Eurocode 2 : background & applications — Eurocode 2 : background & applications — design of concrete buildings, worked examples (JRC)

EUROCODE 2: BACKGROUND & APPLICATIONS DESIGN OF CONCRETE BUILDINGS Worked examples Authors: F.Biasioli, G.Mancini, M.Just, M.Curbach, J.Walraven, S.Gmainer, J.Arrieta, R.Frank, C.Morin, F.Robert Editors: F.Biasioli, M.Poljanšek, B.Nikolova, S.Dimova, A.Pinto 2014 RReeppoorrttEEUURRx2x6xx5x66ENEN

European Commission Joint Research Centre Institute for the Protection and Security of the Citizen Contact information Address: Joint Research Centre, Via Enrico Fermi 2749, TP 480, 21027 Ispra (VA), Italy E-mail: eurocodes@jrc.ec.europa.eu Tel.: +39 0332 78 9989 Fax: +39 0332 78 9049 http://ipsc.jrc.ec.europa.eu/ http://www.jrc.ec.europa.eu/ Legal Notice Neither the European Commission nor any person acting on behalf of the Commission is responsible for the use which might be made of this publication. Europe Direct is a service to help you find answers to your questions about the European Union Freephone number (*): 00 800 6 7 8 9 10 11 (*) Certain mobile telephone operators do not allow access to 00 800 numbers or these calls may be billed. A great deal of additional information on the European Union is available on the Internet. It can be accessed through the Europa server http://europa.eu/. JRC89037 EUR 26566 EN ISBN 978-92-79-36548-5 ISSN 1831-9424 doi:10.2788/35386 Luxembourg: Publications Office of the European Union, 2014 © European Union, 2014 Reproduction is authorised provided the source is acknowledged. Printed in Italy

Contents List of authors and editors....................................................................................................... vii Foreword..................................................................................................................................... ix CHAPTER 1.................................................................................................................................. 1 CONCEPTUAL AND PRELIMINARY DESIGN 1.1 Introduction........................................................................................................................ 3 1.2 Basic data........................................................................................................................... 4 1.2.1. General data and preliminary overview.................................................................... 4 1.2.2. Slab geometry .......................................................................................................... 6 1.3 Actions................................................................................................................................ 9 1.3.1. Densities, self-weight, imposed loads, partial and combination factors ................... 9 1.3.2. Wind ....................................................................................................................... 10 1.3.3. Snow load .............................................................................................................. 12 1.4 Materials ........................................................................................................................... 12 1.4.1. Concrete................................................................................................................. 12 1.4.1.1 Exposure classes and concrete strength class ...................................... 12 1.4.1.2 Exposure classes, structural classes and concrete cover ..................... 13 1.4.2. Reinforcing steel .................................................................................................... 17 1.4.2.1 Steel characteristics ............................................................................... 17 1.4.2.2 Maximum bar diameters ........................................................................ 17 1.5 Conceptual design of slabs ............................................................................................ 18 1.5.1. Slab height ............................................................................................................. 18 1.5.1.1 Slenderness ........................................................................................... 18 1.5.1.2 Slab height determination ...................................................................... 20 CHAPTER 2................................................................................................................................ 23 STRUCTURAL ANALYSES 2.1 Finite element modelling of the building ....................................................................... 25 2.2 Loads, load cases and their combinations ................................................................... 26 2.2.1. Loads ..................................................................................................................... 26 i

2.2.2. 2.2.3. 2.2.4.

Load cases for dead loads ..................................................................................... 26 2.2.2.1 Load case 1 – dead load of the bearing structure.................................. 26 2.2.2.2 Load case 2 – dead load of the interior.................................................. 27 2.2.2.3 Load case 3 – dead load of the facade .................................................. 27 Load cases for wind, snow and service loads 1 and 2 ........................................... 28 2.2.3.1 Load case 51 ......................................................................................... 28 2.2.3.2 Load case 101 ....................................................................................... 28 2.2.3.3 Load case 201 ....................................................................................... 29 2.2.3.4 Load cases 202 to 206........................................................................... 29 2.2.3.5 Load cases 1326, 1336, 1356 and 1366................................................ 30 2.2.3.6 Load cases 10001, 10011, 10021 and 10031........................................ 31 2.2.3.7 Load cases 10101, 10111, 10121 and 10131........................................ 32 Rules for the combination of load cases ................................................................ 32

2.3 Internal forces and moments.......................................................................................... 32 2.3.1. Position of calculated internal forces and moments ............................................... 32 2.3.1.1 Column B2 ............................................................................................. 32 2.3.1.2 Shear wall B1......................................................................................... 32 2.3.1.3 Frame axis 2 – beam ............................................................................. 32 2.3.1.4 Frame axis B – beam............................................................................. 33 2.3.1.5 Punching for flat slab ............................................................................. 33 2.3.2. Results of structural analysis ................................................................................. 33 2.3.2.1 Column B2 - results for ULS and SLS ................................................... 33 2.3.2.2 Shear wall B1, results for ULS and SLS ................................................ 36 2.3.2.3 Frame axis 2 – beam, results for ULS ................................................... 38 2.3.2.4 Frame axis B – Beam, results for ULS................................................... 42 2.3.2.5 Punching for flat slab, results for ULS.................................................... 47

CHAPTER 3................................................................................................................................ 49 LIMIT STATE DESIGN (ULS - SLS) 3.1 Introduction...................................................................................................................... 51 3.1.1. Motivation ............................................................................................................... 51 3.1.2. Materials................................................................................................................. 53 3.2 Ultimate limit state design .............................................................................................. 53 3.2.1. Slab on beams ....................................................................................................... 53

ii

3.2.2. 3.2.3.

3.2.1.1 Static model and cross section of the slab on beams ............................ 53 3.2.1.2 Determination of the bending reinforcement in general ......................... 55 3.2.1.3 Determination of the bending reinforcement for the T-beams ............... 57 3.2.1.4 Design of beams for shear ..................................................................... 59 3.2.1.5 Design of slabs supported by beams ..................................................... 64 Flat slab.................................................................................................................. 68 3.2.2.1 Loads and internal forces for the calculation of the flat slab .................. 68 3.2.2.2 Determination of the bending reinforcement .......................................... 68 3.2.2.3 Punching shear - column B2 .................................................................. 70 3.2.2.4 Column B2 ............................................................................................. 75 3.2.2.5 Design of shear walls ............................................................................. 84 Slab with embedded elements ............................................................................... 86 3.2.3.1 Bending reinforcement - beam axis B ................................................... 87 3.2.3.2 Bending reinforcement of the slab with embedded elements ................ 89 3.2.3.3 Shear capacity ....................................................................................... 91

3.3 Serviceability limit states................................................................................................ 92 3.3.1. SLS deflection - general ......................................................................................... 92 3.3.1.1 Deflection control by calculation ............................................................ 92 3.3.1.2 Tabulated K values and basic ratios (l/d) ............................................... 94 3.3.1.3 Slab on beams ....................................................................................... 94 3.3.1.4 Flat slab ................................................................................................. 95 3.3.1.5 Slab with embedded elements ............................................................... 95 3.3.2. SLS crack width - general ...................................................................................... 96 3.3.2.1 Crack width control – slab with embedded elements ............................. 98

CHAPTER 4.............................................................................................................................. 101 DETAILING OF THE REINFORCEMENT 4.1 Detailing - general.......................................................................................................... 103 4.1.1. Anchorage length ................................................................................................. 103 4.1.2. Lap length ............................................................................................................ 105 4.2 Detailing of structural members................................................................................... 107 4.2.1. Detailing of footing B-2 ......................................................................................... 107 4.2.1.1 Design of the footing ............................................................................ 107 4.2.1.2 Arrangement of the reinforcement ....................................................... 110

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4.2.2. 4.2.3. 4.2.4.

Detailing of beams ............................................................................................... 114 4.2.2.1 Beam A2 – B2 – C2 for the case 1 ...................................................... 115 4.2.2.2 Beam B1 – B2 – B3 for the case 3....................................................... 118 Detailing of slabs .................................................................................................. 121 4.2.3.1 Slab AB12 for case 1 ........................................................................... 121 Detailing of columns ............................................................................................. 123 4.2.4.1 Column B2 for the case 2 .................................................................... 124 4.2.4.2 Column B2 - case 2 ............................................................................. 125

CHAPTER 5.............................................................................................................................. 129 SOME GEOTECHNICAL ASPECTS OF BUILDING DESIGN (EN 1997) 5.1 Introduction.................................................................................................................... 131 5.2 Geotechnical data.......................................................................................................... 131 5.3 Actions on the foundations .......................................................................................... 133 5.3.1. Structural and geotechnical actions ..................................................................... 133 5.3.2. General: the three design approaches of Eurocode 7 ......................................... 134 5.3.2.1 Design approach 1 (DA1) .................................................................... 134 5.3.2.2 Design approach 2 (DA2 and DA2*) .................................................... 136 5.3.2.3 Design approach 3 (DA3) ................................................................... 137 5.3.2.4 Summary for DA1, DA2 and DA3 (for “fundamental” combinations) ... 137 5.4 Column B2 - design of foundation ............................................................................... 138 5.4.1. Bearing capacity (ULS) ........................................................................................ 138 5.4.2. Sliding (ULS) ........................................................................................................ 139 5.5 Comments on settlements (SLS) ................................................................................. 140 5.5.1. Compensated foundation ..................................................................................... 140 5.5.2. Calculation based on the results of Mènard pressuremeter method.................... 141 5.5.3. Adjusted elastic method ....................................................................................... 141

CHAPTER 6.............................................................................................................................. 145 FIRE RESISTANCE ACCORDING TO EN 1992-1-2 6.1 Introduction.................................................................................................................... 147 6.2 Data concerning building.............................................................................................. 147

iv

6.2.1. 6.2.2. 6.2.3. 6.2.4. 6.2.5.

Description of the building .................................................................................... 147 Mechanical material properties ............................................................................ 150 6.2.2.1 General ................................................................................................ 150 6.2.2.2 Concrete .............................................................................................. 150 6.2.2.3 Reinforcing bars................................................................................... 151 Materials’ physical and thermal properties ........................................................... 153 6.2.3.1 Thermal strain of concrete and steel.................................................... 153 6.2.3.2 Concrete specific heat ......................................................................... 154 6.2.3.3 Concrete thermal conductivity.............................................................. 154 6.2.3.4 Density of concrete and reinforcing bars ............................................. 154 Reinforced members’ sections (column, beam and slab) .................................... 155 6.2.4.1 Column B2 ........................................................................................... 155 6.2.4.2 Beam in axis 2 ..................................................................................... 155 6.2.4.3 Slab on beams ..................................................................................... 156 Actions ................................................................................................................. 156

6.3 Tabulated data ............................................................................................................... 157 6.3.1. Scope ................................................................................................................... 157 6.3.2. Column 500/52..................................................................................................... 158 6.3.3. Beam 250/44........................................................................................................ 159 6.3.4. Slab 180/36/49 ..................................................................................................... 160 6.3.5. Summary .............................................................................................................. 162

6.4 Simplified calculation methods .................................................................................... 162 6.4.1. Methodology......................................................................................................... 162 6.4.2. Column ................................................................................................................. 163 6.4.3. Continuous beam ................................................................................................. 166 6.4.3.1 Mid-span .............................................................................................. 166 6.4.3.2 Internal support .................................................................................... 166 6.4.3.3 End support.......................................................................................... 168 6.4.3.4 Summary.............................................................................................. 168 6.4.4. Two-way slab ....................................................................................................... 168

6.5 Advanced calculation methods .................................................................................... 170

6.6 Conclusions ................................................................................................................... 170

v

vi

List of authors and editors

Authors

FOREWARD

Martin POLJANŠEK

Silvia DIMOVA

Borislava NIKOLOVA

Artur PINTO

European Laboratory for Structural Assessment (ELSA) Institute for Protection and Security of the Citizen (IPSC) Joint Research Center (JRC), European Commission European Laboratory for Structural Assessment (ELSA) Institute for Protection and Security of the Citizen (IPSC) Joint Research Center (JRC), European Commission European Laboratory for Structural Assessment (ELSA) Institute for Protection and Security of the Citizen (IPSC) Joint Research Center (JRC), European Commission European Laboratory for Structural Assessment (ELSA) Institute for Protection and Security of the Citizen (IPSC) Joint Research Center (JRC), European Commission

