THE DEVELOPMENT OF A COMPONENT-BASED CONNECTION ELEMENT FOR ENDPLATE CONNECTIONS IN FIRE

Transcription

1 THE DEVELOPMENT OF A COMPONENT-BASED CONNECTION ELEMENT FOR ENDPLATE CONNECTIONS IN FIRE FLORIAN BLOCK 1, IAN BURGESS 2, BUICK DAVISON 3 and ROGER PLANK 4 ABSTRACT This paper describes the development of a component-based element for endplate connections in fire. The reported research is part of an ongoing project aimed at understanding joint behaviour in fire. The paper summarises the derivation of the stiffness matrix of this new element, based a on spring model, and the incorporation of the element into the non-linear finite element program Vulcan. It also states the component characteristics that have been used for the individual zones of deformation in an endplate connection. Furthermore, the additional features of the element, necessary for correct response at elevated temperatures, like the consideration of the temperature distributions across the connection as well as cooling and unloading are summarised. The proposed element is then used to predict the momentrotation curves of connection experiments at ambient and elevated temperatures. Finally, the advantages and limitations of the new high temperature connection element are listed. 1. INTRODUCTION Traditionally, beam-to-column connections are assumed to have sufficient fire resistance due to their cooler temperatures and slower rate of heating than the members to which they are attached, caused by the large concentration of thermal mass in the connection. However, from the full-scale fire tests in Cardington and the subsequent research, it has been observed that connections are more vulnerable than assumed. This is mainly caused by the forces and deformations to which a connection is subjected during a fire, which are significantly different from those normally assumed in design. The internal forces change from moment and shear at ambient temperature to moment, shear and compression due to 1 Research Student, Dept. of Civil & Structural Engineering, University of Sheffield, Sheffield S1 3JD, UK cip2fmb@sheffield.ac.uk 2 Professor, Dept. of Civil & Structural Engineering, University of Sheffield, Sheffield S1 3JD, UK ian.burgess@sheffield.ac.uk 3 Senior Lecturer, Dept. of Civil & Structural Engineering, University of Sheffield, Sheffield S1 3JD, UK j.davison@sheffield.ac.uk 4 Professor, School of Architecture, University of Sheffield, Sheffield S1 3JD, UK r.j.plank@sheffield.ac.uk

2 restrained thermal expansion of the beams in the early and intermediate stages of a fire, and finally to shear and tension in the later stage when applied loads are supported through catenary action. It is obvious that it is very difficult to generate these loading conditions realistically through experiments, except in full-scale testing. Furthermore, a large number of experiments would be required because of the large variety of possible connection details and the structure-connection interaction. Hence, alternative ways to investigate and design connections in-situ for the fire case are required. One such way could be the use of the Component Method to predict the detailed behaviour of a connection, in combination with a structural finite element program which is capable of simulating the non-linear behaviour of structures in fire. However, this combination is only possible if the principles of the Component Method are included into a connection element. 2. COMPONENT JOINT MODELLING The Component Method was initially developed for the ambient temperature design of semi-rigid joints by Tschemmernegg et al. 1 and later introduced into the Eurocode EC The original feature of this method is to consider any joint as a set of individual basic springlike components. For each component the stiffness and maximum force is computed and assembled to form a spring model which represents the behaviour of the whole joint. In order to describe the behaviour of an isolated connection at elevated temperature, the Component Method has been used successfully by a number of researchers. Leston- Jones 3 was the first to apply the method to his cruciform tests; Al-Jabri 4 used the method to model the flexible endplate behaviour of his high temperature experiments, and Spyrou 5 conducted a large number of high-temperature component tests and combined the investigated components using a simple two spring model. Simones da Silva et al. 6 used the component models given in EC3-1.8, in combination with the temperature reduction factors given in EC , to model the above-mentioned cruciform tests. However, none of these studies combined the Component Method directly with whole-frame action. Furthermore, apart from a limited study by Spyrou, the effects of axial load on the connection in fire have not been considered. A first attempt to include high-temperature connection behaviour into the finite element program ADAPTIC using the Component Method was done by Ramli Sulong et al. 8 However, from this publication it was not clear how the model was implemented into the finite element software and what was used for the individual component characteristics. At ambient temperature however, researchers have included the detailed connection response into global frames using component-based connection elements 9, 1, 11. The main characteristic of these connection elements is that they are single finite elements, instead of assemblies of spring elements, which obtain their stiffness matrix from a number of translational springs. These internal springs represent various parts of the connection, as is assumed in the Component Method. The spring characteristics are based on analytical or empirical models. The development of such a connection element for endplate connections at elevated temperatures will be shown in the following section. 3. FORMULATION OF THE CONNECTION ELEMENT The connection element presented in this paper has been included in the non-linear finite element program Vulcan as a 3D finite spring element with zero length. The proposed element is compatible with Vulcan s general beam-column elements, which allow the

