1 Introduction

The most commonly used methods to splice reinforcing steel bars are lap splices, welded splices and mechanical connections. Since the 1970s when grouted splices (to be referred to as splices from this point forward) were invented (Yee 1973), they have been used as a preferred technology to splice bars particularly in precast concrete elements. This type of splice can provide continuity of reinforcement between precast elements and develop quality connections maintaining structural integrity. Figure 1 illustrates a splice of typical configuration. The grout-filled coupling sleeve came in different sizes to splice reinforcing bars of the adjacent precast members. Ribs were produced on the sleeve’s interior surface to increase the resistance force of the bond between the sleeve and grout. A stop located in the sleeve’s midsection ensures appropriate positioning of the two embedded bars. On construction sites, high-strength and non-shrink cementitious grout was cast into the sleeve and around the bars using a low-pressure grout pump. By doing this, lateral confinement to the bars was generated to enhance the grout-bar bond, resulting in a transfer of axial force between discontinued bars.

Fig. 1
figure 1

Grouted splice of typical configuration used in this study.

Efforts have been conducted to investigate the mechanical performance of splices with coupling sleeves of various geometrical and mechanical configurations, mostly under incremental tensile loading. In general, the most common failure modes for splices under incremental tensile loading have been bar fracture and bar pull-out due to grout-bar bond failure (Einea et al. 1995; Jansson 2008; Ling et al. 2012). In these circumstances, the mechanical performance of the splices essentially depends on the bond behavior between grout and bars. Test results have indicated that the mechanical performance of the splices is generally not as strong as that of the connected bars. For example, the tensile strengths and deformation capacities of the splices were lower than those of the connected bars (Einea et al. 1995; Ling et al. 2012; Kim and Lee 2012; Ling et al. 2014; Henin and Morcous 2015), e.g., the ultimate strengths and ultimate strains of 24 specimens appropriately fabricated by Ling et al. (2014) averaged 95 and 62 % of those of the connected bars, respectively. In general, appropriate geometrical configuration of coupling sleeves, adequate embedded length of bars, and high compressive strength of grout help to enhance the grout-bar bond and, therefore, improve the mechanical performance of the splices. On the other hand, limited information is available for the behavior of splices under cyclic loading. Hayashi et al. (1997) studied strength, rigidity, capacity of elongation, and the bond stress-slip relationship of splices under cyclic loading. The loading reversed 20 loops and ranged from 0.95 F ys in tension to −0.5 F ys in compression, where F ys denotes the force corresponding to the yield strength of the connected bars. They found that the mechanical performance of the splices improved with an increase in embedded length of bars and compressive strength of grout.

Despite these efforts, data are still scarce on the mechanical performance of splices, especially those involved in cyclic loadings when the splices are considered for use in earthquake-prone areas. Moreover, stress–strain relationships for splices under various loading schemes have not been available, which has made it difficult to build accurate models to simulate and to understand the behavior of structural components and structures (Ren et al. 2015; Ameli et al. 2015). In fact, most of the technical details for this type of product are private and proprietary with restricted access. In this regard, tests on a total of 36 specimens were conducted in this study to gain new knowledge on the characteristics of the splices under tensile and cyclic loadings. Special attention was paid to the deformation capacities of the splices and the association of the mechanical performance of the splices under different loadings. Stress–strain relationships were presented for splices under various loadings.

Four loading schemes were used in this study, i.e., incremental tensile (IT) loadings and repeated tensile (RT) loadings as well as cyclic loadings at high stress (CH) and at large strain (CL). The second loading scheme (RT) relates to a splice under service loads with a stress level of generally no more than 60 % of the yield strength of the spliced bars. The third and fourth loading schemes (CH and CL) are of special interest because they involve splices under frequent and rare earthquakes, respectively. Commonly, the stress level of the splices under frequent earthquakes did not exceed the bar’s yield strength. However, for the splices under rare earthquakes, the stress level of the splices generally can go beyond the bar’s yield strength and large plastic strain could occur. As a result, 90 % of the bar’s yield strength and two to five times of the bar’s yield strain in tension were defined as high stress and large strain, respectively, along with reversed loading of up to 33 % of the bar’s yield strength in compression.

