Buckling Length REPORT

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Aalborg University

Structural and Civil Engineering, 9th Semester

Department of Civil Engineering

Sohngårdsholmsvej 57

www.bsn.aau.dk

Title: Determination of buckling lengthof columns in multi storey plane

steel frames

Project period: B9K - Trainee, Au-

tumn 2010

By: Sugunenthiran Markandu

Supervisors:Lars Pedersen

Print runs: 4

Number of pages: 76

Appendix: 42 Appendix report and 1

Appendix CD.

Completed: 6 January 2011

Synopsis:

Buckling length of columns in a load-bearing

multi storey steel frame structure, used

as case study, are determined following

approaches given by AISC and DIN 18800.

Additionally the numerical tool Robot is

applied for this issue.

Initially frame design in practice, the differentmethods given by EC 3 are explained where

design based on equivalent column method

is chosen. Hence the concept of effective

buckling length is explained by considering

the fundamental column cases where the in-

fluence of support conditions on the buckling

length of column (K-factor) is elaborated.

Several other buckling analysis are performed

on frames with variance restraint conditions

in Robot in order to determine factors that

influence on K-factor of columns in framed

structure.

K-factor determination charts given by AISC

and DIN 18800 are presented where the

use and limitations of them are explained.

Furthermore theoretical deviation of AISC

charts are elaborated.

Two load cases are considered due to obtain K-

factor of columns in the case study structure

by application of AISC and DIN 18800

approaches. Buckling analyses in Robot areperformed for this issue. The determined

results by employing different approaches are

compared and discussed. Furthermore the

influence of the K-factor on the final result is

examined by performing code check in Robot.

Finally the conclusion is made upon which

method is most suitable for practical use.

The report’s content is freely available, but the publication (with source indications) may only happen by agreement

with the authors.


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Preface

This report is a product of project work made by the author at the 3rd semester of

the candidate program of Structural and Civil Engineering at the Department of Civil

Engineering at Aalborg University.

The project is made during an internship at Rambøll Aalborg, where the author also

participated in other projects and activities. These projects and activities are shortlydescribed in appendix F. The project is completed within the period of 6th of September

to the 07th of January 2011.

The project covers the investigation of different methods to determine the effective

buckling length of columns in a load-bearing multi storey steel frame structure. The

case study used for the current project is a plan steel frame structure part from a project

called "Z-house".

The project report consists of four parts: Pre-analysis of frame design, Case study,

Conclusion and Appendix. The appendix is divided into A, B, C etc., which are found at

the end of the report.

The project report uses the Harvard method of bibliography with the name of the source

author and year of publication inserted in brackets after the text, for example: [Bonnerup

and Jensen, 2007]. The lists of all the sources of reference are found at Bibliography list

in the end of the report.

A resume of this report including important conclusive matters gathered by different

analysis and studies, with the aim to provide a quick overview of this project for staff at

Rambøll and furthermore be a guidance to determine the K-factor in framed structure of

steel in practice, is given in appendix E.

Acknowledgements

I would like to acknowledge the employees at the Building department at Rambøll Aalborg

for daily guidance and for being good colleagues during the internship. I was very pleasant

with my stay at Rambøll Aalborg where I found both the working environment and the

social life in general very much attractive.

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Table of contents

Chapter 1 Introduction 1

1.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Methods of analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Layout of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

I Pre-analysis of frame design 7

Chapter 2 Frame design in practice 9

2.1 Frame classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 EC 3 - formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 1. and 2. order response . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Accounting for P −∆ and P − δ effect in EC 3 . . . . . . . . . . . . 13

2.3 Design approach preferred at Rambøll . . . . . . . . . . . . . . . . . . . . . 14

Chapter 3 Elastic buckling of columns 17

3.1 Euler buckling load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Critical buckling load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Effective length factor (K-factor) . . . . . . . . . . . . . . . . . . . . 23

3.3 Critical buckling load of columns in framed structure . . . . . . . . . . . . . 25

Chapter 4 K-factor determination in practice 27

4.1 AISC - formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1.1 Non-sway frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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Trainee report - Rambøll - Autumn 2010 Table of contents

4.1.2 Sway frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1.3 Assumptions made in AISC specification . . . . . . . . . . . . . . . . 36

4.2 DIN 18800 procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.1 Non-sway frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.2 Sway frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3 Frame base effects on K-factor . . . . . . . . . . . . . . . . . . . . . . . . . 43

II Case study 45

Chapter 5 Case study structure 47

5.1 Load and load cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2 Global analysis in Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Chapter 6 K - factor determination 53

6.1 AISC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.2 DIN 18800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.3 ROBOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.4 Results compairison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.5 Code check using Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.5.1 Results and sensitivity analysis . . . . . . . . . . . . . . . . . . . . . 67

III Conclusion 71

Chapter 7 Conclusion 73

Chapter 8 Putting into perspective 75

IV Appendix 77

Appendix A Buckling analysis in Robot 1

A.1 Buckling analysis in Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

A.2 Convergence test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

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Table of contents 9th semester

Appendix B Factors that influence the K-factor 7

B.1 Bracing effect of bays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

B.2 Bracing effect of storeys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Appendix C Frame Base Effects on K-factor 13

Appendix D K-factor determination using Robot 17

D.1 Global buckling analysis in Robot . . . . . . . . . . . . . . . . . . . . . . . . 17

D.2 Application of Robot to local storey buckling load determination . . . . . . 20

Appendix E Resume of the report 27

Appendix F Overview of other participated project and activities at

Rambøll 35

Appendix G Guide to Appendix CD 39

G.1 K-factor determination in sway frame . . . . . . . . . . . . . . . . . . . . . 39

G.2 Course - Buckling analysis in Robot . . . . . . . . . . . . . . . . . . . . . . 39

Bibliography 41

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Introduction1In this chapter the motivation for this project will be described followed by a

presentation of the problems to be handled. This leads to the problem definition

for the project, which will be answered in the report. Furthermore the objectives

of the project in order to handle the problem are described.

Design of tall buildings using steel frames is a very common method in the modern

industry. Utilising steel frames as the primary load bearing structure allow a long spanning

multiple-storey construction, where the benefit is that steel elements don’t take up a lot

of space. Tall buildings made of steel frames have a lower self weight in comparison with

for instance a solution of reinforced concrete elements. This means that the foundation

cost of the building is lower than else. Furthermore steel elements are easier to handle at

the construction site. These aspects make a construction solution of steel frames simple

and economical, [Thomsen, 1968]. An example on such a construction is shown in figure

1.1.

Figure 1.1. The exclusive project: "Z-house" near Aarhus harbour

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Trainee report - Rambøll - Autumn 2010 1. Introduction

The construction sketch shown in figure 1.1 is the exclusive project named "Z-house". The

house is intended to be build at a location near Aarhus harbour. The building is planned

to consist of 11000 m2 housing area and 14000 m2 for commercial lease. The construction

work is suspended at the moment caused by the economic crisis. But Rambøll Aalborg

has until the date of suspension been the advisor regarding the engineering field related

to the project. The construction engineers involved in the project at Rambøll Aalborg

have chosen the primary load bearing principle of the house to be based on steel frames.

These frames, with different levels in height, are joined in extension to each other in order

to meet the special requirements of the geometry for the Z-house. [Dalsgaard, 2008]

The static model shown to the left in figure 1.2, represents a simplified frame from the

project of Z-house. This static model is used for the case study in the current project. It

is an unbraced, pinned, 10-storey frame consisting of 2 bays. Each storey is with a height

of 3.6 m and a bay span of 8 m. The connections between the columns and beams are

regarded to be rigid, see the illustration to right in figure 1.2. HE400B profiles are used

for the columns in all the storeys. The beams in all the storeys are designed asymmetrichaving a wide lower flange in order to support the concrete floor, the dimensions are shown

in figure 1.2.

