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

Abstract

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.

1. Introduction

As a prestressed tensioning system with tension cables as the main load-bearing members, the cable structure is widely used in suspension bridges [1,2], cable-stayed bridges [3,4,5], housing structures [6,7] or arch bridges with cable-stayed buckling construction [8,9] because of its simple forces and beautiful shape. However, because it is a highly flexible material, it will produce large rotation and displacement under external load. So, the geometric nonlinear effects must be considered in the accurate calculation of the cable structure. The calculation methods for cable structure can be divided into the analytical method for solving linear equations and the finite element method.

The analytical method is only applicable to the simple cable structure, while the finite element method is generally used for complex and variable cable structures. In 1965, Ernst modeled the cable as a two-node truss element to account for the non-linear effects of the cable element. He equated the modulus of elasticity of the truss element with taking into account the cable’s sag effects, employing the Ernst formula [10]. Still, this method was only for short cables with high stresses, and the longer the cable length, the less accurate it was. Later in 1971, Knudson improved this method by equating the cables to multiple truss elements [11], but the method required a sufficient number of truss elements to achieve accuracy. With the advancement of computer technology, some researchers consider the influence of the sag effects of the cable by modeling the cable as a multi-node curvilinear element based on the Lagrangian shape function [12,13,14], which started from the geometry of the cables and improved the computational accuracy of the geometric nonlinearity of the cable structure, but with many nodes, the degrees of freedom increased, and it was not easy to converge. In addition, to avoid the detrimental effect of numerous nodes, researchers have also proposed two-node curved cable elements [15,16], but still in the form of an infinite approximation to the shape of the cable structure.

On the other hand, in order to be closer to the geometry of the cable suspension, researchers have equated the cable to a segment of parabolic form and derived an explicit expression for its tangential stiffness matrix [17,18,19,20,21], which is more accurate and straightforward to calculate, but ignores the sagging effect of the cable and the error increases with the increase of the span. For this case, many researchers have developed two-node suspension curve elements by solving the equilibrium equations satisfying the elastic suspension curve of the cable element [22,23,24,25,26,27,28,29,30,31,32,33,34], which is widely used in the analysis of nonlinearities in cable structures due to the small number of nodes, high accuracy, and fast efficiency. In addition, Chen et al., developed a suspension link element with rigid arm for accurately simulating the rigid arm connection of cable-stayed bridges [35]; Thongchai et al., proposed a scalable cable element based on the variational approach, which considered one end to be free to slide horizontally [36]; Crusells-Girona et al., proposed a cable element based on a hybrid variational formulation of finite deformation curve coordinates, in which the material nonlinearity and geometric nonlinearity are considered to obtain high accuracy with few elements [37]; Andreu et al., proposed a cable network element based on the formula of deformable suspension curve element, which improves the calculation accuracy and efficiency [38]; Vanja and Ivica proposed a three-dimensional suspension curve element, which can perform static and dynamic analysis and can be directly applied to the dynamic calculation of suspension bridges under live load [39]; Chung et al., proposed a three-dimensional elastic catenary cable element considering slip, and verified the correctness of the element by comparing experimental and theoretical data [40]; Riabi and Shooshtari proposed a stiffness matrix and nodal force equation for a three-dimensional cable structure under self-weight based on the suspension curve equation, thus developing a cable element considering material and geometric nonlinearity [41]; Rodrigo et al., investigated the effects of material and geometric nonlinearities on the cable element by introducing temperature effects [42]; Rezaiee-Pajand proposed a three-dimensional thermoelastic catenary cable element without rotating coordinates, which eliminated the solution of the inverse matrix of the catenary equation and improved the computational efficiency [43].

In addition to the above finite element methods, several researchers have adopted new methods to consider the nonlinearity of the cable structure. Croce proposed an exact expression for the equivalent stiffness of the cable element based on the principle of virtual work, which can be used as an approximate expression for the calculation of geometric nonlinearity [44]; Lee and Park developed a three-dimensional catenary cable element based on the Absolute Nodal Coordinates Formula (ANCF) and verified the accuracy of the element by comparing the static and dynamic calculation results [45]; Huttner et al., analyzed the computational efficiency and degree of convergence of nine different dynamic relaxation methods in the geometric nonlinearity of the cable element [46]; Bouaanani et al., investigated the thermoelastic response of the catenary cable element by the finite difference approach [47,48].