2 Experimental Program

2.1 Materials and Specimen Fabrication

The splice specimens consisted of three components of various types, i.e., four sizes of coupling sleeves (5VSA, 7VSA, 8VSA and 10VSA), grout, and two grades of reinforcing steel bars (HRB400 and HRB500). The coupling sleeves and grout used in this study were manufactured by a producer and are commercially available. The sleeves were cast of ductile iron with a specified tensile strength and relative elongation of no less than 600 MPa and 3 %, respectively (JGJ355-2015 2015). The four sizes of the coupling sleeves with specific geometric parameters, as presented in Fig. 1, were used for different diameters of connected bars. The left part with a length of L 2 and right part with a length of L 1 for each coupling sleeve were not similar and thus the product was non-symmetrical. The left and right parts were associated with an outlet port and inlet port, respectively. This profile made it easy to embed bars and ensure the quality of casting grout. The prepackaged grout was mixed per the manufacturer’s instructions with a fluidity of 180–270 mm for easy operation. Tables 1 and 2 present the mechanical properties of the reinforcing steel bars and grout at 28 days obtained by testing. All data in the tables represent the mean values of the results of three specimens. Obviously, the strengths of the bar grade HRB500 are higher than those of HRB400. In fabrication of the splices, the sleeves were first placed vertically by means of fixture. The bars were then carefully inserted into the sleeves from their two ends to make the two bars line up as accurately as possible. Ring-form rubber end seals were affixed to the narrow end of each coupling sleeve to prevent grout leakage. Afterwards, the grout was poured in the sleeves from inlet ports using a grout pump, as illustrated in Fig. 2. Finally, the splice specimens were removed from the fixture after 24 h and horizontally placed at room temperature for about 28 days until testing.

Table 1 Mechanical properties of reinforcing steel bars.
Table 2 Mechanical properties of grout.
Fig. 2
figure 2

Pouring grout in splice specimens.

2.2 Specimens Under Incremental and Repeated Tensile Loadings

Table 3 presents details of 24 IT- and RT-series specimens. For each loading scheme, the test plan included two grades of bars with six different diameters. Table 4 shows the loading procedures by reference to the Chinese code (JGJ355-2015 2015). These loading procedures were also similar to the international acceptance criteria AC133 (2014) but not identical. A 500 kN servo-hydraulic actuator with an accuracy of 2 kN was used for loading. Figure 3 illustrates the test setup, arrangement of the displacement transducers and strain gauges. The measured data included loading forces, sleeve strains, and splice deformations within gauge length of L 3 (L 3 = L + 4d, where L and d denote sleeve length and bar diameter, respectively). The gauge length of L 3 was used in the Chinese code JGJ355-2015 and uniformly adopted for all 36 specimens in this study for the sake of comparison. The loading forces were recorded automatically by a data acquisition device of the actuator. For each coupling sleeve, its strain distribution was investigated by sticking strain gauge rosettes, sp 1 to sp 6, along the length of its external surface. Two displacement transducers with an accuracy of 0.01 mm were used for each specimen and their average values were applied to obtain deformations. As a result, the relative elongations at ultimate loads δ IT of the IT-series splices are calculated by means of the elongations, u IT, and deformed length within the gauge length of L 3, L IT, using Eq. (1):

Table 3 Details of specimens under incremental and repeated tensile loadings.
Table 4 Loading procedures for specimens under incremental and repeated tensile loadings.
Fig. 3
figure 3

Test setup, arrangement of displacement transducers and strain gauges for specimens under incremental and repeated tensile loadings.

