Unstressed Length Research Articles

Main cable structure analysis and construction control of short span suspension bridge within three-tower

The Feng Chu suspension bridge in Xiangyang holds the distinction of having one of the shortest span lengths among global suspension bridges, measuring less than half the length of conventional designs. This paper focuses on the engineering background of the bridge and delves into the analysis of the main cable structure and construction control methods for short-span suspension bridges with three towers. The study examines the impact of elastic modulus deviation in the main cable and the dead-load error of the stiffening beam on key parameters such as the unstressed length of the main cable, pre-deflection of the cable saddle, and mid-span elevation of the empty cable. Additionally, the effects of temperature, span, and cable length on the sag of the empty cable during the erection process are analyzed. Findings reveal that the elastic modulus and dead-load error significantly influence the unstressed length of the main cable, pre-deflection of the saddle, and the shape of the empty cable. This study establishes a relationship between changes in cable length and mid-span sag, providing valuable theoretical guidance for aligning and adjusting the main cable strands. Furthermore, synchronized alignment adjustments for the two mid-span sections are proposed to prevent main cable strand sliding, which may occur due to significant differences in sag between the sections.

Calculation method of temporary cable force in incremental launching construction of large span steel box girder without auxiliary pier

The Fenshui River Bridge is a steel–concrete composite girder bridge with a main span of 90 m. Due to the topographical constraints, the steel box girder was constructed without the use of auxiliary piers, employing incremental launching techniques. This article focuses on the construction technology used for the steel box girder of the Fenshui River Bridge. Firstly, using the influence matrix method, the cable force is determined based on the maximum cantilever state of the structure, with the vertical deformation of the front end of the guide beam and the horizontal deformation of the top of the tower as the control objectives. The unstressed cable length is then calculated based on the mechanical relationship between cable deformation and cable force. A calculation method for adaptive cable force is proposed, which is based on the variation of the stress-free cable length within the adaptive structural system. Next, the finite element analysis method was employed to determine the optimal layout position for the tower. The results indicate that during the incremental launching construction of the steel box girder, the calculated cable forces using the method proposed in this paper are in close agreement with the measured cable forces. At the maximum cantilever state of the structure, the calculated and measured values of the cable force resulted in a percentage difference of 3.96%. The calculated values of deformation and stress in key sections showed a percentage difference of 6.4% for deformation and 6.6% for stress. To maximize the effectiveness of the tower and cable, the tower should be positioned above the bridge pier when the guide beam crosses the maximum span. The findings of this paper can serve as a reference for the construction of similar types of bridges.

Assessment of maximum and minimum hanger forces in a triple-cable suspension bridge under live load: Analytical algorithm with manta ray foraging optimization

Triple-cable suspension bridges (TCSBs) are widely used in sea-crossing, island-linking projects with extra-long spans and ultra-wide decks. Their safe operation requires sufficient resistance of all bridge components (towers, cables, girders, and hangers) to static/dead and cyclic/live loads. Therefore, reliable assessment of maximum and minimum axial forces acting on each hanger is topical. This study proposes an analytical algorithm for assessing the internal forces and deformations of a TCSB. Specific steps and formulae for solving the coefficients are described by taking the estimation of girder response under random eccentric line loading as an example. Parameters of the suspension bridge under dead load are treated as known quantities. Actual engineering scenarios, including girder displacements along the bridge, are analyzed and discussed in detail. Basic unknown quantities are determined via governing equations under conditions of elevation differences’ closure and conservation of main cables’ unstressed lengths, yielding direct solutions for deformations of the main cables, girder, towers, and hangers. The proposed analytical model’s feasibility is verified via the finite element method. Optimal solutions to the above equations are derived by the manta ray foraging optimization (MRFO) algorithm to find the most unfavorable load distributions for all vehicle lanes of the bridge and the maximum and minimum axial forces of random hangers. Finally, the proposed method is successfully applied to a TCSB with a 1600 m main span under two operating conditions. The most unfavorable load distributions and the maximum and minimum axial forces of hangers at the mid-span and at the quarter-span point are estimated. The calculation results are compared against those obtained by the influence surface method, showing their good correlation and superiority of the proposed method in computation workload.

