Gravity-triggered rotational connecting method for automated segmental bridge construction

Abstract

Automated construction has become urgently needed because the construction industry faces labor safety and cost challenges. However, these developments require investments in new equipment to facilitate automation in construction, resulting in even higher capital costs. Therefore, the research proposes a gravity-triggered rotational connecting (GTRC) method for automating segmental bridge construction. In this automated construction method, a segment-to-segment connector is developed to exploit an eccentric moment introduced by gravity and achieve segmental connections. For implementation, a specific rigging method is presented for a conventional telescopic crane to maintain a particular orientation. Meanwhile, crane path planning is also proposed to guide one segment toward the other segment. A combined computational and experimental verification program is established and employs a simply supported bridge as an example for the proposed method. With the designed connector and rigging assembly, the proposed method is computationally and experimentally verified to automate segmental bridge construction.

Graphical abstract

Keywords

automated construction / crane path planning / segment-to-segment connector / rigging design / rotational connection

1 Introduction

Automated construction is becoming increasingly valuable in the assembly of precast components in civil engineering structures. Recently, automated technologies have demonstrated higher construction quality by leveraging precision engineering knowledge, such as precision machining and advanced sensing techniques. Due to problems related to skilled labor shortages and an aging workforce, some countries are encouraging construction companies to adopt automated construction in ongoing and forthcoming projects. Moreover, these automated solutions also offer efficient productivity in repetitive construction tasks, such as building highway viaducts. As a result, these automation developments integrate multidisciplinary technologies into construction, encompassing rigging and craning methods to expedite the assembly of structural components.

Researchers have designed various methods for automating the assembly of structural components. For instance, the advanced technology for large structural systems (ATLSS) introduced a gravity-based connection for the column-beam connection [1]. A six degrees-of-freedom Stewart platform was developed to execute this automated connection for ATLSS. Another example is the SMART system, which includes a steel column connector designed for assembling steel structures [2]. The system employed various machines and tools to construct a steel-frame building. Additionally, Kim et al. [3] used guiding ropes and modified steel beam connectors to automate beam assembly. A wire control machine was developed to erect and assemble steel beams. Liang et al. [4] also proposed a robotic assembly system (RAS) for steel beam assembly. The RAS introduced a new steel beam connector and allowed for the rotation of beams during assembly. While these automated solutions offer the possibility of connecting one structural component to another, they still rely on specific erection and transportation methods and are challenging to realize with currently available machines and tools, such as telescopic cranes.

The rigging assembly should be appropriately designed in accordance with the structural components to be lifted. Traditional rigging assembly design requires iterative refinements during the lifting process. Some commercial software tools provide fundamental rigging analysis [5–7], but rigging design using these software tools typically necessitates manual or semi-manual adjustments of rigging parameters. In contrast to the traditional approach and commercial solutions, MinayHashemi et al. [8] proposed an automated rigging design method. This automated method optimizes the rigging assembly based on user-defined parameters for the structural component. Most rigging design methods in previous studies primarily focused on maintaining the lifted structural component with a specific orientation (e.g., perpendicular or parallel to gravity). In addition to this orientation, a corresponding design method should be developed for the rigging assembly when lifting structural components with the desired orientation.

Lifting path planning plays a vital role in the assembly of structural components. A construction machine lifts and transports the component to its desired position following either a fixed or adjustable path. For example, in the construction of segmental bridges, certain construction methods employ fixed paths established by specific machines for lifting and transporting structural components. In the segmental precasting method [9], a customized overhead gantry is used to lift bridge segments. This overhead gantry moves between two bridge piers to transport a segment into its designated position. Another approach is the incremental launching method, where segments are built at one end and transported on a rail to complete a span [10]. Other construction methods utilize commonly available construction equipment, such as mobile cranes, which can follow adjustable paths. Because of technological advancements, cranes have become more controllable and observable for construction applications [11–13]. Therefore, well-established path-planning methods for cranes exist [14]. Instead of relying on specialized equipment, existing construction equipment is more convenient and adaptable to various construction sites.

This research aims to develop a gravity-triggered rotational connecting (GTRC) method for segmental bridge construction. The rotational connection, linking two consecutive bridge segments, utilizes a combination of tenon and mortise with a circular guider. Gravity then directly forces a segment to rotate by an eccentric moment using a specific rigging approach. To implement the proposed GTRC method, three thrusts are involved: geometric design of the connector, rigging assessment, and crane path planning. The geometric design exploits concentric circles to form a rotational tenon–mortise connector, wherein both tenon and mortise parts are separately mounted on two consecutive segments. The rigging assessment determines the rigging assembly, including the lifting points and movable segment orientation for a telescopic crane. Meanwhile, the crane follows a predetermined path to transport the movable segment into the desired position and trigger the rotational connection. For verification of the proposed GTRC method, a combined experimental and simulation verification program provides an example of a simply supported bridge. The example aims to construct a segmental bridge using the GTRC method. Finite element analysis (FEA) is employed to iteratively modify the cross-section properties of the rotational tenon–mortise connector and evaluate sufficient strength. Additionally, a scaled experiment and crane simulator are employed to verify the rigging design and crane path planning. The proposed GTRC method is successfully implemented and verified by the combined computational and experimental verification program.

