Limit-State Analysis and Design of Cable-Tensioned Structures

Transcription

1 Limit-State Analysis and Design of Cable-Tensioned Structures J. Y. Richard Liew 1, N. M. Punniyakotty 2 & N. E. Shanmugam 3 ABSTRACT This paper deals with application of nonlinear analysis to study the system's limit state behaviour of cable-tensioned steel structures. The cable tendon is modelled as an equivalent truss element with initial pretensioned force applied either as selfequilibrating axial load or as an equivalent thermal load. The modulus of elasticity of the cable material can be modified to account for the constructional stretch and sagging effects in the cables. Pin-jointed space truss systems are modelled using a plastic-hinge nonlinear formulation, which implicitly satisfies the design specifications for ultimate strength of axially loaded members. Rigid space frame systems are modelled using an elasto-plastic nonlinear frame analysis program. The effectiveness of providing pretensioned cables in improving the limit state behaviour of pin- and rigid-connected space frames is demonstrated through several examples. Keywords: Barrel vault, cable-strut structures, limit state design, nonlinear analysis, space structures, tensioned structures. 1 3 Assoc. Prof., Prof., Dept. of Civ. Engrg., National Univ. of Singapore, Department of Civil Engineering 1 Engineering Drive 2, Singapore Tel: , Fax: Sr. Struct. Engr., Sembawang Marine and Offshore Engineering, 60 Admiralty Road West, #01-02, Singapore

2 NOTATION A = Area of cross-section E = Modulus of elasticity of material E eff = Effective modulus of elasticity of cable member E eq = Equivalent modulus of elasticity of cable member H = Horizontal projected length of cable L = Length / Chord length of member P = External load P b = Inelastic buckling load P 0 = Cable pretensioned force r = Radius of gyration T = Cable tension w = Unit weight of cable t = Change in temperature α = Coefficient of linear expansion σ y = Design strength of material 2

3 1 INTRODUCTION Advanced analysis has been shown to be suitable for designing complex structures that are slender and flexible [1-3]. Design of steel structural systems based on direct advanced analysis has become simpler as it only requires the system to be stable and capable of resisting the loads with sufficient factor of safety. This is in line with the modern limit-state design codes which prefer global analysis of systems behaviour to gain "direct" insights to their strength and stability compared to the conventional approaches involving the use of semi-empirical formulae for member stability checks [4-7]. For aesthetic reason, some skylight structures require members to be sleek so that they occupy less physical area to allow passage of natural light. In other structural systems like telecommunication masts, which carry predominantly selfweight and wind loads, small member sizes and hence slender members are essential to achieve an economical design. Since slender members are susceptible to buckling, their effects need to be included in the limit analysis of the overall structural system. Advanced analysis, which can capture both members' and system's instability, would help in identifying the most critical component members governing the limiting strength of the system [1]. The limiting strength of slender structural systems can be enhanced by any one of the following three approaches. The first approach is to apply prestressing forces to the structural system by inducing initial stresses opposing those caused by the design load. For a double-layer grid system, Hanaor and Levy [8] and Levy et. al. [9] carried out experiments to study the effect of prestressing the critical strut members having brittle type buckling characteristics by imposing lack of fit. They found 40% increase in the load carrying capacity of the prestressed configuration compared with the non-prestressed configuration. In the second approach, pretensioned cables are added to the structural system to enhance the buckling strength of the most critical members. The pretensioned force in the cables induces initial stresses in the critical members to counteract the design forces experienced by them so that their failure can be delayed. Belenya [10] reviewed various investigations carried out in this area and presented design and erection concepts for prestressed steel trusses, beams, bridges, masts and towers, and other special-purpose structures. Liew et al [11] proposed the use of nonlinear analysis to study the self-erection of framework using cable tensioning technique. The stability conditions of the structural frameworks during and after erection were investigated. The third approach is to adopt an altogether different structural system which uses only high-strength cables and steel struts to resist tension and compression respectively. This form of structural system is referred to as cable-strut structure or tensegrity system [12-16]. In the present work, the effectiveness of using prestressed cables in improving the limit-state behaviour of steel space frame system is studied using the advanced plastic hinge method [1]. Cable elements are introduced in the space frame system to: (i) reduce the load acting on the critical members, (ii) produce counteracting prestressing force in the most critical members, and/or (iii) reduce the effective unbraced length of critical members by generating lateral restraining forces. 3

