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Chapter 3
Design of steel frames by second-order P-∆ δ analysis
fulfilling code requirements
3.1.1 Compression Resistance............................................................. 78 3.1.2 “Method of guess” for effective length ....................................... 79
3.1.3 Code method for finding effective length ................................... 82
3.1.4 Examples using BS5950.............................................................. 85
3.2 Design of beam-Columns.................................................................. 90
3.2.1 Local capacity Check, clause 4.8.3.2, p.73 ................................. 90
3.2.2 Overall Buckling Check, clause 4.8.3.3, p.73 ............................. 90
3.2.3 Some general questions related to hand design........................... 91
3.3 Design formulae for columns in BS5950.......................................... 92
3.4 Some deficiencies of the Effective length method ........................... 93
3.5 P-∆-only analysis vs. P-∆ δ analysis................................................. 98
3.6 Second-order P-∆-only analysis of finding the bending moment .. 98
3.7 Second-order analysis P-∆ δ analysis .............................................. 99
3.7.1 P-δ-∆ analysis ignoring beam lateral-torsional buckling check 100
3.7.2 P-∆−δ analysis allowing for beam buckling.............................. 101 3.7.3 Design check against local buckling ......................................... 102
3.8 The 2 Analysis Procedures for P-∆ δ analysis ............................. 104
3.8.1 Incremental Load Method determining load resistance ............ 104
3.8.2 Fixed load method for checking against design loads............... 105
3.9 Buckling strength curves in BS5950(2000) .................................... 106
3.10 The Euro-code 3 for second-order analysis and design............. 110
3.11 Limitations and advantages of second-order analysis.............. 112
3.12 EXAMPLES .........................................................................................114
3.13 REFERENCES .....................................................................................129
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3.1 Effective length design method – the unreliable method
The current design is summarised as follows.
3.1.1 Compression Resistance
The compression resistance is equal to
gcc A pP ⋅=
where gA =gross sectional area
c p = compressive strength
Summary of steps to calculate cP :
1. Select section and grade of steel
2. Check section classification (Table 7 in BS5950)
3. Find design strength yP
4. Estimate effective length ( eL ) and calculate Slenderness ratio
( r /Le ).
Linear AnalysisNon-linear Analysis for Simple
Idealised Individual Members
Development of Design Rules
Individual Member Capacity Check
Ouput of Member
Forces and Moments
The Conventional Design Procedure
This part by the
practising engineer This part by the code drafter
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5. Select strut curve from Table 25
6. Find cP from Table 27a-27d
gcc A pP ⋅=
3.1.2 “Method of guess” for effective length
As we can see, the accuracy of the method relies on the effective length
assumed. In many codes, there are methods to determine the effective length
or the second-order analysis is used (Le/L).
In effective length method, the critical problem for assessing the buckling
strength will be the assumption of effective length. Below are the typical
values for effective length factor.
Effective length factor 1
Rotation Fixed
Translation FixedRotation Free
Translation Fixed
Rotation Fixed
Translation Free
Rotation Free
Translation Free
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In practice, it is quite common to approximate the buckling behaviour as
above. However, many design codes including the BS5950 do not allow this
coarse simplification and we need to use more refined method, especially for
complex frames. It has been noted that engineers may artificially assume an
effective length to suit the required design resistance of the column. In
general, we need to consider the behaviour of a column as a member of a
structural system, instead of in isolation which is very dangerous or
uneconomical.
Before the introduction of design methods, we need to first realise the
behaviour of a structure under loads and the terminology.
Advanced analysis : an analysis that sectional capacity check is adequate
for design load capacity. It may allow for one or more than one plastic
hinges in the analysis process and moment re-distribution.is allowed in the
analysis.
Elastic Critical Load Factor λ cr : a factor multiplied to the design load to
cause the structure to buckle elastically. The large deflection and material
yielding effects are not considered here and the factor is an upper bound
solution that cannot be used directly for design.
P-delta effects : refer to the second-order effect. There are two types, being
P-∆ and P-δ .
P-∆ effect : second-order effect due to change of geometry of the structure
P-δ effect : second-order effect due to change of member stiffness under
load and additional moment along a member due to its curvature. A member
under tension is stiffer than under compression.
Linear analysis : an analysis assuming the deflection and stress are
proportional to load. It does not consider buckling nor material yielding.
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Nonlinear analysis : an analysis which does not assume a linear
relationship between load, displacement, stress (σ ) and E. This is a very
board term.
Notional Force : a small force applied horizontally to a structure to
simulate lack of verticality and imperfection. It can also be used to measure
the lateral stiffness so that the elastic critical factor can be determined.
Verticality is considered by application of notionalforces to a vertical frame in an analysis model
Second-order P-∆ only analysis for plotting bending moment: an
analysis used to plot the bending moment and force diagrams based
on the deformed geometry. It considers only the P-∆ effect but not the
P-δ effect. Nearly all commercial software can only do this type of
analysis at the time when this note is written.
φ
P P P PPφ
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Second-order analysis P-∆ δ analysis allowing for section capacity
check : an analysis improved from above and similar to advanced
analysis (Liew, 1992) in AS4100(1995), but stops at first plastic hinge
of which the assumption is more widely accepted in practice. It
considers both the P-∆ and P-δ effects. This new term is to preventconfusion against those considering only one P-delta, but the concept
and methodology is well documented in Euro-code 3 (2003).
3.1.3 Code method for finding effective length Le
1. Calculation λcr by one of the following methods1.1 Application of notional force. λcr can be determined as,
in which δU and δL are the upper and lower story deflections. Themaximum φ among all stories should be used in order to obtain theminimum critical load factor. (This implies that a storey deflection
controls completely the structural buckling strength.).
λcr is defined as the factor multiplied to the design load causing the frame to buckle elastically.
