Optimization of extradosed concrete bridges subjected to seismic action

Highlights

• An optimization algorithm for the design of extradosed concrete bridges is proposed.

• The seismic action governs the design of extradosed concrete bridges.

• The algorithm finds economically and structurally efficient solutions.

• Balanced solutions between the cable suspension and deck bending stiffness.

• The stiffness and mass distribution are rearranged to improve the seismic response.

Abstract

An optimization algorithm is proposed to assist in the design of extradosed concrete bridges under static and seismic loading. This procedure is composed of structural analysis, sensitivity analysis and optimization modules. The finite element method is used for the three-dimensional analysis considering static loading (dead load and road traffic live load), geometrical nonlinearities, time-dependent effects and seismic action by using a modal response spectrum analysis. The design is formulated as a multi-objective optimization problem with objectives of minimum cost, deflections and stresses including service and strength criteria. The solution of the minimax problem is obtained by minimizing a convex scalar function obtained through an entropy-based approach. The design variables are the extradosed cables and tendons forces, the extradosed cables and tendons cross-sectional areas, the deck, towers and piers sizes. The analytical discrete direct method is used to find the structural response to changes in the design variables. A convex optimization strategy with multiple starting points finds minimum cost solutions with an adequate stiffness and mass distribution that satisfy the design criteria under both, static loading and seismic action. Numerical examples concerning the optimization of a real-sized extradosed concrete bridge illustrate the features and capabilities of the proposed method.

Introduction

The structural system of extradosed bridges (EB) combines the concepts of cable suspension (cable-stayed bridges, CSB) and bending of the high stiffness box-girder (prestressed box-girder bridges, PBGB). EB constitute an economic alternative to CSB and PBGB for main spans of 100 to 200 m, being used for both roadway and railway crossings [1]. The Tempul Aqueduct designed by the Spanish engineer Eduardo Torroja and built in 1926 can be considered one of the world’s first prestressed concrete structures, and a precursor of modern concrete cable-stayed bridges and extradosed concrete bridges [2]. The “extradosed” term was first used by the French engineer Jacques Mathivat in 1988 [3] to describe a novel cable arrangement developed for the Arrêt-Darré Viaduct in France. Mathivat was inspired by the design of the Ganter Bridge in Switzerland. This bridge, designed by the Swiss engineer Christian Menn and completed in 1980, has prestressed concrete walls on each side of the deck working as tension members, like rigid cable-stays. Mathivat replaced the walls by cables and, given the small height of the towers, named the cables as extradosed cables instead of cable-stays[4]. To enhance the prestress structural efficiency, the tendons are installed outside and above the main girder and deviated by short towers located at supports. Therefore, they are termed extradosed cables.

Several authors [6], [7], [8], [9], [5], [10], [11], [1], [12] studied the structural behaviour of EB, pointing out the main characteristics of this structural system and indicating preliminary design rules concerning the overall bridge dimensions and the extradosed cables. Fig. 1 depicts the main structural members and the usual dimensions of an EB.

Even though in appearance they are similar, an EB differs from usual CSB with the towers having a height of less than the usual 20% of the main-span observed in CSB. The smaller cable inclination, associated with the lower towers, increases the deck compression forces caused by the cables and decreases the vertical component of the forces at the cable anchorages. However, the extradosed cables not only prestress the deck, but also provide some vertical support as in a CSB. Given that the vertical loads are partially resisted by the main girder, the stress variations in the extradosed cables due to live loads are smaller than those observed in CSB. This is similar to the behaviour noticed in PBGB, where the girder has a crucial influence on the structure stiffness and live loads produce only limited stress variations in the tendons [9], [5]. This leads to an increase of the allowable stresses considered in the extradosed cables design. EB present several advantages for the range of 100 to 200 m of main spans. When compared to PBGB, the height of the main girder may be lower, which reduces the structure self-weight. Compared to CSB, the height of the towers is lower, therefore a reduction in materials and in labour costs of construction can be achieved.

Concerning the dynamic behaviour of these structures the smaller periods and higher deck mass when compared to CSB may lead to higher seismic effects. Otsuka et al. [13] reported increases of 50 – 60% in the shear forces and of 20 – 30% in the bending moments at the piers base. The authors also concluded that CSB are advantageous for a span over 150 m when considering both the superstructure and the substructure costs. For spans of about 80 – 150 m, EB are structurally and aesthetically favourable due to the lower girder depth than that of a PBGB.

