Flexural behavior of high-strength, steel-reinforced, and prestressed concrete beams

Abstract

To study the flexural behavior of prestressed concrete beams with high-strength steel reinforcement and high-strength concrete and improve the crack width calculation method for flexural components with such reinforcement and concrete, 12 specimens were tested under static loading. The failure modes, flexural strength, ductility, and crack width of the specimens were analyzed. The results show that the failure mode of the test beams was similar to that of the beams with normal reinforced concrete. A brittle failure did not occur in the specimens. To further understand the working mechanism, the results of other experimental studies were collected and discussed. The results show that the normalized reinforcement ratio has a greater effect on the ductility than the concrete strength. The cracking- and peak-moment formulas in the code for the design of concrete (GB 50010-2010) applied to the beams were both found to be acceptable. However, the calculation results of the maximum crack width following GB 50010-2010 and EN 1992-1-1:2004 were considerably conservative. In the context of GB 50010-2010, a revised formula for the crack width is proposed with modifications to two major factors: the average crack spacing and an amplification coefficient of the maximum crack width to the average spacing. The mean value of the ratio of the maximum crack width among the 12 test results and the relative calculation results from the revised formula is 1.017, which is better than the calculation result from GB 50010-2010. Therefore, the new formula calculates the crack width more accurately in high-strength concrete and high-strength steel reinforcement members. Finally, finite element models were established using ADINA software and validated based on the test results. This study provides an important reference for the development of high-strength concrete and high-strength steel reinforcement structures.

Keywords

high-strength steel reinforcement / high-strength concrete / flexural behavior / crack width

Introduction

High-performance materials (HPMs), such as high-strength concrete (HSC) and high-strength steel (HSS), have been widely introduced in the construction industry and have been adopted in most of the current reinforced concrete (RC) design codes, including ACI 318-14 [¹], Eurocode 2 [²], NZS3101 [³], and GB50010-2010 [⁴]. Because HPMs reduce the amount of materials used to resist the same structural loads when compared with traditional normal-performance materials, that is, normal-strength concrete (NSC) and normal-strength steel (NSS), the structural self-weight, energy consumption, and carbon content in the structures are all reduced [⁵–⁷]. HSC has additional beneficial properties, such as a low permeability and high durability [⁸]. Prestressing technology is widely adopted in structures with HPMs to improve the anti-cracking and deformation, particularly in large-span structures, because with relatively small cross-sections, cracks and deformations under a service limit state may become the main concern in the design. Compared with ordinary prestressed concrete (PC) components, PC components with HPMs have certain advantages, namely, higher prestressing forces and a lower prestressing loss, a delayed onset of crack occurrence, and enhanced anchorage of the prestressed tendons.

However, two main problems exist in the application of RC and PC components with HPMs. First, HSS and HSC are more brittle than NSS and NSC. Therefore, the ductility of RC and PC components with HPMs is one of the main issues. HSC and/or HSS beams have been viewed by numerous researchers as being prone to brittle failure [⁹–¹¹]. An experimental study on 12 beams (where the compressive strength of concrete ranged from 99.3 to 103.1 MPa, and the yield strength of the longitudinal reinforcement ranged from 400 to 500 MPa) also showed that beams with HSS and HSC exhibited less ductility and more significant cracks than specimens constructed using NSS and NSC [¹²]. Opposite phenomena have also been observed by other researchers. Swamy and Anand [¹³] conducted an experimental study involving 16 HSC prestressed concrete beams (the compressive strength of concrete ranged from 85 to 110 MPa). The results suggest that HSC beams exhibit an inelastic behavior comparable to that of NSC beams. Ashour [¹⁴] tested nine HSC beams (the compressive strengths of the concrete were 48, 78, and 102 MPa, respectively) to investigate their deflection behavior. The best ductility was found when the compressive strength of the concrete was 78 MPa. Pam et al. [¹⁵] confirmed that the ductility of HSC beams can meet the design requirements by controlling the reinforcement ratio. Kaminska [¹⁶] presented a test and reported the deformation capacity of an HSC structure. Lopes and Bernardo [¹⁷] tested five HSC beams with a concrete compressive strength of 64 MPa and four HSC beams with a concrete compressive strength of 83 MPa, and their ductility was compared. The results indicate that, for a given number of longitudinal bars, the ductility slightly increased with the compressive strength. Ghasemi and Shishegaran [¹⁸,¹⁹] bent the reinforcement bars at 45° from 1/3 of the beam length to increase the beam bending capacity. From a literature review, we found no common conclusion regarding the ductility of elements with HPMs. Rabczuk et al. [²⁰–²³] proposed a series of numerical methods to model the crack and fracture of prestressed concrete beams and validated them using relative test results.

