Performance of Steel Bar Lap Splices at the Base of Seismic Resistant Reinforced Concrete Columns Retrofitted with FRPs—3D Finite Element Analysis

Abstract

This paper examines analytically the design criteria for the composite retrofit of reinforced concrete (RC) columns with a short lap splice length of steel rebars inside the critical region. The advanced potential of pseudo-dynamic three-dimensional (3D) finite element (FE) modelling is utilized to investigate critical design parameters for the required carbon fiber-reinforced polymer (FRP) jacketing of RC columns with a rectangular cross-section based on the experimental lateral force-to-drift envelope behavior of characteristic cases from the international literature. The satisfactory analytical reproduction of the experimental results allows for the systematic numerical investigation of the developed stress along the lap splice length. The maximum lateral force and the horizontal displacement ductility of the column, as well as the maximum developed tensile axial force on the longitudinal bars, their variation along the lap, the bar yielding, and the plastic hinge length variation, are considered to determine the seismic behavior of the columns. For the first time, cases of smooth bar slip together with delayed bar yielding or without bar yielding are identified that may be recorded through a “ductile” P-d seismic response. Such pseudo-ductile response cases are revisited through suitably revised redesign criteria for adequate FRP jacketing.

1. Introduction

Existing earthquake-prone columns in RC brick wall-infilled framed structures often need to be repaired or strengthened together with the suitable modification of infills [1,2,3,4,5,6,7] in order to meet modern seismic and energy retrofit requirements. RC columns designed according to early code provisions may suffer detrimental damage during strong seismic excitations, as in many cases the corresponding ductility demands or detailing of reinforcement are inadequate.

A widespread technique to strengthen RC columns is the external application of advanced composite jacketing. Axially loaded concrete specimens, with or without internal steel reinforcement, have been used to investigate the confinement effect of external bonded composite materials (with an organic or inorganic matrix) [8,9,10,11,12,13,14,15]. They suggest that the use of such techniques leads to the increase of lateral pressure on the member and therefore to the increase of axial compressive strength and axial strain ductility of the concrete.

Furthermore, to estimate the effectiveness of FRP confinement on RC columns under seismic loading, numerous experimental tests have been carried out that subjected the foundation of half RC columns to a constant top axial load and imposed top cyclic horizontal displacements. Among the numerous parameters under investigation, columns with square and rectangular cross-sections with lap-spliced longitudinal steel bars [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39] within the critical region are included. Such columns are usually met in existing structures, and when the lap splice length is inadequate, an elastic or inelastic relative slip of the longitudinal bars may limit the shear strength and/or the horizontal deformation capacity.

The available experimental investigations have led to the proposal of equations for calculating the chord rotation at failure (θ_u) of the columns. Modern codes for seismic design and interventions, such as Eurocode 8 part 3 (EC8.3) [40] and Greek Retrofit Code (KANEPE) [41], have adopted such equations. The extensive investigation of the recent experimental and analytical database by Anagnostou et al. [42] (see also [43]) for 261 columns with or without lap splices suggests that the existing design models for RC columns of square or rectangular sections, with or without FRP confinement, provide the predicted shear strength compared against the experimental one, with an average absolute error (AAE) of about 20%. Further, the same study reveals that the AAE of the predicted chord rotation at failure was higher than 35% (for both codes). Therefore, besides the recent research focusing on the effects of external confinement with composites on the seismic behavior of columns [44,45,46], including their deformation capacity [47,48] and the plastic hinge length [49,50,51], further research is necessary to address the unidentified gaps in our knowledge.

Finite element (FE) analyses may provide the suitable analytical framework to improve our understanding, better assess the effects of different design parameters, and identify missing ones in different RC applications [52,53,54,55,56,57,58,59,60,61,62,63]. Three-dimensional (3D) pseudo-dynamic FE inelastic analyses with advanced material models may help with analytically investigating the interactions among different materials, as well as the internal reinforcements and external retrofits [64,65,66,67]. Such advanced analyses may provide critical missing parameters to enrich existing analytical databases through a hybrid experimental-analytical approach to propose improved design models (see its application in RC beams strengthened with FRPs in shear in Rousakis et al. [68] and in FRP-confined RC columns under axial concentric load in [69]). In RC columns with lap-spliced bars and an external FRP retrofit that are subjected to seismic loading, a similar approach is necessary to address the varying bars’ lap-splice performance.

This study utilizes several characteristic published seismic experiments of columns with lap-spliced longitudinal bars [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39], which were retrofitted with FRPs based on earlier investigations of all available experiments and design models in the field of seismic-resistant FRP retrofits ([48,70], among others). Then, these carefully chosen columns were modelled and analyzed pseudo-dynamically with 3D finite elements (FEs). The variable performance of bar lap splices at the base of the columns was thoroughly investigated. The maximum developed tensile axial force on the longitudinal bars, their variation along the lap, the bar yielding, and the plastic hinge length variation were considered to determine the seismic behavior of the columns. The FE analysis-derived missing variables enrich the developed databases. The abovementioned systematic hybrid approach is utilized for the first time to propose more reliable design models for the prediction of the chord rotation at failure of columns with lap spliced bars and external FRP confinement. For the first time, cases of smooth bar slip together with delayed bar yielding or without bar yielding are identified that may be recorded through a “rather ductile” P-d seismic response. Such pseudo-ductile response cases are revisited through suitably revised redesign criteria for adequate FRP jacketing.

