Kinetic energy based model assessment and sensitivity analysis of vortex induced vibration of segmental bridge decks

Abstract

In this paper, semi 3D models for segmental Bridge decks are created in ABAQUS CFD program with the support of MATLAB codes to simulate and analyze vortex shedding generated due to wind excitation through considering the stationary position of the deck. Three parameters (wind speed, deck streamlined length and dynamic viscosity of the air) are dedicated to study their effects on the kinetic energy of the system in addition to the shapes and patterns of the vortices. Two benchmarks from the literature Von Karman and Dyrbye and Hansen are considered to validate the vortex shedding aspects for the CFD models. Good agreement between the results of the benchmarks and the semi 3D models has been detected. Latin hypercube experimental method is dedicated to generate the surrogate models for the kinetic energy of the system and the lift forces. Variance based sensitivity analysis is utilized to calculate the main sensitivity indices and the interaction orders for all the three parameters. The kinetic energy approach performed very well in revealing the rational effects and the roles of each parameter in the generation of vortex shedding and predicting vortex induced vibration of the deck.

Keywords

vortex induced vibration / reynolds number / kinetic energy / vorticity / latin hypercube sampling

Introduction

Wind load is the primary prominence for the safety of long span cable supported Bridges during the stage of design. Vortex induced vibration which is one of the wind-induced phenomenon, is predominantly decisive for the security and serviceability of these structures. The segmental Bridge decks are selected supporting on structural and economic characteristics, where the essential shape of the deck is not requisite to be aerodynamically efficient optimally. Due to this fact, long span Bridges are often undergo vortex induced vibration. Several research studies on the geometries and the vortex shedding mechanisms for bluff bodies have been conducted. Presently the deck cross sections are designed between bluff bodies and streamlined, for example the Tsing Lung Bridge in Hong Kong. The wind flow in the wake region of a bluff body such as segmental Bridge decks is described by vortices which are shed from its trailing edge continuously at a particular frequency, where they are often attributed to Karman vortices. The shape and the pattern of the vortices are occasionally referred to Karman street. These vortices are shedding from the Bridge deck continuously regardless of the wind speed magnitude. The shedding eddies have frequencies varying linearly with the wind speed since the Strouhal number is stable mostly. When the shedding frequency is matching the frequency of particular mode of vibration of the structure, either vertical or torsional, the resonance might occur as a result the lock-in of the Bridge oscillation starts. Within particular limits and if the wind speed continues to increase, the shedding frequency of the vortices keeps unaltered. This situation is named synchronization domain. In a certain case where the range of the wind speeds is wide, this might cause fatigue, discomforts or failure, counting on the oscillations amplitude [1–29].

Various shapes and patterns of vortex shedding exist. These depend on the shape of the deck, diverse vortex shedding mechanisms and Reynolds number (Re). The vortex shedding mechanism might be totally different relying on the shape of the deck cross section. The main reason of vortex induced vibration in a long span Bridge is referred to structural low damping and structural slenderness. This truth is approved in the case of Rio-Niteroi Bridge in Rio de Janeiro that has manifested vortex shedding vibrations even at weak wind speed of 14 m/s, where this event was not counted for in the design stage. Vortex induced vibration can be reduced through selecting suitable cross sections and shapes for Bridge decks [4,5,30–56].

The aerodynamic behavior and the damping of the Bridge decks are affected and altered by the vortex shedding pattern and the shape of the vortices. As a result, the deck reaction is affected. It is worthy to mention that the vortex shedding pattern and the shape of the vortices are affected by the frequency and the amplitude of the oscillations. Hence, the study of the vortices shape and the vortex shedding pattern helps to realize the relationship between the vortex shedding patterns and the structural response at various Reynolds numbers. Despite the fact that the vortex induced vibration is a type of limiting vibration amplitude with the self-excited features, this type of vibration is yet needs prediction [42,57–67].

An analytical study concerning the stability of the vortex patterns in a wake of a stationary rectangular cylindrical body was carried out by Von Karman and Rubach. Based on the two dimensional potential flow theory and assuming that the fluid is irrotational except in concentrated vortices, it was shown that the vortex pattern is stable, if the vortices are organized in unsymmetrical double row pattern (see Fig. 1). This stable configuration of vortices possesses the relations {{\displaystyle h}}_K/{{\displaystyle b}}_K={\mathrm{\pi }}^{-\mathrm{1}}\arccos \mathrm{h}\sqrt{\mathrm{2}}\mathrm{0.281} and \left|\mathrm{\Gamma }\right|=\sqrt{\mathrm{8}}{{\displaystyle u}}_K{{\displaystyle b}}_K, where \mathrm{\Gamma } is the strength of vorticity of a concentrated vortex, {{\displaystyle u}}_K is the speed of the vortices, {{\displaystyle b}}_K is the distance between the vortices in a row and {{\displaystyle h}}_K is the distance between the rows.

Furthermore, it was shown that the vortex trail induces almost steady drag force to the body given by the formula

(1){{\displaystyle \overline{H}}}_K=qh{{\displaystyle C}}_{HK}

in which {{\displaystyle \bar{H}}}_K is the mean value of the induced horizontal force per unit length, q is the kinetic pressure and the associated force coefficient C_HK is expressed as

(2){{\displaystyle C}}_{HK}={\displaystyle \frac{\mathrm{2}\mathrm{\Gamma }}{U^{\mathrm{2}}h}}{\displaystyle \frac{{{\displaystyle h}}_K}{{{\displaystyle b}}_K}}(U-\mathrm{2}{{\displaystyle u}}_K)+{\displaystyle \frac{{\mathrm{\Gamma }}^{\mathrm{2}}}{\mathrm{\pi }{U}^{\mathrm{2}}h{{\displaystyle b}}_K}}\mathrm{.}

Some attempts have been made to obtain analytical expressions for the fluctuating lift force experienced by the bluff cylinder owing to the ideal Karman-Benard vortex trail (Fig.1). Considering infinite vortex trail and circular cross-section, CHEN found that:

(3){{\displaystyle V}}_K=\rho {{\displaystyle u}}_K\mathrm{\Gamma }=qh{{\displaystyle C}}_{VK},

where

(4){{\displaystyle C}}_{VK}={{\displaystyle \left({\displaystyle \frac{\Gamma }{U{{\displaystyle b}}_K}}\right)}}^{\mathrm{2}}{\displaystyle \frac{{{\displaystyle b}}_K}{h}}\tanh {\displaystyle \frac{\pi {{\displaystyle h}}_K}{{{\displaystyle b}}_K}}\mathrm{.}

Here, {{\displaystyle V}}_K is the fluctuating across-wind force per unit length and C_VK is the associated aerodynamic exciting coefficient. The study does not establish the time-dependence of the unsteady exciting coefficient, but SALLET has been able to suggest that the dependence should be sinusoidal with a frequency given by Strouhal’s relation. The aeroelastic actions can be considered in Eq. (2) and Eq. (4) by noting that the spacing between the vortex row h_K, and thus the induced forces, can alter, if the cross-section performs across-wind oscillations. In general, analytical formulations for the aerodynamic exciting coefficients are dependent on the approximations assumed for the near-wake behind the body [68].

