1 Introduction

For facilitating the spanning of natural or man-made barriers, bridges are man-made structures that overcome gravity, while also serving as monuments that symbolize mankind’s continuous progress (Tang, 2018). Thanks to the developments of technology, modern bridges with longer spans and superior structural performance can be erected. Engineers’ search for efficient, lightweight and elegant bridges has always been a relentless endeavor (Schlaich, 2004). Following the continuous evolution of artificial intelligence, how to design and construct bridges in a smart manner has become a research hotspot.

With the development of computers, such scientific computing methods as finite element analysis (FEA) have been booming over recent decades. As a kind of FEA-based optimization approaches, topology optimization techniques have also been rapidly developed, with various topology optimization algorithms proposed and investigated, including the homogenization method (Bendsøe & Kikuchi, 1988), the solid isometric material with penalization (SIMP) method (Andreassen et al., 2011; Bendsøe & Sigmund, 1999; Rietz, 2001), the evolutionary structural optimization (ESO) method (Xie & Steven, 1993, 1997), the level set method (Allaire et al., 2002; van Dijk et al., 2013; Wang et al., 2003), and the truss-based topology optimization (Richardson et al., 2015). Among the above topology optimization algorithms, the ESO method and its enhanced version, the bi-directional ESO (BESO) method (Huang & Xie, 2010a), are two of the most commonly used approaches in architectural and structural design, due to their conceptual simplicity and ease of implementation. Without relying on any initial configuration of forms and structures, these state-of-the-art technologies can inspire designers to conceive innovative bridge structures with efficient, lightweight and versatile features (Xie et al., 2014). Nowadays, the BESO method has become an appropriate approach to exploring highly efficient bridge structures during the design process (Lai et al., 2023). In fact, it is not only an important tool to size the structural members, but also assist designers in finding the most suitable forms of structures from a structural and architectural perspective (Xie, 2022).

Nevertheless, although the topology optimization is getting a growing popularity as a smart technology for architects and engineers, its practical application is still relatively rare in large-scale projects. Over recent years, many formally unique and structurally superior bridge’s conceptual designs have been proposed by using different topology optimization methods, but few of them have been reported to be successfully implemented. A critical reason for this result is that the forms generated with topology optimization typically feature irregular and complex nonlinear geometric appearances, which are difficult or expensive to realize in conventional construction techniques. Since bridges are obviously huge in scale, their construction methods are remarkably distinct from the manufacture of mechanical or industrial products. The successful construction of a bridge according to the form-finding results from topology optimization requires a combination of the currently leading technology and the sophisticated construction logic. Therefore, it is essential to incorporate topology optimization techniques into bridge design when fully considering constructability while satisfying the demands in terms of safety, economy and durability.

Given the above discussion, this study would, inspired by topology optimization outcomes, propose a novel and unique pedestrian bridge design based on the construction requirements that need to be met in real projects as well as the challenges that have to be addressed. Focused on the form-finding process by applying topology optimization algorithms in conceptual design and detailed structural design of the bridge, this paper is organized as follows: Sect. 2 presents the theoretical background of the BESO and multi-material BESO method; Sect. 3 presents the design concept of a five-span pedestrian bridge; Sect. 4 explores the optimization problem and the whole form-finding process; Sect. 5 offers the detailed design considerations; Sect. 6 discusses the structural analysis of the bridge; and finally, Sect. 7 draws the conclusions of this study.

2 Theoretical background

Stiffness is used to characterize the ability of a structure to resist deformation by external forces, so it is an important indicator to reveal the structural performance. As a result, structural optimization usually chooses stiffness as the optimization objective function, because a stiff structure is generally effective in resisting large external loads and frequent vibrations without significant deformation. Since strain energy is inversely related to stiffness, the problem with topology optimization in real engineering structures is about certain compliance minimization, subject to a volume constraint.

2.1 BESO method

As an important branch of topology optimization, the BESO method allows materials to be removed and added simultaneously. Especially, one of its variants, the convergent and mesh-independent BESO method (Huang & Xie, 2007), has been widely adopted for its efficiency and robustness. This improved algorithm is based on the finite element discretization approach, in which the densities of elements are treated as design variables. The mathematical description of this method is as follows:

$$\begin{array}{c}{\text{Minimal}} \, \begin{array}{c} C=\frac{1}{2}{f}^{\text{T}}u\end{array}\end{array}$$
(1)
$$\begin{array}{c}{\text{Subject to}} \, {V}^{*}-\sum_{i=1}^{N} {V}_{i}{x}_{i}=0\end{array}$$
(2)
$$\begin{array}{l} {x}_{i}=1 \,\text{or}\,{ x}_{\text{min}}\end{array}$$
(3)

where \(C\), \(f\), \(\text{u}\), and \(K\) are the mean compliance, the global force vector, the structural displacement vector, and the global stiffness matrix, respectively; \({V}^{*}\) is the prescribed total structural volume; \({V}_{i}\) is the volume of an individual element; N is the total element number; \({x}_{i}\) is the ith design variable, with a candidate value of either 1 for the solid element (presence) or prescribed \({x}_{\text{min}}\) = 0.001 for the void element (absence).