2. Experimental Database

An experimental database has been created by the authors that includes 261 large-scale RC column specimens with a square or rectangular cross-section. All columns were loaded with a constant (or zero) axial compressive load and subjected to cycles of increasing lateral displacements, simulating pseudo-seismic loading [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100]. The columns are sorted according to those with or without seismic-related damage and with or without FRP confinement. The database gathers the geometrical and mechanical characteristics of these columns, as well as the experimental values of maximum lateral force (V_R or P_max) and chord rotation at failure (θ_u). In addition, the yield and ultimate chord rotation, the yield curvature, the yield moment, and the shear strength of the columns are calculated according to EC8.3 [40] and KANEPE [41].

In order to compare the analytical and numerical values to the experimental ones, the Average Ratio, AR, of the analytical-to-experimental values (Equation (1)) and the Average Absolute Error, AAE (Equation (2)), are utilized.

$$AR=\frac{1}{n}·{{\displaystyle \sum }}_{i=1}^n\frac{{\left(x_{anal}\right)}_i}{{\left(x_{exp}\right)}_i}$$

(1)

$$AA{\rm E}=\frac{1}{n}·{\displaystyle \sum }_{i=1}^n\frac{\left|{\left(x_{anal}\right)}_i-{\left(x_{exp}\right)}_i\right|}{{\left(x_{exp}\right)}_i}·100,$$

(2)

In the above equations, x_anal is the analytical value and x_exp is the experimental value of the parameter (e.g., θ_u, V_R). It should be noticed that values of AR close to 1 and values of AAE close to 0% indicate accurate predictions of the values under investigation.

In this study, columns with lap splices are further investigated. RC cantilever columns subjected to seismic loading are selected from the study by Bousias et al. [17] in order to investigate the effect of different parameters, such as the lap splice length and the CFRP jacket layers, through advanced FE analyses. The geometry and the material properties, as well as the axial load, of the nine thoroughly selected columns are presented in

Table 1. Features of selected columns for pseudo-dynamic 3D FE analyses.

Column	fc (MPa)	Geometry			FRP						Longitudinal Bars				Stirrups				v
		bw (m)	h (m)	Ls (m)	Type of FRP	Height of FRP (mm)	tj (mm)	Ej (GPa)	εju	r (mm)	dbL (mm)	Lap-splice Length (dbL)	fyL (MPa)	fuL (MPa)	dw (mm)	fyw (MPa)	s (m)	fuw (MPa)
R-0L0	31.0	0.250	0.500	1.60	-	-	-	-	-		18	-	514	659	8	425	0.2	596	0.26
R-0L1	27.4	0.250	0.500	1.60	-	-	-	-	-		18	15	514	659	8	425	0.2	596	0.23
R-0L3	27.4	0.250	0.500	1.60	-	-	-	-	-		18	30	514	659	8	425	0.2	596	0.28
R-0L4	27.4	0.250	0.500	1.60	-	-	-	-	-		18	45	514	659	8	425	0.2	596	0.28
R-P2L0	32.9	0.250	0.500	1.60	CFRP	600	0.26	230	0.015	30	18	-	514	659	8	425	0.2	596	0.23
R-P2L1	26.9	0.250	0.500	1.60	CFRP	600	0.26	230	0.015	30	18	15	514	659	8	425	0.2	596	0.30
R-P2L3	26.9	0.250	0.500	1.60	CFRP	600	0.26	230	0.015	30	18	30	514	659	8	425	0.2	596	0.28
R-P2L4	26.9	0.250	0.500	1.60	CFRP	600	0.26	230	0.015	30	18	45	514	659	8	425	0.2	596	0.28
R-P5L3	27.0	0.250	0.500	1.60	CFRP	600	0.65	230	0.015	30	18	30	514	659	8	425	0.2	596	0.29

f_c: concrete compressive strength, b_w: width of cross-section, h: depth of cross-section, L_s: shear span length, t_j: total thickness of FRP jacket, E_j: elastic modulus of CFRP, ε_ju: strain at failure of CFRP, r: bend radius of FRP at the corners of the member, d_bL: diameter of longitudinal steel bar, f_yL: yield stress of longitudinal steel bar, f_uL: ultimate stress of longitudinal steel bar, d_w: diameter of transverse steel bar, f_yw: yield stress of transverse reinforcement, s: stirrup spacing, f_uw: ultimate stress of transverse reinforcement, v: normalized axial force.

. More details on the geometry and the material properties of these specimens, as well as the experimental procedure, can be found in [17].

3. 3D Finite Elements Modelling

The analytical models of the characteristic columns are developed in ANSYS Workbench, Explicit Dynamics. Advanced material models for concrete, steel, and composites are suitably calibrated.

