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Design of stadium roofs with a given level of reliability
Engineering Structures 209 (2020) 110245
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Design of stadium roofs with a given level of reliability a,⁎
Anatoliy Orzhekhovskiy , Iurii Priadko
b,c,d
a
, Anton Tanasoglo , Serafim Fomenko
T a
a
Theoretical and Applied Mechanics Department, Donbas National Academy of Civil Engineering and Architecture, Ukraine Beijing International Education Institute, 38 East 3rd Ring North Road, Chaoyang, Beijing 100026, China Huzhou Vocational and Technical College, School of Civil Engineering, No.299, Xuefu Road Modern Teaching District, Huzhou, Zhejiang 313000, China d Theoretical and Applied Mechanics Department, Kyiv National University of Construction and Architecture, Ukraine b c
A R T I C LE I N FO
A B S T R A C T
Keywords: Structural reliability Steel structures Stadium roof Random variable Stochastic parameters Safety factor
The article discusses method for the selecting cross-section of steel rod elements of the roof structure placed above the stands of stadiums. Particular attention is paid to ensuring a given level of structural reliability. An algorithm to determine the actual safety factors for the most critical elements in the roof structure is proposed. Snow load, cross-sectional area of the elements, ultimate stress of the material, settlement of the base, mounting imperfections and defects are considered as the stochastic parameters in calculation of the reliability indicators. All specifications of the distributions for these magnitudes are based on the real experimental data. Since the building regulations do not provide a clear algorithm for calculating the reliability indicators of unique (and any other) building structures, it is possible to apply the proposed methodology both to ensure the design reliability of the whole structure as well as the separate most critical structural elements. It also might be applied to other kinds of spatial steel rod structures whose elements work as trusses. The internal force factor in the elements of such structures is longitudinal force. The method has been tested on the stadium structures of FC Olympic in Donetsk city (Ukraine).
1. Introduction Nowadays, the design of steel cantilever frame structures of roofs above the stadium stands is based on the method of limit states. This method is the basis of such building regulations as EUROCODE (European Union), DBN (Ukraine), SNiP and GOST (Russia) etc. Many controversial issues in determining the structural reliability arise when using the methodology proposed in these documents [1–4]. The method of limit states requires comparing the design stress in element (Qp) with the ultimate stress (Ry). In case of Qp < Ry, it is considered that reliability is ensured. Thus, the reliability of structures in all design cases is approximately the same and depends only on the design scheme and material of the structure. EUROCODE [2], DBN [3] and GOST [4] try to overcome this shortcoming. They rank constructions according to the levels of accidents consequences (economic and human losses) and criticality rating, as well as give the limits of allowable probability of failure for this classification. Moreover, the method of determining the probability of failure is given extremely generalized and does not have a clear algorithm of calculation in particular cases. In this case, the probability of failure of the designed structure remains unknown to the engineer. Therefore, current design standards for unique structures and structures with high criticality rating need to be refined. It is proposed
⁎
in this article to carry out numerical calculation of reliability and survivability at design stage for such kinds of unique structures as steel rod roofs over the stadium stands. Such random variables and factors as design strength of material, mounting imperfections and defects, magnitude of temporary loads (snow load in particular) and settlement of base are taken into account in calculation process. According to the analysis of literature, it is concluded that many authors solve problem of ensuring the reliability of building structures within constructive methods, i.e. the required level is ensured either by using stronger materials or increasing of elements cross sections or applying the most reliable design schemes etc. [5]. This approach may not always be applicable in terms of the economic requirements for the design, as well as matter of the numerical determination of reliability indicators remains open. A number of authors carried out research for particular cases [6,7]. They propose various approaches to improve common methods to determine reliability indicators (in some cases calculation examples are attached). However, these methods are not complete within a complex approach to determine the reliability of structures. A number of authors analyzed the reliability of complex structures and proposed methods for determining its reliability indicators [8–10]. However, they describe techniques and structures that are not suitable for steel rod roofs above the stadium stands. The topic
Corresponding author. E-mail address: [email protected] (A. Orzhekhovskiy).
https://doi.org/10.1016/j.engstruct.2020.110245 Received 10 June 2019; Received in revised form 15 December 2019; Accepted 14 January 2020 0141-0296/ © 2020 Elsevier Ltd. All rights reserved.
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Fig. 1. Roof structure above the stadium stands of FC Olympic in Donetsk city (Ukraine).
reliability is:
̂ > 0; Y(t) = R (t) − S(t)
ˇ
(1)
where: Ř(t)- bearing capacity of a structure or element; Ŝ(t)- loading applied on the structure (stress in structural element); Ŷ(t)- strength reserve or bearing capacity reserve. The probability of failure may be represented as an integral: ∞
Q (t ) = ∫ R (t ) pS (t ) dt; 0
(2)
where: R (t ) - probability distribution function of a random variable R ; pS (t ) - probability density of a random variable. The values stress in material [12], geometric parameters of the cross-sections of structural elements [16], magnitude of temporary loads (snow load in particular) [17] and settlement of the base [18] are considered as random variables. For the purpose of probabilistic calculation, authors chose common method Monte Carlo for studying random processes which has been tested and proved its effectiveness for building structures [19]. Since the system is many times statically indeterminate, it is most advisable to calculate reliability indicators for the most critical elements. An iterative geometrically non-linear calculation of the structure is performed for this purpose. As a result of the calculation, the order of collapse of the structural elements is set. As a criterion of collapse, it is proposed to consider the occurrence of the first limit state in each rod. This condition is chosen because the second limit state will not cause such large-scale economic and human consequences for considered kinds of structures. At the next stage, an iterative calculation for the selected elements is performed, taking into account the specified random variables. Thus, a sample of stresses for the group of the considered rods (Ŝ(t)) is formed. The sample should have a significant amount. Studying this problem, Pichugin S.F. [12] gives the range of numbers in the interval 104–108. The second component of Eq. (1) is the random variable of the strength of the material of construction (the yield strength in this case) Ř(t). It is formed on the basis of the analysis of statistical data obtained at metallurgical factories. The obtained two generalized random variables are processed by the methods of mathematical statistics. Their mathematical laws and density distribution are determined as well. If it is impossible or difficult to perform this task, it is recommended to select the most appropriate distribution and approximate a random variable with an accuracy that satisfies the calculation. Having types of distribution of random variables, their mathematical parameters such as expectation and standard deviation are calculated. Using the density, laws and parameters of distributions, the probabilities of failure for a group of selected elements are calculated. The resulting failure probability is the upper limit of failure [20]. This
Fig. 2. Design scheme of the cantilever frame structure of the roof.