The solution to the stiffness optimization outlined in Eqs. (1–3) can be effectively and efficiently acquired via the BESO technique, which, as a gradient-based methodology, can implement the element sensitivities calculated by differentiating the objective function. The sensitivities of elements reflect the variation in the objective function resulting from alterations in an element’s status, typically ranging from \({x}_{\text{min}}\) (absence) to 1 (presence). The optimization process initiates with an initial full design, where all elements are present in the design domain. In the context of a compliance minimization task, the elements with the highest sensitivities in their present state are transitioned from 1 to \({x}_{\text{min}}\), while those with the lowest sensitivities would be switched from \({x}_{\text{min}}\) to 1. This process represents a quintessential “steepest descent” iterative method, aimed at finding the optimal solution.

Volume control is straightforwardly managed by progressively decreasing the total volume until achieving the compliance with the volume constraint specified in Eq. (2). The overall number of iterations is typically governed by a parameter called the evolutionary volume ratio (ER), which dictates the percentage change in volume per iteration.

Furthermore, termination criteria are applied within this iterative framework to ensure the robustness of the stiffness optimization. Beyond meeting the volume constraint, an additional criterion is used to verify the solution convergence, so as to guarantee the integrity and efficacy of the optimization outcome. More details on the BESO method can be found in the Refs (Huang & Xie, 2010b; Xia et al., 2018; Zuo & Xie, 2015).

2.2 Multi-material BESO method

To overcome the limitations of the existing single-material-based BESO methods, Li and Xie (2021a, 2021b; 2023a) have recently developed an advanced algorithm of multi-material BESO (MBESO) method based on stress levels, which shows great potential and advantages in optimizing the conceptual design of composite structural bridges or buildings (Li et al., 2022, 2023a).

In the MBESO method, material utilization serves as a criterion for determining the importance of different materials across structures. This method can efficiently distribute materials suitable for tension and compression across corresponding regions in a structure. Specifically, steel (M1), which is suitable for tension, is assigned to tensile elements, while concrete (M2), which is suitable for compression, is assigned to compressive elements. This method involves a “two-phase” design that incorporates three kinds of materials: two solid materials (M1 and M2) with different mechanical properties, and one void material (M3). Young’s moduli of these materials are set as E1, E2, and E3, respectively. Young’s modulus of M3 is set as \({E}_{3}={x}_{min}^{p}{E}_{2}\), where \({x}_{\text{min}}\) = 0.001 and p is a penalty factor (typically, p = 3). To express the three materials, two variables, xi and mi, are defined as:

$${x}_{i}=\left\{\begin{array}{l}1 \, \, \text{ solid}\\ {x}_{min}\text{ void}\end{array}\right. , \,i=\text{1,2},\dots ,N$$
(4)
$$m_{i} = \left\{ {\begin{array}{*{20}l} 1 & {({\text{material}} 1, \ {\text{in tension}})} \\ 2 & {({\text{material}} 2,\ {\text{in compression}}),\,i = 1,2, \ldots ,N} \\ 0 & {({\text{void}})} \\ \end{array} } \right.$$
(5)

where xi indicates whether element i in a structure is solid or void; mi indicates the stress state of element i; and N is the total number of elements.

The first invariant of stress tensors \({I}_{1}\) of element i is adopted as the criterion for judging whether an element is in tension or compression. If \({I}_{1,i}\ge 0\), element i will be regarded as being in tension, while \({I}_{1,i}<0\) indicates that it is in compression. Furthermore, for solid elements, if \({I}_{1,i}\ge\) 0, the elements will be assigned with material M1; if \({I}_{1,i}\) < 0, the elements will be assigned with material M2, and material M3 will be assigned to void elements, as indicated below:

$${E}_{i}=\left\{\begin{array}{l}{E}_{1} \,({x}_{i}=1,{I}_{1,i}\ge 0)\\ {E}_{2} \,({x}_{i}=1,{I}_{1,i}<0)\\ {E}_{min} \,({x}_{i}={x}_{min})\end{array}\right.$$
(6)

