COMPARATIVE ANALYSIS OF SECOND ORDER EFFECTS BY …

COMPARATIVE ANALYSIS OF SECOND ORDER EFFECTS BY DIFFERENT

STRUCTURAL DESIGN CODES

João Luis Martins Soares Nogueira

Projeto de Graduação apresentado ao Curso de

Engenharia Civil da Escola Politécnica,

Universidade Federal do Rio de Janeiro, como

parte dos requisitos necessários à obtenção do

título de Engenheiro.

Orientadores:

Sérgio Hampshire de Carvalho Santos

João Carlos de Oliveira Fernandes de Almeida

Rio de Janeiro

Setembro de 2017

ii

COMPARATIVE ANALYSIS OF SECOND ORDER EFFECTS BY DIFFERENT

STRUCTURAL DESIGN CODES


PROJETO DE GRADUAÇÃO SUBMETIDO AO CORPO DOCENTE DO CURSO

DE ENGENHARIA CIVIL DA ESCOLA POLITÉCNICA DA UNIVERSIDADE

FEDERAL DO RIO DE JANEIRO COMO PARTE DOS REQUISITOS

NECESSÁRIOS PARA OBTENÇÃO DO GRAU DE ENGENHEIRO CIVIL.

Examinado por:

__________________________________________________

Prof. Sérgio Hampshire de Carvalho Santos, D. Sc., EP/UFRJ

__________________________________________________

Profa. Maria Cascão Ferreira de Almeida, D.Sc., EP/UFRJ

__________________________________________________

Prof. Ricardo Valeriano Alves, D. Sc., EP/UFRJ

RIO DE JANEIRO, RJ – BRASIL

SETEMBRO DE 2017

iii

Soares Nogueira, João Luis Martins

Comparative analysis of second order effects

by different structural design codes/ João Luis

Martins Soares Nogueira – Rio de Janeiro: UFRJ/

Escola Politécnica, 2017.

XI, 63 p.: il.; 29.7 cm.

Orientadores: Sérgio Hampshire de Carvalho

Santos, D.Sc.; João Carlos de Oliveira Fernandes

de Almeida, Ph.D.

Projeto de Graduação – UFRJ/ Escola

Politécnica/ Curso de Engenharia Civil, 2017.

Referências Bibliográficas: p. 50

1. Second order effects. 2. Edifício de concreto

armado. I. Santos, Sérgio Hampshire de Carvalho

et al.. II. Universidade Federal do Rio de Janeiro,

Escola Politécnica, Curso de Engenharia Civil. III.

Comparative analysis of second order effects by

different structural design codes

iv

AGRADECIMENTOS

Em primeiro lugar tenho de agradecer à minha família. A todos os primos, tios, à minha

avó e especialmente aos meus pais, que não apenas me sustentaram como também

deram todo o amor, suporte e apoio moral durante a minha vida, tornando minha

educação possível.

Um obrigado aos meus orientadores, os professores Sérgio Hampshire e João Almeida.

Não só leram como também corrigiram este trabalho, estando sempre disponíveis

quando precisasse, dando conselhos e tirando todas as dúvidas com paciência, mesmo

com grandes diferenças de horário.

Pelo que aprendi sobre SAP2000 devo agradecer aos meus amigos e colegas Bárbara

Cardoso e Gabriel Saramago. Também tenho que reconhecer a ajuda da Catarina Brito e

do Rodrigo Affonso que foram tirando algumas dúvidas parvas que iam surgindo.

Finalmente, não posso deixar de agradecer a todos os amigos que me acompanharam

nesta jornada de 5 anos, tanto em Portugal como no Brasil. Sem eles isto não teria sido

possível ou, pelo menos, não teria tido tantos momentos divertidos. Eles sabem quem

são, mas enumero desde a Turma IX, Quarteto, MX, Diplomatas, Moita, Geral, os

gringos, o pessoal do Frei e da UFRJ e de outros lugares da vida. Um obrigado.

v

Resumo do Projeto de Graduação apresentado à Escola Politécnica/UFRJ como parte

dos requisitos necessários para a obtenção do grau de Engenheiro Civil

Análise comparativa dos efeitos de segunda ordem segundo diferentes normas de

dimensionamento estrutural


Setembro/2017

Orientadores: Sérgio Hampshire de Carvalho Santos e João Carlos de Oliveira

Fernandes de Almeida

Curso: Engenharia Civil

Neste trabalho fez-se uma comparação entre alguns métodos de análise dos efeitos de

segunda ordem, tanto globais como locais, em pilares de concreto armado, segundo três

normas de dimensionamento estrutural: Norma Brasileira NBR 6118:2014, Europeia

EN 1992-1-1 e Model Code 2010 da fib. Para tal, modelou-se um edifício de concreto

armado de 12 andares e 24 pilares com três níveis de referência no programa SAP2000,

sujeito a carregamentos de vento, carga acidental e peso próprio. No caso da análise

global, enquanto a Norma Brasileira dá a possibilidade de escolha entre o método P-

Delta e um fator multiplicador de momentos fletores, a Norma Europeia apenas

considera um fator multiplicador de forças horizontais. Já o Model Code 2010 não faz

qualquer referência a estes efeitos. Para a análise local, foram utilizados seis métodos,

dois de cada uma das três normas mencionadas. Existem muitas similaridades entre

alguns métodos, mas há considerações particulares de cada norma, que fazem com que

os resultados sejam diferentes.

Palavras-chave: Efeitos de segunda ordem; Edifício de concreto armado; Análise não

linear; Análise comparativa.

vi

Abstract of Undergraduate Project presented to POLI/UFRJ as a partial fulfilment of the

requirements for the degree of Civil Engineer

Comparative analysis of second order effects by different structural design codes


September/2017

Advisors: Sergio Hampshire de Carvalho Santos and João Carlos de Oliveira Fernandes

de Almeida

Course: Civil Engineering

This work compares some methods of analysis of global and local second order effects

on reinforced concrete columns, according to three structural design codes: Brazilian

NBR 6118:2014, European EN 1992-1-1 and fib Model Code 2010. For this, a 12-

storey reinforced concrete building, subjected to wind, accidental and self-weight loads,

with 24 columns at three reference levels was modelled using the software SAP2000.

For the global analysis, while the Brazilian Standard gives the possibility of choosing

between the P-Delta method and a bending moment multiplier, the EN 1992-1-1 only

considers a factor for multiplying horizontal forces. The fib Model Code 2010 does not

make any reference on global second order effects. Finally, the local analysis compared

six different methods, two from each of the structural codes mentioned above. Although

many similarities were found, there are some particularities in each standard

that makes the final results differ among them.

.

Keywords: Second order effects; Reinforced concrete building; Nonlinear analysis;

Comparative analysis

vii

SUMMARY

AGRADECIMENTOS .................................................................................................. iv

SUMMARY ................................................................................................................... vii

LIST OF FIGURES ........................................................................................................ x

LIST OF TABLES ......................................................................................................... xi

1. INTRODUCTION ..................................................................................................... 1

1.1. Context and motivation ...................................................................................... 1

1.2. Objectives and methodology .............................................................................. 1

1.3. Outline of the document ..................................................................................... 2

2. FUNDAMENTAL CONCEPTS ............................................................................... 4

2.1. Second order effects ........................................................................................... 4

2.2. Methods of analysis ............................................................................................ 5

2.2.1. P-Delta analysis ........................................................................................ 6

2.2.2. Computational nonlinear analysis – physical and geometrical ................. 7

2.3. Buckling, slenderness and effective length ........................................................ 8

2.4. Geometrical imperfections ................................................................................. 9

2.5. Creep ................................................................................................................ 10

2.6. General code approach on second order effects ............................................... 11

3. BUILDING ANALYSIS .......................................................................................... 12

3.1. Material definition and assignment .................................................................. 13

3.2. Section properties – definition and assignment ................................................ 15

3.3. Definition of the acting loads ........................................................................... 17

3.4. Definition of the load combinations ................................................................. 18

3.4.1. Combination for the ultimate limit state ................................................. 18

3.4.2. Combination for the serviceability limit state ......................................... 19

4. ANALYSIS ACCORDING TO NBR 6118:2014 ................................................... 20

4.1. Initial considerations ........................................................................................ 20

viii

4.2. Global second order effects .............................................................................. 20

4.2.1. Verification of the ULS – Analysis through the coefficient 𝜸𝒛 .............. 20

4.2.2. Verification of the ULS – Analysis through the parameter 𝜶................. 26

4.2.3. Verification of the SLS – Comparative displacement analysis............... 26

4.3. Local second order effects ................................................................................ 27

4.3.1. Local geometric imperfections................................................................ 28

4.3.2. Slenderness criterion for isolated compressed members ........................ 28

4.3.3. Effects of creep ....................................................................................... 29

4.3.4. Method of the standard-column with approximated stiffness ................. 29

4.3.5. Method of the standard-column with approximated curvature ............... 30

4.3.6. Method of the standard-column associated to the M, N and curvature

diagrams ................................................................................................................. 30

4.3.7. General method ....................................................................................... 30

4.3.8. Comparison of methods .......................................................................... 31

5. ANALYSIS ACCORDING TO THE EN 1992-1-1, EUROCODE 2................... 32

5.1. Initial considerations ........................................................................................ 32

5.1.1. Material properties .................................................................................. 32

5.1.2. Structural properties ................................................................................ 32

5.2. Global second order effects .............................................................................. 33


5.3.1. Local geometric imperfections................................................................ 37

5.3.2. Slenderness criterion for isolated compressed members ........................ 37

5.3.3. Effects of creep ....................................................................................... 38

5.3.4. Method based on the nominal stiffness ................................................... 38

5.3.5. Method based on the nominal curvature ................................................. 39

5.3.6. General method ....................................................................................... 40

5.3.7. Method comparison ................................................................................ 40

ix

6. ANALYSIS ACCORDING TO fib MODEL CODE 2010 ................................... 42


6.1.1. Level I of approximation......................................................................... 43

6.1.2. Level II of approximation ....................................................................... 43

6.1.3. Level III of approximation ...................................................................... 43

6.1.4. Level IV of approximation ...................................................................... 44

6.1.5. Method comparison ................................................................................ 44

7. FINAL CONSIDERATIONS .................................................................................. 46

7.1. Conclusions ...................................................................................................... 46

7.2. Suggestions for future works ............................................................................ 49

8. REFERENCES ......................................................................................................... 50

APPENDIX A – RELATIONSHIP BETWEEN 𝜶 AND NOMINAL BUCKLING

LOAD ..................................................................................................................... 51

APPENDIX B – EXAMPLE OF LOCAL SECOND ORDER EFFECTS

ANNALYSIS FOR NBR 6118:2014 ............................................................................ 52

APPENDIX C – EXAMPLE OF LOCAL SECOND ORDER EFFECTS

ANNALYSIS FOR EN 1992-1-1 ................................................................................. 56

APPENDIX D – EXAMPLE OF LOCAL SECOND ORDER EFFECTS

ANNALYSIS FOR fib 2010 MODEL CODE ............................................................ 61

x

LIST OF FIGURES

Figure 2-1– Resulting bending moments in braced and unbraced elements (CÂMARA,

2015) ................................................................................................................................. 4

Figure 2-2– Modes of deflection of columns in a structure: (a) P-Δ effect (whole

structure); (b) P-δ effect (single column) (NARAYANAN; BEEBY, 2005) ................... 5

Figure 2-3 – P-Delta method for single columns (LONGO, 2017) .................................. 6

Figure 2-4 – P-Delta method for framed structures (LONGO, 2017) .............................. 7

Figure 2-5 – Stress-strain diagrams for and concrete (COMITÉ EUROPÉEN DE

NORMALISATION, 2004).............................................................................................. 7

Figure 2-6 – Stress-strain diagrams for reinforcing steel (COMITÉ EUROPÉEN DE

NORMALISATION, 2004).............................................................................................. 8

Figure 2-7 – Types of geometrical imperfections (COMITÉ EUROPÉEN DE

NORMALISATION, 2004)............................................................................................ 10

Figure 3-1 – Structural model of the building (GOMES, 2017) .................................... 12

Figure 3-2 – Formwork drawing of the building, showing the different classes of

columns (GOMES, 2017) ............................................................................................... 13

Figure 3-3 – Properties of the reinforced concrete used in the model ............................ 15

Figure 3-4 – Properties of the reinforcing steel used in the model................................. 15

Figure 3-5 – Distributed forces due to the wind acting in the structure, in kN/m .......... 18

Figure 5-1 – Global displacements ................................................................................. 33

Figure 5-2 – Effective lengths for isolated members (COMITÉ EUROPÉEN DE

NORMALISATION, 2004)............................................................................................ 36

Figure 6-1 – Values of integration factors, 𝑐𝑖, as a function of the load type and the

boundary conditions (FÉDÉRATION INTERNATIONALE DU BÉTON, 2010) ........ 44

Figure B-1 Concrete interaction diagram for X direction, NBR 6118 (2014) ............... 54

Figure B-2 – Concrete interaction diagram for Y direction, NBR 6118 (2014) ............ 55

Figure B-3 – Cross section detail for the NBR 6118 (2014) .......................................... 55

xi

LIST OF TABLES

Table 3-1 – Properties of the RC used ............................................................................ 14

Table 3-2 – Dimensions of the cross sections of various structural elements ................ 16

Table 3-3 – Wind pressure and force acting in the structure .......................................... 17

Table 4-1 – Model’s structure output: Base reaction forces, in kN ................................ 21

Table 4-2 – Model’s structure output: Base reaction bending moments, in kN.m ......... 21

Table 4-3 – Joint displacements in the various levels of a column, in meters ............... 22

Table 4-4 – Comparison of bending moments in the base of various columns.............. 24

Table 4-5 – Displacements of the various pavements, both absolute and relative, in

meters.............................................................................................................................. 27

Table 4-6 – Comparison of the Brazilian methods. Moments given in kN∙m. ............... 31

Table 5-1 – Comparison of the methods from Eurocode 2. Moments given in kN∙m ... 40

Table 6-1 – Comparison of the methods from MC 2010. Moments given in kN∙m ...... 45

Table 7-1 – Comparison of the different approximated methods for each column, in the

X direction. Moments given in kN∙m ............................................................................. 47


Y direction. Moments given in kN∙m ............................................................................. 48

1

1. INTRODUCTION

1.1. CONTEXT AND MOTIVATION

In the past few decades, due to the increase of urban populational density, allied

with the evolution of building materials and the constructive and designing techniques,

the structures of buildings have become taller and taller, and consequently more and

more slender. As a result, this evolution brought potential problems of instability, the so

called second order effects.

