Extended Tension Chord Model for Boundary Elements of RC Walls Accounting for Anchorage Slip and Lap Splices Presence

Abstract

This paper presents a mechanical model for the simulation of reinforced concrete (RC) wall boundary elements with lap splices, which builds on the tension chord model. The model is composed of an assembly of components, each one accounting for a different source of deformation. Namely: (i) an anchorage-slip element accounting for the strain penetration of the longitudinal reinforcement into the foundation; (ii) a basic tension chord element evaluating the response outside the lap splice zone; and (iii) a lap splice element describing the behaviour within the lap splice region. For an imposed global displacement, the model provides the steel and concrete stress and strain distributions, the crack distribution and opening, as well as the global resisting axial force. For spliced members, the ultimate displacement is computed through a semi-empirical relationship providing the average lap splice strain at failure. Validation is carried out against a series of uniaxial cyclic tests on RC wall boundary elements featuring both continuous and spliced reinforcement; different lap splice lengths and confining reinforcement are considered. Overall, a good match is obtained between numerical and experimental results in terms of crack width, rebar strain distribution along the splices and ultimate displacement.

1 Introduction

Experimental tests on reinforced concrete (RC) walls have shown that the presence of lap splices may lead to a significant reduction of the member strength and ductility capacity (Almeida et al. 2017). The behaviour of lap splices is influenced by several factors, among which lap splice length (l_s) and confining reinforcement play a dominant role. Namely, short and poorly confined lap splices located in regions where inelastic deformations are largest (i.e. plastic hinges) may induce failure of the RC wall prior to yielding of the longitudinal reinforcement. Longer and more confined lap splices may allow the wall to develop its flexural strength, however a decrease in the deformation capacity of the structural member is often still observed. Finally, long and adequately confined lap splices will relocate the plastic hinge above the spliced zone, where the deformation capacity of the flexural hinge is limited by rupture of the reinforcement or crushing of the concrete. In all cases, damage typically starts at the wall edges (boundary elements) where the deformation demand is highest (Tarquini et al. 2017), which are also the regions first attaining failure.

Most studies in the past focused on the strength of lap splices (e.g. Orangun et al. 1977; Zuo and Darwin 2000) while significantly less research is available concerning their deformation capacity. Recently, based on an experimental programme on RC wall boundary elements, Tarquini et al. (2019) proposed an expression to estimate the average strain at failure of lap splices as a function of the lap splice length, confining reinforcement, and casting position. Such expression allows estimating the deformation capacity of RC wall boundary elements with lap splices.

Past mechanical models for lap splices aimed at the characterization of their strength capacity (Priestley et al. 1996; Canbay and Frosch 2005). To the authors knowledge, the analytical model proposed by Tastani et al. (2015) is the only available in the literature describing the state of bond along spliced rebars, which can be used to predict the force–displacement response of lap splices. However, their approach is limited to the elastic response of the reinforcement bars and therefore only applicable to describe the behaviour of very short lap splice lengths (where failure will occur before any inelasticity develops in the rebars).

The present work proposes a mechanical model for the simulation of RC wall boundary elements with lap splices. It represents an extension of the tension chord model (Marti et al. 1998), reviewed in Sect. 2, which adopts the fundamental hypothesis on the constitutive materials (steel and concrete) and bond–slip laws. The model is constituted by an assembly of components, connected in series, which discretize the structural member. The three components, described in Sect. 3 are: (i) the anchorage-slip element; (ii) the basic tension chord element; and (iii) the lap splice element. The solution procedure, which concludes the same section, allows to obtain for an imposed global displacement: the steel and concrete stress and strain distributions along the boundary element, crack location and width, and the total applied axial force. Moreover, the equation proposed by Tarquini et al. (2019) for the lap splice deformation capacity can be directly employed in order to determine the failure of the spliced RC boundary element. The validation of the proposed model is carried out in Sect. 4 in terms of force–displacement, crack widths, and spliced rebar strains. Conclusions are drawn in Sect. 5.

2 Tension Chord Model

The tension chord model was originally developed as a simplified method for the determination of the rotation capacity of flexural plastic hinges in reinforced concrete girders (Sigrist and Marti 1994; Sigrist 1995). Later, it was extended to plane stress analysis problems (Kaufmann 1998; Kaufmann and Marti 1998) and to investigate the deformation capacity of prestressed and non-prestressed RC members (Alvarez 1998). It proved particularly useful in addressing problems of cracking, tension stiffening, and minimum reinforcement in RC members subjected to uniaxial loading (Marti et al. 1998). In the following paragraphs, the main assumptions and theoretical aspects underlying the tension chord model are briefly summarized.

Considering the tension chord element of Fig. 1a, the equilibrium equations for an infinitesimal length dx of concrete and steel volumes can be written as (see Fig. 1b):

$$\frac{{d\sigma_{c} }}{dx} = - \frac{{\tau_{b} \cdot \pi \cdot \phi_{l} }}{{A_{t} \cdot \left( {1 - \rho_{l} } \right)}}$$

(1)

$$\frac{{d\sigma_{s} }}{dx} = \frac{{4 \cdot \tau_{b} }}{{\phi_{l} }}$$

(2)

where σ_c and σ_s are the concrete and steel stresses, τ_b is the bond stress, ϕ_l is the longitudinal rebar diameter, A_t is the gross sectional area, and ρ_l = A_s/A_t is the longitudinal reinforcement ratio, where $A_{s} = \pi \phi_{l}^{2} /4$ is the steel area.

Fig. 1

After (Marti et al. 1998): a Sketch of a basic tension chord element, i.e. the portion of tension chord between two cracks. b Concrete and steel equilibrium for an infinitesimal length dx. c Compatibility requirements.

