Nonlinear Static Analysis of Cable Net Structures by Using Newton-Raphson Method

Nonlinear Static Analysis of Cable Net Structures by Using Newton-Rahson Method Sayed Mahdi Hazheer Deartment of Civil Engineering University Selangor (UNISEL) Selangor, Malaysia hazheer.ma@gmail.com Abstract This study deals with the analysis of cable net structures under static loads. The analysis aroach is based on the minimization of the total otential energy of the cable net assembly, using Newton-Rahson method and the geometric nonlinear behavior of cable nets has taken into consideration. A MATLAB algorithm has been develoed to erform the analysis, the results are then verified with the work of other researcher. Lastly the effect of various arameters such as cable sacing, level of retension and variation of cross-section area of cables on the behavior of cable net structures have been studied. Keywords nonlinear behavior; cable net; analysis, total otential energy. I. INTRODUCTION The structures that either entirely of artially make use of cables as load carrying elements are defined as cable structures. These structures exists in a variety of forms such cable networks, cable reinforced membrane roofs, cable trusses, guyed towers and susension or cable stayed bridges. In general, cables transfers the load to other arts of the structure by going into tension. This is considered as one of advantages that cable structure have, such as in case of susension and cable stayed bridges in which large loads can be suorted in an economical and efficient way by use of members in tension [1]. Such efficiency is considered as one of the major factors that contributes to raid develoment of the cable structures in recent years. Comared to conventional structure, a cable structures under the alication of external loading, undergo large as oosed to small deformation before it reach the equilibrium osition, this behavior called geometrical non-linear behavior. Therefore the geometrical changes caused by large deformation must be taken into account in the analysis of such structures [2]. When the deformations are large, the equilibrium of nodes has to be considered with resect to the deformed geometry of the structures. Since this geometry is not known rior to the analysis, the general linear method of analysis is not alicable and leaving iterative techniques as the only viable aroach. Generally the cable structures can be modeled and analyzed by using two tyes of element i.e. bar element which is reresenting cable and analytical cable elements []. A. Bar Element (reresenting cable) This tye of element is similar to the conventional bar element, but with no flexural resistance. Therefore incaable of carrying comressive force. In contrast to conventional bar elements, bar element which reresenting cable undergo large deformation under the action of alied load, therefore this element ossess geometric stiffness in addition to elastic stiffness. The use of bar element in analysis of structure was common since invention of digital comuters in early 1960 s, which resulted in the develoment several numerical methods for linear analysis of structures. Among the various methods, the stiffness or dislacement method due its efficiency in comuter usage has been widely adoted. The method was originally develoed for analysis of linear structures and was not caable to analyze the structures undergo large as oosed to small dislacement. The modification of the original stiffness method enabled this technique for analysis of structures exhibits geometrically nonlinear behavior []. A full descrition of the derivation the stiffness matrix for structures undergo large deformation is available at [2], in which the global or tangent stiffness matrix is consist of elastic and geometric stiffness matrices. Beside the stiffness method, a more general finite element formulation is derived by Krishna [4]. Further Godbole et al. [5] obtained the tangent stiffness matrix for geometric nonlinear structures based on variational method of finite element. Another aroach in analysis of cable networks is obtained by minimizing the total otential energy of the structural assembly with resect to all dislacements. The equilibrium osition of the structure is reached when the