In summary, the advanced nonlinear theory of catenary cable element has been quite abundant, but most researchers still use the method of preparing finite element program to apply it to engineering examples, and the preparation of the program will lead to a series of shortcomings such as unfriendly pre- and post-processing interface, unable to store a large amount of computational data, low computational efficiency and not having a visualization function. ANSYS, an extensive general purpose finite element program, provides users with a secondary element development platform–UPFs, which facilitates researchers to develop units by combining proposed more advanced finite element algorithms with ANSYS secondary development tools. In order to improve the accuracy and efficiency of ANSYS to calculate the nonlinearity of complex structures, Deng et al., then developed planar beam elements using the more advanced CR algorithm [49], and the authors also developed planar truss elements by using the variational approach [50]. When the pre-stress of the cable structure is high and the network topology is involved, the accurate initialization solution is essential for the convergence of the nonlinear structure. For this reason, in this paper, the initial configuration of the catenary cable element is first derived by dichotomous iteration, and then the algorithm of the element is derived based on the explicit expression of the tangent stiffness matrix of the two-node catenary cable element in the UL column and combined with the secondary development tools (UPFs) in the ANSYS platform. The two-node three-dimensional elastic catenary cable element (user101) is developed. The element only needs to obtain its single stiffness matrix, which is automatically formed in the ANSYS program for the total stiffness and operates the same way as the other elements. The method makes use of the powerful pre- and post-processing and visualization capabilities of the ANSYS program to combine a more accurate algorithm with a commercial program and has been validated by comparison with examples and natural bridges to show that the method improves the efficiency and accuracy of the calculation and has solid practical value.

2. Tangential Stiffness Matrix of Two-Node Catenary Cable Element Based on UL Formulation

2.1. Basic Assumptions

• The stress-strain relationship of the cable conforms to Hooke’s law.
• The cable cannot be subjected to compression and bending but only to tension, being ideally flexible.
• The cross-section of the cable does not change under external loading.
• The self-weight constant load set of the cable is consistent along the length of the cable.

2.2. Tangential Stiffness Matrix of the Catenary Cable Element

Research manuscripts reporting large datasets that are deposited in a publicly available database should specify where the data have been deposited and provide the relevant accession numbers. If the accession numbers have not yet been obtained at the time of submission, please state that they will be provided during review. They must be provided prior to publication. In order to highlight the advancement and accuracy of developing the cable element, this paper adopts the tangential stiffness matrix of the cable element derived based on the displacement pattern consistent with the cable properties and basic assumptions [26], as shown in Figure 1, in the right-angle coordinate system OXYZ, with the suspension chain lead element as ij and the coordinates of the branch point i as (x_i, y_i, z_i) and the coordinates of pivot point j are (x_j, y_j, z_j). u_i, v_i, w_i are node i displacements, and u_j, v_j, w_j are node j displacements.

Displacement pattern of the catenary cable element:

{u} = [N]{u_ij}

(1)

where {u} = {u v w}^T, {u_ij} = {u_i v_i w_i u_j v_j w_j}^T,$\left[N\right]=\left[\begin{array}{ccc}\begin{array}{cc}{\phi }_1& 0\end{array}& \begin{array}{cc}0 & {\phi }_2\end{array}& \begin{array}{cc}0 & 0\end{array}\\ \begin{array}{cc}0& {\phi }_1\end{array}& \begin{array}{cc}0 & 0\end{array}& \begin{array}{cc}{\phi }_2& 0 \end{array}\\ \begin{array}{cc}{\phi }_3& 0\end{array}& \begin{array}{cc}{\phi }_1& -{\phi }_3\end{array}& \begin{array}{cc}0& {\phi }_2\end{array}\end{array}\right]$ is the form function.

We have ${\phi }_1=1-x/L$; ${\phi }_2=x/L$; ${\phi }_3=\frac{D}{2{\beta }^2}\left[\cos \mathrm{h}\beta -\cos \mathrm{h}\left(\frac{2\beta x}{L}-\beta \right)\right]-\beta \sin \mathrm{h}\beta +\left(\frac{2x}{L}-1\right)\beta \sin \mathrm{h}\left(\frac{2\beta x}{L}-\beta \right)$; L is the chord length of the cable element; β = ql/2H; q is the mean load set along the cable curve; l is the length of the horizontal projection of the cable; H is the horizontal force of the cable.

$$D=\frac{\beta \left(\beta -2\sin \mathrm{h}\beta +\sin \mathrm{h}\beta \cos \mathrm{h}\beta \right)}{2\left(\beta \cos \mathrm{h}\beta -\sin \mathrm{h}\beta \right)}$$

(2)

The tangential stiffness matrix of the cable element is then obtained from the UL columnar imaginary work increment equation.