Strain–displacement relations provide $\varepsilon_{s} = du_{s} /dx$ and $\varepsilon_{c} = du_{c} /dx$, where u_s and u_c are the steel and concrete displacements, as shown Fig. 1c, and $\varepsilon_{s}$ and $\varepsilon_{c}$ are the corresponding strains. From compatibility considerations (Fig. 1c), the steel–concrete slip δ can be expressed as the difference between the steel and concrete displacements:

$$\delta = u_{s} - u_{c}$$

(3)

Assuming a linear and bilinear stress–strain law for the concrete and steel in tension (Fig. 2a, b), the following second order differential equation is obtained by combining the previous equations and using the chain rule of differentiation:

$$\frac{{d^{2} \delta }}{{dx^{2} }} = \frac{{4 \cdot \tau_{b} }}{{{\raise0.7ex\hbox{${d\sigma_{s} }$} \!\mathord{\left/ {\vphantom {{d\sigma_{s} } {dx}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${dx}$}} \cdot \phi_{l} }} + \frac{{\tau_{b} \cdot \pi \cdot \phi_{l} }}{{A_{t} \cdot E_{c} \cdot \left( {1 - \rho_{l} } \right)}}$$

(4)

where $d\sigma_{s} /dx$ and E_c are the steel and concrete tangent stiffness. Before steel yielding $d\sigma_{s} /dx = E_{s}$, while after yielding $d\sigma_{s} /dx = E_{sh}$, which stands for the post-yield steel stiffness.

Fig. 2

Constitutive relations employed in the extended tension chord model: a Bilinear steel stress–strain law. b Concrete tensile stress–strain behaviour. c Steel–concrete bond–slip relationship.

Equation (4) can be integrated if the bond–slip (τ_b–δ) relationship is known. In the tension chord model, the latter is assumed to be stepped, rigid-perfectly-plastic (Fig. 2c): the bond stress is constant (τ_b0) up to reinforcement yielding, after which it halves (τ_b1 = τ_b0/2). This assumption for the τ_b–δ law is particularly convenient as it allows the uncoupling of the bond stress from the slip of the reinforcing bar. In the portion between two consecutive cracks of the tension chord (herein referred as basic tension chord element), the steel stress distribution can thus be derived from the equilibrium conditions alone, i.e. without the need to resort to complex numerical integration of the above second order differential equation. As a consequence, the concrete stress, concrete strain, and steel strain distributions can also be obtained, allowing the determination of the crack width and basic tension chord elongation. In the present work, as suggested by Marti et al. (1998), it is assumed τ_b0 = 2·f_ct and τ_b1 = f_ct, where f_ct is the tensile concrete strength. The latter can be computed as a fraction of the concrete cylinder strength f′_c; the calculations presented herein assume f_ct = 0.3·f′^2/3_c (Sigrist and Marti 1994; Sigrist 1995).

2.1 Evolution of Deformation in a Tension Chord Subjected to Increasing Displacement

Consider a tension chord of total length L₀ (Fig. 3a) subjected to an increasing imposed tensile displacement Δ. A qualitative force–displacement response is depicted in Fig. 3b. Before first cracking, no relative slip occurs between the longitudinal steel and concrete, which thus share the same state of deformation, corresponding to state A in Fig. 3. The equivalent member axial stiffness is computed as the sum of the stiffness of the two materials:

$$\frac{{\left( {EA} \right)_{equiv.} }}{{L_{0} }} = \frac{{\left( {E_{c} \cdot A_{c} + E_{s} \cdot A_{s} } \right)}}{{L_{0} }}$$

(5)

where A_c = A_t − A_s is the concrete area. The first crack occurs when the concrete reaches the tensile strength f_ct (state B in Fig. 3), which takes place at the following applied force:

$$N_{fc} = \frac{{\left( {EA} \right)_{equiv.} \cdot f_{ct} }}{{E_{c} }}$$

(6)

Fig. 3

Tension chord subjected to increasing imposed displacements: a Force–displacement response. b Qualitative steel and concrete strain distributions.

At the first forming crack the concrete stress and strain are null (state C in Fig. 3). All the applied force is taken by the longitudinal reinforcement and the steel stress passes from a pre-crack stress σ_s,B to a post-crack stress σ_s,C. Assuming that the rebar remains elastic, which is typically the case for common ranges of longitudinal reinforcement ratios, they are computed as:

$$\sigma_{s,B} = \frac{{E_{s} \cdot f_{ct} }}{{E_{c} }}$$

(7)

$$\sigma_{s,C} = \frac{{N_{fc} }}{{A_{s} }}$$

(8)

For larger imposed deformations, cracks will open one after the other along the tension chord (state D in Fig. 3 refers to the opening of the second crack) up until crack stabilization. The latter corresponds to the situation in which an increase in the imposed deformation results in simple opening of existing cracks; i.e. no new cracks are forming as shown by states E, F and G in Fig. 3. The difference between the rebar force in the post-crack phase, $\sigma_{s,C} \pi \phi_{l}^{2} /4$, and the rebar force in the pre-crack state, $\sigma_{s,B} \pi \phi_{l}^{2} /4$, is transferred through bond action along a rebar surface $\pi \phi_{l} l_{b}$, which allows to express the development length l_b as (see Fig. 3a):

$$l_{b} = \frac{{\phi_{l} \cdot \left( {\sigma_{s,C} - \sigma_{s,B} } \right)}}{{4 \cdot \tau_{b0} }} = \frac{{\phi_{l} \cdot f_{ct} \cdot \left( {1 - \rho_{l} } \right)}}{{4 \cdot \rho_{l} \cdot \tau_{b0} }}$$

(9)

Two contiguous cracks cannot open at a distance greater than 2 l_b, for in such case a region between these cracks would exist where the concrete stress would be larger than the concrete tensile strength f_ct. On the other hand, crack spacing srm cannot be smaller than l_b, for it would be impossible to develop concrete stress equal to f_ct, even assuming a completely asymmetrical contribution of bond stress towards one of the sides. Although several expressions have been proposed in the literature to evaluate the average crack spacing srm (CEB 1978; FIP-CEB 1990; EN 1992-Part 1-1 2004; FIB 2013), in this paper the intermediate value of srm = 1.5·l_b is assumed, which was judged to provide the compromise between simplicity (limited number of parameters) and accuracy (best final agreement with the experimental results).