The vibrations generated by vortex shedding usually occur in slender structures with low damping. The vibrations occur if the vortex shedding frequency coincide with or come close to the natural frequency of the structure. The vortex shedding frequency fs of a non-vibrating body can be derived as suggested by Dyrbye and Hansen (see Fig.2). The time between the vortices at each side is equal to the distance lv divided by the speed of the vortices, U1. The frequency is the inverse of the period, giving that fs= U1/lv. The distance between the vortices, lv, must be proportional to the structure width d, since this is the only relevant length [69].

Previous studies were mostly supporting on the results of Reynolds number and Strouhal number to calculate the vortex induced vibration of the deck and in the same time without considering important parameters that have uncertain effects on this type of vibration. In this study, kinetic energy concept is devoted to reveal the roles and the effects of three parameters (wind speed, deck streamlined length and dynamic viscosity of the air) on the generation of vortex shedding and vortex induced vibration by studying the shapes and patterns of the vortices in conjunction with the energy dissipation concept. In this study the lift force that is responsible of vertical oscillation of the deck is covered only because it is the most critical case in vortex induced vibration phenomenon. Variance based sensitivity analysis with the support of Latin hypercube sampling method are utilized for this research study.

Vortex induced vibration parameters

Reynolds number

Reynolds number is a non dimensionless number that describes the flow around smooth bluff bodies such as a Bridge deck. It is the ratio between the inertia forces and viscous forces:

(5)\mathrm{Re}={\displaystyle \frac{DU}{v}},

in which D is the diameter of the cylinder, U is the flow speed and v is the kinematic viscosity of the fluid. Flow regimes are obtained as the result of many changes of the Reynolds number. The changes of the Reynolds number generate separation flows in the wake region of the bluff body, which are called vortices. At low values of Reynolds number<5, there is no separation occurs. When Reyno ds numbe is further increased, the separation starts to occur and becomes unstable and initiates the phenomenon named vortex shedding at particular frequency. The main parameter of Reynolds number is the wind speed that is directly related to the generation of vortex induced vibrations by affecting the kinetic energy of the flow system that is the source of vortex shedding [12,70–73].

Deck shape

The deck shape of a Bridge is essential to be in such a way to enhance the aerodynamic behavior and to reduce the tendency to vortex shedding. Bienkiewicz [76] studied various cross section samples and Nagao et al. [77] studied the ratios of width and the height of box girders and many shapes of fairing. The results of these studies concluded that the best streamlined deck shape involves better aerodynamic stability of the structure, while Wardlaw et al. [78] discovered that the turbulence can suppress the vortex shedding. Noda et al. displayed the role of the leading edge of the bottom deck separation on the lift force at a certain Reynolds number. A deck shape with a surface with high curvature, the separation point modifies with respect to Reynolds number, which is the source of diverse effects related to vortex shedding. However, there are other flow features, like top deck leading edge separation and bottom deck trailing separation where their aerodynamic behavior are still not clear [11,13,39,41, 43,74–85].

Dynamic viscosity of air

Vortex shedding has an important relation with the dynamic viscosity of air, so it is essential to consider this parameter in the study of vortex shedding patterns and vortices shapes. The contact between the air particles and the deck surface has adhesion force that creates the boundary layer. However, because of the air mass, it undergoes inertial effect according to Newton’s law and Navier-Stokes equations. The link between the inertial forces and the viscous forces is specified by Reynolds number. The boundary layer separation point explains the deck behavior submerged in the wind flow [86,87]. The dynamic viscosity of air represents the resistance of the air to shearing wind flows. This parameter has values that vary linearly with the variation of temperature (see Fig.3).

Vortex shedding phenomena in bridges

When a Bridge deck is subjected to a wind flow perpendicularly, the wind would be retarded when it is with contact with the deck surface and boundary layer being composed. This boundary layer at some parts of the deck, trends to separate from the deck. The separation of the boundary layer generates a force on the deck and pressure on the windward part and suction at leeward part. As a result, vortices would be formed which alter the distribution of pressure on the Bridge deck surface leading to structural deflections of the deck. These vortices might not be symmetric around the deck. Hence diverse lifting forces are generated around the deck. The motion of the deck would be transverse to the direction of the incoming wind. When the shedding frequency of vortices matches the natural frequency of the Bridge, resonance oscillations will occur predominantly. The amplitude of the oscillations relies on the system damping and the fluctuation of the wind. These oscillations might reach lock-in region, consequently result in dangerous extension and failure due to fatigue [6–10,88–95].

Due to fluctuation of wind and variation with respect to time and elevation, the frequency of vortex shedding is not remaining stable for a long time. As a result Bridges are not susceptible to everyday wind excitation. In a critical case, when the wind is unusually stable and long lasting, vortex shedding can result in vertical motion of the deck with amplitude up to decimeters, where the duration of wind excitation in this case ranges between several minutes and some hours. The amplitude of oscillations is an important case that is concerning the users’ comfort, especially when it reaches noticeable level and when the fatigue of the structural elements is expected. A principal sketch of the phenomenon can be seen in Fig.4.

A wind flow around asymmetric bluff body like a Bridge deck will affect the periodic vortex shedding or even suppress it. The motion of the deck which is periodic in nature may occur when the shedding frequency of the vortices is close to natural frequency of the Bridge. The vortex induce vibrations can be averted by assigning the frequencies of the generated vortices widely far from the natural frequency of the Bridge. This step can be done by changing the geometry of the Bridge structure or altering the natural frequencies in the design stage or after construction. Strohal number is the most critical parameter in the subject of vortex shedding, which links between the shedding frequencies of the vortices, buff body diameter and flow speed. Due to nonlinearity in the Bridge structures and aerodynamic forces, Simiu and Scanlan suggested a synthesis sample to take into account the nonlinear influence [94–104].

Finite element model

The Bridge segmental deck model is generated in ABAQUS-CFD in semi 3D model with dimensions 2.6 m height and total width of 22 m and thickness is 2 m as shown in Fig.5. The flow domain size is 140 m length and 40 m height, the position of the deck model in the flow domain should be in such a position so that to facilitate a proper area to show the vortex shedding in the downstream Fig.6 in addition to an appropriate area above and under the deck model to show the boundary layers around the deck with the separation points.

The air density is assigned 1.29 kg/m³ and the dynamic viscosity of the air is assigned in the range (1.6321E–05–1.9821E–05) Pa.s depending on the temperature between –20 C° and 50 C° (see Fig.3). The model part is meshed using CFD element fluid family with FC3D8: A-8 node linear fluid brick. The deck wall assigned with 0.4 element size and the flow domain with 1 element size. A flow step with 100 seconds duration is created to simulate the vortex shedding. Spalart- Allmaras turbulence model has been used in this analysis for turbulent flow situations. Four boundary conditions are defined for the model, fluid B.C for the inlet flow and far fields assigning the air speed value in the horizontal direction only (zero attack angle) and the other two directions with zero magnitudes, fluid B.C for the outlet flow assigning zero pressure, fluid B.C for the front and back of the flow model with zero speed magnitude for the third direction perpendicular to the model z-direction and non-slip fluid B.C for the wall condition of the deck. A job is created and the results are analyzed depending on the case if the model needs to be assigned model turbulence or not, this is to be done by checking the time history of the kinetic energy dissipated, where if the plot is oscillating randomly this means that the model needs to activate the model turbulence in the flow step, as a result this indicates necessity of predefining a fluid turbulence in the initial step. Commonly a very large number of Reynolds number indicates the need for a turbulence model.

