DOI: https://doi.org/10.62204/2336-498X-2025-1-16
TECHNICAL SCIENCES
ANALYSIS OF BRIDGE RELIABILITY BASED ON NORMAL
AND ASYMMETRIC DISTRIBUTION LAWS
Kostiantyn Medvedev,
Candidate of Physical and Mathematical Sciences, Professor,
National Transport University, Kyiv, Ukraine,
kvmedvediev@gmail.com; ORCID: 0000-0002-0704-7093
Yurii Yevseichyk,
Candidate of Physical and Mathematical Sciences, Associate Professor,
National Transport University, Kyiv, Ukraine,
jura_ntu@ukr.net; ORCID: 0000-0002-3507-4734
Maryna Babych,
Senior Lecturer,
National Transport University, Kyiv, Ukraine,
marinababich.ntu@gmail.com; ORCID: 0009-0009-3012-0824
Annotation. In modern calculations of structural reliability, it is traditionally assumed that the distribution laws of random variables, such as material resistance and load effects, obey the normal law (Gauss’ law). This approach is convenient and well-known due to the symmetric nature of the distribution and its mathematical simplicity. However, the results of experimental studies show that in real conditions the distributions of such variables often have an asymmetric nature.
In modern calculations of structural reliability, traditionally, the asymmetry of the distribution of material resistance in most cases has a negligible effect and can be ignored. At the same time, ignoring the asymmetry in load distributions can lead to significant errors in assessing the reliability of structures. In the paper, the lognormal law was used to model asymmetric distributions, which is universal and widely used to describe physical quantities such as material strength, load, river flow, etc.
The results of the study are presented in the form of tables and graphs, reflecting the dependence of the reliability of structures on the safety factor. The results for symmetric (PNN) and asymmetric (PLL, PNL) distribution laws are compared. All calculations were performed using the Mathcad software package, which provides high accuracy and convenience in performing complex mathematical calculations.
The paper also discusses the selection of distribution parameters, which should be based on statistical data for a specific design, materials, and operating mode. It is concluded that the need to consider asymmetry depends on the value of the asymmetry coefficient and the safety factor.
The purpose of the study is to determine the conditions under which taking into consideration the asymmetry of the distribution of random variables is necessary to ensure the accuracy of calculations of the reliability of structures, as well as to justify the cases when the use of symmetric models is acceptable.
Keywords: asymmetric distribution laws, structural reliability, normal distribution law, lognormal distribution, safety factor.
Introduction. In engineering structural reliability calculations, it is traditionally assumed that random variables, such as material resistance and applied loads, obey a normal distribution law (Gaussian law). Although this approach is convenient and widespread, it has a fundamental drawback: the normal distribution is symmetric, while real experimental data indicate a significant asymmetry of such parameters.
Studies [1–3] show that the distribution of the mass of vehicles with extreme loads has a multimodal character with several peaks corresponding to different categories of vehicles. In [4], examples of problems are given where it is necessary to consider joint distributions of loads and bearing capacity reserves, which required the construction of appropriate compositions or differences of distributions.
In this work, the lognormal law was chosen to model the asymmetry of random variable distributions. The calculations performed demonstrate that neglecting asymmetry can significantly distort the assessment of structural reliability, underestimating or overestimating real values. The convolution formulas used for the analysis allow us to obtain analytical expressions for joint distributions. The examples show that distributions for temporary loads can be asymmetric and significantly differ from the normal law, which is critical for ensuring the accuracy of the calculations.
Scope. The results of this study find practical application in many areas related to the design, operation and modernization of bridge structures, in particular:
– when designing bridges and other transport structures.
Refined calculation methods allow taking into consideration the asymmetric distribution of loads which is characteristic of modern transport flow. This provides a more realistic assessment of the reliability of span structures and increases the durability of structures. The use of asymmetric distribution models contributes to the optimization of structural solutions, which allows reducing material consumption without reducing the level of safety.
– operation and monitoring of the technical condition of structures.
The results of the study can be used to create monitoring systems that analyze the variability of loads in real time, identifying potential threats to their development into emergency situations.
Forecasting the residual resource of structures based on data on uneven wear of elements allows timely decisions to be made on repair or strengthening.
– analysis of transport flows and infrastructure planning.
