Catenary Cable Element Research Articles

A fully coupled dynamic water-mooring line system: Numerical implementation and applications

Several numerical challenges exist in the analysis of water-mooring line systems which require robust, yet practical, methods to address this type of fully coupled nonlinear dynamic problems. The present study proposes a novel class of numerical techniques for the formulation and implementation of a fully coupled dynamic system which involves water flows and catenary mooring line system. In particular, the three-dimensional water flow model is replaced by a simplified multilayer shallow water system with mass exchange terms between the layers including frictional forces at the bed topography and wind-driven forces at the water free-surface. Coupling conditions between the multilayer shallow water model and the mooring line system are also investigated in the current work. As numerical solvers we implement a fast finite volume method for the multilayer shallow water equations and a nonlinear dynamic analysis for the mooring line based on elastic catenary cable elements. Efficient calculations of the interaction forces between the shallow water flow and the submerged mooring line system and associated numerical implementations are also discussed. The accuracy and computational advantages of the proposed fully coupled system are verified using a series of well-established benchmark problems and wind-driven flows over both flat and non-flat beds. The computational results obtained show high performance the developed model and demonstrate the ability of the method to simulate fully coupled dynamic water-mooring line systems.

Dynamic Formulation for Cable Stayed Bridge and It’s Validation

Cable stayed bridge is the most sensitive structure because of its higher flexibility. Exact bridge behavior can be obtained by proper dynamic bridge formulation. Cable stayed bridge deck and pylon can be modelled as abeam element. Researchers used to model cables with simple truss element and letter by parabolic cable element; although they are not presenting the exact cable performance under different lateral loads. In this study, a complete comprehensive study of modelling scheme for cable stayed bridge with improved catenary cable element is adopted by Equivalent modulus approach. The Modelling strategy involved finite element modelling with a dynamic stiffness method. A real CS Bridge is modelled in Midas-civil and results are compared. This improved innovative modelling approach agrees well with the bridge analysis results of the research article.

Geometric Nonlinear Analysis of the Catenary Cable Element Based on UPFs of ANSYS

The catenary cable element has more advantages than other nonlinear truss elements but is less used in commercial programs. In this paper, the initial geometric configuration of the element is solved iteratively by the dichotomous method. Then the Updated Lagrangian (UL) Formulation for the two-node catenary cable element is combined with the element secondary development tool provided by ANSYS platform-User Programmable Features (UPFs) to develop a three- dimensional cable element—user101. The algorithm and procedure of this paper are verified through examples and a real bridge. The study shows that the developed cable element user101 is more accurate and faster than the ANSYS self-contained element. The method can effectively use the computational theory of nonlinear cable elements with catenary geometry and combine it with ANSYS commercial program, which saves computation time without reducing accuracy and has good practicality.

Cable structures: An exact geometric analysis using catenary curve and considering the material nonlinearity and temperature effect

Structures formed by cables constitute structural systems of great practical application in engineering, such as suspension bridges, cable-stayed bridges, cable cars, transmission lines in which case they are used as supporting elements that conduct electricity, overcoming the spans between the towers or in stays for the so-called cable-stayed towers, among other applications. Cable structures have a highly non-linear behavior both from a geometric point of view, arising from their initial equilibrium configuration, and from a material point of view, since the constitutive laws of the cables present an elastoplastic behavior. The objective of this work is to present a geometrically exact formulation for static analysis of cable structures considering the catenary configuration. For this, both geometric and material nonlinearities are included in the development of the the catenary cable element formulation. The effect of temperature variation is also considered. A rigorous updated Lagrangian description combined with the corotational approach is employed to obtain the consistent tangent stiffness matrix of the exact cable element. The proposed methodology was implemented in the software program called Advanced Structural Analysis System (ASTRAS), which is an academic software based on the Finite Element Method (FEM). Finally, numerical examples are analyzed using the present procedure and the obtained results are compared with those found in the literature and given by commercial softwares in order to demonstrate the consistency and precision of the formulation.

A general framework for sliding cable analysis with elastic catenary equation

Sliding cables are widespread in engineering applications. Most of the existing sliding cable models are developed basing on conventional straight truss element, which is not applicable for many engineering applications where structures possess long-span and large sag effects. To effectively analyze this kind of system, this paper proposes a multi-node sliding catenary cable element. Due to the usage of analytic form of elastic catenary equation, the proposed element has the capability to accurately analyze both very taut and very sagged cables with minimum degrees of freedom. The formulation is developed within both dynamic relaxation approach and fully implicit approach. Effective numerical strategies in the nonlinear iteration, as well as the related tangent matrices are rigorously given. In particular, the multi-loop strategy permits to integrate the very complex nonlinear element into general finite element procedure, without needing to introduce any additional variable in the system. Multiple examples are presented to demonstrate the validity of the proposed framework. The numerical results also emphasize that sliding effects have a significate influence on the mechanical behavior of the catenary cable system, while can be easily addressed by using the methodology developed in this work.