Although these effects can be dismissed in the most common structures, there are

cases that they assume a considerable magnitude and can damage the structure, which

may cause both material and human losses. For this reason, the consideration of these

effects may take a very important role on the design of the structures and structural

codes usually have chapters dedicated to this problem.

However, even though second order effects are a worldwide problem, not all

structural codes address this issue in the same way and assumptions, simplifications and

approaches are different, as each country has their own particular way of dealing with

engineering problems.

As today’s world is a global one, it is nowadays necessary to compare some

codes and see their differences and similarities. The ones analysed herein are:

1. NBR 6118:2014 – Brazilian Standard;

2. EN 1992-1-1 – European Standard (Eurocode 2);

3. fib Model Code 2010 – International Federation for Structural Concrete.

Since this work was done as the Final Project of Double Degree between IST-

Lisbon (Portugal) and UFRJ (Brazil), the three codes were considered to be the most

important to be analysed for both countries.

From the past few years, UFRJ has been making significant theoretical research

on this topic. This work in particular give continuity to the Final Project of GOMES

(2017).

1.2. OBJECTIVES AND METHODOLOGY

The objective of this work is to analyse and to compare the already mentioned

design codes regarding to:

2

1. Checking the requirements for consideration of second order effects;

2. Analysing and evaluating the influence of these effects on reinforced

concrete (RC) structures;

3. Comparing the different analysis and approaches, showing its similarities and

differences, evaluating if they are too conservative or, on the other hand,

against safety.

To fulfil this objective, the following methodology was followed:

1. Understand the fundamental concepts behind second order effects;

2. Understand the different procedures of analysis for these effects, and

evaluate the sensibilities of each method;

3. Analyse the structural model of a building previously created according to

each code, compare their results.

Following this procedure, this work aims to contribute for a better understanding

of second order effects and its practical applications in different countries.

1.3. OUTLINE OF THE DOCUMENT

The present work is divided in eight chapters, references and appendixes.

Besides this introducing chapter, the remaining chapters are described below.

The second chapter presents the fundamental concepts regarding structural

instability, starting with the definition of second order effects and the methods for

analysing them. This will be followed by other critical definitions on the subject, such

as buckling, slenderness, geometrical imperfections and creep. The chapter concludes

with the basic general approach on second order effects according to different codes.

The third chapter shows and defines the structure in analysis and the structural

model used for this, fully detailing its material and section properties, applied loads and

considered combinations of actions. This model will serve as a base for use in the

remaining chapters.

Chapters four to six analyse the model previously described, each one by a

different code. These three chapters start with the normative definitions, followed by

global and local analyses of the structure.

Finally, conclusions are presented in the seventh chapter, also presenting

proposals for continuing this work.

3

All the references that guided this work are shown on chapter eight.

Appendix A demonstrates the relationship between the instability parameter α

found in the Brazilian code, and the nominal buckling load of a building.

Appendixes B to D show the step by step design according to the different

methods of each design code.

4

2. FUNDAMENTAL CONCEPTS

2.1. SECOND ORDER EFFECTS

It is a well known fact that when an element is subjected to axial force and

bending moments, it deflects. The interaction between this deformation and the axial

force will cause an increase of the bending moment at a given section that, depending

on the slenderness of the element, may take a very important role in the structural

design. This additional effect is what is called second order effect.

Figure 2-1– Resulting bending moments in braced and unbraced elements (CÂMARA,

2015)

In general, structural design codes assume that second order effects can be

ignored if they represent less than 10% of first order moment. However, this rule is not

practical for the structural designer, as it requires calculating previously these effects to

know if they are dismissible. For this reason, regulations have developed simplified

ways to verify if these effects are in fact a problem.

If the case is that second order effects cannot be dismissed, a nonlinear analysis

must be made. This type of analysis must consider both geometrical and physical

nonlinearities, and they will be detailed in the following sub-chapters.

It is important to note that there are two distinct kinds of second order effects:

- Global effects (or P-Δ effects) affect the entire structure and occur in

structures that present significant horizontal displacements when subjected to

vertical and horizontal loads. This kind of structures is called sway

structures.

- Local effects (or P-δ effects) affect isolated elements that suffer significant

displacement when subjected to axial loads, independently from what

5

happens with the structure. Here, the bending moment diagram is decurrent

from a nonlinear behaviour, unlike to what happens to the first order

moment.

Both global and local second order effects will be addressed in this work.

Figure 2-2– Modes of deflection of columns in a structure: (a) P-Δ effect (whole

structure); (b) P-δ effect (single column) (NARAYANAN and BEEBY, 2005)

2.2. METHODS OF ANALYSIS

First order linear elastic analysis, where stresses and forces are calculated

through the equilibrium of the elements in their non-deformed configuration, is the most

usual form of structural analysis, as it is simple and for most typical cases, accurate.

However, in some cases nonlinear analysis is required, since a first order linear analysis

does not provide accurate results. Second order effects fit this category, and for their

analysis nonlinearity shall be considered.

Nonlinear analyses shall consider both geometric and physical nonlinearity.

Geometric nonlinearity deals with the equilibrium in the deformed state, whereas

physical nonlinearity (nonlinearity associated with the materials) considers, beside the

inherent nonlinear behaviour of the materials, the effects of cracking, creep and other

material behaviour of the reinforced concrete structure.

Although this kind of analysis can return accurate results, it is very complex and

requires the use of an appropriate computer software. Another problem is that a

nonlinear analysis needs the knowledge of the reinforcement to be used, which is not

always previously known.

6

Due to these major inconveniences, other methods for evaluating second order

effects have been developed. The following sub-items will detail two of those methods.

2.2.1. P-DELTA ANALYSIS

The P-Delta method is an approximated iterative method that allows for

calculating global second order effects by creating fictitious horizontal forces (binaries)

that cause an effect equivalent to the second order moments. Here, the nonlinear

analysis is replaced by as many linear analyses as needed for convergence, considering

a pre-defined tolerance.

Figure 2-3 – P-Delta method for single columns (LONGO, 2017)

The method will be first explained for a single column then generalized for

framed structures. This sequence is followed:

1. Obtain the displacements in each pavement, 𝑎𝑖, by a first order linear

analysis;

2. Calculate the relative displacements in each storey: ∆𝑎𝑖 = 𝑎𝑖 − 𝑎𝑖+1

3. Calculate the forces acting on each pavement: 𝐻𝑖 = 𝑁𝑖∆𝑎𝑖

ℎ𝑖

4. In each section, fictitious horizontal forces (binaries) are defined as

𝐻𝑖∗ = 𝐻𝑖 −𝐻𝑖−1. These forces shall be added to the original horizontal loads,

𝐹𝑖, making 𝐹∗ = 𝐹𝑖 +𝐻𝑖∗.

5. Recalculate the displacements in each pavement, until the method converges.

For framed structures, the procedure is very similar and the horizontal forces will

be:

𝐻𝑖 =∑𝑁𝑗,𝑖

∆𝑎𝑖ℎ𝑖⇒𝐻𝑖

∗ = 𝐻𝑖 − 𝐻𝑖−1 =∑𝑁𝑗,𝑖∆𝑎𝑖ℎ𝑖−∑𝑁𝑗,𝑖 − 1

∆𝑎𝑖−1ℎ𝑖−1

(1)

7

Where:

- ∑𝑁𝑗,𝑖is the summation of the vertical loads on level 𝑖 and above

Figure 2-4 – P-Delta method for framed structures (LONGO, 2017)

2.2.2. COMPUTATIONAL NONLINEAR ANALYSIS – PHYSICAL AND

GEOMETRICAL

Reinforced concrete (RC) have a physically nonlinear behaviour since the

relationship between loads and material deformations are not linear. This can be easily

observed in the stress-strain design diagrams of concrete and reinforcing steel:

Figure 2-5 – Stress-strain diagrams for and concrete (COMITÉ EUROPÉEN DE

NORMALISATION, 2004)

8

Figure 2-6 – Stress-strain diagrams for reinforcing steel (COMITÉ EUROPÉEN DE


As already said, to properly perform a nonlinear analysis, it is required to use an

appropriated software and to have a previous knowledge of the RC section detailings.

To take the physical nonlinearity into consideration, this software shall actualize the

stiffness of the RC section in each iteration, as a function of the adjusted forces and the

corresponding moment-curvature diagrams. So, the stiffness of the various RC sections

will depend on the actual internal forces.

Because this consideration is very difficult to apply, some codes allow the

reduction of the stiffness of the elements through approximate formulas.

2.3. BUCKLING, SLENDERNESS AND EFFECTIVE LENGTH

As already said, an element deflects when subjected to axial forces and bending

moments, causing an increase of the bending moments due to the interaction between

the lateral deformations and the axial forces.

The effect of pure buckling was first studied by Euler (1707-1783), whom

showed that an ideal double pinned column of length 𝑙 subjected to a concentrated axial

force at its top has a critical buckling load equal to:

𝑃𝑐𝑟 =𝜋2𝐸𝐼

𝑙2 (2)

It is important to notice that the critical buckling load depends on the flexural

stiffness of the column, instead of its material strength. So, to increase the buckling

resistance it is needed to increase its flexural stiffness (moment of inertia).

9

Another important consideration is that pure buckling will not occur in real

structures, since there are imperfections, eccentricities and transverse loads to be

considered.

Another important remark is that when lim𝑙→0𝑃𝑐𝑟 = ∞. This means that long

columns have smaller critical buckling loads and the opposite happens to short columns.

So, the notion of slenderness is necessary for understanding the concept of instability

and second order effects.

The slenderness coefficient is given by the ratio of the effective length 𝑙0 and the

radius of gyration, 𝑖:

𝜆 =𝑙0𝑖

(3)

𝑖 = √𝐼

𝐴

𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑐𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛⇒ 𝑖 = √

𝑏ℎ3

12

𝑏ℎ=

ℎ

√12 (4)

Where 𝐼 is the moment of inertia in the axis perpendicular to the buckling plan.

Effective length (or buckling length) is defined as the distance between the

inflection points of the curvature of the deformed element, in an unstable condition. It

depends on the support conditions of the column. As it will be seen, the Brazilian and

European standards define the effective length differently.

The concept of the effective length of a column is very important, as it can

generalize Euler’s expression for different support conditions:

𝑃𝑐𝑟 =𝜋2𝐸𝐼

𝑙02 (5)

2.4. GEOMETRICAL IMPERFECTIONS

Even though structures are ideally modelled as perfectly straight frames, when

they are executed it is impossible to make them exactly as designed, as it is inevitable to

have some deviation of its geometry and in the position of the acting loads.

For the ULS design, these imperfections must be considered, as they will lead to

additional actions, affecting the stresses on the elements. The methods of calculation of

these actions vary from code to code.

10

Geometrical imperfections can be divided in global (entire structure) and local

(isolated elements).

Figure 2-7 – Types of geometrical imperfections (COMITÉ EUROPÉEN DE


These effects will be detailed in the following chapters, according to each code.

2.5. CREEP

When a RC element is subjected to a long-lasting constant load (permanent or

quasi-permanent loads), its initial deformation is progressively increased in the time,

because of the creep effect. This increase of deformation will cause an increase of

stresses and, consequently, a reduction of the element’s resistance. Because creep is a

consequence of the nonlinear behaviour of RC, its consideration is fundamental when

evaluating second order effects.

Since the evaluation of the effects of creep is dependent on many factors, such as

the composition of the cement, air humidity and age of concrete, structural regulations

have developed simplified ways of calculating its influence.

11

2.6. GENERAL CODE APPROACH ON SECOND ORDER EFFECTS

In general, structural design codes follow a similar approach dealing with global

and local second order effects (NARAYANAN and BEEBY, 2005):

1. Classify the structure according to its stiffness against lateral deformations,

as braced if it contains bracing elements (very rigid elements that can be

assumed to resist to all the horizontal forces) or unbraced, if otherwise. For a

local analysis, the boundary conditions of each element must be defined, to

make possible to correctly evaluate its buckling length;

2. Classify the structure as sway or non-sway, according to its deflection mode.

This step is only valid for global effects.

- Sway structure – the columns are supposed to have significant

deflections (sidesway of the whole structure) and global second order

effects assume an important role;

- Non-sway structure – horizontal displacements are small (there is no

significative sidesway, only occurring the deflection of a single or

some columns) and, generally, global second order effects can be

neglected.

3. Check if the slenderness effects may be disregarded.

4. Design the structural elements, taking the slenderness effects into account if

necessary.

12

3. BUILDING ANALYSIS

This work will use the structural model of a 12-storey reinforced concrete (RC)

building previously created by GOMES (2017), modelled using the software SAP2000

(COMPUTERS AND STRUCTURES, 2016), defining bar elements for the beams and

columns, and shell elements for the slabs.

The building in question is 36.0 meters high, 30.0 meters wide and 16.0 meters

deep. The computational model and formwork drawing can be seen in Figure 3-1 and

Figure 3-2 respectively.

To make the analysed structure as close as possible to a real building, the model

was divided and analysed in three vertical sections with four pavements each, separated

at 12, 24 and 36 meters height.

Figure 3-1 – Structural model of the building (GOMES, 2017)

13

Figure 3-2 – Formwork drawing of the building, showing the different classes of

columns (GOMES, 2017)

A very important remark must be made about the structural design of the

building. Although this structure may not seem reasonable for a European designer, as it

does not contain rigid cores or walls (i.e., bracing elements), this structural design is

quite common in Brazil, and will help to accomplish the objective of this work, which is

to test the approach on second order effects according to different international codes.

