Study on Crack Development of Frame Beams with U-Shaped Engineered Cementitious Composites Cover Layer Under Negative Moments

Abstract

In order to enhance the durability of concrete frame beams, a U-shaped engineered cementitious composites (ECC) protective layer is applied at the end of the frame beams. The bond between the ECC protective layer and the concrete is reinforced by incorporating notches and grooves in the occupancy plate. The development and resistance to cracking of reinforced concrete (RC) frame beams and frame beams with an ECC protective layer were investigated using monotonic loading tests. The test results show that the average value of crack spacing in the negative moment zone of the RC frame beam specimen is in close agreement with the crack spacing calculated according to the GB50010 Code for Design of Concrete Structures. While the dispersion of crack width in the negative moment zone of the RC frame beam specimens is considerable, the distribution pattern of crack width undergoes a gradual change with increasing load. When the maximum crack width calculation method of GB50010 is employed in the negative moment zone of RC frame beams, the crack width should be increased by approximately 1.25 times. Furthermore, the crack spacing and crack width of the ECC protective layer are markedly smaller than those of RC frame beams.

1. Introduction

Concrete frame construction is widely used in large public buildings. As the tensile strength of concrete is much lower than the compressive strength, it is prone to cracking during the normal use stage, which is not conducive to the durability of the structure. In large public buildings such as airports, the span of the frame beams can reach 18 m or more. Under the action of gravity load, the maximum crack width at the end of frame beams often plays a dominant role in the structural design. The calculation formula of concrete crack width in Code for Design of Concrete Structures GB50010 [1] is mainly based on the results of experimental research on the pure bending section of simply supported beams. However, the negative moment zone of frame beams is affected by shear force, which is different from that of flexural members. Whether the calculation method of crack width is applicable or not has not yet formed a relatively consistent view.

Engineered cementitious composite (ECC) has a super tensile deformation capacity. The tensile capacity can be maintained at a high level under uniaxial tension [2,3]. The ultimate tensile strain can exceed 3% and the maximum crack width is only about 0.1 mm. The durability is superior and has broad application prospects in structural engineering [4]. Marshall and Cox [5] and Li and Leung [6] studied the theory of cracking. Han et al. [7] proposed a constitutive model based on the total strain of ductile fiber-reinforced cementitious composites, which can be used to simulate the structural performance of components under cyclic and seismic loads. Kanakubo [8] proposed a method for evaluating the tensile properties of ductile fiber-reinforced cement-based composites after a large number of tests. High ductile concrete has excellent performance and has been widely used in the reinforcement and reconstruction of existing structures [9]. Fischer and Li [10] showed that the interaction between the linear elastic FRP bar and the ECC matrix in the tensile state has a ductile stress-strain behavior, forming a nonlinear elastic bending response characteristic with stable hysteresis behavior and small residual deflection. Leung et al. [11] concluded that ECC can effectively control the propagation of small cracks and the fatigue properties of materials have been greatly improved.

ECC has been demonstrated to enhance the bearing capacity and control crack development in RC beams. Hu et al. [12] used ECC to improve the shear performance of RC beams. Adding ECC to the compression zone can slow down crack development and delay the failure of RC beams. The test results show that the use of ECC improves the shear strength and deformation capacity of RC beams. Sheng et al. [13] compared the effect of ECC and TRECC (a type of ECC which are reinforced with a textile) on RC. The number of cracks in TRECC-RC composite beams was similar to that of ECC-RC composite beams. The effect of the ECC height ratio and textile ratio on the crack distribution of TRECC-RC composite beams is less. Cui et al. [14] investigated the effects of precast ECC plate thickness and reinforcement ratio on the bending behavior of precast ECC-RC beams. The tests show that the cracks in the bending zone of precast ECC plates are uniformly distributed and the average width is smaller than that of normal precast concrete beams. ECC can effectively control the generation and development of bending and shear cracks in precast concrete beams.