Figure 1.2. Two-bay and ten-storey plane frame construction to be used for the case study

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1.1. Problem statement 9th semester

The local coordinate system of the elements is illustrated on a column and beam element

but is valid for all the other respective members in the structure. The frames are intended

to be placed with an individual distance of 6 m in longitudinal direction (parallel with the

z-axis). It shall be mentioned that the stability in the z-axis direction is assumed to stable;

hence only in plane situation is required by Rambøll to be considered. In accordance to

the illustrated local coordinate system for the elements, the geometric and mechanical

parameters of the members are presented in table 1.1.

Profil Length [mm] E [M P a] I z

mm4

f yk [M P a]

Asymmetric beam 8000 210 · 103 776453 · 103 350

HE400B column 3600 210 · 103 576805 · 103 350

Table 1.1. Geometrical and mechanical parameters of members involved in the frame used as

case study, see figure 1.2 for illustration of the case study structure.

Determination of the effective buckling length of the columns in the case study structure

shown in figure 1.2, by employing different analytical and numerical methods is the aim

of this project. The motivation and furthermore why construction engineers at Rambøll

Aalborg are interested on this study is described in the following.

1.1 Problem statement

The stability analysis of a frame shall be performed following the code of practice. Hence

the stability of steel frame structure shall be insured by following the instruction given inEurocode 3. In general the code introduces three different methods in order to analyse

and document the stability of the frame. But basically the design procedure is required

to be based on either 1. or 2. order theory or by a combination of these. A more detailed

description of this is given in chapter 2. [EC3, 2007]

The construction engineers at Rambøll Aalborg prefer to apply the equivalent column

method (based on the 1. order theory) for the stability analyses of frames. This is due to

the fact that the equivalent column method is the traditional way which the engineers are

familiar with. Therefore they find it to be the most secure way of insuring the stability of

the frames as they are able to follow the calculation steps. A more detailed description of

why they prefer the equivalent column method is given in chapter 2.

Applying the equivalent column method requires the designer to determine the effective

buckling length value of the columns based on a global buckling mode of the frame

accounting for the stiffness behaviour of the members and joint and the distribution of

the compressive forces. This means that the objective get complex. Eurocode 3 nor

Danish National Annex suggest any procedure to determine the effective buckling length

value of the columns but refer to some other relevant literature for this objective. It is

hence essential to find and employ a method which gives reliable results and the use of

numerical tools can be relevant. There are hence a number of methods; therefore the

accuracy, usability and limitations may be studied in order to point out one or moresuitable methods in the practical engineering work. [EC3, 2007]

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Trainee report - Rambøll - Autumn 2010 1. Introduction

The description of the problem and requirements from the construction engineers at

Rambøll lead to the following problems which seeks to be investigated and answered

through the project:

•

Point out, one or more methods whereby a quick and reliable estimate of effectivebuckling length of columns in framed steel structure can be determined.

This project focuses on determination of the effective buckling length of columns in frames

applying different methods. Hence the following problem formulation is the main issue of

this project:

Determination of the effective buckling length of columns in steel framed

structures by employing different analytical and numerical methods

1.2 Problem definition

In order to handle the described problem, the following objectives for the project are

made:

• Understand the design requirements and methods for steel frames given in Eurocode

3 and what is meant by 1. and 2. order analysis.

• Understand the concept of the effective buckling length in general.

• Classify whether a given frame is of sway or non-sway type.• Perform analyses in order to determine the parameters that influence buckling length

of columns in a frame.

• Apply different approaches to determine the effective buckling length of columns

and study theirs assumption, usability and limitations.

• Perform analyses in order to verify the reliability of the commercial program Robot

with respect to buckling analysis and examine in what extend it can be applied due

to determine the buckling length of columns in framed structures.

• Determine the effective buckling length of columns in the structure presented as case

study using different analytical and numerical methods.

• Perform code check and sensitivity analysis due to examine the influence of effectivebuckling length value for the final design.

Due to the lack of time available for this project, limitations on the treatment of some

of the described objectives are made. These limitations are described in the respective

chapters. Furthermore the instability problem, lateral torsional buckling of the members

is not included this study.

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1.3. Methods of analysis 9th semester

1.3 Methods of analysis

Analytical approach

Alignment charts given by AISC, American institute of steel construction, and charts

published by the German code DIN 18800 are employed in order to determine theeffective buckling length of columns. Furthermore the theoretical background of the AISC

alignment charts is developed analytically.

Eurocode 3, mentioned EC 3 in the following, is studied in order to understand the

design requirement in practice. The theoretical background is granted by study of several

scientific notes and books on analysis of steel frame structures, references are made

throughout the report.

Numerical approach

The finite element program: "AutoDesk Robot Structural Analysis Professional 2011",mentioned as Robot in the following, is applied in order to model and perform buckling

analysis. Furthermore calculations programs available at Rambøll as Excel and MathCAD

are used in order to set up small programs and MatLab is employed to plot graphs.

The use of the program Robot is enabled by 1 week of training at Rambøll, following

the manuals offered by AutoDesk. Understanding of the methods Robot calculations are

based on, are gathered by studying the Robot manuals.

1.4 Layout of the report

This report is divided into 4 parts exclusive the introduction. Each chapter of this report

starts with an overview of the contents in the actual chapter. The current report consists

of following chapters and appendix:

• Chapter 1: Introduction

Part I - Pre-analysis of frame design

• Chapter 2: Frame design in practice

• Chapter 3: Elastic buckling of columns

• Chapter 4: K-factor determination in practice

Part II - Case study

• Chapter 5: Case study structure

• Chapter 6: K-factor determination

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Part I

Pre-analysis of frame design

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Frame design in practice2In this chapter the frame design in practice following EC 3 is presented. Initially

the discussion and definition on classification of the frame type is given. This is

followed by a description of EC 3 formulation of theory and methods to be applied

in practice design of frames. Finally the design method preferred by Rambøll is

described whereby the cause for the current study of this project is elaborated.

It shall initially be mentioned that buckling analysis in Robot is widely used in this project.

Hence a description on the method Robot uses and input parameters it requires due to

perform buckling analysis and furthermore a convergence test is made, see appendix A.

The reader is strongly suggested to read this document due to get the theoretical background

of buckling analysis in Robot.

The main goal of this chapter is to clarify what is stated in EC 3 regarding the practical

design of frames. Eurocode is in general made to cover a large number of construction

types why it often contains a wide description of the design methods. Therefore it becomes

hard to get an overview of the design requirement for a given construction. Hence this

chapter is made due to enable a brief overview of the requirements in EC 3 that is valid

for frames of the kind presented as case study in chapter 1. But before this objective, the

current chapter is initiated by a classification study on frame types introducing definitions

and terms that are widely used in the stability study of frames and not least in this

chapter.

2.1 Frame classification

When dealing with stability of columns or stability of frames, codes and design books

commonly use the following terms, which is dependence on the deformation fashion that

occurs when the frame is subjected to loading: [University of Ljubljana - Slovenia, 2010a]

•

Sway / unbraced frame, shown to right in figure 2.1.• Non-sway / braced frame, shown to left in figure 2.1.