Considering the multimodal nature of the distribution of vehicle mass contributes to the design of bridges and roads that meet modern requirements for intensity and weight restrictions.
The results of the study can be used in modeling extreme transport scenarios, such as the movement of military equipment or the transportation of heavy loads.
– modernization of existing structures
When reconstructing and strengthening structures, it is important to consider the asymmetric nature of loads to avoid irrational overconsumption of materials and ensure the reliability of the structure during the remaining operational life.
The developed methods allow for effective assessment of the impact of changes in transport flows on structures in operation and to adjust their design parameters.
– development of regulatory documents.
The results of the study can be used to update construction codes and standards for methods for assessing the reliability of structures considering the asymmetry of load distributions.
Proposals for the introduction of new coefficients and calculation rules can be adapted for international standards and national regulations.
– risk management and assessment of catastrophic scenarios.
Models built taking into account asymmetry allow for the assessment of the impact of extreme loads, such as emergency overloads, natural disasters or man-made disasters.
They allow structures to be better prepared for such situations and reduce the likelihood of destruction.
– educational programs and scientific research.
The results obtained can be integrated into training courses for future engineers, contributing to increasing their competence in calculating the reliability of complex structures.
Further scientific developments in the field of load distribution and strength can be based on the approaches proposed in the work.
Objective and methods. The purpose of this study is to highlight the influence of asymmetry in distribution laws on the assessment of structural reliability, as well as to determine the conditions and coefficients under which the use of symmetric distributions can be considered acceptable. Particular attention is paid to studying the influence of asymmetry in load distributions and material resistance on the accuracy of estimating the bearing capacity reserve and determining the conditions under which simplified symmetric models can be used.
The study aims to conduct:
– analysis of the nature of load distribution and resistance based on experimental data;
– assessment of cases of possible ignoring of asymmetry in structural reliability calculations;
– development of recommendations for engineering practice regarding the consideration of asymmetry of distributions. the work used a number of theoretical, experimental and numerical methods;
– study of mathematical models of distributions of random variables, in particular normal and lognormal;
– analysis of convolution formulas for modeling joint distributions of load effects and resistance;
– development of analytical expressions for estimating the probability of limit states of structures.
Results and explanation.
According to the classical reliability theory [6,7], the reliability of a structure (or its element) is the probability that the value of the generalized safety reserve will have a positive value, i.e.:
P = Prob (S˃0) (1)
where P is structural reliability; S is safety reserve.
The safety reserve is defined as the difference of two random values: the generalized resistance of the element R and the generalized load effect E:
S = R-E (2)
According to probability theory, when the laws of distribution of R and E values are known, we can determine the distribution law of safety reserve, which we denote by pₛ. Taking into consideration (1), the reliability of the structure P and the value V=1-P, which is called the probability of failure, are determined as:
(3)
The geometric meaning of the quantities P and V is that their values are equal to the area under the distribution curve of the strength reserve pS(s). In the case when s > 0, this area corresponds to the reliability of the structure P. When s < 0, the area corresponds to the probability of failure V (Figure 1)
In most cases, the values R and E can be considered independent random variables. Then, under random distribution laws of R and E, the mathematical expectation of ms and the mean square deviation of the safety reserve σs are determined by the formulas:
(4)
where mᵣ , mₑ are mathematical expectations, σᵣ, σₑ are standards for the distribution of generalized resistance and load effect, respectively.
We denote by β the quantity called a reliability index:
(5)
The value of β was first proposed in [6]. It plays an extremely important role in the theory of reliability. As it can be seen from Figure 1, the reliability index determines the number of standards that are placed in the interval from s = 0 to s = mₛ, and this is true for any distribution laws. Taking into account (4), the reliability index β can be written in the form:
The formula for determining the reliability index (8) has an advantage over formula (6), because the coefficients of variation can be estimated even with insufficient statistical information regarding the structural resistance and the load effect. In addition, when the load changes or the cross-sectional dimensions of structural elements change, Cᵣ and Cₑ variations remain unchanged.
