## Computer-aided design of tension structures

Ong, C.F. (1992).
*Computer-aided design of tension structures*.
(Unpublished Doctoral thesis, City University London)

## Abstract

This thesis consists of three parts. Part I (chapters 1-4) gives a review and description of the basis for the numerical modelling of tension structures. The discussion in Part I leads to the conclusion of a need for an interactive design procedure for tension structures which is the subject under consideration in Part II (chapters 5-7). In the design of tension structures, an area which requires special attention is the dynamic response often initiated by the action of a natural wind. In Part III (chapters 8 and 9), this area is examined in detail and a strategy is proposed to give an improved modelling of dynamic response. The numerical procedure developed is assessed by comparison with previously reported test results for a pneumatic dome. Chapter 1 gives a general introduction to tension structures, and their main characteristics and behaviour are briefly described. From both the structural and architectural points of view, tension structures (classified as either prestressed cable nets or doubly curved membranes) do offer a number of benefits which arise from their characteristics and behaviour. The different types of cable nets which can be contructed are outlined, and various possible

types of membrane structures and membrane materials which can be used are described. In addition, the form-finding and load analysis stages in the design process of tension structures are briefly discussed. An overview of Parts I, II and III of this thesis is also included. As a result of the flexible nature of a tension structure, large deformations often occur under loads acting normal to the surface. In addition, the coated woven fabrics exhibit material non-linearities, ie. the material properties vary under loads. In other words, a full non-linear structural analysis accounting for both material and geometric non-linearities is required in order to give a realistic modelling of the behaviour of a tension structure. Chapter 2 reviews numerical methods which have been widely reported for dealing with nonlinear structural analysis. From the review, it will be noted that the dynamic relaxation (DR) method is well suited to solving the highly non-linear problems which occur particularly in the case of tension structures. The dynamic relaxation method with a finite element idealisation of the structure is chosen as the solution scheme for all the analysis work in this thesis, and a detailed description of the method is given in chapter 3. Features of the method which are particularly useful for the design of tension structures are: (a) the effective decoupling of the equations of compatibility and equilibrium which allows complex material properties modelling and the use of slip cables, etc., and (b) the use of a 'kinetic' damping procedure which permits gross changes in support geometries to be made during interactive form-finding without the possibility of numerical instability. Although the main surface spanning elements may be purely tensile, many tension structures will employ compression and bending elements for their support. For example, as a means of providing support to a large span tension structure, a compression boundary is considered to be an efficient alternative to tension anchorages. In a sense, a compression boundary is complimentary to the tension elements in the structure as these elements also act as supports to the compression boundary. This gives the advantage of a compression boundary comprising of slender sections. The compression boundary is modelled as a series of beam elements. The moment-curvature equations of a beam element expressed in the form of natural stiffness relations, are developed in chapter 4. In addition, the non-linearities, both geometric and material, and boundary conditions which can be dealt with by the beam elements are considered. An outline is given of the implementation of the beam elements using the dynamic relaxation method. Included in chapter 4 are also the results to test problems which have been set up in order to validate the underlying theory and implementation of the beam elements. As tension structures often exhibit complex surface curvatures, a study of surfaces, their properties and behaviour is appropriate, and useful in the understanding of concepts applied in the design process. This study is the subject of chapter 5 which focusses on the relevant topics of differential geometry. A few useful ideas from differential geometry form the basis of certain procedures implemented into the form-finding and patterning stages being considered in chapter 6. The derivation of the equilibrium equations for a surface when acted on by applied loads is also given in chapter 5. The discussion in chapter 6 is about the stages of form-finding and static load analysis in the design process. A review of the available solution methods for the form-finding problem are given in this chapter. In these methods, the solution can be for either the unknown geometry or unknown stresses, or both. The adopted approach in this thesis is to solve for the unknown equilibrium geometry given the stress distributions, and initial and boundary conditions. The controls which can be used during form-finding to achieve the desired geometries of cable nets and membrane structures are discussed. The equilibrium geometry derived