Predicting Residual Flexural Strength of Corroded Prestressed Concrete Beams: Comparison of Chinese Code, Eurocode and ACI Standard

Abstract

In the present study, 104 sets of flexural tests on corroded prestressed concrete (CPC) beams were gathered from different publications. A flexural strength database for CPC beams was created by incorporating standardized concrete strength and the corrosion rate of prestressed steel. This database enables the analysis of the impact of different factors on the flexural capacity, such as the beam’s width and effective depth, concrete compressive strength, shear span ratio, the prestressed steel’s corrosion level, prestressing ratio (PPR), and the effective prestress. The findings show that the flexural strength of the CPC beams is notably influenced by variations in beam width, shear span ratio, and the prestressed steel’s corrosion level and prestressing ratio compared to other parameters. Furthermore, a comparison was conducted of the Chinese code, Eurocode and ACI standard for evaluating the flexural strength of CPC beams using the established database. It shows that the three design codes overestimate the flexural strength of CPC beams, and it is unsafe to predict flexural strength using these three codes without taking into account the impact of corrosion. Finally, a practical model based on the ACI standard is suggested to provide precise and reliable predictions across a diverse set of test data.

1. Introduction

Compared to normal reinforced concrete (RC) structures, PC structures are characterized by strong crack resistance, good impermeability, high stiffness, high strength, good shear capacity and fatigue resistance [1,2,3]. PC structures are very effective in saving steel, reducing the size of structural cross-section, lowering structural deadweight, preventing cracking and reducing deflection. Therefore, PC structures are extensively utilized in diverse civil engineering including construction, hydraulic engineering and transportation engineering [4]. Prestressed strands in PC are more difficult to corrode than reinforcing bars in RC due to the thicker protective layer. However, the strands in concrete were found to be experience corrosion during the service, which could be attributed to improper protection during transportation and storage of steel strands, insufficient grouting, poor construction quality of PC structures and the external erosive environment [5]. The reduction in cross-sectional area caused by corrosion is more sensitive to prestressed steel. Prestressed steel usually maintains a pre-tension stress throughout the service stage, which accounts for 55–60% of its ultimate tensile strength. A reduction in the section of the prestressed steel will correspondingly increase the pre-tension stress on the section, leading to the fracture and yield of the prestressed steel. Conversely, the tensile strength of prestressed steel is typically 4–5 times that of non-prestressed steel. If the corrosion rate is the same, the load of prestressed steel is 4–5 times that of ordinary steel bars [6]. Therefore, the corrosion of prestressed steel strands at high stress levels is likely to cause their brittle fracture without any warning, resulting in the sudden destruction of the PC structure. For instance, the Ynys-y-Gwas bridge in the UK collapsed suddenly in 1985 as a result of the corrosion of prestressed steel cables [7]. In 2000, a prestressed concrete footbridge in North Carolina of the United States collapsed unexpectedly as a result of the prestressed tendons’ corrosion [8]. In 2019, one of the prestressed steel cables of the Shenzhen Bay Bridge, constructed 12 years prior, exhibited signs of corrosion and failure (Figure 1). Due to this, the bridge had to be closed as an emergency for 28 days, during which the damaged cable was replaced [9].

Deterioration of corroded prestressed steel is intimately connected to the flexural bearing behavior of PC structures. The mechanical performance of corroded steel strands has been studied in detail, and their constitutive relationships are proposed on this basis [10,11,12,13,14,15,16,17,18]. Simultaneously, some researchers have experimentally investigated the flexural behavior of CPC beams, both rectangular sections and T-shaped sections. In the case of rectangular sections, the majority of research initially focused on how the corrosion rates of prestressed strands on the flexural strength of CPC beams [19,20,21]. These studies examined the effects of prestressed strands’ corrosion on the cracking load of concrete, yield and ultimate load of prestressed steel, as well as the failure pattern of CPC beams [19,22]. They also examined the effects of different corrosion rates and positions on the flexural behavior of CPC beams [21]. Secondly, some scholars have investigated the influence of corrosion on the flexural strength of CPC beams caused by prestressed strands in insufficiently grouted ducts [23,24]. Thirdly, bending tests were carried out on post-tension prestressed concrete beams with different corrosion degrees to explore the impact of the deformability between prestressing tendons and concrete on the flexural capacity [25]. Compared with rectangular sections, the research on the flexural strength of CPC T-shaped beams is relatively less. Most of these tests were also used to study the impact of the prestressed steel strand’s corrosion on the flexural strength of the T-shaped beam [26,27,28].

Meanwhile, there were two types of models proposed to evaluate the flexural strength of CPC beams, including the strain compatibility approach and computational intelligence technique. One type of model is proposed based on the strain compatibility approach after the prestressed steel is corroded. In the process of the stress–strain analysis, the section loss and mechanical property deterioration of prestressed steel are usually considered [22,28,29,30,31,32]. In addition, some scholars examined the coordination performance of corroded prestressed strands and concrete in predicting the flexural strength [29] and bond strength degradation [30,31]. The other type of model used a computational intelligence technique to calculate the flexural strength of CPC beams by gathering a relatively large experimental dataset [33].

Based on the above, the prediction models mentioned above often agree well with some experimental results. However, these models are more complicated to use in practical engineering, and whether they are universal is unknown. Moreover, current codes rarely consider the effects of prestressed steel’s corrosion on the flexural strength. The accuracy and applicability of the current main codes for the evaluation of the flexural strength of CPC beams is also unclear. There is thus a clear need to provide a new model that is precise and straightforward to utilize in engineering applications. Therefore, this paper presents a comprehensive database containing 104 flexural experimental tests of CPC beams gathered from the published sources. This database was employed to investigate the effects of various primary parameters on the flexural strength of CPC beams. These parameters included the beam’s width, the effective depth of the beam, the concrete compressive strength, the shear span ratio, the corrosion rate of the prestressed steel, the prestressing ratio (PPR) and the effective prestress. Then, Chinese code, Eurocode and ACI standard were evaluated to investigate the applicability and precision for calculating the flexural strength of CPC beams using the database. In addition, a novel model for predicting the flexural strength of the CPC beams is proposed based on the ACI standard incorporating the prestressed steel’s corrosion. The outcomes of the work presented in this paper can provide guidance for correctly using the standard formula to evaluate the flexural strength of CPC beams.

