Behavior of Existing Box Beams Repaired with High-Strength Mortar Layer and Ultra-High-Performance Concrete (UHPC) Overlay: Experimental, Numerical, and Theoretical Investigations

Abstract

Box beams constructed earlier were prone to inadequate bending capacity owing to low construction standards, overloading, and environmental degradation. To resolve the challenge, three full-scale box slab beams in service for 15 years were strengthened with a high-strength mortar layer and an ultra-high-performance concrete (UHPC) layer in this paper. The flexural performances of unstrengthened beams (control beam) and strengthened beams (mortar beam, UHPC beam) were investigated by in situ four-point bending tests and numerical simulations. The experimental results showed that the cracking of box beams, strengthened with high-strength mortar and UHPC layers, was effectively mitigated. In comparison to the control beam, the cracking load of the mortar beam and the UHPC beam increased by 20%, and the ultimate load increased by 23.5% and 35.3%, respectively. The high-strength mortar layer had little influence on the elastic-phase stiffness of box beams. In contrast, the stiffness of the elastic phase of the box beam, strengthened by the UHPC layer, increased by 32.9%. In the numerical simulations, the load-deflection curves obtained from finite elements and tests coincided well. The characteristic loads showed relatively good agreement with the test results, with errors below 10%. Combined with the tests and numerical analyses, the proposed equations for predicting the ultimate bearing capacities of the control beam, mortar beam, and UHPC beam were presented with a better prediction accuracy.

1. Introduction

Reinforced concrete (RC) box beams are widely used in medium- and small-span bridges due to the advantages of the small member height and the high degree of assembly [1,2]. With the surge in highway traffic demand and environmental degradation, numerous box beams have suffered from various diseases, e.g., concrete spalling, hinge joints, and the cracking of base slabs [3,4]. Once these bridges are demolished, the construction waste produced will inevitably impose a significant burden on the economy and the environment. This runs contradictory to the globally advocated ideals of environmental protection and sustainable development. It was, thus, essential to strengthen box beams while bearing risks. Nevertheless, the currently popular methods (enlarged section method, bonding steel plate method, in vitro prestressing method, and bonding carbon fiber composite material method), are either insufficiently durable or limited in terms of load-carrying performance enhancement [5,6,7]. The risk of the secondary retrofitting of bridges after a certain period of strengthened operation remains relatively high. Hence, a strengthening method with superior durability and significant improvement in load-carrying capacity is urgently demanded.

UHPC is a new cement-based material with outstanding mechanical properties and durability, and it has been recognized as the most innovative engineering material of the past 30 years [8,9,10,11]. The use of ultra-fine mineral powders, highly efficient water reducers, and low water–cement ratios in UHPC, along with the abandonment of coarse aggregates, results in sufficient hydration and the reaction of mixed matrices with a densely packed and low-permeability microstructure [12,13,14]. UHPC, hence, has relatively high compressive strength, modulus of elasticity, and excellent durability. UHPC received much attention for its use in applications requiring the repairing of reinforced concrete (RC) structures [15,16,17,18]. Hable et al. [19] first proposed the concept of using UHPC to repair RC structures and conducted systematic tests and simulations on UHPC–RC composite beams. Zhang et al. [20,21] conducted flexural tests of damaged RC beams that were strengthened with reinforced UHPC layers and prestressed UHPC layers. Safdar et al. [22] conducted experimental investigations and numerical simulations on RC beams strengthened with UHPC layers of different thicknesses. Yang et al. [23] conducted an experimental investigation on damaged RC arches strengthened with UHPC. Results indicated that the initial crack load and ultimate bearing capacity of composite arches strengthened by UHPC were significantly improved, and good effectiveness in strengthening and toughening was observed. Moreover, high-strength mortar has also been gaining the attention of researchers in the field of strengthening owing to its advantages, such as good work coordination with concrete substrate, lower cost, good corrosion resistance, and durability [24]. Researchers have matched high-strength mortar with steel mesh or CFRP mesh to form a composite layer to strengthen RC beams, and research on the flexural and shear properties of strengthened beams was conducted. The bending load capacity, cracking resistance, and plastic-deformation capacity of RC beams were significantly enhanced by the composite strengthening layer [25,26]. Meanwhile, the shear-strength capacity and crack patterns of RC beams were remarkably modified by the composite strengthening layer in the shear experiments [27,28,29].

At present, many researchers have conducted laboratory-scale model tests on the bending performances of UHPC–RC and Mortar–RC composite beams, but full-scale model tests have not been reported. Moreover, the scaling effect can lead to differences in force ratios between the proportional model and the full-scale structure. The flexural performance of full-scale box beams reinforced by UHPC and high-strength mortar has not been studied. To solve the problems of existing research, this paper selected three box beams from a national highway in Guangxi, which had been in service for 15 years, and two of them were strengthened with a reinforced high-strength mortar layer and a reinforced UHPC layer, respectively. This study conducted four-point bending tests, and the structural responses of unstrengthened and strengthened beams under bending load were obtained. After that, three-dimensional FE models were developed to simulate the flexural performances of the strengthened box beams. Lastly, the ultimate bearing capacity, load-deflection curves of unstrengthened beams (control beam), strengthened beams (mortar beam, and UHPC beam), and the stiffness and ductility coefficients of the test beams before and after strengthening were compared. The effects of the reinforced high-strength mortar layer and reinforced UHPC layer on the flexural behaviors of box beams in service were investigated.

