Permit checking of overloaded customized transport vehicle based on serviceability limit state reliability of concrete bridges

Abstract

To increase the authorization efficiency of overloaded customized transport vehicle (CTV), a serviceability limit state (SLS) reliability based permit checking method for concrete bridges is proposed through the optimization towards critical load effect ratio. In this procedure, the SLS reliability of crack width and the SLS reliability of concrete stress in tensile region are analyzed for reinforced concrete (RC) and prestressed concrete (PC) structures, respectively. The durability requirements and a unified reliable level can be concentrated reflected by the optimized critical load effect ratio. The results show that it is unreasonable to take a uniform target reliability index for all routes in permit checking of CTV, a stricter authorization criterion should be adopted for a higher expected authorization frequency. For a specific route level, a fluctuant variation of critical load effect ratio can be found with the increasing of bridge span. By introducing an ultimate limit state (ULS) based safety checking procedure, it is found that the SLS based permit checking criterion is crucial and determinative for the authorization of CTV instead of the ULS.

Keywords

bridge engineering concrete bridges overloaded customized transport vehicle permit checking method SLS reliability

Introduction

Generally, the maximum gross weight and dimensions of vehicles circulate on national highway are legally regulated. However, the movement of indivisible industrial product infrequently requires special overloaded vehicle, of which the gross weight and/or the geometrical dimensions exceed the corresponding legal limits. Meanwhile, the special overload vehicle is usually assembled according to the geometrical dimensions of the carried cargo and thus can be named as customized transport vehicle (CTV; Han et al., 2019).

The cargo carried by the CTV is usually very important for the corresponding economic activity, which leads to a strong demand for passage. However, the passage of CTV may jeopardize the bridges to be passed, which leads to the necessity of permit checking (including routine permit and special permit) before the passage (Choi et al., 2006; Fu and Hag-Elsafi, 2000; Grimson et al., 2008). Routine permits are valid for multiple trips within a period and the authorized CTVs are allowed to be mixed in random traffic. Special permits are employed for heavier vehicles and for a single trip (Minervino and Sivakumar, 1999).

In the past decades, the load rating methods, including allowable stress rating (ASR), load factor rating (LFR), and load and resistance factor rating (LRFR), were frequently adopted in permit checking (Fu and Moses, 1991; Ghosn et al., 2013; Moses, 1990; Phares et al., 2004). In these researches, the safety problems and ULS are mainly concerned. However, some surveys conducted recently indicate that: (1) catastrophic structural failures caused by overloaded vehicle are very rare, while serviceability failures occur more frequently, for example the crack width or stress level exceeds the corresponding limits; (2) most structural defects are related to serviceability conditions rather than strength; and (3) serviceability failure constitutes the largest single source of economic loss of structural maintenance (Honfi et al., 2012; Quan and Gengwei, 2002). Therefore, the serviceability of bridges needs to be paid more attentions before the passage of CTV is authorized, and it is of great significance for the structural durability and maintenance to develop a serviceability based permit checking method.

Recently, many researches have been completed toward the permit checking method of overloaded vehicle based on SLS, among which the method of load effect comparison method is mostly adopted. According to this method, the passage of overloaded CTV is decided to be authorized or not according to the comparison between design vehicle load effect and CTV load effect, in which the most important index is the critical load effect ratio (Correia et al., 2014; Correia and Branco, 2006). In the current engineering practices, the value of critical load effect ratio could not involve the durability requirements in a reasonable manner, or the value fails to produce a uniform SLS reliability level in permit checking. The solution of this problem needs an in-depth SLS reliability analysis procedure.

In the existing researches associated with structural assessment, the reliability theory has been widely employed and the reliability index, which corresponds to a specific failure probability, is usually adopted to evaluate the service performance. The greatest advantage of reliability theory in structural assessment is that it can provide a universe target performance level for all assessment cases (Ghasemi and Nowak, 2017). To date, significant progress has been made in structural SLS reliability analysis, including the theoretical description of SLS, the determination of target reliability index and cognition towards the uncertainty embedded in the key variables (Bian et al., 2017; Hwang et al., 2014; Roubos et al., 2018; Quan and Gengwei, 2002; Stewart, 1996). These achievements make it possible to develop a SLS reliability based permit checking method of overloaded CTV through the optimization on critical load effect ratio.

This paper aims to develop a SLS reliability based permit checking method of overloaded CTV. To achieve this goal, the permit checking method is firstly established according to critical load effect ratio. In the next, a brief review on SLS reliability is conducted and SLS reliability analysis is completed to determine the critical load effect ratio for given bridges. About 12 simply supported concrete bridges including RC and PC structures are taken as prototype bridges to conduct case study. In the last, according to the obtained critical load effect ratios, the permit checking criteria based on SLS and ULS are compared.