of complex structures reliability is also extensively covered in the works [11,12]. Based on these works, an extensive research work was carried out in this direction, which was used in the development of building codes (DBN) for the design of steel structures [13] and for ensuring the reliability of structures [3]. Unfortunately, these authors did not provide a single method to determine reliability indicators. All of the above works are based on classical research in this scientific field [14,15]. They describe in detail a number of methods to determine the probability of failure and safety parameters (as indicators of system reliability), but using simple structures as an example, which causes a number of controversial points when calculating these indicators for complex multi-element systems. This article describes method to determine the cross-sections of rod elements in steel cantilever frame structures of roofs above the stadium stands, taking into account upper limit of reliability which means that failure of one rod will not result failure of the entire system. The described method is also applicable to other spatial steel rod structures whose elements work as trusses. The internal force factor in elements of such structures is longitudinal force.
2. Research methods The task of ensuring a given level of reliability in the structural calculation within the limit state method falls on a number of safety factors. Normally, such factors are indicated incorrectly or not considered at all for unique structures (such as static steel rod roofs over the stadium stands). Thus, reliability requirements for such structures may not be fully met. As we know from the work [12] the basic equation of the structural 2
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A. Orzhekhovskiy, et al.
method satisfy not only the requirements of the limiting states calculation, but also correspond to the specified level of reliability.
Table 1 The cross-sections of the steel rods of the roof structure. №
Cross-sections
Size
1 2 3 4 5
I-beam Channel Roll-formed welded rectangular section Roll-formed welded square section Roll-formed welded rectangular section (purlin on the cantiliever)
][ №16 [ №16 □ 100 × 60 × 3 □ 60 × 2,5 □ 200 × 60 × 4,5
3. Subject of research This article discusses the method of selecting cross-sections of rods in the static steel cantilever frame structures of roofs above the stadium stands within the example of an existing section of the roof that overhangs southern stand of football stadium which belongs to FC Olympic in Donetsk city. The structural scheme of the static roof represents the L - shaped plane cantilever frames (as the main bearing structures) with 6 m pitch center-to-center, which are linked by trussed bracings and purlins on the cantiliever (Fig. 1). All joints are rigid. This means that the structure is many times statically indeterminate system (Fig. 2). Fixing to the foundation is constructively done in such a way that it is advisable to consider it as a rigid joint in the design scheme as well. The crosssections of the structural elements are shown in the Table 1 (see Fig. 3).
means that elements in a system are so arranged that failure of one rod will not result the failure of the entire system. There is also a lower limit of failure, that assumes the sequential inclusion of elements in the work, but the probability of failure in this case is much higher and we will not consider it. The obtained reliability parameters are compared with the normative ones taking into account the reliability class and consequences class of the considered structure [2,3]. If the requirements of reliability are met, the calculation ends. If the reliability is not ensured, the partial reliability indicators are recalculated and the inverse problem is solved, the purpose of which is to determine the required value of the expectation of stresses in the structural elements and the required crosssection areas of the rods. The structure with the newly defined crosssection proportions is recalculated anew according to the algorithm described above. Iterations are carried out until the strength conditions are satisfied by the considered group of elements. The cross-sections of rods in static steel cantilever frame structures of roofs above the stadium stands obtained by the described above
4. Research algorithm Nowadays, authors have not found a universal algorithm for solving the set task. Therefore, a number of programs have been written by means of high-level interpreted programming code MATLAB [21]. All of the following calculations have been implemented within these programs. The testing and debugging of the resulting software package were performed using the LIRA-CAD 2013 software package [22]. It is reasonable to split the created software package into the 2 blocks:
a)
b)
Fig. 3. Geometrical scheme with indication of cross-sections types in structural elements: (a) plane frame; (b) trussed bracing. 3
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Fig. 4. The flowchart to determine the most critical structural elements.
4.1. Block 1
1. Determination of the most critical elements in the structure, which will define its reliability in general. 2. Forming a stresses samples of a group of the most critical elements in the system. Processing of the stresses samples and yield strength using the methods of mathematical statistics. Calculation of the reliability criteria.
Nonlinear design calculation is based on the finite element method (FEM) in the spatial interpretation. The geometric non-linearity of construction is taken into account as follows: 1. The coordinates of the joints are updated at the each iteration, taking into account their movement at the previous one. 2. The stresses arised at the previous iteration due to the displacement of structural joints are taken into account at the stage of forming the element stiffness matrix.
Both of the blocks within example of mentioned above structure are described below.
4
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The elements of the considered roof structure that are shown on the Figs. 5–8 have been failed at the fourth iteration. 4.2. Block 2 Simulation of snow load. Snow load as a random variable was taken with a double exponential Gumbel distribution law [12]. The equation describing the sample of the snow load values for the Donetsk city has been obtained by the method of mathematical inversion.
