Failure analysis—background

The current trend in structural design is to optimise the load-bearing to weight ratio of structures, and cable structures are very much involved in this trend. Structural optimisation leads to non-linear phenomena and parameter sensitivity. For safety reasons, it is very important to know the behaviour of a structure under large loads.

Generally, a structure has different modes of failure: elastic or plastic instability, and material failure at ultimate strength. The first failure type—the elastic instability phenomenon—has been extensively analysed for bar, beam and shell structures at the Department of Structural Engineering at the Royal Institute of Technology, Stockholm. An analysis tool has been developed for these structures, and this tool can also be used to analyse the stability of cable structures. In the following sections the other aspects related to failure analysis of cable structures are reviewed. In the last section, a structural concept is discussed.

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Plasticity

The high strength steel used in modern cables exhibits linear stress-strain character-istics over only a portion of its usable strength. Therefore, in ultimate load analyses of general cable structures, the resulting formulations must consider material non-linearity. Under large external loads, some cables will go into the plastic range while some will stay in the elastic zone and other will lose their pretension. Hence, the cable elements must handle all three cases. In their present form, the elements in this thesis are only valid for the elastic and the slack cases and therefore have to be extended to consider plasticity. In reference 1 the general cable element was extended to include material non-linearity. Since the general cable element forms the basis in the derivation of the analytical cable elements, it is anticipated that it is possible to include material non-linearity also in the analytical elements.

Some works have been concerned with the plastic analysis of cable structures. For example, Ma et al. [73] used four-node isoparametric cable elements in the plastic dynamic analysis of a saddle-shaped cable net. More recently, Atai and Miodu-chowski [6] derived conditions for the stability of cable structures in the plastic state and addressed important issues, such as load history, path-dependency of so-lution and unloading of a cable from a plastic to a slack state. From these works some important conclusions have been drawn and they ought to provide a good starting point for further improvement of the plastic analysis of cable structures.

Parameter and imperfection sensitivity

The load-bearing capacity of general cable structures depends on several parameters, of which the level of prestressing is the most important. For ultimate load analysis it is necessary to determine the sensitivity of the structure to changes in some of these parameters. Also different types of imperfections, e.g. non-straight compression elements or misplaced cables, may have a serious impact on the maximum allowable loads on the structure. Two works that can be referred to in this context are presented below.

Lewis et al. [67] analysed the cladding stiffening effects in prestressed cable roofs.

A parametric study with different cladding-to-net stiffness ratios showed that for ratios found in practice the cladding significantly contributes to the net stiffness.

As a result of the cladding, the prestressing forces could be lowered. However, in this case the composite action between the cladding and cable net must be assured throughout the lifetime of the structure. The connection between the cladding and net must be able to transmit large shear forces, which may give rise to higher cost and thereby decreases the benefits of the cladding-net interaction.

In general, space trusses are regarded as highly redundant structures with the abil-ity to survive the loss of several members without losing overall stabilabil-ity. These structures also have the property to be very sensitive to imperfections. Wada and Wang [123] have investigated the effect that different types of imperfections have on the load-bearing capacity of a double layer space truss. Their investigation included

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variation of member strength, initial imperfection of member length, and errors in the assembly process. The conclusions were that the fabrication errors have minor influence on the capacity of the truss, but the human errors, like assembly errors, have enormous influence on the mechanical behaviour of the structure. For the space truss analysed, if two or more members out of 288 members had errors the structure had a large probability of collapsing. A reduction of the load-bearing capacity may also occur in cable structures if cable connections are not assembled in their right positions.