CHAPTER 1 - Preliminary design of buildings

Francesco BIASIOLI

Dipartimento di Ingegneria Strutturale, Edile e Geotecnica (DISEG), Politecnico di Torino, Italy

Giuseppe MANCINI

Dipartimento di Ingegneria Strutturale, Edile e Geotecnica (DISEG), Politecnico di Torino, Italy

CHAPTER 2 -

Structural analyses Martin JUST Manfred CURBACH

TU Dresden, Institute for concrete structures, Germany TU Dresden, Institute for concrete structures, Germany

CHAPTER 3 -

Limit state design (ULS-SLS)

Just WALRAVEN

Delft University of Technology, The Netherlands

Susanne GMAINER

Vienna University of Technology, Austria

CHAPTER 4 - Detailing of the reinforcement

Jose ARRIETA

PROES S.A., Spain

CHAPTER 5 -

Some geotechnical aspects of building design

Roger FRANK

Université Paris-Est, Ecole des Laboratoire Navier Geotechnical France

Ponts ParisTech, team (CERMES),

CHAPTER 6 -

Fire resistance according to EN 1992-1-2

Caroline MORIN

CERIB, France

Fabienne ROBERT

CERIB, France

vii

Editors

Francesco BIASIOLI Martin POLJANŠEK Borislava NIKOLOVA Silvia DIMOVA Artur PINTO

Dipartimento di Ingegneria Strutturale, Edile e Geotecnica (DISEG), Politecnico di Torino, Italy European Laboratory for Structural Assessment (ELSA) Institute for Protection and Security of the Citizen (IPSC) Joint Research Center (JRC), European Commission European Laboratory for Structural Assessment (ELSA) Institute for Protection and Security of the Citizen (IPSC) Joint Research Center (JRC), European Commission European Laboratory for Structural Assessment (ELSA) Institute for Protection and Security of the Citizen (IPSC) Joint Research Center (JRC), European Commission European Laboratory for Structural Assessment (ELSA) Institute for Protection and Security of the Citizen (IPSC) Joint Research Center (JRC), European Commission

viii

Foreword The construction sector is of strategic importance to the EU as it delivers the buildings and infrastructure needed by the rest of the economy and society. It represents more than 10% of EU GDP and more than 50% of fixed capital formation. It is the largest single economic activity and it is the biggest industrial employer in Europe. The sector employs directly almost 20 million people. Construction is a key element not only for the implementation of the Single Market, but also for other construction relevant EU Policies, e.g. Sustainability, Environment and Energy, since 40-45% of Europe’s energy consumption stems from buildings with a further 5-10% being used in processing and transport of construction products and components. The EN Eurocodes are a set of European standards which provide common rules for the design of construction works, to check their strength and stability against live extreme loads such as fire and earthquakes. In line with the EU’s strategy for smart, sustainable and inclusive growth (EU2020), standardization plays an important part in supporting the industrial policy for the globalization era. The improvement of the competition in EU markets through the adoption of the Eurocodes is recognized in the "Strategy for the sustainable competitiveness of the construction sector and its enterprises" – COM (2012)433, and they are distinguished as a tool for accelerating the process of convergence of different national and regional regulatory approaches. With the publication of all the 58 Eurocodes Parts in 2007, the implementation of the Eurocodes is extending to all European countries and there are firm steps toward their adoption internationally. The Commission Recommendation of 11 December 2003 stresses the importance of training in the use of the Eurocodes, especially in engineering schools and as part of continuous professional development courses for engineers and technicians, which should be promoted both at national and international level. It is recommended to undertake research to facilitate the integration into the Eurocodes of the latest developments in scientific and technological knowledge. In light of the Recommendation, DG JRC is collaborating with DG ENTR and CEN/TC250 “Structural Eurocodes” and is publishing the Report Series ‘Support to the implementation, harmonization and further development of the Eurocodes’ as JRC Scientific and Policy Reports. This Report Series includes, at present, the following types of reports: 1. Policy support documents – Resulting from the work of the JRC in cooperation with partners and stakeholders on ‘Support to the implementation, promotion and further development of the Eurocodes and other standards for the building sector’; 2. Technical documents – Facilitating the implementation and use of the Eurocodes and containing information and practical examples (Worked Examples) on the use of the Eurocodes and covering the design of structures or its parts (e.g. the technical reports containing the practical examples presented in the workshop on the Eurocodes with worked examples organized by the JRC); 3. Pre-normative documents – Resulting from the works of the CEN/TC250 and containing background information and/or first draft of proposed normative parts. These documents can be then converted to CEN technical specifications 4. Background documents – Providing approved background information on current Eurocode part. The publication of the document is at the request of the relevant CEN/TC250 SubCommittee; 5. Scientific/Technical information documents – Containing additional, non-contradictory information on current Eurocode part, which may facilitate its implementation and use, or preliminary results from pre-normative work and other studies, which may be used in future revisions and further developments of the standards. The authors are various stakeholders involved in Eurocodes process and the publication of these documents is authorized by relevant CEN/TC250 Sub-Committee or Working Group. Editorial work for this Report Series is assured by the JRC together with partners and stakeholders, when appropriate. The publication of the reports type 3, 4 and 5 is made after approval for publication from the CEN/TC250 Co-ordination Group. The publication of these reports by the JRC serves the purpose of implementation, further harmonization and development of the Eurocodes. However, it is noted that neither the Commission ix

nor CEN are obliged to follow or endorse any recommendation or result included in these reports in the European legislation or standardization processes. This report is part of the so-called Technical documents (Type 2 above) and contains a comprehensive description of the practical examples presented at the workshop “Design of concrete buildings with the Eurocodes” with emphasis on worked examples. The workshop was held on 20-21 October 2011 in Brussels, Belgium and was organized by the Joint Research Centre of the European Commission together with CEN/TC250/Sub-Committee 2 and Politecnico di Torino, with the support of CEN and the Member States. The workshop addressed representatives of public authorities, national standardisation bodies, research institutions, academia, industry and technical associations involved in training on the Eurocodes. The main objective was to facilitate training on Eurocode 2 through the transfer of knowledge and training information from the Eurocode 2 writers (CEN/TC250 Sub-Committee 2) to key trainers at national level and Eurocodes users. The workshop was a unique occasion to compile a state-of-the-art training kit comprising the slide presentations and technical papers with the worked example of a concrete structure designed following Eurocode 2. The present JRC Report compiles all the technical papers and the worked example prepared by the workshop lecturers. The editors and authors have sought to present useful and consistent information in this report. However, it must be noted that the report does not present complete design example and that the reader may still identify some discrepancies between chapters. The chapters presented in the report have been prepared by different authors therefore are partly reflecting the different practices in the EU Member States. Users of information contained in this report must satisfy themselves of its suitability for the purpose for which they intend to use it. We would like to gratefully acknowledge the workshop lecturers and the members of CEN/TC250 Sub-Committee 2 for their contribution in the organization of the workshop and development of the training material comprising the slide presentations and technical papers with the worked example. We would also like to thank especially Prof. Francesco Biasioli for the contribution, coordination of lecturers and support to the workshop. All the material prepared for the workshop (slides presentations and JRC Report) is available to download from the “Eurocodes: Building the future” website (http://eurocodes.jrc.ec.europa.eu). M. Poljanšek, S. Dimova, B. Nikolova, A. Pinto European Laboratory for Structural Assessment (ELSA) Institute for the Protection and Security of the Citizen (IPSC) Joint Research Centre (JRC) x

CHAPTER 1 CONCEPTUAL AND PRELIMINARY DESIGN Francesco BIASIOLI and Giuseppe MANCINI Dipartimento di Ingegneria Strutturale, Edile e Geotecnica (DISEG), Politecnico di Torino, Italy

Preliminary design of buildings F. Biasioli and G.Mancini

1.1 Introduction

Preliminary design of buildings F. Biasioli and G.Mancini

The series of European standards commonly known as “Eurocodes”, EN 1992 (Eurocode 2, in the following also listed as EC2) deals with the design of reinforced concrete structures – buildings, bridges and other civil engineering works. EC2 allows the calculation of action effects and of resistances of concrete structures submitted to specific actions and contains all the prescriptions and good practices for properly detailing the reinforcement.

EC2 consists of three parts:

o EN 1992-1

Design of concrete structures - Part 1-1 General rules and rules for buildings, Part 1-2 Structural fire design (CEN, 2002)

o EN 1992-2

Design of concrete structures - Part 2: Concrete Bridges – Design and detailing rules (CEN, 2007).

o EN 1992-3

Design of concrete structures - Part 3: Liquid retaining and containment structures (CEN, 2006).

In the following, the principles of Eurocode 2, part 1-1 are applied to the design of a simple design case study - a six-storey building with two underground parking storeys. This is similar to the one used for the Workshop on “Eurocode 8: Seismic Design of Buildings”1. The aim of the exercise is to have two case studies referring to the same building, carrying the same vertical loads but two different sets of horizontal actions (EC2: vertical loads + wind; EC8: vertical loads + earthquake).

Design team

The design process has been divided between different authors, some of whom were involved in the preparation and/or assessment of Eurocode 2.

The parts and the authors are:

o F. Biasioli/G. Mancini: Basis of Design /Materials /Durability/ Conceptual design

o M. Curbach:

Structural Analysis

o J. Walraven:

Limit states design 1 (ULS - SLS)

o J. Arrieta:

Detailing of reinforcement and members

o R. Frank

Foundation Design

o F. Robert

Fire design

This document deals with the definition of actions, the assessment of durability, the selection of materials and the “conceptual design” of the geometry of the structure.

Conceptual design has been defined as “….choosing an appropriate solution among many possible ones, in order to solve a particular problem taking into account functional, structural, aesthetical and sustainability requirements…” 2. Using a sort of “reverse engineering” of a number of EC2 equations, suitable hints and rules to be used during this design stage are proposed.

1 Details on Eurocode 8 workshop: see http://eurocodes.jrc.ec.europa.eu/showpage.php?id=335_2 2 H. Corres Peiretti - Structural Concrete Textbook - fib bulletin 51, Lausanne 2012

1.2 Basic data

Preliminary design of buildings F. Biasioli and G.Mancini

1.2.1. General data and preliminary overview The building (6 floors + 2 underground parking levels) is ideally located in an urban area (terrain category IV for wind actions) not close to the sea, at 300 m AMSL (Above Mean Sea Level). The ground floor is occupied by offices open to public, 1st to 5th floor are for dwellings and the roof is not open to public. The dimensions in plan are (30,25 x 14,25) m, the plan surface is 431 m2 and the height is h = 25 m (Figures 1.2.3 – 1.2.4). A conventional working life of 50 years is assumed for design. The structure consists mainly of reinforced concrete frames. Compared to the EC8 example previously cited, the size of the building and the layout of columns remain unchanged, but the geometry of some vertical elements (columns and walls) has been changed: shear walls from the staircase area are substituted by columns as well as two new walls have been added along outer alignments no.1 and 6 (see Figure 1.2.3 and Figure 2.1.1. in Chapter 2.1).

Fig.1.2.1 Slab deformation The reason for this is that in the EC8 example, the internal stairs wall and lift core were used as bracing elements. Assuming that horizontal loads (due to wind or earthquakes) are uniformly applied to the slab (if due to the limited stiffness of columns this is idealized as a beam supported by the core only), Figure 1.2.1 gives evidence of the slab deformations in both X- and Y-directions. To reduce these and to increase the building’s torsional rigidity, peripheral walls should be added (see Figure 1.2.2). As the stiffness of the slab in the X-direction is relevant, as well as the stiffness of the core in the same direction, walls in the X-direction may not be present. The inner columns are founded on square spread foundations of dimensions (B·L) = (2·2) m; the outer columns and the shear walls are supported by a peripheral diaphragm retaining wall of width 0,6 m, height 9 m, embedded 3 m below the 2 levels of the parking (see Chapter 5). Fig.1.2.2 Peripheral walls

Preliminary design of buildings F. Biasioli and G.Mancini Fig.1.2.3 EC2 Reinforced concrete building – sections 5

1.2.2. Slab geometry

Preliminary design of buildings F. Biasioli and G.Mancini

Slab bays have equal spans in X- and Y-directions: in X-direction such equality never gives the most efficient structural solution, as long-term deflection of the outer bays governs the depth h of the slab. With some lack of optimization - the X-direction spans of the outer bays, wherever possible, should be not greater than 90% of those of the adjacent inner bays. To cover different building solutions commonly used in Europe, three slab alternatives have been considered. The first (A-A) solution is a two-way concrete solid slab with depth3 h = 18 cm supported by (0,25 x 0,32) m protruding beams in the X- and Y-directions (Figure 1.2.4). Justification of the geometry of structural elements is given later. This solution reduces both depth and reinforcement of the slab, but has the inconvenience of protruding beams, which may hinder matching the internal walls sequence, especially for dwellings. For a safe preliminary evaluation of the quantity of materials the slab “voids” due to stairs and lifts may be assumed as not present. The resulting extra volume of concrete takes into accounts deformations of formworks and any loss of concrete during casting (pump filling, etc.). The overall quantity of concrete is (30,25x14,25)·0,18+(6·14,25+3·30,25)·0,32·0,25 = 77,6 + 14,1 = 91,7 m3.

