MANUAL

for

ADVANCED DESIGN REINFORCED CONCRETE MEMBERS subjected to FLEXURE and SHEAR FORCE

FIRST Volume

Second EDITION

Dumitru MOLDOVAN

Horia CONSTANTINESCU

Tehnoredactare: Dumitru MOLDOVAN

Descrierea CIP a Bibliotecii Naionale a Romniei

MOLDOVAN, DUMITRU

Manual for advanced design : reinforced concrete members subjected to

flexure and shear force / Dumitru Moldovan, Horia Constantinescu. - Iai : Stef,

2014

Bibliogr.

Index

ISBN 978-606-575-401-0

I. Constantinescu, Horia

658.512.2

MANUAL

for

ADVANCED DESIGN

Dumitru Moldovan, Horia Constantinescu: Manual for Advanced Design

Copyright Dumitru Moldovan, Horia Constantinescu 2012

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The current edition is a teaching book that represents the personal opinions of the

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Preface

The present book on reinforced concrete design according to (SR EN 1992-1-1:2006) (Eurocode 2:

Design of concrete structures: 2006 ed.) is part of a complex project on safe design, superior casting

productivity and cost attractive members and structures. The herein book, either e-book or in-print, is

based on the very first book the authors have jointly contributed. Since it is the first volume of the fore

mentioned project, the 1 is used to represent that. Since it is the second edition of the hereto mentioned

volume the .2 is used to represent that.

Apart from the modifications that incurred in the styles defined and used in writing this book, some

modifications have been implemented to better outline the desired explanations, as plain text and/or

figures and tables with corresponding notes and keys. Extensive re-writing of the design examples

provided in later chapters incurred also.

Above all, this book is for the use of civil engineering students, especially those at their first contact

with structural concrete and its design. Nonetheless, this does not exclude other interested parties to read,

comment, refer to the present work or address suggestions to the authors to better organise the information

provided or improve specific points.

Concrete is in the authors opinion, one of the oldest and greatest inventions. Still, there are some

aspects regarding this material that may be approached differently, in order to unlock its full potential, or,

in a more general definition, to achieve sustainability.

The first step towards that goal is to proper understand the limits of the material, plain and (especially)

reinforced. By doing so, it is possible to design members and structures that are safe and cost attractive,

avoiding waste of energy, materials and manpower.

That being said, it is the authors great pleasure to (re)introduce their first book (volume) that addresses

the design process. Simply, get MAD!

Cluj-Napoca, 15th July 2012

Manual for Advanced Design vii

Acknowledgments

The authors wish to thank the following:

ASRO (Asociaia de Standardizare din Romnia): permission no. LUC/12/154/7.02.2012

DIN (Deutsches Institut fr Normung e.V.)

Prof. PhD Eng. Dan Paul GEORGESCU

Prof. PhD Eng. Zoltan KISS

For permission to cite corresponding works as presented in the REFERENCE chapter.

All support is gratefully acknowledged.

ix Manual for Advanced Design

Contents List

Preface ________________________________________________________________________ vii

Acknowledgments ________________________________________________________________ ix

Contents List _____________________________________________________________________ xi

[Chapter 1] Flexural Design _________________________________________________________ 15

[Section (1) A] Actions and Loads ____________________________________________________ 16

[Note (1-A) a] Combinations of Actions _____________________________________________ 17

[Note (1-A) b] Limit States _______________________________________________________ 18

[Section (1) B] Flexural Design Model _________________________________________________ 19

[Note (1-B) a] Neutral Axis _______________________________________________________ 20

[Note (1-B) b] Model Assumptions _________________________________________________ 20

[Note (1-B) c] Stress Block _______________________________________________________ 21

[Section (1) C] Predesign Evaluation __________________________________________________ 24

[Note (1-C) a] Sizing of the Cross Section ____________________________________________ 26

[Section (1) D] Singly Reinforced Rectangular Section (SRRS) _____________________________ 28

[Note (1-D) a] Theoretical Model for SRRS __________________________________________ 28

[(1-D-a) Discussion] Check Yielding of Tension Reinforcement ________________________ 29

[Note (One-D) b] Practical Calculation for SRRS ______________________________________ 31

[Section (1) E] Doubly Reinforced Rectangular Section (DRRS) ____________________________ 33

[Note (1-E) a] Theoretical Model for DRRS __________________________________________ 33

[Note (One-E) b] Practical Calculation for DRRS ______________________________________ 34

[(1-E-b) Discussion] Check Yielding of Compression Reinforcement ____________________ 35

[(1-E-b) Discussion] Check Yielding of Tension Reinforcement ________________________ 35

[Section (1) F] Tee/Flanged Sections (FS) ______________________________________________ 37

[Note (1-F) a] Theoretical Model for SRFS ___________________________________________ 37

[Note (1-F) b] Practical Calculation for SRFS _________________________________________ 39

[Note (1-F) c] Theoretical Model for DRFS __________________________________________ 41

[Note (1-F) d] Practical Calculation for DRFS ________________________________________ 42

[(1-F-d) Discussion] Check Yielding of Compression Reinforcement ____________________ 43

[(1-F-d) Discussion] Check Yielding of Tension Reinforcement ________________________ 43