2 Gravity-triggered rotational connecting method

The proposed GTRC method automates segment-to-segment connections for segmental bridge construction. The implementation requires the connector design, rigging assessment, and crane path planning, as seen in Fig.1. The proposed method exploits a rotational interlock mechanism for the geometric design of the tenon–mortise connector. Gravity triggers the rotational mechanism by an eccentric moment and connects the tenon part to the mortise part. Moreover, this GTRC method conducts the rigging assessment to initialize the lifted orientation of the segment. With the designed rigging form, the GTRC method plans a crane path for the lifting, transportation, and rotational connection of the segment. Finally, a telescopic crane follows the designed path and accomplishes the segment connection for the segmental bridge construction.

2.1 Connector design

The GTRC method employs a circular guider to establish a tenon–mortise connector, as illustrated in Fig.2(a). Two parallel lugs attached to an adaptor plate form the tenon part, while the mortise part consists of two parallel slots fixed on an adaptor plate. These adaptor plates are mounted on the top of the cross-section on bridge segments (Fig.2(b)). Meanwhile, the tenon and mortise parts are installed on a movable and fixed (or in-place) segment, respectively. To design this connector, four key parameters, such as the length, height, width, and depth (i.e., L_c, H_c, W_c, t_c in Fig.2) determine the basic geometric properties of the connector mechanics.

As for the rotational connection, the concentric circular guider required sophisticated geometry to be fulfilled. The geometry originates from a 2D shape developed on the x–y plane in the Cartesian coordinate system, as shown in Fig.2(b). The x–y plane lies on the central longitudinal cross-section of the connected segments. The origin of the coordinate system is located at the top corner of the movable segment on the cross-section. Moreover, the center of gravity of the movable segment, C_{\mathrm{G}}, is assumed to lie on the x–y plane. C_{\mathrm{G}} combined with the connector geometry automates the gravity-triggered rotational connection. Note that the GTRC method needs a post-connection in Fig.2(c) or post-tensioning confinement in Fig.2(d) to reinforce the designed connection for the positive moment resistance [15,16] after two bridge segments are connected. With the proposed connector, two horizontally adjacent components of a structure are well linked together, i.e., for a segmental bridge.

The proposed connector is geometrically categorized into a rotatable joint arc (\widehat{P_1{P}_2{P}_3}), upper connecting arc (\widehat{P_3{P}_4}), lower connecting arc (\widehat{P_4{P}_5}), bottom connecting arc (\widehat{P_6{P}_7}), and a line segment (\overline{P_7{P}_8}), dividing this connector into the tenon and mortise parts from a rectangular area, as shown in Fig.3. The connector length and height, L_c and H_c, are user-defined and describe the allowable installation space between two consecutive segments.

A design procedure is detailed in the following. The rotatable joint is designed by the user-defined r₁ and d₁, which define the size of the joint. The center of the arc, C_1, is located at

(1){\mathit{c}}_1=\left[\begin{array}{c}{L}_{\mathrm{c}}/2\\ 0\end{array}\right],

where this center is on the perpendicular bisector of the rectangular area. r_1 must be greater than d_1; otherwise, the joint submerges below the connecting surface and becomes non-rotatable. Thus, the central angle of the arc can be calculated by

(2){\theta }_1=2\text{π}-2{\mathrm{s}\mathrm{i}\mathrm{n}}^{-1}\left(d_1/r_1\right),

where d_1 is positive (i.e., C_1 lies inside the rectangular area). The relationship between r₁ and d₁ and Eq. (1) determines the geometry of the rotatable joint. The connector design point, P₁, P₂, and P₃, are located at p₁, p₂, and p₃, respectively, defined by

(3){\mathit{p}}_1={\left[\begin{array}{cc}{L}_{\mathrm{c}}/2-{\left(r_1^2-d_1^2\right)}^{0.5}& 0\end{array}\right]}^{\mathrm{T}},

(4){\mathit{p}}_2={\left[\begin{array}{cc}{L}_{\mathrm{c}}/2+{\left(r_1^2-d_1^2\right)}^{0.5}& 0\end{array}\right]}^{\mathrm{T}},

(5){\mathit{p}}_3=\mathit{R}\left({\theta }_1\right)\left({\mathit{p}}_1-{\mathit{c}}_1\right)+{\mathit{c}}_1,

where

(6)\mathit{R}\left({\theta }_1\right)=\left[\begin{array}{cc}\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_1& -\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_1\\ \mathrm{s}\mathrm{i}\mathrm{n}{\theta }_1& \mathrm{c}\mathrm{o}\mathrm{s}{\theta }_1\end{array}\right].

The user-defined r_2 and {\theta }_2 design the upper connecting arc (\widehat{P_3{P}_4}), where r_2 is positive and defines the location of C_2 at

(7){\mathit{c}}_2={\mathit{p}}_3+{\left[\begin{array}{cc}0& -r_2\end{array}\right]}^{\mathrm{T}}.

For this arc,the central angle should satisfy

(8)

where

(9)\psi ={\mathrm{s}\mathrm{i}\mathrm{n}}^{-1}\left[{\alpha }_x/{\left({{\alpha }_x}^2+{{\alpha }_y}^2\right)}^{0.5}\right],

(10)-0.5\text{π}\le \psi \le 0.5\text{π}.

{\alpha }_x and {\alpha }_y are defined by

(11){\left[\begin{array}{cc}{\alpha }_x& {\alpha }_y\end{array}\right]}^{\mathrm{T}}=\left({\mathit{c}}_2-{\mathit{c}}_1\right)/r_2.