ECC has higher tensile strain and durability and is widely used in reinforcement works [15,16]. The ductility increases significantly when the ECC thickness exceeds the critical value (30% of beam height) [17]. However, this may require more material resulting in higher economic costs. Ge et al. [18] proposed the theoretical formulas for internal force and bending moment of ECC-layered reinforced RC beams and analyzed the influence of the influential parameters, such as ECC substitution rate and reinforcement rate, on the yield moment, ultimate moment, and ductility of the beams. Tian et al. [19] investigated the flexural synergistic effect of ECC-reinforced RC beams and analyzed the damage mode, crack development mode, and crack width of the specimens. An effective model was proposed to predict the crack extension mode of ECC-RC. ECC is also often used in combination with RC and FRP for better performance. Zhu et al. [20] used high-strength ECC for the reinforcement of RC members and found that 20 mm thickness of ECC increased the bearing capacity by 8.3%, and the use of FRP further enhanced the bearing capacity and slowed down the early crack development. The failure model of FRP changed from debonding to fracture after the introduction of ECC [21]. By taking advantage of the large tensile strain of ECC, the cracks can be effectively controlled, avoiding the debonding of FRP after cracking of RC. Cong et al. [22] conducted an experimental study on the bending behavior of ECC-CFRP beams, and when the ECC thickness is small, the CFRP failure mode is still debonding. The contact relationship between ECC and tensile reinforcement can significantly affect the bending performance of RC beams, rather than the ECC thickness. The proposed load capacity prediction model can better predict the load capacity of two after reinforcement. Wu et al. [23] placed the ECC layer on top of over-reinforced beams and investigated the effect on their bearing capacity and ductility. The test results show that the ECC layer can effectively improve the brittle fracture characteristics of over-reinforced beams, and the ductility is improved.

Currently, a large number of research works have focused on the effect of ECC on the bearing performance of members. In addition, the study of ECC in beam-column joints has also attracted the attention of scholars [24,25,26]. Qudah and Maalej [26] used ECC to improve the seismic behaviors of the beam-column joints. The test results showed that the use of ECC materials in the beam-column joint region significantly improved the shear strength and cracking response of the joints. The specimens with ECC materials all showed little spalling, while the conventional concrete specimens showed extensive spalling of the concrete. Said and Razak [27] investigated the performance of ECC in the joint region by cyclic loading tests on full-size beam-column specimens. The crack spacing and crack width of the ECC were significantly smaller than those of the RC specimens, and the ECC had a higher ultimate bearing capacity and ductility, which were higher than those of the RC specimens. Shamass and Cashell [28] used fiber-reinforced polymer rebars, such as carbon fibers and glass fiber-reinforced plastics, which can be an effective, sustainable, and durable solution to improve the durability of reinforced concrete structures in corrosive environments.

In general, crack widths can be reduced by reducing the stresses on the reinforcement, which can be achieved by increasing the longitudinal force reinforcement on the top surface of the beam end. However, the use of dense reinforcement not only results in increased steel consumption but also presents challenges in concrete vibration, potentially leading to difficulties in ensuring construction quality. Furthermore, augmenting the negative reinforcement at the extremity of the frame beam will have a deleterious effect on the seismic structure of strong columns and weak beams. Therefore, it is of practical significance to seek a method to control the maximum crack width at the end of concrete frame beams by making full use of the superior anti-cracking performance of ECC materials [29,30].

By the characteristics of the distribution of wider cracks in frame beams at the top and side faces of beam ends, this paper proposes to set up a U-shaped ECC protective layer installed at the ends of large-span concrete frame beams. The bonding force between the ECC protective layer and RC is enhanced by setting concave points and grooves in the occupancy plate and forming point bonds, strip bonds, and other structures on the surface of RC. In order to investigate the stress performance of frame beams with an ECC protective layer, two normal reinforced concrete specimens and two RC specimens with an ECC protective layer were designed. The distribution patterns of crack spacing and crack width of the frame beams under static loading were investigated by monotonic step-by-step loading tests.

2. Frame Beam with ECC Protective Layer

2.1. Status of ECC Protective Layer

In the normal use stage of long-span concrete frame beams, where the crack width at the end of the beam exceeds the limit of the GB50010 [1], the ECC protective layer is set at the end of the frame beam. This allows the full potential of the exceptional crack resistance of ECC materials to be realized, thereby ensuring the durability of the primary structure required for the design’s working life [31]. The ECC protection layer is set on the top surface of the beam end, and the side of the beam end can be set with full height or partial height of the ECC protection layer according to the distribution of the force cracks. The occupancy plate of the ECC protection layer is used as a temporary formwork, which has less influence on the RC pouring of the frame beams, and the RC portion and the ECC protection layer are constructed in segments.