S = 521.48 − 270.413·ln(−1.0·ln(R + 0.0375))
(3)
where R is generator of the uniform random variable within [0… 0,935]. The necessary parameters of the Gumbel distribution are obtained by analyzing the statistics of annual maximum snow load during the 40 years. Analyzing the Eq. (3) it is clear that the termln(R + 0.0375) < 0 must be met, so that the value of the snow load does not move into the range of complex numbers. In case of using the maximum value of R = 0.935 this will not happen. This equation has a snow load ranging from 200 to 1500 Pa, which corresponds to the nature of the load for current region. Modeling the strength parameters of the material and the geometric parameters of the cross-sections. The generation of the geometric parameters of cross-sections and strength parameters is based on the analysis of rolled metal products in two metallurgical factories: Pipe Producing Factory in Lugansk city and Comintern Metallurgical Factory in Dnepr city [23]. A sample of the strength properties of cross-section as well as a sample of geometric parameters of all sections have been subjected to χ2-analysis to determine the possibility of approximating the distribution of random variables by the normal distribution law of Pearson criterion [14]. The analysis showed that the distribution of each random variable differs from the normal one by no more than 5% (a significance level of 0.05 has been set). Therefore, these random variables might be considered within normal distribution law (Table 2 and 3). Simulation of geometric imperfections and settlement of the base. Settlement of the foundations and geometrical imperfections of the structure have been taken into account by the geodetic survey analyzing the roof above the southern stand of the football stadium which belongs to FC Olympic in Donetsk city (Figs. 9 and 10). The survey of the structure's geometry took place in two cycles and gave the results displayed on the graphs and histogram (Figs. 11–13). Analyzing the information received, there is a significant excess of deviations that allowed by regulatory documents (Figs. 11 and 12). The data are taken into account at the stage of formation of local stiffness matrices in the design scheme. Representing the deviations of the geometry of points as a random variable, several distribution laws were considered for approximation (χ2, lognormal and normal distribution). Normal distribution law has been taken as an approximating one (Fig. 13). Additionally, the considered random variable was subjected to χ2 analysis by the Pearson criterion [14]. The analysis showed that the distribution law of deviations of a random variable at the considered points differs from the normal one by no more than 5%. Therefore, the normal distribution law might be considered as an acceptable one. Knowing the distribution laws of all considered random variables and using a generator of random variables, a sample of 10,000 random variables has been formed. Further, applying the deterministic calculation based on the finite element method, the samples of stresses in the eight elements were determined. The obtained random variables of the stresses and yield strengths were analyzed using the methods of mathematical statistics (χ2 analysis, Shapiro – Uilk criterion) [14]. Since the obtained samples of stresses in the rods and the yield strength are approximated by a normal distribution law, the reliability index β
Fig. 5. Structural elements that failed at the 4th calculation cycle (marked with red colors). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Structural elements that failed at the 5th calculation cycle (marked with red colors). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. The order of failure of structural elements at the each stage of loading.
A snow load is applied to the joints of the considered roof structure and than an iterative calculation is made until the specified amount of elements are failed. A load of 1 kN is additionally applied at the each calculation cycle. In case if any element has failed in the previous cycle, it is excluded from system, after which system is recalculated again without additional load. Thus, phenomenon of avalanche destruction is taken into account (Fig. 4). 5
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Fig. 8. The position of the rods that determine the reliability of the structure.
Table 2 Experimentally obtained stochastic parameters of the cross-section dimensions of roll-formed welded sections. No
1 2 3 4
Cross-section
60 × 40 × 3 120 × 60 × 3 100 × 100 × 3 100 × 60 × 3
Number of measurements
250 250 250 250
Wall thickness (mm)
Section height (mm)
Section width (mm)
Expectation
Standard deviation
Coefficient of variation (%)
Expectation
Standard deviation
Coefficient of variation (%)
Expectation
Standard deviation
Coefficient of variation (%)
2,96 2,98 2,93 2,89
0,25 0,19 0,26 0,23
8,52 6,63 9,10 8,00
58,98 118,07 99,13 97,51
1,48 3,98 5,60 5,63
2,54 3,4 5,65 5,77
38,77 58,39 98,34 57,79
2,15 2,79 5,85 1,71
5,54 4,77 5,95 2,96
Table 3 Experimentally obtained strength parameters of the roll-formed welded sections. No
1 2 3 4
Cross-section
Expectation
60 × 40 × 3 120 × 60 × 3 100 × 100 × 3 100 × 60 × 3
Standard
Numerical standard (μ)
σ¯B
σ0,2 ¯
σB̂
̂ σ0,2
482,58 443,67 428,85 418,54
317,44 332,21 323,63 379,03
22,38 31,07 28,53 25,84
35,15 22,07 31,88 29,46
3,15 3,15 3,15 3,15
γc = 1 −
(
σ¯ p 1 − μσ k σ
− σ0p
R¯ yn (1 − μr kr )
);
Standard value (MPa)
n Rym
n Run
Ryn
Run
206,72 262,69 223,21 286,23
411,50 345,80 338,98 337,14
225 225 225 225
370 370 370 370
ordinary deterministic design calculation.
(failure range) [2] was calculated as a probability parameter. The safety factor γc [3] is proposed to be calculated as follows [12]:
γm
Exponential value (MPa)
The flowchart of the calculation is presented on Fig. 14. (4)
5. The research results
where:
The obtained values of β were compared with regulatory ones [2,3]. Regulations requirements are not met (β ⩾ βiex ). The required value of the mathematical expectation of the cross-sectional area in the elements is calculated. New cross-sections are set, after which the second block of the software is fully recalculated, taking into account the gained safety factors. The calculation results are summarized in Table 4. Analyzing the results it is concluded: (1) the symmetry of the structure is obvious; (2) the calculated β and γc values for structures with high consequences class differ significantly from those recommended by the DBN regulations and Eurocode. The following are the designations of variables to define P(t) and β specified on Fig. 14, whereas γc was explained in (4):
γm - material reliability factor; σ p - standard deviation of stresses arising in the considered structural element; μσ , μr - mathematical expectation of the stress values in the considered structural element and the mathematical expectation of a random variable of the material strength (the yield strength); k σ , kr - variation coefficients of two considered random variables; R yn - the standard value of the designed material strength of construction; σ0p - the designed value of the stresses in the element obtained by 6
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Fig. 9. Scheme of geodetic survey for monitoring the structures of the stadium stands.