total otential energy is minimum. Buchholdt [1] excessively investigated the nonlinear analysis of cable net structures as rocess of energy minimization and obtained a system of nonlinear equations by using the rincile of stationary otential energy. Buchholdt [1] suggested various iterative rocedures to obtain numerical solutions, i.e. the method of steeest descent, the method of conjugate gradients and the Newton-Rahson method. Alication of the Newton-Rahson method in solving the nonlinear system of equations led to the derivation of the tangent stiffness matrix, which is the summation of elastic and geometric stiffness matrices.the same aroach of H. A. Buchholdt [1] has been considered by G. D. Stefanou and S. E. Moossavi Nejad [6], the analysis is based on conjugate gradient method. A Cable structures with fixed boundaries has been analyzed and a scaling matrix has been introduced to increase the rate of convergence. G. D. Stefanou [7] included the

dynamic analysis as well, the dynamic forces due to gust and wind loads has been introduced. W. J. Lewis and M. S. Jones [8], investigated the nonlinear static analysis of retensioned cable networks using the minimum otential energy rincile. And the solution are obtained by dynamic relaxation method. Further Lewis [9], studied the efficiency of the stiffness matrix and dynamic relaxation method in analysis of retensioned cable nets and in-jointed frame structures. The studies carried on various cable net examles. W. J. Lewis [9] concluded that the stiffness matrix solution did not converge for the case of saddle shaed cable net. This is due to numerical illconditioning caused during matrix inversion and can be overcome by introducing a diagonal scaling matrix suggested by H. A. Buchholt [1]. B. Analytical Cable Element This tye of cables are the geometrically reresentations of a curve formed by freely hanging cables susended from its ends, which obeys Hooke s law and has negligible flexural resistance. These elements includes arabolic, elastic catenary and associated catenary cable elements. Thai and Kim [10], Andreu et al. [11] formulated elastic catenary cable elements for nonlinear analysis of cable structures. The tangent stiffness matrix and nodal forces of element which includes self-weight and retension force are formulated based on exact analytical exressions of elastic catenary. Thai and Kim [10] obtained the tangent stiffness matrix and internal force vector in exlicit form for static and dynamic loading. The material nonlinearity effect are consider as well at the two ends of the elements. The nonlinear static and dynamic equation of motions are solved based on Newton- Rahson and Newmark direction integration incrementaliterative rocedures. Salehi [12] roosed continuous and discrete catenary cable elements for analysis of threedimensional cable structures. The tangent stiffness matrices of the cable elements are obtained under various loading condition such as concentrated, distributed and thermal loads. At first the stiffness matrix and internal force vector of catenary cable is derived exlicitly and then by transforming the continuous equations into a discrete form, discrete cable element is introduced. In discrete modeling each cable member is divided into several straight axial cable elements, this allowing the concentrated loads to be alied along the element length. II. ANALYSIS OF CABLE NET STRUCTURE A. Total Potential Energy (TPE) of a Structure A unified aroach for both linear and nonlinear static analysis of structure is obtained by minimizing the total otential energy of the structural assembly with resect to all dislacements. The equilibrium osition of the structure is attained when the total otential energy is minimum [1]. By considering a two dimensional structure as shown in Figure 1, having only one free joint with two degree of freedom. The Total Potential Energy (TPE), W of the structure for different dislacements of the free joint can be drawn as a contour ma as shown in Figure 2, in which all oints on each of the contour line reresent dislacements of the free joint for which W is constant. The equilibrium condition in the ith direction of joint j can be exressed as W x ji Fig. 1. Two-dimensional retensioned cable system with one free joint [1]. Fig. 2. Contour lines of total otential energy surface for a structure, as shown above in Fig 1 [1] The minimum osition of the TPE, W is determined by moving down the TPE surface along a descent vector v with a distance S, referred to as ste length, until W is minimum at that direction where, dw ds From this oint a new descent vector is calculated and the rocess reeated until the equilibrium osition is reached. The minimization rocess is diagrammatically illustrated in Figure 2 for which the dislacement vector at the (k+1)th iteration mathematically can be exressed as x k+1 x k + S k V k