2. Experimental Database of the Flexural Capacity of CPC Beams

2.1. Set up of the Experimental Database

This section collected experimental data from 104 sets of CPC beams under concentrated loads from published sources. Of these, 86 sets used rectangular sections, while 18 sets used T-shaped sections. These experimental data are integrated into a database and shown in

Table 1. Experimental database of flexural strength of CPC beams collected from existing studies.

Reference	Specimen	b (mm)	h (mm)	Concrete Strength (MPa)	Ast (mm2)	fyt (MPa)	Asc (mm2)	fyc (MPa)	Ap (mm2)	fpt (MPa)	Ep (N/mm2)	η	λ	Mexp (kN·m)
Rinaldi et al. [20] (9 sets)	NO.7	200	300	34	157	400	157	400	226.19	1976	195,000	0	3.57	73.20
	NO.8	200	300	34	157	400	157	400	226.19	1976	195,000	0.2	3.57	43.95
	NO.9	200	300	34	157	400	157	400	226.19	1976	195,000	0.2	3.57	52.95
	NO.2	200	300	41.5	157	400	157	400	226.19	1976	195,000	0	3.57	86.70
	NO.3	200	300	41.5	157	400	157	400	226.19	1976	195,000	0.14	3.57	39.45
	NO.1	200	300	41.5	157	400	157	400	226.19	1976	195,000	0.2	3.57	30.90
	NO.4	200	300	47.4	157	400	157	400	226.19	1976	195,000	0	3.57	95.70
	NO.6	200	300	47.4	157	400	157	400	226.19	1976	195,000	0.07	3.57	91.20
	NO5	200	300	47.4 (fcu,150)	157	400	157	400	226.19	1976	195,000	0.2 (ηwt)	3.57	32.70
Li and Yuan [19] (9 sets)	PRB1	150	200	33.0	57	284	226	375	98.7	1913	195,000	0.0173	3.96	27.7
	PRB2	150	200	33.0	57	284	226	375	98.7	1913	195,000	0.0219	3.96	27.5
	PRB3	150	200	33.0	57	284	226	375	98.7	1913	195,000	0.0224	3.96	25.1
	PRB4	150	200	33.0	57	284	226	375	98.7	1913	195,000	0.0287	3.96	25.0
	POB0	150	200	35.2	57	284	226	375	98.7	1913	195,000	0	3.96	28.9
	POB1	150	200	35.2	57	284	226	375	98.7	1913	195,000	0.0094	3.96	25.8
	POB2	150	200	35.2	57	284	226	375	98.7	1913	195,000	0.0151	3.96	25.3
	POB3	150	200	35.2	57	284	226	375	98.7	1913	195,000	0.0198	3.96	26.3
	POB4	150	200	35.2 (fcu,150)	57	284	226	375	98.7	1913	195,000	0.0112 (ηwt)	3.96	25.8
Zhang et al. [24] (8 sets)	PCB1	150	220	31.82	101	235	226	335	140	1910	195,000	0.7790	3.70	11.4
	PCB2	150	220	31.82	101	235	226	335	140	1910	195,000	1	3.70	9.9
	PCB3	150	220	31.82	101	235	226	335	140	1910	195,000	0.0635	3.70	31.5
	PCB4	150	220	31.82	101	235	226	335	140	1910	195,000	0.0982	3.70	30.6
	PCB5	150	220	32.35	101	235	226	335	140	1910	195,000	0.5510	3.70	15.9
	PCB6	150	220	32.35	101	235	226	335	140	1910	195,000	0.4804	3.70	19.5
	PCB7	150	220	34.28	101	235	226	335	140	1910	195,000	0.3589	3.70	22.8
	PCB8	150	220	34.28 (fcu,150)	101	235	226	335	140	1910	195,000	0.0128 (ηsn)	3.70	34.5
Li et al. [34] (7 sets)	SCC0-2	150	250	34.37	308	360	100.6	254	201.1	622.83	205,000	0	2.23	46.53
	SCC2-1	150	250	34.65	308	360	100.6	254	201.1	622.83	205,000	0.0067	2.23	44.28
	SCC2-2	150	250	32.42	308	360	100.6	254	201.1	622.83	205,000	0.0108	2.23	42.48
	SCC2-3	150	250	33.73	308	360	100.6	254	201.1	622.83	205,000	0.0120	2.23	42.03
	SCC2-4	150	250	30.47	308	360	100.6	254	201.1	622.83	205,000	0.0138	2.23	41.8
	SCC1-3	150	250	33.65	308	360	100.6	254	201.1	622.83	205,000	0.0122	2.23	42.03
	SCC4-1	150	250	35.07 (fcu,150)	308	360	100.6	254	201.1	622.83	205,000	0.0113 (ηsn)	2.23	51.03
Zhang et al. [29] (8 sets)	CB0	150	220	34.1	101	235	226	335	140	1910	195,000	0	3.70	37.8
	CB1	150	220	33.7	101	235	226	335	140	1910	195,000	0.7370	3.70	13.5
	CB2	150	220	33.7	101	235	226	335	140	1910	195,000	0.4600	3.70	21
	CB3	150	220	33.7	101	235	226	335	140	1910	195,000	0.6170	3.70	15.9
	CB4	150	220	33.7	101	235	226	335	140	1910	195,000	0.8470	3.70	11.7
	CB5	150	220	32.4	101	235	226	335	140	1910	195,000	0.1210	3.70	33
	CB6	150	220	32.4	101	235	226	335	140	1910	195,000	0.1950	3.70	31.5
	CB7	150	220	34.3 (fcu,150)	101	235	226	335	140	1910	195,000	0.2700 (ηsn)	3.70	28.8
Jeon et al. [14] (4 sets)	RB	150	220	37.46	157	400	265.46	400	140	1865	194,500	0	3.66	49.17
	QB	150	220	37.46	157	400	265.46	400	140	1865	194,500	0.0482	3.66	48.3
	MB1	150	220	37.46	157	400	265.46	400	140	1865	194,500	0.0667	3.66	43.92
	MB2	150	220	37.46 (fcu,150)	157	400	265.46	400	140	1865	194,500	0.0751 (ηsn)	3.66	40.68
Zeng et al. [35] (9 sets)	L1	150	300	44	402	365	226	360	140	1950	203,000	0	3.33	91.12
	L2	150	300	44	402	365	226	360	140	1950	203,000	0.087	3.33	86.68
	L3	150	300	44	402	365	226	360	140	1950	203,000	0.136	3.33	78.04
	L4	150	300	44	402	365	226	360	140	1950	203,000	1	3.33	47
	L5	150	300	44	402	365	226	360	140	1950	203,000	0.203	3.33	55.28