Studies also showed that owing to the brittleness of the HSC, the cracking behavior of the beams was different from that of the NSC beams. Researchers have also found that the method for calculating the cracking of NSS and NSC beams is unsuitable for HSS and HSC beams [²⁴,²⁵]. The error in cracking calculations is more significant for HSC elements given the higher brittleness of HSC compared with NSC, and the existing maximum crack width equation is no longer suitable for HSC beams [¹²,²⁶–²⁸]. Padmarajaiah and Ramaswamy tested eight full and seven partial PC beams with HSC. The crack widths were analyzed and compared, and an analytical model was proposed to calculate the crack width considering the effect of the magnitude of the longitudinal reinforcement strain and the interfacial bonding stress between the concrete and reinforcement [²⁹,³⁰]. Overall, studies on crack width calculations for components with HPMs are still insufficient.

To solve these problems, 12 PC beams with HSS and HSC under static loads were tested in this study. Furthermore, the results of other existing experimental studies were collected. Based on the tested and collected results, (1) the experimental results in terms of the failure mode, strength, and ductility of the 12 specimens were discussed, (2) the major factors influencing the flexural ductility of the beams were evaluated, and (3) an improved crack width calculation method for flexural components with HPMs was formulated.

Experimental program

Design of test specimens

To evaluate the influence of (1) the compressive strength of concrete, (2) the prestressing ratio, and (3) the amount of longitudinal reinforcement on the flexural behavior, 12 simply supported RC beams were fabricated for testing. Two concrete strength grades were adopted, i.e., C80 and C100, with cubic compressive strengths of 80 and 100 MPa, respectively. HRBF500 reinforcement (i.e., hot-rolled, ribbed, fine grain reinforcement with a yield strength of 500 MPa) and HRB 400 reinforcement (i.e., hot-rolled, ribbed reinforcement with a yield strength of 400 MPa) were adopted for longitudinal mild steel reinforcement and stirrups, respectively. An 1860 strength-grade strand (i.e., the ultimate tensile strength was 1860 MPa) was used for the prestressed tendons. For all beams, the cross-section dimensions were 200 mm × 450 mm, the length was 5800 mm, and the test span was 5400 mm. The thickness of the concrete cover (i.e., the distance between the outermost surface of the reinforcement and the closest outer surface of the beam) was 25 mm. Prestressed tendons were arranged linearly at the lower part of the beam, and the distance between the centroid of the strands and the bottom surface of the beams was 110 mm. The control stress for prestressing (σ_con) was 1395 MPa. The compressive and tensile mild steel reinforcements were arranged symmetrically on the beams.

Figure 1 and Table 1 summarize the main parameters of each specimen, where A_s, A_s', and A_p are the total section areas of the tensile, compressive mild steel reinforcement, and prestressed tendons, respectively. The stirrups in the pure bending sections of each beam were all C8@200 (where C8 indicates that the stirrups use an 8-mm diameter HRB 400 reinforcement, and @200 means that the spaces of the stirrups are 200 mm), although they differed in terms of the shear force in the bending-shear sections for different specimens, as shown in Table 1. The prestressing ratio (λ) in Table 1, which represents the ratio between the strength of the prestressed tendons and the mild steel reinforcement, is defined as follows:

(1)\mathit{\lambda }={\displaystyle \frac{f_{\mathrm{p}\mathrm{y}}{A}_{\mathrm{p}}}{f_{\mathrm{y}}{A}_{\mathrm{s}}+f_{\mathrm{p}\mathrm{y}}{A}_{\mathrm{p}}}},

where f_y and f_py are the yield strengths of the mild steel reinforcement and prestressed tendons, respectively. Because the prestressed tendons and the mild steel reinforcement have different yield strengths, to compare the ratios of reinforcement in different specimens, a normalized reinforcement ratio (ρ_s) is defined as follows:

(2){\mathit{\rho }}_{\mathrm{s}}={\displaystyle \frac{f_{\mathrm{y}}{A}_{\mathrm{s}}+f_{\mathrm{p}\mathrm{y}}{A}_{\mathrm{p}}}{f_{\mathrm{y}}bh}},

where b and h are the width and depth of the cross-section, respectively (Fig. 1).