S – random variable of stresses in the structural elements; m- sample size. 6. Conclusion 1. A study of limit states method has revealed that it does not provide required level of reliability for unique structures such as static steel cantilever frame structures of roofs above the stadium stands. Therefore, a new design method has been devised which apply direct methods of the reliability theory for determination of the crosssections in structural elements, determination of reliability indicators and subsequent specification of β and γc. 2. The recommended by regulations general measures for providing structural reliability, in particular Eurocodes (grading by consequences classes (CC) of structure, reliability classes (RC) etc.) do not always provide the required level of reliability βiex for structures with high consequence for loss of human life or economic, social or environmental consequences. The reliability of individual key structural elements could be most easily provided by adjusting the values of the safety factor γc. However, their values are proposed without taking into account the level of consequences class of structure in the existing regulations, so that they need to be clarified. The method described in this article solves this issue and takes into account an individual approach while ensuring the required level of reliability of the building structure. 3. Since the building regulations do not provide a clear algorithm for calculating the reliability indicators of unique (and any other) building structures, it is possible to apply the proposed methodology for the structural rod roof system placed above the stadium stands both to ensure the design reliability of the whole structure as well as the separate most critical structural elements. 4. The described above method is devised for the static steel cantilever frame structures of roofs above the stadium stands. However it might be applied to other spatial steel rod structures whose elements work as trusses. The internal force factor in the elements of such
Fig. 10. The scheme of control points on the structures of the stands.
SR - standard deviation of the random variable of material yield strength shown in Tables 2 and 3 (sample obtained experimentally); SQ - standard deviation of the random variable of stresses in the structural elements; ∼ R - mathematical expectation of the random variable of material yield strength; ∼ Q - mathematical expectation of the random variable of stresses in the structural elements; P(t) - random variable of failure probability; F(S) - random variable of structural strength reserve; F¯ (S) - mathematical expectation of structural strength reserve; 7
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Fig. 11. The deviation of the elevations in point «A».
Fig. 12. The deviation of the elevations in point «B»
Fig. 13. Histogram of the deviations distribution of the structure joints coordinates with an approximating curves.
8
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Fig. 14. Flowchart of the algorithm for determining the reliability criteria in static steel cantilever frame structures of roofs above the stadium stands. Table 4 The results of determination of the cross-sectional area in the structural elements considering the requirements of reliability. Element’snumber
Initial cross-section Cross-section
32 33 60 61 88 89 116 117
□ □ □ □ □ □ □ □
100 100 100 100 100 100 100 100
× × × × × × × ×
60 60 60 60 60 60 60 60
× × × × × × × ×
3 3 3 3 3 3 3 3
Determined cross-section β
γc
Cross-section
4,4255 4,6404 4,0775 4,2966 4,0775 4,2966 4,4255 4,6404
0,9729 0,9742 0,9708 0,9722 0,9708 0,9722 0,9729 0,9742
□ □ □ □ □ □ □ □
9
100 100 100 100 100 100 100 100
× × × × × × × ×
60 60 60 60 60 60 60 60
× × × × × × × ×
5 5 5 5 5 5 5 5
Standard value β
γc
γc (DBN)
βiex (DBN)
4,9827 5,1573 4,7634 4,8135 4,7634 4,8135 4,9827 5,1573
0,9734 0,9748 0,9712 0,9725 0,9712 0,9725 0,9734 0,9749
0,9
4,76
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structures is the longitudinal force.
[7] Chen C, Zhu Y, Yao Y. An evaluation method to predict progressive collapse resistance of steel frame structures. J Constr Steel Res 2016;122:238–50. https://doi. org/10.1016/j.jcsr.2016.03.024. [8] Zhang H, Ellingwood B, Rasmussen K. System reliabilities in steel structural frame design by inelastic analysis. Eng Struct 2014;81:341–8. https://doi.org/10.1016/j. engstruct.2014.10.003. [9] Gordini M, Habibi M, Tavana M. Reliability analysis of space structures using Monte-Carlo simulation method. Structures. 2018;14:209–19. https://doi.org/10. 1016/j.istruc.2018.03.011. [10] Podofillini L, Sudret B, Stojadinovic B. Safety and reliability of complex engineered systems: ESREL 2015. CRC Press; 2015. p. 730. [11] Perelmuter AV. Selected problems of reliability and safety of building structures. Moscow: ASV 2007;254:rus. [12] Pichugin SF. Reliability of steel structures in industrial buildings. Poltava: ASMI, 2009. 452 p. (rus). [13] DBN V.2.6-163: 2010. Steel structures. Design, manufacturing and installation regulations. [Building codes]. Kyiv: Minbud Ukrayini, 2011. 202 p. (ukr). [14] Haldar A, Mahadevan S, Stojadinovic B. Probability, reliability, and statistical methods in engineering design. Wiley; 1999. p. 320. [15] Nowak A, Collins K. Reliability of structures. second ed. CRC Press; 2012. p. 407. [16] Mushchanov VF, Annenkov AN, Orzhekhovsky AN. The probabilistic nature of the geometric imperfections of the spatial cantilever frame roof structures above the stadium stands. J Metal Struct 2014;20(3):169–78. Donbas National Academy of Civil Engineering and Architecture. [17] Taras A, Huemer S. On the influence of the load sequence on the structural reliability of steel members and frames. Structures 2015;4:91–104. https://doi.org/10. 1016/j.istruc.2015.10.007. [18] Nour A, Slimani A, Laouami N. Foundation settlement statistics via finite element analysis. Comput Geotech 2002;8:641–72. https://doi.org/10.1016/S0266-352X (02)00014-9. No. 29. [19] Mushchanov VF, Orzhekhovsky AN, Zubenko AV, Fomenko SA. Refined methods for calculating and designing engineering structures. Mag Civ Eng 2018;2:101–15. https://doi.org/10.18720/MCE.78.8. [20] Gorokhov EV, Mushchanov VF, Priadko IN. Ensuring the required level of reliability during the design stage of latticed shells with a large opening. J Civil Eng Manage 2015. No.;21(3):282–9. https://doi.org/10.3846/13923730.2015.1005020. [21] Hahn B. Essential MATLAB for engineers and scientists. 6th ed. Academic Press; 2016. p. 428. [22] Gorodeckiy D.A., Barabash M.S. LIRA-CAD 2013. Tutorial. Kyiv-Moscow, 2013. 376 p. (rus). [23] Mushchanov VF, Orzhekhovsky AN. An experimental researching of the strength and geometric parameters of roll-formed welded rectangular section produced in Ukraine. J Metal Struct 2013;19(3):173–81. Donbass National Academy of Civil Engineering and Architecture.
