Information Paper for the Calculation of Elastic Crticial Load Factor for Steel Structures

Structural Engineering Branch, ArchSD Page i of ii File code : LamdaCritical.doc

Information Paper on Calculation of Elastic

Critical Load Factor for Steel Structures

CTW/MKL/CYK

Issue No./Revision No. : 1/- Issue/Revision Date : November 2012

INFORMATION PAPER

Determination of Elastic Critical Load Factor

for

Steel Structures

Structural Engineering Branch

Architectural Services Department

November 2012

Structural Engineering Branch, ArchSD Page ii of ii File code : LamdaCritical.doc



CTW/MKL/CYK


TABLE OF CONTENTS

1. Introduction ............................................................................................................. 1

2. Determination of Elastic Load Factor ..................................................................... 1

3. Effects of Horizonal Load ....................................................................................... 3

4. Conclusions ............................................................................................................. 4

5. References ............................................................................................................... 4

Annex A Example 1 – Calculation of λcr by Deflection Method

Annex B Example 2 – Calculation of λcr by Eigenvalue Analysis

Annex C Example 3 – Considerations in the Determination of λcr by Eigenvalue Analysis

Annex D Example 4 – Effects of Horizontal Load on λcr

Structural Engineering Branch, ArchSD Page 1 of 4 File code : LamdaCritical.doc



CTW/MKL/CYK


1. Introduction

1.1 Code of Practice for Structural Use of Steel 2011 (the “Code”) published by Buildings

Department clause 6.3.2 on the elastic critical load factor, λcr, has been revised to clarify

the determination of λcr by the deflection method. However, there had not been any

elaboration on the use of the eigenvalue analysis method though the correct

determination of λcr is a critical step in the design and analysis of steel structures.

1.2 The purposes of this paper are:

a) to summarize the available methods in the Code for the calculation of λcr;

b) to give examples to calculate λcr and to provide guidance on the correct

interpretation of λcr found by different methods; and

c) to study the effect of horizontal load on the calculation of λcr.

2. Determination of Elastic Critical Load Factor

2.1 In the Code, λcr of a steel frame is defined as the ratio by which the factored loads

would have to be increased to cause elastic instability. λcr is an important parameter that

is required to classify a steel frame into “non-sway”, “sway” and “ultra-sensitive sway”.

Clause 6.3.3 of the Code classifies frames with λcr ≥10 as non-sway, and specifies that

the P- effect is insignificant for such frames. For sway frames with λcr <10, they are

classified as either sway with 5≤ λcr <10 or ultra-sensitive sway with λcr <5, and clause

6.6.1 of the Code specifies that both the P- and P-δ effects should be considered in

their design. In accordance with clause 6.3.3 to 6.3.5 of the Code, λcr can be calculated

either by the eigenvalue analysis for general structures or the deflection method for

geometrically regular and rectangular frame.

2.2 Deflection method

In the deflection method, a frame is classified into either sway (or ultra-sensitive sway)

or non-sway by considering the magnitude of the horizontal deflection of each storey

due to the application of notional horizontal loads at each storey which are taken

typically as 0.5% of the factored dead plus live loads on and above the floor considered.

For clad structure provided that the stiffening effect of masonry infill wall panels or

diaphragms of profiled steel sheeting is not explicitly taken into account, the frame is

considered as non-sway if the deflection of every storey is less than 1/2000 of the storey

height under the action of the notional horizontal loads. With such limit on deflection,

equation (6.1) of the Code therefore gives the following equation to calculate λcr of a

steel frame, except for frames with sloping members having moment-resisting

connections in which the calculation of λcr should refer to clause 8.11 of the Code:

NV

N

δ

h

F

Fcrλ

where FV is the factored dead plus live loads on and above the floor considered;

FN is the notional horizontal force taken typically as 0.5% of FV for building

frames;




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h is the storey height;

and δN is the notional horizontal deflection of the upper storey relative to the lower

storey due to the notional horizontal force FN.

An example is included in Annex A to show how to determine the λcr for a typical

geometrically regular and rectangular frame by deflection method. Further example on

the calculation of λcr can also be found in NCCI: Calculation of alpha-cr (King 2005)

(available: http://www.access-steel.com/, accessed: 18 September 2012).

2.3 Eigenvalue analysis

2.3.1 The determination of λcr by eigenvalue analysis is usually calculated by commercially

available computer software (e.g. SAP2000, NIDA). In eigenvalue analysis, the

numbers of modes to be found can be specified in the computer program and each

eigenvalue found is associated with a buckled mode shape. λcr should be taken as the

eigenvalue corresponding to the first “sway” buckling mode. An example is included in

Annex B to demonstrate the determination of λcr by eigenvalue analysis with NIDA and

SAP2000. It should be noted that the eigenvalue determined by the computer

programme may either be a “sway buckling mode” or “non-sway buckling mode” (or

“local buckling mode”). Figure 1 has therefore been added in the Code in its 2011

version in order to distinguish sway buckling mode from local column buckling mode.

