Nonlinear analysis of cable structures using the dynamic relaxation method

Abstract

The analysis of cable structures is one of the most challenging problems for civil and mechanical engineers. Because they have highly nonlinear behavior, it is difficult to find solutions to these problems. Thus far, different assumptions and methods have been proposed to solve such structures. The dynamic relaxation method (DRM) is an explicit procedure for analyzing these types of structures. To utilize this scheme, investigators have suggested various stiffness matrices for a cable element. In this study, the efficiency and suitability of six well-known proposed matrices are assessed using the DRM. To achieve this goal, 16 numerical examples and two criteria, namely, the number of iterations and the analysis time, are employed. Based on a comprehensive comparison, the methods are ranked according to the two criteria. The numerical findings clearly reveal the best techniques. Moreover, a variety of benchmark problems are suggested by the authors for future studies of cable structures.

Keywords

nonlinear analysis / cable structure / stiffness matrix / dynamic relaxation method

Introduction

In the dynamic relaxation method (DRM), a fictitious mass and damping forces are added to the structural static equations to create a simulated dynamic system. Achieving the steady-state response of this dummy dynamic system, which is solution of the static equations, is the basis of the DRM. This approach is an explicit procedure for solving the simultaneous equations in which vector operations are used in all relationships due to the assumption of a diagonal mass matrix. Furthermore, the internal forces of each element are calculated from the multiplication of the node displacement and the element stiffness matrix. By superposition of the internal forces of all elements, a total internal force vector is obtained. The residual force is determined by the difference between the calculated internal forces and the applied external force. In the DRM, the residual force of the structure is assumed to be zero in the static state. Hence, assembly of the structural stiffness matrix is not required. Therefore, this method is suitable for solving problems with extreme geometric and material nonlinearities. Moreover, the damping matrix is considered to be a multiple of the mass matrix. The damping factor, mass matrix, time step, and first displacement vector are the unknown factors of the DRM. Estimating an appropriate value for each parameter can increase the rate of convergence, improve the numerical stability, and reduce the analysis duration [¹].

Thus far, a number of techniques have been proposed to estimate the DRM factors. Day [²] first utilized this method. Bonce used Rayleigh’s principle to determine the critical damping [³]. Using Gerschgorin’s approach for finding the mass matrix was suggested by Cassel and Hobbs [⁴]. In 1983, Underwood [¹] introduced an efficient relationship for the DRM. Qiang [5] used a novel damping and time step. In 1989, Zhang and Yu [⁶] applied the modified DRM to obtain the steady-state response. Furthermore, Zhang et al. [⁷] proposed a new technique to calculate the damping factor. In their method, all degrees of freedom of a node have the same damping.

In recent years, many studies have been performed in this topic. Rezaiee-Pajand [⁸] suggested a new DRM procedure that utilizes a Taylor series. Kadkhodayan et al. [⁹] minimized the residual force to estimate the time step. Rezaiee-Pajand and Alamatian [¹⁰] proposed new mass and damping matrices obtained using the Gerschgorin circle theorem. In another investigation, Rezaiee-Pajand and Sarafrazi [¹¹] calculated the minimum eigenvalues using a power iterative scheme. Their method increased the convergence rate of the DRM. These investigators also proposed a novel time step by setting the critical damping of the system to zero [¹²]. Comparison of well-known DRM procedures was carried out by Rezaiee-Pajand et al. [¹³]. Alamatian [¹⁴] used displacement-based methods for calculating the buckling load, and for tracing the post-buckling regions. Rezaiee-Pajand and Rezaee [¹⁵] suggested a new time step for the kinetic DRM based on analysis of the displacement error. The structural buckling limit load was computed by Rezaiee-Pajand and Estiri [¹⁶] using the DRM. These researchers also found the equilibrium path in the DRM by minimizing external work [¹⁷]. In another investigation, they mixed the DRM with the load factor and displacement increments [¹⁸].‏ Further, two comparative works were conducted by Rezaiee-Pajand and Estiri [¹⁹,²⁰] on three-dimensional (3-D) frames and bending plates using the DRM. In a comprehensive study, Rezaiee-Pajand et al. [²¹,²²] reviewed the proposed formulations of the DRM and their application. Recently, Labbafi et al. [²³] compared the viscous and kinetic DRMs in the form-finding of membrane structures. Furthermore, Rezaiee-Pajand and Mohammadi-Khatami [²⁴] suggested a new method for viscous DRM in order to shorten the analysis time. In another study, Rezaiee-Pajand et al. [²⁵] created some new DRM techniques in which the DRM factor was replaced.

Many different methods have been proposed to analyze the cable structures. In 1978, Ozdemir [²⁶] proposed a finite-element approach to solve the cable problem. A numerical method for analyzing 3-D cable structures was also used by Pevrot and Goulois [²⁷]. Monforton and El-Hakim [²⁸] applied the energy search method for analysis of this type of structure. Jayaraman and Knudson [²⁹] presented a small strain elastic catenary element to analyze static and dynamic cable problems. To analyze pretensioned cable networks, the DRM was utilized by Lewis et al. [³⁰]. Kmet and Kokorudova [³¹] presented a technique for solving cable truss problems having distinct forms of supports. Shape-finding of cable structures having slack and prestressed elements was carried out by Deng et al. [³²]. Andreu et al. [³³] suggested a new catenary element to analyze cable structures. Yang and Tsay [³⁴] conducted a geometric nonlinear analysis of cable structures with a two-node cable element. Additional research studies were proposed by other investigators [³⁵–⁴²]. Moreover, studies addressing the stiffness matrix and numerical methods were recently conducted [⁴³–⁴⁵].