Concluding Remarks on Chapter One _________________________________________________ 45

Manual for Advanced Design xi

Contents List

[Chapter 2] Shear Force Design ______________________________________________________ 47

[Section (2) A] Theoretical Model for Shear Force _______________________________________ 48

[Note (2-A) a] Variable Angle Truss Model __________________________________________ 48

[(2-A-a) Discussion] Section Through Strut ________________________________________ 49

[(2-A-a) Discussion] Section Through Tie _________________________________________ 50

[Section (2) B] Practical Calculation for Shear Force ______________________________________ 53

[Note (2-B) a] Shear Force Evaluation_______________________________________________ 53

[(2-B-a) Discussion] Constructive Reinforcement ___________________________________ 54

[(2-B-a) Discussion] Check of Strut Failure ________________________________________ 56

[(2-B-a) Discussion] Check of Tie Failure _________________________________________ 56

[(2-B-a) Discussion] Establish Spacing/Size of Stirrups ______________________________ 56

[Note (2-B) b] Anchorage Length __________________________________________________ 58

[(2-B-b) Discussion] Establish Tension Force for Longitudinal Bars ____________________ 58

[(2-B-b) Discussion] Calculations for the Anchorage Length __________________________ 58

[Note (2-B) c] Check Shear between Web and Flange __________________________________ 60

Concluding Remarks on Chapter Two _________________________________________________ 63

[Chapter 3] Worked Examples _______________________________________________________ 65

Project Specifications ______________________________________________________________ 66

[Section (3) A] Predesign Analysis ____________________________________________________ 68

[Note (3-A) a] Durability Requirements _____________________________________________ 68

[Note (3-A) b] Bearing Members and Partial Loads Evaluation ___________________________ 69

[Section (3) B] Singly Reinforced Rectangular Section (SRRS) _____________________________ 71

[Note (3-B) a] Sizing of the Cross Section ____________________________________________ 71

[Note (3-B) b] Area of Reinforcement for Flexure _____________________________________ 74

[(3-B-b) Discussion] Place Bars in Two Layers, see [(3) Figure 3] ______________________ 76

[(3-B-b) Discussion] Bundled Bars Horizontal Placing, see [(3) Figure 4] _______________ 78

[(3-B-b) Discussion] Bundled Bars Vertical Placing, see [(3) Figure 5] _________________ 79

[(3-B-b) Discussion] Decrease the Height of the Cross Section _________________________ 80

[Note (3-B) c] Area of Reinforcement for Shear Force __________________________________ 82

[(3-B-c) Discussion] Establish Elementary Length(s) ________________________________ 85

[(3-B-c) Discussion] Layout for Stirrups __________________________________________ 88

[(3-B-c) Discussion] From support towards midspan, see [(3) Figure 8] __________________ 89

[(3-B-c) Discussion] From midspan to supports, see [(3) Figure 9] ______________________ 91

[(3-B-c) Discussion] Alternative layout and placing _________________________________ 92

[Note (3-B) d] Anchorage Length __________________________________________________ 93

Manual for Advanced Design xii

Contents List

[Section (3) C] Doubly Reinforced Rectangular Section (DRRS) ____________________________ 95

[Note (3-C) a] Load Evaluation ____________________________________________________ 95

[Note (3-C) b] Area of Reinforcement for Flexure _____________________________________ 96

[(3-C-b) Discussion] Place Bars on Two Layers, see [(3) Figure 10] _____________________ 97

[(3-C-b) Discussion] Bundled Bars Vertical Placing, see [(3) Figure 11] _______________ 100

[(3-C-b) Discussion] Modify Cross Section and Reinforcement Layout _________________ 102

[Note (3-C) c] Area of Reinforcement for Shear Force _________________________________ 109

[Note (3-C) d] Shear Reinforcement Layout _________________________________________ 111

[Note (3-C) e] Anchorage Length _________________________________________________ 112

[Section (3) D] Tee Section (TS)_____________________________________________________ 113

[Note (3-D) a] Area of Reinforcement for Flexure ____________________________________ 115

[(3-D-a) Discussion] Single Reinforcement _______________________________________ 115

[(3-D-a) Discussion] Double Reinforcement ______________________________________ 118

[Note (3-D) b] Area of Reinforcement for Shear Force _________________________________ 121

[(3-D-b) Discussion] Establish Elementary Length(s) _______________________________ 125

[(3-D-b) Discussion] Layout for Stirrups _________________________________________ 128

[(3-D-b) Discussion] From support towards midspan, see [(3) Figure 15] ________________ 129

[(3-D-b) Discussion] From midspan to supports, see [(3) Figure 16] ____________________ 131

[Note (3-D) c] Anchorage Length _________________________________________________ 133

[Section (3) E] Fast Track to Design (FTD) ____________________________________________ 135

[Note (3-E) a] Singly Reinforced Rectangular Section (SRRS) __________________________ 135

[(3-E-a) Discussion] Sizing of the Cross Section ___________________________________ 136

[(3-E-a) Discussion] Static Analysis _____________________________________________ 136

[(3-E-a) Discussion] Area of Reinforcement for Flexure _____________________________ 136

[(3-E-a) Discussion] Exact Calculation of the Area of Reinforcement for Flexure _________ 137

