50 J.M. Rotter described derive from large displacements; small displacement ideas and small displacement analysis lead to serious misinterpretation of both test results and appropriate design measures. Under fire loading, the dominant phenomenon in determinate structures is material degradation. In highly redundant structures, the single most important factor is the effect of thermal expansion. Where this leads to high stresses, damage occurs to the material (plasticity or concrete cracking). Where instead, large displacements develop in a post-buckling mode, the expansion is accommodated without so much damage, loads are carried by membrane action and the performance is considerably improved. Large displacements are commonly associated with bending failures, but here they may be beneficial, occurring with membrane thrusts, or with membrane tensions, depending on the thermal regime. A key conclusion is that the design criteria must not be based on limitation of deflections during the fire. The effects of high temperatures on structures are best interpreted in the context of Eqns 2-4, which permit the roles of expansion and material degradation to be properly identified and which decouple the displacement and stress fields. Thermal expansion often couples with large displacements to produce effects which appear counter-intuitive to the conventionally trained structural engineer. These findings are of fundamental importance to our understanding of composite frames in fire. They have major implications for the development of design philosophies and procedures. ACKNOWLEDGEMENTS The support of DETR for funding this research through the PIT scheme is gratefully acknowledged. The author is most grateful for many discussions and calculations provided by Dr Asif Usmani and Dr Abdel Sanad of Edinburgh University and Dr Mark O'Connor and Dr Xiu Feng of British Steel. REFERENCES 1. ABAQUS (1997) "Abaqus Theory Manual and Users Manual", Version 5.7, Hibbit, Karlsson and Sorensen Inc., Pawtucket, Rhode Island, U.S.A. 2. Euler, L. (1744) "Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti", Lausanne & Geneva, Reprinted 1952 in Leonhardt Euleri Opera Omnia, Series 1, Vol. 24, Bern. 3. ENV 1994-1-2 (1995) "Design of Composite Steel and Concrete Structures: Structural Fire Design", Eurocode 4 Part 1.2, CEN, Brussels. 4. Kirby, B.R. (1997) "British steel technical European fire test programme - Design, construction and results", in Fire, Static and Dynamic Tests of Building Structures, eds G.S.T. Armer and T. O'Dell, Spon, London, ppl 11-126. 5. Martin. D,M. (1995) "The behaviour of a multi-storey steel frame building subject to natural fires", British Steel Technical Report No. 2. 6. Moore, D.B. (1997) "Full scale fire tests on complete buildings", in Fire, Static and Dynamic Tests of Building Structures, eds G.S.T. Armer and T. O'Dell, Spon, London, pp3-15. 7. Newman, G.M. (1997) "Design implications of the Cardington fire research programme", in Fire, Static and Dynamic Tests of Building Structures, eds G. Armer and T. O'Dell, Spon, pp 161-168. 8. Rotter, J.M., Sanad, A.M., Usmani, A.S. and Gillie, M. (1999) "Structural performance of redundant structures under local fires" Proc., Interflam '99,8th Int. Fire Science and Engg Conf., Edinburgh, 29 June- 1 July, Vol. 2, pp 1069-1080. 9. Sanad, A.M., Rotter, J.M., Usmani, A.S. and O'Connor, M.A. (1999) "Finite element modelling of fire tests on the Cardington composite building" Proc., Interflam '99,8th Int. Fire Science and Engg Conf., Edinburgh, 29 June - 1 July, Vol. 2, pp 1045-1056. Design Formulas for Stability Analysis of Reticulated Shells S. Z. Shen Harbin University of Civil Engineering and Architecture 202 Haihe Road, Harbin 150090, China ABSTRACT The aim of the paper is to propose some kind of design formulas for stability analysis of single-layer reticulated shells, reflecting the recent advances in theoretical study but simple in form for the convenience of practical application. For this purpose a comprehensive parametrical analysis of stability behaviors of single-layer reticulated shells of different types with various geometric and structural parameters has been carried out based upon complete load-deflection response analysis with consideration of the effects of initial imperfections and unsymmetrical distribution of loads More than 2800 examples of reticulated shells of prototype were analyzed, and the plentiful results obtained were thoroughly studied. As a result, practical formulas for predicting limit loads of reticulated domes, reticulated vaults with different supporting conditions, as well as reticulated shallow shells, obtained by regression analysis, were proposed. KEYWORDS Stability analysis, Complete load-deflection analysis, Limit load, Design formula, Reticulated shells, Reticulated domes, Reticulated vaults, Reticulated shallow shells, Reticulated saddle shells INTRODUCTION