According to the related scientific literature, no comprehensive studies have previously been carried out on the use of various stiffness matrices of the cable element for the DRM. Because the DRM is capable of analyzing structures having extreme nonlinear behavior, this work aims to investigate the solution process of the cable structure. It should be added that one of the effective steps in the analysis of the cable is proper estimation of the element stiffness matrix. To deeply explore this issue, all well-known formulations, which are matched to the analytical DRM formulae and lead to six different stiffness matrices of the cable elements, are utilized throughout this study. Aside from these issues, the literature review demonstrates that only a limited number of the benchmark cables are accessible; unfortunately, most of them are not significant problems. To compensate for this shortcoming, several applicable and interesting cable structures are designed and solved by the authors, which can be utilized in future investigations. To achieve accurate assessments, 16 diverse cable examples, with nonlinear behavior, are analyzed in this study. Both available benchmarks and suggested structures are solved. Based on the criteria of the analysis time and required number of iterations, the merits and weaknesses of the cable’s stiffness matrices, in combination with the DRM, are discovered.

Stiffness matrix of cable element

Based on different assumptions, researchers have proposed various relationships to determine the stiffness matrix of a cable element. In the following subsections, six well-known methods, which are matched with assumptions and analytical dynamic relaxation formulae, are presented for estimating this matrix.

Simple truss element

Using a tension truss bar for estimating the cable stiffness matrix is one of the simplest methods. Figure 1 shows a bar element that only tolerates the tension force T. This force is calculated as follows [⁴²]:

(1)T={\displaystyle \frac{EA}{s_{\mathrm{0}}}}(r-s_{\mathrm{0}}),

where E, A, r, and s₀ are the modulus of elasticity, the cross-sectional area, the distance between two end joints in the chord direction, and the slack length of the element, respectively. It should be added that the tension force of the cable should be a nonnegative value. Therefore, if the value of T in Eq. ‎(1) is negative, it is set to zero. With these data, the stiffness of each element is defined in the following form:

(2)S_k=S_k^E+S_k^G={\displaystyle \frac{E_k{A}_k}{s_{\mathrm{0}k}}}+{\displaystyle \frac{T_k}{r_k}},

where k, S_k^E, andS_k^G show the counter, elastic stiffness and geometric stiffness of the members, respectively. As a common process, the stiffness of the nth degree of freedom is obtained by assembling the stiffness of all elements:

(3)S_n={\displaystyle \sum _k{S}_{k,n}},

where S_k,n and S_n are the element stiffness and the stiffness of the nth degree of freedom. After finding the structural displacements, the nodal forces are determined by tension force decomposition in the direction of each degree of freedom.

Element of Torkamani & Naserian

To estimate the stiffness matrix more accurately by utilizing a truss bar element, researchers have used higher-order truss elements. For the first time, this element was suggested by Torkamani and Shieh [⁴⁶] to analyze the plane truss structure of a nonlinear finite element. Rezaiee-Pajand and Naserian modified this technique to use it in the analysis of a 3-D truss. In this study, these elemental matrices are utilized to determine the cable stiffness matrix [⁴⁷,⁴⁸]. These investigators calculated the higher-order stiffness matrix by including second-order axial strain terms in the elastic constitutive equation of the 3-D truss [⁴⁸]. According to the obtained formulations, the local element stiffness matrix consists of the following five matrices: K₀ is the conventional linear stiffness matrix for uncoupled axial behavior, and is constant; K_P is the initial stiffness matrix, which depends on the tension force at the beginning of each incremental load step; K₁ and K₂ are the linear functions of the incremental element displacements; and K₃ is a quadratic function of the incremental element displacements.

Element of Deng et al.

In this procedure, cables are regarded to be perfectly flexible, and they are devoid of any flexural rigidity. As shown in Fig. 2, loads on the cable, which must include at least the structural weight, are distributed uniformly along the curve of the cable, which is assumed to be a parabola [³²].

In this scheme, the cable tensile force can be obtained from the next constitutive relationship:

(4)g(T,r,l,c,s_{\mathrm{0}},q,E,A)={\displaystyle \frac{l^{\mathrm{2}}T}{\mathrm{2}rw}}\left(\ln \left(-{\displaystyle \frac{\mathrm{2}c}{l}}+{\displaystyle \frac{rw}{lT}}+{\displaystyle \frac{\sqrt{b}}{lT}}\right)-\ln \left(-{\displaystyle \frac{\mathrm{2}c}{l}}-{\displaystyle \frac{rw}{lT}}+{\displaystyle \frac{\sqrt{a}}{lT}}\right)\right)+{\displaystyle \frac{c}{\mathrm{4}rw}}(\sqrt{a}-\sqrt{b}),

+{\displaystyle \frac{\mathrm{1}}{\mathrm{8}T}}(\sqrt{a}+\sqrt{b})-s_{\mathrm{0}}-{\displaystyle \frac{T}{EA}}\left({\displaystyle \frac{l^{\mathrm{2}}}{r}}+{\displaystyle \frac{c^{\mathrm{2}}}{r}}+{\displaystyle \frac{w^{\mathrm{2}}r}{\mathrm{12}{T}^{\mathrm{2}}}}\right)=\mathrm{0},

where l is the horizontal distance between the two end joints of the cable; c is the vertical separation between two end joints; w= qs₀ is the result of the acting load on the cable; q is the intensity of the uniformly distributed load; and the factors a and b are calculated as follows:

(5)a=w^{\mathrm{2}}{r}^{\mathrm{2}}+\mathrm{4}{c}^{\mathrm{2}}{T}^{\mathrm{2}}+\mathrm{4}{l}^{\mathrm{2}}{T}^{\mathrm{2}}+\mathrm{4}crwT,

b=w^{\mathrm{2}}{r}^{\mathrm{2}}+\mathrm{4}{c}^{\mathrm{2}}{T}^{\mathrm{2}}+\mathrm{4}{l}^{\mathrm{2}}{T}^{\mathrm{2}}-\mathrm{4}crwT\mathrm{.}

Considering the computed T force from the above equations, the stiffness of each degree of freedom can be determined by Eqs. ‎(2) and ‎(3).