Test setup and measurements

The tested beams were simply supported, and two concentrated loads were applied at the trisected points, as shown in Fig. 2. A two-stage loading control method was adopted: Before yielding of the specimens, the vertical loading process was force-controlled, and after yielding of the specimens, the loading process was controlled through displacement. The test terminated when the load decreased to 85% of the peak load (JGJ 101-2015). In the test, three linear variable differential transformers (LVDTs) were placed at mid-span and at the two supports to measure the deflection. To measure the strain, a force transducer was placed under the hydraulic jack to measure the load, and electrical resistance strain gauges were attached to the mild reinforcements in the specimens.

Material properties

Before the experiment, the strength and elastic modulus of the two types of concrete were tested in accordance with the Chinese code (GB/T50081-2002) [³¹]. The average axial strengths (f_c) of C80 and C100 obtained from prism specimens (150 mm × 150 mm × 300 mm) were 56.4 and 72.1 MPa, respectively, and the average elastic moduli were 40.7 and 44.0 GPa, respectively. The mechanical properties of the mild steel reinforcements and prestressed tendon were tested in accordance with the Chinese code (GB/T228.1-2010) [³²] and are summarized in Table 2.

Experimental results

Failure modes

All specimens exhibited similar damage processes and failure modes during the test. During the initial loading stage, the beams behaved as elastic elements, and no cracks were observed. When loaded to 30%–40% of the peak load, vertical flexural cracks first occurred at the tensile edge in the pure bending section. With increasing load, new cracks occurred, and existing cracks developed and propagated gradually upward. After yielding of the longitudinal mild steel reinforcement and the prestressed tendons, the crack width and deflection of the specimens rapidly developed. Finally, the concrete in the compression zone was crushed and the beams lost their load-bearing capacity. Owing to the brittle feature of HSC, the compressive concrete was suddenly crushed with a loud noise, which was different from the relatively slow crushing process of NSC. Despite the sudden damage to the concrete, all specimens notably deformed as a result of the yielding of the reinforcement, and we concluded that all beams behaved in a typical ductile failure mode, similar to properly reinforced NSC beams. A typical ultimate failure mode of the specimen is shown in Fig. 3, and the crack distributions of all specimens are illustrated in Fig. 4.

Mid-span bending moment-deflection curves

For clarity, all specimens are divided into four groups. As the grouping principal, for each group, all design parameters are the same, except for the amount of mild steel reinforcement and thus the normalized reinforcement ratio ρ_s. The mid-span bending moment–deflection curves of each group are shown in Fig. 5. The curves show that the loading process of the specimens can be divided into four stages. (1) In stage 1, before the cracking of the concrete, the curves are basically linear. (2) In stage 2, with the development of cracks, the stiffness of the specimens notably dropped. (3) In stage 3, after a yielding of the mild steel reinforcement and the prestressed tendons, the deflection increases rapidly, and the curves flatten. Finally, (4) in stage 4, after a crushing of the compressive concrete, the curves descend, indicating a loss of load-bearing capacity. As mentioned above, the crushing of the concrete was quite sudden; thus, the descending part of the curves was relatively steep. Notably, owing to the extremely hot weather during the fabrication stage, obvious defects formed on the top surface of specimens PBC-2 and PBC-3. As a result, a crushing of the compressive concrete occurred quite early in these specimens, and the third stage of the curves was extraordinarily short.