CRediT authorship contribution statement Anatoliy Orzhekhovskiy: Conceptualization, Methodology, Software, Writing - review & editing. Iurii Priadko: Data curation, Writing - original draft, Supervision. Anton Tanasoglo: Software, Validation, Visualization. Serafim Fomenko: Formal analysis, Investigation. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.engstruct.2020.110245. References [1] Summerville N. Basic reliability: an introduction to reliability engineering. AuthorHouse 2004;136:p. [2] Eurocode – Basis of structural design: EN 1990:2002+A1. [European Standard]. Brussels: Management Centre, 2002. 116 p. [3] DBN V 1.2-14-2009. General principles of reliability and constructive safety provision for buildings, structures and foundations. [Building codes]. Kyiv: Minbud Ukrayini, 2009. 49 p. (ukr). [4] GOST R 54257-2010. Reliability of building structures and foundations. Basic requirements. [Building codes]. Moscow: Standardinform, 2011. 20 p. (rus). [5] Yun X, Gardner L. Numerical modelling and design of hot-rolled and cold-formed steel continuous beams with tubular cross-sections. Thin-Walled Struct 2018;132:574–84. https://doi.org/10.1016/j.tws.2018.08.012. [6] Kudzys A, Lukoseviciene O. Conventional stochastic sequences in reliability assessments and predictions of structural members. Procedia Eng 2013;57:642–50. https://doi.org/10.1016/j.proeng.2013.04.081.
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Design of stadium roofs with a given level of reliability a,⁎
Anatoliy Orzhekhovskiy , Iurii Priadko
b,c,d
a
, Anton Tanasoglo , Serafim Fomenko
T a
a
Theoretical and Applied Mechanics Department, Donbas National Academy of Civil Engineering and Architecture, Ukraine Beijing International Education Institute, 38 East 3rd Ring North Road, Chaoyang, Beijing 100026, China Huzhou Vocational and Technical College, School of Civil Engineering, No.299, Xuefu Road Modern Teaching District, Huzhou, Zhejiang 313000, China d Theoretical and Applied Mechanics Department, Kyiv National University of Construction and Architecture, Ukraine b c
A R T I C LE I N FO
A B S T R A C T
Keywords: Structural reliability Steel structures Stadium roof Random variable Stochastic parameters Safety factor
The article discusses method for the selecting cross-section of steel rod elements of the roof structure placed above the stands of stadiums. Particular attention is paid to ensuring a given level of structural reliability. An algorithm to determine the actual safety factors for the most critical elements in the roof structure is proposed. Snow load, cross-sectional area of the elements, ultimate stress of the material, settlement of the base, mounting imperfections and defects are considered as the stochastic parameters in calculation of the reliability indicators. All specifications of the distributions for these magnitudes are based on the real experimental data. Since the building regulations do not provide a clear algorithm for calculating the reliability indicators of unique (and any other) building structures, it is possible to apply the proposed methodology both to ensure the design reliability of the whole structure as well as the separate most critical structural elements. It also might be applied to other kinds of spatial steel rod structures whose elements work as trusses. The internal force factor in the elements of such structures is longitudinal force. The method has been tested on the stadium structures of FC Olympic in Donetsk city (Ukraine).
1. Introduction Nowadays, the design of steel cantilever frame structures of roofs above the stadium stands is based on the method of limit states. This method is the basis of such building regulations as EUROCODE (European Union), DBN (Ukraine), SNiP and GOST (Russia) etc. Many controversial issues in determining the structural reliability arise when using the methodology proposed in these documents [1–4]. The method of limit states requires comparing the design stress in element (Qp) with the ultimate stress (Ry). In case of Qp < Ry, it is considered that reliability is ensured. Thus, the reliability of structures in all design cases is approximately the same and depends only on the design scheme and material of the structure. EUROCODE [2], DBN [3] and GOST [4] try to overcome this shortcoming. They rank constructions according to the levels of accidents consequences (economic and human losses) and criticality rating, as well as give the limits of allowable probability of failure for this classification. Moreover, the method of determining the probability of failure is given extremely generalized and does not have a clear algorithm of calculation in particular cases. In this case, the probability of failure of the designed structure remains unknown to the engineer. Therefore, current design standards for unique structures and structures with high criticality rating need to be refined. It is proposed
⁎
in this article to carry out numerical calculation of reliability and survivability at design stage for such kinds of unique structures as steel rod roofs over the stadium stands. Such random variables and factors as design strength of material, mounting imperfections and defects, magnitude of temporary loads (snow load in particular) and settlement of base are taken into account in calculation process. According to the analysis of literature, it is concluded that many authors solve problem of ensuring the reliability of building structures within constructive methods, i.e. the required level is ensured either by using stronger materials or increasing of elements cross sections or applying the most reliable design schemes etc. [5]. This approach may not always be applicable in terms of the economic requirements for the design, as well as matter of the numerical determination of reliability indicators remains open. A number of authors carried out research for particular cases [6,7]. They propose various approaches to improve common methods to determine reliability indicators (in some cases calculation examples are attached). However, these methods are not complete within a complex approach to determine the reliability of structures. A number of authors analyzed the reliability of complex structures and proposed methods for determining its reliability indicators [8–10]. However, they describe techniques and structures that are not suitable for steel rod roofs above the stadium stands. The topic
Corresponding author. E-mail address: [email protected] (A. Orzhekhovskiy).
https://doi.org/10.1016/j.engstruct.2020.110245 Received 10 June 2019; Received in revised form 15 December 2019; Accepted 14 January 2020 0141-0296/ © 2020 Elsevier Ltd. All rights reserved.
