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An approach for adjusting the tensile force coefficient in equivalent static cable-loss analysis of the cable-stayed bridges
Structures 25 (2020) 720–729
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An approach for adjusting the tensile force coefficient in equivalent static cable-loss analysis of the cable-stayed bridges Mohammad Ali Fathali, Ehsan Dehghani, Seyed Rohollah Hoseini Vaez
T
⁎
Department of Civil Engineering, Faculty of Engineering, University of Qom, Qom, Iran
A R T I C LE I N FO
A B S T R A C T
Keywords: Cable-stayed bridges Cable rupture Cable rupture duration Linear static analysis Time history analysis Optimization
The sudden rupture of cable is one of the factors that can threaten the bridge safety; causing progressive collapse due to severe vibrations along with numerous changes in the internal forces of all bridge members. Therefore, analyzing the bridge behavior after the sudden cable rupture and estimating the maximum response of the bridge members is of great importance. According to Post-Tensioning Institute (PTI), the sum of the static analysis responses of two models can be used instead of dynamic analysis of cable rupture. In one of these models, the ruptured cable should be removed and twice its tensile force should be applied at the two ends of the cable. In this study, an approach proposed for investigating the tensile force coefficient (TFC) of the ruptured cable in the model of cable rupture effect. For this purpose, an optimization problem was defined to minimize the difference between the static and the maximum dynamic analysis results. The performance of the proposed approach is illustrated with a three-span cable-stayed bridge. This problem was solved for the rupture of each of the bridge cables under examination at different durations of cable rupture using ECBO meta-heuristic algorithm. Moreover, this meta-heuristic algorithm was used to adjust the tensile forces of the cables. Based on the results, a general form of equations was found between cable rupture duration and TFC, which the maximum of it, is in accordance with the PTI coefficient. According to this equation, increasing the cable rupture duration reduces this coefficient.
1. Introduction The occurrence of some failures in a structure can cause a rapid spread of damage to other members and parts. As such, the failure proceeds rapidly and leads to eventual collapse of all the structure or some parts of it, called progressive collapse. Due to its sudden occurrence, progressive collapse of the structures must be prevented. On cable-stayed bridges, this type of failure can occur due to the cable rupture. Cable rupture can be due to a variety of reasons, such as reduced cross-sectional area due to corrosion (cables are more susceptible to corrosion than other bridge components), fatigue, increased axial force due to lateral effects, reduced resistance due to fire or intentional attack. Sudden rupture of the cable can threaten the safety of the bridge; as the loss of a cable causes severe vibrations on the bridge, causing many changes in the internal forces of all structural members. Cable rupture and its consequences on structures such as submerged floating tunnel [1], suspen-dome [2], suspended arch bridges [3] and cable-stayed bridges [4,5] have been studied. Hoang et al. investigated the vibration performance of the cable-stayed bridge after the sudden rupture of a cable due to lateral force using numerical analysis and
⁎
model experiments [6]. In another study, Wu et al. evaluated the behavior of the self-anchored suspension bridge caused by hanger breakage using nonlinear analysis and considering the effect of the corroded hangers [7]. Starossek investigated and categorized the types of progressive collapse and introduced the loss of a cable in the cablestayed bridge leading to the zipper-type progressive collapse [8]. RuizTeran and Aparicio investigated and compared the dynamic amplification factors (DAF) related to deflections, flexural moments and shear forces in the case of oscillation caused by cable rupture in cable-stayed bridges [9]. In the another study, they performed a parametric investigation of the responses of two different types of under-deck cablestayed bridges in sudden cable rupture using a linear static analysis method and DAF [10]. They considered parameters such as the cable rupture duration, the number of ruptured cables, and the amount of traffic live load. In another parametric study, Mozos and Aparicio investigated the influence of characteristics such as stay-cables layout and deck stiffness on the dynamic response caused by sudden cable rupture [11,12]. They also studied the effect of cable rupture duration on the response of cable bridges in another study [13]. Considering the geometrical and material nonlinearity effects, Cai et al. investigated the
Corresponding author. E-mail addresses: [email protected], [email protected] (S.R. Hoseini Vaez).
https://doi.org/10.1016/j.istruc.2020.03.054 Received 27 November 2019; Received in revised form 17 March 2020; Accepted 25 March 2020 2352-0124/ © 2020 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
Structures 25 (2020) 720–729
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Fig. 1. Schematic of the cable-stayed bridge.
estimate maximum responses for numerical studies such as progressive collapse. The rest of the study is organized as follows: Section 2, presents a description of the numerical modeling and linear static analysis of cable-loss. Section 3 is allocated to the definition of the objective function of the proposed approach and the explanation of ECBO metaheuristic algorithm. Section 4 provides the results of adjustment of the cables tensile force and the tensile force coefficient (TFC) of ruptured cable. Finally, conclusions are made in Section 5.