Fig.1.2.4 Slab h = 0,18 m on (0,25 x 0,32) m beams 3 Justification of the height for the three solutions are given later in the chapter. 6

Preliminary design of buildings F. Biasioli and G.Mancini The second (B-B) solution is a flat concrete solid slab with depth h = 24 cm spanning in both X- and Y-directions (Figure 1.2.5). This solution, very common in a number of countries, has the advantage of the absence of protruding beams. In recent years it has been improved by the availability of advanced scaffolding and formwork systems (lightweight elements with dropheads allowing easy scaffolding and early striking) as well as reinforcing systems (ordinary steel “carpets” tailor designed, or post-tensioned unbonded tendons in the case of longer spans and/or heavier weights). The design focuses in the case of flat slabs are: 1) deflection (in this case also governed by the outer bays) 2) punching resistance (columns C7 and C10). Ignoring as before the “voids” of stairs and lifts, the overall quantity of concrete (dimensions in m) is (30,25·14,25)·0,24=103,4 m3. Fig.1.2.5 Flat Slab h = 0,24 m

Preliminary design of buildings F. Biasioli and G.Mancini The third (C-C) solution is a slab of total height h = 0,23 m with embedded lighting clay elements4. Ribs spanning in Y-direction are supported by protruding T-beams in X-direction (lateral: (0,25 x 0,30) m; central: (0,25 x 0,17) m. Two beams in Y-direction (0,25 x 0,30) m are provided at both slab ends (Figure 1.2.6). Fig.1.2.6 Slab with embedded elements h = 0,23 m In Figure 1.2.7 an example of a clay lighting element is given: it forms every 0,50 m a T-section with web height h = 0,18 m and width bw = 0,12 m. The flange depth is hf = 0,05 m. These elements require a supporting scaffolding plan, an alternative being filigree concrete slabs with embedded EPS which have embedded temporary reinforcement therefore can be laid down on discrete supports. 4 Besides clay, eembedded lighting elements could also be made of EPS (expanded polystyrene), concrete, plastics or wood.

Preliminary design of buildings F. Biasioli and G.Mancini

Fig.1.2.7 Clay lighting element h = 0,23 m The unit concrete quantity for 1 m element (2 ribs) is (2·0,12·0,18) + 0,05 = 0,093 m3/m2. The overall quantity of concrete, evaluated as before, and disregarding stairs and lifts voids is (30,252·0,40)·[14,25-(2·0,50+0,60)]·0,093+[2·(0,40 ·0,23+0,25·0,30)]·(14,25-2·0,50)+[2· (0,50·0,23+ +0,25·0,30)+(0,60·0,23+0,25·0,17)]·30,25=34,6+4,4+17,0=56,0 m3. Compared to the two other, this is the solution requiring the least amount of concrete.

1.3 Actions

Actions have been obtained from the following parts of EN1991: o EN1991-1.1 Densities, self-weight and imposed loads o EN1991-1.2 Fire actions o EN1991-1.3 Snow loads o EN1991-1.4 Wind loads Due to the limited dimensions of the building, thermal actions were not considered, nor were impact and explosion actions. Where available, gamma partial safety factors are taken as the suggested values in EC2.

1.3.1. Densities, self-weight, imposed loads, partial and combination factors

Self-weight G1 Reinforced concrete: Embedded clay elements h = 0,18 m: Permanent loads G2 Finishing, pavement, embedded services, partitions: Walls on external perimeter (windows included): Variable loads Qi

γG = 1,35 (unfavourable) 25 kN/m3 0,75 kN/m2 γG = 1,35 ÷ 1,0 3,0 kN/m2 8,0 kN/m γQ = 1,50 ÷ 0

Preliminary design of buildings F. Biasioli and G.Mancini

Table 1.3.1 Variable loads

Type Dwelling Stairs, office open to public Snow Parking

qk 2 (kN/m ) 2,00 4,00 1,70 2,50

Ψ0 0,70 0,50 0,70

Ψ2 0,30 0,00 0,60

1.3.2. Wind

The combination factors considered are:

Ψ0 = 0,60,

Ψ2 = 0,0

As already outlined, the building is ideally located in an urban area (terrain category IV) not close to the sea, at 300 m AMSL (Above Mean Sea Level). The assumed values (EN 1991-1.4) are:

o Basic wind velocity

vb = cdir cseason vb,0

cdir = cseason = 1,0

vb,0 = 30 m/s vb = vb,0 = 30 m/s

o Terrain category IV

z0 = 1 zmin = 10 m

o Terrain factor

 z0 0,07

 10 0,07

kr  0,19    0,19    0,234 m/s

 0,05 

 0,05 

o Orography factor co = 1,0

o Turbulence intensity

kl = 1,0

lv (z)  kl co (z) ln z z0

1 ln z z0

o Exposure factor ce(z) taking into account turbulence z ≤ 10 m

z



z

2 10 



10 

c z  c z   k c ln 7  c ln  0,23 1 ln 7  1 ln  1,13 const.

e

e min r 0 z  o z 

1

1

0

0

z > 10 m

z

z

ce z  kr co ln  7  co ln   0,053 lnz 7  lnz.

z0 

z0 

o Basic velocity pressure

Preliminary design of buildings F. Biasioli and G.Mancini

qb  12 ρvb2  12 1,25  302 10-3  0,563 kN/m2

o Peak velocity pressure

z ≤ 10 m z > 10 m

qp (ze) = ce (z) qb = ce (z) 0,563 kN/m2 qp (ze) = ce (zmin) qb = ce (10) 0,563 kN/m2

o Wind pressure on external surfaces

cpe = + 0,8 cpe = - 0,4

Fig.1.3.1 Wind pressure on external surfaces o Structural factor cs cd = 1,0 (framed buildings with structural walls less than 100 m high) Wind pressure on external surfaces we = qp (ze) cpe cs cd = qp (ze) (0,8 - (- 0,4))·1 = 1,2 qp (ze) kN/m2 o z > 10 m we (ze) = 1,2·ce (ze) 0,56 = 0,035·ln (ze) [7 + ln (ze)] kN/m2 we (19) = 1,04 kN/m2 o z ≤ 10 m we (10) = 0,0357·ln (10) [7 + ln (10)] = 0,77 kN/m2

Fig.1.3.2 Wind pressure on external surfaces – example 11

1.3.3. Snow load

Preliminary design of buildings F. Biasioli and G.Mancini

The snow load is described in Chapter 2.

1.4 Materials 1.4.1. Concrete 1.4.1.1 Exposure classes and concrete strength class EC2 requires (2.4) that “…the structure to be design such that deterioration over its design working life does not impair the performance of the structure below that intended, having due regard to its environment and the anticipated level of maintenance…”. Environmental influences are therefore considered for assessing the durability of concrete structures. EC2 basically refers to a a) a 50-years design working life, b) “normal” supervision during execution, c) “normal” inspection and maintenance during use. Quality management procedures to be adopted during execution are described in EN13670. For what concerns deterioration of concrete and corrosion of reinforcing steel due to potentially aggressive environment, the designer has to identify the (anticipated) conditions of the environment where the structure will be located “…in order to take adequate provision for protection of the materials used in the structure…”. Environmental exposure conditions are classified by “exposure classes”. An example is given in Figure 1.4.1.

Fig.1.4.1 Environmental exposure classes5 5 Legend : Ambiente di terra = ground environment; ambiente marino = seaside environment; gelo e pioggia = frost and water; aria marina e gelo = sea airborne and frost; senza gelo = no frost; acqua salmastra = salty water; acqua di mare = sea water

Preliminary design of buildings F. Biasioli and G.Mancini ”Deemed to satisfy” rules related to the different exposure classes of the structural members are given in: EN206-1 Annex F (concrete standard) for a) minimum concrete strength class and b) concrete composition; and in EN 1992-1 for c) minimum concrete cover to reinforcement and, for more critical exposure classes, d) maximum allowed crack width. In both EC2 and EN206 concrete strength is used as indirect measure of concrete durability, on the basis of the assumption: higher strength → less porous concrete → higher durability. Complementary information is given in EN206 National Annexes about the maximum water/cement ratio and minimum cement content per m3 of concrete. The result is a large variation of requirements in different EU countries6.

1.4.1.2 Exposure classes, structural classes and concrete cover

Exposures classes are identified by the letter X followed by the initial letter (in English) of the deterioration mechanism to which they refer: o Corrosion of reinforcement due to Carbonation (XC) or to chlorides from De-icing agents, industrial wastes, pools (XD), or from Sea water (XS) o Deterioration of concrete due to Freeze/thaw action (XF) or chemical Attack (XA). According to EC2, chapter 4, to design the minimum concrete cover required for all reinforcement (stirrups included) the sequence is: 1. Identify the exposure class(es) for the different structural elements 2. Identify the MINIMUM strength class for each exposure class(es) (EC2 Informative annex E and EN206, Annex F - use multiple classes only if separate casting sequences are provided (e.g. foundations vs. walls, columns vs. slabs etc.); 3. Identify the MINIMUM cover for both durability (“dur”) and bond (“b”)

cmin = max [cmin,b; (cmin,dur - cdur,add); 10 mm]

(1.1)

Concrete surface protections, if any, are taken into account by the cover reduction term cdur,add. 4. Identify the nominal reinforcement concrete cover cnom (Figure 1.4.2) to be used in drawings and for reinforcement detailing

cnom = max [(cmin + c); 20 mm]

(1.2)

Besides bond and steel corrosion protection, cnom has also to take into account resistance to fire. c = 0 - 10 mm is the «execution tolerance».

hd d'

d' Fig.1.4.2 Nominal concrete cover cnom

long/2 staffe d' c nom

6 CEN/TR 15868:2009 - Survey of national requirements used in conjunction with EN206-1:2000

Preliminary design of buildings F. Biasioli and G.Mancini

As it is not possible to generalise or take into account the provisions valid in all countries, in the design example the following exposure classes and related concrete strength classes have been used:

XC1

for internal slabs, beams and foundations

C25/30

XC2

for columns

C30/37 > C25/30

Even when the environment is the same for slabs and columns, the concrete class used for columns should preferably be higher than the one used for slabs and beams, as suggested by the Eurocode 8 “capacity design” rule: in the case of seismic structures, to guarantee building stability and avoid the “soft storey” mechanism, plastic hinges required for energy dissipation have to occur in horizontal elements only and never in vertical ones (Figure 1.4.3).

NO

YES

Fig.1.4.3 Soft-storey mechanism

To identify the minimum concrete cover for durability cmin,dur once the environmental class and the related concrete strength class have been identified, the “structural class” may be chosen using Eurocode 2, Table 4.3N. Eurocode 2 considers structural class S4 as the “default” one, to be modified on the basis of criteria listed in Table 1.4.1.