Equation (7) yields the center of the circle, C_2, on the extension of \overline{P_1{P}_3}. The \theta₂ constraint in Eq. (8) ensures the tenon fits the mortise. Any \theta₂ greater than the upper bound in Eq. (8) results in a non-rotatable connection. Thus, C_2 and \theta₂ determine the position of P₄ by

(12){\mathit{p}}_4=\mathit{R}\left({\theta }_2\right)\left({\mathit{p}}_3-{\mathit{c}}_2\right)+{\mathit{c}}_2.

The defined r_3 and the user-defined {\theta }_3 design the lower connecting arc (\widehat{P_4{P}_5}) and satisfy

(13)r_3=\overline{C_1{P}_4},

(14)

where

(15)a_1=\text{π}-2{\mathrm{s}\mathrm{i}\mathrm{n}}^{-1}\left(d_1/r_1\right),

(16)a_2={\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}\left|{\delta }_y/{\delta }_x\right|+\frac{\text{π}}{2}.

Equation (14) provides the lower and upper bound for \theta₃. The lower bound fulfills the connection between the tenon and the mortise, while the upper bound limits the maximum rotatable angle to ensure construction safety and connector workability. Equation (15) is the upper bound of the rotating angle to prevent any tenon–mortise collision on the upper surface. Equation (16) calculates the other upper bound, a_2, preventing the unwanted segment rotation (i.e., disconnecting rotation). Therefore, a_2 considers the position difference as

(17){\left[\begin{array}{cc}{\delta }_x& {\delta }_y\end{array}\right]}^{\mathrm{T}}={\mathit{c}}_{\mathrm{G}}-{\mathit{c}}_1,

where C_{\mathrm{G}} is the center of gravity of the movable segment located at

(18){\mathit{c}}_{\mathrm{G}}={\left[\begin{array}{cc}{x}_g& y_g\end{array}\right]}^{\mathrm{T}}.

This position is assumed to be known from the digital model of the segment (i.e., CAD drawing). Given \theta₃, the position of P₅ can also be calculated by

(19){\mathit{p}}_5=\mathit{R}\left({\theta }_3\right)\left({\mathit{p}}_4-{\mathit{c}}_1\right)+{\mathit{c}}_1.

The user-defined r₄ and \theta₄ design the bottom connecting arc and satisfy

(20)

(21){\theta }_4>0.

The arc starts from P₆ defined by

(22){\mathit{p}}_6={\mathit{c}}_1+r_4({\mathit{p}}_5-{\mathit{c}}_1)/\left|{\mathit{p}}_5-{\mathit{c}}_1\right|.

Also, P₆ lies on the extension of C_1\overline{P_5} and ends at P₇ determined by

(23){\mathit{p}}_7={\mathit{c}}_1+\mathit{R}\left({-\theta }_4\right)({\mathit{p}}_6-{\mathit{c}}_1).

P₈ is defined by

(24){\mathit{p}}_8={\left[\begin{array}{cc}{\mathit{p}}_{7,x}& -H_{\mathrm{c}}\end{array}\right]}^{\mathrm{T}}.

This point is right on the edge of the connector side length, and \overline{P_7{P}_8} is perpendicular to this side. P₁, P₂, and P₈ are located on the edge of the rectangular area of length, L_{\mathrm{c}}, and height, H_{\mathrm{c}}, in Fig.3, while other connector design points (i.e., P₃, P₄, P₅, P₆, and P₇) are distributed inside the area.

After determining the geometric parameters by this procedure, the additional parameters, including t_{\mathrm{c}} and W_{\mathrm{c}} are user-defined. The connector depth, t_{\mathrm{c}}, is positive, and the connector width, W_{\mathrm{c}}, should satisfy

(25)W_{\mathrm{c}}>2t_{\mathrm{c}}.

All geometric parameters are iteratively adjusted and checked through structural analysis to ensure the mechanical capacity of the proposed tenon–mortise connector.

2.2 Rigging assessment

For the proposed tenon–mortise connector, a specific tilt angle, {\theta }_{\mathrm{t}}, on the movable segment is required and implemented by locating three lifting points for the rigging. The rigging method employs three tight cables, as shown in Fig.4. With a telescopic crane, one end is hung on the hook, while the other is fixed on the movable segment. This research assumes both C_{\mathrm{G}} and the middle lifting point are located on the x–y plane (i.e., z = 0), dividing the movable segment into half. Moreover, the middle cable has a length of l and is the shortest one. The other two cables with the same length maintain the segment orientation. The three-cable rigging setup provides a stable rigging mechanism for the movable segment.

The initial tilt angle, {\theta }_{\mathrm{t}0}, starting the rotational connection is defined by

(26)

The lower bound prevents over-rotation, which results in two consequences. One of the consequences causes the tenon–mortise collision on the upper surface of the connector (i.e., {\theta }_{\mathrm{t}0}\le -a_1), while the other leads to the unconnected rotation (i.e., {\theta }_{\mathrm{t}0}\le -a_2 for the clockwise rotation in Fig.4(a)). The upper bound of Eq. (26) allows the rotatable space for the circular guider. The user-defined {\theta }_{\mathrm{t}0} in the allowable range in Eq. (26) results in a reachable joint connection between two segments.