Analysis of tensegrity structures

During the last decade new structural principles have been developed, the struc-tural behaviour of which are not yet fully understood. The most interesting of the new structural principles is that of self-stressed systems, also known as tensegrity systems. Self-stressing is interesting as it allows cable-strut structures to be built without the need for supporting structures to equilibrate the stresses in the initial configuration. In addition to this advantage, tensegrity structures have other bene-fits that make them interesting for research; for example, they are lightweight and earthquake resistant. There is also a strong desire from architects to implement the tensegrity principle in buildings because of the pure shapes it produces. Although the research activity in this area is high, see for example references 46, 80, 81, 126, there are still several questions that need to be answered before full scale applica-tion can be a reality. In this secapplica-tion, analysis aspects of tensegrity structures will be discussed.

Morphology studies. From the invention of the tensegrity concept in the late 1940’s until the beginning of the 1990’s most research projects have dealt with the geo-metrical shapes of tensegrity networks. According to Hanaor [46] “It appears that the morphological study of tensegrity networks has reached a degree of saturation, whereby the range of conceivable patterns exceeds by far the likely range of appli-cations.” Nevertheless, double-layer tensegrity grids have been developed by joining tensegrity simplexes [79]. Wang [126] concludes that “the future work will be con-centrated on applying other simplexes in space structures.”

Initial equilibrium configurations. Several different shapes of tensegrity structures are available from the morphology studies. But, form-finding with geometrical meth-ods does not guarantee mechanical equilibrium and solutions must be checked with a numerical method [81]. Motro et al. [82] applied both the dynamic relaxation method and the force density method to solve the initial equilibrium problem. They anticipated that the latter method is more suitable for large systems, but that more work has to be done to check its efficiency. One possible way to modify the force den-sity method to apply to tensegrity systems might be to adopt Mollaert’s approach given in reference 75. In that approach, the compression and tension members are separated to ensure a solution out of the plane. Recently, Bruno [15] used a method based on the minimisation of the potential energy to find the equilibrium configuration of simple two-dimensional tensegrity systems. A more mathematical

CHAPTER 6. CONCLUSIONS AND FURTHER RESEARCH

approach was developed by Roth and Whiteley [101] and extended by Connelly and Whiteley [25]. This approach has been used by Burkhardt [17] to analyse quite complicated tensegrity domes.

Mechanism elimination. Finding a configuration which satisfies equilibrium is not enough to solve the initial equilibrium problem for a tensegrity system; the internal mechanisms for that configuration must be identified, classified and, if possible, elim-inated by prestressing [80]. The method by Calladine and Pellegrino [20, 93] can be used to find the mechanisms and determine the stability of the initial configuration.

Recently, Tomka [120] introduced a technique called the method of stabilising force to analyse the stability of cable structures. The advantage of this method is that in addition to the qualitative result (stable or unstable), obtained by the method by Calladine and Pellegrino, quantitative conclusions can also be drawn concerning the measure of stability.

After an acceptable solution has been found, the behaviour of the tensegrity system has to be studied under the effect of external loads. In particular, the stability of the self-stressing configurations should be studied [80]. It is of cardinal importance to know if mechanisms can reappear as a result of external loadings. According to Motro [80] this subject “still remains a fairly open matter.”

Construction. Besides the theoretical aspects mentioned above, the analysed struc-tures must be possible to build. In the present state, there has not been much application of the tensegrity principle in the construction field. The reason for this is that several fundamental technical problems still need to be solved [46, 80]. The main problems are:

• suitable prestressing procedures,

• efficient node systems, and

• incorporation of cladding.

Finding a suitable construction and prestressing procedure is quite difficult. These procedures tend to be cumbersome and uneconomic because general tensegrity sys-tems are geometrically complex and lack rigidity prior to prestressing [80]. The prestressing methods must be reliable and assure the level and permanence of the tension that has been put in. The efficiency of the node systems is very much related to the construction procedure and the prime objective is to have compact connec-tions. Concerning the roof covering, a flexible membrane is preferable because of the flexibility of the tensegrity frameworks. It is important that the membrane forms an integral part of the design, as it is not a trivial matter to obtain a correct stress distribution in the membrane [46]. According to Hanaor [46] “none of the studies carried out to date, consider the surface membrane.”

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