3 simulation of whole steel and composite frames. Currently, the following parts of a real joint are included in the element: 1. Endplate in bending 2. Column flange in bending 3. Bolts in tension 4. Column web in compression The first three components form the tension zone of the connection and are combined as two T-stubs in series. An additional shear spring had to be included in order to transfer the vertical load from one node to another. This shear spring is assumed to be rigid at present, but the formulation of the element allows the implementation of slip and shear failure of the bolts. It was important to position this shear spring vertically in order to uncouple the vertical and horizontal stiffnesses of the element. The shear zone in the column web is not yet included, which theoretically limits the use of the element to internal joints with equal moments in which the column web does not experience shear deformations. The assumed position of the connection element can be seen in Fig. 1(a) below. (a) 4 2, 3 1, 3 (b) k 1 lc,1 lt,2 lt,1 V i, w i i j V j, w j lc,2 5 lt,3 M i, φ i N i, u i k 2 M j, φ j N j, u j φ w u mm Fig. 1 - Position of the connection element within a joint (a) and a basic spring model (b) In order to include the element into Vulcan, the component behaviour had to be formulated following the principles of the finite element method. Therefore, the behaviour of the connection is represented in the form: F = K C u (1) However, due to the non-linear behaviour of the individual connection components in fire and also to the highly non-linear behaviour of the connected structural members, it is necessary to solve equation (1) iteratively using the tangent stiffness K C, incremental forces F and displacements u. Therefore, equation (1) is rewritten as: F = K C u (2) with T F = N x i Vy i Vz i M x i M y i M z i N x j Vy j Vz j M x j M y j M,,,,,,,,,,, z, j (3) and T u = u φ φ φ φ φ φ i vi wi x, i y, i z, i u j v j wj x, j y, j z, j (4) k 3

4 During the iterative process, Vulcan assumes u based on the previous step s stiffness and the connection element has to recalculate its stiffness matrix in accordance with the proposed displacements and also the updated incremental force vector F. Both are then returned to the main routines of the program, and a convergence check, based on the out-of-balance forces, is performed. If convergence is reached, the equilibrium of the next load or temperature is calculated, otherwise the incremental displacement is varied until equilibrium is reached. The derivation of the tangent stiffness matrix of the connection element is shown in the next section. 3.1 Basic spring model and derivation of the stiffness matrix In order to derive the stiffness matrix of the connection element, the simplest case of the element with only three springs is considered, as shown in Fig. 1(b). As in the original component method it is assumed that a joint will primarily deform in its plane, and all other out-of-plane and torsional degrees of freedom (DOF) are assumed to be rigidly connected. This reduces the problem to three DOF per node. If now each of these DOF is moved individually, as shown in Fig. 2, and the resulting spring forces are calculated, it is possible, with the help of some geometrical considerations and the spring characteristics, to derive the stiffness matrix of the basic connection element. δ 1,i F 1 = k 1 δ 1,i δ 1,i F 1 = k 1 δ 1,i F 2 = k 2 δ 2,i u i = 1 w i = 1 φ i = 1 l 1 δ 2,i F 3 = k 3 δ 3,i F 3 = k 3 δ 3,i l 3 δ 3,i Fig. 2 - Stiffness derivation process for node i of the basic connection element The resulting stiffness matrix for the basic two dimensional connection element, is shown in equation (5) below. δ 3,i ( k + k ) ( l k l k ) ( k + k ) ( l k l k ) k2 k2 K C = k k l k l k l k + l k l k l k l k + l 2 k ( l1k 1 l3 k 3 ) ( l1 k 1 + l3 k 3 ) ( l1k 1 l3k 3 ) ( l1 k 1 + l3 k 3 ) ( k + k ) ( l k l k ) ( k + k ) ( l k l k ) ( ) ( ) ( ) ( 3 3 ) (5) However, most connections will have more than one bolt row and therefore more than one tension spring. Furthermore it is possible that, due to restrained thermal expansion, the whole connection will be under compression, which requires the implementation of a

