1. Introduction
In accordance with guidelines [
1,
2,
3,
4], a check of structural reliability is based on an idealized concept of limit states and their verification by means of a semi-probabilistic method of partial safety factors. A building structure is characterized by two excluding states, namely a fitness state, in which the object meets the set requirements, and an unfitness state, when it does not satisfy those requirements. The conditions that separate the fitness state from the unfitness state are termed limit states. The method mentioned above involves an analysis of the bearing capacity mobilization of members or sections (ULS—ultimate limit state) and of serviceability criteria (SLS—serviceability limit state). Uncertainties in materials parameters and loads are taken into account solely in the form of related, representative values that are called partial safety factors. Those factors are understood to be related quantiles of probability distributions that describe random variables. The role of partial safety factors is to ensure the required level of reliability of the structure. The fully probabilistic methods are extensions of semi-probabilistic method of limit states. These methods are divided on two groups: approximated and simulated methods. The first order reliability method (FORM) [
5,
6,
7] and the second order reliability method (SORM) [
8,
9,
10,
11] are approximated approaches. Importance sampling [
12,
13,
14], artificial intelligence [
15,
16], and the Monte Carlo method [
17,
18,
19] are simulation approaches. A probabilistic approach makes it possible to provide a description that is more accurate and closer to the actual reality with respect to structure materials, geometric parameters, and loads. This is done by giving the type of probability distribution and distribution parameters.
In this paper, the first order reliability method was used to assess the limit states of the structure. The measure of structural reliability was the Hasofer-Lind reliability index [
5]. The previously defined Cornell index [
6] had a significant disadvantage—no invariance and it resulted from the position of the point for linearization as average values. This made the reliability measure difficult to use when comparing the safety of different structures. In 1978, the design point approximation algorithm was proposed by Rackwitz and Fiessler [
7]. In [
20], Hohenbichler and Rackwitz came up with an idea to apply the Rosenblatt transformation [
21] to transform dependent random variables into a standard space. Der Kiureghian and Liu [
22] employed the Nataf transformation to that end. The use of first- or second-order functions to approximate the limit state function at the design point leads to the first order reliability method (FORM) or the second order reliability method (SORM).
Early applications of reliability analysis methods assumed that the limit function was an explicit function of random variables. The presented dependence can only be implemented on simple examples. Unfortunately, in practical implementations, this dependence is implicit, which requires the use of a numerical procedure, for example, the finite element method (FEM). The development of numerical methods contributed to the use of very desirable tools in the theory of structure reliability, such as the methods of implicit dependence of the boundary state functions on basic random variables. An example is the perturbative method proposed in the works of Hisada and Nakagiri [
23], Liu, Belytschko, and Mani [
24], and Shinozuki [
25] for tasks of the linear theory of elasticity. This method consists in expanding the stiffness matrix into a power series with respect to the random fluctuations of its parameters. However, the perturbative method is not intended to calculate the failure probability, but only to obtain statistical response moments. Calculating the probabilities would require introducing probability distributions of random variables into the calculations and estimating the higher moments of the statistical responses. Similar to the perturbative method, also using the Neumann expansion (the inverse matrix to the stiffness matrix is expanded in series), only information about the statistical moments of the random variable (being the response of the system) is obtained. Li and Der Kiureghian [
26], and also Matthies [
27] proposed a method that allows to estimate the probability of failure in a situation where the design is calculated using the finite element method.