4 Case studies are presented to demonstrate how one or more of the above will affect the maximum strength of space frame systems by the provision of additional cables and strut members. 2 MODELLING OF PRETENSIONED CABLE When the lateral load effect is negligibly small, the cable element may be modelled as an equivalent truss element [11] whose length is equal to the chord length of the cable as shown in Fig. 1. The initial pretensioned force in the cable, P 0, can be specified either as self-equilibrating axial load or as an equivalent thermal load. In the latter case, the pretensioned force, P 0, can be converted to an equivalent change in temperature, t, by using the following equation [11]: t = P 0 / (E α A) (1) where, A is the cross sectional area, α is the coefficient of thermal expansion and E is the modulus of elasticity of the cable material. When the lateral load effect is significant, the geometric change in cable length must be included in the analysis of cable structures. The total apparent change in the length of a cable is the result of the sum of the three distinct actions: (i) the elastic strain in the cable material which is linear and governed by the modulus of elasticity; (ii) change in the sag of the cable which is strictly a geometric effect, independent of material stress and varies in a nonlinear manner with the axial tensile force in the cable; and (iii) the relative movement of individual strands in the cable to rearrange themselves into a more consolidated cross-section under load. This rearrangement is known as constructional stretch. The constructional stretch is permanent and occurs below a specified tensioned force. This permanent deformation is usually removed in the manufacturing process by prestretching the cable to a load larger than the working load. The deformations which are non-permanent can be compensated for by using a reduced effective modulus of elasticity of the cable, E eff, which is independent of tension force in the cable. Since the variation of the sag with the axial force in the cable is nonlinear, the axial stiffness of the cable will vary with the increase in axial force. To account for the effects of material deformation, constructional stretch and the change in sag in the cable members, the actual modulus of elasticity of the cable material, E, is replaced with an equivalent modulus of elasticity, E eq, proposed by Buchholdt [17] as: Eeff Eeq = 2 (2) ( wh) AEeff T where, w = unit weight of the cable, H = horizontal projected length of the cable, A= cross-sectional area, and T = tensile force in the cable. When modelling the cable element as equivalent truss element, the elastic, geometric and higher-order stiffness matrices that are essential for nonlinear analysis are computed using the equivalent modulus of elasticity of the cable given by Eqn. (2). 3 MODELLING OF INELASTIC BEAM-COLUMN ELEMENT Analysis of rigid-jointed space frames is based on an advanced plastic hinge analysis program [1]. The main feature of the advanced analysis formulation is to use one element per member to model each structural component and to obtain a realistic representation of material and geometric nonlinear effects. The analysis operates on 4

5 element stress resultants, i.e., forces, bi-moments and torsion. The beam-column is subject to end forces acting on three transitional and three rotation degrees of freedom at each end node. The effects of large displacements and coupling between lateral deflection and axial strain are included by using nonlinear strain relations. The elastic tangent stiffness matrices are calculated from closed form expressions, with no numerical integration over the element cross section or over the element length. They contain the influence of axial force acting on lateral deformations of the member (P-δ effect). The detailed formulation of the three-dimensional inelastic beam-column is reported in Ref. [18] and will not be repeated here. For the tie members, the axial load-elongation relationship is obtained based on bilinear elastic-plastic stress-strain curve of the material. The axial load deformation relationships obtained for the strut and tie elements model the large displacement inelastic behaviour of an axially loaded member. The ultimate strength of axially loaded member is calculated directly based on the design code requirement, and hence no separate check is required for member stability and strength. A detailed description on the strut and tie model for advanced analysis of space structures can be found in Ref. [19]. 4 ANALITICAL PROCEDURE Analysis of cable-tensioned structures may be performed in two stages. In the first stage, only cable tensioning forces are applied, and incremental nonlinear analysis is carried out to obtain the load-displacement behaviour of the structure. Each increment of pretensioned forces is transformed to the global coordinate system as internal resistance vector. By solving the incremental-iterative equilibrium equation for the unbalanced forces, the new equilibrium configuration of the system, which satisfies the force equilibrium at each node, can be found. The next increment of pretensioned force is then applied and the same iterative process is repeated. This procedure is continued till the desired magnitude of pretensioned forces is applied to all cable members [11]. In the second stage, external loads are applied on the prestressed structure. Again, incremental nonlinear analysis is carried out until the factored design load level is reached. The structure is said to satisfy the ultimate design limit state if the load factor associated with the limit of resistance obtained from the advanced analysis is higher than the design load factor. The following criteria are assumed as premature failure conditions for the pretensioned system: (i) cable slackening caused by compressive force before the service load is reached, (ii) tensile force in the cable reaching the Minimum Breaking Load (MBL) before the full design loads are applied, or (iii) system's limiting capacity is reached before the full design loads are applied. The slackening or compressive failure of a cable is a gradual process and impairs the serviceability of the structural system. On the other hand, tensile failure of a cable is an explosive event involving release of strain energy and ought to be classified as an ultimate limit state for all intents and purposes. Hence, the structural system is considered to reach its limit states of design when any of the cables in the system reaches either compression or tension failure in their respective limit states. 5