effects of progressive collapse with the simultaneous sudden rupture of multiple cables on nonlinear responses and bridge behavior, after which they compared the results with the rupture of one cable [14]. Zhou and Chen developed a framework to assess the reliability of a long-span cable-stayed and traffic system under cable rupture conditions [15]. Based on finite-element method, Zhou and Chen presented a new platform for non-linear dynamic simulation of cable rupture. Also, they attempted to improve the following in the proposed new framework: modeling the cable loss process, considering realistic traffic and wind loads through the bridge-traffic-wind interaction model based on finite-element, and adding material as well as geometrical nonlinearities with various sources [16,17]. On the basis of the provided platform, they further carried out a numerical study of cable rupture in long-span cable bridges and investigated post-rupture bridge performance [18]. Shoghijavan and Starossek studied the maximum flexural moment created on the girder after cable rupture using an analytical method based on system differential equations and obtained an approximate function for the cable rupture scenario [19]. Bedon et al. used inverse techniques for an experimental and theoretical investigation on the Pietratagliata cable-stayed bridge with damage phenomena in the cables, including analytical, experimental and numerical comparisons to assess the effects (and prevention) of cable rupture [20]. Also, in some other studies, the seismic performance of other components of cable-stayed bridges such as pylons [21] has been investigated. According to the Post-Tensioning Institute (PTI) [22], instead of using dynamic analysis, the elastic analysis of two different models along with superposition principle can be used to investigate the behavior of the bridge after the sudden cable rupture. The first model is the main bridge model (before the cable rupture) in which all dead and live loads are applied. The second model is a model in which the ruptured cable is removed and twice the tensile force of the ruptured cable is inserted inversely at its two ends. This model considers the effects of the cable rupture. In this study, an approach proposed to obtain the most appropriate tensile force coefficient (TFC) of the ruptured cable in the equivalent static cable-loss analysis of the cable-stayed bridges. For this purpose, an objective function is formulized so that the difference between the results of the dynamic analysis of the cable rupture and the equivalent static analysis is minimized. Also, the bending moment values of the deck at the location of the cables and the pylons connections to the deck should be equal in the main bridge modeling. For this purpose, another optimization problem is defined for adjusting the amount of tensile forces of the cables. To solve these optimization problems, the enhanced colliding bodies optimization (ECBO) algorithm is used. A three-span cable-stayed bridge considered for evaluating the performance of the proposed approach. In order to investigate the effect of each cable on the bridge performance, the problem of obtaining the most appropriate TFC is solved for each ruptured cable with rupture duration of 0.01 s. It is clear that the shorter the duration of the cable rupture, the stronger the vibration in the bridge. In this regard, the main optimization problem is solved for the different durations of cable rupture to evaluate the overall validity of the proposed approach. Then a suitable cable rupture duration is suggested for cable ruptures to
2. Description of the numerical modeling and analysis 2.1. Numerical modeling Fig. 1 indicates the geometry of the cable-stayed bridge under study. The distance of the cable connection to the deck is 10 m. For the bridge deck, a rectangular concrete section of 0.8 m thick and 10 m wide is considered. Also, the pylon cross-section is defined as a concrete pipe section with a diameter of 3.0 m and a thickness of 0.3 m. Table 1 indicates the nominal properties of materials. Optimization techniques for multi-objective analyses on bridges confirmed that the mechanical properties of materials (including the modulus of elasticity of concrete) are sensitive to loading rate and manifest in a modulus variation under dynamic loads [23]. Because the main purpose of this study is to present an approach for obtaining the most appropriate TFC, the sensitivity of the mechanical properties of the materials to the loading rate has not been considered in order to simplify. The deck distributed gravity load is 39.2266 kN/m. The list of the cross-sectional area of the cables is as reported in Table 2. An open-source software based finite-element method (OpenSees software) was used for two-dimensional (2D) modeling and analysis of cable-stayed bridge [25]. Since cables are tensile members, the cables are modeled using “Truss” elements with initial axial strain assigned by “Initial Strain” material. As the analysis is linear, the “elastic beamcolumn” element is used to model the deck and pylons. Also, a “zerolength” element was used to model the neoprene performance in the transfer of forces between the deck and the pylon; such that the flexural moment is not transmitted, and the horizontal and vertical forces are transferred by defining elastic behavior using an “Elastic Uniaxial” material with low (3 kN/mm) and high (108 kN/mm) stiffness, respectively [26]. To model the cable rupture in dynamic analysis, the considered cable for rupture was removed from the bridge model and instead its axial force was applied proportional to a time series at both ends. The Table 1 Material property [12].
721
Parameter
Steel (Gr270 [24])
Concrete
Modulus of elasticity, E (GPa) Weight per unit volume, γ (kN/m3) Compressive Strength, fc' (MPa) Yield strength, fy (MPa) Ultimate strength, fu (MPa)
190 76.93 – 1690 1860
36.342 24.50 40 – –
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force (TFC) in the second model is investigated.
Table 2 The list of the cross-sectional area of the cables. Cable
Cross-section (mm2)
Number of strands*
C1-C18 C2-C17 C3-C16 C4-C15 C5-C14 C6-C13 C7-C12 C8-C11 C9-C10
18,000 6300 11,400 18,000 18,000 12,000 18,000 14,100 17,700
120 42 76 120 120 80 120 94 118
3. The proposed approach 3.1. General form of an optimization problem The general form of an optimization problem is according to Eq. (3):
find X = {x1, x2 , ...,x n } to minimize f (X) subject to gi (X) ⩽ 0, i = 1, 2, ...,nc where x min ⩽ x ⩽ x max
* Cross-sectional area of each strand = 150 mm2.
(3)
time series of applying the axial force is such that it first applies the force completely and then it reaches zero after the considered duration for the cable rupture.
in which, X represents the random variables vector with minimum and maximum vectors of xmin and xmax, respectively. Moreover, f indicates the objective function, and the problem is aimed at minimizing the value of this function. The function of gi indicates the function of the ith constraint of the problem.
