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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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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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Trainee report - Rambøll - Autumn 2010 2. Frame design in practice
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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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.
[Galambos and Surovek, 2008]
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.
[Galambos and Surovek, 2008]
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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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.
[Galambos and Surovek, 2008]
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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4.1. AISC - formulation 9th semester
( 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:
[Galambos and Surovek, 2008]
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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4.1. AISC - formulation 9th semester
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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Trainee report - Rambøll - Autumn 2010 4. K-factor determination in practice
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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4.1. AISC - formulation 9th semester
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.
[Galambos and Surovek, 2008]
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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Trainee report - Rambøll - Autumn 2010 4. K-factor determination in practice
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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Trainee report - Rambøll - Autumn 2010 4. K-factor determination in practice
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