Advances in Steel Structures - part 18 docx

150 CONCLUSIONS B.H.M. Chan et al. From the numerical example illustrated above shows that the ultimate strengths and the deformations of semi-rigid steel frames can be load-sequence dependent when both the geometric and material non-linearities are accounted for. Analysis based on proportional load approach can result in an under-estimation of the load-carrying capacity of structures. REFERENCES Chan, S.L. (1988). Geometric and Material Nonlinear Analysis of Beam-Columns and Frames using the Minimum Residual Displacement Method. Int. J. Num. Meth. in Engrg, 26, 267. Chan, S.L. and Chui, P.P.T. (1997). A generalised design-based elastoplastic analysis of steel flames by section assemblage concept. Engrg. Struct., 19:8, 628. EC3 (1993). Eurocode 3: Design of steel structures: Part 1.1 General rules and rules for buildings, European Committee for Standardization, Brussels. ECCS (1983). Ultimate Limit State Calculation of Sway Frames with Rigid Joints, European Convention for Constructional Steelwork, Rotterdam. Lui, E.M. and Chen, W.F. (1988). Behavior of braced and unbraced semi-rigid frames. Int. J. Solids. Struct., 24:9, 893. SECOND-ORDER PLASTIC ANALYSIS OF STEEL FRAMES Peter Pui-Tak Chui ~ and Siu-Lai Chan 2 Ove Arup & Partners (Hong Kong) Ltd., HONG KONG 2 Dept. of Civil & Structural Engineering, The Hong Kong Polytechnic University, HONG KONG ABSTRACT A second-order refined-plastic-hinge method for determining the ultimate load-carrying capacity of steel frames is presented. Member imperfection and residual stress in hot-rolled I- and H-sections are considered. Second-order effect due to the geometrical nonlinearity is accounted for. In the present inelastic model, gradual degradation of section stiffness is allowed for simulating a more realistic and smooth transition from the elastic to fully plastic states. The developed model has been verified to be valid through a benchmark calibration frame. INTRODUCTION It has been long recognized that the second-order effects due to geometrical changes and inelastic material behaviour can dominate the load-carrying capacity of steel structures significantly, as shown in Fig. 1. However, the first-order elastic analysis is usually employed to estimate the member forces in conventional engineering design. In pace with the advent in computer technology, the sophisticated analysis is feasible. Recently, a refined method of analysis, which is called the Advanced Analysis, has been coded in the Australian limit states standard for structural steelwork (AS4100 1990). The basis of the Advanced Analysis is to consider initial imperfections and second- order effects so as to estimate the member forces and the overall structural behaviour accurately. This should result in more economical and safe selection of member size. The existing models for second-order plastic analysis can be broadly categorized into two types, namely the plastic-zone (Ziemian 1989) and the plastic-hinge (Gharpuray and Aristizabal-Ochoa 1989) models. In the plastic-zone method, the beam-column members are divided into many very fine fibres. Its results are generally considered as the exact solutions. However, it is much costly and, therefore, its solutions are usually used for calibrating of various plastic-hinge models. In the plastic-hinge method, a plastic hinge of zero-length is assumed to be lumped at a node. This eliminates the tedious integration process on the cross-section and permits the use of less elements per member. Therefore, it reduces computational time significantly. Although it can only predict approximately the strength and stiffness of a member, it is more suitable and practical in engineering design practice. In this paper, a refined-plastic-hinge model is proposed and studied. 151 152 P. P T. Chui and S L. Chan FUNCTIONS OF YIELD SURFACES In the present refined-plastic-hinge analysis, a function is employed to mathematically describe a limiting surface which is used to check whether or not the interaction point for axial-force and bending-moment lying outside this yield surface. As the name implies, a full-yield surface and an initial-yield surface are here used to define the ultimate strength surface and the initial yield surface respectively on the plane of normalized force diagram for a cross-section. The functions of these surfaces employed in this paper are defined as follows. Full- Yield Surface A full-yield surface is a strength surface of a section to control the combination of normalized axial force and moment. In other words, it represents the maximum plastic strength of the cross-section in the presence of