Serviceability based permit checking method

The overloaded CTV as shown in Figure 1 is mainly focused, of which the following characters can be summarized: (1) a special permit checking for single trip is required before passage request is authorized; (2) the random traffic is close when passing a bridge structure, the passage process is strictly controlled and escorted; (3) the gross vehicle weight is very large and can reach to hundreds of tons; and (4) the distribution of cargo weight is mainly by hydraulic balancing system, which will lead to the randomness of axle weight and also the randomness of CTV load effect.

Figure 1.

Passing scenario of a special permitted CTV.

In addition, the concrete bridges are mainly focused herein due to the more prominent challenge of durability embedded in this structural type (Kušter Marić et al., 2020; Memon et al., 2019; Sun et al., 2020). For a planned route to be passed by CTV, the passage request can be authorized only when the following equation is satisfied:

\forall i (i = 1, 2, \dots, n) ϕ_{i, ratio} = \frac{μ (M_{CTV, i})}{M_{DVL, i}} \leq ϕ_{i, critical}

(1)

where n denotes the total number of concrete bridges to be considered in permit checking. $ϕ_{i, ratio}$ and $ϕ_{i, critical}$ denotes the load effect ratio and critical load effect ratio of the ith bridge. The former researches had pointed out that serious durability problems of concrete bridges are mainly caused by bending deformation or bending cracks (Germaniuk et al., 2016; Han et al., 2018a; Ma et al., 2020). Hence, in equation (1), the bending moments are employed to calculate $ϕ_{i, ratio}$ . $M_{CTV, i}$ , and $M_{DVL, i}$ denote the bending moments of the ith bridge under CTV load and design vehicle load, respectively.

As mentioned before, the axle weight of CTV is random in nature and leads to the randomness of bending moment. Hence, $M_{CTV, i}$ is deemed to be a random variable in the following analysis. In addition, the existing researches had pointed out that the variability of CTV load effect is pretty weak (Han et al., 2018a, 2019). Thus in equation (1), for the simplicity of engineering application and the comparability between two sides of equation (1), the mean value of $M_{CTV, i}$ , that is, $μ (M_{CTV, i})$ , is employed to calculate the load effect ratio for each bridge. To pursue an uniform failure probability of SLS for all bridges considered in permit checking, a SLS reliability analysis procedure is introduced in the following sections to determine the $ϕ_{i, critical}$ .

A brief review on structural SLS reliability

Formulations of SLS reliability

In the researches associated with structural design or assessment, the definition of SLS and the corresponding reliability analysis are mainly aimed at maintaining the durability and extending the service life (Zięba et al., 2019). The general description of performance function of SLS can be given as (Hwang et al., 2016):

g (R, S) = θ_{R} R - θ_{Q} Q

(2)

where g, R, and Q denote SLS performance function, generalized resistance, and generalized load effect, respectively. $θ_{R}$ and $θ_{Q}$ denote model uncertainty coefficients regarding R and Q, respectively.

Up to now, many mechanical indices including deformation, material stress, crack width, vibration parameters, etc. have been employed as generalized load effect (Ellingwood and Tallin, 1984). The failure of SLS indicates that the mechanical index under external loads exceeds the corresponding allowable value, for instance, when deformation is selected for SLS reliability analysis, the failure of SLS can be deemed to occur once a deformation exceeding the corresponding limit (Galambos and Ellingwood 1986; Ghasemi and Nowak, 2016; Honfi et al., 2012; Stewart, 1996; Xu et al., 2018; Val and Chernin, 2009).

The SLS reliability of concrete stress was investigated by Hwang et al. (2016) and Yao et al. (2017). The crack width is frequently used in the SLS reliability analysis of RC structures. Qin and Zhao calibrated the embedded reliability level of a crack width computational formula (Quan and Gengwei, 2002). A time-dependent reliability analysis was conducted by Li and Lawanwisut et al. based on a stochastic model of crack width (Li et al., 2005). Some other indices, for example, carbonation depth, have also been reported to be adopted in SLS reliability analysis (Zhao and Jin, 2002). In the researches mentioned above, the generalized resistances and calculation methods of generalized load effects are usually taken from the structural design codes (Ghasemi and Nowak, 2016).

According to the researches listed above and also the reliability theory of engineering structure, the SLS and the corresponding failure criterion can be defined once if the following requirements can be satisfied: (1) an unambiguous physical connotation of failure phenomenon; (2) the explicit variables of generalized resistance and generalized load effect. Therefore, it is feasible to establish the SLS function of concrete bridge under overloaded CTV. In addition, it should be noted that in this research, the SLS is introduced as a boundary condition in the developing of permit checking method of overloaded CTV, and cannot make a direct connection with bridge maintenance and the corresponding service life.

