Flexural Response of UHPC Wet Joints Subjected to Vibration Load: Experimental and Theoretical Investigation

Abstract

This study aims to investigate the flexural performance of ultra-high-performance concrete (UHPC) wet joints subjected to vibration load during the early curing period. The parameters investigated included vibration amplitude (1 mm, 3 mm, and 5 mm) and vibration stage (pouring—final setting, pouring—initial setting, and initial setting—final setting). A novel simulated vibration test set-up was developed to reproduce the actual vibration conditions of the joints. The actuator’s reaction force time-history curves for the UHPC joint indicate that the reaction force is stable during the initial setting stage, and it increases linearly with time from the initial setting to the final setting, trending toward stability after 16 h of casting. Under the vibration of 3 Hz-5 mm, cracks measuring 14 cm × 0.2 mm emerge in the UHPC joint. It occurs during the stage from the initial setting to the final setting. The flexural performance of wet joint specimens after vibration was evaluated by the four-point flexural test, focusing on failure modes, load-deflection curves, and the interface opening. The results show that all specimens with joints exhibited bending failure, with cracks predominantly concentrated at the interfaces and the sides of the NC precast segment. The interfacial bond strength was reduced by vibrations of higher amplitude and frequency. Compared with the specimens without vibration, the flexural strength of specimens subjected to the vibration at 3 Hz-3 mm and 3 Hz-5 mm were decreased by 8% and 19%, respectively. However, as the amplitude and frequency decreased, the flexural strength of the specimens showed an increasing trend, as this type of vibration enhanced the compactness of the concrete. Additionally, the calculation model for the flexural strength of UHPC joints has been established, taking into account the impact of live-load vibration. The average ratio of theoretical calculation values to experimental values is 1.01, and the standard deviation is 0.04, the theoretical calculation value is relatively precise.

1. Introduction

Ultra-high-performance concrete (UHPC), a new type of cement-based composite material widely used in various building structures, has always attracted significant attention regarding its durability issues [1,2,3,4,5]. Particularly when facing extreme loads such as earthquakes, explosions, impacts, and overload, many serious problems have been exposed. Due to the limitations in the mixing, transportation, and curing of UHPC, wet joints are inevitable in cast-in-place UHPC structures. Similarly, for precast NC structures, there will inevitably be UHPC components or parts that require on-site casting. In the early stages of concrete joints, when live loads are applied to engineering structures, the deflection of the wet joint connecting the two parts may differ. This differential deflection could adversely affect the interfacial bond strength between precast elements and longitudinal joints before they attain the necessary tensile strength and plasticity. Wet joints are particularly susceptible to flexural failure under the action of live-load vibrations. The impact of live load vibrations on the flexural strength of the joint has not yet been clearly defined. Consequently, investigating the flexural performance of both the integral interfaces and wet joint interfaces of UHPC is of paramount importance.

In order to explore the influence of complex and harsh environments on wet joints and to find countermeasures for dealing with the durability problems of UHPC wet joints subjected to live-load vibrations, numerous studies have been conducted. It was found by Deng et al. [6] that a T-shaped girder wet joint for a lightweight steel-UHPC composite structure showed better crack resistance than the traditional I-shaped wet joint. Graybeal et al. [7] investigated the structural performance of cast-in-place UHPC connections by static and cyclic tests. Similar studies were performed by Haber et al. [8] and Varga et al. [9], and the results showed that UHPC connections have superior mechanical performance than those of NC. In response to the above phenomena, Graybeal et al. [10] designed and deployed the cast-in-place UHPC connections. In addition, Lee et al. [11] and Hash et al. [12] investigated the flexural behavior of prefabricated reinforced concrete specimens connected with UHPC. Their studies indicated that vibrations applied during the initial and final setting stages of the joint can influence the splitting tensile strength of the concrete. Zhang et al. [13] investigated the shear strength of UHPC-NC specimens; the results show that the effect of surface preparation on the shear strength of the UHPC–stone interface is most significant. It was found by Guan et al. [14] that vibrations that started before the initial setting increased the early compressive strength of NC. Additionally, vibrations applied from the initial to the final setting stage increased the concrete’s early strength. However, all types of vibrations started after the final setting was observed to decrease its early compressive strength and reduce the long-term strength. The study of Zhang et al. [15] and Yang et al. [16] indicates that continuous live-load vibrations have a significant effect on the cracking load and ultimate load of composite materials, but have no significant impact on their deformation performance. A vibration table was used by Wu et al. [17] to simulate the vibration caused by the live load during the construction of joints, which studied the impact of vibration on the compressive and splitting tensile strength of the concrete. Huang et al. [18] conducted a vibration table test to study the seismic performance of two prestressed beams with UHPC connections; it found that UHPC connections can provide strong resilience even under high-intensity earthquake ground motions. Wang et al. [19] and Leng et al. [20] prepared UHPC by a vibration mixing method and pointed out that at the early stage of hydration, the position of steel fibers in the UHPC matrix can move freely, which significantly affects the flexural response. The result shows that with the increase in amplitude and frequency, the flexural response of UHPC specimens decreases sharply due to the distribution and orientation of steel fibers. Zhang et al. [15] conducted vibration tests on 324 newly cast thin plate specimens with different vibration frequencies and durations of vibration as parameters, to clarify the impact of traffic vibration on the flexural response of PVA-ECCs, followed by four-point bending tests. The results indicate that continuous traffic vibration has a significant adverse effect on the flexural response of the composite material.