5 compression spring at the location of each beam flange. Hence the stiffness matrix in equation (5) is expanded to three dimensions and n bolt rows, as shown in equation (6). In which K 11 K 15 K 11 K 15 K 33 K 33 K 51 K 55 K 51 K 55 K C = K 11 K 15 K 11 K 15 K 33 K 33 K 51 K 55 K 51 K 55 n 2 K 15 = K 51 = lt, i k T, i + lc, i k C, i (7), (8) K = k + k 11 T, i C, i i = 1 i = 1 K n 2 i = 1 i = 1 n = k s K 55 = lt, i k T, i + lc, i k C, i i = 1 i = 1 (6) (9), (1) It should be noted that the lever arm l of an individual spring is measured from the reference axis, which is sited at the position of the node, positively defined in accordance with the local z-axis of the element. The governing equation (2), together with the stiffness matrix (6), forms the backbone of the connection element, but it is equally important to specify the forcedisplacement-temperature (F-δ-θ) behaviour of each of the individual components. 3.2 Tension spring characteristics Traditionally, the tension components in an endplate connection are represented as a T-stub of a certain width. At ambient temperature, the force-displacement behaviour of T- stubs has been extensively studied. At elevated temperatures however, only Spyrou has investigated this component and derived simplified models, predicting multi-linear F-δ-θ relationships up to bolt failure for the three possible failure modes of a T-stub as shown in Fig. 3. The simplified models are solely based on the geometrical and material properties and the temperature of the component; these have been introduced into the connection element. 1 st failure mode 2 nd failure mode 3 rd failure mode Fig. 3 - The three failure modes of a T-stub As mentioned previously, a certain width has to be specified for the T-stub. This width ensures that the isolated T-stub behaves in similar fashion to the bolt row it represents

6 in either the endplate or column flange. Zoetemeijer 12 specified a number of yield line patterns, which can be used to calculate this equivalent width. These have been adopted by EC From here, the yield line patterns for extended and flush-endplates and column flanges, as shown in Fig. 4, have been taken and implemented into the connection element. (a) (b) Fig. 4 - Considered yield line pattern in column flanges and flush-endplates (a) and in extended endplates (b) If the distance between bolt rows is below a certain limit, and if the bolt rows are not separated by stiffeners, it is possible that two or more bolt rows fail together in a common yield line pattern, which reduces the strength and stiffness of the tension components. However, due to the complexity of the calculation procedure and the considerable programming effort involved, group effects have not yet been considered in the connection element. In an unstiffened endplate connection, two T-stubs in series have to be considered per bolt row, representing the endplate and the column flange respectively. In the new element, the F-δ-θ curves for both of these T-stubs springs are calculated initially and then, in order to satisfy equilibrium in the bolt row, the total displacement of the bolt row is distributed in accordance with the stiffness of each spring, ensuring equal forces in both springs. With the resulting tangent stiffness of both springs, an equivalent stiffness for this bolt row is calculated, and the resistance is then defined by the weaker T-stub. Further, it is assumed that each T-stub shares the equivalent bolt length equally, as is assumed as in EC Compression spring characteristics The characteristics of the compression spring (i.e. the behaviour of a column web in compression) has been studied extensively at ambient temperature. At elevated temperature, this component was also investigated by Spyrou and later by the first author of this paper 13. In the latter study, the effects of axial load in the column section on the compression zone were investigated at elevated temperature, resulting in a simplified model for this component. This model for the F-δ-θ relationship of the compression spring has been implemented into the connection element. 3.4 Relocation of the reference axis If a composite structure is modelled in Vulcan, the position of the nodes of the slab elements as well as the beam elements is normally located at the mid-plane of the concrete slab. In order to move the beam elements into the correct position, an offset equal to the distance between the centre of the slab and the centre of the beam is used. To be fully compatible with the other elements, the same principle has to be used with the connection element. Therefore, the same offset as for the beam elements is subtracted from the lever arm of each spring, and so the reference axis of the element is moved to the middle of the slab element. This simple modification allows the connection element to be used to model