[k] = [k₀][k_σ]

(3)

where [k₀] is the elastic stiffness matrix and [k_σ] is the geometric stiffness matrix, expressed as follows [26]:

$$\left[k_0\right]=\frac{EA}{L}\left[\begin{array}{cc}{k}_{\tau }& -k_{\tau }\\ -k_{\tau }& k_{\tau }\end{array}\right]$$

(4)

$$\left[k_{\sigma }\right]=\frac{N}{L}\left[\begin{array}{cc}{k}_{\eta }& -k_{\eta }\\ -k_{\eta }& k_{\eta }\end{array}\right]$$

(5)

where $\left[k_{\tau }\right]=\left[\begin{array}{ccc}{\tau }_1& 0& 0\\ 0& 0& 0\\ 0& 0& {\tau }_2\end{array}\right]$,$\left[k_{\eta }\right]=\left[\begin{array}{ccc}\eta & 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right], \eta =D^2\left(\frac{1}{6}+\frac{\sin \mathrm{h}2\beta -2\beta \cos \mathrm{h}2\beta +2{\beta }^2\sin \mathrm{h}2\beta }{8{\beta }^3}\right)$, ${\mathsf{\tau }}_1=1+D\left(\frac{\sin \mathrm{h}2\beta }{4{\beta }^2}-\frac{\cos \mathrm{h}2\beta }{2\beta }\right)+D^2\left(-\frac{1}{24}+\frac{\cos \mathrm{h}4\beta }{64{\beta }^2}+\frac{\sin \mathrm{h}4\beta }{256{\beta }^3}\right)$, τ₂ = sinh2β/4β − 0.5.

The tangential stiffness matrix [k] in the local coordinate system is converted to the tangential stiffness matrix [K] in the general coordinate system to obtain the equilibrium equation for the cable element in the general coordinate system [51]:

[K]{ΔU} = {P} − {F}

(6)

where {ΔU} is the displacement increment in the general coordinate system of the demand solution, {P} is the external load vector in integral coordinates, {F} is the force at the end of the cable in necessary coordinates, and the expression for the force at the end of the rope in its local coordinate system is as follows [26]:

{f} = {−H 0 q(s − ccothβ)/2 H 0 q(s + ccothβ)/2}

(7)

The iterative solution of the tangential stiffness matrix of the cable element can be divided into two types: (1) solving for the stress-free length s₀ of the cable according to the initial shape of the cable element; (2) solving for the constraint equation of the cable element by the known s₀ to obtain β, which can be substituted into Equation (3) to obtain the tangential stiffness matrix of the cable element.

2.3. Iterative Solution of a Catenary Cable Element with Known Pretension T_j or T_i

As can be seen from the tangential stiffness matrix expression of the catenary cable element in the previous section, the most critical thing is to solve for the tangential stiffness matrix β, and the need to determine the value of β requires the solution of the horizontal force H. In the initial tensioning of the cables of a cable-stayed bridge or the buckled anchor cables of a cable-stayed arch bridge with buckled construction, the cable end force T_j or T_i is often known, so the initial state horizontal force H⁰ and the unstressed cable length s₀ can be obtained by seeking the cable end. For the catenary cable element shown in Figure 2, its differential cable equation under uniformly distributed vertical loads along the ties is:

$$y=\frac{2H^0}{q}\sin \mathrm{h}\left(\frac{qx}{H^0}+D_1\right)\sin \mathrm{h}\left(\frac{qx}{2H^0}\right)$$

(8)

where $D_1={\sin \mathrm{h}}^{-1}\left(\frac{{\beta }^{0}c/l}{\sin \mathrm{h}{\beta }^0}\right)+{\beta }^0$, β⁰ = ql/2H⁰, H⁰ is the horizontal force in the initial state of the cable, c and l are the vertical and horizontal lengths in the initial tensioned state of the cable. The formula for calculating the cable length is:

$$s=\sqrt{c^2+{\left(\frac{l\sin \mathrm{h}\beta }{\beta }\right)}^2}$$

(9)

The conditions for the equilibrium of forces along the x-axis lead to the following:

$$\begin{gathered} T^0\cos {\theta }^0-H^0=0 \\ {T}^0=\cos \mathrm{h}\left(\frac{qx}{H^0}+D_1\right)H^0 \end{gathered}$$

(10)

From the above equation, the tensile forces on the ties at points i and j, respectively, are:

$$\begin{gathered} T_i^0=H^0\cos \mathrm{h}{D}_1 \\ {T}_j^0=H^0\cos \mathrm{h}\left(\frac{ql}{H^0}+D_1\right) \end{gathered}$$