In summary, existing research on the impact of live-load predominantly builds upon studies of NC, with a strong emphasis on using a miniature vibration table and being only applicable to standard-sized specimens. However, the investigation into the impact of live-load vibration on UHPC remained inadequately comprehensive. To clarify the effects of live-load vibration, a four-point bending test was carried out in this study, specifically focusing on the role of vibration stage and amplitude. The key indexes such as the failure mode, load-deformation relationship, and interface opening were analyzed. At the same time, it introduces a live-load vibration coefficient to correct the calculation model for the bending strength of UHPC joints, and the model’s calculated values match the test results.

2. Experimental Program

2.1. Specimen Design and Test Parameters

To investigate the effects of different vibration conditions on structural bending strength and failure modes, this study designed and produced UHPC-NC composite specimens with dimensions of 2000 mm × 350 mm × 200 mm. The joint size is 300 mm × 350 mm × 200 mm, with rebars arranged in an interlaced pattern in the joint. The spacing of rebars is 72.5 mm, and the cover thickness is 30 mm. The precast section size is 850 mm × 350 mm × 200 mm, the spacing of longitudinal rebars is 145 mm, and the spacing of transverse rebars is 150 mm. The configuration of the joint specimen is shown in Figure 1.

The dynamic response parameters of engineering structures caused by live loads include frequency, amplitude, damping, stiffness [21], etc. In this study, the simulation of live-load vibration focuses on amplitude, while also incorporating the vibration stage as two parameters.

The grouping scheme of the test specimens is shown in

Table 1. Specimen design parameter.

Test Parameters	Test Piece Number	Vibration Amplitude	Vibration Stage	Experimental Age
Not vibration	N-1	/	/	28 d
Vibration amplitude	F-1	1 mm	Pouring-final setting	28 d
	F-2	3 mm	Pouring-final setting	28 d
	F-3	5 mm	Pouring-final setting	28 d
Vibration stage	J-1	3 mm	Pouring-initial setting	28 d
	J-2	3 mm	Initial setting-final setting	28 d

. First, the vibration amplitude represents the differential deflection between the staged constructed structures. In this study, the vibration amplitude was designed following a real construction field. The maximum value of the first-order fundamental frequency and the maximum deflection of the background structure of this study are calculated by MIDAS/Civil (2022 v1.2) finite element analysis software and are 2.3 Hz and 5 mm, respectively. In summary, based on the suggestion by Ng et al. [22], the vibration amplitude varies from 1 mm to 5 mm.

Secondly, the impact of early live-load vibration on concrete joints is not yet clear. Consequently, an investigation into the impact of vibration exposure during the early stages of joint maturation was undertaken. To achieve this, the early maturation phase of the joint was delineated by the initial setting time and final setting time points, and enforced vibrations were applied at various developmental stages. The method of penetration resistance measured the setting time of the concrete, in accordance with the GBT50080-2016 [23].

“N” represents the specimen that is not vibrated, “J” represents the vibrated stage, and “F” represents the vibration amplitude. The specimen will be cured for 28 days for joint bending performance testing after the vibration.

The specimen fabrication process is shown in Figure 2. First of all, according to the design size of the specimen and the layout of the reinforcement, the integral formwork was designed. Subsequently, the prefabricated parts were cast synchronously with ordinary concrete. Immediately after pouring, the specimens were covered with wet burlap and a plastic sheet. The specimens were then cured at room temperature for 28 days. After the curing process, the surface at the junction of the wet joint and the pre-cast section was made rough to improve the adhesion of the joint. Finally, the joint UHPC for the live-load vibration test was cast.

2.2. Material Properties

The matrix ratio of UHPC in this study is shown in

Table 2. Material mix proportion (Unit: kg/m³).

Component	Cement	Silica Fume	Quartz Sand	Coarse Aggregate	Water	Mixed Steel Fiber
Mass ratio	771.2	192.8	848.3	231.4	173.5	170.1

, the composition includes cement, silica fume, quartz sand, slag, mixed steel fibers, and coarse aggregate. The percentage by volume of mixed steel fibers is 2%, and the maximum diameter coarse bone particle size is 4 mm. The material of the precast sections is the grade of C60, and the cement is made of PO 42.5 normal portland cement, fine aggregate using river sand, coarse aggregate using crushed stone with a diameter of 4–18 mm, a water cement ratio of 0.3, and C60-grade concrete. The mixes used are shown in Table 2.

