Optimization of hanger arrangement in pedestrian tied arch bridge with sparse hanger system

Abstract

The hanger arrangement has a decisive influence on the mechanical behavior of the tied arch bridge with network hanger system. Many investigations on highway or railway tied arch bridges show that the arch bridges with dense network hangers are superior to those with vertical hangers under larger live load. However, numerous dense inclined hangers lower the esthetic effect of the bridge, especially for pedestrian tied arch bridges. Consequently, the sparse inclined hanger system is recommended in the design of pedestrian tied arch bridges. However, the amount of possible schemes of the hanger arrangement grows rapidly with the number of hangers increasing beyond 10, rendering great difficulties in searching for proper schemes. In this article, a dimensionless optimization approach based on genetic algorithm is proposed in searching for hanger arrangement schemes. Numerical analysis indicates that the proposed method is effective in the optimization of pedestrian tied arch bridge with sparse inclined hanger system, and some of the feasible hanger arrangement solutions show more excellent mechanical properties.

Keywords

genetic algorithm hanger arrangement pedestrian tied arch bridge sparse inclined hanger system

Introduction

Relying on the artistic configuration and unique structural advantages, tied arch bridges have great advantages in small and medium span bridges. In the design of tied arch bridges, the arrangement of hangers is particularly important, as it directly affects the mechanical and esthetic properties of the arch bridge.

Among the existing tied arch bridges, most of them are arranged with vertical hangers. It was realized that the stiffness of tied arch bridge could be greatly improved by replacing vertical suspender with inclined suspender. This idea was originally proposed by Nielsen, a Danish engineer. Therefore, the tied arch bridge with inclined hanger is also called the Nielsen system. Next, Per Tveit (1966, 1987) introduced the concept of network arch bridges, which can be defined as those with inclined hangers with multiple intersections. Compared to the conventional arch bridge with vertical hangers, the great advantage of the network arch bridge lies in the decrease of internal forces under dead load and live load. Thus, a steel bridge with vertical hangers may require up to 4.5 times more material than a corresponding network bridge (Per Tveit, 1999).

Several researchers have studied a lot to improve the structural behavior of the tied arch bridges with dense network hangers. Some network arch solutions with aesthetical advantages and better structural behavior have been designed by Per Tveit (2001, 2008). Brunn and Schanack proposed a new hanger arrangement (radial arrangement) for railway bridges with concrete decks. They adopted a hypothesis for optimization of hanger arrangement that the arch should be a part of a circle and the hanger should be arranged in such a way that hanger intersections lie on the radii of the arch circle (Brunn and Schanack, 2003). More studies on structural performance in bridges with vertical, fan, and network patterns were performed by De Zotti et al. (2007). Numerical analyses of about 400 configurations are developed (De Zotti et al., 2007). After that, Teich (2012) analyzed five different arrangement types and concluded that the radial arrangement and arrangement with constantly decreasing slope of hangers are the best choices. He also recommended using up to 50 hangers, since their efficiency of improvement reduces significantly above 50 in his dissertation. Pellegrino et al. (2010) focused on decreasing the fatigue stressing of vertical, fan, and network arrangements without increasing the bending moments in the arch and the girder. Using a three-step optimization algorithm, Bruno et al. (2016) proposed a new design methodology, which evaluated the optimum configuration of network arch bridge schemes. Recently, studying network arch timber bridges, Ostrycharczyk and Malo (2017) introduced a new network pattern as a modification of a radial pattern to improve the performance of the network arch. Involving conventional scheme of network arch bridge, the identification of some topological parameters as well as shape configurations and all sizing parameters of structural members, Belevicius et al. (2018) sought the minimum weight of network arch bridges. By using genetic algorithm (GA), Tan and Yao (2018) combined ANSYS with MATLAB, and optimized the hanger arrangement of tied arch bridge to obtain some layout forms of sparse and non-uniform hanger, which have good mechanical performance and varied configuration.