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Fig. 1. Roof structure above the stadium stands of FC Olympic in Donetsk city (Ukraine).
reliability is:
̂ > 0; Y(t) = R (t) − S(t)
ˇ
(1)
where: Ř(t)- bearing capacity of a structure or element; Ŝ(t)- loading applied on the structure (stress in structural element); Ŷ(t)- strength reserve or bearing capacity reserve. The probability of failure may be represented as an integral: ∞
Q (t ) = ∫ R (t ) pS (t ) dt; 0
(2)
where: R (t ) - probability distribution function of a random variable R ; pS (t ) - probability density of a random variable. The values stress in material [12], geometric parameters of the cross-sections of structural elements [16], magnitude of temporary loads (snow load in particular) [17] and settlement of the base [18] are considered as random variables. For the purpose of probabilistic calculation, authors chose common method Monte Carlo for studying random processes which has been tested and proved its effectiveness for building structures [19]. Since the system is many times statically indeterminate, it is most advisable to calculate reliability indicators for the most critical elements. An iterative geometrically non-linear calculation of the structure is performed for this purpose. As a result of the calculation, the order of collapse of the structural elements is set. As a criterion of collapse, it is proposed to consider the occurrence of the first limit state in each rod. This condition is chosen because the second limit state will not cause such large-scale economic and human consequences for considered kinds of structures. At the next stage, an iterative calculation for the selected elements is performed, taking into account the specified random variables. Thus, a sample of stresses for the group of the considered rods (Ŝ(t)) is formed. The sample should have a significant amount. Studying this problem, Pichugin S.F. [12] gives the range of numbers in the interval 104–108. The second component of Eq. (1) is the random variable of the strength of the material of construction (the yield strength in this case) Ř(t). It is formed on the basis of the analysis of statistical data obtained at metallurgical factories. The obtained two generalized random variables are processed by the methods of mathematical statistics. Their mathematical laws and density distribution are determined as well. If it is impossible or difficult to perform this task, it is recommended to select the most appropriate distribution and approximate a random variable with an accuracy that satisfies the calculation. Having types of distribution of random variables, their mathematical parameters such as expectation and standard deviation are calculated. Using the density, laws and parameters of distributions, the probabilities of failure for a group of selected elements are calculated. The resulting failure probability is the upper limit of failure [20]. This
Fig. 2. Design scheme of the cantilever frame structure of the roof.
of complex structures reliability is also extensively covered in the works [11,12]. Based on these works, an extensive research work was carried out in this direction, which was used in the development of building codes (DBN) for the design of steel structures [13] and for ensuring the reliability of structures [3]. Unfortunately, these authors did not provide a single method to determine reliability indicators. All of the above works are based on classical research in this scientific field [14,15]. They describe in detail a number of methods to determine the probability of failure and safety parameters (as indicators of system reliability), but using simple structures as an example, which causes a number of controversial points when calculating these indicators for complex multi-element systems. This article describes method to determine the cross-sections of rod elements in steel cantilever frame structures of roofs above the stadium stands, taking into account upper limit of reliability which means that failure of one rod will not result failure of the entire system. The described method is also applicable to other spatial steel rod structures whose elements work as trusses. The internal force factor in elements of such structures is longitudinal force.
2. Research methods The task of ensuring a given level of reliability in the structural calculation within the limit state method falls on a number of safety factors. Normally, such factors are indicated incorrectly or not considered at all for unique structures (such as static steel rod roofs over the stadium stands). Thus, reliability requirements for such structures may not be fully met. As we know from the work [12] the basic equation of the structural 2
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method satisfy not only the requirements of the limiting states calculation, but also correspond to the specified level of reliability.
Table 1 The cross-sections of the steel rods of the roof structure. №
Cross-sections
Size
1 2 3 4 5
I-beam Channel Roll-formed welded rectangular section Roll-formed welded square section Roll-formed welded rectangular section (purlin on the cantiliever)
][ №16 [ №16 □ 100 × 60 × 3 □ 60 × 2,5 □ 200 × 60 × 4,5
3. Subject of research This article discusses the method of selecting cross-sections of rods in the static steel cantilever frame structures of roofs above the stadium stands within the example of an existing section of the roof that overhangs southern stand of football stadium which belongs to FC Olympic in Donetsk city. The structural scheme of the static roof represents the L - shaped plane cantilever frames (as the main bearing structures) with 6 m pitch center-to-center, which are linked by trussed bracings and purlins on the cantiliever (Fig. 1). All joints are rigid. This means that the structure is many times statically indeterminate system (Fig. 2). Fixing to the foundation is constructively done in such a way that it is advisable to consider it as a rigid joint in the design scheme as well. The crosssections of the structural elements are shown in the Table 1 (see Fig. 3).
means that elements in a system are so arranged that failure of one rod will not result the failure of the entire system. There is also a lower limit of failure, that assumes the sequential inclusion of elements in the work, but the probability of failure in this case is much higher and we will not consider it. The obtained reliability parameters are compared with the normative ones taking into account the reliability class and consequences class of the considered structure [2,3]. If the requirements of reliability are met, the calculation ends. If the reliability is not ensured, the partial reliability indicators are recalculated and the inverse problem is solved, the purpose of which is to determine the required value of the expectation of stresses in the structural elements and the required crosssection areas of the rods. The structure with the newly defined crosssection proportions is recalculated anew according to the algorithm described above. Iterations are carried out until the strength conditions are satisfied by the considered group of elements. The cross-sections of rods in static steel cantilever frame structures of roofs above the stadium stands obtained by the described above
4. Research algorithm Nowadays, authors have not found a universal algorithm for solving the set task. Therefore, a number of programs have been written by means of high-level interpreted programming code MATLAB [21]. All of the following calculations have been implemented within these programs. The testing and debugging of the resulting software package were performed using the LIRA-CAD 2013 software package [22]. It is reasonable to split the created software package into the 2 blocks:
a)
b)
Fig. 3. Geometrical scheme with indication of cross-sections types in structural elements: (a) plane frame; (b) trussed bracing. 3
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Fig. 4. The flowchart to determine the most critical structural elements.
4.1. Block 1
1. Determination of the most critical elements in the structure, which will define its reliability in general. 2. Forming a stresses samples of a group of the most critical elements in the system. Processing of the stresses samples and yield strength using the methods of mathematical statistics. Calculation of the reliability criteria.