The elastic modulus of UHPC material and C60 concrete is determined by Code for Design of Concrete Structures GB50010-2010 [24]. The mechanical properties of UHPC and C60 obtained from the test are shown in

Table 3. Mechanical properties of UHPC and C60.

Category	Cube Compressive Strength fcu/MPa	Cube Tensile Strength fsu/MPa	Flexural Strength ft/MPa	Elastic Modulus/GPa
UHPC	142.9	10.2	13.8	50.4
C60	61.8	/	6.5	36

. The joint specimen is reinforced with HRB400, the diameter of the longitudinal rebar is 12 mm, and the diameter of the stirrup is 6 mm. The tensile strength of 6 mm rebar and 12 mm rebar is 421.3 MPa and 435.1 MPa, respectively, and their elastic modulus is 2.15 GPa.

2.3. Simulated Vibrating Device and Test Setup

2.3.1. Simulated Vibrating Device

To simulate the live-load vibration to the fresh casting UHPC wet joints, a simulated vibration device was designed. In this study, a simulated vibration device was designed to simulate the live-load vibration of newly poured UHPC wet joints. The device in this study consists of three fixture components: left, middle, and right U-shaped steel formwork, precast RC blocks, and the actuator. The prefabricated RC blocks in the U-shaped steel formwork are clamped with left, right, and central clamp. The middle fixture is connected to the actuator that connected to the reaction beam, achieving simultaneous simulation of the deformation vibration of two joints [25]. The layout of the simulated vibrating device is illustrated in Figure 3a,b.

2.3.2. Load Settings and Data Acquisition

Figure 4 shows the process of the joint live-load vibration test. After the calibration of the device, UHPC will be poured at the joint of the specimen. Once the joint has been poured, the actuator’s amplitude and frequency will be aligned with the predetermined test parameters. Meanwhile, the vibration time will be recorded. When the vibration is completed, the joint specimen will be cured for 28 days before the bending strength tests.

The SIControl (V2.0) software is designed to facilitate the real-time acquisition and storage of critical experimental parameters, including frequency, amplitude, and actuator reaction force. Furthermore, it is capable of plotting the actuator reaction force as a function of time, thereby rendering the test procedure tangible and observable. The loading setup and data acquisition of the software are shown in Figure 4f. The study of the reaction forces exerted on the actuator during vibration and the surface of the joint after vibration found that the reaction forces on the actuator at the UHPC joint during vibration showed no significant changes at the initial setting stage. From the initial to final setting, the maximum and minimum reaction forces increased linearly with time, and the load growth tended to level off at 16 h after concreting. Under the live load vibration of 3 Hz-5 mm, the UHPC joint developed cracks measuring 14 cm in length and 0.2 mm in width. The cracks in the joint specimens under live load vibrations occurred during the initial to final setting stages of the joint material casting.

2.4. Four-Point Flexural Test and Instrumentation

2.4.1. Measuring Point Layout

The strain measurement points are mainly located in the rebars and concrete of the precast slab area near the joints, and the distribution of measurement points is shown in Figure 5. Linear variable displacement transducers (LVDTs) were used to monitor the interface opening and deflection of the specimen. The accuracy of the LVDT is 0.001 mm. Monitoring the strain of steel and concrete was to paste strain gauges on the surface of the specimen. Figure 6. shows the layout of the measuring points. In addition, the crack width of the specimen was recorded using a crack observation instrument with an accuracy of 0.02 mm.

2.4.2. Loading Program

The loading device for the bending test of the joint specimen is shown in Figure 7. The specimens were loaded step by step with a hydraulic jack. In the initial stages, the specimen was preloaded to 10% of its ultimate load to eliminate the effects of minor gaps between the distribution beam and the specimen.

During formal loading, force control was initially used, with each increment of load staged at 10% of the ultimate load, maintained for a duration of 2 min to facilitate the observation of the material response and the emergence of test phenomena. Subsequently, the loading intensity was escalated to 90% of the estimated ultimate load at a rate of 3 mm/min. Thereafter, a displacement control protocol was implemented, with the loading rate modulated to 0.05 mm/min. When the load on the specimen could no longer be increased, the specimen was considered to have failed. The test set-up is shown in Figure 8.

3. Test Results and Discussion

3.1. Failure Mode and Crack Pattern

Figure 9 shows the failure modes and crack patterns of each specimen. All specimens exhibited the same kind of failure, characterized by basically no cracks or failure phenomena in the joint; the main crack initiated at the bottom of the prefabricated section and the interface opening was apparent.

During the elastic state, there were no visible cracks or interface openings. As the load increased, the first crack was observed at the bottom of the right prefabricated section. As depicted in Figure 9, the precast section below the left support of the distribution beam cracked, and the crack width was 0.11 mm and developed upward. However, the cracks in F-2 and F-3 appeared earlier than the F-1. This shows that the size of the vibration amplitude has a bad effect on the cracking load. In addition, the pure bending area in the middle of the prefabricated part will form the main crack as the load increases. Subsequently, the tensile reinforcement yielded, and the interface concrete was crushed, which is accompanied by the decrement of the rigidity.