Strength

The tested cracking moment M_{\mathrm{c}\mathrm{r}}^{\mathrm{t}}, yielding moment M_{\mathrm{y}}^{\mathrm{t}}, peak moment M_{\mathrm{p}}^{\mathrm{t}}, and ratios {\displaystyle \frac{M_{\mathrm{c}\mathrm{r}}^{\mathrm{t}}}{M_{\mathrm{p}}^{\mathrm{t}}}} and {\displaystyle \frac{M_{\mathrm{y}}^{\mathrm{t}}}{M_{\mathrm{p}}^{\mathrm{t}}}} are listed in Table 3. The yielding moment is the moment according to the yielding point on the moment–deflection curve obtained using the geometrography method [³³]. Here, PCB-2 and PCB-3 were not included in the table and analysis owing to the unreliable test results that were caused by the aforementioned defects. From the data in Table 3, the following conclusions can be drawn.

(1) The ratios of the cracking and peak moment of the specimens were within the range of 0.33 to 0.47, and the mean value was 0.38, which was obviously larger than that for the RC beams owing to the effect of the prestressing force. The ratios of yielding and peak moment were approximately 0.75–0.88, and the mean value was 0.80.

(2) For the specimens in groups 2 and 4 with more prestressed tendons, the cracking bending moment was much higher than that in groups 1 and 3.

(3) A comparison between the specimens with different reinforcement ratios ρ_s indicated that the yielding and peak moment both increased with ρ_s, whereas ρ_s had little effect on the cracking moment.

(4) The peak moment of the beams increased slightly with the concrete strength. Taking group 2 with C80 concrete and group 4 with C100 concrete as examples, the peak moment increased by 5.7%–14.8% from group 2 to group 4.

The cracking moment and peak moment of all specimens were calculated using the formulas provided in the Chinese Code for Design of Concrete Structures (GB 50010-2010). The calculated results forM_{\mathrm{c}\mathrm{r}}^{\mathrm{c}}, M_{\mathrm{p}}^{\mathrm{c}}, and the ratios of the experimental and calculated results {\displaystyle \frac{M_{\mathrm{c}\mathrm{r}}^{\mathrm{t}}}{M_{\mathrm{c}\mathrm{r}}^{\mathrm{c}}}} and {\displaystyle \frac{M_{\mathrm{p}}^{\mathrm{t}}}{M_{\mathrm{p}}^{\mathrm{c}}}}, respectively, are listed in Table 3. The mean values of these two ratios were 1.06 and 1.05, respectively, indicating that the calculation methods for the PC beam in GB 50010-2010 were still suitable for the beams with HPMs and could provide accurate estimates.

Ductility

The yielding displacement Δ_y (i.e., the displacement according to the yielding point on the moment–deflection curve), the ultimate displacement Δ_u (i.e., the displacement when the bending moment dropped to 85% of theM_{\mathrm{p}}^{\mathrm{t}} or when the test terminated), and the ductility ratio µ (=Δ_u/Δ_y) are listed in Table 4.

From the table, we concluded the following:

(1) For the specimens in one group with the same number of prestressed tendons, an increase in the mild steel reinforcement increased both the prestressing ratio λ and the normalized reinforcement ratio ρ_s, and thus decreased the ductility ratio.

(2) Comparisons between the results of the specimens with the same λ but different ρ_s also showed that the ductility ratio varied inversely with ρ_s.

(3) The ductility ratios of the specimens were within the range of 2.83 to 7.49, indicating that all specimens with HSC and HSS had satisfactory deformation capabilities. Note that the limit of the normalized reinforcement ratio ρ_s proposed in the Chinese Code for the Design of Concrete Structures (GB 50010-2010) is 2.5%, which is close to the maximum ρ_s value of 2.21% in this study.

(4) Comparisons between the results of group 2 with C80 concrete and group 4 with C100 concrete indicated that the concrete strength slightly benefited the ductility of the beams. Although an increase in the compressive strength of the concrete could shorten the descending part of the moment–deflection curve, it could enlarge the sectional curvature according to the peak load of the beam by decreasing the depth of the compression zone.

Effects of concrete strength and ρ_s on ductility of beams with HPMs

To further study the influence of concrete strength and the normalized reinforcement ratio ρ_s on the ductility of the beams with HPMs, some experimental results obtained by other researchers were collected and are listed in Table 5 [¹⁴,¹⁵,³⁴]. Fig. 6 shows the variations in the ductility ratio µ with f_c obtained through regression analyses, where µ slightly increases with f_c. All data points were quite discrete instead of concentrated near the trend line because the test specimens had different normalized reinforcement ratio ρ_s values, which had a greater effect on the ductility than the concrete strength.