Table 1.4.1 Recommended structural classification (EC2 Table 4.3N)

Criterion

Structural Class

Exposure Class according to Table 4.1 (Eurocode 2)

X0

X1

XC2/XC3

XC4

XD1

XD2 / XS1 XD3 / XS2 / XS3

Design Working Life of 100 years

increase class by 2

increase class by 2

increase class by 2

increase class by 2

increase class by 2

increase class by 2

increase class by 2

Strength Class 1) 2) Member with slab geometry (position of reinforcement not affected by construction process) Special Quality Control of the concrete production ensured

≥ C30/37 reduce class by 1 reduce class by 1 reduce class by 1

≥ C30/37 reduce class by 1 reduce class by 1 reduce class by 1

≥ C35/45 reduce class by 1 reduce class by 1 reduce class by 1

≥ C40/50 reduce class by 1 reduce class by 1 reduce class by 1

≥ C40/50 reduce class by 1 reduce class by 1 reduce class by 1

≥ C40/50 reduce class by 1 reduce class by 1 reduce class by 1

≥ C45/55 reduce class by 1 reduce class by 1 reduce class by 1

Preliminary design of buildings F. Biasioli and G.Mancini

Assuming a 50 years working life and no special concrete production Quality Control, for this example the structural classes are

o Slabs:

concrete C25/30

S(4 – 1) = S3 reduction for slab geometry

o Beams:

concrete C25/30

S4

no reduction

o Columns:

concrete C30/37

S4

no reduction

On the basis of the environmental and structural classes the minimum concrete cover for durability may be identified (Table 1.4.2 below is the one used in Eurocode 2, national tables may be different):

Structural Class S1 S2 S3 S4 S5 S6

Table 1.4.2 Values of minimum cover (Eurocode 2 Table 4.4N)

Environmental Requirement for cmin,dur (mm)

Exposure Class according to Table 4.1 (Eurocode 2)

X0

XC1 XC2 / XC3

XC4

XD1 / XS1 XD2 / XS2

XD3 / XS3 30 35 40 45 50 55

o cmin,dur - slabs o cmin,dur - beams o cmin,dur - columns

(XC1/S3) = 10 mm (XC1/S4) = 15 mm (XC2/S4) = 25 mm

Assuming c,dev = 5 mm for controlled execution, the calculated nominal cover to reinforcement cnom is:

o slabs o beams o columns

cnom = cmin,dur + c,dev = max(15+5; 20) = 20 mm cnom = max(20+5; 20) = 25 mm cnom = max(25+5; 20) = 30 mm

Resulting cover values have always to be rounded upwards to the nearest 5 mm. For earth retaining walls and foundations cnom = 40 mm is common, due to the difficulty of any visual inspection to detect deterioration On the basis of the EC2 nationally determined parameters and EN206 national provisions, the whole procedure can be easily implemented in an Excel™ spreadsheet (Figure 1.4.4).

Preliminary design of buildings F. Biasioli and G.Mancini

Concrete cover

Parameters 1 Exposure class 2 Freeze/thaw 3 Strenght class 4 Service life 5 Slab or similar? 6 Quality control? 7 Max bar diam. (mm) 8 ∆cdur,st 9 ∆cdur,γ 10 ∆cdur,add

Suggested C30/37 0 0 0

User defined XC3 ‐ C30/37 50 NO NO 16 0

11 Δctoll

A) Recommended

12 Structural class

S4

13 cmin,dur

14 cmin,b

15 cmin

16 cnom

0 X0 XC1 XC2 XC3 XC4 XS1 XS2 XS3 XD1 XD2 XD3

Cmin,dur 5 10 15 20 25 30 35 40 45 50 55 60

Fig.1.4.4 Excel procedure for calculating the minimum concrete cover

Alternatively, national tables with all information related to the required concrete composition may be prepared. An example valid for Italy is given in Figure 1.4.5.

Fig.1.4.5 Minimum concrete cover (Copriferro) for durability, concrete strength class and concrete composition (composizione del calcestruzzo) (Italy data)

1.4.2. Reinforcing steel

Preliminary design of buildings F. Biasioli and G.Mancini

1.4.2.1 Steel characteristics Medium ductility S500 B (grade 500 class B) reinforcing steel has been adopted. In the idealised and design stress-strain diagrams the lower elasto-plastic design curve B without stress-hardening has been used (Figure 1.4.6).

. Fig.1.4.6 Reinforcing steel – design stress-strain diagrams

Assuming partial safety coefficients γs = 1,15 for Ultimate Limit State (ULS - persistent and transient design situation) and γs = 1,0 for Serviceability Limit States (SLS), the characterizing values of the diagram are:

o Strength

fyk  500 N/mm2; Es = 200 kN/mm2; fyd = 500 / 1,15 = 435 N/mm2;

(fy,max ≤ 1,30 fyk , fyk ≤ 650 N/mm2) εs,yd = fyd / Es = 435/200 = 2,17‰

o Ductility

k = (ft / fy) k  1,08

εuk  5%

εud = 0,90εuk  4,5%

1.4.2.2 Maximum bar diameters The design of the geometry of concrete structures, especially of concrete buildings, is increasingly governed by considerations of Serviceability Limit States (SLS - deformation, cracking, stress limitation) rather than those of Ultimate Limit States (ULS). It is therefore important to identify in EC2 the limiting values for the different SLSs, if any, to be considered in design. For crack widths up to a maximum of 0,30 mm – the upper limit for all environmental classes according to EC2, Table 7.1N - the SLS of cracking may be verified without calculation by limiting either the diameter of reinforcing bars as a function of steel stress, or their maximum spacing. For a S500 B steel and various concrete classes stress Table 1.4.3 gives maximum bar diameters as a function of steel stress ratio σs / fyk evaluated in a cracked section under the quasi permanent (QP) load condition – bold values are EC2 ones.

Preliminary design of buildings F. Biasioli and G.Mancini

Table 1.4.3 Maximum bar diameters for crack control

Steel 500B

Concrete class

C20/25

C25/30

C30/37

C35/45

fct,eff

2,3

2,6

2,9

3,4

σs

σs / fyk

Φl,max for crack width wk=0,30 mm

0,32

0,34

0,36

0,38

0,40

0,42

0,44

0,46

0,48

0,52

0,56

Note: EC2 values up to fyk; 25 mm for σs = 200 MPa

C40/50 3,6 38 36 34 32 28 26 24 22 20 16 14

In conceptual design commonly used bar diameters are first selected, then the related maximum limiting values of σs,QP / fyk are identified. In this case they are:

o Slabs:

ϕ14 mm

C25/30

σs,QP / fyk ≤ 0,48

o Beams:

ϕ16 mm

C25/30

σs,QP / fyk ≤ 0,42

o Columns:

ϕ20 mm

C30/37

σs,QP / fyk ≤ 0,44

These limiting ratios will be considered in design (see forward).

1.5 Conceptual design of slabs 1.5.1. Slab height 1.5.1.1 Slenderness The design of slabs has to fulfil both Serviceability Limit States (SLS) and Ultimate Limit States (ULS) requirements (in this exact order!). In general the height “h” of slabs is controlled by the deflection limits (EC2 7.4). In the case of flat slabs, punching frequently also governs. In EC2 the deemed-to-satisfy rule for verifying SLS deflection is based on the limitation of elements’ “slenderness” by setting maximum “slenderness ratios” (lef /d) of the “effective span” lef (axis-to-axis distance in the case of supporting beams, or centre-to-centre distance of columns in the case of flat

Preliminary design of buildings F. Biasioli and G.Mancini

slabs) to the “effective depth”, d, (distance of the centroid of the tensile forces from the most compressed concrete fibre). For flat slabs with spans ≤ 8,5 m and slab and beams with spans ≤ 7 m, as in this example, EC2, paragraph 7.4.2 gives the formula

lef

310  l 

500 As,prov  l 

 Ks    Ks



d

σs  d 0

fyk As,req  d 0

(1.3)

K takes into account the type, restraints and relative position of each element in the structural system. It transforms the “effective span” into the so-called “normalized span”

ln = lef / K

(1.4)

of an ideal simply supported (SS) element which has the same deflection as the actual element. For a slab of constant height h subjected to the same permanent G2 and variable Q loads, the bay governing the whole slab height, the one with the maximum “normalized span”, can be easily identified as the one having the greatest normalized span ln.

Fig.1.5.1 K values vs. structural systems

As already mentioned K values demonstrate that continuous slabs or beams with spans all equal, as in this example, never represent a good structural solution: the span of the external slab (or beam) should never be longer than (1,3/1,5)·100 = 87% of the adjacent internal slab span. The “shape” factor s, takes into account the geometry of the slab transverse section, in particular the variation of its section moment of inertia. EC2 assumes s = 1,0 for R(rectangular) sections, as solid slabs, or compact T-sections with b/bw ≤ 3; s = 0,80 for T-sections with b/bw > 3, as is the case of slabs with embedded elements (Figure 1.2.7) where b / bw = 500 / 120 = 4,8 > 3. (l/d)o is the “reference span to depth ratio”. In the case of slabs with ρ ≤ ρ0 EC2 gives:

l 

ρ

   11 1,5 fck  3,2 fck

 d 0

ρ0

 ρ 3   1  ρ0 

ρ0  103 fck

(1.5)

ρ = As / (bd) is the geometrical reinforcement ratio of the steel area As in tension. For a number of selected concrete classes (l/d)o and ρ0 values calculated with the formulae above are given in Table 1.5.1.

Preliminary design of buildings F. Biasioli and G.Mancini

Table 1.5.1 (l/d)o and ρ0 values for selected concrete classes

C20/25 C25/30 C30/37 C32/40 C35/45

ρ0 (%)

0,45

0,50

0,55

0,57

0 ,59

(l/d)o

In Figure 1.5.2 the (l/d)o hyperbolic formula above is represented on the left side: bullet points calculated for ρ = ρ0 divide (lightly-reinforced) slabs from (medium- to heavily-reinforced) beams.

Fig.1.5.2 (l/d)o function of geometrical reinforcement ratio ρ

In the case of low reinforcement ρ ratios the EC2 formula gives very high (l/d)oo slenderness values which, if not properly assessed, could lead to excessively slender slabs. The maximum (l/d)o value should therefore be limited – in the diagram and for this example (l/d)o,max = 36. Assuming ln = lef /k, fyk = 500 N/mm2 and As,prov = As,req the minimum effective depth dmin is:

lef

500 As,prov  l  ln  l 

ln

 Ks

    s    dmin 

d

fyk As,req  d 0 d  d 0

l d 0 s

(1.6)

As stated before, Eqn (1.6) confirms that the governing bay in a continuous slab is the one characterized by the maximum normalized length, ln.

1.5.1.2 Slab height determination Identification of the slab height is an iterative process based on the successively better estimations of the (initially unknown) self-weight G1 of the slab. A (safe) preliminary evaluation of slab self-weight G1 may be based on the height calculated for the ρ = ρ0 condition. For concrete class C25/30 Table 1.5.1 gives ρ0 = 0,50% then (l/d)o = 20. For the three different slab types the first trial minimum effective depths dmin for deflection control are:

Preliminary design of buildings F. Biasioli and G.Mancini

Table 1.5.2 Minimum effective depth for deflection control

lef,x

lef,y

lef

K

ln (l/d)0 s dmin

m

m

m

m

m

Slab on beams 6,0 7,125 6,0 1,3 4,62 20 1,0 0,23

Flat slab

6,0 7,125 7,125 1,2 5,94 20 1,0 0,30

Slab with emb. el. - 7,125 7,125 1,3 5,48 20 0,8 0,27

Due to the high reinforcement ratios ρ0, these effective depths dmin are no doubt conservative. On the basis of the conventional reinforced concrete unit weight (25 kN/m3) and the actual or “equivalent” (for ribbed or slabs with embedded lighting elements – see below) solid slab depth h (m):

G1 = 25h

(kN/m2)

For slabs with lighting embedded clay elements of width (38+12) cm with 5 cm concrete topping, the equivalent height results to be 51 ÷ 55% of the height of a solid slab (see Table 1.5.3) . As an example, if h = 0,23 m

G1 = 25h = 25·(0,54·0,23) = 3,10 kN/m2

Table 1.5.3 Equivalent height for slabs with lighting embedded elements

hle [m] 0,16 0,18 0,20 0,22 0,24

h=hle + 0,05 [m] 0,21 0,23 0,25 0,27 0,29

G1 [kN/m2] 2,89 3,08 3,27 3,46 3,69

heq = G1/25 [m] 0,116 0,123 0,131 0,138 0,148

heq / htot 0,55 0,54 0,52 0,51 0,51

On the basis of the already determined ϕl = 14 mm (SLS cracking), and cnom = 20 mm: o for flat slabs with bars in X- and Y-direction h = dmin + cnom + ϕl = dmin + 20 + 14 = (dmin + 34) mm o for slabs with embedded lighting elements h = dmin + cnom + 1/2ϕl = (dmin + 27) mm As the sustained permanent load G2 is known, the total permanent load G is: G = G1 + G2. If the variable load Q and the related factor for the quasi-permanent (QP) load condition ψ2 are known, an improved normalized slenderness ratio may be calculated with the formula:

 ln 

λs

  3

 d  G  ψ2Q

(1.7)

The formula is based on SLS deformation, and the ULS design of simply supported slabs of “normalized span” ln subjected to uniform loads G and Q. λs depends on the concrete class and the shape factor s only. Values for slabs are given in Table 1.5.4.

