接上文：

However, Eq. (5) is mostly applied for building structures which mainly bear static loads, and the main factor that affects the strength of concrete in Eq. (5) is the durability of concrete. As for bridge structures, live load effects is also quite significant. In addition to the factor of durability, the material fatigue can also cause concrete strength decay, and its effect can not be ignored in practical engineering. J. L. Zhang et al (2004) tested the concrete strength of more than 10 old bridges located in the Central South and the South China regions by means of hammer, core samples drilled and ultrasonic wave methods, and 703 useful data were obtained, and modified Eq. (5) based on the obtained data, and suggested the formula for concrete bridges given by

In fact, Eq. (6) is revised by means of the bridge structure in situ measured data, which is under the dual roles of durability and fatigue and close to the actual bridge structure conditions. As the bridge adopted in this article lacks actual traffic statistical data and it also located in South China’s Pearl River Delta region, Eq. (6) is so adopted here to revise the resistance R in Eq. (1).

As for the changing law of concrete tensile strength, combined with Eq. (4) and Eq. (6), this paper suggests:

In the formula: μt0 and σt0 are the mean and standard deviation of concrete cube tensile strength under 28 days curing respectively; μt(t) and σt(t) is the time-varying equations of the mean and standard deviation respectively after the concrete cube services t years.

2.3 Structural load effects probability density function

K. M. Jo etc. (2005) assume that the bridge load effects probability density function also obey normal distribution. So, the load effects probability density function can be expressed as the following formula:

In the formula: fs(s) is the Gauss distribution function of the concrete member load effects; μs is the component load effects mean; σ2s is the component load effects variance.

According to the previous discussion, the resistance and load effects of the bridge are both obey normal distribution. Therefore, the reliability of the concrete bridge can be calculated according to Eq. (2).

Because the bridge member resistance has two probability density functions fRc(r) and fRt(r), therefore, according to Eq. (1), there are two reliability indexes βc and βt responding to the load probability density function fs(s) . In view of this, the calculation methodology in this paper is: if |μs-μc|>|μs-μt|, calculate reliability index βc according to Eq. (2); if not, then, calculate the reliability index βt , of which the meaning is shown in the calculation schematic diagram below:

Fig. 1 Reliability index calculation schematic diagram of the prestressed concrete bridges

Then, as the reliability index calculated by the above proposed methodology only reflects the local reliability state and the time-varying characters around the embedded sensors, so, we call this point time-varying reliability.

3. Illustration of the SHMS of the bridge used in this article and initial data processing

3.1 The long-term heath monitoring system installed on the bridge

The superstructure of the bridge main beam is a continuous box-beam system with a total of eight main piers and 7 main spans. The first span is 145.4 m long and the sixth span is 87 m long, and the 4 center spans are all 144 m long. The cross section of box girder is a single-box and single-chamber. The heights, thickness of base plate and thickness of web plate vary from 8 m to 2.8 m, 1 m to 0.32 m and 0.9 m to 0.45 m respectively in cross sections from the supporting base to the mid-span.

The cross sections with the measuring points of the health monitoring system in girder locate near piers, in mid-span and in 1/4 span, and there are total 20 sections. The embedded locations of strain variety sensor (The sensor is show in Fig. 2) in each section are illustrated in Fig. 3 with given numbers. With the given name of cross section and number, a sensor in the SHMS can be located in the girder uniquely, such as a sensor is named 3-4MID-1, which means it locates in the top plate center of the mid-span cross-section between pier 3# and pier 4#. The measuring time interval of each sensor is 1 hour. The parameters of JMZX-215 type strain gauge are shown in Table 2. So far, monitoring of the bridge is still continuing and data for the past few years has been acquired.

Table 2 Basic performance parameters of JMZX-215 type strain gauge

Fig. 2 JMZX-215 intelligent string-type digital strain gauge installed inside the bridge before casting

Fig. 3 Positions of the embedded sensors in bridge cross section

3.2 The initial monitoring data

At present, the bridge has been monitored more than 4 years. Here, the data collected from the sensors named 3G1H-1, 3-4MID-1, 4Z9H-1 and 3-4MID-2 are selected as examples, of which the selected monitoring time range is from March 2006 to April 2010. In fact, there are tens of thousands of data collected from the health monitoring system. The monitored data should be pre-processed firstly to delete some singular values which may be induced by strong thunders and other unexpected factors. The principle of deleting the singular values is: firstly, find out the difference between the values of each sampling point and its previous sampling point; then, if the value of the difference is greater than 200 micro-strains (engineering experience value (2007)), the signal value of this sampling point is regarded as singular value and will be removed. Fig. 4 shows the shape of the original data after the singular values are deleted.

Fig. 4 The data profile collected from the SHMS

3.3 Processing of the strain data

As the monitored strain can not be directly used for reliability calculation, it must be carried on some necessary processing to transform into stress, and then can be used to calculate the reliability index. The steps are as follows:
(1) Take the sensor initial setting value after the casted concrete is solidified. Because the sensors are embedded before the concrete casting, the concrete hydration heat will produce initial strain in sensors. So, this value should be subtracted from the monitored strain value of each sensor, of which the goal is to get setting values of the sensors after the concrete is solidified.

(2) Subtract the shrinkage and creep strain values from the sensor monitoring strain value. By use of the finite element technology, build simulation model of the bridge according to each construction stage until to the bridge closure (For example, use finite element calculation software MIDAS etc.), and modify the finite element model by the field test data. Then, based on the FEM model of the bridge, calculate and extract the shrinkage and creep values corresponding to the embedded sensor position. Then, subtract this value from the sensor measuring strain values.
(3) Subtract the thermal expansion strain value from the sensor monitoring strain value. Due to the variation of environmental temperature, the monitored strain values include thermal strain. It is best to choose temperature digital strain senor, as mentioned above, which can simultaneously monitor temperature. So, it is easy to remove the thermal strain from the monitored strain.

After the monitored data is processed according to the above method, the stress data can be conversed from the strain data by the following formula:

In the formula: E is the concrete elastic modulus.

In case of a limited number of measurements of SHMS, Bayesian methodology (2014) can be used to update the structural resistance and load effects. Nevertheless, continuous monitoring over a long-term period can increase the reliability of the assessment and prediction of structural performance. In general, the damage development speed of bridges is very slow. Considering this reason and the data sample size etc., this paper determines each statistical time section of the monitoring data is 6 months, and so the data sample size of each statistical time section will reach 4000, which is enough for load effects statistics, and the derived load effects include the influence of environmental temperature (include extreme weather), the structure shape, the traffic loads (include heavy loads) and resistance changing with time during the bridge operation etc.

According to the climate characteristics of the bridge which locates in Chinese Pearl River Delta area, then, the time statistics section has two kinds: one is called summer section, from May to October; another is called winter section, from November to April of the next year. In this paper, the statistical time starting point is 2006 May and the end point is 2010 April, and each statistical time section is named in a series A, B, C, D, E, F, G, H.

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