The maximum force carried by the concrete, N_c,max, depends on the value of srm and can be computed as:

$$N_{c,max} = \frac{srm}{2} \cdot \tau_{b0} \cdot \phi_{l} \cdot \pi$$

(10)

In between two cracks, the steel and concrete stress and strain distributions can be obtained by solving analytically Eq. (4). The crack width w is then calculated as the integral along srm of the difference between the steel and concrete strains—Eq. (11)—while the integral of the steel strains gives the total elongation of the basic tension chord—Eq. (12).

$$w = \mathop \int \limits_{{ - \frac{srm}{2}}}^{{\frac{srm}{2}}} \varepsilon_{s} - \varepsilon_{c} dx$$

(11)

$$\Delta_{TC} = \mathop \int \limits_{{ - \frac{srm}{2}}}^{{\frac{srm}{2}}} \varepsilon_{s} dx$$

(12)

For the assumed material and bond–slip relationships, in-between cracks the steel and concrete stress and strain distributions remain linear up to yielding of the longitudinal reinforcement (state G in Fig. 3). After yielding, the reduction of the steel tangent stiffness (from E_s to E_sh) and bond stress (from τ_b0 to τ_b1) causes the shift from linear to bilinear stress and strain distributions, as represented in state H of Fig. 3a.

3 Mechanical Model for Boundary Elements of RC Walls with Lap Splices

The mechanical model proposed in the following is developed to simulate the response of RC wall boundary elements with lap splices, subjected to a uniaxial tension–compression loading. It is an extension of the classical tension chord model (Marti et al. 1998) described in the previous section. The basic tension chord element is used in series with an anchorage-slip element and a newly developed lap-splice element; they are described in the next three Sects. 3.1 to 3.3. Section 3.4 deals instead with the iterative solution strategy employed to obtain local deformations and forces for an imposed global displacement.

3.1 Anchorage-Slip Element

The deformation of RC walls due to strain penetration of the longitudinal reinforcement in the foundation can represent an important contribution to the total member displacement (Zhao and Sritharan 2007; Sousa et al. 2018). The anchorage-slip element presented herein allows to estimate the anchorage-slip displacement Δ_anc as well as to determine the steel and concrete stress and strain distributions along the anchorage length l_anc. The input parameter is the steel strain at the loaded end, ε_ac, where the subscript ac stands for ‘at crack’ as it corresponds to the location of the RC wall-foundation interface crack. It is assumed that the rebar capacity cannot crack the foundation and that concrete strains are negligible compared to steel strains, which is typically the case. The elongation Δ_anc can again be computed as the integral of the steel strains:

$$\Delta_{anc} = \mathop \int \limits_{{l_{anc} }}^{{}} \varepsilon_{s} dx$$

(13)

Building on the material and bond assumptions described in the previous section, analytical expressions for Δ_anc can be derived, which depend on the anchorage type (straight or bent rebars), anchorage length, and imposed free-end deformation ε_ac. For a review of the different cases, the reader is referred to Feng and Xu (2018), where a full description of the equations for the calculation of Δ_anc can be found. The current work only considers the anchorage configuration featured by the test units used to carry out the model validation in Sect. 4; i.e., rebars bent inside the foundation and an anchorage length l_anc longer than the development length required to achieve the ultimate steel stress (l_ult). For bent rebars, l_anc can be evaluated as a function of the straight portion of the anchored length l₀ (Sezen and Setzler 2008):

$$l_{anc} = l_{0} + 5 \cdot \phi_{l}$$

(14)

while l_ult, composed by elastic and plastic components (l_y and l_p respectively), can be computed as:

$$l_{ult} = l_{y} + l_{p} = \frac{{f_{y} \cdot \phi_{l} }}{{4 \cdot \tau_{b0} }} + \frac{{\left( {f_{u} - f_{y} } \right) \cdot \phi_{l} }}{{4 \cdot \tau_{b1} }}$$

(15)

where f_y and f_u are the steel yield and ultimate strength. The analytical expression to calculate the slip Δ_anc in case of l_anc > l_ult is given by the following equation:

$$\Delta_{anc} = \left\{ {\begin{array}{*{20}l} {\frac{{\varepsilon_{ac} }}{2} \cdot l_{ac} } & \quad {for\, \varepsilon_{ac} < \varepsilon_{y} } \\ {\frac{{\varepsilon_{y} }}{2} \cdot l_{y} + \frac{{\varepsilon_{y} + \varepsilon_{ac} }}{2} \cdot \left( {l_{ac} - l_{y} } \right) } & \quad {for \, \varepsilon_{ac} > \varepsilon_{y} } \\ \end{array} } \right.$$

(16)

where l_ac is the development rebar length required to attain the strain at crack ε_ac, and is computed as:

$$l_{ac} = \left\{ {\begin{array}{*{20}l} {\frac{{\varepsilon_{ac} \cdot E_{s} \cdot \phi_{l} }}{{4 \cdot \tau_{b0} }} } & \quad {for \, \varepsilon_{ac} < \varepsilon_{y} } \\ {\frac{{f_{y} \cdot \phi_{l} }}{{4 \cdot \tau_{b0} }} + \frac{{\left( {\varepsilon_{ac} - \varepsilon_{y} } \right) \cdot E_{sh} \cdot \phi_{l} }}{{4 \cdot \tau_{b1} }} } & \quad {for \, \varepsilon_{ac} > \varepsilon_{y} } \\ \end{array} } \right.$$

(17)

The previous expressions underline that pre- and post-yielding cases are distinguished; for each case, a sketch of the qualitative steel stress and strain distributions is included in Fig. 4.