Notional force is (1) to simulate lack of verticality of frames and taken as
0.5% of the factored dead and imposed loads applied horizontally to the
structure and (2) to calculate the elastic critical load factor λcr . This percentage of notional force may vary for other types of structures like
scaffolding where imperfections are expected to be more serious. In Hong
Kong Code, λ is calculated as,δ
λ N
N
F
HF=
2 Check the value of effective length by the following procedure
3.1.3.1 Non-sway frame
When λcr ≥10 for 2000 version it is a non-sway frame. P-∆ effect can be
ignored here and only P-δ effect is needed. The effective length of
members in frames can be designed by chart in Figures E.1, E.2 and E.3
in BS5950(2000) or by E.6 and λcr directly.
h
δδ
indexswaytheis φ where200φ
1λ
LU
cr
−==
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Design chart method Annex E (p.103, BS5950)
Determine k1 and k2 as jointat thememberstheallof stiffnessTotal
jointat thecolumnstheof stiffnessTotal
=k
To calculate the capacity of the column )AP(K cc ⋅= , the effective length
of the column is needed to determine and can be evaluated as follows
)(
)(
2
1
R L Lc
LC
TRTLuc
uc
K K K K
K K k
K K K K
K K k
++++
=
++++
=
With these values of k 1 and k 2 , the effective length ratio (L
Le )
can be obtained from Figure 24 for sway frames or
Figure 23 for non-swayed frames.
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3.1.3.2 Sway-sensitive frames
When 4 < λcr < 10, it is a sway sensitive frame.
A structure should have sufficient stiffness so that the second-order moment
due to vertical load and lateral deflection will not be so great as to affect the
structural safety. P-∆ effect is to account for the effect of global sway of a
frame and it is particularly important in sway-sensitive frames. For a frame
with large sway or weak in lateral sway stiffness, we must consider theadditional moment or instability effect due to sway. When a structure is
under vertical loads, the member and complete global stiffness are reduced
and therefore their sway stiffness is weakened. This leads to the importance
of considering the P-∆ effect in some structures.
Moment amplification method
Application of an amplification factor k amp below to enlarge the moments
and forces obtained from a linear analysis.
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1.0 5.115.1
≥−
=cr
cr
ampk λ
λ for clad frame without considering the favourable
effect of cladding in analysis
1−= cr cr
ampk λ λ for unclad frames or the favourable effects of cladding has
been considered.
The above considers the P-∆ effect such that the effective length of the
column is then taken as its true length (see portal frame example later).
Elastic Critical Load Method by E.6
ccr
E F
EI L
λ
π 2= (3)
3.1.3.3 Sway very sensitive frames
When λcr < 4, only second-order analysis method can be used.
In a second-order analysis method, both P-∆ and P-δ effects are considered by the analysis part. A linear analysis program cannot be used here.
3.1.4 Examples using BS5950
The 4-storey frame shown is designed. All members are 203x203x60 UC
with the following properties.
Area = 76. cm2
, Iy = 2047cm4
, Iz =6103cm4
, Zy = 199cm3
, Zz=582cm3
. r y=5.12 cm, r z= 8.96 cm, T
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Using the method of sway index, the elastic buckling load factor, λcr , iscalculated in case 1 as follows.
Deflections (mm)& sway indices iφ Storey Case 1 (Bent aboutminor axis, no
bracing)
Case 2 (Bent about
major axis, no
bracing)
Case 3 (Bent about
minor axis, fully
braced)
1 6.3 / 0.00151 2.1 / 0.00053 0.1 / 0.000025
2 15.25 / 0.00230 5.3 / 0.00080 0.2 / 0.000025
3 24.63 / 0.00236 8.7 / 0.00085 0.4 / 0.00005
4 32.73 / 0.00202 11.6 / 0.0007 0.6 / 0.00005
Table 1 Deflections at various levels of the 4-storey frame
h
iii
1−−= δ δ
φ
Case1 Unbraced case by Annex E, Equations 16 and 20 in this note.
The maximum φs is 0.00236 and the λcr is = 1/200/0.00236 = 2.12
Using NIDA, λcr is calculated as 2.136.
The effective length = m x
x x x
F
EI L
ccr
E 25.6000,50012.2
102047000,205 422===
π
λ
π
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Braced and unbraced 4-storey frames
However, since λcr is less than 4.0 here, the effective length method can nolonger be used in the Euro-code 3, the BS5950(2000) or the Hong Kong
Steel Code 2004. There are two options to solve this problem. The first is touse the major principal axis of members to resist loads, which is considered
as case 2. The other option is to add bracings members which is designated
as case 3.
Case2 Unbraced case by Annex E, Equation 20 in this note.
Referring to Table 1, the selected φs is 0.00085 and the λcr is =1/200/0.00085 = 5.9 > 4 and < 10, sway sensitive frame.
Using computer, λcr
is 6.3
The effective length =
m x
x x x
F
EI L
ccr
E 47.6000,5009.5
106103000,205 422===
π
λ
π
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L/r = 6.47/89.6 = 72.2,
From Table 24b, BS5950, permissible axial force = pcA = Pc = 197.6x7600
= 1,520 kN
Design load factor = 1657/500 = 3.0
Case 3 Fully braced case by Annex E and chart (see Figure )
From Table of deflection, the frame is non-sway and the beam is bent under
single curvature.
From Table E.3, consider column in the second level as the most critical.
8.05.2/2
5.01 ==
++
+=
L I
L I
L I
L
I
L
I
k
8.05.2/2
5.02 ==
++
+=
L
I
L
I
L
I L
I
L
I
k
From Chart Figure E.1 for non-sway frame, Le/L = 0.855,
Thus effective length = 0.855x4 =3.42m
L/r = 3420/51.2 = 66.8From Table 24c, pc=187.4 N/mm
2
Pc=187.4x7600 = 1424. kN
At design load, the axial force in column is 428.7 kN,
Permissible load factor = 1424/428.7 =3.3
Question : For braced frames, the notional force goes into the bracing and
then the support etc. Can we use the sway index method to classify whether
the frame is sway or not and then use the to find the effective length by the
elastic critical load method in session 3.1.3 ?
No, we will miss the column buckling mode and the effective length is not
for the critical mode.