3.2. Load-Deflection Curve Responses

3.2.1. Effects of the Vibration Properties

Table 4. Summary of critical results.

Specimen Set	Specimen Number	$\mathbf{Cracking} \mathbf{Loads} {\mathit{P}}_{\mathit{c}\mathit{r}}(\mathbf{k}\mathbf{N})$	$\mathbf{Ultimate} \mathbf{Load} {\mathit{P}}_{\mathit{u}}(\mathbf{k}\mathbf{N})$	$\mathit{\lambda }$	$\mathbf{Peak} \mathbf{Displacement} {\mathit{\delta }}_{\mathit{u}}\mathbf{(}\mathbf{m}\mathbf{m}\mathbf{)}$	$\mathit{\kappa }$
N	N-1	47	143.2	1.00	24.9	1.00
F	F-1	55	168.1	1.17	30.1	1.21
	F-2	50	131.4	0.92	29.3	1.24
	F-3	47.5	116.7	0.81	30.4	1.22
J	J-1	50	148.1	1.03	22.8	0.92
	J-2	46	143.7	1.00	24.9	1.00

Note: When NC and UHPC develop cracks, the crack width is generally around 0.05 mm. Therefore, the load corresponding to a crack width of 0.05 mm is defined as the cracking load.

summarizes the test results; the $\lambda$ stands for the increase in ultimate load and the $\kappa$ stands for the increase in peak displacement. To evaluate the impact of different vibration amplitudes, the load-deflection curves in the span during the test were plotted according to the test data. To facilitate the comparison, the average results for each group were selected for analysis in each group, as shown in Figure 10a. By analyzing Figure 10a, it can be found that the highest flexural strength of UHPC joint specimens under different vibration amplitudes is 3 Hz-1 mm (F-1), followed by 3 Hz-3 mm (F-2), and finally, 3 Hz-5 mm (F-3). The ultimate load of the F-1 specimen increased by 17% compared to the specimen that did not vibrate, while the ultimate load of the F-2 specimen decreased by 8% compared to the specimen that did not vibrate, and the ultimate load of the F-3 specimen decreased by 19% compared to the specimen that did not vibrate. This proves that low-amplitude live-load vibrations can be used to make the joints more compact. In addition, in order to improve the flexural strength of the joint specimens, the bond strength of the joint interface and the prefabricated segment can be increased. In contrast, high-amplitude live-load vibrations will destroy the bonding performance between the prefabricated segment and the joint interface, therefore reducing the flexural capacity of the joint specimen.

The load-deflection curves at mid-span for different vibration stages of the specimens are shown in Figure 10b. By analyzing Figure 10b, it can be observed that the highest flexural strength of UHPC joint specimens under different vibration stages is pouring—initial setting (J-1) followed by initial setting—final setting (J-2), and finally pouring—final setting (F-2). The ultimate load of specimen J-1 increased by 3% compared to the specimen that did not vibrate, the ultimate load of specimen J-2 was equal to that of the specimen that did not vibrate, and the ultimate load of specimen F-2 decreased by 8% compared to the specimen that did not vibrated. This indicates that the timing of live-load vibration has no significant effect on the flexural strength of the specimens. The load-displacement curve trends for the vibration stage groups are essentially consistent, with no obvious discrepancy in the initial cracking load and the yield load.

3.2.2. Ductility and Stiffness

In general, the stiffness is used to measure the performance of the RC beam structure [26], and the bending stiffness K is used as the calculation target in this study. The displacement stiffness, as presented in Equation (1), was used to assess the stiffness of the specimens in this study.

$$K={\displaystyle \frac{P_y}{{\delta }_y}}$$

(1)

where the $P_y$ is the yield load, the ${\delta }_y$ is the yield displacement.

The ductility is used to measure the plastic deformation capacity of the specimen, the ductility of RC structure is generally defined as the ratio of the ultimate displacement ${\delta }_u$ to the yield displacement ${\delta }_y$, and the calculation method for the ductility is shown in Equation (2).

$$\mu ={\displaystyle \frac{{\delta }_{\mathrm{u}}}{{\delta }_y}}$$

(2)

The ductility and stiffness of the specimens obtained from Equations (1) and (2) are tabulated in

Table 5. Summary of ductility and stiffness.

Specimen Number	${\mathit{P}}_{\mathit{y}}$ (kN)	${\mathit{\delta }}_{\mathit{y}}$ (mm)	${\mathit{\delta }}_{\mathit{u}}$ (mm)	K (kN/mm)	$\mathit{\mu }$
N-1	75.98	3.79	27.34	20.05	7.21
F-1	83.01	3.81	41.59	21.78	10.91
F-2	77.52	4.69	31.48	16.41	6.71
F-3	75.13	4.82	33.44	15.56	6.93
J-1	75.74	3.60	27.90	20.83	7.75
J-2	78.45	3.87	33.43	20.15	8.64

.