In [
28], Pelisetti and Schueller provided an overview of various software tools that enable rational treatment of the effects of uncertainty. The article covered issues that include the underlying algorithms, and also the interaction with deterministic FE codes. In addition, other topics were discussed, including the availability of graphical user interfaces, distributed processing, verification functions, and the possibility of probabilistic modeling. The work in [
29] describes the problems that can be solved with two additional tools introduced by Ansys Inc.: ANSYS Probabilistic Design System and the ANSYS DesignXplorer. In both modules, an extremely efficient method can be used in which an accurate high-order response surface, using single finite element analysis, variation technology, is provided. Der Kiuregian et al. [
30] discussed the Calrem/Ferum and OpenSees programs for the analysis of structural reliability, while the work in [
31] presents an overview of the current state of COSSANe software. In [
32], Thacker et al. discussed NESSUS probabilistic analysis software, which allows the user to combine advanced probabilistic algorithms with analytical equations, commercial finite element analysis programs, and their “own” standalone deterministic analysis codes to calculate probabilistic responses or system reliability. Gollwitzer et al. [
33] described examples of analyses using a combination of the Strurel probabilistic analysis system, which includes Comrel, Sysrel, Costrel, and Statrel, and the PERMAS general purpose finite element program. The authors of [
34,
35] described the general-purpose probabilistic software PHIMECA and Proban. A summary of the possibilities of the program for probabilistic analysis and designing ProFES is presented in [
36]. This program allows to perform a wide range of probabilistic function assessments, including finite element analysis in a graphical environment. UNIPASS is a probabilistic evaluation software that provides the ability to model uncertainty, define probability distribution variables, and probabilistic models and responses [
37]. The use of reliability analysis and of the finite element method has been described by many researchers (Lee and Ang [
38], Engelstad and Reddy [
39], Mahadevan and Mehty [
40], Zabojszcza and Radoń [
41,
42], Mochocki et al. [
43], Kubicka et al. [
44]).
When creating a program that enables reliability analysis through the use of the finite element method, the best situation is to have access to the FEM source code and a reliability program. In general, however, this is not possible; therefore, at the expense of efficiency loss, existing MES packages are combined with a reliability program, using various types of interfaces. The paper presents a combination of the Numpress reliability program with the MES3D proprietary module. There is resistance among engineers to use numerical probabilistic methods, the complexity of which is hidden inside computer programs. Additional effort required from the user of the program is the necessity to characterize data with two parameters, i.e., the expected value and the standard deviation, instead of the one parameter used in deterministic methods. For this reason, it is necessary to provide engineers with algorithms that will enable the analysis of a structure, taking into account random factors. According to the authors, the FORM method may fill this gap in the future.
3. Results
3.1. Deterministic Static and Strength Analysis
The geometry of a steel lattice dome, consisting of 120 nodes and 220 elements, is presented in
Figure 4 and in
Table 1. The bars of the structure are modeled as frame elements. It was assumed that the structure elements are made of RO159x11 steel pipes with a yield point of fy = 235 MPa and Young’s modulus E = 210 GPa. The structure of the dome is based on 20 reinforced concrete columns, 5 m high, at the ends of which there is a reinforced concrete wreath that closes and stiffens the structure, and simulating the work of the grate.
The considered dome was loaded with a permanent load (self-weight of the bar structure and covering), a snow load determined for zone 3 according to [
1], and a wind load according to [
2] (
Figure 5). The combination of loads applied to the structure was consistent with [
3]. The deformation of the structure is shown in
Figure 5c.
In the MES3D program, self-weight can be added automatically as a load evenly distributed on the bar or planar elements. In the calculations, it is also possible to take into account the additional weight of the covering elements, fittings, etc. In a similar way, the snow load is taken into account—in this case, it was collected from adjacent areas and added to the load on latitudinal elements.
In the case of circular domes, for wind loads, the PN-EN 1991-1-4 standard recommends assuming surface loads acting in the direction normal to the surface of the sphere. These values are constant in each arc resulting from the intersection of the canopy with the surface perpendicular to the wind direction (
Figure 6 [
2]).
However, even in the case of simple circular domes, the above definition causes a lot of problems in modeling; therefore, in practical implementations, various simplifying procedures are used, e.g., the surface is divided into zones, assuming constant load values in them.
For the dome under consideration, based on the geometry and geographic location, the values of the pressure coefficients at points A, B and C were determined. Values in the remaining points in the direction of the wind axis were determined using parabolic approximation. It was assumed that the load was applied to flat surface elements formed by the dome bars (
Figure 4) and inside them it had a constant value defined for the element’s center of gravity. Triangular shell elements were used to transfer the load from the element surface to the nodes. For four-node areas, the load was divided into two equal parts, and then transferred to mesh nodes through triangular elements with different meshing (
Figure 7). This modeling method does not introduce additional errors related to the use of triangular elements.
A static-strength analysis, on the basis of which the structure was dimensioned, was made according to [
4] using MES3D software with spatial frameworks. In order to improve the accuracy of calculations, all elements were further divided into 10 parts.