In most reliability calculations, distribution laws for the random variables R and E are assumed to be in the form of a normal distribution:
This law is the most studied, so it is widely used in probability theory. The main advantage of this distribution law is stability: the sum or difference of a normal distribution of random variables is also a value under a normal distribution law. Thus, if the random variables R and E have a normal distribution (9), then the safety reserve S will also have a distribution according to the same law, and its parameters mₛ and σₛwill be determined according to (4):
Considering (10), the reliability of Pₙ structure under normal distribution laws of R and E can be written in the form:
Figure 1 shows a graphical representation of dependence (10). As it can be seen, the normal law is symmetric with respect to the mathematical expectation m, which in this case coincides with the mode M (the value with the greatest probability). In the general case, the distribution law of a random variable can be asymmetric (Figure 2), for which m ≠ M.
The application of formula (11) for calculating reliability is quite simple and convenient, which led to the widespread use of the normal distribution law for resistance and load effect in the form of (9). But this law has two significant shortcomings. The first is that the argument under a normal distribution varies from -∞ to +∞, although in their physical essence both resistance and load effect are purely positive quantities. The second disadvantage is that the normal law is symmetric.
It is known that the asymmetry of the distribution is characterized by the coefficient of skewness A, which in the case of a discrete series of values is equal to:
The distribution of a random variable can have both positive (A>0) and negative (A<0) skewness (Figure 2).
Under positive skewness (Figure 2 a), the mathematical expectation m1 will be greater than the corresponding mode М1. This means that negative deviations from the average value (for example, reduced load effects) will be more often repeated than positive ones. For the distribution with the coefficient A<0 (Figure 2 b), the situation will be the opposite. That is, load effects that are greater than average will be more often repeated. Asymmetry can be characterized both by the value of A itself and by its relation to the coefficient of a variable A/C.
Experimental studies showed that both resistance and load effects are random variables, which in most cases are distributed according to the laws of positive skewness.
In the work, the lognormal distribution was chosen to approximate the asymmetric law.
Further in the text, x will denote the value of the random variable R (resistance), y – the value of the random variable E (load effect), and s – the random value of safety reserve S.
The lognormal distribution has the form:
The asymmetry coefficient for it is equal to A =3C + C3.
Distributions (13) are defined only for positive values of the argument, therefore they do not contradict the physical essence of resistance and load effects.
For the convenience of calculations and analysis of the obtained results, we introduce dimensionless quantities:
In dimensionless quantities (14), the lognormal distribution laws for loads and generalized resistance will take the form (the sign of the dimensionless quantity is omitted here and further):
For the case when the strength is distributed according to the normal law, the density is denoted by Pᵣₙ
According to probability theory, if the random variables R and E are distributed according to the laws pᵣ(x) and pₑ(у), the law for the distribution of the random variable S=R-E is determined by the equation:
The area of integration D is the intersection of three areas:
– changes in the x argument (resistance)
– changes to the y argument (load effect)
– regions (x – y) < s.
The areas of change in x and y are determined by the accepted distribution laws. In the case of normal laws, the argument will vary from -∞ to +∞, and in the case of asymmetric distribution laws (15) and (16) from 0 to + ∞.
Using the rule of differentiation of integral functions and taking into consideration the domain D for a combination of lognormal distributions, we obtain the general density of the distribution, which we denote by pₛLL:
Similarly, we obtain the distribution density for a combination of normal and lognormal laws:
Table 1 shows the results of calculations of safety and reliability indexes that depend on the safety factor ξ for the selected coefficients of variation of the generalized resistance and load effect CR = 0,1, CЕ=0,35(the corresponding asymmetry coefficients АR = 0.3 and AE=0.904).
In table 1 Δ denotes the percentage of the relative deviation of the reliability PNL and PLL (22) from the value of PNN, which is calculated according to formula (11).
As it can be seen from the analysis of the results presented in Table 1, for reliability indexes β > 2,5 the deviations of the reliability PNL and PLL from PNN are sufficiently small. This means that the asymmetry of the load effect distribution can be neglected with sufficient accuracy in this case. When β decreases from 2,5 to 1,5 percent, the deviation of the reliability PNL and PLL from PNN increases and can reach Δ ≈ 1,5%.
Since the structural reliability, according to the current DSTU [8], can vary in a small range (approximately from 1 to 0,95), the error, which is 1,5%, is quite significant.
Figure 3 presents graphs of PNN , PLL and PNL dependencies on the value of the reliability index β.
