Kablolu taşıyıcı sistemlerin nonlineer statik analizi için bir yöntem

Abstract

Sunulan bu çalışmada, öngergilendirilmiş uzay kablolu taşıyıcı sistemin, etkiyen dış yükler altındaki nonlineer statik cevabını incelemek amacıyla sayısal bir hesap yöntemi geliştirilmiştir. Dış yüklerin sisteme düğüm noktalarından etkitildiği bu çözüm yönteminde, düğüm noktasına ait denge denklemleri dinamik esaslar doğrultusunda ifade edilmiştir, öngergilendirilmiş kablolu taşıyıcı sistemin her hangi bir düğüm noktasındaki iç ve dış kuvvetler arasında meydana gelen farkın oluşturduğu artık kuvvetlerin, sistemde meydana geldiği varsayılan sönümlü titreşimden doğan kütle atalet -kuvvetleri ve sö nüm kuvvetleri toplamınca karşılandığı kabulüne dayanan Dinamik Rölaksasyon Yaklaşımı kullanılarak statik dengeye, sözü edilen dinamik denge denklemlerinin Sabit İvme İntegrasyon Tekniği ile nümerik in- tegrasyonları ile ulaşılmıştır. Yürütülen sayısal uygulamalara da yanılarak elde edilen sonuçların gerek deney ölçümlerine, gerekse farklı integrasyon teknikleri ile ulaşılan çözümlere olan uyumlulu ğu gösterilmiştir. Rijitlik matrisinin kullanılmaması nedeniyle geliştirilen bil gisayar algoritmasına ait bellek gereksiniminin minimize edildiği bu çalışmada, çözümün stabil olarak en optimum bilgisayar süresinde gerçekleştirilebilmesi için noktasal kütlenin seçimine bağlı olarak kritik sönüm katsayısının ve integrasyon adımının maksimum değeri nin saptanması gerekmektedir. Bu nedenle gözönüne alınan dört fark lı sınır şartlarına sahip öngergilendirilmiş hiperbol ik-paraboloid kablolu taşıyıcı sistemler için geniş çaplı bir parametrik çalışma ya başvurulmuş ve çeşitli dış yüklemeler ve dış yük miktarları göz önüne alınarak sistemin açıklığı, kabloların düzenleme sıklığı, yüzey eğriliği, kablo enkesiti ve elastisite modülü gibi yapısal parametrelerin çözüme ve çözüm parametrelerine etkisi incelenmiştir. Bu parametrik çalışmaya dayanılarak geliştirilen bilgisayar algoritması ile hakim periyot saptanmış ve Newmark tarafından tanımlanan yakınsama oranı ifadesi yardımıyla, dört farklı hiperbolik parabo loid kablolu taşıyıcı sistem için farklı karakter gösteren yakınsa ma eğrileri formüle edilmiştir, önerilen formüller yardımıyla, kri tik sönüm katsayısına karşı gelen maksimum integrasyon adımının tesbiti bir otomatiğe bağlanarak, çözümün zaman açısından da opti mizasyonu sağlanmıştır. Dinamik Rölaksasyon'a dayanılarak hazırlanan genel bilgisayar algoritmasına, optimum çözüm parametrelerinin tesbitini içeren ön programcığın ilavesi ile nonlineer statik analiz, minimum bellek gereksinimi ile en optimum çözüm süresinde gerçekleştirilmiş ve yü rütülen sayısal örneklerle önerilen çözüm tekniğinin uyumluluğu ve etkinliği gösterilmiştir.