Fig. 4

Anchorage-slip element for bent anchored bars with l_anc> l_ult: a Sketch of the anchorage detail. b Qualitative steel strain profiles. c Qualitative steel stress profiles.

The upper and lower domain boundaries of applicability for Eqs. (16) and (17) are defined by the minimum steel strain required to have crack stabilization (ε_cs, see Sect. 3.4 for its calculation) and the ultimate steel strain (ε_ult), respectively. In fact, for ε_ac< ε_cs the RC wall boundary element remains uncracked, implying Δ_anc = 0. On the other hand, steel rupture will occur for ε_ac> ε_ult, thus resulting in the total loss of the member axial load capacity (N = 0).

3.2 Basic Tension Chord Element

As introduced in the previous section, the basic tension chord element represents the portion of a tension chord enclosed between two consecutive cracks (spaced srm apart). For an arbitrary value of ε_cs < ε_ac < ε_ult, where ε_ac is the steel strain at crack, the steel and concrete stress and strain distributions can be determined by equilibrium considerations. The crack width as well as the total chord elongation are obtained by applying Eqs. (11) and (12), respectively. Three cases can be distinguished: (i) ε_ac < ε_y; (ii) ε_ac > ε_y and l_ac,p < srm/2; and (iii) ε_ac > ε_y and l_ac,p > srm/2, where l_ac,p is the length required to develop the plastic strain ε_p = ε_ac - ε_y, computed as:

$$l_{ac,p} = \frac{{A_{s} \cdot E_{sh} \cdot \left( {\varepsilon_{ac} - \varepsilon_{y} } \right)}}{{\tau_{b1} \cdot \pi \cdot \phi_{l} }}$$

(18)

For each case, the material stress and strain distribution is depicted in Fig. 5, while the expression for calculating the basic chord elongation Δ_TC and the crack width w are provided by Eqs. (19) and (20), obtained by solving Eqs. (11) and (12):

$$\Delta_{TC} = \left\{ {\begin{array}{*{20}l} { \left( {\varepsilon_{ac} - \frac{{N_{c,max} }}{{2 \cdot A_{s} \cdot E_{s} }}} \right) \cdot srm } & { for\,case \left( i \right)} \\ {\varepsilon_{ac} \cdot l_{ac,p} + \varepsilon_{y} \cdot \frac{srm}{2} + \varepsilon_{srm/2} \cdot \left( {\frac{srm}{2} - l_{ac,p} } \right) } & {for \, case \left( {ii} \right)} \\ { \left( {\varepsilon_{ac} - \frac{{N_{c,max,p} }}{{2 \cdot A_{s} \cdot E_{sh} }}} \right) \cdot srm} & {for\, case \left( {iii} \right)} \\ \end{array} } \right.$$

(19)

$$w = \left\{ {\begin{array}{*{20}l} {\left( {\varepsilon_{ac} - \frac{{N_{c,max} }}{{2 \cdot A_{s} \cdot E_{s} }} - \frac{{N_{c,max} }}{{2 \cdot A_{c} \cdot E_{c} }}} \right) \cdot srm } & { for \, case \left( i \right)} \\ {\Delta_{TC}^{{case\left( {ii} \right)}} - \varepsilon_{c,y} \cdot \frac{srm}{2} - \varepsilon_{c,srm/2} \cdot \left( {\frac{srm}{2} - l_{ac,p} } \right)} & { for \, case \left( {ii} \right)} \\ {\left( {\varepsilon_{ac} - \frac{{N_{c,max,p} }}{{2 \cdot A_{s} \cdot E_{sh} }} - \frac{{N_{c,max,p} }}{{2 \cdot A_{c} \cdot E_{c} }}} \right) \cdot srm} & { for \, case \left( {iii} \right)} \\ \end{array} } \right.$$

(20)

where ε_srm/2 and ε_c,srm/2 are the steel and concrete strains evaluated at srm/2, Eqs. (21) and (22); ε_c,y is the concrete strain in correspondence of the steel yield point, Eq. (23); and N_c,max,p is the maximum concrete force occurring at srm/2 when the steel is in the post-yield state, Eq. (24).

$$\varepsilon_{srm/2} = \varepsilon_{y} - \frac{{\tau_{b0} \cdot \pi \cdot \phi_{l} \cdot \left( {\frac{srm}{2} - l_{ac,p} } \right)}}{{A_{s} \cdot E_{s} }}$$

(21)

$$\varepsilon_{c,srm/2} = \varepsilon_{c,y} + \frac{{\tau_{b0} \cdot \pi \cdot \phi_{l} \cdot \left( {\frac{srm}{2} - l_{ac,p} } \right)}}{{A_{c} \cdot E_{c} }}$$

(22)

$$\varepsilon_{c,y} = \frac{{\tau_{b1} \cdot \pi \cdot \phi_{l} \cdot l_{ac,p} }}{{A_{c} \cdot E_{c} }}$$

(23)

$$N_{c,max,p} = \sigma_{c,max,p} \cdot A_{c} = \frac{srm}{2} \cdot \tau_{b1} \cdot \phi_{l} \cdot \pi$$

(24)

Fig. 5

Basic tension chord element: a Sketch. b Qualitative steel and concrete strain profile. c Qualitative steel and concrete stress profile.