2. Experimental Program

2.1. Specimens

2.1.1. Unstrengthened Beam (Control Beam)

The beam used in the test was a non-prestressed box beam with 15 years of service on a national highway in Guangxi. The length of box beams is 1596 cm, the height of the cross-section is 90 cm (including 15 cm pavement), and the widths of the top and bottom plates are 94 cm and 104 cm, respectively. HRB300 grade steels with diameters of 10 mm and 20 mm were used in the top plate, HRB335 grade steels with a diameter of 25 mm were used in the bottom plate, and hoop steel bars were made of HRB300 grade steels with a diameter of 8 mm. The cross-section dimensions and steel-bar arrangements are shown in Figure 1. Owing to environmental degradation and long-term loading, lateral cracks developed in the bottom slabs, and concrete spalling at the edge of the bottom were observed in the test beams, as shown in Figure 2a,b. For convenience of expression, the unstrengthened beam is named C.

2.1.2. Strengthened Beams (Mortar Beam and UHPC Beam)

In this paper, the box beams in service were strengthened with reinforced high-strength mortar and UHPC, and they were named as mortar beams and UHPC beams, respectively. To fit the actual engineering situation, the strengthening position was set at the bottom of the box beam. In the strengthening layer, the longitudinal steel bars were HRB400 with a diameter of 10 mm, and the transverse steel bars were HRB400 with a diameter of 8 mm. At the same time, L-shaped steel bars with a diameter of 8 mm were implanted at the interface, as shown in Figure 3. To guarantee the bonding properties of strengthening interfaces, a 6.5 cm depth of steel bars was implanted in the bottom plate of test beams, and a 2.5 cm depth of steel bars was implanted in the high-strength mortar layer and the UHPC layer. The configuration of the strengthening layer and the arrangement of steel bars are shown in Figure 4a–d.

2.2. Material Properties

It was extremely challenging to peel regular cubes from the box beams used for 15 years. Therefore, cylindrical specimens were drilled at the ends with less damage for the compressive strength experiment. Then, the beam end was crushed and the steel bars in the beam were intercepted for the tensile-strength experiment. The mechanical behavior experiments of concrete and steel bars were conducted per JGJ/T 384-2016 [30] and GB/T 228.1-2010 [31], respectively.

The UHPC mix proportion is shown in

Table 1. UHPC mix proportion (kg/m³).

Constituents	UHPC
Cement	1000
Silica fume	100
Fly ash	300
Basalt sand	1100
Water	240
Steel fiber	170

. High-strength mortar is a commercial material purchased from the Hunan Ruifu Technology Development Company. The tests on the mechanical properties of UHPC and high-strength mortar were conducted according to GB/T 31387-2015 [32] and JGJ/T70-2009 [33], respectively. Figure 5 shows the general experiment arrangement of material properties.

According to GB/T50152-2012 [34], based on the measured cylindrical compressive strength of concrete, the axial compressive strength and elastic modulus of concrete were calculated using Equations (1) and (2).

$$f_{cu}=\alpha f_c$$

(1)

$$E_{\mathrm{c}}=\frac{{10}^5}{2.2+\frac{34.7}{f_{cu}}}$$

(2)

where $f_{cu}$ denotes the axial compressive strength, $f_c$ denotes the cylindrical compressive strength, $\alpha$ denotes the conversion coefficient and is taken as 0.8 in this study, and $E_c$ denotes the elastic modulus of concrete. The mechanical properties of the materials tested are shown in

Table 2. Mechanical properties of materials.

Material	Cube Strength (MPa)	Axial Compressive Strength (MPa)	Yield Strength (MPa)	Cracking Strength (MPa)	Elastic Modulus (GPa)
Concrete	66.8	50.3			3.5 × 104
UHPC	142.5			7.0	4.22 × 104
High-strength Mortar					0.8 × 104
HRB300			335.0		2.0 × 105
HRB335			376.0		2.1 × 105
HRB400			435.0		2.2 × 105

.

It is worth mentioning that the axial compressive strength of concrete measured in this study was 50.3 MPa, while the axial compressive strength of the standard value of C30 is 20.1 MPa. The value was 2.5 times the standard value of the axial compressive strength of C30, which was specified in the specification. This is attributed to the fact that the concrete-hydration reaction was sufficiently conducted during the 15 years of service, and a dense microstructure was formed. Consequently, the compressive strength of concrete significantly increased.

