as P-Ⅲ,
and products of several random variables as log normal, etc.
Zeng Yu-hong
Hu Min-liang
Hydraulic and Electric Institute of Wuhan University, Wuhan 430072, China
Abstract:
The reliability of railway bridge is calculated in this paper at first. Railway
bridge is considerable important structure on railway, and can’t be repaired
soon once broken. Three items are included in calculating: The first is to
examine the reliability of bridges’ structures with the extreme state
equation, and accurate results can be got when the achievements in civil
engineering are consulted. The second is to calculate the reliability of the
bridges’ discharging ability, which has been discussed in Ref [6], so no more
will be talked about in this paper. The last is to calculate the reliability of
a bridge to resist pier scour. The extreme state equation has considered the
depths of both normal scour and partial scour here. To be simple, the statistic
distribution patterns of variables related to geometrical size are regarded as
normal, coefficients as triangular,
as P-Ⅲ,
and products of several random variables as log normal, etc.
On the basis of analyzing the reliability of railway bridges, calculation methods about systems are discussed. A section, a rail bureau, or the whole Jingguang railway line can be all regarded as a system. Series and parallel systems are the two kinds of basic patterns, which can form complex systems. After the reliability of the element (bridge) was known, methods to calculate reliability of systems have been suggested in this paper.
Finally,
risk damage, which has connected with risks and damages, has been put forward,
and it can be written as
initially. Then human power and
resources can be concentrated to benefit those regions with a maximum
.
Keywords: reliability, bridge, flood risk, railway system
In China, the appearance of floods on plane railways is frequent. To keep the railway unblocked, threat of flood must be eliminated. Only by estimating the disasters scientifically, can human-power and material resources be used on those serious regions.
First, reliability of bridges, culverts and embankments must be assessed before analyzing flood disasters, which has been elaborated in Ref [6] initially. On the base of bridge analysis, calculating methods of these simple railway systems and regionalization of flood disasters will be discussed.
Railway bridges are the most important structure on railways, and they are commonly used when railways run across rivers and mountains. On the other hand, bridges are erceted over the rivers, thus affected by those mutative natural factors such as floods and river channels, faced by serious threats, and can’t be repaired soon once broken.
For those important bridges, “the hydrologic examination” must be done. For example, examination of Ming-Bridge on the Jingguang line includes three items:
(1) Examining the bridge’s structure.
(2) Determining its design and practical discharging capacity.
(3) Determining its ability to resist the pier scour.
Some “safety factors” are recommended in the examination, and its result is a simple “yes” or “no”. The content of hydrologic examine need not to be changed for it’s the experiential result. After statistic analysis, Ref [2] provides 10 reasons for bridge floods and the most important ones are small apertures, shallow foundations and scanty bridge heights, which are in accordance with hydrologic examination.
Reliability theory believes that “safety factors” are not absolutely safe, and its result is not the simple “yes” or “no”. The bridge’s reliability “the probability of a bridge performing its set functions under the set condition and period”, which has considered random variables such as discharging, water level, velocity, is more tally with the bridge’s practical conditions, thus more scientifically.
The equation of bridge structures at extreme states can be concluded as:
(1)
In which
is load on the structure, including
last-load, live load, wind load, and snow load. So it can be expressed as
(2)
where
are called random variables, in the
same way
is the resistance, which includes
intensity of materials and sizes:
(3)
in which
are all random variables, so
is the function of
. If
, the structure is safe,
, lose efficiency,
, the structure is at a extreme state. So the equation is called a extreme state
equation.
Calculating the reliability of the structure is to calculate the probability (or reliability) by
(4)
Distribution of variables can consult achievements in civil engineering.
Calculating method about reliability of a bridge’s discharging has been discussed in Ref [6], so no more will be talked about here.
The extreme state equation about a bridge’s ability to resist the pier scour can be concluded as:
(5)
In which
is pier foundation depth,
is the depth of normal scour,
is the depth of partial scour, and
is the minimal safe depth.
can be got by:
(6)
In which
is the maximum flood flow, and its
distribution pattern is P-Ⅲ.
is coefficient about stability of
river and flow.
is the sand diameter, generally
.
are connected with geology.
For cohesionless soil,
(7)
For
clayey soil,
(8)
In which
is the incipient motion velocity of
sediment on the riverbed,
(9)
is the incipient velocity of the
sediment around the pier,
(10)
is the pier width,
is the pier size,
is the sand diameter, generally
,
is the exponent, and
(11)
is the coefficient of sands,
)
(12)
are all coefficients connected with
pier shapes, support sizes, and water depths.
is the angle of flow to the pier.