[(3-E-a) Discussion] Shear Force Reinforcement ___________________________________ 138

[(3-E-a) Discussion] Anchorage Length __________________________________________ 138

[Note (3-E) b] Even Faster Track to Design _________________________________________ 139

[Section (3) F] Preparing the end term EXAM __________________________________________ 140

[Note (3-F) a] Requirements _____________________________________________________ 140

[(3-F-a) Discussion] The first 10 minutes _________________________________________ 140

[(3-F-a) Discussion] 10 to 20 minutes ___________________________________________ 141

[(3-F-a) Discussion] 20 minutes to 35 minutes _____________________________________ 142

[(3-F-a) Discussion] The last 25 minutes _________________________________________ 144

Concluding Remarks on Chapter Three __________________________________________________ 147

Appendix _________________________________________________________________________ 149

References ________________________________________________________________________ 213

Manual for Advanced Design xiii

Contents List

Appendix _______________________________________________________________________ 149

[Appendix 1] Actions and loads arrangements __________________________________________ 151

[Appendix (1) A] Design life _____________________________________________________ 151

[Appendix (1) B] Variable actions _________________________________________________ 152

[Appendix (1) C] Service Limit States ______________________________________________ 161

[Appendix 2] Durability Requirements ________________________________________________ 162

[Appendix (2) A] Exposure Classes ________________________________________________ 162

[Appendix (2) B] Cement ________________________________________________________ 167

[Appendix (2) C] Concrete Cover _________________________________________________ 171

[Appendix 3] Materials Properties ___________________________________________________ 173

[Appendix (3) A] Concrete ______________________________________________________ 173

[Appendix (3) B] Steel __________________________________________________________ 175

[Appendix 4] Reinforcement ________________________________________________________ 177

[Appendix (4) A] Bars __________________________________________________________ 177

[Appendix (4) B] Welded Wires __________________________________________________ 178

[Appendix 5] Fire Resistance _______________________________________________________ 181

[Appendix (5) A] Slabs _________________________________________________________ 182

[Appendix (5) B] Beams ________________________________________________________ 186

[Appendix 6] Selected Service Requirements ___________________________________________ 189

[Appendix (6) A] Deflection Control without Calculation ______________________________ 189

[Appendix (6-A) a] Code Provisions ____________________________________________ 189

[Appendix (6-A) b] National Practice ____________________________________________ 191

[Appendix (6) B] Crack Control without Calculation __________________________________ 193

[Appendix (6-B) a] Stress Limitation ____________________________________________ 193

[Appendix (6-B) b] Crack Width Limitation ______________________________________ 194

[Appendix (6-B) c] Minimum Reinforcement Area _________________________________ 194

[Appendix (6-B) d] Size or Distance Limitation ____________________________________ 198

[Appendix 7] Design Tables ________________________________________________________ 199

[Appendix 8] Bond Requirements ____________________________________________________ 203

[Appendix 9] Shear Force Calculations _______________________________________________ 205

[Appendix (9) A] Shear Reduction in the Vicinity of Supports ___________________________ 205

[Appendix 10] Elementary diagrams for shear force and flexure ____________________________ 209

[Appendix 11] Beam reinforcement drawing ___________________________________________ 211

References ______________________________________________________________________ 213

Manual for Advanced Design xiv

15 Manual for Advanced Design Flexural Design [1]

[Chapter 1] Flexural Design

The aims and objectives of the herein chapter may be summarized in the infra list:

Brief introduction to the concepts of actions and loads as defined in the Eurocodes;

Brief introduction to the design model of reinforced concrete members subjected to flexure;

Adequate supplementary explanations as well as recommended practice provisions related to the above.

Yielding, also referenced as ductility in the seismic action vocabulary, is the

property of materials such as steel to exhibit large plastic deformations thus in turn

allowing structural members, namely slabs, beams and columns to provide visual

pointers for the imminence of failure. Although there may be many requirements the

structural engineer has to accommodate within a structure, the fundamental criterion,

which may not explicitly be written down in all the design codes and the corresponding

provisions is the avoidance of loss of human lives. As this condition is always designed

for, there will be a line at which some trade-offs with the economic side of any

construction will come first so the structural engineer should use the so called

experience/self-judgement/etc. to decide the limits of the fore mentioned exchange.

It is well established that under the name fundamental combination of actions as

will be detailed here-after, the resistive capacity of any member is about two times the

maximum stress load that member may experience in its service life. Therefore there

are no pre-established limits for any trade-offs as mentioned in the previous paragraph.

Still, CAUTION IS ADVISED as there are possible un-designed for events, such as

explosions, impacts (special design hypothesises not usually taken in consideration in

current design practice for normal structures) and in some limits seismic action that

may put to work that very one-time reserve and therefore make the difference in-

between collapse and survival of the structure.

The reserve of resistive capacity makes the corresponding design, especially for

bending, rather straight forward if no special requirements impose specific conditions.

Since the yielding condition is usually satisfied, the corresponding design model for

bending hasnt changed since it was first proposed in the early 1930s by Charles

Whitney and any section that fulfils the strength condition will generally satisfy

ordinary service conditions as well.