2.3. Test Setup and Instrumentations

The test beams, with 300 cm pure bending, were subjected to four-point bending loads. Due to challenges in the transportation of beams, a self-balancing loading device was designed on-site to facilitate the conduct of tests, as shown in Figure 6. Two old beams were poured into a whole as a reaction beam, and supporting piers were poured at both ends of the reaction beam. The experimental beams were supported on the supporting piers using rubber bearings. To ensure the transverse uniform distribution of force, the distribution steel plate was placed at the bottom of the jack. The distribution beam was installed above the jack and below the reaction beam. The finishing rolling steel bars were used to connect the two distribution beams to ensure that the device was self-balanced and achieved loading.

Two LVDTs were set horizontally at the mid-span, loading point, and end of the experimental beam, totaling 10 LVDTs. The average value was taken as the displacement of the corresponding position. Vertical compression existed in rubber bearings used for simple supported box beams. Therefore, the displacements of the mid-span and loading point were equal to the corresponding displacements subtracting the displacements of the ends. The LVDT arrangement is shown in Figure 6b.

The level and rate of loading were controlled by the oil meter. Before formal loading, the experimental beams were preloaded with 60 kN to eliminate the gap between the beam and the device. After preloading, a graded loading scheme with an increment of 40 kN was adopted. After the steel bars yielded, the load increment was reduced to 10 kN.

3. Test Results

3.1. Failure Process

For the control beam, the initial crack was observed in the web when the load was 200 kN (i.e., cracking load, 29% $P_{uc}$, where $P_{uc}$ is the ultimate bearing capacity of the control beam), with a deflection of 10.49 mm (L/1474). The specimen was about to exit the elastic stage. With a further increase in load to yield load (i.e., 94% $P_{uc}$), the width, height, and quantity of cracks in the bottom and web plates increased, and the mid-span deflection reached 44.29 mm (L/349). The number of cracks remained constant when the ultimate load was reached (i.e., 680 kN). The web crack rapidly extended to the top plate, forming the main crack; the mid-span deflection was 99.56 mm at this moment. The crack distribution in the pure bending section of the control beam is shown in Figure 7. It is interesting to note that the concrete in the compression area was not crushed when failure of the control beam occurred. This phenomenon indicated that the strength of concrete with an adequate hydration reaction was significantly improved.

For the mortar beam, when the load reached 240 kN (i.e., cracking load, 38% $P_{um}$, where $P_{um}$ is the ultimate bearing capacity of the mortar beam), vertical cracks developed along the web of the box beam and high-strength mortar layer, and the bottom plate had no cracks due to the constraint of high-strength mortar. The mid-span deflection was 11.72 mm (L/1319). During the process of increasing the load to 720 kN (i.e., yield load, 85% $P_{um}$), the number of cracks in the high-strength mortar layer of the bottom plate of the box beam increased rapidly, and the cracks on the web slowly developed towards the top plate. When the load reached the yield load, the mid-span deflection increased to 50.64 mm (L/305). Accompanied by the crushing of the concrete in the compression area, the ultimate load (i.e., 840 kN, 100% $P_{um}$) was reached. The mid-span deflection reached 128.90 mm (L/120), and the mortar beam lost its bearing capacity. When the mortar beam reached the ultimate load, the high-strength mortar layer severely cracked, and the compressed concrete was crushed. The final crack distributions are shown in Figure 8 and Figure 9.

For the UHPC beam, when the load reached 240 kN (i.e., cracking load, 26% $P_{uu}$, where $P_{uu}$ is the ultimate bearing capacity of the UHPC beam), three vertical cracks developed in the web of the box beam, and one micro-crack appeared in the UHPC layer. However, due to the high tensile strength of UHPC, there were no cracks in the bottom plate of the box beam, and the mid-span deflection reached 8.42 mm (L/1836). The UHPC beam was about to enter the elastic-plastic stage. During the process of increasing the load from the cracking load to 800 kN (i.e., yield load, 87% $P_{uu}$), the number of cracks in the bottom plate and UHPC layer of the box beam continuously increased, and the cracks on the web slowly developed towards the top plate. When the load reached the yield load, the mid-span deflection increased to 55.65 mm (L/278). During the process of increasing the load to 880 kN (i.e., ultimate load, 100% $P_{uu}$), the number of cracks remained unchanged, and the mid-span cracks rapidly developed towards the top plate. When the deflection reached 119.81 mm (L/129), the UHPC beam lost its bearing capacity. When the UHPC beam failed, the UHPC layer cracked at the ultimate load, and the compressed area of the box beam was not crushed. The final crack distribution is shown in Figure 10.

3.2. Load-Deflection Curves

Figure 11 shows the load-deflection curves of the test beams at the loading point and mid-span. Under the same load level, the mid-span deflection value was greater than the loading point. As shown in Figure 11, the test beams can be divided into three stress stages from the initial loading to failure. (a) Elastic stage: At the initial stage of loading, the stress of the box beam was similar to the homogeneous elastic solid, the stiffness of the specimen was greater, and the load increased linearly with the deflection. (b) Elastic-plastic stage: the cracking load was reached, and the stiffness of the specimen was reduced. Hence, the load-deflection curves no longer showed a linear relationship. (c) Plastic stage: After the load reached the yield load, the steel bars reached the yield strength. The stiffness of the specimen further degraded, while the load increased slowly.

