At present, in the Technical Regulations for Reinforced Concrete Composite Structures (JGJ 138-2001 GB) [
13] issued by the Ministry of Construction of China, the calculation formula of the normal section load-carrying capacity of simply supported composite beams is based on the assumption of a plain section, and the assumption of the full section yield of the shaped steel. In the Design Regulations for Steel-reinforced Concrete Structures (YB9082-97) [
14], the simple superposition method is used to calculate the bending capacity of the normal section of PSRC beams. It can be seen that the research on steel-reinforced concrete composite structures in China started late. Through experimental research and theoretical analysis, several calculation methods for the crack width of steel-reinforced concrete composite beams were proposed [
15].
3.1. Calculation of Average Crack Spacing
Because the crack problem itself is very complicated, there are many influencing factors. In this paper, the average crack width and the average crack spacing of the five test beams before the tensile flange yield are obtained by analyzing the positive and negative cracks in the pure bend section of the five test beams with few experimental data at home and abroad. To be consistent with the expression form of Technical Regulations for steel-reinforced concrete Composite Structures and Code for Design of Concrete Structures [
13,
16], the average crack spacing of pre-stressed steel-reinforced concrete simply supported composite beams is still calculated according to Formula (1).
As the tension longitudinal bar is located below the tension flange of the shaped steel, according to domestic test data [
14], the crack width at the horizontal part of the longitudinal tension reinforcement bar is generally higher than that at the lower flange of the shaped steel, so it is reasonable to take the crack width at the horizontal part of the longitudinal tension reinforcement bar as the checking object of the crack width. In the calculation formula of average crack spacing, it is reasonable to take the protective layer
of the outermost tensile longitudinal bar as the concrete protective layer thickness.
Due to the existence of shaped steel, the average crack spacing of section steel-concrete beams is larger than that of the same section reinforced concrete beams. Therefore, the influence of the shaped steel on the equivalent diameter and effective reinforcement ratio of the tensile longitudinal bar should be reasonably considered in the formula of the average crack spacing. The coefficient before the term in the formula for calculating average crack spacing includes the ratio of longitudinal reinforcement to concrete bond stress. Therefore, the bond strength between steel and concrete should be considered when calculating the average crack spacing of steel-reinforced concrete beams. It can be seen that the main factors affecting the crack spacing of PSRC beams concrete include not only the thickness of the concrete protective layer but also the bond strength between the shaped steel and the concrete.
According to the literature [
17], the bond strength between shaped steel and concrete is only 0.45 times that between circular steel bars and concrete. Many domestic scholars have done a lot of research on the bond strength of steel and concrete and reached their conclusions, but they agree with it. Here, we take the 0.45 × 0.7 = 0.315 reduction coefficient of bonding property between shaped steel and concrete and introduce it into the calculation formula of average crack spacing of steel-concrete composite beams. When considering the effective diameter of the tension flange part of the web of the shaped steel and the tension reinforcement, the calculation is considered according to the following Formulas (2) and (3).
The effective reinforcement ratio of the tension flange, part of the web, and the tension reinforcement is calculated according to the following Formula (4).
For the problem of how to consider the influence of the H-shaped steel web on beam cracks in the tensile area, refer to the literature [
13]. However, since the PSRC test beams are all bonded post-tensioned pre-stressed beams, the tensile height of the H-shaped steel web can be reached to the bellows epithelium.
where
is the concrete protective layer thickness of the longitudinal tensile reinforcement;
is the effective diameter of the tension flange and partial web of H-shaped, as well as the tensioned non-pre-stressed reinforcement and pre-stressed reinforcement;
is the effective reinforcement ratio of the tension flange and partial web of the H-shaped steel and the tensioned non-pre-stressed reinforcement and pre-stressed reinforcement;
, , , is the section area of longitudinal tension non-pre-stressed steel bar, pre-stressed steel bar, H-shaped steel tension flange, and web;
is the sum of longitudinally tensioned non-pre-stressed bar, pre-stressed bar, H-shaped steel tension flange, and partial web perimeter;
, is the thickness of H-shaped steel flange and web;
is the influence coefficient of an H-shaped steel web;
is the number of longitudinally stretched steel bars of type ;
is the relative bond characteristic coefficient of the tensile bar. Deformed steel υ = 1.0, smooth steel υ = 0.7, steel strand υ = 0.4, bond coefficient between steel and concrete υ = 0.315;
is the H-shaped steel web height.
According to the above calculation formula for average crack spacing, the calculated values of the average crack spacing of the 5 simply supported composite beams in this test are compared to the measured values, as shown in
Table 7.
The actual average crack spacing of 5 PSRC simply supported beams in this test is 106.04 mm, which is calculated according to the above formula of average crack spacing. The average ratio between the calculated value and the measured value is lcr,t/lcr,c, , and the standard deviation is , and the coefficient of variation is .