6 5 VERIFICATION STUDIES Two numerical examples, (i) a two-dimensional cable truss and (ii) a threedimensional saddle net, are considered to verify the cable element formulation. The inelastic beam-column formulations have been verified elsewhere [1, 18-20] and will not be repeated herein. 5.1 Two Dimensional Cable Truss The two-dimensional cable truss as shown in Fig. 2 consists of an upper and a lower cable, and 14 vertical hangers [20]. The span of the structure is 3.03m, and the height at the supports is 0.81m. Vertical hangers are equally spaced along the span. All nodes are prevented from lateral displacement in the out-of-plane direction. A single point vertical load of N is applied at the lower node of the fifth hanger from the left support. In Fig. 3, the vertical and the horizontal displacements of upper and lower chords obtained by the present analysis are compared with those obtained by Broughton and Ndumbaro [20] using a geometric nonlinear analysis program. Close agreement between the results is observed. 5.2 Three Dimensional Saddle Net The three dimensional saddle net [21] of plan dimension 50 m 40 m consists of 142 pretensioned cable elements spaced at 5 m 5 m grid as shown in Fig. 4(a). The modulus of elasticity and area of cross-section of all cables are 147 kn/mm 2 and 306 mm 2, respectively. A pretensioned force of P o = 60 kn is applied to all the cables. The external loading consists of 1 kn force indicated as P along the positive X and Z directions at all the free nodes on one-half of the net as shown in Fig. 4(a). The displacements obtained from the present method are compared in Table 1 with those obtained by Lewis [21] using Dynamic Relaxation Method. The results for displacements and cable forces predicted by the present nonlinear analysis method are very close to those obtained by Lewis [21]. The maximum errors in the computation of displacements are within ±2%. To study the effect of cable pretensioned force on the system's limit load, loaddisplacement analyses are carried out on the cable net system by varying the cable pretensioned force viz. P o = 10, 20, 30 and 40kN. The results are shown in Fig. 4(b). External loads [denoted by P in Fig. 4(a)] are applied incrementally until failure is detected in any one of the cables represented by cable slackening or the cable reaching the minimum breaking load (MBL). The resulting limit loads, P, for the four load cases are, 1.23, 3.07, 2.98 and 2.88kN respectively. In the case of P o = 10kN, slackening of cables is detected at P= 1.23kN. In all the other three cases, the cable joining the nodes 2 and 12 attained its MBL at the system's limit load. The analyses show that P o = 20kN is the optimum pretensioned force that can be applied to achieve the maximum load-carrying capacity for this particular structure. From Fig. 4(b), it can be observed that the limit load values decrease if the cable tensioned forces are increased beyond the optimum value of 20kN. When the pretensioned force is very high, the cable reaches the MBL at an early stage of loading. The centre node deflection due to initial prestressing force is in the upward direction as indicated by the negative displacement values when the applied force is zero (i.e., P = 0). The initial upward displacement increases with the increase in pretensioning force in the cables. Considering the displacement pattern due to live 6

7 load, wind load etc., the magnitude of pretensioned force in each cable can be designed to match the desire final configuration of the structure. 6 NUMERICAL EXAMPLES In the following sections, three examples viz. (i) a shallow spherical dome (ii) a barrel vault and (iii) a pretensioned column are presented to study the limit-state behaviour of cable-tensioned systems. The advantages of using cable-tensioning technique in improving the performance of various space frame systems are illustrated. 6.1 Shallow Dome The shallow spherical dome, similar to the one used by Yang and Kuo [22], is shown in Fig. 5(a). A total of 24 members form a spherical surface with a 10 m diameter base and maximum rise of m at the centre. All the supports are assumed to be simply supported. CHS mm are used for all structural members with Modular of Elasticity E = 205 kn/mm 2 and design strength σ y = 275 N/mm 2. The members are assumed to be either weld-connected (rigidly jointed) or mechanically connected (pin-jointed). The dome is subject to the following two load cases, (i) all the loads are concentrated as a point load applied at the crown joint or (ii) the loads are applied as distributed point loads acting at the unsupported nodes Pin-Jointed Dome The dome is idealised as pin-jointed space frame and elastic-plastic nonlinear analysis is carried out for both single- and multiple-point load cases. The loaddisplacement curves are plotted in Figs 6 and 7 with respect to the vertical degree of freedom at the crown joint. The limit loads for the single- and multiple-point load cases are 52.7 kn and kn ( kn) respectively as shown in Table 2. It is found that under single concentrated load case, the limit load is attained due to the buckling of top six members whereas under multiple concentrated loading, the limit load is attained due to the buckling of bottom twelve members Rigid-Jointed Dome The dome is idealised as a rigid-jointed space frame and nonlinear inelastic analyses are performed for both the single- and multiple-point load cases. The corresponding load-displacement curves with respect to the vertical degree of freedom at the crown joint are given in Fig. 8. The limit loads obtained are 64.7 kn and kn ( kn) for single- and multiple-point load cases representing a capacity increase of 22% and 41% respectively compared to pin-jointed dome. These increases are due to joint rigidity which enhances the overall stability of the dome structure Dome with Cable-Strut System A set of cable-strut system is proposed as shown in Fig. 5b. It consists of a vertical strut of CHS 48.3 mm 3.2 mm x 4000mm long connecting the crown joint and six cables at m height as shown in Fig. 5(b). The properties of each cable are: 1 19 stainless steel strands of 10 mm diameter with MBL = 71.1 kn and E = kn/mm 2. The cable and strut system is studied for two cases: (i) no pretensioned force is applied to the cables and (ii) with a pretensioned force of 10 kn applied at each cable. The analyses are carried out for both single- and multiple-point load cases with pinned and rigid joint idealisations for the same dome. The respective load-displacement curves for the vertical degree of freedom at crown joint are shown in Figs. 6 and 7 for pin-jointed dome under single- and multiple-load points respectively. The results for 7