(11)

The derivation is now made using the known T_i as an example and is obtained from Equation (11):

$$D_1=\pm \ln \left(\frac{T_i^0}{H^0}+\sqrt{{\left(\frac{T_i^0}{H^0}\right)}^2-1}\right)$$

(12)

From Equation (12), we have:

$$c=\frac{H^0}{q}\cos \mathrm{h}\left(\frac{ql}{H^0}\right)\cos \mathrm{h}{D}_1+\frac{H^0}{q}\sin \mathrm{h}\left(\frac{ql}{H^0}\right)\sin \mathrm{h}{D}_1-\frac{H^0}{q}\cos \mathrm{h}{D}_1$$

(13)

Substituting Equation (12) into Equation (13) gives:

$$\frac{l\sqrt{{\left(\frac{T_i^0}{ql}\right)}^2{\left(\frac{ql}{H^0}\right)}^2-1}}{\frac{ql}{H^0}}\sin \mathrm{h}\left(\frac{ql}{H^0}\right)+\frac{T_i^0}{q}\cos \mathrm{h}\left(\frac{ql}{H^0}\right)=c+\frac{T_i^0}{q}$$

(14)

Let: a = ql/H⁰, the left-hand side of Equation (14) be f(a), then:

$$f\left(a\right)=\frac{l}{a}\sqrt{{\left(\frac{T_i^0}{ql}\right)}^2{a}^2-1}\sin \mathrm{h}a+\frac{T_i^0}{q}\cos \mathrm{h}a$$

(15)

When a ≥ ql/T_i⁰ (that is T_i⁰ ≥ H), f(a) is a single increasing function (f’(a) ≥ 0). Therefore, the equation f(a) = c + T_i⁰/q has a single root and it can be obtained by the dichotomy method. For this purpose, it is first necessary to determine two lower and upper limits, a₁ and a₂, that will encompass the value of a. In this paper, the initial lower and upper limits a₁ and a₂ are determined in the following way.

Because f(a) ≥ T_i⁰cosha/q; so, c + T_i⁰/q ≥ coshaT_i⁰/q; which is:

$$-\ln \left(\frac{ql}{T_i^0}+1+\sqrt{{\left(\frac{ql}{T_i^0}+1\right)}^2-1}\right)\le a\le \ln \left(\frac{ql}{T_i^0}+1+\sqrt{{\left(\frac{ql}{T_i^0}+1\right)}^2-1}\right)$$

(16)

Make:

$$a_2=\ln \left(\frac{ql}{T_i^0}+1+\sqrt{{\left(\frac{ql}{T_i^0}+1\right)}^2-1}\right)$$

(17)

Then, because, f(a) < $\frac{\mathrm{l}}{a}\sqrt{{\left(\frac{T_i^{0}a}{ql}\right)}^2}\sin \mathrm{h}a+\frac{T_i^0}{q}\exp \left(a\right)$, so, f(a) < T_i⁰exp(a)/q, which is: c + T_i⁰/q < T_i⁰exp(a)/q, which is: a > ln(ql/T_i⁰ + 1), in addition, a ≥ ql/T_i⁰, therefore:

$$a_1=\max \left(\frac{ql}{T_i^0}, \ln \left(\frac{ql}{T_i^0}\right)+1\right)$$

(18)

From Equations (17) and (18) are the initial upper and lower limits found. And by using the dichotomy method to solve the equation and thus obtain a value of a with sufficient accuracy. Following from a = ql/H⁰, H⁰ is obtained, and D₁ is received from Equation (12), which in turn gives the cable equation and the unstressed cable length s₀ in the initial state of the tension cable.

2.4. Iterative Solution of the Tangential Stiffness Matrix of a Catenary Cable Element with Known S₀

β can be obtained by solving the constraint equation for the unit conforming to the hanging chain trail if s₀ is known. This means that:

$$f\left(\beta \right)=\sqrt{c^2+{\left(\frac{l\sin \mathrm{h}\beta }{\beta }\right)}^2}-\frac{q{l}^2}{4EA\beta }\left\{1+\frac{\cot \mathrm{h}\beta }{\beta }\left[{\sin \mathrm{h}}^2\beta +2{\left(\frac{\beta c}{l}\right)}^2\right]\right\}$$

(19)

Newton’s method can be used to substitute the solved βⁱ into Equation (3) obtain the cable’s tangential stiffness matrix [kⁱ].