The stability analysis is known as the key problem for the design of reticulated shells. The stability character of a complicated structure with numerous degrees of freedom such like reticulated shells can be revealed clearly and accurately by complete load-deflection response analysis, in which the 51 52 S.Z. Shen structural response under loading is regarded as a continuous process rather than some individual structural properties such as critical load, buckling mode and etc The complete load-deflection curves give a more perfect picture about the behaviors of the structure. The deformation shapes varying with the loading process, and the possible buckling of different orders and of different characters (over-all or local buckling, bifurcation or limit point), with the corresponding critical loads and buckling modes, can be revealed in their proper order by the complete-process analysis. With the development of non-linear finite element analysis and methods for tracing equilibrium path, it can be said that the problem of stability evaluation of reticulated shells on the basis of complete load- deflection response analysis has been well solved from the viewpoint of theoretical side. However, engineers working in design practice still feel puzzled when dealing with stability problems of reticulated shells. The theoretical method as discussed above seems to them too complicated for direct application. So it's desirable to propose some kind of design formulas, reflecting the recent advances of theoretical study but simple in form for the convenience of practical application. For this purpose a comprehensive parametric analysis of stability behaviors of different types of single-layer reticulated shells with varying geometric and structural parameters has been carried out based upon complete load-deflection analysis with consideration of the effects of initial geometric imperfections and unsymmetrical distribution of loads. The "Consistent Mode Method" is proposed for the imperfection analysis. This method assumes the geometric imperfection of a reticulated shell to be distributed in consistence with the buckling mode of first order of the structure, which is supposed to be very likely the most unfavorable for the expected limit load of the reticulated shell. More than 2800 examples of reticulated shells of prototype were analyzed, and the plentiful results obtained were thoroughly studied. As a result, practical formulas for predicting limit loads, obtained by regression analysis respectively for different types of reticulated shells, rather simple for application but based upon accurate theoretical procedure as described, were proposed. The complete-process analysis was carried out on the basis of geometrically non-linear finite element method, without consideration of material non-linearity, because it would be too time-consuming and hence very difficult at present to carry out such a large-scale parametric analysis with consideration of both geometric and material non-linearity. Besides, the reticulated shells under service condition are working in elastic range, and the material non-linearity would lead to some decrease of safety reserve in load-capacity of the structure; the latter effect could be assessed by some independent study [Wang]. A special computer program for complete load-deflection response analysis of complicated structures based upon non-linear finite element method, compiled by the author's team, was used for the parametric analysis and was proved to be effective. THE PLAN OF PARAMETRIC ANALYSIS The parametric analysis was carried out for single-layer reticulated domes, vaults, elliptical paraboloid shells ( EP shells, or shallow shells ) and hyperbolic paraboloid shells ( HP shells, or saddle shells ). For the purpose of practical application, all the reticulated shells analyzed are of prototype with member sections determined by calculation as in practical design. As the usual case in China, circular steel tube members and welded hollow spherical joints are used for these structures. Design Formulas for Stability Analysis of Reticulated Shells 53 The reticulated domes analyzed have net systems of Kiewitt type ( K8 and K6 ), Schwedler type and geodesic type. For the Kiewitt dome K8, which was taken as the typical system to be studied, four different spans ( L = 40, 50, 60 and 70m ) and four different raise-span ratios ( f/L = 1/5, 1/6, 1/7 and 1/8 ) with four sets of member sections for each size, i.e. 64 different domes were analyzed. The effects of initial imperfections of consistent mode and with a maximum value equal to L/1000 were analyzed for each of the domes; besides, for part of the domes the effects of imperfections with different values from L/1000 to L/100 were systematically studied. It's assumed that the dead load ( g ) is uniformly distributed over full span, while the live load ( p ) could also be distributed