Nowadays, due to their economy and suitability to cover large areas at relative low cost, pretensioned cable.net structures shown in Figure 1-2 are increasingly used. Because of significant geometric changes under external loadings, these structures have nonlinear load deformation characteristics. For this reason, the usual assumption of infinitesimal deformation leads to erroneous results when applied to these structures. "Thus, thâre has been considerable research [4-41], [51-70] into the area concerning nonlinear static analysis methods suitable to this kind nf structures. Each of these methods takes into account the deformed geometry of the cables under the external static load and employs iterative solution procedures such as Newton-Raphson of Modified Newton-Raphson etc. In these methods the structure is assumed to be loaded and the internal forces are present in it from the beginning of the calculations. Thus unknown forces are then found by forming and storing the overall stiffness matrix, then assembling and solving a large amount of simultaneous linear equations for each iteration step. Therefore these methods require considerable large amount of computer memory. In this study an efficient and simple numerical solution procedure is proposed which is based on "dynamic relaxation method" and conveniently applicable to the nonlinear analysis of aforementioned cable net structures. Proposed method is conceptually more powerful than the aforementioned conventional iterative methods. The procedure deals directly with nonlinear ity and accomodates the problem of slack tensile cables without forming and storing the overall stiffness matrix of the system. The method is essentially based on a step-by-step integration of the dynamic equilibrium equation which corresponds to the critically damped forced vibrations of the system under the external time dependent loads as is seen in Fig. 1-6. Two principal parameters control the convergency and the efficiency of the nonlinear static analysis. These are the damping coefficient and the time increment selected. Proposed study deals directly with these parameters and gives an algorithm for automatic evaluation of the critical damping value and refined expressions for the most appropriate integration time steps using Newmark's relation between period and convergence rate about the constant acceleration integration method. XX In order to demonstrate the efficiency and accuracy of the proposed method, various numerical results have been produced and compared with the others obtained for the same structures employing the existing conventional nonlinear solution methods and experimental procedures. MATHEMATICAL FORMULATION Figure 3-2 represents a typical node point and geometry of a cable net before and after applying the external loads P, P and r-r- j a xq yq P. The nodes p and q deflects through u, v, w and u, v zq ^ n &p'p'p q q and w respectively, along the x, y and z-axes. The initial coordinates of the nodes p and q z respectively. It is assumed that are x, y, z and x, y and p yp' p q Jq - the cables are straight between two nodes, - cable elements are connected by frictionless pins, - flexural and shearing capacities of cables are neglected, - external loads are applied at the nodes only. Depending on these descriptions with reference to Fig. 3-2, static equilibrium equation for this node point q will be given as in Eq. 1 xq F. (x + u q q v = R xq (la) 'yq zq F. - - (y + v - y - v ) *q q *p p s. 1 = R. yq F. 1 (z + w q q p p = R zq (lb) (lc) in which F. is the internal member force, R, R and R are 1 xq yq ^q the nodal residual forces. L. is the current length of the member and is computed as follows 3i=[(Xq + Uq 2 2 x - u ) + (y + v - y_ - v ) + p p Jq q Jp p + (zq + Wq " Zp " V 1/2 (2) XXI Eq. 1 can be expressed in matrix form as is given below, P - F = R (3) /¦V /v ^ in which P represents external load vector, F represents displacement dependent internal force vector aricf R represents current nodal residual force vector. ~ For the non-zero residual force vector, corresponding to the equilibrium equation of the static configuration of the system can be regarded as a consequence of a previous dynamic state. Employing the d'Alembert equation of dynamic equilibrium, residual force vector can easily be stated as is given below M.Ü + C.Û = R (4) in which U, y are the joint acceleration and velocity vector of the structure?' M is the concentrated mass vector selected and C is the damping coefficient vector which is chosen to be critical, so that the oscilattions will die out in the quickest possible time and displacement function converges to the static deflection value. If the equilibrium position is reached then the residual force vector R,, nodal acceleration vector Ü and nodal velocity vector y will be equal to zero. By that time nodal deformation vector U will correspond to the real static deflection vector under the external loads considered. CHOICE OF DYNAMIC RELAXATION PARAMETERS The majority of numerical procedures depending on the dynamic relaxation principals employs the central finite difference formulation in the time-wise integration of Eq.4 which is only conditionally stable [34]. The present study is based on an explicit integration of the equation given in Eq.4 employing the constant acceleration integration procedure [43]. As mentioned before, two principal parameters control the convercency and efficiency