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Trainee report - Rambøll - Autumn 2010 2. Frame design in practice

Figure 2.1. Non-sway/braced frame to left and sway/unbraced frame to right. [University of

Ljubljana - Slovenia, 2010a]

Sway frame is defined as a frame which is not restrained from deflecting laterally and

non-sway is hence a frame which is restrained from deflecting laterally. But this doesn’t

means that the structure example shown in figure 2.1 to right and left always is classified

as sway and non-sway frame, respectively. If the restraint or the bracing of the braced

structure is very flexible, then the frame may be classified as sway frame. Likewise if thestiffness of the elements in the unbraced structure is sufficiently large, then the frame may

be classified as non-sway frame. [University of Ljubljana - Slovenia, 2010a]

In fact the definition given above of non-sway frame has no real significance and is only

valid in an "engineering" sense. Because there is no structure, whether it is braced

or unbraced that doesn’t displace laterally. But it is a question on how small the

displacements are thus to be considered equal zero in an engineering sense. But eventually

the reason for defining whether the frame is a sway or non-sway type is due to argue for

adopting conventional analysis on non-sway frames or if the 2. order analysis (on sway

frames) shall be performed. Further description on this matter is given in this chapter

2.2. [University of Ljubljana - Slovenia, 2010a]

A more precise definition of a non-sway frame is hence a structure which, from the points of

view of stability, can be considered to have small inter-storey displacements. Therefore the

local column buckling is independent from the global frame buckling, why the instability

problem can be uncoupled, [University of Ljubljana - Slovenia, 2010a]. EC 3 indirectly

provides the following criterion in order to define whether the frame can be considered as

sway or non-sway type. A frame may be classified as non-sway if αcr factor for a given

load case satisfies the criterion given in equation 2.1. [EC3, 2007]

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2.2. EC 3 - formulation 9th semester

αcr = F crF Ed

≥ 10 (2.1)

αcr Critical buckling factor, by which the design loading have to be increasedto cause elastic instability in the global mode

F Ed The vertical design load on the structure

F cr Elastic critical buckling load for global instability mode based on initial elastic

stiffness

It is hence seen that the definition of a frame as sway or non-sway type depends on the

magnitude of vertical loads; which is understandable since even a very flexible structure

doesn’t have any 2. order effects if the vertical loads are equal to zero. Therefore the

classification of sway or non-sway type is not general for a given frame, but is just validfor a specific vertical load case. If equation 2.1 is satisfied, the global buckling can be

neglected when carrying out the check against column buckling, further description on

this matter is given in the following. [University of Ljubljana - Slovenia, 2010a]

2.2 EC 3 - formulation

In stability analysis of frames, flexure is the primary means for unbraced rigid frames by

which they resist the applied load. Therefore it may be essential to account for so called

2. order effects. The effect of deformed geometry (2. order analysis) of a structure shallbe included if they significantly increase the action effects. Therefore influence of 2. order

effects shall be specified and evaluated. In the following the formulation given in EC 3 on

this matter is described. Initially what is meant by 1. and 2. order response is illustrated.

[EC3, 2007]

2.2.1 1. and 2. order response

EC 3 suggests design procedure of frames based on either 1. or 2. order analysis. Before

going onto further details with the design regulations, a description on what is assumedand accounted for in 1. and 2. order analysis is given in the following. [University of

Ljubljana - Slovenia, 2010b]

• 1. order analysis

– Assumes small deflection behaviour.– Resulting forces and moments do not account for the additional effect due to

the deformation of the structure under loading.

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• 2. order analysis

– Large displacement theory :

∗ Resulting forces and moments take full account of the effects due to the

deformed shape of both the structure and its members.

– Stress stiffening :

∗ Effect of element axial loads on structure stiffness: Tensile loads stiffening

an element and compressive loads softening an element.

In the following two cases, symmetric and asymmetric loading on an unbraced in-plane

frame is considered in order to illustrate what is meant by the 2. order effect. Figure

2.2 to left shows an undeformed frame with uniformly distributed load. For this case the

primary deflection due to load P < P cr will be symmetrical until the bifurcation point is

reached, illustrated in the middle in figure 2.2. A detailed description on the critical load

P cr and the bifurcation point is given in chapter 3. When the critical load is reached thedeflection pattern changes to fail by side-sway buckling, shown on the illustration to right

in figure 2.2. This behaviour is sketched in a load - lateral deflection curve, see figure 2.4,

where elastic behaviour is assumed. [Galambos and Surovek, 2008]

Figure 2.2. Symmetric deflection of the frame due to symmetric loading until bifurcation point

is reached, hereafter deflection pattern changes to fail by side-sway buckling

Consider the frame in figure 2.3, which is in addition to the previous case, subjected to

a lateral load H . This frame doesn’t have any bifurcation point where the deflection

pattern changes, but it deflects laterally from the start of loading. The P −∆ behaviour

of this case can be described based on either 1. or 2. order deflection, see figure 2.4.

In 1. order analysis, the load - deflection response is based on the undeformed structure

where equilibrium is formulated on the deformed structure; hence it results in a linear load

deflection curve. In the 2. order analysis, a load increment gives a incremental deflection,

which is a little more than in the previous load increment. Hence slope of the 2. order

curve decreases as the load increases, why it results in a non linear curve in figure 2.4.

[Galambos and Surovek, 2008]

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2.2. EC 3 - formulation 9th semester

Figure 2.3. Unsymmetrical deflection (side-sway buckling) of the frame due to lateral loading

H .

Figure 2.4. Load - lateral deflection curve P − ∆ for symmetric and unsymmetrical loadingincluding 1. and 2. order analysis. [Galambos and Surovek, 2008]

Figure 2.3 also shows the element deflection δ , due to the axial loading. Hence to provide

a complete stability analysis of frame both the P −∆ and P − δ effects may be included.

Such an analysis is called 2. order P − ∆− δ analysis. [S.L Chan & C.K. Lu, 2006]

2.2.2 Accounting for P −∆ and P − δ effect in EC 3

EC 3 states the criterion given in equation 2.1 for the safety factor αcr; if (αcr ≥ 10),

the 2. order effect is assumed to be neglectable and the calculations can be performed

using 1. order elastic analysis. For critical value lower than three, αcr ≤ 3, a precise 2.

order analysis shall be performed. For intermediate values, 3 ≤ αcr < 10, EC 3 suggests

to multiply the horizontal loads due to wind and imperfections by an amplification factor

given by the equation 2.2. [EC3, 2007]

Afactor = 1

1−

1

αcr

(2.2)

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The global critical value αcr for the structure is directly obtainable by performing buckling

analysis in Robot. Hence it can be verified if 2. order effect shall be included. Anyhow EC

3 suggests three approaches in order to account for P − ∆− δ . Without going to details,

it can briefly be said that EC 3 differentiate between three kind of analysis in order to

demonstrate the structural stability of frames: [EC3, 2007]

1. Complete P −∆−δ analysis method: Analysis where 2. order effects in individual

members (P − δ effect) and relevant member and global imperfections are totally

accounted for in the global analysis (P − ∆ effect) of the structure.

• No individual stability check for the members is necessary.

2. Partly P − ∆ − δ and partly equivalent column method: Analysis where

2. order effects in individual members (P − δ effect) or certain individual member

imperfections are not fully accounted for in the global analysis (P −∆ effect) but 2.

order effect of global imperfections are included.

• Individual stability check for the members following the instruction given in EC

3, section 6.3: "Buckling resistance of members" is necessary, where buckling

length equals to the system length is used.

3. Equivalent column method: Analysis where only 1. order analysis, without

considering imperfections, is accounted for in the global analysis.

• Stability of the frame is accessed by a check with the equivalent column method

according to the instruction given in EC 3, section 6.3. The buckling length

values should be based on a global buckling mode of the frame accounting for

the stiffness behavior of the members and joints, the presence of plastic hingesand the distribution of compressive forces under the design load.

As the different design approaches stated in EC 3 are explained, the approach that the

construction engineers at Rambøll prefer to use and the reason for it is described in the

following.

2.3 Design approach preferred at Rambøll

The construction engineers at Rambøll Aalborg prefer to use the design approach based on

the equivalent column method. This is due to the fact that the equivalent column method

is the conventional method they are familiar with, as described in the introduction, chapter

1.

On the other hand the numerical tool Robot, available at Rambøll, is able to perform

a complete P − ∆ − δ analysis, and hence no individual element check is required, why

this approach obviously seems to be a quick method. But the problem connected to

this method the engineers call attention to, is that the global-frame and local-element

imperfections of the structure shall be included when performing the analysis.