Analysis method of reliability index

In the existing researches related to SLS, first order reliability method (FORM), second order reliability method (SORM) and Monte Carlo (MC) simulation method were adopted (Honfi et al., 2012; Wiśniewski et al., 2004). For the SLS reliability analysis with complicated performance function, Qin and Zhao provided detailed calculation steps of reliability index by FORM (Quan and Gengwei, 2002). By incorporating the advantages of FORM, importance sampling and artificial neural network, Cheng developed a more efficient method for the SLS reliability analysis (Cheng, 2013). In the present study, MC simulation and importance sampling are combined in the following sections to conduct SLS reliability analysis.

Statistical properties of random variables

The uncertainties and randomness related to SLS reliability analysis are mainly embedded in parameters of: (1) model uncertainty; (2) geometrical dimensions; and (3) material properties. With the continuous progress in the researches of material science and meanwhile the accumulation of field test data, more and more statistical information is applicable for the reliability analysis (Val and Chernin, 2009). For example, the detailed statistical properties of random variables used to calculate the deformation, concrete stress and crack width were provided in the research of Honfi et al. (2012)Hwang et al. (2014) and Li et al. (2005) respectively.

According to the suggestion of Sun and Huang, the model uncertainty coefficient can be regarded to obey normal distribution when the test data is scarce (Sun and Huang, 2006). In the following analysis, the listed statistical properties of basic random variables in the former researches are selected and cited to complete the SLS reliability analysis.

Target reliability index of SLS

Attributed to the different potential failure consequences, the target reliability index of SLS is obvious lower than that of ULS (Lenner et al., 2019; Roubos et al., 2018). According to the suggestion of ISO, the lifetime target reliability index can be taken as 0 and 1.5 when SLS is reversible and irreversible, respectively (ISO2394, 1998). Table 1 provides the annual target reliability indices of irreversible SLS specified by Joint Committee on Structural Safety (JCSS), Eurocode and that suggested by Stewart (EN1990, 2000; Stewart, 1996; Vrouwenvelder, 1997). The suggested value by Stewart has been widely adopted in American engineering practices. However, the target reliability indices mentioned above are mainly for structural design instead of performance assessment. A detailed analysis procedure is still necessary to determine the target reliability index of SLS for the permit checking of overloaded CTV.

Table 1.

Annual target reliability indices of SLS.

Cost of safety measures	JCSS	Eurocode	Stewart (1996)
High	1.3	2.9	2.76
Normal	1.7
Low	2.3

Concrete bridges considered in permit checking

The bridge structural composition of a route that an overloaded CTV plans to pass is usually very complicated. However, it is unnecessary to take all the bridges into account during permit checking, because only a few of them are crucial and determinative for permit checking result. In this research, simply supported concrete bridges are considered preferentially, because their ratios of design live load effect to dead load effect are higher than other bridge types, which indicates that their mechanical performances are more sensitive to vehicle load level and would induce higher requirements on CTV load configurations.

To be more specific, 12 simply supported concrete bridges are selected and the detailed information is provided in Table 2, in which three RC slab bridges, four PC slab bridges, and five PCT beam bridges are included. These bridges are specified in the standard design drawings published by Chinese Ministry of Transport, and are frequently adopted in Chinese highway bridge constructions (Han et al., 2018b).

Table 2.

Detailed information of the selected bridge.

Bridge ID	Girder type	Span length (m)	Girder Height (m)	Number of prestress tendons	$M_{DVL}$ (kN · m)	Dead load effect (kN · m)	Sectional resistance (kN · m)
RCS6	RC solid slab	6	0.32	/	92.73	52.00	414.8
RCS8	RC hollow slab	8	0.42	/	127.60	100.00	562.9
RCS10	RC hollow slab	10	0.50	/	166.21	168.60	807.4
PCS10	PC hollow slab	10	0.60	8 $ϕ^{s}$ 15.2	202.04	169.60	995.1
PCS13	PC hollow slab	13	0.70	12 $ϕ^{s}$ 15.2	274.55	385.10	1460.4
PCS16	PC hollow slab	16	0.80	14 $ϕ^{s}$ 15.2	353.59	545.70	2038.3
PCS20	PC hollow slab	20	0.95	18 $ϕ^{s}$ 15.2	474.69	950.00	2845.3
PCT20	PC T beam	20	1.50	18 $ϕ^{s}$ 15.2	1068.67	1539.90	4803.4
PCT25	PC T beam	25	1.70	26 $ϕ^{s}$ 15.2	1560.93	2568.40	7690.8
PCT30	PC T beam	30	2.00	31 $ϕ^{s}$ 15.2	2124.80	3949.90	10738.4
PCT35	PC T beam	35	2.30	37 $ϕ^{s}$ 15.2	2738.09	5713.30	14802.0
PCT40	PC T beam	40	2.50	51 $ϕ^{s}$ 15.2	3348.13	8260.80	20740.6