	L6	150	300	44	760	370	226	360	140	1950	203,000	0.058	3.29	110.68
	L7	150	300	44	402	365	226	360	140	1950	203,000	0	3.33	84.6
	L8	150	300	44	402	365	226	360	140	1950	203,000	0.26	3.33	48.32
	L9	150	300	44 (fcu,150)	402	365	226	360	140	1950	203,000	0.017 (ηwt)	3.33	75.84
Ma et al. [31] (5 sets)	B0	150	220	31.8	101	290	226	375	140	1910	193,000	0	3.69	37.8
	B1	150	220	31.8	101	290	226	375	140	1910	193,000	0.0270	3.69	33
	B2	150	220	31.8	101	290	226	375	140	1910	193,000	0.0620	3.69	31.5
	B3	150	220	31.8	101	290	226	375	140	1910	193,000	0.0910	3.69	28.8
	B4	150	220	31.8 (fcu,150)	101	290	226	375	140	1910	193,000	0.1060 (ηwt)	3.69	21
Youn and Kim [36] (3 sets)	PC1	150	300	31	265.46	400	265.46	400	98.7	1834	195,000	0	/	45.6
	PC2	150	300	31	265.46	400	265.46	400	98.7	1834	195,000	0.0881	/	44.6
	PC3	150	300	31 (fck)	265.46	400	265.46	400	98.7	1834	195,000	0.1963 (ηsn)	/	39.8
Yu et al. [22] (6 sets)	L1	150	300	44.2	402.1	365	226.2	359	140	1959	203,000	0	3.33	91.1
	L2	150	300	44.2	402.1	365	226.2	359	140	1959	203,000	0.087	3.33	86.7
	L3	150	300	46.5	402.1	365	226.2	359	140	1959	203,000	0.136	3.33	78.0
	L4	150	300	44.2	402.1	365	226.2	359	140	1959	203,000	1	3.33	47.0
	L5	150	300	45.4	760.3	368	226.2	359	140	1959	203,000	0.058	3.29	110.7
	L6	150	300	42.5 (fcu,150)	402.1	365	226.2	359	140	1959	203,000	0.017 (ηwt)	3.33	75.8
Jeon et al. [21] (5 sets)	CB1	280	380	37.46	157	400	265.46	400	138.7	1883	195,000	0.0949	6.02	181.75
	CB2	280	380	37.46	157	400	265.46	400	138.7	1883	195,000	0.0884	6.02	222.02
	CB3	280	380	37.46	157	400	265.46	400	138.7	1883	195,000	0.0689	6.02	230.45
	CB4	280	380	37.46	157	400	265.46	400	138.7	1883	195,000	0.068	6.02	226.92
	RB5	280	380	37.46 (fcu,150)	157	400	265.46	400	138.7	1883	195,000	0 (ηsn)	6.02	226.31
Liu et al. [37] (5 sets)	B2	150	250	45.1	308	335	308	335	197.4	1860	195,000	0.071	3.34	59.79
	B3	150	250	43.6	308	335	308	335	197.4	1860	195,000	0.043	3.34	67.17
	B5	150	250	45.5	308	335	308	335	197.4	1860	195,000	0.102	3.34	57.63
	B7	150	250	46.7	308	335	308	335	197.4	1860	195,000	0.022	3.34	68.52
	B9	150	250	45.6 (fcu,150)	308	335	308	335	197.4	1860	195,000	0 (ηwt)	3.34	69.66
Xu et al. [38] (8 sets)	PCB-1	150	250	51.2	226	400	226	400	140	1661	202,000	0	3.04	57.8
	PCB-2	150	250	51.2	226	400	226	400	140	1661	202,000	0.0322	3.04	56
	PCB-3	150	250	51.2	226	400	226	400	140	1661	202,000	0.0611	3.04	53.5
	PCB-4	150	250	51.2	226	400	226	400	140	1661	202,000	0.0895	3.04	50.9
	PCB-5	150	250	51.2	308	400	226	400	140	1661	202,000	0.0623	3.02	64.5
	PCB-6	150	250	51.2	402	400	226	400	140	1661	202,000	0.0635	3.00	67.5
	PCB-7	150	250	59.9	226	400	226	400	140	1661	202,000	0.0582	3.04	55.5
	PCB-8	150	250	44.5 (fcu,150)	226	400	226	400	140	1661	202,000	0.0753 (ηwt)	3.04	53.4
Qiu [26] (8 sets)	A0	89(265)	240(75)	55.7	100.6	350	100.6	350	140	1860	195,000	0	4.09	61.63
	A1	89(265)	240(75)	55.7	100.6	350	100.6	350	140	1860	195,000	0.0033	4.09	61.71
	A2	89(265)	240(75)	55.7	100.6	350	100.6	350	140	1860	195,000	0.0124	4.09	59.93
	A3	89(265)	240(75)	55.7	100.6	350	100.6	350	140	1860	195,000	0.0237	4.09	56.7
	A4	89(265)	240(75)	55.7	100.6	350	100.6	350	140	1860	195,000	0.0452	4.09	53.72
	A5	89(265)	240(75)	55.7	100.6	350	100.6	350	140	1860	195,000	0.0615	4.09	49.39
	A6	89(265)	240(75)	55.7	100.6	350	100.6	350	140	1860	195,000	0.0747	4.09	47.86
	A7	89(265)	240(75)	55.7 (fcu,150)	100.6	350	100.6	350	140	1860	195,000	0.0831 (ηwt)	4.09	44.54
Yang [27] (5 sets)	B01	80(320)	250(80)	45.2	100.6	350	402.4	350	140	1860	195,000	0	3.07	59.2
	CB2	80(320)	250(80)	45.2	100.6	350	402.4	350	140	1860	195,000	0.0085	3.07	57.2
	CB3	80(320)	250(80)	45.2	100.6	350	402.4	350	140	1860	195,000	0.0358	3.07	54.9
	CB4	80(320)	250(80)	45.2	100.6	350	402.4	350	140	1860	195,000	0.0648	3.07	51.4
	CB5	80(320)	250(80)	45.2 (fcu,150)	100.6	350	402.4	350	140	1860	195,000	0.0842 (ηwt)	3.07	47.5
Zhou et al. [28] (5 sets)	A0	100(320)	250(80)	51.75	100.6	415.3	402.4	415.3	140	1980.8	195,000	0	3.06	220.33
	A1	100(320)	250(80)	51.75	100.6	415.3	402.4	415.3	140	1980.8	195,000	0.0313	3.06	207.51
	A2	100(320)	250(80)	51.75	100.6	415.3	402.4	415.3	140	1980.8	195,000	0.0798	3.06	186.08
	A3	100(320)	250(80)	51.75	100.6	415.3	402.4	415.3	140	1980.8	195,000	0.1184	3.06	160.56
	A4	100(320)	250(80)	51.75 (fcu,150)	100.6	415.3	402.4	415.3	140	1980.8	195,000	0.1353 (ηsn)	3.06	152.74