Jayaraman element

In this method, the cable stiffness matrix and nodal forces are determined by assuming 2-D elastic behavior for the cable. By utilizing the transformation matrices, these values are then calculated in 3-D space. Based on Fig. 3, the relationships for 2-D space can be obtained from the following equations:

(6)H=(x_j-x_i)\sin \beta +(y_j-y_i)\cos \beta ,

(7)V=z_j-z_i,

(8)\sin \beta =x_j-x/{[{(x_j-x_i)}^{\mathrm{2}}-{(y_j-y_i)}^{\mathrm{2}}]}^{\raisebox{1ex}{$\mathrm{1}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{2}$}\right.},\cos \beta ={(\mathrm{1}-{\sin }^{\mathrm{2}}\beta )}^{\raisebox{1ex}{$\mathrm{1}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{2}$}\right.}

(9)F_{\mathrm{1}}=-{\displaystyle \frac{qH}{\mathrm{2}\lambda }},

(10)F_{\mathrm{2}}={\displaystyle \frac{q}{\mathrm{2}}}\left[-V{\displaystyle \frac{\cos \mathrm{h}\lambda }{\sin \mathrm{h}\lambda }}+s_{\mathrm{0}}\right]\mathrm{.}

As illustrated in Fig. 3, H and V indicate the horizontal and vertical distance between the two end joints of the cable in the prior step, and F_i shows the nodal forces for the first step. If H= 0 or the slack length of the cable (s₀) is less than (H² + V²)^1/2, then l is equal to 0.2. Otherwise, the value of l can be approximated from the following relation:

(11)\lambda ={\left[\mathrm{3}\left({\displaystyle \frac{s_{\mathrm{0}}^{\mathrm{2}}-V^{\mathrm{2}}}{H^{\mathrm{2}}}}-\mathrm{1}\right)\right]}^{\mathrm{1}/\mathrm{2}}\mathrm{.}

Furthermore, the distances and the nodal forces in the subsequent steps are obtained as follows:

(12){\left\{\begin{array}{l}{F}_{\mathrm{1}}\\ F_{\mathrm{2}}\end{array}\right\}}^{i+\mathrm{1}}=K^i{\left\{\begin{array}{l}\delta H\\ \delta V\end{array}\right\}}^i,

(13)F_{\mathrm{3}}=-F_{\mathrm{1}},

(14)F_{\mathrm{4}}=-F_{\mathrm{2}}+q{s}_{\mathrm{0}},

(15)HAX=-F_{\mathrm{1}}\left[{\displaystyle \frac{s_{\mathrm{0}}}{EA}}+{\displaystyle \frac{\mathrm{1}}{q}}\ln \left({\displaystyle \frac{F_{\mathrm{4}}+T_j}{F_{\mathrm{2}}-T_i}}\right)\right],

(16)VAX={\displaystyle \frac{(T_j^{\mathrm{2}}-T_i^{\mathrm{2}})}{\mathrm{2}EAq}}+{\displaystyle \frac{T_j-T_i}{q}},

where HAX and VAX are, respectively, the same horizontal and vertical distance between two end joints of the element in the subsequent step. By obtaining the value of nodal forces in each step, the tensile forces of the nodes have the following values:

(17)T_i={(F_{\mathrm{1}}^{\mathrm{2}}+F_{\mathrm{2}}^{\mathrm{2}})}^{\mathrm{1}/\mathrm{2}},

(18)T_j={(F_{\mathrm{3}}^{\mathrm{2}}+F_{\mathrm{4}}^{\mathrm{2}})}^{\mathrm{1}/\mathrm{2}},

where T_i and T_j are the tensile force of the ith and jth nodes, respectively. In addition, the stiffness matrix is obtained as follows:

(19)[\mathit{K}]=\mathit{F}{\mathit{E}}^{-\mathrm{1}},

[\mathit{F}\mathit{E}]=\left[\begin{array}{ll}f{e}_{\mathrm{11}}& f{e}_{\mathrm{12}}\\ f{e}_{\mathrm{21}}& f{e}_{\mathrm{22}}\end{array}\right],

f{e}_{\mathrm{11}}={\displaystyle \frac{H}{F_{\mathrm{1}}}}+{\displaystyle \frac{\mathrm{1}}{q}}\left[{\displaystyle \frac{F_{\mathrm{4}}}{T_j}}+{\displaystyle \frac{F_{\mathrm{2}}}{T_i}}\right],f{e}_{\mathrm{12}}=f{e}_{\mathrm{21}}={\displaystyle \frac{F_{\mathrm{1}}}{q}}\left[{\displaystyle \frac{\mathrm{1}}{T_j}}-{\displaystyle \frac{\mathrm{1}}{T_i}}\right],f{e}_{\mathrm{22}}=-{\displaystyle \frac{s_{\mathrm{0}}}{EA}}-{\displaystyle \frac{\mathrm{1}}{q}}\left[{\displaystyle \frac{F_{\mathrm{4}}}{T_j}}+{\displaystyle \frac{F_{\mathrm{2}}}{T_i}}\right],

where K and FE are the stiffness and flexibility matrices, respectively. If the values of factors \delta H=H-HAX and \delta V=V-VAX, which are shown in Fig. 3, are less than the preset tolerance, the stiffness matrix is computed in 3-D space [²⁹]. Otherwise, the processes resumes from Eq. (6).