Both the P- and P-δ effects are only required for sway frame, and hence only λcr for

sway buckling mode is required.

Non-sway buckling mode Sway buckling mode

Figure 1 Sway and non-sway buckling modes

2.3.2 In the determination of λcr by eigenvalue analysis, it is therefore important to study the

form of each buckling mode to see if it is a sway mode or a non-sway mode. King (2005)

commented that when using eigenvalue analysis in finding the first sway-mode, “it is

important to study the form of each buckling mode to see if it is a frame mode or a local

column mode. In frames where sway stability is ensured by discrete bays of bracing

(often referred to as “braced frames”), it is common to find that the eigenvalues of the

column buckling modes are lower than the eigenvalue of the first sway mode of the

frame. Local column modes may also appear in unbraced frames at columns hinged at

both ends or at columns that are much more slender than the average slenderness of

columns in the same storey.” Similarly, Rathbone (2002) noted that “where the columns

are axial load predicated, many of the lower buckling modes will be [local] column

buckling modes. It is the [sway] buckling mode of the whole structure that is important”

to include second-order effects. Therefore, λcr should be taken as the factor

corresponding to the first sway buckling mode rather than the local buckling mode.

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2.3.3 Therefore, if the first mode of the eignevalue analysis is a local buckling mode, then the

lowest eigenvalue found does not correspond to the first sway buckling mode. Instead,

the eigenvalue analysis finds the eigenvalue for the local buckling mode. In such case,

PSE should take care in using the eigenvalue analysis and not just use the lowest

eigenvalue which may only be local buckling mode, and it is not the original intent to

use it to define sway sensitivity. Should eigenvalue analysis be adopted, PSE should

scan the output to see which eigenvalue corresponds to the first sway buckling mode.

Alternatively, the deflection method can be used to calculate λcr for the first sway

buckling mode. An example is included in Annex C to illustrate the correct

determination of λcr for a typical portal frame by eigenvalue analysis. Further real

example can also be found in SEB Information Paper - A Case Study of Using Metal

Scaffold System for Demountable Grandstand: The Opening Ceremony of Hong Kong

2009 East Asian Games (available: http://asdiis/sebiis/2k/resource_centre/).

3. Effects of Horizontal Load

3.1 Clause 6.3.2.2 of the Code states that “… of a geometrically regular and rectangular

frame subjected to gravitational loads or gravitational loads plus horizontal load (e.g.

wind), the elastic critical load factor for a sway frame may be calculated as …”, and this

means that the deflection method is applicable to frames with or without horizontal load.

However, it should be noted that equation (6.1) of the Code is only a function of vertical

load and the storey height and is irrelevant of the horizontal load. Horne (1975)

remarked that “[i]t should be noted that the elastic critical load of a frame is affected to

an entirely negligible effect by horizontal loading, and only the vertical loads need be

considered.” In order to verify the accuracy of the deflection method and Horne’s

remark, the values of λcr of a simple portal frame (at Annex D) under various

combinations of horizontal and vertical loading by both deflection method and

eigenvalue analysis have been analysed.

3.2 From the example at Annex D, the following can be observed that:

a) The results from both deflection method and eigenvalue analysis are similar when

λcr is less than 10.

b) When λcr becomes larger, the differences between the deflection method and

eigenvalue analysis also become larger but this is not a concern as it has already

been a non-sway frame with λcr 10.

c) Under the same applied vertical load, λcr determined by deflection method remains

unchanged while that determined by eignevalue analysis remains of the same order

of magnitude under the action of different magnitude of horizontal loads.

d) For the case where the frame is subject to horizontal load only (from horizontal load

of small to large magnitude), the values of λcr so obtained are always substantially

larger than 10, i.e. non-sway frame.

Therefore, all of the above observations agree with Horne’s remark that “the elastic

critical load of a frame is affected to an entirely negligible effect by horizontal loading.”

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4. Conclusions

4.1 In using eigenvalue analysis for the determination of λcr, PSE should use the eigenvalue

corresponding to the first sway buckling mode, not the local buckling mode, as λcr.

Alternatively, the deflection method can be used to calculate λcr for geometrically

regular and rectangular frame structure.

4.2 Horizontal loading has negligible effect on λcr of a frame, and hence only the vertical

loads need to be considered in the determination of λcr.