3.3 Lap-Splice Element

As outlined above, the strength and/or deformation capacity of RC walls may be sensibly reduced by the presence of lap splices above the foundation level, where the seismic demand is maximum. Adequate detailing, i.e. providing appropriate lap-splice length and confining reinforcement, is crucial in order to attain the desired member ductility. The component presented in this subsection allows to account for the presence of lap splices in RC wall boundary elements and to estimate: (i) the steel strain distribution in the pair of spliced rebars; (ii) the crack width along the lap-splice length as well as the width contribution to the splice-end cracks originating from lap-splice deformation; (iii) the total lap-splice displacement and the failure point.

This component builds on the same hypothesis regarding material and bond behaviour assumed for the tension chord model described in Sect. 2. Once crack stabilization is attained along the lap-splice length (Fig. 6a), the resisting force is transferred from the anchored to the free end (unloaded) rebar through concrete bond. It is herein assumed that the concrete remains undeformed while transferring the force from one bar to another. Although the applied tensile load is partly resisted by the concrete, causing the formation of splitting cracks, the previous simplification represents a reasonable approximation up to the point of lap splice failure. In fact, as pointed out by Tastani et al. (2015), neglecting the concrete strain contribution does not result in large model errors since the maximum tensile strain carried by normal strength-concrete up to tensile failure is less than 5% of the yield strain of the reinforcement. Nevertheless, this hypothesis implies a slight overestimation of the crack width as the latter is computed from the steel deformations alone, i.e. the tension stiffening effect due to concrete strains is ignored. A qualitative sketch of the steel stress and strain distribution for the couple of spliced rebars is represented in Fig. 6b, c for two different levels of strain at the interface crack ε_ac (pre- and post-yielding). From the top interface crack downwards, the steel stress is progressively transferred from the top-anchored to the bottom-anchored rebar. The stress transfer stops when equilibrium is reached with the two rebars attaining the same force level. An analogous if mirrored physical phenomenon occurs on the bottom half of the lap splice.

Fig. 6

Lap-splice element: a Sketch. b Qualitative steel strain profile. c Qualitative steel stress profile.

The crack spacing srm within the lap-splice zone is assumed the same as outside, which is supported by experimental observations. The width of the cracks located within the lap-splice region (for instance w_lap,1 and w_lap,2 in Fig. 6a) is computed by integrating, along the corresponding influence length (e.g. l₁ and l₂ in Fig. 6a), the envelope (i.e., the maximum) of the strains along the spliced rebars. For a given crack, the influence length is taken as the sum of the two half distances from the contiguous upper and lower cracks. Similarly, the integral of the strain envelopes along l_top and l_bot (see Fig. 6a) provide the portion of the top and bottom interface crack widths due to deformations originating within the lap splice region. The total width of these cracks is then obtained by summing up the contributions due to deformations occurring within and outside the lap splice region (the next subsection provides further details). The steel strain envelope is considered for the calculation of the crack width because, along the lapped zone, the spliced rebars are in general not equally stressed; the more stressed bar governs crack width (Tastani et al. 2015). Closed-form expressions for the calculation of the splice-internal crack widths and the contribution to the interface cracks given by deformations within the lap-splice region are relatively complicated to obtain as they depend on a large number of variables. Therefore, in the present study these widths are computed numerically.

The total lap splice displacement (Δ_ls) is calculated as the integral of the steel strain envelopes along the entire lap splice length: the expressions for both the elastic (ε_ac < ε_y) and post-yield (ε_ac > ε_y) cases are as follows:

$$\Delta_{ls} = \left\{ {\begin{array}{*{20}l} {\frac{1}{4}\varepsilon_{ac} \left( {2 \cdot l_{s} - l_{ac} } \right) } & \quad { \varepsilon_{ac} < \varepsilon_{y} } \\ {\left( {\varepsilon_{ac} - \varepsilon_{y} } \right) \cdot l_{ac,p} + \left( {\varepsilon_{y} + \varepsilon_{eqF} } \right) \cdot l_{y,eqF} + 2 \cdot \varepsilon_{eqF} \cdot \left( {\frac{{l_{s} }}{2} - l_{ac,p} - l_{y,eqF} } \right)} & \quad {\varepsilon_{ac} > \varepsilon_{y} } \\ \end{array} } \right.$$

(25)

where ε_eqF, Eq. (26), is the strain of both top and bottom anchored rebars when they share the same force, and l_y,eqF is the distance between the yield point and the first point in which ε_eqF is reached, see Fig. 6b.

$$\varepsilon_{eqF} = \frac{{E_{sh} \cdot \left( {\varepsilon_{ac} - \varepsilon_{y} } \right) + E_{s} \cdot \varepsilon_{y} }}{{2 \cdot E_{s} }}$$

(26)

$$l_{y,eqF} = \frac{{A_{s} \cdot \left( {E_{s} \cdot \varepsilon_{y} - E_{sh} \cdot \left( {\varepsilon_{ac} - \varepsilon_{y} } \right)} \right)}}{{2 \cdot \tau_{b0} \cdot \pi \cdot \phi_{l} }}$$

(27)

Based on experimental data on uniaxial cyclic tests on spliced RC members, Tarquini et al. (2019) determined an expression for the average strain at lap splices failure (ε_ls). The latter, which depends on lap splice length, amount of confinement reinforcement and casting position, is used in the proposed model to define the deformation capacity of lap splices. The ultimate lap-splice displacement (Δ_ls,ult) is therefore straightforwardly obtained by multiplying ε_ls by the nominal lap-splice length l_s; failure of the lap-splice element is assumed to take place at this point, with the complete loss of the axial load capacity. It is worth noticing that the above-mentioned failure criterion typically precedes the attainment of non-zero stress values at the lap splice free ends (l_ac > l_s). In such a case, pull-out rather than lap splice failure might be expected.