By analyzing Figure 11, it can be seen that the ductility values of specimen F-1 has increased by 51.3% compared to the N-1 specimen; however, the vibration decreased the ductility of the specimens F-2 and F-3. This result implies that the low amplitude has a positive effect on the ductility, but the high amplitude has a negative effect and improves on the ductility of the specimen. Additionally, the ductility of J-series specimen increased compared to the specimen without vibration. The stiffness of specimen F-1 increased by 8.6% compared to the specimen that did not vibrate, while the specimen F-2 was decreased by 18.1% compared to the specimen that did not vibrate, and the specimen F-3 decreased by 22.3% compared to the specimen that did not vibrate. However, the vibration stage does not significantly improve based on the stiffness of the specimens.

3.3. Load-Crack Width Relationship

The maximum crack width of all regions at each load level was called the maximum crack width in the curve. During the loading process of specimens F-1 (3 Hz-1 mm), F-2 (3 Hz-3 mm), and F-3 (3 Hz-5 mm), cracks began to appear and propagate as the load increased. The F-1 specimen developed a crack on the surface of the bottom at 55 kN, and the specimen failed due to a through-crack at the right interface at 168.1 kN. The F-2 specimen developed a crack on the surface of the bottom at 50 kN and failed due to the crushing of the precast section and the interface opening at 131.4 kN. The F-3 specimen cracked on the bottom surface at 47.5 kN and failed due to diagonal cracking, crushing, and the interface opening at 116.7 kN. According to the failure modes, it was observed that the precast section below the right support of the distribution beam experienced rapid crack propagation, crushing of the precast section, and a sudden opening of the right interface, indicating that the vibration amplitude significantly affected crack development and specimen failure. No cracks were observed in the UHPC during loading, and the specimens failed in a ductile manner. The vibration did not obviously affect the cracking load of the specimens, and specimens with and without vibration showed similar behavior in the linear elastic range. The maximum crack width of the amplitude group specimen is shown in Figure 12a.

The failure modes of specimens J-1, J-2, and F-2 indicate that in the specimens vibrated during the pouring to initial setting period, cracks first appeared at the bottom of the precast section below the right support of the distribution beam and gradually extended upward. At a load of 148.1 kN, the main crack expanded rapidly, leading to concrete crushing and interface opening, resulting in the ultimate failure of the specimen. Specimens that vibrated during the initial to final setting period exhibited similar failure patterns, with crack development and interface opening eventually leading to specimen failure. The specimens that vibrated during the pouring to final setting period had the same failure characteristics as the F-2 specimens, so further description is omitted. From the failure modes, all specimens showed rapid development of cracks in the precast section below the right support of the distribution beam, with concrete crushing above and a sudden opening of the right interface. No cracks were observed in the UHPC during loading, indicating a ductile failure of the specimens. The maximum crack width of vibration stage group specimens is shown in Figure 12b.

3.4. Load-Interface Opening

Figure 13a compares the load-interface opening curves of specimens under different vibration amplitudes. It indicates that specimen F-1 exhibited significant crack widths and interface opening widths at ultimate load, while the F-2 and F-3 vibration specimens showed faster interface opening speeds at the yield load, indicating that vibration significantly affects crack development and interface bonding. For the yield stage, an increase in vibration amplitude decreased the interface bonding strength, leading to an increased rate of interface opening. Consequently, the rotational angle of the interface increased, which decreased the post-yield flexural stiffness of the specimens. However, for specimens with lower vibration amplitudes, vibration increased the post-yield flexural stiffness. This is because low-amplitude and low-frequency vibrations increase the compactness of the UHPC and also enhance the interface bonding strength.

Figure 13b compares the load-interface opening curves of specimens under different vibration stages. Interface opening width curves of specimens were shown under different vibration stages. The specimen vibrated during pouring to initial setting had a crack of left opening of 0.54 mm and a maximum crack width of 0.77 mm at the yield load, while the specimen that vibrated during pouring to final setting had a crack of the left opening of 0.40 mm and a maximum crack width of 0.56 mm at the yield load; the specimen that vibrated during the initial to final setting had a crack of left opening of 0.34 mm and a maximum crack width of 0.21 mm at the yield load. These results indicate that different vibration stages are related to the crack development speed and joint opening speed of UHPC wet joints, and that vibration significantly affects crack development and interface bonding.

However, for J-2, F-2, and F-3 specimens, the load become larger at the greatest crack width. This is because during the loading process of UHPC joint specimens, the steel fibers within it can effectively hinder further propagation of cracks. Once the cracks reach a certain extent, the bridging action of the fibers or changes in the material’s microstructure may cause the cracks to cease expanding, but the specimen can still bear larger loads.

3.5. Load-Strain Curves

3.5.1. Load-Steel Strain Curves

To obtain the load-strain development curve of the specimen, the strain gauge should be laid on each longitudinal bar in the wet joint, so as to take the average strain value; the curve is as shown in Figure 14. Due to the consistent pattern of load-steel strain curves for joint specimens under different vibration stages and amplitude factors, the load-steel strain curves of the F-2 specimen from the vibration stage group (pouring-final setting) and the F-3 specimen from the vibration amplitude group (3 Hz-5 mm) are selected for analysis here.