. The concrete compressive strength in the database is represented by the prismatic compressive strength (f_ck) and cube crushing strength (f_cu,150) obtained from 150 mm cube tests, respectively. The prestressed steel’s corrosion rate is represented by the weight loss ratio (η_wt) and section loss ratio (η_sn) in the database, respectively. The definitions of η_wt and η_sn are given in Section 2.3. A summary of the key characteristics of the selected experimental CPC beams is presented

Table 2. Important characteristics of the selected experimental CPC beams.

	Compressive Strength of the Concrete	Yield Strength of the Longitudinal Tensile Reinforcement	Yield Strength of the Longitudinal Compression Reinforcement	Ultimate Tensile Strength of the Prestressed Reinforcement	Section Loss Ratio of the Prestressed Reinforcement	Shear Pan Ratio	Prestressed Reinforcement Type
	fcu,150 (MPa)	fyt (MPa)	fyc (MPa)	fpt (MPa)	ηsn	λ
Range	30.47~59.9	235~415.3	254~415.3	622.83~1980.8	0~100%	2.32~6.02	12.7 mm and 15.2 mm steel strands, 16 mm prestressed steel bar

. It is noted that the typical failure pattern of CPC beams is the wire rupture of corroded prestressed strands or concrete crushing, as shown in Figure 2 [27].

2.2. Unification of Compressive Strength of Concrete

Different test standards are employed to assess the concrete compressive strength, as listed in Table 1, including the conversion between f_cu,150 and f_ck, as suggested by GB50010-2010 [39]. In the following contrast analysis of Chinese Code, Eurocode and ACI Standard, various compressive strengths of concrete are also used in three national standards. For the purpose of a comparative study, different compressive strengths of concrete must be converted to each other. The conversion between f_cyl,150 (the concrete compressive strength obtained from cylinder tests of 150 × 300 mm specimens) and f_cu,150 are provided in Eurocode 2 [40].

2.3. Unification of Corrosion Rate of Prestressed Steel

To define the prestressed steel’s corrosion rate, different researchers used different indices, e.g., the weight loss ratio (η_wt) and the section loss ratio (η_sn). The η_wt and η_sn can be calculated according to Equations (1) and (2) [39].

$${\eta }_{wt}=\left(m_0-m_r\right)/m_0\times 100\%$$

(1)

$$\begin{array}{l}{\eta }_{sn}=\left(A_0-A_{\min }\right)/A_0\times 100\%=\left(d_0^2-d_{\min }^2\right)/d_0^2\times 100\%\\ =\left[d_0^2-{\left(d_0-X\right)}^2\right]/d_0^2\times 100\%\end{array}$$

(2)

where m₀ and m_r is the initial and residual mass of the prestressed steel before and after corrosion, respectively; d₀ and A₀ are the diameter and sectional area of the prestressed steel, respectively; d_min and A_min are the minimum residual diameter and sectional area of the prestressed steel after corrosion, respectively; and X is the maximum corrosion depth of the prestressed steel.

Meanwhile, the conversion relationship between η_sn and η_wt proposed by Xue [41] is adopted in this study. At the same time, this study employs the conversion correlation between η_sn and η_wt, as suggested by Xue [41]:

$${\eta }_{sn}=0.936{\eta }_{wt}+0.0045$$

(3)

3. Effects of Parameters on the Flexural Strength of CPC Beams

This study firstly used the experimental database to examine the impact of the main parameters on the flexural strength. The normalized ultimate flexural stress, M_n,exp, M_n,exp = M_n,exp/(bh₀f_cu,150), is defined to analysis the influence of each main parameter. The beam’s effective depth (h₀) [39] can be calculated as follows:

$$h_0=h-a_s$$

(4)

where a_s is the distance from the resultant force of the prestressed steel and the ordinary tensile steel bar to the edge of the beam section in the tension zone.