The lifting point, P, in the x−y coordinates is determined by

(27)\mathit{p}=\mathit{R}\left(\text{π}/2-{\theta }_{\mathrm{t}0}\right){\left[\begin{array}{cc}\eta & 0\end{array}\right]}^{\mathrm{T}}+{\mathit{c}}_{\mathrm{G}},

where \eta is a length that locates \mathit{p} on the movable segment when {\theta }_{\mathrm{t}0} = 0° (i.e., the movable segment is connected to the fixed one). Three lifting points (i.e., the ends of cables on the movable segment) are located at the same position in the x–y coordinates, as shown in Fig.4(a). The middle lifting point is located at z = 0 (Fig.4(b)), while the other two lifting points are respectively situated at z = {\mathrm{\Delta }}_{\mathit{\text{z}}} and z = -{\mathrm{\Delta }}_{\mathit{\text{z}}}, where {\mathrm{\Delta }}_{\mathit{\text{z}}} is a user-defined variable. The proposed lifting points can intrinsically tilt the movable segment with an inclination of {\theta }_{\mathrm{t}0}.

2.3 Crane path planning

The crane path planning for the segment connection needs both transportation and rotational connection phases. The transportation phase moves the movable segment into the position for the rotational connection phase. Then, the rotational connection phase activates the gravity-triggered connection and guides the movable segment to accomplish the connection. With a telescopic crane, the GTRC method develops the crane operation following the designed path for the segment connection.

The crane hook in the transportation phase follows a designed path containing three consecutive linear path segments (i.e., the black densely dashed lines, S₁–S₃ in Fig.5). The path segments are composed of a start point, Q_s, and three critical via points (i.e., Q_a, Q_b, and Q_c) determined by the user-defined distances, l_sa, l_ab, l_bc, and cable length, l. The path segment, S₁, determined by Q_s and Q_a, lifts the movable segment with a distance of l_sa. The path segment, S₂, connecting Q_a and Q_b, with a length of l_ab moves the movable segment toward the fixed segment at the fixed elevation (i.e., the same position in the y-axis). The path segment, S₃, defined by Q_b and Q_c, has a length of l_bc and lowers the movable segment until the rotatable joint center reaches C₁. The combined path with the three consecutive linear path portions starts from Q_s and ends at Q_c. Q_c is located at a higher position than the lifting point, P, when the movable segment is tilted by {\theta }_{\mathrm{t}0} in Fig.5. Moreover, the distance between P and Q_c is the cable length, l. Finally, the other two critical via points and the start point, Q_s, can be obtained using the known path lengths and the position of Q_c. These three linear path segments compose a path transporting the movable segment to the position for the rotational connection.

The rotational connection phase, including a linear path segment, S₄, and a curved path segment, S₅, shown in Fig.6, accomplishes the gravity-triggered rotational connection. The path segment, S₄, following behind the path segment, S₃, lowers the movable segment to start the rotational connection. Meanwhile, this segment tilts by the rigging cables. The inclined cables apply a lateral force on the movable segment to prevent the tenon connector from detaching from the mortise connector. The following path segment, S₅, lowers the movable segment for the rotational connection without other inclinations of the rigging cables. To form the combined path composed of S₄ and S₅ for the rotational connection phase, two critical via points, Q_c and Q_d, and the endpoint, Q_f, are required. Q_d is located directly below Q_c with a user-defined distance of l_cd. Q_f is the crane hook position where {\theta }_{\mathrm{t}} reaches 0° for the complete rotational connection. The lifting point, P, is directly below Q_f by a distance of l subtracted by l_cd. Q_f and Q_d are connected by a curved path S₅ shaped with an arc of the circle with a radius of r_p centered at C_c, as illustrated in Fig.6. The arc subtends an angle determined by {\theta }_{\mathrm{t}0}. The center of the arc, C_c, is located at the position directly above C₁ at a distance of l subtracted by l_cd. r_p is the distance between the lifting point, P, and the rotatable joint center. S₅ satisfies the complete rotational connection without considering the segment rotation caused by S₄. The crane moves the hook on the predefined paths, S₄ and S₅, allowing gravity to rotate the movable segment for the segment connection.

Each predefined path segment can be broken into multiple via points through linear or rotation interpolation. The position of the via point, q_n, on a path segment is derived by evenly dividing the path segment into the user-defined number, N_p, of short paths. For a linear path segment (i.e., S₁–S₄), the via points are defined by the linearly interpolated points between the endpoints of the linear path segment, e.g., the linearly interpolated points between Q_s and Q_a for S₁. As for the curved path segment, S₅, the via points, Q_n, are defined by

(28){\mathit{q}}_n=\mathit{R}\left(\varnothing \right)({\mathit{q}}_{\mathrm{d}}-{\mathit{c}}_{\mathrm{c}})+{\mathit{c}}_{\mathrm{c}},

where

(29)\varnothing ={(N_{\mathrm{p}}-n)\theta }_{\mathrm{t}0}/N_{\mathrm{p}},

n=0,1,2,3,\dots ,N_{\mathrm{p}},

q_d and {\mathit{c}}_{\mathrm{c}} are the positions of Q_d and C_c, respectively. The via points ensure the crane hook moves smoothly and correctly from one critical via point to another on the desired path.