Internal forces and strength utilization of individual bars of the structure were specified (
Figure 8,
Figure 9 and
Figure 10). Based on the static-strength analysis, it was observed that the most stressed bar was bar number 212. The maximum vertical displacement concerns node 58. The values of the internal forces, the capacity for the most stressed element of the structure (bar no. 212 in the meridian line), and the maximum horizontal and vertical displacement for node 58 are collated in
Table 2. In the dimensioning of the structure, the case that turned out to be decisive was the one in which the permanent load, governing snow load and accompanying wind load were combined. As regards the selection of cross-sections, it was decided by the stability condition of the bent and compressed element.
The limit values in
Table 2 have been determined in accordance with the standard PN-EN 1993-1-1. Eurocode 3: Design of steel structures. Part 1-1: General rules and rules for buildings.
Figure 8 shows the distribution of axial forces with the distinction of the bar 212, for which the value is 445.444 kN.
The value of the My bending moment with respect to the y-y axis for bar no. 212 was: My = 9.55 kNm.
Figure 9 shows diagrams of the bending moment with respect to the y-y axis.
The value of the Mz bending moment with respect to the z-z axis for bar no. 212 was: Mz = 12.45 kNm.
Figure 10 shows diagrams of the bending moment with respect to the z-z axis.
3.2. Computation of the Hasofer-Lind Reliability Index
The next stage involved the analysis of the limit states of the single layer ribbed dome, in which a full probabilistic description was utilized. In the literature, many studies can be found in which the deterministic approach was adopted to describe the ultimate and serviceability limit states [
48,
49,
50,
51,
52,
53,
54]. According to the deterministic approach, loads, material, and geometry features of the structure are described with code-specified characteristic values and relevant safety factors. The probabilistic approach offers a description of the above-mentioned quantities that is more accurate and closer to reality. It is possible to achieve that by providing the type and parameters of distribution.
The most frequently applied distribution in the probability theory is the normal distribution. It can be adopted to describe, e.g., the randomness of the location of structure nodes or of a permanent load multiplier. In actual systems, distributions of some loads significantly diverge from the Gaussian distribution. As a result, the adoption of the assumption of their being normal may lead to gross errors in the assessment of structure reliability. Weather-related loads are highly non-Gaussian in character. The Gumbel distribution was used to describe snow and wind loads. The modulus of elasticity and cross-sectional area were described by means of a log-normal distribution. In the study, a comparative analysis of two cases was conducted. For Case 1, all adopted random variables had a normal distribution. Case 2 accounted for a more accurate description of the quantities of concern. The adopted random variables were independent. Their descriptions are shown in
Table 3 and
Table 4.
The normal distribution is characterized by the following density function:
Function (19) is defined by two parameters, namely by mean value:
and by variance:
The Gumbel distribution is characterized by the following density function:
where
,
are parameters of the Gumbel distribution. The mean value and standard deviation are equal to
and
.
The lognormal distribution is characterized by the following density function:
where
.
In this work, it necessary to transform the normal distribution into a Gumbel or log-normal distribution; to achieve that, Method of Moments was used [
55].
As regards the two-parameter distributions utilized in the paper, the method of probabilistic moments is based on the use of conformity conditions of the mean value and the variance. For the two-parameter distributions A and B, from the conformity conditions of the mean value and the variance, the following equations were developed:
Serviceability limit state SLS was considered as the condition of non-exceedance of the allowable vertical displacement:
where w—external variable describing the maximum displacement of the structure, D—dome diameter.
Ultimate limit state (ULS) was defined as condition of non-exceedance of the capacity value, resulting from the stability criterion of the bent and compressed element:
where N [kN]—axial force, My [kNm]—bending moment with respect to the y-y axis, Mz [kNm]—bending moment with respect to the z-z axis, N
b,Rd [kN]—buckling resistance of the compressed element, M
y,Rk [kNm]—bending resistance with respect to the y-y axis, M
z,Rk [kNm]—bending resistance with respect to the z-z axis, χ—buckling coefficient, k
zy, k
zz—interaction coefficient according to [
4].