3.1. Materials

The response of a part is determined by the material properties assigned to the part. Material properties, depending on their application, may be linear, non-linear, or temperature-dependent.

3.1.1. Concrete

The Riedel–Hiermaier–Thoma (RHT) [101] plasticity model for brittle materials, which is also suitable for modelling the dynamic loading of concrete, was used. The RHT constitutive model is a combined plasticity and shear damage model which enables the reproduction of monotonic, repeated, or cyclic imposed displacements. The deviatoric stress in the material is limited by a generalized failure surface (Equation (3)) which can be used to represent the aspects of the response of geological materials: pressure hardening, strain hardening, strain rate hardening in tension and compression, third invariant dependence for compressive and tensile meridians, strain softening (shear induced damage), and coupling of damage due to porous collapse.

$$f\left(p,{\sigma }_{eq},\theta ,\dot{\epsilon }\right)={\sigma }_{eq}-Y_{TXC}\left(p\right)·F_{CAP}\left(p\right)·R_3\left(\theta \right)·F_{RATE}\left(\dot{\epsilon }\right)$$

(3)

The RHT model is suitably calibrated to provide the behavior of concrete under pseudo-seismic loading, as well as the effects of pre-damaged concrete. The values for the compressive strength of concrete for each column are included in Table 1. The material parameters for concrete modeling are presented in

Table 2. Basic concrete material parameters.

Property	Value
Density (kg·m−3)	2314
Tensile Strength ft/fc	0.1
Shear Strength fs/fc	0.18
Intact Failure Surface Constant A	1.6
Intact Failure Surface Exponent n	0.61
Tension/Compression Meridian Ratio Q2.0	0.6805
Brittle to Ductile Transition BQ	0.0105
Hardening Slope	2
Elastic Strength/ft	0.7
Elastic Strength/fc	0.53

.

3.1.2. Longitudinal and Transverse Steel Bars

The elastic-plastic behavior of the steel bars (longitudinal and transverse) and the strain hardening of the material after yield strength were simulated using the multilinear isotropic hardening model, a plasticity model usually used in large strain analysis. The multilinear hardening behavior is described by a piece-wise linear stress—total strain curve, starting at the origin and defined by sets of positive stress and strain values, according to the information provided by the experimental tests. The first stress-strain point corresponds to the yield stress. Subsequent points define the elastic-plastic response of the material. In addition, the Young’s modulus of steel was considered equal to 200 GPa and Poisson’s ratio equal to 0.3. The steel material parameters of the model are presented in

Table 3. Basic steel material parameters.

	Property	Value
	Density (kg·m−3)	7850
Isotropic elasticity	Young’s Modulus (MPa)	2 × 105
	Poisson’s Ratio	0.3
	Bulk Modulus (Pa)	1.6667 × 1011
	Shear Modulus (Pa)	7.6923 × 1010
	Plastic Strain	Stress
Multilinear isotropic hardening (Steel stirrups)	0	425
	0.05	550
	0.1	580
	0.195	596
Multilinear isotropic hardening (Longitudinal bars)	0	514
	0.05	630
	0.1	650
	0.17	659

. The impactor (steel plate) is modelled as a purely elastic material.

3.1.3. Fiber Reinforced Polymer Jacket

The elastic behavior of the FRP jacket was simulated with the orthotropic elasticity model. For elastic modulus of the carbon fibers alone of 230 GPa per fiber layer thickness, the elastic modulus was considered to be 59.16 GPa to include the properties of the impregnation epoxy resin. The thickness of the jacket was suitably increased to account for the combined fiber and resin thickness per layer (with a similar axial rigidity value for fiber thickness multiplied by the fiber modulus of elasticity or for combined thickness multiplied by the modulus of elasticity of the fiber-resin jacket). This approach provides more reliable stress-strain behavior and the local fracture of the FRP jacket.

Table 4. Basic CFRP jacket parameters.

Property	Value
Density (kg·m−3)	1.451 × 10−9
Young’s Modulus X direction (MPa)	59,160
Young’s Modulus Y direction (MPa)	59,160

presents the material parameters of CFRP used for the orthotropic elasticity model.

3.2. Geometry

The geometry of the RC columns was accurately reproduced. The models consist of different parts of solid body for the concrete, the FRP jacket, and the impactor. Line bodies were used for the longitudinal and transverse reinforcements (Figure 1).

3.3. Connections

The interaction between the different parts of the FE model need to be defined for an accurate prediction of the behavior and failure mode of the member [102,103,104,105]. In the context of the present numerical study, the contact region between concrete and FRP was defined as “bonded” for all columns and no sliding or separation between faces or edges was allowed. This approach is in line with other analytical research work [60,106], and is supported by the experimental evidence [9,12]. No slip was observed during RC column tests confined with FRP jackets. In cases where the fracture of the concrete core occured at the neighboring area, this was captured by the advanced concrete model. For all bodies, the body interaction type was defined as “frictionless” if it activated frictionless sliding contact between any exterior node or face of the selected bodies. In addition, all elements of the line bodies (internal steel reinforcement), which are contained within any solid body (concrete) in the model, are defined to be converted to discrete reinforcement (using the “reinforcement” body interaction type, typically used for RC applications).

