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
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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.
159