Figure 7 describes the variations in the ductility ratio µ with the normalized reinforcement ratio ρ_s, and where the regression curve is an exponential function. Because the other influential factor f_c only had a slight effect on µ, the discreteness of the data was relatively unclear. In addition, µ decreased rapidly with ρ_s when ρ_s was small, but with an increase in ρ_s, particularly when ρ_s was larger than 2.2%, the descending speed of µ gradually slowed. The upper limit of ρ_s proposed in the Code for Design of Concrete Structures (GB 50010-2010) is 2.5%, and the corresponding value of µ obtained from the regression curve was 2.8, which is close to 3, indicating that this limit is reliable.

Modifications to the crack width analysis method in GB 50010-2010

Maximum crack width analysis methods

Two theories about flexural cracking in RC structures are well acknowledged: the bond–slip theory and the no-slipping theory. Two typical methods used to calculate the maximum crack width of the PC and RC components, based on a physical model that comprehensively considers these two theories, are introduced below.

Method in GB 50010-2010

In Chinese code GB 50010-2010, the maximum crack width w_max at the bottom surface under a short-term load can be calculated as follows:

(3){\omega }_{\max }={\tau }_{\mathrm{s}}{\omega }_{\mathrm{m}}={\tau }_{\mathrm{s}}{\alpha }_{\mathrm{c}}\psi {\displaystyle \frac{{\sigma }_{\mathrm{s}}}{E_{\mathrm{s}}}}{l}_{\mathrm{c}\mathrm{r}},

where w_m is the average crack width under short-term loading, τ_s is the amplification coefficient, α_c is the coefficient that reflects the contribution of elongation of concrete between cracks, and ψ is the coefficient considering the nonuniformity of tensile reinforcement strain between cracks. For the RC components, σ_s is the tensile reinforcement stress at the crack section, and for the PC components, σ_s is the equivalent stress of the prestressed tendons and the mild steel reinforcement, assuming that these two types of reinforcement are located together at the action point of the resultant force. In addition, E_s is the elastic modulus of the reinforcement, and l_cr is the average crack spacing, which can be calculated through the following equation:

(4)l_{\mathrm{c}\mathrm{r}}=\mathrm{1.9}{c}_{\mathrm{s}}+\mathrm{0.08}{\displaystyle \frac{d_{\mathrm{e}\mathrm{q}}}{{\rho }_{\mathrm{t}\mathrm{e}}}},

where c_s is the distance from the edge of the outmost tensile reinforcement to the beam tensile edge, d_eq is the equivalent diameter of the reinforcement, and ρ_te is the reinforcement ratio calculated according to the effective tensile section area.

Method in EN 1992-1-1

In the European code EN 1992-1-1: 2004, the equation to estimate the crack width is shown in the following:

(5){\mathit{\omega }}_k=s_{\mathrm{r},\max }({\mathit{\epsilon }}_{\mathrm{s}\mathrm{m}}-{\mathit{\epsilon }}_{\mathrm{c}\mathrm{m}}),

where s_r,max is the maximum crack spacing, and ε_sm and ε_cm are the mean values of the reinforcement and concrete strain between cracks, respectively. Here, ε_sm–ε_cm can be calculated from Eq. (6):

(6){\epsilon }_{\mathrm{s}\mathrm{m}}-{\epsilon }_{\mathrm{c}\mathrm{m}}={\displaystyle \frac{{\sigma }_{\mathrm{s}}-k_{\mathrm{t}}{\displaystyle \frac{f_{\mathrm{t},\mathrm{e}\mathrm{f}\mathrm{f}}}{{\rho }_{\mathrm{p},\mathrm{e}\mathrm{f}\mathrm{f}}}}(\mathrm{1}+{\alpha }_{\mathrm{e}}{\rho }_{\mathrm{p},\mathrm{e}\mathrm{f}\mathrm{f}})}{E_{\mathrm{s}}}}\ge \mathrm{0.6}{\displaystyle \frac{{\sigma }_{\mathrm{s}}}{E_{\mathrm{s}}}},

where σ_s is the reinforcement stress, α_e is the ratio E_s/E_c (E_s and E_c are the elastic moduli of steel reinforcement and concrete, respectively), ρ_p,eff is the effective reinforcement ratio, and k_t is a factor considering the loading duration, which is 0.6 for short-term loading and 0.4 for long-term loading. The maximum crack spacing may be calculated using the following:

(7)s_{\mathrm{r},\max }=k_{\mathrm{3}}\cdot c+{\displaystyle \frac{k_{\mathrm{1}}{k}_{\mathrm{2}}{k}_{\mathrm{4}\varphi }}{{\mathit{\rho }}_{\mathrm{p},\mathrm{e}\mathrm{f}\mathrm{f}}}},

where k₁ and k₂ are coefficients that account for the bonding state of the reinforcement and the distribution of strain, respectively; and k₃ and k₄ are empirical coefficients, the recommended values of which are 3.4 and 0.425, respectively.

For the 12 test specimens in this study, the maximum crack widths were calculated using these two methods, and Fig. 8 shows a comparison between the tested and calculated maximum crack widths. The superscripts t and c denote the tested and calculated results, respectively. Note that for each specimen, the maximum crack width was measured at four stress levels. As a result, we recorded a total of 48 observations and 48 data points for each calculation method.

Figure 8 shows that most of the GB 50010 points are below the diagonal line, indicating that the method produces conservative results. For the EN 1992 method, the diagonal line is almost at the center of all data points, although the points are more discrete than with the other method.

The calculated crack widths were divided by the measured widths to further evaluate the accuracy of the various calculation methods, and the mean value, standard deviation (SD), and coefficient of variation (CV, i.e., the ratio of the mean value to the standard deviation) are presented in Table 6. The EN1992 method produced a mean value closer to 1 but had a larger CV. For the GB 50010-2010 method, the CV was smaller, but the mean value was only 0.883, showing that this method overestimates the crack widths, and the accuracy needs to be improved.

Modifications to the crack width Eq. in GB 50010-2010

The analysis in Section 5.1 shows that the computation equation of the crack width in GB 50010-2010 no longer fits the case of the PC beams with HSS and HSC. Some empirical coefficients were modified in the calculation of the crack spacing and the maximum crack width to improve the accuracy of the equations.

Crack spacing

The final crack spacing of the 12 tested beams was calculated using Eq. (4), and the tested versus calculated results are shown in Fig. 9. The tested values (l_cr^t) for the specimens were less than the calculated values (l_cr^c), and the maximum error was approximately 20%. In addition, to the experiments conducted in this study, the test results obtained by other researchers were collected, and the tested and calculated crack spacings are as listed and compared in Table 7. The results show that the tested crack spacing was mostly less than the calculated value. Therefore, some modifications to the crack spacing calculation in Eq. (4) were necessary.

Based on Eq. (4), the parameters m₁ and m₂ in the expression l_cr = m₁× c + m₂× d_eq/ρ_te can be obtained from the linear regression, where m₁ = 1.6 and m₂ = 0.08. Therefore, the crack spacing can be calculated using Eq. (8). Comparisons between the calculated results of Eqs. (4) and (8) show that the Eq. (8) can produce more reasonable results (Table 7).

l_{\mathrm{cr}}^m=\mathrm{1.6}{c}_s+\mathrm{0.08}{\displaystyle \frac{d_{\mathrm{eq}}}{{\rho }_{\mathrm{te}}}}.

Amplification coefficient between maximum and average crack widths

A linear relationship exists between ω_max and ω_m, expressed in Eq. (3). The amplification coefficient τ_s is usually considered an eigenvalue with 95% confidence and can be determined after derivation of the probability distribution. For the 12 specimens in this study, the widths of 703 cracks under several stress levels were recorded, and the distribution of ω_i/ω_m (where ω_i is the width of every crack, and ω_m is the mean value of the crack widths) is presented in Fig. 10. Here, ω_i/ω_m almost follows a normal distribution N (1.0, 0.267²), the mean value of which was 1.0, and the standard deviation was 0.267. Therefore, to ensure a 95% confidence, the modified amplification coefficient can be described as τ_s^m = 1+ 1.645 × SD = 1+ 1.645 × 0.267= 1.44. This value is less than the limit (i.e., 1.66) specified in GB50010-2010.