$$ \delta_{{\rm {IT}}} = \frac{{u_{{{{\rm IT}}}} }}{{L_{ 3} }},\;\text{with}\;u_{{{{\rm IT}}}} = L_{{\text{IT}}} - L_{3} $$
(1)

Similarly, the relative elongations at ultimate loads δ RT of the RT-series specimens are calculated by means of the elongations, u RT, and deformed length within the gauge length of L 3, L RT, using Eq. (2):

$$ \delta_{{\rm {RT}}} = \frac{{u_{{{{\rm RT}}}} }}{{L_{ 3} }},\;\text{with}\;u_{{{{\rm RT}}}} = L_{{\text{RT}}} - L_{3} $$
(2)

The relative elongations δ IT and δ RT gave mean values similar to the ultimate strains of bars. The loading force was controlled with a stress rate of about 30 MPa/s before bar yielding and beyond that transferred to displacement control with a strain rate of about 0.002 s. Deformations within the gauge length rather than L 3 were also measured for another research purpose, which is not presented herein.

2.3 Specimens Under Cyclic Loading at High Stress and at Large Strain

Table 5 presents details of the CH- and CL-series specimens. For each loading scheme, the test plan included two grades of bars with three typical diameters, i.e., small, medium and large diameters of 16, 25 and 32 mm, respectively.

Table 5 Details of specimens under cyclic loadings at high stress and at large strain.

Table 6 shows the loading procedures and to be measured residual deformations u 20, u 4 and u 8 (JGJ355-2015 2015). A 2000 kN servo-hydraulic actuator with an accuracy of 2 kN was used and illustrated in Fig. 4, because this machine can apply loading in a cyclic way. The ways to obtain these data and loading rules were identical to those under incremental and repeated tensile loadings. The relative residual deformations δ 20, δ 4 and δ 8 of the splices can be calculated using Eqs. (3)–(5), respectively:

$$ \delta_{{\text{20}}} = \frac{{u_{{\text{20}}} }}{{L_{ 3} }} $$
(3)
$$ \delta_{\text{4}} = \frac{{u_{\text{4}} }}{{L_{ 3} }} $$
(4)
$$ \delta_{\text{8}} = \frac{{u_{\text{8}} }}{{L_{ 3} }} $$
(5)

where u 20 is the residual deformation of the CH-series specimens while u 4 and u 8 denote the residual deformations of the CL-series specimens. The relative residual deformations describe the effect of cyclic loading on the deformations of the splices.

Table 6 Loading procedures for specimens under incremental and repeated tensile loadings.
Fig. 4
figure 4

Test setup for specimens under cyclic loadings at high stress and at large strain.

3 Test Results

3.1 Specimens Under Incremental and Repeated Tensile Loadings

Table 7 summarizes the test results of the IT- and RT-series specimens.

Table 7 Test results for specimens under incremental and repeated tensile loadings.

3.1.1 Failure Modes

Two failure modes were observed and illustrated in Fig. 5, i.e., bar fracture and bar pull-out due to grout-bar bond failure. For the bar pull-out failure, a block of cone-shaped grout with an angle of about 45° was pulled out of the bar. Table 7 presents the failure mode for each specimen. These two failure modes were also observed in previous studies (Einea et al. 1995; Jansson 2008; Ling et al. 2012). Other failure modes found in these three references, e.g., sleeve tensile failure and grout-sleeve bond failure, did not occur in this study. This was probably because, in the current study, the coupling sleeve and grout-sleeve bond provided relatively strong resistance compared to the resistance offered by the connected bars and grout-bar bond. As a result, sleeve tensile failure and grout-sleeve bond failure were prevented.

Fig. 5
figure 5

Two failure modes of specimens under incremental and repeated tensile loadings.