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When the structure is under a set of more realistic loads due to beam
reactions and distributed evenly at the four levels, how to check the column
strength with variable axial force along its length ?
Using the maximum portion, of course. But it is a waste of material. Second-order analysis does not have this problem.
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3.2 Design of beam-Columns
A structural member subjected to the action of axial load and moment is
called a beam-column.
3.2.1 Local capacity Check, clause 4.8.3.2, p.73
At the point of maximum moment (local !), the following equation must be
satisfied.
For slender, semi-compact, compact
2.3.8.4,73.,1 clausepM
M
M
M
p A
F
cy
y
cx
x
yg
≤++ (4)
F = axial load
gA = gross cross-sectional area
yx M,M = applied moment about xx and yy axes
cycx M,M = moment capacity about xx and yy axes in the absence of axial
load
3.2.2 Overall Buckling Check, clause 4.8.3.3, p.73
(5)
mLT = equivalent uniform factor (Table 18)
c p = compressive strength clause 4.7.5, p.57
bM = buckling resisting moment about the major axis. Taking into
account
the compactness of the section (slender, semi-compact etc.)
yZ = elastic modulus about the yy axis.
1≤++yy
y
b
xlt
cg Zp
mM
M
Mm
p A
F
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3.2.3 Some general questions related to hand design
Is the statement below correct ?
Assuming the effective length equal to true length always gives us a safe
design ?
It is only true for columns with both ends immovable. Whenever the support
moves, the effective length factor can be larger than 1.
How to determine effective length accurately ? Very difficult by judgement
and subjective. Argument between engineers.
When using this method, the greatest uncertainty will lie on the
determination of effective length. Professor Bolton indicated the error can be
very large. The second-order moment may not be second-order in
consequence or in magnitude. Buckling is a type of failure due to second-
order effects coupled with weak lateral stiffness. The frequent collapse of
scaffolding in various places shows the importance of buckling in collapse.
A structure is therefore required to be checked against sway effects which
should then be accounted for in the analysis.
Correct assumption of effective length is important. For slender elastic
structures, a 20% error in effective length can lead to an over-estimation of
capacity by about 40% since buckling load is inversely proportional to the
square of slenderness ratio as follows.
2
2
⎟ ⎠
⎞⎜⎝
⎛ =
r
L
EAP
e
cr
π (6)
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in which Pcr is the buckling load andr
Le is the slenderness ratio, Le is the
effective length and r is the radius of gyration.
3.3 Design formulae for columns in BS5950
The buckling strength or the load capacity of a column is dependent
on its length, boundary conditions, second-moment of area from cross
sectional geometry, section shape variation (I, Channel, box etc.), residual
stress and imperfections.
The formula for the buckling strength curves is given by (Ayrton and
Perry, 1886),
c E c E c y p p p p p p η =−− ))(( (7)
in which py = the design strength or yield stress,
pc= compressive strength,
pE = Euler’s buckling stress = 2
2
⎟ ⎠
⎞⎜⎝
⎛ r
L
E π
η = curve-fitting parameter for straightness
= 2r
y∆
(analytical)
= 01000
)( 0 ≥− λ λ a
(empirical from BS5950)
in which λ0= y p
E 22.0 π
,
y = distance of extreme fibre from the centroidal axis
L = Length
r = radius of gyration
From test results of different slenderness, section type and
manufacturing type, η can be found to fit the experimental curves.
For curves (a) in BS5950, a = 2.0
For curves (b) in BS5950, a = 3.5
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For curves (c) in BS5950, a = 5.5
For curves (d) in BS5950, a = 8.0
3.4 Some deficiencies of the Effective length method
It can be seen that most practical columns or struts cannot be assumed as
above which is an ideal condition. Unfortunately, an error in effective length
leads to a large error in buckling strength since their relationship is not
linearly proportional.
However, we can hardly classify a practical column as above since its
interaction with other members is not considered. In general, we need to
consider the behaviour of a column as a member of a structural system,
instead of in isolation which is very dangerous or uneconomical.
Methods suggested by BS5950 for sway sensitive frames include the swayamplification method, using the elastic critical load factor and the design
chart in Appendix E. For non-sway frames, only the methods using the
elastic critical load factor and the design chart in Appendix E are referred.
Elasto-plastic Buckling Analysis
Conventional linear
Deflection
Load
Pc
Py
Pe
Figure 1 Typical behaviour of a steel portal
Limit Point Analysis
Elastic Buckling Load
δ
Second-order Elastic
Analysis
Local beam lateral-torsional or local plate buckling
P
δP
2x0.5%P
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In design of frames by BS5950(2000), we learnt that a frame is needed to be
classified as non-sway, sway-sensitive and sway very-sensitive frames
according to the value of λcr . The lecture provided us a background on thecodified method.
The hand method has a number of shortcomings as follows.
The calculation of deflections at each storey is relatively inconvenient and
the method may not be applicable to some irregular structures.
It cannot be used when λcr is less than 4, which is not uncommon, especially
for temporary structures.
For a multi-storey sway sensitive bare steel frame, the P-∆ effect can hardly be considered accurately. A similar problem may exist for design of other
frames of which the P-∆ effect cannot be considered in detail.
W
WDo you expect the two circled
columns have the same effective length ?
If not, how can we use Appendix E.2 to distinguish
the differences ?
(Assume all members are the same size for simplicity)
φ
φ4
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The effective length can be determined by the method of using the elastic
critical load asccr
E F
EI L
λ
π 2= . But when a less critical or non-critical member
under smaller axial is designed, the effective length is very long since Fc is
very small. Is it reasonable ?
About the amplification method, the amplification cannot be used for non-
sway frames and, more importantly, it is inconvenient to use for all
members, especially the inclined members, in a large frame.
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Question
We were taught to ignore the compression member since the tension
takes most load. Can we consider this by a linear analysis ? NO !
Linear analyiss tells you that the tension and compression members
take the same load, which is incorrect.
It can only be considered by a second-order analysis allowing for P-δ effect (i.e. change of member stiffness).