A lightweight material with strong plasticity, such as an extruded sheet, is selected for the ECC protective layer occupying the plate which is convenient for forming and splicing. In addition, by setting array depressions on the surface of the occupation plate, a large number of protrusions are formed on the surface of RC. It increases the bonding force between the ECC protective layer and RC and avoids the separation or concentrated cracks between the ECC protective layer and RC. Furthermore, the mechanical bite force between the stirrups of the frame beam and the ECC protective layer can be used to enhance the integrity of the component [32]. The secondary construction of the ECC protective layer is carried out by the plastering method, which is simple and convenient. Due to the small range of the ECC protective layer, it has little effect on the structural cost. The size of the frame beam with the ECC protective layer is the same as that of the RC member. In addition to reducing the maximum crack width, other mechanical properties of the frame beam with the ECC protective layer are the same as those of the RC frame beam.

2.2. Structural Construction of ECC Protective Layer

(1) Distribution point key

The distribution point key is located on the main contact surface between the ECC protective layer and RC, which is arranged in a plum-blossom shape. A large number of distributed point keys can significantly improve the bonding force between the ECC protective layer and the internal concrete and still maintain a close fit after a large deformation of the frame beam. The distribution of concave points on the inner side of the occupying plate is shown in Figure 1.

(2) Stirrup key

The stirrup key is located between the ECC protective layer and the contact surface of the RC. By setting a groove with the same diameter and spacing as the stirrup of the frame beam in the occupation plate, the stirrup part (not more than half) protrudes from the surface of the RC when the RC is poured first. Through the mechanical interlocking between the ECC protective layer and the stirrup, the force transmission capacity between the ECC protective layer and the internal concrete is enhanced. The stirrup keys on the inner side of the occupying plate are shown in Figure 2.

(3) Beam end T-shaped key

The joint surface between the ECC protective layer and the frame column is located at the maximum negative moment at the beam end. By bending the ECC protective layer inward and outward on the column surface, the T-shaped key at the beam end is formed to enhance the anchorage performance of the ECC protective layer at the beam end and delay the occurrence of cracks at the end of the ECC protective layer. The inner anchorage depth of the T-shaped key at the beam end is the same as the thickness of the ECC protective layer, and the width of the outer extension section is 3 to 5 times the thickness of the ECC protective layer. The T-shaped key occupying the plate at the beam end is shown in Figure 3.

(4) Long round key

The long round key is located at the interface between the ECC protective layer and the surrounding RC, forming an oblong bulge on the side of the RC through the occupying plate to enhance the bonding force between the ECC part of the secondary construction and the RC. And it can restrain the gap between the two materials. The long round key around the placeholder is shown in Figure 4.

3. Test Survey

3.1. Design and Manufacture of Specimen

In the test, two RC frame beam specimens and two specimens with the ECC protective layer at the beam end were designed. The geometric dimensions and reinforcement of RC frame beam specimens are shown in Figure 5. The geometric dimensions of specimens KJ-1 and KJ-2 are the same as those of reinforcement.

The specimens with the ECC protective layer at the end of the beam are shown in Figure 6 and Figure 7. The geometric dimensions and reinforcement of the specimens KJ-E1 and KJ-E2 are the same as those of the RC specimens. The length of the ECC protective layer is 1.5 times the height of the beam. Among them, the side protection range of the KJ-E1 beam is the same as that of beam height and the side protection range of the KJ-E2 beam is the upper 2/3 beam height. The fabrication of frame beam specimens with the ECC protective layer is shown in Figure 8.

Locate and support the wooden square template according to the specimen detail drawing after the reinforcing steel work is finished, and the template is fixed and supported with scaffolding outside, as shown in Figure 9a. When pouring concrete, for specimens KJ-1 and KJ-2, it is sufficient to pour concrete at one time after supporting the formwork; however, for specimens KJ-E1 and KJ-E2, it is necessary to place ECC occupancy boards before supporting the formwork before proceeding with the subsequent pouring work, as shown in Figure 9b. After the normal concrete is well cured, painting operation is carried out, and then the ECC protective layer is processed, successively removing the placeholder plate and plastering ECC operation, painting the surface of the ECC protective layer after its curing is completed, and finally drawing the reference line for the four specimens as a whole, as shown in Figure 9c,d.