As the bottom plate of the box beam had cracked during service, the cracking load in this study was defined according to the first inflection point of the load-deflection curve. The critical loads and corresponding deflections of the test beams are shown in

Table 3. Critical loads and corresponding deflections.

Test Beams	${\mathit{P}}_{\mathit{c}\mathit{r}}/\mathbf{k}\mathbf{N}$	${\mathit{\delta }}_{\mathit{c}\mathit{r}}/\mathbf{m}\mathbf{m}$	${\mathit{P}}_{\mathit{y}}/\mathbf{k}\mathbf{N}$	${\mathit{\delta }}_{\mathit{y}}/\mathbf{m}\mathbf{m}$	${\mathit{P}}_{\mathit{u}}/\mathbf{k}\mathbf{N}$	${\mathit{\delta }}_{\mathit{u}}/\mathbf{m}\mathbf{m}$
Control-Beam	200	10.49	640	44.29	680	99.56
Mortar-Beam	240	11.72	720	50.64	840	128.90
UHPC-Beam	240	8.42	800	55.66	920	119.81

Where $P_{cr}$, $P_y$, and $P_u$ are the cracking load, yield load, and ultimate load, respectively, and ${\delta }_{cr}$, ${\delta }_y$, and ${\delta }_u$ denote the corresponding mid-span deflection values.

.

3.3. Critical Load

Figure 12 shows the comparison of the critical loads of the control beam, the mortar beam, and the UHPC beam. The results show that both the reinforced high-strength mortar layer and the reinforced UHPC layer can improve the crack resistances of the box beams, increasing the crack resistance from 200 kN to 240 kN, with an increase of 20%. This is mainly due to the increase in the height of the section, which led to an increase in the moment of inertia of the section. Meanwhile, the reinforcing layer shared the load in the tension areas. This caused the tensile stress in the section of the tension areas to be reduced, and the cracking of the beam was delayed. Compared with the control beam, the yield loads of the mortar beam and the UHPC beam increased by 12.5% and 25%, respectively. Compared with high-strength mortar, UHPC had higher tensile strength and could bear a higher load during the loading process, delaying the yield of steel bars. Therefore, the reinforced UHPC layer had better performance in improving the yield load. High-strength mortar has no strain-hardening characteristics and cannot maintain the stress state after cracking. The contribution of high-strength mortar to the bearing capacity of box beams mainly came from the steel bars. However, both the UHPC and steel bars of the reinforced UHPC layer can bear the load. Therefore, the reinforced UHPC layer is superior to the reinforced high-strength mortar in improving the ultimate bearing capacity. The enhancement of the ultimate load by the two methods of strengthening was 23.5% and 35.3%, respectively.

3.4. Stiffness

To simplify the calculation, the three loading stages of the test beams are regarded as linear processes. The slope of the load mid-span deflection curve at the corresponding stage is the bending stiffness, which is calculated as follows:

$$K_e=\frac{P_{cr}}{{\delta }_{cr}}, K_{ep}=\frac{P_y-P_{cr}}{{\delta }_y-{\delta }_{cr}}, K_p=\frac{P_u-P_y}{{\delta }_u-{\delta }_y}$$

(3)

where $K_e$, $K_{ep}$, and $K_p$ are the bending stiffnesses of the elastic stage, elastic-plastic stage, and plastic stage, respectively. The results are listed in

Table 4. Bending stiffnesses of the test beams.

Test Beam	${\mathit{K}}_{\mathit{e}}/(\mathbf{k}\mathbf{N}/\mathbf{m}\mathbf{m})$	${\mathit{K}}_{\mathit{e}\mathit{p}}/(\mathbf{k}\mathbf{N}/\mathbf{m}\mathbf{m})$	${\mathit{K}}_{\mathit{p}}/(\mathbf{k}\mathbf{N}/\mathbf{m}\mathbf{m})$
Control Beam	19.1	13.0	0.7
Mortar Beam	20.5	12.3	1.5
UHPC Beam	28.5	11.9	1.8

.

According to Table 4, the bending stiffness of the test beams gradually decreased with the increase in load. Moreover, in different stages, the high-strength mortar layer and the UHPC layer had varying degrees of influence on the flexural stiffness of box beams. In the elastic stage, the stiffness of the mortar beam was close to that of the control beam because the elastic modulus of the high-strength mortar was low, and the additional bending moment of inertia to the section was small. Compared with high-strength mortar, the elastic modulus of UHPC was higher. The stiffness of the UHPC beam is, hence, significantly improved compared to the control beam and mortar beam, which is 32.9% higher than the stiffness of the control beam in the elastic stage. The strengthening layer gradually cracked and the section bending moment of inertia added to it gradually failed. Therefore, in the elastic-plastic stage, the bending stiffnesses of the three test beams are equivalent. After entering the plastic stage, the load is mainly carried by the steel bars. Both the mortar beam and the UHPC beam were enhanced by the steel bars within the strengthening layer, and their stiffnesses were equivalent. Moreover, they are considerably higher compared to the control beam.