There are various methods for determination of the decent direction vector [v] in eqn. (.), along which the TPE can be minimized. The Most common methods used are: Newton- Rahson method, the method of conjugate gradient and the method of steeest descent [1]. The Newton-Rahson method requires more comuter storage, but has the advantage of faster convergence in terms of iterations and to a higher degree of accuracy as comared to other methods. B. The Gradient Vector of Total Potential Energy To obtain the gradient vector of TPE, The total otential energy (TPE) of a cable assembly at a given osition x in dislacement sace is given by Substituting equation (10) into equation (6) yields the following exression for the sth element in gradient vector. gs = n=1 (t jn (X ni + x ni X ji x ji )) F s Where, t jn = (T jn0 + EA L jn0 e jn ) L jn0 W U + V Where U is the strain or otential energy due to internal forces and V the otential energy due to alied force. If the unloaded structure shown in Figure (a) is taken as the datum, equation (4) may be written as W (U 0 + T 0 e + EA n=1 N n=1 F n x n 2L 0 e 2 )n Where U 0 is the initial strain energy due to retension, is the number of cable links, T 0 is the initial retension force, e is the elongation of cable link due to alied load, F n is the alied load vector, x n is the dislacement vector due to alied load, N is the total degree of freedom of all joints. Now the gradient vector of the TPE is obtained by differentiating equation (5) with resect to x s, the gradient vector g of the total otential energy for the sth element yields, (a) gs = (T 0 + EA e n n=1 F L 0 e)n x s s By observing the link jn in Figure (a) and (b) the exression for e n x s obtained as: 2 = (X ni X ji ) 2 i=1 L nj0 (L jn0 + e jn0 ) 2 = (X ni + x ni X ji x ji ) 2 i=1 Where L jn0 is the re-tensioned but unloaded length of link jn, and X are the joint co-ordinates in the X 1, X 2 and X axes. Subtracting equation (7) from equation (8) yields e jn 1 L i=1 jn0 ((X ni X ji )(x ni x ji ) + 1 2 (x ni x ji ) 2 ) Differentiating equation (9) with resect to x s, the dislacement in the ith direction at a joint j, since x s =x ji yields e jn x s = 1 L jn0 (X ni + x ni X ji x ji ) Fig.. Coordinates and force for link jn in a re-tensioned but not loaded cable assembly. (b) Coordinates and force for link jn in a re-tensioned and loaded cable assembly, at a stage of the iterative rocess where the extension of link jn is ejn.[1]. C. Minimization of the TPE To minimize the total otential energy W, we must move in the descent vector v k direction of the k th iteration and a distance S k, until the minimum value of W at that direction is reached. Now by substituting equation () into (5) the exression for total otential W = (U 0 + T 0 e + EA n=1 e 2 N 2L n=1 F n (x + 0 )n SV) n

In order to calculate S we must obtain an exression for W in terms of S. By substitution of equation () into (9) the exression for the elongation e n, is obtained as e n = e jn = 1 2L i=1 jn0 Sv ji ) + (x ni + Sv ni x ji Sv ji ) 2 ) Equation (14) can be written as; (2(X ni X ji )(x ni + Sv ni x ji e n = e jn = (a 1+a 2 S+a S 2 ) L jn0 dw/ds = 0. To locate the stationary oint along the decent direction v k for which W is a minimum, equation (16) is differentiated with resect to S and ut equal to zero, which resulting a cubic equation as follows: 4C 4 S + C S 2 + 2C 2 S + C 1 = 0 Only the smallest ositive root of equation (18) corresonding to the nearest stationary oint along the descent direction is required, therefore the comlete solution of the equation is not necessary. This root can be obtained from Newton s aroximation formula, which can be written