 .
Figure 26.Ultimate deflection comparison between FEM and EXP.
Figures 27 and 28 clearly illustrate this match for two types of slabs, whereas Figure 29 demonstrates the final debonding stage of CFRP for BN slab in which the locations of slippage matched exactly the experimental observations.

Figure 27 .
Figure 27.General comparison between EXP and FEM for CFRP strain for BN slab.

Figure 26 .
Figure 26.Ultimate deflection comparison between FEM and EXP.
Figures 27 and 28 clearly illustrate this match for two types of slabs, whereas Figure 29 demonstrates the final debonding stage of CFRP for BN slab in which the locations of slippage matched exactly the experimental observations.

Figure 26 .
Figure 26.Ultimate deflection comparison between FEM and EXP.
Figures 27 and 28 clearly illustrate this match for two types of slabs, whereas Figure 29 demonstrates the final debonding stage of CFRP for BN slab in which the locations of slippage matched exactly the experimental observations.

Figure 27 .
Figure 27.General comparison between EXP and FEM for CFRP strain for BN slab.

Figure 27 .
Figure 27.General comparison between EXP and FEM for CFRP strain for BN slab.

Figure 28 .
Figure 28.Comparison between EXP and FEM for the concrete strain of BS1 slab.

Figure 29 .
Figure 29.Final stage of CFRP laminates debonding in BN slab model.

Figure 28 .
Figure 28.Comparison between EXP and FEM for the concrete strain of BS1 slab.

Figure 28 .
Figure 28.Comparison between EXP and FEM for the concrete strain of BS1 slab.

Figure 29 .
Figure 29.Final stage of CFRP laminates debonding in BN slab model.

Figure 29 .
Figure 29.Final stage of CFRP laminates debonding in BN slab model.

Figure 32 .
Figure 32.FEM simulation comparison between equivalent test loads and real uniformly distributed load for slab BS1.

Figure 32 .
Figure 32.FEM simulation comparison between equivalent test loads and real uniformly distributed load for slab BS1.

Figure 36 .
Figure 36.Comparison between field laminate results with BS1.

Figure 36 .
Figure 36.Comparison between field laminate results with BS1.

Table 1 .
Summary of specimen's type.

Table 1 .
Summary of specimen's type.

Table 1 .
Summary of specimen's type.

Table 2 .
Cracking, ultimate loads and deflections for tested slabs.

Table 2 .
Cracking, ultimate loads and deflections for tested slabs.

Table 4 .
Experimental and FEM of cracking load, ultimate load, and ultimate deflection.

Table 4 .
Experimental and FEM of cracking load, ultimate load, and ultimate deflection.

Table 4 .
Experimental and FEM of cracking load, ultimate load, and ultimate deflection.