3.1.2 Ultimate Loads

Failure modes have revealed that the ultimate load of a specimen depends on two factors, i.e., the ultimate tension of the bar and the resistance force provided by the grout-bar bond, of which is weaker. Table 7 also indicates that the ultimate loads P u of the specimens using the HRB500 reinforcing steel bar were greater than those that used the HRB400 bar. In general, this observation was explicable when associated with the two failure modes. The failure mode of the bar fracture involved the specimens using the HRB400 and HRB500 reinforcing steel bars with their diameter of 14 mm. The failure mode of the bar pull-out included specimens with a diameter of 22, 25 and 32 mm. For the bar fracture, the ultimate loads depended on the ultimate strengths of the connected bars. Therefore, the ultimate loads of the specimens using the HRB500 reinforcing steel bars were greater than those of the HRB400 bars because the ultimate strength of a HRB500 bar is higher than that of a HRB400 bar with identical diameter. For the bar pull-out failure, the average bond strength of the HRB400 bar is less than that of the HRB500 bar because the low yield strength of the HRB400 bars comes along with a low average bond strength (Steuck et al. 2009). The local bond stresses were reduced by two effects related to the yielding of reinforcing steel bars. First, the large bar strains associated with yielding increase the slip of the bar from the grout, which can exceed the slip corresponding to peak bond stress and thus reduce the bond. Second, plastic deformation reduces the bar’s area which also negatively affects the bond. In addition, the stresses of bars at ultimate loads averaged 92 and 91 % of the ultimate stresses of the embedded bars for the IT- and RT-series specimens, respectively. This meant that repeated loading of about 0.6F ys had a minor effect on the ultimate loads of the specimens.

3.1.3 Deformations

The relative elongations at ultimate loads δ IT and δ RT presented in Table 7 ranged from 3.7 to 11.0 % with a mean value of about 6.6 %, which was significantly less than the ultimate strains of the embedded bars, which varied from 11 % (0.11) to 15 % (0.15) with a mean value of about 12 % as presented in Table 1. This indicated a weak deformation capacity of the splices compared to the bars in ultimate state. Moreover, the relative elongations averaged 6.8 and 6.4 % for the IT- and RT-series specimens, respectively, which showed no obvious differences between the specimens under the two load types.

3.1.4 Load-Deformation Curves

Load-deformation responses of the specimens were obtained and typical curves are presented in Fig. 6. Four horizontal lines from the bottom up in each figure correspond to yield loads of bars HRB400 and HRB500 and the ultimate loads of bars HRB400 and HRB500, respectively. Three stages can be recognized from the curves, i.e., elastic, yield and hardening stages. Generally, no undesired brittle failure behavior without yield or hardening stages could be found in this study. Such failure is characterized by bar pull-out failure before yielding of the spliced bars in certain specimens, as noted in (Ling et al. 2012). In this study, specimens failed after yielding and at least close to 0.83 (see Table 7) of the ultimate strengths of the connected bars. As a result, the failure was ductile even though the splices underwent bar pull-out. This is also evident from the flattened hardening stages in the load-deformation curves in Fig. 6. Additionally, the unloaded and reloaded paths almost overlapped for the RT-series specimens. Typically, the residual deformations were about 0.09 mm for specimens RT-H400D25 and RT-H500D25, which was approximately 5 % of the deformations when splices yielded. Therefore, the residual deformations in RT-series specimens can be ignored for the tested splices.

Fig. 6
figure 6

Typical load-deformation curves for specimens under incremental and repeated tensile loadings.

3.1.5 Behavior of Coupling Sleeves

Mechanical behavior of the coupling sleeves was investigated in terms of strain development and peak strain distributions at ultimate loads on their external surfaces both in the longitudinal and transverse directions. As an example, Fig. 7 presents the peak strain distribution on the surface of specimen IT-H400D25. The values on the x axis denote the distances from the narrow end of the sleeve to the observed points. First, it was found that along the longitudinal direction, all coupling sleeves were in tension. The peak strain of each specimen appeared in the midsection and decreased gradually towards the two sleeve ends. Second, in the transverse direction, the strains were relatively small compared to those in the longitudinal direction. Peak strains were generally located in the midsection but were compressive. Located outside of the central parts, the sleeves were in tension or compression. Finally, the principal strain distributions of all sleeves were calculated based on the strain analysis theory of continuum mechanics by means of strain gauge rosettes. The principal stresses did not exceed their specified tensile strength of 600 MPa with an assumption of the elasticity modulus of 200 × 103 MPa.