Preliminary design of buildings F. Biasioli and G.Mancini

s for s = 1,0 s for s =0,8

Table 1.5.4 λs for selected concrete classes

C20/25 53 49

C25/30 57 53

C30/37 60 56

C35/45 63 59

C40/50 65 61

The iterative process can be easily implemented in an Excel™ spreadsheet, as outlined below. The reduction of the initial height is evident, two or three iterations are enough to identify the final height.

Table 1.5.5 Iterative reduction of the height

dmin dmin+d’ coeff heq

G1

G2

Qk W2

tot

ls Ln/d

ln

dmin

m m m kN/m2 kN/m2 kN/m m

1st iteration

Slab on beams 0,23 0,26 1,00 0,26 6,62 3,0 2,0 0,30 10,22 57 26 4,62 0,18 -23%

Flat slab

0,30 0,33 1,00 0,33 8,27 3,0 2,0 0,30 11,87 57 25 5,94 0,24 -20%

Slab with l,el, 0,27 0,30

2nd iteration Slab on beams 0,18 0,21

Flat slab

0,24 0,27

0,55 0,17 4,14 1,00 0,21 5,26 1,00 0,27 6,82

3,0 2,0 0,30 7,74 53 27 5,48 0,21 -25% 3,0 2,0 0,30 8,86 57 27 4,62 0,17 -5% 3,0 2,0 0,30 10,42 57 26 5,94 0,23 -4%

Slab with l,el, 0,21 0,23

3rd iteration Slab on beams 0,17 0,20

Flat slab

0,23 0,26

0,55 0,13 3,20 1,00 0,20 5,06 1,00 0,26 6,56

3,0 2,0 0,30 6,80 53 28 5,48 0,20 -4% 3,0 2,0 0,30 8,66 57 28 4,62 0,17 -1% 3,0 2,0 0,30 10,16 57 26 5,94 0,23 -1%

Slab with l,el, 0,20 0,22 0,55 0,12 3,08 3,0 2,0 0,30 6,68 53 28 5,48 0,20 -1%

The height values so obtained, based as they are on the Eurocode 2 equations, automatically guarantee the respect of SLS deformation and have not to be further checked.

CHAPTER 2 STRUCTURAL ANALYSES Martin JUST and Manfred CURBACH TU Dresden, Institute for concrete structures, Germany

Structural analyses M.Just and M.Curbach

Structural analyses M.Just and M.Curbach 2.1 Finite element modelling of the building The example building is a six-storey building with two underground parking levels, described in 1.2.1, Chapter 1. At the broad side there are three rows of columns (axes A, B, C (and D in levels -1 and -2)), in length there are six rows (axes 1 to 6). These columns support the slabs. Three different slab types (see Chapter 1) have been considered regarding the slabs: a) Flat slab with height h = 21 cm, directly supported by the columns b) Slab on beams, 2 spans, h = 18 cm, loads flow through beams into columns c) Slabs with embedded lighting elements h = 23 cm, loads flow through the coffered ceiling beams into supporting beams, which lead into the columns Figure 2.1.1 shows the finite element model of the building, modelled with SoFiSTiK® software: Fig.2.1.1 Finite element model of the building (type flat slab)

Structural analyses M.Just and M.Curbach 2.2 Loads, load cases and their combinations

2.2.1. Loads

Several multiple loads are present: the dead load of the construction, the interior (finishing, pavement, etc.) and the facade, service loads and two environmental loads – wind and snow. The following tables list the loads, their classes and the factors for combination:

Class Dead load Environmental Load 1 Environmental Load 2 Service Load 1 Service Load 2

Table 2.2.1 Loads

Load name

Value of load

dead load of construction dead load of interior dead load of facade

Variable values 3,0 kN/m² 8,0 kN/m

wind (below 1000m above sea level) snow on roof or external area dwelling (level 1-6) stairs, office (level 0) parking (level -1, -2, external area)

0,77 kN/m² below 10 m 1,09 kN/m² at 19 m between 10 m and 19 m linear rising 1,70 kN/m² 2,00 kN/m² 4,00 kN/m² 2,50 kN/m²

Ψ0 Ψ2 - 0,6 0 0,5 0 0,7 0,3 0,7 0,6

2.2.2. Load cases for dead loads 2.2.2.1 Load case 1 – dead load of the bearing structure Dead loads of the structure are automatically calculated by FEM-Software on the basis of the geometry and unit weight of materials. Figure 2.2.1 shows the dead load for one storey.

Fig.2.2.1 Calculated dead load (values in kN/m and KN/m²) 26

Structural analyses M.Just and M.Curbach 2.2.2.2 Load case 2 – dead load of the interior The dead load of the interior is modelled in load case 2. Therefore an area load is set on all plate elements. Figure 2.2.2 shows the dead load for one storey. Fig 2.2.2 Dead load of the interior (values in KN/m²) 2.2.2.3 Load case 3 – dead load of the facade In load case 3 the dead loads from the facade are modelled. In slab types a) and b) the loads are area set on the outer row of finite elements of the slabs, which have a size of (0,5 x 0,5) m, and a load value of 16 kN/m². In slab type c) they are set on the outer beams as line loads. In this case, the load value is as given in Table 2.2.1. Figure 2.2.3 shows the dead load for one storey for type a). Fig.2.2.3 Dead load of the facade (values in KN/m²)

Structural analyses M.Just and M.Curbach 2.2.3. Load cases for wind, snow and service loads 1 and 2 2.2.3.1 Load case 51 Load case 51 contains the loads from wind in global X-direction of the building (parallel to the longer side). The loads are set on the columns, therefore the area loads have been converted to line loads. The facade is assumed to carry the loads as a two-span girder to the columns. Figure 2.2.4 shows the resulting loads. Fig.2.2.4 Arrangement of the loads for load case 51 (values in kN/m) 2.2.3.2 Load case 101 Load case 101 contains the loads from wind in global Y-direction of the building (perpendicular to the longer side). As in load case 51 loads are set on the columns and therefore the surface loads are converted to line loads. The facade is assumed to carry the loads as a six-span girder to the columns. Figure 2.2.5 shows the resulting loads. Fig.2.2.5 Arrangement of the loads for load case 101 (values in kN/m)

Structural analyses M.Just and M.Curbach 2.2.3.3 Load case 201 In load case 201 the snow on the roof is modelled as an area load on all plate elements of the roof slab as shown in Figure 2.2.6. Fig.2.2.6 – Arrangement of the loads for load case 201 (values in kN/m²) 2.2.3.4 Load cases 202 to 206 Load cases 202 to 206 contain the field by field snow loads on the external area. In load case 202 the load is set on the field between axes 1 and 2, in 203 between 2 and 3 and so on. The arrangement of the snow loads for load case 202 is shown in Figure 2.2.7. Fig.2.2.7 Arrangement of the loads for load case 202 (values in kN/m²) 29

Structural analyses M.Just and M.Curbach 2.2.3.5 Load cases 1326, 1336, 1356 and 1366 Load cases 1326, 1336, 1356 and 1366 contain the service load 1 on the roof in variable arrangements, as in the following figures. Figure 2.2.8 shows a combination chosen to maximize moment and shear action effects in the beam along axis 2, Figure 2.2.9 - a combination where all fields are loaded and Figure 2.2.10 shows a combination to maximize bi-axial moments in the columns. Fig.2.2.8 Arrangement of the loads for load case 1326 (values in kN/m) Fig.2.2.9 Arrangement of the loads for load case 1336 (values in kN/m)

Structural analyses M.Just and M.Curbach Fig.2.2.10 – Arrangement of the loads for load case 1356 (values in kN/m) Figure 2.2.11 shows a combination chosen to maximize moment in field 2 of the beam in axis B. Fig.2.2.11 – Arrangement of the loads for load case 1366 (values in kN/m) More combinations have been examined, but these were not governing the design. 2.2.3.6 Load cases 10001, 10011, 10021 and 10031 These load cases contain the service load 1 for the levels 0 to 5 in variable arrangements. Load case 10001 is similar to load case 1356. Levels 0, 2 and 4 have the arrangement as in Figure 2.2.10; levels 3 and 5 have the contrary arrangement. Load case 10011 is similar to load case 1366. Levels 0, 2 and 4 have the same arrangement (Figure 2.2.11). The Levels 3 and 5 have the opposite arrangement (load on fields between axis 1 - 2, 3 - 4 and 5 - 6). Load case 10021 is similar to load case 1326. All levels from 0 to 5 have the same arrangement, which is shown in Figure 2.2.8. Load case 10031 equals load case 1336 – the load acts on all fields in all levels.

Structural analyses M.Just and M.Curbach 2.2.3.7 Load cases 10101, 10111, 10121 and 10131 These load cases contain the service load 2 for the levels -1 and -2 and for the external area in variable arrangements, which are the logical continuation of the arrangements from levels 0 ÷ 6. For example load case 10101 belongs to load case 10001. The parking loads were set in their own load cases, because they have another exposure class then the loads from dwelling and offices, as shown in 2.2.1.

2.2.4. Rules for the combination of load cases

All dead load cases are assumed to act at every time. The load cases 51 and 101 are assumed to never act at the same time. The snow load cases 201 to 206 can act at the same time, if authoritative. From load cases 1326, 1336, 1356 and 1366 only one can act at a time, this applies for 10001, 10011, 10021 and 10031, also for 10101, 10111, 10121 and 10131. Furthermore one load case from 1326, 1336, 1356 and 1366 and one load case from 10001, 10011, 10021 and 10031 together have the predominant influence, because they are in the same class. For the ultimate limit state (ULS) only one combination is calculated (general combination):

γGG  γQ,1Q1  γQ,i (ψ0,iQi )

(2.1)

with γG = 1,35 and γQ,1 = γQ,i = 1,5. For the serviceability limit states (SLS) the two following combinations are calculated: o characteristic combination:

G  Q1  (ψ0,iQi )

(2.2)

o quasi permanent combination:

G  (ψ 2,iQi )

(2.3)

2.3 Internal forces and moments 2.3.1. Position of calculated internal forces and moments 2.3.1.1 Column B2 The internal forces for column B2 are calculated right above the foundation at the bottom end of the column. Also the values at the top of the column (in the parking level “-2”) that are linked to the values at the bottom end are calculated. For the design of the foundation data are calculated at the sole of the foundation. 2.3.1.2 Shear wall B1 The internal forces for shear wall B1 are calculated at the bottom end of the wall (ground level). 2.3.1.3 Frame axis 2 – beam The calculated internal forces and moments represent the beam in axis 2 on the 2nd floor for the slab type “Slab on beams.”

Structural analyses M.Just and M.Curbach 2.3.1.4 Frame axis B – beam The calculated internal forces and moments represent the beam in axis B on the 2nd floor for the slab type “slab with embedded elements.” The results are only shown for the fields 1 to 3 (while 4 and 5 are mirrored results of 1 and 2). 2.3.1.5 Punching for flat slab The given forces Vd are the design values that represent the shear forces which flow into the columns as normal forces at the columns A1 and B2. The values represent a flat slab situated in level 2. 2.3.2. Results of structural analysis With the exception of punching (see 2.3.1.5) all internal forces and moments are calculated as characteristic values then combined in varying combinations as shown in 2.2.4. For each of these combinations, different superpositions have been calculated: Data units are [kN] and [m]. o The maximum internal moment My and the according values for internal forces N, Vy and Vz and for the internal moment Mz are calculated. This superposition is called “max My”. o The maximum internal moment Mz and the according values for internal forces N, Vy and Vz and for the internal moment My are calculated. This superposition is called “max Mz”. o The maximum internal force Vy and the according values for internal forces N and Vz and for the internal moments My and Mz are calculated. This superposition is called “max Vy”. o The maximum internal force Vz and the according values for internal forces N and Vy and for the internal moments My and Mz are calculated. This superposition is called “max Vz”. o The maximum internal force N and the according values for internal forces Vy and Vz and for the internal moments My and Mz are calculated. This superposition is called “max N”. o The minimum internal moment My and the according values for internal forces N, Vy and Vz and for the internal moment Mz are calculated. This superposition is called “min My”. o The minimum internal moment Mz and the according values for internal forces N, Vy and Vz and for the internal moment My are calculated. This superposition is called “min Mz”. o The minimum internal moment Vy and the according values for internal forces N and Vz and for the internal moments My and Mz are calculated. This superposition is called “min Vy”. o The minimum internal moment Vy and the according values for internal forces N and Vy and for the internal moments My and Mz are calculated. This superposition is called “min Vz”. o The minimum internal moment N and the according values for internal forces Vy and Vz and for the internal moments My and Mz are calculated. This superposition is called “min N”. In case of the flat slab, only the maximum shear force VEd at two columns has been calculated. 2.3.2.1 Column B2 - results for ULS and SLS Results from the superposition in the ultimate limit state for the design of the column are in table 2.3.1. The combination is given in Eq. (2.1). The column local Y-axis equals the global X-axis parallel to the longer side of the building, the local Z-axis equals the (negative) global Y-axis perpendicular to the same side. Tables 2.3.1 to 2.3.3 show the combined values for the bottom end of the column, tables 2.3.4 to 2.3.6 the related values for the top of the column in parking level “-2”.