Nonlinear design calculation is based on the finite element method (FEM) in the spatial interpretation. The geometric non-linearity of construction is taken into account as follows: 1. The coordinates of the joints are updated at the each iteration, taking into account their movement at the previous one. 2. The stresses arised at the previous iteration due to the displacement of structural joints are taken into account at the stage of forming the element stiffness matrix.
Both of the blocks within example of mentioned above structure are described below.
4
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The elements of the considered roof structure that are shown on the Figs. 5–8 have been failed at the fourth iteration. 4.2. Block 2 Simulation of snow load. Snow load as a random variable was taken with a double exponential Gumbel distribution law [12]. The equation describing the sample of the snow load values for the Donetsk city has been obtained by the method of mathematical inversion.
S = 521.48 − 270.413·ln(−1.0·ln(R + 0.0375))
(3)
where R is generator of the uniform random variable within [0… 0,935]. The necessary parameters of the Gumbel distribution are obtained by analyzing the statistics of annual maximum snow load during the 40 years. Analyzing the Eq. (3) it is clear that the termln(R + 0.0375) < 0 must be met, so that the value of the snow load does not move into the range of complex numbers. In case of using the maximum value of R = 0.935 this will not happen. This equation has a snow load ranging from 200 to 1500 Pa, which corresponds to the nature of the load for current region. Modeling the strength parameters of the material and the geometric parameters of the cross-sections. The generation of the geometric parameters of cross-sections and strength parameters is based on the analysis of rolled metal products in two metallurgical factories: Pipe Producing Factory in Lugansk city and Comintern Metallurgical Factory in Dnepr city [23]. A sample of the strength properties of cross-section as well as a sample of geometric parameters of all sections have been subjected to χ2-analysis to determine the possibility of approximating the distribution of random variables by the normal distribution law of Pearson criterion [14]. The analysis showed that the distribution of each random variable differs from the normal one by no more than 5% (a significance level of 0.05 has been set). Therefore, these random variables might be considered within normal distribution law (Table 2 and 3). Simulation of geometric imperfections and settlement of the base. Settlement of the foundations and geometrical imperfections of the structure have been taken into account by the geodetic survey analyzing the roof above the southern stand of the football stadium which belongs to FC Olympic in Donetsk city (Figs. 9 and 10). The survey of the structure's geometry took place in two cycles and gave the results displayed on the graphs and histogram (Figs. 11–13). Analyzing the information received, there is a significant excess of deviations that allowed by regulatory documents (Figs. 11 and 12). The data are taken into account at the stage of formation of local stiffness matrices in the design scheme. Representing the deviations of the geometry of points as a random variable, several distribution laws were considered for approximation (χ2, lognormal and normal distribution). Normal distribution law has been taken as an approximating one (Fig. 13). Additionally, the considered random variable was subjected to χ2 analysis by the Pearson criterion [14]. The analysis showed that the distribution law of deviations of a random variable at the considered points differs from the normal one by no more than 5%. Therefore, the normal distribution law might be considered as an acceptable one. Knowing the distribution laws of all considered random variables and using a generator of random variables, a sample of 10,000 random variables has been formed. Further, applying the deterministic calculation based on the finite element method, the samples of stresses in the eight elements were determined. The obtained random variables of the stresses and yield strengths were analyzed using the methods of mathematical statistics (χ2 analysis, Shapiro – Uilk criterion) [14]. Since the obtained samples of stresses in the rods and the yield strength are approximated by a normal distribution law, the reliability index β
Fig. 5. Structural elements that failed at the 4th calculation cycle (marked with red colors). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Structural elements that failed at the 5th calculation cycle (marked with red colors). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. The order of failure of structural elements at the each stage of loading.
A snow load is applied to the joints of the considered roof structure and than an iterative calculation is made until the specified amount of elements are failed. A load of 1 kN is additionally applied at the each calculation cycle. In case if any element has failed in the previous cycle, it is excluded from system, after which system is recalculated again without additional load. Thus, phenomenon of avalanche destruction is taken into account (Fig. 4). 5
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Fig. 8. The position of the rods that determine the reliability of the structure.
Table 2 Experimentally obtained stochastic parameters of the cross-section dimensions of roll-formed welded sections. No
1 2 3 4
Cross-section
60 × 40 × 3 120 × 60 × 3 100 × 100 × 3 100 × 60 × 3
Number of measurements
250 250 250 250
Wall thickness (mm)
Section height (mm)
Section width (mm)
Expectation
Standard deviation
Coefficient of variation (%)
Expectation
Standard deviation
Coefficient of variation (%)
Expectation
Standard deviation
Coefficient of variation (%)
2,96 2,98 2,93 2,89
0,25 0,19 0,26 0,23
8,52 6,63 9,10 8,00
58,98 118,07 99,13 97,51
1,48 3,98 5,60 5,63
2,54 3,4 5,65 5,77
38,77 58,39 98,34 57,79
2,15 2,79 5,85 1,71
5,54 4,77 5,95 2,96
Table 3 Experimentally obtained strength parameters of the roll-formed welded sections. No
1 2 3 4
Cross-section
Expectation
60 × 40 × 3 120 × 60 × 3 100 × 100 × 3 100 × 60 × 3
Standard
Numerical standard (μ)
σ¯B
σ0,2 ¯
σB̂
̂ σ0,2
482,58 443,67 428,85 418,54
317,44 332,21 323,63 379,03
22,38 31,07 28,53 25,84
35,15 22,07 31,88 29,46
3,15 3,15 3,15 3,15
γc = 1 −
(
σ¯ p 1 − μσ k σ
− σ0p
R¯ yn (1 − μr kr )
);
Standard value (MPa)
n Rym
n Run
Ryn
Run
206,72 262,69 223,21 286,23
411,50 345,80 338,98 337,14
225 225 225 225
370 370 370 370
ordinary deterministic design calculation.