Fig. 7
figure 7

Peak strain distribution on the external surface of the coupling sleeve of specimen IT-H400D25.

These observations can be interpreted as follows. Two spliced bars were disconnected near the middle section of a sleeve. As a result, the middle part of the sleeve carried almost all the accumulative tensile loading transferred from the embedded bar and, therefore, resulted in peak stress. In the transverse direction, the strain condition depended on the combination of two factors. The first one was the Poisson effect because sleeves were tensile loaded longitudinally resulting in transverse shrinkage. The second factor was split expansion in the radial direction induced by the filled grout due to bar-grout bond, which caused the sleeves to dilate transversely. Consequently, different sleeve configurations influenced the two factors, resulting in various strain distributions in the transverse direction which can be different from the observation in this study (Einea et al. 1995; Ling et al. 2012).

3.2 Specimens Under Cyclic Loadings at High Stress and at Large Strain

Table 8 summarizes the test results of the CH- and CL-series specimens.

Table 8 Test results for specimens under cyclic loadings at high stress and at large strain.

3.2.1 Failure Modes

Bar pull-out failure was found for all specimens. Only specimen CH-H400D16 formed an exception which failed in bar fracture. In most cases, the failure mode of a specimen was identical to that under tensile loadings.

3.2.2 Ultimate Loads

The ultimate loads presented in Table 8 were compared with those of the IT-series specimens as illustrated in Fig. 8. It is evident that the ultimate loads under different loading schemes were almost identical, indicating that the strength degradation of the bar-grout bond of specimens under cyclic loading, if any, was insignificant.

Fig. 8
figure 8

Comparison of the ultimate loads under different loadings.

3.2.3 Deformations

The relative residual deformations δ 20, δ 4 and δ 8 presented in Table 8 scattered in ranges of 0.06 to 0.60, 0.03 to 0.12 %, and 0.07–0.31 %, respectively. This implied appreciable residual deformations concerning grout-bar bond for the CH- and CL-series specimens, compared to yield strains of about 0.25 % for the embedded bars.

3.2.4 Load-Deformation Curves

Figures 9 and 10 present the typical load-deformation curves for specimens CH-H400D16 and CL-H400D16, respectively. First, it was found that the load-deformation responses featured a significant pinching effect because the curves contained approximately horizontal sections near the origin, and the hysteresis loops were severely ‘pinched’. The pinching effect is an important issue associated with RC structural members under cyclic loading, generally resulting in reduced ductility and energy dissipation capacities. The pinching effect is believed to be caused by opening and closing of concrete cracks, and by the concrete-bar bond behavior in loading, unloading, and reverse loading as well (Kim 2016). This effect has been observed in a previous study (Hayashi et al. 1997) and is also similar to the behavior of a reinforcing steel bar anchored in concrete under cyclic loading (Model Code 2010). Second, the stiffness degraded gradually, which led to an increase in residual deformation. Finally, the shape of the load-deformation curves for specimens under cyclic loadings were similar to those under incremental tensile loading. These findings formed a basis for presenting the stress–strain relationship of the splices under cyclic loadings.

Fig. 9
figure 9

Load-deformation curves for specimens CH-H400D16 under cyclic loading at high stress.

Fig. 10
figure 10

Load-deformation curves for specimens CL-H400D16 under cyclic loading at large strain.