3.4. Mesh

Explicit mesh was followed to form an accurate 3D grid. The solid bodies for concrete and the CFRP jacket were modelled with eight-node hexahedral elements (element type Hex8). They are well suited to transient dynamic applications including large deformations, large strains, large rotations, and complex contact conditions. The line bodies for steel bars and stirrups were modelled with 2-node line elements that allow for large displacements and the resultant elasto-plastic response.

Regarding the element size, as it gets smaller, more divisions are generated, and the accuracy is improved. However, a greater number of elements are created, and the required time for one analysis becomes unprofitable for the large-scale model of an RC column. Based on the mesh convergence study conducted by the authors, a mesh size of 25 mm (for all individual parts of the model) was chosen as the most favorable to provide satisfactory accuracy regarding the seismic behavior of RC columns at an acceptable analysis time.

3.5. Boundary Conditions and Loading

The cantilever column specimens were fix-supported at the base. A constant pressure was exerted on the top section, reproducing the constant axial load during the experimental test (the values for normalized axial load, v, are included in Table 1). After the application of the axial load, an increasing horizontal displacement (monotonic loading) was imposed at the steel impactor (Figure 2). The monotonic response (pseudo-dynamic analysis) can serve as a rather reliable envelope force-to-deformation curve of the cyclic one and thus save valuable computational time. Therefore, the 3D FE analyses were performed by imposing an increasing horizontal displacement.

4. Numerical Results and Discussion

4.1. Lateral Force-to-Drift (%) Curves

The numerical lateral force-to-drift (P-θ) curves of columns R-0L0, R-0L1, R-0L3, R-0L4, R-P2L0, R-P2L1, R-P2L3, R-P2L4, and R-P5L3 are presented against the experimental ones in Figure 3.

The displacement and load at ultimate were considered at the fracture of the steel reinforcement, at the failure of concrete under compression, at the fracture of the FRP jacket (not local), or when the limit lateral bearing load reached 80% of the maximum one (KANEPE, EC8.3).

The numerical and experimental results of all columns for the maximum and ultimate lateral force (P_max, P_u) and the ultimate displacements (δ_Pu) are presented and compared in

Table 5. Comparative experimental and numerical results of columns R-0L0, R-0L1, R-0L3, R-0L4, R-P2L0, R-P2L1, R-P2L3, R-P2L4, and R-P5L3.

Column	Results	Pmax (kN)	AE (%)	Pu,exp (kN)	δPu (mm)	δPu AE (%)	δu,exp (mm)	Pδu (kN)	Pδu AE (%)
R-0L0	Experimental	196.50	0.79	157.20	40.00	6.93	40.00	157.20	2.06
	numerical	194.95		157.20	37.23		40.00	153.96
R-0L1	Experimental	148.00	5.63	123.40	30.40	10.79	30.40	123.40	12.77
	numerical	139.67		123.40	27.12		30.40	107.64
R-0L3	Experimental	173.75	7.67	139.00	30.40	1.78	30.40	139.00	1.64
	numerical	187.07		139.00	30.94		30.40	141.28
R-0L4	Experimental	205.00	6.32	164.00	39.00	22.92	39.00	164.00	19.62
	numerical	192.05		164.00	30.06		39.00	131.82
R-P2L0	Experimental	217.00	0.33	173.60	67.20	14.02	67.20	173.60	5.67
	numerical	217.73		173.60	76.62		67.20	183.44
R-P2L1	Experimental	171.50	3.13	137.20	52.20	15.31	52.20	137.20	11.84
	numerical	176.86		137.20	60.19		52.20	153.45
R-P2L3	Experimental	211.50	1.13	169.20	70.10	5.34	70.10	169.20	7.29
	numerical	209.11		169.20	66.36		70.10	156.86
R-P2L4	Experimental	208.50	1.24	166.80	87.30	10.27	87.30	166.80	4.46
	numerical	211.08		166.80	96.27		87.30	174.24
R-P5L3	Experimental	225.25	1.04	180.00	89.30	8.81	89.30	180.00	4.73
	numerical	222.90		180.00	97.17		89.30	188.52

. The experimental results refer to the average values of the two directions, push and pull.

The divergence between the experimental and numerical values of loads at maximum and ultimate, as well as of displacements at ultimate (based on the experimental load at ultimate or the experimental displacement), is low. Therefore, the analytical results can be further elaborated to draw reliable conclusions for the mechanical behavior of RC columns with lap-spliced longitudinal bars. Based on the unique features of the concrete material model, the developed 3D FE columns may be utilized to investigate the effects of steel corrosion or the preloading of columns as well.

4.2. Analytical Response of the Longitudinal Bars

The maximum axial tensile force of the bars throughout the loading, as well as the variation of the axial force (or of the stress) along the lap splice, are examined.

4.2.1. Axial Force of Longitudinal Bars at the Base of Column

The axial force developed on the longitudinal bars in tension at the base of the column, F_max (or the developed stress, f_s), is provided. The comparative curves of the tensile axial force of the bar versus the horizontal displacement of the columns (F-δ curves) without FRP confinement are depicted in Figure 4.