axial force. Based on the British Standard BS5950 (1985), the Steel Construction M/Mp = 1-2.5(P/Py) 2 M/Mp = 1.125(1-P/Py) when P / Py < 0.2 (1) when P / Py > 0.2 Institute (1988) has recommended a full-yield surface of hot-rolled 1-section for compact section bending about the strong axiS, as, in which M and P are moment and axial force acting on the section, Mp is the plastic moment capacity of the section under no axial force and Py is the pure crush load of the section. Initial- Yield Surface The European Convention for Constructional Steelwork (ECCS 1983) has provided a detailed and comprehensive information with regard to appropriate geometric imperfections, stress-strain relationship and residual stress for uses in the plastic zone analysis. The pattern of ECCS residual stress for hot-rolled I- and H-sections is shown in Fig. 2. The residual stress will result in the early yielding of a section and the initial-yield surface can be defined as, Mer = Z e ( Oy - Ore s -P/A) (2) in which Mer is the reduced moment elastic capacity under axial force P, Ze is the elastic modulus, (Yy is the yield stress, Crre s is the residual stress and A is the cross-section area. In case of no residual stress and axial force, the M~r will become the usual maximum elastic moment (i.e. Mer = Zr Cry). As the normalized force point is within the initial yield surface, the member behaves elastically. The effect of residual stress on the moment-curvature relationship is illustrated in Fig. 3. PROPOSED PLASTICITY METHOD In the traditional plastic-zone (P-Z) method, beam-colunm members are divided into a large number of elements and sections are further subdivided into many fibres. The solutions by this method are generally considered as the exact solutions. However, the computation time required is much heavier and it is usually for research study, but not for practical design purpose. To simplify the inelastic analysis, a refined-plastic-hinge method is proposed because of its efficiency. Second-Order Plastic Analysis of Steel Frames Refined-Plastic-Hinge (R-P-H) Method 153 The proposed refined-plastic-hinge method is a plastic-hinge based inelastic analysis approach considering the stiffness degrading process of a cross-section under gradual yielding for the transition from the elastic to plastic states. In the proposed method, material yielding is allowed at nodal section only and can be represented by a pseudo-spring. The stiffness of the spring is dependent on the current force point on the thrust-moment plane. When the force point does not exceed the initial-yield surface, the section remains elastic and the spring stiffness is infinite. If the point reaches on the full-yield surface, the section will form a fully plastic hinge and the value of the spring stiffness will be zero. To avoid computer numerical difficulties, the limiting values of oo and zero will be assigned as 101~ and 10I~ respectively. When the force point lies between the surfaces, section will be in partial yielding and the function of the spring stiffness, t, is proposed to be given by, t - 6EI IMpr-M I when Mer<M<Mpr (3) L IM-M rl in which EI is the flexural rigidity, L is the element length, and Mer and Mpr are the reduced initial- and full-yield moments in the presence of axial force, P, shown in Fig. 4. Movement Correction of Force Point After a fully plastic hinge is formed at a section, correction of forces must be considered to insure the force point is not outside the maximum strength of section. As the axial force increases, the moment capacity will be reduced and hence the value of bending moment would decrease. If the force point is outside the full-yield surface, the point is assumed to shift orthogonally back onto the yield surface. ELEMENT STIFFNESS Assuming the section spring stiffness at the ends of an element to be t~ and t 2, an incremental form of element stiffness can be expressed (see Fig. 4) as, / AeMI/ tl -tl 0 0 AIM1[ = -t I 4EI/L +t 1 2EI/L 0 AiM: / 0 2EIIL 4EI/L +t 2 -t 2 AoM~) 0 0 -t~ t~ Ae01/ AiOl/ Ai02/ Ae 02) (4) in which the subscript "1" and "2" are referred to the node 1 and node 2, AeM and AiM are the incremental nodal moments at the junctions between the spring and the global node and between the beam and the spring and, Ae0 and Ai0 are the incremental nodal rotations corresponding to these moments. It is assumed that the loads are applied only at the global nodes and hence both AiM1 and AiM2 are equal to zero, we obtain, 154 P. P T. Chui and S L. Chan Ai01) 1 A i02 ) = ~ -2EI/L 4EI/L +t 1 tAeO2) (s) in which 13 = (4EI/L+t0(4EI/L+t2) - 4(EI/L) 2. Eliminating the internal degrees of freedom by substituting the