Figure 14a shows the load-steel strain curve of the F-2 vibration specimen. Since the cracks in the UHPC widened joint mainly appear in the precast section, the strain of the longitudinal tensile rebar in the precast section exhibits distinct characteristics of an elastic phase, crack development phase, and yielding phase. In the elastic phase, the strain at each measurement point increases linearly with the load. During the crack development phase, the tensile strain at points SN1′, SN2′, and SN3′ is approximately twice that of the tensile strain at points SU1′, SU2′, and SU3′ in the joint, indicating that the rate of tensile strain increase in the longitudinal tensile rebar of the precast section is much greater than that in the joint. Under the ultimate load, the longitudinal tensile rebar inside the precast section yielded, while the longitudinal compressive rebar in the joint did not, due to the staggered arrangement of longitudinal rebar in the UHPC joint, with a rebar ratio approximately twice that of the precast sections on both sides of the joint, indicating that when the anchorage length in the UHPC joint is 28 cm, the rebar in the precast section can bear the load adequately, and the anchorage length is sufficient.

Figure 14b shows the load-steel strain curve for the F-3 vibration specimen. The curve of the F-3 specimen is similar to that of the F-2, and since the cracks in the UHPC-widened joint mainly appear in the precast section, the strain of the longitudinal tensile rebar in the precast section also exhibits distinct characteristics of an elastic phase, crack development phase, and yielding phase. After the specimen was cracked, the strain of the steel bar increased rapidly, and some concrete in the tensile zone failed. The slope of the load-strain curve did not change obviously when the strain exceeded the yield strain of the steel bar. Until the specimen yielded, the strain continued to increase, but the load remained basically unchanged. This phenomenon proves that the stress-strain relationship of steel bars in UHPC is different from that of bare bars.

3.5.2. Load-Concrete Strain Curve

Also, the load-concrete strain curves of the F-2 specimen from the vibration stage group (pouring-final setting) and the F-3 specimen from the vibration amplitude group (3 Hz-5 mm) are selected for analysis.

Figure 15 shows the load-concrete strain curves for the pouring-final setting vibration specimens. Figure 15a,b represents the load-concrete strain curves for the left and right precast sections, respectively. The compressive strain at the top of the right precast section is much greater than that at the top of the left precast section. The compressive strain at the top of the right precast section corresponding to the ultimate load has already exceeded the ultimate compressive strain, which matches the experimental phenomenon of the top crushing of the pure bending section of the right precast section when the joint specimen fails. The strain at the lateral measurement points of the precast section conforms to the plane assumption during the elastic stage of the specimen. The measurement points at the bottom of the precast section undergo tensile strain. As the concrete of the precast section reaches the ultimate tensile strain, the specimen cracks, and the strain gauges are subsequently damaged.

Figure 15c shows the load-concrete strain curve for the joint. Since the UHPC joint did not crack throughout the loading process, it can be seen that the strain of almost all measurement points increases linearly, and the lateral measurement points conform to the plane section assumption throughout the loading process.

Figure 16 shows the load-concrete strain curves for the F-3 vibration specimen. Figure 16a and Figure 16b represent the load-concrete strain curves for the left and right precast sections, respectively. The compressive strain at the top of the right precast section is significantly greater than that at the top of the left precast section. The compressive strain at the top of the right precast section corresponding to the ultimate load has already exceeded the ultimate compressive strain, which corresponds with the experimental observation of the top crushing in the pure bending region of the right precast section when the joint specimen fails. The strain at the lateral measurement points of the precast section conforms to the plane assumption during the elastic stage of the specimen. The measurement points at the bottom of the precast section experience tensile strain, and as the concrete of the precast section reaches its ultimate tensile strain, the specimen begins to crack, leading to the subsequent damage of the strain gauges.

Figure 16c shows the load-concrete strain curve for the joint. Since the UHPC joint did not crack throughout the entire loading process, it can be observed that the strain at almost all measurement points changes linearly, and the lateral measurement points conform to the plane assumption throughout the loading process.

4. Prediction of the Flexural Strength Considering Vibration

The flexural strength of UHPC wet joints was calculated by the strut-and-tie model. The basic assumptions of the strut-and-tie model configuration are as follows:

(1)

In the strut-and-tie model, the tensile members within the structure are subjected to axial tensile forces, while the compression members are subjected to axial compressive forces. It is posited that the nodes within the framework do not sustain bending moment loads and are in compliance with the equilibrium equations of forces;

(2)

The compression members are composed of concrete from a certain area, the shape of which is not uniform, and the random distribution of coarse and fine aggregates in concrete leads to a decrease in the strength of the compression member concrete. Therefore, the strength of the compression members cannot be simply taken as the compressive strength of concrete; instead, a strength influence factor should be considered for correction. According to the ACI318-2011 code [27], the calculation method for the effective strength of compression member concrete is shown in Equation (3).

$$f_{cu}=0.85{\beta }_n{f}_c^{\prime }$$

(3)

where $f_{cu}$ is the effective strength of compression member concrete and ${\beta }_n$ is the strength influence factor. When all members in the node area are concrete compression members, the value is taken as 1.00; when there is one tensile member in the node area, the value is taken as 0.75; when there are two tensile members, the value is taken as 0.6. $f_c$ is the design compressive strength of concrete, obtained from the basic mechanical performance tests of concrete materials.