Element of Thai and Kim [³⁶]

To accurately simulate realistic behavior of cable structures, the cable element is derived based on the exact analytical solution of the elastic catenary element. The basic assumptions behind this approach are perfect flexibility, uniformly distributed self-weight, and a constant cross-sectional area. The relationships of the method are expressed briefly as follows [³⁶].

Figure 4 shows the cable element suspended between two end joints, i and j, which have the Cartesian coordinates (0, 0, 0) and (l_x, l_y, l_z), respectively. Based on the assumptions of Thai et al., the projected lengths of the cable can be expressed in terms of the nodal forces as follows:

l_x=-{\displaystyle \frac{F_{\mathrm{1}}{s}_{\mathrm{0}}}{EA}}-{\displaystyle \frac{F_{\mathrm{1}}}{q}}\{\ln [\sqrt{F_{\mathrm{1}}^{\mathrm{2}}+F_{\mathrm{2}}^{\mathrm{2}}+{(q{s}_{\mathrm{0}}-F_{\mathrm{3}})}^{\mathrm{2}}}+q{s}_{\mathrm{0}}-F_{\mathrm{3}}]-\ln (\sqrt{F_{\mathrm{1}}^{\mathrm{2}}+F_{\mathrm{2}}^{\mathrm{2}}+F_{\mathrm{3}}^{\mathrm{2}}}-F_{\mathrm{3}})\},

(20)l_y=-{\displaystyle \frac{F_{\mathrm{2}}{s}_{\mathrm{0}}}{EA}}-{\displaystyle \frac{F_{\mathrm{2}}}{q}}\{\ln [\sqrt{F_{\mathrm{1}}^{\mathrm{2}}+F_{\mathrm{2}}^{\mathrm{2}}+{(q{s}_{\mathrm{0}}-F_{\mathrm{3}})}^{\mathrm{2}}}+q{s}_{\mathrm{0}}-F_{\mathrm{3}}]-\ln (\sqrt{F_{\mathrm{1}}^{\mathrm{2}}+F_{\mathrm{2}}^{\mathrm{2}}+F_{\mathrm{3}}^{\mathrm{2}}}-F_{\mathrm{3}})\},

l_z=-{\displaystyle \frac{F_{\mathrm{1}}{s}_{\mathrm{0}}}{EA}}+{\displaystyle \frac{q{s}_{\mathrm{0}}^{\mathrm{2}}}{\mathrm{2}EA}}+{\displaystyle \frac{\mathrm{1}}{q}}\{\sqrt{F_{\mathrm{1}}^{\mathrm{2}}+F_{\mathrm{2}}^{\mathrm{2}}+{(q{s}_{\mathrm{0}}-F_{\mathrm{3}})}^{\mathrm{2}}}-\sqrt{F_{\mathrm{1}}^{\mathrm{2}}+F_{\mathrm{2}}^{\mathrm{2}}+F_{\mathrm{3}}^{\mathrm{2}}}\}\mathrm{.}

The stiffness matrix and the internal forces of the element can be derived using an iterative procedure.

By differentiating both sides of Eq. ‎(20), the following flexibility matrix can be obtained:

f{e}_{\mathrm{11}}=-\left({\displaystyle \frac{s_{\mathrm{0}}}{EA}}+{\displaystyle \frac{\mathrm{1}}{q}}\ln \left({\displaystyle \frac{T_j+F_{\mathrm{6}}}{T_i-F_{\mathrm{3}}}}\right)\right)+{\displaystyle \frac{F_{\mathrm{1}}^{\mathrm{2}}}{q}}\left[{\displaystyle \frac{\mathrm{1}}{T_i(T_i-F_{\mathrm{3}})}}-{\displaystyle \frac{\mathrm{1}}{T_j(T_j+F_{\mathrm{6}})}}\right],

(21)f{e}_{\mathrm{12}}=f{e}_{\mathrm{21}}={\displaystyle \frac{F_{\mathrm{1}}{F}_{\mathrm{2}}}{q}}\left[{\displaystyle \frac{\mathrm{1}}{T_i(T_i-F_{\mathrm{3}})}}-{\displaystyle \frac{\mathrm{1}}{T_j(T_j+F_{\mathrm{6}})}}\right],

f{e}_{\mathrm{13}}=f{e}_{\mathrm{31}}={\displaystyle \frac{F_{\mathrm{1}}}{q}}\left[{\displaystyle \frac{\mathrm{1}}{T_j}}-{\displaystyle \frac{\mathrm{1}}{T_i}}\right],f{e}_{\mathrm{23}}=f{e}_{\mathrm{32}}={\displaystyle \frac{F_{\mathrm{2}}}{q}}\left[{\displaystyle \frac{\mathrm{1}}{T_j}}-{\displaystyle \frac{\mathrm{1}}{T_i}}\right],f{e}_{\mathrm{33}}=-{\displaystyle \frac{s_{\mathrm{0}}}{EA}}-{\displaystyle \frac{\mathrm{1}}{q}}\left[{\displaystyle \frac{F_{\mathrm{6}}}{T_j}}+{\displaystyle \frac{F_{\mathrm{3}}}{T_i}}\right]\mathrm{.}

The stiffness matrix is determined by computing the inverse of the flexibility matrix FE, as follows:

(22)\mathit{S}=\left[\begin{array}{ll}-\mathit{K}& \mathit{K}\\ \mathit{K}& -\mathit{K}\end{array}\right],

\mathit{K}=\mathit{F}{\mathit{E}}^{-\mathrm{1}},

where S is the stiffness matrix. Moreover, the relation between the nodal forces and the tension forces at nodes i and j have the following form:

F_{\mathrm{4}}=-F_{\mathrm{1}},

(23)F_{\mathrm{5}}=-F_{\mathrm{2}},

F_{\mathrm{6}}=-F_{\mathrm{3}}+q{s}_{\mathrm{0}},

T_i=\sqrt{F_{\mathrm{1}}^{\mathrm{2}}+F_{\mathrm{2}}^{\mathrm{2}}+F_{\mathrm{3}}^{\mathrm{2}}},