Note that the dead load effect and sectional resistance are regarded as random variables in reliability analysis and the values provided in Table 2 are standard values. By taking four bridges as examples, Figures 2 and 3 provide the mid-span cross sections and the corresponding sectional reinforcement arrangements, respectively.

Figure 2.

Mid-span cross sections (unit: cm): (a) RCS6, (b) RCS8, (c) PCS10, and (d) PCT20.

Figure 3.

Sectional reinforcement arrangements (unit of diameter: mm): (a) RCS6, (b) RCS8, (c) PCS10, and (d) PCT20.

SLS reliability analysis of concrete bridges

Limit state functions

In this research, the irreversible SLSs of concrete bridges associated with cracking control, including SLS of crack width and SLS of concrete stress in tensile zone, are chosen to conduct reliability analysis and the results are further employed to determine the critical load effect ratio of each prototype bridge.

RC slab bridges

For the bridges of RCS6, RCS8, and RCS10, SLS is deemed to occur once a crack width exceeds the corresponding allowable value, and the limit state function is given as:

Z_{RCS} = [W] - θ_{cr} W_{cr}

(3)

where $Z_{RCS}$ denotes the performance function of RC slab bridges. $θ_{cr}$ denotes the model uncertainty coefficient. $[W]$ denotes the allowable value of crack width and is deemed to be constant, and $W_{cr}$ is the maximum crack width induced by CTV load effect.

In general, $[W]$ of RC structures is in the range of 0.1∼0.5mm (Thoft-Christensen, 2001). Table 3 provides the $[W]$ specified by several design codes (ACI318, 1999; BS8110, 1997; EN1990, 2000; Germaniuk et al., 2016; Harries et al., 2012). Here, the permissible crack width of 0.20mm specified by Chinese design code is adopted. By comparing with the values provided in Table 3, it can be known that Chinese bridge engineering practices have a more strict control towards the cracking of RC structures.

Table 3.

Allowable concrete crack width (unit: mm).

Code	EN 1990	BS 8110	ACI 318	AASHTO		Polish bridge standards
Code	EN 1990	BS 8110	ACI 318	Class 1: typical exposure	Class 2: more aggressive	Polish bridge standards
[W]	0.30	0.30	0.30	0.43	0.32	0.20

According to Chinese design code, $W_{cr}$ is calculated as:

W_{cr} = C_{1} C_{2} C_{3} \frac{σ_{ss}}{E_{s}} (\frac{30 + d}{0.28 + 10 ρ})

(4)

where $C_{1} = 1.0$ denotes the geometric coefficient of rebar. $C_{2}$ denotes the influence coefficient of long-term effect and is taken as 1.0 in permit checking. $C_{3}$ is the coefficient associated with mechanical character and is taken as 1.0 for RC slabs. $σ_{ss}$ is the stress of rebar and is calculated as:

σ_{ss} = \frac{M_{s}}{0.87 A_{s} h_{0}}

(5)

where $M_{s} = M_{CTV} + M_{DL}$ is the bending moment effect of short-term combination, and $M_{DL}$ is the dead load effect. $A_{s}$ and $h_{0}$ denote total area of tensile rebar and the sectional effective height, respectively. $E_{s}$ denotes the elasticity modulus of tensile rebar. d and $ρ$ are the diameter and reinforcement ratio of tensile rebar, respectively. According to Chinese bridge design code, the applicability of equations (4) and (5) is not affected by the magnitude of load effect, and can be adopted in the crack width analysis of RC structure under overloaded CTV. Substitute equations (4) and (5) to equation (3) and leads to:

Z_{RCS} = [W] - θ_{cr} \cdot (\frac{M_{CTV} + M_{DL}}{0.87 E_{s} A_{s} h_{0}}) \cdot (\frac{30 + d}{0.28 + 10 ρ})

(6)

PC slab bridges

The hollow slab of PCS bridge is partially prestressed member, which indicates that the sectional bearing capacity is provided by prestress tendons and meanwhile the rebars. According to the design requirements, the concrete of this member can be in tensile, but the normal tensile stress is strictly restricted and cracking is not allowed. The limit state function is defined as:

Z_{PCB} = σ_{pc} - σ_{st} + 0.7 f_{tk}

(7)

where $Z_{PCB}$ is the SLS performance function of PCS bridge. $σ_{pc}$ denotes the concrete compressive stress under the sectional prestress and is calculated by:

σ_{pc} = \frac{N_{p}}{A_{n}} + \frac{N_{p} \cdot e_{pn}}{I_{n}} y_{n}

(8)

where $N_{p}$ is the resultant force of prestress tendons and rebars and is calculated as:

N_{p} = (1 - ω) σ_{con} A_{p}

(9)

where $ω$ is the loss percentage of prestress. The introducing of $ω$ is to simplify the complicated analysis procedure of prestress loss. $σ_{con}$ is the controlled tension stress of prestress tendons and $A_{p}$ is the corresponding sectional area. $A_{n}$ is the net sectional area, which excludes the channel area of prestress tendons. $I_{n}$ denotes the inertia moment of net section, which is calculated according to the design geometrical parameters of cross section. $e_{pn}$ denotes the distance from center of gravity of net section to the point of resultant force. $y_{n}$ is the distance from center of gravity of net section to the bottom concrete, as shown in Figure 4. $σ_{st}$ denotes the concrete tensile stress under the bending moment effect of short-term combination and is calculated as:

σ_{st} = \frac{M_{DL}}{W_{n}} + \frac{M_{CTV}}{W_{0}}

(10)

where $W_{n}$ and $W_{0}$ denotes the elastic resistance moment of the net section and transformed section, respectively. $f_{tk}$ is the characteristic value of tensile strength of concrete.

Figure 4.

Illustration of the variables in SLS limit state function of PC slab.

Note that for the bridge with span length reaches to 20 m, the strands are generally arranged in curves instead of straight line. In this condition, $e_{pn}$ is a variable value. However, in the permit checking of overloaded CTV by bending moment effect, the strands around critical cross section are usually arranged in straight line, for example the mid-span cross section of simply supported bridge. Thus, in this study, for a specific cross section, $e_{pn}$ can be regarded as constant. Substitute equations (8) to (10) to equation (7) and leads to:

\begin{matrix} Z_{PCB} = \frac{(1 - ω) σ_{con} A_{p}}{A_{n}} + \frac{(1 - ω) σ_{con} A_{p} \cdot e_{pn}}{I_{n}} y_{n} \\ - \frac{M_{DL}}{W_{n}} - \frac{M_{CTV}}{W_{0}} + 0.7 f_{tk} \end{matrix}

(11)

PC T-beam bridges

The T beams are fully prestressed members, which indicates that the sectional bearing capacity is only provided by prestress tendons, and the rebars are mainly used to improve the sectional ductility. For this kind of member, the concrete is not allowed to be in tensile. The limit state function is proposed as:

Z_{PCT} = σ_{pc} - σ_{st}

(12)

where $Z_{PCT}$ is the SLS performance function of PC T beam bridge. The determination of $σ_{pc}$ and $σ_{st}$ is the same as equation (7). Substitute equations (8) to (10) to equation (12) and leads to:

\begin{matrix} Z_{PCT} = \frac{(1 - ω) σ_{con} A_{p}}{A_{n}} + \frac{(1 - ω) σ_{con} A_{p} \cdot e_{pn}}{I_{n}} y_{n} \\ - \frac{M_{DL}}{W_{n}} - \frac{M_{CTV}}{W_{0}} \end{matrix}

(13)

Note that during the deducing of equation (6), (11), and (13), plane cross-section assumption and the derivation process is not applicable for the shear controlled components. In addition, the assumption of linear elasticity is employed and it requires a further discussion for the component in plastic stage.

Importance sampling based reliability analysis

For the reliability analysis of limit state functions illustrated by equations (6), (11), and (13), Qin and Zhao had already proposed an FORM based analysis procedure (Quan and Gengwei, 2002). Alternatively, the MC simulation can be adopted, and in this condition, the probabilistic characteristics of all the random variables should be firstly explicated.

To ensure the analysis efficiency, the direct importance sampling is combined with MC simulation method, and the failure probability of SLS, $P_{f, SLS}$ , can be given as:

P_{f, SLS} = \Pr {Z (x) \leq 0} = \int_{DF} f_{X} (x) d x

(14)

where $x = (x_{1}, \dots, x_{m})$ denotes the m-dimensional vector of input variables and $f_{X} (x)$ is the corresponding joint probability density function (PDF). DF is the abbreviation of domain of failure.