8 rigid-jointed dome under single- and multiple-load points are shown in Figs. 8, 9 and 10. The corresponding limit loads obtained from the analysis are sumerised in Table 2. If the cables are not prestressed, slackening occurs in all the cables when multiple point loads are applied. Hence, this condition is not considered for this particular load case. With the provision of the cable-strut system, there is a substantial increase in the limit load of the system and a marginal decrease in vertical displacement for single concentrated loading case. However, such beneficial effect is not observed in the case of multiple-point load case [Figs. 7 and 10]. The reason is that in single concentrated load case, a portion of the concentrated load at the crown is transferred to the cables through the central strut resulting in a reduction of compressive forces on the top six critical members. However, in multiple point loads case, the bottom twelve members are the critical members and the provision of cables and strut does not reduce the loads at the bottom members. It should be noted that prentensioned cables provide cambering effect against deflection under gravity load. They also increase the overall stiffness of the structure. 6.2 Single-Layer Barrel Vault System The span length, depth (in the longitudinal direction) and central rise of the barrel vault are 29.97m, 21m and 1.973m respectively. It consists of 212 tubular members, with E = 205 kn/mm 2 and σ y = 275 N/ mm 2, arranged as shown in Figure 11. The boundary nodes along the two straight edges are constrained to move in any directions but free to rotate. All members are rigidly connected to each other. The structure is subject to a uniformly distributed dead load (DL) of 0.50 kn/ m 2 and live load (LL) of 0.75 kn/m 2 per plan area. Wind load (WL) computed using BS: 6399 Part 2 [24] with basic wind speed of 30 m/s is considered. The wind load may act in the longitudinal (θ = 90 ) or transverse direction (θ = 0 ) as shown in Figs. 12 a & 12b. Considering the two external wind pressure coefficients ±0.2 for zone D and wind acting in transverse (θ = 0 ) and longitudinal (θ = 90 ) directions, there are four wind load cases. The net wind pressures are indicated as Zones A, B, C and D in Fig. 12. A notional load (NL) of 0.5% of (1.4DL + 1.6LL) is considered [25] to account for the initial imperfection effects at the system level. The loads are applied as nodal loads. Nonlinear analyses are carried out for the following load combinations [26]: DL + 1.6LL + NL [DL + LL + WL(θ = 0 ; D = 0.2)] [DL + LL + WL(θ = 0 ; D = +0.2)] (5) [DL + LL + WL(θ = 90 ; D = 0.2)] [DL + LL + WL(θ = 90 ; D = +0.2)] In the nonlinear analysis, load factor of 1.0 means that the applied loads correspond to the particular factored load combination, whereas load factor greater than 1.0 represents the safety margin of which the system s limit load is larger than the required design load Barrel Vault with Arch Edges Unsupported Nonlinear inelastic analysis is carried out for the barrel vault in which the arch edges are free to translate. CHS 273mm x 12.5 mm is selected for arch members and CHS 193.7mm x 12.5 mm for all other members. The limit load factors for the system under various load combinations are shown in Table 3. The most critical load combination is Load Case 3, in which the central arch members placed are subject to 8

