Non Linear Dynamic Analysis of Cable Structures

Transcription

1 DINAME Proceedings of the XVII International Symposium on Dynamic Problems of Mechanics V. Steffen, Jr; D.A. Rade; W.M. Bessa (Editors), ABCM, Natal, RN, Brazil, February 22-27, 2015 Non Linear Dynamic Analysis of Cable Structures Wagner de Cerqueira Leite 1, Reyolando M.L.R.F. Brazil 1 1 Universidade Federal do ABC - UFABC, wagner.leite@ufabc.edu.br, reyolando.brasil@ufabc.edu.br Abstract: A time domain analysis using a vector approach is performed for a non-linear cable structure under the action of wind forces. These forces are divided in two parts, one concerning the average wind velocity, considered permanent, and another related to the fluctuation of the velocity about its average value, of random nature. The program developed for this paper solves the equations of motion through the Central Difference Method. Displacements and axial forces for permanent loading are obtained using Dynamic Relaxation. Dynamic wind loadings, of random nature, are simulated by applying harmonic loads on the structure with randomly set phase angles and amplitudes corresponding to spectral density function bands. The computer processing will result series of displacements and cable tensions for determining the characteristic design values using a probability density function. Keywords: cable structures, nonlinear dynamics, random vibrations, geometric nonlinearities INTRODUCTION Cable elements combine small self-weight with high stiffness to tension loads. These characteristics make them interesting for application in various types of structures, such as the cable network analyzed in this paper, cable-stayed bridges, antennas, and others. In a cable network which constitutes the structure of a roof that covers a large area, as the one that will be analyzed in the present paper, the displacements produced by loadings are of such magnitude that the geometric nonlinearity must be considered. The loads on this roof are the self-weight of the structure and the pressures resulting from the incident wind. Such pressures can be divided into two parts: a parcel of average wind, admitted as part of permanent loading, and another parcel of floating pressures about its mean value. The effects produced by the floating wind portion on this structure are significant, from which it is mandatory its consideration, which must include its random variation in time. In this paper, a program for the analysis of cable structure employing the Central Difference Method for solving the equations of motion is used. Elements Definition and Equations of Motion In the analysis of cables structures, one can adopt either a matrix or a vector approach. In this paper, we adopt a vector approach. We considered the ends of cable elements as nodes n i of the structure,, where NN is the number of nodes of the structure. The dynamic loads are applied at these nodes as concentrated forces with components referred to a right-handed axes coordinate system, and the masses and the damping as nodal lumped masses and dampers. Knowing the coordinates for each node on the system the lengths of the cable elements are defined from the coordinates of its initial and final nodes. For a cable element, where NE is number of cable elements, with initial and final nodes e, the length is given by Eq. (1): (1) e 3 e 2 j NF e 1 L j, A j, E j NI Figure 1- The cable element

2 Non Linear Dynamic Analysis Of Cable Structures L j, A j and E j are, respectively, length, sectional area and the Young s modulus of the material of a cable j. With this, the direction cosines of the cable axis, related to the axes system, are determined. It is admitted that the stress-strain diagram of Fig. 2 is valid, where is the yield stress: Figure 2- Stress-strain relationship The software developed for this paper considers the cable element as a bar element only when the element is tensioned. It compares, on each time step, the length of the cables with the original (unstressed) length. If the distance between ends of a cable is less than its original length, the program does not consider that cable in the structure (puts out) during the step. Defined the initial geometry of the structure to be analyzed, the displacements of the structural nodes are obtained at successive instants (steps). In this manner, series of displacements will be generated. For this, the Eq. (2) of motion is solved for each node at each time step. This equation is written for the nodes at the three orthogonal directions in the space,. Let be the displacement components for a node, referred on axes. Thus, and, where upper lines indicate derivation in time, are the velocity and acceleration components, are the nodal inertial forces, and are the nodal damping forces. The normal forces in the cable elements j that are hinged on the node are the restoring forces of Eq. (2), which may be written in referential axis directions indicated by a k index, where k is equal to 1, 2 or 3, in accordance with the axis ( ) to which k corresponds: The j index in Eq. (3), in turn, assumes values corresponding to the cables hinged on the node. The equilibrium equations thus determined, in number of 3NN, are uncoupled, since lumped nodal masses and dampers, and, were adopted. A numerical method to solve these equations is the Central Difference Method, as presented by Brasil (1997). Let us consider the two expansions of Eq. (4) in Taylor series at the neighborhood of a nodal displacement, where a displacement component on a referential axis direction, and in a given instant t, being supposed known: (2) (3) (4) Neglecting higher order terms, the subtraction between the two expansions (4) will result an approximation to nodal velocities: and the sum of expansions (4) results an approximation to accelerations: Rewriting the generic equilibrium equation (2) for a node in k direction and time t, where is the sum of force components in k direction from cables hinged at node : (5) (6) (7)