Results of vorticity and kinetic energy

Wind speed effect

When the input parameter (wind speed) denoted by X1 increases, the vortex shedding in the downstream of the deck is obviously being better generated and regulated in shape and the frequency of shedding increases. When the wind speed v= 0.5 m/s, the vortex shedding is in the way to generate and the vortices are simply shedding but without regular style (see Fig.7).The vorticity field of this case shows the wind flow around the deck and the start of vortex shedding in the wake region (see Fig.8). When the wind speed increases to v =12.5 m/s, the vortex shedding appears very well and the asymmetric pattern of the vortices can be seen very well (see Fig.9) with bigger frequency of shedding. The vortices shape and style in this case are bigger and wider than the previous case (see Fig.10). While when the wind speed is v =25 m/s, the vortex shedding can be seen more organized with larger frequency of shedding (see Fig.11). The vorticity field shows bigger vortices and different style of shedding (see Fig.12). All these cases indicate that increasing the wind speed will help to better generate the vortex shedding in the wake region, also the shape and the patterns of the vortices change to different styles and the frequency of shedding increases, as a result enhancing the start of the deck vibration.

Regarding the kinetic energy of the system, when the wind speed increases, the kinetic energy increases in a nonlinear way. When the wind speed is v = 0.5 m/s, the kinetic energy starts increasing till 18 seconds from the simulation, and after that it starts to decrease in a semi linear way due to energy dissipation which caused generation of very simple vortex shedding that can be seen in Fig.13. While for wind speed v = 12.5 m/s, the kinetic energy increases till two seconds from the simulation and suddenly drops down till seven seconds from the simulation and again starts to rise until 13 seconds with an oscillatory style in the same time decreasing to a stable oscillatory position starting from 20 seconds from the simulation till the end of the simulation indicating the start of the deck vibration with a certain frequency (see Fig.14). Furthermore, when the wind speed is higher v = 25 m/s, the kinetic energy increases highly in the same way as previous case of v = 12.5 m/s, but the sudden drop and rise starts in earlier position in the same time with a larger frequency of oscillation (see Fig.15). The overall comparison between the three kinetic energy cases can be seen in Fig.16.

Deck streamlined length

The input parameter X2 is the deck streamlined length. When the wind speed is constant and just this length is changed, many situations of vortex shedding from the wake region and vortices shapes and patterns are recognized. When this length L = 0 m, the vortex shedding is generated in an primary establishing pattern showing the two rows of the asymmetric rows of the vortices (see Fig.17), but when the length L =1 m, the vortex shedding modifies to better shapes and patterns of vortices in the two rows and the vortex trail can be recognized and seen better (see Fig.18). While when this length L = 2 m, the vortex shedding disappears and the vortices shapes and patterns are destroyed (see Fig.19).

This means that a certain streamlined shape of the deck helps to avoid the vortex shedding generation in the wake region and as a result avoiding the vortex induced vibration of the deck. The complex process of separation of the wind flow from the deck edges, reattachments of them and again separation of the flow to establish vortex shedding are dependable on many factors that arrange better situation to avoid vortex shedding not only the streamlined shape of the deck itself only.

The results of the kinetic energy for all the three deck streamlined length cases proof energy dissipation in a harmonic form but in different rates. When the deck streamlined length is L = 0 m, the kinetic energy starts to increase till 25 seconds from the wind flow simulation and it continues to be stable for five seconds, then it decreases in light oscillatory pattern indicating energy dissipation due to vortex shedding and vibration of the deck till the end of 100 seconds of the simulation. In the same way for the deck streamlined length L = 2 m, but only after 30 seconds of the simulation the start of energy dissipation is taking place in a smaller rate than the previous case (see Fig. 20). Furthermore, for the deck streamlined length case L = 1 m, the kinetic energy starts to increase until 15 seconds from the simulation and continues to be stable for the next five seconds, after that the kinetic energy decreases in a semi smooth way showing larger energy dissipation than the previous two cases. This indicates that a certain streamlined shape of the deck is critical for earlier vortex shedding generation as a result earlier vibration of the deck.

Dynamic viscosity of air

The input parameter X3 is the dynamic viscosity of air. For a certain stable wind speed, the vortex shedding from the downstream of the deck is generated in an organized style and in asymmetric two rows of vortices. When this parameter increases, the vortices shapes and patterns are affected with a very small rate which can be distinguished from the simulation results a little hardly as shown in Figs. 21–23, where the style of the vortex shedding and the asymmetric shape of the vortices are not changing almost. This indicates that the vortex shedding is affected by this input parameter with a small amount.

Considering kinetic energy of the system, the two cases of air dynamic viscosity m= 1.632E–05 N/m²·s and m= 1.757E–05 N/m²·s have the same effect on the vortex shedding from the deck where at the beginning of the wind flow simulation. The kinetic energy increases to higher values till 50 seconds and after that the kinetic energy values decreases and increases in an oscillatory pattern until the end of the simulation, this is an indication that the oscillation of the deck starts after 50 seconds due to energy dissipation because of vortex shedding in the wake region and the kinetic energy becomes not stable. While in the case of air dynamic viscosity case m= 1.882E–05 N/m²·s, the kinetic energy of the system starts to increase till five seconds, but after that it begins to decrease in a simpler oscillatory non stable pattern but in a higher range (see Fig. 24). This behavior means that the vortex shedding begins in earlier position but with the same pattern like the previous two cases for the same wind speed due to the increase in the air dynamic viscosity but with higher vibration amplitudes. There is a nonlinear behavior of the deck vibration in relation with the air dynamic viscosity of the air.

Validation of semi 3D models

The finite element models are validated through a comparison process against experimental data. The calculated results of the vortex shedding generated due to wind flow assignation in the semi 3D models of the segmental Bridge decks are compared with two benchmarks results of the vortex shedding from the literature regarding the vortices shapes and patterns, where the generation of the vortices are indirectly associated with the kinetic energy of the system due to wind flow. Furthermore, the shapes and patterns of the vortices are affected by the motion of the deck, and the wind speed strongly takes a main role in this process.

Von Karman benchmark

The axis of validation is the ratio of the distance between two rows of vortices and the distance between two vortices in one row for a certain vortex trail. The value of this ratio in the vortex shedding for a bluff body stated by Von Karman is 0.282, where the same ratio value for the present semi 3D CFD models are between 0.291 and 0.296 considering five wind speed situations (5 m/s, 10 m/s, 15 m/s, 20 m/s and 25 m/s). There is a good agreement between the two results where the range of difference is narrow and falls between 0.009–0.014 (see Fig. 25) and Table 1, where this ratio values in the CFD models are stable for many wind speed situations approximately.