2.2. Adjustment of the cables tensile force
3.2. Formulating the objective function to adjusting the TFC
The purpose of adjusting the cables tensile force is to equate the flexural moment value of the deck in the location of the cables (except back-stay cables) and the pylons connections to the bridge deck with the following equation:
In this study, an unconstrained optimization problem with fR objective function is defined by Eq. (4) in order to scrutinize the tensile force coefficient of the ith ruptured cable:
Mt =
NC − 1
fR =
L2
w. 12
l=1
(4)
where, Zc is the function of calculating the relative differences of the maximum tensile force of jth cable in the two states of static analysis proposed by PTI and time history analysis of cable rupture and calculated according to Eq. (5):
ZCj =
(FCs )j − (FCd )j (FCd )j
(5)
where, (FCs)j indicates the jth cable tensile force in static analysis and (FCd)j indicates the maximum tensile force produced in jth cable due to ith cable rupture, obtained from time history analysis. Moreover, NC represents the number of cables. In Eq (4), ZDk is the function of calculating the relative differences of the maximum flexural moment of the deck in the kth location of cables or pylon connections to the deck in the two states of static analysis proposed by PTI and time history analysis of cable rupture, and calculated according to Eq. (6):
(2)
i=1
k=1 j
ND
∑ |Mi − Mt|
NP
∑ ZDk + ∑ ZPl
j=1
(1)
In the Eq. (1), Mt is the target flexural moment of the deck in the location of cable or pylon connection to the deck, while w indicates the sum of distributed gravity loads per unit length of the deck and the unit weight of the deck length, and L is the distance of cable connection to the deck. Adjusting the tensile force of the cables to achieve a defined goal, needs a great amount of trial and error, and different ways have been suggested for it. In this study, an unconstrained optimization problem which is solved using the meta-heuristic algorithm is defined to adjust the tensile force of the cables. The objective function is the optimization problem defined as follows:
Min F =
ND
ZCj +
∑
where, Mi is the flexural moment of i connection of the deck to the cables or pylon, while ND indicates the number of connections of the deck to the cables (except back-stay cables) and pylon. The variables in this problem are the initial strain values applied from the [0, 1] interval for each cable. th
ZDk =
(MDs )k − (MDd )k (MDd )k
(6)
in which, (MD )k indicates the flexural moment of the deck in the location of kth connection of cables or pylons to the deck in the static analysis and (MDd)k indicates the maximum flexural moment produced at the kth cables or pylons connection to the deck due to ith cable rupture, obtained from time history analysis. Moreover, ND represents the number of connection of cables or pylons to the deck. Also in Eq. (4), ZPl is the function of calculating the sum of the relative differences of the maximum flexural moment of the lth pylon end and its connection to the deck in the two states of static analysis proposed by PTI and the cable rupture time history analysis. ZPl is calculated by Eq. (7): s
2.3. Linear static analysis of cable-loss Sudden cable rupture due to different reasons such as corrosion and intentional explosion leads to severe vibrations in the bridge. In other words, sudden cable rupture leads to considerable changes and excessive oscillations in the internal forces of the members and their deformation. To accurately analyze the behavior of the bridge after cable rupture, a dynamic time history analysis must be performed. However, according to PTI, linear static analysis can be used instead of dynamic analysis, and the results of the analysis of the following two models can be summarized using the superposition principle:
ZPl =
- First model: Modeling of the bridge with dead and live loads applied to it; - Second model: Modeling of the bridge without a ruptured cable and applying twice its tensile force at the two ends inversely.
(MPse )l − (MPde )l (MPsD )l − (MPdD )l + d (MP e )l (MPdD )l
(7)
are respectively flexural moments of lth in which, (MPe )l and pylon end and its connection to the deck in static analysis, while (MPed)l and (MPDd)l are respectively the maximum flexural moments produced at lth pylon end and its connection to the deck due to ith cable rupture, obtained by the time history analysis. Moreover, NP represents the s
In this study, the applied coefficient on the ruptured cable tensile 722
(MPDs)l
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Fig. 2. The flowchart of the proposed approach.
in which, mk and fit(k) are the values of the mass and the objective function of the ith colliding body (CB), respectively; and nCB is the number of algorithm population.
number of pylons. The variable of the defined optimization problem is the tensile force coefficient of the ruptured cable in the static analysis second model, considered at the interval of [1,5]. Fig. 2 represents the flowchart of this approach.
nCB ⎞ vi = 0 , ⎛i = 1, 2, ..., 2 ⎠ ⎝
3.3. Enhanced colliding bodies algorithm
vi = x i − x i − (nCB
Enhanced colliding bodies optimization (ECBO) algorithm [27] was introduced in order to improve the performance of colliding bodies optimization (CBO) algorithm. CBO algorithm is inspired by the physical laws governing collision of objects [28]. These algorithms have been used in many studies [29–33]. In ECBO algorithm, the population sorted in an ascending order based on their value of the objective function. This sorted population is divided into two equal groups: stationary and moving colliding bodies (CBs). The stationary group is the first half of the sorted population, and the moving group is the second half. Fig. 3 indicates the pseudo-code of ECBO algorithm. The parameters of this algorithm are: nCB (number of ECBO population), tmax (maximum number of ECBO iterations), nCM (size of the Colliding Memory (CM)) and Pro (within interval (0, 1)). The equations used in this algorithm are as follows:
1 ⎞ mk = ⎜⎛ ⎟ ⎝ fit (k ) ⎠
⎛i = ⎝
nCB nCB + 1, + 2, ...,nCB ⎞ 2 2 ⎠
mi + (nCB 2) + εmi + (nCB vi′ = ⎜⎛ mi + mi + (nCB 2) ⎝
2) ⎞ ⎟
vi + (nCB
2) ,
⎠
(10)
⎛i = 1, 2, ..., nCB ⎞ 2 ⎠ ⎝
mi − εmi − (nCB 2) ⎞ nCB nCB + 1, + 2, ...,nCB ⎞ vi′ = ⎜⎛ ⎟ vi , ⎛i = + m m 2 2 ⎝ ⎠ i i ( nCB 2) − ⎠ ⎝
(11)
(12)
th
In Eqs. (9) to (12), vi and vi' are the velocity of i CB before and after the collision, respectively; ε is the coefficient of restitution; and xi is the current position of ith CB.
nCB ⎞ x inew = x i + rand°vi′, ⎛i = 1, 2, ..., 2 ⎠ ⎝ x inew = x i − (nCB xinew
2)
+ rand°vi′, ⎛i = ⎝
nCB nCB + 1, + 2, ...,nCB ⎞ 2 2 ⎠ th
(13)
(14)
is the new position of i CB. Here, rand is a vector with a where, dimension of the number of problem variables (N), consisting of random numbers in the interval of (1, −1).
nCB
1 ⎞ ⎛ ⎜ ∑ fit (i) ⎟ = i 1 ⎠ ⎝
2) ,
(9)
(8) 723
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Fig. 3. The pseudo-code of the ECBO algorithm. Table 3 The axial force of cables before cable rupturing. Cable
C1-C18
C2-C17
C3-C16
C4-C15
C5-C14
C6-C13
C7-C12
C8-C11
C9-C10
Cable axial force before cable rupturing (kN)
9141
3852
2914
2459
2455
2916
3557
4297
5092
Fig. 4. Flexural moment of deck before cable rupturing. Table 4 The TFC values of the ruptured cables (cable rupture duration = 0.01 s). Ruptured cable
C1
C2
C3
C4
C5
C6
C7
C8
C9
TFC
1.6656
1.6993
1.5137
1.3443
1.4015
1.5113
1.3067
1.4330
1.7445
4. Results and discussion
of the population and the maximum iterations of algorithm are considered to be respectively 60 and 300 in order to solve the problem of cables tensile force adjustment. In Table 3, the tensile force obtained for each cable reported. Fig. 4 indicates the flexural moment of deck before cable rupturing.