The analyses suggest that the axial force developed on the bars in tension at the lap region at the base of column (F_max) of unretrofitted column without lap splices (R-0L0) is 145.94 kN at a displacement of 37.80 mm (at ultimate or failure), higher than the analytical yield force F_y = 130.8 kN. The presence of inadequate lap splices with 15 d_bL length at the base of an unretrofitted RC column results in the decrease of F_max by 40% in column R-0L1 (F_max = 87.95 kN at 8.57 mm, lower than the analytical yield force F_y = 130.8 kN), 5% in column R-0L3 (F_max = 138.33 kN at 16.64 mm), and 2% in column R-0L4 (F_max = 143.30 kN at 31.36 mm) compared to R-0L0. Both columns with considered adequate lap splice lengths (R-0L3, R-0L4) may face some temporary or detrimental slip after steel bar yielding.

As for the columns externally confined with two layers of FRP (Figure 5), the analyses suggest that the F_max is 147.89 kN (at 57.78 mm) in column R-P2L0, 85.49kN (at 7.29 mm) in column R-P2L1 (lower than the analytical yield force, F_y = 130.8 kN), 134.63 kN (at 55.90 mm) in column R-P2L3, and 157.42 kN (at 96.27 mm) in column R-P2L4. It is observed that the F_max-to-displacement curves of R-P2L0 (without lap splices) and R-P2L4 (45 d_bL lap splice length) are almost identical (also with R-0L0), confirming that a lap splice length of 45 d_bL provides an equivalent tensile bar response (Figure 5). In contrast, the analyses suggest that the F_max obtained for column R-P2L1 is lower than the F_y, although the column is externally confined with FRPs. The two layers of CFRP jacketing, in this case, are inadequate to prevent the lap-spliced bars from slipping, despite the pseudo-ductile behavior observed in Figure 3f. This pseudo-ductility is attributed to the pseudo-ductile bar contribution (see Figure 5). However, the response of the column is considered to be inadequate, as there is slip and it fails to reach bar yielding. This bar response is similar to the one in R-0L1 (Figure 4). For R-P2L3, the maximum tensile force of the bars exceeds the yield one, but at a horizontal displacement of 40 mm. The column seems to suffer initially by the controlled slip of the bars, starting before the displacement at the yielding of the bars of columns R-0L0 or R-P2L0 (Figure 5). However, at around 60 mm displacement, the bars slip again. The revealed deficient performance of the lap-spliced bars in R-P2L3 is in accordance with the lower bearing horizontal load of around 180–200 kN at the pseudo-yielding point (Figure 3). The corresponding bearing horizontal load for R-P2L0 at real bar yielding is around 190–225 kN (Figure 3).

The above conclusions coincide with observations made based on the experimental results of Bournas & Triantafillou [19]. They tested RC columns with lap-spliced bars (20 d_bL = 280 mm = 0.67 l_b,min or 40 d_bL = 560 mm = 1.29 l_b,min) without FRP confinement (L20d_C, L40d_C) or with FRP confinement (L20d_R2, L40d_R2). They provided the steel strain histories of starter bars at the column base during testing, measured with strain-gauges (Figure 6). The lap-splice length equal to 20 d_bL is inadequate for developing the yield strain (or yield stress) of the longitudinal bars (ε_yL = 523/200,000 = 2.62‰), while the slippage of lap-spliced bars is observed. Although a lap-splice length of 40 d_bL is adequate to develop the yield stress of the longitudinal bars, it is not sufficient to prevent slippage, which is observed after yielding. The application of two layers of CFRP delays the slippage of the lap-spliced bars for the inadequate lap-splice length of 20 d_bL, happening at higher drifts of 3.75%. However, longitudinal bars do not really surpass yielding strain, while this marginal yielding of bars initiates at a higher column drift of 3.44%. As for the adequate lap-splice length of 40 d_bL, FRP confinement does not affect the drift at the yielding of the lap-spliced bars, while no slippage is observed at higher drifts up to column failure.

Getting back to the 3D FE analyses, the axial force of the tensile longitudinal bar at the base of the column (F) to the top displacement (δ) curve changes slope when F = F_y (Figure 7a). However, for inadequate lap splice lengths, this change occurs at a lower force (F < F_y). In this case, the axial force is defined as pseudo-yield force (F_py) and the corresponding displacement as pseudo-yield displacement (δ_py), as shown in Figure 7b. This pseudo-ductility or delay in real tensile bar yielding development may explain the divergences in prediction in some cases in the yield flexural moment or chord rotation of the columns with lap-spliced bars, despite the fact that they are more or less affected by the mechanical characteristics of the steel bar under tension. These cases urge an assessment of the performance of such columns as acceptable or not in redesign.

4.2.2. Stress along the Lap Splice Length

Melek et al. [107] carried out experimental tests on six RC columns with short lap splices (20 d_bL = 508 mm = 0.78 l_b,min). They placed strain gauges to monitor the strain histories of the lap-spliced bars (Figure 8a), which recorded at about 3% lateral drift when most of them were damaged due to the damage to the columns. They expected a triangular strain distribution before bond deterioration (Figure 8b).