equation (5) into (4), the final incremental stiffness relationships for the element can be formulated as, EA/L 0 0 0 tl -t12(K22 + t2)/13 tlt2K12/13 Ae01 0 tlt2K21/13 t2-t~(K11+t1)/~ t ao~ (6) in which A is section area, AP is axial force increment and AL is axial deformation increment. NUMERICAL EXAMPLE The two-bay six-storey European calibration frame subjected to proportionally applied distributed gravity loads and concentrated lateral loads has been reported by Vogel (1985). The frame is assumed to have an initial out-of-plumb straightness and all the members are assumed to possess the ECCS residual stress distribution (ECCS 1983). The paths of load-deformation curves shown in Fig. 5 are primarily the same by the plastic-zone and the plastic-hinge analyses. The maximum capacity is reached at a load factor of 1.11 for Vogel's plastic-zone method (Vogel 1985), 1.12 for Vogel's plastic-hinge method (Vogel 1985), and 1.125 for the proposed refined-plastic hinge method. The maximum difference between these limit loads is less than 1.4%. This example shows the adequacy of the plastic hinge method for large deflection and inelastic analysis of steel frames. The same frame has also been studied by the Cornell University inelastic program: the CU- STAND (Hsieh et al. 1989). The force diagrams of the frame with key values at specified locations and at the maximum load of the frame are plotted in Fig. 6. The ultimate load factors are 1.13 for the CU-STAND and 1.125 for the present study. The force distribution and the plastic hinge location obtained by the analyses are essentially similar. The CU-STAND hinge analysis detects a total of 19 plastic hinges while the present study detects 16 plastic hinges. The difference may be explained by the fact that the present limit load, which is less than that obtained by Hsieh et al. (1989), is not high enough to produce further fully plastic hinges at these three locations. Referring to the figure, the present bending moments at the three locations are very close to the fully plastic moment capacity of section just before structural collapse. CONCLUSIONS A plastic-hinge based approach for inelastic analysis of steel frames, the refined-plastic-hinge methods, is presented. The inelastic behaviour of a beam-column member can be simulated by a spring model allowing for degradable stiffness of sections between the elastic and plastic states. From the example, the inelastic behaviour of frame controls the ultimate load and should be Second-Order Plastic Analysis of Steel Frames 155 considered. Generally speaking, based on the simplified numerical model employed, the proposed refined-plastic-hinge analysis is more suitable and practical in design practice when compared with the plastic-zone analysis. ACKNOWLEDGEMENTS The authors gratefully acknowledge that the work described in this paper was substantially supported by a grant from the Research Grant Council of the Hong Kong Special Administration Region on the project "Static and Dynamic Analysis of Steel Structures (B-Q 193/97)". The support of the first author by Ove Arup and Partners(Hong Kong) Ltd. is also acknowledged. REFERENCES 1. British Standard Institution (1985), BS5950: Part I." Structural Use of Steelwork in Building, BSI, London, England. 2. European Convention for Constructional Steelwork (1983), Ultimate Limit State Calculation of Sway Frames with Rigid Joints, ECCS, Technical Working Group 8.2, Systems, Publication No. 33. 3. Gharpuray, V. and Aristizabal-Ochoa, J.D. (1989), "Simplified Second-Order Elastic Plastic Analysis of Frames", J. of Computing in Civil Engng., 3:1, pp.47-59. 4. Standards Australia (1990), AS4100-1990 Steel Structures, Australian Institute of Steel Construction, Sydney, Australia. 5. Steel Construction Institute (1988), Introduction to Steelwork Design to BS5950: Part 1, SCI Publication No. 069, Berkshire, England. 6. Ziemian, R.D. (1989), Verification Study, School of Civil and Environmental Engng., Cornell Univ., Ithaca, N.Y. 7. Vogel, U. (1985), "Calibrating frames", Stahlbau, 54, October, pp.295-311. 8. Hsieh, S.H., Deierlein, G.G., McGuire, W. and Abel, J.F. (1989), "Technical manual for CU-STAND", Structural Engineering Report No. 89-12, School of Civil and Environmental Engineering, Cornell University, Ithaca, N.Y., U.S.A. 156 P. P T. Chui and S L. Chan 8eooncl.Order Bmtk~ Unur Analym / (Flint-order Butlr Bmtlr Bifurcation Load Plastlo Umlt Load Bastlc-Pl~Ic Analysis 8eaond-order Plutlc-hlnge Atolls Aotual Beh~our oo% Local and/or L~eml Torsional bucldlng 8eoond-order Plmtlc Zone Generalised Displacement Fig. 1 General Analysis Types of Framed Structures D I i I 0.5 0.5 I 0.5 B _1 ~/~=os D/B< 12 03 ~i'~ "-~ o a o~a I 3 03 ~/~=oa D/B > 12 Fig. 2 ECCS residual stress distribution for hot-rolled I-ssctlons M/Mp ~ Idealized elastic-perfectly plasUc behaviour or ~ r W'ithout residual stresses 9 IT ," ,,.