Superposition max My max Mz max Vy max Vz max N min My min Mz min Vy min Vz min N

Structural analyses M.Just and M.Curbach

Table 2.3.1 Results for column B2 (bottom end) - ULS

Nd [kN] -4517,82

Vy,d [kN] 0,23

Vz,d [kN] -4,05

My,d [kNm] 4,21

Mz,d [kNm] -0,31

Considered load cases

Q1

Qi

203 - 206, 1356, 10111

-4827,82 4,46 1,88 -2,43

4,45

10111

51, 203- 206, 10011

-4827,82 4,46 1,88 -2,43

4,45

10111

51, 203- 206, 10011

-5139,33 -2,46 2,96

-3,62

-2,08

10031, 10101

-4408,94 -1,83 2,27

-2,73

-1,38

202 – 205

-5300,62 -2,48 2,96

-3,64

-2,12

201, 1326, 10031, 10101

-5407,83 -5358,27

-4,65 -4,81

-1,43 -1,46

1,17 -2,09

-4,85 -4,70

10121 10121

101, 201, 202, 1326, 10021 201, 202, 1356, 10021

-4467,29 0,25 -4,05 4,20

-0,29

202- 206, 10111

-5697,49 -4,53 1,54

-2,36

-4,49

10031 & 1336

201, 10121

Table 2.3.2 gives results from Superposition for serviceability limit states SLS. The first combination is given in Eqn. (2.2).

Table 2.3.2 Results for column B2 (bottom end) for SLS – characteristic combination

Superposition max My

Nd [kN] -3339,34

Vy,d [kN] -0,10

Vz,d [kN] -2,63

My,d [kNm] 2,70

Mz,d [kNm] -0,45

Considered load cases

Q1

Qi

203 - 206, 1356, 10111

max Mz

-3546,00 2,72 1,31 -1,73 2,72 10111

51, 203- 206, 10011

max Vy

-3546,00 2,72 1,31 -1,73 2,72 10111

51, 203- 206, 10011

max Vz

-3753,68 -1,89 2,04 -2,52 -1,63 51

10031, 10101

max N

-3266,75 -1,47 1,58 -1,93 -1,17 51

202 – 205

min My

-3861,20 -1,91 2,04 -2,53 -1,66 51 201, 1326, 10031, 10101

min Mz min Vy

-3932,68 -3,35 -0,89 0,67 -3,48 10121 -3899,64 -3,46 1,04 -1,50 -3,38 10121

101, 201, 202, 1326, 10021 201, 202, 1356, 10021

min Vz

-3305,65 -0,08 -2,63 2,69 -0,44 101

202- 206, 10111

min N

-4125,78 -3,27 1,09 -1,68 -3,23 10031 & 1336

201, 10121

The second combination is given in Eqn. (2.3).

Structural analyses M.Just and M.Curbach

Table 2.3.3 Results for column B2 (bottom end) for SLS – quasi permanent combination

Superposition max My

Nd [kN] -3316,43

Vy,d [kN] -0,60

Vz,d [kN] 0,63

My,d [kNm] -1,04

Mz,d [kNm] -0,60

Considered load cases Qi 1356, 10111

max Mz

-3419,06 1,28

0,68

-1,11

1,18

10011, 10111

max Vy

-3419,06 1,28

0,68

-1,11

1,18

10011, 10111

max Vz

-3482,90 -2,71 0,82

-1,31

-2,64

10031, 10101

max N

-

-

-

-

-

Not applicable

min My

-3526,55 -2,83 0,92

-1,39

-2,73

1326, 10031, 10101

min Mz

-3596,67 -3,32 0,95

-1,35

-3,16

1326, 10021, 10121

min Vy

-3582,47 -3,32 0,95

-1,35

-3,15

1356, 10021, 10121

min Vz

-3301,99 -0,59 0,63

-1,04

-0,60

10111

min N

-4075,03 -3,20 1,07

-1,66

-3,17

1336, 10031, 10121

The values for the design of the foundation equal those given above, but dead load of the foundation has been included. The following tables give the values for the top of the column (in the parking level “-2”) corresponding to the superposition uses for the bottom end.

Table 2.3.4 Results for column B2 (top end) for ULS

Superposition max My

Nd [kN] -4492,51

Vy,d [kN] 0,23

Vz,d [kN] -4,05

My,d [kNm] -7,92

Mz,d [kNm] 0,60

Considered load cases

Q1

Qi

101 203 - 206, 1356, 10111

max Mz

-4802,51 4,46 1,88 3,18 -8,95 10111 51, 203- 206, 10011

max Vy

-4802,51 4,46 1,88 3,18 -8,95 10111 51, 203- 206, 10011

max Vz

-5114,02 -2,46 2,96 5,28 5,24

10031, 10101

max N

-4383,62 -1,83 2,27 4,10 4,05

202 – 205

min My min Mz min Vy

-5275,31 -5382,52 -5332,96

-2,48 -4,65 -4,81

2,96 -1,43 1,46

5,28 -3,10 2,32

5,28

10,09 10121

9,75 10121

201, 1326, 10031, 10101 101, 201, 202, 1326, 10021 201, 202, 1356, 10021

min Vz

-4441,97 0,25 -4,05 -7,93 0,56 101

202- 206, 10111

min N

-5672,18 -4,53 1,54 2,30 9,13 10031 & 1336

201, 10121

Results from Superposition for serviceability limit state. The first combination is given in Eqn. (2.2).

Structural analyses M.Just and M.Curbach

Table 2.3.5 Results for column B2 (top end) for SLS – characteristic combination

Superposition max My

Nd [kN] -3320,59

Vy,d [kN] -0,10

Vz,d [kN] -2,63

My,d [kNm] -5,20

Mz,d [kNm] 0,91

Considered load cases

Q1

Qi

101 203 - 206, 1356, 10111

max Mz

-3527,25 2,72 1,31 2,20 -5,45 10111 51, 203- 206, 10011

max Vy

-3527,25 2,72 1,31 2,20 -5,45 10111 51, 203- 206, 10011

max Vz

-3734,93 -1,89 2,04 3,61 4,01

10031, 10101

max N

-3248,00 -1,47 1,58 2,82 3,22

202 – 205

min My min Mz min Vy

-3842,45 -1,91 2,04 3,61 4,04

201, 1326, 10031, 10101

-3913,93 -3,35 -0,89 -1,98 7,24 10121

101, 201, 202, 1326, 10021

-3880,89 -3,46 1,04 1,63 7,01 10121 201, 202, 1356, 10021

min Vz

-3286,90 -0,08 -2,63 -5,20 0,89 101

202- 206, 10111

min N

-4107,03 -3,27 1,09 1,62 6,60 10031 & 1336

201, 10121

The second combination is given in Eqn. (2.3).

Table 2.3.6 Results for column B2 (top end) for SLS – quasi permanent combination

Superposition

Nd

Vy,d

Vz,d

[kN]

[kN] [kN]

max My

-3297,68 -0,60 0,63

max Mz

-3400,31 1,28 0,68

max Vy

-3400,31 1,28 0,68

max Vz

-3464,15 -2,71 0,82

max N

-

-

-

min My

-3507,80 -2,83 0,92

min Mz

-3577,92 -3,32 0,95

min Vy

-3563,72 -3,32 0,95

min Vz

-3283,24 -0,59 0,63

min N

-4065,28 -3,20 1,07

My,d [kNm] 0,85 0,93 0,93 1,17 1,38 1,49 1,50 0,84 1,56

Mz,d [kNm] 1,19 -2,66 -2,66 5,49 5,74 6,83 6,83 1,18 6,43

Considered load cases Qi 1356, 10111 10011, 10111 10011, 10111 10031, 10101 Not applicable 1326, 10031, 10101 1326, 10021, 10121 1356, 10021, 10121 10111 1336, 10031, 10121

2.3.2.2 Shear wall B1, results for ULS and SLS Results from the ULS superposition for the design of the column. The combination is given in Eqn. (2.1). Figure 2.3.1 shows as an example the superposition for the shear wall for “max Mz“.

Structural analyses M.Just and M.Curbach

Fig.2.3.1 – Superposition “max Mz” in ULS

Note: The local Y-axis of the column equals the global X-axis, parallel to the longer side of the building. The local Z-axis of the column equals the (negative) global Y-axis, perpendicular to the same side.

Table 2.3.7 Results for Shear Wall B1 (Bottom End) for ULS

Superposition max My

Nd [kN] -2392,60

Vy,d [kN] -22,90

Vz,d [kN] 10,17

My,d [kNm] 66,59

Mz,d [kNm] -35,53

Considered load cases

Q1

Qi

10021

max Mz

-2190,99 3,16 11,44 -29,59 -10,75 51

1336, 10111

max Vy

-2143,66 3,21 11,57 -29,44 -10,73 51

203, 10111

max Vz

-2178,68 -18,13 12,01 65,73 -25,93 101

205, 1356 ,10131

max N

-2143,70 2,85 10,91 -29,24 -11,78 51

-

min My min Mz min Vy min Vz min N

-2338,46 -2493,57 -2523,28 -2353,11 -2591,47

0,73 -24,85 -24,88 -21,59 -24,54

10,39 7,69 7,64 3,94 5,40

-30,71 38,22 38,61 -5,27 -3,55

-15,60 -39,31 -39,34 -32,94 -38,32

1366 & 10031 1366 & 10021 1366 & 10001 1326 & 10021

201, 205 ,206, 1336, 10001, 10131 101 101, 201 - 201, 10121

Results from the superposition for the SLS. The first combination is given in Eqn. (2.2).

Structural analyses M.Just and M.Curbach

Table 2.3.8 Results for shear wall B1 (bottom end) for SLS – characteristic combination

Superposition max My

Nd [kN] -1754,69

Vy,d [kN] -16,62

Vz,d [kN] 7,23

My,d [kNm] 44,11

Mz,d [kNm] 25,64

Considered load cases

Q1

Qi

10021

max Mz

-1620,29 0,76 8,08 -20,41 -9,11

1336, 10111

max Vy

-1588,74 0,79 8,17 -19,91 -9,10

203, 10111

max Vz

-1612,08 -13,43 8,46 43,54 -19,23 101

205, 1356 ,10131

max N

-1588,76 0,55 7,73 -19,78 -9,80

-

min My min Mz min Vy min Vz min N

-1718,61 -0,86 7,38 -1822,01 -17,91 5,58 -1841,82 -17,93 5,55 -1728,37 -15,74 3,08 -1887,27 -17,71 4,06

The second combination is given in Eqn. (2.3).

