Determination of prestressing levels for cable trusses as a function of their stability

The stability of prestressed cable trusses subjected to static and dynamic loads is analysed in the paper. The stiffness of the elastic cable truss system was varied by changing the prestressing force. Modelling results for different levels of tension show that, in terms of satisfying dynamic resistance conditions, a much greater tensile force is required for cable grids with diagonal members, compared to those with vertical rods. The conclusions derived constitute recommendations for the use of calculation methods, as well as for the determination of prestressing forces under which stability criteria are fulfilled.


Bestimmung von Vorspannkräften in Seiltragwerken im Hinblick auf die Stabilität
Cable trusses are double-layered catenary systems made of steel ropes.They form a system in a vertical plane together with the bearing and the stabilizing catenary, which are connected with either diagonal bars (type D, figure 2.A) or vertical bars (type V, Figure 2.B).By tensioning the lower stabilizing catenary the stressing forces are introduced in the entire system.In this way, the stiffness of these girders is achieved, i.e. their swinging in relation to asymmetrical and dynamic loads is prevented or limited.Because of geometrical characteristics of catenaries and the rope structure, the bearing elements are elastic, making this system unstable if there are no tensile forces in the bars.This is particularly true for the lower stabilizing catenary which must have a tensile force even in the case when the girder is exposed to maximum gravity load caused by its own load, as well as by snow and wind (g+s+w) Known methods for the stabilisation of these systems involve application of load to girders either by a weight, by prestressing, or by combining with elements resistant to compression and bending.Structures in which the stabilisation by prestressing is applied will be analysed in this paper.The prestressing is a stabilization method that replaces the weight that would otherwise oppose asymmetrical external loads.The prestressing force introduced into a girder is calculated through analytical relations as a consequence of an assumed "contact force" between the bearing cable and the prestressing cable.The inventor of these systems introduced the "contact force" as an additional fictive load (v) which will satisfy the necessary condition related to the tensioning of the stabilised catenary [1].Professor Balgac also gave his analytical formulae for calculating "contact forces" [17,18] in form of simple expressions which connect the additional load with geometrical characteristics of girders (span, depth, cross section) and the elastic modulus of the material.The use of cable trusses has been improved during the past several decades by making appropriate changes to their basic form.Principal developments have been made in two directions: main girders in suspended systems, and primary semi-truss roof girders with a circular cable (Figure 1).From the very first use of cable trusses, the dynamic influence of wind has been checked for all large-size suspended roof structures.In the 1960s, David Jawerth calculated for these structures the time periods of own vibrations according to original expressions for elastic systems [1].Meteorological data about the speed, frequency, and direction of wind acting on the structure, and the duration of gusts, were analysed.On the basis of wind tunnel results, von Karman made nomograms in order to establish a relationship between forces in the girder support, rope diameter, Strowhale number, length of bar between nodes, and the speed of wind at which the resonance due to harmonic excitation may occur.By comparing the calculated own frequencies with the data from the von Karman nomogram, Jawerth determined whether the resonance would occur and, if so, at what wind speeds.In the early 1980s, following a long period of disinterest after 1960s, Massimo Majowiecki [2] introduced a new use of cable nets (Figure 2.B), and his German colleagues accepted such structural solutions, and even contributed to the new concept (Figure 2A, Schlaich, Bergerman and Partner).The use of computers in the process of calculation has contributed to the development of numerical procedures.Basic dynamic characteristics of cable nets were calculated using the subspace method, i.e. the Jacobi iterative algorithm.Majowietcki and Schlaih treated wind as a gust of sinusoid distribution of intensity which appears periodically, i.e. as a time-dependent load, in order to determine maximum movements and maximum forces in cable structure bars over a certain time interval.[2][3][4][5][6].Bearing all this in mind, the authors of the paper analysed, in addition to force and displacement, the frequency spectrum of cable truss responses [7].Two models were exposed to identical static and dynamic load.Diagrams of real-time changes of frequencies show that the occurrence of resonance is possible, and provide an answer to the question of stability, Determination of prestressing levels for cable trusses as a function of their stability i.e. the minimum prestressing level that guarantees fulfilment of previously-set stability criteria.The aim of the paper is to estimate stability of cable nets as related to the fulfilment of known parameters (intensity of forces and displacement) caused by static and dynamic load, and a special emphasis is placed on the analysis of the frequency spectrum of concordance of the own and imposed oscillations.The result of the research provides an answer to the tensioning level of cable nets, as needed to meet known additional stability criteria for two basic models of these systems.

