Analytical and Numerical Reliability Analysis of Certain Pratt Steel Truss

Abstract

The main aim of this paper was to propose a new reliability index for steel structure assessment and to check it using the example of a popular Pratt truss girder. Structural analysis was completed in the finite element method system Autodesk ROBOT, and probabilistic analysis was implemented in the computer algebra software MAPLE. The stochastic finite element method (SFEM) was contrasted here with the Monte Carlo simulation and the girder span was selected as the input structural uncertainty source. Both methods were based on the same structural polynomial response functions determined for extreme deformation, for extreme stresses and also for the structural joint exhibiting the largest effort. These polynomials were statistically optimized during the additional least squares method experiments. The first four basic probabilistic characteristics of the structural responses, the first-order reliability method (FORM) index, and as the new proposition for this index were computed and presented. This new index formula follows the relative probabilistic entropy model proposed by Bhattacharyya. The computer analysis results presented here show a very strong coincidence of both probabilistic numerical techniques and confirms the applicability of the new reliability index for the input coefficient of variation not larger than 0.15. These studies should be continued for other engineering systems’ reliability and, particularly, for large-scale and multiscale computer simulations. The results presented in this paper may serve in different applied sciences, from biology through to econometrics, experimental physics and, of course, various branches of engineering.

1. Introduction

Stochastic computational mechanics is still a very extensively explored research area with many concurrent methods. However, many of them do not have widely accessible computational implementations, especially in the context of the finite element method [1,2]. One could recall here not only the classical Monte Carlo simulation [1,3] or the stochastic perturbation technique [1,4,5,6] in its various order implementations. Moreover, still very popular is the Bayesian approach [7], Karhunen–Loeve decomposition and polynomial chaos expansion [8], stochastic kriging models [9], different analytical and semi-analytical techniques [1,10] and fuzzy stochastic analysis [11]. It should be mentioned that various computational studies concerning numerical error for stochastic methods are available [1,12]. Stochastic analysis has been implemented together with acceleration [13] and reduction algorithms [14], and is recommended for the homogenization approach in micromechanics [1]. The aforementioned numerical approaches may be used even for a solution to the insufficient material data problem in solid mechanics [15]. Unfortunately, practicing engineers must use some additional software or must create new small programs to determine the reliability indices mandatory for modern engineering structures. A very specific role is in the distribution and usage of the computer algebra systems such as MAPLE, MATLAB or MATHEMATICA, where many stochastic methods can be and are implemented. Let us note that the finite element method (FEM) modeling in the presence of any uncertainty still remains a separate part of the engineer’s work. It is well known that a huge portion of mathematical effort in the development of analytical formulas for structural engineering analyses has been lost in the last few years thanks to a common and fast methodology of “computing everything”. However, many relatively simple structures may be quite efficiently designed using classical formulas, extended relatively easily with mathematical software towards reliability assessment.

The guidelines for structural safety are based upon the very simplified reliability index formulated by Cornell many years ago. It belongs to the first-order reliability method, so that even if the engineers follow Eurocode 0 statements [16], their analyses have rather limited importance. This index is based upon the assumption that structural resistance R and the effect of external loadings E on the given structure both have Gaussian distributions, and that the limit function has linear character. Furthermore, such a designing code does not offer any algorithm for how to efficiently compute the first two probabilistic moments for the functions R and E. On the other hand, an application of the second-order reliability method (SORM) for practicing engineers is usually impossible because it demands knowledge about a curvature of the limit state surface; this curvature cannot be directly determined using the FEM software. This gave the authors a motivation to look over mathematical works concerning probabilistic distance (divergence), which could be applied to measure structural safety using some relative entropy in between the distributions of R and E. It should be mentioned here that statistical divergence (or diversity) has been studied in different areas of the applied sciences, particularly in biology [17], econometrics [18], statistical physics [19] and even accidentally in civil engineering, where it served in some seismic analyses [20].

Therefore, this paper aims to present an application of analytical and numerical probabilistic analysis in a reliability assessment of exemplary steels and popular civil engineering structures. The Pratt steel truss girder has been selected to demonstrate such an approach due to its popularity in academic analyses during civil engineering academic courses and a huge number of the existing old steel bridges all around the world. A novelty in this work is in the proposition of the alternative reliability index determination and also in the comparison of the stochastic analytical and numerical approaches. The proposed approach is based upon (i) polynomial response [1,21,22] determination via the series of FEM experiments for varying design parameters and the least squares method (LSM) [23]; (ii) two various probabilistic numerical techniques, namely the Monte Carlo simulation as well as the iterative generalized stochastic perturbation technique; and (iii) FORM [24] and relative entropy-based reliability indices. The main goal of this work was achieved using civil engineering-oriented FEM system Autodesk ROBOT (deterministic FEM series) and computer algebra software MAPLE 2019.2 (LSM, probabilistic procedures and reliability indices determination). Future experiments with dynamic excitations of linear and nonlinear large-scale structures, as well as a time-dependent reliability index of structures subjected to corrosion and/or aging [25], was also considered.