9 the maximum force and they become the most critical members which govern the failure of the system. Figure 13a shows the deformed shape of the structure at the limit load under Load Case 3. Figure 13b shows the global load-displacement curve for the central node (denoted as "C"). The member axial load-displacement curves for the arch members are shown in Fig. 14. The limit load of the barrel vault system is governed by inelastic buckling of arch members (labelled as 79, 80, 89 and 90) as shown in Fig. 14. The minimum steel weight for the barrel vault of which the arch edges are free to translate is 41.3 tons Barrel Vault with Arch Edges Stiffened by Tensioned Cables A set of pre-tensioned radial cables are provided along the two arch edge, as shown in Figure 15, to enhance the buckling resistance of the arch members. The cable support point is located at the eaves level and is anchored to the adjacent structure to prevent in-plane movement. The cables provide planar restraints to the arch members so as to increase their load carrying capacities. In the process of prestressing, the unbalanced force components in the cable would displace the system in a direction opposite to that due to wind uplift resulting in an overall reduction of final deflections due to service wind loads. Figure 16a shows an application of radial tensioned cables for stiffening a barrel vault skylight system used in actual construction. Figure 16b shows the detailed design of the cable-support point. Considering a pretensioned force of 80kN in each cable and various tubular sections for the members, nonlinear analyses were performed for the barrel vault system with all possible load combinations. A minimum proportional load factor of 1.08 is obtained for the most critical load combination: 1.2[DL + LL + WL] when CHS x 8 mm section is chosen for all the members. The limiting load is reached when the cables slacken which triggers the collapse of the overall system. The displaced configuration of the structure at the limit of resistance under load case 3 is shown in Figure 17a. The global load-displacement curve for the central node C and axial load-displacement curves for the critical arch and cable members are presented in Figs. 18 and 19a-c for the load combination 3. The total steel weight of the pretensioned system is only 22.5 tons, which is 45% lighter than the structure with unsupported arch edges. From the cable axial load-displacement plots shown in Figs. 19b, it is observed that the application of external loads on the system induces additional tension force on the left side cables (denoted as member 213 and 222 in Fig. 17). However, it induces compressive force on the right side cables (members 219 and 228) causing release of pretensioned force as shown in Fig. 19c. Thus, the load carrying capacity of the system is controlled by the cable slackening leading to the loss of stiffness of the arched members. Figure 19a shows that arch members 80 and 90 with a size of CHS mm can provide an axial resistance of 390 kn compared to CHS mm with an axial resistance of 295 kn under unsupported arch edge condition. Thus, the pretensioned cables act as effective restraints to the arch members and enable the use of smaller size member to carry more axial compressive load than the case in which the arch members are unsupported. Larger pretensioned forces may be required to prevent cable slackening so as to enhance the load carrying capacity. For example, instead of applying 80 kn pretensioned force in the cables, a 100 kn pretensioned force is applied, the resulting system's limit load is found to be increased by 7%. Since the applied initial pretensioned force cannot exceed the maximum breaking load of the cable, the load 9

10 carrying capacity of the system cannot be increased unboundedly by increasing the cable pretensioning force. The location of the cable-support points is also an important parameter for design consideration. It is observed that the load-carrying capacity increases marginally if the cable support points are lowered and vice versa. For example, if the cable support point is lower by 0.75 m, the limit load factor becomes 1.09 for the load combination 3. If it is moved up by the same amount, the limit load factor reduces to Prestressed Column A 20m long tubular strut of CHS mm subject to axial compression is considered. The material properties of the column are, E = 205 kn/mm 2 and σ y = 275 N/mm 2. Considering the full length, the slenderness ratio of the column is L/r = 660, which is very slender compared to the maximum slenderness ratio stipulated by steel design code. In order for the column to resist compression load, intermediate lateral restraints are required. A system of pre-tensioned cables and struts is proposed, as shown in Fig. 20, to increase the resistance of the slender column against axial compression. By providing three square panels with their diagonal members rigidly connected to the column, a 20m long column is separated into four segments each of 5 m long. Such a cable-tensioned column has been used in actual construction for supporting an entrance canopy, as shown in Fig. 21. The column is under tension when it is supporting the weight of the canopy and is in compression when it is subject to wind uplift. The pretensioned column is analysed with two initial geometric conditions: a) a perfectly straight column; and b) column with an initial out-of-straightness imperfection as shown in Fig. 23a Perfectly Straight Column In the case of perfectly straight column, there is theoretically no lateral deformation until bifurcation occurs. That is, the buckling load cannot be obtained directly from the load-displacement analysis. However, the buckling load is implicitly obtained when the first negative diagonal element is encountered in the structural stiffness matrix. Since the inelastic buckling load of the column depends mainly on the magnitude of the pretensioned force, P 0, buckling analyses are carried out assuming that the column is subject to various magnitude of prestressed forces. The inelastic buckling capacity, P b, obtained from the analyses is plotted against the pretensioned force P 0 as shown in Fig. 22. It is observed that P b increases with P 0 only up to a value of P 0 = 1.2 kn and decreases thereafter. As the pretensioned force is increased, the lateral restraining forces induced by the cables acting on the column also increase. This results in shorter unbraced length of the column and the corresponding increase in the inelastic buckling load. However, the buckling load reduces when the pretensioned force is greater than 1.2kN. The reason for this is that the pretensioned forces induce additional axial load in the column. The perfectly straight column reaches its peak capacity of 47.3kN when the pretensioned force is 1.2kN Column with Initial Imperfections The stiffened column is assumed to have an initial out-of-straightness in a form of a half sine-wave with maximum out-of-straightness of span/500 at the mid length. Load-displacement analysis is performed by applying a prestressed force of 1.2 kn to each cable. The limit load of the column is found to be 0.5 kn. This is 10