Forward iterative algorithm for fabrication geometry of steel girders in cable-stayed bridges

This paper proposes a forward iterative algorithm designed for determining the fabrication geometry of the steel girder in cable-stayed bridges during the segment installation process. The expression of fabrication geometry control parameters and geometric dimensions for steel girder fabrication is deduced based on the installation geometry. Based on the equivalent plane algorithm for spatial stay cables, this paper establishes a forward iterative algorithm to simulate the construction process of cable-stayed bridges, employing the unstressed cable length as the control calculation parameter. The algorithm is programmed into a finite element method software, enabling a precise and well-converged simulation of the cable-stayed bridge construction process. This method has been successfully applied in a kilometer-scale cable-stayed bridge, facilitating the determination of internal forces and girder geometry throughout the construction process. This also streamlines the acquisition of fabrication parameters for individual girder segments. The application results demonstrate that the research presented in this paper provides an efficient and accurate calculation method for determining the fabrication parameters of steel girders in cable-stayed bridges.

Analytical assessment of the full-bridge response to the vertical live load: An algorithm for hybrid cable-stayed suspension bridges

Hybrid cable-stayed suspension bridges combine the advantages of cable-stayed and suspension bridges, mitigating their deficiencies. Due to its super spanning ability, this novel form of bridge enjoys a bright application prospect as a large-span structure. This study introduces an analytical algorithm to estimate the full-bridge response of a hybrid cable-stayed suspension bridge with a vertical uniformly distributed load applied to the main beam. This method is based on the assumption that all parameters, such as the geometric configuration, internal force distribution, and material properties, of the whole bridge under a dead load are known; then, it analyzes the full-bridge response under a live load. First, the minimum and independent basic unknown parameters of the full-bridge response are determined. Next, the governing equations are derived and solved based on the conservation of unstressed length of each section of cable, closure of span lengths and elevation difference in different spans, and conditions for stress balance of the main beam. Thus, the values of basic unknown parameters are obtained. Finally, they are substituted into the governing equations to estimate the full-bridge response, including each bridge component’s internal forces and deflections. The proposed analytical method involves no iterative procedures when dealing with nonlinear problems but only focuses on the bridge’s state with and without loading. The results are obtained directly by solving the equations, which have the advantages of high efficiency, simplicity, and clear physical meaning. Finally, the feasibility and effectiveness of the proposed method are verified by a finite-element-based calculation of an exemplary asymmetric cable-stayed suspension cooperation system bridge with a main span of 1400 m.

An analytical method for adjusting dead load allocation between side and central cables in a cable-stayed bridge with three cable planes

Mechanical performance of cable-stayed bridges, especially those with super-wide main beams, can be significantly improved by adopting the three cable planes' scheme. However, broad application prospects of the novel bridge scheme require a proper adjustment of the ratio of the main beam's dead load allocated to the side and central cables. Otherwise, the central cables will bear an excessive dead load, deteriorating the overall bridge appearance and torsional stiffness. To this end, a simple analytical method was proposed for dead load allocation, reducing the share of the main beam's dead load borne by the central cable and re-allocating it to the side and central cables in any ratio by adjusting the unstressed lengths of the stay cables. This method takes full advantage of the convenience of adjusting the unstressed lengths of the stay cables and involves calculating the stay cables based on the catenary theory, which ensures high calculation accuracy. The proposed method first converts the spatial model of cable-stayed bridges with three cable planes into a planar model, employing the principle of virtual work to determine the elevation differences between the two anchor points of the three stay cables. Then, the unstressed lengths of stay cables are calculated based on the catenary theory. Finally, the proposed algorithm's feasibility is proved by applying the finite element method to a floating-type cable-stayed bridge with three cable planes with a main span of 1120 m and a bridge deck width of 75 m.

Direct simulation of the tensioning process of cable-stayed bridge cantilever construction

The construction analysis of cable-stayed bridges plays a crucial role in the security of such structures because stresses and deflections attained during construction might exceed those in service. Although some commercial software are capable of simulating this, their main drawback is that they typically apply the stage superposition principle, owing to which the intermediate stages cannot be analyzed independently. Moreover, under this approach, the corrections of the tensioning process due to deviations in the cable stresses measured on site are more difficult. In this study, we conducted a direct simulation of the construction process of cable-stayed bridges built using the cantilever erection method by analyzing independent finite element models. To assure the achievement of a certain target state after construction (objective service stage, (OSS)), the prestressing forces in these models were modeled as imposed strains based on the unstressed length concept. Thus, the construction process can be simulated faster when time dependent phenomena are neglected. Moreover, this will lead to the direct simulation of the construction stages and OSS. To illustrate the application of the proposed method, a cable-stayed bridge was analyzed.