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2.3. Design approach preferred at Rambøll 9th semester

This means that the imperfections shall be calculated, which is maybe not the main time

consuming process, but implementing them in Robot is a very time consuming process.

This practically means that the geometry shall be adjusted including the imperfections,

by offsetting the element nodes. The other problem is to place the imperfections thus

it reflects the most unfavorable situation for a given load case. This objective gets very

complicated, as in practice a large number of load combinations shall be checked and it is

hard to point out which one is more critical in forehand, even for an experienced engineer.

All these complications committed to the P −∆−δ analysis method, makes the engineers

in practice to prefer the well known equivalent column method following EC 3, which is

also available in Robot. The other design approach, where partly P − ∆ − δ and partly

equivalent column method is applied, also consist of complications as described before,

why this method neither is preferred.

Using the equivalent column method, requires to determine the effective buckling length

of the columns based on a global buckling mode of the frame accounting for the stiffness

behavior of the members and joints, the presence of plastic hinges and the distribution

of compressive forces under the design load. No suggestion is given in EC 3 or Danish

National Annex, in terms of how to determine the buckling length of columns in frames.

The Danish National Annex refers to some other relevant literature for this objective.

This leads to the reason for the scope of this project as described in chapter 1. It shall be

mentioned that presence of plastic hinges are not included this study as only the elastic

behaviour of the structure is considered.

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Elastic buckling of

columns3

In this chapter the concept of effective buckling length is explained based ona study of elastic buckling of planar columns. The expression of Euler load

is derived for the basic case, pin-ended column. Critical buckling load and

thereby the effective buckling length factor (K-factor) is determined for some

other fundamental cases.

The basic and essential question in a study of the stability of a given structure goes on

whether it is stable or instable. The definition of a stable elastic structure is that "a small

increase in load causes small increase in displacement" where the instability is defined as"a small increase in load causes large displacement". The condition of stability refers to

the state of equilibrium of the system which can be illustrated as: [Galambos and Surovek,

2008]

Figure 3.1. Illustrations indicating the state of equilibrium of a system. [Aalborg University, -]

The illustration (a) in figure 3.1 indicates a stable equilibrium where the "element" can

be disturbed but will return to the initial position. Contrary to this, the illustration

(b) indicates an unstable equilibrium where the "element" will fail if it is disturbed.

Illustration (c) represents the neutral equilibrium, where the element will find a new

position of equilibrium if it get disturbed. These illustrations on state of equilibrium are

the basic for understanding the stability condition of a structure. [Galambos and Surovek,

2008]

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Trainee report - Rambøll - Autumn 2010 3. Elastic buckling of columns

Structural engineers are familiar with so called Euler load P E , of an axial loaded column.

This is the critical buckling load P cr, of a pin-ended column, referred as the basic case

in the buckling analysis. More explanation on this matter will be given later in this

chapter. Considering figure 3.1, the state of stability at the level of critical buckling load

is recognised as the upper limit of the condition shown at the illustration (a), meaning

that further increase in load will lead to instability of the column where unstable state

shown at the illustration (b) occurs. Illustration (c) represents the "loading path" from

no load on the column till the critical buckling load where the column keeps on finding

new positions that establish equilibrium of the system as the load increases.

3.1 Euler buckling load

Having illustrated the state of equilibrium of the system, the next step is to determine the

critical buckling load of a compression member with a given support conditions. In the

stability study of compression elements, the Euler load is used as the reference which is

determined from the Euler buckling equation. It is of the greatest important to understand

the derivation of the Euler buckling equation where for instance the influence of the

boundary conditions on the critical load can be demonstrated. Thereby it becomes easier

to understand the behaviour of a column and perform analysis of frames where the columns

are connected to beams that act as supports. Hence derivation of the Euler buckling

equation based on the basic case, a pin-ended column, is performed in the following.

Bernoulli-Euler beam theory is applied in the following, where the internal forces are

assumed to act in accordance to the undeformed plane, in other words plane cross

section remains plane. Hence a perfectly straight, pin-ended, Bernoulli-Euler bar withthe buckling stiffness E · I , subjected to a point load P is considered, see figure 3.2.


Figure 3.2. Pin-ended column with buckling stiffness E ·I , suspected to point load P . [Bonnerup

and Jensen, 2007]

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First derivative of v(z) is the slope of the deflection v

(z) and is given in equation 3.14.

The second derivative v

(z), given in equation 3.15, is the curvature κ, used to define

the moment which fulfils the constitutive condition, see equation 3.2. Third derivative

v

(z) is the derivative of the curvature, see equation 3.16, which is utilized to define the

shear force by differentiating the moment - curvature relation given in equation 3.2. Using

these derivatives, the boundary conditions for various support conditions is formulated.


v

= B + C · k · cos(k · z) − D · k · sin(k · z) (3.14)

v

= −C · k2 · sin(k · z) − D · k2 · cos(k · z) (3.15)

v

= −C · k3 · cos(k · z) + D · k3 · sin(k · z) (3.16)

An example of a fundamental case is a cantilever column shown in figure 3.4, where it’s

base end is fixed and the top end is free. The critical buckling load for this case will bedetermined in the following.

Figure 3.4. Cantilever column subjected to axial load

The boundary conditions for the present case are:

• Zero moment at z = 0 : v

(0) = 0

• Zero shear at z = 0 : v

(0) + k2 · v

(0) = 0

•

Zero deflection at z = L : v(L) = 0• Zero slope at z = L : v

(L) = 0

By applaying these boundary conditions to equation 3.13 and it’s derivates, the following

four simultaneous equations are obtained:

v

(0) = 0 =A(0) + B(0) + C (0) + D(−k2)

v

(0) + k2 · v

(0) = 0 =A(0) + B(k2) + C (0) + D(0)

v(L) = 0 =A(1) + B(L) + C (sin(k · L)) + C (cos(k · L))

v

(L) = 0 =A(0) + B(1) + C (k · cos(k · L)) −D(k · cos(k · L))

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3.2. Critical buckling load 9th semester

These equation can be presented in the following matrix form:

0 0 0 −k2

0 k2 0 0

1 L sin(k · L) cos(k · L)

0 1 k · cos(k · L) −k · cos(k · L)

A

B

C

D

= 0 (3.17)

The coefficients A, B,C and D define the deflection of the buckled bar, why one or more

of them have value other than zero. Thus, the determinant of the coefficient must be equal

to zero, in order to obtain nontrivial solution to the eigenvalue problem. [Galambos and

Surovek, 2008]

0 0 0 −k

2

0 k2 0 0

1 L sin(k · L) cos(k · L)

0 1 k · cos(k · L) −k · cos(k · L)

= 0 (3.18)

Solution to the problem given in equation 3.18, or in other words solution to the critical

buckling load P cr for the case shown in figure 3.4 is hence found to be contained in the

following eigenfunction.

cos(k · L) = 0 (3.19)

The eigenfunction in equation 3.19 has infinite number of roots or eigenvalues as n goes

from one to infinity. But as described earlier only the first defection mode n = 1 is of

interest. Hence the lowest critical buckling load is determined:

k · L =

P

E · I · L = n ·

π

2 ⇒ P cr =

π2 · E · I

4 · L2 (3.20)

The critical buckling load for the present case is thus reduced by 25 % in comparison

with the Euler buckling load for pin-ended column case, see equation 3.11. Therebythe influence of the support conditions on the critical buckling load is demonstrated.

This is done by employing the general governing differential equation, applying boundary

conditions and solving the eigenvalue problem.

3.2.1 Effective length factor (K-factor)

Having determined the Euler buckling load P E , for pin-ended column, representing the

basic case and critical buckling load P cr, of a cantilever column presented in figure 3.4,

the next step is to define the relation between these given by the effective length factorK , see equation 3.21. [Galambos and Surovek, 2008]

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K 2 = P E P cr

=π·E ·I L2

π·E ·I 4·L2

= 4 (3.21)

Hence the effective length factor, denoted as "K-factor" in the following, is found to be:K = 2, for a cantilever column. K-factor is the ratio between the buckling length and the

actual column length, see figure 3.5.