-20,76 -12,35

25,19 25,45 -3,80

-28,16 -28,18 -23,91

-2,65 -27,50

1366 & 10031 1366 & 10021 1366 & 10001 1326 & 10021

201, 205 ,206, 1336, 10001, 10131 101 101, 201 - 201, 10121

Table 2.3.9 Results for shear wall B1 (bottom end) for SLS – quasi permanent combination

Superposition max My

Nd [kN] -1664,22

Vy,d [kN] -14,79

Vz,d [kN] 4,31

max Mz

-1609,81 -13,30 4,87

max Vy

-1596,30 -13,29 4,89

max Vz

-1603,16 -13,36 5,05

max N

-1596,29 -13,49 4,55

My,d [kNm] -2,72 -3,01 -2,96 -3,03 -2,85

Mz,d [kNm] -22,04 -18,92 -18,92 -19,09 -19,50

Considered load cases Qi 10021 1336, 10111 10011 1356, 10131 -

min My min Mz min Vy min Vz min N

-1643,80 -14,02 4,63 -1664,02 -14,80 4,27 -1664,02 -14,80 4,27 -1630,17 -14,16 4,12 -1677,96 -14,66 4,60

-3,31 -2,83 -2,74 -3,12 -2,85

-20,39 -22,06 -22,07 -20,82 -21,63

1336, 10001, 10131 1336, 10031 1366, 10021 1366, 10001 1326, 10021, 10121

2.3.2.3 Frame axis 2 – beam, results for ULS Figures 2.3.2 and 2.3.3 show the results for the superposition of “max My”:

54.37 75.12 97.32

11.37 31.67

75.45 89.80 102.99

-31.12 -6.56

-81.88 -57.82

-37.28 -15.30 2.34 21.50 42.11

-112.19 -145.77 -124.67

-101.15 -77.10

-56.49

-9.60

50.19 -8.00 7.19 -69.70 -132.79 -98.58

19.11

58.21

88.46 68.36

42.52 65.62

-66.90 12.39 -1.16 54.32

66.92 66.46

Structural analyses M.Just and M.Curbach Fig.2.3.2 Vz values for the superposition of “max My” Fig.2.3.3 My values for the superposition of “max My” For the superposition of “min My”, two cases have been considered. The first one is the minimum internal moment at the end support; the second is the minimum internal moment at the middle support. This was necessary to reduce the data output, because the file size of FEM-results grew too big otherwise. Figures 2.3.4 and 2.3.5 show the results for the superposition of “min My” for end support: Fig.2.3.4 Vz values for superposition “min My”

54.37 75.12 97.32

11.37 31.67

75.45 89.80 102.99

-31.12 -6.56

-81.88 -57.82

-37.28 -15.30 2.34 21.50 42.11

-112.19 -145.77 -124.67

-101.15 -77.10

Structural analyses M.Just and M.Curbach Fig.2.3.5 My values for the superposition of “min My” Figures 2.3.6 and 2.3.7 show the results for the superposition of “min My” for the middle support: Fig.2.3.6 Vz values for the superposition of “min My” Fig.2.3.7 Values for My for the superposition of “min My” Figures 2.3.8 and 2.3.9 show the results for the superposition of “max Vz” for shear forces at the end support:

-56.49

-9.60

50.19 -8.00 7.19 -69.70 -132.79 -98.58

19.11

58.21

88.46 68.36

42.52 65.62

-66.90 12.39 -1.16 54.32

66.92 66.46

Structural analyses M.Just and M.Curbach Fig.2.3.8 Values for Vz for the superposition of “max Vz” Fig.2.3.9 Values for My for the superposition of “max Vz” Figures 2.3.10 and 2.3.11 show the results for the superposition of “min Vz” for shear forces at the middle support: Fig.2.3.10 Values for Vz for the superposition of “min Vz” 41

54.37 75.12 97.32

11.37 31.67

75.45 89.80 102.99

-31.12 -6.56

-81.88 -57.82

-37.28 -15.30 2.34 21.50 42.11

-112.19 -145.77 -124.67

-101.15 -77.10

-56.49

19.11

42.52 -9.60 65.62

50.19 -8.00 7.19 -69.70 -132.79 -98.58

58.21

88.46 68.36

-66.90 12.39 -1.16 54.32

66.92 66.46

189.35 148.00

185.56 193.28 111.06 63.78

202.09 165.83

78.38 40.23

19.76 12.58

26.95

36.81

-36.17

-19.82 -27.01

-16.23

-59.87 -97.37

-67.15 -116.46

-162.67 -198.25

-208.44 -196.31

Structural analyses M.Just and M.Curbach Fig.2.3.11 Values for My for the superposition of “min Vz” 2.3.2.4 Frame axis B – Beam, results for ULS For the superposition of “max My”, two cases have been considered. The first case is the maximum internal moment in field one, the second is the maximum internal moment in the second field. This was necessary to reduce the data output, because the file size of FEM-results grew too big otherwise. Figures 2.3.12 and 2.3.13 show the results for the superposition of “max My” for the first field: Fig.2.3.12 Values for Vz for the superposition of “max My” Fig.2.3.13 Values for My for the superposition of “max My” Figures 2.3.14 and 2.3.15 show the results for the superposition of “max My” for the second field:

87.67 111.92 98.37 19.66 -46.06 -227.28 -28.56 -5.37 14.39 21.58 5.35 -5.46 -28.32 -231.30 -46.57

-182.78 -81.50

-170.62

131.61 -6.68

2.15 78.79 166.33 177.24

200.89 164.45

190.55 198.74 106.47 56.62

183.52 143.92 82.28 46.30

14.16

35.21 27.49 21.35

-14.19 -21.38 -27.03 -34.77

-49.82 -87.44 -156.70 -198.09 -33.16 -82.24 -172.07

-197.07

121.84 156.63 139.36 35.28 -43.84 -223.75 -29.63 -8.25 1.52 15.69 9.49 -8.29 -29.40 -227.77 -44.36

-189.75 -85.10

-167.27

-12.72

-3.98

90.74

52.76 117.11 124.90

Structural analyses M.Just and M.Curbach Fig.2.3.14 Values for Vz for the superposition of “max My” Fig.2.3.15 Values for My for the superposition of “max My” For the superposition of “min My”, three cases have been considered. The first case is the minimum internal moment at the end support, the second is the minimum internal moment at the first internal support and the third is the minimum internal moment at the second internal support. This was necessary to reduce the data output, because the file size of FEM-results grew too big otherwise. Figures 2.3.16 and 2.3.17 show the results for the superposition of “min My” for the end support: Fig.2.3.16 Values for Vz for the superposition of “min My” 43

182.98

185.30

199.01 162.83

147.00 54.51 4.71

151.39 67.21 23.63

15.33 10.29

22.52

34.92

-34.99

-17.51 -24.70

-13.92

-38.74 -84.00 -173.65

-18.38 -67.23 -158.95

-195.62

-198.25

Structural analyses M.Just and M.Curbach Fig.2.3.17 Values for My for the superposition of “min My” support: Figures 2.3.18 and 2.3.19 show the results for the superposition of “min My” for the first internal Fig.2.3.18 Values for Vz for the superposition of “min My” Fig.2.3.19 Values for My for the Superposition of “min My” support: Figures 2.3.19 and 2.3.20 show the results for the superposition of “min My” for the second internal

50.90 116.71 124.26

-10.39

-230.37

86.62

-23.50

-266.16 -117.29

84.65

110.93 96.35 14.97 -66.86 -312.21 -38.00

-15.72 2.50

8.84 -6.82

-17.45 -42.22

-54.91

-298.11

245.86 194.94 83.74 47.78

-53.34 -91.06

-215.58 -270.73

257.86 202.98

80.35 42.24

-62.43 -100.23 -221.24 -267.34

42.58

25.41

18.22 8.67 -15.86 -19.45 -26.67

-51.64

265.46 218.83

-163.68 -1.10 71.27 159.13 170.82 128.43 -6.31 -180.88 0.19 -72.65 121.65 150.99 132.69 26.86 -46.06 -227.28 -26.84 -6.94 8.39 13.74 -0.38 -10.03 -32.14 -222.58 -40.91

Structural analyses M.Just and M.Curbach Fig.2.3.19 Values for Vz for the superposition of “min My” Fig.2.3.20 Values for My for the superposition of “min My” Figures 2.3.21 and 2.3.22 show the results for the superposition of “max Vz” for the end support: Fig.2.3.21 Values for Vz for the superposition of “max Vz” 45

248.85 197.69 115.53 68.28

-47.53 -85.12

-154.26 -195.62

256.43 201.56

106.31 55.84

-59.57 -97.36

-162.66 -198.25

42.12

25.19

18.00 10.81 -13.09 -16.68 -23.87

-31.79

266.83 220.44

49.85 117.53 125.80

-12.95

-235.79

89.76

-17.19

-257.56 -112.98

86.34

111.48 96.24 13.30 -70.13 -317.36 -37.63

-18.03

-0.11 6.08 -8.97 -19.28 -41.17 -57.46

-302.19

248.85 197.69 85.65 49.53

-51.74 -89.44

-213.33 -268.40

256.43 201.56

79.30 41.13

-63.96 -101.82

-223.18 -269.07

36.81

25.12

17.93 8.58 -15.24 -18.84 -26.02

-46.34

266.83 220.44

Structural analyses M.Just and M.Curbach Fig.2.3.22 Values for My for the superposition of “max Vz” support: Figures 2.3.23 and 2.3.24 show the results for the superposition of “min Vz” for the first internal Fig.2.3.23 Values for Vz for the superposition of “min Vz” Fig.2.3.24 Values for My for the superposition of “min Vz” support: Figures 2.3.25 and 2.3.26 show the results for the superposition of “min Vz” for the second internal

0.80 56.15 117.94 167.89

-159.66

119.24

-23.50

51.90

-266.16 -76.42

108.64

150.23 88.52

22.48

-66.86 -312.21

-30.99

-5.60 9.74 15.88 -0.66 -11.63 -40.54 -218.60 -38.41

180.72 140.92 79.67 43.78

-70.48 -120.45

-215.58 -270.73

186.50 145.33

76.03 38.02

-83.37 -132.98 -221.24 -267.34

35.64

22.53

15.34 8.15

-16.54 -23.73

-31.45 -51.64

197.65 161.29

65.76 123.07 170.68 119.17

-12.95

-235.79

53.23

-83.99 -240.23

117.06

-11.69

150.76 98.18 19.66 -46.06 -227.28 -33.43

-17.22

7.07

-7.01 -16.25 -33.76 -57.46

-302.19

185.30 144.13

182.98 142.96 81.42 45.45

199.01 162.82

74.78 36.72

15.26 8.07

22.45

29.61

-27.40 -46.34

-16.07 -23.26

-85.31 -134.99 -223.19 -269.07

-68.35 -118.30

Structural analyses M.Just and M.Curbach Fig.2.3.25 Values for Vz for the superposition of “min Vz”

-213.33 -268.40

-317.36

-257.56 -72.65

-163.68

-222.58 -40.91

-13.07 -39.54

13.74 -2.34

-7.75 7.51

-43.52 -24.06

-70.12

-17.19

-1.10

20.26

54.47

150.93 87.33

109.94

123.38

55.19 118.68 169.92

Fig.2.3.26 Values for My for the superposition of “min Vz”

2.3.2.5 Punching for flat slab, results for ULS Results from the superposition for the ultimate limit state are in Table 2.3.10. The combination is given in Eqn. (2.1).

Position A1 B2

Table 2.3.10 Flat Slab - shear forces at columns A1 and B2

Vd [kN] 176,48 693,02

Considered load cases

Q1

Qi

10021

1356, 201

10021

202, 101

The serviceability limit states action effects have not been considered for the flat slab.

Structural analyses M.Just and M.Curbach

CHAPTER 3 LIMIT STATE DESIGN (ULS - SLS) Just WALRAVEN1 and Sausanne GMAINER2 1Delft University of Technology, The Netherlands 2Vienna University of Technology, Austria

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

3.1 Introduction

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

3.1.1. Motivation The aim of this paper is the design of a six-storey building with two underground parking levels according to Eurocode 2. The building is described in details in 1.2.1, Chapter 1. The view of the ground floor and the main sections of the building are given and shown in Figure 3.1.1 and Figure 3.1.2.

Fig. 3.1.1 View of the building in plan On the basis of the effects of the actions from the structural analysis (Chapter 2), the following consider the ultimate limit state for typical bending, shear, axial and punching cases in design procedures. To satisfy also the serviceability limit state criteria the calculation for limiting the crack width and the deflection for the critical members are presented. Three different types of horizontal slabs are considered: o Slabs on beams (hslab = 0,18 m, hbeam = 0,40 m) o Flat slab (h = 0,21 m) o Slabs with embedded lighting elements (h = 0,23 m, T-beams: h = 0,40 m) Furthermore the verification for the columns and walls is carried out.

Limit state design (ULS-SLS) J. Walraven and S. Gmainer Fig. 3.1.2 Section S1 and S2 of the building 52

3.1.2. Materials

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

In Table 3.1.1 the assumed materials data and ULS safety factors are shown.

Concrete class o Beams and slabs: C25/30 o Columns: C30/37 Environmental class XC2-XC3 cnom = 30 mm γc = 1,5

Table 3.1.1 Materials data Steel class o Grade 500 class B γs = 1,15

3.2 Ultimate limit state design 3.2.1. Slab on beams Figure 3.2.1 shows the considered beam (axis 2).

Fig. 3.2.1 Plan view of the slab on beams in axis 2 3.2.1.1 Static model and cross section of the slab on beams The static model for the main bearing beam and the cross section S2’ are in Figure 3.2.2: a continuous beam with a T-cross section where the effective width has to be defined.

