NONLINEAR DYNAMIC RESPONSE AND DESIGN OF CABLE NETS
National Technical University of Athens School of Civil Engineering Department of Structural Engineering Laboratory of Metal Structures Doctoral Thesis NONLINEAR DYNAMIC RESPONSE AND DESIGN OF CABLE NETS by Isabella Vassilopoulou Structural Engineer at ODOTECHNIKI LTD Supervisor: Dr. Charis J. Gantes, Associate Professor NTUA Athens, November 2011 AbstractThe research presented in this thesis aims at investigating the response of cable nets subjected to dynamicloads, focusing on the dynamic phenomena that characterise nonlinear structures. First, a simple cable net isstudied, consisting of two crossing cables and the equation of motion is derived. Neglecting small terms ofits equation of motion, a simplified single-degree-of-freedom (SDOF) cable net is assumed, which is provedto be similar to a Duffing oscillator with a cubic nonlinear term of the displacement. The analytical solution ofits steady-state response, found in the literature, is adopted for this simple cable net and the occurrence offundamental and secondary resonances, such as superharmonic and subharmonic resonances, is verified forthis system. The response diagrams are plotted for different resonant conditions showing bending of theresponse curve, hardening behaviour and dependence on the initial conditions. This response is confirmedby solving numerically the equation of motion as well as using finite element software and performing time-history analyses, considering also the geometric nonlinearity of the cable net. With this investigation, animportant first step towards understanding the dynamic response of cable nets is achieved. Although doublecurvature renders cable nets stiffer than simple cables and a weakly nonlinear behaviour would be expected,nonlinear dynamic phenomena, established for simple cables, are also detected for these systems.Proceeding to multi-degree-of-freedom (MDOF) systems, a saddle-form cable net with circular plan view isassumed, similar to the roof of the Peace and Friendship Stadium in Faliro, Greece. The cable net boundaryis considered either as rigid, with cable ends modelled as pinned, or as flexible, simulating the deformableedge ring. The first symmetric and antisymmetric vibration modes and the corresponding natural frequenciesare calculated. A parametric analysis shows that changing the sag-to-span ratio of the net and themechanical characteristics of the cables, regarding their axial stiffness and their pretension, the sequence ofthe first modes changes. A non-dimensional parameter λ2, similar to the one used for simple cables todescribe this phenomenon, is also introduced for cable nets in this study. It is confirmed that this parameterdetermines the sequence of their vibration modes, as in simple cables. For specific values of this parametertwo or more vibration modes have equal frequencies although they have different shapes, leading to internalresonances. Thus, knowing the important role of this parameter, it is possible to choose appropriately themechanical and geometric characteristics of the cable net in order to avoid internal resonances. Semi-empirical formulae are also proposed to estimate the frequencies of the first vibration modes of the systemwith satisfactory accuracy compared to modal analysis results. Modelling the ring is proved to influencesignificantly the symmetric vibration mode of the net, due to the ring’s in-plane mode, which induces asymmetric oscillation to the net. On the other hand, the antisymmetric modes of the net remain unalteredirrespectively of whether the cable supports are considered as fixed or as flexible.
Having the analytical solution of the simple cable net, the concept of an equivalent SDOF system forestimating the dynamic response of a MDOF system is then explored. The transformation of thecharacteristics from the large system to the smaller one is obtained by similarity relations adopted from apreliminary method used at the first steps of this research, which is extended here for this purpose.Response diagrams are plotted for both SDOF and MDOF systems, based on the analytical solutions andconducting time-history analyses, respectively. The two responses are compared for several geometries andcable initial stresses in order to define the field of application of this method, showing a good agreement.The main advantage of this method is that it can be used to define with small error and minimumcomputational time the loading amplitude and frequency for which nonlinear phenomena develop. It is alsonoted that, in order to have a superharmonic or a subharmonic resonance, large amplitudes of the load arerequired. Especially for subharmonic resonances, large initial conditions are also necessary. The combinationof these two conditions leads to cable tensile failure during the transient response at the beginning of theanalysis. Thus, it is unlikely for cable nets to experience subharmonic resonance.Next, the influence of the spatial load distribution on the response of a cable net subjected to harmonicloads is investigated. Three different spatial load distributions are assumed: a symmetric one, and twoantisymmetric ones with respect to one or both horizontal axes. Response diagrams are plotted for loadingfrequencies either close to the natural frequency, leading to fundamental resonances, or smaller than theeigenfrequency, accounting for superharmonic resonances. The bending of the response curve, whichindicates a hardening nonlinear behaviour, is more intense when the net is loaded antisymmetrically ratherthan symmetrically. As a result, the initial conditions influence the steady-state response for a large range ofthe loading frequency. The behaviour of the net, when it is uniformly loaded, is altered significantly if thedeformability of the boundary ring is also taken into account in the simulation. On the other hand, thepresence of the ring does not alter the response of the net for antisymmetric loading, as also noted for theantisymmetric modes.In order to analyse the behaviour of such structures to actual dynamic loads such as wind actions, the windpressure distribution on this kind of surfaces is defined based on the recommendations of Eurocode 1. Thesaddle-form roof is divided into zones and pressure coefficients are provided for each zone according to thewind direction. The proposed wind pressure distribution is also compared with experimental results in orderto verify the accuracy of the assumptions made. It is proved that the approach adopted in this thesis resultsin slightly larger pressure coefficients in some cases, but the spatial distribution of the wind pressure issatisfactory. Finally, a measured wind record and an artificial one are considered and nonlinear time-historyanalyses are performed to detect nonlinear resonant phenomena for the wind action, as well. The dynamicbehaviour of the cable nets is compared with the static one, which is calculated according to the quasi-staticprocedure recommended by Eurocode 1. Large oscillation amplitudes are also observed in the responsespectra for frequencies equal to the eigenfrequencies, although the main frequencies of the wind are muchsmaller than the eigenfrequencies of the cable nets, while for frequencies close to the natural frequencies,the amplitude of the wind load is small. This leads to the conclusion that the small frequencies with largeamplitudes of the wind load cause superharmonic resonances to the net, while a weak excitation withfrequency near the eigenfrequency enforces the system to experience a fundamental resonance, althoughdamping is considered. As a result, large differences between static and dynamic responses are observed forall cable nets, while, as the parameter λ2 increases, the oscillation amplitude becomes smaller. The quasi-static methods cannot predict these nonlinear dynamic phenomena and thus they cannot be considered asaccurate for the analysis and design of such structures.