4.1. Adjustment of the cables tensile force According to the load considered and the Eq. (1), the target moment value at the cables connection to the deck is 1897.76 kN.m. The values 724
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Fig. 5. Comparison of the flexural moment distribution in the deck for the rupture of cables a) C1, b) C4, c) C5 and d) C9, in the responses states of dynamic analysis (Dyn) and static analysis (calculated TFC, PTI TFC).
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Fig. 6. Comparison of the cable axial force for the rupture of cables a) C1, b) C4, c) C5 and d) C9, in the responses states of dynamic analysis (Dyn) and static analysis (calculated TFC, PTI TFC).
726
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Fig. 7. Comparison of the flexural moments of the pylons for the rupture of cables a) C1, b) C4, c) C5 and d) C9, in the responses states of dynamic analysis (Dyn) and static analysis (calculated TFC, PTI TFC).
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Fig. 8. The scatter plot of the TFCs against the cable rupture durations for the rupture of cable C1.
than or equal to 2. As a result, the maximum value of this equation for a very small rupture duration (less than 0.01 s) can be 2, which is in accordance with the proposed coefficient in the PTI. The first term of Eq. (15) indicates that as the cable rupture duration increases, the TFC generally decreases; because the longer the cable rupture duration, the lower the vibration will be in the cable-stayed bridge. Also, it can be find that there is no significant changes in the TFC for the cable rupture duration at the interval of 0.001 to 0.01 s. Accordingly, a cable rupture duration of 0.01 s is appropriate for dynamic analysis of the sudden cable rupture.
Table 5 The linear fitting equation between the duration of cable rupture and TFC. Ruptured cable
Equation
C1 C2 C3 C4 C5 C6 C7 C8 C9
TFC TFC TFC TFC TFC TFC TFC TFC TFC
= = = = = = = = =
−0.51 −0.95 −0.28 −0.33 −0.40 −0.45 −0.24 −0.32 −0.60
× × × × × × × × ×
RD RD RD RD RD RD RD RD RD
+ + + + + + + + +
1.66 1.85 1.50 1.32 1.38 1.46 1.31 1.36 1.77
5. Conclusion An abrupt rupture of the cable in structures such as cable-stayed bridges can disrupt the operation of the bridge by causing severe oscillations in the whole bridge. In this study, an approach is proposed to estimate the most appropriate TFC of the equivalent static analysis suggested by PTI. TFC is the coefficient of the tensile force of a ruptured cable in one of the equivalent static analysis models of tearing the cable. For this purpose, an optimization problem is defined in which the objective function is the difference between the equivalent static analysis responses and maximum responses of dynamic analysis in each bridge components (cables, pylons, and deck). This problem was solved for rupturing each of the cables of a three-span cable-stayed bridge by considering different durations of the cable rupture. As the results shown, the obtained TFCs by this approach makes the acceptable difference between the dynamic and equivalent static analysis results. This result shows the efficiency of the defined objective function considering the numerous terms. Also, a linear fit was obtained between the TFCs for each cable and cable rupture durations. Based on the resulting equations, a general form was found between TFC and cable rupture duration, in which the maximum of it is in accordance with the PTI coefficient. In this form, the coefficient of the cable rupture duration is negative, which is in accordance with this concept that the bridge vibration decreases with increasing cable rupture duration. Also, it can be found that for the cable rupture duration at the interval of 0.001 to 0.01 s there are no significant changes in the TFC.
4.2. Adjustment of the TFC of ruptured cable To solve the optimization problem of TFC adjustment, the population and the maximum iterations of algorithm are considered 20 and 100, respectively. Due to the symmetry of the bridge, the TFC of the ruptured cable is calculated for rupture of the cables C1 to C9. The TFC values of the ruptured cables in the second model of static analysis are presented in Table 4, considering a rupture duration of 0.01 s. Figs. 5–7 show the flexural moment distribution in the deck, the cable axial force and the flexural moments of the pylons for the rupture of cables C1, C4, C5 and C9, in the responses states of dynamic analysis and static analysis. According to these figures, it can be observed that the result of static analysis gets closer to the maximum result of dynamic analysis in most cases, indicating appropriate definition of the objective function considering the numerous terms in the objective function. To investigate the effect of the cable rupture duration, the optimization problem has been solved by considering cable rupture duration in an interval of 0.001 to 1.0 s for each cable rupture. For example, Fig. 8 shows the resulting TFCs against the cable rupture duration considered for the rupture of cable C1. Regarding TFC values for each cable rupture at different duration of cable rupture, a linear fit between the results is considered to obtain a formula between the cable rupture duration and the TFC for each cable rupture. For example, the fit line for the TFC of the rupture of cable C1 is indicated in Fig. 8. The equations of the fit lines for each cable rupture are presented in Table 5. In these equations, RD is the duration of cable rupture. Considering the linear fit between the results, the following equation in general form can be defined between the cable rupture duration (RD) and the TFC:
TFC = −a × RD + b
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.