Many investigations on highway or railway tied arch bridges show that the arch bridges with dense network hangers are superior to those with vertical hangers under larger live load. However, numerous dense inclined hangers lower the esthetic effect of the bridge, especially for pedestrian tied arch bridges. On the other hand, too dense an arrangement of hangers is not economical for a pedestrian bridge because the cost of hangers is usually three times higher than that of ordinary steel. Consequently, sparse inclined hanger system is recommended in the design of pedestrian tied arch bridges for economical and esthetic reasons. However, as the hangers become sparse, the mechanical behavior of the bridge is significantly different from that of a traditional network arch bridge and whether the better hanger arrangement (the radial arrangement and arrangement with constantly decreasing slope of hangers) in the original dense hanger system is still applicable requires further verification.

In this article, a dimensionless optimization approach based on GA is proposed in searching for better hanger arrangement schemes. Numerical analysis indicates that the proposed method is effective in optimization of sparse network hanger arch bridge, and some of the feasible hanger arrangement solutions show more excellent mechanical properties.

The relationship between the number of hangers and the load capacity of a pedestrian network arch bridge

Basic parameters

The reference bridge is a pedestrian tied arch bridge having a 4 m wide and span of 100 m, the arch rise is 17 m, and hangers change from 6 to 50 per arch. The steel arches are circular, consisting of two connected arch ribs each with a diameter of 600 mm and thickness of 8.5 mm. The bridge has a concrete deck supported by two longitudinal steel I-beams with a height of 800 mm and a width of 600 mm and 40 transverse beams. The bending moment of inertia ratio I_arch/I_girder is 0.18 in the vertical plane and 4.6 in the horizontal plane.

Considering dead loads, the self-weight is automatically taken into account by the finite element software ANSYS according to geometry and materials (the total self-weight is about 1235 kN), while the other dead loads (road pavement, concrete slabs, and guard rail) are equal to 18.8 kN/m. Live load is movable uniformly distributed, which equals to 3.5 kN/m² and is arranged in the most unfavorable case through secondary development of ANSYS according to General Code for Design of Highway Bridges and Culverts in China (Ministry of Communication (MOC) of China, 2015).

The analysis is performed with ANSYS, considering 100 different possible positions (100 load positions in longitudinal direction are defined as 100 steps of 1 m) of the most unfavorable live load, according to the specifications mentioned above.

Considering the effect of internal force on different parts of the structure, bending moment is often one of the most unfavorable factors for arches. For the girder, too large bending moment will still cause an increase in the amount of steel, and the fatigue performance of the hangers depends mainly on the magnitude and variation of the axial force. Integrally considering the mechanical behavior of the arches, the girder, and the hangers, the following mechanical evaluation parameters are adopted:

Max M_a: the maximum absolute value of the bending moment in the arches;

Max M_g: the maximum absolute value of the bending moment in the girder;

Ave N: the average axial force of the hangers;

Max ΔN: the maximum variation of axial force occurring in the most unfavorable hanger of each model.

Two kinds of traditional hanger arrangement of network arch bridge

Since the network arch bridge system was proposed by Per Tveit, many scholars have studied the optimization of the hanger arrangement of the system. Among them, there are two kinds of hanger arrangement widely recognized, constant-change-of-slope arrangement and radial arrangement. These two traditional hanger arrangements are both based on equidistant nodes along the circular arch. To investigate the suitable number of hangers for pedestrian network arch bridge, in this section two relevant rules are taken into account:

Constant-change-of-slope arrangement, for which the variables describing the arrangement are the starting angle α and the change of the inclination Δα from one hanger to the next (Figure 1). The configuration taken as a reference is the arrangement in which $α = 50 ° and Δ α = - 0.3 °$ . This is found to be the best one according to the study of Brunn and Schanack (2003).

Radial arrangement, in which the hangers cross symmetrically the radii with the constant angle $φ$ , which is marked gray in Figure 2. In this article, $φ = 30 °$ is taken as a reference as this seems to be the best one from the multiparametric study of De Zotti et al. (2007).

Figure 1.

Constant-change-of-slope arrangement with 44 hangers (Brunn and Schanack, 2003).

Figure 2.

Radial arrangement with 44 hangers (Brunn and Schanack, 2003).

Influence of the number of hangers

To examine the influence of the number of hangers in pedestrian network arch bridge with two kinds of traditional hanger arrangement mentioned above, 46 bridges were calculated varying the number of hangers (23 bridges for each hanger arrangement). Considering the symmetry of the structure, the number of hangers changes from 6 to 50 with a step of 2.

