Abstract of PhD Thesis by Isabella Vassilopoulou Structural Engineer at ODOTECHNIKI LTD Download full text: http://www.odotechniki.gr/main/index.php/en/downloads

1. 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
Abstract
The research presented in this thesis aims at investigating the response of cable nets subjected to dynamic
loads, focusing on the dynamic phenomena that characterise nonlinear structures. First, a simple cable net is
studied, consisting of two crossing cables and the equation of motion is derived. Neglecting small terms of
its equation of motion, a simplified single-degree-of-freedom (SDOF) cable net is assumed, which is proved
to be similar to a Duffing oscillator with a cubic nonlinear term of the displacement. The analytical solution of
its steady-state response, found in the literature, is adopted for this simple cable net and the occurrence of
fundamental and secondary resonances, such as superharmonic and subharmonic resonances, is verified for
this system. The response diagrams are plotted for different resonant conditions showing bending of the
response curve, hardening behaviour and dependence on the initial conditions. This response is confirmed
by 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, an
important first step towards understanding the dynamic response of cable nets is achieved. Although double
curvature 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 is
assumed, similar to the roof of the Peace and Friendship Stadium in Faliro, Greece. The cable net boundary
is considered either as rigid, with cable ends modelled as pinned, or as flexible, simulating the deformable
edge ring. The first symmetric and antisymmetric vibration modes and the corresponding natural frequencies
are calculated. A parametric analysis shows that changing the sag-to-span ratio of the net and the
mechanical characteristics of the cables, regarding their axial stiffness and their pretension, the sequence of
the first modes changes. A non-dimensional parameter λ2, similar to the one used for simple cables to
describe this phenomenon, is also introduced for cable nets in this study. It is confirmed that this parameter
determines the sequence of their vibration modes, as in simple cables. For specific values of this parameter
two or more vibration modes have equal frequencies although they have different shapes, leading to internal
resonances. Thus, knowing the important role of this parameter, it is possible to choose appropriately the
mechanical 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 system
with satisfactory accuracy compared to modal analysis results. Modelling the ring is proved to influence
significantly the symmetric vibration mode of the net, due to the ring’s in-plane mode, which induces a
symmetric oscillation to the net. On the other hand, the antisymmetric modes of the net remain unaltered
irrespectively of whether the cable supports are considered as fixed or as flexible.

2. Having the analytical solution of the simple cable net, the concept of an equivalent SDOF system for
estimating the dynamic response of a MDOF system is then explored. The transformation of the
characteristics from the large system to the smaller one is obtained by similarity relations adopted from a
preliminary 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 and
conducting time-history analyses, respectively. The two responses are compared for several geometries and
cable 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 minimum
computational time the loading amplitude and frequency for which nonlinear phenomena develop. It is also
noted that, in order to have a superharmonic or a subharmonic resonance, large amplitudes of the load are
required. Especially for subharmonic resonances, large initial conditions are also necessary. The combination
of these two conditions leads to cable tensile failure during the transient response at the beginning of the
analysis. 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 harmonic
loads is investigated. Three different spatial load distributions are assumed: a symmetric one, and two
antisymmetric ones with respect to one or both horizontal axes. Response diagrams are plotted for loading
frequencies either close to the natural frequency, leading to fundamental resonances, or smaller than the
eigenfrequency, accounting for superharmonic resonances. The bending of the response curve, which
indicates a hardening nonlinear behaviour, is more intense when the net is loaded antisymmetrically rather
than symmetrically. As a result, the initial conditions influence the steady-state response for a large range of
the loading frequency. The behaviour of the net, when it is uniformly loaded, is altered significantly if the
deformability of the boundary ring is also taken into account in the simulation. On the other hand, the
presence of the ring does not alter the response of the net for antisymmetric loading, as also noted for the
antisymmetric modes.
In order to analyse the behaviour of such structures to actual dynamic loads such as wind actions, the wind
pressure distribution on this kind of surfaces is defined based on the recommendations of Eurocode 1. The
saddle-form roof is divided into zones and pressure coefficients are provided for each zone according to the
wind direction. The proposed wind pressure distribution is also compared with experimental results in order
to verify the accuracy of the assumptions made. It is proved that the approach adopted in this thesis results
in slightly larger pressure coefficients in some cases, but the spatial distribution of the wind pressure is
satisfactory. Finally, a measured wind record and an artificial one are considered and nonlinear time-history
analyses are performed to detect nonlinear resonant phenomena for the wind action, as well. The dynamic
behaviour of the cable nets is compared with the static one, which is calculated according to the quasi-static
procedure recommended by Eurocode 1. Large oscillation amplitudes are also observed in the response
spectra for frequencies equal to the eigenfrequencies, although the main frequencies of the wind are much
smaller 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 large
amplitudes of the wind load cause superharmonic resonances to the net, while a weak excitation with
frequency near the eigenfrequency enforces the system to experience a fundamental resonance, although
damping is considered. As a result, large differences between static and dynamic responses are observed for
all 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 as
accurate for the analysis and design of such structures.