Prestressing and stability criteria
Prestressing is actually a substitution for load, i.e. it is its equivalent.The load was originally used as a real load for the stabilisation of catenary girders.The load caused by dynamic forces (wind or seismic action) induces inertia forces which exert a negative influence on the elastic girder of the catenary.The same effect is achieved by prestressing, albeit without inertia forces, which makes these lightweight roof structures susceptible to wind load, but not to seismic action.The authors of the paper have recognized the need to establish a correlation between the minimum prestressing force and the additional fictive equivalent for which the girder would meet stability criteria.

Stability criteria current practice
Basic stability criteria have been defined by structural engineers and researchers in their papers.They are also summarized in provisions of the AISI Manual for Structural Applications of Steel Cables for Buildings.2010 [8] and the Eurocode 3 [9]: Criterion 1: Standardized wire cables, protected against corrosion, relaxed.Criterion 2: The effective strength of a cable must be bigger than the biggest forces value in cables multiplied by an appropriate constant (1.6 up to 2.7 depending on the loading phase -ultimate limit state).Criterion 3: Maximum allowed structural movements must not exceed the prescribed values (serviceability limit state).Criterion 4: None of the cable elements must unstressed i.e., all of them must be tensioned.Criterion 5: Conditions for achieving resonance and fake resonance must be avoided, in case the hanging roofs are exposed to dynamic load (wind, explosion or earthquake).