Figure 3.5. Effectiv buckling length of a pin-ended (left) and cantilever (right) column. [DelftUniversity of Technology, -]

Figure 3.5 shows the buckling length of a pin-ended and cantilever case where the buckling

length is defined as the horizontal length between the points of inflection of the deformed

shape of the column. Point of inflection is the point at which the secound derivative of

the buckled shape changes sign.

Multiplication of K-factor by the actual column length L, the equivalent or effective column

length is determined, which is replaced in the Euler buckling equation instead of L. Thismatter is analysed in Robot by determining the critical buckling load of a cantilever

column of length 5 m/2 = 2.5 m, see figure 3.6.

Figure 3.6. Cantilever column of length 2.5 m modelled in Robot in order to perform buckling

analysis.

The critical buckling load is expected to have the same magnitude as found earlier for the

pin-ended column of 5 meter length, given the same stiffness parameters, see table 3.1.

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3.3. Critical buckling load of columns in framed structure 9th semester

Thereby it can be evaluated if buckling analysis in Robot provides results in accordance

with the theory. Hence the critical buckling load for the case shown in figure 3.6 is

determined to P cr = 47819.8 kN in Robot, which is in accordance with the theory.

Some other fundamental cases than already studied and the point of inflection of the

deformed shape are shown in figure 3.7. The K-factor for the cases are:

• Fixed-ended: Both ends are fixed - K = 0.5

• Fixed-pinned: One end is pinned, the other end is fixed - K = 0.7

Figure 3.7. Effective buckling length of a fixed-ended (left) and fixed-pinned (right) column.

[Delft University of Technology, -]

3.3 Critical buckling load of columns in framed structure

In what was done, the definition of K-factor is given and explained. It was demonstrated

that K-factor is just a method of mathematically reducing the problem of evaluating

the critical buckling load for columns in structures to that of equivalent pin-ended braced

columns. Determination of the K-factor of the columns in complex frame buckling problem

is the scope of this project.

As the bases in buckling analysis of columns are clarified, some more complex models are

studied aiming towards the scope of this project. Hence the parameters that influence on

the K-factor of columns in framed structures are studied in more detail, see appendix B,

where two analyses are made:

• Bracing effect of bays: Determine chances in degree of bracing of the exterior

column by the other members of the storey as the number of its bays increases

stepwise from 1 to 8.

• Bracing effect of storeys: Determine chance in degree of bracing on the interior

column in a two-bay frame as the number of storeys increases stepwise from 1 to 4.

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Hence the important conclusive matters from the analyses are included here, but detail

description and results need to be found in appendix B:

Bracing effect of bays

Scientists have made research of steel framed construction on this matter and have

concluded the following statement which is also what was verified in the study made

in appendix B and is hence also the conclusion of the performed analysis:

"In general, the critical buckling load which produces failure by side-sway can be distributed

among the columns in a storey in any manner. Failure by side-sway will not occur until

the total frame load on a storey reaches the sum of the potential individual critical column

loads for the unbraced frame. There is one limitation, the maximum load an individual

column can carry is limited to the load permitted on that column for the braced case,

K = 1. Side-sway is a total storey characteristic, not an individual column phenomenon."

[Joseph A. Yura, 2003]

Bracing effect of storeys

Analyses made in terms to determine the bracing effect of storeys on a considered storey

showed that only the adjoining storeys of a considered storey have remarkably effect on its

columns K-factor. Thereby it is evaluated to be sufficient to consider only the adjoining

storeys when determination of K - factor of columns in a given storey.

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K-factor determination in

practice4

In this chapter the AISC alignment charts in order to determine K-factor of columns in sway and non-sway frames and the theoretical background in

derivation of them are presented. Furthermore the approach given in DIN 18800

for K-factor determination is presented.

Design of framed structures can among others be dealt by the concept of effective length

or K-factor. Definition of K-factor is given in chapter 3. Figure 4.1 illustrates the physical

of effective buckling length of a column in a rigid connected frame.

Figure 4.1. Illustration on physical of effective buckling length of a column in a rigid sway frame.

[G.Johnston, 1976]

Studies made in chapter 3, clarify the influence of support conditions on the K-factor,

which are illustrated for the fundamental cases. Those analyses were based on idealised

support conditions. This assumption will not be the case for the columns in framed

structure, as they interact with other members.

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Trainee report - Rambøll - Autumn 2010 4. K-factor determination in practice

This interaction makes it necessary to consider the connecting members when designing

columns of frames. EC 3 and Danish National Annex refer to other specific literature for

K-factor determination. This chapter presents the approaches given in the following two

codes of practice:

• AISC: American Institute of Steel Structure

• DIN18800: German code for the design of structural steel

Descriptions on the use of the charts provided by the mentioned codes are given in

subsequent sections of this chapter. The background of the AISC approach is elaborated

due to get the theoretical understanding of the charts. The aim is to apply these procedures

to determine K-factor of columns in the case study structure, which is done in chapter 6.

4.1 AISC - formulation

AISC provides so-called alignment chart for sway and non-sway frames whereby the K-

factor of a column is determined based on the joint stiffness of the column ends. In

the following the background of the charts and the applied model and assumptions are

elaborated.

4.1.1 Non-sway frame

A general case of column subjected to compression and restrained by elastic springs attheir ends is considered. This situation reflects a column restrained by beams of finite

stiffness. For non-sway frame case, it is assumed that the column ends do not translate

with respect to each other. The static model of the actual case is shown in figure 4.2.


Figure 4.2. Static model applied for non-sway frame - column with rotational spring. [ Galambos

and Surovek, 2008]

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4.1. AISC - formulation 9th semester

Consider the rigid connection at the columns top; the column and adjoining beam are

perpendicular to each other, meaning there is no deflection. Hence the slope of the column

and beam, denoted θT , are equal.

The total spring constant from restraint in top and bottom of the column is denoted αT

and αB, respectively. Hence the moment from elastic restrained beam at the top canbe expressed as M = αT · θT . Further explanation on the spring constant quantity is

given later, but only the symbol is used now. This expression for moment is rewritten

to M = αT · v

(0), where v(z) is the lateral deformation as a function of z. Moment

at the columns top, emerged from the change in column slope can be expressed as

M = −E · I C · v

(0), where I C is the moment of inertia of the column. Hence from

the equilibrium condition the following relation is established:

αT · v

(0) − E · I C · v

(0) = 0 (4.1)

The above given condition is also valid for point B, at the distance LC (length of the

column). But in point B, the sign for moment from elastic restrained beam is negative,

hence the relation becomes: [Galambos and Surovek, 2008]

−αB · v

(LC )− E · I C · v

(LC ) = 0 (4.2)

These are 2 of the 4 boundary conditions needed in order to solve the differential equation

for the system. The remaining 2 boundary conditions are governed by requiring no lateral

displacement at the top and bottom, given by:

v(0) = 0

v(LC ) = 0

The 4 boundary conditions are applied the general solution for the governing differential

equation, given in equation 3.13, chapter 3. Thus the determinant of the coefficient A,B,C

and D becomes as given in equation 4.3, where the variable k is earlier defined in equation

3.5, chapter 3. [Galambos and Surovek, 2008]

0 =

1 0 0 1

1 LC sin(k · LC ) cos(k · LC )

1 αT αT · k E · I C · k2

0 −αB a43 a44

(4.3)

a43 = −αB · k · cos(k · LC ) + E · I C · k2 · sin(k · LC )

a44 = αB · k · sin(k · LC ) + E · I C · k2 · cos(k · LC )

The eigenfunction of the model is determined by solving the determinant in equation 4.3.