The crane can move the hook on the predefined path segments by a specific crane configuration. Given the user-defined crane parameters (i.e., L_{\mathrm{t}}, {\theta }_{\mathrm{h}}, d, and h) in Fig.7, the boom length, D, and the boom pitch angle, {\theta }_{\mathrm{p}}, can be written by

(30)D=L_{\mathrm{t}}\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{\mathrm{h}}+\sqrt{{\left(L_{\mathrm{t}}\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{\mathrm{h}}\right)}^2-\left({L_{\mathrm{t}}}^2-{\left|{\mathit{q}}_n-{\mathit{p}}_{\mathrm{c}}\right|}^2\right)},

(31){\theta }_{\mathrm{p}}={\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}\left({\beta }_y/{\beta }_x\right)-{\mathrm{s}\mathrm{i}\mathrm{n}}^{-1}\left({\left|{\mathit{q}}_n-{\mathit{p}}_{\mathrm{c}}\right|}^{-1}{L}_{\mathrm{t}}\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{\mathrm{h}}\right),

where

{\left[\begin{array}{cc}{\beta }_x& {\beta }_y\end{array}\right]}^{\mathrm{T}}={\mathit{q}}_n-{\mathit{p}}_{\mathrm{c}},{\mathit{p}}_{\mathrm{c}}={\left[\begin{array}{cc}d& h\end{array}\right]}^{\mathrm{T}},

(32)

L_t is the distance from the crane tip to the hook. The angle between the boom and hook is {\theta }_{\mathrm{h}}. The crane boom rotational center, P_c, is located at {\mathit{p}}_{\mathrm{c}} and composed of the position, d, in the x-coordinate and the position, h, in the y-coordinate. {\theta }_{\mathrm{p}} is confined due to the boundary of the crane configuration. The calculated crane configuration sequence moves the hook onto the predefined path.

The GTRC method develops an algorithm to estimate the current tilt angle, {\theta }_{\mathrm{t}}, during the construction. For the ideal transportation phase (i.e., ideal S₁–S₃), the tilt angle is {\theta }_{\mathrm{t}0}. In contrast, the tilt angle can be estimated using the following algorithm for the rotational connection phase. This research involves estimating the pose of the movable segment on the path through the proposed algorithm, followed by verification of the designed crane path using a combined computational and experimental verification program.

Algorithm 1: Tilt angle estimation for S₄ and S₅

Input: {\mathit{q}}_n, {\theta }_{\mathrm{t}0}, c₁

For n\leftarrow {\theta }_{\mathrm{t}0} to 0 do

Find {\theta }_{\mathrm{t}}=n, such that \overline{Q_{n}P}=l

End

Output: {\theta }_{\mathrm{t}}

2.4 Overall design and construction process

The overall design and construction process are organized and illustrated in the flowchart in Fig.8. Twenty-three user-defined variables are summarized in Tab.1, while nine design checks shown in the flowchart are needed. The initial steps in Fig.8(a) identify the segment information for the connector design. The following steps in Fig.8(b) represent the design sequence for the connector geometry. If a connector design fails in the strength check, the connector needs to be redesigned for geometric and/or material properties. After the connector design passes the strength check, the lifting points on the segment are determined in Fig.8(c). The rest of the steps establish the construction implementation with a telescopic crane. The crane builds up a segmental bridge segment by segment. The rotational connection followed by a post-connection is applied to each segment. The rotational connection provides the strength to support the movable segment. Thus, the crane can release the movable segment and prepare the next rotational connection while constructing the post-connection. In addition, the GTRC method redesigns the crane path until the corresponding crane operations are feasible. Consequently, the tenon–mortise connector, lifting points, and crane path designed by the flowchart implement the rotational connection for each segment.

3 Example of gravity-triggered rotational connecting method for a segmental bridge

The research uses a combined computational and experimental verification program to apply the GTRC method to a segmental bridge. The segmental bridge, spanning 30 m in length, is a simply supported bridge modified from the design by Yeh et al. [17]. A steel foundation on the ground acts as the counterweight to support the suspension bridge segments. The bridge segment is a box girder in Fig.9(a) with the height, length, width, cross-sectional area and distributed load (i.e., H_{\mathrm{s}}, L_{\mathrm{s}}, W_{\mathrm{s}},A_{\mathrm{s}\mathrm{c}}, and w_{\mathrm{d}\mathrm{l}}) listed in Tab.2. Each reinforced concrete segment weighs 195.43 kN.

With the CAD drawing, the research uses commercial 3D modeling software (i.e., AUTOCAD®) to determine the position of C_{\mathrm{G}} (i.e., {\mathit{c}}_{\mathrm{G}} in simulation) shown in Tab.2. For the scaled experiment, the position of C_{\mathrm{G}} needs to consider the additional equipment attached to the movable segment (e.g., temporary cable anchors). As a result, compared to the calculated {\mathit{c}}_{\mathrm{G}} in the simulation, the experimental {\mathit{c}}_{\mathrm{G}} is closer to the movable segment’s upper surface. The allowable space between the consecutive segments has a length and height of L_c and H_c for the connector design. The segment connection is built and verified by the proposed computational and experimental verification program.

The crane path is generated to execute the connection between the fixed and movable segments. In Fig.9(b), the crane path simulation is constructed in Unity® with the physics engine. The cable simulator, Phyllo in VirtualMethod of Unity®, implements the rigging method between the hook and segment. In addition, a telescopic crane simulator (i.e., Rusik3Dmodels of Unity®) with the proposed crane operations is employed to implement the rotational connection computationally. In the experimental verification, a 1:10 scale bridge segment and a 1:10 structural connector made of acrylic are exploited, while a robot arm mimics the telescopic crane, as shown in Fig.9(c). Both simulation and experiment provide a verification program for the proposed GTRC method, particularly for the rigging and crane path.