3 Wagner de Cerqueira Leite, Reyolando M. L. R. F. Brasil and replacing Eq. (5) and Eq. (6) in equation (7), one arrives at the pseudo static equilibrium equation: (8) Equation (8) will be used to determine the displacements of the nodes at successive time steps that corresponds to successive time increments. Note that because the calculation process depends only on values of displacements available, obtained in previous steps, is an explicit method. But as it requires, it is not self-initiated. Initial Step (Start of the Calculation Process) To start the process, acceleration at the initial time is obtained from the equation of motion and initial conditions,,. Then, by supposing constant acceleration ( ) in the first interval, it results: (10) Updating of geometry and forces to the next step With the displacements thus obtained, the new coordinates for the nodes are calculated in a given time and, therefore, the new lengths of the elements are determined with Eq.1, from which one obtains the length variations of the cables,. Also new direction cosines for the axes of the elements are calculated. It is admitted that normal forces in compressed cables are negligible. Considering this, cables in compression at one given step of calculation must not be considered in the structure, making its normal forces equal to zero during the step. To detect a cable in compression, its length, the distance between cable ends, is compared with the length of this cable when its normal force is equal to zero (untensioned length ). The following algorithm gives an estimate of the normal force at the time : If the j element is a cable, and is less than or equal to the cable unstressed length, then the normal force in the cable sections,, is zero. otherwise: and then: The program developed accepts bar elements in the structure. Normal compression forces in bar elements, when it occurs, may be significant. Thus, if an element is not a cable, but a bar, the following condition must be added to the algorithm: (only for a bar element) If the j element is a cable element and, then the length is updated for the next iterations: Projecting the normal forces applied to node n i in the direction k through its direction cosines, we obtain and proceed to a new iteration, writing the equilibrium equation for time. Static Displacements. Dynamic Relaxation. The following procedure is based on an analogy between static and dynamic structural analysis. The static analysis of structures of nonlinear behavior in general is iterative. From the nonlinear equations of nodal equilibrium, successive calculations of the displacements of the nodes are made until its convergence to the equilibrium position. Here, in a dynamic analysis, the structure, considered to be critically damped, is submitted to an instant loading until displacements converge to the end position of static equilibrium. This convergence is checked by observing the time after which increasing of the displacement becomes very small and can be neglected. (11) (12) (13)

4 Non Linear Dynamic Analysis Of Cable Structures To determine the critical damping of the structure, we need to determine its circular natural frequency of vibration. Hence, the critical nodal damping C ci on a node n i is: An estimate of the natural frequency may be made by an analysis of the structure in time domain through the application of a load from a given instant whose intensity is held constant over a time interval, being this load removed in an instant later and then allowing the free vibration of the structure, while the nodal damping is made equal to zero. The resulting time history is then typical of a periodic function with constant amplitude. Plotting this curve, one can estimate the period T through measurement made directly on the curve. The cyclic frequency, f, is and the circular frequency is Substituting in Eq. (14), one gets the critical damping C c. Next, displacements in time for the structure are computed, with estimated nodal critical damping and permanent loads. Plotting this time history it is possible to verify if the critical damping value is correct. (14) (15) (16) Displacements Underdamped Static equilibrium critical damping overdamped Figure 3: Static analysis via dynamic relaxation Time If the estimated damping value is slightly below the real critical damping (underdamped structure), displacements will oscillate, tending to the equilibrium position. If the value of the damping is above the real (overdamped structure), convergence occurs without exceeding the position of static equilibrium. In both cases, however, the convergence will occur on a longer time than that for critical damping. Cable Roof Submitted to Turbulent Wind Here, we apply the central difference method to the analysis of a hyperbolic paraboloid shaped cable network which constitutes the structure of a roof, subjected to wind forces. Figure 4 corresponds to that presented by Paik (1975), where the lengths are in feet, as in the original reference. Figure 4: The sample structure