Following the guidelines previously described in second chapter, the first thing

to do is classifying the structure. As this building does not have bracing elements, the

structure is classified as sway and unbraced, which means that its design should begin in

considering the sidesway of the entire structure (global effects), and only after that, in

the design of the individual columns for resisting to the local effects.

3.1. MATERIAL DEFINITION AND ASSIGNMENT

The RC used in the model above has the properties defined in Table 3-1. These

properties are defined according to Brazilian Standard NBR 6118:2014.

14

Table 3-1 – Properties of the RC used

Concrete

Compressive strength 𝑓𝑐𝑘 = 35 𝑀𝑃𝑎

Initial tangent modulus of elasticity 𝐸𝑐𝑖 = 1.0 × 5600 × √35

= 33 130 𝑀𝑃𝑎

Coefficient 𝛼𝑖 𝛼𝑖 = (0.8 + 0.2 ×

35

80)

= 0.888

Secant modulus of elasticity 𝐸𝑐𝑠 = 0.888 × 33130.05

= 29 403 𝑀𝑃𝑎

Secant modulus of elasticity for global

stability analysis

𝐸𝑐𝑠∗ = 29402.9 × 1.10

= 32 343 𝑀𝑃𝑎

Coefficient of thermal dilatation 𝛼 = 1 × 10−5℃−1

Poisson coefficient 𝜐 = 0.2

Specific weight of columns and beams 𝛾 = 25 𝑘𝑁/𝑚³

Specific weight of slabs

(with a correction for floor finishing) 𝛾𝑠𝑙𝑎𝑏𝑠 = 38.33 𝑘𝑁/𝑚³

Reinforcement

Steel

Minimum yield stress 𝑓𝑦 = 414 𝑀𝑃𝑎

Minimum tensile stress 𝑓𝑢 = 621 𝑀𝑃𝑎

Secant modulus of elasticity 𝐸𝑐𝑠 = 210 000 𝑀𝑃𝑎

Specific weight 𝛾 = 76.97 𝑘𝑁/𝑚³

Note that the specific weight of the concrete of the slabs was majored by

2 𝑘𝑁/𝑚2 so the mass of the masonry and cladding can be considered (1 𝑘𝑁/𝑚2each):

𝛾𝑠𝑙𝑎𝑏𝑠 =𝛾 × ℎ𝑠𝑙𝑎𝑏 + 1 + 1

ℎ𝑠𝑙𝑎𝑏=25 × 0.15 + 2

0.15= 38.33 𝑘𝑁/𝑚³ (6)

Figure 3-3 and Figure 3-4 show how some of the properties above were

introduced on SAP2000:

15

Figure 3-3 – Properties of the reinforced concrete used in the model

Figure 3-4 – Properties of the reinforcing steel used in the model

3.2. SECTION PROPERTIES – DEFINITION AND ASSIGNMENT

The design of the structural elements followed the considerations below.

- The beams widths were set as 𝑏𝑏𝑒𝑎𝑚 = 15 𝑐𝑚, changing its height according

to their span;

- The thickness of the columns was set as ℎ𝑐𝑜𝑙𝑢𝑚𝑛𝑠 = 20 𝑐𝑚 and their widths

were defined according to the estimated area of influence. To simplify the

design, the columns were categorized in four groups, in each reference level,

16

grouping the ones that present similar levels of stress, resulting in 12 types of

columns. Each group is represented in Figure 3-2. The reinforcement of the

columns is distributed along the longer faces, assuming 𝑑′ = 4.0 𝑐𝑚;

- The slabs were defined with the height of ℎ𝑠𝑙𝑎𝑏𝑠 = 15 𝑐𝑚, applying the same

loading per area with a majored specific weight, as shown above.

All these calculations can be found in the work of GOMES (2017), and the

dimensions of the cross sections are detailed in Table 3-2. The number below the group

indicates the number of columns of that group.

Table 3-2 – Dimensions of the cross sections of various structural elements

Element Reference Level Width, b (m) Height, h (m)

Columns

Group 1

(4)

0.0 0.55 0.20

12.0 0.40 0.20

24.0 0.30 0.20

Columns

Group 2

(8)

0.0 0.80 0.20

12.0 0.50 0.20

24.0 0.30 0.20

Columns

Group 3

(8)

0.0 0.95 0.20

12.0 0.80 0.20

24.0 0.55 0.20

Columns

Group 4

(4)

0.0 1.20 0.20

12.0 0.80 0.20

24.0 0.40 0.20

Longitudinal Beam 0.15 0.45

Transversal Beam 0.15 0.80

It is important to notice that although the building in analysis will be the same

one for every code, the definition of some properties such as the elasticity modulus is

different from code to code. So, the model will suffer slight changes in each analysis, to

take these different properties into account.

17

3.3. DEFINITION OF THE ACTING LOADS

Although the definition of the wind action and partial safety coefficients used in

the load combinations defined below vary according to the code in use (Brazilian, or

European standards), the acting forces must be the same to make a comparison possible.

The loads acting on the building are:

- Permanent loading, referring to the self-weight of the structure, masonry and

cladding, that are considered directly through the weight of the concrete, as

defined before;

- Accidental loading according to the NBR 6120 (1980): 𝑞 = 2.00 𝑘𝑁/𝑚² ;

- Wind action, as previously calculated by GOMES (2017), according to

NBR 6123 (1990), considering the wind in the city of Rio de Janeiro, with a

basic velocity of 𝑉0 = 35 𝑚/𝑠. This action is resumed in Table 3-3.

Table 3-3 – Wind pressure and force acting in the structure

Pavement Velocity

[𝑚/𝑠]

Dynamic Pressure

[𝑁/𝑚²]

Effective Pressure [𝑁/𝑚²]

Windward Leeward

1 – 4 𝑉𝑘.12 = 25.90 𝑞12 = 411.21 Δ𝑝𝐴.12 = 411.21 Δ𝑝𝐵.12 = − 164.48

5 – 8 𝑉𝑘.24 = 28.70 𝑞24 = 504.92 Δ𝑝𝐴.24 = 504.92 Δ𝑝𝐵.24 = −201.97

9 – 12 𝑉𝑘.36 = 30.80 𝑞36 = 581.52 Δ𝑝𝐴.36 = 581.52 Δ𝑝𝐵.36 = −232.61

Pavement Distributed force[𝑁/𝑚]

Windward Leeward

Lobby q𝐴,𝑇 = 616.82 q𝐵,𝑇 = −246.72

1 – 4 q𝐴.12 = 1233.63 q𝐵.12 = −493.44

5 – 8 q𝐴.24 = 1514.76 q𝐵.24 = −605.91

9 – 11 q𝐴.36 = 1744.56 q𝐵.36 = −697.83

12 q𝐴.36 = 872.28 q𝐵.36 = −348.92

The forces above are shown in Figure 3-5, as applied in the model, in 𝑘𝑁/𝑚.

18

Figure 3-5 – Distributed forces due to the wind acting in the structure, in kN/m

3.4. DEFINITION OF THE LOAD COMBINATIONS

Eurocode 0 (COMITÉ EUROPÉEN DE NORMALISATION, 2002) states that a

structure must be designed to have adequate structural resistance, serviceability and

durability. This means that the structure must fulfil the requirements of both ultimate

and serviceability limit states.

It also stablishes that “for each critical load case, the design values of the effects

of actions shall be determined by combining the values of actions that are considered to

occur simultaneously, and each combination of actions should include a leading

variable action and an accidental action”. The acting combinations of actions for both

limit states will be the ones defined in the Brazilian Standard NBR 6118 (2014)

disregarding the effects of indirect actions such as temperature and shrinking.

As the main scope of this work is to analyse the effects of the wind forces in the

structure, the combination that defines accidental load as the main load will be

disregarded. The combination in which the wind is the main action will be the one to be

analysed. However, in a real building design, all the relevant load combinations would

be accordingly considered.

3.4.1. COMBINATION FOR THE ULTIMATE LIMIT STATE

As said, the ultimate limit state (ULS) is the one that concerns to the safety of

people, and/or the safety of the structure. For RC buildings, this means the exhaustion

of the resistance of the structural elements.

19

The usual combination of actions for the ULS is given by the following

expression of NBR 6118 (2014):

𝐹𝑑 =∑𝛾𝑔𝑖𝐹𝐺𝑖,𝑘

𝑚

𝑖=1

+ 𝛾𝑞 [𝐹𝑄1,𝑘 +∑𝜓0𝑗𝐹𝑄𝑗,𝑘

𝑛

𝑗=2

] (7)

Considering the respective partial factors for the actions:

𝐹𝑑,𝑈𝐿𝑆 = 1.4 ∙ 𝑔 + 1.4 ∙ (𝑞𝑣 + 0.7 ∙ 𝑞1) (8)

Where:

- 𝑔 – permanent load due to self-weight, masonry and cladding;

- 𝑞𝑣 – variable wind load;

- 𝑞1 – variable accidental load (considering the building occupation as office).

3.4.2. COMBINATION FOR THE SERVICEABILITY LIMIT STATE

Serviceability limit state (SLS) is the one that concerns the behaviour of the

structure and individual structural members under normal use, the comfort of people

and the appearance of the construction. This means, deformations, damages or even

vibrations that are likely to adversely affect the appearance, durability, comfort of users

or the functional effectiveness of the structure shall be avoided.

For the effects of wind as predominant variable load, the combination in analysis

must be the frequent combination, corresponding to actions that are repeated several

times during the lifespan of the structure. The frequent combination of actions for the

SLS are given by the NBR 6118:2014 as follows:

𝐹𝑑,𝑠𝑒𝑟𝑣 =∑𝐹𝐺𝑖,𝑘

𝑚

𝑖=1

+ 𝜓1𝐹𝑄1,𝑘 +∑𝜓2𝑗𝐹𝑄𝑗,𝑘

𝑛

𝑗=2

(9)

Again, adopting the respective partial factors for actions:

𝐹𝑑,𝑆𝐿𝑆 = 𝑔 + 0.3 ∙ 𝑞𝑣 + 0.3 ∙ 𝑞1 (10)

Where the nomenclature is the same defined in 3.4.1.

20

4. ANALYSIS ACCORDING TO NBR 6118:2014

4.1. INITIAL CONSIDERATIONS

In this chapter, the previously defined structural model will be analysed

according to the Brazilian standard for concrete structures, NBR 6118:2014

(ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS, 2014). The analysis

encompasses the global second order effects and also the local ones.

Following the methodology previously described, the first step is evaluating

whether the global second order effects can or cannot be dismissed, which is done

through the item 15.5 of the code mentioned above.

4.2. GLOBAL SECOND ORDER EFFECTS

The global second order effects in the analysed building are herein evaluated,

according to NBR 6118. The effects of global imperfections are considered as not

decisive for the design and therefore not presented.

4.2.1. VERIFICATION OF THE ULS – ANALYSIS THROUGH THE

COEFFICIENT 𝜸𝒛

A normative criterion of NBR 6118 uses the coefficient 𝛾𝑧 for classifying the

structure as a sway or non-sway structure, evaluating in this way the global stability of

the building.

The coefficient 𝛾𝑧 is determined from a first order linear analysis, as defined in

item 15.5 of NBR 6118. Approximate stiffness reduction factors are to be considered

according to item 15.7.3 of the same standard, for taking into account the physical

nonlinearity:

- Slabs: (𝐸𝐼)𝑠𝑒𝑐 = 0.3 𝐸𝑐𝐼𝑐;

- Beams: (𝐸𝐼)𝑠𝑒𝑐 = 0.4 𝐸𝑐𝐼𝑐 (in the general case where 𝐴𝑠′ ≠ 𝐴𝑠);

- Columns: (𝐸𝐼)𝑠𝑒𝑐 = 0.8 𝐸𝑐𝐼𝑐.

To take these reduction factors into consideration another model has been

created, with the same properties as described in the previous item, but reducing the

stiffness of the structural elements, as defined above.

21

Also, the elasticity modulus used in this analysis is increased in 10%, as it is

shown in the material definition in item 3.1 of this work: 𝐸𝑐𝑠∗ = 32343 𝑀𝑃𝑎

The coefficient 𝛾𝑧 is calculated from the following expression:

𝛾𝑧 =1

1 −∆𝑀𝑡𝑜𝑡,𝑑

𝑀1,𝑡𝑜𝑡,𝑑

(11)

Where:

- 𝑀1,𝑡𝑜𝑡,𝑑 is the first order moment, the sum of the products of the horizontal

forces applied to a storey and the height of that storey, with respect to the

base of the structure;

- ∆𝑀𝑡𝑜𝑡,𝑑 is the increase of moments with respect to the first order analysis,

given by the sum of the products of all the vertical design forces acting on

the structure by the horizontal displacements of their respective points of

application.

The structure is classified as non-sway if 𝛾𝑧 ≤ 1.1. If 1.1 ≤ 𝛾𝑧 ≤ 1.3, the code

allows for an approximate consideration of the global second order effects, through the

multiplication of the horizontal loads by the factor 0.95𝛾𝑧.

Running a SAP2000 analysis, it is possible to obtain the forces and

displacements in the model, for the ULS combination. The main results are shown in

the following tables.

Table 4-1 – Model’s structure output: Base reaction forces, in kN

OutputCase Case Type Step Type Global FX Global FY Global FZ

ELU Combination 0.00 -1004.10 70 010

P-DELTA NonStatic Max 0.00 -1004.10 70 010

P-DELTA NonStatic Min 0.00 -1005.45 70 010

Table 4-2 – Model’s structure output: Base reaction bending moments, in kN.m

Output Case Case Type Step Type Global MX Global MY Global MZ

ELU Combination 20 202 0.00 0.00

P-DELTA NonStatic Max 23 892 0.00 0.00

P-DELTA NonStatic Min 23 892 0.00 0.00

22

It is important to notice that the resultant of horizontal forces do not change with

the P-Delta effect, only the moments.