The details presented in this chapter are relevant for the above reason for the so

called normal sections, slabs having a maximum thickness of [200 mm] and beams

having the height greater than the breadth up to an imposed limit (five times). Other

cases, flat slabs (thickness bigger than [200 mm]) or shallow beams (height smaller

that the breadth) or deep beams (height bigger than five times the breadth) are not

referenced here-in as such members are considered special and therefore not

currently used in usual applications.

[1] Flexural Design Manual for Advanced Design 16

[Section (1) A] Actions and Loads

It was considered appropriate to introduce those explanations here to underline both

the concept of actions and load arrangements as presented in the Eurocodes, see ( SR

EN 1990:2004/A1:2006), (SR EN 1991-1-1:2004/NA:2006) or [Appendix 1] Actions

and loads arrangements as well as the values considered in the worked example as

presented infra.

The term action(s) is used to express the various types of loads that may act on a

given structure. The most common are listed below:

[-] Permanent actions refer to actions for which the variation in magnitude with time

is negligible. This category consists of the self-weight of the

members/elements/other features resting or attached to the structural members (the

so called dead load). Partitioning that is/are unlikely to change positions over

long periods of time may be considered permanent. Other situations that create a

relatively fixed position of the action (i.e., mechanical devices or machines of

important weight) are also part of this category. In this book only the self-weight of

the structural members will be considered; Variable actions are those for which

the variation in magnitude is dependent on time. This category groups relatively all

other types of actions and are mainly given by specific functions of the structure

(the so called live loads) as well as by the location of the structure (climatic

actions such as wind, snow, volume changes, etc.). Only those imposed loads that

are specific of the current service life will be considered here-in;

[-] Accidental actions are actions of short duration but of significant magnitude that

may or may not occur on a given structure during the intended service life. This

category consists most frequently of seismic and fire action. Other cases may be

considered, depending on the specific functional requirements of the structure such

as accidental explosions, plane impacts, terrorist attacks, etc. Since this discussion

requires extensive explanations on dynamic loads and is beyond the intended

purpose of the current book, this aspect will not be considered next;

[-] Imposed deformations usually occur during the intended service life and are

frequently given by settlement of the subgrade (the soil under the foundations of

the structure) under the influence of internal and/or external factors, most often

because of water circulation or water season cycles in combination with poor

subgrade properties or because of temperature changes determined by a specific

function of the/or part of the structure. Since these usually occur locally in the

structure, the change in the footings level that affect columns may cause some

horizontal members (mainly beams and/or slabs) to develop local deformations that

affect first the usability of all affected members and may even permanently

negatively impact the structure. Special attention must therefore be given to the

subgrade properties and characteristics. Since this is beyond the intended purpose

of the current book, this aspect will not be considered here-in.

Each action is referred to by its characteristic value that in turn is defined by one of

the following alternatives:

[-] Mean value, generally used for permanent actions;

[-] An upper value with an intended probability of not being exceeded or a lower one

with an intended probability of being exceeded;

[-] A value with an intended probability of being achieved (normally used for variable

actions with known statistical distributions, such as wind or snow);

17 Manual for Advanced Design Flexural Design [1]

[-] A nominal value that is used for some variable and accidental actions.

Furthermore, each variable action may be referred to by four representative values:

[-] THE PRINCIPAL REPRESENTATIVE VALUE kQ is the characteristic value

and this can be determined statistically or, where there is insufficient data, a

nominal value may be used;

[-] THE COMBINATION VALUE is obtained by applying the factor 0 to the

characteristic value thus giving 0 kQ and is intended to take into account the

reduced probability of the simultaneous occurrence of two or more variable

actions;

[-] THE FREQUENT VALUE is obtained by applying the factor 1 to the

characteristic value thus giving 1 kQ and is intended to take into account the

possibility of the characteristic value to be exceeded only for a short period of time

and is used primarily for the Serviceability Limit States (SLS) and also the

accidental load for the Ultimate Limit State (ULS);

[-] THE QUASI-PERMANENT VALUE is obtained by applying the factor 2 to

the characteristic value thus giving 2 kQ and is intended to take into account

the possibility of the characteristic value to be exceeded for a considerable period

of time; alternatively it may be considered as an average loading over time. It is

used for the long-term effects for the SLS and also accidental and seismic loads for

ULS;

Each of the above factors is established based on semi-probabilistic methods

and is specific to every type of imposed loads. Further information on derivation of the

factors can be found in [Appendix C] of the Eurocode (SR EN 1990:2004).

[Note (1-A) a] Combinations of Actions

The term combination of actions is used to name the situation when, at different

limit states, various actions act together, case in which it is mandatory to establish the

magnitude of each of those actions. It should not be confused with load cases, a term

that refers to the arrangement of the variable actions to give the most unfavourable

conditions and may be consulted in the material Eurocodes, such as (SR EN

1990:2004).

To determine the adequate value of actions used in design several processes can be

used:

[-] Identify the design situation:

[x] Persistent: referring to conditions of normal use;

[x] Transient: referring to temporary conditions of the structure (i.e.,. during

construction or repair);

[x] Accidental: involving exceptional conditions of the structure or its exposure,

including fire, explosion, impact, etc.;

[x] Seismic: when the structure is subjected to a seismic event;

[1] Flexural Design Manual for Advanced Design 18

[-] Identify all realistic actions;

[x] Determine the partial factors for each applicable combination of actions;

[x] Arrange the actions to produce the most critical conditions.