3.1 Connector design

Tab.3 lists the user-defined variables and the corresponding geometric parameters for a tenon–mortise connector of the bridge segment. The connector geometry is designed using {\mathit{c}}_{\mathrm{G}} in simulation and the defined rectangular area with a length and height of L_c and H_c. The user-defined variables are individually and sequentially determined during the design process.

The rotatable joint arc shown in Fig.10(a) is determined by a given L_{\mathrm{c}}, the user-defined d_1 and r_1 satisfying r_1 > d_1, and Eqs. (1)–(6). Equations (1)–(6) determine {\mathit{c}}_1, {\theta }_1, {\mathit{p}}_1, {\mathit{p}}_2, and {\mathit{p}}_3. The inappropriate d_1 or r_1 (i.e., r_1\le d_1) results in the non-rotatable joint arc, i.e., r_1= 0.04 m and d_1=0.04 m in Fig.10(b).

The upper connecting arc shown in Fig.10(c) is defined by the given {\mathit{p}}_3, {\mathit{c}}_1, the user-defined r_2 and {\theta }_2, and Eqs. (7)–(12). The center of the arc, {\mathit{c}}_2, is defined by Eq. (7) and r_2. By Eqs. (9)–(11), {\theta }_2 satisfies the constraint in Eq. (8). The inappropriate {\theta }_2 (e.g., {\theta }_2 = 211.70° shown in Fig.10(d)) outside the range in Eq. (8) results in the non-rotatable joint. Thus, the endpoint position of the arc, {\mathit{p}}_4, and r_3 are defined by Eqs. (12) and (13), respectively.

The lower connecting arc shown in Fig.10(e) is defined by the given d_1, r_1, {\mathit{c}}_1, the simulation {\mathit{c}}_{\mathrm{G}}, r_3, Eq. (19), and the user-defined {\theta }_3 satisfying Eq. (14). The upper bound of {\theta }_3 is the minimum value between a_1 = 119.16° and a_2 = 107.31° defined by Eqs. (15) and (16), respectively. The connector with {\theta }_3 greater than the upper bound (i.e., 107.31° in this example) results in the automated disconnection, e.g., {\theta }_3 = 107.31° in Fig.10(f). The endpoint position of the arc, {\mathit{p}}_5, is defined by Eq. (19). In addition, considering the experimental {\mathit{c}}_{\mathrm{G}} for the scaled experiment, {\theta }_3 is also in the range defined by Eq. (14). Thus, the connector design is feasible for both simulation and experiment.

The bottom connecting arc and the line segment shown in Fig.10(g) are defined by given r_3, {\mathit{c}}_1, {\mathit{p}}_5, H_{\mathrm{c}}, Eqs. (22)–(24), and the user-defined r_4 and {\theta }_4 satisfying Eqs. (20) and (21). r_4 is set to 0.39, greater than r_3 which is 0.255 m, and {\theta }_4 is set to 56.29° greater than 0. Either r_4 or {\theta }_4 set to the value outside the range defined by Eqs. (20) and (21) eliminates the circular guide of the connector, i.e., the bottom connecting arc with r_3=0.255 m and r_4= 0.2 m in Fig.10(h) and the arc with {\theta }_4 = –15° in Fig.10(i). The rest of the connector design points (i.e., {\mathit{p}}_6, {\mathit{p}}_7, and {\mathit{p}}_8) are defined by Eqs. (22)–(24).

The connector design in Fig.10(j) shows all connector design points, and the positions of the points are listed in Tab.4. Those design points, except for P_1, P_2, and P_8 on the side of the rectangular area, are inside the rectangular area. After W_{\mathrm{c}} and t_{\mathrm{c}} satisfy Eq. (25), the connector for the segment is realized and shown in Fig.10(k).

3.2 Finite element analysis

FEA examines the proposed connector design for the moment and shear resistance. The critical moment and shear can be obtained from a cantilever beam with the distributed load (i.e., segment weight) in Fig.11(a). The mortise connector mounted on the fixed segment is the fixed end of the cantilever beam. The movable segment connected to the fixed one with the tenon connector represents the hanging beam for the cantilever beam analysis. According to the equivalent static load method for the dynamics introduced during assembly, the allowable strength is more than two times higher than the desired strength, ensuring the requirement for the connection [18]. Due to the nonlinear contact surfaces of the connector, the controlled displacement or rotation is applied to analyze the moment and shear resistance.

For the shear resistance analysis, the mortise connector is fixed on the side connected to the fixed segment. A downward displacement is applied to the tenon connector until the connector made of A36 steel reaches the ultimate von Mises stress in Fig.11(b). FEA shows the connector can resist the shear force under 3660 kN, about 19 times higher than the required capacity (i.e., 195.43 kN) for the bridge example.

To verify the moment resistance for the connector, the research uses the controlled rotation in FEA. The counter-clockwise rotation is applied to the tenon part, while the mortise part is fixed at the right vertical side in Fig.11(c). In Fig.11(d), the clockwise rotation applied to the mortise part shows the stress distribution on the tenon part, which is fixed at the left vertical side. The rotation increases until the connector meets the ultimate von Mises stress in Fig.11(c) and Fig.11(d). As a result, the maximum moment strength for the connector is 2040 kN·m, about 5 times greater than the strength requirement (i.e., 426 kN·m) for this bridge. The FEA verifies that the connector is strong enough to support the hanging segment as a temporary connection.

