Advances in Steel Structures - part 18 docx

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 capacit

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

S E C O N D - O R D E R P L A S T I C A N A L Y S I S OF S T E E L F R A M E S

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 / M p = 1 - 2 5 ( P / P y ) 2

M / M p = 1 1 2 5 ( 1 - P / P y )

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,

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,

L I M - 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,

AIM1[ = -t I 4EI/L +t 1 2EI/L 0

AiM: / 0 2EIIL 4EI/L +t 2 -t 2

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

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,

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

0.5

~/~=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 ,"

, , Wi th residual ~ /stresses

/%, o-< : (Ty = yield stress

Fig, 3 Moment-curvature relationship for I-ssctlon

with and without residual stresses Section spring of stiffness, t2

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

IPESO0

/ -I '"'=~ ~' I

/ ,, L ~,L~M_ F~ ~IWN/m

/ -I "'= ~ I

I E = 205 KN/mm < ~ ;" -"-'_~, ~_'-"

/ ~ = 1/450 (_P!astic zone) /7~ ~/'~/7 7-/

= 1/300 (Plastic hinge) l< 2xe 112m l I

D Plastic zone (Vogel 1985) 9

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 ~ [2 ] L / [~:6:3] [147~ f [147.6] [147.6] 142.7

4O7 I I 879 II 4 I 146.5 [147.6]A [147.5] 145.5 [147.5] /

L/~.4"/'q- J~30.g 7

I 154.-g - [230.3]/ 125.4P~.4]

/

/ 112.5 [2~.s]

(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

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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

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