3.2. Material Properties

The mechanical properties of concrete materials were tested by 150 mm × 150 mm × 150 mm cube test blocks, with six specimens per batch. Test blocks and test pieces were under the same conditions of maintenance for 28 days. After the completion of maintenance operations, test blocks were placed in the universal pressure testing machine for concrete cube compressive test. According to the measured value of cubic compressive strength obtained from GB/T 50152-2012 [33], the cube strength f_cu, axial compressive strength f_c, axial tensile strength f_t, and elastic modulus E_c of concrete materials can be calculated, as shown in

Table 1. Mechanical properties of concrete materials.

Specimen Number	fcu/MPa	fc/MPa	ft/MPa	Ec/MPa
KJ-1, KJ-2	45.01	34.21	3.21	33,660.2
KJ-E1, KJ-E2	46.08	35.02	3.25	33,864.4

.

According to the JC/T 2461-2018 Standard test method for the mechanical properties of ductile fiber reinforced cementitious composites [34], three 100 mm × 100 mm × 100 mm cube specimens were used in the compressive test of ECC materials, and three dog-bone specimens were used in the tensile test_. The mechanical properties of ECC materials were measured as shown in

Table 2. Mechanical properties of ECC materials.

Specimen Number	f1 /MPa	f2 /MPa	ft /MPa	fc /MPa	fcu /MPa
KJ-E1, KJ-E2	18.32	8.54	6.0	43.09	56.70

Note: f₁ and f₂ are the flexural strength and equivalent flexural strength of ECC materials, respectively.

. It can be seen from Table 2 that the strength indexes of ECC materials are higher than those of normal concrete.

The nominal diameter range of steel reinforcement is 6 mm to 20 mm, and the steel reinforcement grade is HRB400. According to GB/T 228.1-2021 [35], three specimens were used for each steel reinforcement diameter. The measured cross-sectional area, yield strength f_y, ultimate strength f_u, and elongation δ of each nominal diameter steel bar are presented in

Table 3. Mechanical properties of HRB400 reinforced bars.

Diameter/mm	Cross-Sectional Area/mm2	fy/MPa	fu/MPa	δ/%
6	28.86	435.67	658.77	13.7
8	49.24	412.77	653.30	14.7
10	76.38	418.55	657.32	14.7
12	100.51	491.68	675.90	16.7
14	139.84	445.80	632.48	21.0
18	233.28	487.69	700.01	20.3
20	288.89	489.07	690.78	22.3

.

3.3. Test Loading and Measurement

The test was carried out in the large-structure laboratory of the China Academy of Building Research. The bottom of the specimen is supported on the fixed hinge. The axial pressure of 1700 kN is applied on the top of the column during the test and the axial compression ratio of the column is 0.3. Two 100 t tension and compression actuators were used to apply loads on both ends of the frame beam at the same time. The test-loading device is illustrated in Figure 10.

The test loading is the crack resistance test of the frame beam. According to the relevant requirements of GB/T 50152-2012 Concrete Structure Test Method Standard [36], a concentrated force P is applied synchronously downward at both ends of the specimen and monotonically loaded step by step according to the order of 5 kN, 10 kN, 15 kN, 20 kN, 40 kN, 60 kN, 80 kN, 100 kN, and 120 kN (KJ-1 loaded to 100 kN). The loading system for the four specimens was identical and is illustrated in Figure 11.

Displacement meters are arranged along the loading direction and the length direction of the frame beam. The displacement meters D-1, D-2, and D-3 are used to measure the vertical deformation of the frame beam and the frame column. Furthermore, the displacement meters D-4 to D-9 are used to measure the horizontal deformation of the frame beam. In addition, steel strain gauges were set up in the column longitudinal reinforcement, beam end longitudinal reinforcement, and beam stirrup on the front and back of the specimen. The loading of the specimen and the arrangement of the measuring points are shown in Figure 12. The crack width is collected and recorded by the crack width gauge and the measurement accuracy is 0.01 mm.