After the von Mises stress of each element is obtained from FEA, the material utilization of each element \({u}_{i}\) in the design domain is defined as:

$${u}_{i}=\left\{\begin{array}{l}\frac{{\sigma }_{i}^{vm}}{[{\sigma }_{m1}]} \, {I}_{1,i}\ge 0\\ \frac{{\sigma }_{i}^{vm}}{[{\sigma }_{m2}]} \, {I}_{1,i}<0\end{array}\right., \,i=\text{1,2},\dots ,N$$
(7)

where \({\sigma }_{i}^{vm}\) is the von Mises stress of element i, and [\({\sigma }_{m1}\)] and [\({\sigma }_{m2}\)] are the allowable stresses of the two materials of M1 and M2, respectively.

The filter scheme, stabilization process, and convergence criterion of the MBESO method are similar in principle to those of the BESO method. It should be noted that in the filter scheme of the MBESO method, both the material utilization \({u}_{i}\) and the first invariant of stress tensors \({I}_{1}\) need to be filtered, so as to avoid the checkerboard pattern and realize bi-directional switching of elements. More details can be found in Refs (Li & Xie, 2021a; Li et al., 2023b).

3 Conceptualization of a novel pedestrian bridge

3.1 Project profile

A pedestrian bridge was planned to be built in Shenzhen, China, so as to improve the accessibility for commuters to cross the rails adjacent to a high-speed train station. Based on the on-site boundary conditions and various constraints, this bridge was designed as a five-span construction, with a span arrangement of (41 + 85 + 152 + 85 + 41) m and a width of 17 m. Due to a considerable difference in elevation between the bridge deck and the ground on one end of the bridge, a two-level variant spiral bicycle ramp is provided to connect a plaza on the ground and the bridge deck.

3.2 Construction requirements and challenges

There is a commercial center in the high-speed rail station near this pedestrian bridge, and this center is one of the most prosperous urban areas in Shenzhen. The construction requirements and challenges of this project include: (1) complex land boundaries and constraints surrounding the project, (2) a high level of landscape and implementation requirements for the bridge imposed by the owner, and (3) a road and several high-speed railways in operation being under the bridge, so disruptions during the construction shall be as much minimized as reasonably possible.

3.3 Design concept

In order to minimize the maintenance burden of the cross-bridge crossing the operating rails and protect the overhead electrical contact-line system, the railway administration disallowed adoption of any steel or steel–concrete composite structure for the bridge. Given that the project is located in a central business district surrounded by numerous high-rise buildings, the bridge should blend in with the environment and adopt a low-profile form according to the urban design specification. Therefore, the pattern of cable-supported bridges with lofty pylons is unsuitable, while a bridge with concrete girder and arch is an economical and reasonable solution with minor maintenance burden. However, Shenzhen is a seaside city with a thick cover of soft soil foundation. So, if a deck arch bridge is adopted, the arch seats cannot be located directly on the rock foundation, as shown in Fig. 1(a), because enormous long-term horizontal thrusts from the arch is prone to cause soil creep, thus endangering the bridge itself and the railways under it. A possibly appropriate solution is a prestressed concrete (PC) girder bridge with variable cross-section, as shown in Fig. 1(b). However, as the main span is up to 152 m, the girder depth at the main pier should be 11 m if the conventional design specifications are followed. In addition, like in deck arch bridges, variable cross-section PC girder bridges will incur excessive elevation lift of the bridge deck, thus significantly increasing the bicycle-climbing length along the ramp of this bridge. Thus, conventional structural solutions are not a good choice.

Fig. 1
figure 1

The concept development process: a deck arch bridge with flat arch, b variable-section PC girder bridge, and c variable-depth spinal-shaped PC girder bridge

Full size image

How can we reduce the girder depth, but still meet the bearing capacity requirements for such a long-span bridge? This study proposes a solution of setting part of the load-bearing structure above the bridge deck based on the variable-section girder bridge, forming a distinctive combination consisting of a spine structure and variable-section girder, as shown in Fig. 1(c). This solution can lower the deck's elevation by two meters compared to a conventional PC girder, thus significantly reducing the ramp length. However, if no improvement is made, this unique structure will seem too bulky. Therefore, this study optimizes the superstructure of the main bridge and piers of the bicycle ramp by using structural optimization techniques, finally delivering a strikingly elegant bridge with high efficiency.

4 Form finding with topology optimization

4.1 Spinal-shaped girder

4.1.1 Topology optimization settings

Given that the load of the bridge is primarily concentrated in the vertical plane, the structural model can be simplified to a plane stress issue, which can be addressed more efficiently. The SSG uses the 2D MBESO algorithm for form finding. Compared with 3D topology optimization, the 2D topology optimization method can present additional details with more FEA elements, making it more suitable in this case.