over half-span (uniformly as well ); three different ratios of live load to dead load were considered: p/g = 0, 1/4 and 1/2. According to this plan, near 500 examples of K8 domes were analyzed, and, if including the similar study for K6 domes, Schwedler domes and geodesic domes, the non-linear complete-process analysis was carried out for 840 reticulated domes. The reticulated vaults might have three kinds of supporting conditions: supported along the boundary, supported along two longitudinal edges, or supported at two end cross-sections by means of rigid diaphragms. The triangular net system, consisting of longitudinal and two sets of diagonal members, as the most popular one is assumed for the reticulated vaults. The ratio of length to wave-span ( width ) of the vault ( L/b ) is a main factor effecting the structural behavior, and different ratios: L/b = 1.0, 1.4, 1.8, 2.0, 2.2, 2.6 and 3.0 were considered in the parametric analysis, keeping the wave- span of the vaults unchanged" b - 15m. Different raise-span ratios (f/b) and several sets of member sections were assumed, and effects of different initial imperfections and unsymmetrical load distributions were studied. Besides, relatively long vaults supported at two ends may be provided with intermediate diaphragms, and the effects of these diaphragms were analyzed. In sum, 1220 examples of complete load-deflection response analysis were carried out for single-layer reticulated vaults, including 350 examples for vaults with boundary supporting, 54 examples for vaults supported along two longitudinal edges and 816 examples for vaults supported at two ends. The elliptical paraboloid reticulated shells are usually used for rectangular or square plans, supported along four sides by means of rigid diaphragms. The surface of a elliptical parapoloid is formed by a vertical parabola (as the generatrix), moving along another vertical parabola in the transverse direction. In engineering practice the parabolas are usually replaced with circle arcs, and the EP shells are often called as known as the shallow shells. Three kinds of plan dimensions (30"30m, 40"40m and 30"45m), three different raise-span ratios (f/L = 1/6, 1/7 and 1/8) and four sets of member sections for each size were considered. The raise-span ratio f/L is defined for each of the two directions, and equal ratios are assumed for both directions. Two kinds of net systems: triangular system and orthogonal system with diagonals were compared. As before, the effects of initial imperfections and unsymmetrical distributions of loads were studied. There were in all 783 examples of reticulated shallow shells to be analyzed. The complete load-deflection behavior of hyperbolic paraboloid reticulated shells has its specific characteristic. In this paper 14 HP shells of regular rhombic (square) plan with diagonal length equal to 60m (taken as the span of the shell) were analyzed with consideration of the effects of different net systems, different raise-span ratios and different rigidities of edge beams. According to the plan of parametrical analysis as described above, more than 2800 examples of 54 S.Z. Shen reticulated shells of different types were analyzed. For each of the examples the load-deflection curve drawn for the joint with maximum deflection at the end of iteration was taken to represent the analyzed structure. From the viewpoint of practical application, the critical point of first order and the related structural properties (critical load, buckling mode, and etc.), as well as the effects of different factors to these properties, are of primary interest. So it's usually sufficient to take the beginning part of the load-deflection curve (just ensuring a certain post-buckling path to be reserved ) for investigation. After this part, the load-deflection curve could be varied and colorful, theoretically very interesting but less practical significance because of the too large deflections. Due to the limited length of the paper just some examples of the curves obtained will be shown in the later sections. STABILITY OF RETICULATED DOMES The buckling of reticulated domes in most cases has a form of local concave on the surface as shown in Fig.l, starting from snap-through of some joint and gradually expanding its area to become a concave. The concave emerges at different place for different type of reticulated domes: it starts from some joint of a main rib for Kiewitt domes, from some joint of the third ring (from bottom) for Schwedler domes, and from some joint on the triangular surface for geodesic domes. The first buckling of a dome is characterized as a limit point of the load-deflection curve, and the corresponding critical load is taken as the limit load of the dome. Figure 1: Buckling modes of reticulated domes Because of the excellent 3-dimensional behavior of dome structure the unsymmetrical distribution of load shows very little