3.3 Rigging assessment

The respective {\theta }_{\mathrm{t}0} satisfying Eq. (26) and {\mathrm{\Delta }}_{\mathit{\text{z}}} is determined for the simulation and scaled experiment. The lower and upper bounds of {\theta }_{t0} using {\mathit{c}}_{\mathrm{G}} in the simulation are -a_2 = -107.31° and -{\theta }_3 = −88° by Eq. (26), respectively. The initial tilt angle outside the range of Eq. (26) results in rotational connection failures, e.g., the initial tilt angle equal to the lower bound is shown in Fig.12(a), or the initial tilt angle greater than the upper bound is shown in Fig.12(b). As for the experimental {\mathit{c}}_{\mathrm{G}}, the lower bound of {\theta }_{\mathrm{t}0} becomes −119.16°. Therefore, {\theta }_{\mathrm{t}0} is set to −91.5° (Fig.12(c)) and −90.0° (Fig.12(d)) for the simulation and scaled experiment, respectively. By {\theta }_{\mathrm{t}0} = −91.5°, the position of three lifting points in the x–y coordinates is the same (i.e., \mathit{p}=[-3.6684\mathrm{m}-0.7674\mathrm{m}{]}^{\mathrm{T}}) and calculated by Eq. (27) for the movable segment in the simulation. The spacing between the lifting points is {\mathrm{\Delta }}_{\mathit{\text{z}}} = 3.33 m for the simulation. Similarly, for the scaled experiment, the position of the lifting points in the x–y coordinates (i.e., \mathit{p}={[-0.366\mathrm{m}-0.060\mathrm{m}]}^{\mathrm{T}}) is calculated using {\theta }_{t0} = −90°, and the spacing, {\mathrm{\Delta }}_{\mathit{\text{z}}}, is set to 0.67 m. To link the crane hook and the lifting points on the segment’s cross-section using cables, the temporary cable anchors are utilized in the simulation and scaled experiment (Fig.13(a) and 13(b), respectively). The temporary cable anchors are implemented as C-clamps mounted on the top flange of the segment in the scaled experiment. The research specifies the lifting points using the defined {\theta }_{\mathrm{t}0} and {\mathrm{\Delta }}_{\mathit{\text{z}}} in the verification program.

3.4 Crane path planning

With the planned operations, a telescopic crane simulator follows the designed path for the rotational connection in simulation. The user-defined cable length, l, and the user-defined length of each path segment (l_{\mathrm{s}\mathrm{a}}, l_{\mathrm{a}\mathrm{b}}, l_{\mathrm{b}\mathrm{c}}, and l_{\mathrm{c}\mathrm{d}}) determine the critical path points (Q_{\mathrm{s}}, Q_{\mathrm{a}}, Q_{\mathrm{b}}, Q_{\mathrm{c}}, Q_{\mathrm{d}}, and Q_{\mathrm{f}}) in Tab.5. By Eqs. (28)–(31), each path segment turns into the crane operation sequence with N_p = 50 in Fig.14 using the crane parameters in Tab.5. The boom pitch angle, {\theta }_{\mathrm{p}}, among the crane operations varies in the range of the value defined by Eq. (32). The maximum magnitude of the pitch angle occurs when the crane hook moves to Q_b. The designed path and the corresponding crane operations help the movable segment connect to the fixed one by the GTRC method in simulation.

The crane operations affect the segment tilt angle, {\theta }_{\mathrm{t}}, during the transportation and rotational connection phases. The tilt angle measurement on the designed path is compared to the estimation using Algorithm 1 proposed in the GTRC method in Fig.15. The sawtooth waves are found and measured in S₁, S₂, and S₅ in simulation because the movable segment oscillates when the crane hook abruptly moves and stops. Each path segment is divided into multiple short paths with the defined N_p. Due to the discrete crane controls, the higher density of via points on a path segment (i.e., greater N_p) results in a smoother motion of the movable segment (e.g., S₃ and S₄).

A robot arm moves the movable segment on the precise path to validate the rotational connection in the scaled experiment. The path is designed using the user-defined length and the defined critical via points in Tab.6. The robot arm uses linear motions to follow the path between two consecutive critical points. As for S_5, this curved path is divided into 20 short linear paths for the robot arm motion (i.e., N_p = 20 for S_5 in the scaled experiment). This experiment measures the tilt angle, {\theta }_{\mathrm{t}}, of the movable segment and the corresponding hook position (i.e., the tool center point of the robot arm). This scaled experiment physically evaluates the proposed rotational connection based on the designed connector and path.

The proposed GTRC method estimates {\theta }_{\mathrm{t}} for the transportation and rotational connection phases in the scaled experiment. The rotational connection starts from the beginning of S₄ (i.e., Q_c) and ends before the end of S₅ (i.e., Q_f) as predicted in Fig.16. As seen in the results, the minor fabrication errors in the scaled models cause a deviated tilt angle of −0.6° at the end of S_5 where y=0.5 m and {\theta }_t= 0° in Fig.16(b). In addition, during the early stage of S_4, the tenon connector’s joint exhibits a slight deviation from the desired position as it first engages with the mortise connector. Subsequently, as the tenon connector is lowered (i.e., y becomes smaller), the tenon connector progressively aligns itself with the mortise connector’s joint and ultimately rests at the desired position. In short, the experiment still verifies the connector design and the rotational connection with the proposed rigging assembly and planned path.