(15)
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cable-stayed bridges under stochastic traffic and wind. Eng Struct 2015;105:299–315. Shoghijavan M, Starossek U. An analytical study on the bending moment acting on the girder of a long-span cable-supported bridge suffering from cable failure. Eng Struct 2018;167:166–74. Bedon C, Dilena M, Morassi A. Ambient vibration testing and structural identification of a cable-stayed bridge. Meccanica 2016;51:2777–96. Xu Y, Zeng S, Duan X, Ji D. Seismic experimental study on a concrete pylon from a typical medium span cable-stayed bridge. Front Struct Civ Eng 2018;12:401–11. PTI (Post-Tensioning Institute). Recommendations for stay cable design, testing and installation. 5th ed. Phoenix, AZ, USA: Cable-Stayed Bridges Committee; 2007. Chisari C, Bedon C, Amadio C. Dynamic and static identification of base-isolated bridges using Genetic Algorithms. Eng Struct 2015;102:80–92. ASTM A416/A416M-18. Standard specification for low-relaxation, seven-wire steel strand for prestressed concrete. West Conshohocken, PA, USA: ASTM International; 2018. Mazzoni S, McKenna F, Scott MH, Fenves GL. Open system for earthquake engineering simulation (OpenSees): version 2.5.0. University of California, Berkeley, California, USA: Pacific Earthquake Engineering Research (PEER) Center; 2016. Mazzoni S, McKenna F, Scott MH, Fenves GL. OpenSees Command Language Manual. University of California, Berkeley, California, USA: Pacific Earthquake Engineering Research (PEER) Center; 2005. Kaveh A, Ilchi Ghazaan M. Enhanced colliding bodies optimization for design problems with continuous and discrete variables. Adv Eng Softw 2014;77:66–75. Kaveh A, Mahdavi VR. Colliding bodies optimization: a novel meta-heuristic method. Comput Struct 2014;139:18–27. Kaveh A, Ilchi Ghazaan M. A comparative study of CBO and ECBO for optimal design of skeletal structures. Comput Struct 2015;153:137–47. Gholizadeh S, Aligholizadeh V. Reliability-based optimum seismic design of RC frames by a metamodel and metaheuristics. Struct Des Tall Spec Build 2019;28:e1552. Fathali MA, Hoseini Vaez SR. Optimum performance-based design of eccentrically braced frames. Eng Struct 2020;202:109857. Hoseini Vaez SR, Mehanpour H, Fathali MA. Reliability assessment of truss structures with natural frequency constraints using metaheuristic algorithms. J Build Eng 2020;28:101065. Kaveh A, Izadifard RA, Mottaghi L. Optimal design of planar RC frames considering CO2 emissions using ECBO, EVPS and PSO metaheuristic algorithms. J Build Eng 2020;28:101014.
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An approach for adjusting the tensile force coefficient in equivalent static cable-loss analysis of the cable-stayed bridges Mohammad Ali Fathali, Ehsan Dehghani, Seyed Rohollah Hoseini Vaez
T
⁎
Department of Civil Engineering, Faculty of Engineering, University of Qom, Qom, Iran
A R T I C LE I N FO
A B S T R A C T
Keywords: Cable-stayed bridges Cable rupture Cable rupture duration Linear static analysis Time history analysis Optimization
The sudden rupture of cable is one of the factors that can threaten the bridge safety; causing progressive collapse due to severe vibrations along with numerous changes in the internal forces of all bridge members. Therefore, analyzing the bridge behavior after the sudden cable rupture and estimating the maximum response of the bridge members is of great importance. According to Post-Tensioning Institute (PTI), the sum of the static analysis responses of two models can be used instead of dynamic analysis of cable rupture. In one of these models, the ruptured cable should be removed and twice its tensile force should be applied at the two ends of the cable. In this study, an approach proposed for investigating the tensile force coefficient (TFC) of the ruptured cable in the model of cable rupture effect. For this purpose, an optimization problem was defined to minimize the difference between the static and the maximum dynamic analysis results. The performance of the proposed approach is illustrated with a three-span cable-stayed bridge. This problem was solved for the rupture of each of the bridge cables under examination at different durations of cable rupture using ECBO meta-heuristic algorithm. Moreover, this meta-heuristic algorithm was used to adjust the tensile forces of the cables. Based on the results, a general form of equations was found between cable rupture duration and TFC, which the maximum of it, is in accordance with the PTI coefficient. According to this equation, increasing the cable rupture duration reduces this coefficient.
1. Introduction The occurrence of some failures in a structure can cause a rapid spread of damage to other members and parts. As such, the failure proceeds rapidly and leads to eventual collapse of all the structure or some parts of it, called progressive collapse. Due to its sudden occurrence, progressive collapse of the structures must be prevented. On cable-stayed bridges, this type of failure can occur due to the cable rupture. Cable rupture can be due to a variety of reasons, such as reduced cross-sectional area due to corrosion (cables are more susceptible to corrosion than other bridge components), fatigue, increased axial force due to lateral effects, reduced resistance due to fire or intentional attack. Sudden rupture of the cable can threaten the safety of the bridge; as the loss of a cable causes severe vibrations on the bridge, causing many changes in the internal forces of all structural members. Cable rupture and its consequences on structures such as submerged floating tunnel [1], suspen-dome [2], suspended arch bridges [3] and cable-stayed bridges [4,5] have been studied. Hoang et al. investigated the vibration performance of the cable-stayed bridge after the sudden rupture of a cable due to lateral force using numerical analysis and
⁎
model experiments [6]. In another study, Wu et al. evaluated the behavior of the self-anchored suspension bridge caused by hanger breakage using nonlinear analysis and considering the effect of the corroded hangers [7]. Starossek investigated and categorized the types of progressive collapse and introduced the loss of a cable in the cablestayed bridge leading to the zipper-type progressive collapse [8]. RuizTeran and Aparicio investigated and compared the dynamic amplification factors (DAF) related to deflections, flexural moments and shear forces in the case of oscillation caused by cable rupture in cable-stayed bridges [9]. In the another study, they performed a parametric investigation of the responses of two different types of under-deck cablestayed bridges in sudden cable rupture using a linear static analysis method and DAF [10]. They considered parameters such as the cable rupture duration, the number of ruptured cables, and the amount of traffic live load. In another parametric study, Mozos and Aparicio investigated the influence of characteristics such as stay-cables layout and deck stiffness on the dynamic response caused by sudden cable rupture [11,12]. They also studied the effect of cable rupture duration on the response of cable bridges in another study [13]. Considering the geometrical and material nonlinearity effects, Cai et al. investigated the
Corresponding author. E-mail addresses: [email protected], [email protected] (S.R. Hoseini Vaez).
https://doi.org/10.1016/j.istruc.2020.03.054 Received 27 November 2019; Received in revised form 17 March 2020; Accepted 25 March 2020 2352-0124/ © 2020 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
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Fig. 1. Schematic of the cable-stayed bridge.
estimate maximum responses for numerical studies such as progressive collapse. The rest of the study is organized as follows: Section 2, presents a description of the numerical modeling and linear static analysis of cable-loss. Section 3 is allocated to the definition of the objective function of the proposed approach and the explanation of ECBO metaheuristic algorithm. Section 4 provides the results of adjustment of the cables tensile force and the tensile force coefficient (TFC) of ruptured cable. Finally, conclusions are made in Section 5.