Figures 3 and 4 show the relation between four evaluation parameters in the pedestrian network arch bridge and the number of hangers for different hanger arrangements, respectively. Although the arrangement of the hangers is different, the two diagrams show the similar results that increasing the number of hangers noticeably changes the evaluation parameter curves, which means the maximum bending moment of the arches and the girder are reduced and the fatigue performance of the hangers is improved. After reaching a certain number of hangers, the four curves flatten out. For constant-change-of-slope arrangement, this happens at about 16 hangers and for radial arrangement at about 20 hangers. For more than 26 hangers, the mechanical behavior improvement effect of increasing the number of hangers is insignificant, but causes extra costs as the cost of hangers is usually three times higher than that of ordinary steel.

Figure 3.

Diagram of evaluation parameters with constant-change-of-slope arrangement.

Figure 4.

Diagram of evaluation parameters with radial arrangement.

Observing Figures 3 to 5, it can be found that, compared with the constant-change-of-slope arrangement, although radial arrangement reduces the value of ave N and max $Δ N$ , it also increases max M_a and max M_g. In structures with 18 hangers, max M_g in the radial arrangement reached 1.5 times that in the constant-change-of-slope arrangement (Figure 5). Therefore, the radial arrangement system dramatically increases the max bending moment of the girder while slightly improving the fatigue performance of the hangers.

Figure 5.

Results of different hanger arrangements with 18 hangers.

Based on all results from the first part of the parameter studies, the best range for the number of hangers seems to be between 16 and 20 for a 100 m-long pedestrian network arch bridge with traditional hanger arrangement. In this article, the hanger arrangement systems beyond this range are defined as follows: sparse hanger system (the number of hangers is less than 16) and dense hanger system (the number of hangers is greater than 20). As mentioned in section “Introduction,” many scholars have conducted a lot of research on the dense hanger system and achieved fruitful results, but the study of the sparse hanger system is less. In the two kinds of traditional hanger arrangement, after the number of hangers is reduced to less than 16, the internal forces of the structure will be greatly increased (Figures 3 and 4). This is because when the hangers become sparse, the mechanical behavior of the structure is sensitive to the rationality of the arrangement of any hanger. However, the hangers are arranged in a strict and uniform manner in geometry in the two kinds of hanger arrangement, so it is difficult to arrange the hangers in the most reasonable position. In this article, the idea of discretization is used to study the sparse hanger system. That is, taking the position of hanging points as variables and the mechanical behavior of the structure as the objective function, under the condition of 12 hangers, several better sparse hanger systems can be obtained by applying GA.

Formulation of the optimization problem

The discrete concept of hanger arrangement

In the past, research on the optimization of hanger arrangement on network arch bridges was usually based on the assumption that hanger distribution was in accordance with certain rules or shapes. Undoubtedly, such assumptions are reasonable for the dense hanger system. The continuation of such assumptions under sparse hanger system will miss better arrangements of hangers. Therefore, this article proposes a discrete concept of the optimization of hanger arrangement for a sparse hanger system.

A certain number of possible hanging points on the arches and the girder are set first (the distance between two adjacent possible hanging points is 5 m). The upper hanging points (on the arches) and the lower hanging points (on the girder) are taken as variables, and the location of each hanger is determined by the location of the upper and lower hanging points.

Based on this concept, once the specific number of the hangers is determined, the exact arrangement of the hangers can be determined by the information about the hanging points. According to the results in section “Influence of the number of hangers,” 12 hangers are taken to study the sparse hanger system in this article. Hence, in the engineering sense, all kinds of hanger arrangements in the sparse hanger system can be compared to obtain better results. However, most discrete optimization problems are hard to handle because a derivative of the objective function is not available in general, and as a result, it is not easy to obtain the minimum value of the objective function. To overcome this difficulty, a dimensionless optimization approach based on GA is used in this study. Because GA selects initial sets of design variables randomly and searches the whole design variable space in parallel from the sense of probability, the possibility of an optimal design is very high (Goldberg, 1989; Jenkins, 1992). And a dimensionless objective function is proposed to make it more efficient.