Figure 9a,b show the strain distribution of corner and middle bars, respectively, for column S20MI tested by Melek et al. [107]. The yield stress of the longitudinal bars is f_yL,main = 510 MPa for the main bars and f_yL,starter = 521 MPa for the starter bars. For the starter bars, the strain distribution is triangular indeed (blue), while for the main bars it has a curved form (orange). A different response of the corner and middle longitudinal bars in tension during cyclic loading can also be observed.

The analytical model proposed by Chowdhury & Orakcal [108] (calibrated with the experimental results of Melek et al. [107] research work) for inadequate lap splices (20 d_bL = 508 mm = 0.78 l_b,min) predicts the increment of stress in both the starter and the main bars, for monotonic (Figure 10a) and cyclic (Figure 10b) analyses, as the drift ratio increases up to the lateral load capacity of columns (about 0.5% drift level). For both bars, the stress increases as the drift level increases. The stress of the starter bar along the lap splice length resembles a parabolic distribution at lower lateral drifts, which changes into triangular at higher lateral drifts. The stress distribution of the main bar has a wavy form.

3D FEA can provide the developed tensile (as well as compressive) stresses along lap-splice length for both starter and main bars at points of interest (depending on finite element size) and at any displacement/drift level during analysis. The effect of the steel stirrups, as well as of the FRP jacket, on developed stresses can also be examined. Herein, the developed tensile stress on bars at the lap-splice region is examined, as it constitutes a critical parameter to better understand the seismic behavior of RC columns.

Figure 11 shows the developed stress along the bar in tension of column R-0L0 without lap-spliced bars and without FRP confinement at about (a) one third of the yielding force of tensile bars, (b) the yielding force of tensile bars, (c) the maximum lateral force of the column and (d) failure. It is observed that initially, as the developed stress remains lower than the yield stress (f_yL = 514 MPa), it has a wavy form. For higher lateral displacements and after yielding, it has a triangular form. Moreover, a longer part of the steel bar reaches the yield stress and the critical region within which the plastic hinge develops is extended up to 300 mm.

Similarly, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18 show the numerical stress distribution on reinforcing bars (starter and main), as well as the total developed stress, along the lap splice length at different displacement levels for columns R-0L1, R-0L3, and R-0L4 well as for R-P2L1, R-P2L3, R-P2L4, and R-P5L3.

Around the horizontal displacement level of the column without laps, at which the yielding of the tensile reinforcement occurs (around 10 mm), the developed stress along the lap splice length is maximum at the starter bars at the column section adjacent to the joint and zero at the end of starter bars (and vice versa for the main column bars, especially after starter bar yielding (f_yL = 514 MPa and yield strain ε_sL = 2.57‰). Both bars seem to develop the same stress at about two thirds of the lap splice length away from the bottom section, even in cases of inadequate lap splice length. Moreover, the shape of the curves representing the stress along lap splice up to yielding for starter and main bars validates more or less the analytical model provided by Chowdhury & Orakcal [108]. FE analyses suggest that in most cases the developed stress at the base of the column is not fully transferred to the end of the lap splice, as the tensile strain of the bars may be lower far from the bottom section due to lower moment values along the column. Therefore, the loading along the lap-splice length is obviously not symmetrical.

In cases where the lap splice length is adequately long (45 d_bL = 810 mm = 1.53 l_b,min) and no slippage is observed, the total bearing tensile stress by both bars at any section (gray line in Figure 12 and Figure 13) is lower at the end of the lap splice in accordance with the developed moments, similar to the column without laps (Figure 11). For higher column drift, the same pattern is valid (see R-0L4 and R-P2L4), as the bars achieve hardening behavior in response to the increased bearing stress. Further, it is interesting to note that in column R-0L0, the plastic hinge length is around 300 mm, defined by the length the tensile steel bars surpass the yield stress. In the case of R-0L4 with adequate lap-splice, the plastic hinge length is extended to 400 mm, as we consider the sum of the stress developed in both bars at an identical column length (gray line in Figure 12). Similarly, in R-P2L4, the FRP jacketing further extends the plastic hinge length to around 620 mm.

In cases where the lap splice length is 30 d_bL (540 mm = 1.02 l_b,min), the bars develop their yielding, but clearly, a slip occurs at ultimate drift (see R-0L3, Figure 15, and R-P2L3, Figure 16). When the column is confined with two layers of FRP, slip occurs at a far higher ultimate drift. Further, in the case of R-0L3, the plastic hinge length is extended to 325 mm, as we consider the sum of the stress developed in both bars at an identical column length (gray line in Figure 14). Similarly, in R-P2L3, the FRP jacketing further extends the plastic hinge length to around 450 mm. In R-P5L3, the corresponding plastic hinge length is developed at higher drifts but extends to 540 mm.

In cases where the lap splice length is inadequate (15 d_bL = 270 mm = 0.51 l_b,min), the total tensile stress by both bars at any section remains rather constant all along the lap-splice, and lower than the yield stress of the bars. This pattern is not affected by external FRP confinement. The total stress of the main bar may be higher than the starter’s for different horizontal drifts of the column. No plastic hinge length may be defined based on steel yielding.