3.5. Ductility

This paper used the displacement ductility coefficient to evaluate the ductility performance of the test beams, calculated by Equation (4):

$$\Delta =\frac{{\delta }_u}{{\delta }_y}$$

(4)

where ${\delta }_y$ denotes the mid-span deflection under yield load, and ${\delta }_u$ denotes the mid-span deflection under ultimate load.

The ductility coefficients of the control beam, mortar beam, and UHPC beam were calculated as 2.25, 2.71, and 2.15, respectively. It revealed that reinforced high-strength mortar can improve the plastic-deformation ability of box beams, while reinforced UHPC weakens the plastic-deformation ability of box beams. The ductility of the reinforced high-strength mortar layer was better than the reinforced UHPC layer.

4. Finite Element Model

4.1. Element Type and Meshing

Based on the bending tests, numerical analysis was conducted using Abaqus finite element software (2020 version). The C3D8R solid element was used to simulate concrete. The steel bars were simulated using T3D2 truss elements. The bond slip between the steel bars and concrete was ignored. The embedded was chosen for the connection of concrete and steel bars. Due to the symmetry of the model, and to save computational effort, only 1/4 of the structure was modeled. A simply supported beam boundary condition was used for the box beam, i.e., hinged at one end and rolling at the other. The mesh dimensions used for the parts were 25 cm × 25 cm. The details of finite element modeling are shown in Figure 13.

4.2. Material Constitutive Properties

The test was a failure test, and the material was damaged during loading. Therefore, the concrete plastic damage (CPD) model in Abaqus was used to simulate the nonlinear behaviors of concrete, UHPC, and high-strength mortar [35,36,37]. The relevant parameters of the yield function and flow criterion are shown in

Table 5. Parameters related to the yield function and flow criterion.

Material	$\mathit{\psi }$	$\mathit{e}$	${\mathit{\sigma }}_{\mathbf{b}0}/{\mathit{\sigma }}_{\mathbf{c}0}$	$\mathit{K}$	$\mathit{\mu }$
Concrete	36°	0.1	1.16	0.6667	0.0005
UHPC	38°	0.1	1.16	0.6667	0.00005
High-strength mortar	26°	0.1	1.16	0.667	0.00005

Where $\psi$ denotes the dilation angle, $e$ denotes the eccentricity of flow potential, ${\sigma }_{\mathrm{b}0}/{\sigma }_{\mathrm{c}0}$ denotes the ratio of uniaxial to biaxial compressive strength, $K$ denotes the ratio of the second stress invariant on the compression and tensile meridian plane, and $\mu$ denotes the viscosity coefficient.

.

4.2.1. Concrete

The constitutive relationship of concrete was referred to as GB 50010-2010 [35], as shown in Figure 14.

(1)

Concrete under uniaxial compression:

$$\sigma =(1-d_c)E_c\epsilon$$

(5)

$$d_c=\left\{\begin{array}{cc}1-\frac{{\rho }_{c}n}{n-1+x^n}& x\le 1\\ 1-\frac{{\rho }_c}{{\alpha }_c{(x-1)}^2+x}& x>1\end{array}\right. ({\rho }_c=\frac{f_{c,r}}{E_c{\epsilon }_{c,r}}, n=\frac{E_c{\epsilon }_{c,r}}{E_c{\epsilon }_{c,r}-f_{c,r}}, x=\frac{\epsilon }{{\epsilon }_{c,r}})$$

(6)

where ${\alpha }_c$ denotes the parameter value of the descending section of the stress–strain curve of concrete under uniaxial compression, $f_{c,r}$ denotes the uniaxial compressive strength of concrete, ${\epsilon }_{c,r}$ denotes the peak compressive strain of concrete, $d_c$ denotes the damage factor of concrete under uniaxial compression, and $E_c$ denotes the elastic modulus of concrete.

(2)

Concrete under uniaxial tension:

$$\sigma =(1-d_t)E_c\epsilon$$

(7)

$$d_t=\left\{\begin{array}{cc}1-{\rho }_t\left[1.2-0.2x^5\right]& x\le 1\\ 1-\frac{{\rho }_t}{{\alpha }_t{(x-1)}^{1.7}+x}& x>1\end{array}\right. (x=\frac{\epsilon }{{\epsilon }_{t,r}}, {\rho }_t=\frac{f_{t,r}}{E_c{\epsilon }_{t,r}})$$

(8)

where ${\alpha }_t$ denotes the parameter value of the descending section of the stress–strain curve of concrete under uniaxial tension, $f_{t,r}$ denotes the uniaxial tensile strength of concrete, ${\epsilon }_{t,r}$ denotes the peak tensile strain of concrete, and $d_t$ denotes the damage factor of concrete under uniaxial compression.

4.2.2. UHPC

Referring to [36,37], the constitutive relationship of UHPC is shown in Figure 15.