The SSG involves a series of topology optimization calculations, with material parameters mostly remaining the same across these cases. For the two materials of steel and concrete, the parameters used for calculation the tension and compression are: Et = 200 MPa for steel, and Ec = 50 MPa for concrete, while the allowable stresses for the two materials are: [σs] = 200 MPa and [σc] = 50 MPa. And the Poisson’s coefficient of these two materials is µ = 0.3. The magnitude of the line load on the flat bridge deck and the curved pedestrian walkway is set to q1: q2 = 2:1. In different cases, the total number of elements in the design domain varies, but the size of individual elements remains the same, with four-node rectangular elements measuring 0.1 m × 0.1 m in length. If the design domain is a complete rectangle, the total number of the finite elements is 292,340, which is sufficient to showcase small structural details. Other topology optimization parameters are set as follows: the evolution rate er is set to 2%, and the filter radius is set to r = 0.5 m.

4.1.2 Evolution of the initial design domain

As a simple way, adjusting the initial design domain can directly influence the topology optimization’s results. For a new structural form, like the bridge explored in this study, an ideal optimized structure can be obtained by continuously adjusting the design domain until a suitable design domain is acquired.

Figure 2 shows a series of topology optimization results achieved by modifying the design domain boundaries. In Fig. 2(a), the optimized structure is obtained by using the rectangular design domain, with the bridge deck as a non-design domain and fixed boundaries at the bridge piers. The compressed materials, as indicated in blue in Fig. 2, are radially distributed around the pier, providing support to the bridge deck, while the tensioned materials, as indicated in red, are mainly concentrated above the bridge deck, so as to resist bending. The force transmission path and the synergistic mode between the two materials are reasonable, but some components do not meet the functional requirements for the bridge. For example, the inclined branches almost touch the ground can obstruct the traffic under the bridge deck. Therefore, the design domain boundaries in these local areas need to be adjusted. In Fig. 2(b), the redundant space under the bridge deck is removed; thus, the optimized structure presents a simple form, with the blue compressed materials forming tree-shaped columns, while the tensioned members above the bridge deck forms a bow-shaped inverted beam string structure system. As to the functional requirements of pedestrians, the design domain after setting the non-design domain above the bridge deck is shown in Fig. 2(c); and accordingly, the optimized result also includes the pedestrian area. Furthermore, by restricting the space above the bridge deck, the cable-like tensioned members above the pedestrian walkway can be removed and merged into a single tension cable, as depicted in Fig. 2(d).

Fig. 2
figure 2

Various design domains, support and loading conditions, and corresponding topology optimization results: a a complete rectangular design domain, b a design domain with the space below the bridge deck removed, c a design domain with added non-design space for pedestrian walkways, and d a design domain with the space above the pedestrian walkway removed

Full size image

4.1.3 Driving topology optimization by setting non-design domain

As demonstrated in Fig. 2, setting non-design domains ensures that the topology optimization results can meet the functional requirements of structures, such as retaining the bridge deck and pedestrian walkway in the results. In addition, by setting non-design domains appropriately, it is possible to drive the evolution direction of topology optimization to an ideal design reference prototype. Characterized by thin and numerous branches, the structures shown in Fig. 2 are not suitable for the subsequent design and construction of concrete components for this bridge. Therefore, in order to achieve both high structural efficiency and ease of construction simultaneously, this study increases the filter radius, despite a small loss in the structural efficiency (i.e., around 2%, as demonstrated by Xie, 2022).

In Fig. 3(a), the design domain is further adjusted based on the optimized result in Fig. 2(d) by adding a lower chord arch to the bridge as a non-design domain, so as to increase the target volume fraction to 0.8 and adjust the filter radius to r = 1.0 m. The optimized result shows a stable tree-shaped structure, with four inclined columns radiating from the piers and elegant curved lines of tensioned materials. In order to obtain diverse designs, the area with vertical columns is added in Fig. 3(b) as a non-design domain. Due to the existence of this non-design domain, the materials within a certain range around the central column would converge towards the vertical columns in the topology optimization process, creating an optimized structure with three columns. Both of these structural forms have certain design reference values, and it is evident that setting non-design domains in very small areas can lead to completely different topology optimization structures.