Using the modified crack spacing in Eq. (8) and the amplification coefficient, the maximum crack widths of the 12 specimens were calculated. The mean values, standard deviations, and CVs of ω_max^t/ω_max^c (the subscripts t and c denote the tested and calculated values, respectively) are listed in Table 8. The results show that the mean value of the modified Eq. was closer to 1.0, and the SV and CV were relatively smaller, indicating that the modified Eq. has a satisfactory precision.

Finite element analysis

Finite element model building

Fig. 11 shows the finite element model of the test specimens established with the general software Automatic Dynamic Incremental Nonlinear Analysis (ADINA). The solid and rebar elements were used to build the concrete and prestressed tendons, respectively. Steel stirrups and steel longitudinal reinforcements were modeled by the truss element and embedded in concrete. The prestress in prestressed tendons was exerted by the initial strain, and the values of the initial strain in each model are as shown in Table 9. The boundary condition of the finite element model was the same as that of the test. The displacements of the X-, Y-, and Z-directions on one side and the Y- and Z-directions on the other side were limited.

Material parameters

The critical parameters of the concrete in ADINA are shown in Table 10. Bilinear and multilinear materials were used to model the common reinforcements, including steel stirrups, steel longitudinal reinforcements, and prestressed tendons, as shown in Fig. 12. Each parameter of common reinforcements and prestressed tendons can be calculated by Eqs. (9) and (10), respectively:

(9)\{\begin{array}{l}\sigma =\epsilon E_{\mathrm{s}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\epsilon \le \hspace{0.17em}{\epsilon }_{\mathrm{y}}\hfill \\ \sigma =f_{\mathrm{y}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\epsilon >\hspace{0.17em}{\epsilon }_{\mathrm{y}}\hspace{0.17em}\hfill \end{array}

(10)

where the ε is the elastic strain; E_s is the reinforcements elastic modulus; and f_y is the yield strength of the reinforcements. The coefficients (k₁, k₂, k₃, and k₄) in Eq. (10) are k₁ = 1.0, k₂ = 2.2, k₃ = 5.33 and k₄ = 1.082, respectively.

Finite element analysis results

Mid-span bending moment-deflection curves

Figure 13 compares the tested and predicted mid-span bending moment–deflection curves. In general, the tested curves correlated well with the predicted results. It should be noted that in the finite element model, the slip between the concrete and common reinforcements was not considered, which indicates that a significant bond behavior can be generated between the HSS and HSC. However, some defects existed in PCB2 and PCB3. The predicted results were higher than those obtained after the steel yielding stage.

Steel longitudinal reinforcement strain and the crack distribution

Figs. 14 and 15 show the steel longitudinal reinforcement strain of all specimens and the crack distributions of PCB-9 and PCB-10, respectively. The changes in the steel longitudinal reinforcement strain and the crack distribution obtained from the finite element analysis are similar to the test results. The aforementioned analyses indicate that the finite element model established in ADINA can reflect the real capacity of the test specimens.

Conclusions

From the flexural behavior testing of the 12 HSS and HSC prestressed beams considered in this study, and the results of other existing experimental studies that were collected, the following conclusions can be drawn.

(1) The failure modes of HSS and HSC prestressed beams are similar to those of ordinary reinforced concrete, exhibiting the classical four stages of failure. A brittle failure did not occur in the specimens with more brittle concrete.

(2) The formulas for the cracking and peak moments of all specimens provided in the Chinese Code for Design of Concrete Structures (GB 50010-2010) are suitable for HSS and HSC prestressed beams. The analysis shows that the normalized reinforcement ratio has a greater effect on the ductility of HSS and HSC members than the concrete strength.

(3) The maximum crack width calculated by Eq. in GB 50010-2010 may overestimate the crack width, whereas the European code EN1992-1-1:2004 is more accurate but has a larger standard deviation. Two factors in the Eqs. were modified, which are the average crack spacing and the amplification coefficient of the maximum crack width to the average, as proposed in the context of GB 50010-2010.

(4) The simulation results for the prestressed concrete beam test are consistent with the test results and phenomena, implying that the established finite element model in ADINA can reflect the real capacity of the test specimens.

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