Economy can be gained since the capacity of compression member is
considered.
Force
Compression Member
Tension Member
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What is the effective length for the back chord member in
the out-of-plane direction ?
Buckled Shape of a Bow-Shaped Truss
Suction wind making back chordin compression. What is its effective length ?
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3.5 P-∆-only analysis vs. P-∆ δ analysisAccording to many research papers, the Hong Kong Steel Code 2004 and the
LRFD (1996), there exist two P-delta effects as P-∆ and P-δ effects. We can
carry out a P-∆-only analysis and a more refined and much better P-∆−δ analysis for design.
3.6 Second-order P-∆-only analysis of finding the bending moment
The second-order analysis is a new method referenced and recommended by
various codes including the BS5950(2000), Eurocode, AS4100 etc. In the
analysis, the instability and second-order effects are allowed for in the
determination of the strength of a steel frame.
In Australia, the second-order analysis is carried out to determine the
bending moment allowing for the P-∆ effect so that the complex checking of
sway and non-sway mode etc can be skipped. The member is then designed
as non-sway, usually with its true length equal to the effective length. This is
an improved method that the P-∆ effect is computed in the software. But itstill requires the manual checking of member strength to the design code. It
is similar to, but more accurate than the sway amplification method.
However, the method is useless for non-sway frames and cannot consider
member imperfection. In design, we need to plot the bending moment
allowing for P-∆ effect of a sway frame and then use the design chart for
buckling resistance check of non-sway frames to check the member
resistance.
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3.7 Second-order analysis P-∆ δ analysis
If we consider both the P-δ and P-∆ effects, we then need not assume an
effective length and the load capacity of a structure can then de determined by checking the section strength of the member. For example, we can obtain
the same buckling load as Table 24 of BS5950 for columns with any
boundary condition WITHOUT assuming an effective length which, in
general frame, is unknown.
Second-order analysis allowing for P-∆
andP-δ effects ALLOWING
for member & frame imperfections
Simple section capacity check for
all members in the software
according to steel grade andsection type used
Additional check for beam lateral-torsional buckling
for laterally unrestrained beams
Reduced sectional
dimensions/design strength
for slender sections
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3.7.1 P-δ-∆ analysis ignoring beam lateral-torsional buckling check
With other terms readily obtained from a linear analysis, Nida can check the
strength of every member by the following section capacity check.
1)()(
≤=+∆+
++∆+
+ ϕ δ δ
z y
z z z
y y
y y y
y Z p
PP M
Z p
PP M
A p
P (8)
where
P = axial force in member
py = design strength
Zy, Zz = effective modulus about principal axes
My, Mz = moment about principal axes
P P
The P-∆ and P-δ Effects
δ
∆
If we consider both P-∆ and P-
δ effects, we need not worry
about the effective length and
the design is more efficient
and accurate.
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ϕ = material consumption factor. If ϕ >1, member fails in design strength
check and if ϕ
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p b can be calculated from section and material properties and slender
determined from beam boundary conditions (see Appendix B.2, BS5950
[2000]). Sx is the plastic modulus used for plastic and compact sections and
elastic modulus for semi-compact and slender sections. Note that the uniform moment fact, mLT, is taken as 1 for destabilising load.
In case where the loads are normal (i.e. loads are applied at the shear centre),
separated section and member capacity checks are needed. For member
capacity check, the “mLT” factor is less than 1 and taken from BS5950 as,
1≤=+∆+
++∆+
+ ϕ δ δ
z y
z z z
b
y y y LT
y Z p
PP M
M
PP M m
A p
P (10)
where mLT is determined under various shapes of bending moment diagram.
For example of a beam under general condition, mLT can be determined by
sampling the bending moment along a beam as,
max
432 15.05.015.02.0 M
M M M m LT
+++= (11)
where M2, M4 and M3 are respectively bending moments at quarter points
and at mid-span.
For section capacity check, Equation 2 can be used.
3.7.3 Why second-order analysis is important for column buckling onlyOne may wonder why second-order analysis is needed for column buckling
check, but not quite necessary for beam or local plate buckling checks.
For section local buckling check, either the effective width or the effective
stress can be used.
The local plate and lateral-torsional buckling of beams are localised effects
and their checking in design codes is more on isolated members and
therefore their design is simpler than flexural column buckling. Column
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buckling is more a system interactive behaviour that its buckling strength is
affected sensibly by member far away from it. As a result, frame
classification is needed and the effective length method is not applicable
when the elastic critical load factor is less than 4.
The effect of lateral-torsional buckling is more local and their checking in
design codes is more on isolated members which can be carried out by a
simple procedure or programming. For uncommon slender section, the
sectional properties or design strength can be revised to prevent this
occurrence.
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3.8 The 2 Analysis Procedures for P-∆ δ analysis
There are two procedures.
3.8.1 Incremental Load Method determining load resistance
Increment the load step by step until any member fails. The
incremental load can be approximately 2% - 10% of the design load, as the
accuracy requirement, of the guessed design load. This exact incremental
load value is unimportant and only affects the number of load cycles causing
the structure to fail. But sometimes it cannot be too large to prevent
divergence.
Method to t race the complete equilibrium path beyond buckling
Displacement, u
Appl ied Load, F
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3.8.2 Fixed load method for checking against design loads
Apply the design load to check if any member fails. Sometimes, we
need to divide the load into a number of increments and use the arc-length
with minimum residual displacement method to prevent early divergence for
post-buckling analysis.
Iteration Method by 2 load increments to reach the design load
Equilibrium Path
Load, F
Displacement, u
Divergence Load
T
KO T
F0
F1
u0
1u
Design Load Level
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3.9 Buckling strength curves in BS5950(2000)
The major differences between limit state code BS5950 and allowable stress
code BS449 regarding column buckling are :
1. BS5950 includes section shape variation (i.e. the use of four compressivestrength tables)
2. BS5950 allows for locked-in stresses (i.e. residual stresses) and3. It also allows for stocky column effect
The buckling strength or the load capacity of a column is dependent
on its length, boundary conditions, second-moment of area from cross
sectional geometry, section shape variation (I, Channel, box etc.), residualstress and imperfections.