Dyrbye and Hansen benchmark

Another validation base is the distance between two vortices in one row in the vortex trail. The results of this distance considering the present semi 3D CFD models for the previously mentioned five wind speed situations are between 11.22–11.31 m, while this distance is suggested by Dyrbye and Hansen to equal 4.3 times the height of the bluff body, and in this case equals to 11.18 m. It is quite obvious that there is a good agreement between the two results and the gap range is between 0.04–0.13 m (see Fig. 26 and Table 2). The results of this value are better validated for low wind speeds.

Sensitivity analysis

The sensitivity analysis is a famous component of analysis for complex systems. The sensitivity analysis specifies the contribution of individual uncertain parameters to the uncertainty in the results of the analysis. Where the uncertainty analysis appoints the determination of the uncertainty in the results of the analysis that originates from the uncertainty in the analysis parameters. Analyses involve the consideration of models of the form: y=(x). Where y=[y1,y2,….,ynY] is a vector of analysis result and x=[x1,x2,….,xnX] is a vector of imprecisely known analysis inputs. In general the model f can be quite large and involved (e.g., a system of nonlinear partial differential equations requiring numerical solution or possibly a sequence of complex, linked models as is the case in a probabilistic risk assessment for a nuclear power plant or a performance assessment for a radioactive waste disposal facility the vector y of analysis results can be of high dimension and complex structure (e.g., the elements of y might be several hundred temporarily or spatially dependent functions); and the vector x of analysis inputs can also be of high dimension and complex structure e.g., several hundred variables, with some variables corresponding to physical properties of the system under study and other variables corresponding to parameters in probability distributions or perhaps to designators for alternative models. The uncertainty in the elements of x is characterized by a sequence of probability distributions. D1,,…., where Dj is a probability distribution characterizing the uncertainty in Xj [105–107].

Latin hypercube sampling

Latin hypercube sampling is one of the popular methods of experimental design. In order to allocate P samples, the range for each parameter is divided into P bins, which for n design variables, yields a total number of Pn bins in the design space. The samples are randomly selected in the design space so that each sample is randomly placed inside a bin, and for all one dimensional projections of the P samples and bins, there is exactly one sample in each bin. Uniform sampling increases the realization efficiency while randomizing within the strata prevents the introduction of a bias and avoids the extreme value effect associated with simple stratified sampling [108–110].

Sobol's sensitivity indices

Sobol’s sensitivity indices are ratios of partial variances to total variance, and for independent variables satisfy the relationship:

(6)\mathrm{1}={\displaystyle \sum _i{{\displaystyle S}}_i+{\displaystyle \sum _i{\displaystyle \sum _{j>\mathrm{1}}{{\displaystyle S}}_{ij}+}}}{\displaystyle \sum _i{\displaystyle \sum _{j>\mathrm{1}}{\displaystyle \sum _{k>\mathrm{1}}{{\displaystyle S}}_{ijk}+\cdots }}}

None of the sensitivity indices in Eq. (6) may be negative; therefore none may exceed one. The sum of S_j over all inputs j cannot exceed one, and does not equal one unless all the interaction orders are zero. By contrast, the sum of T_j over all j is never less than one, and equals one only if all interaction orders are zero, because this sum includes every main effect once and every interaction order multiple times [111]. The variance contributions to the total output variance of individual parameters and parameter interactions can be determined. These contributions are characterized by the ratio of the partial variance to the total variance, the Sobol sensitivity indices:

First order sensitivity index,

(7){{\displaystyle S}}_i={\displaystyle \frac{{{\displaystyle V}}_i}{V}}\mathrm{.}

Second order sensitivity index

(8){{\displaystyle S}}_{ij}={\displaystyle \frac{{{\displaystyle V}}_{ij}}{V}}\mathrm{.}

Total sensitivity index

(9){{\displaystyle S}}_{Ti}={{\displaystyle S}}_i+{\displaystyle \sum _{j\ne \mathrm{1}}{{\displaystyle S}}_{ij}}+\cdots

The first order index, S_i, is a measure for the variance contribution of the individual parameter X_i to the total model variance. The partial variance V_i in Eq. (7) is given by the variance of the conditional expectation {{\displaystyle V}}_i=V[E(Y/{{\displaystyle X}}_i)] and is also called the main effect’ of X_i on Y. It can be described as the fraction of the model output variance that would disappear on average when X_i would be fixed to a value in its range because V(Y)=E[V(Y/{{\displaystyle X}}_i)]+V[E(Y/{{\displaystyle X}}_i)] . The impact on the model output variance of the interaction between parameters X_i and X_j is given by S_ij and S_Ti is the result of the main effect of X_i and all its interactions with the other parameters (up to the Pth order).

The calculation of S_Ti can be based on the variance that results from the variation of all parameters, except X_i.

(10){{\displaystyle S}}_{Ti}=\mathrm{1}-{\displaystyle \frac{{{\displaystyle V}}_{\sim i}}{V}}

For additive models and under the assumption of orthogonal input parameters, S_Ti andS_i are equal and the sum of all S_i (and thus all S_Ti ) is 1. For non-additive models interactions exist: S_Ti is greater than S_i and the sum of all S_i is less than 1. On the other hand, the sum of all S_Ti is greater than 1. By analyzing the difference betweenS_Ti andS_i , one can determine the impact of the interactions between parameter X_i and the other parameters [112,113].

Surrogate model results and discussion

The regression coefficients were calculated for the actual kinetic energy and the lift force supporting on Latin hypercube sampling method, and in order to formulate the surrogate models for the predicted kinetic energy and lift force, 250 and 200 samples were used considering convergence process between the sensitivity indices respectively. Quadratic and interaction orders are used to construct the surrogate model for the case of kinetic energy for the system. The calculated coefficient of determination R² between the actual and the predicted kinetic energy was 99.99% (see Fig.27) which is an excellent representation of the predicted kinetic energy, in the same time means that just 0.01% of the system response still unexplained.

The following response surface plots in Figs. 28–30 which have been generated using MATLAB codes, are the relation between each two aerodynamic parameters X1X2, X1X3, X2X3 and the predicted kinetic energy of the system due to wind excitation and vortex shedding.

Quadratic and interaction orders are used to build the surrogate model to calculate the lift force generated in the deck. The coefficient of determination R² calculated between the actual and the predicted lift force was R² = 99.99% which is a very good approximation for the prediction of lift force, where only 0.0 1% of the system response remains unexplained (see Fig.31).

The next response surface plots express on the relation between each two aerodynamic parameters X1X2, X1X3, X2X3 and the predicted lift force generated in the deck as shown in Figs. 32–34 where these figures have been generated using MATLAB codes.

Sensitivity analysis results and discussion

The main orders of sensitivity indices for each aerodynamic parameter in addition to its interaction orders were calculated considering the convergence results recommending the use of 250 samples to calculate the kinetic energy of the system and 200 samples to calculate the lift force for vortex induced vibration of the deck (see Fig.35 and Fig.36). Supporting on the calculated results, the total sensitivity indices for each aerodynamic parameter have been calculated (see Table 3).