To clearly distinguish the effects of different parameters on the flexural strength, the rectangular and T-shaped section beams are discussed separately in the process of analyzing the influence of these parameters.

3.1. Beam’s Width and Effective Depth

M_n,exp was analyzed in relation to the beam’s width b and effective depth h₀, as shown in Figure 3 and Figure 4. The trend lines are plotted for different corrosion rates after a regression analysis. The test results were classified into four groups according to the section corrosion loss ratio of prestressed steel (η_sn), namely η_sn < 5%, 5% ≤ η_sn < 10%, 10% ≤ η_sn < 20%, and η_sn ≥ 20%. The findings presented in Figure 3 and Figure 4 show that M_n,exp tends to increase gradually with the increase of b and h₀. The trend of normalized ultimate flexural stress with the increase in beam width is more obvious than that with the increase in the effective height of the CPC beams. Compared with rectangular beams, the beam width and effective depth have a greater influence on the flexural strength of T-shaped CPC beams. It is noted that because the test data of the T-shaped CPC beam of η_sn ≥ 20% were not collected, there are the test data of η_sn < 20% in Figure 3b and Figure 4b. Moreover, it is evident that for the same beam width or effective depth, whether rectangular or T-shaped, the CPC beams with more severe corrosion of prestressed steel have a lower flexural strength.

3.2. Compressive Strength of Concrete and Shear Span Ratio

Figure 5 illustrates the relationships between M_n,exp and concrete compressive strength, as indicated by f_cu,150. Based on the trend line in Figure 5a, it is evident that as the concrete compressive strength increases, M_n,exp rises first and then declines, with the index R² of 0.066 (for the test data of η_sn < 20%) and 0.422 (for the test data of η_sn ≥ 20%) in the rectangular section. The changing trend is similar for the CPC T-beams, as shown in Figure 5b. When f_cu,150 is larger than 40 MPa in rectangular sections and 50 MPa in T-shaped section, the normalized ultimate flexural stress shows a noticeable decrease. This downtrend is even more pronounced for beams with a corrosion section ratio exceeding 10%.

Figure 6 illustrates the variations in the normalized ultimate flexural stress with the shear span ratio (λ), which is the ratio of the minimum distance a (a is the shear span) from the concentrated load point on the beam to the edge of the support to the effective depth h₀ of the section for simply supported beams. As depicted in Figure 6a, the flexural strength of CPC rectangular beams experiences a significant increase as λ rises within the range of 3.5. Once λ > 3.5, there is a significant increase in the flexural strength as λ increases. This trend is similar to the relationship between the shear strength and λ of uncorroded concrete beams. When λ is to 2.5, the shear strength of the concrete beam is in the valley of shear failure [42,43]. However, the flexural strength of CPC T-shaped beams decreases as the λ increases, as shown in Figure 6b. This difference may be due to the fact that the test data of CPC T-beams are relatively small, and only two shear span ratios are used for the regression analysis.

3.3. Corrosion Rate of Prestressed Steel

Figure 7 shows the variation in M_u,exp of the CPC beams with the corrosion level of the prestressed steel, represented by η_sn. As depicted in Figure 7a, the M_u,exp of the CPC rectangular section (Square dots in the Figure 7a) decreases with the increase in the corrosion rate of the prestressed steel. A clear downward trend is observed when the section loss ratio is below 60%. However, this downward trend becomes less pronounced when the section loss ratio exceeds 60%. Figure 7b indicates that the M_u,exp of the CPC T-shaped section decreases always as η_sn rises within the range of 15%, which is consistent with the CPC rectangular section in Figure 7a. For the test data obtained from the same experiment (Triangle dots in the Figure 7b), this trend is more obvious. As shown in Figure 7b, the corresponding coefficient of the determination R² of Zhou et al. [28] is as high as 0.985.

3.4. Prestressing Ratio (PPR) and Effective Prestress

The prestressing ratio (PPR) and effective prestress are important factors affecting the flexural behavior of PC beams. PPR is the ratio of the material strength of prestressed steel to that of non-prestressed steel [35]:

$$\mathrm{PPR}=\frac{A_P{f}_{pyt}}{A_P{f}_{pyt}+A_{st}{f}_{yt}}$$

(5)

where f_pyt and f_yt are the yield strength of the tensile prestressed and non-prestressed reinforcement, respectively. A_p and A_st are the cross-sectional areas of the tensile prestressed and non-prestressed reinforcement, respectively.

Figure 8 illustrates the correlation between the M_u,exp and the PPR. Because the T-shaped beams in the data sources did not give PPR, the data of the CPC beams in Figure 8 are all rectangular sections. The normalized ultimate flexural stress decreases as the PPR increases, with an index R² of 0.346. Additionally, it is evident that the CPC beams with a higher section loss ratio exhibit lower flexural strength at the same PPR.

Figure 9a,b illustrates the correlation between M_u,exp and the effective prestress of CPC rectangular and T-beams, respectively. Compared with other influencing factors, the discrete relationship between the effective prestress and flexural strength is greater. Based on the regression trend in Figure 9, M_u,exp initially increases and then decreases as the effective prestress, including the rectangular section and T-shaped section. When effective prestress is about 900 MPa and 1100 MPa, M_u,exp reaches the maximum value in the rectangular section and in the T-shaped section, respectively. When the same effective prestress is applied, the corrosion of the prestressed steel will also reduce the flexural strength of the CPC beam.