Fig. 3
figure 3

Driving topology optimization direction by setting non-design domain: a the optimization result without setting the middle vertical column as a non-design domain, and b the optimization result with the middle column set as a non-design domain

Full size image

4.1.4 Setting different filter radii

In addition to employing non-design domains, adjusting the filter radius in topology optimization can provide an effective means to influence the direction of optimization and deliver diverse structural designs. By expanding the settings demonstrated in Fig. 3(a) and increasing the filter radius to r = 3.0 m, an entirely different topologically optimized structure can be yielded, as demonstrated in Fig. 4. The structure obtained with a smaller filter radius would feature slender and more members, including four inclined columns. On the other hand, the structure produced with a larger filter radius would deliver fewer and sturdier members, incorporating only two inclined columns. When designing structures for practical engineering applications, an appropriate topology optimization outcome should be selected by giving comprehensive consideration to other factors, such as the construction complexity and the aesthetic value.

Fig. 4
figure 4

Obtaining diverse topology optimization results by adjusting the filter radius: a design domain, support, and load conditions; b topology optimization result obtained with the filter radius r = 1.0 m; and c topology optimization result obtained with the filter radius r = 3.0 m

Full size image

4.2 Tree-shaped branching pier

In this project, a spiral bicycle ramp is needed on one end of the pedestrian bridge. Since the ramp is located near a plaza, a light appearance is required, so as not to overwhelm the visual senses. For this purpose, this ramp is designed with a Fibonacci-type spiral shape, which is both appealing and interesting. The superstructure of the ramp is a continuous girder with equi-sections, and the substructure utilizes multiple piers with spacing of 8 m to 12 m to bolster the entire spiral ramp. However, if conventional vertical piers are used for this ramp, it would not only provide a monotonous appearance, but would also result in a bulky superstructure.

To narrow down the span of the superstructure and provide more uniform force, this study designs the shape of the piers in the BESO method, while implementing 3D topology optimization by using the Ameba software (Zhou et al., 2018). The piers are intended to be made of steel. Therefore, single-material topology optimization is sufficient for structural optimization. The material has a Young’s modulus of Et = 200 MPa and a Poisson’s ratio of μ = 0.3. For a conservative consideration, a uniform load of 100 kN/m2 is applied to the ramp deck above, and the material density is set at 7800 kg/m3.

4.2.1 Initial process of the topology optimization

In view of the sufficiently large radius of the spiral bicycle ramp, the design domain is simplified for topology optimization. A segment covering a 12 m long region and supported by one column is set as the design domain for topology optimization, with a height of 12 m. To simplify the design process, this cambered segment of the bicycle ramp is represented as a straight line, and the upper surface with a slope is simplified into a horizontal surface. Furthermore, given the symmetry of the simplified design domain, only one-quarter of the structure is set for calculation with symmetric boundaries.

Initially, this study performed topology optimization based on different design domains, with various shapes of tree-shaped branching piers (TSBP) obtained, as shown in Fig. 5. In addition, by changing the filtration radius, the TSBP with different numbers of branches is acquired, as shown in Fig. 6.

Fig. 5
figure 5

Topology optimization and corresponding form-finding outcomes for the bicycle ramp’s piers based on different design domains: a design domain, and b topology optimization outcomes

Full size image
Fig. 6
figure 6

Form-finding outcomes after post-design for the TSBP with different numbers of branches: a and b six branches; c and d ten branches

Full size image

Figure 6 illustrates the visual effects of the TSBP with six and ten branches. Obviously, if with more branches, the TSBP would appear cumbersome and disorderly against the overall visual effect of the bicycle ramp. In addition, too many intersecting nodes and overly free-form shapes would exacerbate the difficulty of fabrication and raise up the construction cost. Therefore, it is necessary to adapt a topology optimization approach with full consideration of constructability.

4.2.2 Form-finding based on comprehensive considerations

To facilitate the construction process and lower the construction costs, the TSBP are designed with basically identical branches, which are different from each other only in the height of “trunks”. Additionally, fewer branches mean that the members would intersect with fewer nodes. This would not only provide a simpler appearance, but also facilitate the manufacturing process.

Following the above thoughts, the topology optimization is readjusted, with a wide design domain utilized, as shown in Fig. 7(a). The bridge deck area is set as a non-design domain with a thickness of 200 mm (as depicted in the red area in the figure). The target volume fraction for the topology optimization is set at 0.2. Due to symmetry, a quarter of the design domain is discretized into a mesh containing approximately 70,000 elements. The filter radius for the topology optimization is set as r = 500 mm.

Fig. 7
figure 7

Topology optimization process for the TSBP: a design domain, support, and load conditions; b topology optimization results; c results after the post-design process

Full size image

Based on this quarter of the design domain, significant topology optimization results are acquired, as illustrated in Fig. 7(b). And the complete topology optimization results are presented in Fig. 7(c). It can be observed that the topology optimization results acquired are highly streamlined, with only one single main trunk branching into four smaller branches to support the bridge deck.