3. User Element Secondary Development Process

The user element is most notably the development of the nested programs UECxxx.F and UELxxx.F; the introduction of these two subroutines can be found in the literature [52] and will not be repeated here. The compilation process of the subroutine UECxxx.F, which provides the element property parameters, can be found in the literature [50], and this paper focuses on the compilation process of the core subroutine UELxxx.F:

(1)

Enter the variables to be used in the cable element;

(2)

Obtain basic information about the cable element: modulus of elasticity E, area A, Poisson’s ratio μ, density den, initial strain ε, temperature Δt;

(3)

Call the function to calculate the initial chord length L₀ of the cable element and the transformation matrix [T₀], horizontal and projection lengths l₀ and c₀ based on the initial position of the cable element;

(4)

Solve for β₀ with sufficient accuracy according to Equations (17) and (18) where the end force T_i or T_j is known and substituted into Equation (9) to derive the unstressed length s₀ of the cable and the initial state horizontal force H⁰;

(5)

Calculation of the tangential stiffness matrix that is the elastic stiffness matrix [k₀] and the geometric stiffness matrix [k_σ], for the cable element in the structural coordinate system: when the first iteration is performed, the elastic stiffness matrix is calculated as in the linear analysis and the geometric stiffness matrix is 0; if it is not the first iteration, it is calculated according to Equation (3);

(6)

Calculation of the mass matrix [M];

(7)

Calculation of the external load vector P in the structural coordinate system;

(8)

Solve the equilibrium equation for the cable element according to Equation (6) for the unbalanced load, which is the external load vector {P}, at the first iteration;

(9)

Superimpose to the total displacement vector {U} of the cable element, based on the displacement increments {ΔU} obtained from the solution, and update the chord lengths of the cable element, the transformation matrix [T], and the horizontal and vertical projection lengths l and c.

(10)

Solve the constraint Equation (19) according to Newton’s downhill method to obtain β, substitute β into Equation (7) to obtain the cable end force, which is NR recovery force {F}, and the unbalanced load {ΔP};

(11)

The convergence of the force is judged by comparing the magnitude of the L2-van of the unbalanced force with the tolerance value TOLER; if it is less than or equal to TOLER, the program is considered to have converged. Otherwise, it returns to Step (5) to continue the operation; where the tolerance value TOLER is a minimal value, and the default is 0.001;

(12)

Calculation of strains and stresses in the rope element from the total displacement vector, followed by calculation of axial forces, etc.;

(13)

Outputs the variables saved by the cable element.

The calculation process is shown in Figure 3 below:

4. Verification and Numerical Examples

In order to verify the correctness and superiority of the development element, the following calculation example is used.

4.1. Single Cable Structures Subjected to Concentrated Loads

Figure 4 shows a single cable structure (f₀/L = 0.1) with a load set of q = 46.12 N/m, an area of 548.4 mm², a self-weight drape of f₀ = 30.48, a modulus of elasticity of the cable E = 131 × 105 kPa and a span L = 304.8 m, calculate the displacement at point C at the position shown when subjected to a concentrated load of P = 35.586 kN. In the analysis, the cable structure was divided into two cable elements, AC and CB, and the load was applied in one step. The calculation results are shown in

Table 1. Comparison of displacement at point C (unit: m).

Researcher (s)	Present	Jayaraman & Knudson [31]	Tibert [21]	Rezai-Pajand [43]
Element type	Catenary elements	Truss	Elastic parabola	Catenary elements
Number of elements	2	10	2	2
Vertical displacement uz at point C	−5.62637	−5.471	−5.601	−5.626
Lateral displacement ux at point C	−0.859393	−0.845	−0.866	−0.859

, and the load-displacement curve at point C under 20 load steps is shown in Figure 5.

As can be seen from the calculation results in Table 1 and Figure 5, the results of this paper using the developed two-node catenary cable element user101 are very close to those found in other literature, and a total of two elements were divided as proof of the correctness and feasibility of the developed element.

4.2. Verification and Analysis of Tensioned Cable Structures

A calculation model for the construction phase of an asymmetric cable-stayed bridge is shown in Figure 6, with a cable-stayed cable with rigid arms on each side of the tower, elements 5 and 6, for a total of eight nodes and nine elements. ①-④ in Figure 6 indicate main beam elements 1-4, ⑤-⑥ in Figure 6 indicate cable-stayed cable elements 5-6, ⑦ in Figure 6 indicates lower tower column element 7. ⑧-⑨ in Figure 6 indicate upper tower column elements 8-9. The modulus of elasticity E = 1.097×10⁷ MPa and other basic calculation parameters are shown in

Table 2. Basic calculation parameters.