Main Cable Shape of Short Span in Three-Tower Suspension Bridge

The main cable is one of the essential components of the suspension bridge; accurate calculation of the main cable shape is crucial in constructing a suspension bridge. This paper solves the challenge of the main cable shape for a “short span” in a three-tower suspension bridge by the analytical method of the ideal bridge shape and empty cable shape of the main cable. Feng Chu suspension bridge Xiangyang is taken as a prototype. This paper discusses the main cable shape and anchor-span tension control method in the construction stage of the main cable analyzes the deflection and stress characteristics of the main tower in the hoisting stage of the stiffening beam. The influence of pushing the cable saddle on the main tower provides accurate control of the main cable shape. Design sag and span of the main cable were taken as the known quantities. The sectional catenary equation is used to deduce and iteratively calculate the completed bridge alignment. The completed bridge alignment was taken as the target, the iterative calculation method of main cable bridge alignment and empty cable alignment compiles the corresponding program with MATLAB software. The calculation values of the unstressed length of the main cable, the pre-deflection of the saddle, and the mid-span sag of the empty cable are compared with the design values. The results show that the calculation accuracy of the program is high.

Influence and experiment of cable-net manufacturing errors on surface accuracy of mesh reflector antennas

Cable-net structures are of substantial importance in the construction of large mesh reflector antennas. Owing to the inevitable errors in their manufacturing process, the reflector surface accuracy deteriorates. This study makes a comprehensive investigation of random manufacturing errors during constructing the mesh reflector antennas, and analyze its influence on reflector surface accuracy. Firstly, the sensitivity of reflector surface accuracy with respect to the random errors of the unstressed cable length is mathematically deducted. Secondly, a non-button connecting method is proposed and analyzed to reduce manufacturing errors. Thirdly, two physical experiment models based on 2.62-meter mesh reflector antenna are made. Finally, numerical examples and experimental tests are given to demonstrate the effectiveness of the proposed method. Compared with the traditional method, the proposed method can effectively reduce the influence of the manufacturing errors on the reflector surface accuracy. Moreover, the reduction in the sizes of the nodes also reduces the risk of entanglement of the mesh reflector antenna during the deployment process, and thereby improves the deployment reliability.

A new type of wind-resistance cable net for narrow suspension bridges and wind-resistance cable element for its calculation

In this paper, a new-style cable net structure consisting of a wind-resistance main cable, wind-resistance secondary cable and pulleys is proposed to improve the wind-resistance stability of narrow suspension bridges. In the improved structure, the wind-resistance secondary cable which slides on pulleys to achieve uniform distribution of cable force. However, the calculation for this structure is very difficult due to the sliding cable, which means the unstressed length of the cable between the pulleys is not fixed. To this end, a multi-node wind cable element considering the cable sliding on nodes is proposed for the wind-resistance secondary cable. Additionally, an analytical model of the wind cable element is established according to the structural features. The unstressed state method, virtual work principle and influence matrix method are all used to deduce and verify the tangent stiffness matrix of the element. As a result, the exact nodal forces can be calculated according to the static equilibrium conditions and the proposed element implemented for the nonlinear finite element analysis of the suspension bridge. Numerical and bridge examples are used to demonstrate that the new element can successfully simulate the sliding of the cable on the pulleys, and be applied to the nonlinear finite element calculation like any normal element. By means of this method, a wind-resistance secondary cable with any cable section connected by pulleys to the wind-resistance main cable and stiffening girder, can be accurately simulated with only one wind cable element. Hence, the calculation problem of the wind-resistance cable is resolved and overall calculation accuracy and efficiency are improved.

Feasibility Study of Super-Long Span Bridges Considering Aerostatic Instability by a Two-Stage Geometric Nonlinear Analysis

To meet the need of constructing fixed cross strait links, super-long span bridge with a main span over 2 000[Formula: see text]m is considered as a candidate for their ability to cross deep and wide straits. To this end, some super-long span bridges with proper cable and girder systems were previously proposed and studied. The major design considerations are aimed at adopting new cable material, increasing the entire rigidity of the bridge, stabilizing the dynamic characteristics, strengthening the deck sections, etc. In this paper, a brief review of main cable and girder system is first given of the concepts previously proposed for the design of super-long span bridges. Then some typical examples are studied, focused on various issues related to the design of super-long span bridges, including composite cable, the unstressed length and tension force of the main cable, the stiffness and mass effects of the deck on critical wind speed, and the critical wind speed of various cable systems. The most challenges in super-long span bridges are to solve aerostatic and aerodynamic instability at required design wind speed. In this connection, the wind-induced aerostatic instability of super-long span bridges is studied by a two-stage geometric nonlinear analysis for dead loads and wind loads. The developed program adopted herein for geometric nonlinear analysis was verified and confirmed before. The proposed methods (i.e. composite cable, slotted girder, increasing deck stiffness and mass, cable layout, etc.) obtained for all the examples are in agreement with this study, which indicates applicability of the design approaches presented.