effects of progressive collapse with the simultaneous sudden rupture of multiple cables on nonlinear responses and bridge behavior, after which they compared the results with the rupture of one cable [14]. Zhou and Chen developed a framework to assess the reliability of a long-span cable-stayed and traffic system under cable rupture conditions [15]. Based on finite-element method, Zhou and Chen presented a new platform for non-linear dynamic simulation of cable rupture. Also, they attempted to improve the following in the proposed new framework: modeling the cable loss process, considering realistic traffic and wind loads through the bridge-traffic-wind interaction model based on finite-element, and adding material as well as geometrical nonlinearities with various sources [16,17]. On the basis of the provided platform, they further carried out a numerical study of cable rupture in long-span cable bridges and investigated post-rupture bridge performance [18]. Shoghijavan and Starossek studied the maximum flexural moment created on the girder after cable rupture using an analytical method based on system differential equations and obtained an approximate function for the cable rupture scenario [19]. Bedon et al. used inverse techniques for an experimental and theoretical investigation on the Pietratagliata cable-stayed bridge with damage phenomena in the cables, including analytical, experimental and numerical comparisons to assess the effects (and prevention) of cable rupture [20]. Also, in some other studies, the seismic performance of other components of cable-stayed bridges such as pylons [21] has been investigated. According to the Post-Tensioning Institute (PTI) [22], instead of using dynamic analysis, the elastic analysis of two different models along with superposition principle can be used to investigate the behavior of the bridge after the sudden cable rupture. The first model is the main bridge model (before the cable rupture) in which all dead and live loads are applied. The second model is a model in which the ruptured cable is removed and twice the tensile force of the ruptured cable is inserted inversely at its two ends. This model considers the effects of the cable rupture. In this study, an approach proposed to obtain the most appropriate tensile force coefficient (TFC) of the ruptured cable in the equivalent static cable-loss analysis of the cable-stayed bridges. For this purpose, an objective function is formulized so that the difference between the results of the dynamic analysis of the cable rupture and the equivalent static analysis is minimized. Also, the bending moment values of the deck at the location of the cables and the pylons connections to the deck should be equal in the main bridge modeling. For this purpose, another optimization problem is defined for adjusting the amount of tensile forces of the cables. To solve these optimization problems, the enhanced colliding bodies optimization (ECBO) algorithm is used. A three-span cable-stayed bridge considered for evaluating the performance of the proposed approach. In order to investigate the effect of each cable on the bridge performance, the problem of obtaining the most appropriate TFC is solved for each ruptured cable with rupture duration of 0.01 s. It is clear that the shorter the duration of the cable rupture, the stronger the vibration in the bridge. In this regard, the main optimization problem is solved for the different durations of cable rupture to evaluate the overall validity of the proposed approach. Then a suitable cable rupture duration is suggested for cable ruptures to
2. Description of the numerical modeling and analysis 2.1. Numerical modeling Fig. 1 indicates the geometry of the cable-stayed bridge under study. The distance of the cable connection to the deck is 10 m. For the bridge deck, a rectangular concrete section of 0.8 m thick and 10 m wide is considered. Also, the pylon cross-section is defined as a concrete pipe section with a diameter of 3.0 m and a thickness of 0.3 m. Table 1 indicates the nominal properties of materials. Optimization techniques for multi-objective analyses on bridges confirmed that the mechanical properties of materials (including the modulus of elasticity of concrete) are sensitive to loading rate and manifest in a modulus variation under dynamic loads [23]. Because the main purpose of this study is to present an approach for obtaining the most appropriate TFC, the sensitivity of the mechanical properties of the materials to the loading rate has not been considered in order to simplify. The deck distributed gravity load is 39.2266 kN/m. The list of the cross-sectional area of the cables is as reported in Table 2. An open-source software based finite-element method (OpenSees software) was used for two-dimensional (2D) modeling and analysis of cable-stayed bridge [25]. Since cables are tensile members, the cables are modeled using “Truss” elements with initial axial strain assigned by “Initial Strain” material. As the analysis is linear, the “elastic beamcolumn” element is used to model the deck and pylons. Also, a “zerolength” element was used to model the neoprene performance in the transfer of forces between the deck and the pylon; such that the flexural moment is not transmitted, and the horizontal and vertical forces are transferred by defining elastic behavior using an “Elastic Uniaxial” material with low (3 kN/mm) and high (108 kN/mm) stiffness, respectively [26]. To model the cable rupture in dynamic analysis, the considered cable for rupture was removed from the bridge model and instead its axial force was applied proportional to a time series at both ends. The Table 1 Material property [12].
721
Parameter
Steel (Gr270 [24])
Concrete
Modulus of elasticity, E (GPa) Weight per unit volume, γ (kN/m3) Compressive Strength, fc' (MPa) Yield strength, fy (MPa) Ultimate strength, fu (MPa)
190 76.93 – 1690 1860
36.342 24.50 40 – –
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force (TFC) in the second model is investigated.
Table 2 The list of the cross-sectional area of the cables. Cable
Cross-section (mm2)
Number of strands*
C1-C18 C2-C17 C3-C16 C4-C15 C5-C14 C6-C13 C7-C12 C8-C11 C9-C10
18,000 6300 11,400 18,000 18,000 12,000 18,000 14,100 17,700
120 42 76 120 120 80 120 94 118
3. The proposed approach 3.1. General form of an optimization problem The general form of an optimization problem is according to Eq. (3):
find X = {x1, x2 , ...,x n } to minimize f (X) subject to gi (X) ⩽ 0, i = 1, 2, ...,nc where x min ⩽ x ⩽ x max
* Cross-sectional area of each strand = 150 mm2.
(3)
time series of applying the axial force is such that it first applies the force completely and then it reaches zero after the considered duration for the cable rupture.
in which, X represents the random variables vector with minimum and maximum vectors of xmin and xmax, respectively. Moreover, f indicates the objective function, and the problem is aimed at minimizing the value of this function. The function of gi indicates the function of the ith constraint of the problem.