(1)

UHPC under uniaxial compression:

$$\sigma =\left\{\begin{array}{cc}{f}_c\frac{n\xi -{\xi }^2}{1+(n-2)\xi }& 0<\epsilon <{\epsilon }_{cp}\\ f_c\frac{\xi }{2{(\xi -1)}^2+\xi }& \epsilon >{\epsilon }_{cp}\end{array}\right. (\xi =\frac{\epsilon }{{\epsilon }_{cp}}; n=\frac{E_c}{E_{cs}})$$

(9)

where $f_c$ denotes the compressive strength of the UHPC prism, ${\epsilon }_{cp}$ denotes the peak compressive strain, taken as 0.0035 (the failure strain is taken as 0.01), $E_c$ denotes the elastic modulus of UHPC, $E_{cs}$ denotes the secant elastic modulus at the peak, taken as 35.8 MPa, and $n$ denotes the elastic modulus ratio, taken as 1.19.

(2)

UHPC under uniaxial tension:

$$\sigma =\left\{\begin{array}{cc}\frac{f_{ct}}{{\epsilon }_{ca}}\epsilon & 0<\epsilon \le {\epsilon }_{ca}\\ f_{ct}& {\epsilon }_{ca}<\epsilon <{\epsilon }_{pc}\\ \frac{f_{ct}}{{(1+w/w_p)}^p}& 0<w\end{array}\right.$$

(10)

where $f_{ct}$ denotes the tensile strength of UHPC, ${\epsilon }_{ca}$ denotes the peak tensile strain, taken as 0.0002, ${\epsilon }_{pc}$ denotes the limit tensile strain, taken as 0.01, $w$ denotes the crack width, $p$ denotes the curve parameter, taken as 0.95, and $w_p$ denotes the crack width when the tensile stress decreases to $2^{-p}{f}_{ct}$, taken as 1 mm.

4.2.3. High-Strength Mortar

The constitutive model of high-strength mortar under compression was obtained from Tongji University [38], and its stress–strain relationship was shown in Equation (11):

$$\left\{\begin{array}{cc}\frac{\sigma }{f_m}=\frac{\epsilon /{\epsilon }_m}{0.3{\left(\frac{\epsilon }{{\epsilon }_m}\right)}^2+0.4\left(\frac{\epsilon }{{\epsilon }_m}\right)+0.3}& \epsilon /{\epsilon }_m\le 1\\ \frac{\sigma }{f_m}=\left(1.1-0.1\frac{\epsilon }{{\epsilon }_{\mathrm{m}}}\right)& \epsilon /{\epsilon }_m>1\end{array}\right.$$

(11)

where $f_m$ is the compressive strength of high-strength mortar. In addition, since the elastic modulus can be taken as the secant modulus at $\sigma$ = 0.4$f_m$ of the stress–strain curve, the peak strain ${\epsilon }_m$ can be inferred accordingly.

Compared to the compressive strength, the tensile strength of high-strength mortar is relatively small. The tensile constitutive curve of high-strength mortar is consistent with concrete, and the mechanical parameters of high-strength mortar material are used in the calculation.

In the CDP model of high-strength mortar, when the material stress exceeded the elastic limit, it entered the elastic–plastic stage, and the specified stiffness-damage factor is calculated using the following Equation (12) [39]:

$$d=1-\sqrt{\frac{\sigma }{E_m\epsilon }}$$

(12)

where $d$ is the compression-stiffness-damage factor and $E_m$ indicates the elastic modulus of high-strength mortar.

4.2.4. Steel Bars

This test is a bending test, and the stress level of the steel bars generally did not exceed the yield strength. Therefore, an ideal elastic–plastic model was used to describe the mechanical behavior of the steel bars, as shown in Figure 16.

4.3. Validation of Finite Element Analysis

The load mid-span deflection curves obtained by the tests and Abaqus finite element analysis were compared, as shown in Figure 17. The simulation curves were in good agreement with the experimental curves, and the finite element model can successfully simulate the elastic stage, elastic–plastic stage, and plastic stage.

Figure 18 compares the critical loads obtained by the tests and the finite element model. It can be found that the cracking loads of the three beams obtained by the finite-element analysis were greater than those obtained by the tests. The errors were more than 10% because the finite-element model did not consider the initial defects of the test beams and did not consider the bond slip between the strengthening layer and the concrete interface. However, the absolute errors of the yield loads and ultimate loads were within 10%. This revealed that the finite-element model established in this study can accurately predict the yield loads and ultimate loads of the test beams. Based on the load-deflection curves and critical loads, the simulation results in this study were in good agreement with the experimentation results.

The stress state of steel bars under the ultimate load was crucial for the derivation of the ultimate bearing capacity. In order to avoid secondary damage to box beams, this test did not paste the strain gauge on the longitudinal steel bars after the beam bottom plate was broken. Therefore, this study used the finite-element model to obtain the stress levels of steel bars in the ultimate state, as shown in Figure 19:

It can be seen from Figure 19 that when the control beam was in the ultimate state, the stress of the bottom longitudinal steel bars reached 376 MPa, and they yielded in tension. When the mortar beam and the UHPC beam were in the ultimate state, the stress of the steel bars in the original box bottom plate reached 376 MPa. Meanwhile, the stress of the steel bars in the high-strength mortar layer and the UHPC layer reached 435 MPa, both of which yield in the tension.