Theoretical solution for multi-span continuous cable structures considering sliding

This paper proposes a theoretical solution for multi-span continuous sliding cable structures, which were usually analyzed by numerical simulations. Firstly, the analytical equation for the unstressed length of single-span cable based on the undeformed configuration is derived, in which the tension forces at the ends can be accurately calculated. Then, according to the tension balance condition at sliding joints and the assumption of invariant total unstressed cable length (ITUL method), multivariate transcendental equations for the sliding analysis of multi-span cable structures with and without considering friction are established and solved by multiple Newton-Raphson iterations. Moreover, a theoretical method based on the assumption of invariant total cable length (ITL method) is also proposed and the applicability of the ITL method is investigated. Finally, four typical examples are analyzed by the ITUL method, and the accuracy of the ITL method as well as some existing numerical simulations is evaluated. The result shows that the ITUL method proposed in this paper is more accurate and reliable than the ITL method. The ITUL method can provide a theoretical basis for the design and analysis of multi-span sliding cable structures.

Cable spring effect and its longitudinal restraint stiffness on towers

On cable-stayed or suspension bridges, the cable spring concept is used to describe the influence of the main cable on the deflection of the towers. A formula for the longitudinal constraint stiffness of a cable spring was derived based on the catenary equation and two examples (construction stage and completion stage) were assessed introduced for verification. The results showed that the proposed catenary stiffness formula provides satisfactory accuracy. The stiffness value is related to the horizontal and vertical components at one end of the main cable and the weight of the cable when the unstressed length is an invariant constant. Furthermore, the stiffness value is closely related to the sag–span ratio of one span. Thus, in a two-tower suspension bridge, the longitudinal restraint stiffness of a side-span cable to the tower is much larger than the corresponding stiffness of the mid-span. In a three-tower suspension bridge, with two main spans and two side-spans, the cable spring effect plays an important role in the total stiffness of the mid-tower. The developed equations can be used for preliminary estimates of longitudinal displacement.

A Theory for the Design of Multi-Stable Morphing Structures

Multi-stable structures can provide desired reconfigurability and require relatively simple actuation. This paper considers general bar and plate structures connected by frictionless hinges that are to be made locally stable in a set of chosen target configurations by attaching extensional and rotational, linear-elastic springs to the structure. The unstressed lengths and angles of the springs, as well as their stiffnesses, are the unknown design parameters to be determined. A set of equilibrium and stability conditions to be satisfied in each of the target configurations of the structure are derived. Solutions of these equations provide specific values of the spring properties that correspond to local energy minima in all of the target configurations. The formulation is fully general and is applicable to structures of any complexity. A simple example is used to illustrate the design process for a bi-stable origami structure and a physical prototype is also presented.

A Simplified Calculation Method of Length Adjustment of Datum Strand for the Main Cable with Small Sag

In order to overcome the complicated iterative process of the cable length adjustment based on catenary theory and large error of length adjustment for cable with a small sag based on the parabola theory, this paper firstly develops a direct and simply calculation method based on parabola theory, considering the influence of elastic elongation on the cable unstressed length, which can apply for datum strand of the main cable and catwalk bearing rope with a small sag to improve the construction accuracy of the datum strand of suspension bridges. Then, the applicability of the proposed cable length adjustment formula under different conditions of the sag‐span ratio is analyzed and compared with other calculation methods based on the theory of catenary or parabola. Finally, numerical examples are presented and discussed to illustrate the accuracy and efficiency of the proposed analytical method.

A New Method for Analysis of Sliding Cable Structures in Bridge Engineering

The Sliding Cable Structure (SCS) is widely used in bridge construction by cable crane method. The SCS generally consists of cables and fixed pulleys and the cables will slide around the pulleys when the cables are loaded until the balanced state is achieved. In this paper, an efficient Angle Bisector Method (ABM) is proposed for the mechanical analysis of SCS. The constraint effect of pulley on the cable is simulated by fixed hinge and the constraint force perpendicular to the angle bisector of cable elements adjacent to the pulley is therefore zero at balanced state. By utilizing “Unit Length Influence Matrix” (ULIM), the adjustment of unstressed length of each sub-cable can be determined. As the angle bisector at each pulley will change when the unstressed lengths of cables are changed, iterative procedures are necessary to find the convergence solutions. The proposed method is firstly verified by two examples and then applied to the analysis of cable crane during construction of the world’s longest steel truss cable-stayed bridge, namely Yachihe Bridge. Satisfactory numerical solutions can be obtained only with several iterations and are in good agreement with field measurements from the cable crane load tests, which confirms the accuracy and validity of the proposed method.