Anchored rebar joints primarily facilitate load transfer through the cohesive interaction between the rebar and the surrounding concrete matrix. Based on the “circuitous force flow” transfer mechanism of anchored rebar joints [28], the principle of circuitous force transfer in the post-cast wet joint composed of anchored rebar, transverse rebar, and UHPC joint can be idealized as the calculation diagram of the strut-and-tie model, which is displayed in Figure 17. In the wet joint, circuitous force are transferred between the anchored rebar, transverse rebar, and joint. The anchored rebar and transverse rebar act as tension members, represented by solid lines in the diagram; the joint between the anchored rebars acts as a compression member, also represented by solid lines in the figure; and the compression member and tension members are in a state of force equilibrium at node C. In the diagram, B represents the concrete compression member, and T represents the steel tension member.

On the basis of the wet joint strut-and-tie model (STM) of the anchor rebar shown in Figure 17, the final bearing capacity of the STM of joint is controlled by the yield of the anchor rebar tension rod AC, the crushing of the diagonal concrete compression rod BC, and the yield of the transverse rebar tension rod AB, which means it depends on the compressive strength of the joint and the yield strength of the rebar. The ideal failure mode for wet joints of anchor rebars is that after the anchor rebar tension rod AC yields, the concrete compression rod BC is crushed or the transverse rebar tension rod AB yields, resulting in ductile failure of the joint. On the contrary, if the yield of the anchoring steel rod occurs after the concrete compression rod is crushed or the transverse steel rod yields, the anchoring rebar cannot exert its bearing capacity, and the failure of the joint is brittle failure rather than ductile failure.

As shown in Figure 17, the internal force expressions for the concrete compression rod AC, the anchoring steel rod AB, and the transverse steel rod BC within the wet joint STM are derived from the force equilibrium condition at node A, which are calculated according to Equations (4)–(6).

$$F_s={\displaystyle \frac{T}{2\cos \theta }}$$

(4)

$$F_h=T$$

(5)

$$F_l={\displaystyle \frac{T}{2}}\tan \theta$$

(6)

where $T$ is the tensile force of anchoring rebar tie rod; $F_s$ is the pressure of concrete compression rod; $F_h$ is the tensile strength of anchoring rebars and tie rods; $F_l$ is the tensile force of transverse rebar tie rod; $\theta$ is the angle between two opposite anchoring rebars at adjacent positions, $\cos \theta ={\displaystyle \frac{2l}{\sqrt{4l^2+s^2}}}$; l is the overlap length of two opposite anchoring rebars at adjacent positions; and s is the spacing between two adjacent anchoring rebars in the same direction.

The final bearing capacity of the wet joint of anchored rebars is controlled by the ultimate compressive strength of the concrete compression rod and the ultimate tensile bearing capacity of the rebar tension rod. The ultimate compressive strength of the concrete compression rod is calculated according to Equation (7).

$$F_s=f_{cu}{A}_{strut}=0.85{\beta }_n{f}_c^{\prime }{A}_{strut}$$

(7)

where $F_s$ is the ultimate compressive strength of concrete compression rod; $A_{strut}$ is the cross-sectional area of the concrete compression bar, $A_{strut}=D{W}_s={\displaystyle \frac{Dl\sin \theta }{2}}$; D is the height of the concrete compression rod taken as the diameter of the anchoring plate at the end of the anchoring rebar; and $W_s$ is the width of the concrete compression bar.

The ultimate compressive strength of the concrete compression rod and the ultimate tensile bearing capacity of the anchored steel rod can be used to determine the STM ultimate bearing capacity of the wet joint of the anchored steel rod. They are calculated according to Equations (8)–(10).

$$T_{us}=2F_s\cos \theta =2F_s{\displaystyle \frac{2l}{\sqrt{4l^2+s^2}}}={\displaystyle \frac{3.4f_c^{\prime }{A}_{strut}l}{\sqrt{4l^2+s^2}}}$$

(8)

$$T_{uh}=F_h=f_{yh}{A}_h$$

(9)

$$T_{ul}={\displaystyle \frac{2f_{lh}{A}_l}{\tan \theta }}={\displaystyle \frac{4f_{lh}{A}_{l}l}{s}}$$

(10)

where $T_{us}$ is ultimate tensile bearing capacity of concrete compression bar; $T_{uh}$ is the ultimate tensile bearing capacity of anchored rebars and tie rods; $T_{ul}$ is the ultimate tensile bearing capacity of transverse rebar tie rods; $f_{yh}$ is the yield strength of anchored rebars; $f_{lh}$ is the yield strength of transverse rebars; $A_h$ is the cross-sectional area of anchored rebars; and $A_l$ is the cross-sectional area of transverse rebars.

