chapter 3_design of steel frames by second-order analysis fulifilling code requirements

8/20/2019 Chapter 3_Design of Steel Frames by Second-Order Analysis Fulifilling Code Requirements

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Chapter 3, M.Sc. Lecture Note

Professor S.L. Chan © 2004 77

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


27/54



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


30/54



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,


42/54



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



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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


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


48/54



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.


49/54



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.


50/54



Front View of the Ceremony Hallomputer model


51/54



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


52/54



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.


53/54



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.

Clough, R.W. and Penzien, J. (1993), “Dynamics of Structures”, 2nd edition, Civil

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.

chapter 3_design of steel frames by second-order analysis fulifilling code requirements

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