11 because the cables with a pretensioned force of 1.2 kn is not sufficient to provide adequate lateral restraints to the imperfect column. Subsequently, the nonlinear analyses were carried out with increased pretensioned forces. A limit load of 39 kn was achieved with a pretensioned force of 10 kn. The deformed configuration and the load-displacement curve are shown in Figs. 23(b) and (c). The study shows that the geometrical imperfections have significant effects on the limit load of the stiffened column. Numerous studies in the past tend to focus on buckling analysis of structures without considering the effect of structural initial imperfection. The present investigation shows that the ultimate behaviour of a prestressed column is very much affected by the structural initial imperfection. Higher pretensioning forces may be required to provide adequate restraints to members of imperfect geometry if their compression resistance is to be fully realised. It should also be noted that the magnitude of pretensioned force in the cables lying in same plane should be the same in order to avoid initial imperfections that would otherwise affect the ultimate resistance of the column. 7 CONCLUSIONS The examples presented in this paper illustrate that the proposed nonlinear analysis tools can be used for limit-load analysis of cable-tensioned structures. With careful consideration of member and system imperfection effects, it is possible to determine system capacity with good accuracy. This is in line with the modern design codes which allow the use of advanced analysis for designing steel structures. Investigations have been carried out on the use of prestressing in various space structures, such as cable-net, dome, barrel vault and 3-D column. Their design limitstate behaviours are studied considering various structural arrangements. Computation procedure was developed and the effect of the prestressed value was analysed. The following observations are drawn from the case studies presented in this paper: (1) The assumed cable and strut system is found to be effective in increasing the limit load of the pin- and rigid-jointed spherical domes by 47% and 37%, respectively. However, the proposed cable-strut system is designed to resist the single-point load applied at the crown joint and may need to be modified for other load cases. (2) In the case of barrel vault system, a set of pretensioned radial cables act as lateral restraints to the arch members and lead to a steel weight reduction of 45% under the same loading condition. (3) A pretensioned stiffening system is effective in providing lateral restraints to slender columns. The presence of physical imperfections in the column reduces the ultimate resistance quite substantially and hence their consideration in the design of such structures is essential. For the shallow dome, the provision of pretensioned cable-strut system is to reduce the forces acting on the critical members by altering the load path. In the case of the barrel vault and slender column, the force in the cable creates a stress opposite in the direction to that from the load in the most stressed members of the system. Pretensioning is an effective means to reduce steel consumption and cost of structures and to enhance their strength and rigidity. Effective use of pretensioning in slender space frame design requires the development of innovative system, ones 11

12 different from conventional non-prestensioned structures. In general pretensioned structures are of more complicated arrangement whose behaviour is to a great extent affect by nonlinearity of deformation and stresses, initial imperfections, fabrication technique and erection sequence. It is good practice to allow for these factors in analysis and design. ACKNOWLEDGEMENTS The investigation presented in this paper is part of the research programs on computer aided second-order inelastic analysis for frame design being carried out in the Department of Civil Engineering, National University of Singapore. The work is funded by research grants (RP920651) made available by the National University of Singapore. 12

13 Table 1 Comparison of displacements in saddle net (1) (2) Node By Lewis[27] (mm) By present analysis (mm) No. X Y Z X Y Z Maximum error = [(2) (1)] 100/(1)% = ±2% Type of joints Table 2 Limit load capacity of shallow spherical dome Loading details Limit load capacity (kn) of dome with No cables Non-pretensioned cables Pin-jointed crown all joints Rigidjointed crown Pretensioned cables all joints Table 3 Limit load factors for barrel vault system Arch BCs. Member size, mm (CHS) Weight (Tons) Limit load factors for load combination Free & edge Cables