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( πK

)2 · GT · GB

4 − 1 +

GT + GB2

·

1 −

πK

tan( πK

)

+

2 · tan

π2·K

πK

= 0 (4.7)

Figure 4.3. Subassembly rigid frame for non-sway case, where single curvature bending of the

beam, with the slope θ at both ends is assumed. [Galambos and Surovek, 2008]

In equation 4.7, the K-factor K = πk·L

is adapted and the flexibility parameters GT and

GB are introduced, which are determined by equation 4.8. [G.Johnston, 1976]

GT =

I C LC

I BT LBT

(4.8)

GB =

I C LC I BBLBB

Summation of all members rigidly connected to the joint and laying in the plane

in which buckling of the column is being considered.

I C , LC I C is the moment of inertia and LC the corresponding unbraced length of

the column of consideration.

I BT , LBT I BT is the moment of inertia and LBT the corresponding unbraced length of

the beam at columns top.

I BB , LBB I BB is the moment of inertia and LBB the corresponding unbraced length of the beam at columns end.

Having determined GT and GB, the AISC Alignment chart for non-sway frame given in

figure 4.4 is used to determine the K-factor of the columns. Otherwise equation 4.7 is

used in a numerical solver, where K-factor is determined by iteration.

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The variables T T and T B in equation 4.9, account for the translation stiffness where:


T T =

β T · L3

E · I

T B = β B · L

3

E · I

The AISC Specification, assumes that a sway frame consists of subassembly type of frames

where the top of the column is able to translate with respect to the bottom, see figure 4.6.

Furthermore it is assumed that the bottom column cannot translate where translational

restraint is infinite large T B = ∞, and the top column is free to translate T T = 0. These

are hence applied the equation 4.9 by substituting T T = 0 into the first row and dividing

the third row by T B and then equating T B to ∞, which yields: [Galambos and Surovek,

2008]

0 =

0 k · L2 0 0

0 RT RT · k · L (k · L)2

1 1 sin(k · L) cos(k · L)

0 RB a43 a44

(4.10)

a43 = RB · k · L · cos(k · L) − (k · L)2 · sin(k · L)

a44 = −RB · k · L · sin(k · L)− (k · L)2 · cos(k · L)

Figure 4.6. Subassembly rigid frame for sway case, where reverse curve bending of the beam,

with the slope θ at both ends is assumed. [Galambos and Surovek, 2008]

The rotational stiffness for this case is found by assuming equal rotation in magnitude

and direction at near and far ends of the restraining beam but producing reverse curve

bending, see figure 4.6. This means αT = 6 · E ·I BT

LBT and αB = 6 ·

E ·I BBLBB

[Shanmugam

and Choo, 1995]. Substituting these values into equation 4.10, and after some algebraic

manipulation, the eigenfunction given in equation 4.11 is derived. [Galambos and Surovek,

2008]

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πK

tan

πK

−

πK

2· GT ·GB − 36

6 · (GT + GB) = 0 (4.11)

This equation is the basic for the sway alignment chart shown in figure 4.7,that relatesthe flexibility parameters GT and GB with the K-factor.

Figure 4.7. AISC - Alignment chart for K-factor determination in a rigid sway frame. [Galambos

and Surovek, 2008]

Figure 4.8 shows the members involved in K-factor determination of the column marked

with red, for both the sway and non-sway frames.

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Figure 4.8. The members enclosed by the dashed lines are involved in K-factor determination

of the column marked with red. [Galambos and Surovek, 2008]

For a column base connected to footing by a frictionless hinge, GB is theoretically infinite

but 10 is suggested to be used in design practice. If the column base is rigidly attached,GB approaches the theoretical value of zero, but should not be taken lower than 1,

[G.Johnston, 1976]. Having introduced the theoretical background in AISC Specifications

for K-factor determination, the inherent assumptions are summarized and discussed in

the following.

4.1.3 Assumptions made in AISC specification

Mathematical solution to a practice problem is found by putting up a model and make

a number of assumptions, otherwise it is impossible to determine a solution. Hence theobtained results would not be the exact, but the better the mathematical model describes

the practical problem, the better the final results becomes. Hence the mathematical model

and assumptions adopted in the AISC Specifications due to determine the K-factor are

discussed in the following. The alignment charts are based on the following assumptions:

1. Behaviour is purely elastic.

2. All members have constant cross section.

3. All joints are rigid.

4. For the non-sway frame case, rotations at the far ends of restraint beams are equal

in magnitude but opposite in sense to the joint rotations at the far ends (singlecurvature bending).

5. For the sway frame case, rotations at the far ends of the restraint beams are equal in

magnitude and in the same sense as the joint rotations at the column ends (reverse

curvature bending).

6. All columns in the frame buckle simultaneously.

7. Only the members shown at figure 4.8 is accounted in K-factor determination.

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Purely elastic behaviour

The assumption that the behaviour is purely elastic is not valid when the load increases

thus yielding of the column occurs. This means E C reduces and the beams provide more

relative restraint to the columns. Hence it causes a lower G - factor and consequently a

lower K-factor, see the charts in figure 4.4 and 4.7. Thus the alignment charts provideconservative values regarding this matter. [Galambos and Surovek, 2008]

Constant cross section

The assumption that all members have constant cross section is not valid around a joint

where for instant the column dimension changes. This is often seen in tall buildings that

the dimension of the columns in the upper storeys is smaller than in the lower storeys.


Rigid joints

AISC assumes rigid joins, which require perpendicular shape between beam and columnis maintained under deformation. The joints shall be able to transfer moment. This

assumption put requirement for the performance of the joints in practice. An example of

rigid and pinned connections are given to the left and right in the figure 4.9, respectively.

Pinned connections are theoretically only able to transfer axial and shear forces. It is

hence important to establish the joints in practice as assumed. [University of Ljubljana -

Slovenia, 2010b]

Figure 4.9. Examples of rigid (left) and pinned (right) connections. [University of Ljubljana -

Slovenia, 2010b]

Single/reverse curvature bending

Single curvature bending for the non-sway and reverse curvature bending for sway frame

is assumed. These assumptions are only fully valid for a perfectly symmetric deformation

which requires symmetric geometry and loading conditions. The restraint of the columns

by beams is affected by the far-end rotation of the beams. Hence the following modification

of the beam length L

B is suggested in order to account for the variation from the

assumptions: [G.Johnston, 1976]

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4.2. DIN 18800 procedure 9th semester

It is hence obtained that only the adjoining storeys of the considered storey are found

to influence the K-factor. Therefore it is evaluated to be acceptable, only to include the

adjoining storeys in K-factor determination of the case study structure, as suggested in

AISC Specifications. It shall be mentioned that this evaluation is based on the analysed

case (limited storeys); therefore this is not necessary valid in general for frames with

various number of storeys.

As the AISC specifications including the deviation of the charts and its assumptions are

presented and discussed, another method for determining K-factor, provided by German

code DIN18800 is presented in the following.

4.2 DIN 18800 procedure

The procedure presented in DIN 18800 is given in the following, where only the practical

use of it is explained. DIN 18800 also suggests two charts; one for non-sway and one forsway frames. The original text by DIN 18800 in German is translated to English by the

author. It should be mentioned that K-factor is denoted as (β ) in the following due to

keep the same denotation given by DIN 18800. [DIN-Standards and Regulations, 1989]

Common for both the non-sway and sway frames, are the two parameters C O and C U that

is determined by using the equation 4.13, and the indices are illustrated at figure 4.10.