2.2. Adjustment of the cables tensile force
3.2. Formulating the objective function to adjusting the TFC
The purpose of adjusting the cables tensile force is to equate the flexural moment value of the deck in the location of the cables (except back-stay cables) and the pylons connections to the bridge deck with the following equation:
In this study, an unconstrained optimization problem with fR objective function is defined by Eq. (4) in order to scrutinize the tensile force coefficient of the ith ruptured cable:
Mt =
NC − 1
fR =
L2
w. 12
l=1
(4)
where, Zc is the function of calculating the relative differences of the maximum tensile force of jth cable in the two states of static analysis proposed by PTI and time history analysis of cable rupture and calculated according to Eq. (5):
ZCj =
(FCs )j − (FCd )j (FCd )j
(5)
where, (FCs)j indicates the jth cable tensile force in static analysis and (FCd)j indicates the maximum tensile force produced in jth cable due to ith cable rupture, obtained from time history analysis. Moreover, NC represents the number of cables. In Eq (4), ZDk is the function of calculating the relative differences of the maximum flexural moment of the deck in the kth location of cables or pylon connections to the deck in the two states of static analysis proposed by PTI and time history analysis of cable rupture, and calculated according to Eq. (6):
(2)
i=1
k=1 j
ND
∑ |Mi − Mt|
NP
∑ ZDk + ∑ ZPl
j=1
(1)
In the Eq. (1), Mt is the target flexural moment of the deck in the location of cable or pylon connection to the deck, while w indicates the sum of distributed gravity loads per unit length of the deck and the unit weight of the deck length, and L is the distance of cable connection to the deck. Adjusting the tensile force of the cables to achieve a defined goal, needs a great amount of trial and error, and different ways have been suggested for it. In this study, an unconstrained optimization problem which is solved using the meta-heuristic algorithm is defined to adjust the tensile force of the cables. The objective function is the optimization problem defined as follows:
Min F =
ND
ZCj +
∑
where, Mi is the flexural moment of i connection of the deck to the cables or pylon, while ND indicates the number of connections of the deck to the cables (except back-stay cables) and pylon. The variables in this problem are the initial strain values applied from the [0, 1] interval for each cable. th
ZDk =
(MDs )k − (MDd )k (MDd )k
(6)
in which, (MD )k indicates the flexural moment of the deck in the location of kth connection of cables or pylons to the deck in the static analysis and (MDd)k indicates the maximum flexural moment produced at the kth cables or pylons connection to the deck due to ith cable rupture, obtained from time history analysis. Moreover, ND represents the number of connection of cables or pylons to the deck. Also in Eq. (4), ZPl is the function of calculating the sum of the relative differences of the maximum flexural moment of the lth pylon end and its connection to the deck in the two states of static analysis proposed by PTI and the cable rupture time history analysis. ZPl is calculated by Eq. (7): s
2.3. Linear static analysis of cable-loss Sudden cable rupture due to different reasons such as corrosion and intentional explosion leads to severe vibrations in the bridge. In other words, sudden cable rupture leads to considerable changes and excessive oscillations in the internal forces of the members and their deformation. To accurately analyze the behavior of the bridge after cable rupture, a dynamic time history analysis must be performed. However, according to PTI, linear static analysis can be used instead of dynamic analysis, and the results of the analysis of the following two models can be summarized using the superposition principle:
ZPl =
- First model: Modeling of the bridge with dead and live loads applied to it; - Second model: Modeling of the bridge without a ruptured cable and applying twice its tensile force at the two ends inversely.
(MPse )l − (MPde )l (MPsD )l − (MPdD )l + d (MP e )l (MPdD )l
(7)
are respectively flexural moments of lth in which, (MPe )l and pylon end and its connection to the deck in static analysis, while (MPed)l and (MPDd)l are respectively the maximum flexural moments produced at lth pylon end and its connection to the deck due to ith cable rupture, obtained by the time history analysis. Moreover, NP represents the s
In this study, the applied coefficient on the ruptured cable tensile 722
(MPDs)l
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Fig. 2. The flowchart of the proposed approach.
in which, mk and fit(k) are the values of the mass and the objective function of the ith colliding body (CB), respectively; and nCB is the number of algorithm population.
number of pylons. The variable of the defined optimization problem is the tensile force coefficient of the ruptured cable in the static analysis second model, considered at the interval of [1,5]. Fig. 2 represents the flowchart of this approach.
nCB ⎞ vi = 0 , ⎛i = 1, 2, ..., 2 ⎠ ⎝
3.3. Enhanced colliding bodies algorithm
vi = x i − x i − (nCB
Enhanced colliding bodies optimization (ECBO) algorithm [27] was introduced in order to improve the performance of colliding bodies optimization (CBO) algorithm. CBO algorithm is inspired by the physical laws governing collision of objects [28]. These algorithms have been used in many studies [29–33]. In ECBO algorithm, the population sorted in an ascending order based on their value of the objective function. This sorted population is divided into two equal groups: stationary and moving colliding bodies (CBs). The stationary group is the first half of the sorted population, and the moving group is the second half. Fig. 3 indicates the pseudo-code of ECBO algorithm. The parameters of this algorithm are: nCB (number of ECBO population), tmax (maximum number of ECBO iterations), nCM (size of the Colliding Memory (CM)) and Pro (within interval (0, 1)). The equations used in this algorithm are as follows:
1 ⎞ mk = ⎜⎛ ⎟ ⎝ fit (k ) ⎠
⎛i = ⎝
nCB nCB + 1, + 2, ...,nCB ⎞ 2 2 ⎠
mi + (nCB 2) + εmi + (nCB vi′ = ⎜⎛ mi + mi + (nCB 2) ⎝
2) ⎞ ⎟
vi + (nCB
2) ,
⎠
(10)
⎛i = 1, 2, ..., nCB ⎞ 2 ⎠ ⎝
mi − εmi − (nCB 2) ⎞ nCB nCB + 1, + 2, ...,nCB ⎞ vi′ = ⎜⎛ ⎟ vi , ⎛i = + m m 2 2 ⎝ ⎠ i i ( nCB 2) − ⎠ ⎝
(11)
(12)
th
In Eqs. (9) to (12), vi and vi' are the velocity of i CB before and after the collision, respectively; ε is the coefficient of restitution; and xi is the current position of ith CB.
nCB ⎞ x inew = x i + rand°vi′, ⎛i = 1, 2, ..., 2 ⎠ ⎝ x inew = x i − (nCB xinew
2)
+ rand°vi′, ⎛i = ⎝
nCB nCB + 1, + 2, ...,nCB ⎞ 2 2 ⎠ th
(13)
(14)
is the new position of i CB. Here, rand is a vector with a where, dimension of the number of problem variables (N), consisting of random numbers in the interval of (1, −1).