of proposed procedure. In order to describe their effect, a large amount of parametrical study has been performed. If the damping coefficient can be selected equal to the critical damping value, the vibration of the system will die out rapidly within a certain time interval as is shown in Fig. 4-5. It is clearly seen from this figure, for each curve representing a certain mass value, there is only one minimum point independent of the time increment selected as is shown in Figs. 5-3, 5-14, 5-26 and 5-37. XXII In Figures 5-7 and 5-15 the relation between the critical damping value and maximum allowable time increment is plotted. In paranthesis iterations required to obtain the static solution are given. For each selected critical damping value, employing the corresponding maximum allowable time increment, approximately same amount of iteration cycle number is needed until reaching the static equilibrium. From Equations 3 and 4 it is obvious that if there's no nonlinearity, Equation 4 represents free vibration under the external loads. Depending on this fact, for the lightly nonlinear cable net systems, one needs to know the natural frequency of the cable net to obtain the critical damping value. However, for moderately nonlinear systems, it is not possible to get most appropriate damping value in this way (Fig. 6-2). After several numerical applications it is shown, that the best way to reach this critical damping value, is to run the system without damping coefficient until 1 finding out the first peak point of the motion defined by Eq. 4. Defining the period of the vibration in that way it is possible to get the natural frequency w and C critical damping value of each node point as well, as is seen in Fig. 6-5. In order to figure out which frequency values obtained will represent the actual dominant frequency of the nonlinear undamped motion defined by Eq. 4, various applications have been performed and plotted (Figs.. 6-3 to 4). In Fig. 6-5 each node point is shown in brackets and the iterations corresponding to the selected damping value and required to obtain the static solution are given in paranthesis. It is clearly seen from this figure that, if the damping values corresponding only to the node points which are loaded by the external concentrated loads are used in the static solution procedure proposed in this work, then lesser iteration cycle is needed for the stable static solution than that required for the other damping coefficient values. Depending on this fact and trying to find out the first peak point of the nonlinear undamped motion of loaded node points represented by Eq.4, it is possible to get the most suitable period and frequency for the proposed solution procedure, as is given below : 1 = 4^ (5) 6i = 2tt/T (6) The appropriate critical damping value C = 2.m.» (7) cr works better in the solution procedure proposed herein. XXIII The selection of the tism increment is very important from the computational viewpoint. Its value must be as large as possible without loosing numerical stability if the computer time economy is aimed. The majority of numerical procedures depending on dynamic relax ation principals employs the central, forward or backward difference integrations to solve Eq.4, which is conditionally stable [33], [35], [36], [37]. In this proposed procedure "constant acceleration method" is tak en into account which is known as stable for all periods and time steps [48]. However, since the accuracy of results depends on the integration tine step selected, for this problem the equation of the motion behaves not well, if this time step exceeds a certain value. In his well known paper "A Method of Computation for Structural Dynamics" Newmark expressed convergence rate p as a ratio between the error in derived and assumed acceleration as is* seen in Eq.8 error in derived acceleration,0v p = - (a) error in assumed acceleration and presented the following relation between the time increment At and period T of their motion as is given below At 9* V 4p (9) 2t Considering this relation and employing a trial and error method, numerous numerical applications have been performed for four typical hyperbolic -paraboloid cable nets as is given in Figures 1-7 a to d, in order to obtain the most appropriate relation between the values of convergence rate and period. A) TYPE I (Figure l-7a) In Fig. 6-8 the relation between the period of the undamped mo tion defined by Eq. 4 and convergence rate is shown for five sag- span ratios as is seen in Table 5-1 considering a constant span and cable spacing. The values represent the applied loads in kN. After a certain value of applied external loads an instability problem oc curs and convergence rate values fall down leaving the typical re sponce curve. In general, an increase of the applied load value leads to lower periods but higher convergence rates. Increasing the roof curvature, the curves are moving right down by increasing their periods, but decreasing their convergence rates. Same response can easily be seen in Fig. 6-9 if the span is held larger than the previous one while the cable spacing is kept constant. If the cable spacing becomes larger, then the curves are moving upright with an increase in their period and convergence rate as well, as is seen in Fig. 6-11. The change in cross sectional area and modulus of elasticity has a great influence on these curves, too. If their value is increased, their