The formula for the buckling strength curves is given by,
c E c E c y p p p p p p η =−− ))((
in which py = the design strength or yield stress,
pc= compressive strength,
pE = Euler’s buckling stress = 2
2
⎟ ⎠
⎞⎜⎝
⎛ r
L
E π
η = curve-fitting parameter for straightness
=2r
y∆(analytical)
= 01000
)( 0 ≥− λ λ a
(empirical from BS5950)
in which λ0= y p E
2
2.0 π ,
y = distance of extreme fibre from the centroidal axis
L = Length
r = radius of gyration
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From test results of different slenderness, section type and
manufacturing type for hot-rolled cold-formed etc. , η can be found tofit the experimental curves.
For curves (a) in BS5950, a = 2.0For curves (b) in BS5950, a = 3.5
For curves (c) in BS5950, a = 5.5
For curves (d) in BS5950, a = 8.0
With slenderness ratio, λ, and steel grade, the design bucklingcapacity can be determined from Table 24 as,
Pc=Apc
in which A is the cross sectional area and pc is the compressive strength.
Nida uses a curved element with initial imperfection at mid-span denoted as
δ0 which can be assigned by the users. This value is given by (see PerryEquation),
λ r
y
L
δη ⋅⋅= 0 ( )0λ λ a001.0 −= λ a001.0≈
Rearranging terms will give:
r y
a001.0
L
δ 0 =
From above, it can be seen that the δ0/L value depends on the section type,
axis of bending and the geometry of the section. In other words, for the same
type of section and axis of bending, the value of δ0/L is maximum if the
section has the minimum value of y/r. Therefore, in order to obtain the lower
bound solution of δ0/L for each section type and axis of bending, a section
having the smallest value of y/r (the critical section) is used. Table 1
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summarizes the critical section for each section type and axis of bending and
its corresponding value of δ0/L calculated according to Equation above.
TABLE 1
δ0/L FOR CRITICAL SECTIONS OF VARIOUS TYPES OF SECTION
AND AXIS OF BENDING
Axis of Bending
x-x y-yType of
SectionSection
δ0 /L·1000 Section δ0 /L·1000
UB 305x165x40 1.697 127x76x13 1.685
UC 356x368x129 3.000 356x406x634 2.860
CHS 508.0x10.0 1.389 - -
SHS 300x300x6.3 1.598 - -
RHS 300x200x6.3 1.513 500x200x8.0 1.732
Channel Any axis: 152x89 4.474
The following lower bound δ0/L values were obtained.
δ0/L =1.75 for UB, CHS, SHS
3.0 for UC and
4.5 for channel
They are lower bound solutions to the BS design curves. Economy can
further be gained for fine tuning of δ0/L by calibration with the design
curves in BS or, in fact, any other national design codes.
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With the above information, the buckling design curves of various critical
sections are plotted using Nida. Figure shows an example of buckling design
curve of a section against the BS5950(2000) curve “a”. Similar good results
can be obtained for other buckling curves or in fact buckling curves in other
national codes by adjusting the δ0.
0.0
50.0
100.0
150.0
200.0
250.0
300.0
0 50 100 150 200 250 300 350
Slenderness
B u c k l i n g S t r e n g t h
( N / m m
2 )
NAF-NIDA BS5950 Euler
Buckling design curves (Curve a)
UB (x-x)
CHS
SHS
RHS
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3.10 The Euro-code 3 for second-order analysis and design
The code is the most comprehensive code in dealing with second-order
analysis and design. Below is the abstract of some of its clauses related to
steel structure design. Basically, one must allow for P-∆ and P-δ effects andtheir imperfections in design.
P-∆ and P-δ effects for any structure in compression
P-∆ effect P-δ effect
1 Geometry update by a nonlinear
analysis or
2 Amplify moment by a factor
1−λ
λ with
δ λ
v
N
F
HF=
1 Member curvature update
by use of curved element or
2 Buckling strength formulaeassuming members of Le =1
and
3 Amplify moment by1−λ
λ
withcFL
EI2
2π λ = not explicitly
required in Eurocode 3 but
needed in LRFD (B1 factor)
and HKSC2004.
Where,
λ is the elastic critical load factorFc is design axial force,
H is the storey height
δ is the relative storey drift or lateral deflection)FV is the vertical force (i.e. factor design vertical loads)
F N is the notional force (i.e. normally 0.5% of Fv.
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Simulation of imperfections present in all practical structures
P-∆ imperfection P-δ imperfection
1 Eigen-buckling mode with amplitude
equal to building tolerance or
. Notional force or
. Inclined structural geometry or
1 Use of curved element withmid-span imperfection or
2 Several elements to modelcurved geometry or
3 Use of buckling strengthformulae
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3.11 Limitations and advantages of second-order analysis
The advantages of the method are as follows.
Economical design since the designed structure will be lighter. It can be
viewed as a MATERIAL OPTIMISATION process by re-arranging the
material correctly. We normally over-estimate the effective length for
about 80% of members.
The design is safer. Some members will not be over-designed whilst
others are under-designed. We can identify the key members for safer
design and the under-designed 20% member strength may lead tocollapse.
Quick design output, design is completed simultaneously with analysis.
Accurate in output since the determination of buckling effect is rigorous,
but not by manual judgement which varies from one engineer to another.
Change of stiffness or stiffening and weakening effects of tension and
compression members are considered in full.
Wider application, it accounts for complex cases, such as change of
stiffness in the presence of axial force, sloping bracing members, snap-
through instability, pre-tensioned structures etc.
More reliability e.g. effect of adding bracing members can be seen
directly.
Interactive behaviour can be considered. A system design instead of a
member design approach is used.
Lesser chance of human error when using the design codes.
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Disadvantages
Super-imposition cannot be applied. It becomes more complicated for
many load cases.