as Where, S i+1 = S i+1 dw/ds d 2 W/dS 2 a 1 = ((X ni X ji ) + 1 i=1 (x 2 ni x ji )) (x ni x ji ) a 2 = i=1 ((X ni X ji ) + (x ni x ji )) (v ni v ji ) a = 1 (v 2 ni v ji ) 2 i=1 Substituting the exression for e jn into equation (1) yields a olynomial of fourth order in stelength S for the total otential energy as following W = C 4 S 4 + C S + C 2 S 2 + C 1 S + C 0 Where, C 4 = ( EAa 2 n=1 2L 0 )n C = ( EAa 2a n=1 L 0 )n Or S i+1 = S i 4C 4S i +C S i 2 +2C2 S i +C 1 12C 4 S i 2 +6C S i +2C 2 E. The Newton-Rahson Method The descent direction vector v k can be found by various gradient methods in which the minimum total otential energy W can take lace along. However, to minimize the total otential energy based on Newton-Rahson method, let δx k reresent the change in dislacement vector at the end of k th iteration. The gradient vectors of TPE at x k and x k+1 dislacement resectively can be written as g k = δw/δx k g k+1 = δw/δx k+1 Exanding the gradient at x k in terms of a Taylor series yields W = W + ( W ) δx δx k+1 x k x k x k + 1 ( ( ( W ))) δx k 2! x k x k x k δx k + k C 2 = (t 0 a + EA(a 2 +2a 1 a ) n=1 2L 0 ) 1 ( ( ( W! x k x k x k x k ))) δx k δx k δx k C 1= (t 0 a 2 + EAa 1a 2 i=1 L n=1 F n v n 0 )n C 0= (U 0 + t 0 a 1 + EAa 1 2 2L N i=1 n=1 F n x n 0 )n The exression for tension coefficients for (k + 1)th iteration stage is obtained by calculating the gradient vector of total otential energy at x k+1 dislacement yields; N The otential energy of the loading is linear in x, thus ignoring the ignoring the cubic and higher order terms, equation (2) can be written as g k+1 = g k + ( 2 U x k 2) δx k By assuming g k+1 is equal to zero equation (24) can be reduced to t jn = t jn0 + EA L (a 1 + a 2 S + a S 2 ) jn jn0 g k = ( 2 U x k 2) δx k D. Determination of the Stelength The stationary oint in the direction v k at which W is a minimum is said to be the equilibrium osition of the assembly. Hence, the ste-length S can be found from the condition that Where the Hessian matrix 2 U/ x k 2 is recognized as the stiffness matrix at oint x k at dislacement sace. Therefore the change in the dislacement vector at the at the end of kth iteration given by

δx k = K k 1 g k Where δx k is considered as a descent vector along which the minimization takes lace Finally, by referring to equation (26) the tangent stiffness matrix for retensioned link at the kth iteration is given by [1] K k = EA T k L 0 [ G T kg k T G k G k T G k G k T G k G ] + T k [ I I ] L k 0 I I Where I is a unit matrix of dimension ( ) and G k = 1 L 0 [X + x k Y + y k Z + z k ] T F. Iterative Procedures To minimize the total otential energy in order to obtain the equilibrium osition the dislacement vector equation () is solved iteratively, and it is necessary to determine the stelength S the direction vector [v] at each iteration Before the iteration began calculate the following a) The tension coefficients from equation (12). b) The retensioned length of the cable elements using equation (7) c) Determination of the gradient vector of TPE equation from equation (11). d) Calculate the Euclidean norm of the gradient vector. e) Calculate the elemental and global stiffness matrix from equation (27) and hence the descent vector using equation (26). f) Calculate the coefficients (a) and (c) from equations (15) and (16). g) Calculation of the otimum ste-length from equation (20). h) Udate the tension coefficients using equation (17) i) Udate the dislacements vector from equation (). j) Reeat of the iteration rocess by returning to ste (c) until the gradient vectors are zero or negligible. k) Calculation of the new length and internal forces of cable elements. The flow chart diagram is shown in Fig. 4. III. RESULTS AND DISCUSSIONS A. Numerical Varifications The minimization of total otential energy based on Newton- Rahson method have been rogramed in MATLAB. The nonlinear static analysis of cable net structures has been carried out and the relative joint dislacements and tension forces of the cable net structures has evaluated. The flow chart of the roosed rogram is illustrated in Fig. 4. The accuracy and efficiency of the rogram has been checked on several examles of cable net structure. For the verification urose, all the results obtained from analysis are comared with the results obtained by Lewis [9]. By observing the results in TABLE I TABLE VI, it can be seen that the results obtained by the current study are very close to those obtained by Lewis [9], therefore the current method is suitable for the analysis of small and large cable net structures. 