C O = 1

1 +(α·K O)K S+K S,O

(4.13)

C U = 1

1 +

(α·K U )K S+K S,U

The K parameters given with indices in equation 4.13, are illustrated in figure 4.10. The

respective K value is in general determined by K = I /L, where I and L are the moment of

inertia and length of the member. The α values, known as the rotational stiffness factor,

shall be applied as: [DIN-Standards and Regulations, 1989]

•

α = 4 for the case where beams far end is fixed• α = 1 for case where beams far end is pinned

Furthermore for the pinned-base and fixed-base case the prescribed value C U = 1 and

C U = 0 are suggested, respectively. Parameters C O and C U are comparable with the

flexibility parameters GT and GB given in the AISC formulation. But the difference in

DIN 18800 from AISC formulation is the number of elements that is included for the K-

factor determination. DIN 18800, consider the elements shown in figure 4.10, to contribute

to restraint the storey marked with red.

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The representative K-factor (β ) is determined using the charts; Thereafter the K-factor

β j for each of the columns are found corresponding to the normal force and stiffness

distribution of the columns in the storey, see equation 4.14. [DIN-Standards and

Regulations, 1989]

Figure 4.10. Elements included in K-factor determination of the storey marked with red,

suggested by DIN 18800.

β j =

N · K jN j · K S

· β for j = 1, 2 .. n (4.14)

N =

N j for j = 1, 2 .. n

K = I /LK j = I j/L j for j = 1, 2 .. n

K S =

K j for j = 1, 2 .. n

β Representative K-factor of the storey

β j K-factor of the individual columns in the storey

n Number of columns in the storey

N j Normal force distribution factor, indicating the factor the column in question is

loaded in comparison to the other columns in the storey

N Sum of the normal force distribution factors of the columns in the storey

K S Sum of stiffness factors for the individual columns K jI j , L j Moment of inertia and length of the individual columns in the storey

This idea is also what is concluded in appendix B, that the side-sway is a total storey

characteristic and not an individual column phenomenon; hence all the columns and beams

in a storey contribute to the total restraints of the storey.

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4.2.2 Sway frame

In the same manner the representative K-factor β , of a storey in a sway frame is found

by reading the chart in figure 4.12. Thereafter β j , K-factor of the individual columns are

determined by applying equation 4.14.

Figure 4.12. K-factor β , represent for given storey, determination chart for a sway frame given

by DIN 18800. [DIN-Standards and Regulations, 1989]

A calculation seat with illustrations and explanations is made in MathCAD and enclosed

the Appendix CD, G.1.

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Part II

Case study

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Case study structure5

In this chapter, the geometric and mechanical parameters of the members,

required to determine the K-factor of the columns in the frame presented as

case study structure in the introduction are given. Two different vertical load

cases are considered in order to determine whether the case study structure is

sway or non-sway type by buckling analysis in Robot.

The frame to be used in this chapter and the following chapter is the one presented as case

study structure in the introduction, see figure 1.2 in chapter 1. The geometric parameters

of this structure are summarized in figure 5.1. It consists of two bay and ten storeys,

with rigidly connected members and pinned supported base. The local coordinate system

applied for the each of the column and beam elements of the structure is illustrated, where

z-axis is shown to be out-of-plane. It shall be mentioned that the stability in longitudinal

direction, z-axis, is assumed to be stable, hence only in-plane situation is considered.

The frame to be used in this chapter and the following chapter is the one presented as

the case study structure in the introduction, see figure 1.2 in chapter 1. The geometric

parameters of this structure are summarized in figure 5.1. It consists of two bay and ten

storeys, with rigidly connected members and pinned supported base. The local coordinate

system applied for the each of the column and beam elements of the structure is illustrated

in the figure mentioned above, where z-axis is shown to be out-of-plane. It is assumed that

the stability in longitudinal direction, z-axis, to be stable; hence only in-plane situation isrequired to be considered by Rambøll.

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Trainee report - Rambøll - Autumn 2010 5. Case study structure

Figure 5.1. The case study structure consisting 10 storeys in total

The frames have an individual distance of 6 meters between each other in the longitudinal

(z-axis) direction, (not included in the figure 5.1). The geometric and mechanical

parameters of the members are presented in table 5.1 in accordance to the local coordinate

system shown in the figure.

Profil Length [mm] E [M P a] I z

mm4

f yk [M P a]

Asymmetric beam 8000 210 · 103 776453 · 103 350

HE400B column 3600 210 · 103 576805 · 103 350

Table 5.1. Geometrical and mechanical parameters of members involved in the frame used as

case study, see figure fig:framedetailcasestud for illustration of the frame.

5.1 Load and load cases

Global buckling analysis of the frame is performed in order to determine whether it is a

sway or non-sway type. Hence only the vertical loads are considered. The vertical loads

are limited to account for permanent and imposed loads on the construction.

Permanent load

The permanent load of a floor including the installations and partition walls, is determined

to be 6.2 kN/m2. This load is set to act uniformly distributed on the beams, calculated

as G1 = 6.2 kN/m2 · 6 m = 37.2 kN/m. For simplification, the same load is assumed to

act on the upper beams to account for load from the roof-floor.

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5.1. Load and load cases 9th semester

The facade of Z-house is of glass and weighs 1 kN/m2. This load is assumed to

act as centric point load on the exterior columns at each storey, calculated as G2 =

1 kN/m2 · 6 m · 3.6 m = 21.6 kN .

Imposed load

The primary use of construction is assumed to be office related, which falls into category

B in Eurocode definitions for the use of the construction. Hence the characteristic value

of imposed load is taken as 2.5 kN/m2. Thus the uniformly distributed load on the beams

is N = 2.5 kN/m2 · 6 m = 15 kN/m. This load is also set to act at the top beams of the

frame to account for platform roof.

In accordance with the Danish National Annex for EC 1, the total imposed loads from

several storeys may be multiplied by the reduction factor αn given in equation 5.1, where

n is number of storeys and ψ0

is a factor, that depends on the category, that is 0.6 for

office areas.

αn = 1 + (n − 1) · ψ0

n = α10 =

1 + (10 − 1) · 0.6

10 = 0.64 (5.1)

Load cases

Z-house is categorised as high consequence class, CC3. Two load combinations consisting

permanent and imposed loads are considered for the current analysis:

• LC 1 : 1.1 · 1.0 ·G + 1.1 · 1.5 · αn ·N

• LC 2 : 0.9 ·G + 1.1 · 1.5 · αn · N

Load combination LC 1, consists of loads that are to be applied symmetrically around

the interior columns of the frame, see to the left in figure 5.2. Load combination LC 2,

consists of loads that are to be applied asymmetrically around the interior columns of the

frame, see to the right in figure 5.2 where the imposed load is only applied on the bays

to the right. The choices of the two combinations are based on the advice by Rambøll, to

establish two situations where the normal force in the columns varies the most compared

to each other.

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Trainee report - Rambøll - Autumn 2010 5. Case study structure

Figure 5.2. Loads corresponding to LL 1 (left) and LL 2 (right)

5.2 Global analysis in Robot

Robot offers the feature to perform global buckling analysis of a frame. In appendix A

the theoretical background in Robot calculations and the required input parameters are

described. A global analysis on the frame is performed, subjecting the frame to each of

the load cases presented in figure 5.2. Hence the critical global buckling load P cr and thecritical global load factor αcr of the frame can be determined to define whether the frame

is sway or non-sway type based on the definition given in equation 2.1, chapter 2.

Results

The base columns of the frame are pinned and loaded the most; hence the base storey

causes the global failure of the frame, but in general the global failure mode shall be

considered due to point out the storey that causes the global failure of the frame. In

appendix D more description on the global analysis in Robot and interpretation of theresults are given. Results from the current buckling analyses in Robot, determined for the

base storey, are presented in table 5.2 and 5.3 for LL 1 and LL 2, respectively.

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5.2. Global analysis in Robot 9th semester

Robot performs the buckling analysis by an iterative process, where it factorises the

applied load and requires equilibrium. If the equilibrium state is found, it continuously

increase the factor until the equilibrium is no longer obtainable. Hence the iterative

process results in a critical load factor αcr, which is directly multipliable by the internal

normal forces in the column, whereby the critical buckling load is determined. This fact

is notable by considering for instant the results in table 5.2.