3.3 Discussion

The ultimate load of a splice was less than that of the connected bars, i.e., with averages of about 92, 91, 90 and 93 % of the ultimate load of the connected bar for the IT-, RT-, CH- and CL-series specimens, respectively. The main reasons are believed to be due to the failure modes, misalignment and offset of the embedded bars (Jansson 2008; Ling et al. 2012). The dominant failure mode of the naked plain bars in tensile tests was bar fracture, while the failure mode of most of the splices was bar pull-out associated with relatively low ultimate strengths. The misalignment and offset generated secondary moments, which led to nonuniform stress distribution in the critical section of the connected bars. This caused local and earlier failure compared to the case of purely axial loading. As a result, the ultimate strengths and ultimate strains of the splices decreased. Figure 11 illustrates an example of misalignment. The measured misalignment angles in this study ranged from 0.23° to 3.27° and the offsets were not available due to measurement difficulties.

Fig. 11
figure 11

Misalignment of two spliced bars.

Additionally, the ultimate deformation capacities of the splices were not as strong as those of the connected bars, as mentioned in Sects. 3.1.3 and 3.2.3. The reasons for these differences are believed to be the relatively weak deformation capacity of the sleeve region, misalignment, and offset of the bars.

Furthermore, appreciable distinctions of ultimate deformation were found between the splices with two different failure modes. For IT- and RT-series specimens with bar fracture, the relative elongations at ultimate loads ranged from 5.2 to 8.3 % with an average of 7.0 %. However, for IT- and RT-series specimens with bar pull-out, corresponding values varied from 3.7 to 11.0 % with an average of 6.3 %. These results were consistent with the understanding that the deformation capacities of bars were better than those of the sleeve regions (JGJ355-2015 2015). Based on these findings, it was suggested to pursue the failure mode of bar fracture instead of bar pull-out in the product design process to make full use of the deformation capacities of the bars.

Besides, experimental results from tests on all 36 specimens indicated that each splice, in tension, developed more than 100 % of the specified tensile strength of the connected bar, and also more than 125 % of the specified yield strength of the connected bar. This means that all specimens met the requirements of Type 2 splices according to AC133 (2014).

Finally, the mechanical performance of the splices, e.g., the ultimate strengths and ultimate deformation capacities, was relatively weak compared to that of the reinforcing steel bars. This may lead to performance variation in the plastic hinge zones of precast structural members in which grouted splices are used. Particularly, in earthquake-prone areas, the plastic hinge zones must resist earthquake-induced deformations and stresses. The performance of the plastic hinge zones partially depend on the mechanical performance of the reinforcement and the grouted splices (Ameli et al. 2015). However, this issue is beyond the scope of this study and needs further research.

4 Stress–Strain Relationships

4.1 For Splices Under Tensile Loadings

Under incremental and repeated tensile loadings, the splices behaved similar to the connected bars. Load-deformation curves of the splices and bars generally consisted of three stages, i.e., an elastic stage, a yield plateau and a hardening stage. Moreover, almost no strength and stiffness degradations were found for the RT-series splices compared to the IT-series splices where stress was no more than 0.6 f ys. However, primary differences existed between the two curves of the splices and bars. As presented in Table 9, the elasticity moduli, yield strengths, yield plateau lengths, ultimate strengths, and ultimate strains (i.e., relative elongations at ultimate loads) of the splices were reduced on average by about 9, 8, 63, 8, and 44 % compared to those of the bars, respectively, when a gauge length of L 3 was considered for comparison. As a result, the stress–strain relationship for the splices under incremental and repeated tensile loadings can be expressed in a form similar to that for reinforcing steel bars with consideration of the above differences.

Table 9 Comparison of parameters between splices and bars.

4.2 For Splices Under Cyclic Loadings at High Stress and at Large Strain

Pinching effect and stiffness degradation were the main characteristics for the CH- and CL-series splices. These features were similar to those of bond stress-slip behavior between embedded reinforcing steel bars and concrete under cyclic loading. The analytical model for bond stress-slip behavior proposed in (Teng and Zou 1996) appropriately considers these characteristics and, therefore, was used as a basis for developing a stress–strain relationship of the tested splices.