Based on the above investigation, the impact of the main parameters on the flexural strength of CPC rectangular and T-beams is almost the same. The main parameters include the beam width (b), the beam effective depth (h₀), the concrete compressive strength (f_cu,150), the shear span ratio (λ), the corrosion rate of the prestressed steel (η_sn), the prestressing ratio (PPR) and the effective prestress (f_pe). Compared with other influencing parameters, the flexural strength of the CPC beams is notably influenced by variations in the beam width, shear span ratio, section corrosion loss rate of the prestressed steel and prestressing ratio. In particular, the corrosion of prestressed steel will reduce its mechanical performance and affect the flexural strength of PC beams. Moreover, other factors such as concrete strength, effective height, prestressed steel and so on will be affected by the corrosion of prestressed steel, thereby affecting the flexural strength of the CPC beam. Obviously, the flexural strength of the CPC beam is also affected by the corrosion of prestressed steel in the analysis of other influencing factors. Therefore, it is crucial to take into account the critical corrosion impact on the prestressed steel in the model when attempting to predict the flexural strength of CPC beams.

4. Comparison between the Test Results and Predictions of Chinese Code, Eurocode and ACI Standard

In China, the design code for reinforced concrete structures is represented by the GB50010-2010 [39]. In turn, EN 1992-1-1:2004+A1 [40] and ACI 318-19 [44] are the more internationally accepted standards. Therefore, the following mainly discusses the predictions of the flexural strength by the Chinese Code, Eurocode and ACI Standard.

4.1. GB50010-2010 (Chinese Code: Design Code for Concrete Structure, 2010)

The flexural strength calculation for a rectangular section is identical to that of the first type of T-shaped sections. All the T-shaped test beams in the database belong to the first type of T-shaped sections. The standard formula provided below mainly focuses on the prediction of the flexural strength for rectangular sections. Figure 10 shows the section stress distribution diagram in normal section analysis of GB50010-2010 [39]. The calculation of the flexural strength for rectangular sections and the first type of T-shaped sections in GB50010-2010 is expressed as follows [39]:

$${\alpha }_1{f}_{ck}bx=f_{yt}{A}_{st}-f_{yc}{A}_{sc}+f_{py}{A}_P+\left({\sigma }_{P0}^{\prime }-f_{py}^{\prime }{A}_P^{\prime }\right)$$

(6)

$$M_{cal}={\alpha }_1{f}_{ck}bx\left(h_0-\frac{x}{2}\right)+f_{yc}{A}_{sc}\left(h_0-a_s^{\prime }\right)-\left({\sigma }_{P0}^{\prime }-f_{py}^{\prime }\right)A_P^{\prime }\left(h_0-a_p^{\prime }\right)$$

(7)

where α₁ is the ratio of the equivalent rectangular compressive stress to the maximum stress of the concrete compressive (f_ck); f_ck is the axial compressive strength of concrete; f_yt and f_yc are the design values of the tensile and compressive strengths of the non-prestressed reinforcement, respectively; x is the concrete depth of the compression zone; A_st and A_sc are the sectional areas of the longitudinal tensile and compressive non-prestressed reinforcement, respectively; A_p and $A_P^{\prime }$ are the sectional areas of the longitudinal tensile and compressive prestressed steel, respectively; ${\sigma }_{P0}^{\prime }$ is the stress of the prestressed steel when the normal stress of the concrete at the resultant point of the longitudinal prestressed steel in the compression zone is equal to zero; b is beam width in the rectangular section or flange width in the first type of the T-shaped section; M_u is the calculated value of the flexural strength of the beam; h₀ is the effective depth of the cross-section; and $a_s^{\prime }$ and $a_p^{\prime }$ are the distances from the resultant point of the longitudinal non-prestressed and prestressed steel in the compression zone to the compression edge of the cross-section, respectively. It should be pointed out that the compression zone of PC beams in the database is not equipped with prestressed steel, so the last item of Equations (6) to (7) is zero when calculating its flexural strength.

The concrete depth of the compression zone should meet the following conditions [39].

$$x\le {\xi }_b{h}_0$$

(8)

$$x\ge 2a^{\prime }$$

(9)

where ξ_b is the height of the relative limit compression zone of concrete; that is, the limit of compression failure and tensile failure of reinforced concrete components, and a′ is the distance from the resultant point of all longitudinal compression reinforcements to the compression edge of the cross-section. When the longitudinal prestressed steel is not configured in the compression area or the prestressed steel’s stress in the compression zone is the tensile stress, a′ in Equation (9) is replaced by a′_s.

Equation (8) ensures that the component does not undergo over-reinforcement failure. When the condition is not satisfied, take x = ξ_bh₀. When Equation (9) is satisfied, the compressive reinforcement of the component will yield, and the stress of the compression reinforcement takes its yield strength in the calculation. If Equation (9) is not satisfied, then take x = 2a′.

4.2. EN 1992-1-1:2004+A1 (Eurocode 2: Design of Concrete Structure-Part 1-1: General Rules and Rules for Buildings, 2014)

Figure 11 shows section stress distribution diagram in the normal section analysis of EN 1992-1-1:2004+A1 [40]. The calculations of the flexural strength of the rectangular sections and the first type of the T-shaped section are the same as that of the Chinese code. Eurocode 2 uses Equations (10) and (11) [40] to calculate the flexural strength:

$${\eta }_1{f}_{ck}\left({\lambda }_{1}x\right)b=A_{st}{f}_{yt}\left(d-\frac{{\lambda }_{1}x}{2}\right)-A_{sc}{\sigma }_s^{\prime }$$

(10)

$$M_{cal}=A_p{f}_{pt}\left(d_p-\frac{{\lambda }_{1}x}{2}\right)+A_{st}{f}_{yt}\left(d-\frac{{\lambda }_{1}x}{2}\right)-A_{sc}{\sigma }_s^{\prime }\left(\frac{{\lambda }_{1}x}{2}-d^{\prime }\right)$$

(11)

where η₁ and λ₁ are the same as α₁ and β₁ in the Chinese code; d, d_p and d′ are distances from extreme compression fiber to centroid of longitudinal tension reinforcement, prestressed steel and longitudinal compression reinforcement, respectively; and σ′_s is stress of the longitudinal compression reinforcement. If σ′_s ≥ f_yc, take σ′_s = f_yc; if σ′_s < f_yc, then σ′_s is calculated according to Equation (12):

$${\sigma }_s^{\prime }={\epsilon }_{cu2}{E}_s\frac{x-d^{\prime }}{x}$$

(12)

where ε_cu2 is the ultimate compressive strain of the concrete and E_s is the modulus of elasticity of the reinforcing steel.