2. Structural Analysis

The numerical experiments in this work were entirely focused on the simple Pratt truss (Figure 1) schematically presented in Figure 2, whose height was adopted as h = 1.50 m, whose number of segments equaled 12, and whose span l was treated as the input random variable, having in turn triangular, uniform and also Gaussian probability distributions. The random length was initially represented by the discrete set of values ranging from 15.00 up to 21.00 m every 0.60 m to recover polynomial response functions via the series of traditional FEM experiments. They connectextreme deformations and reduced stresses with input uncertainty sources (e.g., parameter l in this study). Each truss FEM model in this series was uploaded with the same constant load q redistributed throughout the upper chord nodal points as the concentrated forces. Its characteristic value was set as q_k = 15 kN/m, while the design load was proposed for an illustration as q_d = 20 kN/m—the adoption and combination of the safety indices had a marginal influence on this study so it was postponed here by arbitrary load values. All the cross-sections of structural components were assumed as hot-finished square hollow steel profiles made of structural steel S235, whose material parameters were assumed according to the design code Eurocode 3 [26]; structural parameters assigned to the individual elements have been presented in

Table 1. Structural parameters of steel structural elements in the given truss.

Parameter	Upper Chord	Diagonals	Lower Chord
Member length ly	0.90 × l	1.00 × l	0.90 × l
Member length lz	1.00 × l	1.00 × l	1.00 × l
Buckling length coefficient y	0.08 × l	0.90 × l	0.08 × l
Buckling length coefficient z	0.16 × l	1.00 × l	0.50 × l

below. The buckling lengths of the members making up the girder were assumed on the basis of annex BB.1 to EN-1993-1-1: 2005 [26]. In-plane buckling of the chords was assumed to be 1/12 of the length of the truss. Diagonals reduce the buckling length to 0.08 × l. In the case of out-of-plane buckling of the upper chord, the buckling length was reduced by using every second purlin to support the transverse roof bracing to 0.16 × l. The out-of-plane buckling coefficient for the lower chord was assumed with the assumption of vertical bracing in the center of the spar. The buckling length of the diagonals was reduced to 0.90 × l due to the degree of fastening of the bars in the chords, i.e., welded joints. Let us note that the proposed cross-sections of structural elements were determined based on initial FEM calculations, and designing procedures provided for the mean value of the truss span l. This structure was initially designed by verification of both ultimate and serviceability limit states (ULS and SLS) in all truss elements and also the bearing capacity of the truss connections. Finally, the following profiles were proposed: (i) upper chord—SHS 140 × 140 × 8, (ii) diagonals—SHS 90 × 90 × 8, and (iii) lower chord—SHS 120 × 120 × 8.

The numerical experiments were completed assuming geometrical nonlinearities and P-delta effect in the truss structural behavior, so that incremental Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm was engaged to determine the resulting deformations and stresses. The FEM discretization was completed with the use of 49 linear two-noded truss finite elements. The following additional parameters were used in the FEM analysis: (i) load increment number equaled 5, (ii) maximum iteration number for one increment equaled 40, (iii) increment length reduction number was set as 3, (iv) increment length reduction factor was equal to 0.5, (v) the maximum number of line searches equaled 0, (vi) control parameter for the line search method was as 0.5, (vii) the maximum number of the BFGS corrections was automatically set as equal to 10, while (viii) relative tolerance for the residual forces and displacements was predefined as 0.0001. Of course, all system matrices were updated after each subdivision.

All the finite element method experiments were carried out in the civil engineering system Autodesk ROBOT, where linear truss elements were selected, one finite element was equivalent to a single structural element and each node had two degrees of freedom (vertical and horizontal displacements). This structure was simply supported at both ends in the lower chord, while the connections were all designed as perfectly non-deformable (due to the welds). The results of numerical modeling in both deterministic and stochastic contexts were contrasted with the analytical approach briefly presented below.