4. Theoretical Analysis of Maximum Crack Width in Negative Moment Zone

4.1. Calculation of Flexural Bearing Capacity of Normal Section

The strength of the concrete, the stress of the steel bar, and the thickness of the protective layer have a direct influence on the crack width of concrete beams. The measured average value of the test is applied to the standard formula. According to the material test results of RC and steel bars in Section 3.2, the specific parameters of the test are obtained. The diameter of the steel bar in the longitudinal tension zone and compression zone is 17.23 mm and the tensile strength is 487.69 MPa. The thickness of the protective layer of the frame beam section is 20 mm, the diameter of the outer stirrup is 7.92 mm, and the spacing of the steel bars in the longitudinal tension zone is 25 mm.

The flexural bearing capacity of the normal section is obtained by GB 50010 [1].

$$a_{\mathrm{s}}=c+d_{\mathrm{stirrup}}+2d_{\mathrm{bar}}/3+d_{\mathrm{bar}}/2=48 \mathrm{mm}$$

(1)

$$h_0=h-a_{\mathrm{s}}=352 \mathrm{mm}$$

(2)

$$\rho =A_{\mathrm{s}}/(b{h}_0)\times 100\%=1.06\%$$

(3)

$${\rho }_{\min }=\max \left\{0.20\%,45{\displaystyle \frac{f_{\mathrm{t}}}{f_{\mathrm{y}}}}\%\right\}<\rho$$

(4)

$$x=f_{\mathrm{y}}(A_{\mathrm{s}}-A_{\mathrm{s}}^{\prime })/{\alpha }_{1}b{f}_{\mathrm{c}}=28.12 \mathrm{mm}<2{a_s}^{\prime }$$

(5)

$$M=f_{\mathrm{y}}{A}_{\mathrm{s}}(h_0-{a_{\mathrm{s}}}^{\prime })=215.22 \mathrm{kN}\cdot \mathrm{m}$$

(6)

where $a_{\mathrm{s}}$ is the distance from the resultant force point of the longitudinal reinforcement in the tensile zone to the edge of the tensile zone of the section, ${a_{\mathrm{s}}}^{\prime }$ is the distance from the resultant force point of the longitudinal reinforcement in the compression zone to the edge of the compression zone of the section, $h_0$ is the effective height of the section, $\rho$ is the reinforcement ratio of the section, $f_{\mathrm{c}}$ is the axial compressive strength value of concrete, $A_{\mathrm{s}}$ and $A_{\mathrm{s}}^{\prime }$ are the cross-sectional area of the longitudinal steel bars in the tension zone and the compression zone, respectively, and M is the calculated value of section bending moment.

4.2. Shear Bearing Capacity Calculation of Inclined Section

The shear force value $P_{\mathrm{u}}$ is smaller than $V_{\mathrm{cs}}$, which is the calculated value of the bearing capacity of the concrete and stirrups on the inclined section of the specimen.

$$P_{\mathrm{u}}=M/L=153.73 \mathrm{kN}$$

(7)

$$V_{\mathrm{cs}}={\alpha }_{\mathrm{cv}}{f}_{\mathrm{t}}b{h}_0+f_{\mathrm{yv}}{h}_0{A}_{\mathrm{sv}}/s=262.88 \mathrm{kN}>P_{\mathrm{u}}$$

(8)

where ${\alpha }_{\mathrm{cv}}$ is the shear bearing capacity coefficient of inclined section concrete, $f_{\mathrm{t}}$ is the axial tensile strength value of concrete, $f_{\mathrm{yv}}$ is the tensile strength value of the stirrup, and $A_{\mathrm{sv}}$ is the total cross-sectional area of each limb of the stirrup configured in the same section [37].