5. References

British Standard Institution (2005), BS EN 1993:2005 – Eurocode 3: Design of Steel

Structures (London: BSI).

British Standard Institution (2000), BS 5950-1:2000: Structural Use of Steelwork

Design of Steelwork in Building (London: BSI).

Buildings Department (2011), Code of Practice for the Structural Use of Steel 2011

(Hong Kong: Buildings Department) (available: http://www.bd.gov.hk/english/

documents/, accessed: 18 September 2012).

Buildings Department (2007), Explanatory Materials to Code of Practice for the

Structural Use of Steel 2005 (Hong Kong: Buildings Department) (available:

http://www.bd.gov.hk/english/documents/, accessed: 18 September 2012).

Horne, M R (1975), “An Approximate Method for Calculating the Elastic Critical

Loads of Multi-storey Plane Frames,” The Structural Engineer, 6(53), pp. 242-248

King, C (2005), NCCI: Calculation of Alpha-cr (Ascot: SCI) (available:

http://www.access-steel.com/, accessed: 18 September 2012).

Rathbone, A J (2002), “Second-order Effects – Who Needs Them?” The Structural

Engineer, 80(21), pp. 19-21 (available: www.istructe.org, accessed: 18 September

2012).

http://www.bd.gov.hk/english/%20documents/

http://www.bd.gov.hk/english/%20documents/

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Annex A




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Issue No./Revision No. : 1/- Annex A Issue/Revision Date : October 2012

Example 1

Calculation of cr by Deflection Method

This example shows the procedures to determine the value of cr of a 3-storey portal frame at

5m c/c (Figure A1) by using the deflection method.

Design Loadings:

Dead Load = 7.5 kN/m2

Live Load = 5.0 kN/m2

Steel Members:

Steel Grade: S355

Beam Size: 356×171×51 kg/m UB

Column Size: 254×254×107 kg/m UC

Design Load Case = 1.4 DL + 1.6 LL

Factored Load = (1.4 7.5 + 1.6 5.0) 5 = 92.5 kN/m

Load per floor = 92.5 12 = 1110 kN

Therefore, Notional Horizontal Force = 0.5% of vertical load = 0.5% 1110

= 5.55 kN

Figure A1

Annex A




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Issue No./Revision No. : 1/- Annex A Issue/Revision Date : October 2012

The structrue was analsyed with QSE, and the displacement of each floor obtained is shown

in Figure A2:

Figure A2 Deformed Shape

1 = 5.585 mm h1 = 3.0m

2 = 4.230 mm h2 = 3.0m

3 = 2.704 mm h3 = 3.0m

Using equation (6.1) of the Code,

cr,1 = 230.4585.5

3000

1110

55.5

= 11.07 10

cr,2 = 704.2230.4

3000

21110

255.5

= 9.83 10

cr,3 = 704.2

3000

31110

355.5

= 5.55 10

Therefore, the value of cr should be taken as the smallest of cr,1, cr,2 and cr,3, i.e. 5.55.

Annex B




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Issue No./Revision No. : 1/- Annex B Issue/Revision Date : October 2012

Example 2

Calculation of cr by Eigenvalue Analysis

The same example at Annex A is now analysed by the eigenvalue analysis with NIDA and

SAP2000.

Design Loadings:

Dead Load = 7.5 kN/m2

Live Load = 5.0 kN/m2

Steel Members:

Steel Grade: S355

Beam Size: 356×171×51 kg/m UB

Column Size: 254×254×107 kg/m UC

The computer analysis by NIDA and SAP2000 is carried out in accordance with the following

steps:

(A) NIDA

Step 1:

The geometry of the structure to be analyzed is input in the computer model with the factored

design load case of 1.4 DL +1.6 LL.

Annex B




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Step 2:

In the analysis option, select the “Eigen-Buckling Analysis” as the analysis type.

Step 3:

After the completion of analysis, the load factor corresponding to the first buckling mode for

the load combination is found to be 5.36.

Annex B




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Step 4:

Check the mode shape of the model to confirm if the structure deforms in a sway buckling

mode.

Therefore, λcr should be taken as 5.36 (the load factor corresponding to the first sway buckling

mode) which is about the same value as that obtained from the deflection method at Annex A.

(B) SAP2000

Step 1:

The geometry of the structure to be analyzed is input in the computer model with the factored

design load case of 1.4 DL +1.6 LL.

Annex B




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Step 2:

In the definition of load cases, select the “Buckling” as the load case type.

Step 3:

After the completion of analysis, the load factor corresponding to the first buckling mode for

the load combination is found to be 5.29. Check the mode shape of the model to confirm if the

structure deforms in a sway buckling mode.


mode) which is about the same value as that obtained from the deflection method in Annex A

and NIDA.