It is known that a deflection of the periodic and regular truss having two parallel chords were determined using elementary knowledge following the rules of the strength of materials and, particularly, fourth-order ordinary differential equations relevant to the Euler–Bernoulli beam and its analogy to the truss structures with parallel upper and lower chords. One may demonstrate that deflection of such a truss can be determined using the following function u(x) including a distance $0\le x\le l$ of the given point from its support

$$u\left(x\right)=\frac{q{l}^4}{24EJ}\left[\frac{x}{l}-2{\left(\frac{x}{l}\right)}^3+{\left(\frac{x}{l}\right)}^4\right],$$

(1)

where q is a characteristic value of the load, l is the theoretical span of the truss, E is the Young modulus, J is the inertia moment of a system of the upper and lower chords. It is very important that this model postpones a rigidity contribution of both columns and diagonals during bending analysis of the truss. The center of gravity of the truss chords is calculated as (see Figure 3)

$$d=\frac{A_1 h}{A_1+A_2},$$

(2)

where A₁ is the total cross-sectional area of the upper chord, and A₂ is a lower chord cross-sectional area. The resulting inertia moment of both truss chords according to the Steiner theorem equals

$$I=I_1+A_1{\left(h-d\right)}^2+I_2+A_2 d^2,$$

(3)

where I₁ is the moment of inertia of an upper chord, while I₂—inertia moment of the lower chord.

Having analytical formula for the deflection function and the given probability density function of the truss span, one may derive the basic four probabilistic characteristics of the deformation line u(x) of such a truss following classical definitions available from the theory of probability [1]. It yields

$$E\left[u\left(x\right)\right]=0.0417\frac{qx}{EI}\left(E^3\left[l\right]+3E\left[l\right]{\sigma }^2\left(l\right)-2E\left[l\right]x^2+x^3\right),$$

(4)

$$\begin{array}{l}Var\left(u\left(x\right)\right)=\\ 0.001736{\sigma }^2\left(l\right)\frac{q^2{x}^2}{E^2{I}^2}\left(9E^4[l]+36E^2[l]{\sigma }^2\left(l\right)-12E^2[l]x^2+15{\sigma }^4\left(l\right)-12{\sigma }^2\left(l\right)x^2+4x^4\right)\end{array},$$

(5)

$$\begin{array}{ll}{\mu }_3\left(u\left(x\right)\right)=& 0.001302\frac{E\left[l\right]{\sigma }^4(l)q^3{x}^3}{E^3{I}^3}\\ & \left(45{\sigma }^4(l)+48E^2[l]{\sigma }^2(l)-24{\sigma }^2(l)x^2+9E^4\left[l\right]-12E^2[l]x^2+4x^4\right)\end{array},$$

(6)

$$\begin{array}{l}{\mu }_4\left(u\left(x\right)\right)=9.04224\cdot {10}^{-6}\frac{{\sigma }^4(l)q^4{x}^4}{E^4{I}^4}\left(3465\right.{\sigma }^8(l)+17280E^2[l]{\sigma }^6(l)-2520x^2{\sigma }^6(l)+\\ +11934E^4[l]{\sigma }^4(l)-8136E^2[l]{\sigma }^4(l)x^2+840{\sigma }^4(l)x^4+2160E^6[l]{\sigma }^2(l)-3240E^4[l]{\sigma }^2(l)x^2+\\ \left.+1440E^2[l]{\sigma }^2(l)x^4-160{\sigma }^2(l)x^6+81E^8[l]-216E^6[l]x^2+216E^4[l]x^4-96E^2[l]x^6+16x^8\right)\end{array}$$

(7)

where E[l] and σ(l) stand for the expectation and standard deviation of the truss girder length. Such a characterization of probabilistic structural response is definitely wider than that coming from numerical methods, because it is able to draw the diagrams of expectations and variances throughout the girder length as the continuous function of their parameters and, furthermore, this is exact from the probability theory point of view. The engineers do not need to contrast these results with Monte Carlo simulation results and the only error comes from the approximate deterministic model. The reliability index β was determined for the numerical and analytical results of truss deflection (fourth-order polynomial of the truss span l) and the numerical results of stresses in joints (third-order polynomial of l appeared to be optimal here). The FORM reliability index was based upon the limit function g defined as

$$g\left(l\right)=R\left(l\right)-E\left(l\right),$$

(8)

where R means general structural resistance, while E denotes the overall effect of external actions. The reliability index β is with this notation defined as [16]

$$\beta \left(l\right)=\frac{E\left[g\left(l\right)\right]}{\sigma \left(g\left(l\right)\right)},$$

(9)

where E[g] is the expected value of the limit state function, σ(g) is its standard deviation. It should be underlined that this definition is derived using the assumption that the limit function has a linear form, which is not necessarily true in many engineering applications, so the second-order reliability method (SORM) is frequently preferred. After analytical transformations with the function g, the reliability index β takes the following form:

$$\beta \left(l\right)=\frac{E\left[g\left(l\right)\right]}{\sigma \left(g\left(l\right)\right)}=\frac{E\left[R\left(l\right)-E\left(l\right)\right]}{\sigma \left(R\left(l\right)-E\left(l\right)\right)}=\frac{E\left[R\left(l\right)\right]-E\left[E\left(l\right)\right]}{\sqrt{Var\left(R\left(l\right)-E\left(l\right)\right)}}=\frac{E\left[R\left(l\right)\right]-E\left[E\left(l\right)\right]}{\sqrt{Var\left(R\left(l\right)\right)+Var\left(E\left[l\right]\right)}}.$$