1) Examle 1: A hyerbolic araboloid net, shown in Fig. 5., is consists 1 cable segments and 12 free joints (6 degrees of freedom). The two sets of cables running in x and y- directions roduces a model of hyerbolic araboloid surface. The structure is subjected to a concentrate load of 15.7 N at all internal nodes, excet nodes 17, 2 1 and 22, the cross-section area of the cables are 0.785 mm 2 and the Young s modulus is 124800 N/mm 2. All the cables are re-stressed with 200 N force. This structure has been numerically analyzed by several other researchers, such as Thai and Kim [10], Salehi [12], Andreu [11] and the results obtained by them are in good argument with the results from Lewis [9]. The exerimental studies on this structure are reorted by Lewis [8]. The result from the analysis for dislacement and tension forces are summarized in TABLE I and TABLE II resectively and comared with the results obtained by Lewis. 2) Examle 2: A satial cable structure as shown in Fig. 6., has a lan dimension of 24 m 16 m comosed of 8 cable segments saced at 4 m 4 m grid. The structure is symmetrical about the centerlines in both x and y axes and the z-coordinates for a quarter of the structure are given in TABLE III. The initial geometry of the structure is achieved by means of the retension force of 90 kn in the x-direction and 0 kn in the y-direction. The structure is subjected to a vertically concentrated load of 6.8 kn at all internal nodes. The cross-sectional area of cables is 50 mm 2 in the x-direction and 120 mm 2 in the y-direction, and the elastic modulus of all cables is160 kn/mm 2. Thai and Kim [10], has also been analyzed this structure and comared the results with Lewis [9]. The comarison of nodal dislacements and tension forces are resented in TABLE III and IV. ) Examle : The saddle net shown in Fig. 7. which has 6 free joints (189 degrees of freedom) and a lan dimension of 40 m 50 m is consists of 142 cable segments saced at 5 m 5 m grid. The structure is symmetrical about the centerlines in both x and y axes and the z-coordinates for a quarter of the structure are given in TABLE V. The value of retension force for all the cable segments is 60 kn and cross-sectional area and elastic modulus of all cables are 06 mm 2 and 147kN/mm 2 resectively. The saddle net is subjected to the external loads of 1kN in the x- and z-directions at all the free nodes on one-half of the net. They are nodes 11-15, 22-26, -7, 44-48, 55-59, 66-70 and 77-81. This structure has also been analyzed by Thai and Kim [10] and Salehi [12]. The comarison of nodal dislacements and tension forces are resented in TABLE V and VI

Start Inut: elastic modulus, crosssection area of cables, retension force, osition of nodes, Force vector, node boundary condition For iter =1 to 100 iter==1 No Yes Calculate: initial length of cables eqn (7), tension coefficients eqn (12) Calculate gradient vector eqn (11) Calculate Euclidean norm, R1_1 R1_1<1e- Yes No Calculate: Elemental and Global stiffness matrices eqn (27), imose boundary condition, calculate decent vector eqn (26) Udate Coordinates Calculate New Length Calculate: coefficients (a) and (c) from eqn (15-16), determine stelength eqn (20) Udate tension coefficient (17) Calculate Tension Forces Outut: dislacement vector, tension forces Udate dislacement vector eqn () End Fig. 4. Flow chart diagram for the iterative rocedure

Fig. 5. Hyerbolic araboloid net TABLE I. COMPARISON OF DISPLACEMENTS OF HYPERBOLIC PARABOLOID NET. Node Lewis [9] Present no. x (mm) y (mm) z (mm) x (mm) y (mm) z (mm) 5-1.01-0.95-19.9 0.6 0.40-19.5 6-2. -1.02-25.4 1.22 0.74-25.5 7-2.8-0.78-22.96 1.96 0.82-2.04 10-1.15-2.22-25.58 0.66 1.1-25.60 11-2.77-2.56 -.80 1.9 2.09 -.62 12 -.5-1.99-29. 2.71 2.0-29.08 15-1.04 -.1-25.42 0.77 2.17-25.6 16-2.45 -.45-1.10 1.90 2.95-0.6 17-2.48-2.4-21.27 1.90 2.02-20.8 20-0.8 -.2-21.15 0.70 2.55-20.57 21-1.6-2.9-19.77 1.08 2.4-18.79 22-1.7-2.06-14.27 1.19 1.76-1.41 TABLE II. COMPARISON OF TENSION TORCES OF HYPERBOLIC PARABOLOID NET. Lewis result Present result Cable link Tension (kn) Tension (kn) 1-5 0.2282 0.221 2-6 0.2497 0.2500-7 0.2400 0.254 4-5 0.244 0.245 9-10 0.2766 0.272 14-15 0.2667 0.2599 19-20 0.297 0.218