In relation with the kinetic energy, the total order sensitivity index of aerodynamic parameter (wind speed) is 0.9997, which is the biggest aerodynamic parameter, while the total order sensitivity index of X2 is 0.0036. Furthermore, the total order sensitivity index of X3 is 0.0040, this means that the kinetic energy of the system is 99.97% due to variation in the wind speed, and it is 0.36% due to the variation in the deck streamlined length. In the other hand, it is 0.40% due to the variation in the dynamic viscosity of the air. While the interaction index between the aerodynamic parameters X1 and X2 is 0.0023 and between X1 and X3 is 0.0027, while between X2 and X3 is 0.0001, which means that there is a small interaction between the aerodynamic parameters taking part in the variation of the kinetic energy of the system due to vortex shedding.

While considering the lift force, the total order sensitivity index of X1 is 0.8375, which is the biggest aerodynamic parameter, while the total order sensitivity index of X2 is 0.1707. Furthermore, the total order sensitivity index of X3 is 0.0294, this means that the lift force is dependable 83.75% on the wind speed variation, while it is 17.07% due to the deck streamlined length variation, and it is 2.94% due to the variation in the dynamic viscosity of the air. While the interaction index between X1 and X2 is 0.0157 and between X1 and X3 is 0.0150, in the other hand, between X2 and X3 is 0.0012. This proofs that the surrogate model is non-additive because there are small interactions between the aerodynamic parameters especially between wind speed and deck streamlined length once and between wind speed and dynamic viscosity of the air in other side take part in the variation of the lift force generated in the deck, while not appreciable interaction between the deck streamlined length and the dynamic viscosity of the air is seen to affect the lift force.

Convergence of the results

The process of global sensitivity analysis supporting on Sobol's sensitivity indices requires sufficient samples of experiments to identify the predicted effect of the aerodynamic parameters on the vortex induced vibration of the deck through two related outputs. The most efficient number of samples is being identified through the convergence of the sum of first orders and total sensitivity indices of the aerodynamic parameters. All the sensitivity indices (first orders, interaction orders and total orders) for each aerodynamic parameter have been calculated using m MATLAB codes. Two outputs have been considered in the process of converges, the kinetic energy and the lift force. Fig.35 and Fig.36 show the relation between the number of samples and the sum of first orders sensitivity indices, in the same time between the number of samples and the sum of total sensitivity indices of the three aerodynamic parameters.

For the case of kinetic energy (see Fig.35), the two curves of the sum of first orders and total orders of sensitivity indices at the beginning are starting without stability for coinciding to reach convergence, this situation continues till 250 samples. After this stage the two curves are starting to converge at the 250 number of samples, where the two curves continue to remain in a stable. In the same way, for the lift force (see Fig.36) the two curves are reaching convergence at 200 samples, and the coinciding pattern continues in a stable situation.

The convergence results of the two output cases predetermine utilizing 250 and 200 samples of experiments to efficiently calculate the predicted rational effects of each aerodynamic parameter on both the variation of the kinetic energy and lift force respectively.

Conclusions

Below are the conclusions which have been formulated:

1) The wind speed is the most effective parameter that affects the generation of vortex shedding, where increasing the wind speed alters the shapes and patterns of the vortices and changes the oscillatory behavior of the deck, this by increasing the shedding frequency and increasing the lift forces.

2) The deck streamlined length parameter has a nonlinear effect that represents a certain deck shape falls between a bluff shape and a streamlined shape which is identified to be the optimum case that supports the generation of vortex shedding and the oscillation of the deck despite of that the streamlined shapes help to eliminate the vortex shedding.

3) The dynamic viscosity of air parameter has a nonlinear effect on the vortex shedding and the oscillation of the deck, where increasing this parameter to high values will enhance the generation of the vortex shedding due to energy dissipation mechanism.

4) Latin hypercube sampling method established efficient surrogate models that are used to calculate the predicted kinetic energy of the system and the lift forces to analyze the vortex induced vibration.

5) Kinetic energy concept involved in detecting superb results that represents the behavior of the segmental Bridge deck during wind excitation in the stages of vortex shedding and oscillation of the deck.

6) The rational effects of wind speed parameter were 99.97% and 83.75% on the kinetic energy of the system and the lift forces respectively, which is the maximum role among the other parameters that should be considered for uncertainties in the generation of vortex shedding and oscillation of the deck due to vortex induced vibration. While the rational effects of the deck streamlined length parameter were 0.36% and 17.07% for the same outputs, which is considered not important for the kinetic energy of the system and in the same time has an appreciable effect on the lift forces, and consequently a fair consideration should be given to this parameter due to its effect on the generation of vortex shedding and the vibration of the deck. In the other hand, the dynamic viscosity of air parameter has rational effects 0.40% and 2.94%, which is an indication that this parameter has a very small effect on the kinetic energy of the system and a small effect on the lift forces, hence uncertainties related to this parameter on the vortex shedding and oscillation of the deck is very small especially in the high values only.

References

[1] Sarwar M W, Ishihara T. Numerical study on suppression of vortex-induced vibrations of box girder bridge section by aerodynamic counter measures. Journal of Wind Engineering and Industrial Aerodynamics, 2010, 98(12): 701–711

[2] Frandsen J B. Numerical bridge deck studies using finite elements.Part I: flutter. Journal of Fluids and Structures, 2004, 19(2): 171–191

[3] Chen A, Ma R, Wang D. Aerodynamic Problems of Cable Stayed Bridges Spanning over one thousand meters. Engineering and Science, 2009, 7(3): 16–22

[4] Fujino Y, Siringoringo D. Vibration Mechanisms and Controls of Long-Span Bridges: A Review. Structural Engineering International, 2013, 3(3): 248–268

[5] Jose MaTerres Nicoli. Different Mechanisms of Vortex Induced Vibration of Bridges. PhD [dissertation], Granada: Department of Mechanical and hydraulic engineering structures, University of Granada, 2008.

[6] Diana G, Resta F, Belloli M, Rocchi D. Marco Belloli, Daniele Rocchi. On the vortex shedding forcing on suspension bridge deck. Journal of Wind Engineering and Industrial Aerodynamics, 2006, 94(5): 341–363

[7] Belloli M, Fossati F, Giappino S, Muggiasca S. Vortex induced vibrations of a bridge deck: Dynamic response and surface pressure distribution. Journal of Wind Engineering and Industrial Aerodynamics, 2014, 133: 160–168

[8] Flamand O, De Oliveira F, Stathopoulos-Vlamis A, Papanikolas P. Conditions for occurrence of vortex shedding on a large cable stayed bridge; Full scale data from monitoring system. 6th European and African Conference on Wind Engineering, Robinson, Cambridge, 2013.

[9] Zhang M M, Cheng L, Zhou Y. Closed-loop-controlled vortex shedding and vibration of a flexibly supported square cylinder under different schemes. Physics of Fluids, 2004, 16(5): 1439–1448

[10] Grouthier C, Michelin S, Bourguet R, Modarres-Sadeghi Y, de Langre E. On the efficiency of energy harvesting using vortex-induced vibrations of cables. Journal of Fluids and Structures, 2014, 49: 427–440

[11] Zhang X. Wind Effect on Long Span Bridge. M.Sc.[dissertation]. Massachusetts: Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, 2012.