effect to the limit load. For comparison, the load-deflection curves for three different distributions of loads ( p/g = 0, 1/4 and 1/2 ), taking the total load ( p+g ) as the ordinate, have been put together for each of the domes. It's surprise to find that these three curves nearly coincide one with another. Meanwhile, the reticulated domes are very sensitive to the initial geometric imperfections. As an example, the load-deflection curves for a Kiewitt dome with L=60m,f/L=l/8 and with nine different values of initial imperfections ( the maximum value of imperfections r = 0, 3, 6, 10, 20, 30, 40, 50 and 60 cm, respectively ) are shown in Fig.2a. It can be indicated that the imperfections studied attain a rather big value (up to L/100), and the presented study is primarily of theoretical interest. The nine corresponding curves are put together for comparison. It's noticed that the curves vary with the increase of imperfections in a good regularity. Then, if studying the load capacity of the domes, the Design Formulas for Stability Analysis of Reticulated Shells 55 variation of limit load with the increase of imperfection values is shown in Fig.2b. It's seen that the limit load drops rapidly at beginning, reaches a minimum value (approximately 50% of the limit load of the corresponding perfect dome) as r=20 cm (i.e. L/300). Afterwards, the curve somewhat lifts again, which seems inconsistent with the normal idea we might have. In fact, as the initial imperfections go beyond some limit, the dome seriously deviates from its spherical shape and would become a" distorted" structure somewhat different from the original one. It can be seen from Fig.2a the character of the load-deflection curves gradually varies with the increase of imperfection values: the limit buckling for normal domes changes into bifurcation buckling for the domes with overlarge imperfections. Besides, the" distorted" domes are less rigid, the deflections develop rapidly, and the possible increase of critical load is meaningless in practice. Figure 2 : a. Load-deflection curves of a dome with different imperfection values b. Limit loads varying with increase of initial imperfection The Schwedler domes with initial imperfections behave very similarly to Kiewitt domes, only the limit load reaches the minimum value more rapidly ( as r = L/1000 - L/500 ). The response of geodesic domes is somewhat different: the limit load, as well as the rigidity of the dome, drops continuously with the increase of imperfection value within the studied range ( up to L/100 ), which demonstrates the special significance of error control in erecting geodesic domes. For practical purpose, it seems suitable to appoint a value of L/500 - L/300 as the acceptable maximum error of erection for reticulated domes, and to assume the limit load of the practical domes with imperfections equal to 50% of that of the corresponding perfect structures. The geodesic domes can also satisfy such an agreement. How to make use of the large number of results obtained from the parametrical analysis for the purpose of practical design? As one of the possible ways, it's considered preferable to propose some appropriate formulas for predicting limit loads of reticulated shells by regression analysis of the data obtained from the parametrical analysis. For reticulated domes such a formula is perhaps not so difficult to work out, because there exists analytical formula of linear theory for predicting limit loads of continual thin domes, the form of which could be taken as a reference. The formula for predicting limit loads of reticulated domes is then suggested in the form as follows : 56 S.Z. Shen ~/BD qcr=g ~ (1) R 2 in which: R radius of curvature of the dome ( m ); B the equivalent membrane rigidity of the dome ( kN/m ); D the equivalent bending rigidity of the dome ( kN.m ); and K coefficient, determined by regression analysis. The rigidity of reticulated shells is not uniform over the surface. So the proper position for calculating the value of B and D should be in consistence with the buckling mode of domes. For example, the buckling of Kiewitt dome occurs, as described above, at some joint of a main .rib, i.e., the limit load of the dome is primarily determined by the rigidity of the area round this joint. So B and D should be calculated according to the net size and member sections in this area. Similarly, for Schwedler dome or geodesic dome the joint of the third ring or the joint on the triangular surface should be taken as the calculated position, respectively. The formulas for calculating B and D are given in the Appendix to the paper. Besides, the reticulated shells are usually an-isotropic, and B and D in Eqn. 1 could be considered as the mean value of the rigidities in both main directions. Due to the limited length of the paper the process of regression analysis is neglected, just indicating that the coefficients