Table 4-3 – Joint displacements in the various levels of a column, in meters

Pavement Output Case Case Type Step Type U2

1 P-Delta NonStatic Max 0.00885

ELU Combination 0.00717























Analysing the results above it is possible to see that the second order effects

cause a displacement amplification of approximately 1.22.

23

It is now possible to calculate 𝛾𝑧:

𝑀1,𝑡𝑜𝑡,𝑑 =∑𝐹𝑝𝑎𝑣𝑒𝑚𝑒𝑛𝑡𝑖 × ℎ𝑝𝑎𝑣𝑒𝑚𝑒𝑛𝑡𝑖

12

𝑖=1

= 14430.1 𝑘𝑁𝑚 (12)

∆𝑀𝑡𝑜𝑡,𝑑 =∑𝛿𝑝𝑎𝑣𝑒𝑚𝑒𝑛𝑡𝑖 ×𝑊𝑝𝑎𝑣𝑒𝑚𝑒𝑛𝑡𝑖

12

𝑖=1

= 2681.6 𝑘𝑁𝑚 (13)

The weight of each floor can be found by dividing the total base reaction by the

number of storeys, and dividing again by 1.4 to consider the safety factor 𝛾𝑓.

𝛾𝑧 =1

1 −2681.6

14430.1

= 1.228 (14)

As 𝛾𝑧 = 1.228 > 1.1, the structure is sway. However, since 𝛾𝑧 = 1.228 < 1.3,

the Brazilian code allows for the approximation defined previously. For the sake of the

comparisons on this work, both procedures will be used.

With the results of Table 4-2 it is possible to verify that, in the analysed case, the

approximation defined by the NBR 6118 (2014) is accurate:

20202.11 × 0.95 ∙ 𝛾𝑧 = 23572.57 ≅ 23892.10 (15)

The error in this approximation is roughly 1.33%, showing that, in the analysed

case, this procedure provides a good approximation.

4.2.1.1. COMPARATIVE ANALYSIS OF FORCES

With the P-Delta analysis complete, it is possible to perform a comparison

between the results obtained considering separately the ULS, P-Delta and the Brazilian

approximation 𝑈𝐿𝑆 × 0.95𝛾𝑧.

Choosing four columns, one column of each group, and comparing the moments

at the base and the normal forces in each one, the results are shown in Table 4-4.

24

Table 4-4 – Comparison of bending moments at the base of various columns

GROUP 1 – BENDING MOMENT M2, kN∙m

PAVEMENT P-Delta ELU 𝐸𝐿𝑈 × 0.95𝛾𝑧 ERROR

1 54.7 46.6 54.4 -0.52%

2 62.5 55.1 64.3 +2.83%

3 57.7 51.7 60.3 +4.28%

4 55.1 50.1 58.5 +5.84%

5 49.3 43.9 51.2 +3.76%

6 46.1 42.2 49.2 +6.24%

7 42.3 39.5 46.1 +8.22%

8 39.0 37.1 43.3 +9.93%

9 35.5 33.1 38.6 +8.06%

10 30.8 29.7 34.6 +10.91%

11 26.0 25.6 29.9 +12.92%

12 22.7 22.7 26.5 +14.12%



1 67.5 55.6 64.9 -3.84%

2 57.7 44.3 51.7 -10.29%

3 52.0 41.0 47.8 -8.14%

4 46.0 36.7 42.9 -6.81%

5 38.0 30.0 35.0 -8.12%

6 31.9 25.6 29.9 -6.43%

7 25.7 21.1 24.6 -4.00%

8 19.8 16.5 19.3 -2.34%

9 16.5 13.6 15.9 -3.66%

10 10.1 8.5 9.9 -1.96%

11 4.0 3.3 3.9 -2.89%

12 -2.2 -2.3 -2.7 +19.98%

25



1 95.8 82.8 96.7 +0.86%

2 111.9 101.6 118.5 +5.58%

3 103.7 95.3 111.2 +6.74%

4 99.9 93.2 108.8 +8.18%

5 101.4 92.4 107.8 +5.93%

6 95.5 89.3 104.2 +8.32%

7 89.9 85.5 99.8 +9.90%

8 85.8 82.9 96.7 +11.31%

9 73.9 70.1 81.8 +9.69%

10 68.9 67.3 78.5 +12.18%

11 61.7 61.2 71.4 +13.56%

12 59.7 59.7 69.6 +14.24%



1 93.7 77.6 90.5 -3.37%

2 68.3 49.8 58.1 -14.84%

3 61.9 46.6 54.4 -12.09%

4 53.5 40.8 47.6 -11.09%

5 50.5 38.3 44.7 -11.56%

6 41.2 31.7 37.0 -10.30%

7 32.3 25.5 29.7 8.12%

8 23.6 18.7 21.8 7.52%

9 17.9 14.2 16.6 7.32%

10 9.6 7.5 8.8 9.16%

11 2.1 1.1 1.3 35.70%

12 -6.4 -6.6 -7.7 17.41%

The mean absolute error in all columns is 8.77%, ranging from 6.54% in Group 2

to 12.37% in Group 4. This shows that although the criterion of NBR 6118 (2014) is not

very accurate, for a simple analysis it provides a good approximation.

26

As an additional comment, it can be seen that the NBR 6118:2014 criterion gives

better results in the lower storeys that in the upper ones, probably because the horizontal

displacements are relatively higher in the upper floors.

4.2.2. VERIFICATION OF THE ULS – ANALYSIS THROUGH THE

PARAMETER 𝜶

NBR 6118 (2014) gives another parameter to evaluate the necessity to consider

global second order effects. The instability parameter 𝛼 classifies the structure as non-

sway if a parameter 𝛼 is smaller than a reference value 𝛼1:

𝛼 = 𝐻𝑡𝑜𝑡√∑𝑁𝑘∑𝐸𝑐𝐼𝑐

≤ 𝛼1 = {0.2 + 0.1𝑛 𝑖𝑓 𝑛 ≤ 30.60 𝑖𝑓 𝑛 ≥ 4

(16)

Where:

- 𝑛 is the number of storeys;

- 𝐻𝑡𝑜𝑡 is the total height of the structure;

- ∑𝑁𝑘 is the characteristic value of the vertical acting loads;

- ∑𝐸𝑐𝐼𝑐 is the stiffness of a column representing all the horizontal stiffness of

the building.

Although this parameter will not be used herein, it is important to register that

this verification is associated with a limitation of the total vertical load in the building

with respect to a fraction of its nominal buckling load (see Appendix A).

4.2.3. VERIFICATION OF THE SLS – COMPARATIVE DISPLACEMENT

ANALYSIS

To verify the SLS according to the Brazilian standard, it is necessary to check

the displacements of the building, considering only first order effects, using its stiffness

without reduction.

The check is done considering the two sets of requirements defined by the

standard:

𝛿

𝐻≤

1

1700≅ 0.000588 (17)

𝛿𝑖 − 𝛿𝑖−1ℎ

≤1

850≅ 0.0118

(18)

27

Where:

- 𝛿 is the maximum storey displacement;

- 𝛿𝑖 is the displacement of a given storey 𝑖;

- 𝐻 is the building height, 36 meters;

- ℎ is the difference of heights between consecutive storeys, 3 meters.

Since the model considers the pavements as rigid diaphragms, the slabs have the

same displacements in the XY plan, in relation to the Z axis. Thus, it is possible to

choose any column to perform this analysis. The analysis of the horizontal

displacements in the Y direction (axis 2) is shown in Table 4-5.

Table 4-5 – Displacements of the various pavements, both absolute and relative, in

meters

Pavement OutputCase CaseType 𝛿 𝛿𝑖 − 𝛿𝑖−1ℎ

1

ELS Combination

0.001245 0.00042

2 0.002541 0.00043

3 0.003746 0.00040

4 0.004859 0.00037

5 0.006195 0.00045

6 0.007378 0.00039

7 0.008400 0.00034

8 0.009265 0.00029

9 0.010333 0.00036

10 0.011115 0.00026

11 0.011603 0.00016

12 0.011801 0.00007

As it is shown in Table 4-5, the requirements for the SLS are fulfilled.

4.3. LOCAL SECOND ORDER EFFECTS

As long as the global second order effects were considered, it is necessary to

verify the local effects, for all the columns. All the steps of the analysis will be exposed,

but explicit calculations will be shown only for one column, presented in Appendix B.

28

The Brazilian standard has some restrictions about the slenderness of the

elements. It states that columns have a limit slenderness of 𝜆 ≤ 200, except in some

cases where the normal force is very low.

Depending on the slenderness of the elements it is possible to use some

simplified methods, as shown in the following. A general (“exact”) method is required

when the slenderness is higher than 𝜆 > 140.

NBR 6118:2014 allows for a reduction in the length of the column regarding the

theoretic centre-to-centre of slabs value. The effective length of the columns 𝑙𝑒 is

defined as a function of the free length 𝑙𝑓𝑟𝑒𝑒, distance between faces of beams

(𝑙𝑓𝑟𝑒𝑒 = 𝑙𝑐𝑜𝑙𝑢𝑚𝑛 − ℎ𝑏𝑒𝑎𝑚), height of beams ℎ𝑏𝑒𝑎𝑚 and height of the column itself

ℎ𝑐𝑜𝑙𝑢𝑚𝑛, considering these dimensions in the same plane in which the second order

displacements take place:

𝑙𝑒 = min{𝑙𝑓𝑟𝑒𝑒 + ℎ𝑐𝑜𝑙𝑢𝑚𝑛; 𝑙𝑓𝑟𝑒𝑒 + ℎ𝑏𝑒𝑎𝑚} (19)

This effective length may be different in the two directions, X and Y.

4.3.1. LOCAL GEOMETRIC IMPERFECTIONS

In framed structures, the effects from local geometrical imperfections can be

considered through a minimum first order moment acting on the columns:

{𝑀1𝑑,𝑚𝑖𝑛,𝑋 = 𝑁𝑑𝑒𝑎,𝑋 = 𝑁𝑑(0.015 + 0.03ℎ)

𝑀1𝑑,𝑚𝑖𝑛,𝑌 = 𝑁𝑑𝑒𝑎,𝑌 = 𝑁𝑑(0.015 + 0.03𝑏) (20)

4.3.2. SLENDERNESS CRITERION FOR ISOLATED COMPRESSED

MEMBERS

According to the NBR 6118:2014, local second order effects can be neglected if

the following condition is met:

𝜆 =𝑙𝑒𝑖≤ 𝜆1 =

25 + 12.5𝑒1

ℎ

𝛼𝑏 (21)

Where 35 ≤ 𝜆1 ≤ 90 and:

- 𝜆 is the element slenderness, previously described;

- 𝑒1 is the first order eccentricity, 𝑒1 = |𝑀𝐴

𝑁|;

29

- 𝛼𝑏 is a coefficient that varies according to the moments in the boundary;

0.40 ≤ 𝛼𝑏 = 0.60 + 0.40𝑀𝐵𝑀𝐴≤ 1 (22)

The values of 𝑀𝐴, 𝑀𝐵 are taken from SAP2000 analysis, being ⌊𝑀𝐴⌋ ≥ ⌊𝑀𝐵⌋.

4.3.3. EFFECTS OF CREEP

According to the Brazilian standard, it is only required to consider the effects of

creep for local second order effects when 𝜆 > 90. In this way, these effects can be

neglected for the columns analysed in this work.

4.3.4. METHOD OF THE STANDARD-COLUMN WITH APPROXIMATED

STIFFNESS

This method can be only used for columns with 𝜆 ≤ 90, rectangular constant

section and symmetrically constant reinforcing. The geometrical nonlinearity is

considered by assuming a sinusoidal deformation (standard-column approximation) and

the physical nonlinearity is taken into account by considering approximate values for

the effective stiffness of the columns.

The design moment is calculated through a multiplier of the first order moment:

𝑀𝑆𝑑,𝑡𝑜𝑡 =

𝛼𝑏𝑀1d,A

1 −𝜆2

120𝜅

𝜈

≥ 𝑀1d,A (23)

Where:

- 𝜈 is the normal dimensionless force, 𝜈 =|𝑁𝑑|

𝐴𝑐𝑓𝑐𝑑;

- 𝜅 is the dimensionless stiffness, 𝜅𝑎𝑝𝑝𝑟𝑜𝑥 = 32 ∙ 𝜈 ∙ (1 + 5𝑀𝑆𝑑,𝑡𝑜𝑡

ℎ𝑁𝑑).

It is possible to obtain directly the total design moment by finding out the root of

the following second-degree equation:

𝐴𝑀𝑆𝑑,𝑡𝑜𝑡2 + 𝐵𝑀𝑆𝑑,𝑡𝑜𝑡 + 𝐶 = 0,where {

𝐴 = 5ℎ

𝐵 = 𝑁𝑑ℎ2 −

𝑁𝑑𝑙𝑒2

320− 5ℎ𝛼𝑏𝑀1d,A

𝐶 = −𝑁𝑑ℎ2𝛼𝑏𝑀1d,A

(24)

30

4.3.5. METHOD OF THE STANDARD-COLUMN WITH APPROXIMATED

CURVATURE

Like in the previous method, this method can be only used for columns with 𝜆 ≤

90, rectangular constant section and symmetrically constant reinforcement. The

geometrical nonlinearity is also considered by assuming a sinusoidal deformation and

the physical nonlinearity is taken into account by an approximated value assumed for

the curvature in the critical section. All the necessary parameters have been defined

already defined.

𝑀𝑑,𝑡𝑜𝑡 = 𝛼𝑏𝑀1d,A +𝑁𝑑𝑙𝑒2

10

1

𝑟≥ 𝑀1d,A (25)

1

𝑟=

0.005

ℎ(𝜈 + 0.5)≤0.005

ℎ (26)

4.3.6. METHOD OF THE STANDARD-COLUMN ASSOCIATED TO THE M, N

AND CURVATURE DIAGRAMS

This method is an improvement of the two previous methods. It can be used in

columns up to 𝜆 ≤ 140, using for the curvature of the critical section the values

obtained in the M, N and 1/r diagrams, for each case.