[12] Peter A. Irwin. Bluff body aerodynamics in wind engineering. Journal of Fluids and Structures, 2008, 96: 701–712

[13] Fariduzzaman. Leonardo Gunawan, Lavi R. Zuhal, Amrinsyah Nasution. Low Wind Speed Heaving Resonance. 3rd International Symposium on Advanced Fluid/Solid Science and Technology in Experimental Mechanics, Taiwan, China, 2008.

[14] Budarapu P R, Sudhir Sastry Y B, Natarajan R. Design concepts of an aircraft wing: composite and morphing airfoil with auxetic structures. Frontiers of Structural and Civil Engineering, 2016, 10(4): 394–408

[15] Sudhir Sastry Y B, Budarapu P R, Madhavi N, Krishna Y. N. Madhavi Y. Krishna. Buckling analysis of thin wall stiffened composite panels. Computational Materials Science, 2015, 96B: 459–471

[16] Sudhir Sastry Y B, Budarapu P R, Krishna Y, Devaraj S. Studies on ballistic impact of the composite panels. Theoretical and Applied Fracture Mechanics, 2014, 72: 2–12

[17] Zi G, Rabczuk T, Wall W A. Extended Meshfree Methods without Branch Enrichment for Cohesive Cracks. Computational Mechanics, 2007, 40(2): 367–382

[18] Rabczuk T, Belytschko T. Application of particle methods to static fracture of reinforced concrete structures. International Journal of Fracture, 2006, 137(1–4): 19–49

[19] Rabczuk T, Bordas S, Zi G. A three-dimensional meshfree method for continuous multiplecrack initiation, nucleation and propagation in statics and dynamics. Computational Mechanics, 2007, 40(3): 473–495

[20] Bordas S, Rabczuk T. Zi. G.: Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by extrinsic discontinuous enrichment of meshfree methods without asymptotic enrichment. Engineering Fracture Mechanics, 2008, 75(5): 943–960

[21] Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A geometrically non-linear three dimensional cohesive crack method for reinforced concrete structures. Engineering Fracture Mechanics, 2008, 75(16): 4740–4758

[22] Rabczuk T, Bordas S, Zi G. On three-dimensional modelling of crack growth using partition of unity methods. Computers & Structures, 2010, 88(23–24): 1391–1411

[23] Rabczuk T, Zi G. A meshfree method based on the local partition of unity for cohesive cracks. Computational Mechanics, 2007, 39(6): 743–760

[24] Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37–40): 2437–2455

[25] Rabczuk T, Areias P M A, Belytschko T. A meshfree thin shell method for nonlinear dynamic fracture. International Journal for Numerical Methods in Engineering, 2007, 72(5): 524–548

[26] Nguyen-Thanh N, Valizadeh N, Nguyen M N, Nguyen-Xuan H, Zhuang X, Areias P, Zi G, Bazilevs Y, De Lorenzis L, Rabczuk T. An extended isogeometric thin shell analysis based on Kirchho-Love theory. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 265–291

[27] Areias P, Msekh M A, Rabczuk T. Damage and fracture algorithm using the screened Poisson equation and local remeshing. Engineering Fracture Mechanics, 2016, 158: 116–143

[28] Areias P M A, Rabczuk T, Camanho P P. Finite strain fracture of 2D problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 2014, 72: 50–63

[29] Areias P, Rabczuk T, Dias-da-Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137

[30] Dahl K. Aeroelastic behavior of very long span suspension bridges. M.Sc.[dissertation].  Stavanger: Faculty of Mechanical and Structural Engineering and Materials Science, University of Stavanger, 2013.

[31] Patil A S. Mitigation of Vortex Induced Response in Long Span Bridges. M.Sc.[dissertation].  Florida: Department of Civil and Environmental Engineering. The Florida State University, 2010.

[32] Tang J, Viieru D, Shyy W. Dragos Viieru, Wei Shyy. Effects of Reynolds Number and Flapping Kinematics on Hovering Aerodynamics. AIAA Journal, 2008, 46(4): 967–976

[33] Abdi I A, Abdulrahim A, Haser S, Uzol O, Kurç O. Experimental and Numerical Investigation of Wind Effects on Long Span Bridge Decks. 10th International Congress on Advances in Civil Engineering, Ankara, TurkeyOctober, 2012.

[34] Adelino V. Lopes, Alvaro Cunha, Luis M. C. Simoes. CFD based evaluation of the lock-in phenomena of a bridge under wind load. III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering, Lisbon, Portugal, 2006.

[35] Bosman E. Wind induced Vibration of fr UHSC Bridge Decks. M.Sc.[dissertation]. Delft: Faculty of Civil Engineering and Geosciences, Delft University of Technology, 2012.

[36] Abraham S. Corriols,  Guido Morgenthal. Computational analysis of VIV observed on existing bridges. The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7), Shanghai, China, 2012.

[37] Francisco J. Huera Huarte. Multi-mode Vortex Induced Vibrations of Flexible Circular Cylinder.  PhD [dissertation], London: Department of Aeronautics, Imperial College London, 2006.

[38] Chen W L, Li H, Hu H. An experimental study on the unsteady vortices and turbulent flow structures around twin-box-girder bridge deck models with different gap ratios. Journal of Wind Engineering and Industrial Aerodynamics, 2014, 132: 27–36

[39] Hansen S O, Srouji R G, Bjørn I K B. Vortex-induced vibrations of streamlined single box girder bridge decks. 14 TH International conference on wind engineering ICWE 2014, Robinson, Cambridge, 2013.

[40] Blackburn H, Henderson R. Lock-in Behavior in Simulated Vortex –Induced Vibration. Journal of Experimental Thermal and Fluid Science, 1996, 12(2): 184–189

[41] Harold R. Bosch,  Tarun Dhall. Aerodynamic Investigation of Generic Bridges Deck Section Models. BBAA VI International Colloquium on: Bluff Bodies Aerodynamics & Applications, Milano, Italy, 2008.

[42] Schewe G, Larsen A. Reynolds number effects in the flow around a bluff bridge deck cross section. Journal of Fluids and Structures, 1998, 74–76: 829–838

[43] Larsen A, Walther J H. Discrete vortex simulation of flow around five generic bridge deck sections. Journal of Wind Engineering and Industrial Aerodynamics, 1998, 77–78: 591–602

[44] Amiri F, Milan D, Shen Y, Rabczuk T, Arroyo M. Phase-field modeling of fracture in linear thin shells. Theoretical and Applied Fracture Mechanics, 2014, 69: 102–109

[45] Areias P, Rabczuk T. Finite strain fracture of plates and shells with configurational forces and edge rotation. International Journal for Numerical Methods in Engineering, 2013, 94(12): 1099–1122

[46] Chau-Dinh T, Zi G, Lee P S, Song J H, Rabczuk T. Phantom-node Method for Shell Models with Arbitrary Cracks. Computers & Structures, 2012, 92–93: 242–256

[47] Nguyen-Thanh N, Kiendl J, Nguyen-Xuan H, Wuchner R, Bletzinger K U, Bazilevs Y, Rabczuk T. Rotation free isogeometric thin shell analysis using PHT-splines. Computer Methods in Applied Mechanics and Engineering, 2011, 200(47-48): 3410–3424