The 3D FE numerical results suggest increased tensile stresses of the bars at the steel stirrups levels for different columns, as they tend to resist the opening of the potential longitudinal crack along the spliced bars that leads to their relative slip. The CFRP retrofit seems to improve the axial force transfer mechanism between the bars and to result in a better bar stress distribution. The total stress received by the bars increases for more layers of FRP. The plastic hinge length is remarkably extended for a higher lap-splice length or for higher FRP jacketing.

Furthermore, EC8.3 takes into consideration the height of the FRP jacket compared to the lap splice length for calculating the plastic part to chord rotation. Herein, the height of FRP (h_f) is 600 mm (Table 1). It barely exceeds the lap splice length of 30 d_bL (h_f = 600 mm > l_o = 540 mm), while it is shorter than a lap splice length of 45 d_bL (h_f = 600 mm < l_o = 810 mm). The height of the provided FRP confinement seems to affect the transfer mechanism of stress along the lap splice, as the diagrams in Figure 13 show a sudden stress decrease at 600 mm from the column bottom, at the end of the FRP jacket.

Based on the above observations, it is suggested that the critical position to place strain gauges on the starter and the main bar along the lap splice length to measure tensile stress under seismic loading is at the mid-distance between two consecutively steel stirrups and at the steel stirrups level. Further, the whole lap-splice length has to be covered with strain gauges in both bars, as the sum of the stress at both bars is crucial.

4.2.3. Proposed Modifications

In previous research work conducted by the authors [42], it was suggested that the characteristic stress-strain behavior of internal steel bars, exhibiting hardening, should be taken into account (regarding columns with FRP confinement). In [64], it was proposed that f_y must be multiplied by an additional relation c (Equation (4)), regarding the RC columns with insufficient lap splices (0 < l_b/l_b,min < 1).

$$c=\left(-4·\alpha +6.2\right)·\frac{l_b}{l_{b,min}}+\left(4.5·\alpha -5.95\right)$$

(4)

The relation c takes values between 0.5 and 1 (0.5 ≤ c ≤ 1) and provides the maximum developed stress at the lap splice region based on the provided lap splice length and the provided confinement by stirrups and the FRP jacket. It identifies the cases where the bar does not reach the yield stress in order to avoid FRP strengthening (an unsuitable retrofit method for such inadequate lap-splices—alternative suitable methods could be proposed). Similarly, FRP strengthening is avoided for l_b < 0.5 l_b,min. If the bars reach the yield stress (c = 1), then the hardening of the steel bar is taken into consideration (1.25·f_y). The (Equation (4)) is inserted in the yield curvature to calculate M_y and θ_y, and in the mechanical ratios of reinforcement to calculate θ_u and therefore μ_δ for adequate FRP strengthening (at least two layers). Obtained values of μ_δ lower than 3.5 are avoided and higher FRP strengthening is recommended. Further, it should be secured that the shear force required to develop the full flexural strength of the column after the tensile yielding of the bars is lower than the shear capacity of the column. In cases where the lap length is adequate (higher than l_b,min), c equals 1. It should be noted that, based on the proposed modification, the l_b,min is different from that in seismic codes.

4.2.4. Comparison of Analytical and Experimental Values

The original database of 261 columns revealed an AAE of 22.99% and AR 0.82 for V_R and an AAE of 37.48% and AR 0.97 for θ_u, according to KANEPE. The corresponding values, according to EC8.3, are 19.40% and 0.87 for V_R and 41.45% and 0.91 for θ_u.

Regarding the 61 gathered columns with lap splices and FRP confinement (included in the total database), the design relations of KANEPE predict the V_R with an AAE of 20.83% and an AR of 0.80, while they predict the θ_u with an AAE of 38.86% and an AR of 0.64. As for EC8.3, the corresponding values are 18.73% and 0.83 for V_R and 46.11% and 0.54 for θ_u. The above performance of the existing design models suggests that the error of prediction of the existing models is significant. As was already proposed in Section 4.2.3, the variable performance of the lap splices with inadequate length should be incorporated into the models to better address the variable beneficial effects of different quantities of external FRP confinement and their upper and lower limits.

Therefore, in this paper, the use of relation c (Equation (4)) reduces the absolute error (AE) of predictions remarkably, especially for FRP retrofitted RC columns with an inadequate lap splice length.

Table 6. Comparison of analytical and experimental values of chord rotation at failure.