T_j=\sqrt{F_{\mathrm{4}}^{\mathrm{2}}+F_{\mathrm{5}}^{\mathrm{2}}+F_{\mathrm{6}}^{\mathrm{2}}}\mathrm{.}

Yang method

Yang et al. proposed a catenary two-node element for estimating the stiffness matrix of cables. This element is assumed to be perfectly flexible and linearly elastic, with the self-weight uniformly distributed along the length of the curve. Figure 5 shows the 2-D and 3-D views of this element. The basic relationships of this approach are given by Ref. [³⁴]

(24)l={\displaystyle \frac{F_{\mathrm{1}}{s}_{\mathrm{0}}}{EA}}+{\displaystyle \frac{F_{\mathrm{1}}}{q}}\left[{\sin \mathrm{h}}^{-\mathrm{1}}\left({\displaystyle \frac{F_{\mathrm{2}}}{F_{\mathrm{1}}}}\right)-{\sin \mathrm{h}}^{-\mathrm{1}}\left({\displaystyle \frac{F_{\mathrm{2}}-W}{F_{\mathrm{1}}}}\right)\right],

It is evident from Fig. 5 that l and h are the horizontal and vertical length of the element, respectively. In addition, the elements of the flexibility matrix for the 2-D mode are obtained as follows:

f{e}_{11}={\displaystyle \frac{s_{\mathrm{0}}}{EA}}+{\displaystyle \frac{\mathrm{1}}{q}}\left[{\sin \mathrm{h}}^{-\mathrm{1}}\left({\displaystyle \frac{F_{\mathrm{2}}}{F_{\mathrm{1}}}}\right)-{\sin \mathrm{h}}^{-\mathrm{1}}\left({\displaystyle \frac{F_{\mathrm{2}}-W}{F_{\mathrm{1}}}}\right)\right]+{\displaystyle \frac{\mathrm{1}}{q}}[-{\displaystyle \frac{F_{\mathrm{2}}}{\sqrt{{F_{\mathrm{1}}}^2-{F_2}^2}}}+{\displaystyle \frac{F_{\mathrm{2}}-W}{\sqrt{{F_{\mathrm{1}}}^2-{(F_2-W)}^2}}}],

(25)f{e}_{12}=f{e}_{21}={\displaystyle \frac{\mathrm{1}}{q}}\left[{\displaystyle \frac{F_1}{\sqrt{{F_{\mathrm{1}}}^2-{F_2}^2}}}-{\displaystyle \frac{F_1}{\sqrt{{F_{\mathrm{1}}}^2-{(F_2-W)}^2}}}\right],f{e}_{22}=\frac{s_0}{EA}+{\displaystyle \frac{\mathrm{1}}{q}}[{\displaystyle \frac{F_{\mathrm{2}}}{\sqrt{{F_{\mathrm{1}}}^2-{F_2}^2}}}-{\displaystyle \frac{F_{\mathrm{2}}-W}{\sqrt{{F_{\mathrm{1}}}^2-{(F_2-W)}^2}}}],

f{e}_{33}=\frac{S_0}{EA}={\displaystyle \frac{\mathrm{1}}{q}}[{\sin \mathrm{h}}^{-1}\left({\displaystyle \frac{F_2}{F_1}}\right)-{\sin \mathrm{h}}^{-1}\left({\displaystyle \frac{F_2-W}{F_1}}\right)],f_{13}=f_{31}=f_{23}=f_{32}=0.

After these calculations are made, the stiffness matrix of the cable element can be determined directly as the inverse of the flexibility matrix:

(26){\mathit{S}}_L=\left[\begin{array}{cc}-K& K\\ K& -K\end{array}\right],

K=F{E}^{-\mathrm{1}}={\left[\begin{array}{lll}f{e}_{\mathrm{11}}& f{e}_{\mathrm{12}}& \mathrm{0}\\ f{e}_{\mathrm{21}}& f{e}_{\mathrm{22}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& f{e}_{\mathrm{33}}\end{array}\right]}^{-\mathrm{1}},

where S_L is the local stiffness matrix. Therefore, the transformation matrix should be applied to update the stiffness matrix from the local coordinate system to the global one:

(27)T{R}_{\mathrm{6}\times \mathrm{6}}=\left[\begin{array}{ll}tr& \mathrm{0}\\ \mathrm{0}& tr\end{array}\right],

tr=\left[\begin{array}{lll}\cos \theta & \mathrm{0}& \sin \theta \\ \mathrm{0}& \mathrm{1}& \mathrm{0}\\ -\sin \theta & \mathrm{0}& \cos \theta \end{array}\right],

where TR is the transformation matrix. Moreover, the nodal and tensile forces of the two end nodes are determined as follows:

(28)T_i=\sqrt{F_{\mathrm{1}}^{\mathrm{2}}+{(F_{\mathrm{2}}-W)}^{\mathrm{2}}},T_j=\sqrt{F_{\mathrm{1}}^{\mathrm{2}}+F_{\mathrm{2}}^{\mathrm{2}}}\mathrm{.}

Dynamic relaxation method

The basic relationships of this process are as follows:

(29){\stackrel{.}{\mathit{X}}}_i^{n+\frac{\mathrm{1}}{\mathrm{2}}}={\displaystyle \frac{\mathrm{2}-{\mathit{C}}_{ii}^n{h}^n}{\mathrm{2}+{\mathit{C}}_{ii}^n{h}^n}}{\stackrel{.}{\mathit{X}}}_i^{n-\frac{\mathrm{1}}{\mathrm{2}}}-{\displaystyle \frac{\mathrm{2}{h}^n}{{\mathit{M}}_{ii}^n(\mathrm{2}+{\mathit{C}}_{ii}^n{h}^n)}}({\mathit{R}}_i^n),i=\mathrm{1},\mathrm{2},\mathrm{...},ndof