[48] Nguyen-Thanh N, Rabczuk T, Nguyen-Xuan H, Bordas S. A smoothed finite element method for shell analysis. Computer Methods in Applied Mechanics and Engineering, 2008, 198(2): 165–177

[49] Thai H C, Nguyen-Xuan H, Bordas S, Nguyen-Thanh N, Rabczuk T. Isogeometric analysis of laminated composite plates using the higher-order shear deformation theory. Mechanics of Advanced Materials and Structures, 2015, 22(6): 451–469

[50] Thai C H, Ferreira A J M, Bordas S, Rabczuk T, Nguyen-Xuan H. Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory. European Journal of Mechanics. A, Solids, 2014, 43: 89–108

[51] Phan-Dao H, Nguyen-Xuan H, Thai-Hoang C, Nguyen-Thoi T, Rabczuk T. An edge-based smoothed finite element method for analysis of laminated composite plates. International Journal of Computational Methods, 2013, 10(1): 1340005

[52] Thai C H, Nguyen-Xuan H, Nguyen-Thanh N, Le T H, Nguyen-Thoi T, Rabczuk T. Static, free vibration and buckling analysis of laminated composite Reissner-Mindlin plates using NURBS-based isogeometric approach. International Journal for Numerical Methods in Engineering, 2012, 91(6): 571–603

[53] Nguyen-Xuan H, Rabczuk T, Bordas S, Debongnie J F. A smoothed finite element method for plate analysis. Computer Methods in Applied Mechanics and Engineering, 2008, 197(13-16): 1184–1203

[54] Budarapu P R, Javvaji B, Sutrakar V K, Roy Mahapatra D, Zi G, Rabczuk T. Crack propagation in Graphene. Journal of Applied Physics, 2015, 118(6): 064307

[55] Areias P, Rabczuk T, Camanho P P. Initially rigid cohesive laws and fracture based on edge rotations. Computational Mechanics, 2013, 52(4): 931–947

[56] Ghorashi S, Valizadeh N, Mohammadi S, Rabczuk T. T-spline based XIGA for Fracture Analysis of Orthotropic Media. Computers & Structures, 2015, 147: 138–146

[57] Liu Z. On The Investigation of Bridge Buffeting Simulation Techniques. PhD [dissertation], Nottingham: School of Civil Engineering, University of Nottingham, 2011.

[58] Borna A, Habashi W G, Mc Clure G, Nadarajah S K. CFD-FSI simulation of vortex-induced vibrations of a circular cylinder with low mass-damping. Journal of Wind and Structures, 2013, 16(5): 411–431

[59] Yang S W, Budarapu P R, Mahapatra D R, Bordas S P A, Zi G, Rabczuk T. A meshless adaptive multiscale method for fracture. Computational Materials Science, 2015, 96B: 382–395

[60] Budarapu P R, Sudhir Sastry Y B, Javvaji B, Mahapatra D R. Vibration Analysis of Multi-walled Carbon Nanotubes Embedded in Elastic Medium. Frontiers of Structural and Civil Engineering, 2014, 8(2): 151–159

[61] Budarapu P R, Narayana T S S, Rammohan B, Rabczuk T. Directionality of sound radiation from rectangular panels. Applied Acoustics, 2015, 89: 128–140

[62] Budarapu P R, Gracie R, Yang S W, Zhaung X, Rabczuk T. Efficient Coarse Graining in Multiscale Modeling of Fracture. Theoretical and Applied Fracture Mechanics, 2014, 69: 126–143

[63] Budarapu P R, Gracie R, Bordas S P A, Rabczuk T. An adaptive multiscale method for quasi-static crack growth. Computational Mechanics, 2014, 53(6): 1129–1148

[64] Zhuang X, Augarde C, Mathisen K. Fracture modelling using meshless methods and level sets in 3D: framework and modelling. International Journal for Numerical Methods in Engineering, 2012, 92(11): 969–998

[65] Ren H, Zhuang X, Cai Y, Rabczuk T. Dual-Horizon Peridynamics. International Journal for Numerical Methods in Engineering, 2016, 108(12): 1451–1476

[66] Areias P, Rabczuk T, Msekh M. Phase-field analysis of finite-strain plates and shells including element subdivision. Computer Methods in Applied Mechanics and Engineering, 2016, 312: 322–350

[67] Areias P, Reinoso J, Camanho P, Rabczuk T. A constitutive-based element-by-element crack propagation algorithm with local mesh refinement. Computational Mechanics, 2015, 56(2): 291–315

[68] Kiviluoma R. Frequency-domain approach for calculating wind-induced vibration and aeroelastic stability characteristics of long-span bridges. PhD [dissertation], Espoo: Helsinki University of Technology, 2001.

[69] Knut Inge Edvardsen. Numerical analysis of vortex shedding. M.Sc.[dissertation]. Stavanger: Faculty of Science and Technology, University of Stavanger, 2010.

[70] Muhammad Tedy Asyikin. CFD Simulation of Vortex Induced Vibration of a Cylindrical Structure.  M.Sc.[dissertation]. Trondheim: Department of Civil and Transport Engineering. Norwegian University of Science and Technology, 2012.

[71] Wu T. Ahsan Kareem. An overview of vortex-induced vibration (VIV) of bridge decks. Frontries of Structural and Civil Engineering-Springer, 2012, 6(4): 335–347

[72] Kazutoshi M, Masafumi T, Tooru I. Reynolds Number Effects on the Steady and Unsteady Aerodynamic Forces Acting on the Bridge Deck Sections of Long-Span Suspension Bridge. IHI Engineering Review, 2007, 40(1): 12–26

[73] Sutardi S. The Rate of Turbulent Kinetic Energy Dissipation in a Turbulent Boundary Layer on a Flat Plate. IPTEK. Journal of Proceeding Series, 2014, 1: 63–66

[74] Naimul Haque M. Hiroshi Katsuchi, Hitoshi Yamada,Mayuko Nishio. Flow field analysis of a pentagonal-shaped bridge deck by unsteady RANS. Engineering Applications of Computational Fluid Mechanics, 2016, 10(1): 1–16

[75] Yuh-Yi Lin. Chii-Ming Cheng, Jong-Cheng Wu, Tsang-Lien Lan and Kuo-Ting Wu. Effects of Deck Shape and Oncoming Turbulence on Bridge Aerodynamics. Tamkang Journal of Science and Engineering, 2005, 8(1): 43–56

[76] Bienkiewicz B. Wind Tunnel Study of Effects of Geometry Modification on Aerodynamics of a Cable Stayed Bridge Deck. Journal of Wind Engineering and Industrial Aerodynamics, 1987, 26(3): 325–339

[77] Nagao F, Utsunomiya H, Oryu T, Manabe S. Manabe, S.: Aerodynamic Efficiency of Triangular Fairing on Box Girder Bridge. Journal of Wind Engineering and Industrial Aerodynamics, 1993, 49(1–3): 565–574

[78] Wardlaw R L, Tanaka H, Utsunomiya H. Utsunomiya, H.: Wind Tunnel Experiments on the Effects of Turbulence on the Aerodynamic Behavior of Bridge Road Decks. Journal of Wind Engineering and Industrial Aerodynamics, 1983, 14(1–3): 247–257