Calculation analysis
Because of their multiple static indetermination the calculations of such complex structures are highly complex.Two typical approaches are used for determining the stress and deformation: numerical [11][12][13][14][15][16] (Transient Stiffness Method, Force Density Method and Dynamic Relaxation Method) and analytical [1,17,18].Movements of nodes and forces in bars are obtained, for given geometrical parameters and load values, through calculation of these geometrical nonlinear structures.In order to meet stability criteria from Section 2.1, it is necessary to define minimum stressing forces caused by load influencing the roof girder and the additional load which causes the "contact force".This "contact force" is the direct consequence of the additional fictive loading "v".The contact force is the inner force between the bearing cable and the stabilizing cable, and it acts through filling rods (diagonal or vertical).Through decomposition of cable trusses into layer cables in the phase of the greatest gravity load, the upper bearing cable assumes all external loads and an additional fictive load (g + s + w + v), while the lower stabilizing cable is affected only by an additional fictive load (v), when the contact force between the cables is the smallest, just like the force in the stabilizing cable.
Through the unloading of cable trusses the contact force between the cables increases (v + k), and the girder itself elastically moves upwards for the change in depth Δf when the depth of the stabilizing cable increases (f p + Δf), and that of the bearing cable decreases (f n -Δf).The unloading causes an increase of forces in the stabilizing cable and a decrease in the bearing cable.A reverse process occurs if load is exerted on the girder [7].
In fact, the need to determine displacements and forces in cable truss ropes, in accordance with stability criteria from section 2.1, has prompted the authors to conduct an analysis based on analytical and numerical methods presented in this paper, during which it would be possible to control the contact force in accordance with recommended additional fictive load values for cables.Numerical procedures, i.e. the Force Density Method and the Dynamic Relaxation Method, quickly lead to the convergence of the iterative procedure with a minimum of input data.The intensity of prestressing force is one of significant input data.This data is inserted based on recommendations that are not precisely defined, and the result of calculation concerning fulfilment of stability criteria is unknown.The use of these methods most often results in insufficient structure tensioning forces, and the calculation procedure must be repeated by increasing the prestressing level, until all stability conditions from Section 2.1.are met.That is why the authors have chosen a combination of the analytical and numerical approach, i.e. the Transient Stiffness Method [7].The analytical method uses conditions of balance and the change of catenary length at load, thus forming systems of nonlinear differential equations.Through approximations and eliminations of small higher-order values, the equations are made linear and are reduced to an easily solved problem, with the calculation error from 6 % to 10 %.The shortcomings of both methods are turned to their advantage through appropriate combination of their results.
Mirko Aćić, Ljubomir M. Vlajić, Dragan Kostić The imprecision due to approximation, and the length of analytical procedure, are not necessarily shortcomings, as the procedure can be conducted using computer programmes, and so the time needed can be reduced significantly while, in this relation, the imprecision can be tolerated as the data obtained by analytic procedure will be treated numerically, using the Transient Stiffness Method.The shortcoming of this numerical method is the multitude of necessary input data about configuration of the system and prestressing forces in bars.This, however, is obtained in the scope of the analytical method and so, in this case, this is not a shortcoming but rather a quality in the combined use of the two methods.
A modified analytical procedure developed by professor Balgac [7,17,18] was used in order to obtain the data about the system configuration and the forces in the so called referential position.This procedure was adapted to the cable truss calculation, due to some limiting elements that had to be substituted by general ones, so that the calculation can be applied to a larger number of problems.The calculation adapted in this way was used to write programme modules for computer, according to which the cable-truss referential position parameters can rapidly be found with great accuracy.The quality of analytical procedure proposed by Professor Balgac lies in the condition according to which the stabilizing cable must keep in itself the stressing force even under the highest gravity load, and the vertical component of this force, the so called "contact force", will be exerted on the bearing cable through some fictive load "v".The presence of "contact force" guarantees stability of the roof.The static calculation must be followed by dynamic analysis so that fulfilment of criterion 5 from Subsection 2.1 can be checked.Cable structures behave as geometrically nonlinear structures in both static and dynamic way.However, the linearization is made in the dynamic analysis in order to simplify the problem and reduce the calculation time.
The modal analysis is used to determine Eigen frequencies and Eigen shapes of the oscillation of structures.It is also used as the basis for other detailed dynamic analyses such as: transient analysis, harmonic analysis, and spectral analysis.
For the problem with prestressing, geometrical nonlinearity, possible great deformations, and an increase in stiffness with an increase in tensioning force, the most favourable method is the subspace method, according to the instructions for the use of the program package "ANSYS MultiPhysics", Houston 2003.
The transient dynamic analysis, also known as the "time history" analysis, is the method by which the dynamic response of a structure subjected to a time-dependent load is obtained (F(t)).This type of dynamic analysis was used in order to determine time-dependent movements and forces in the structure, and to see whether the structure reacts to any combination of static, transient, or harmonic loads.The change of load over time, and inertia or damping effects, exert a significant influence on the analysis results.
Basic movement equations are solved by the Newmark"s time integration in defined time "spots".The so called full method was used for solving the problem of transient vibrations caused by wind action.The transient stiffness method, or the method of final displacements, as it was called on the day it was created, was used for static and dynamic calculations.The cable truss model was made of the so-called link finite elements.They are spatial elements with alternating tension and compression, without stiffness to bending.Every element has two nodes, each of them with three degrees of freedom (u, v, w).They are appropriate for nonlinear analysis, analysis of big deformations, presence of prestressing forces, increase of model stiffness caused by an increase of their inner forces, and for dynamic analysis with possible damping.External forces act in nodes, just like concentrated masses in dynamic analysis.

Estimation of prestressing level as a function of cable truss stability
The level of prestressing of cable trusses is estimated through analysis of results obtained by analytical and numerical static analyses, and by modal, harmonic and transient analysis.
Structural analysis is conducted for defined loads (dead load, snow and wind), which are increased by an additional fictive load "v", in order to obtain the "contact force".The span of cable trusses was chosen according to the needs of a covered universal arena, bearing in mind the number of spectators (<5000) corresponding to the needs of a small town in Serbia.The chosen span was L = 60,00 m (Figure 2).Geometrical and physic characteristics of the chosen cable truss models, i.e. structural scheme of girder configuration, span, distance between cable trusses, and their support and loading system, are given in Table 1.According to the manufacturer"s catalogue, the elasticity modulus of selected cables amounts to E = 165 kN/mm².
Determination of prestressing levels for cable trusses as a function of their stability

Dynamic wind load on high-rise structures
Realistically expected loads are defined according to prevailing technical regulations, whereas the wind is analysed as a horizontal laminar motion, i.e. as a random and continuous process, which is defined according to principles used in mathematical statistics, turbulent flow mechanics, and structural theory, [4,6,10,16].On the basis of meteorological and statistical data, the wind was treated as a load whose intensity varies at each moment in time, actually as a stochastic excitation which is divided into a harmonious force with the frequency of 0-0.4 Hz, and a dynamic impulse force repeating at specific time intervals after a period of "lull" (Figure 3 and Figure 4).The dynamic force of wind is present in nodes of the upper cable truss layer, according to distribution shown in Figure 4.b.During the "lull" period the girder is influenced by the harmonious wind excitation force ranging from 0 Hz to 0.4 Hz (Figure 4).Determination of prestressing levels for cable trusses as a function of their stability