4.3. ACI 318-19 (ACI Standard: Building Code Requirements for Structural Concrete, 2019)

Figure 12 shows the section stress distribution diagram in a normal section analysis of ACI 318-19 [44]. When the longitudinal tensile and compressive reinforcement reach the yield stress, the flexural strength of the PC rectangular beam and the first type of T-shaped section is calculated as follows [44]:

$$0.85f_{ck}ab=f_{yt}{A}_{st}-f_{yc}{A}_{sc}+f_{ps}{A}_P$$

(13)

$$M_{cal}=f_{yt}{A}_{st}\left(d-\frac{a}{2}\right)-f_{yc}{A}_{sc}\left(d^{\prime }-\frac{a}{2}\right)+f_{ps}{A}_p\left(d_p-\frac{a}{2}\right)$$

(14)

where a is depth of the equivalent rectangular stress block, d_p is the distance from the extreme compression fiber to the centroid of the prestressed steel and f_ps is the stress in the prestressed steel at the nominal flexural strength, which can be calculated using strain compatibility method and an approximate formula. When f_se is larger than 0.5 f_pu, the approximate formula used in the ACI Standard is as follows [44]:

$$f_{ps}=f_{pu}\left\{1-\frac{{\gamma }_p}{{\beta }_1}\left[{\rho }_b\frac{f_{pu}}{f_{ck}}+\frac{d}{d_p}\left(\omega -{\omega }^{\prime }\right)\right]\right\}$$

(15)

where γ_p is factor for the type of prestressed steel, β₁ is the factor that correlates to the depth of the equivalent rectangular compressive stress block to the depth of the neutral axis, ρ_b is ratio of A_p to bd, ω is tensile reinforcement index, given by ω = ρf_yt/f_ck, ω′ is compression reinforcement index, given by ω′ = ρ‘f_yt/f_yc, and ρ and ρ′ are the ratios of A_st to bd and A_sc to bd.

When the reinforcement index ω_T exceeds the required maximum reinforcement index of 0.36β₁, the nominal flexural strength is calculated according to the following formulas [44].

$$\begin{array}{cc}\mathrm{Rectangular}\mathrm{sections}\hfill & \\ & \hfill M_n=0.85a{d}_p^2\left(0.36{\beta }_1-0.08{\beta }_1^2\right)\end{array}$$

(16)

$$\begin{array}{l}\text{T-shaped} \mathrm{section}\\ M_n=f_{ck}b{d}_p^2\left(0.36{\beta }_1-0.08{\beta }_1^2\right)+0.85f_{ck}\left(b-b_w\right)h_f\left(d_p-0.5h_f\right)\end{array}$$

(17)

4.4. Calculation of the Flexural Capacity without Considering the Effects of Strand Corrosion

Figure 13 depicts a comparison between the experimental results and predictions of the three design codes. Additionally, statistic indicators reflecting the accuracy of the calculation are provided in

Table 3. Comparison of prediction accuracy of three specifications for the flexural capacity of CPC beams.

Types of Code	AAE	MSE	SD
GB50010-2010 [39]	0.6186	1146.8869	0.5304
EN 1992-1-1:2004+A1 [40]	0.3868	371.4661	0.4676
ACI 318-19 [44]	0.2829	238.2815	0.3263
The new proposal	0.1990	178.5013	0.2764

Note: AAE, MSE and SD are the average absolute error, mean square error and standard deviation, respectively.

. The influence of prestressed steel corrosion is not considered in the calculation of flexural strength by using Chinese code, Eurocode and ACI standard. As depicted in Figure 13, the calculated flexural strength of the CPC beams with the same arrangement of reinforcement is identical, resulting in a vertical line in the comparison diagram between the calculated and the experimental value, which is not consistent with the actual situation. Furthermore, as the corrosion rate of prestressed steel increases, the ratio of the experimental results to the prediction decreases, indicating a growing deviation between the calculated results of each specification and the test results. This is mainly due to the fact that these three specification formulae do not take into account the reduction in the cross-sectional area and its mechanical properties of the prestressed reinforcement, as well as the reduction in the cooperative working capacity of the prestressed reinforcement and the concrete caused by the corrosion of the prestressed reinforcement. As a result, the flexural capacity of CPC beams is overestimated. As the degree of corrosion of the prestressed reinforcement increases, the deteriorating effect of the corrosion of the prestressed reinforcement on the flexural behavior of the PC beam becomes more obvious.

In the three codes, the average value of the calculated flexural strength is greater than the average test results. The Chinese code is the largest, and the calculated value is 17.0% higher than the test value. Eurocode follows, with an average of 10.9%, and the American standard is the smallest, with an average of 4.8%. It can be concluded that it is unsafe to directly predict the flexural strength of CPC beams by using the three standard formulas without taking into account the impact of corrosion. Furthermore, the deviation between the flexural bearing capacities of the CPC beam, calculated using the ACI standard and the test value, is the smallest. The satisfactory predictive performance of the test results according to the ACI standard is also confirmed by the statistical evaluation given in Table 3.