Figure 12 presents the profile of stress–strain curves for splices under cyclic loadings at high stress and at large strain. The thick solid skeleton lines are composed of three line segments (segment OF′, F′ G′ and GW), which can be determined from test results of the splices under incremental tensile loading, as described in Sect. 4.1 for the splices used in this study. For the first loading, elastic behavior is assumed if the stress was not beyond 0.9 f y and then unloading. Otherwise, reversed characteristics are expressed as follows:

Fig. 12
figure 12

Stress–strain curves for specimens under cyclic loadings at high stress and at large strain.

  1. (1)

    First loading beyond 0.9f y (segment OA)

    The splice is first elastically loaded to point A of stress σ A and strain ε A with elasticity modulus E, in which σ A is greater than 0.9 f y:

    $$ \varepsilon_{\text{A}} \text{ = }\frac{{\sigma_{\text{A}} }}{E} $$
    (6)
  2. (2)

    First positive unloading (segment AB)

    For the first positive unloading from point A to point B along with the stresses reducing to 0, a modulus E 2 in unloading was obtained from tests.

  3. (3)

    First negative loading (segment BCD A′)

    The splice is negatively loaded from point B to C with the modulus E 2. Afterwards, the strain decreases rapidly from point C to D due to the pinching effect. The splice is then negatively loaded from point D to control point A′ (stress level was controlled by no more than—0.33 f y with a modulus of E 3 in the tests). The relationships between these parameters are:

    $$ \sigma_{\text{C}} = - \alpha {}_{\text{f}}\sigma_{\text{A}} $$
    (7)
    $$ \varepsilon_{\text{C}} \text{ = }\varepsilon_{\text{A}} \text{ + }\frac{{\sigma_{\text{C}} - \sigma_{\text{A}} }}{{E_{\text{2}} }} $$
    (8)
    $$ \sigma_{\text{D}} \text{ = }\sigma_{\text{C}} $$
    (9)
    $$ \varepsilon_{\text{D}} \text{ = }\varepsilon_{\text{C}} - {{\Delta }}\varepsilon_{\text{ - }} $$
    (10)
    $$ \varepsilon_{{\text{A}^{\text{'}} }} \text{ = }\varepsilon_{\text{D}} \text{ + }\frac{{\sigma_{{\text{A}^{\text{'}} }} \text{ - }\sigma_{\text{D}} }}{{E_{\text{3}} }} $$
    (11)

    where α f denotes the stress coefficient due to negative friction, and \( \Delta \varepsilon_{ - } \) means horizontal strain length under negative loading due to the pinching effect.

  4. (4)

    Negative unloading and reloading (segment ABCDA″)

    Similar to positive unloading and negative loading (segment AABCD A′), the relations between the related parameters are:

    $$ \upsigma_{{\text{C}^{{\prime }} }} { = - \alpha }_{\text{f}}^{{\prime }}\upsigma_{{\text{A}^{{\prime }} }} $$
    (12)
    $$ \varepsilon_{{\text{C}^{\text{'}} }} \text{ = }\varepsilon_{{\text{A}^{\text{'}} }} \text{ + }\frac{{\sigma_{{\text{C}^{\text{'}} }} \text{ - }\sigma_{{\text{A}^{\text{'}} }} }}{{E_{\text{4}} }} $$
    (13)
    $$ \sigma_{{\text{D}^{\text{'}} }} \text{ = }\sigma_{{\text{C}^{\text{'}} }} $$
    (14)
    $$ \varepsilon_{{\text{D}^{\text{'}} }} \text{ = }\varepsilon_{{\text{C}^{\text{'}} }} \text{ + }\Delta \varepsilon_{\text{ + }} $$
    (15)
    $$ \varepsilon_{{\text{A}^{{\text{''}}} }} \text{ = }\varepsilon_{{\text{D}^{\text{'}} }} \text{ + }\frac{{\sigma_{{\text{A}^{{\text{''}}} }} \text{ - }\sigma_{{\text{D}^{\text{'}} }} }}{{E_{\text{1}} }} $$
    (16)

    where \( \alpha_{\text{f}}^{\text{'}} \) denotes the stress coefficient due to positive friction, \( {{\Delta }}\varepsilon_{\text{ + }} \) means horizontal strain length under positive loading due to the pinching effect, E 4 is the modulus in negative unloading, and E 1 is the modulus when reloading.