4.3.7. GENERAL METHOD

This is the “exact” method of analysis, mandatory for 𝜆 > 140. It considers the

RC’s physical and geometrical nonlinear behaviour by considering the actual moment-

curvature relations in the sections of a discretized column, in the way that has been

previously described in item 2.2.2.

If the case is that 𝜆 > 140, in addition to considering the effects of creep, it is

also required to take into consideration an additional multiplier for the forces:

𝛾𝑛1 = 1 + [0.01(𝜆 − 140)

1.4] (27)

31

4.3.8. COMPARISON OF METHODS

This work will compare the two approximated methods for 12 columns: one

column of each group, in each reference level. The results are found in the table below

and the explicit calculations for column G1-C0 are found in Appendix B.

The results below correspond only to the situation of minimum moments, since

second order effects are not to be considered for the actual moments acting in the

columns, as the eccentricity due to the local geometric imperfections is not added to the

first order eccentricity (see also Appendix B).

Table 4-6 – Comparison of the Brazilian methods. Moments given in kN∙m.

Column Nominal Stiffness Nominal Curvature Error (%)

X Y X Y X Y

G1-C0 40.9 48.7 50.4 56.7 18.81% 14.16%

G1-C12 25.9 27.9 32.1 34.4 19.53% 18.74%

G1-C24 12.1 12.6 15.1 16.0 19.53% 21.18%

G2-C0 67.5 95.7 81.0 104.6 16.67% 8.45%

G2-C12 43.3 49.9 51.7 58.0 16.19% 14.01%

G2-C24 40.9 42.5 44.3 47.8 7.75% 11.12%

G3-C0 81.4 127.5 97.3 136.6 16.40% 6.64%

G3-C12 53.0 75.1 65.8 83.0 19.53% 9.43%

G3-C24 25.5 30.4 31.7 35.6 19.53% 14.55%

G4-C0 127.3 231.7 145.6 241.4 12.53% 4.04%

G4-C12 82.8 117.5 95.2 126.7 12.99% 7.27%

G4-C24 40.9 44.2 47.1 51.6 13.21% 14.34%

Average 16.05% 11.99%

Analysing the results of the table above, it is possible to see that the Nominal

Curvature method is the most penalising method, producing bending moments 4% to

20% higher than the ones obtained by the Nominal Stiffness method.

32

5. ANALYSIS ACCORDING TO THE EN 1992-1-1, EUROCODE 2

5.1. INITIAL CONSIDERATIONS

As it was said before, since Eurocode 2 defines some properties differently from

the NBR 6118:2014, the structural model described before must suffer some changes

for properly represent the Eurocode’s approach on second order effects. These changes

will be detailed in this subchapter.

5.1.1. MATERIAL PROPERTIES

According to the Eurocode 2, the modulus of elasticity is given by:

𝐸𝑐,𝐸𝐶2 = 22 (𝑓𝑐𝑘 + 8

10)0.30

= 22 × (35 + 8

10)0.30

≅ 34.08 𝐺𝑃𝑎 (28)

However, for this kind of analysis, Eurocode states that the design modulus of

elasticity should be used:

𝐸𝑐𝑑 =𝐸𝑐𝑚𝛾𝑐𝑒

=34.08

1.2= 28.4 𝐺𝑃𝑎 (29)

As far as the structural design is concerned, two other factors will be different:

• The partial safety coefficient of the concrete is 𝛾𝑐𝑜𝑛𝑐𝑟𝑒𝑡𝑒,𝐸𝐶2 = 1.50,

instead of 1.40 of the NBR 6118 (2014);

• The Rüsch effect coefficient will take the value of 𝛼𝐸𝐶2 = 1.00, instead

of the 0.85 from the NBR 6118 (2014).

However, neither of these two changes will affect the computational model. So,

the structural model for this analysis will have the same properties as described before,

apart from the elasticity modulus, that will assume the value above.

5.1.2. STRUCTURAL PROPERTIES

As stated, the building in analysis does not have bracing members such as walls

or cores. For this reason, the calculation of the bracing members stiffness, ∑𝐸𝑐𝑑𝐼𝑐 can

be done by two different ways, that will be compared later:

- By the summation of the stiffness of every column in each direction. The

inertia will be given by the summation of the inertia of the individual

33

columns, considering an average section width, multiplied by the number of

columns with those properties:

𝐼𝑏𝑟𝑎𝑐𝑖𝑛𝑔 =∑𝑛𝑔𝑟𝑜𝑢𝑝𝑏𝑎𝑣𝑒𝑟𝑎𝑔𝑒 × 0.20

3

12= 0.1193 𝑚4 ⇒

⇒∑𝐸𝑐𝑑𝐼𝑐 = 28 400 000 × 0.1193 = 3 387 000 𝑘𝑁 ∙ 𝑚2

(30)

- Through an analogy to a cantilever beam subjected to the loads described

above, analysed through the application FTOOL (ALIS, 2017):

𝛿 =

𝐹𝐿3

3𝐸𝐼⇒ 𝐸𝑐𝑑𝐼𝑐 =

𝐹𝐿3

3𝛿=251.25 × 363

3 × 0.01698= 230120141 𝑘𝑁𝑚2

With Ftotal=251.25 kN, the displacement is 16.98 mm, vs. 0.547 m.

(31)

Figure 5-1 – Global displacements

5.2. GLOBAL SECOND ORDER EFFECTS

In the section 5.8.3.3 of EN 1992-1-1 it is possible to find information on how to

deal with global second order effects, as well as an expression that allows the designer

to ignore those effects, only valid for structures that fulfil the five conditions below:

1. Structure reasonably symmetrical;

2. Global shear deformations are negligible;

34

3. The rotations of the basis can be dismissed;

4. The stiffness of the bracing members is reasonably constant along the height;

5. The total vertical load increases by approximately the same amount per

storey.

It is important to notice that although this building is not braced, for the sake of

this work, this formula will be used comparing the two different values of stiffness

calculated above.

𝐹𝑉,𝐸𝑑 ≤ 𝑘1𝑛𝑠

𝑛𝑠 + 1.6∙∑𝐸𝑐𝑑𝐼𝑐𝐿2

(32)

Where:

- 𝐹𝑉,𝐸𝑑 is the total vertical load, 70010 𝑘𝑁, given by the total vertical base

reaction through SAP2000;

- 𝑘1 is a coefficient that takes the value 0.31;

- 𝑛𝑠 is the number of storeys, 12;

- ∑𝐸𝑐𝑑𝐼𝑐 is the stiffness of the bracing elements;

- 𝐿 is the building height, 36 meters.

So, using the first value of stiffness, ∑𝐸𝑐𝑑𝐼𝑐 = 3386898 𝑘𝑁𝑚2:

0.31 ×12

12 + 1.6×3386898

362= 714.8 𝑘𝑁 < 𝐹𝑉,𝐸𝑑 (33)

As the condition is not met, global second order effects cannot be ignored. Using

the expressions found in the Appendix H of the same standard, it is possible to calculate

a horizontal load for which the structure should be designed.

𝐹𝐻,𝐸𝑑 =𝐹𝐻.0𝐸𝑑

1 −𝐹𝑉,𝐸𝑑

𝐹𝑉,𝐵

(34)

Where:

- 𝐹𝐻.0𝐸𝑑 is the first order horizontal resulting forces;

- 𝐹𝑉,𝐵 is the nominal buckling load, that (disregarding shear deformations), is

equal to 𝐹𝑉,𝐵𝐵, nominal buckling load, 𝐹𝑉,𝐵𝐵 = 613.9 𝑘𝑁.

It is important to notice the similarity between this factor and the 𝐹𝑚𝑎𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛

from the NBR 6118 (2014) previously analysed. So, the expression above can be seen

35

as a magnification factor for the horizontal forces due to wind, imperfections etc, on

each floor, that will be named 𝐹𝑚𝑎𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛:

𝐹𝑚𝑎𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 =1

1 −𝐹𝑉,𝐸𝑑

𝐹𝑉,𝐵

(35)

Thus,

𝐹𝑉,𝐵 ≅ 𝐹𝑉,𝐵𝐵 = 𝜉∑𝐸𝐼

𝐿2= 7.8 ∙

𝑛𝑠𝑛𝑠 + 1.6

∙1

1 + 0.7𝑘∙0.40 × ∑𝐸𝑐𝑑𝐼𝑐

𝐿2=

= 7.8 ∙12

12 + 1.6∙

1

1 + 0.7 × 0∙0.40 × 3386898

362⇔𝐹𝑉,𝐵 = 7194.4 𝑘𝑁

(36)


1 −70010

7194.4

= −0.11 (37)

As the factor is negative, it is plausible to say that the result is not valid. This

occurs since the actual stiffness of the building, corresponding to the frames composed

by the beams and columns is much higher than the one corresponding only to the

columns. So, the same steps are repeated, but now using the other previously calculated

value of stiffness, 𝐸𝑐𝑑𝐼𝑐 = 230120141 𝑘𝑁𝑚2 (Equation 31):

0.31 ×12

12 + 1.6×230120141

362= 48568.4 𝑘𝑁 < 𝐹𝑉,𝐸𝑑 (38)

Just like above, global second order effects cannot be ignored. Using the same

expressions as before, changing only the stiffness:

𝐹𝑉,𝐵𝐵 = 7.8 ∙12

12 + 1.6∙

1

1 + 0.7 × 0∙0.40 × 230120141

362= 488817 𝑘𝑁 (39)


1 −70010

488817

= 1.167 (40)

The result is now plausible and this value means that the analysis of the building

considering global second order effects must include the multiplication of horizontal

forces by the factor of 1.167.

This shows that stiffness is one of the most important parameters and its correct

evaluation is fundamental for an accurate analysis.

36


Just like what was described in item 4.3, this item will deal with local second

order effects, following EN 1992-1-1, aided by WESTERBERG (2004). Similar to the

Brazilian standard, Eurocode 2 gives the designer three methods of analysis to choose

from, which will be detailed in the following items. However, the complete calculations

will be shown only in Appendix C.

Unlike the NBR 6118:2014, Eurocode 2 does not put any slenderness restriction

in the method of analysis, and creep must be considered in all methods, unless some

conditions are met.

Another factor that changes with this standard is the definition of the effective

length. For isolated members, Eurocode 2 defines:

Figure 5-2 – Effective lengths for isolated members (COMITÉ EUROPÉEN DE


When considering the structure, the effective length for unbraced columns is

given by:

𝑙0 = 𝑙.max {√1 + 10𝑘1𝑘2𝑘1 + 𝑘2

; (1 +𝑘1

1 + 𝑘1) (1 +

𝑘21 + 𝑘2

)} (41)

Where 𝑘𝑖 is the relative flexibility of rotational restraints at end 𝑖,

𝑘𝑖 =(∑𝐸𝐼

𝐿)𝑐𝑜𝑙𝑢𝑚𝑛𝑠

(4∑𝐸𝐼

𝐿)𝑏𝑒𝑎𝑚𝑠

37

5.3.1. LOCAL GEOMETRIC IMPERFECTIONS

For isolated unbraced members, the effect of geometric imperfections may be

considered as a transverse force, 𝐻𝑖, or as an additional eccentricity, 𝑒𝑖. To be consistent

to the Brazilian approach, only the latter will be used:

𝑒𝑖 =𝜃𝑖𝑙0

2, where 𝜃𝑖 = 𝜃0𝛼ℎ𝛼𝑚 (42)

- 𝜃0 is the basic value, recommended to be 𝜃0 =1

200;

- 𝛼ℎ is the reduction value for height, 𝛼ℎ =2

√𝑙;

- 𝛼𝑚 is the reduction value for number of members, 𝛼𝑚 = √0.5 (1 +1

𝑚);

Considering 𝑙 = 3 and 𝑚 = 1 (isolated element);

𝜃𝑖 = 𝜃0𝛼ℎ𝛼𝑚 =1

200∙2

√3∙ √0.5 ∙ (1 +

1

1) ⇒ 𝑒𝑖 =

𝑙0

200√3 (43)

Just like in item 4.2, global geometric imperfections are not to be considered.

5.3.2. SLENDERNESS CRITERION FOR ISOLATED COMPRESSED

MEMBERS

The local second order effects can be dismissed if the following condition is met:

𝜆 =𝑙0𝑖≤ 𝜆𝑙𝑖𝑚 =

20 ∙ 𝐴 ∙ 𝐵 ∙ 𝐶

√𝑛 (44)

Where:

- 𝑖 is the radius of gyration of the uncracked concrete section;

- 𝐴 =1

1+0.2𝜑𝑒𝑓 but if 𝜑𝑒𝑓 is not known ⇒𝐴 = 0.7;

- 𝐵 = √1 + 2𝜔 but if 𝜔 is not known ⇒𝐵 = 1.1;

- 𝐶 = 1.7 − 𝑟𝑚 but if 𝑟𝑚 is not known ⇒𝐶 = 0.7;

- 𝑛 is the relative normal force, 𝑛 =𝑁𝐸𝑑

𝐴𝑐𝑓𝑐𝑑;

- 𝜑𝑒𝑓 is the effective creep ratio;

- 𝜔 is the mechanical reinforcement ratio, 𝜔 =𝐴𝑠𝑓𝑦𝑑

𝐴𝑐𝑓𝑐𝑑= 𝜌

𝑓𝑦𝑑

𝑓𝑐𝑑;

38

- 𝐴𝑠 is the total area of longitudinal reinforcement;

- 𝑟𝑚 is the ratio of first order end moments, 𝑟𝑚 = 𝑀01

𝑀02, and |𝑀02| ≥ |𝑀01|;

Using the default values of A, B and C, the limit slenderness is:

𝜆𝑙𝑖𝑚 =20 × 0.7 × 1.1 × 0.7

√𝑛=10.78

√𝑛 (45)

5.3.3. EFFECTS OF CREEP

Eurocode 2 allows to ignore the effects of creep if three conditions are met:

1. 𝜑(∞,𝑡0) ≤ 2

2. 𝜆 ≤ 75

3. 𝑀0𝐸𝑑

𝑁𝐸𝑑≥ ℎ

To make a better comparison between the codes, the effects of creep will be

neglected for this analysis, making 𝜑𝑒𝑓 = 0, even though the conditions above are not

going to be verified.