From above, factors connected with
scour depths, such as topography, geography, physical properties and natures of
the river, are all complex random variables, so the scour depth is a more
complex random variable, yet this condition can be simplified according to the
theory of reliability. First, distribution patterns of variables related to
geometrical size, such as width
, height
, pier-size
and sand diameter
can all be considered as normal.
Second, some complex variables related to geography and river natures can be
regarded as normal too, since their distributions are difficult to define, and
not important enough.
Of course, distribution patterns of
some important parameters can’t be simplified as normal, for example,
distribution of
is P-Ⅲ,
and coefficients are regarded as triangular:
(13)
Third,
and
are functions of several random
variables, their distribution patterns can be regarded as log normal according
to theory of reliability. Following methods can get their statistical
parameters: Imaging
is the function of independent
random variables
, written as:
(14)
so its mean
and standard deviation
can be calculated as:
(15)
(16)
(17)
After knowing distribution patterns of variables, the reliability of a bridge to resist pier scour can be calculated.
The reliability of element-bridge analyzed, reliability of systems and reliability-risk damage relationships should be studied. At last, regionalization of flood disaster can be done.
System is a relative concept, and its define is connected with the range and purpose of analysis. For example, a bridge is a system if we regard the beams and piers as its components. Another example, a stretch including several bridges is a system too, and its components are bridges, embankments and sections. Expanding the scale, whole rail authority or even the Jingguang line can all be regarded as a system, and exemplary sections as components.
The purpose of reliability analysis is to identify the system’s reliability after formal analysis of its components. Sometimes the opposite should be done, which is called the distribution of reliability. This paper will mainly discuss the former.
Series and parallel systems are the two basic patterns. A series system is like a bridge consisting of a group of piers, and the failure of one pier will cause the system failure. Double-lines on railway is a parallel system, for it fails only when both lines have failed.
If
is the reliability of the
component, then the series system’s reliability is:
(18)
the reliability of the parallel system is:
(19)
If the system is a complex series redundant one, it can’t be defined as series or parallel simply, then network analysis must be used. The system will be divided into a group of series subsystems, and each subsystem is a parallel one. Exact analytical methods such as the minimal cut, and minimum path, can be consulted in references.
Reliability is the probability of the
components accomplishing their set functions under prescriptive conditions and
in prescriptive periods. Risk is just the opposite. While considering flood
disasters, it’s not enough to know the risks only, for the critical degree is
connected with flood damages. So risk-damage
, which means potential damages, can be described as:
(20)
in which
is the risk,
is the damage.
Using the analysis methods mentioned above, flood disaster of a chosen section, which is on the Jingguang line, Huabei-plain, can be regionalized. The section is 190.4 km long, wholly double lines, and its damage is mainly caused by floods. One especially big bridge, 29 big or medium bridges, some culverts, embankments and lines form a parallel system. The purpose is to know its reliability and distribution of flood disasters. Calculating done by computer shows that this section has a low reliability.
A single-line on the railway can be regarded as a series system and double-line as a parallel one. Obviously, the double-line can increase the system’s reliability, yet the single-line will do the opposite. Up and down line on the railway is more complex, but they can be calculated as series or parallel systems too.
Loss
will only conclude direct financial
loss, which can be got from statistics, while indirect loss, which can’t be
described with the number, will be ignored in initial analysis if its influence
is not very obvious.
At last, damage
can be got by
. Then human power and resources can be concentrated to benefit those regions
with a maximum
.
(1) Reliability of a bridge’s discharging and resisting pier scour can be calculated
(2) Using the calculating method of reliability and risk-damage suggested above, flood disasters of plain railway can be regionalized
(3) Statistical analysis is very important to improve calculating precision
References
[1] Primary severity-evaluation of flood disasters to railways, China Institute of Railroad Research, 1995.1.
[2] Investigation of flood damages and preventive measures about railroad bridges on medium rivers, China Institute of Railroad Research, Zhengzhou Rail Bureau, 1995.5.
[3] Hydrologic examination of Ming-bridge on Jingguang line, China Institute of Railroad Research, 1994.6.
[4] Zhao Guofan, et al: Reliability of engineering structures, Hydraulic and Electric Publishing House, Beijing, 1984.12.
[5] Jiang Ronghan: Reliability-analysis of engineering system, Hunan University Publishing House, Changsha, 1987.12.
[6] Hu Minliang: Flood risk analysis of railway bridge, Proceedings of the 10th symposium on hydrodynamics, Beijing, Ocean Publishing House, 1996.9.