(30){\mathit{X}}_i^{n+\mathrm{1}}={\mathit{X}}_i^n+\mathrm{\Delta }{\mathit{X}}_i^{n+\mathrm{1}}={\mathit{X}}_i^n+h^{n+\mathrm{1}}{\stackrel{.}{\mathit{X}}}_i^{n+\frac{\mathrm{1}}{\mathrm{2}}},i=\mathrm{1},\mathrm{2},\mathrm{...},ndof

where \mathit{X} and \dot{\mathit{X}} are the displacement and velocity vectors, respectively; M, C, and h are the fictitious mass matrix, the damping matrix, and the time step, respectively; the subscript i is the node counter, and the superscript n is the iteration counter; and R is the residual force vector, which is calculated as

(31)\mathit{R}=\mathit{M}\stackrel{\mathrm{..}}{\mathit{X}}+\mathit{C}\stackrel{.}{\mathit{X}}=\mathit{P}-\mathit{F},

where \ddot{\mathit{X}}, P, and F are acceleration, external load, and internal force vectors, respectively. The residual force vector consists of the mass and damping forces, which are added to the static equation. Thus, the static solution is obtained by setting the residual force vector to zero. To achieve this goal, it is necessary to make appropriate choices for the artificial DRM factors.

According investigations in this work, the Zhang procedure is used to analyze the cable structure using the DRM. The proposed relationships of the Zhang method are presented as follows. Zhang and Yu suggested one of the simplest relationships for estimating the damping matrix [⁷]:

(32)\mathit{C}=\mathrm{2}\sqrt{{\displaystyle \frac{{\mathit{X}}^T\mathit{F}}{{\mathit{X}}^T\mathit{M}\mathit{X}}}}\mathit{M}

In addition, the mass matrix is calculated as follows:

(33){\mathit{M}}_{ii}={\displaystyle \frac{h^{\mathrm{2}}}{\mathrm{4}}}{\displaystyle \sum _{j=\mathrm{1}}^{ndof}\left|{\mathit{S}}_{ij}\right|},

where Mii is the ith element of the mass matrix. In Zhang’s procedure, the time step is equal to one, and the first displacement vector is considered to be zero.

Numerical example

To estimate the stiffness matrix of the cable, six different methods are briefly explained. In this section, 16 examples are analyzed by combining the aforementioned techniques and the DRM. These methods, which are used for the present analyses, are displayed in Table 1. Because Torkamani and Naserian proposed 2- and 3-D formulations of the same cable element, respectively, these two schemes are combined as one technique.

To compare the abilities of cable stiffness matrices using the DRM, the authors have written a program in the Fortran language. The structures are analyzed considering nonlinear geometric and linear elastic material behavior. For all cables, the loads are applied in 10 successive steps. The residual force error is assumed to be 10⁻⁴ for all examples. The static path in each example is drawn for a special node. To draw the charts more generally, the horizontal and vertical axes of the graph are demonstrated as dimensionless, by dividing the load (P) per rigidity (AE), and displacement (D) per the biggest length of the structure, respectively. The results of all six methods are shown for each numerical example. Based on the obtained results, the outcomes are compared with the number of iteration and analysis time criteria. After finding the solutions, based on the analysis time (E_t) and number of iterations (E_i), all procedures are graded. Finally, the rank of each technique is available for each example.

(34)E_t={\displaystyle \frac{t_{\max }-t}{t_{\max }-t_{\min }}},

(35)E_i={\displaystyle \frac{i_{\max }-i}{i_{\max }-i_{\min }}},

where t and i represent the analysis time, and the number of iterations, respectively. Since the number of iterations and the analysis duration of the T&N technique are not homologous to the other procedures, this method is ignored in the comparison studies. However, the related outputs are inserted in the tables.

Simple cable net

Figure 6 shows the geometry for this example. This net consists of 12 nodes and 12 members. The load values, P, which are applied on the intermediate nodes, are indicated in the figure. The pretension force is 23.687 kN in inclined elements, and 24.283 kN in horizontal ones. The values of the H and P/AE parameters are 91.44 m and 29 × 10⁻⁴. Figure 7 illustrates the static equilibrium path of the internal nodes. The analysis results are arranged in Table 2. According to the obtained outcomes, the DNG and TEL techniques provide the lowest analysis time and number of iterations, respectively. Since this structure is rather small, the total number of the iterations for the HTI and TEL procedures is very close to that of the DNG.

Hyper net

The initial configuration for this structure is presented in Fig. 8. With the original dimensions, this problem was considered by Lewis et al. [49]. The value of load P is 15.7 N. For all members, the cross-sectional area and the pretension force are 0.785 mm² and 200 N, respectively, and the dimensionless quantity P/AE is 16.03 × 10⁻⁵. Furthermore, the value of factor H is 2.5 m. Figure 9 shows the static equilibrium path of node D. The analysis results are given in Table 3. Based on the outcomes, the TEL method provides the least number of iterations and lowest response time. In addition, although the number of iterations for both the HTI and DNG procedures is nearly the same, the analysis time for the DNG technique is significantly less.

Inverted cable dome

Figure 10. Shows the initial geometry of the inverted dome. This structure has 121 nodes and 312 members. The cross-sectional area, pretension force, and self-weight of all elements are 146 mm², 1000 N, and 1.46 N/m, respectively. In addition, the load P of 35 kN is applied on the undermost level. The values of the factors H and P/AE are 34 m and 25 × 10⁻⁴. Figure 11 illustrates the static equilibrium path of the lowest node. The outcomes are given in Table 4. Based on these results, the TEL method has the best performance compared with the other procedures. It should be mentioned that the DNG technique failed to analyze this example. For the JYR, HTI, and YNG processes, although the number of iterations for achieving the answers are close to each other, they are different in terms of analysis duration.