~/ With residual stresses /%, o-< : (Ty = yield stress o ~ +/+y Fig, 3 Moment-curvature relationship for I-ssctlon with and without residual stresses Section spring of stiffness, t2 Node1 ~ Node 2~.M2 P "' 2,.0, Fig. 4 Internal forces of an element with end-ssction springs accounting for cross-ssction plsstlflcatlon employed by the present study Second-Order Plastic Analysis of Steel Frames 1.2 1.0 0.9 0.8 ,,<: ,.z 0.7 0 ,.~ 0.6 _9 0.5 0.4 0.3 0.2 0.1 0.0 0 Umiting load factor,)~ 1.11 1.12 1.125 _ ~kN/m IPESO0 2 LS?2 I = / -I '"'=~ ~' I I ~L L ~.l.~., ~.,m / -~ F ;~-~ ~TM~ ~ / ,, .L ~,L~M_ F~ ~IWN/m / -I "'= ~ I I E = 205 KN/mm < ~ ;" -"-'_~, ~_'-" / ~= ~ ~mr~ ~ ~ ~ ~' ~i / ~ = 1/450 (_P!astic zone) /7~ ~/'~/7 7-/ = 1/300 (Plastic hinge) l< 2xe 112m __l I 9 D Plastic zone (Vogel 1985) 0 Plastic hinge (Vogel 1985) Refined-plastic hinge (this study) (5 6 (cm) I I I I l I I 5 10 15 20 25 30 35 Fig. 5 Inelastic load-deflection behsvlour of Vogel six-storey frame ~ 81.4 255 II 547 II ~ / f 142.7 . [2 ] L [~:6:3] [147~ [147.6] [147.6] 4O7 I I 879 II 4 I 146.5 [147.6]A 145.5 [147.5] [147.5] / !~8.; 152.4 [1 L/~.4"/'q-__J~30.g 7 I 154 g [230.3]/ 125.4P~.4] / / 112.5 [2~.s] $] / ~ [3o,.1] 7 ~69] ~914] ~ ,j/W~.111.6 / [~2204.7 ] / ~10~:59] (a) Axial force (kN) (b) Bending moment (kN-m) Values: Symbols: This study, k u =1.125 0 Plastic hinge location by CU-BTAND [CU-STAND, X u =1.1 3] ~ Plastic hinge location by this study 9 Common plastic hinge location by CU-STAND and this study Fig, 6 Comparslon of member forces of Vogel frame by Cornell studies and this study 157 This Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left Blank STUDY ON THE BEHAVIOUR OF A NEW LIGHT-WEIGHT STEEL ROOF TRUSS P. Makel~iinen and O. Kaitila Laboratory of Steel Structures, Helsinki University of Technology, P.O.Box 2100, FIN-02015 HUT, Finland ABSTRACT The Rosette thin-walled steel truss system presents a new fully integrated prefabricated alternative to light-weight roof truss structures. The trusses will be built up on special industrial production lines from modified top hat sections used as top and bottom chords and channel sections used as webs which are jointed together with the Rosette press-joining technique to form a completed structure easy to transport and install. A single web section is used when sufficient and can be strengthened by double-nesting two separate sections or by using two or several lateral profiles where greater compressive axial forces are met. A series of laboratory tests have been carried out in order to verify the Rosette truss system in practice. In addition to compression tests on individual sections of different lengths, tests have also been done on small structural assemblies, e.g. the eaves section, and on actual full-scale trusses of 10 metre span. Design calculations have been performed on selected roof truss geometries based on the test results, FE-analysis and on the Eurocode 3, U.S.(AISI) and Australian / New Zealand (AS) design codes. KEYWORDS Rosette-joint, truss testing, light-weight steel, roof truss, cold-formed steel, steel sheet joining. INTRODUCTION The Rosette-joining system is a completely new press-joining method for cold-formed steel structures. The joint is formed using the parent metal of the sections to be connected without the need for additional materials. Nor is there need for heating, which may cause damage to protective coatings. The Rosette technology was developed for fully automated, integrated processing of strip coil material directly into any kind of light-gauge steel frame components for structural applications, such as stud wall panels or roof trusses. The integrated production system makes prefabricated and dimensioned frame components and allows for just-in-time (JIT) assembly of frame panels or trusses without further measurements or jigs. 159 . Rosette-joint, truss testing, light-weight steel, roof truss, cold-formed steel, steel sheet joining. INTRODUCTION The Rosette-joining system is a completely new press-joining method for cold-formed. second-order refined-plastic-hinge method for determining the ultimate load-carrying capacity of steel frames is presented. Member imperfection and residual stress in hot-rolled I- and H-sections. simplify the inelastic analysis, a refined-plastic-hinge method is proposed because of its efficiency. Second-Order Plastic Analysis of Steel Frames Refined-Plastic-Hinge (R-P-H) Method 153