Horizontal lines show reliability levels, which according to [8] correspond to the numbers of technical conditions: from state 1 (in operation) to state 5 (out of operation). As can be seen from the graphs above, when β < 2,5 and the asymmetry of the distribution law for the load is not taken into consideration, an number of the technical state in which the structure actually is can be overestimated . For example, if β=2,2 under the normal distribution and the reliability PNN = 0,9862, this corresponds to the third technical state; at the same time, under the lognormal distribution, the reliability will be PNL = 0,9724, which corresponds to the fourth technical state of the structure. Such a situation with the definition of technical state, the number of which determines the level of bridge maintenance, is quite dangerous. Especially when it comes to structures that are in 3rd or 4th technical state.
From the above table and graphs it can also be seen that the asymmetry of the material resistance at the index АR = 0,3 practically does not affect the value of the calculated reliability.
The calculations performed at different values of the asymmetry coefficient showed that at А<0,6 (both for resistance and for load effects) the influence of asymmetry on the value of reliability can be neglected.
Conclusions. The paper shows that the asymmetry of the distribution of random variables that describe the load effect and resistance of structures significantly affects the accuracy of reliability calculations. In particular, when the asymmetry coefficient A > 0.6, it is necessary to take into account the asymmetry, since ignoring this factor can lead to significant errors in determining the operational state of the structure.
For cases where the asymmetry coefficient A < 0,6, the influence of the asymmetry on the calculation results is insignificant (the discrepancy does not exceed 0,5%). In such conditions, the use of symmetric models, in particular the normal distribution law, is appropriate and justified in order to simplify calculations.
It is shown that at high values of the reliability index β > 2,5, reliability calculations can be performed using a simple formula (11), which is based on the assumption of a symmetric distribution of quantities. This simplification is justified, since the errors remain within acceptable accuracy.
In cases where A > 0,6 and β < 2,5 , taking into account asymmetry becomes critically important. Under such conditions, it is necessary to use models that take into account the asymmetry of the distribution of random quantities of resistance and load effects. Ignoring this factor can lead to a significant underestimation or overestimation of the reliability of the structure.
The approach proposed in the work allows for accurate modeling of resistance and load effect distributions based on the lognormal law. The use of convolution formulas ensures the construction of joint distributions that adequately describe real operating states.
The results of the study can be used to improve regulatory methods for assessing reliability, develop more accurate recommendations for determining the bearing capacity margin and analyzing the durability of structures.
An important practical conclusion is the possibility of using simplified models in certain conditions when A < 0,6 or β > 2,5. This allows for optimization of calculations without significant loss of accuracy, which is especially important for engineering practice.
References:
- Yu, Yang, “An Enhanced Bridge Weigh-in-motion Methodology and A Bayesian Framework for Predicting Extreme Traffic Load Effects of Bridges” (2017). LSU Doctoral Dissertations. 4140. https://repository.lsu.edu/gradschool_dissertations/4140
- Xuejing Wang, Xin Ruan, Joan R. Casas , Mingyang Zhang Probabilistic model of traffic scenarios for extreme load effects, Structural Safety, 106 (2024) 102382, ISSN 0167-4730, https://doi.org/10.1016/j.strusafe.2023.102382
- Thomas Braml, Christian Kainz Practical concepts for the use of probabilistic methods in the structural analysis and reassessment of existing bridges – presentation of latest research and implementation, Acta Polytechnica CTU Proceedings 36:47–58, 2022, https://doi.org/10.14311/APP.2022.36.0047.
- Pichugin S.F. Calculation of the reliability of building structures / S.F. Pichugin – Poltava: ASMI LLC, 2016 – 520 p.
- DSTU-N B EN 1990:2002 Eurocode 0. Fundamentals of structural design (EN 1990:2002, IDT) – K.: Minregion, 2013. – 8 p.
- Rzhanytsyn A.R. Theory of calculation of building structures for reliability / A.R. Rzhanytsyn. – Moscow: Stroyizdat, 1978 – 239 p.
- Gnedenko B.V. Mathematical methods in the theory of reliability / B.V. Gnedenko, Y.K. Belyaev, A.D. Solovyev. – Moscow: Nauka, 1965. – 524 p.
- DSTU 9181:2022 “Guidelines for assessment and forecasting of the technical state of road bridges”. Ministry of Regional Development of Ukraine, K.: 2022.