(10)

It is assumed, of course, that random functions R and E are uncorrelated here, which reflects engineering practice very well. The expected value of admissible deformation in the case for the SLS and ULS correspondingly equal

$$E\left[R\left(l\right)\right]=\frac{l}{250}, E\left[R\left(l\right)\right]=235 MPa,$$

(11)

while the variance of admissible values equals in the SLS

$$Var\left(R\left(l\right)\right)={\sigma }^2\left(l\right),$$

(12)

since R and l are connected by a linear transform. It is simply equal to 0 in the case of the ULS analysis; Equation (12) holds true in the case of the random function E. Because of the limitations of the FORM analysis, let us adopt the following proposition for the reliability index estimation [27]:

$${\beta }^{\prime }={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\sqrt{p_R(x) p_E(x)} dx},$$

(13)

where p_R(x), p_E(x) define probability function associated with structural resistance and probability function related to structural effort, respectively. It follows a more general idea of the distance (divergence) in between two different probability distributions proposed in [27]. One could alternatively use the Kullback–Leibler approach [28]; however, this second distance has non-symmetric properties and may exhibit some unwanted properties. It can be derived analytically that the new reliability index can be expressed for two Gaussian distributions with the given expectations and standard deviations (μ_R, μ_E—means of the structural resistance and σ_R, σ_E—their standard deviations) in the following, relatively simple, algebraic form:

$${\beta }^{\prime }=\frac{1}{4}\frac{\left(E\left[R\right]-E\left[E\right]\right)}{{\sigma }^2\left(R\right)+{\sigma }^2\left(E\right)}+\frac{1}{2}\ln \left(\frac{{\sigma }^2\left(R\right)+{\sigma }^2\left(E\right)}{2\sigma \left(R\right)\sigma \left(E\right)}\right).$$

(14)

Let us underline that the assumption about Gaussian distributions for the structural resistance and effort should be justified from the engineering point of view and this is the case of the basic structural demands included in Eurocode 0 [16].

3. Numerical Results for Structural Elements and Their Discussion

Numerical analysis was focused in turn on the following issues: (i) determination of polynomial response functions of the structural effort and of the extreme deformation as the functions of the truss span; (ii) determination of the basic probabilistic characteristics of structural responses with the use of FEM series and of the analytical formulas; (iii) determination of the reliability indices using both analytics and numerical analysis, including the new reliability index formula; and (iv) stochastic analysis of the most efforted structural joint in the truss.

Three different probability distributions were selected for this analysis—triangular, uniform and Gaussian—to study the Monte Carlo statistical approach and the iterative generalized stochastic perturbation technique of the 10th order. The detailed analytical derivations of the perturbation-based formulas for the first four central probabilistic moments could be found in [1] for the Gaussian PDF, whereas those relevant to triangular and uniform PDFs are presented in [29]. The results of stochastic analysis have all been collected for some range of the input uncertainty, i.e., $\alpha \left(l\right)\in \left[0.00,0.15\right]$, which has been taken as relatively wide to study the convergence (and possible divergence) of all three tested probabilistic numerical techniques.

The first results are presented in Figure 4 and they concern structural effort variations in all groups of the structural elements determined as the functions of the truss span. A numerical model was created in such a way that none of the designed steel profiles exceed the limit effort (100%), even for the extremely large span of this structure. Figure 4 clearly documents that all these efforts almost linearly depend upon this span—the largest effort was noticed within the welds, then, a slightly smaller one was observed for upper and lower chord members, while the smallest was in the truss diagonals. One may conclude that the numerical values of this effort ranged from about 65% up to 95% for the structural joints (whereas l varies from 15.0 up to 21.0 m), whereas the effort for lower and upper chord elements varied from 45% up to more than 85% at the same time. This enables us to conclude that the ULS-based reliability index should be based upon the joints’ rational designing, while the SLS, traditionally, upon this truss midspan deflection.