By introducing an indicator function as follows:

I (x) = {\begin{matrix} 1, Z (x) \leq 0 \\ 0, Z (x) > 0 \end{matrix}

(15)

The equation (14) can also be expressed as:

P_{f, SLS} = \int I (x) f_{X} (x) d x = Ex p_{f} [I (x)]

(16)

where $Exp (\cdot)$ denotes the expectation operator. To further increase the calculation efficiency of $P_{f, SLS}$ by MC simulation, optimization algorithm is introduced to determine the most probable point (MPP) located on the limit state surface. In the next, an importance sampling density $h_{X} (x)$ can be established by taking MPP as sampling center, and $h_{X} (x)$ satisfies the normalization condition:

\int h_{X} (x_{1}, x_{2}, \dots, x_{m}) d x_{1} d x_{2} \dots d x_{m} = 1

(17)

Then the failure probability $P_{f, SLS}$ can be rewritten as:

P_{f, SLS}^{IS} = \int I (x) \frac{f_{X} (x)}{h_{X} (x)} h_{X} (x) d x = Ex p_{h} [I (x) \frac{f_{X} (x)}{h_{X} (x)}]

(18)

Given a set of independent and identically distributed samples ${x^{(1)}, x^{(2)}, \dots, x^{(N)}}$ drawn from $h_{X} (x)$ , the failure probability can be estimated as (Xiao et al., 2020):

{\hat{P}}_{f, SLS}^{IS} = \frac{1}{N} \sum_{i = 1}^{N} [I (x^{(i)}) \frac{f_{X} (x^{(i)})}{h_{X} (x^{(i)})}]

(19)

In general, the theoretical optimal importance sampling density, that is, $h_{X}^{*} (x) = I (x) f_{X} (x) / P_{f, SLS}$ , is not implementable since it involves the failure probability which is unknown beforehand. Here, the importance sampling density is approximately regarded to be normal distributed, and the coefficient of variation (COV) is taken as 1.0–2.0 times of that of original variable (Cheng and Cai, 1997). Meanwhile, the expectation of importance sampling density is taken as the point of maximum likelihood of $f_{X} (x)$ , and can be determined by solving a mathematical programming problem:

{\begin{matrix} max f_{X} (x) \\ s . t . Z (x_{1}, \dots, x_{m}) = 0 \end{matrix}

(20)

Probabilistic characteristics of random variables

According to the existing researches, Table 4 provides the probabilistic characteristics of the main random variables involved in the SLS reliability analysis procedure by MC simulation (Han et al., 2018a; Quan and Gengwei, 2002; Sun and Huang, 2006). In Table 4, $κ$ denotes the ratio of mean value to the corresponding standard value. Note that the probabilistic characteristics of geometric parameters are mainly taken from the national standard of China, which are determined based on huge volumes of field test data collected from engineering practices (GB/T50283, 1999).

Table 4.

Probabilistic characteristics of random variables.

Type of variable	Description of variable	Distribution type	$κ$	COV
Model uncertainty	$θ_{cr}$	Normal distribution	0.9500	0.3400
Geometric parameters	Sectional height	Normal distribution	1.0064	0.0255
	Sectional width	Normal distribution	1.0013	0.0081
	Flange thickness	Normal distribution	1.0320	0.1019
	Effective height	Normal distribution	1.0124	0.0229
	Hole diameter of RCS	Normal distribution	0.9769	0.0329
	Diameter of rebar	Normal distribution	1.0000	0.0100
	Concrete cover thickness	Normal distribution	1.0178	0.0496
Material parameters	$E_{s}$	Normal distribution	1.0000	0.0600
	$f_{tk}$	Normal distribution	1.0000	0.1100
	$σ_{con}$	Normal distribution	1.0000	0.0365
Load effects	Dead load effect	Normal distribution	1.0148	0.0431
	CTV load effect	Lognormal distribution	1.0000	0.0368

For the probabilistic characteristics of random variables which are not specified in Table 4, a modeling procedure by incorporating MC simulation and fitting analysis is introduced. For example, Figure 5(a)–(d) shows the fitting analysis towards the MC simulation results of $y_{n}$ and $I_{n}$ by taking PCS10 and PCT20 as examples. Furthermore, Figure 5(e)–(h) provide the $κ$ and COV of $y_{n}$ and $I_{n}$ of all the PC concrete bridges.

Figure 5.

Probabilistic characteristics of I_n and y_n: (a) y_n of PCS10, (b) I_n of PCS10, (c) y_n of PCT20, (d) I_n of PCT20, (e) k of PCS, (f) COV of PCS, (g) k of PCT, and (h) COV of PCT.