The design domain of the TSBP is then refined to facilitate the construction process, with the rescaled conical design domain shown in Fig. 8(a). The results obtained based on the settings described above are displayed in Fig. 8(b). As can be seen, the optimized topology features a straightforward four-branch design, with two branches parallel to the bicycle path and the other two vertical to it. This structural configuration guarantees the rigidity against vertical loads, while providing adequate lateral stiffness to withstand lateral disturbances.

Fig. 8
figure 8

Topology optimization process for the TSBP with a refined design domain: a design domain, support, and load conditions; b topology optimization results; c results after the post-design process

Full size image

Based on the topology optimization results, the post-designed TSBP is shown in Fig. 8(c), which involves designing four identical branches. This design can help maintain structural efficiency, while reducing construction costs.

5 Structural design

5.1 Overall design

In terms of design and construction, this bridge is required to meet commuting functions, while delivering on the effect of novelty, aesthetics and uniqueness. Given such engineering constraints and requirements, its superstructure is designed to be a variable-depth spinal-shaped PC girder, with an elegant curving facade in the center of the deck, from which one pathway cantilevers on either side. By making arch-shaped holes in the webs of the SSGs which are inefficient areas, the bridge can not only reduce its self-weight, but also seem lighter. Besides, people can reach the other side of the deck through these holes, and the interior of the SSGs creates a space for pedestrians to get shelter from the sun and rain. The stairway above the SSGs gradually leads up to the top at the main pier, offering scenic views of the surroundings. The segregated twin pathways safely separate the faster commuter bicyclists from the slower pedestrians. Figures 9 and 10 show the rendering, general arrangement, and typical cross-sections of the main bridge.

Fig. 9
figure 9

Rendering of the proposed bridge: a aerial view, b close-range perspective view, c view from the bridge deck, and d view from the stairway

Full size image
Fig. 10
figure 10

General arrangement and typical cross-section of the main bridge (unit: cm)

Full size image

5.2 Construction method

Due to the stringent requirements of rail operations, the construction of a newly built bridge overhead must be completed at the fastest speed within the time window specified by the railway administration department. To minimize the safety risk and disruption to the railways during the bridge construction, this project adopted the horizontal swiveling construction method for the superstructures. To ensure weight balance during rotation, the superstructure was arranged symmetrically along the main pier. First, each section of the superstructure was cast with temporary support along the direction of the railways, until the maximum cantilever state was reached to allow horizontal rotation. Afterward, the hinges were sealed immediately when the superstructure was in place, so as to avoid large displacements caused by small rotations of the ball hinges. Finally, the spinal-shaped superstructure was closed with the side span cast-in-place sections, and the mid-span rigid hinge was installed. Figure 11 illustrates the on-site installation method of the bridge.

Fig. 11
figure 11

Schematic diagram of the on-site installation method: a cast-in-place of the superstructures, b horizontal swiveling process, and c finalization of horizontal swiveling in place

Full size image

5.3 Structural design

5.3.1 Superstructure of main bridge

Based on the form-finding approach in the MBESO method, this study manually makes slight simplification and adjustment to the shape of the bridge, with full consideration given to the constructability of related structures. The superstructure (as shown in Fig. 12) is designed as a thin-walled concrete structure with a regular curved surface. Both the main girders and the lower chords adopt the single-box three-cell structure with aligned webs, which naturally transition through a converging section. The abreast SSGs are set above the deck and aligned with the middle webs of the main girders. A top slab is set between the two SSGs to strengthen the integrity of the structure, while the pedestrian steps can be set on the slab. The top of the SSGs adopts the shape of an inverted horseshoe, where most of the longitudinal prestressed tendons of this bridge are arranged, with only a few parts to be configured in the top and bottom slabs of the main girder. Vertical prestressed tendons are arranged in the webs, so as to resist the principal tensile stresses. In the lateral direction, given the considerable cantilever length of the girder flange, the lateral prestressed tendons are configured in the top slabs.

Fig. 12
figure 12

Schematic diagram of the superstructure: a schematic along the axial direction, b schematic along the lateral direction, and c typical cross-section of the superstructure (unit: cm)

Full size image

5.3.2 Horizontal swiveling structure

The horizontal swiveling structure of this bridge consists of components for rotation and traction, as well as a balancing system, as shown in Fig. 13(a). The core component is a ball hinge, which is installed below the upper bearing caps to deliver horizontal rotation around a positioning steel shaft in the center of the main pier under the action of continuous tension jacks, thus driving the overall horizontal swiveling of the superstructure.