period and convergence rate are decreas ing with moving the aforementioned curve down right as is shown in Figs. 6-12 to 13. Depending on these findings, the XXIV approximate relation between convergence rate p and period T is ex pressed with Eq. 10, by employing k = 1,05 x 10~2 1/s2 for symmetrical and k = 0,75 x 10" L 1/s2 for antimetrical and â concentrated loads. P =k (l-ez) (1+ç ) a AE -T/4.5T 14, 2T /T 36 e ° + e ° (10) 9 = f/L (11) ç = 1/L (12) m : nodal concentrated mass (kg) I : cable spacing (m) L : cable span (m) f : dip of cable (m) E : modulus of elasticity (N/cm ) A : cross-sectional area of cable (cm ) Tq: normalization constant ( = Is) The approximation of this relation is demonstrated in Figs.6-29a to A. By employing Eq.10 it is possible to calculate the most suit able time increment from Eq. 9. As mentioned before, after applying a certain value of external concentrated load, called maximum allowable external load, an insta bility problem occurs. In order to give a refined expression for it, symmetrical loading case is taken into account and the influence of the structural properties on it is shown in Figs. 6- 30a to e. As a result of these findings maximum allowable external load value which can be applied to the system within a single step without causing instability in the solution can be defined as follows by employing k = 1,05 x 10a ; ke = 1,5 cm2 and k = 4 N/cm2. p r e E -1/lOç 1 1-8 pmav " kJK + A) (k + ) e ( ) (13) max p f e 1Q 7 ç The approximation of this formula can be seen in Tables 6-la to b in the third column; next column represents actual external load values calculated parametrically. The comparison of both values as relative percentages can be seen on the last column. E) TYPE II (Figure l-7b) Figure 6-14a represents the relation between the period and con vergence rate for five sag-span ratios. An increase of the applied load values gives lower periods but higher convergence rates. Increasing the roof curvature, the curves are moving right down, re sulting in longer periods but lower convergence rates. It is clearly seen from Fig. 6-14c, that with the decreasing values of the span length without a change in the cable spacing, the periods tend to become shorter, while the convergence rate values become higher. But if the cable spacing becomes larger, the curves are moving left upright with a decrease in their period and an increase in their convergence rate, as is seen in Fig. 6-15. The response curves are not affected by the change of cable cross sectional areas. However the increasing values of the modulus of elasticity causes steeper decrement in these curves. XXV Depending on these findings the approximate relation between convergence rate and period is expressed by employing kb = 2,25 x 10~2 i/s2 as is given below m. I - (0,5+6 ) (1+AJ) ± A^E (1+6J) P=kb -(1+AJ) j^ (İ4Ç) -T/b? b T0/T" 80e l + & (14) X =YAk (15) b1 = 2,985 (l+e^2) (16) b2 = 12,85/ (1+e^) (17) where A. and A^ represent cross sectional areas of interior and edge cables respectively. The curves representing this relation are dem onstrated in Figs. 6-32a to I. From the evaluation of these curves, it is possible to notice, how close they represent the real response characteristiscs. In Figure 6-33a to e the effect of structural properties on the maximum allowable external load values which can be applied to the system within a single step without causing instabil ity in the solution, is demonstrated. As a result, maximum allowable external concentrated load value is given below by employing k = 1,632 x 10^N and kQ = 108 N/cm2. p, 1 -1/105 1 1-6 1 1-8 P"_^k(l +- ) (1+x )Q+ )e ( - ) (1+ - ) (18) max p ke 100 ç 5 105 In Tables 6- 2a to b the values obtained from the above expres sion is given in the 3rd column and the real values calculated para metrically are seen in the next column. Last column represents the comparison of both values as relative percentages. C) TYPE III (Figure l-7c) In Figure 6-17 the relation between the period of the motion defined by Eq.4 and convergence rate is shown for five sag-span ra tios. The values represent the applied loads in kN, those in boxes, the so called critical external load values, correspond to the occur rence of slack tensile cables. If the roof curvature is small, in this case, any increment in the loading values reduces the periods, but increases the convergence rates. On the other hand, when the loading case has the same charac teristic as mentioned before, the change in load values does not affect the period as demonstrated in Figs. 6-17 to 18, if the roof curvature is quite steep. After applying a certain value of it the period slightly leads to an increase in its value. The response curves corresponding to the greater roof curvatures are present on the left hand side of the aforementioned figure. If the span is held larger than the previous one, the curves are moving right down increas ing their periods, but decreasing their convergence rates as is seen in Fig. 6-1 7b. If the cable spacing becomes large without a change of its current span, the curves are tending to move right with a decrease of their periods, but an increase of their convergence rates, as is shown in Fig. 6-19. XXVI The change of cross-sectional area and modulus of elasticity has a great influence on periods. An increase of these parameters leads to a decrement of their periods as demonstrated in Figs. 6. 