Fig. 6. Satial net TABLE III. COMPARISON OF DISPLACEMENTS OF SPATIAL NET. Node Lewis [9] Present z-coord No. x (mm) y (mm) z (mm) x (mm) y (mm) z (mm) 1-1000.0 - - - - - - 2-2000.0 - - - - - - -000.0 - - - - - - 6 0.0 - - - - - - 7-819.5 0.42-5.14-0.41 0.40-5.0-29.45 8-1409.6 0.47-2.26-17.7 0.9-2.2-17.10 9-1676.9-2.27 0.00.62-2.6 0.00.20 1 0.0 - - - - - - 14-687.0 0-4.98-4.49 0.00-4.92-42.8 15-1147.8 0-2.55-44.47 0.00-2.55-44.24 16-117.6 0 0.00-41.65 0.00 0.00-42.06 TABLE IV. COMPARISON OF TENSION FORCES OF SPATIAL NET Lewis result Present result Cable link Tension (kn) Tension (kn) 1-7 25.989 26.0677 7-14 25.9866 26.0710 2-8 19.8156 20.157 8-15 19.7211 20.058-9 2.8401 24.870 9-16 22.5546 2.0760 6-7 104.869 105.0174 7-8 10.1241 10.754 8-9 102.2507 102.8761 1-14 125.804 126.551 14-15 124.5757 125.1256 15-16 12.8600 124.4052

Fig. 7. Saddle net TABLE V. COMPARISON OF DISPLACEMENTS OF SADDLE NET Node no z-coord Lewis [9] Present x (mm) y (mm) z (mm) x (mm) y (mm) z (mm) 1 168.0 2 240.0 192.0 4 648.0 5 800.0 10 0.0 11 101.6-4.49 15.51-81.95-4.44 15.52-81.44 12 184.0-5.62 11.4-61.55-5.48 11.48-60.87 1 2407.1-4.28 7.0 -.52-4.05 7. -2.51 14 2751.0 -.21 5.28-17.97 -.04 5.2-17.8 15 2865.6-2.9 4.09-11.2-2.52 4.09-9.82 21 0.0 - - - - - - 22 792.0 -.55 14.40-97.28 -.51 14.46-97.0 2 1408.0-4.52 11.22-7.02-4.40 11.4-72.77 24 1848.0 -.04 7.19-1.9-2.85 7.5-1.64 25 2112.0-2.19 5.6-10.2-1.40 5.5-4.7 26 2200.0-0.67 4.76 11.7-0.42 4.76 11.69 2 0.0 - - - - - - 649.1-1.72 11.69-92.44-1.70 11.77-92.98 4 1154.0-2.14 9.51-66.85-2.08 9.60-67.75 5 1514.6-1.19 6.25-19.81-1.10 6.4-21.14 6 171.0-0.26 4.88-14.7-0.10 4.97 12.48 7 180.1-0.49 4.64-46.55 0.60 4.67 4.67 4 0.0 44 60.0 0.00 10.82-90.92 45 1072.0 0.00 8.98-66.0 46 1407.0 0.00 6.01-18.50 47 1608.0 0.00 4.77 16.92 48 1675.0 0.00 4.58 40.91

TABLE VI. COMPARISON OF TENSION FORCES OF SADDLE NET Lewis result Present result Cable link Tension (kn) Tension (kn) 1-11 75.5151 75.5221 2-12 79.2964 79.28-1 70.22 70.2881 4-14 61.956 60.949 5-15 54.1575 54.198 10-11 55.102 54.8725 21-22 60.79 60.4575 2-65.5596 65.085 Fig. 9. x Cable net with cable sacing of 0. m 4-44 67.1878 66.118 B. Parametric Studies Having verified the rogram, therefore, in this section it has been lanned to investigate the effect of various arameters such as cable sacing, level of retension and cross-section area of cables on the behavior of cable net structures. For the urose of arametric studies the hyothetical flat cable net shown in Fig. 8. is considered. The structure has 9 free joints (27 degrees of freedom) and a lan dimension of 2.4 m 2.4 m is consists of 24 cable segments and is symmetrical about the centerlines in both x and y axes and subjected external load of 15 N at all the internal joints. The cross-sectional area of the cables are 2 mm 2 and the Young s modulus is 24800 N/mm 2 and all cables are re-tensioned with 200 N force. 1) Cable Sacing: To investigate the effect of cable sacing on node dislacement, the structure shown in Fig. 8. initially has been analyzed on its original configuration. Then the sacing of cables in both X and Y directions has been modified as shown in Fig. 9 and 10, while the structure boundary dimensions remain fixed. The result of investigation resented in TABLE VII and Fig. 11. Column (a, b and c) reresent the dislacement of nodes for the cable net configuration shown in Fig. 8, 9 and Fig. 10 resectively. TABLE VII. Node Fig. 10. x Cable net with cable sacing of 0.9 m INFLUENCE OF CABLE SPACING ON THE DISPLACEMENT OF NODES Dislacement -z(mm) no. a b c 5 21.00 28.55 10.88 6 25.98 1.26 15.90 7 21.00 28.55 10.88 10 25.98 1.85 15.90 11 2.90 4.95 27.08 12 25.98 1.85 15.90 15 21.00 26.85 10.88 16 25.98 29.5 15.90 17 21.00 26.85 10.88 Fig. 8. x Cable net with cable sacing of 0.6 m From TABLE VII and Fig. 11 it can be seen that the sacing of cables does affect the behavior of cable net structures under the alied load, node dislacements are increased when the cable sacing is reduced and when the sacing of cable increased the corresonding node dislacements are reduced.