5. Calculation of the Ultimate Bearing Capacity

5.1. Computational Simplification and Assumptions

To simplify the calculation, according to the principle of equal area and equal bending moment of inertia, the section of the box beam was equivalent to the I-shaped section [40], as shown in Figure 20.

At the same time, the following assumptions were made for the calculation:

(a)

Plane-section assumption: the strain of the section is linearly distributed along the height.

(b)

Without considering the contribution of concrete in the tensile area to bearing capacity: ignoring the tensile strength of concrete.

(c)

Without considering the bond slip between the UHPC layer and the bottom of the box beam: the full section strain is continuous.

5.2. Control Beam

According to GB 50010-2010 [35], the stress distribution of concrete in the compression area is shown in Figure 21. According to experimental phenomena and the finite-element analysis, when the unstrengthened beam reached its ultimate state, the concrete in the compression area was not crushed (i.e., the strain of the concrete at the edge of the compression area was less than ${\epsilon }_{cu}$), and the steel bars in the tension area reached yield load. If the concrete in the compression area was not crushed, the complex stress distribution cannot be equivalent to a simple rectangular distribution according to the stress correction coefficient of the specification. To facilitate the calculation, it was assumed that the concrete at the edge of the compression area just reached its axial compressive strength at the ultimate bearing state, i.e., the strain of the concrete at the edge of the compression area reached ε_cu = 0.002 [41].

Figure 22 shows the cross-section stress distribution of the control beam under the ultimate bearing state, where $h_{co}$, $h_c$, and $b$ are the height of the section, the height of the compression area, and the effective width of the section, respectively, $F_c$ denotes the pressure provided by the concrete in the compression area, $H_c$ denotes the height from the neutral axis, $f_{st}$ and $A_{st}$ denote the yield strength and total area of the steel bars in the tensile area, and $F_s$ denotes the tensile force provided by the steel bars in the tensile area.

According to the static equilibrium conditions of the section and the assumption of the plane section, the combined force in the compression area was obtained:

$$\left\{\begin{array}{c}\frac{h}{h_c}=\frac{{\epsilon }_c}{{\epsilon }_0}\\ {\sigma }_c(h)=f_c[1-{(1-\frac{h}{h_c})}^2]\\ F_c={\displaystyle {\int }_0^{h_c}{\sigma }_c(h)·b·dh=\frac{2f_{c}b{h}_c}{3}}\end{array}\right.$$

(13)

The distance between the resultant force provided by the concrete in the compression area and the neutral axis was obtained:

$$H_c=\frac{{\displaystyle {\int }_0^{h_c}{\sigma }_c(h)·h·dh}}{{\displaystyle {\int }_0^{h_c}{\sigma }_c(h)·dh}}=\frac{5h_c}{8}$$

(14)

According to the static equilibrium conditions of the section, the height of the compression area was obtained:

$$\left\{\begin{array}{c}{F}_s=f_{st}{A}_{st}\\ F_c=F_s\\ h_c=\frac{3f_{st}{A}_{st}}{2f_{c}b}\end{array}\right.$$

(15)

The ultimate bending bearing capacity was obtained as follows:

$$\left\{\begin{array}{l}{M}_u=0.5P_u(0.5L_s-0.5L_b)\hfill \\ M_u=f_{st}{A}_{st}(h_{co}-a_s-h_c+H_c)\hfill \end{array}\right.$$

(16)

where $L_s$ is the calculation span, and $L_b$ is the pure bending length.

5.3. Mortar Beam

When the mortar beam reached its ultimate bearing state, the concrete in the compression area was crushed, and the high-strength mortar layer cracked. Due to the lack of strain-hardening properties of high-strength mortar, the stress could not be maintained after cracking. The contribution of high-strength mortar was not considered in the calculation. In addition, the tensile steel bars in box beams and high-strength mortar layers reached yield strength. Due to the crushing of concrete in the compression area, the stress distribution of concrete in the compression area can be simplified into a simple rectangular distribution according to the code GB 50010-2010 [35], as shown in Figure 23, where the stress coefficient $\alpha$ is taken as 1.

In Figure 23, $h_m$ and $b_m$ are the height and width of the high-strength mortar layer, respectively, $f_{sm}$ and $A_{sm}$ denote the yield strength and total area of tensile steel bars in the high-strength mortar layer, $a_s$ denotes the distance between the bottom concrete surface and the center of gravity of the steel bars, and $a_m$ denotes the distance between the bottom high-strength mortar surface and the center of gravity of the steel bars in the strengthening layer.