K calculated for different types of reticulated domes are very close one to another. This demonstrates that the formula in the form of Eqn.1 really reflects the characteristic features of the stability behavior of reticulated domes, and that it's correct to select the position for calculating B and D according to the buckling mode of different domes. It's finally suggested that the limit load of practical reticulated domes of different types with initial imperfections to be controlled within a limit less than L/500 can be determined by a unified formula as follows: ~BD qcr = 1.05 R T- ( 2 ) STABILITY OF RETICULATED VAULTS Vaults supported along the boundary a. Vault supported along boundary b. Vault supported on longitudinal edges Figure 3 : Buckling modes of reticulated vaults Design Formulas for Stability Analysis of Reticulated Shells 57 The buckling mode of reticulated vaults supported along the boundary in most cases has the form of a concave with three half-waves in the cross-section as shown in Fig.3a. For relatively long vaults (L/b >~ 2.6) unsymmetrical mode with two half-waves (Fig.3b) is also possible, as in the case of vaults supported on two longitudinal edges. It demonstrates the restricting effect of the end diaphragms for vaults with L/b<2.6. For short vaults with L/b~< 1.4, such restricting effect becomes rather strong, and the buckling may have a mode of even higher order with four half-waves in the cross-section. The effect of length-span ratio L/b to the limit load of reticulated vaults supported along the boundary is very obvious, as shown in Fig.4. The limit load drops rapidly with the increase of L/b at beginning, but gradually reaches a limit, in most cases as L/b = 2.6, but for high vaults with f/b = 1/2 the curve becomes even more slowly, usually as L/b>~ 3.0. Figure 4 : Limit load of reticulated vaults supported along boundary with increase of L/b The reticulated vaults supported along the boundary are not so sensitive to the initial imperfections. Systematical analysis shows that the reduction in limit load at most consists of 20%, even as the range of initial imperfections studied approaches a value as big as b/100. The unsymmetrical distribution of loads nearly does not affect the stability behavior of reticulated vaults of this type. As revealed by comparative analysis, the limit load defined as the total load p+g does not decrease under unsymmetrical loading, only with an exception for short vaults of L/b~< 1.2. For practical application, the effect of unsymmetrical loading to the limit loads of these short vaults can be considered by a coefficient K2 calculated as: K2 = 0.6 + 0.4 / ( 1 +2 p/g ) (applicable as p/g = 0-~2) (3) It is somewhat difficult to derive the regression formula for the limit loads of reticulated vaults, because there does not exist any theoretical form that could be referred to like the case with domes. Anyway, some preliminary forms can be assumed based upon the ideas obtained from the parametrical analysis. After repeated comparison by trial and error method, the following formula is finally suggested for predicting the limit loads of reticulated vaults supported along the boundary: 58 S.Z. Shen 911 0-4 B22 029 qcr = 72.0 R3 L/b) 3 + 1.95 x 1 R(L/b) + 75.0 (R + 3f)b 2 ( 4 ) in which the indexes 11 and 22 indicate the longitudinal and transverse direction, respectively. The effect of initial imperfections has been considered in the formula. For short vaults with L/b 1.2 coefficient K2 as given by Eqn.3 should be multiplied to consider the effect of possible unsymmetrical distribution of loads. Vaults supported on longitudinal edges To study the vault supported along the boundary, it can be imagined that, with the increase of length of the vault, the effect of the end diaphragms to the behavior of center part of the vault would decrease, and the behavior of the vault in general is gradually close to that of a vault supported only on two longitudinal edges. It's seen now from Eqn.4 that the limit load decreases with L/b increasing, and only the third term of the formula will be retained as L/b approaches infinitive. It leads to a very interesting question: if the third term of the formula can be used to evaluate the limit load of the vault supported on two longitudinal edges. This theoretical deduction was proved by the complete-process analysis of 54 examples of reticulated vaults of such kind. It's then concluded that the limit load of reticulated vaults supported on two longitudinal edges can be predicted by the formula as follows: qcr =75.0 D22 (R +3f)b2 (5) The effect of unsymmetrical distribution of loads need not be considered for vaults of this type. Vaults supported at two ends The vault supported at two ends has free longitudinal edges, but strengthened by edge beams with certain rigidity. Such a vault is behaving like a huge beam with curve cross-section supported at two end