3.4 Model implementation: iterative procedure and failure criteria

The components described in the three previous subsections can be connected in series in order to simulate the response of RC wall boundary elements with lap splices subjected to increasing tensile loading. Figure 7a illustrates the assembly of an anchorage, lap splice, and several basic tension chord elements, to which a global top displacement Δ_tot is imposed. Given the steel and concrete material properties, outputs of the model are the resisting axial force, crack spacing and widths (inside and outside the lap-splice region), steel and concrete strain distributions, and the ultimate displacement. Iterations are required to solve the nonlinear problem, unless global forces are imposed, wherein a straightforward non-iterative solution is available. A flowchart depicting the steps involved in the iterative procedure is illustrated in Fig. 7b and discussed in the following paragraph.

Fig. 7

RC boundary element model: a Assembly of the components for a RC member featuring lap splice, anchorage, and multiple basic tension chord (TC_i) elements. b Flowchart of the iterative procedure.

Up to first cracking, perfect bond exists between steel and concrete, which therefore share the same strain:

$$\varepsilon_{s} = \varepsilon_{c} = \frac{{\Delta_{tot} }}{{L_{0} }}$$

(28)

where L₀ represents the total length of the boundary element (see Fig. 7a). First cracking occurs at a displacement level $f_{ct} L_{0} /E_{c}$ with N_fc given by Eq. (6). Between first cracking and crack stabilization (identified by the subscript ‘cs’), cracks open one after the other with the axial force that is assumed constant and equal to N = N_fc. In reality, small force drops occur due to the stiffness reduction caused by each crack opening, as shown in Fig. 3. The displacement at crack stabilization Δ_cs is identified by a steel strain at crack equal to:

$$\varepsilon_{cs} = \frac{{N_{fc} }}{{E_{s} \cdot A_{s} }}$$

(29)

The above does not apply to cracks located within the lap-splice region where the steel area contributing to the axial stiffness is double. Such cracks open at an imposed axial force N_fc,lap > N_fc; however, for common longitudinal reinforcement ratios the difference between the two forces is relatively small and can be neglected (e.g., for the case-study of Sect. 4, N_fc,lap = 140 kN while N_fc = 130 kN).

For imposed displacements larger than Δ_cs, each component of the boundary element (anchorage, lap splice, and basic tension chord) can be solved separately for a given strain at crack ε_ac. This quantity is initially estimated as ε_ac = Δ_tot/L₀ which is then used to compute the resulting total boundary element displacement. The latter, identified as Δ_comput, is obtained by summing up the resulting displacement of each element, evaluated through Eqs. (16), (19), and (25). The computed displacement Δ_comput is then compared to the externally imposed Δ_tot: if their difference is smaller than a user-defined tolerance (in the following applications tol = Δ_tot/1000 is used), convergence is attained, otherwise an updated estimate of ε_ac is calculated (see Fig. 7b) and a new iteration is performed. At convergence, the steel/concrete stress/strain distributions can be retrieved from each element, as well as the crack widths (see Fig. 7a). Finally, the total imposed axial force is calculated as:

$$\left\{ {\begin{array}{*{20}l} {N = \varepsilon_{ac} \cdot E_{s} \cdot A_{s} } &\quad {\varepsilon_{ac} < \varepsilon_{y} } \\ {N = \varepsilon_{y} \cdot E_{s} \cdot A_{s} + \left( {\varepsilon_{ac} - \varepsilon_{y} } \right) \cdot E_{sh} \cdot A_{s} } &\quad {\varepsilon_{ac} > \varepsilon_{y} } \\ \end{array} } \right.$$

(30)

Without a specific criterion defining the failure of the boundary element, the procedure above can be performed for any imposed displacement Δ_tot up to a strain at crack equal to the ultimate steel strain (ε_ac = ε_ult). The latter represents a reasonable failure criterion only in case of continuous reinforcement and monotonic tensile loading. In case of cyclic loading and continuous reinforcement, the ultimate steel strain will result in an overestimation of the member displacement capacity and a value of ε_s = 0.6 ε_ult can be used (Priestley et al. 2007). In case lap splices are also present, the attainment of the aforementioned ultimate displacement of the lap splice element, Δ_ls,ult, signals the member failure, which consists in the total and sudden loss of the axial load carrying capacity.

4 Validation of the Proposed Model

Results from a recently concluded experimental programme on RC wall boundary elements with lap splices (Tarquini et al. 2018) are used to validate the proposed mechanical model. The 24 test units (TUs), of which 22 with lap splices and two reference units with continuous reinforcement, shared the same geometry, illustrated in Fig. 8. They differed in terms of lap-splice length, confining reinforcement and loading history, which constituted the variable parameters of the test programme. The testing machine was a uniaxial press with a fixed top and a mobile bottom actuator to which the TUs were connected by means of rigid steel profiles. The instrumentation included load-cells as well as LVDTs and LED grids to evaluate global and local displacement values. Namely, LEDs were also directly glued on the pair of spliced rebars, allowing a direct monitoring of rebar strains.