[79] Gosteev Yu A, Obuhovskiy A D, Salenko S D. Influence of the shape on aerodynamic characteristics of girder bridges. Magazine of Civil Engineering, 2014, 5(49): 63–72

[80] Cai Y, Zhu H, Zhuang X. A continuous/discontinuous deformation analysis (CDDA) method based on deformable blocks for fracture modelling. Frontiers of Structural & Civil Engineering, 2014, 7(4): 369–378

[81] Valizadeh N, Natarajan S, Gonzalez-Estrada O A, Rabczuk T, Bui T Q, Bordas S P A. Tinh Quoc Bui, Bordas S.P.A.: NURBS-based finite element analysis of functionally graded plates: static bending, vibration, buckling and flutter. Composite Structures, 2013, 99: 309–326

[82] Rabczuk T, Belytschko T. A three dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29-30): 2777–2799

[83] Rabczuk T, Eibl J, Stempniewski L. Simulation of high velocity concrete fragmentation using SPH/MLSPH. International Journal for Numerical Methods in Engineering, 2003, 56(10): 1421–1444

[84] Rabczuk T, Eibl J, Stempniewski L. Numerical analysis of high speed concrete fragmentation using a meshfree Lagrangian method. Engineering Fracture Mechanics, 2004, 71(4-6): 547–556

[85] Rabczuk T, Eibl J. Modeling dynamic failure of concrete with meshfree particle methods. International Journal of Impact Engineering, 2006, 32(11): 1878–1897

[86] Jurado J A, Sanchez R, Hernandez S, Nieto F, Kusano I. A review of cases of vortex shedding excitation in bridges: Sectional models testing. The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7), Shanghai, China; 2012.

[87]  Maria Isabel Ribeiro. 2004. Gaussian Probability Density Functions: Properties and Error Characterization. Institute for Systems and Robotics, Instituto Superior Tcnico, 2004.

[88] Meinhardt C, Yin X, Luo Y, Gao X. Enhancement of the Chong Qi Bridge Project girder sections to reduce vortex shedding induced oscillations by applying passive Tuned Mass Damper Systems.  Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014, Porto, Portugal; 2014.

[89] Rabczuk T, Samaniego E, Belytschko T. Simplied model for predicting impulsive loads on submerged structures to account for fluid-structure interaction. International Journal of Impact Engineering, 2007, 34(2): 163–177

[90] Rabczuk T, Areias P M A, Belytschko T. A simplied meshfree method for shear bands with cohesive surfaces. International Journal for Numerical Methods in Engineering, 2007, 69(5): 993–1021

[91] Rabczuk T, Samaniego E. Discontinuous modelling of shear bands using adaptive meshfree methods. Computer Methods in Applied Mechanics and Engineering, 2008, 197(6–8): 641–658

[92] Rabczuk T, Gracie R, Song J H, Belytschko T. Immersed particle method for fluid-structure interaction. International Journal for Numerical Methods in Engineering, 2010, 81(1): 48–71

[93] Rabczuk T, Belytschko T. Cracking particles: a simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343

[94] FlamandOlivierDe Oliveira, FabriceStathopoulos-Vlamis, ArisPapanikolas, PanayotisPanagis, Akis. Using Non Continuous Records From Full Scale Monitoring System For Fatigue Assessment, La Cité Nantes, France, 2014.

[95] Andrianne T. Experimental and Numerical Investigations of the Aeroelastic Stability of Bluff Structures. PhD [dissertation], Liege: Department of Aerospace and Mechanics, University of Liege; 2012.

[96] Dol S S, Kopp G A, Martinuzzi R J, 0. Gregory A. Kopp, Robert J. Martinuzzi. The suppression of periodic vortex shedding from a rotating circular cylinder. Journal of Wind Engineering and Industrial Aerodynamics, 2008, 96(6–7): 1164–1184

[97] Stevens R, Simonsen K. Investigation on Long Span Suspension Bridges During Erection, The Great Belt East Bridge. M.Sc.[dissertation]. Aalborg: Department of Civil Engineering, Aalborg University, 2008.

[98] Diego A A M. Stochastic Structural Control of Bridges Subject to Wind-Induced Vibrations Using Separated Surfaces. M.Sc.[dissertation]. Colombia: Faculty of Engineering, National University of Colombia, 2003.

[99] Rabczuk T, Belytschko T, Xiao S P. Stable particle methods based on Lagrangian kernels. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12–14): 1035–1063

[100] Rabczuk T, Akkermann J, Eibl J. A numerical model for reinforced concrete structures. International Journal of Solids and Structures, 2005, 42(5–6): 1327–1354

[101] Rabczuk T, Eibl J. Numerical analysis of prestressed concrete beams using a coupled element free Galerkin/nite element method. International Journal of Solids and Structures, 2004, 41(3–4): 3–4, 1061–1080

[102] Rabczuk T, Belytschko T. Adaptivity for structured meshfree particle methods in 2D and 3D. International Journal for Numerical Methods in Engineering, 2005, 63(11): 1559–1582

[103] Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31

[104] Amiri F, Anitescu C, Arroyo M, Bordas S, Rabczuk T. XLME interpolants, a seamless bridge between XFEM and enriched meshless methods. Computational Mechanics, 2014, 53(1): 45–57

[105] Curtis B. Storlie, Jon C. Multiple predictor smoothing methods for sensitivity analysis: Description of techniques. Journal of Reliability Engineering and System Safety, 2008, 93(1): 28–54

[106] Saltelli A. Global sensitivity analysis: An introduction. European Commission, Joint Research Centre of Ispra, Italy, 2004.

[107] Haruko M. Wainwright, Stefan Finsterle, Yoojin Jung, Quanlin Zhou, Jens T. Birkholzer. Making sense of global sensitivity analyses. Computers & Geosciences, 2014, 65: 84–94

[108] Pan F, Zhu J, Ye M, Yakov A. Pachepsky, Yu-Shu Wue. Sensitivity analysis of unsaturated flow and contaminant transport with correlated parameters. Journal of Hydrology (Amsterdam), 2011, 397(3–4): 238–249

[109] Koziel S, Ciaurri D E, Leifsson L. Comput. Optimization, Methods and Algorithms, SCI 356, Springer-Verlag Berlin, Heidelberg, 2011, 33–59.

[110] Ashok Bakkiyaraj R, Kumarappan N. Reliability Evaluation of Composite Electric Power System Based On Latin Hypercube Sampling. International Journal of Electrical, Robotics. Electronics and Communications Engineering, 2013, 7(4): 245–250

[111] Glen G, Isaacs K. Estimating Sobol sensitivity indices using correlations. Journal of Environmental Modelling & Software, 2012, 37: 157–166

[112] Nossent J, Elsen P, Bauwens W. Sobol sensitivity analysis of a complex environmental model. Journal of Environmental Modelling and Software, 2011, 26(12): 1515–1525

[113] Zhang X Y, Trame M N, Lesko L J, Schmidt S. Sobol Sensitivity Analysis: A Tool to Guide the Development and Evaluation of Systems Pharmacology Models. CPT: Pharmacometrics & Systems Pharmacology, 2015, 4(2): 69–79

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