Pretension Force (N) Dislacement (mm) Pretension Force (N) 40.00 5.00 0.00 25.00 20.00 15.00 10.00 5.00 0.00 5 6 7 10 11 12 15 16 17 Node no a b c 650 600 550 500 450 400 50 00 250 200 150 00.0 400.0 500.0 600.0 700.0 Maximum Member Force (N) Fig. 11. Variation of node dislacement to different cable sacing 2) Level of Pretension: To study the influence of retension, the structure shown in Fig. 8 has been exosed to five different level of retension force, namely 200, 250, 00, 50, 400 and 600 N in both X and Y direction cables. The resonse of structure resect to maximum dislacement and maximum member force is determined and summarized in TABLE VIII and grahically resented in Fig. 12 and 1. TABLE VIII. Pretension force RESPONSE OF CABLE NET TO DIFFERENT LEVEL OF PRETENSION Maximum Tension Force Maximum Node dislacement Element number number N -z (mm) N 200 11 2.90 25.6 250 11 29.92 5.4 00 50 11 11 27.18 24.71 12-1 9-10 16-20 2-6 85.1 420.2 400 11 22.5 458.2 600 11 16.19 629.9 650 600 550 500 450 400 50 00 250 200 150 15.00 20.00 25.00 0.00 5.00 Maximum Dislacement (mm) Fig. 12. Influence of cable retension on maximum dislacement Fig. 1. Influence of cable retension on maximum member force Observing values in TABLE VIII, shows that the maximum dislacement occurred at node number 11 and further it is seen that while retension force increased from 200 to 00 (50%), dislacement reduces by 17.4%, and when the retension is increase by 100%, the reduction in maximum dislacement is 1.5%. By further increase of retension force to 200 % of its initial value, i.e. 600 N, the dislacement decreased by 50.8% (16.19 mm). For the case of member tension forces, when the retension force of cables increased by 50% the maximum tension in cables increased from 25.6 to 85.1 N i.e.18.%, and when the cross-section area increased by 100%, the maximum tension force increased to 458.2 N, which is 40.7% of its initial value. By further increase of retension force to 200 % of its initial value, i.e. 600 N, the maximum tension force increased by 9.5% i.e. 629.9 N. ) Cross-Section Area of Cables: To investigate the effect of variation in cross-section area of cables on the behavior of cable net structure shown in Fig. 8. The structure have been analyzed under different cross-section area of cables, such as 2, 2.5,,.5, 4 and 6 mm 2 while the retension force ket constant as 200 N. the results of maximum joint dislacement and maximum member forces are obtained and resented in TABLE IX and grahically resented in Fig. 14 and 15. TABLE IX. Cross-section area of cable RESPONSE OF CABLE NET TO DIFFERENT CROSS-SECTION AREA OF CABLES Maximum Tension Force Maximum Node dislacement Element number number mm^2 -z (mm) N 2 11 2.90 25.6 2.5 11 1.7 12-1 42.9 11 0.12 9-10 16-20 58..5 11 29.08 2-6 72.2 4 11 28.18 85.0 6 11 25.51 428.0

Cross-section area of cable (mm^2) Cross-section area of cable (mm^2) 7 6 5 4 2 1 0 25.00 27.00 29.00 1.00.00 5.00 Fig. 14. Influence of cable cross-section area on maximum dislacement 7 6 5 4 2 Maximum dislacemet (mm) 1 20.0 70.0 420.0 470.0 Maximum member force (N) Fig. 15. Influence of cable cross-section area on maximum member force From the table it can be seen that by increase of crosssection area of cables the dislacement of nodes are reduced and the tension forces in cables increased. For instance when the cross section area increased from 2 mm 2 to mm 2 (50%), the dislacement reduced from 2.9 mm to 0.12 mm which is about 8.4 % and when area increased to 4 mm (100%), the dislacement is reduced to 28.18 mm which is 14.5 % decrease comared to its initial value. By further increasing of crosssection area of cable to 200 % of its initial value, i.e. 