Rectangular grid

Figure 12 and Fig. 13 clearly demonstrate 2-D and 3-D views of this structure, respectively. This grid consists of 121 nodes and 320 members. The cross-sectional area, pretension force, and weight per unit length of each cable element are 15.7 cm², 23.5 kN and 1.46 N/m, respectively. The intermediate node of line F is subjected to a concentrated load P, which is equal to 122.8 kN. In this problem, the value of H is 30 m and P/AE is 10 × 10⁻³. The structure geometry is symmetric with respect to axes F and N. Table 5 shows the z-coordinates for a quarter of the structure. It should be added that every border node coordinate is zero. To evaluate the effect of the boundary conditions, the structure is analyzed for three different boundary conditions. In the first one, all border nodes are located on the hinge support. If the intermediate nodes of the K axis are released, the second case is formed. Finally, the nodes of the L and M axes have a joint support. Figure 14 illustrates the static equilibrium path of node no. 61 for all three conditions.

• First case

As previously noted, in this case all border nodes are on the pinned supports. The outcomes are given in Table 6. Based on the results, the DNG and JYR methods yield the answers by the least number of iterations and lowest analysis time, respectively.

• Second case

In this circumstance, only the nodes on directions L, M, and N are fixed. Table 7 shows the required number of iterations and the analysis time of the procedures. It is apparent that the HTI method reduces the number of iterations compared with the other techniques. Furthermore, the JYR and YNG processes have the same required number of iterations. With respect to the analysis time, the HTI method is the fastest one. It is worth noting that although the DNG procedure is the best method in terms of both iteration and time for the first case, this scheme cannot converge to the answers in the second case.

• Third case

In this situation, all nodes except for those on the L and M axes are free. Table 8 displays the outcomes for this problem. According to the obtained results, the TEL method achieved the first rank in terms of both the iterations and analysis duration. It is notable that the HTI and DNG approaches, which obtained the best results in previous cases, cannot converge to the responses in this case. In general, it can be concluded that the superiority of a method for analyzing a structure with a special boundary condition cannot be extended to be the best for all other boundary conditions.

Spatial net

This problem was previously considered by other investigators, and it consists of 31 nodes and 38 members [³⁵,⁴⁵]. Figure 15 shows the 3-D view of the structure. The cross-sectional area and pretension force of members are 350 mm² and 90 kN in the x-direction, and 120 mm² and 30 kN in the y-direction, respectively. Moreover, the spatial net is subjected to a vertically-concentrated load of 6.8 kN at all internal nodes. The value of parameter H is 24 m, and P/AE is 12.1 × 10⁻⁵. Figure 16 displays the static equilibrium path of node D. The obtained outcomes are listed in Table 9. According to the results, all procedures can converge to the true response. Although the analysis time is very short for this numerical example, nevertheless, the TEL method solves it with the lowest time and iterations.

Saddle net

This structure with 95 nodes and 195 members is one of the most outstanding numerical examples, and its experimental results are available in Ref. [⁴⁹]. The cross-sectional area of the elements is 306 mm². The initial geometry of this structure is achieved by a pretension force of 60 kN for each member. As shown in Fig. 17, the configuration is symmetric with respect to the middle axis. Table 10 presents the z-coordinate for a quarter of the nodes. According to Fig. 17, a load of 1 kN is applied to half of the intermediate nodes, in both the x- and z-directions. Furthermore, the values of parameters H and P/AE are 50 m and 22.2 × 10⁻⁶, respectively. Figure 18 illustrates the static equilibrium path of node 11. Table 11 clearly displays the ranking and comparison of the methods. It is observed that the TEL technique gives the best results based on both criteria.

Simple cable

To check the validity of the computer program, this benchmark problem is solved. The cross-sectional area of the member in this example is 548.4 mm² [³⁶,⁴⁵]. The value of parameter H is 304.8 m, and P/AE is 49.5 × 10⁻⁵. Other geometric properties are illustrated in Fig. 19. The static equilibrium path of the point load is drawn in Fig. 20. Table 12 shows the analysis results of this problem for all methods. The outcomes indicate that the HTI procedure analyzed the structure with the lowest number of iterations, and that the HTI, JYR, and YNG techniques are the fastest methods. Because the results of the T&N method were not compared with the procedures for the other examples, no comparison was provided for this example.

Tetrahedral grid

Figure 12 shows the x-y view of the structure. Other geometric properties are demonstrated in Fig. 21, for the B, C, D, E, F, I, and J directions. The value of the concentrated load is 3.14 kN. The cross-sectional area of the cables is 157 mm², and their pretension force is 23.5 kN. The values of parameters H and P/AE are 30 m and 25 × 10⁻⁴, respectively. This example is analyzed for three boundary conditions. In the first case, only the internal nodes are free. The second case is formed by putting the nodes of the A, L, and M directions on the pinged supports. Finally, only the nodes of the L and M directions are fixed. Figure 22 illustrates the static equilibrium path of the center node.

• First case

As mentioned previously, in this situation, only the border nodes are fixed. Table 13 demonstrates the obtained results for this structure and a comparison of all methods. According to the outcomes, the TEL and DNG approaches cannot analyze this structure. It should be noted that the TEL scheme was the best procedure in most of the previous examples. For this problem, the minimum time and iterations are obtained by the JYR and TEL techniques, respectively.

• Second case

In this circumstance, the nodes of the A, L, and M directions are on the pinned supports. All the obtained outcomes are listed in Table 14. It can be seen that only the HTI method can analyze this structure.