The ultimate tensile bearing capacity of the entire joint specimen is related to the number of anchoring rebars on one side of the joint and is calculated according to Equation (11).

$$T_u=N\times \min \left({\displaystyle \frac{2.72f_c^{\prime }{A}_{strut}l}{\sqrt{4l^2+s^2}}},f_{yh}{A}_h,{\displaystyle \frac{4f_{lh}{A}_{l}l}{s}}\right)$$

(11)

In this study, the post poured wet joint bears a pure bending load, and the rebars and concrete in the joint are well bonded. The bonding between the precast section of the joint specimen and the post poured wet joint is reliable. The calculation method of the bending capacity of a single reinforced rectangular section bending member can be referred to calculate the bending capacity of the wet joint. The calculation diagram of the bending capacity of the wet joint is shown in the following Figure 18.

The position x of the neutral axis of concrete and the bending capacity of the joint $M_u$ can be obtained based on two equilibrium conditions: the sum of circuitous force in the horizontal direction on the section is equal to zero and the sum of moments at the point of resultant force in the compressed area is equal to zero. They are calculated according to Equations (12) and (13).

$$x={\displaystyle \frac{C_u}{0.85f_c^{\prime }b}}={\displaystyle \frac{T_u}{0.85f_c^{\prime }b}}$$

(12)

$$M_u=T_u\xi \left(d_z-{\displaystyle \frac{x}{2}}\right)=T_u\xi \left(d_z-{\displaystyle \frac{T_u}{1.7f_c^{\prime }b}}\right)$$

(13)

where $C_u$ is the resultant force of concrete in the compression zone of the section; $T_u$ is the joint force of rebars in the tensile zone of the cross-section; $b$ is joint width; $d_s$ is the effective height of the cross-section; and $\xi$ is the impact coefficient of live-load vibration, taken as 1 when considering no live-load vibration. When considering live-load vibrations $\xi =k\alpha +k_1$, where $\alpha$ is the amplitude of the live-load vibration, $k$ and $k_1$ are constants, $k$ is the intercept, and $k_1$ is the influence coefficient of the live-load vibration amplitude. By substituting the experimental data into Formula (13), it can be calculated that the $k$ is 1.208 and the $k_1$ is −0.125; therefore, the $\xi =-0.125\alpha +1.208$.

In summary, the final bending capacity of UHPC joint specimens was calculated by substituting each group of specimens into Formula (13). The comparison results with the experimental values are shown in

Table 6. The summary of calculated values and experimental values.

Test-Piece	Mc	Mtes	Mc/Mtes
W-1	28.17	28.64	0.98
F-1	32.59	33.62	0.97
F-2	27.53	26.28	1.05
F-3	22.46	23.34	0.96

Note: Mc is the calculated value, Mtes is the experimental value, both in kN·m.

.

As shown in Table 4, the ratio of the theoretical calculation value to the experimental value of the ultimate bending capacity of each specimen is between 0.96 and 1.05, with an average value of 0.99 and a standard deviation of 0.035. The accuracy of the theoretical calculation value is high, which can provide theoretical support for the calculation of joint flexural capacity under live-load vibration.

5. Conclusions

The following conclusions were obtained:

• After the specimens were destroyed, UHPC joints barely showed any cracks but the initial and main cracks were observed within the pure bending zone of the NC. These cracks were particularly evident at the top of the precast section, where the concrete was crushed. It indicated that UHPC wet joints have excellent crack resistance;
• The vibrations of higher amplitude and frequency will reduce the interfacial bond strength of the wet joints. Compared with the specimens without vibration, the post-yield flexural stiffness and flexural strength of the specimens reduced by 24.39%. Additionally, the flexural strength of specimens subjected to the vibration at 3 Hz-3 mm and 3 Hz-5 mm were decreased by 8% and 19%, respectively;
• Vibration with lower amplitude or frequency was found to increase the flexural strength of joint specimens. As the amplitude and frequency of vibration decrease, the live-load vibration shows an increasing trend in the flexural strength of the specimens. This is due to the fact that such vibration helps to enhance the compactness of the concrete;
• Low amplitude vibration has a positive effect on ductility. However, compared to specimens without vibration, the stiffness of the 3 Hz-1 mm specimen increased by 8.6%, whereas the stiffness of the 3 Hz-3 mm and 3 Hz-5 mm specimens decreased by 18.1% and 22.3%, respectively;
• A calculation model for the flexural strength of UHPC joints was established, considering the impact of live-load vibrations. A live-load vibration coefficient was introduced to refine the calculation formula. The average ratio of theoretical calculation values to experimental values is 1.01, with a standard deviation of 0.04, indicating a high level of accuracy.