Regarding the soil-structure interaction, Yi et al. [14] observed that the amplification of ground motion occurs for soft soil conditions and within the high frequency range. Given that the most significant modes of EB exist in the low frequency range, the structural response in this range is not affected by the soil conditions and no amplification of member forces was observed. He et al. [15] also observed that, if the soil-structure interaction is disregarded, although the displacements can be underestimated, a reduction in the piers and towers internal forces can be observed.

Benjumea and Chio Cho [16] studied the effect of the type of connection between the deck and piers on the seismic response of EB. A monolithic connection is advantageous in zones with low seismicity and the height of the piers does not affect the structural response of the deck under traffic loads and low intensity earthquakes. For moderate and high seismicity zones a better structural performance is observed when the deck is supported on piers. For this connection the effect of the piers height is negligible. Benjumea and Chio Cho [17] also observed that the seismic vulnerability of EB during cantilever construction is higher than in service. Additionally, the authors found that extradosed cables and towers are the most vulnerable members and that the most critical erection stages are before the bridge closure.

A search in the Web of Science electronic database was conducted using the following terms: (“extradosed”) AND (“optimization” OR “optimum” OR “optimal” OR “synthesis” OR “minimum cost” OR “least cost”). These terms were searched in the article’s title. The search revealed only two papers [18], [19] that concern the extradosed cable forces optimization. From a recent survey article by the authors [20], 4 additional papers concerning optimization of EB can be identified. These works refer to the optimization of the non-stayed segment length [21], [23], the side-to-main-span ratio optimization [22] and the extradosed cable forces optimization for the complete bridge under permanent loading [24].

The design of EB is a challenging task aiming to find an adequate balance between the extradosed cables suspension effect and the stiffness of the main girder, which depends on the cross-sectional dimensions and the two sets of prestressing forces (extradosed cables and tendons). Moreover, the construction stages, the geometrical non-linear effects and the time-dependent effects need to be considered. The seismic action adds more complexity to the design problem because the mass and stiffness distribution optimise the bridge dynamic response. Due to the complexity and the large amount of information involved, the design of these structures may be favoured by the use of optimization techniques to find economical and structurally efficient solutions under both, static and dynamic loading. To the best of the authors’ knowledge, the optimization aiming at cost minimization of extradosed concrete bridges was only reported in a recent work by the authors [25].

The research work reported in this article aims to develop an optimization-based computational method to assist in the design of extradosed concrete bridges under static loading and seismic action. A previously developed computer program for the optimization of concrete cable-stayed bridges [26] and reinforced concrete frames [27] was generalized to solve the current optimization problem. This program comprises two modules: a structural analysis module plus a sensitivity analysis and optimization module.

The finite element method is used in the first module to obtain the three-dimensional structural response under static loading (dead load and road traffic live load), considering the second-order effects and the time-dependent effects. A modal response spectrum approach is employed to access the structural response under seismic action defined according to Eurocode 8 [28]. In the second module the design of extradosed concrete bridges is formulated as a multi-criteria optimization problem which is solved by the minimization of a convex scalar function obtained through an entropy-based approach. The design variables considered are the extradosed cables cross-sectional areas and prestressing forces, tendons areas and prestressing forces, deck, piers and towers sizes. Design objectives of minimum cost, deflections and stresses, considering both, service and strength criteria are considered. The goal of the sensitivity analysis is to access how the change in each design variable will affect the different design objectives. The analytical discrete direct method is used for the sensitivity analysis which provides the derivatives needed for the gradient-based optimization algorithm used. Fig. 2 shows the flowchart of the developed computer program. For a given initial design (starting point), the program starts reading the problem data and for each stage, the algorithm computes the structural response (displacements, stresses, internal forces) and the corresponding sensitivities with respect to each design variable. This information is used as input for the gradient-based optimization algorithm to improve the current design seeking the optimum solution. The design variables are updated and the analysis and optimization process is repeated until changes in the design variables and cost of the structure become small. Numerical examples concerning a 150 m main-span extradosed bridge are presented to illustrate the features and applicability of the developed method.