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
Column	Lap Length (dbL)	Layers CFRP	θu,exp	θu,pred (KANEPE)	AE (%)	θu,pred (EC8.3)	AE (%)	θu,prop (Proposed- Including Equation c)	AE (%)
R-P2L0	0	2	0.044	0.0422	4.1	0.0419	4.9	0.0419	4.8
R-P2L1	15	2	0.034	0.0121	64.5	0.0167	50.9	0.0352	3.5
R-P2L3	30	2	0.047	0.0231	51.0	0.0227	51.7	0.0442	5.9
R-P2L4	45	2	0.056	0.0303	45.9	0.0298	46.8	0.0467	16.5
R-P5L3	30	5	0.056	0.0256	54.3	0.0252	55.0	0.0632	12.8
AAE(%)					44.0		41.9		8.7
AR					0.56		0.58		0.98

gathers the experimental and analytical (according to KANEPE-columns 5 & 6 of the table, EC8.3-columns 7 & 8 and proposed model-columns 9 & 10) values of chord rotation at failure, as well as the comparison of the analytical values to the experimental ones, for the examined columns with lap splices and CFRP confinement. Some indicative AR and AAE values for the five RC columns are also included at the last line of Table 6 to assess the improvement in predictions when using the proposed model including relation c (Equation (4)).

On the basis of the above observations, the divergences in shear capacity are explained by the fact that no yielding occurs, or the yielding of the reinforcing bars has occurred belatedly. The predicted error of chord rotation is higher, as it is affected by the plastic hinge length. The use of relation c better predicts whether the steel bars yield or not. The yielding affects the base shear and the flexural capacity of the RC column. In addition, the plastic hinge length needs to be further investigated because it affects the chord rotation for cases involving the yielding of the reinforcing bars.

5. Conclusions

This study follows a hybrid approach to enrich the developed database of FRP-confined RC columns under axial load and top lateral cyclic imposed displacements with analytically derived missing design parameters. 3D FE models of characteristic RC columns are developed and analyzed pseudo-dynamically to accomplish this task. Suitably calibrated models for concrete, steel, and FRP confinement are used. The numerical horizontal load—drift (%) curves compare well with the experimental ones (reported in experiments by Bousias et al. [9]) in terms of the general response and characteristic load and displacement values at maximum and ultimate. Therefore, the analytical results are further elaborated to retrieve for the first time a rather consequent advanced analytical insight of the variable response of the rebars at the lap-splice region.

Based on the 3D FE analytical investigation of the lap splice region, it is concluded for the considered columns that:

• In most cases, the developed stress at the base of the column is not fully transferred to the end of the lap splice, as the tensile strain of the bars may be lower far from the bottom section (because of lower moment values). Both bars (starter and main) seem to develop the same stress at about two thirds of the lap splice length away from the bottom section (even in cases of inadequate lap length).
• In cases where the lap splice length is adequately long (45 d_bL), the total bearing tensile stress by both bars at any section is lower at the end of the lap in accordance with the developed moments, similar to the column without laps. For higher column drift, the same pattern is valid, as the bars achieve the hardening behavior of increased bearing stress. Further, if we consider the sum of the developed stress for the lapped bars at any point, the plastic hinge length is extended from 300 mm for R-0L0 without lap, to 400 mm for R-0L4, and to more than double (620 mm) for FRP jacketing at the lap region in R-P2L4.
• In cases the lap splice is 30 d_bL the bars develop their yielding, but clearly a slip occurs at ultimate drift. When a column is confined with two layers of FRP, it initially suffers a temporary controlled slip of the bars, starting before the displacement at yielding of the columns R-0L0 or R-P2L0. After delayed bar yielding, the detrimental slip occurs at a far higher ultimate drift. The plastic hinge length is lower than in R-0L4 but higher than in R-0L0 (325 mm). Again, the plastic hinge length extends to around 450 mm or 540 mm for two or five layers of FRP jacketing, based on which is higher for higher lap length or FRP jacketing.
• In cases where the lap splice length is inadequate (lap length of 15 d_bL), the total tensile stress of both bars at any section remains rather constant all along the lap-splice and lower than the yield stress of the bars. This pattern is not affected by external FRP confinement. The total stress of the main bar one may be higher than the starter’s for different horizontal drifts of the column. No plastic hinge length may be defined in these cases, as no bar yielding occurs (despite pseudo-yielding).
• Increased tensile bar stresses occur at the steel stirrups levels for different columns, as they tend to resist the opening of the potential crack along the spliced bars that leads to their relative slip. The CFRP retrofit seems to improve the axial force transfer mechanism and to result in a better bar stress distribution. The sum of the tensile stress received by both lapped bars increases for more layers of FRP.
• The height of the provided FRP confinement seems to affect the transfer mechanism of stress along the lap splice. If the FRP does not cover all lap splice regions, then a sudden bar stress decrease at the end of the FRP jacketing occurs.
• For the first time, cases of smooth bar slip together with delayed bar yielding or without bar yielding are identified that may be recorded through a “ductile” P-d seismic response. Such pseudo-ductile response cases are revisited through suitably revised redesign criteria for adequate FRP jacketing that identify if the lapped bar will yield.
• The authors propose that composite jacketing that leads to pseudo-ductile P-d behavior while the lapped bar does not yield due to slip (temporary or not) should be considered as inadequate in redesign, and additional FRP layers should be provided as per the framework presented in Section 4.2.2.
• The high potential of advanced dynamic 3D FEA should be further utilized to extend investigations in several challenging real cases of deficient existing columns with different cross-section geometry, multiple side steel bars inside the lap region, and different detailing of existing steel stirrups or of composite jacketing to assist successful redesign.