Parameters	Lower Tower	Upper Tower	Main Beams	Tensioning Cable
A/m2	0.9290	0.2787	0.7432	0.1022
I/m4	1.7262	0.1726	0.3884	0

.

The load set of the diagonal cables q = 4.37 kN/m and the relative positions of the rigid arm elements are shown in

Table 3. Rigid arm position parameters.

Element Number	Δai	Δbi	Δaj	Δbj
11	0.5	−0.53	2.104	0
12	2.104	0	0.5	−0.053

.

The calculation is divided into two working conditions:

(1)

The two cables are synchronized for initial tensioning at the end of the tower, with the initial tension of 27,154 kN for the left-hand cable, that is, element No. 5, and 33,037 kN for the right-hand cable, that is, element No. 6, anchored after tensioning is completed;

(2)

A concentrated force couple of 8 × 10⁶ kN·m is applied simultaneously at each of the two cantilevered ends of the main beam.

Figure 7 and

Table 4. Deformation value of square frame under concentrated tension.

	Initial Tensioning			Concentration Couple
	Δv1*/m	M2/kN·m	M3/kN·m	Δv1/m	M2/kN·m	M3/kN·m
Development element user101	0.1764	612,145	628,180	−0.1211	3,620,453	3,234,527
ANSYS element	0.1847	624,478	649,296	−0.1248	3,630,024	3,432,105
Chen [34]	0.1763	612,039	628,102	−0.121	3,610,762	3,234,485
Wang [53]	0.1783	613,973	629,048	/	/	/
BDCMS * [54]	0.1805	624,395	649,100	−0.125	3,629,264	3,431,046

* (1) Δv₁ denotes the vertical displacement of node 1, M₂ and M₃ are the end section bending moments of element 2 and element 3, respectively; (2) BDCMS considers only the drape effect of the cable using the Ernst formula.

show that the displacement results of the catenary cable element user101 developed in this paper based on ANSYS secondary development and the cable element developed by the program prepared in the literature [34] differ by 0.06%. The bending moment results only differ by 0.02% under the action of working condition one. The displacement results only differ by 0.08%, and the bending moment results only differ by 0.24% under working condition two, thus verifying the accuracy of the cable element developed in this paper. The accuracy of the cable element developed in this paper is thus confirmed.

4.3. Engineering Example Validation and Analysis

The Shatuo Bridge in Guizhou Province, China, is a reinforced concrete cantilevered arch bridge with a net span of 240 m, a net vector height of f = 40 m, a net vector-to-span ratio of ƒ/L = 1/6, and an arch axis factor of m = 1.85. The arch ring section form is a single-box double-chamber structure, box width 10 m, height 4.5 m, where the thickness of the top plate from the foot of the arch is 80 cm to L/4 section of 35 cm, and then to the top of the arch L/2 section of 50 cm gradual; bottom plate thickness from the foot of the arch is 80 cm to L/4 section of 35 cm gradual, L/4 section to the top of the arch bottom plate thickness; web thickness from the foot of the arch 65 cm to L/4 section of 40 cm at the foot of the arch and then tapered to 50 cm at the L/2 section of the top plate, as shown in Figure 8. The main arch circle is divided longitudinally into 37 sections and is constructed by hanging basket cantilever casting. The site layout of the buckling anchor cables for the Shatuo Special Bridge is shown in Figure 9. ④-⑤ and ⑧-⑨ in the figure indicate No. 4-5 piers and No. 8-9 piers. ⑥-⑦ in the figure indicate No. 6-9 archs. The Scene picture of Shatuo Bridge is shown in Figure 10a.

In order to verify the correctness and practicality of the developed elements, this paper uses the measured data from the Shatuo Special Bridge for comparison with the ANSYS theoretical model data. APDL parametric modeling is used for the Shatuo Bridge; as shown in Figure 10b, the whole bridge is divided into 1808 elements and 1011 nodes, of which the 3D beam element Beam188 simulates the central arch circle, abutment pier, and buckling tower, the buckling anchor cables are simulated by the development element user101 and truss element link10 respectively. The buckling anchor cables are simulated by the virtual rigid beam Beam44 between the main arch circle and the abutment pier. Boundary conditions: Solidified form at the foot of both arches, anchor cable anchorage position is solidified. Loads: 90 t for the hanging baskets plus formwork and 40 t for the hangers plus formwork. Both were acting as concentrated loads on the arch ring. In order to verify the correctness of the development element user101, the finite element model was corrected by reference [55], and the updated values of each material parameter are shown in

Table 5. Various material parameters.