7 composite construction. However, shear connector elements and a fairly dense slab element mesh have to be used in order to allow the correct slippage between the beam and the slab and also the correct approximation of cracking in the concrete. 3.4 Temperature effects The degeneration of the strength and stiffness of the connection material with increasing temperatures is included in the connection element by using the temperaturedependent strength reduction factors for mild steel given in EC3-1.2 for the column and the endplate. For the bolts, however, the temperature reduction factors derived by Kirby 14 have been used. A comparison between the T-stub experiments by Spyrou and the connection element has shown that the EC3-1.2 reduction factors for bolts gave very conservative results. The Young s modulus of the bolts is reduced in accordance with the temperature reduction factors for mild steel as concluded by Spyrou. In order to specify the temperatures of the individual components, a single timetemperature curve is used in combination with a temperature pattern. This pattern consists of temperature multipliers allowing the specification of the temperatures of the column flange, bolts and endplate for each bolt row individually, and also for the column flange and the column web in the two compression zones. This technique gives the user the capacity to consider any temperature distribution across the connection taken from experiments, analysis or design codes. The effects of cooling on beam-to-column connections can be critical for the survival of a structure, due to the large tensile forces which are developed in the beams when the thermal strain is recovered, leaving the post-fire beams considerably shorter than they were originally due to permanent deformations 15. This effect has been accounted for in the beam element in Vulcan, which makes it possible to expose the connection element to the correct forces during cooling of the structure. However, when high-strength bolts are heated above their annealing temperature around 6 C and allowed to cool down naturally, the strengthening effects of quenching and tempering used during manufacture of the bolts vanishes, and the bolt material returns to its base material which is considerably weaker as Kirby has observed. Unfortunately, to date there is no experimental data available on bolts which are loaded during cooling, and therefore this effect can not yet be included in the connection element. It is assumed that the bolts, as well as the material of the endplate and the column, regain their full strength when cooled down to ambient temperature. 3.5 Unloading of the connection element As mentioned previously, when a structure starts to heat up the beams introduce compressive force, in addition to the pre-existing moment, into the connections. This causes the tension zone of the connection to unload until, if enough restraint is present, the connection is fully compressed. If in a later stage of the fire the beams cannot support the applied load in bending any more, catenary action introduces tensile forces into the connection. If this force is large enough, both compression zones in the connection unload and eventually all bolt rows are in tension. A similar effect is caused by cooling of the beam as described above. In order to respond correctly to such a changing combination of loads, a connection element for elevated temperatures needs a robust loading-unloading-reloading approach. Therefore, the classic Massing rule with memory effects has been included into the new element. The response of the connection element simulating a symmetric connection with two bolt rows under alternating axial loads can be seen in Fig. 5 below.

8 Axial Force [kn] (a) Axial Force [kn] Load steps Axial Displacement [mm] Fig. 5: Cyclic loading (a) and response (b) of the new element due to axial load The difficulty with this approach is that the tensile and compressive forces in the connection do not act in the same place. Therefore, it is necessary, for correct prediction of the internal forces of the springs, that the compression springs are only allowed to act in compression, whereas the tension springs are allowed to act in both tension and compression, but only until the endplate at the height of the beam flange adjacent to the tension spring contacts the column. It is then assumed that all compression force is taken by the much stiffer compression spring. In reversal, when a compression spring is plastically deformed and unloads until the endplate loses contact with the column, the adjacent tension spring has to start taking load from this deformed position. (b) 4. VALIDATION AT AMBIENT AND ELEVATED TEMPERATURES The connection element has been validated against a number of experiments, at ambient and elevated temperatures, found in literature. Unfortunately, there are no available connection tests including axial load in the beams at elevated temperature, and therefore this novel aspect of the element could not be tested. 4.1 Ambient Temperature Experiments In order to validate the ambient temperature behaviour of the connection element two different types of tests were used. The first test series, by Girão Coelho 16, was designed to investigate the rotational capacity of the endplate side of extended endplate connections. Therefore a short beam (IPE3) was connected to a heavy column section (HE34M) by three bolt rows (M2-8.8) and endplates of varying thickness. In the first (FS1a) and second (FS2a) tests endplate thicknesses t p of 1mm and 15mm were used, respectively.