Where there is ONLY ONE VARIABLE ACTION (i.e., imposed load) in a combination, the MAGNITUDE OF THE ACTIONS can be obtained by

MULTIPLYING them by the APPROPRIATE PARTIAL FACTORS. Where THERE

IS MORE THAN ONE VARIABLE ACTION in a combination, it is necessary to

identify THE LEADING ACTION ,1kQ (subscript 1) and OTHER

ACCOMPANYING ACTIONS ,k iQ (subscript i). The ACCOMPANYING ACTION is always taken as THE COMBINATION VALUE (defined in supra list).

[Note (1-A) b] Limit States

The Ultimate Limit State consists of the following categories:

[-] EQU (Equilibrium), Loss of Equilibrium of the structure. This situation may

appear in the case of a one span simply supported beam with an end overhang

WHEN ONLY the overhang is loaded, an unbalanced case load that CAN CAUSE

the opposite end of the beam to lift from its support if the support closer to the

overhang allows the beam to rotate freely;

[-] STR (Strength), Internal failure or excessive deformation of the structure or

structural member. This situation is the most common category and is the one to

be considered in this book;

[-] GEO (Geological), Failure due to excessive deformation of the ground. This

category should be considered when imposed deformations occur not based on a

temperature gradient but on a settlement of the subgrade as briefly presented supra;

[-] FAT (Fatigue), Fatigue failure of the structure or structural members. This

category should be considered whenever regular actions upon the structure have a

cyclic behaviour (i.e., heavy cutting machinery used in some industries to produce

the stock length of materials).

The different combinations for each of these Ultimate Limit States are presented in [Appendix 1] Actions and loads arrangements as provided by the Eurocodes.

For persistent and transient design situations under the STR limit state, the Eurocode defines three possible combinations, which are given in Expressions (6.10), (6.10a) and

(6.10b), see (SR EN 1990:2004). The engineer may use either (6.10) or the less

favourable of (6.10a) and (6.10b);

It may appear that there is considerably more calculation required to determine the appropriate load combination. Still; with experience the engineer will be able to

determine this by inspection;

Expression (6.10) is always equal to or more conservative than the less favourable of Expressions (6.10a) and (6.10b). Expression (6.10b) will normally apply when the

permanent actions are not greater than 4.5 times the variable actions (except for storage

loads (category E) where Expression (6.10a) always applies). Therefore, for a typical

concrete frame building, Expression (6.10b) will give the most structurally economical

combination of actions;

19 Manual for Advanced Design Flexural Design [1]

For members supporting one variable action the combination ,1 ,11,35 1,50k kG Q

derived from (Expression 6.10b) can be used to design the corresponding

reinforcement, should the permanent actions be smaller than 4,5 times the variable

actions (except for storage loads);

For the Serviceability Limit State care should be taken NOT TO CONFUSE THE COMBINATIONS OF CHARACTERISTIC, FREQUENT AND QUASI-

PERMANENT with the representative values that have the same titles.

[Section (1) B] Flexural Design Model

Knowledge in general and engineering in particular works with models that may be

considered to showcase the following levels of understanding:

[ 1 > The global level, at which one perceives everything as one unitary object (in

constructions, a structure);

[ 2 > The system level, at which one perceives the object to be compound of different

groups of elements (in constructions, all the members with similar functions, i.e.

the slabs, the beams or the columns);

[ 3 > The element level, at which one perceives a particular component (in

constructions, a slab, a beam, or a column);

[ 4 > The macroscopic level, at which one perceives the major structure of that

particular component (in constructions, in the case of reinforced concrete, the

cross section of a member as a compound section of concrete and reinforcement);

[ 5 > The microscopic level, at which one perceives the properties and interaction of

the different constituents (in constructions, in the case of concrete, the

constituents of the mix cement, aggregates, water, etc.);

[ 6 > The atomic level, at which one perceives the properties and interaction of atoms

(in constructions, in the case of concrete, yet to be established).

By only addressing the macroscopic level (4th) it is possible for the engineer to

predict the behaviour of the superior levels of knowledge (3rd, 2nd and 1st) in limits

deemed satisfactory. Of course, as knowledge progresses it is possible to minimize

those limits by use of advanced computer calculations or a more fundamental

approach, namely intuition. Please bear in mind that progress, technically speaking, is a

two-step process: it always starts with a brain-storm that should always be

accompanied by hard work.

Reinforced concrete is (as presented) subjected to the previous as well, that is why,

before all, its relevant to discuss the design process from a theoretical point of view,

by explaining the milestones in flexure, which is (generally speaking) a state of loading

in which the same cross section will have opposite stresses, of tension in one part and

in compression for the rest. The transition area in-between is called the neutral axis (nil

stress or better said very close to nil stress). Its position on the height of the cross

section is variable, depending on the values opposing maximum stresses reach.

Shear behaviour will be presented in the next chapter, [Chapter 2] Shear Force

Design.