Accumulative Traction Construction Analysis of the FAST Cable-net Structure

Considering the distinctive structural features and complex construction aspects of the five-hundred-meter aperture spherical radio telescope (FAST), this paper proposes the accumulative traction of serial cable construction technology (ATSC). This method makes full use of the guiding function and load-bearing capacity of guide ropes, which are simple and inexpensive, and the setting of catwalks is avoided. The entire ATSC process is divided into several static cases, and the NDFEM form-finding method is introduced to obtain the structural response of each case. Based on the principle of the constant total unstressed length of guide-rope elements, two strategies and a varying multiplication coefficient, λ, are introduced, and the “element length recursive multiplication” strategy is found to excel in terms of the analysis efficiency. Additionally, the total unstressed length of the guide-rope elements is set to three different values to determine the configuration trends. The results show that the most unfavorable configuration of guide ropes exists in initial cases and that the optimal unstressed length of the guide rope lies at an intermediate value between the total unstressed length of the serial main cables and the straight-line distance between the two endpoints of the guide rope for FAST.

The Key Measures on Safety and Linear Control in Catwalk Erection

Multiple factors affect internal force and linear of catwalk, meanwhile, there are many safety and quality risks caused by these factors during its erection. So the control of the catwalks during its erection is very important. Based on the Puli grand bridge, the key control technology of catwalks erection is studied in this paper . The unstressed length of track cable in catwalks and the ultimate unbalance horizontal force on top of bridge tower during erection are calculated, which are part of precondition for safety control. The effects of temperature and tower deviation, which are main factors influenced the linear of catwalk track rop, are analyzed in this paper. Then the safety and linear control measures are put forward. The construction result shows that the safety control methods and linear adjustment measures are effective.

Geometry control simulation for long-span steel cable-stayed bridges based on geometrically nonlinear analysis

The aim of implementing geometry control simulation to a cable-stayed bridge is to ensure that the specific designed geometric configuration will be satisfied after its construction. This paper presents a methodology, along with its software development, to simulate the different construction stages associated with cable stayed bridges and provide the erection configuration for each stage which ensures that the final designed geometric configuration is satisfied. To achieve this goal, a numerical simulation and control technique which considers geometric nonlinear effects is adopted. Multiple short truss elements are used to model the sag effect of long cables; additionally, the simulation of multiple tensioning stages associated with long cables is performed by successively adjusting the unstressed lengths of the cables. Beam–column and large displacement effects are considered using geometric nonlinear analysis based on the updated Lagrangian method. Three activation modes for new generated elements are considered during the geometry control analysis: tangent mode, parallel mode, and designated position mode. In order to demonstrate the capabilities of the proposed approach, a finite element computation software, denoted as BGCS, is developed and applied to the Sutong Bridge in order to simulate its geometric configuration throughout the construction process. The comparison of the simulation results with and without geometric nonlinearity shows that the effects of geometric nonlinearity cannot be neglected during the geometry control simulation associated with long-span steel cable-stayed bridges.

An extended force density method for form finding of constrained cable nets

The force density method (FDM) is a classical method used in linear and nonlinear form. The linear approach presents a quick tool for finding cable net new shapes by solving a set of linear equilibrium equations for certain topology, boundary conditions and assumed cables force density. The nonlinear approach was introduced to solve cable nets under constraints (assigned certain distance between nodes, limit force or unstressed length in some elements). Any type of constraint introduces nonlinearity.This paper studied the prestressed cable nets and the loaded cable nets. For prestressed cable nets, coordinate constraints to all nodes of the cable net are introduced to modify the shape after graphically examining the preliminary shape. This preliminary shape resulted from linear analysis of assumed distribution of cable force densities. For analyzing cable nets under different load cases, the first load case is analyzed to achieve the coordinate constraints assigned to nodes. Analysis results are node coordinates, cable forces and lengths. Young’s modulus and areas of cables are used to calculate the unstressed length of all cables using materialization equations, those lengths are used as constraint in the analysis of other load cases. Forces in all cables under different load cases/combinations are calculated. By using this approach, design of cable net under static load is simplified.