• Third case

The third case is formed by setting the M and L directions on the hinged support. Based on the given results in Table 15, the TEL and DNG procedures cannot find the solutions. In performing numerical processes, the HTI technique reduces the number of iterations and analysis time significantly. Generally, it can be concluded that this scheme can solve a variety of structures with different boundary conditions.

Annular roof

In this section, a structure with 180 nodes and 396 cable elements is investigated. Figure 23 shows the geometric properties of this annular roof. The cross-sectional area and pretension force of all cables are 800 mm² and 1 kN, respectively. In this structure, all border nodes are fixed and the external load P acts on the all internal nodes. The values of parameters H and P/AE are 50 m and 35.25 × 10⁻⁵, respectively. The static equilibrium path of node D is displayed in Fig. 24. Table 16 presents the obtained results. For this example, the DNG method is unable to analyze this annular roof. Moreover, the HTI and TEL procedures converge to answers with the lowest iterations and fastest time, respectively.

Rhombus net

The initial geometry of this structure is achieved by the horizontal component of pretension force of 222.5 kN. Figure 25 shows this structure, which has been investigated by other researchers [²⁸]. The rhombus net consists of 41 nodes and 65 members, and is symmetric with respect to the middle axis. In Table 17, the z-coordinates for the numbered nodes are given. The cross-sectional area of the cables is 645 mm². Except for node 7, each joint is subjected to a vertical load of 4.45 kN. For node 7, the vertical load is 66.8 kN, and the horizontal load is 44.5 kN. The values of parameters H and P/AE are 97.6 m and 6.25 × 10⁻⁴. For node 7, Fig. 26 illustrates the static equilibrium path of the vertical displacement. The outcomes are displayed in Table 18. Based on the obtained results, and in terms of both iteration and time criteria, the TEL method is the best procedure for analyzing this example.

Cylindrical net

This structure consists of 143 nodes and 358 members. As shown in Fig. 27, the cylindrical net is symmetric with respect to the z-axis. All nodes on the edge borders are fixed. The cross-sectional area and pretension force of all elements are 550 mm² and 5 kN, respectively. This structure is analyzed under two types of loads. For the symmetric load, the central node of the net is subjected to a vertical load of 5 kN. For the asymmetric load, in addition to the vertical force, a horizontal force of the same size acts on the x-axis. The values of the H and P/AE parameters are 30 m and 7 × 10⁻⁷, respectively. Figure 28 illustrates the static equilibrium path of the central node. The results of the analysis are summarized as follows.

• Symmetric load

Based on the outcomes listed in Table 19, only the TEL, T&N, and HTI methods can solve this example. Furthermore, the HTI procedure analyzes the structure with the lowest iterations and time.

• Asymmetric load

For this case, the results of the analysis are given in Table 20. Once again, the JYR, DNG, and YNG techniques cannot converge to the true response. It should be noted that the HTI method significantly reduces the analysis time and the number of iterations.

Method ranking

For each numerical example, all well-known schemes for estimating the stiffness matrices of the cable element were compared. It is noteworthy that the T&N procedure was ignored in this comparison, because its outcomes were inconsistent with the other techniques. Based on their gained ranks in each example, these procedures can be rated generally. To accomplishing this goal, the number of times that method i gains the rank j, is denoted as G_ij. For example, the HTI procedure achieves first place eight times in terms of the number of iterations; therefore, the value of G_i1 is 8. In this rate, 5 points are given to the first place, and 1 point to the fifth rank. It should be added that the techniques which were not able to analyze and converge to the true responses are not rated. According to the mentioned scheme, the overall scores of the methods are calculated using the following relation:

(36)S_i=\mathrm{100}\times {\displaystyle \sum _{j=\mathrm{1}}^{\mathrm{5}}\frac{G_{ij}(\mathrm{6}-j)}{\mathrm{80}}},

where S_i is the rating of method i. If one approach could achieve the first rank in all examples, its score would be 100. The overall scores and ranking of the techniques based on the number of iterations and the analysis time are given in Table 21 and Table 22.

Conclusions

In this study, several stiffness matrices were presented for nonlinear analysis of the structural cables. These matrices were used by dynamic relaxation schemes. This investigation aimed to compare the ability and advantage of six different stiffness matrices of cable elements to determine the most efficient method of analysis. For this purpose, the DRM parameters (damping and mass matrices) were chosen from Zhang and Yu technique. By utilizing available and designed benchmark problems, 11 various cable structures were solved. To perform comprehensive investigations, 16 interesting numerical examples were presented. All of these cables had linear elastic material and nonlinear geometrical behavior. To compare the performance of these techniques, two criteria were used: the number of iterations and the analysis time to converge on the accurate responses. Because of the required number of iterations and the analysis time, the T&N procedure was incompatible with the other approaches, and its scoring and ranking were ignored. The results revealed that based on the number of iterations, the HTI and TEL methods achieved the first and second rank, respectively. However, based on the analysis time, the TEL and JYR procedures achieved the first and second rank, respectively. Because the presented cable structures were designed to be practical, significant, and general benchmark problems, future researchers can benefit from these outcomes in their studies.

It is worth emphasizing that all the formulations used in this study can be divided into two major categories, truss and parabolic elements. The TEL and T&N procedures are part of the truss element class, and the others are in the category of the parabolic cable element. It should be noted that even with the simplicity of the assumptions, which are used in the ordinary truss element, the obtained displacement responses of the TEL and T&N techniques have less than a 1% error compared with the other methods. Interestingly, this simple matrix reduced the analysis time for the TEL method. On the contrary, because the T&N procedure uses a higher-order stiffness matrix, its results are not consistent with the other techniques with regard to iteration and time criteria.

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