[1] Flexural Design Manual for Advanced Design 20

[Note (1-B) a] Neutral Axis

In these authors opinion the design process in flexure is very interconnected with

the position of the neutral axis (height in compression) though the actual design seems

to provide no direct link to this, meaning this variable isnt as highlighted as others, for

most formulas are pairing the neutral axis with other variables (i.e., ductility condition

for plastic analysis presented in the third volume of this project).

In order to explain, the reader is invited to assume the position of a lab technician

on the point of testing a beam in flexure by gravitational loading (perpendicular to its

geometric longitudinal axis).

Also assume that:

[-] You have marked with vertical lines the lateral faces of your beam over its full

length;

[-] Pretend that each line/slice (therefore each cross section) may be extracted from the

beam itself without affecting the structural behaviour of the element, similar to the

way in which you may pick out from a book set your favourite book to read;

[-] In terms of stresses that develop during loading, name the effect of pushing

compression (consider it to be occurring on the front cover of the book) while

the effect of pulling tension (consider it to be occurring on the back cover of the

fore-mentioned book);

[-] At first, prior to the loading, there is nil stress in each and every point over the

height of the cross section. Since the neutral axis is only a limited area in the cross

section, its safe to asses that the neutral axis is outside of the cross section. Since

we assumed the loading to be gravitational in nature, that would mean that the

neutral axis is somewhere below the extreme lower fibre of the cross section.

[-] As the loading begins, the extreme lower part of the cross section will develop

tension stresses while the extreme upper part of the cross section will develop

compression stresses, in other words the neutral axis starts to move from the

extreme lower edge to the upper edge of the cross section, until the resistive

capacity of the cross section is reached.

[-] Failure will occur, theoretically, when the entire cross section will be in tension (a

ripping apart effect). In fact, for normal strength concrete, there will be a small

area at the top of the cross section that will crush under compression, therefore

causing the collapse.

[-] From a mechanical (static) point of view, that is the same with the element

becoming a mechanism with a hinge located at the top of the cross section. In this

case, the collapse is instantaneous (even under own weight).

[Note (1-B) b] Model Assumptions

Lets convert the beam described previously to the material we call reinforced

concrete. The present design model assumes that cracked concrete does not contribute

to the bearing capacity, although it is well known that between cracks, the bond of

concrete to steel leads to a reduction in the tensile stresses in the reinforcement. As the

concrete grade increases so does its tensile strength making it logical to assume that, in

some degree, the previous assumption may deflect the model from the actual

behaviour. Since this is still under debate, no further details will be provided here-in.

21 Manual for Advanced Design Flexural Design [1]

Other assumptions used in the model are:

[ 1 > Concrete is considered to be on the brink of failure (the so called 3rd state);

[ 2 > Bernoullis hypothesis of plane sections and the compatibility of strains in

concrete and steel for the same fibre in the cross section;

[ 3 > Hookes Law allows for strains and stresses to be considered a ratio of the elastic

properties of the material;

[ 4 > The General Stress block for the compressed area of concrete is replaced by a

simplified rectangular stress block;

[ 5 > The reinforcement yields prior to the crushing of concrete.

[Note (1-B) c] Stress Block

In comparison with a uniaxial load, whether its compression or tension, flexure

determines different fibres over the height of the cross section to be subjected to

different stresses, not only as per value but also as per nature (compression on top of

the cross section and tension at the bottom in the case of gravitational loads). That is

why, although flexure can be considered an eccentric compression, the compressive

stress in concrete subjected to flexure is not the same as in pure compression.

First, in pure compression, all fibres are under about the same stress.

Testing has shown that there is a reduction in the stress value with the increase of the distance from the centre of geometry of the sample to its edges (further details are

available in literature not cited here-in).

This is not the case for flexure where eccentricity introduces variation in values, some fibres being subjected to higher stresses than others. Therefore, different

longitudinal layers of fibres have a tendency to slip from each other.

This is in these authors opinion a positive effect as it will lead to:

[-] An increase in the strains an element can develop due to a decrease in the speed of

strain development over time;

[-] A delayed failure of concrete due to a roll-over mechanism which transmits the

stress from the fibres under the maximum effort to the less loaded fibres closer to

the neutral axis.

Second, the longitudinal splitting effect is in opposition with the compression stress

which will lead to a reduction in the amount of stress the most compressed fibres will

bear (further details are available in literature not cited here-in).

The calculation model for flexure consists of two ideal forces in mechanical

equilibrium, see [(1) Figure 1]:

[ 1 > A compressive force in concrete, C ;

[ 2 > A tensile force in the reinforcement, T ;

[ 3 > A lever arm in-between, z .

[1] Flexural Design Manual for Advanced Design 22

The resistive flexural capacity for a given cross section is written as:

RdM C z T z

C T

Since the final form of the above should be a formula for the area of reinforcement,

one should:

[-] Evaluate the compressive resultant:

[x] Establish a function for the stress variation based on strain values (which are

very easy to measure in experimental tests) such as:

0

( )cu

cF b

( ) is the stress function of strains.

[x] Establish/find the limit for integration;

Cross section Strain distribution General stress block

(a) Assume the cross section to be rectangular:

(i) h is the height and b is the width;

(b) Selected parameters:

(i) x is the height in compression and d is the depth of the section.

Since the above is difficult to evaluate precisely, a simplified stress block replacing

the real distribution of stresses while being easier to evaluate was deemed necessary.