Annex C




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Issue No./Revision No. : 1/- Annex C Issue/Revision Date : October 2012

Example 3

Considerations in the Determination of cr by Eigenvalue Analysis

The value of λcr of a single storey portal frame with a diagonal bracing (Figure C1) is

determined by both the deflection method and eigenvalue analysis.

Design Loading:

1000kN at ¼ th span of the first bay (factored load)

Steel Members:

Steel Grade: S355

Beam Size: 152 × 152 × 23 kg/m UC

Column Size: 152 × 152 × 23 kg/m UC

Bracing Size: 88.9 × 6.3mm CHS

Figure C1

By Deflection Method

Notional horizontal defelction due to the notional horizontal force (0.5%×1000 = 5kN) is

0.149mm (using computer software NIDA)

Therefore, λcr = 3000/200/0.149 = 100.6

5 kN

3m

3m 3m

1000 kN

Annex C




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Issue No./Revision No. : 1/- Annex C Issue/Revision Date : October 2012

By Eigenvalue Analysis

Mode Mode Shape Sway Mode

The ratio by

which the load has

to be increased to

cause elastic

instability

1st

Lateral deflection at top of frame = 0.72mm

Non-sway

buckling mode/

Local column

buckling mode

7.1

2nd


Non-sway

buckling mode/

Local column

buckling mode

25.1

3rd


Non-sway

buckling mode/

Local column

buckling mode

27.4

4th


Sway buckling

mode 99.0


mode) which is of about the same value as that obtained from the deflection method.

Annex D




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Issue No./Revision No. : 1/- Annex D Issue/Revision Date : October 2012

Example 4

Effects of Horizontal Load on cr

The value of λcr of a single storey portal frame (Figure D1) is determined by both the

deflection method and eigenvalue analysis under different combinations of horizontal and

vertical loads.

Steel Members:

Steel Grade: S355

Beam Size: 254 × 146 × 373 kg/m UB

Column Size: 203 × 203 × 46 kg/m UC

Figure D1

Case

Horizontal

Load, Fh

(kN)

Vertical

Load, Fv

(kN)

λcr(Deflecton)

(Deflection

Method)

λcr(Eigenvalue)

(Eigenvalue

Analysis)

Ratio

(λcr(Deflection)/

λcr(Eigenvalue))

1 50 400 8.09 7.5 1.08

2 100 400 8.09 7.5 1.08

3 200 400 8.09 7.4 1.09

4 400 400 8.09 7.2 1.12

5 50 50 64.7 59 1.12

6 100 50 64.7 56 1.17

7 200 50 64.7 48 1.36

8 400 50 64.7 34 1.90

9 10 10 332.93 295 1.13

10 20 10 332.93 278 1.20

11 30 10 332.93 258 1.29

12 40 10 332.93 239 1.40

13 50 10 332.93 219 1.52

4.5m 1.5m

3m

D

B

A

Horizontal

Load, Fh

Vertical Load, Fv

C

Annex D




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Issue No./Revision No. : 1/- Annex D Issue/Revision Date : October 2012

From the analysis results, it can be observed that:

a) The results of both deflection method and eigenvalue analysis are similar when λcr is

less than 10.

b) When λcr becomes larger, the differences obtained from the deflection method and

eigenvalue analysis also become larger but this is not a concern as it has already been

a non-sway frame for λcr 10.

c) Under the same applied vertical load, λcr as determined by deflection method remains

unchanged while that as determined by eignevalue analysis remains of the same order of

magnitude under the action of different magnitude of horizontal loads.

Values of λcr of the same structure under the action of horizontal load only (Figure D2) has

also been determined as follows:

Figure D2

Case

Horizontal

Load, Fh

(kN)

Vertical

Load, Fv

(kN)

λcr(Eigenvalue)

(Eigenvalue

Analysis)

Lateral

Deflection

(mm)

Deflection/

Height

14 10 0 1783 9 1/333

15 20 0 891 18 1/167

16 30 0 594 27 1/111

17 40 0 446 36.1 1/83

18 50 0 356 45.1 1/67

19 100 0 178 90.2 1/33

20 200 0 89 180.3 1/17

21 400 0 45 360.6 1/8

In the above cases where the frame is subject to horizontal load only (from horizontal load of

small to large magnitude), λcr so obtained are always substantially larger than 10, i.e. non-

sway frame and therefore the horizontal load will not affect sway stability for frame

classification.

6m

3m

D

B

A

Horizontal

Load, Fh

C

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Information Paper for the Calculation of Elastic Crticial Load Factor for Steel Structures