It is a new method which requires us to learn and be familiar with.However, with the changing technology and globalisation, it appears that
we cannot avoid using better and new methods else we cannot compete
with our counterparts.
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3.12 Examples
Example 1 Testing of a simple truss of 4.8m span
The truss of nominal size 4.8m wide x 1 m deep shown in Figure below
was tested. One end of the truss is allowed to slide freely along the
longitudinal x-axis and to rotate about all axes by simply placing the
member onto the supporting platform. The other supporting end is welded to
a flat plate fixed onto the support so that torsional twist and displacements in
all directions are prevented. All members of the truss are made of 48.3x3.2
Circular Hollow Section (CHS) and grade S275 steel.
1198 1199 1200 1201
978
All members are 48.3x3.2 CHS
Geometry of the Tested Truss
Applied Load
unit in mm
y x
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A point load at the mid-span bottom of the truss was applied to the truss
until buckling, which was indicated by an excessive deflection of the top
chord. Deflections at several nodal locations were measured against the load.
This loading arrangement made the top chord in compression and buckled
laterally.
In the design of the truss, a simple question will be raised. What is the
effective length of the top chord against buckling in out-of-plane direction ?
A simple widely used assumption for this effective length determination is
the distance between chord for in-plane buckling and the distance between
support for buckling out-of-plane.
When using this conventional approach of assuming the
distance between supports as effective length, it is then taken as 4.798m and
the slenderness ration (Le/r) for the tubular sections of 48.3x3.2 CHS of
grade 43 steel is 299.9. From BS5950, the permissible stress is 21 N/mm2
and the permissible load in top chord is equal to pyA or 9.513 kN. The
applied load generating this compressive load is then calculated as 7.8 kN.
In the experiment, the tested buckling load of about 34
kN is much higher than the design load calculated from the conventional
method of 7.8 kN by 4.4 times. This shows the uneconomical output by the
conventional design method following strictly to the design code.
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The experimental load versus deflection plot for the
lateral deflection at mid-span node is also shown in Figure below, together
with the computational results. In the theoretical analysis, the nodal co-
ordinates are taken from previous Figure, with allowance of initial
imperfection. For the first case, one end was assumed free to rotate
longitudinally and the second case assumed this twist is restrained about the
longitudinal x-axis. A 0.5% notional force is further applied in order to fulfil
the code requirement. Nevertheless, it was noted that the notional force is
unimportant for buckling analysis when the member initial imperfection was
considered since both of them are disturbances to activate buckling. The
objective of this notional force is to simulate the imperfection like the out-
of-plumbness in a frame.
0 20 40 60 80 100
40
35
30
25
20
15
10
5
0
Lateral Deflection at Middle Node "A" (mm)
L o a d , P
( k N )
Load versus Deflection of Simple Truss
Theory (Both ends
Theory (Only one end
Experimentrestrained against twist)
Buckled Shape
Undeformed Shape
restrained against twist)
Buckled shape
A
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It can be seen in the Figure that the theory simulating the
actual condition is close to the tested results. The computed applied force, P,
by Nida is 32 kN at a lateral displacement of 107 mm. The calculated
buckling strength results is less than the tested load of 34.2 kN. It was
difficult to determine precisely the elastic buckling load of the truss since the
elastic load capacity increases exponentially with displacement. This
uncertainty is eliminated when using section capacity check.
The buckled shape of the truss is plotted in Figure below.
It can be seen that the bottom tension member deflects whilst the top
compression member with the complete truss twists, demonstrating the
system buckles simultaneously. This contribution by the torsional stiffness
of the tension member stiffens the compression member against buckling
significantly and its consideration will, therefore, make the design more
economical.
When we assume the truss is restrained against twist, the design
buckling strength is 39.5 kN. It can be seen in the Figure that the deviation
between the two sets of computational results increases when the deflection
entered the non-linear range, demonstrating the stiffening-tension member
effect activated when the structure behaved non-linearly. Linear analysis
cannot therefore reveal this phenomenon for a planar truss.
When we use the concept of effective length, we encounter a problem of
varying axial force in the buckling chord. This effect is not considered in
most national codes which consider only the geometrical and boundary
conditions. By conventional analysis using the maximum load in top chord,
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we can obtain the same result as our buckling analysis if the effective length
is assumed as 2.311 m or the effective length factor is taken as 0.482. In this
case, the buckling stress from BS5950 is then equal to 86 N/mm2 and the
permissible buckling load is then 39 kN, which can be produced by an
applied point load of 32 kN.
Following the conservative assumption of using the distance between
support equal to 4.7985 m as the effective length, the buckling stress from
BS5950 [1990] is 21 N/mm
2 and the buckling applied load is only 7.8 kN. It
differs from our computer and test result by about 4 times !
This example demonstrates the versatility and accuracy of the computer
method in predicting the buckling load of a tubular truss against out-of-plane
buckling. It further illustrates the significance of the torsional effect in
buckling and the stiffening-tension member effect.
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Example 2 the design of a simple 4-storey frame
Using the proposed NIDA, the buckling strength for the frame is 885 kN
which is determined as the load level violating section capacity check (see
Figure above). The equilibrium path is also plotted in the Figure above. The
advanced analysis indicates the maximum elasto-plastic buckling load as915 kN. From this comparison, the proposed method predicts a load capacity
of 11.6% above the conventional design method, but still well within the
theoretical ultimate load by elasto-plastic large deflection analysis. This
indicates the method is economical and safe.
However, since λcr is less than 4.0 here, the above method can no longer beused in the new BS5950(2000). There are two solutions for this problem.
The first is to use the major principal axis of members to resist loads, which
is considered as case 2. The other option is to add bracings members which
is designated as case 3.
Case2 Unbraced case by Annex E, Equation 20 in this note.
Referring to Table 1, the selected φs is 0.00085 and the λcr is =1/200/0.00085 = 5.9 > 4 and < 10, sway sensitive frame.