The substitution of one with the other is based on two conditions:

[-] The volume of stresses must be correctly evaluated;

[-] The position of the compression centroid in the real and simplified diagram must be

the same.

Those conditions are used to calculate two reduction coefficients, and .

The notations used in general with reinforced concrete and their meaning are presented

in [(1) Figure 2] and the subsequent list. The cross section is considered to be

rectangular both for the shapes simplicity and for the fact that this particular shape is

the most common in constructions.

with [1-1]

[1-2]

with [1-3]

(1) Figure 1

Forces

in equilibrium

Note(s)

23 Manual for Advanced Design Flexural Design [1]

[1-4]

[1-5]

(1) Figure 2

Design model

parameters

Note(s)

With this model in mind, the previous equation may be re-written as:

cdC x f b

sl ydT A f

In (Expression [1-4]) terms have been grouped to outline the simplified stress block while in (Expression [1-5]) subscripts dependent on the part of the cross section (lower

or upper) have been omitted to outline the formula for the resultant in the

reinforcement (whether in tension or compression).

Cross section Strain distribution General stress

block

Simplified stress

block

(a) Concretes properties in compression given by its design strength cdf while the

maximum strain in compression in concrete is denoted by cu ;

(b) Reinforcements properties in tension are given by its design d strength at yielding y

denoted ydf , also considered the maximum stress;

(c) Main reinforcement ,1slA as longitudinal l steel s bars in tension , T t placed in the bottom part 1 of the cross section:

(i) ,1sl is the ultimate tensile strain;

(ii) stf is the stress when the reinforcement doesnt yield;

(iii) 1d is the axis distance from the extreme bottom fibre;

(d) Main reinforcement ,2slA as longitudinal l steel s bars in compression , C c placed in the top part 2 of the cross section:

(i) ,2sl is the ultimate tensile strain;

(ii) scf is the stress when the reinforcement doesnt yield;

(iii) 2d is the axis distance from the extreme top fibre;

(e) Simplified stress block parameters are:

(i) Coefficient for reducing the maximum allowable compressive stress in concrete c :

1.00 50

50 1 50 90

200

ck

ckck

if f MPa

fif f MPa

as per [Expression 3.21; 3.22] in (SR

EN 1992-1-1:2006);

[1] Flexural Design Manual for Advanced Design 24

(ii) Coefficient for reducing the height in compression :

0.80 50

50 0.8 50 90

400

ck

ckck

if f MPa

fif f MPa

as per [Expression 3.19; 3.20] in

same reference.

In addition, by writing equilibrium for the horizontal forces, the above (Expression

[1-4] and Expression [1-5]) give the height in compression as:

sl yd

cd

A fT C x

b f

This may be used to calculate the position of the neutral axis x ONLY after the

reinforcement has been calculated. Bear in mind that UNLESS OTHERWISE

STATED there is NO need to actually calculate the height in compression (further

guidance will be provided on the matter as described in subsequent sections).

[Section (1) C] Predesign Evaluation

The reader is advised to refer to the code provisions (SR EN 1992-1-1:2006), (SR EN 1992-1-2:2006) and the Appendixes at the end of this book to make use of the

Tables and other referenced information as indicated;

It is assumed that the maximum flexural moment EdM and the maximum shear

EdV are known so no information on static calculation is provided.

Any structure must fulfil two fundamental requirements:

[-] To be designed in such a manner that it does not collapse under normal loading

conditions, partially or totally, and that any partial collapse does not impair the

unaffected part of the structure causing a domino effect to bring the structure

down (to be therefore redundant), collectively named Ultimate Limit States

Design (ULSD);

[-] To be designed in such a manner that it does not impair on the intended use of that

structure, partially or totally, collectively named Service Limit States Design

(SLSD).

To address the first supra limit state its enough to provide for a cross section

defined primarily by its width and height the corresponding reinforcement.

Sizing of the cross section is subject to certain conditions, which will be detailed

herein. To address the second supra limit state its enough to check that the proposed

section fulfils additional requirements which in the case of RC members are deflection

and crack width. For Normal Strength Concrete, by obeying certain limitations as per

code provisions, a section proposed for ULSD will check for SLSD also. For this

reason SR EN 1992-1-1 states firmly that in all the cases when given limitations are

respected there is no need to check for SLSD conditions.

Note(s)

[1-6]

x

b h

25 Manual for Advanced Design Flexural Design [1]

(1) Figure 3

Concrete cover

parameters

Note(s)

Therefore the first step in any design is to propose a cross section which would

best provide safety by bearing the loads acting upon it and which is also economical

and easy to cast. That is why, today, as mankind struggles to find better management

plans for the depleting resources at our disposal, engineers in general and civil

engineers in particular are called upon to find ways unexplored before to achieve that

end (i.e. construction industry consumes about 40% of the overall energy produced

worldwide). In order to achieve SUSTAINABILITY (by its generalised meaning) its

primordial to insure DURABILITY for each member and therefore the structure itself.

DURABILITY is plainly said, the response of a member subjected to exposure

conditions due to climatic conditions (rain, snow, etc.) or processes (wanted or

accidental) which take place inside the structure or outside it that may negatively affect

steel or concrete. It is achieved by providing a minimum concrete cover to protect the

reinforcement from corrosion or the adverse effects of fire. Of course, the concrete

grade is the most important factor to be considered, as it will be explained herein.