Combining the discrete concept of hanger arrangement with GA, the method of chromosome coding will be defined as follows (Figure 6). First, the arch and the girder are divided into 20 equal parts in longitudinal section. According to the symmetrical characteristic, the arch generates 19 possible hanging points, while the girder generates 10 possible hanging points, and each possible hanging point is followed by the corresponding number 1 to 19 and 21 to 30. Then, the exact hanging points of hangers can be located by these numbers. For a sparse hanger system with 12 hangers, the design variables consist of 6 upper hanging points (the numbers of which vary from 1 to 19) and 6 lower hanging points (the numbers of which vary from 21 to 30). Finally, an integer string composed of hanging point numbers makes the chromosome of an individual. Figure 7 shows a randomly generated individual whose chromosome is {2 5 5 7 10 15 23 23 26 28 26 28}. It should be noted that the chromosome is transformed into a binary vector during the genetic optimization process.

Figure 6.

The possible hanging points on the arch and the girder.

Figure 7.

A randomly generated individual.

Design constraints

In order to satisfy the design of tied arch bridge under dead load and live load, according to the General Code for Design of Highway Bridges and Culverts in China, the following design constraints related to stresses in hangers and vertical deflections of the girder are considered in the optimization process.

(a) Hanger stress.

g_{1} = σ_{i}^{\max} - σ_{Max} \leq 0 i = 1, 2, \dots, N_{h}

(1)

where $σ_{i}^{\max}$ is the maximum tensile stress in hanger i under the dead load and live load; $σ_{Max}$ is the maximum allowable tensile stress of the hangers, the standard value of tensile strength of steel wire for hanger is 1670 MPa ( $σ_{k}$ ). According to Specifications for Design of Highway Steel Bridge in China, the strength partial coefficient of hanger materials is 3 ( $λ$ ), thus $σ_{Max} = σ_{k} / λ = 557 MPa$ ; $N_{h}$ is the total number of hangers, which is defined as 12.

(b) Vertical deflections of the girder.

g_{2} = {| δ_{i} |}^{\max} - δ_{Max} \leq 0 i = 1, 2, \dots, N_{HP}

(2)

where ${| δ_{i} |}^{\max}$ is the maximum absolute value of the vertical deflection of the girder at the possible hanging point i under the dead load and the live load; $δ_{Max}$ is the maximum allowable vertical deflection of the girder, which is taken as L/600; and $N_{HP}$ is the number of possible hanging points of hangers on the girder, which is defined as 10.

Dimensionless objective function

In section “The relationship between the number of hangers and the load capacity of a pedestrian network arch bridge,” we have already calculated the mechanical evaluation parameters of pedestrian network arch bridge with two kinds of traditional hanger arrangement as the number of hangers changes from 6 to 50. On the whole, constant-change-of-slope arrangement (with $α = 50 ° and Δ α = - 0.3 °$ ) is better. When the total number of hangers equals 12, the mechanical evaluation parameters are listed as follows: max M_a = 585.4 (kN.m), max M_g = 908.1 (kN.m), ave N = 437.7 (kN), and max ΔN = 181.6 (kN).

In order to normalize the objective function and compare the optimal sparse hanger system with the traditional hanger arrangement, adopting the idea of dimensionless, the objective function in this study is expressed as

F = (\frac{\max M_{a}}{A} + \frac{\max M_{g}}{B} + \frac{ave N}{C} + \frac{\max Δ N}{D}) / 4

(3)

where max M_a, max M_g, ave N, and max ΔN are defined in section “Basic parameters” as A = 585.4; B = 908.1; C = 437.7; D = 181.6.

In fact, the dimensionless objective function integrally considers the mechanical behavior of the whole structure and greatly accelerates the process of optimization. Usually, if the value of F is lower than 1, it shows that the corresponding sparse hanger system is better than the traditional hanger arrangement.

Penalized objective function

As we know, GA is suitable only for the unconstrained optimum design problem. Therefore, the objective function of the optimization problem in hand (equation (3)) is penalized by adding a penalty function, consistent with constraint violations. The penalized objective function F_p for a problem with n constraints is defined as follows

F_{p} = F + \sum_{i = 1}^{n} α_{i} δ_{i}

(4)

{\begin{matrix} δ_{i} = g_{i}, for g_{i} > 0 \\ δ_{i} = 0, for g_{i} \leq 0 \end{matrix}

where F_p is the penalized objective function. F is the original objective function defined in equation (3). $δ_{i}$ is a violation factor for the i-th constraint, g_i is the design constraint defined before, and $α_{i}$ is a penalty parameter imposed for violation of constraint i, which depends on the type of the optimization problem. In this study, the values of the penalty parameter $α_{i}$ are assumed to be 0.1 and 100 for the stress constraint and the deflection constraint, respectively.