Having verified the deterministic safety of the designed Pratt truss, the polynomial response function was recovered via the series of FEM experiments with varying truss span. Discretization of the span computational domain, the corresponding series of the extreme truss deflections and third-order polynomial basis determined using the LSM procedure are shown in Figure 5. It can be seen that LSM fitting corresponds almost perfectly to the FEM data and that the approximating function is very smooth and regular. It should be mentioned that extreme deformation corresponds to the last equilibrium determined from the BFGS incremental path. Analytical representation was necessary for further perturbation-based derivations and it was obtained on the basis of numerical results delivered in

Table 2. Truss deflection and its span interdependence—numerical results.

Length l [m]	15.00	15.60	16.20	16.80	17.40	18.00	18.60	19.20	19.80	20.40	21.00
Deflection u [cm]	1.42	1.63	1.86	2.13	2.41	2.73	3.08	3.46	3.88	4.34	4.83

as

$$u(l)=0.0001522l^3-0.003769l^2+0.04322l-0.17150.$$

(15)

It should be mentioned that this relatively low order of polynomial basis causes, that higher than the fourth-order derivatives of the function u(l) simply vanish, so that the Taylor series expansions inherent in the stochastic perturbation technique are very short and contain a few terms only.

Probabilistic results concerning extreme deflection of the truss were further collected (in Figure 6, Figure 7, Figure 8 and Figure 9) and these are: expected value, coefficient of variation, skewness and kurtosis. They were collected for Monte Carlo analysis (abbreviated here as MC), perturbation method (PM), separately for the FEM approach (NC) and following analytical expressions (AC). Three different symmetric probability distributions were compared and these are triangular (TD), uniform (UD) and Gaussian (normal) distribution (ND). These distributions have the same mean value and are all symmetric, while the ranges for TD and UD were calculated from Gaussian distribution standard deviation with the use of the well-known three-sigma rule. In this work, the values obtained from the 10th order iterative stochastic perturbation method and the Monte Carlo Simulation were compared assuming a random sample size equal to n = 10⁵.

A general conclusion which can be drawn from these data is that all probabilistic characteristics monotonously increase together with an additional increase in the input uncertainty level. This is quite an expected result, as is the fact that the first two moments reach the largest values when the input distribution has a uniform PDF. The second general conclusion is that the results of both probabilistic methods coincide almost perfectly for all of the first four probabilistic characteristics, which is a very promising result in the context of the relatively large input uncertainty level and the fact that a nonlinear BFGS solver was used. It should be noticed at this point that a situation may be more complex for large-scale engineering structures and the input CoV interval permitting such a perfect agreement may become somewhat smaller, i.e., $\alpha \left(l\right)\in \left[0.00,0.10\right]$; this should be verified through more detailed FEM experiments. The next important information, which can be rather useful in civil engineering practice, is that expectations computed via analytical formulas (Figure 6) are remarkably smaller than those obtained from the FEM experiments. This may follow the fact that analytics is available in the linear elastic regime only, whereas the FEM analysis is conducted including geometrical nonlinearity; nevertheless, the first approach seems to be insufficient in all these cases, where nonlinear effects may play a remarkable role during exploitation time.

Higher order statistics such as skewness (Figure 8) and kurtosis (Figure 9) differ from 0, so that the resulting distributions of extreme displacements are slightly different from the Gaussian PDF, and this divergence increases while increasing the input uncertainty level. It is noticeable that the largest values of both coefficients are obtained for the Gaussian PDF of the truss span, slightly smaller values are obtained for the triangularly distributed span and the smallest are obtained in the case of the uniform distribution. Skewness keeps positive values for the entire variability interval of truss span uncertainty for all chosen probability distributions, while kurtosis values are more complex. They are all positive for the normal PDF, start from negative and tend towards positive while having a triangularly distributed parameter l and are all negative when the uniform distribution is considered. These higher-order statistics are usually postponed during the final reliability assessment; however, these results demonstrate that the initial choice of PDF type remarkably affects the resulting stochastic structural response. The first two probabilistic moments are contained in Figure 6 and Figure 7 and resulting from the Gaussian distribution of the truss span were used to calculate final reliability values in the SLS state.

Having computed the probabilistic response of the Pratt truss, its reliability indices were computed in the computer algebra software MAPLE 2019 according to Equations (9) and (14), sequentially. This was also carried out for three different probability distributions: triangular, uniform and Gaussian, to see an influence of the PDF choice on their final values. Numerical values relevant to Equation (9) have been collected in

Table 3. Reliability indices β in the SLS via numerical and analytical methods based on the Monte Carlo simulation method and the stochastic perturbation technique for the triangular, uniform and Gaussian PDFs.