Static analysis of adopted models
A cable truss model, 60 m in span, is considered for various categories of dead weight of roof cover (ultra lightweight = 0.15 kN/m 2 , very lightweight = 0.30 kN/m 2 , lightweight = 0.50 kN/m 2 and medium weight = 0.75 kN/m 2 roof covers).At that, each cable truss "D" and "V" is exposed to realistic load combinations (6 combinations of static and 4 combinations of dynamic load), while its stiffness level is varied.This stiffness level is defined by the "contact force" and by an additional fictive load ranging from v = 0.05 to 0.70 kN/m 2 .The stability of adopted models under the influence of static load is analysed in accordance with criteria given in Section At that, some of the analysed models: -can under some conditions be considered as "statically stable" structures, because they are fully compliant with the criterion 2, and partly with criteria 3 and 4 (U).-can be accepted as stable structures for static load because they are fully compliant with criteria 2, 3 and 4 (StSt -statically stable).
Table 3 clearly presents statically stable models according to roof load categories and prestressing (additional fictive load) levels.No model can be considered an unstable structure with respect to static loads.In order to be confirmed as comprehensively stable, those models that are marked as conditionally statically stable (C) and statically stable (StSt) must fulfil the dynamic stability conditions, through analysis of modal forms, frequencies, superposition of modal and harmonic vibrations, and response to time-dependent load.

Modal analysis of adopted models
The modal analysis defines basic dynamic characteristics of a structure (own frequencies and own shapes) which are not dependent on the loading or support movements, or damping, but are functionally related to initial parameters: configuration of the system (position of nodes in space and their connection with rods), material properties, and inner forces due to prestressing.Consequently, modal parameters differ for balance states g + w and g + s + w.The method of inverse iterations (Subspace Method) was chosen for calculating modal parameters because it is highly accurate (full matrices [K] and [M] are used).From the programming standpoint, it enables superposition of tone forms in subsequent dynamic analysis (harmonious superposition of modal shapes and transient superposition of modal shapes).Modal analysis was made using ANSYS on a spatial model consisting of "link" elements.The program module CABL-T [7], based on prof.Balgacs"s modified analytical procedure, was used in the preparatory phase to determine the system configuration and forces in rods in a reference position (prestressing phase).Out of 10 eigen forms obtained, only the first to fourth forms are tones in vertical plane (xOy), while the others occur in horizontal plane (xOz), or the eigen vectors are negligibly small (order of 1•E -7 ).It is obvious that the frequencies increase and periods of oscillations decrease with an increase in the prestressing level, i.e. with an increase in the model tensioning level.Eigen frequency values are higher for model "D" than for model "V", and for the load category "g + w" as compared to "g + s + w", as shown in Table 4 for the first two oscillation tones.The modal form participation factor is the largest for the first and second eigen forms of vibration in the xOy plane.Through analysis of similarity of modal forms, the common first and second characteristic modal forms are summarised depending on variable factors: additional fictive load v (which is due to prestressing) and static load phases g + w and g + s + w, as presented in the Table 5.   * Additional fictive load was increased during the calculation until the correspondence between common eigen oscillation forms in phases "g+w" i "g+s+w" was reached.
Table 5.Typical shapes of vertical eigen forms for model types "D" and "V" (Y correspondence of own shapes with shapes given in the table)  Yi -there is a similarity with i-combination diagrams from Figures 5 to 7 The transient analysis yielded frequency change diagrams, i.e. the frequency spectrum of models in the time interval of dynamic excitation (0 to 185 seconds).The frequency change diagram presented in Figure 5 shows a model oscillating in the first characteristic form with rhythmical vibrations.The frequency change diagram given in Figure 6 presents a model that "wanders about" in search for its characteristic oscillation form, and so, generally, this structure cannot be considered as dynamically resistant.