14 REFERENCES 1. Liew J.Y.R. and Tang, L.K., Advanced Plastic Hinge Analysis for the Design of Tubular Space Frames, Engineering Structures, Elsevier Science, 22, 2000, Kitipornchai, S., Advances in nonlinear analysis of spatial and thin-walled structures, Int Num Meth Eng., 1995, 38(5). 3. Akio Hori and A Sasagawa, Large Deformation of Inelastic Large Space Frame II: Application, J. Structural Engineering, ASCE, 126(5), 2000, Liew J Y R, Chen W.F. and Chen, H., Advanced inelastic analysis of frame structures, Journal of Constructional Steel Research, UK, 55:(1-3), July 2000, White, D. W., Liew, J. Y. R. and Chen, W. F., Toward Advanced Analysis in LRFD, Plastic Hinge Based Methods for Advanced Analysis and Design of Steel Frames - An Assessment of The State-Of-The-Art, Structural Stability Research Council, Lehigh Univ., Bethlehem, PA., March 1993, Ziemian R.D., McGuire, W., and Deierlein, G.G. Inelastic Limit States Design Part II: - three Dimensional Frame Study, J. Structural Engineering, ASCE, 118(9), Chan, S.L. and Zhou, Z. H., On the Development of a Robust element for Secondorder NnliinearIntegrated Design and Analysis (NIDA), J Constructional Steel Research, Elsevier, 47(1/2), 1998, ,. 8. Hanaor, A. and Levy, R., Imposed lack of fit as a means of enhancing space truss design, Space Structures, Elsevier App. Science, 1, 1985, Levy, R., Hanaor, A. and Rizzuto, N., Experimental investigation of prestressing in double-layer grids, J. Space Structures, 9(1), 1994, Belenya, E., Prestressed load-bearing metal structures, MIR publishers, 1977, 463pp. 11. Punniyakotty, N M, Liew J.Y.R. and Shanmugam, N.M., Erection of Steelworks by Cable-Tensioning Technique, J Structural Engineering, ASCE, Volume 126, Issue 3, 2000, Eekhout, M., Advanced glass space structures, Space Structures 4, Thomas Telford, London, 1993, McClleland, N.C. and Perry, J.C., Glass support structure interaction, analysis and design, The Third Int. Kerensky Conf. on Global Trends in Structural Engineering, Singapore, 1994,

15 14. Takeuchi, T., et. al., A practical design and construction of tension rod supported glazing, Proc. of the IASS-ASCE Int. Symposium on Spatial, Lattice and Tension structures, Georgia, 1994, Pellegrino, S., A class of tensegrity domes, J. of Space Structures, 7(2), 1992, Campbell, D. M., Chen, D., Gossen, P.A. and Hamilton, K. P., Effects of spatial triangulation on the behaviour of Tensegrity domes, Proc. of the IASS-ASCE Int. Symposium on Spatial, Lattice and Tension Structures, Georgia, 1994, Buchholdt, H. A., An Introduction to Cable Roof Structures, Cambridge Univ. Press, Cambridge, New York, 1985, 257pp. 18. Liew, R J Y, Chen, H, Shanmugam, N E and Chen W F, Spatial instability and second-order plastic hinge analysis of space frame structures, Research Report, CE029/99. Singapore: National University of Singapore, April 1999, 50 pp. 19. Liew, J. Y. R., Punniyakotty, N. M. and Shanmugam, N. E., Advanced analysis and design of spatial structures, J. Constr. Steel Res., Elsevier Science, 42(1), 1997, Broughton, P. and Ndumbaro, P., The Analysis of Cable & Catenary Structures, Thomas Telford, London, 1994, 88pp. 21. Lewis, W. J, A comparative study of numerical methods for the solution of pretensioned cable networks, Proc. Int. Conf. On Non-Conventional Structures, Vol. 2, Civil Comp. Press, Edinburgh, 1987, Yang, Y. B. and Kuo, S. R., Theory & Analysis of Nonlinear Framed Structures, Prentice Hall, Singapore, 1994, 579 pp. 23. Yamada, S. and Taguchi, T., Nonlinear buckling response of single layer latticed barrel vaults, Proc. of the IASS-ASCE Int. Symposium on Spatial, Lattice and Tension Structures, Georgia, 1994, BS6399: Part2, Loading for buildings Part 2: Code of Practice for Wind Loads, British Standards Institution, London, BS5950 Part1, Structural use of steelwork in building Part 1: Code of Practice for Design in Simple and Continuous Construction: Hot Rolled Section, British Standards Institution, London,