Fig. 13
figure 13

Schematic diagram of the horizontal swiveling structure and rigid hinge expansion device: a decomposition of the horizontal swiveling structure, and b rigid hinge expansion device

Full size image

5.3.3 Rigid hinge

Since the bearing caps of the bridge’s main piers are equipped with horizontal swiveling structures, no space is available to install movable bearings. To reduce such adverse effect as temperature changes on the bridge while ensuring the continuity of the deck, this study disconnects the main girders at the mid-span and installs a rigid hinge expansion device, as shown in Fig. 13(b). This solution releases the longitudinal degrees of freedom of the main girders and limits the remaining degrees of freedom.

5.3.4 Bicycle ramp

On one end of this bridge, there is a considerable difference in elevation between the bridge deck and the ground; in addition, there are some constraints in the site area. To address these problems, this study provides a two-level variant Fibonacci spiral-shaped bicycle ramp with a 3.5% slope, as shown in Fig. 14(a). The TSBP obtained from the BESO method results in a more reasonable force distribution for the superstructure, which can considerably shorten the girder depth and lighten the weight of the structure. The superstructure of the bicycle ramp adopts a continuous curved steel girder with a depth of 30 cm, supported by a series of organic TSBPs. This design conforms to the principle of the minimum transfer path and reflects the convergence process of force flow from top to bottom and from dispersion to concentration. Thanks to the efficient load transfer, the elegant and rational bicycle ramp structure creates an enjoyable and dynamic traversing experience, with scenic views in any direction, as shown in Fig. 14(b) and (c).

Fig. 14
figure 14

Rendering of the bicycle ramp: a aerial view, b view on the ground, and c close-range view

Full size image

To deliver on an organic shape, the TSBPs require a natural and harmonious transition at the intersection of the main and branch tubes. Therefore, this study makes reference to the unique merits of cast steel members by using tree-shaped cast steel nodes to replace the nodes where the main and branch tubes intersect, through integral casting. These cast steel nodes are joined using butt welds rather than coherent welds, effectively dispersing the welds and mitigating stress concentration typically associated with steel tube coherent welding.. The structure of the continuous steel girder and the TSBP is shown in Fig. 15.

Fig. 15
figure 15

Structure of the continuous steel girder and the TSBP: a details of steel girder segments, and b TSBP decomposition

Full size image

6 Structural analysis

6.1 Finite element model

Detailed finite element models were established for the SSG and TSBP to verify the structural performance of the proposed design for the pedestrian bridge. The superstructure of the main bridge, i.e. the SSG, was simulated in Midas/FEA NX with solid elements, except for prestressed tendons, which were simulated with tendon elements (as shown in Fig. 16). The models were discretized into 3,408,842 solid and 22,062 tendon elements, with a mesh size of 10 cm and 50 cm, respectively. The constraints of the SSG include: fixed constraints at the bottom of the bearing cap, and simple constraints at the side supports. Since rigid hinges are provided in the mid-span, symmetric restraint and release of axial restraint were used. In addition, the TSBP was simulated in Abaqus CAE (as shown in Fig. 17), and the models were discretized into 75,339 shell elements, with a mesh size of 10 mm and 50 mm, respectively.

Fig. 16
figure 16

Detailed finite element model of SSG: a meshing details, and b prestressed tendon details

Full size image
Fig. 17
figure 17

Detailed finite element model of TSBP

Full size image

6.2 Design parameters

The relevant parameters of the design loads are as follows:

  1. 1.

    Table 1 summarizes the parametric values of major construction materials according to relevant design specifications. A second-stage dead load of 16.8 kN/m is applied to the superstructure, including the deck pavement and railings.

  2. 2.

    Prestressing loads: The prestressed strand’s tension control stress value is 0.72 times its standard tensile strength, i.e., 1,339.2 MPa. For the effective prestressing force in the tendons, this study takes full account of various prestress losses, such as: ① concrete shrinkage and creep, ② tendon relaxation, ③ anchorage deformation, ④ friction between tendons and pipes, etc., all calculated according to related specifications.

  3. 3.

    Temperature effect: An overall temperature variation range of ± 25 °C. The gradient temperature for the superstructure is set at 14 °C for the positive difference and 7 °C for the negative difference.

  4. 4.

    Live load: The pedestrian load is set at 3.5 kN/m2, including a full crowd on the whole bridge deck and a full crowd on one side of the deck.