In the next, according to the analysis of sections “Limit state functions” and “Importance sampling based reliability analysis,” and meanwhile the probabilistic characteristics of random variables, an analysis program of SLS reliability of the concrete bridges is compiled. For all reliability analysis by MC simulation, the samples are all 100,000.

SLS reliability based critical load effect ratios

Selection of target reliability index

Annual target reliability index

In China, some calibration researches had been finished towards the structural SLS reliability, and it was found that the reliability index is larger than 0.8. However, the corresponding reference period is not explicit. Here, the annual target reliability index ( $β_{T, annual}$ ) of 1.3 recommended by JCSS is adopted for CTV permit checking analysis considering the higher potential safety measure costs of bridge in operation stage. For a specific engineering practice, $β_{T, annual}$ is recommended to be determined according to the different bridge maintenance levels, and meanwhile different requirements based on the balance between economic development and structural durability.

Target reliability index for single special permit checking

To further determine the target reliability index for a single special permit checking, that is, $β_{T, single}$ , according to $β_{T, annual}$ , the expected annual average times (AAT) of special permit checking of a route should be considered. The special permits occur within a year are regarded as independent events, and the relationship between $β_{T, single}$ and $β_{T, annual}$ can be deduced as:

β_{T, single} = Φ^{- 1} [{(Φ (β_{T, annual}))}^{\frac{1}{AAT}}]

(21)

To date, for the focused special permit checking of this research, which has been illustrated in section “Serviceability based permit checking method,” there is no explicit stipulation on the critical weight of CTV. Here, the engineering practices of CTV permit checking of Shaanxi province in China are referenced and the critical weight is determined as 200 t. In addition, the statistical information of authorized CTVs of Lianhuo expressway in Shaanxi province, as shown in Figure 6, is employed to illustrate the value of AAT. According to Figure 6, AAT = 23.75, which can deem to be a very high level considering the following reasons:

(1) The military industry in Shaanxi province is very developed, which leads to a strong demand of heavy cargo transportation and a high volume of overloaded CTV;

(2) The Lianhuo expressway, which runs through Shaanxi province and connects the east and west regions of China, as shown in Figure 6, is one of the main channels of heavy cargo transportation;

(3) The statistical data employed in Figure 6 is collected from 2012 to 2015, during which the authorization frequency of CTV in Shaanxi province is very high.

Figure 6.

Statistical information of AAT of Lianhuo expressway.

According to equation (21), the AAT directly affects the value of $β_{T, single}$ once $β_{T, annual}$ is determined. Obviously, it is unreasonable to take a uniform AAT for all the routes. Herein, three levels of routes are proposed, as shown in Table 5, according to AAT of special permits, and meanwhile by incorporating the analysis results of Figure 6. Then $β_{T, single}$ of the three levels of route can be derived as 2.32, 2.57, and 2.71, respectively, which are also provided in Table 5.

Table 5.

Route level definition according to AAT of special permit.

Route level	Level I	Level II	Level III
AAT	0< AAT ≤ 10	10< AAT ≤20	20< AAT ≤30
$β_{T, single}$	$β_{T, single, I} = 2.32$	$β_{T, single, II} = 2.57$	$β_{T, single, III} = 2.71$

Loss percentage of prestress

For the post-tensioned prestressed members, the factors induce prestress loss can be divided as three categories: (1) factors occurred in tension stage; (2) relaxation of prestress tendons; and (3) shrinkage and creep of concrete (Caro et al., 2013; Guo et al., 2018; Páez and Sensale, 2018).

Figure 7 provides the prestress loss percentages of the PC bridges induced by the factors of first two categories according to the specified models of Chinese design code. In addition, according to the research of Biswal et al., the prestress loss percentage induced by shrinkage and creep of concrete can reach to 27% after a service time of 50 years (Biswal and Ramaswamy, 2017; Biswal and Reddy, 2019). However, the service time of bridge that the CTV to pass is usually very short. For example, in many provinces of China, the passage request of CTV is directly refused without a permit checking procedure if the planed route includes bridge whose technical condition is not good enough. Therefore, a very low level of prestress loss percentage induced by shrinkage and creep of concrete can be considered. In summary, the value of $ω$ is taken as 20%–25%.

Figure 7.

Prestress loss percentage of PC bridges according to Chinese design code.

Results analysis

SLS reliability analysis of all the selected bridges is conducted by the compiled program. Figure 8 provides the analysis results of SLS reliability index by taking the RC bridges, PCS20 and PCT25 as examples, in which the load effect ratio indicates the times of $μ (M_{CTV})$ to the $M_{DVL}$ . $β_{T, single}$ of the three levels of route are also depicted in Figure 8. Then the critical load effect ratio for each bridge can be determined according to the coordinates of line intersections.