4.3. Calculation of Maximum Crack Width

According to the GB50010 [1], the maximum crack width ${\omega }_{\max }$ can be calculated by the following formula:

$$\psi =1.1-0.65{\displaystyle \frac{f_{\mathrm{t}}}{{\rho }_{\mathrm{te}}{\sigma }_{\mathrm{s}}}}=0.84$$

(9)

$$E_{\mathrm{s}}=2.0\times {10}^5 \mathrm{MPa}$$

(10)

$$d_{\mathrm{eq}}=32 \mathrm{mm}$$

(11)

$$c_{\mathrm{s}}=60+7=67 \mathrm{mm}$$

(12)

$${\omega }_{\max }={\alpha }_{\mathrm{cr}}\psi {\displaystyle \frac{{\sigma }_{\mathrm{s}}}{E_{\mathrm{s}}}}{l}_{\mathrm{cr}}={\alpha }_{\mathrm{cr}}\psi {\displaystyle \frac{{\sigma }_{\mathrm{s}}}{E_{\mathrm{s}}}}\left(1.9c_{\mathrm{s}}+0.08{\displaystyle \frac{d_{\mathrm{eq}}}{{\rho }_{\mathrm{te}}}}\right)=0.35 \mathrm{mm}$$

(13)

where $\psi$ is the uneven coefficient of strain in longitudinal tensile reinforcement between cracks, ${\rho }_{\mathrm{te}}$ is the reinforcement ratio of tensile reinforcement calculated from effective tensile concrete cross-sectional area, ${\sigma }_{\mathrm{s}}$ is the calculated stress value of the reinforcement, ${\alpha }_{\mathrm{cr}}$ is the stress characteristic coefficient, and $c_{\mathrm{s}}$ is the distance from the outer edge of the outermost longitudinal tension reinforcement to the bottom edge of the tension zone.

On the whole, for the reinforced beam, the greater the reinforcement ratio of the beam section, the stress of the steel bar, and the crack spacing, the greater the maximum crack width. According to the calculation results of the above formula, the theoretical maximum crack width appears at the intersection section of the beam and column at the farthest distance from the loading point. The calculated theoretical width is 0.35 mm, which is 0.10 mm different from the experimental result of 0.25 mm. The formula needs to be further modified.

The crack spacing $l_{\mathrm{cr}}$ is the data that can be directly obtained in the test. It is closely related to the calculation method of the maximum crack width. At the same time, the stress characteristic coefficient ${\alpha }_{\mathrm{cr}}$ of the component is the value taken considering the maximum crack width under long-term load. However, this paper only aims at short-term load, so it is necessary to modify the stress characteristic coefficient ${\alpha }_{\mathrm{cr}}$ of the component. Therefore, this section focuses on the influence of crack spacing $l_{\mathrm{cr}}$ and component stress characteristic coefficient ${\alpha }_{\mathrm{cr}}$ on the maximum crack width.

5. Study on Fracture Development

5.1. Development of Cracks

In the crack resistance test stage, the change of the crack development of the frame beam specimen with the loading is shown in Figure 13, Figure 14, Figure 15 and Figure 16. For the frame beam specimens KJ-1 and KJ-2, when loaded to P_cr = 5 kN (0.05 P_u), the first crack appears in the specimen and the crack width is 0.01 mm. When P = 20 kN (0.13 P_u), several bending cracks appear on the side near the column and the crack direction is perpendicular to the beam axis. When P = 40–60 kN (0.26–0.40 P_u), the number of cracks increases. The height and width of cracks increase, while the cracks on the left and right sides are approximately symmetrically distributed. When P = 80–100 kN (0.53–0.67 P_u), the cracks near the loading end incline to the loading point, the number and width of cracks increase further, and secondary oblique cracks appear locally. When P = 120 kN (0.8P_u), the new cracks are connected with the previous shear oblique cracks and the crack width increases significantly. The obvious difference between specimens with the ECC protective layer (KJ-E1 and KJ-E2) and comparison specimens (KJ-1 and KJ-2) is that the cracks within the range of the ECC protective layer are more intensive and compact, and the spacing of the cracks is significantly smaller than that of the comparison specimens (KJ-1 and KJ-2). The ECC protective layer did not detach from the RC during the loading stage of the beam. It shows that the interface construction of the ECC protective layer in Section 2 achieves a reliable connection between the ECC protective layer and the bonding surface of RC, which plays a greater degree of crack resistance in the negative moment zone of RC frame beams.

At the stage of the crack resistance test, the final failure of the frame beam specimen is shown in Figure 17. The variation of cracks in KJ-E1 and KJ-E2 specimens with the ECC protective layer with loading is the same as that of specimens KJ-1 and KJ-2. The cracks in the range of the ECC protective layer are denser, and the crack spacing is much smaller than that of the RC specimens. However, the crack width is significantly reduced, and the ECC protective layer and internal components have good working performance [38]. When the ECC protective layer is 2/3 beam height (KJ-E2), the maximum crack width outside the protection range is not more than 0.2 mm, which can also achieve a better protection effect.