(failure range) [2] was calculated as a probability parameter. The safety factor γc [3] is proposed to be calculated as follows [12]:
γm
Exponential value (MPa)
The flowchart of the calculation is presented on Fig. 14. (4)
5. The research results
where:
The obtained values of β were compared with regulatory ones [2,3]. Regulations requirements are not met (β ⩾ βiex ). The required value of the mathematical expectation of the cross-sectional area in the elements is calculated. New cross-sections are set, after which the second block of the software is fully recalculated, taking into account the gained safety factors. The calculation results are summarized in Table 4. Analyzing the results it is concluded: (1) the symmetry of the structure is obvious; (2) the calculated β and γc values for structures with high consequences class differ significantly from those recommended by the DBN regulations and Eurocode. The following are the designations of variables to define P(t) and β specified on Fig. 14, whereas γc was explained in (4):
γm - material reliability factor; σ p - standard deviation of stresses arising in the considered structural element; μσ , μr - mathematical expectation of the stress values in the considered structural element and the mathematical expectation of a random variable of the material strength (the yield strength); k σ , kr - variation coefficients of two considered random variables; R yn - the standard value of the designed material strength of construction; σ0p - the designed value of the stresses in the element obtained by 6
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Fig. 9. Scheme of geodetic survey for monitoring the structures of the stadium stands.
S – random variable of stresses in the structural elements; m- sample size. 6. Conclusion 1. A study of limit states method has revealed that it does not provide required level of reliability for unique structures such as static steel cantilever frame structures of roofs above the stadium stands. Therefore, a new design method has been devised which apply direct methods of the reliability theory for determination of the crosssections in structural elements, determination of reliability indicators and subsequent specification of β and γc. 2. The recommended by regulations general measures for providing structural reliability, in particular Eurocodes (grading by consequences classes (CC) of structure, reliability classes (RC) etc.) do not always provide the required level of reliability βiex for structures with high consequence for loss of human life or economic, social or environmental consequences. The reliability of individual key structural elements could be most easily provided by adjusting the values of the safety factor γc. However, their values are proposed without taking into account the level of consequences class of structure in the existing regulations, so that they need to be clarified. The method described in this article solves this issue and takes into account an individual approach while ensuring the required level of reliability of the building structure. 3. Since the building regulations do not provide a clear algorithm for calculating the reliability indicators of unique (and any other) building structures, it is possible to apply the proposed methodology for the structural rod roof system placed above the stadium stands both to ensure the design reliability of the whole structure as well as the separate most critical structural elements. 4. The described above method is devised for the static steel cantilever frame structures of roofs above the stadium stands. However it might be applied to other spatial steel rod structures whose elements work as trusses. The internal force factor in the elements of such
Fig. 10. The scheme of control points on the structures of the stands.
SR - standard deviation of the random variable of material yield strength shown in Tables 2 and 3 (sample obtained experimentally); SQ - standard deviation of the random variable of stresses in the structural elements; ∼ R - mathematical expectation of the random variable of material yield strength; ∼ Q - mathematical expectation of the random variable of stresses in the structural elements; P(t) - random variable of failure probability; F(S) - random variable of structural strength reserve; F¯ (S) - mathematical expectation of structural strength reserve; 7
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Fig. 11. The deviation of the elevations in point «A».
Fig. 12. The deviation of the elevations in point «B»
Fig. 13. Histogram of the deviations distribution of the structure joints coordinates with an approximating curves.
8
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Fig. 14. Flowchart of the algorithm for determining the reliability criteria in static steel cantilever frame structures of roofs above the stadium stands. Table 4 The results of determination of the cross-sectional area in the structural elements considering the requirements of reliability. Element’snumber
Initial cross-section Cross-section
32 33 60 61 88 89 116 117
□ □ □ □ □ □ □ □
100 100 100 100 100 100 100 100
× × × × × × × ×
60 60 60 60 60 60 60 60
× × × × × × × ×
3 3 3 3 3 3 3 3
Determined cross-section β
γc
Cross-section
4,4255 4,6404 4,0775 4,2966 4,0775 4,2966 4,4255 4,6404
0,9729 0,9742 0,9708 0,9722 0,9708 0,9722 0,9729 0,9742
□ □ □ □ □ □ □ □
9
100 100 100 100 100 100 100 100
× × × × × × × ×
60 60 60 60 60 60 60 60
× × × × × × × ×
5 5 5 5 5 5 5 5
Standard value β
γc
γc (DBN)
βiex (DBN)
4,9827 5,1573 4,7634 4,8135 4,7634 4,8135 4,9827 5,1573
0,9734 0,9748 0,9712 0,9725 0,9712 0,9725 0,9734 0,9749
0,9
4,76
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structures is the longitudinal force.
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CRediT authorship contribution statement Anatoliy Orzhekhovskiy: Conceptualization, Methodology, Software, Writing - review & editing. Iurii Priadko: Data curation, Writing - original draft, Supervision. Anton Tanasoglo: Software, Validation, Visualization. Serafim Fomenko: Formal analysis, Investigation. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.engstruct.2020.110245. References [1] Summerville N. Basic reliability: an introduction to reliability engineering. AuthorHouse 2004;136:p. [2] Eurocode – Basis of structural design: EN 1990:2002+A1. [European Standard]. Brussels: Management Centre, 2002. 116 p. [3] DBN V 1.2-14-2009. General principles of reliability and constructive safety provision for buildings, structures and foundations. [Building codes]. Kyiv: Minbud Ukrayini, 2009. 49 p. (ukr). [4] GOST R 54257-2010. Reliability of building structures and foundations. Basic requirements. [Building codes]. Moscow: Standardinform, 2011. 20 p. (rus). [5] Yun X, Gardner L. Numerical modelling and design of hot-rolled and cold-formed steel continuous beams with tubular cross-sections. Thin-Walled Struct 2018;132:574–84. https://doi.org/10.1016/j.tws.2018.08.012. [6] Kudzys A, Lukoseviciene O. Conventional stochastic sequences in reliability assessments and predictions of structural members. Procedia Eng 2013;57:642–50. https://doi.org/10.1016/j.proeng.2013.04.081.
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