3.5 Discussions for the bridge example

The center of gravity of the movable segment, C_G, is considered in designing the connector (i.e., the constraint of {\theta }_3) and determining the feasible range of the initial tilt angle, {\theta }_{\mathrm{t}0}. In the connector design process, the position of C_G (i.e., {\mathit{c}}_{\mathrm{G}}) is required to check the feasibility of {\theta }_3. Moreover, the constraint for the user-defined variable, {\theta }_{\mathrm{t}0}, considers the given {\mathit{c}}_{\mathrm{G}} and {\theta }_3 in Eq. (26). Any changes of C_G can significantly influence the constraints of {\theta }_3 for the connector design and the initial tilt angle, {\theta }_{\mathrm{t}0}, for the movable segment.

The geometric parameters of the connector are highly relevant to the allowable stress in the FEA. Instead of adjusting all design parameters, increasing t_c is more beneficial to strengthen the connector. However, the connector space constraint, W_c, should be considered as the limit in Eq. (25) when increasing t_c.

Unlike the connector design and the rigging assembly, the crane path requires customization for each segment in the segmental bridge construction. Varied path lengths result in the movable segment reaching different locations for the rotational connection. The critical factor in crane path planning is the crane boom rotational center position, {\mathit{p}}_{\mathrm{c}}, or the crane position. To prevent that the boom pitch angle, {\theta }_{\mathrm{p}}, exceeds the constraint in Eq. (32), the telescopic crane should be placed at a greater distance from the segmental bridge. However, increasing the distance reduces the crane’s lifting capacity. Moreover, the maximum extendable length of the crane boom limits the workspace for the segmental bridge construction. Therefore, simulation and experimentation are crucial for applying the GTRC method to segmental bridge construction. The overall planned paths are utilized to analytically select the most appropriate telescopic crane for the segmental bridge construction.

To optimize the efficiency of the connection between two segments, certain user-defined variables in the GTRC method play a crucial role. Regarding the connector geometry design variables, r_1 determines the difficulty of placing the connector joint in the desired position while {\theta }_3 defines the length of the lower connecting arc. Thus, a longer arc consumes more time for the rotational connection. In rigging, having {\theta }_{\mathrm{t}0} closer to −{\theta }_3 saves more time for the rotational connection. In crane path planning, l_{\mathrm{b}\mathrm{c}}, l_{\mathrm{a}\mathrm{b}}, and l_{\mathrm{s}\mathrm{a}} determine the transportation length and time of the movable segment. A shorter transportation length results in a shorter transportation time. Moreover, to decrease the transition time between the via points on a path segment, a smaller N_{\mathrm{p}} is needed. However, a smaller N_{\mathrm{p}} compromises crane path accuracy, potentially leading to a rough rotational connection.

4 Conclusions

This research developed a GTRC method utilizing a telescopic crane for the segmental bridge construction. The proposed tenon–mortise connector, incorporating a circular guider, facilitated the connection between two consecutive bridge segments. The segment equipped with the connector undergoes preparation using the proposed rigging to attain the desired pose for the rotational connection. While gravity drove the rotational connection through the eccentric moment, the telescopic crane regulated the connection progress through the proposed rigging and crane path planning. The GTRC method encompassed connector design, rigging assessment, and crane path planning, all of which are implemented. The parametric design allowed for variations in the geometry of the rotational tenon–mortise connector, which comprised circles and line segments. To manipulate the bridge segment using the crane, the GTRC method calculated three lifting points on the movable segment during rigging assessment. The crane followed the critical via points, determined by the user-defined lengths, to transport and rotate the movable segment. In a practical application using a segmental bridge example, a combined computational and experimental verification program, including FEA, a simulation, and a scaled experiment, was conducted to verify the GTRC method. The program utilized the connector examined through FEA and the rigging assembly. Furthermore, the program implemented the segmental bridge construction using the proposed crane path planning in both simulation and the scaled experiment. The proposed GTRC method, designed for automating segmental bridge construction, was successfully applied and verified in both the simulation and scaled experiment.

To implement a rotational connection for a movable segment, the proposed flow chart systematically broke down the three major design components of the GTRC method into specific steps. These components comprised connector design, rigging assessment, and crane path planning. The connector design established a rotational connection mechanism, ensuring the strength of the temporary connection. Rigging assessment identified the lifting points that tilt the movable segment, initiating the rotational connection. Crane path planning enabled the crane to adjust its configuration, positioning the movable segment for the rotational connection. Modifications to the connector geometry or rigging necessitated a corresponding adjustment to the crane path. The proposed equations specified these three major design components in the GTRC method, providing a comprehensive overview in the proposed flow chart.

The GTRC method underwent application in a segmental bridge example as part of the computational and experimental verification program. Connector designs that ignored the proposed design constraints were demonstrated as potential design failures in the verification program. In contrast, the proposed connector design successfully met the design constraints and exhibited the necessary structural strength for the temporary connection. Adhering to the rigging design constraints ensured that the defined initial tilt angle of the movable segment prevented disconnecting rotation and tenon–mortise collisions. Furthermore, the GTRC method accurately estimated the tilt angle of the movable segment at any position along the designed crane path during both simulation and the scaled experiment. The combined verification results confirm the successful application of the GTRC method to the segmental bridge example, showcasing its efficacy in both computational and experimental settings. In the future research plan, the proposed GTRC method will be experimentally verified by a full-scale bridge, and the stresses and strains will be investigated during the segmental transition and construction.

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