9 Moment [knm] Girao Coelho - FS1a Connection element Rotation [mrad] Rotation [mrad] Fig. 6 - Comparison of the new element with an extended endplate tests by Girão Coelho Moment [knm] Girao Coelho - FS2a Connection element From Fig. 6 above, it can be seen that the new element compares reasonably well with the experimental results of both tests. However, in both tests the rotation of the connection is slightly under-predicted, which may be explained by the fact that the original yield-line patterns for the conversion between the T-stub and the real endplate have been calibrated to the ultimate resistance but are used throughout the whole test. It should be noted that for these tests the whole effective bolt length has been allocated to the endplate side of the connection, as the column flange is extremely stiff, and so the datum of the T-stub displacement in the endplate is assumed to be at the centre of the nut on the column-flange side of the connection. The second group of experiments has been designed to fail in the compression zone of the column web. Firstly, the ambient-temperature cruciform test by Leston-Jones on flush endplate connections (t p = 8mm) with three bolt rows (M16-8.8) connecting small beam (UC152x152x23) and column (UB 254x12x22) sections was modelled. The test failed by plastic buckling of the column web and large deformations of the column flange in tension. In Fig. 7(a) the good correlation between the response of the connection element and the experimental M-Φ data can be seen. Moment [knm] (a) (a) Test - Leston-Jones Connection element Moment [knm] Bailey and Moore - Test 1 N =.8 Npl Connection element - with axial load effect Connection element - without axial load effect Rotation [mrad] Rotation [mrad] Fig. 7 - Comparison of the new element with a flush endplate test by Leston-Jones (a) and an extended endplate test Bailey and Moore (b) (b) (b)

10 The second example is again a cruciform test, conducted by Bailey and Moore 17, but this time using reasonably large beam (UB457x191x74) and column (UC254x254x17) sections connected by a very stiff extended endplate (t p = 3mm) and four bolt rows (M3-8.8). The test was designed to investigate the influence of axial column load (N =.8 N pl ) on the compression zone. As the connection element is able to account for this effect two runs were conducted, the first without consideration of the axial load and the second one with the axial load considered. From Fig. 7(b), the excellent comparison between the proposed element and the experiment can be seen, if the reducing effects of the axial column load are included. In general, it can be said that the proposed connection element in Vulcan compares accurately with experimental data at ambient temperature. However, the rotational capacity is always conservatively predicted, which suggests that further refinement of the tension zone behaviour is required as the predicted rotation was limited in most cases by the fracture of the bolts, which did not always occur in the tests. 4.2 Elevated-Temperature Experiments As part of the validation process of the new connection element, the elevatedtemperature connection tests conducted by Leston-Jones have been modelled. The size of the connected sections, and the connection itself, are the same as in the ambient-temperature test discussed in the previous section. However, instead of loading the cruciform assembly until failure of the connections occurs, a constant moment was applied to the connection and then the temperature was increased by approximately 1 C/min until failure occurred or the test had to be terminated due to spatial constraints. As a temperature distribution in the connection, the average temperature multipliers for each component over the duration of the whole test were used. In total four tests with applied connection moments ranging from 5kNm to 2kNm have been compared with the response of the proposed element, which can be seen in Fig. 8 below. Steel temperature [ C] knm 15 knm Test 1 Connection element Connection rotation [mrad] Steel temperature [ C] knm 2 knm Test Connection element Connection rotation [mrad] Fig. 8 - Comparison of the new element with the high-temperature tests on flush endplate connections by Leston-Jones In general, a very good correlation is seen between the tests and the predictions of the proposed element. However, the new element under-predicts the rotations found in the test slightly, which is probably because of the non-consideration of group effects between the bolt