20 to 21. Depending on these findings the relation between period and convergence rate is given as follows, enveloping only the convergence rates corresponding to the critical external concentrated load values, by employing k =1,9 1/s^ m £ -T/2,88T p = k (l-8Z)e (19) c AE (1+e. 0 Various approximations of Eq.20 are demonstrated in Figs. 6-35 a to I. The effect of the structural parameters on the critical external concentrated load values, representing the symmetrical loading case and corresponding to the occurrence of slack tensile cables, can be seen in Fig. 6-36. The expression defining the influence of these characteristics is given below by employing k = 5,2 x 10 N; kf = 10 cm2 and ke = 10 b oN/cm2. cr 1-e3 A E P""^ Kr 5 (10 ) (1 ) (20) cr cr 1-9 k, k f e D) TYPE IV (Figure l-7d) As mentioned for Type III, same response can be clearly seen between the periods and convergence rates for different spans and five sag-span ratios shown in Figures 6-23a-b and for various cable spacings without a change of the span as is seen in Fig. 6-24. If the values of modulus of elasticity become larger, then the periods and convergence rates get smaller as is seen in Fig. 6-25. An increase of cross sectional area of interior cables without a change in the cross-sections of edge cables causes lower periods, but higher convergence rates as is shown in Fig. 6-26. Due to this fact convergence rate is expressed as follows by employing k, = 7,5 I/S2 m. a (1+9 )., T/, Q " p = k,. (1+A3) e"T/1'9'To (21) Aj^.E (1+9. Ç) The approximation of it is demonstrated in Figs. 6-38a to I. Critical external concentrated load value for symmetrical loading case is expressed as a result of the,findings as seen in ^ Figs. 6- 39a to e by employing k = 4,6 x 10 N and k = 10^ N/cm. LI c ? E P^^k (1+2,59) ç(l-2XZ) (1 ) (22) cr cr k e Calculated real values from the parametrical study can be seen in 4th column of the Tables 6-4a to b and the obtained values from the given expression above in the 3rd column. The comparison of both values is represented in the last column as a relative percent age for various applications. XXVII Considering the convergence rate expressions for each type of hyperbolic -paraboloid pretensioned cable nets it is possible to calcu late the most suitable tiine increment from the Eq. 9. PRINCIPAL FEATURES AND DESCRIPTION OF SUBROUTINES OF THE COMPUTER PROGRAM The main program reads the data such as structural and geometri cal properties, pretension and external load values, tolerances, con centrated mass value etc. Then the optimization subprogram OPTIMA is called in order to find out the most suitable solution parameters. Following this, main program calls the subroutine NONSTA for nonlinear static analysis. SUBROUTINE OPTIMA 1- Begins with the choice of damping coefficient value equal to zero. Obtains the time increment from Eqs. 5-9 to 12 for selected type of cable net. Performs numerically integration of Eq.4. 2- Evaluates the dynamic response obtained in step 1 in order to find out the first peak point of the motion corresponding to Eq.4 for the loaded node points and controls whether the velocity due to it is zero. 3- From the time corresponding to this peak point calculates period, frequency and critical damping value from Eqs. 5, 6 and 7 respec tively. 4- Employing the Eqs. 10, 14, 19 and 2l obtains convergence rate for the given type of cable net. 5- Employing this calculated rate in Eq.9 obtains the maximum allow able time increment for the integration. SUBROUTINE NONSTA 6- Using these economical solution parameters, begins to integrate numerically the dynamic equations employing constant acceleration iteration procedure. 7- For time t = 0 sets the initial conditions as follows; U(t) and U(t) are equal to zero 8- Acceleration is obtained from the residual force. R = P-F(t) (23) Û'(t + At7 = Tr - C U (t)l / M (24) 9- Computes the displacement vector at time t+ At as, U(t+At) = U(t) + U(t)ût + [~Ü(t+At) + Ü(t)l At2/4 (25) 10- (Computes current displacement dependent force vector F(t + At) and length from Eq.l and 2 employing the displacement'yvalues obtained in the previous step. 11- Computes acceleration vector at time t + At as, Ü(t+ At) =fp - C.Û (t+ At)- F(t+At)l /M (26) XXVIII 12- Computes velocity vector at time t+ At as U(t+At) = U(t) + U(t+At) + U(t) r-J -1 At/2 (27) 13- If the dynamic equilibrium is reached then checks for the static equilibrium. 14- If the accuracy is not achieved then considers a new time incre ment and returns to step 9 till the members of residual force vector R, are close enough to zero so that the static equilibrium is satisfied. NUMERICAL EXAMPLES The computer program prepared for the numerical procedures pro posed in this study was applied to the hyperbolic-paraboloid cable nets shown in Figs. 8-1 to 2, tested experimentally by Jensen [3] and by Lewis [35] respectively and in Fig. 8-3 designed and computed by Krishna [6] using Newton-Raphson method. The results are summarized in Tables 8-3 to 9. The optimum results for various numerical examples are compared with the solution previously used by Uzgider [44] employing conven tional Newton-Raphson iterative method and summarized in Tables 8-10 to 25. CONCLUSION The numerical method proposed in this study is proved to be more efficient and simple than those existing in literature. Namely the memory requirement is reduced to %53 and CPU (Central Processing Unit) time to 7o63 of that required when Newton-Raphson method [44J is employed.