Coupled nonlinear and time-dependent analysis for long span cable-stayed bridges

This paper proposes a new cable element and a novel formation procedure of unbalanced loads for cable-stayed bridges. The proposed method couples the effects of geometric nonlinearity and concrete creep/shrinkage. An improved catenary cable element with arbitrary rigid arms is introduced to reduce the degree of freedoms in simulations. An explicit expression of tangent stiffness of the new cable element is derived. Then, an incremental stiffness equation under co-rotational coordinate system is established by introducing a recurrence equation of creep loads. Following this, a new formation method is proposed to evaluate the unbalanced loads on the current deformed equilibrium configuration. This method is illustrated and validated by two actual long span cable-stayed bridges in Yangtze River. The results indicate that the coupled algorithm can significantly enhance the computational efficiency and accuracy for the bridge analysis during construction.

Nonlinear Analysis of Spatial Cable of Long-Span Cable-Stayed Bridge considering Rigid Connection

To determine the cable sag effect and solve the rigid connection problem at cable ends of a long-span cable-stayed bridge, a two-node spatial catenary cable element with arbitrary rigid arms is developed. Using the finite rotation formula of the space vector and a differential method, the incremental relation between displacement and force at both ends of the rigid arm is given. Then, explicit expression of the tangent stiffness matrix of the element with arbitrary rigid arms is derived based on the catenary equations. Two numerical examples are provided to verify the validity of the new element. A long-span cable-stayed bridge application model is established, and the cables are simulated using three methods. The results show that the rigid end effect has influence on displacements, bending moments and rigidity and should not be ignored. The catenary cable element with arbitrary rigid arms can be used to simulate the geometric nonlinear mechanical behavior of the cables and can well solve the rigid connection problem.

An Improved Analytical Algorithm on Main Cable System of Suspension Bridge

This paper develops an improved analytical algorithm on the main cable system of suspension bridge. A catenary cable element is presented for the nonlinear analysis on main cable system that is subjected to static loadings. The tangent stiffness matrix and internal force vector of the element are derived explicitly based on the exact analytical expressions of elastic catenary. Self-weight of the cables can be directly considered without any approximations. The effect of pre-tension of cable is also included in the element formulation. A search algorithm with the penalty factor is introduced to identify the initial components for convergence with high precision and fast speed. Numerical examples are presented and discussed to illustrate the accuracy and efficiency of the proposed analytical algorithm.

Dynamic responses of a cable-stayed bridge under a high speed train with random track irregularities and a vertical seismic load

The dynamic analysis of train-bridge systems has been a popular research topic for a long time; however, studies on cable-stayed bridges subject to train and seismic loads remain limited. Cable-stayed bridges can experience large vibrations under external periodical loads due to the high flexibility caused by their long decks and cables. Previous approaches to modeling the cables have limitations in accuracy, principle, or calculation efficiency, making them unsuitable for dynamic analysis with numerous time steps for long-span bridges with many cables. Furthermore, track irregularities and earthquakes bring additional excitations to the train-bridge system and threaten its structural and running safety. Thus, an accurate and fast approach to modeling and analyzing cable-stayed bridges under train loads and other excitations is needed. In this study, an improved parabolic cable element is included for dynamic analysis. This cable element facilitates a faster calculation while maintaining an accuracy similar to that of a catenary cable element. The coupled equation-of-motion of the train and cable-stayed bridge system is derived and solved via the time integration method. The effects of railway track quality and seismic load are investigated through the dynamic responses of train-bridge systems by considering various classes of tracks and levels of seismic intensities, respectively. The impact factors of the vertical displacement of the deck and the tensions in cables are used to represent the behaviors of the bridge, while the maximum accelerations of vehicle bodies are used to represent the behaviors of the train. Track irregularities and seismic load significantly increase the responses of the bridge and train. With increased train speed, the negative effects of poor quality track on the responses of the train-bridge system increased; however, with increased seismic loads the effects were found to decrease.