16 List of Figures Figure 1 Axial cable element Figure 2 Two-dimensional cable truss Figure 3 Comparison of results for cable truss Figure 3 (a) Vertical displacement at lower chord locations Figure 3 (b) Horizontal displacement at upper and lower chord locations Figure 4 Three-dimensional saddle net Figure 4 (a) Structure and loading details Figure 4 (b) Effect of cable pretension on system limit load Figure 5 Shallow spherical dome structural configurations Figure 5 (a) Without cable and strut system Figure 5 (b) With cable and strut system Figure 6 Pin-jointed dome under vertical load at crown Figure 7 Pin-jointed dome under uniform loads at free nodes Figure 8 Load-displacement curves for rigid-jointed dome without cables and strut Figure 9 Load-displacement curve for rigid-jointed dome with non-pretensioned cables and strut for vertical load at crown Figure 10 Load-displacement curves for rigid-jointed dome with pretensioned cables and strut Figure 11 Single layer barrel vault system Figure 12 Wind loads on barrel vault system Figure 12 (a) Wind in longitudinal (θ = 90 ) direction Figure 12 (b) Wind in transverse (θ = 90 ) direction Figure 13 Barrel vault with unsupported arch edges Figure 13 (a) Displaced configuration at limit load condition Figure 13 (b) Global load-displacement curve for central node C Figure 14 Axial load-displacement curves for barrel vault with unsupported edges for 1.2 [DL+LL+WL (θ = 0 ; D = +0.2)] loads Figure 15 Barrel vault with radial cables on arch edges Figure 15 (a) Isometric view Figure 15 (b) Side elevation

17 Figure 16 (a) General arrangement of arch and radial cables Figure 16 (b) Details for radial cables support point Figure 17 Displace configuration at limit load Figure 18 Global load-displacement curve at central node C Figure 19 Axial load-displacement curves for barrel vault with radial cables on arch edges for 1.2 [DL+LL+WL (θ = 0 ; D = +0.2)] loads Figure 19 (a) For members 80 and 90 Figure 19 (b) For cable members 213 and 222 Figure 19 (c) For cable members 219 and 228 Figure 20 Figure 21 Figure 22 Cable-strut stiffened column Application of cable-strut stiffened members for supporting an entrance canopy Effect of pretension in cables of stiffened column without any member imperfection Geometrically imperfect cable-strut stiffened column with P 0 = 10 kn Figure 23 Figure 23 (a) Initial configuration Figure 23 (b) Displaced configuration at limit load Figure 23 (c) Axial load-displacement curve

18 Figure 1 Axial cable element Figure 2 Two-dimensional cable truss

19 Figure 3 (a) Vertical displacement at lower chord locations Figure 3 (b) Horizontal displacement at upper and lower chord locations Figure 3 Comparison of results for cable truss

20 Figure 4 (a) Structure and loading details Figure 4 (b) Effect of cable pretension on system limit load Figure 4 Three-dimensional saddle net

21 Figure 5 (a) Without cable and strut system Figure 5 (b) With cable and strut system Figure 5 Shallow spherical dome structural configurations

22 Figure 6 Pin-jointed dome under vertical load at crown Figure 7 Pin-jointed dome under uniform loads at free nodes

23 Figure 8 Load-displacement curves for rigid-jointed dome without cables and strut Figure 9 Load-displacement curve for rigid-jointed dome with non-pretensioned cables and strut for vertical load at crown

24 Figure 10 Load-displacement curves for rigid-jointed dome with pretensioned cables and strut

25 Figure 11 Single layer barrel vault system

26 Figure 12 (a) Wind in longitudinal (θ = 90 ) direction Figure 12 (b) Wind in transverse (θ = 90 ) direction Figure 12 Wind loads on barrel vault system

27 Figure 13 (a) Displaced configuration at limit load condition Figure 13 (b) Global load-displacement curve for central node C Figure 13 Barrel vault with unsupported arch edges

28 Figure 14 Axial load-displacement curves for barrel vault with unsupported edges for 1.2 [DL+LL+WL (θ = 0 ; D = +0.2)] loads

29 Figure 15 (a) Isometric view Figure 15 (b) Side elevation Figure 15 Barrel vault with radial cables on arch edges

30 Figure 16 (a) General arrangement of arch and radial cables Figure 16 (b) Details for radial cables support point

31 Figure 17 Displace configuration at limit load Figure 18 Global load-displacement curve at central node C

32 (a) For members 80 and 90 (b) For cable members 213 and 222 (c) For cable members 219 and 228 Figure 19 Axial load-displacement curves for barrel vault with radial cables on arch edges for 1.2 [DL+LL+WL (θ = 0 ; D = +0.2)] loads

33 Figure 20 Cable-strut stiffened column

34 Figure 21 Application of cable-strut stiffened members for supporting an entrance canopy Figure 22 Effect of pretension in cables of stiffened column without any member imperfection

35 (a) Initial configuration (b) Displaced configuration at limit load Figure 23 (c) Axial load-displacement curve Geometrically imperfect cable-strut stiffened column with P 0 = 10 kn