According to the static equilibrium condition, the height of the compression area was obtained.

$$\left\{\begin{array}{l}{F}_c=F_s+F_{sm}\hfill \\ \alpha f_{c}b{h}_c=f_{st}{A}_{st}+f_{sm}{A}_{sm}\hfill \\ h_c=\frac{f_{st}{A}_{st}+f_{sm}{A}_{sm}}{\alpha f_{c}b}\hfill \end{array}\right.$$

(17)

The ultimate bearing capacity was obtained by the following formula:

$$\left\{\begin{array}{l}{M}_u=0.5P_u(0.5L_s-0.5L_b)\hfill \\ M_u=f_{st}{A}_{st}(h_{co}-a_s-h_c/2)+f_{sm}{A}_{sm}(h_{co}+h_m/2-h_c/2)\hfill \end{array}\right.$$

(18)

5.4. UHPC Beam

When the UHPC beam reached the ultimate state of bearing capacity, the concrete in the compression area was not crushed, and the UHPC layer cracked. However, due to the strain-hardening characteristics of UHPC, it can still maintain stress after cracking. In addition, the steel bars of box beams and UHPC layers reached the yield strength. Due to the smaller height of the UHPC layer compared to the full section, the influence of the stress-distribution form of the UHPC layer on the ultimate bearing capacity can be ignored. For the convenience of calculation, the stress distribution of the UHPC layer was simplified to a rectangular distribution, as shown in Figure 24. The combined force of concrete in the compression area and its distance to the neutral axis were the same as those calculated by the control beam.

In Figure 24, $h_u$, $b_u$, and $f_{ut}$ denote the height, width, and tensile strength of the UHPC layer, respectively, and $f_{su}$ and $A_{su}$ are the yield strength and total area of tensile steel bars in the UHPC layer.

The height of the compression area was obtained by static equilibrium condition:

$$\left\{\begin{array}{l}{F}_c=F_s+F_{su}+F_u\hfill \\ \frac{2f_{c}b{h}_c}{3}=f_{st}{A}_{st}+f_{su}{A}_{su}+f_{ut}{A}_u\hfill \\ h_c=\frac{3(f_{st}{A}_{st}+f_{su}{A}_{su}+f_{ut}{A}_u)}{2f_{c}b}\hfill \end{array}\right.$$

(19)

The ultimate bearing capacity can be obtained by the following formula:

$$\left\{\begin{array}{l}{M}_u=0.5P_u(0.5L_s-0.5L_b)\hfill \\ M_u=f_{st}{A}_{st}(h_{co}-a_s-h_c+H_c)+(f_{su}{A}_{su}+f_{tu}{A}_u)(h_{co}+h_u/2-h_c+H_c)\hfill \end{array}\right.$$

(20)

As shown in

Table 6. Comparison of the experimental and predicted values in the ultimate bearing capacity.

Test Beam	${\mathit{P}}_{\mathit{t}}/\mathbf{k}\mathbf{N}$	${\mathit{P}}_{\mathit{p}\mathit{r}\mathit{e}}/\mathbf{k}\mathbf{N}$	${\mathit{P}}_{\mathit{t}}/{\mathit{P}}_{\mathit{p}\mathit{r}\mathit{e}}$
Control Beam	680	621	1.095
Mortar Beam	840	704	1.193
UHPC Beam	920	825	1.115

Where $P_t$ is the ultimate bearing capacity obtained from the test, and $P_{pre}$ is the ultimate bearing capacity predicted by the formula.

, the ratios of the ultimate bearing capacity experimental values to the predicted values for the control beam, mortar beam, and UHPC beam were 1.095, 1.193, and 1.115, respectively. This indicates that the bearing capacity prediction formula established in this paper had high computational accuracy.

Conclusively, our combined experimental and theoretical analysis verified that high-strength mortar and UHPC effectively enhanced the load-bearing capacity of existing box beams. Nevertheless, the sample amount in this study was constrained by time and practical challenges. Future studies should expand on this work, including tests on varying material thicknesses and interface treatments, to gain more generalized insights on full-scale members.

6. Conclusions

This study aimed to investigate the flexural behaviors of unstrengthened beams (control beam) and strengthened beams (mortar beam and UHPC beam). Simulating this bending experiment using Abaqus finite element software, the simulated values with the experimental values were compared. The major conclusions are summarized as follows:

• Reinforced high-strength mortar layer had little effect on the stiffness of box beams, but it can effectively improve the ductility of box beams. The reinforced UHPC layer increased the stiffness of the box beams in the elastic stage by 32.9%, but it also slightly reduced the ductility of the box beams.
• Reinforced high-strength mortar layer and reinforced UHPC layer can effectively improve the cracking strengths of box beams, and the cracking load increased by 20%. Moreover, the reinforced high-strength mortar layer and the reinforced UHPC layer can effectively improve the ultimate bearing capacities of box beams. Compared with the control beam, the ultimate bearing capacity of the mortar beam and the UHPC beam was increased by 23.5% and 35.3%, respectively.
• The results of nonlinear finite-element simulations were in good agreement with the experimental results. The established model can effectively simulate the flexural failure process of box beams in service and strengthened box beams and accurately predict the yield load and ultimate load.
• The prediction formula for the ultimate bearing capacity derived from the combination of experimental phenomena and finite-element analysis in this study had high accuracy. The ratios of the experimental values to the predicted values of the control beam, mortar beam, and UHPC beam were 1.095, 1.193, and 1.115, respectively. The established prediction method can accurately predict the flexural bearing capacities of the box beams after strengthening.