Evaluation of load and resistance factors for the reliability-based design of the main cables of earth-anchored suspension bridges

This paper presents load and resistance factors for the reliability-based design of the main cables of earth-anchored suspension bridges for the Strength I Limit State defined in the AASHTO LRFD Bridge Design Specifications. Reliability analyses are performed for three suspension bridges in Korea with main span lengths ranging from 650 m to 1545 m, and the reliability indices and the Most Probable Failure Points (MPFP) of random variables are evaluated. The strength, load and stiffness parameters are taken into account as random variables. The advanced first-order second-moment method is employed to evaluate the reliability indices and the MPFP of the random variables. The elastic catenary cable element is utilized to discretize the main cables. Load factors are proposed based on the normalized MPFP of each load component considered in the Strength I Limit State. The resistance factor that yields the reference target reliability index for the main cable is found as 0.53. Two alternative resistance factors, which give lower and higher reliability indices than the reference value, are also proposed. The proposed load and resistance factors lead to a uniform reliability level for a main cable regardless of the length of the main span of a suspension bridge.

Three-Dimensional Parabolic Cable Element for Static Analysis of Cable Structures

AbstractThis paper presents a three-dimensional inclined parabolic cable element for static analysis of cable structures. After a catenary cable element is briefly reviewed, the three dimensional parabolic cable considered in this paper is presented. The derivation of the stiffness matrix for the element is described by equating the total differentiations of the cable length and its elastic elongation, and defining the stiffness as the ratio of the infinitesimal force increment to the infinitesimal length increment. The static equilibrium equation for a two-node cable element and the procedure for nonlinear structural analysis using the Newton-Raphson iterative method are presented as well. The proposed element can consider an inclined cable without the need for a transformation, and has explicit and simple stiffness formulas compared with those of a catenary element. To validate the proposed parabolic cable element, four example models subjected to static loadings are analyzed using the proposed element,...

A Numerical Method to Material and Geometric Nonlinear Analysis of Cable Structures

In this article, a numerical method is presented for computing the stiffness matrix and compatibility relations of cables under uniform distributed loads on any direction. Both the geometrical and material nonlinearities, including softening and yielding of material, are taken into account and the catenary cable element with gauss integration scheme is employed. The proposed formulation includes the effect of uniform distributed loads on any direction and 3D nodal forces to assess the geometric and possible material nonlinearity of cables. The derived equations are then applied to the analysis of cable structures. The accuracy of the responses obtained by the proposed method is evaluated by several benchmark solutions available in the literature. Results of numerical examples indicate the capability of the proposed stiffness matrix in prediction of the elastic and inelastic responses of cables. The proposed formulation is therefore recommended for cable elements to be used in the analysis of cable structures.

Improved methods for determining the 3 dimensional initial shapes of cable-supported bridges

The TCUD method (Kim and Lee, 2001) can minimize undesirable lateral displacements in cable bridge structures by including the unstrained lengths of the cable members as unknowns and introducing additional constraint conditions. The axial deformations in the girder and the main tower deviating from the target configuration are, however, unavoidable when self-anchored cable bridges are considered. On the other hand, the IFM (initial force method) can be used to adjust the axial deformations to conform to the initial shape but it is critical that the unstrained lengths of the cables are accurately evaluated. In this study, improved schemes for the TCUD method and the IFM for finding the initial shape of 3 dimensional cable-supported bridges under dead load, are presented. For this purpose, the TCUD method and the initial force method are firstly reviewed. Two improved algorithms, which can overcome shortcomings of the previous initial-shaping analysis methods, are developed based on an elastic catenary cable element and a nonlinear space frame element. Finally, the accuracy and effectiveness of the proposed analysis algorithms are demonstrated through the numerical application to 3 dimensional initial shape problems for self-anchored suspension and cable-stayed bridges.

Three-Dimensional Geometric Nonlinear Analysis of Composite Cable-Stayed Bridges Using a Refined Double-Beam Model

In this paper, a refined double-beam finite-element model is constructed for composite cable-stayed bridges. Using this refined model, the authors aimed at performing more accurate three-dimensional (3D) analyses of cable-stayed bridges by considering all the geometric nonlinearities, including large 3D rotation issues, using catenary cable elements for cables, and taking into account member eccentricities between cable anchor points with the deck and member neutral axes. To consider member eccentricities and avoid using rigid links to connect the eccentricities, a validated two-step procedure is developed to transfer member forces and stiffness from the fluctuating neutral axis lines to a straight reference line. Through a numerical example, the effect of member eccentricity is discussed, and it is clearly shown that member eccentricity has a larger influence on displacements and bending moments than on modal responses. Through a detailed comparison example with previous research and existing software, the refined model presented in this paper demonstrates higher accuracy.