	Modulus of Elasticity E (MPa)	Poisson’s Ratio ν	Density ρ (kg/m3)
Main arch ring	3.95 × 104	0.2	2602
Abutment pier	3.25 × 104	0.2	2500
Buckle tower	2.14 × 105	0.3	7850
Buckle anchor cables	2.03 × 105	0.3	7850

.

Table 6. The initial tension and area of the anchor cable.

West Bank Buckle Number	Initial Tension (kN)	West Bank Anchor Cable Number	Initial Tension (kN)	East Bank Buckle Number	Initial Tension (kN)	East Bank Anchor Cable Number	Initial Tension (kN)	Area of Buckled Anchor Cable (m2)
X1	1402	XMS1	920	D1	1402	DMS1	914	0.00264
X2	1282	XMS2	1103	D2	1282	DMS2	1088	0.00306
X3	1303	XMS3	1282	D3	1303	DMS3	1267	0.00306
X4	1303	XMS4	1289	D4	1303	DMS4	1239	0.00306
X5	1353	XMS5	1439	D5	1353	DMS5	1411	0.00306
X6	1501	XMS6	1668	D6	1501	DMS6	1649	0.00264
X7	1402	XMS7	1347	D7	1402	DMS7	1291	0.00264
X8	1498	XMS8	1490	D8	1498	DMS8	1441	0.00348
X9	1449	XMS9	1449	D9	1449	DMS9	1425	0.00348
X10	1697	XMS10	1676	D10	1697	DMS10	1614	0.00445
X11	1749	XMS11	1708	D11	1749	DMS11	1656	0.00445
X12	1906	XMS12	1854	D12	1906	DMS12	1770	0.00445
X13	2155	XMS13	2208	D13	2155	DMS13	2135	0.00445
X14	2197	XMS14	2301	D14	2197	DMS14	2218	0.00445
X15	2260	XMS15	2135	D15	2260	DMS15	2166	0.00445
X16	2256	XMS16	2362	D16	2256	DMS16	2272	0.00348
X17	2101	XMS17	2321	D17	2101	DMS17	2280	0.00348
X18	2003	XMS18	2011	D18	2003	DMS18	2093	0.00348

shows the number, area and initial tensioning force value of the buckled anchor cable of Shatuo Bridge.

For verifying the advancement and accuracy of the development element user101, the measured values of the maximum vertical downward deflection, the measured values of the maximum cable force, and the measured values of the maximum tensile stress in the top and bottom slab of the arch were compared with the calculated values simulated by element user101 and element link10 respectively. The following data and graphs were obtained. The bridge was divided into 63 construction phases. The newly developed element user101 was used to simulate the buckled anchor cables in just 37 min, while the ANSYS element link10 was used to simulate a total of 43 min, an improvement of 6 min in calculation efficiency.

Table 7. Maximum vertical deflection of each segment of the arch ring.

Location of Measurement Points	Experiment/mm	Numerical Values (Development Element User101)/mm	Numerical Values (Ansys Element Link10)/mm
Section 2	−6	−6.1(1.7) *	−5.5(8.3)
Section 3	−11	−10.3(6.4)	−9.5(13.6)
Section 4	−14	−14.3(2.1)	−15.2(8.6)
Section 5	−16	−17.2(7.5)	−14.4(10.3)
Section 6	−24	−26.0(8.3)	−25.3(5.4)
Section 7	−31	−33.6(8.4)	−29.6(4.5)
Section 8	−37	−39.0(5.4)	−39.2(5.8)
Section 9	−48	−51.7(7.7)	−49.6(3.3)
Section 10	−56	−59.8(6.8)	−52.8(5.8)
Section 11	−67	−67.9(1.3)	−69.3(3.5)
Section 12	−78	−82.7(6.0)	−82.2(5.4)
Section 13	−85	−89.3(5.1)	−89.1(4.8)
Section 14	−94	−97.5(3.7)	−97.9(4.1)
Section 15	−98	−103.7(5.8)	−93.1(5.0)
Section 16	−109	−110.9(1.7)	−113.9(4.5)
Section 17	−124	−124.9(0.7)	−120.5(2.8)
Section 18	−136	−139.6(2.6)	−127.8(6.0)

* Numbers in parentheses indicate the absolute error percentage with respect to experimental results.

and Figure 11 show that the cable structure simulated by the user101 element can be matched well with the experimental results. And the relative error of the newly developed element user101 in calculating the main arch deflection of each casting section is 8.4% at most, while the maximum relative error of element link10 is 13.6%. This is because the link10 element, as a two-node truss element, does not take into account the sag effect of the cable, while the sag effect of the buckled anchor cable will cause the main arch to deflect more.