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
P P
2x0.5% P
500
1,000
V e r t i c a l F o r c e P ( k N )
Lateral Drift at Top (m)
4 @ 4 m = 1 6 m
4m
885kN
915kN
722kN
Design strength by conventional method
Design strength by NIDA
Elasto-plastic buckling strength
by method in Chan and Chui (2000)
The 4-storey moment frame
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Using NIDA, λcr is 6.3
The effective length =
m x
x x x
F
EI L
ccr
E 47.6000,5009.5
106103000,205 422===
π
λ
π
L/r = 6.47/89.6 = 72.2,
From Table 24b, BS5950, permissible axial force = 197.6x7600 = 1,520 kN
Design load factor = 1657/500 = 3.0
Design Load Factor by NIDA = 3.2
Case 3 Fully braced case by Annex E and chart
Obviously the frame is non-sway and the beam is bent under single
curvature.
From Table E.3, consider column in the second level as the most critical.
8.05.2/2
5.0
1 ==
++
+=
L
I
L
I
L
I L
I
L
I
k
8.05.2/2
5.02 ==
++
+=
L
I
L
I
L
I L
I
L
I
k
From Chart Figure E.1 for non-sway frame, Le/L = 0.855,
Thus effective length = 0.855x4 =3.42m
L/r = 3420/51.2 = 66.8From Table 24c, pc=187.4 N/mm
2
Pc=187.4x7600 = 1424. kN
At design load, the axial force in column is 428.7 kN,
Permissible load factor = 1424/428.7 =3.3
Using NIDA, λcr is 20.00
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Design load Factor by NIDA = 3.4 using imperfection parameter 1.75/1000.
When the structure is under a set of more realistic loads due to beam
reactions and distributed evenly at the four levels, the manual approach
becomes more complicated to use with its result uncertain. This is because
the column is under a variable axial force and the most critical section is not
obvious. Here, the buckling mode is unsymmetrical and most design codes
do not consider this variable axial load case. Using the proposed method, the
computational and design effort is the same as in the above case and can be
completed very easily and conveniently. The calculated total load taken by
the structure in the case 1 is revised to 1120 kN which is considerably larger
than the above load of 885 kN.
Example 3 Design of a simple portal by amplification method
The portal frame shown in Figure below is analysed and compared with
the design code used in association with the hand method of analysis. It is
under a lateral load and a vertical force at top of one of its column. The
section used for both columns and beam is 356x368x153 H-section and
grade 43 steel.Properties of 356x368x153 H-section are as follows.
A = 195 cm2, I = 48,500 cm
4, r = 15.8 cm, Z = 2680 cm
3
30 m
1 0 m
60kN
1000kN
Mo men t Joints
Pinned Joints
A ll m em ber s 305x3 05x1 98 U C , G ra de S 27 5
The Porta l Fram e
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Hand Moment Amplification Method
Vertical Reaction on the left = R L = 60 x 10,000 / 30,000 = 20 kN
Vertical Reaction on the right = R R = 1000 + 20 = 1,020 kN
Horizontal reaction of the left = HL = HR = 60/2 = 30 kN
MA = MD = 0
M b = Mc =30 x 10 = 300 kN-m
Buckling analysis :-
k a = 1.0, k B = (1/10)/(1/10+1.5x1/30) = 0.67
(Le/L)AB = 2.9 from BS5950
Similarly, Le/LCD= 2.9
Nof = 2xπ2EI/(2.9L)
2 = 2xπ2x200,000x485x106/(2.9x10,000)2 = 2277 kN
λ = 2277/(-20+1020) = 2.27
Using NIDA, λcr is 2.25
Amplified Moment = M* = M λ/(λ-1) = 300x2.35/1.35 = 522.2 kN-m
λ from sway index method is 2.5
For column of Euler buckling length of 1.0 L = 10 m
Column slenderness = 10,000/158 = 63.3
From Table 24, BS5950, pc = 214.4 N/mm2
Axial Force = Pc = Apc = 19500x214.4 = 4180.8 kN
Combined Load Check:
F/Pc + M/Mr = 1,000/4,180.8 + 522.2/275/2680/10-3
= 0.948 < 1.0, O.K.
NIDA output / results
σmax / σys = 249.8/275 = 0.908 < 1.0, O.K.
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Example 4 Lift shaft under vertical loads
Check the strength of an indoor lift shaft below. All gaps are filled by 12+12
laminated glass panels. Try Grade 50 150x150 SHS for mullions and
120x120 for transoms.
If the levels for weakest columns are allowed to be strengthened by cross
bracing, what will be the ultimate design load factor ?
80kN80kN
50kN 50kN 50kN
110kN
3m
3m
1.66m
2.4m
1.35m
2.4m
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Example 5 Two story planar frame
Determine the frame buckling load factor for the rigid-jointed frame shown
in Figure below and also determine the design load factor. The numbers
shown in brackets are the multiples of I = 10.0 x 106 mm4 and the axial force
N* = 200 kN for each member. For example, the values of (5,3) for member
DG correspond to IDG = 50.0 x 106 mm
4 and N *DG = 600 kN. If cross bracing
of 1,000 cm4 are added to the smaller bay (i.e. D-B, A-E, G-E and D-H),
what will be the design load factor ?
Here we need to use a linear analysis program to find deflections at each
story. Then determine λc (1.43 from Professor Trahair) to confirm swayframe. Then use Appendix E to find each column capacity and then compare
these with applied loads.
Using P-∆−δ analysis, a few seconds after the completion of analysis modelwill complete the work more economically.
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Example 6 Stability design for the Macau Return Ceremony Hall
The Macau Return to China ceremony hall was constructed to house the
ceremony taken place for handing over of Macau from Portugal to China in
1999. The
dimensions of the structure are 134m in length, 57m wide and 28.3 m high.
All member connections are welded and the columns are pinned to the pile
cap foundations. Square hollow sections with width ranging from 150mm to
450mm were used and all steel stress is 250 MPa. The photographed
elevation of structure is shown in Figure 4 and the computer plan and
elevation are depicted in Figures 5 and 6. The structure is modeled by
10,315 members and 3,750 nodes. The total weight of steel is about 1300
tons. In the analysis, the first cycle assumed the members are perfectly
straight and their directions of deflections are determined and recorded. In
the second cycle for actual analysis, the member initial imperfections are
assumed to be in the same direction as these member deflections in the first
cycle. This is conservative, but represents a consistent approach to that
adopted in the design code which always assumes a weakening effect of
imperfection.