Nonlinear analysis methods based on the unstrained element length for determining initial shaping of suspension bridges under dead loads

Two nonlinear analysis methods are newly presented for finding the initial shape of suspension bridges under dead load. For this, the extended tangential stiffness matrices of the nonlinear frame element as well as the elastic catenary cable element are derived by adding unstrained element lengths to the unknown. A G.TCUD (generalized target configuration under dead loads) method and an unstrained element length method are then proposed to get the minimized bending moment as well as to remove both lateral and axial deformations in the main girder and the tower. Finally the accuracy and effectiveness of the proposed schemes are demonstrated through two numerical examples.

Nonlinear analysis of cable structures under general loadings

In this paper, two new elements for three-dimensional finite element analysis of cable structures are proposed. The tangent stiffness matrices of the cable elements are derived under spatially concentrated and distributed static and thermal loads. The first element, which is called Continuous Catenary Cable (CCC) element and extends the classic catenary cable element, presents the explicit forms of the stiffness matrix and internal force vector of the cable. The second element, Discrete Catenary Cable (DCC) element, is introduced by transforming the continuous equations of the CCC element into discrete formulation, giving the capability of dividing the cable into several straight elements with axial behavior. The DCC element, having all the features of the CCC element, has the advantages of including more three-dimensional loadings such as point loads along the member and non-uniform distributed lateral loads, and incorporating various geometrical and material nonlinearities such as cable cross-section variation and cable material yielding. Thereafter, employing the proposed elements, we have presented a simple algorithm for the analysis of pretensioned cables. The proposed elements are then used for nonlinear analysis of cable structures, and the response is compared with those obtained by other researchers. The results of numerical examples indicate the capability and robustness of the proposed elements in predicting the deformation response of cable structures.

Stabilization process analysis of cable dome structure

In this paper, a stabilization process of cable dome structures, Sky-parasol, was simulated by using dynamic relaxation method (DRM) with kinetic damping technique and curved catenary cable element. Using the algorithm of this paper, an initially drooped catenary shape of cable dome can be obtained by deleting some diagonal and under cable from the initial design shape for the erection process analysis. This process can be understood as collapse process analysis. Next, the analysis model was prepared by inserting the diagonal and under cables for erection process analysis from the resulting shape of collapse process analysis. The element force of cable and change of the shape of cable dome are obtained from the erection process analysis by increasing the incremental stress-jacking force of diagonal cables. By using the method of this study, it is possible to simulate the characteristics of cable dome for shape finding and determination of member forces during stabilization process analysis.

Nonlinear analysis of cable-supported structures with a spatial catenary cable element

This paper presents a spatial catenary cable element for the nonlinear analysis of cablesupported structures. An incremental-iterative solution based on the Newton-Raphson method is adopted for solving the equilibrium equation. As a result, the element stiffness matrix and nodal forces are determined, wherein the effect of self-weight and pretension are taken into account. In the case of the initial cable tension is given, an algorithm for form-finding of cable-supported structures is proposed to determine precisely the unstressed length of the cables. Several classical numerical examples are solved and compared with the other available numerical methods or experiment tests showing the accuracy and efficiency of the present elements.

Initial Equilibrium Configuration Determination and Shape Adjustment of Cable Network Structures

This article presents a rigorous static analysis of cable network structures, which relates the nodal position, the unstressed cable length and the external load using the elastic catenary cable element in order to take into consideration the cable curve effect and the fact that cables can only be tensioned. The incremental relationship between nodal position and cable length is employed for solving the unstressed cable length in the initial equilibrium configuration determination problem and for accomplishing the shape adjustment operation with an iterative method. Detailed procedures of investigating the problems of the initial equilibrium configuration and shape adjustment of the cable network supporting structure for a large reflector are presented to demonstrate the validity and effectiveness of the method.

Nonlinear Analysis and Optimum Design of Guyed Masts

In this paper the nonlinear analysis and design optimization of guyed masts is addressed. The mast is modeled as a 3D truss and is supported by catenary cable elements that have nonlinear elastic behavior. For nonlinear static analysis, an innovative procedure is proposed that divides the structure into linear and nonlinear parts and analyzes them separately. The proposed method satisfies the equilibrium and compatibility by establishing and solution of a set of nonlinear equations. The optimization problem employs the sizes of members, initial cable tensions and the positions of anchor on the ground and tie level of cables on the mast as design variables. To facilitate the optimization solution, a compatible sensitivity analysis procedure is proposed. Sensitivities of objective function, displacement and strength constraints in the mast and cables, subjected to a variety of load combinations including dead, wind and ice loads are calculated. Numerical examples are provided to show the nonlinear analysis procedure and the applicability of the algorithm to optimum design of practical guyed masts.