diaphragms. With the increase of length of the vault, the member forces in the vault, and hence the cross-section of the members, increase as well, that is not like the vault supported along the boundary. So, if keeping the other parameters unchanged, the member sections determined by calculation as in practical design are different for vaults with different length. Under this condition, the parametrical analysis shows that the limit loads of the vaults are rather stable for different values of length-width ratio L/b. That is, the limit load does not depend evidently upon the ratio L/b. The buckling mode of the vaults has more likely a form of overall deformation of the surface together with the bending and torsion of edge beams. The raise-width ratio f/b has obvious effect to the limit load of the vaults: the vault with bigger f/b ratio shows higher stability load-capacity. The vaults supported at two ends are not so sensitive to initial imperfections. As revealed by systematical analysis, if taking b/300 as the acceptable maximum value of initial imperfection, the reduction in limit load does not exceed 18%. The limit load is evidently affected by unsymmetrical distribution of loads. Such effect becomes sufficiently developed as early as p/g = 0.5, and the further reduction in limit load is not evident for Design Formulas for Stability Analysis of Reticulated Shells 59 the bigger values of ratio p/g. For practical application the effect of unsymmetrical loading can be considered by the coefficient K2 determined as follows: K2 = 1.0- 0.2 L/b ( L/b = 1.0-2.5 ) ( 6 ) K2 = 0.5 ( L/b = 2.5"`3.0 ) The intermediate diaphragms for relatively long vaults supported at two ends may be arranged at an interval roughly equal to the width b in order to increase the overall rigidity of the surface and hence to raise the limit load of the vaults. On the bases of systematical comparison it's suggested for practical application that the effect of intermediate diaphragms can be considered by a coefficient K3 calculated by Eqn.7. For vaults with intermediate diaphragms unsymmetrical distribution of loads does not affect the limit load any more. K3 = 1.52- 0.12 L/b (applicable as L/b = 1.4 3.0) (7) After repeated comparative analysis the regression formula for predicting the limit load of reticulated vaults supported at the ends is proposed as follows: qcr - 0.063 + 0.138 + 0.083 ~ ( 8 ) in which the factor CL 0.96 + 0.16(1.8 - L/b )4 ; ih and Iv the horizontal and vertical linear rigidity of the edge beam, respectively, which can be calculated as ( for latticed beams as usually used ): Ih,v = E(Alr~2+A2rRR)/L, in which A~ and A2 are the cross-section areas of two chords of the latticed beam, r~ and r2 are the corresponding radiuses of inertia. The effect of initial imperfections has been included in the formula. The effect of unsymmetrical loading should be considered by the coefficient K2 given by Eqn.6. For vault with intermediate diaphragms the limit load determined by Eqn.8 should be multiplied by coefficient K 3 given by Eqn.7, but without consideration of coefficient K2 9 STABILITY OF RETICULATED SHALLOW SHELLS The stability behaviors of shallow shells with triangular net system and with orthogonal net system are somewhat different each from other. In the comparative analysis the corresponding shells of these two kinds were designed to have equal weight. Under this condition, the limit load of the shells with triangular system is higher than that of the other. The buckling of the shells with orthogonal system more likely has a form of local concave on the surface, but for shells with triangular system there appears more evident character of overall deformation, i.e., more obvious deformations arise in a much wider range of the surface. The shells with triangular system show higher sensitivity to initial imperfections. According to the comparative analysis, if the maximum value of initial imperfection is controlled as L/500"`L/300, the reduction in limit load of shells with triangular system and with orthogonal system can be taken in practical application as 35% and 25%, respectively. The shallow shells are very sensitive to unsymmetrical distribution of loads. As an example, the limit loads of a shell with a plan of 30"30m and with orthogonal system varying with the increase of ratio . sufficient to take the beginning part of the load-deflection curve (just ensuring a certain post-buckling path to be reserved ) for investigation. After this part, the load-deflection curve could. element modelling of fire tests on the Cardington composite building" Proc., Interflam '99,8th Int. Fire Science and Engg Conf., Edinburgh, 29 June - 1 July, Vol. 2, pp 104 5-1 056. Design. effects which appear counter-intuitive to the conventionally trained structural engineer. These findings are of fundamental importance to our understanding of composite frames in fire. They have major