Table 1 Parametric values of major construction materials
Full size table

6.3 Static response

The following factors are taken into consideration in the structural analysis: the effect of pedestrian load, concrete shrinkage and creep, second-phase dead load, temperature effect, and their combined effects. Figure 18 demonstrates the stress distribution at the ultimate limit state in each structural component and clarifies the load transfer path of the superstructure.

Fig. 18
figure 18

Stress distribution of the superstructure (unit: stress in MPa): a maximum principal tensile stress distribution of external surface, b maximum principal tensile stress distribution of inner surface (along the axial direction), c maximum principal compressive stress distribution of external surface, and d maximum principal compressive stress distribution of inner surface (along the axial direction)

Full size image

As can be seen in Fig. 18, the maximum principal tensile stress is 1.8 MPa and the maximum principal compressive stress is 18.7 MPa. Stresses are concentrated at only a few spots in the corners of the structure, but these regions are minor and the stresses dissipate fast. Except for the localized stress concentration, the principal compressive and tensile stresses in the superstructure are all below the design values of material strength, with sufficient safety reserves.

Since the maximum mid-span vertical deflection under the live load is 125.6 mm (the sum of the absolute values of positive and negative deflection), the deflection-to-span ratio is 1/1209 (less than 1/600), which is sufficient to meet the requirements.

As seen in Fig. 19, there is only a minor stress concentration at the top of the branching point, but the stresses spread quickly. The stress levels in the rest region are moderate. The von Mises stress and the principal compressive and tensile stresses in the TSBP are lower than the allowable stresses for cast steel materials, proving that the structure of the bridge is safe and reliable.

Fig. 19
figure 19

Stress distribution of the TSBP (unit: stress in MPa): a von Mises stress, b maximum principal tensile stress distribution of external surface, and c maximum principal compressive stress distribution of external surface

Full size image

6.4 Dynamic response

The structural dynamic response characteristics of the bridge were analyzed based on FEM, and the structural self-weight and the second-phase dead load were converted into mass in the FEM. The natural vibration frequency of the bridge for the first-to-four and first-to-eight orders is shown in Fig. 20 and Table 2.

Fig. 20
figure 20

Vibration modes of the proposed bridge: a first order, b second order, c third order, and d fourth order

Full size image
Table 2 First-to-eighth order vibration modes of the bridge
Full size table

With the ever-increasing spans of pedestrian bridges and their fundamental frequency decreases, issues with pedestrian-induced vibration and walking comfort become increasingly prominent. Therefore, vibration comfort level shall be evaluated for different crowd density conditions in accordance with the sensitive frequency range evaluation guidelines (Ricciardelli & Demartino, 2016) and relevant codes (AASHTO, 2009). The sensitive frequency range is 1.25 HZ to 4.60 Hz for vertical vibrations and 0.50 HZ to 1.20 Hz for transverse vibrations. When the self-oscillation frequency of a pedestrian bridge is outside the above sensitive frequency range, the bridge can be considered as naturally meeting the requirements on pedestrian-induced vibration. Otherwise, it is necessary to check whether the acceleration response of the pedestrian bridge under the crowd walking load would satisfy the walking comfort condition. Therefore, the second-, sixth-, and seventh-order modes need to be verified for the pedestrian-induced vibration acceleration response. The verification results are shown in Table 3, which shows that the peak acceleration satisfies the specification limit value in each mode.

Table 3 Pedestrian-induced vibration acceleration response
Full size table

7 Conclusions

This study designed an innovative long-span pedestrian bridge over high-speed rails by using the BESO/MBESO methods. In the topology optimization process, several strategies were adopted to deliver an optimized conceptual design, including: setting various design domains; fixing and prohibiting part of the design space; and specifying different filter radii. After manually simplifying and slightly adjusting the form-finding result acquired in the BESO/MBESO methods, this study conducted a detailed design of the structure, with full consideration given to constructability. In addition to its unique and elegant appearance, this bridge demonstrates excellent structural performance, both statically and dynamically, as indicated in the detailed FEA analyses. The stresses within its structure generally remain below the material strength design values, thus ensuring adequate safety reserves. Furthermore, its structural stiffness and acceleration response to pedestrian-induced vibrations meet the requirements of relevant design codes.

As topology optimization techniques are developing quickly, it will be feasible for engineers to use them as intelligent tools to help design innovative bridges with high structural performance. Currently, although these techniques are not yet able to directly consider all of the complex factors in engineering design (e.g., the construction logic and the constructability of large-scale projects), designs do not need to follow a fixed routine. Designers can provide valuable insights from the outcomes of topology optimization. On this basis, by rethinking of structural solutions based on human's cognition and wisdom, and by simplifying complexity, an elegant structure may be retrieved near its rationality.