5.3.4. METHOD BASED ON THE NOMINAL STIFFNESS

This first method considers the effects of cracking, material nonlinearity and

creep to modify the RC’s flexural stiffness and magnify the first order moment,

calculating a final moment for which the element must be designed.

The flexural stiffness is given by:

𝐸𝐼 = 𝐾𝑐𝐸𝑐𝑑𝐼𝑐 + 𝐾𝑠𝐸𝑠𝐼𝑠 (46)

Considering 𝜌 =𝐴𝑠

𝐴𝑐> 0.01:

- 𝐾𝑐 is a factor for effects of cracking, creep and similar effects,

𝐾𝑐 =0.30

1+0.50𝜑𝑒𝑓;

- 𝐼𝑐 is the moment of inertia of concrete cross section;

- 𝐾𝑠 is a factor for contribution of reinforcement, 𝐾𝑠 = 0;

- 𝐸𝑠 is the design value of the modulus of elasticity of reinforcement,

𝐸𝑠 = 200 𝐺𝑃𝑎;

39

- 𝐼𝑠 is the moment of inertia of reinforcement, about the centre of area of the

concrete.

The simplified values of 𝐾𝑐, 𝐾𝑠 shown above are adequate as a first

approximation. However, for this work, they will be used as the final value, for simple

comparison purposes. Note that since 𝐾𝑠 = 0, the inertia of the reinforcement needs not

be calculated.

With the nominal stiffness calculated, it is now possible to estimate the final

moment for which the column should be designed:

𝑀𝐸𝑑 = 𝑀0𝐸𝑑 [1 +𝛽

(𝑁𝐵𝑁𝐸𝑑⁄ ) − 1

] (47)

Where,

- 𝑀0𝐸𝑑 is the first order moment at the base of the column;

- 𝛽 is a factor that depends on the distribution of the moments, 𝛽 =𝜋2

𝑐0;

- 𝑁𝐸𝑑 is the design value of the axial load;

- 𝑁𝐵 is the buckling load, based on normal stiffness, 𝑁𝐵 =𝜋2

𝑙02 𝐸𝐼

Adopting 𝛽 = 1, the expression above can be simplified to:

𝑀𝐸𝑑 =𝑀0𝐸𝑑

1 − (𝑁𝐸𝑑

𝑁𝐵⁄ )

(48)

It is important to remark that EN 1992-1-1 states that because end moments

differ, it shall be considered that 𝑀0𝑒 = 0.6𝑀02 + 0.4𝑀01 ≥ 0.4𝑀02, |𝑀02| ≥ |𝑀01|.

However, this consideration is only applied to non-sway structures, where the maximum

bending moment is found on the middle, and not on the extremes (CÂMARA, 2015).

5.3.5. METHOD BASED ON THE NOMINAL CURVATURE

This first method uses the effects of cracking, material nonlinearity and creep to

modify the RC’s flexural stiffness and magnify the first order moment, calculating a

final moment for which the element must be designed, as described in item 5.3.4.

40

Instead of magnifying the first order moment, this second method adds to the

first order moment, already known, a second order moment considering the column

deflection, based on the effective length and an estimated maximum curvature:

𝑀𝐸𝑑 = 𝑀0𝐸𝑑 +𝑀2 (49)

- 𝑀0𝐸𝑑 is the first order moment, the same as in 5.3.4;

- 𝑀2 is the nominal second order moment, 𝑀2 = 𝑁𝐸𝑑𝑒2;

- 𝑒2 is the deflection, 𝑒2 = (1

𝑟)𝑙02

𝑐, 𝑐 = 𝜋2 ≅ 10 for constant sections;

- 1

𝑟 is the curvature,

1

𝑟= 𝐾𝑟𝐾𝜑 (

1

𝑟0), where

1

𝑟0=

𝜀𝑦𝑑

0.45𝑑=

𝑓𝑦𝑑

𝐸𝑠

0.45𝑑;

- 𝐾𝑟 is a correction factor depending on the axial load, 𝐾𝑟 =𝑛𝑢−𝑛

𝑛𝑢−𝑛𝑏𝑎𝑙≤ 1,

where 𝑛𝑢 = 1 + 𝜔 ≅ 1.5 (adopted), and 𝑛𝑏𝑎𝑙 = 0.4;

- 𝐾𝜑 is a factor taking account creep, 𝐾𝜑 = 1 + 𝛽𝜑𝑒𝑓 ≥ 1𝜑𝑒𝑓=0⇒ 𝐾𝜑 = 1.

Note that 𝜔 = 0.5 is equivalent to 𝜌 ≅ 2.68%. For a more precise result, 𝑛𝑢

should be calculated by an iterative process.

5.3.6. GENERAL METHOD

Just like is defined in NBR 6118:2014, this method is the “exact” one, where a

computational nonlinear analysis shall be performed.

5.3.7. METHOD COMPARISON

As it was done in 4.3.8, the two approximated methods will be compared for 12

different columns, being the calculations for column G1-C0 shown in Appendix C.

Table 5-1 – Comparison of the methods from Eurocode 2. Moments given in kN∙m

Column Nominal Stiffness Nominal Curvature Error (%)

X Y X Y X Y

G1-C0 115.3 84.1 99.7 101.3 13.52% 16.94%

G1-C12 57.4 68.9 80.2 135.2 33.13% 49.02%

G1-C24 37.6 39.3 50.3 65.4 32.40% 39.89%

G2-C0 94.3 105.9 139.0 140.1 42.16% 24.46%

G2-C12 57.7 94.0 87.2 206.1 31.34% 54.38%

G2-C24 26.9 30.7 43.6 65.4 54.31% 53.12%

41

G3-C0 129.4 138.0 186.7 170.9 41.52% 17.61%

G3-C12 123.4 203.8 171.6 388.6 23.62% 47.56%

G3-C24 81.9 97.5 108.2 166.1 26.91% 41.30%

G4-C0 149.9 169.9 215.9 204.3 38.89% 15.97%

G4-C12 93.0 262.0 141.9 474.2 18.69% 44.75%

G4-C24 39.9 53.3 63.5 105.9 44.26% 49.68%

Average 33.40% 37.89%

Just like in 4.3.8, the Nominal Curvature method is the most penalising method.

However, here the bending moments are more than 30% higher, in average, than the

ones obtained by the Nominal Stiffness method.

This huge difference can probably be explained by the fact that in this work only

the basic approximations were used for calculating the moment magnifying factor for

the Nominal Stiffness method. If other iterations were used, perhaps the results would

have been better.

42

6. ANALYSIS ACCORDING TO fib MODEL CODE 2010

Although fib Model Code does not define criteria for global second order effects,

it deals with the local effects using an interesting approach, giving the designer the

possibility of choosing between four levels of approximation, depending on the wanted

accuracy. This work will only show the results for the first two levels of approximation.

There is a lot of similarities between fib and Eurocode 2. The effective length is

calculated the same way, and fib uses the same coefficients found in the method based

on the nominal curvature from Eurocode 2.


2010 Model Code states that the dimensioning value of the bending moment is

given by:

𝑀𝑑 = −𝑁𝑑𝑒𝑑 (50)

Where,

- 𝑒𝑑 is the maximum eccentricity, 𝑒𝑑 = 𝑒0𝑑 + 𝑒1𝑑 + 𝑒2𝑑;

- 𝑒0𝑑 is the eccentricity due to imperfections, 𝑒0𝑑 = max {𝛼𝑖𝑙0

2;𝑑

30};

- 𝑒1𝑑 is the first order eccentricity, 𝑒1𝑑 =𝑀1𝑑

−𝑁𝑑;

- 𝑒2𝑑 is the eccentricity due to the deformation of the compression member,

𝑒2𝑑 = 𝑘𝑑𝑙02

𝑐0;

- 𝛼𝑖 is a coefficient that can be estimated as 1

200≥ 𝛼𝑖 =

0.01

√𝑙0≥

1

300;

- 𝑘𝑑 is the maximum design curvature, 𝑘𝑑 =𝜀𝑠𝑑−𝜀𝑦𝑑

′

𝑑−2𝑐;

- 𝑐0 is an integration factor accounting for the curvature distribution along the

member;

- 𝑑 is the height of the column section;

- 𝑐 is the concrete cover, assumed to be herein 2.50 cm.

Because the different approximations change only the eccentricity 𝑒2𝑑, the

remaining calculations are the same for every level of accuracy.

43

6.1.1. LEVEL I OF APPROXIMATION

This first approximation assumes that:

{

𝑐0 = 𝜋

2 ≅ 10

휀𝑠𝑑 =𝑓𝑦𝑑

𝐸𝑠

휀′𝑠𝑑 = −𝑓𝑦𝑑

𝐸𝑠

⇒ 𝑒2𝑑 ≅𝑓𝑦𝑑

𝐸𝑠

𝑙02

5(𝑑 − 2𝑐) (51)

6.1.2. LEVEL II OF APPROXIMATION

The value of the maximum design curvature can be improved by taking into

account the normal force. The coefficients used are the same as in the nominal curvature

method of Eurocode:

𝑘𝑑 = (𝑛𝑢 − 𝑛

𝑛𝑢 − 𝑛𝑏𝑎𝑙)휀𝑦𝑑

0.45𝑑 (52)

If creep and shrinkage are to be considered, an additional curvature should be

taken into account: 𝑘𝑑,∞ =|𝜀c∞|

𝑑. However, as it has been done in this work, these

effects will be neglected.

6.1.3. LEVEL III OF APPROXIMATION

This third approximation combines uses the 𝑘𝑑 from the Level II, at the same

time refining the value of the integration factor as a summation of various integration

factors as a function of the load type and boundary conditions:

𝑐0 = 𝜋2𝑁

𝑁𝑐𝑟+∑ 𝑀𝑖𝑛𝑖=1

∑𝑀𝑖

𝑐𝑖

𝑛𝑖=1

(1 −𝑁

𝑁𝑐𝑟) (53)

The values of 𝑐𝑖 can be found in Figure 6-1:

44

Figure 6-1 – Values of integration factors, 𝑐𝑖, as a function of the load type and the

boundary conditions (FÉDÉRATION INTERNATIONALE DU BÉTON, 2010)

6.1.4. LEVEL IV OF APPROXIMATION

This final level of approximation will be the same as the general methods

described in NBR 6118 (2014) and EN 1992-1-1.

6.1.5. METHOD COMPARISON

Same as before, the comparison of the bending moments from the first two

approximated methods is shown in the table below.

45

Table 6-1 – Comparison of the methods from MC 2010. Moments in kN∙m

Column Level I Level II Error (%)

X Y X Y X Y

G1-C0 102.3 108.4 90.6 104.1 10.82% 4.13%

G1-C12 80.1 127.4 73.5 120.1 5.23% 6.02%

G1-C24 48.0 59.8 46.8 60.0 2.04% 0.26%

G2-C0 147.7 170.3 125.4 162.6 13.11% 4.72%

G2-C12 93.8 210.7 78.2 177.6 7.38% 18.61%

G2-C24 44.1 64.2 38.9 57.7 8.05% 11.22%

G3-C0 200.1 221.8 168.7 213.9 14.16% 3.70%

G3-C12 172.6 363.0 157.1 350.8 4.27% 3.48%

G3-C24 104.5 150.2 100.7 153.5 2.56% 2.18%

G4-C0 252.7 324.4 193.5 306.6 18.24% 5.79%

G4-C12 166.3 533.6 126.0 407.3 7.56% 31.01%

G4-C24 74.4 119.7 56.1 91.9 15.33% 30.19%

Average 9.06% 10.11%

As expected, the Level II method is less conservative than the Level I, returning

values 10% smaller, in average. It is also plausible to assume that a Level III

approximation would return even lower values for bending moments, closer to the real

values, but still in the safe side.

46

7. FINAL CONSIDERATIONS

7.1. CONCLUSIONS

After this comparative analysis, it is possible to conclude that the three codes

herein analysed have many processes in common, and some particularities that must be

considered when designing.

Out of the three codes, fib Model Code is the only one that does not considers the

global second order effects. Regarding this same topic, while NBR 6118 demands

verification for both limit states, EN 1992-1-1 only cares about the ULS.

For the global ULS analysis, NBR 6118 allows the designer to choose between a

P-Delta computational analysis and a moment multiplying parameter 𝛾𝑍, as long as

some conditions are met. However, Eurocode 2 does not consider the P-Delta method,

only considering the effect of a horizontal force multiplier.

The Brazilian parameter 𝛾𝑍 = 1.228 turned out to be a good approximation for

the global second order effects, as the consequent results present no significant

differences from the P-Delta analysis. The European multiplier was similar, with a value

about 5% lower, 𝐹𝑚𝑎𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 = 1.167. This small difference may be due to different

values of elasticity modulus, or to the difficulty of properly defining the stiffness of the

building.

After this analysis, it became evident that concrete stiffness is a very important

factor, that must be properly defined. Also, because the building did not have any rigid

cores or walls, allied with the fact that all columns were oriented in the same direction,

which made it unlike the most common buildings found in Europe, some important

considerations were not clearly found in Eurocode 2.

As far as local second order effects are concerned, it was shown that NBR 6118

considers geometric imperfections very differently from the two other codes. It

evaluates the effects of these imperfections through a separate analysis, instead of

adding the moments due to the imperfections to the first order moments.

Also, the effective length of the columns is a very important parameter, since it

is important in the definition of other fundamental factors. Although this is considered

in the same way in EN 1992-1-1 and fib Model Code, NBR 6118 defines this

differently, taking into consideration only the dimensions of the adjacent beams.

47

A similarity on how to deal with end moments for defining the first order

moment at the centre of the columns was found between NBR 6118 and EN 1992-1-1.

The latter, in its item 5.8.8.2(2) presents the parameter 𝑀0𝑒 while NBR 6118 in its item

15.8.2 presents 𝛼𝑏. However, while NBR uses this criterion independently from the

boundary conditions of the column, Eurocode 2 only applies this approximation when

the column in analysis is non-sway.