  5. (5)

    Loops for unloading hysteresis after point A″ outside hardening stage (after A”)

    The loop paths following “positive unloading–negative loading–negative unloading–positive reloading” are similar to that of segment AABCD ABCDA″.

  6. (6)

    Loops for loading hysteresis after point A″ outside hardening stage (after A″)

    The loading continues to follow the skeleton lines. The coordinate of point E at intersection can be derived from the two straight-lines OF′ and DA″ in the strain–stress coordinate system.

  7. (7)

    Unloading and reloading in hardening stage (segment HIJKLMHIJKN)

    The loop paths are similar to that of ABCDABCDE with the moduli of E 2, E 3, E 4 and E 1, respectively. Assuming that the yield strength in the negative loading process decreases due to the Bauschinger effect:

    $$ \sigma_{\text{L}} \text{ = }\sigma_{\text{M}} \text{ = }\sigma_{\text{H}} \text{ - 2}f_{\text{y}} $$
    (17)

    the parameters used in the stress–strain relationship, i.e., f y, f u, E, E h, Δε y, E 1, E 2, E 3, E 4, α f, \( \alpha_{\text{f}}^{\text{'}} \), \( \Delta \varepsilon_{ - } \), \( \Delta \varepsilon_{\text{ + }} \) are obtained from test results and are presented in Table 10.

    Table 10 Parameters for stress–strain relationships of specimens under cyclic loadings at high stress and at large strain.

As an example, Fig. 13 faces the modeled and tested stress–strain curves for specimens CH-H400D16 and CL-H400D16. For each specimen, the parameters for skeleton lines (f y, f u, E, E h, Δε y) and the mean values of other parameters (E 1, E 2, E 3, E 4, α f, \( \alpha_{\text{f}}^{\text{'}} \), \( {{\Delta }}\varepsilon_{ - } \), \( {{\Delta }}\varepsilon_{\text{ + }} \)) were used. In general, the proposed model can appropriately describe the mechanical behavior of splices under cyclic loadings.

Fig. 13
figure 13

Modeled and tested stress–strain curves for splices under cyclic loading at high stress and at large strain.

5 Conclusions

Experimental studies were performed on 36 specimens of grouted splices under four loading schemes involving incremental tensile and repeated tensile loadings, cyclic loadings at high stress and at large strain. The following conclusions can be drawn based on this research:

  1. (1)

    No appreciable differences were found between the mechanical performance of splices under incremental and repeated tensile loadings. The repeated loading of about 0.6F ys had a minor effect on the ultimate loads and relative elongations of the splices. The unloading and reloading paths almost overlapped for specimens under repeated tensile loading, which indicated the splices worked well under service loads. In addition, different sleeve configurations influenced the sleeves’ strain distributions in the transverse direction.

  2. (2)

    Mechanical behavior of splices under cyclic loading has a pronounced pinching effect and stiffness degradation. The shape of the load-deformation curves for specimens under cyclic loadings were similar to those under incremental tensile loading. This finding formed a basis for presenting the stress–strain relationship of the splices under cyclic loading. Appreciable residual deformations were found for specimens under cyclic loadings. Degradation of the ultimate strengths and ultimate deformation capacities were not observed compared to specimens under tensile loadings.

  3. (3)

    Two stress–strain relationships, which considered the main aspects of the mechanical performance, were presented for the splices under tensile loadings and under cyclic loadings.