6 mm 2, the dislacement decreased by 22.5% (25.51 mm). For the case of member tension forces, when the crosssection area of cables increased by 50% the maximum tension in cables increased from 25.6 to 58. N i.e.10%, and when the cross-section area increased by 100%, the maximum tension force increased to 85 N, which is 18% of its initial value. By further increasing of cross-section area of cable to 200 %, the tension forces increased by 1.4% (428 N). Fig. 12-1 and 14-15, shows the influence of retention forces and cross-section areas of cables on maximum dislacement and maximum member forces resectively. By observing these figures, it s clear that the dislacement reduces and member forces increases non-linearly by increasing the retension force and cross-section area of cables. The nonlinear change of dislacement and member forces is the indication of non-linear behavior of cable net structures. Further it can be seen that the effect of cross section and retension forces on maximum dislacements are redominant in the zone of small values of areas of cross- section and retension forces. IV. CONCLUSIONS & RECOMMENDATIONS A. Conclusions From the results of this research can be concluded that the behavior of cable net structures does influence by cable sacing, level of retension force and the variation of the crosssection areas of cables. The joint dislacement are reduced when the cables sacing are increased and the joint deflection increased by reducing the cable sacing. The increase of retension forces reduce the joint dislacements significantly. However its effect is redominant to smaller values as comared to larger values of retension force. The tension forces in the cable elements generally increase with of retension. Increasing the cross-section areas of cables reduce the joint dislacements but the effect is not areciable comared to increase of retension force The reduction in deflection values are more for smaller values of cross-section area, however it s not economical to reduce the deflection this way as the cable self-weight increases and hence require more retension as well. Change in cross-sectional area of cables does not affect the tension forces in cables significantly. B. Recommendations This study is focusing on only the static analysis of cable net structures, the future work can include the dynamic analysis. There are various factors affecting the behavior of cable nets such as temerature changes and the cable net suorting boundaries, these factors can be considered in the future studies. Besides the current method, there are various methods available for the analysis of cable net structures such as Finite element method, dynamic relaxation method and stiffness method, a study can be carried out in the efficiency of these numerical methods for the analysis of cable nets. REFERENCES [1] H. A. Buchholdt, An Introduction to Cable Roof Structures, 2nd ed., Thomas Telford, UK, 1999. [2] W. J. Lewis, Tension Structure form and Behavior, 1st ed., Thomas Telford, UK, 200. [] K. A. Kashani, Develoment of Cable Elements and Their Alications in the Analysis of Cable Structures, Phd thesis, University of Manchester, UK, 198. [4] P. Krishna and P. N. Godbole., Cable-Susended Roofs, 2nd ed., McGraw Hill Education Private Limited, India, 201. [5] P.N. Godbole., Boundary Effects in Susended Cable Roofs, ASCE Journal of Structural Engineering, Vol. 110 (5), May, 1984,. 1099-111. [6] G. D. Stefanout and S. E. Moossavi Nejad., A General Method for the Analysis of Cable Assemblies with Fixed and Flexible Elastic Boundaries, Comuters & Structures, Vol. 55 (5), 1995,. 897-905.

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