5 The equation of the unloaded surface is: or: Self weight + covering: 5 psf = 240 N/m2 Prestressing on x direction: 60 k = N Prestressing on y direction: 60 k = N Initial nodal coordinates without loadings, and dead load displacements in feet Wagner de Cerqueira Leite, Reyolando M. L. R. F. Brasil Cables section area: 1.4 in 2 = m² (17) (18) Young modulus of cable material: ksi = N/m² The structure of Fig. 4 was the same presented by Esquilland and Saillard, (1963), and Paik (1975), and Table 1 and its results to dead load displacements for this structure are those presented originally by Paik (1975). Table 1-Dead load displacements according Paik (1975) We consider that to each node, from 1 to 25, it is applied the weight of an area whose horizontal plane projection is a 40 ft ( m) sides square. The paraboloid surface area is given by: That for the geometry given by Eq. (18) results: The term inside the integration may be simplified, if considered as a function of two variables developed in a Taylor series,. Considering terms of the series to the derivative of order 2, one obtains: Finally, by integrating between the limits x 1 and x 2, and y 1 and y 2 one has the contribution area to a node: (19) (20) (21) Thus, on each node, self-weight is equal to the contribution area multiplied by the weight per unit area,, and the nodal mass, equal to the weight divided by the acceleration of gravity, is 3,632 kg. Self weight displacements To obtain these displacements via dynamic relaxation, first the cyclic frequency and critical damping must be estimated. Thus, we performed an analysis considering damping equal to zero. A plotting of displacements is given in

6 Displacement (m) Non Linear Dynamic Analysis Of Cable Structures Fig. 5, and it can be seen that the period T value is around 2s. Thus the cyclic frequency can be estimated by and so: The central difference method is conditionally stable. For convergence of the results it was necessary to use a very small time increment, seconds. The steps number used was equal to 10,000. Fortunately, when damping is added to the structure, convergence is possible for larger time increments, and thus, a smaller number of steps is needed (22) 1.00 Next, a new analysis was performed with the estimated critical damping. The examination of the displacements curves from this processing indicated a good convergence, as displayed in Fig. 6, for node 13: Displacement (m) Time (s) Figure 5 - Displacements at node 13, structure without damping Time (s) Figure 6 - Displacements at node 13, structure with critical damping Obtained self-weight displacements were very close to those of Paik (1975) shown in Tab. 1. Wind forces We determine wind forces from the wind pressures, that are proportional to the square of the wind velocities modified by drag coefficients C p. These coefficients C p are the result of testing a model of the structure in a wind tunnel. We used the coefficients C p given by Esquilland and Saillard, (1963) and that was used, too, by Paik (1975). According the NBR6123 Brazilian standard Forças Devidas Ao Vento Nas Edificações (Wind Forces on Buildings), for a V k characteristic wind speed, we get a q k characteristic wind pressure given by: Thus, the pressure over a structural point is given by: The pressure p on a structural node will be multiplied by its contributing area to result the wind force on the node. The wind velocity, and so its pressure and force, is composed of two parcels, one being a time average velocity or pressure, and the other is a floating contribution about the average (wind gust). It will be considered that the average parcel renders a permanent load, which will be applied to the structure with its self-weight to determine the initial geometry, while the floating parcel gives the dynamic force. The structure analyzed in this paper will be submitted to a wind force of maximum intensity. This force corresponds to a peak wind velocity, or pressure peak, which is believed to occur at a given instant. The design standards, however, only provide average wind velocities in time intervals. Thus, it was decided to adopt as a peak (23) (24)