The results of the local analysis presented previously in this work can be

summarized in Table 7-1 and Table 7-2:


X direction. Moments in kN∙m

Column NBR 6118 (2014) EN 1992-1-1 fib MC 2010

Stiffness Curvature Stiffness Curvature Level I Level II

G1-C0 40.9 50.4 115.3 99.7 102.3 90.6

G1-C12 25.9 32.1 57.4 80.2 80.1 73.5

G1-C24 12.1 15.1 37.6 50.3 48.0 46.8

G2-C0 67.5 81.0 94.3 139.0 147.7 125.4

G2-C12 43.3 51.7 57.7 87.2 93.8 78.2

G2-C24 40.9 44.3 26.9 43.6 44.1 38.9

G3-C0 81.4 97.3 129.4 186.7 200.1 168.7

G3-C12 53.0 65.8 123.4 171.6 172.6 157.1

G3-C24 25.5 31.7 81.9 108.2 104.5 100.7

G4-C0 127.3 145.6 149.9 215.9 252.7 193.5

G4-C12 82.8 95.2 93.0 141.9 166.3 126.0

G4-C24 40.9 47.1 39.9 63.5 74.4 56.1

48


Y direction. Moments in kN∙m

Column NBR 6118 (2014) EN 1992-1-1 fib MC 2010

Stiffness Curvature Stiffness Curvature Level I Level II

G1-C0 48.7 56.7 84.1 101.3 108.4 104.1

G1-C12 27.9 34.4 68.9 135.2 127.4 120.1

G1-C24 12.6 16.0 39.3 65.4 59.8 60.0

G2-C0 95.7 104.6 105.9 140.1 170.3 162.6

G2-C12 49.9 58.0 94.0 206.1 210.7 177.6

G2-C24 42.5 47.8 30.7 65.4 64.2 57.7

G3-C0 127.5 136.6 138.0 170.9 221.8 213.9

G3-C12 75.1 83.0 203.8 388.6 363.0 350.8

G3-C24 30.4 35.6 97.5 166.1 150.2 153.5

G4-C0 231.7 241.4 169.9 204.3 324.4 306.6

G4-C12 117.5 126.7 262.0 474.2 533.6 407.3

G4-C24 44.2 51.6 53.3 105.9 119.7 91.9

It can be seen that for both NBR 6118 and Eurocode 2 standards, the Nominal

Curvature method returns more conservative values than the Nominal Stiffness method,

i.e., the Nominal Curvature methods will result in higher safety factors, when compared

with more refined methods.

Brazilian and European codes present very different results. This discrepancy

occurs because NBR 6118 does not add first order moments with the ones due to

geometric imperfections and also due to the different criteria for defining the column

effective length. Considering these two parameters, the bending moment for the column

G1-C0 analysed in Appendix B would be (Curvature Method):

𝑀𝑑,𝑡𝑜𝑡,𝑋 = (0.4 × 55.21 + 30.16) + 1436 ×

3.162

10× 0.0245

= 87.37 𝑘𝑁 ∙ 𝑚

(54)

The value presented above would be closer to the values of bending moments

found with the other methods.

49

As said in item 5.3.7, the difference between the two European methods may be

due to the basic approximations made in the Nominal Stiffness methods. It can also be

seen that the Level II fib approach is very similar to the curvature method of EN 1992-

1-1, and that bending moments present lower, more economic values, as long as

hypotheses are refined from Level I to Level II

In terms of reinforcement, for the NBR 6118 it was chosen 8Φ12.5mm

(𝐴𝑆 = 9.82 𝑐𝑚2) while both fib and Eurocode 2 reinforcements had to be significantly

increased, for 6Φ25 mm (𝐴𝑆 = 29.45 𝑐𝑚2). In all three cases, the biaxial bending is

decisive for designing the reinforcement design.

Finally, it is possible to see that the choice of 𝜔 = 0.5 was a good

approximation, as 𝜌 =0.002945

0.11= 2.68%.

7.2. SUGGESTIONS FOR FUTURE WORKS

The first suggestion is to check if the results obtained with the approximated

methods are close to the exact ones. This confirmation can be done using the same

software, through a physically nonlinear analysis of the building.

Other suggestion is to analyse the effects of creep and shrinkage and check if

they can produce any significant changes in the results. A comparison between these

three codes and the American Standard ACI 318M-14 is also suggested.

Finally, because stiffness is a very important factor, a parametric analysis of

similar buildings with different stiffness can be an interesting research. Adding cores,

rigid walls or other bracing elements will turn the building more “European”, fact that

may change some the conclusions herein stated.

50

8. REFERENCES

ALIS. FTOOL. . in: https://www.alis-sol.com.br/Ftool/ Rio de Janeiro, Brazil, 2017

ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 6118 - Projeto de

estruturas de concreto - Procedimento. Rio de Janeiro, Brazil, 2014.

CÂMARA, J, N. Estruturas de Betão I - Folhas de apoio às aulas. Lisbon, Portugal,

2015.

COMITÉ EUROPÉEN DE NORMALISATION. European Standard EN 1990:2002

Eurocode - Basis of structural design. Brussels, Belgium, 2002.

COMITÉ EUROPÉEN DE NORMALISATION. European Standard EN 1992-1-

1:2004 Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules

for buildings. Brussels, Belgium, 2004.

COMPUTERS AND STRUCTURES, Inc. SAP2000 v19. Berkeley, California, USA,

2016

GOMES, B. C.. Estudo dos fatores de redução de rigidez da NBR 6118:2014 para

análise aproximada da não linearidade física. Projeto de Graduação, Escola

Politécnica, UFRJ, 2017.

INTERNATIONAL FEDERATION FOR STRUCTURAL CONCRETE (fib) – Model

Code for Concrete Structures 2010. Lausanne, Switzerland 2013.

LONGO, H. I.. Efeitos de Segunda Ordem em Estruturas de Edificações. EP/UFRJ, Rio

de Janeiro, Brazil, 2017.

NARAYANAN, R. S.; BEEBY, A. W. Designers Guide to EN 1992-1-1 and EN 1992-

1-2. London, England, 2005.

SANTOS, S. H. C.. Apostila de Concreto Armado III. EP/ UFRJ, Rio de Janeiro, Brazil,

2017.

TIMOSHENKO, S. P., GERE, J. Theory of elastic stability. Stanford, California, USA,

1963.

WESTERBERG, B.. Second order effects in slender concrete structures - Background

rules in EC2. Stockholm, Sweden, 2004

51

APPENDIX A – RELATIONSHIP BETWEEN 𝜶 AND NOMINAL

BUCKLING LOAD

In a structure of a building, the vertical loads do not act like a concentrated force

at the top but like an evenly distributed load. In this case, it is demonstrated

(TIMOSHENKO; GERE, 1963) that the critical value of the buckling load is given by:

(𝑞𝑙)𝑐𝑟 =7.837𝐸𝐼

𝐻2⇒𝐸𝐼 =

(𝑞𝑙)𝑐𝑟 ∙ 𝐻2

7.837

In the item 4.2.2, it was introduced an instability parameter α used in the

Brazilian standard. During the writing of this work it was noted that there was a

relationship between this parameter and the buckling load.

By putting 𝛼 = 0.50 it is possible to show that

𝛼 = 𝐻𝑡𝑜𝑡√𝑁𝑘𝐸𝑐𝑠𝐼𝑐

⇒ 0.50 = 𝐻√𝑁𝑘𝐸𝐼⇔ 0.25 = 𝐻2

𝑁𝑘𝐸𝐼

Replacing that in the equation above:

0.25 =𝐻2𝑁𝑘(𝑞𝑙)𝑐𝑟∙𝐻

2

7.837

⇔ (𝑞𝑙)𝑐𝑟 = 31.35𝑁𝑘⇔𝑁𝑘 = 0.032(𝑞𝑙)𝑐𝑟

So, by putting 𝛼 = 0.50 the total vertical load will be approximately 3% of the

resulting distributed buckling load.

It is very important to notice the difference between this case and the one of a

cantilever column with only a concentrated top load. The critical load is three times

lower when compared with the distributed load:

𝑃𝑓 =𝜋2𝐸𝐼

(2𝐻)2⇒ 0.25 =

𝜋2

4𝑃𝑓𝑁𝑘⇔𝑃𝑓 = 𝜋

2𝑁𝑘⇔𝑁𝑘 ≅ 0.10 ∙ 𝑃𝑓

52

APPENDIX B – EXAMPLE OF LOCAL SECOND ORDER

EFFECTS ANALYSIS FOR NBR 6118:2014

As stated before, this and the two following appendixes will show the explicit

calculations for local second order effects, for column G1-C0, with 𝑏 = 0.55 𝑚 and

ℎ = 0.20 𝑚, making 𝐴𝑐 = 0.11 𝑚2.

According to the Brazilian standard, it is initially necessary to calculate the

effective length and the slenderness coefficient in each direction:

{𝑙𝑓𝑟𝑒𝑒,𝑋 = 3 − 0.80 = 2.20 𝑚

𝑙𝑓𝑟𝑒𝑒,𝑌 = 3 − 0.45 = 2.55 𝑚⇒ {𝑙𝑒,𝑋 = min{2.20 + 0.80; 2.20 + 0.20} = 2.40 𝑚

𝑙𝑒,𝑌 = min{2.55 + 0.45; 2.55 + 0.55} = 3.00 𝑚

{

𝜆𝑋 = 2.40

√12

0.20= 41.57 > 35

𝜆𝑌 = 3.00√12

0.55= 18.90 < 35

NBR 6118 (2014) considers 35 to be the upper limit for very short columns, so

there is no need to check for second order effects due to the wind forces in the Y

direction, only in the X direction.

Through a SAP2000 computational analysis it was possible to obtain the forces

in column G1-C0, which gives the following relations:

{

𝑁𝑑 = −1436 𝑘𝑁𝑀𝐵 = 54.71 𝑘𝑁 ∙ 𝑚𝑀𝐴 = −55.21 𝑘𝑁 ∙ 𝑚𝑀𝑦 = 15.87 𝑘𝑁 ∙ 𝑚

(𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛)

⇒

{

𝑒1 =

𝑀𝐴𝑁𝑑=55.21

1436= 0.0384 𝑚

𝑀𝐵𝑀𝐴=54.71

−55.21= −0.99

𝛼𝑏 = 0.60 + 0.40𝑀𝐵𝑀𝐴= 0.204

𝛼𝑏≥0.40⇒ 𝛼𝑏 = 0.40

With the values above it is possible to calculate the limit slenderness in the X

direction:

𝜆1,𝑋 =25 + 12.5

0.038

0.20

0.40= 68.45 > 𝜆𝑋 = 41.57

So, as 𝜆1,𝑋 > 𝜆𝑋, local second order effects due to the wind forces may be

disregarded in both directions. The check for biaxial bending is necessary, as shown in

the end of this item.

53

It is necessary to evaluate the local geometrical imperfections, through the

minimum bending moment:

{𝑀1𝑑,𝑚𝑖𝑛,𝑋 = 1436 × (0.015 + 0.03 × 0.20) = 30.16 𝑘𝑁 ∙ 𝑚

𝑀1𝑑,𝑚𝑖𝑛,𝑌 = 1436 × (0.015 + 0.03 × 0.55) = 45.24 𝑘𝑁 ∙ 𝑚

This result gives the eccentricity due to imperfections:

{𝑒𝑖𝑚𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑖𝑜𝑛𝑠,𝑋 =

30.16

1436= 0.021 𝑚

𝑒𝑖𝑚𝑝𝑒𝑟𝑓𝑒𝑐𝑡𝑖𝑜𝑛𝑠,𝑌 =45.24

1436= 0.032 𝑚

The limit slenderness for the minimum bending moments is:

{

𝜆1,𝑋 =

25 +12.5

0.20∙ 0.021

1.00= 26.31 ⇒ 𝜆1,𝑋 = 35

𝜆1,𝑌 =25 +

12.5

0.55∙ 0.032

1.00= 25.72 ⇒ 𝜆1,𝑌 = 35

For the minimum moment, the coefficient 𝛼𝑏 is always equal to the unity, 𝛼𝑏 =

1.

As the limit slenderness is very low in both directions, local second order effects

must be considered in X and in Y.

The design moment according to the method of the standard-column with

approximated stiffness can be obtained as the root of a second-degree equation, with the

coefficients found below:

𝑋 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛: {

𝐴 = 5 × 0.20 = 1.00

𝐵 = 1436 × 0.202 −1436 × 2.402

320− 5 × 0.20 × 1.00 × 30.16 = 1.44

𝐶 = −1436 × 0.202 × 1.00 × 30.16 = −1732.3

𝑌 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛:{

𝐴 = 5 × 0.55 = 2.75

𝐵 = 1436 × 0.552 −1436 × 3.002

320− 5 × 0.55 × 1.00 × 45.24 = 269.62

𝐶 = −1436 × 0.552 × 1.00 × 45.24 = −19650.8

So, the design bending moments according to this method are:

𝐴𝑀𝑆𝑑,𝑡𝑜𝑡2 + 𝐵𝑀𝑆𝑑,𝑡𝑜𝑡 + 𝐶 = 0⇔ {

𝑀𝑆𝑑,𝑡𝑜𝑡,𝑋 = 40.91 𝑘𝑁 ∙ 𝑚

𝑀𝑆𝑑,𝑡𝑜𝑡,𝑌 = 48.70 𝑘𝑁 ∙ 𝑚

54

Now, analysing by the method of the standard-column with approximated

curvature, the normal dimensionless force is:

𝜐 =1436

0.11 ×35000

1.40

= 0.522

The curvature in the critical section will be:

{

(1

𝑟)𝑋=

0.005

0.20 × (0.522 + 0.5)= 0.0245 ≤

0.005

0.20= 0.025 ⇒ (

1

𝑟)𝑋= 0.02454

(1

𝑟)𝑌=

0.005

0.55 × (0.522 + 0.5)= 0.0089 ≤

0.005

0.55= 0.009 ⇒ (

1

𝑟)𝑌= 0.0089

Thus, the design moment according to this method will be:

{

𝑀𝑑,𝑡𝑜𝑡,𝑋 = 1.00 × 30.16 + 1436 ×2.402

10× 0.0245 = 50.39 𝑘𝑁 ∙ 𝑚

𝑀𝑑,𝑡𝑜𝑡,𝑌 = 1.00 × 45.24 + 1436 ×3.002

10× 0.0089 = 56.73 𝑘𝑁 ∙ 𝑚

With the final forces defined it is possible to design the column reinforcement.