Table 8. Comparison of calculated and measured values of the maximum cable force of the arch ring buckles.

Buckle Cable Number	Experiment /kN	Numerical Values (Development Element User101)/mm	Numerical Values (ANSYS Element Link10)/mm
X1	1501	1499(0.1) *	1479(1.5)
X2	1434	1412(1.5)	1398(2.5)
X3	1541	1499(2.7)	1487(3.5)
X4	1550	1495(3.5)	1506(2.8)
X5	1689	1621(4.0)	1619(4.1)
X6	1843	1895(2.8)	1786(3.1)
X7	1707	1657(2.9)	1632(4.4)
X8	1834	1871(2.0)	1772(3.4)
X9	1779	1757(1.2)	1706(4.1)
X10	2129	2150(1.0)	2018(5.2)
X11	2135	2048(4.1)	2039(4.5)
X12	2338	2270(2.9)	2235(4.4)
X13	2683	2576(4.0)	2555(4.8)
X14	2772	2680(3.3)	2613(5.7)
X15	2812	2659(5.4)	2658(5.5)
X16	2765	2610(5.6)	2646(4.3)
X17	2517	2426(3.6)	2439(3.1)
X18	2180	2074(4.9)	2092(4.0)

* Numbers in parentheses indicate the absolute error percentage with respect to experimental results.

and Figure 12 show that the relative error between the maximum buckling force and the measured value of the buckling anchor cable calculated using the newly developed element user101 is 5.6%. In comparison, the maximum relative error of element link10 is 5.7%. The maximum buckling force simulated by element link10 is smaller than the measured value and is smaller than the maximum buckling force simulated by element link10, which is due to the increase in the sag effect caused by the larger buckling length, resulting in a smaller buckling force and an increase in the lower deflection of the arch.

Figure 13 shows that the maximum tensile stress in the top slab is more significant, and the maximum tensile stress in the bottom slab is less than that in the link10 element and is more in line with the measured converted tensile stress values due to the buckling effect of the buckling anchor cable causing a greater deflection in the cast section of the arch ring and a smaller buckling anchor cable force.

In summary, by comparing the deflection values of the above arch casting sections, the maximum cable force of the buckled anchor cable, and the maximum tensile stress of the top and bottom slabs, it can be concluded that the use of the user101 element to simulate the buckled anchor cable is more matching with the experimental results, with higher accuracy and faster calculation efficiency than the link10 element.

5. Conclusions

With the emergence and development of more advanced nonlinear mechanics theories, researchers have realized combining theory and experiment through computer programming. Still, due to the shortcomings of programming, it cannot be applied to large and complex structures, while ANSYS, as a large general-purpose software, can solve this problem well. This research mainly combines the nonlinear mechanic’s theories with the UPFs technology of ANSYS, integrates the advantages of both, and applies the more advanced nonlinear mechanics theories in ANSYS for computational analysis.

• For the nonlinear analysis of cable structures, higher computational accuracy and efficiency are always the goals to be pursued, and when the pretension of the cable structure is high and involves network topology, the accurate initialization solution is essential for the convergence of the nonlinear structure, and this paper adopts the dichotomous method to iterate the initial configuration of the cable unit with pretension to accelerate the convergence and thus improve the computational efficiency.
• Through the comparison and verification between the example and the engineering model, it can be seen that, according to the characteristics of large span cable-stayed bridges or arch bridges with cable-stayed suspension construction, the initial geometric configuration of the cable is first solved iteratively by the dichotomous method under the known initial tension of the cable, and then the equilibrium equation of the cable element is solved iteratively according to the nonlinear geometric algorithm of the two-node catenary cable element, so the catenary cable element user101 developed in this paper can be directly applied to the ANSYS model for analysis and calculation;
• A new type of element, user101, developed by using the nonlinear finite element algorithm of the catenary cable element combined with the ANSYS secondary development platform (UPFs), has a certain improvement in nonlinear calculation accuracy and efficiency compared to the existing commercial program ANSYS developed truss element, in order to save calculation time without reducing accuracy in the finite element calculation.
• More advanced algorithms can be combined with commercial procedures in subsequent work to develop more novel elements and also provide a new way of thinking for researchers to improve the usefulness of nonlinear theory.