Optimization process of hanger arrangement

The whole optimization process is conducted using two computer programs on a common desktop with 4G RAM. The first program is ANSYS, which is employed to develop a 3-D nonlinear finite element model (FEM) for the bridge. The second program is a MATLAB code based on GA, which is used to obtain the minimum objective function. The general procedure, which involves interaction between the FEM and GA for finding the optimum sparse hanger system of pedestrian network arch bridges, is given as follows:

An initial population based on the discrete concept of hanger arrangement mentioned above is created. The population has 40 individuals (40 arch bridges with 12 hangers) selected randomly between the lower and upper bounds of each design variable. Each chromosome defines the hanger arrangement of the arch bridge.

A 3-D nonlinear FEM is created for each individual (bridge) by assigning appropriate element types and proper boundaries. The arches and girder are built with beam elements, and the hanger is constructed with link elements.

Each structure is analyzed under dead loads and live loads with the FEM program ANSYS. The displacements and internal forces are calculated for all elements of the bridge.

The stress and deflection constraints are checked. If any of these constraints is not satisfied, the result of this specific bridge configuration will be diminished by applying the penalized objective function given by equation (4).

The initial population is sorted in ascending order according to the value of the objective function, such that the first-ranked candidate (bridge) has the minimum value of F_p.

A new population is generated by applying the selection, crossover, and mutation operators. The selection is proportional. The crossover type is one-point crossover whose crossover rate is 0.7. And the mutation operator uses discrete mutation whose mutation rate is 0.08. Besides, the generation gap rate is 0.9, which means the offspring number is 36 instead of 40. These operators are applied on the high-ranked functions evaluated in the previous step to produce a new generation with better fitness (smaller F_p).

The previous population is replaced with the newer one, containing new individuals with better fitness, in addition to the best individual found so far.

Steps 2–7 are repeated for a certain number of generations until the optimum solution nearly does not change again. In this study, these steps are repeated for 200 generations to reach the optimum solution.

The individual with the highest fitness (smallest values of the objective function) obtained at step 8 is delivered as the final solution.

All the previous steps are summarized in the flowchart shown in Figure 8. And the computational time required to get the optimum sparse hanger system in one process is usually less than 6 h. As there are many local optimal hanger arrangements in sparse hanger system, a large quantity of sparse hanger systems with good mechanical behavior can be obtained after a certain number of genetic optimization processes.

Figure 8.

Flowchart of the genetic optimization process.

Results and discussion

A series of ideal schemes

Based on the same parameters as the pedestrian bridge described in section “Basic parameters”, except the number of hangers (which is taken as 12), a large number of sparse hanger systems with good mechanical behavior have been obtained after hundreds of genetic optimization processes. However, not all the results conform to the design requirements for artistic criterion. The ideal sparse hanger system is the one that presents both good mechanical behavior and artistic configuration. After artificial screening, it is easy to get a series of ideal schemes (sparse hanger system) whose objective function values are 0.806, 0.831, 0.849, 0.859, 0.864, and 0.888, respectively. According to the chromosome of each scheme, the corresponding hanger arrangement can be determined exactly (Figure 9). Figure 10 shows the ideal schemes obtained in terms of evaluation parameters compared to the constant-change-of-slope arrangement with $α = 50 ° and Δ α = - 0.3 °$ .

Figure 9.

A series of ideal schemes obtained by genetic optimization processes.

Figure 10.

Percentage reduction of evaluation parameters compared to the constant-change-of-slope arrangement with $α = 50 ° and Δ α = - 0.3 °$ .

Observing Figure 10, it is obvious that all evaluation parameters reduced in different degrees in obtained schemes. In general, the smaller the value of the objective function is, the more the mechanical property of the structure improves. For different parameters, the magnitude of reduction is different, with max M_a decreasing the most, followed by max M_g and ave N, and max $Δ N$ decreasing the least. Carlo Pellegrino et al. also point that it is difficult to reduce max $Δ N$ in their study. Nevertheless, all the evaluation parameters are reduced by more than 14% in scheme 1 compared to the constant-change-of-slope arrangement with $α = 50 ° and Δ α = - 0.3 °$ .