This will be done by an Excel spreadsheet (SANTOS, 2016), for the most unfavourable

values of the bending moment. The interaction diagrams are given below:

Figure B-1 Concrete interaction diagram for X direction, NBR 6118:2014

0

10

20

30

40

50

60

70

80

-3000 -2500 -2000 -1500 -1000 -500 0 500

Md

(kN

.m)

Nd (kN)

Interaction Curve (Nd x Md)

As=0

As,given

Nd,Md

55

Figure B-2 – Concrete interaction diagram for Y direction, NBR 6118:2014

Although the minimum reinforcement could be sufficient for each direction

separately, biaxial bending requires a bigger area of reinforcement. The chosen

reinforcement was 8Φ12.5 mm (𝐴𝑆 = 9.82 𝑐𝑚2). The section detailing is shown below:

Figure B-3 – Cross section detail for the NBR 6118 (2014)

With the chosen reinforcement, the resistant moments will be:

{𝑀𝑅𝑑,𝑋 ≅ 70 𝑘𝑁 ∙ 𝑚

𝑀𝑅𝑑,𝑌 ≅ 193 𝑘𝑁 ∙ 𝑚

Verifying the biaxial bending for the results of SAP 2000 (applied moments),

since for minimum moments due to geometrical imperfections this check is not

necessary according to NBR6118:

(55.21

70)1.2

+ (15.87

193)1.2

= 0.80 ≤ 1 (OK)

0

50

100

150

200

250

-3000 -2500 -2000 -1500 -1000 -500 0 500

Md

(kN

.m)

Nd (kN)


As=0

As,given

Nd,Md

56

APPENDIX C – EXAMPLE OF LOCAL SECOND ORDER

EFFECTS ANALYSIS FOR EN 1992-1-1

This analysis according to EN 1992-1-1 will use the same column and

considerations considered in Appendix B. However, the buckling length definition is

different in the European standard.

For the X direction:

𝑘1,𝑋 = 0 → rigid rotational restraint

𝑘2,𝑋 =(∑

𝐸𝐼


(4∑𝐸𝐼


=

0.55×0.203

12

1

3× 2

0.15×0.803

12

4

6

= 0.057

𝑙0,𝑋 = 𝑙 × max {√1 + 10 × 0; (1 +0

1 + 0) (1 +

0.057

1 + 0.057)} = 3 × 1.05 = 3.16

𝜆𝑋 = 𝑙0,𝑋√12

ℎ= 3.16 ×

√12

0.20= 54.78

For the Y direction:

𝑘1,𝑌 = 0→ rigid rotational restraint

𝑘2,𝑌 =(∑

𝐸𝐼


(4∑𝐸𝐼


=

0.20×0.553

12

1

3× 2

0.15×0.453

12

4

6

= 2.43

𝑙0,𝑌 = 𝑙 × max {√1 + 10 × 0; (1 +1

1 + 0)(1 +

2.43

1 + 2.43)} = 3 × 1.71 = 5.13

𝜆𝑌 = 𝑙0,𝑌√12

ℎ= 5.13 ×

√12

0.55= 32.29

Through a SAP2000 computational analysis the forces in the column were

obtained:

{

𝑁𝑑 = −1424 𝑘𝑁𝑀01 = 49.72 𝑘𝑁 ∙ 𝑚𝑀02 = −50.85 𝑘𝑁 ∙ 𝑚

The first order eccentricity is calculated as:

57

𝑒1 =𝑀01𝑁=49.72

1424= 0.0349 𝑚

The relative normal force is the same in both directions:

𝑛 =𝑁𝐸𝑑𝐴𝑐𝑓𝑐𝑑

=1424

0.11 ×35000

1.50

= 0.555

Considering the default values of the coefficients A, B and C, the limit

slenderness will be given by:

𝜆𝑙𝑖𝑚 =10.78

√𝑛=10.78

√0.555= 14.5

So, as 𝜆𝑋, 𝜆𝑌 > 𝜆𝑙𝑖𝑚, local second order effects must be considered in both

directions.

As defined before, the eccentricity due to local geometrical imperfections are:

{

𝑒𝑖,𝑋 =

𝑙0,𝑋

200√3=3.16

200√3= 0.0091 𝑚

𝑒𝑖,𝑌 =𝑙0,𝑌

200√3=5.13

200√3= 0.0148 𝑚

So, the moment due to the local geometrical imperfections will be:

{𝑀𝑖,𝑋 = 𝑁𝐸𝑑 × 𝑒𝑖,𝑋 = 1424 × 0.0091 = 13.0 𝑘𝑁 ∙ 𝑚

𝑀𝑖,𝑌 = 𝑁𝐸𝑑 × 𝑒𝑖,𝑌 = 1424 × 0.0148 = 21.1 𝑘𝑁 ∙ 𝑚

Thus, the final first order moment will be:

{𝑀0,𝑋 = 𝑀𝑖,𝑋 +𝑀0𝑒 = 13.0 + 49.72 = 62.7 𝑘𝑁 ∙ 𝑚

𝑀0,𝑌 = 𝑀𝑖,𝑌 +𝑀0𝑒 = 21.1 + 49.72 = 70.8 𝑘𝑁 ∙ 𝑚

Regarding the method based on the nominal stiffness, it is initially necessary to

calculate the column flexural stiffness:

{𝐸𝐼𝑋 = 0.30 × 28400000 × 0.000367 + 0 = 3124 𝑘𝑁 ∙ 𝑚

2

𝐸𝐼𝑌 = 0.30 × 28400000 × 0.00277 + 0 = 23625 𝑘𝑁 ∙ 𝑚2

58

The total moment are:

{

𝑀𝐸𝑑,𝑋 =

62.7

1 − (1424 10×3124

3.162⁄ )

= 115.3 𝑘𝑁 ∙ 𝑚

𝑀𝐸𝑑,𝑌 =70.8

1 − (1424 10×23625

5.132⁄ )

= 84.1 𝑘𝑁 ∙ 𝑚

Considering the method based on the nominal curvature, the deflection in the

centre of the column will be given by:

{

𝑒2,𝑋 =

1.50 − 0.555

1.50 − 0.40× 1 ×

435

200000

0.45 × 0.15×3.162

10= 0.0260 𝑚

𝑒2,𝑌 =1.50 − 0.555

1.50 − 0.40× 1 ×

435

200000

0.45 × 0.50×5.132

10= 0.0214 𝑚

{𝑀2,𝑋 = 𝑁𝐸𝑑𝑒2,X = 1424 × 0.0260 = 36.97 𝑘𝑁 ∙ 𝑚

𝑀2,𝑌 = 𝑁𝐸𝑑𝑒2,Y = 1424 × 0.0214 = 30.48 𝑘𝑁 ∙ 𝑚

So, the total moment will be:

{𝑀𝐸𝑑,𝑋 = 13.0 + 49.7 + 36.97 = 99.7 𝑘𝑁 ∙ 𝑚

𝑀𝐸𝑑,𝑌 = 21.1 + 49.7 + 30.48 = 101.3 𝑘𝑁 ∙ 𝑚

Using the Excel worksheet in order to define the column reinforcement,

considering in this case 𝛾𝑐𝑜𝑛𝑐𝑟𝑒𝑡𝑒,𝐸𝐶2 = 1.50 and 𝛼𝐸𝐶2 = 1.00., the interaction diagrams

are shown as follows.

59

Figure C-1 – Concrete interaction diagram for X direction, for EN 1992-1-1

Figure C-2 – Concrete interaction diagram for Y direction, for EN 1992-1-1

Due to the biaxial bending, the chosen reinforcement was 6Φ25 mm (𝐴𝑆 =

29.45 𝑐𝑚2), verified below. The section detailing is shown below.

Figure C-3 – Cross section detail for the EN 1992-1-1

0

20

40

60

80

100

120

140

-4000 -3000 -2000 -1000 0 1000 2000

Md

(kN

.m)

Nd (kN)


As=0

As,given

Nd,Md

0

50

100

150

200

250

300

350

400

-3500 -2500 -1500 -500 500 1500

Md

(kN

.m)

Nd (kN)


As=0

As,given

Nd,Md

60

With this reinforcement, the resistant moments will be:

{𝑀𝑅𝑑,𝑋 ≅ 122 𝑘𝑁 ∙ 𝑚

𝑀𝑅𝑑,𝑌 ≅ 336 𝑘𝑁 ∙ 𝑚

Verifying the biaxial bending, the design is finished.

𝑁𝐸𝑑𝑁𝑅𝑑

=𝑁𝐸𝑑

𝐴𝑐𝑓𝑐𝑑 + 𝐴𝑠𝑓𝑦𝑑=

1424

0.11 ×35000

1.50+ 0.0022 ×

500000

1.15

→ 𝑎 ≅ 1.25

(99.7

122)1.25

+ (101.3

340)1.25

= 1

61

APPENDIX D – EXAMPLE OF LOCAL SECOND ORDER

EFFECTS ANNALYSIS FOR FIB 2010 MODEL CODE

This analysis according to fib 2010 Model Code will use the same column and

considerations from Appendix B, and the same buckling length and slenderness as in

Appendix C. The forces will also be the same as those found in Appendix C.

{𝑁𝑑 = −1424 𝑘𝑁𝑀01 = 49.72 𝑘𝑁 ∙ 𝑚𝑀02 = −50.85 𝑘𝑁 ∙ 𝑚

⇒ 𝑛 = 0.555

For both directions, the 𝛼𝑖 coefficient can be estimated as:

{

𝛼𝑖,𝑋 =

0.01

√3.15= 0.0056

𝛼𝑖≤1

200⇒ 𝛼𝑖,𝑋 = 0.005

𝛼𝑖,𝑌 =0.01

√5.13= 0.0044

And as the effective height of the section are 𝑑𝑋 = 0.15 m and 𝑑𝑌 = 0.50 m, the

eccentricity due to imperfections are:

{𝑒0𝑑,𝑋 = max {

0.005 × 3.16

2;0.20

30} = max{0.0079; 0.0067} = 0.0079 𝑚

𝑒0𝑑,𝑌 = max {0.0044 × 5.13

2;0.55

30} = max{0.0113; 0.0183} = 0.0183 𝑚

With these values, the first order eccentricity are as follows:

𝑒1𝑑 =𝑀1𝑑𝑁𝑑

=49.72

1424= 0.0349 𝑚

For the Level I of approximation:

{

𝑒2𝑑,𝐼,𝑋 ≅

435

200000∙

3.162

5 × (0.20 − 2 × 0.025)= 0.0290 𝑚

𝑒2𝑑,𝐼,𝑌 ≅435

200000∙

5.132

5 × (0.55 − 2 × 0.025)= 0.0229 𝑚

{𝑒𝑑,𝐼,𝑋 = 0.0079 + 0.0349 + 0.0290 = 0.0718 𝑚

𝑒𝑑,𝐼,𝑌 = 0.0183 + 0.0349 + 0.0229 = 0.0761 𝑚

{𝑀𝑑,𝐼,𝑋 = 1424 × 0.0718 = 102.3 𝑘𝑁 ∙ 𝑚

𝑀𝑑,𝐼,𝑌 = 1424 × 0.0761 = 108.2 𝑘𝑁 ∙ 𝑚

62

For the Level II of approximation:

{

𝑒2𝑑,𝐼𝐼,𝑋 = (

1.50 − 0.555

1.50 − 0.40)

435

200000

0.45 × 0.20

3.162

10= 0.0208 𝑚

𝑒2𝑑,𝐼𝐼,𝑌 = (1.50 − 0.555

1.50 − 0.40)

435

200000

0.45 × 0.55

5.132

10= 0.0198 𝑚

{𝑒𝑑,𝐼𝐼,𝑋 = 0.0079 + 0.0349 + 0.0208 = 0.0636 𝑚

𝑒𝑑,𝐼𝐼,𝑌 = 0.0167 + 0.0349 + 0.0198 = 0.0731 𝑚

{𝑀𝑑,𝐼𝐼,𝑋 = 1424 × 0.0636 = 90.6 𝑘𝑁 ∙ 𝑚

𝑀𝑑,𝐼𝐼,𝑌 = 1424 × 0.0731 = 104.1 𝑘𝑁 ∙ 𝑚

The Excel worksheet used on Appendix C is also used here to define the column

reinforcement, considering the second level of approximation. The interaction diagrams

are given below.

Figure D-1 – Concrete interaction diagram for X direction, for fib Model Code 2010

0

20

40

60

80

100

120

140

-4000 -3000 -2000 -1000 0 1000

Md

(kN

.m)

Nd (kN)


As=0

As,given

Nd,Md

63

Figure D-2 – Concrete interaction diagram for Y direction, for fib Model Code 2010

The chosen reinforcement is 6Φ25 mm, same as EN 1992-1-1, and the section

detail is shown in Figure C-3. The resistant moments are also the same as in Appendix

C.

Verifying the biaxial bending:

𝑁𝐸𝑑𝑁𝑅𝑑

=𝑁𝐸𝑑

𝐴𝑐𝑓𝑐𝑑 + 𝐴𝑠𝑓𝑦𝑑=

1436

0.11 ×35000

1.50+ 0.0022 ×

500000

1.15

→ 𝑎 ≅ 1.25

(90.6

122)1.25

+ (104.1

340)1.25

= 0.92

0

50

100

150

200

250

300

350

400

-3500 -2500 -1500 -500 500 1500

Md

(kN

.m)

Nd (kN)


As=0

As,given

Nd,Md

Documents

COMPARATIVE ANALYSIS OF SECOND ORDER EFFECTS BY …