The original structure was designed to withstand a 3-second gust wind speedof 6 month return period. After the ceremony, the Macau Government
considered extending the life of the structure to 50 years. A wind tunnel test
was then carried out in China with pressure determined for re-analysis.
Based on this pressure, the structure was then re-designed and checked by
the present method.
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Front View of the Ceremony Hallomputer model
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The results of analysis are indicated graphically in Figure above with
color showing the stress level of each member. A warning colour like RED
indicates the buckling capacity of the member has been reached, yellow
implies the strength factor is between 0.8 and 1 and other colors show
different load level for each member. Deflections can also be determined at
serviceability as well as at the ultimate limit loads. In the analysis, a number
of members were noted to have been under-sized in strength for a 50 year
design life. A proposal for strengthening the structure at minimum cost wassubmitted. This included addition of several inclined members at corners to
increase the moment capacity of the roof trusses and re-fabrication of
column lower end to reduce moment transfer to pile caps due to push-and-
pull action of the four vertical hollows making up the columns.
The design for the complete structure analysis is completed
simultaneously with the analysis which requires 3 iterations for
convergence. Unlike most commercial software for steelwork design
requiring input of effective length, the present method computes the P-δ andP- ∆ effects automatically in strength determination. Also, the former doesnot consider variation of stiffness in the presence of axial force. A re-design
is quick and convenient whilst the design output is material saving.
In the examples, the present method is demonstrated to be a
viable tool for fast, accurate and economical performance-based design
superior to the conventional design procedure since it can consider complex
Deformed shape withcolors indicating the
external load to buckling
strength factor of each
member
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Professor S.L. Chan © 2004 128
geometrical configurations and loading conditions. The proposed method
meets the current design practice and assumption of limiting the design load
as the load causing the formation of the first plastic hinge or the first yield
load. Consequently, the NIDA approach can be immediately used in daily
design and applied to the design of the practical and large size structure in
the next example.
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3.13 References
American Institute of Steel Construction (1986), Load and resistance factor design, specification for structural steel buildings, AISC, Chicago.
AS-4100, Australian Standard for Steel Structures (1990), Sydney.
Bathe, K.J. (1982), Finite element procedures in engineering analysis, Prentice-HallInc., Englewood Cliffs, N.J.
BS5950, British Standards Institution (2000), Structural use of steel in building, Part
1, U.K.
Brush, D.O. and Almroth, B.O. (1975), Buckling of bars, plates and shells, McGraw-
Hill, Inc.
Chajes, A. (1974), Principle of structural stability theory, Civil Engineering andEngineering Mechanics Series, Prentice-Hall Inc., Englewood Cliffs, N.J.
Chan, S.L. (1990), Strength of Cold-formed Box Columns with coupled Local and
Global Buckling, The Structural Engineer, vol. 68, No. 7, April, pp. 125-132.
Chan, S.L. and P.P.T. Chui (2000),"Non-linear Static and Cyclic analysis of semi-
rigid steel frames", Elsevier Science, pp.336.
Chan, S.L. and Zhou, Z.H. (1994), A Pointwise Equilibrating Polynomial (PEP)
Element for Nonlinear Analysis of Frames, Journal of Structural Engineering,
ASCE, Vol. 120, No. 6, June, pp.1703-1717.
Chan, S.L. and Kitipornchai, S. (1987a), Geometric nonlinear analysis of asymmetric thin-walled beam-columns, Journal of Engineering Structures, 9, pp.243-254.
Chan, S.L. and Kitipornchai, S. (1987b), Nonlinear finite element analysis of angle and tee beam-columns, Journal of Structural Engineering, ASCE, 113(4), pp.721-739.
Chen, W.F. and Chan, S.L. (1994), Second-order inelastic analysis of steel frames by
personal computers, Journal of Structural Engineering, vol.21, no.2, pp.99-106.
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Engineering Series, McGraw-Hill.
Horne, M.R. (1949), Contribution to The design of steel frames by Baker, J.F.,Structural Engineer, 27, pp. 421
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Liew J.Y.R. (1992), “Advanced analysis for frame design”, Ph.D. Thesis, Purdue
University, West Lafayette, IN.
Merchant, W. (1954), The failure load of rigidly jointed frameworks as influenced by stability, The Structural Engineer, 32, pp.185-190.
Narayanan, R. (1985), Plated structures - stability and strength, Elsevier Applied
Science, N.Y.
Peng, J.L., Pan, A.D.E. and Chan, S.L., ”Simplified models for analysis and design
of modular falsework”, Journal of Constructional Steel Research, Vol.48, No.2/3,
1998, pp.189-210.
Rankine, W.J.M. (1863), A manual of civil engineering, 2 nd edition, Charles Griffin and Comp. London.
Introduction to Steelwork design to BS5950:Part 1 (1998), The Steel ConstructionInstitute.
Timoshenko, S.P. and Gere, J.M. (1961), Theory of elastic stability, 2nd
edition,
McGraw-Hill, New York.
Trahair, N.S. (1965), Stability of I-beam with elastic end restraints, Journal of the
Institution of Engineers, Australia, 38, pp.157-
Trahair, N.S. and Chan, S.L., “Out-of-plane Advanced Analysis of Steel
Structures”, research report, Centre for Advanced Structural Engineering,
Department of Civil Engineering, Sydney University, 2002 (to appear).
Yau, C.Y. And Chan, S.L. (1994), “Inelastic and stability analysis of flexibly
connected steel frames by the spring-in-series model”, Journal of Structural
Engineering, ASCE, pp.2803-2819.
Zienkiewics, O.C. (1977), “The Finite Element Procedure”, 3rd Edition, McGraw-
Hill.
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