7 Wagner de Cerqueira Leite, Reyolando M. L. R. F. Brasil velocity the average for the shortest time interval of NBR6123 standard, which is 3 seconds, V 3. This peak speed V 3 includes the mentioned average and the floating parcels (gusts). It will be assumed that the peak average velocity is the standard average for a much longer time interval, adopted equal to 10 minutes or 600s, which will be denoted V 600. Having the ratio between the average and the peak velocities, one has also the ratio between the floating and the peak parcels. The NBR6123 standard provides, for flat terrain, group II construction, category IV terrain, a ratio between the velocities V 600 and V 3 equals to: from where: In the same conditions, V 3, the peak velocity, on a point at z meters height above the ground is, according to NBR6123: Then, the peak pressure is: and, therefore, the average parcel of the peak pressure is: and the floating parcel is: To have the correspondent forces on the nodes, it is necessary to multiply these pressures by the node contribution area A = 148,69 m 2, resulting to the average parcel of wind: and the floating parcel or gust: The drag coefficients C p for this structure, obtained for different wind directions from wind tunnel tests, are given by Esquilland and Saillard (1963), and used by Paik(1975). In a real structural design, it will be necessary to consider all these coefficients in the structural analysis. It will be considered in this paper, for simplicity, the coefficients in just one of these directions. Table 2 - Drag coefficients according Esquilland and Saillard, 1963 Node C p (25) (26) (27) (28) (29) (30) (31) (32) Node C p Random Nature of Wind Simulation Forces that simulate the random nature of real forces (simulated sample function) are constituted by the superposition of harmonic functions, with each harmonic function corresponding to a frequency band of a spectral density function concerning the real forces. These forces thus defined are applied directly to the nodes of the structure on the vector approach adopted in this paper. The Power Spectral Density function adopted is the reduced spectrum of the National Building Code of Canada, which is a slightly modified power spectrum of Davenport to the wind speed. is the average wind speed at a height z = 10 m in the open. This Power Spectral Density function of wind velocity can be used to determine the amplitudes of the harmonics instead of the spectral density function of the wind pressure on a point of the structure, given that the two functions differ only by a constant, in which it is built in the drag coefficient of the structure to this point. (33)

8 Non Linear Dynamic Analysis Of Cable Structures Considering that the floating parcel of peak pressure is equals to a sample function constituted of an infinite number of harmonic functions, it is possible to represent this variable parcel through the Fourier integral: where: (34) (35) The variance or mean square deviation of p (t), calculated over a sufficiently large time interval t, is equal to: (36) Making T tend to infinity, one can write: where S p (n) is the spectral density function of p (t), with S p (n) representing the contribution of a frequency interval dn for the average quadratic deviation. Instead of an infinite number of functions that can reproduce it perfectly, in this process p (t) is represented by a k finite number of harmonic functions with periods covering the range of duration of gusts, from 0.5s to 600s. According Franco (1993) one of these functions, the rth, must have the resonance period T of the structure. The other functions periods must be multiples or submultiples of T by powers of two, so that in a logarithmic scale, it results bands of equal length of spectrum for each function, as shown Fig 7. (37) Figure 7 - Spectrum band division The equation, then, becomes: (38) where m is the number of harmonic functions and bands of the spectrum, and: C k 2 S n dn k (39) The C k value is calculated by integrating numerically within the band (interval) of the spectrum corresponding to it. Taking p as the maximum amplitude of the variable parcel of the pressure, given by Eq. 30, to each one of its harmonic components in the sample function corresponds a maximum pressure amplitude: ' C k pk p' m ck p' (41) C k 1 k Values for phase angles for each of the harmonic functions that constitute the sample function are taken from a series of randomly generated numbers. Thus, the harmonic functions that comprise variable wind pressure will overlap randomly, in accordance with the phase angle combinations. (40)