5. A Proposal for a Flexural Strength Model for CPC Beams on the Basis of the ACI Standard

As discussed in Section 4.4, none of the three codes can predict the flexural capacity of CPC beams well without considering the corrosive effects of steel strands. Therefore, there is a clear need to propose a new model that is precise and straightforward to utilize in predicting the flexural capacity of CPC beams. Since the ACI standard has the smallest deviation in predicting the flexural strength of CPC beams, as discussed in Section 4, a calculation model is proposed for evaluating the flexural strength of CPC beams on the basis of the ACI standard. The calculation model incorporates the impact corrosion on the reduction in the cross-section and the degradation in mechanical performance of the prestressed steel.

5.1. Deterioration of Mechanical Properties of Corroded Prestressed Reinforcement

Corrosion can cause the degradation of the mechanical performance of the prestressed steel, as well as brittle fracture under high stress. Based on previous research findings [22,45], the model for the degradation of mechanical performance of corroded prestressed steel can be expressed as follows:

$$f_{pu\eta }=\left(1-0.75{\eta }_p\right)f_{pu0}$$

(18)

$$E_{p\eta }=\left(0.1+0.9e^{-26{\eta }_p}\right)E_p$$

(19)

$${\epsilon }_{su\eta }=e^{-2.556{\eta }_p}{\epsilon }_{su0}$$

(20)

where f_pu0 and f_puη are the ultimate tensile strength of prestressed steel before and after corrosion; E_p and E_pη are the elastic modulus of prestressed steel before and after corrosion; ε_su0 and ε_suη are the ultimate strain of prestressed steel before and after corrosion; and η_p is the section corrosion rate of prestressed steel.

5.2. Proposal of New Model Based on ACI Standard

Considering the decrease in sectional area and degradation in ultimate tensile strength due to the corrosion of prestressed steel, the stress in corroded prestressed steel is determined as follows:

$$f_{ps\eta }=f_{pu\eta }\left\{1-\frac{{\gamma }_p}{{\beta }_1}\left[{\rho }_{b\eta }\frac{f_{pu\eta }}{f_c^{\prime }}+\frac{d}{d_p}\left(\omega -{\omega }^{\prime }\right)\right]\right\}$$

(21)

where f_psη is stress of corroded prestressed steel. Because the stress–strain curve of corroded prestressed steel is not universally recognized, f_psη is calculated using the approximate method based on the ACI standard. ρ_bη is ratio of A_pηs to bd, which considers the decrease in the sectional area due to corrosion.

The flexural strength of CPC beams of the rectangular section and first type of T-shaped section can be calculated as follows:

$$0.85f_{ck}ab=f_{yt}{A}_{st}-f_{yc}{A}_{sc}+f_{ps\eta }{A}_{ps\eta }$$

(22)

$$M_{cal\eta }=f_{yt}{A}_{st}\left(d-\frac{a}{2}\right)-f_{yc}{A}_{sc}\left(d^{\prime }-\frac{a}{2}\right)+f_{ps\eta }{A}_{ps\eta }\left(d_p-\frac{a}{2}\right)$$

(23)

where A_psη is the sectional area of the corroded prestressed steel and M_calη is the calculated flexural strength of the CPC beams.

5.3. Verification of Proposed Model

To validate the proposal, the above-mentioned test database of CPC beams is still used for the comparative analysis. In Figure 14, a comparison is presented between the flexural bearing capacities predicted by the proposal and the actual experimental results. After considering the corrosion effect of the prestressed steel, the calculated values of the model for Li and Yuan [19], Zhang et al. [24], Zhang et al. [29], Zeng et al. [35], Ma et al. [31], Yu et al. [22], Liu et al. [37], Xu et al. [38], Qiu [26] and Yang [27] are more closer to the test values. Evidently, the proposal demonstrates better accuracy and consistency in predicting the flexural strength of CPC beams. Moreover, the statistical evaluation in Table 3 also verifies the prediction of the test results by the proposal. Additionally, the mean and standard deviation for the total prediction/test ratio is 1.004 and 0.206.

From Figure 14, it is evident that the discrepancies between the model prediction and test results of Rinaldi et al. [20] and Zhang et al. [29] have a larger deviation than other datasets. This may be because the prestressed steel used in the Rinaldi et al. [20] test is a single prestressed steel, but the degradation law of the mechanical performance is based on the corroded steel strand. Both of the degradation laws are not necessarily similar. In the test data of Zhang et al. [29], the corrosion rate of prestressed steel is up to 84.74%, while the mechanical performance degradation law of prestressed steel in this paper is fitted below the corrosion rate of 21%, which is much lower than the corrosion rate of the test beam. The mechanical performance degradation law used in the proposed model may not fully adapt to the very severe corrosion situation, resulting in calculation deviation.

6. Conclusions

In this research, 104 sets of flexural tests on CPC beams were gathered from a range of published sources. Various significant factors influencing the flexural strength were examined, and the applicability and accuracy of the Chinese code, Eurocode and ACI standard in predicting the flexural strength of CPC beams were assessed. Subsequently, a novel prediction model was developed, and its validity was confirmed. The key findings and contributions are outlined as follows.

(1) A standardized database with 104 sets of CPC test data is established, in which the concrete compressive strength and the corrosion rate of prestressed steel are unified.

(2) The influence of the main parameters on the flexural strength of CPC rectangular and T-beams is almost the same. Compared with other influencing parameters, the beam width, shear span ratio, section corrosion rate of the prestressed steel and prestressing ratio have an obvious influence on the flexural strength of CPC beams.

(3) The established database of flexural tests shows that the residual flexural strength of CPC beams is overestimated by 17.0%, 10.9% and 4.8% in the Chinese code, Eurocode and ACI standard, respectively, compared to the actual flexural test results. Therefore, it is unsafe to evaluate the flexural strength of CPC beams directly using the three standard formulas without considering the influence of corrosion.

(4) Compared to experimental flexural results, the ACI standard provides accurate and effective predictions for the residual flexural strength of CPC beams by only considering the reduction in cross-section and the deterioration in mechanical performance caused by the corrosion of prestressed steel.