nCB
1 ⎞ ⎛ ⎜ ∑ fit (i) ⎟ = i 1 ⎠ ⎝
2) ,
(9)
(8) 723
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Fig. 3. The pseudo-code of the ECBO algorithm. Table 3 The axial force of cables before cable rupturing. Cable
C1-C18
C2-C17
C3-C16
C4-C15
C5-C14
C6-C13
C7-C12
C8-C11
C9-C10
Cable axial force before cable rupturing (kN)
9141
3852
2914
2459
2455
2916
3557
4297
5092
Fig. 4. Flexural moment of deck before cable rupturing. Table 4 The TFC values of the ruptured cables (cable rupture duration = 0.01 s). Ruptured cable
C1
C2
C3
C4
C5
C6
C7
C8
C9
TFC
1.6656
1.6993
1.5137
1.3443
1.4015
1.5113
1.3067
1.4330
1.7445
4. Results and discussion
of the population and the maximum iterations of algorithm are considered to be respectively 60 and 300 in order to solve the problem of cables tensile force adjustment. In Table 3, the tensile force obtained for each cable reported. Fig. 4 indicates the flexural moment of deck before cable rupturing.
4.1. Adjustment of the cables tensile force According to the load considered and the Eq. (1), the target moment value at the cables connection to the deck is 1897.76 kN.m. The values 724
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Fig. 5. Comparison of the flexural moment distribution in the deck for the rupture of cables a) C1, b) C4, c) C5 and d) C9, in the responses states of dynamic analysis (Dyn) and static analysis (calculated TFC, PTI TFC).
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Fig. 6. Comparison of the cable axial force for the rupture of cables a) C1, b) C4, c) C5 and d) C9, in the responses states of dynamic analysis (Dyn) and static analysis (calculated TFC, PTI TFC).
726
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Fig. 7. Comparison of the flexural moments of the pylons for the rupture of cables a) C1, b) C4, c) C5 and d) C9, in the responses states of dynamic analysis (Dyn) and static analysis (calculated TFC, PTI TFC).
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Fig. 8. The scatter plot of the TFCs against the cable rupture durations for the rupture of cable C1.
than or equal to 2. As a result, the maximum value of this equation for a very small rupture duration (less than 0.01 s) can be 2, which is in accordance with the proposed coefficient in the PTI. The first term of Eq. (15) indicates that as the cable rupture duration increases, the TFC generally decreases; because the longer the cable rupture duration, the lower the vibration will be in the cable-stayed bridge. Also, it can be find that there is no significant changes in the TFC for the cable rupture duration at the interval of 0.001 to 0.01 s. Accordingly, a cable rupture duration of 0.01 s is appropriate for dynamic analysis of the sudden cable rupture.
Table 5 The linear fitting equation between the duration of cable rupture and TFC. Ruptured cable
Equation
C1 C2 C3 C4 C5 C6 C7 C8 C9
TFC TFC TFC TFC TFC TFC TFC TFC TFC
= = = = = = = = =
−0.51 −0.95 −0.28 −0.33 −0.40 −0.45 −0.24 −0.32 −0.60
× × × × × × × × ×
RD RD RD RD RD RD RD RD RD
+ + + + + + + + +
1.66 1.85 1.50 1.32 1.38 1.46 1.31 1.36 1.77
5. Conclusion An abrupt rupture of the cable in structures such as cable-stayed bridges can disrupt the operation of the bridge by causing severe oscillations in the whole bridge. In this study, an approach is proposed to estimate the most appropriate TFC of the equivalent static analysis suggested by PTI. TFC is the coefficient of the tensile force of a ruptured cable in one of the equivalent static analysis models of tearing the cable. For this purpose, an optimization problem is defined in which the objective function is the difference between the equivalent static analysis responses and maximum responses of dynamic analysis in each bridge components (cables, pylons, and deck). This problem was solved for rupturing each of the cables of a three-span cable-stayed bridge by considering different durations of the cable rupture. As the results shown, the obtained TFCs by this approach makes the acceptable difference between the dynamic and equivalent static analysis results. This result shows the efficiency of the defined objective function considering the numerous terms. Also, a linear fit was obtained between the TFCs for each cable and cable rupture durations. Based on the resulting equations, a general form was found between TFC and cable rupture duration, in which the maximum of it is in accordance with the PTI coefficient. In this form, the coefficient of the cable rupture duration is negative, which is in accordance with this concept that the bridge vibration decreases with increasing cable rupture duration. Also, it can be found that for the cable rupture duration at the interval of 0.001 to 0.01 s there are no significant changes in the TFC.
4.2. Adjustment of the TFC of ruptured cable To solve the optimization problem of TFC adjustment, the population and the maximum iterations of algorithm are considered 20 and 100, respectively. Due to the symmetry of the bridge, the TFC of the ruptured cable is calculated for rupture of the cables C1 to C9. The TFC values of the ruptured cables in the second model of static analysis are presented in Table 4, considering a rupture duration of 0.01 s. Figs. 5–7 show the flexural moment distribution in the deck, the cable axial force and the flexural moments of the pylons for the rupture of cables C1, C4, C5 and C9, in the responses states of dynamic analysis and static analysis. According to these figures, it can be observed that the result of static analysis gets closer to the maximum result of dynamic analysis in most cases, indicating appropriate definition of the objective function considering the numerous terms in the objective function. To investigate the effect of the cable rupture duration, the optimization problem has been solved by considering cable rupture duration in an interval of 0.001 to 1.0 s for each cable rupture. For example, Fig. 8 shows the resulting TFCs against the cable rupture duration considered for the rupture of cable C1. Regarding TFC values for each cable rupture at different duration of cable rupture, a linear fit between the results is considered to obtain a formula between the cable rupture duration and the TFC for each cable rupture. For example, the fit line for the TFC of the rupture of cable C1 is indicated in Fig. 8. The equations of the fit lines for each cable rupture are presented in Table 5. In these equations, RD is the duration of cable rupture. Considering the linear fit between the results, the following equation in general form can be defined between the cable rupture duration (RD) and the TFC:
TFC = −a × RD + b
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.
(15)
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