α [-]	MCS-NC	MCS-AC	PM-NC	PM-AC
0.025	37.93	38.28	37.93	38.28
	37.40	37.83	37.40	37.83
	38.11	38.43	38.11	38.43
0.050	18.95	19.12	18.95	19.12
	18.66	18.88	18.66	18.88
	19.05	19.21	19.05	19.21
0.075	12.61	12.73	12.61	12.73
	12.40	12.55	12.40	12.55
	12.69	12.79	12.69	12.79
0.100	9.44	9.53	9.44	9.53
	9.26	9.37	9.26	9.37
	9.50	9.58	9.50	9.58
0.125	7.53	7.61	7.53	7.61
	7.36	7.46	7.36	7.46
	7.59	7.66	7.59	7.65
0.150	6.26	6.32	6.26	6.32
	6.09	6.17	6.09	6.17
	6.31	6.37	6.31	6.37

, where an influence of the input uncertainty of the truss span in the form of the coefficient of variance α has been also included. Each cell of this table includes in turn the reliability index for these three PDFs—the first one corresponds to the triangular probability function, the middle to the uniform one and the lowest to the Gaussian distribution. The abbreviations in this table heading are consistent with the notation of the previous figure, i.e., MCS-NC denotes the results corresponding to the Monte Carlo simulation method while using the finite element method analysis. The data collected in this table confirm the well-known trend of the reliability index, which usually exponentially decreases together with an additional increase in the cumulated input uncertainty level. Monte Carlo simulation and the iterative generalized stochastic perturbation technique both return practically the same results, which, taking into account decisively smaller time and computer power effort of this second technique, means that the perturbation-based SFEM is preferred in this case.

As is expected, the lowest values of this index were returned for the uniform distribution of the truss span, slightly larger values were obtained for the triangular PDF of this span and the extremely high reliability index was obtained in the case of the Gaussian distribution. It is observable that an arbitrary assumption of the Gaussian character of most structural parameters without a precise verification may lead to some inaccuracies or even to unsafe designing of the engineering structures. A very interesting observation can be made while comparing numerical and analytical analyses—these second series return the results each time with no more than a few percent larger than the FEM-based analysis. Nevertheless, this difference is almost negligible when the uncertainty level belongs to the interval $\alpha \in \left[0.05,0.10\right]$, which is highly expected in most of the statistical parameters.

Moreover, it is widely known that standard EN 1990 [16] defines the three reliability classes RC1, RC2 and RC3, which are associated with the consequences classes CC1, CC2 and CC3, describing the possible consequences of a failure of the entire structure or just its member. The table below (

Table 4. Recommended minimum values for reliability index β.

Reliability Class	Minimum Values for β
	1 Year Reference Period	50 Years Reference Period
RC3	5.2	4.3
RC2	4.7	3.8
RC1	4.2	3.3

) contains the minimum values of the reliability index β recommended by EN 1990 (Table B2) [16], which were recalled to determine the admissibility level of input randomness for the given parameter of the Pratt truss span. The results of this study are presented by inserting numerical values from Table 3 and Table 4 into a single graph—see Figure 10 below.

Truss girders are structures that are expected to last at least 50 years and the described construction has been assigned to the third reliability class that allows resisting even if the coefficient of variance of the truss span slightly exceeds the value of α > 0.150. An intersection of the reliability curves delivered by numerical and analytical approaches with the lines representing target values of the reliability index shows that this admissible value equals $\alpha =0.175$. This value is enormously large for any measuring method for the span, so one may conclude that the presented method generally has no limitations coming from the engineering practice while investigating statistics in the truss spans. Taking into account the well-known limitations of the FORM approach, the new reliability index has been proposed following the relative entropy (probabilistic divergence) model proposed by Bhattacharyya [27]. Its values computed using the first two probabilistic moments have been collected above in

Table 5. A contrast of the estimated reliability indices β′ in the SLS by numerical and analytical methods based upon the Monte Carlo simulation method and the stochastic perturbation technique for the triangular, uniform and Gaussian probability distributions.

α [-]	MCS-NC	MCS-AC	PM-NC	PM-AC
0.025	11,426.46	11,579.79	11,426.45	11,579.77
	11,422.07	11,575.61	11,421.98	11,575.53
	11,428.21	11,581.44	11,427.93	11,581.19
0.050	716.04	725.84	716.04	725.84
	714.92	724.76	714.91	724.75
	716.46	726.23	716.42	726.20
0.075	143.47	145.57	143.47	145.57
	142.94	145.05	142.94	145.05
	143.67	145.75	143.65	145.74
0.100	47.21	48.00	47.21	48.00
	46.87	47.65	46.86	47.65
	47.34	48.12	47.33	48.11
0.125	20.92	21.34	20.92	21.34
	20.64	21.05	20.64	21.05
	21.02	21.44	21.01	21.44
0.150	11.46	11.74	11.46	11.74
	11.19	11.46	11.19	11.46
	11.55	11.85	11.55	11.84