It should be noted that the optimized results can be divided into two types: mixed systems containing vertical hangers (schemes 1, 2, 3, and 5) and approximate network systems (schemes 4 and 6) that do not contain vertical hangers. Although the latter are far superior to the traditional network system in mechanical properties, they are still not as good as the former. According to Carlo Pellegrino et al., vertical arrangements are better than network ones if the fatigue in the hangers is considered as the only parameter (Pellegrino et al., 2010). And in the sparse hanger system, the effect of vertical hangers arranged near the springs become more prominent. For example, except for the four vertical hangers near the springs, the arrangement of the eight inclined hangers of scheme 1 is exactly the same as scheme 6. By looking back at Figure 10, it is found that the four mechanical parameters of scheme 1 are much lower than that of scheme 6. This means that the properly arranged vertical hangers not only improve the fatigue performance of the system, but also reduce the bending moment of the arch and the girder.

On the other hand, comparing four different mixed systems, it is easy to find that the arrangements of their four vertical hangers are exactly the same, but the arrangements of the inclined hangers are obviously different. It should be pointed out that these four systems represent only a small part of the mixed systems. By changing the arrangement of the middle eight inclined hangers, it is possible to obtain designs with novel configuration and good mechanical behavior. For instance, in scheme 2, the middle inclined hangers constitute two approximate five-pointed star shapes that are unique in style. In short, these ideal schemes not only show much better mechanical behavior than that of the traditional hanger arrangement, but also enrich the diversity of design in esthetics to a certain extent.

The comparison between the ideal schemes obtained and the traditional hanger arrangements in mechanical behavior

In order to illustrate the importance of the dimensionless optimization approach based on GA, lots of tied arch models with two kinds of traditional hanger arrangement are calculated, and their objective functions are shown in Figures 11 and 12.

Models (A): the parameters are identical to those of the pedestrian bridge with ideal schemes obtained including the number of hangers (12). The hangers are arranged by constant-change-of-slope arrangement, which is defined according to Figure 1, with $α$ varying from $50 °$ to $80 °$ and $Δ α$ varying from $0 °$ to $2 °$ (Figure 11).

Models (B): the parameters are identical to those of the pedestrian bridge with ideal schemes obtained including the number of hangers (12). The hangers are arranged by radial arrangement, which is defined according to Figure 2, with $φ$ varying from $5 °$ to $50 °$ (Figure 12).

Figure 11.

The objective function values of models (A) under live load.

Figure 12.

The objective function values of models (B) under live load.

The minimum objective function value of models (A) and models (B) are 1.005 and 1.023, respectively. It is noticed that both of them are almost 25% larger than the minimum objective function value of ideal schemes obtained (0.806) and over 13% larger than the maximum objective function value of ideal schemes obtained (0.888). Since the models (A) and (B) considered contain most of the tied arch models with two kinds of traditional hanger arrangement, it can be concluded that the mechanical properties of the ideal schemes obtained are much better than those of traditional hanger arrangements. Therefore, the optimization approach in this article is very efficient on tied arch bridge with sparse hanger system.

Conclusion

Focusing on a 100-m long pedestrian tied arch bridge with sparse hanger system, a dimensionless optimization approach based on GA is developed. This approach is utilized to optimize the arrangement of sparse hanger system in mechanics, and enrich the diversity of design in esthetics. The following conclusions can be drawn from the results:

Too dense hanger system is not suitable for the pedestrian tied arch bridge owing to the high cost of hangers and the low esthetic effect.

After the number of hangers is reduced to less than 16, the internal forces state of the bridge with traditional hanger arrangement will deteriorate unless the optimal hanger arrangement is adopted.

Investigations show that the dimensionless optimization method is effective in searching for the optimal hanger arrangement schemes. The optimal results have very excellent mechanical performance under live load.

The ideal sparse hanger systems can be divided into two types: mixed systems containing vertical hangers and approximate network systems. The former are better than the latter in mechanics as the properly arranged vertical hangers can improve the fatigue performance of the system and reduce the bending moment of the arch and the girder.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Yuanbin Yao

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