11 rows in the column flange. Also, as was observed at ambient temperature, the rotational capacities of the connections are underestimated due to bolt failure. 5. DISCUSSION The advantages of the connection element in its present form are evident, as it opens up the possibility to combine the Component Method, and therefore detailed connection behaviour, with the overall frame action of a structure at ambient and elevated temperatures. It can deal with the changing combinations of moment and axial loading which a connection undergoes during a natural fire. Furthermore, to simulate the connection response only the geometrical and material properties of the connection are required. However, there are also a number of disadvantages associated with the connection element. The greatest is that the accuracy of response of the element depends on the accuracy of F-δ-θ models for the individual components, which do not exist for all components at present. Also, in practice the required connection details will most likely not have been designed by the time the structural fire design of a building has to be performed, which adds another iteration to the building design process. In addition, the computational time for the analysis of a structure in fire will be increased. At present, there are still a number of limitations to the use of the connection element, which restrict its use to internal endplate joints due to a lack of calibration of the remaining components in fire. However, in theory it is possible to include all the missing components: Group effects in the bolt rows, Shear deformation in the column-web, Shear deformation in the beam-end zone, Local buckling of the bottom flange of the beam, Bolt behaviour during cooling. 6. CONCLUSION In this paper, the development of a high-temperature component-based connection element has been described. The proposed element is based on the principles of the Component Method and was implemented into Vulcan, a non-linear finite element program specially developed for the analysis of steel and composite structures in fire. This combination of detailed connection behaviour with a realistic simulation of structures in fire can be used to increase the understanding of the complex connection-structure interaction. However, at present this paper can only describe a practical way to develop such a connection element and only preliminary, but good, validations can be shown. Further improvements of the existing component characteristics, and investigation of the hightemperature behaviour of the remaining components, as well as more high-temperature experiments are required before the element can be used for practical design. Nevertheless, when the remaining research is done, the safety and economy of steel and composite framed buildings could be increased significantly, as it will be possible to assess the robustness of buildings against structural collapse more accurately, which will allow an optimisation of structural design and the applied fire protection.

12 REFERENCES [1]. Tschemmernegg, F., Tautschnig, A., Klein, H., Braun, Ch. and Humer, Ch., Zur Nachgiebigkeit von Rahmenknoten Teil 1, Stahlbau 56, Heft 1, S , [2]. CEN, EC3: Design of Steel Structures, Part 1.8: Design of joints, European Committee for Standardization, Document BS EN :25. [3]. Leston-Jones, L.C., The influence of semi-rigid connections on the performance of steel framed structures in fire, PhD Thesis, University of Sheffield, [4]. Al-Jabri, K.S., The behaviour of steel and composite beam-to-column connections in fire, PhD Thesis, University of Sheffield, [5]. Spyrou, S., Development of a component based model of steel beam-to-column joints at elevated temperatures, PhD Thesis, University of Sheffield, 22. [6]. Simoes da Silva, L., Santiago, A. and Villa Real, P., A component model for the behaviour of steel joints at elevated temperatures, Journal of Constructional Steel Research 57, pp , 21. [7]. CEN, EC 3: Design of Steel Structures, Part 1.2: Structural Fire Design, European Committee for Standardization, Document BS EN :25. [8]. Ramli Sulong, N.H., Elghazouli, A.Y., Izzuddin, B.A., Analytical modelling of steel connections at elevated temperature, in the proceeding of Eurosteel 25 4 th European Conference on Steel and Composite Structures, Vol C, pp , 25. [9]. Li, T.Q., Choo, B.S. and Nethercot, D.A., Connection element method for the analysis od semi-rigid frames Journal of Constructional Steel Research 32, pp , [1]. Lowes, L.N. and Altoontash, A., Modeling reinforced-concrete beam-column joints subjected to cyclic loading, Journal of Structural Engineering, Vol. 129, No 12, pp , 23. [11]. Bayo, E., Cabrero, J.M., Gil, B., An effective component-based method to model semi-rigid connections for global analysis of steel and composite structures, Engineering Structures 28, pp , 26. [12]. Zoetemeijer, P., A design method for the tension side of statically loaded, bolted beam-to-column connections, Heron 2, pp. 1-59, [13]. Block, F.M., Development of a component based finite element for steel beam-tocolumn connections at elevated temperatures with special emphasis on the compression zone, PhD Thesis (in preparation), University of Sheffield, 26. [14]. Kirby, B.R., The behaviour of high-strength grade 8.8 Bolts in fire, Journal of Constructional Steel Research 33, pp. 3-38, [15]. Bailey, C.G., Burgess, I.W. and Plank, R.J., Analyses of the effects of cooling and fire spread on steel-framed buildings, Fire Safety Journal 26, pp , [16]. Giaro Coelho, A.M. Characterisation of the ductility of bolted end plate beam-tocolumn steel connections, PhD thesis, University of Coimbra, Coimbra, Portugal, 24 [17]. Bailey, C.G. and Moore, D.B., The influence of local and global forces on column design, Final Report, PII Contract No. CC1494, BRE, Garston, UK, 1999.