5.2. Crack Spacing

According to the GB50010 [1], the crack spacing $l_{\mathrm{cr}}$ can be calculated by the following formula:

$$l_{\mathrm{cr}}=1.9c_{\mathrm{s}}+0.08{\displaystyle \frac{d_{\mathrm{eq}}}{{\rho }_{\mathrm{te}}}}$$

(14)

where $c_{\mathrm{s}}$ is the distance from the outer edge of the outermost longitudinal tensile steel bar to the bottom of the tensile zone, $d_{\mathrm{eq}}$ is the equivalent diameter of the longitudinal reinforcement in the tensile zone, for the ribbed reinforcement $d_{\mathrm{eq}}={\displaystyle \sum n_{\mathrm{i}}}{d_{\mathrm{i}}}^2/{\displaystyle \sum n_{\mathrm{i}}}{d}_{\mathrm{i}}$, ${\rho }_{\mathrm{te}}$ is the reinforcement ratio of the tensile steel bar calculated according to the effective tensile concrete area and the flexural member ${\rho }_{\mathrm{te}}=A_{\mathrm{s}}/(0.5bh)$, where $A_{\mathrm{s}}$ is the cross-sectional area of the tensile longitudinal steel bar.

When the specimen is loaded, the bending moment of the frame beam at the column side section (referred to as the beam end) is the largest. The bending moment of the section at the loading point is zero. The tensile stress ${\sigma }_{\mathrm{s}}$ of the steel bar in the tensile zone can be calculated by the following formula (l is the distance from the loading point to the calculated section).

$${\sigma }_{\mathrm{s}}={\displaystyle \frac{Pl}{0.87\times h_0{A}_{\mathrm{s}}}}$$

(15)

For the frame beam specimen in this paper, the tensile stress of the steel bar at the beam end section is the largest, which changes linearly along the length direction of the beam. The tensile stress of the steel bar at the loading point section is reduced to zero.

When the load P = 60 kN–120 kN (0.4P_u–0.8P_u), the change of the crack spacing of the frame beam with the tensile stress ${\sigma }_{\mathrm{s}}$ (along the length direction of the beam) is shown in Figure 17. It can be seen from Figure 18 that when P ≥ 0.4 P_u, although the dispersion of the crack spacing value is large, the crack spacing at the beam end remains unchanged. The average value of the crack spacing is close to the crack spacing $l_{\mathrm{cr}}$ = 102 mm obtained by Equation (14). In the area where the tensile stress ${\sigma }_{\mathrm{s}}$ of the steel bar is small, the crack spacing is slightly larger. As the load and the tensile stress of the steel bar gradually increase, the crack spacing further decreases. When P ≥ 0.53P_u, the average crack spacing remains unchanged along the length direction of the negative moment zone of the frame beam, which is very close to the crack spacing obtained according to the GB50010 [1].

5.3. Crack Width

According to the GB50010 [1], the maximum crack width w_max of flexural members combined depends on the load standard. And considering the long-term effect can be calculated according to the following formula:

$$w_{\max }={\tau }_l{\tau }_{\mathrm{s}}{w}_{\mathrm{m}}$$

(16)

where w_m is the average crack width, τ_s is the expansion coefficient of short-term crack width, and τ_l is the expansion coefficient considering the long-term effect which is 1.5.

The short-term maximum crack width w_smax can be calculated by the following formula:

$$w_{\mathrm{smax}}={\alpha }_{\mathrm{c}}{\tau }_{\mathrm{s}}\psi {\displaystyle \frac{{\sigma }_{\mathrm{s}}}{E_{\mathrm{s}}}}{l}_{\mathrm{cr}}$$

(17)

where ${\alpha }_{\mathrm{c}}$ is the influence coefficient of the concrete elongation between cracks on the crack width, taken as 0.77. $\psi$ is the strain non-uniformity coefficient of longitudinal tensile reinforcement between cracks, which is calculated by Equation (18).

$$\psi =1.1-0.65{\displaystyle \frac{f_{\mathrm{tk}}}{{\rho }_{te}{\sigma }_{\mathrm{s}}}}$$

(18)

When $\psi$ < 0.2, take $\psi$ = 0.2. When $\psi$ > 1.0, take $\psi$ = 1.0.