As can be seen from
Table 7, the calculated average crack spacing of PSRC simply supported composite beams proposed in this paper is slightly smaller than the measured value. Compared to the load-carrying capacity calculation, fracture calculation has many influencing factors and greater discreteness, so it is often difficult to achieve high accuracy. The difference between the calculated value and the measured value is about 8%, and the dispersion is small. We believe that the formula is feasible. From the above factors, it can be seen that in bonded pre-stressed projects, the effect of grouting bellows on crack distribution is close to that of deformed steel bars with the same diameter, but it has little effect on the stress of cracked steel bars.
3.2. Calculation of Average Crack Width
The factors influencing the calculation of the average crack width of PSRC beams are mainly related to the performance of the beam concrete, such as the influence of the concrete elongation between cracks on the crack width. In addition, it is closely related to the performance of the shaped steel. According to the usual crack width calculation method [
12,
16], after determining the calculation formula of the crack spacing, the crack width of the component can be calculated according to the following formula:
where
is the effect coefficient of concrete self-elongation between cracks on crack width;
is the longitudinal stress value of H-shaped steel tension flange and part of the web.
Since the stress increment of the stress reinforcement is not large from pre-compression to depressurization, the stress
of the reinforcement in the crack section is calculated according to the following formula:
where
is the bending moment calculated according to the combination of load effects;
is the effective height of the section;
is effective pre-stress;
is the distance from the pre-stressing point to the centroid of the section.
Since the stress increment from pre-compression to depressurization reinforcement is not large, the stress of reinforcement
in the crack section is calculated according to Formula (6). To be consistent with the expression form of literature [
12], the coefficient of the internal force arm is still taken as 0.87. For the strain non-uniformity coefficient of the steel bars of pre-stressed steel-concrete simply supported composite beams, the calculation formula is still referred to the literature [
15], namely:
where
is the non-uniform strain distribution coefficient of longitudinal tensile non-pre-stressed bars, pre-stressed bars, lower flange of the shaped steel, and part of the web between cracks;
is the concrete axis standard tensile strength.
The influence coefficient of concrete self-elongation between cracks on crack width can be calculated by the following formula:
When , and are determined according to Formulas (8), (6) and (1) respectively, can be calculated by Formula (9) from the measured crack spacing and average crack width . For the calculation of the PSRC simply supported composite beams, the result is .
In summary, the short-term av
average crack width of PSRC simply supported composite beams can be calculated as follows:
According to the above formula for calculating the average crack width, the calculated and measured values of the average crack width of the 5 PSRC beams before the tensile flange of the shaped steel yield are shown in
Figure 5. The crack width values of the test beam were read with 100× and 24× reading magnifiers. The average value of crack widths under different loads in the pure bend section of the beam and the main crack widths in the shear span section are taken.
It can be seen from
Figure 5 that the average crack width of the 5 test beams presents a good linear correspondence between the calculated value and the measured value. According to the calculation results of the testing machine, the average value of the calculated value
and the ratio of the measured value
of the average crack width of the 5 test beams before the tensile flange yield are
, the standard deviation is
, and the coefficient of variation is
.
Through the analysis of the test results, it is found that the concrete cracking of PSRC beams under tension edge has little effect on the trend of the deflection curve. Through the analysis of test data, formula fitting, and test results, it was found that the crack width of PSRC beams showed a good linear relationship with the load. The total calculated values are slightly smaller than the measured values, indicating that the formula can be improved by further accumulation of test data.
3.3. Calculation of Maximum Crack Width
Due to the non-uniformity of material quality and the randomness of cracks, the dispersion of crack spacing, and crack width is relatively large [
18]. To obtain the maximum crack width can be determined by multiplying an expansion factor by the average crack width. The ratio between the crack width of each crack and the average crack width of the specimen under various loads was calculated from the 128 cracks in the pure bending section of the pre-stressed steel-concrete simply supported composite beam from the crack formation to the first-grade load before the tensile flange yield. Through fracture distribution fitting, the expansion coefficient reflecting uneven fracture distribution with a guaranteed rate can be calculated as 1.67. The maximum crack width should also be considered. Under the combination of long-term loading effects, the average strain of the tensile steel bar between cracks will continue to increase due to the stress relaxation and slip creep of the concrete in the tensile area, and the crack width will also increase due to the concrete shrinkage. Therefore, the short-term maximum crack width must be multiplied by the crack expansion coefficient of the long-term effect of the load. The same value 1.5 in the Code for Design of Concrete Structures can be adopted. In this way, the maximum crack width formula is obtained:
The significance of the crack propagation coefficient in determining the maximum crack width lies in the consideration of both the beam test value and the combination of long-term effects of load. The dual effects of short-term effects and long-term effects are considered. It can be seen from Formula (1) that the force characteristic coefficient of the component in the formula for calculating the maximum crack width of the pre-stressed steel-concrete composite beam is consistent with the value of the literature [
13]. It shows that the application of prestress has little effect on the maximum crack width of pre-stressed steel-concrete composite beams.
Through the analysis of the research results of the crack width and crack spacing of the PSRC beams, the calculation method of the crack width and crack spacing of the PSRC beams should be further improved through a large number of experimental data. At the same time, it is necessary to improve the performance of concrete, improve the toughness of concrete materials, and increase the bond strength between the shaped steel and concrete by improving the concrete compactness, to improve the crack resistance of the PSRC beams.