from the form-finding stage has to be subsequently evaluated for its performance under loads at the load analysis stage. After an equilibrium geometry which behaves satisfactorily under loads has been achieved, the corresponding cutting patterns are developed in the case of membrane structures. Recent advances in interactive computer graphics technology have made it possible to develop a fully interactive CAD system for tension structures. The development of such a CAD system is the subject of discussion in chapter 7. The CAD system integrates together the form-finding, load analysis and fabrication patterning stages, resulting in a continuous design process. It demonstrates how the various concepts discussed in Part I of this thesis fit together within an interactive environment implemented with an effective and functional user interface. It is illustrated in chapter 7 how such a user interface has been achieved. The CAD system fully exploits the capabilities offered by the available computer hardware such that the computations involved during analysis of the structure, in generation of surface shaded graphic images and so on, can be executed at very high speeds. As a result, the CAD system can respond quickly to the user and is thus consistent with the interactive nature of the design process. The discussion in chapter 7 also provides an insight into the various procedures involved throughout the design process. The CAD system has produced a number of benefits of which the main one is the saving in design time which has been achieved. As the CAD system is highly user friendly, only a short learning period is required, thus enabling it to be used more widely among designers. The CAD system also serves as a useful tool for the communication of ideas between the engineer and the architect. In the design of a tension structure unlike that of a conventional building, there is often close cooperation between the engineer and the architect right from the early stages of conceptual development. In chapter 8, the possible loads which may act on a tension structure during its service life are considered. These loads are applied to the structure at the load analysis stage in the design process. In most cases, the design loads are those due to snow and wind. An accurate assessment of the loads is essential in order to achieve a structurally sound and economic design. In addition, it should be possible to represent the loads in a form which can be easily applied in the structural analysis. The considerations which are involved in the assessments of the snow and wind loads are outlined. The dynamic responses of tension structures which result from the action of a natural wind or initiated by other means is considered in chapter 9. A tension structure when set into motion causes additional mass, stiffness and damping terms to be generated. A brief review of the available solution schemes for the dynamic analysis of structures in general is outlined. These solution schemes include the mode superposition method and the direct step-by-step time integration methods. In this chapter, the explicit central difference time integration method is the adopted solution scheme for the dynamic analysis of tension structures. The creep effects which arise from the visco-elasticity present in most membrane materials are also considered. A strategy based on the Voigt-Kelvin model is used to calculate for the accumulated creep strains. An outline is given of an incremental procedure to allow for the on/off element buckling which occurs during dynamic analysis. The internal air stiffening effect due to changes in internal volume and pressure of an air supported structure as it deforms during dynamic analysis,, sralso considered. When a membrane undergoes vibrations, the surrounding air attached to the membrane is mobilised into motion as well. This in turn gives rise to the added mass effects. It is the main aim in chapter 9 to develop a means of modelling the added mass effects and for this purpose, an approach based on the potential flow theory has been formulated. The basic assumptions and concepts of the potential flow theory which are relevant to the approach are outlined. In the approach, it is required to solve the Laplace equation and the solution can be in terms of either a distribution of sources or doublets over the membrane. In this case, a distribution of sources are used to calculate the kinetic energy of the surrounding air when set into motion. An indication of the added mass effects is given by the ratio of the kinetic energy of the surrounding air to the kinetic energy of the vibrating membrane. The mechanics which are involved in the determination of this ratio known as the added mass coefficient, are outlined. The proposed approach is applied to investigate the dynamic response of an air-supported dome and some encouraging results are obtained. A summary of the work discussed in previous chapters is given in chapter 10 which also includes the main conclusions of this thesis. The discussions in previous chapters suggest that there are various areas in which further research work can be pursued and these areas are outlined in chapter 10.

Publication Type: | Thesis (Doctoral) |
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Subjects: | T Technology > TA Engineering (General). Civil engineering (General) |

Departments: | School of Science & Technology > Engineering |

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