In order to determine the expansion coefficient of the short-term crack width in the negative moment zone, the maximum crack width of the four main cracks closest to the column end is divided by the average crack width. The following 80 data are obtained, as shown in

Table 4. Statistics of the expansion coefficient of short-term crack width in the negative moment region.

Load P/kN	Expansion Coefficient of Short-Term Crack Width/Average Crack Width
	Crack 1				Crack 2				Crack 3				Crack 4
	E-N	E-S	W-N	W-S	E-N	E-S	W-N	W-S	E-N	E-S	W-N	W-S	E-N	E-S	W-N	W-S
40	1.60	0.80	0.96	0.64	0.40	1.19	1.19	1.63	0.97	1.24	1.10	0.69	1.26	1.05	1.05	0.63
60	1.30	0.86	0.86	0.97	0.80	0.91	1.03	1.26	1.26	1.05	1.05	0.63	1.42	0.77	1.03	0.77
80	1.27	0.91	0.91	0.91	0.96	0.96	0.87	1.22	1.07	0.89	1.16	0.89	1.00	0.73	1.18	1.09
100	1.33	0.63	0.89	1.14	0.75	1.38	0.88	1.00	0.92	0.80	1.29	0.98	0.98	0.70	1.05	1.26
120	1.18	0.52	0.85	1.46	0.78	1.12	0.88	1.22	0.71	0.71	1.38	1.19	0.75	0.91	1.01	1.33

.

In order to determine the value of τ_s with a 95% guarantee, 80 data collected from the test were analyzed mathematically, and the statistical histogram of crack width was plotted as shown in Figure 19. At the same time, K-S analysis was performed, p value = 0.2 > 0.05, and the data obeyed the normal distribution N (1.0, 0.25). The Quantile–Quantile diagram of the expansion coefficient of short-term crack width is shown in Figure 20. According to the statistical results, the value ${\tau }_s$ with a 95% guarantee rate is: ${\tau }_s=1.0+1.645\times 0.25=1.41$.

Under the action of concentrated load P at the end of the beam, the change of crack width with the tensile stress σ_s (along the length direction of the beam) of the steel bar is shown in Figure 21. The dispersion of the width of the frame beam is large. When P ≥ 0.4P_u, the average crack width at the beam end is close to the maximum crack width w_smax obtained from Equation (17). The average crack width in most other regions is much larger than the maximum crack width obtained from Equation (17). Although the average crack width outside the beam end is still greater than the maximum crack width obtained by Equation (17), the difference between the two gradually decreases as the load and the tensile stress of the steel bar gradually increase. For the frame beam, the maximum crack width at the beam end plays a controlling role. After amplifying the short-term maximum crack width obtained by Equation (17) by 1.25 times, it is in good agreement with the test results of the crack width at the beam end.

When the load P = 0.8P_u, the distribution of crack width along the length direction of the beam of RC specimen KJ-2 and specimen KJ-E1, and KJ-E2 with the ECC protective layer, is shown in Figure 22. The crack width of the ECC protective layer is significantly smaller than that of the RC specimen.

6. Conclusions

(1) The force transfer ability between the ECC protective layer and the internal concrete can be significantly improved by forming distributed point keys, stirrup keys, long round keys, and T-shaped keys at the beam end on the concrete surface.

(2) When P ≥ 0.4P_u, the average value of the crack spacing in the negative moment zone of the frame beam specimen remains unchanged, which is very close to the crack spacing calculated according to the GB50010.

(3) The discreteness of crack width in the negative moment zone of frame beam specimens is large. And the distribution law of crack width gradually changes with the increase of load. When the calculation method of the maximum crack width of flexural members in code GB50010 is applied to the negative moment zone of frame beams, the crack width should be enlarged by about 1.25 times.

(4) The crack spacing in the ECC protective layer is much smaller than that of RC frame beams, and the crack width is significantly reduced. When the ECC protective layer is 2/3 beam height, the maximum crack width outside the protection range is not more than 0.2 mm, which can achieve a better protection effect.