, in the same manner as the results contained in Table 4. Despite the different range of variability caused by the fact that Equation (9) contains a linear form of the expectations of the functions R and E, one may see the same properties and interrelations in between different methods and input distributions. This new approach is more convenient in the engineering sense, as it measures a distance in between two random distributions and no specific form of the limit function is required. Its value has been scaled to compare with the existing designing algorithm and this has been performed using the following semi-empirical formula:

$${\beta }^{″}=\frac{\sqrt{{\beta }^{\prime }}}{2}.$$

(16)

Such a formula has been established by a comparison of the main components in Equations (10) and (14) containing mean values of the structural response and its resistance. It is important to mention that this proposed scaling enables keeping the same practical demands and values and to have a better mathematical and engineering justification of the entire procedure. It is evident after scaling that the new reliability index (

Table 6. A contrast of the estimated reliability indices β″ in the SLS by numerical and analytical methods based upon the Monte Carlo simulation method and the stochastic perturbation technique for the triangular, uniform and Gaussian probability distributions.

α [-]	MCS-NC	MCS-AC	PM-NC	PM-AC
0.025	53.45	53.80	53.45	53.80
	53.44	53.79	53.44	53.79
	53.45	53.81	53.45	53.81
0.050	13.38	13.47	13.38	13.47
	13.37	13.46	13.37	13.46
	13.38	13.47	13.38	13.47
0.075	5.99	6.03	5.99	6.03
	5.98	6.02	5.98	6.02
	5.99	6.04	5.99	6.04
0.100	3.44	3.46	3.44	3.46
	3.42	3.45	3.42	3.45
	3.44	3.47	3.44	3.47
0.125	2.29	2.31	2.29	2.31
	2.27	2.29	2.27	2.29
	2.29	2.32	2.29	2.32
0.150	1.69	1.71	1.69	1.71
	1.67	1.69	1.67	1.69
	1.70	1.72	1.70	1.72

) is more demanding and does not allow for such a large input uncertainty interval; nevertheless, the importance of the input PDF choice remains the same. Analytical and numerical approaches here return almost the same result, so one could recommend inserting them directly into the engineering designing codes to give practicing engineers a fast and widely available alternative to the variety of rather complex stochastic finite element method implementations.

4. Stochastic Response of the Decisive Structural Connection and Its Discussion

Additionally, the load capacity of structural connection nodes, designed according to the statements included in the designing code [30], was studied in the context of stochastic structural response and reliability. This part of the numerical study was based on the reduced von Mises stresses in their welds and some exemplary connections have been schematically shown in Figure 11, whereas the nodes with extreme stresses accounted for in further SFEM analysis have been marked in Figure 12 with the dashed line circles.

Quite similarly to the SLS analysis in the previous section, one first needs to determine the response function of the reduced stresses with respect to the truss span via the LSM fitting on the basis of the results contained in

Table 7. Extreme reduced stresses in the external weld in the truss column.

Length l [m]	15.00	15.60	16.20	16.80	17.40	18.00	18.60	19.20	19.80	20.40	21.00
Stress σ [MPa]	239.22	248.81	258.79	267.98	277.56	287.15	296.74	306.31	315.89	325.47	335.05

. This is obtained in the form of the third-order polynomial having the following analytical form:

$${\sigma }_{red}=0.007761l^3-0.421976l^2+23.5512l-45.2738$$

(17)

Furthermore, probabilistic characteristics of the structural response are determined with the Monte Carlo simulation and the stochastic perturbation method; these are: expectations (Figure 13), coefficients of variance (Figure 14), skewness (Figure 15) and kurtosis (Figure 16). This is conducted for three various probability distributions, but unfortunately, any analytical method is available here. Figure 13 documents that the expected values of the reduced stresses remarkably and nonlinearly decrease, contrary to the extreme deformations, while increasing input uncertainty. Furthermore, some small differences in between the Monte Carlo simulation and the stochastic perturbation method can be noticed for all PDFs, which additionally increase together with an increase in the input CoV. They are quite remarkable in Figure 13, but their realistic range is smaller than one thousandth, so they can be postponed in engineering practice. Variances of the reduced stresses quite expectedly increase in Figure 13 while approaching the extreme values of the input α(l) and now the differences between the statistical and perturbation approaches are invisible. The largest resulting statistical scattering is obtained for the uniformly distributed truss span, next, for the triangular PDF; and the smallest for the Gaussian distribution. This regularity partially validates the entire methodology and, specifically, the efficiency of the iterative generalized stochastic perturbation method for symmetric non-Gaussian distributions also.