Fig. 8

Experimental programme used for model validation (Tarquini et al. 2018): a Photo of test unit LAP-P1 in test stand; Example of a reinforcement layout of (b) a TU with lap splices (LAP-P1); and (c) a TU with continuous reinforcement (LAP-C1).

The same reinforcing steel was used for all the TUs. Although different castings were performed, the concrete showed a limited variability in the cylinder compressive strength f’_c. All details regarding material properties can be found in Tarquini et al. (2018); Table 1 reports the material parameters relevant to the mechanical model that were used to run the analyses shown in the next two subsections.

Table 1 Material parameters used in the simulation, obtained from material tests (see Tarquini et al. 2018).

4.1 TU with Continuous Reinforcement

The mechanical model is compared in this section with the test unit LAP-C1 featuring continuous longitudinal reinforcement and a confinement reinforcement ratio ρ_t ≈ 0.3%. The experimental vs numerical force–displacement curves are shown in Fig. 9a with grey and black solid lines, respectively. The applied axial force N is reported on the vertical axis while the total vertical displacement Δ is given on the horizontal axis. A black dashed line indicates the displacement at which the experimental specimen failed, defined as the displacement at which a loss of axial load capacity of 20% was observed. For LAP-C1, this occurred in compression, due to core concrete crushing. The numerical pushover curve reproduces satisfactorily the tensile experimental backbone curve. Namely, the stiffness evolution as well as the maximum force are well predicted. Due to the failure in compression of the specimen, no failure criterion was defined for the mechanical model, which thus does not show decay in strength. Three coloured dots identify post-cracking (Δ = 3 mm), plastic (Δ = 15 mm), and pre-failure (Δ = 35 mm) states, which are used in the following plots to compare local-level quantities.

Fig. 9

Example of a test unit with continuous reinforcement (TU LAP-C1): a Experimental vs numerical force–displacement response. b Simulated steel strain profile. c Simulated steel and concrete force profiles. d Photo of crack pattern. e Numerical crack pattern obtained for different srm to l_b ratios. f Experimental vs numerical crack width.

Simulated steel and concrete strain distributions along the boundary element are depicted in Fig. 9b with solid and dashed lines, respectively. Steel strains were not directly measured in TUs with continuous reinforcement and therefore a comparison between numerical and experimental data is not possible. The numerically obtained distributions are well established and have been presented in several studies on the tension chord model (e.g. Kaufmann 1998; Marti et al. 1998; Muttoni and Fernandez 2010). Steel strains are maximum at crack locations, minimum midway between cracks, and increase from the post-cracked state A to the pre-failure state C. A similar trend can be observed for the steel force distribution (N_s) illustrated in Fig. 9c; on the other hand, concrete forces (N_c) show an opposite behaviour with N_c = 0 at crack and N_c = N_c,max midway between cracks. Furthermore, due to the reduction in bond strength after the occurrence of reinforcement yielding, N_c between cracks decreases with the spread of inelasticity.

The discretization of the boundary element is shown in Fig. 9e and consists in two anchorage and seven basic tension chord elements, corresponding to a crack spacing srm = 1.5·l_b. The same figure also includes, for comparison purposes, the minimum and maximum theoretical crack spacing (srm = l_b and srm = 2·l_b), whereas the real crack pattern observed during the test is reported in Fig. 9d.

Finally, experimental and numerical crack widths are compared, for the three levels of displacement A, B and C, in Fig. 9f. Experimental values, computed as displacement difference between markers located on the same vertical line, are represented with a dotted line, whereas numerical values are displayed in the form of bar plots at crack location. It is found that the model describes satisfactorily the experimental results for all considered displacement levels, concerning both cracks along the TU as well as foundation/top beam interface cracks. Namely, as it will be shown in Fig. 11b, the model error associated to the simulation of the average crack width is smaller than 5%.

4.2 TUs with Lap Splices

In Fig. 10, results from the mechanical model are compared against experimental data obtained from three TUs with lap splices differing in terms of both lap splice length (l_s) and confining reinforcement ratio (ρ_t). In particular, LAP-P4 had the shortest lap-splice length, l_s = 350 mm (25·Ø_l) and ρ_t ≈ 0.3%, followed by LAP-P16, l_s = 560 mm (40·Ø_l) and ρ_t ≈ 0.2%, whereas the longest lap-splice length was part of LAP-P5, l_s = 840 mm (60·Ø_l) and ρ_t ≈ 0.1%. Differently from the model used to simulate the TU with continuous reinforcement, the one used in the present subsection included a lap-splice element above the bottom anchorage element.

Fig. 10

Examples of test units with spliced reinforcement: Experimental vs numerical comparison for TU LAP-P4 (l_s = 25 Ø_l), LAP-P16 (l_s = 40 Ø_l) and LAP-P5 (l_s = 60 Ø_l): a, d, g Force–displacement response. b, e, h Crack width. c, f, i Steel strains within the lap-splice region for the bottom anchored rebar.

From shorter to longer lap splice lengths, the comparison in terms of global force–displacement is shown in in plots (a), (d) and (g). In all three cases, the numerical pushover follows satisfactorily the tensile backbone curve of the cyclic experimental results. Moreover, the ultimate displacement capacity is also rather well predicted (relative error below 20%); the numerical failure is triggered by the attainment of the ultimate lap-splice displacement. For each state (A, B, C) represented by a coloured dot in the force–displacement curves, plots (b), (e) and (h) display the numerical vs experimental crack width. A good match between the two quantities can be observed, the model being able to capture the crack evolution as well as the different openings of cracks within and outside the lap-splice region. Namely, the former are considerably smaller than the latter and the difference in crack width increases with the spread of inelasticity. The distribution of rebar steel strains along the lap splice length is also adequately simulated by the model. In plots (c), (f) and (i), the bottom-anchored rebar of the spliced pair is considered for comparison (the top anchored rebar would show similar but mirrored results). A good fit is again apparent for the three lap-splice lengths, which validates the model assumption of neglecting the concrete deformations within the lap-splice region. No comparison could be performed close to the two splice-end cracks, where steel deformations are maximum, since no strain measure was available at that location. It should also be noted that the effects of cyclic steel strength degradation and cyclic bond strength degradation are not taken into account in the model, which can contribute to explain some of the differences observed with respect to the experimental results, namely between states B and C. However, in view of the overall good agreement, it is believed that these effects were relatively limited.