LC 1 Left base column Interior base column Right base column

Normal force [kN ] 2584 4555 2584

Critical coefficient 4.826 4.826 4.826

Critical force [kN ] 12471 21982 12471

Table 5.2. Buckling analysis results determined in Robot for the load case: LC 1

LC 2 Left base column Interior base column Right base columnNormal force [kN ] 1586 3303 2180

Critical coefficient 6.637 6.637 6.637

Critical force [kN ] 10528 21926 14470

Table 5.3. Buckling analysis results determined in Robot for the load case: LC 2

The critical load factor αcr = 4.826 is found to be lowest for the considered load cases.

But both load cases indicate the actual frame as sway frame type, as the critical load

factor is lower than 10. Hence the charts for sway frame suggested by AISC and DIN

18800 along with analysis in Robot are used to determine the K-factor of the columns in

the following chapter.

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K - factor determination6

In this chapter, K-factor determination of columns in the case study structure

is performed following the approaches given in AISC and DIN 18800. The

results from buckling analysis of subassembly models in Robot, representing base,

intermediate and top storey cases, are also used for the determination of K-factor.

The results from AISC and DIN 18800 approaches are compared to the results

from Robot and discussed. Finally, code check according to EC 3 by employing

Robot is performed, where a sensitivity analysis of the K-factor influence for the

final result is made.

The case study structure is classified as a sway frame type in chapter 5, which means

the effective buckling length of the columns become larger than the system length,

K > 1.Practical methods according to AISC and DIN 18800 to determine the K-factors

are presented in chapter4 and are applied to the case study structure. In addition, Robot

is also employed for this objective. In appendix D, it is shown that buckling analysis in

Robot is only applicable for global analysis of the structure, resulting in the global critical

parameters. But it is further demonstrated that subassembly models can be used in Robot

to represent the local storey, whereby the critical buckling load of the storey is obtainable

and is used to determine the respective K-factor of the columns, see appendix D.

6.1 AISC

In order to apply the AISC Alignment chart for sway frame, the flexibility parameters

shall be calculated. This is done for 6 different column restraint types, numbered 1 to 6

in figure 6.1. These are representative for the other columns that have identical restraint

conditions at theirs ends.

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Trainee report - Rambøll - Autumn 2010 6. K - factor determination

Figure 6.1. Case study frame, divided into 6 different representative column restraint types.

In equation 6.1, calculation example of the flexibility parameters is shown for column

number 4, where equation 4.8 given in chapter 4 is used and the input parameters are

given in table 5.1, chapter 5. In equation 6.1, L

B = 1.5 · LB, is applied to account for

beams connection at the far end, which is fixed for the actual case. For the example shown

in equation 6.1, the same value of the flexibility parameters for both the top and bottom

connections are obtained due to the identical restraint conditions.

GT = I C LC I BT

LBT =

2 · 576805·103 mm4

3600 mm

2 · 776453·103 mm4

1.5·8000 mm= 2.476 (6.1)

GB =

I C LC I BBLBB

= 2 · 576805·10

3 mm4

3600 mm

2 · 776453·103 mm4

1.5·8000 mm

= 2.476

Thus the flexibility parameters are determined for the different column types as illustrated

in figure 6.1. Hence the respective K-factors are determined by using the equation 4.11

given in chapter 4. The results are given in table 6.1.

Column : Type 1 Type 2 Type 3 Type 4 Type 5 Type 6

GT 4.952 2.476 4.952 2.476 2.476 1.238GB 10 10 4.952 2.476 4.952 2.476

K - factor 2.552 2.192 2.219 1.706 1.954 1.497

Table 6.1. Flexibility parameters and K-factors for different column types as illustrated in figure

6.1, determined in accordance to AISC formulation.

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There are large number of load cases in practice, but in order to stick to the scope of this

project, only the load cases presented in figure 5.2, chapter 5 are employed, but modified

by including the wind load V , hence:

•

LC 3 : 1.1 · 1.0 ·G + 1.1 · 1.5 · αn · N + 1.1 · ψ0 · 1.5 · V • LC 4 : 0.9 · G + 1.1 · 1.5 · αn ·N + 1.1 · 1.5 · V

The wind pressure q max = 0.8 kN/m2 is assumed, which results in the following uniformly

distributed load values:

• V = 0.8 kN/m2 · 6 m · 0.7 = 3.36 kN/m for cf = 0.7

• V = 0.8 kN/m2 · 6 m · 0.3 = 1.44 kN/m for cf = 0.3

The loads included in load cases LC 3 and LC 4 are applied to the structure as shown infigure 6.2. These load distributions are suggested by Rambøll in order to get two situations

where distribution of the compressive forces in columns varies the most.

Figure 6.2. Load distribution, suggested by Rambøll to be applied for load cases LC 3 and LC

4

By applying these load cases and performing static analysis in Robot, the distribution of the compressive forces and hence the αLoad factors are determined. Results are given in

table 6.3 and table 6.4 for LC 3 and LC 4 load cases, respectively. Columns numbered

1 − 3, 4 − 27 and 27 − 30 in the table, belongs to the base, intermediate and top storey

case, respectively.

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6.2. DIN 18800 9th semester

Having obtained the load factors for the different columns in each storeys, the next step is

to determine the respective K-factor for the columns by applying the equation 4.14 given

in chapter 4. An example hereupon is provided in equation 6.3 for columns: 4, 5 and 6,

in load case LC 3. These columns belongs to the category for an intermediate storey, for

which kind the representative K-factor β storey = 1.7, is determined, see table 6.2.

β column =

N j · K j

N j · K S · β storey for j = 1, 2 and 3 (6.3)

β col nr 4 =

4.06 · 576805·103 mm43600 mm1 · 3 · 576805·10

3 mm4

3600 mm

· 1.7 = 1.977

β col nr 4 =

4.06 · 576805·103 mm4

3600 mm

1.88 · 3 · 576805·103 mm4

3600 mm

· 1.7 = 1.440

β col nr 4 =

4.06 · 576805·103 mm43600 mm1.17 · 3 · 576805·10

3 mm4

3600 mm

· 1.7 = 1.826

Similarly the K-factors for the other columns are determined.

N j and N j value to be

inserted in equation 6.3 and the determined K-factors are given in table 6.3 and 6.4 for

LC 3 and LC 4, respectively.

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Load case Normal Load Total load β storey β columnLC 3 force [kN ] factor N j factor

N j K-factor

Col nr 1 2334 1.00 3.299

Col nr 2 4554 1.95 4.17 2.8 2.362

Col nr 3 2834 1.21 2.994

Col nr 4 2155 1.00 1.977

Col nr 5 4062 1.88 4.06 1.7 1.440

Col nr 6 2526 1.17 1.826

Col nr 7 1946 1.00 1.960

Col nr 8 3588 1.84 3.99 1.7 1.444

Col nr 9 2230 1.15 1.831

Col nr 10 1725 1.00 1.947

Col nr 11 3121 1.81 3.93 1.7 1.447

Col nr 12 1939 1.12 1.836

Col nr 13 1495 1.00 1.934

Col nr 14 2661 1.78 3.88 1.7 1.450

Col nr 15 1649 1.10 1.842

Col nr 16 1257 1.00 1.923

Col nr 17 2208 1.76 3.84 1.7 1.451

Col nr 18 1360 1.08 1.849

Col nr 19 1011 1.00 1.914

Col nr 20 1760 1.74 3.80 1.7 1.451

Col nr 21 1075 1.06 1.856

Col nr 22 758 1.00 1.909Col nr 23 1317 1.74 3.78 1.7 1.448

Col nr 24 792 1.04 1.867

Col nr 25 499 1.00 1.909

Col nr 26 875 1.7

Documents

Buckling Length REPORT