Skewness and kurtosis attached here for the completeness of probabilistic modeling (Figure 15 and Figure 16) are all very close to 0, so these reduced stresses could be used in reliability analysis of Gaussian with relatively small error, so that the first two moments could be sufficient. Of course, this approximation is the most efficient when input uncertainty has a Gaussian character. Now, analogously to Figure 14, two stochastic approaches return the same values of both coefficients, which also confirms the applicability of the stochastic perturbation method for higher statistics.

Finally, the reliability indices are contained in

Table 8. A contrast of the reliability indices β in the SLS by numerical and analytical methods based upon the Monte Carlo simulation method and the stochastic perturbation technique for a various probability distribution.

	Distribution
	Triangular		Uniform		Normal
α [-]	MCS-NC	PM-NC	MCS-NC	PM-NC	MCS-NC	PM-NC
0.025	5.79	5.79	4.75	4.75	6.34	6.33
0.050	2.9	2.9	2.37	2.37	3.17	3.16
0.075	1.93	1.93	1.58	1.58	2.11	2.11
0.100	1.45	1.45	1.18	1.18	1.58	1.58
0.125	1.15	1.15	0.94	0.94	1.27	1.26
0.150	0.96	0.96	0.78	0.78	1.05	1.05

and also compared in Figure 17 with the corresponding normative values [16]. Generally, one can observe that their values are remarkably smaller than for the structural elements analyzed before, see cf. Table 6. This is consistent with traditional engineering practice, where the higher importance of structural joints in the optimal designing of metal structures is considered. The highest reliability indices occur when the input truss span is assumed to have the Gaussian PDF; then, slightly smaller values are noticed while triangular distribution is assumed; whereas extremely small indices are noticed for the uniformly distributed span. Furthermore, two probabilistic methods return the same results correspondingly, which confirms the applicability of the stochastic perturbation technique in reliability-oriented designing procedures of structural joints. Contrary to the previous analysis, a range of admissible input statistical scattering of the truss span is very small (α < 0.025); one can conclude that the truss span must be measured very precisely and that any geometrical imperfection in this context may be critical for the very optimally designed connections in the Pratt trusses.

5. Conclusions

(1)

The numerical analysis delivered in this work proves that structural reliability of the Pratt truss structures subjected to geometrical uncertainty can be efficiently modeled using analytical formula describing its deflection line. This is due to a coincidence of the first four probabilistic characteristics and reliability indices obtained via a Monte Carlo simulation and the stochastic perturbation-based FEM analysis. This is a very promising and important result because the deflection line for such a truss can be easily implemented in any computer algebra system with probabilistic libraries, so that some analytical expressions for the expectations and standard deviations could be inserted into engineering design codes to perform reliability assessment, without a rather complex implementation and time-consuming simulations of various SFEM realizations. It would enable further applications of many correlated uncertainty sources to have any desired probability density function. It can also be seen that this approach can be extended with many existing analytical models in civil engineering to develop stress formulas as well as deflection lines for some typical beams, plane frames and other girders, to make analytical reliability analysis more popular. Let us note that a difference in between numerical and analytical results in the case of the reliability indices increases rather slowly together with input statistical scattering, meaning it remains unbiased by the experimental statistics.

(2)

It was documented in this work that it is possible to use the relative entropy (probabilistic divergence) proposed by Bhattacharyya [27] in traditional civil engineering reliability analysis, using the existing designing codes. Furthermore, it was demonstrated that this entropy may be efficiently rescaled to the FORM index as a half of this entropy square root. This relatively simple formula may apply for the input coefficient of variation not larger than $\alpha \le 0.15$ while using either the Monte Carlo simulation or the iterative generalized stochastic perturbation technique. This idea needs further numerical experiments, not only in the case of steel structures, but also for other branches and problems in engineering where input uncertainty may play a remarkable role. The new reliability index may be useful in the Stochastic Reliability-Based Design Optimization (SRBDO) with triple perturbation-based, semi-analytical and Monte Carlo simulation analysis of the first two probabilistic moments, analogously to the study presented in [31].

(3)

It was shown that the admissible uncertainty level cannot be too large in the welded steel structures because of the relatively high sensitivity of optimally designed connections of structural elements to the truss span. It was demonstrated that structural effort in the range of 90% demands an input coefficient of variation α < 0.025. The remaining open research question in this context is the possible correlation of the few uncertainty sources inherent in the functions R and/or E. There are some engineering examples showing that this correlation may remarkably decrease the overall structural reliability index in some specific cases. Nevertheless, it should be included directly into the final formulas such as in Equations (9) and (14) presented in this paper. The numerical approaches shown above are capable of capturing and discussing this issue as well. It would be very instructive to discuss these issues for structural stability problems, which are of paramount importance in the area of steel structures [32].