Figure 8.

Analysis results or SLS reliability index of different load effect ratios: (a) RCS bridges, (b) PCS20, and (c) PCT25.

For the three levels of route, Figure 9 provides the critical load effect ratios of the RC and PC bridges. For the PC bridges, the upper limit of critical load effect ratio corresponds to $ω = 20 %$ and the lower limit corresponds to $ω = 25 %$ . The following conclusions can be drawn from Figure 9:

(1) The increasing of route level leads to a decrease of critical load effect ratio, in other words, to pursue a uniform SLS reliability level, a stricter authorization criterion should be adopted if the expected authorization frequency increases;

(2) For a specific route level, a fluctuant variation of critical load effect ratio can be found with the increasing of bridge span, which is the result of SLS definition, sectional reinforcement ratio, number of prestress tendons, sectional geometric dimensions and bridge span, etc.;

(3) For all the simply supported concrete bridges, the variation ranges of critical load effect ratio of the three route levels are [1.06, 1.66], [1.01, 1.62], and [0.98, 1.60], respectively. The minimum value corresponds to PCT30 and PCT35, which can be mainly attributed to the stricter SLS definition and meanwhile the prestress tendons seem to be insufficient;

(4) The prestress loss percentage has an obvious influence on the analysis results of critical load effect ratio.

Figure 9.

Reliability based critical load effect ratios: (a) Route level I, (b) Route level II, and (c) Route level III.

ULS safety checking based on critical load effect ratio

It can be seen from Figure 9 that the maximum critical load effect ratio reaches to 1.66. To investigate whether the SLS based permit checking criterion is crucial or the ULS based permit checking criterion, the critical load effect ratios correspond to $ω = 20 %$ in Figure 9(a), and together with the load and resistance information provided in Table 2, are employed to conduct a ULS safety checking.

The reliability based safety checking formulation, which had been proposed in the former research of the authors, is written as (Han et al., 2018a):

η = [γ_{G} \cdot M_{G, k} + γ_{CTV} \cdot M_{CTV, k}] / [(1 - ξ_{e}) \cdot R_{k} / γ_{R}]

(22)

where $η$ denotes the safety index, $η < 1.0$ indicates that the sectional safety is satisfied. $γ_{G}$ , $γ_{CTV}$ , and $γ_{R}$ are the reliability based partial factors of dead load effect, CTV load effect, and resistance, respectively, and were calibrated as 1.047, 1.188, and 1.101, respectively. $M_{G, k}$ , $M_{CTV, k}$ , and $R_{k}$ denote the standard value of dead load effect, CTV load effect, and sectional resistance, respectively. $(1 - ξ_{e})$ denotes the reduction coefficient of sectional resistance and is taken as 0.95.

Figure 10 provides the ULS safety checking results of all the concrete bridges. It can be found that the safety indices range from 0.604 to 0.846, which indicates that the sectional safety of all the concrete bridges can be satisfied. In other words, the SLS based permit checking criterion is crucial and determinative for the authorization of CTV instead of ULS based permit checking criterion.

Figure 10.

Safety indices of all the concrete bridges.

Conclusion

The present research aims to develop a SLS reliability based special permit checking method of overloaded CTV. To achieve this goal, the critical load effect ratio is introduced to establish the permit checking method, and further, the critical load effect ratio is determined through a detailed SLS reliability analysis. In the reliability analysis, the SLS of crack width and the SLS of concrete tensile stress are focused for RC and PC concrete bridges, respectively. According to the analysis results of critical load effect ratios, a ULS safety checking procedure is introduced to find out whether the SLS based permit checking criterion or the ULS based permit checking criterion is crucial for authorization of CTV. The following conclusions are drawn from this research:

It is unreasonable to take a uniform target reliability index for all routes in permit checking of CTV. For a specific route, the expected annual average times of special permit should be considered in the determination of SLS target reliability index for a single special permit, and a stricter authorization criterion should be adopted for a higher expected authorization frequency.

The SLS reliability based critical load effect ratio ranges from 0.98 to 1.66 for simply supported concrete bridges. The prestress loss percentage has a great influence on the critical load effect ratio. The bridge technical conditions of a route should be closely integrated in the formulation of SLS reliability based permit checking method;

The SLS reliability based permit checking criterion is crucial and determinative for the authorization of CTV instead of the ULS based permit checking criterion;

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by National Project of Key R&D Plan of China (2019YFB1600702), Natural Science Foundation of China (Project 51878058), and Basic research program of natural science in Shaanxi province of China (2020JQ-665).

ORCID iD

Yangguang Yuan

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