9 ˆf nk y y Wagner de Cerqueira Leite, Reyolando M. L. R. F. Brasil This combination of harmonic functions is a simulation of a possible actual sample originated from wind measurements. Such a combination of harmonic functions is called "synthetic wind." In this manner, the synthetic wind is given by: Changing the phase angles by values of another set of random numbers in a harmonic functions combination, we get another synthetic wind, simulating another sample function. In this paper, 20 sets of k=12 generated random number ( phase angles) were adopted, and so 20 different computer analysis were performed, each one to one different synthetic wind, resulting wind displacements and normal forces in the cables of the structure. These synthetic winds must be applied to the most unfavorable point of the structure, where they will induce the largest displacements or stresses. This point is named gust center. At other points in structure, the wind will consist of harmonics with amplitudes correlated to the center gust amplitudes through a narrowband correlation coefficient Coh(r,n k ) which is a function of the distance between the gust center and the considered point, and of n k frequency relative to the harmonic or its related band. This will result in smaller amplitudes for these points. ˆf Coh r, n e (43) with: 2 z 2 z 1 V z y 2 1 C C where C z = 7 to 10, and C y = 12 to 16. According Franco (1993), the lower values C z = 7 and C y = 12 can be adopted on the safety side. The coefficient Coh(r,n k ) tends to zero with the distance r increasing. Here, in a simplified way, node 13, which suffered the largest static displacement, was chosen as the gust center. For actual design purposes, a more comprehensive study should be conducted, considering several cases, each considering the gust center taken as a different node of the structure. The program implemented in this study allows not only the generation of internal forces and displacements time histories for each sample function (synthetic wind), but also records the maximum values resulting from action of a given sample function (a given combination of phase angles in harmonic functions). The maximum stresses and displacements obtained from the application of different synthetic winds are then adjusted to a maximums probability distribution, the Gumbel distribution: where: and: is the mean, and the standard deviation. k Table 3 shows the maximum displacements determined for node 13, and the maximum tensions for the cable element (20) from the processing: Table 3-Maximum displacements for node 13 and tensions for element (20) from processing (42) (44) (45) Comb Displ. (m) Tension (N) Comb s Displ. (m) Tension (N) Hence, for the displacements: (46)

10 Non Linear Dynamic Analysis Of Cable Structures and the characteristic value for design is the one with 5% probability of be exceeded, that is x = 1.773m For tensions: (47) and among the combinations, the one whose displacement is the closest to the characteristic value of the displacements is combination 15. Regarding tensions, the combination is the 19. For this reason, it is said that the characteristic wind for the displacement is the synthetic wind that corresponds to combination 12. For tensions in the element 20, the characteristic wind corresponds to combination 19. CONCLUSION The analysis of a cable structure of non-linear behavior subjected to wind loads of random nature has been presented. To perform this analysis, a program using the Central Difference Method to solve the equations of motion was developed. A method for considering the random nature of wind is proposed. Time displacements series and time forces series were generated for a sample structure. Characteristic values to be used in the design of the structure were obtained. With the increasing use of cable structures, it is intended with the submitted paper contribute to the analysis procedures of such structural systems. ACKNOWLEDGMENTS The authors acknowledge support by CAPES and CNPq, both Brazilian research-funding agencies. REFERENCES Buchholdt, H.A.,1999, Introduction To Cable Roof Structures, Cambridge University Press, Second Edition. Brasil, R.M.L.R., 1997, Integração Explícita Aplicada à Análise Dinâmica de Treliças Espaciais de Comportamento Elasto-Plástico, Boletim Técnico da Escola Politécnica da USP, BT/PEF/9701, São Paulo. Esquillan, N. ; Saillard, Y., 1963, Hanging Roofs, Boletim Técnico da Escola Politécnica da USP, BT/PEF/9701, São Paulo. Franco, M., 1993, Direct Along-Wind Analysis Of Tall Structures, Boletim Técnico da Escola Politécnica da USP, BT/PEF/9303, São Paulo. Geschwinder, L.F. ; West, H.H., 1980, Forced Vibrations Of Cable Networks, Journal of the Structural Division, Proceedings of The American Society of Civil Engineers. Vol. 106, ST 9, pp Lewis, W.J.; Jones, M.S., 1984, Dynamic Relaxation Analysis Of The Non - Linear Response Of Pretensioned Cable Roofs, Computers & Structures, Vol. 18, No. 6, pp Newland.D.E., 1993, An Introduction to Random Vibrations, Spectral & Wavelet Analysis: Third Edition, Courier Dover Publications. Paik, J.K., 1975, Statistical Analysis Of Cable Nets Under Wind Loads, Ph.D. Thesis, New York University