Since exposure conditions pair up with fire safety conditions to impose a concrete cover and even minimum dimensions for the cross section and are furthermore

generally valid the first answer in the previously proposed endeavour should be to

correctly calculate concrete cover.

Any cross section has (generally speaking) two types of reinforcement, for flexure longitudinal to the axis of the member and for shear transverse to the same axis,

which means that there will be two types of concrete cover to be checked against the

required thickness.

Notations Deviation(s) in placing

(a) The nominal concrete cover is minnom devc c c as per [Expression 4.1, 4.4.1.1] in (SR

EN 1992-1-1:2006);

(b) The minimum concrete cover is

min,

min min, , , ,

max

10

b

dur dur dur st dur add

c

c c c c c

mm

,

as per [Expression 4.2, 4.4.1.2] in (SR EN 1992-1-1:2006) with:

(i) min,bc [mm] is the concrete cover based on bond conditions [App. (2) Table 5];

(ii) min,durc [mm] is the concrete cover based on exposure condition [App. (2) Table 6];

(iii) ,durc [mm] is the safety cover, , 0 durc mm ;

(iv) ,dur stc [mm] is the reduction due to using stainless steel, , 0 dur stc mm ;

[1] Flexural Design Manual for Advanced Design 26

(v) ,dur addc [mm] is the reduction due to additional concrete protection,

, 0 dur addc mm ;

(vi) devc [mm] is the deviation (tolerances) in actual pouring.

The following design workflow may prove useful:

[ 1 > Exposure conditions are explained in:

[ a > (SR EN 1992-1-1:2006, pp. 43-48) or [Appendix (2) C] Concrete Cover;

[ b > Recommended values for pre-design purposes: exposure conditions XC2 and

structural class, S4;

[ 2 > Choose the material (concrete and steel grade), if no values are imposed:

[ a > (SR EN 1992-1-1:2006, pp. 24-42 and 190-191) or [Appendix 3] Materials

Properties;

[ b > Recommended values for pre-design purposes: C30/37 for concrete and S500

for steel;

[ 3 > Choose the steel bar size (herein diameter will be referred to by size to

account for deformed bars that have ribs outside their diameter as opposed to

plain bars):

[ a > Recommended values:

[ i > ,max 6 [ ]sl mm for slabs;

[ ii > ,max 25 [ ]sl mm for beams;

[ iii > ,max 28 [ ]sl mm for columns;

[ iv > 8 [ ] 6 12 (14) [ ]sw mm mm for stirrups;

[ 4 > Calculate nominal concrete cover for both stirrups ,nom swc and the longitudinal

reinforcement ,nom slc ;

[ 5 > Calculate design concrete cover (see previous [(1) Figure 3]):

,

,

max

nom sw

vnom sl sw

cc

c

[ 6 > Check if design concrete cover is at least the minimum cover after casting:

min,v dur devc c c

[Note (1-C) a] Sizing of the Cross Section

It is well known that a board set flat over two supports will bend downward when

pressed upon in direct ratio to the height of the board. Therefore, it is necessary to

avoid excessive deflections. This condition and other requirements as presented in the

subsequent list:

[ 1 > Rigidity conditions (see [Appendix 6] Selected Service Requirements);

[ 2 > Fire safety conditions (see [Appendix 5] Fire Resistance);

Note(s)

[1-7]

[1-8]

27 Manual for Advanced Design Flexural Design [1]

(1) Table 1

Recommended

Slab Thickness

Note

(1) Table 2

Recommended

height-to-width

ratios

[ 3 > Technological conditions (usually likely expressed as span-to-depth ratios).

60 [mm] for roofs 80 [mm] for industrial structures

70 [mm] for civil structures 100 [mm] for pavements

Slab thickness

(1) The indicated values are minimal. In actual practice higher values should be cast.

Similar conditions applied to beams will lead to establishing the height of the cross section h as a multiple of [50 mm] if the result is less than [800 mm] or as a multiple of

[100 mm] otherwise. The width of the cross section b is also a multiple of [50/100

mm] accordingly.

1,5 h/b 3,0

for rectangular sections

2,0 h/b 4,0

for tee sections

bmin 200 [mm]

Height-to-width ratios

The above are true for members with no connections to other structural/non-structural members that may prevent the member to fall over prior to fixing in its final position in

the structure (mainly precast elements). This is ALWAYS a transitory state for precast

members; more details are available in (KISS & ONE, 2010, pp. 332-342);

For all other members a thinner web is recommended 3h b as for reinforced

concrete members the height is more relevant to design than the width. Still, a maximum

ratio should be 5h b because a higher ratio may lead to a cross section too thin to

withstand shear or in danger of developing lateral buckling or to a particular type of

beam, the so called deep beam which has other reinforcement particularities because of

the specific development of principal stresses. This should be avoided as the best solution

(section and corresponding reinforcement) should be reached in one run (with the first

chosen section).

After calculating the concrete cover and establishing the dimensions for the cross

section, design may proceed to the main step, design of the corresponding reinforcement,

first for flexure and second for shear force.