Discussion of results
Dynamic analysis results were obtained using the ANSYS programme, through modal analysis, harmonic analysis of superposition of modal and harmonic oscillations, and transient analysis of the model"s behaviour during the time-dependent loading.Abundant results were obtained through numerous analyses on "D" and "V" models for four roof weight categories, and static and dynamic service loads.Based on a comprehensive analysis of existing structures, recommendations given by some authors [1,11,17,18], and according to appropriate results, the authors of this papers specified and additionally elaborated the stability criteria 3, 4 and 5 from Section 2.1 of this paper.The following criteria were used in the analysis of dynamic resistance of cable trusses: 1.None of the cable elements should be unstressed, i.e. cable elements in each loading phase must be stressed.Minimum force intensity should be no less than 20% of the force present in the rope during prestressing.2. The own modal shapes should be as simple as possible.3. Maximum allowed structure node amplitudes, resulting from the superposition of modal and harmonic oscillations, should be limited to δ max = L/200, where δ max is the maximum displacement of the structure in relation to the design position (phase of load by dead-weight, g); 4. Conditions that enable resonance through harmonic excitation, which leads to large deformations, must be avoided, i.e. modal frequencies must be influenced by the level of prestressing so that they do not correspond to the frequency of imposed harmonic excitations; 5. Due to transient excitation -occasional wind gusts -the model should be quickly calmed down to the level of lower eigenfrequencies (to the level one if possible).Transient analysis results should be used to define frequency changes diagrams.Possible shapes of these diagrams point to the model which oscillates in the first own shape with rhythmical amplitudes (Figure 8.a), and to the model which "wanders about" looking for its own shape of oscillation (Figure 8.b) and so, from the dynamic standpoint, it cannot be considered as a sufficiently resistant structure.
The above mentioned dynamic-stability criteria were used to form Table 7, which shows numerical and descriptive indicators of the level of fulfilment of individual criteria.
Results obtained point to minimum prestressing forces that are due to the existence of the "contact force", i.e. of an additional fictive load "v" that should influence the cable-truss stabilizing rope in the phase of the largest gravity load, in order to obtain the overall stability of the Determination of prestressing levels for cable trusses as a function of their stability analysed girder.Design prestressing forces that should be introduced into the cable truss via the stabilizing cable in order to obtain a fully stable structure, for the adopted models "D" and "V" and defined load values, taking into account additional fictive load values from In dieser Arbeit wird die Stabilität vorgespannter Seiltragwerke unter statischen und dynamischen Kräften untersucht.Die Steifigkeit des elastischen Tragwerks wird durch die Annahme verschiedener Vorspannkräfte abgeändert.Die Resultate der für verschiedene Vorspannstufen analysierten Modele zeigen, dass für Seiltragwerke mit diagonalen Elementen bedeutend höhere Spannkräfte erforderlich sich, um die Bedingungen der dynamischen Beständigkeit zu erfüllen, als für Träger mit vertikalen Seilelementen.Abschließend werden Empfehlungen zur Anwendung von Berechnungsmethoden und zur Reihenfolge der Berechnungsschritte dargestellt, sowie die zur Erfüllung der Stabilitätskriterien erforderlichen Vorspannkräfte erläutert.Schlüsselwörter: Seiltragwerke, Stabilität, Steifigkeit, windverursachte Vibrationen, eigene und erzwungene Schwingungen Determination of prestressing levels for cable trusses as a function of their stability 1. Introduction

Figure 1 .
Figure 1.a) Main girder in suspended systems; b) Primary semi-truss girders

Figure 2 .
Figure 2. Models of cable trusses: a) Type D with diagonal filling rods; b) Type V with vertical filling rods

Figure 3 .
Figure 3. Dynamic effect of Košava wind for the 10 minute average wind velocity, according to Schlaich [6]

Figure 4 .
Figure 4. Wind action on structure, depending on wind direction: a) Harmonious force of wind suction; b) Wind gust force

Figure 8 .
Figure 8. Frequency change diagram: a) dynamically resistant model; b) dynamically non-resistant model

Table 1 Structural characteristics of the chosen cable models
* Y -tension forces are present in bars, i.e. there are no compression (unstressed) rods.The ratio of the influence of the smallest force in the bearing or stabilizing cable rods (filling rods) to the prestressing condition is given in brackets

Table 8 ,
are presented in Figure9.Tearing forces exist in rods i.e. there are compression unstressed rods.The ratio of the minimum force in stabilising-cable rods to the level of prestressing is given in parentheses.