In order to further validate the proposed mechanical model, the entire set of RC wall boundary elements tested by Tarquini et al. (2018) was simulated. The results are summarized in Fig. 11 in terms of relative error concerning the prediction of: (a) the failure displacement; (b) the average crack width outside the lap splice region; and (c) the average crack width within the lap splice region. The relative error η is expressed by the following equation:

$$\eta = \frac{{\upsilon_{num} - \upsilon_{exp} }}{{\upsilon_{exp} }}$$

(31)

where $\upsilon_{num}$ and $\upsilon_{exp}$ represent the considered numerical and experimental quantities, respectively.

Fig. 11

Relative error for the simulation of all test units: a Ultimate element displacement. b Average crack width outside the lap-splice region. c Average crack width in the lap-splice region.

From Fig. 11a, it is apparent that the relative error between the experimental and numerical ultimate displacement is smaller than 25%, for all specimens. Black dots are used to individuate the TUs depicted in Fig. 10 (i.e., LAP-P4, LAP-P16 and LAP-P5) and to demonstrate that the good model performance was not restricted to those cases. A remark is due to the fact that the database used for the current validation is the same employed to calibrate the deformation capacity of lap splices (Tarquini et al. 2019), and therefore acceptable matches were more likely regarding the ultimate displacement. However, to the authors’ knowledge, no other experimental tests are available in the literature investigating the displacement capacity of lap splices. Moreover, the predictive equation proposed in Tarquini et al. (2019) was derived from the experimental data using a semi-empirical approach, i.e. including assumptions on the mechanical behaviour of the structural member (e.g., the contribution factor to the interface crack α = 0.5). The performed validation hence strengthens the validity of such hypotheses, moreover confirming the dependability of the adopted mechanical approach.

Figure 11b, c show the relative average crack width error, for cracks located outside and within the lap-splice region, respectively. For each TU, the numerical and experimentally-measured crack widths of all cracks located in either of these two regions is computed and then averaged at each experimental peak tensile displacement. Again, the TUs used for validation in Fig. 10 are depicted with black lines; the points corresponding to the states A, B, and C, at which the crack widths were compared, are reported with the same marker notation. Results for all other TUs are represented in grey. For both cracks located outside and within the lapped zone, an error η_w < 20% (approximately) can be observed for all TUs and displacement levels. Exception is made for cracks along the lap-splice zone and low imposed displacements (Δ_tot < 3 mm), where the error can arrive to 40%. This deviation can be partly attributed to the fact that experimental cracks, at such small tensile demands, may still evolve along the member (i.e. new cracks can open at different locations) while the points in which they are evaluated are fixed.

Finally, the relative contribution of the three model elements to the total ultimate displacement (at tensile failure) was computed. The results are plotted in Fig. 12, which shows that the different contributions do not vary significantly for the considered specimens. It represents approximately 20%, 60% and 20% for the anchorage, tension chord and lap splice elements respectively. It is also noted that the contribution from the anchorage region is fairly constant, which is expected since it does not depend on the lap splice length, transverse reinforcement or loading history.

Fig. 12

Relative contribution of the three model elements to the total ultimate displacement (at tensile failure), as predicted by the model for all the test units.

5 Conclusions

Reinforced concrete walls often have lap splices above the foundation level, where the expected seismic demand is maximum. A proper detailing of lap splices is of fundamental importance for the cyclic response of the structural member as it can result in a significant reduction of its strength and deformation capacity. Boundary elements represent the most strained region of the RC wall, and therefore the one where lap splice failure is triggered.

In this paper, a mechanical model for the simulation of RC wall boundary elements with lap splices is presented. It is an extension of the tension chord model for continuous reinforcement. The model comprises three different types of elements connected in series: an anchorage element accounting for the strain penetration of the reinforcement into the foundation, a lap-splice element describing the deformation occurring within the spliced region and a basic tension chord element modelling the response outside the lapped zone. The lap-splice element is a novel contribution of the authors. It assumes that the force is transferred from the anchored to the free end (unloaded) rebar through concrete bond. It is further hypothesized that in the lap splice region the concrete transfers stresses from one rebar to the other but does not contribute significantly to the stiffness of the lap splice element (i.e. the tension stiffening effect is neglected). The model is highly versatile and allows any combination and number of the above-mentioned elements, the solution of which requires an iterative method. A new stable solution procedure is therefore proposed to solve the global nonlinear problem. The average lap splice strain at failure suggested by Tarquini et al. (2019) is herein used to determine the ultimate displacement of boundary elements with lap splices.

Finally, the model was validated against a set of 24 tests on RC wall boundary elements with both continuous and spliced longitudinal reinforcement. Different lap-splice lengths and confining reinforcement were also considered. Comparisons were made in terms of force–displacement response, crack width and strain distribution along the pair of spliced rebars. In all cases, and despite the fact that cyclic steel strength degradation and cyclic bond strength degradation are not accounted for in the model, a good match is found between numerical and experimental results; relative errors regarding the ultimate displacement and crack widths are on average below 20%.