The values of w, N, M
y and M
z are external variables that are computed by the MES3D software. For the applied functions of ULS and SLS, the Hasofer–Lind reliability index was computed using the FORM method. The FORM method was employed as a primary research tool. In order to validate the correctness of computation the Monte Carlo method was used (
Table 5 and
Table 6). The obtained values of the probability of failure are within the range 10
−3–10
−5. The necessary number of simulations was 10
9. Due to the large number of simulations needed, only SLS was verified.
3.3. Analysis of the Sensitivity of the Hasofer-Lind Index to Changes in Random Parameters
Additionally, based on the analysis of the elasticity index, which random variables most affect the failure probability in the ultimate limit state and serviceability limit state was estimated. When constructing the mathematical model of the task, the design engineer has to make a decision of which design parameters should be treated as random ones, and which as deterministic ones. With respect to ULS, the analysis of the elasticity index made it possible to estimate that random variables describing the permanent load, wind load, and snow load had the highest impact on the assessment of the structure safety (
Figure 11). When SLS was considered, wind load and Young’s modulus were of the greatest relevance (
Figure 12).
3.4. Discussion of the Results
In the study, two cases were examined that differed with respect to the type of probability distribution of the adopted random variables. In both cases, the possibility of stability failure was accounted for. Instability resulted from the exceedance of the allowable displacement (SLS) and the exceedance of the capacity condition of bent and compressed bar (ULS). Substantial differences in the value of ] reliability index β were observed, depending on the case considered and the limit state. The maximum value of the reliability index, namely 4.334, was obtained for SLS, when all the random variables of concern were accounted for and normal probability distribution was assumed. For Case 2, even when the limit state was the same, this value was reduced by 17.21%. In both cases, for ULS, the values of the reliability index were lower than those obtained for SLS. For Case 1 and Case 2, the index values amounted to 2.738 and 2.492, respectively. Based on these results, it can be observed that the identification of variables decides, to a great extent, the solution to the problem. For that reason, it is essential to examine the sensitivity of the reliability index to changes in the probabilistic characteristics of the random variables under consideration. That is done by means of specifying the elasticity index. Substantial differences in the determined values of the elasticity index were observed for the cases of concern. However, the same variables are decisive in both limit states for both cases. As regards ULS, the decisive effect of loads (random variables P1, P2, P3), especially of the dead load (variable P1), on the values of the reliability index can be seen. For SLS, however, the decisive variables are Young’s modulus—E and wind load (random variable P3). The results obtained with the Monte Carlo method agreed with the results obtained with the FORM method.
4. Conclusions
Two global parameters determine the safety of a structure: load and bearing capacity. Each of these parameters are random. The likelihood of failure is an objective, probabilistic measure of the safety of a structure. However, this measure is still not accepted by engineers. They prefer a security measure with a deterministic overtone, which is adopted in the semi-probabilistic method of limit states. It assesses the safety of a structure on the basis of the quantiles of the values of the characteristic loads and bearing capacity, as well as partial safety factors. The safety factors were separately calibrated for loads and bearing capacities. The random nature of load variability was taken into account by increasing them with an appropriate factor. The bearing capacity was assessed by its reduction. In the design of the limit state method, all possible design situations should be considered and it must be shown that none of the limit states was exceeded. This structural safety assessment is qualitative. The fully probabilistic methods are extensions of semi-probabilistic method of limit states. These methods allow the quantitative assessment of structure reliability.
Therefore, it is necessary to provide engineers with algorithms that enable the analysis of structures, taking into account random factors, which is not an easy task. Commonly available programs for static strength analysis using the finite element method are based on the deterministic limit state method. It is a good idea to try to combine existing design software with reliability analysis programs. The authors presented this path by combining MES3D with Numpress and showing other solutions of this type that are available.
In reliability analyses, it is crucial to adopt a computational model that represents real conditions as closely as possible. Incomplete statistical data or inappropriate assumptions may lead to substantial differences in the reliability index values. The study indicate that for the reliability analysis it is important to appropriately select not only random variables but also the type of their probability distribution.
It is highly advisable to conduct the analysis of the sensitivity of the assumed random variables. A important moment in the design process is to decide which parameter to treat as deterministic and which as random. The FORM method gives a quick answer to this question by analyzing the sensitivity of the reliability index.