Limit state design (ULS-SLS) J. Walraven and S. Gmainer According to chapter 5.3.2.1 (EC 2) the effective width beff is beff   beff ,i  bw  b , where beff ,i  0, 2bi  0,1l0  0,2l0 and beff ,i  bi .

(3.1) (3.2)

Fig. 3.2.2 Static system in axis 2 and cross section Inserting zero points of bending moments (see Figure 3.2.3) into Eqn.(3.1) and (3.2) leads to: Fig. 3.2.3 Zero points of bending moments Cross section of the T-beam at mid-span beff ,1  0,2b1  0,1(0,85l )  0,2·2875  0,1·(0,85·7125)  1181 mm  b1  2875 mm beff ,1  beff ,2  1181 m m beff  2·1181 250  2611 mm Cross section of the T-beam at intermediate support beff ,1  0,2·b1  0,1·(0,30·l )  0,2·2875  0,1·(0,30·7125)  789 mm  b1  2875 mm beff ,1  beff ,2  789 m m beff  2·788,8  250  1828 mm The internal forces for the beam axis 2 are presented in Figure 3.2.5. These maximum design moments MEd and shear forces VEd are the maximum values from different load cases in Chapter 2.

Limit state design (ULS-SLS) J. Walraven and S. Gmainer Fig. 3.2.4 Slab on beams with their effective width

Fig. 3.2.5 Internal forces MEd and VEd of the axis 2 (combination of maximum values from various load cases)

3.2.1.2 Determination of the bending reinforcement in general

The determination of the bending reinforcement based on the method with the simplified concrete design stress block (chapter 3.1.7 – EC 2) is illustrated in Figure 3.2.6 assuming

λ  0,8

η  1,0

as fck  50MPa

Fig. 3.2.6 Stress block according to EC 2 55

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

For the calculation of the bending reinforcement a design diagram can be derived (see Figure 3.2.7), which can generally be used for a rectangular compression zone. Bending of the section (according to Figure 3.2.6) will induce a resultant tensile force Fs in the reinforcing steel, and a resultant compressive force in the concrete Fc which acts through the centroid of the effective compressed area . For equilibrium, the ULS design moment MEd has to be balanced by the resisting moment MRd so that: MEd  Fc z  Fs z (3.3)

where z is the lever arm between the resultant forces Fc and Fs. In Eqn. (3.3) the following expressions apply:

Fc  fcd bx

(3.4)

zdx 2 Substituting Eqn. (3.4) and Eqn. (3.5) in Eqn. (3.3):

(3.5)

MEd  fcd bd ²(1 zd ) zd  2

(3.6)

or

MEd  (1 z ) z  2

bd ²fcd

dd

(3.7)

This can be written as

MEd  K bd ²fcd

(3.8)

where

K  (1 z ) z  2 dd

(3.9)

From equation 3.9 it follows that

z  0,5  0,25  0,5K , d

(3.10)

or written in another way:

z  0,5(1 1 2K ). d

(3.11)

These equations are valid under the assumption that the reinforcing steel yields before the concrete crushes. In order to define the limit of validity the “balanced” section is considered. It is mostly assumed that the balanced situation is reached for a depth of the compressed area equal to x = 0,45d.

Limit state design (ULS-SLS) J. Walraven and S. Gmainer The corresponding compression force is Fc,bal  fcd bx  fcd  b  0,8  0, 45  d  0,36bdfcd whereas the inner lever arm is: zbal  d  0,8  0,45d 2  0,82d Combining Eqn.(3.11) and (3.12) gives: Mbal  0,295bd 2fcd then 2 Mbal  0,295  K ' bd fcd From Eqn. (3.10) for this value of K’: z  0,82 d Taking this as a limit the resulting design diagram is in Figure 3.2.7.

(3.12) (3.13) (3.14) (3.15) (3.16)

Fig. 3.2.7 Ratio z/d as a function of K up to limit value K’ = 0,295 3.2.1.3 Determination of the bending reinforcement for the T-beams Cross section at mid-span The calculation for the bending reinforcement is done first for the cross section at mid-span (see Figure 3.2.8). The maximum ULS bending moment in span AB is MEd = 89,3 kNm. The effective depth d is 372 mm (see Chapter 2).

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

K  2 MEd  89,3 106  0,0148  0,295  bd fcd 2611,2  3722  25 1,5

 z  0,5(1 1 2K )  0,5  (1 1 2  0,0148)  0,9925 d Asl  1 ( MEd  NEd ) fyd z

1 89,3 106

Asl 

(

)  556 mm²

435 372  0,9925

This means 4ϕ14 = 616 mm² or 5ϕ12 = 565 mm².

(3.17)

Fig. 3.2.8 T-beam cross-section at mid-span

Cross section at intermediate support The cross section at intermediate support B is in Figure 3.2.9. The maximum ULS bending moment at the face of the support is MEd = 132,9 kNm.

MEd

132,9 106

K 2 

 0,230  0,295 

bd fcd 250  3722  25

1,5

z  0,5(1 1 2K )  0,5  (1 1 2  0,230)  0,867 d

1 132,9 106

Asl 

(

)  947 mm²

435 372  0,867

So 7ϕ14 = 1078 mm² or 9ϕ12 = 1018 mm² can be spread over the effective width EC2 suggests that part of the reinforcement is concentrated around the web zone

Fig. 3.2.9 Cross-section of T-beam - intermediate support B 58

Limit state design (ULS-SLS) J. Walraven and S. Gmainer 3.2.1.4 Design of beams for shear Control of shear capacity of the beams The shear force may be determined at distance d from the support, where VEd,red (see Figure 3.2.10) is : 129,9 3,36  VEd,red (3,36  0,372) VEd,red  115,52kN.

Fig. 3.2.10 Reduced shear force - support A

The first check according to Chapter 6.2.2 (EC 2) is the verification VEd ≤ VRd,c assuming a section without shear reinforcement. If this inequality does not hold, shear reinforcement is required.

VRd,c  [CRd,ck(100ρl fck )3 ]bwd

(3.12)

where

CRd,c  0,18  0,18  0,12 γc 1,5

k  1 200  2,0 d

k  1

200  1,73  2,0 372

ρl  Asl,prov  0,02 bw d

ρl  565 250  372  0,0061  0,61%

VRd,c  [0,12 1,73  (100  0,0061 25)3 ]  250  372  47902N  47,90kN

VRd,c  47,90 kN  VEd,red  115,52 kN 

Shear reinforcement is required. Figure 3.2.11 shows the model for the shear capacity at stirrup yielding (VRd,s) and web crushing (VRd,max).

Fig. 3.2.11 Shear capacity at stirrup yielding and web crushing 59

Limit state design (ULS-SLS) J. Walraven and S. Gmainer The formulas for yielding shear reinforcement and web crushing are:

VRd,s  Asw s zfywd cot VRd,max  bw zνfcd 1 cot  tan

(3.18) (3.19)

where

fywd is the design value for the stirrup steel yielding,

ν is a reduction factor for the concrete compressive strength of the struts in the stress field, Figure 3.2.11 right,

 is the compression strut angle, to be chosen between 45° and 21,8° (1 ≤ cot ≤ 2,5).

As the geometry of the concrete section is given, the minimum shear reinforcement for the T-beam at support A is determined as follows:

VRd,s  VEd  Asw  VEd s zfywd cot

asw  Asw s  115,52 1000 0,9  0,372  435  2,5  317 mm²/m

Assuming double shear stirrups: ϕ6/175 mm = 339 > 317 mm²/m Minimum shear reinforcement (Chapter 9.2.2.(5) – EC 2):

asw,min  0,08 fck bw  0,08  25  0,25  0.002m²  200mm²

fyk

Maximum longitudinal distance of the stirrups (chapter 9.3.2.(4) – EC 2): sl,max  0,75d (1  cot α ) for α  90 sl,max  0,75  372  279  175 mm

The web crushing criterion checks the upper value of the shear capacity with:

ν  0,6(1 fck )  0,6  (1 25 )  0,54

It results that

VRd,max  0,25  0,9  0,372  0,54  25 1,5  1 2,5  0,4  0,25976MN  259,8 kN > VEd,red  115,5 kN

Figure 3.2.12 marks the stirrups layout at the area near support A. Opposite to the design of bending reinforcement, where the “shift rule” requires a movement of the MEd -line in the unfavourable direction, for shear the opposite applies. As shown in Figure 3.2.11 the shear force at a distance x from the support, is carried by the stirrups over a distance zcotΘ at the left side of x. A practical approach is to move the VEd -line over a distance zcotΘ in the “favourable” direction (towards the support) and “cover” it with the resisting shear due to the reinforcement.

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

Fig. 3.2.12 Stirrup configuration - support A

Shear between web and flanges of the T-section In the T-beam a check of interface shear should be done according to Chapter 6.2.4 (EC 2), see Figure 3.2.13. The strut angle Θf is defined by: o 1,0 ≤ cot Θf ≤ 2,0 for compression flanges (45° ≥ Θf ≥ 26,5°) o 1,0 ≤ cot Θf ≤ 1,25 for tension flanges (45° ≥ Θf ≥ 38,6°).

No transverse tension ties are required if the shear stress at the interface meets the condition:

νEd  Fd  kfctd (hf x)

(3.20)

where ∆Fd is the increment of the longitudinal force in the flange and fctd is the design value of the concrete tensile strength.

Fig. 3.2.13 Shear between web and flanges of T-sections according to EC 2 The recommended value is k = 0,4.

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

fctd  αct fctk,0,05 γc

(3.21)

where αct is a factor that considers sustained loading influences on the concrete tensile strength and unfavourable influences due to the type of loading. It is recommended to assume αct = 1,0. For C25/30:

fctk,0,05  1,8 N/mm²

 fctd  1,0 1,8 1,5  1,2N/mm²

If νEd  0,4fctd  no transverse reinforcement is required. A check for interface shear at the T-beam in axis 2 (see Figure 3.2.1) for support A and C and at the intermediate support B is necessary. The cross-sections considered are in Figures 3.2.14 and 3.2.15.

Fig. 3.2.14 T-beam for check of transverse shear reinforcement - supports A and C

1 (beff  bw ) V

Ed ,red

1  (2611,2  250) 2

115520

νEd,A 





 0,87N/mm²

beff

zhf

2611, 2

0,9  372 180

ν Ed,A  0,87 N/mm²  0, 4fctd  0, 4  1,2  0, 48 N/mm² 

Transverse shear reinforcement at support A is required! Furthermore it has to be checked if transverse shear reinforcement at support C and the intermediate support are necessary. The reduced shear forces at distance d of the supports are determined and assembled in the following equations.

1 (beff  bw ) V

Ed ,red ,C

1  (2611,2  250) 2

89830

νEd,C 





 0,67N/mm²

beff

zhf

2611, 2

0,9  372 180

ν Ed,C  0,67 kN/cm²  0, 4fctd  0, 4 1,2  0, 48 N/mm² 

Transverse shear reinforcement at support C is required.

Fig. 3.2.15 T-beam for check of transverse shear reinforcement - support B 62

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

1 (beff  bw ) V

Ed ,red ,B,left

1  (1827,6  250) 2

101830

νEd,B,left 





 0,73N/mm²

beff

zhf

1827,6

0,9  372 180

ν Ed,B,left  0,73 N/mm²  0, 4fctd  0, 4  1,2  0, 48 N/mm² 

Transverse shear reinforcement at support Bleft is required.

1 (beff  bw ) V

Ed ,red ,B,right

1  (1827,6  250) 2

131410

νEd,B,right 





 0,94N/mm²

beff

zhf

1827,6

0,9  372 180

ν Ed,B,left  0,94 N/mm²  0, 4fctd  0, 4  1,2  0, 48 N/mm² 

Transverse shear reinforcement at support Bright is required. Figures 3.2.16 and 3.2.17 show the zones of beam axis 2 where transverse reinforcement is required.

Fig. 3.2.16 Zones of beam axis 2 where transverse reinforcement is required

Transverse reinforcement Near support A:

A 1 (beff  bw ) V 1

st 2

Ed ,red



s

beff

zfyd cotf

1  (2611,2  250)

Ast 2

115520

s  2611,2  0,9  372  435  2,0  0,18 mm²/mm

→ e.g. bars: ϕ8/250 mm = 0,20 mm²/mm

Near support C:

1  (2611,2  250)

Ast 2

89830

s  2611,2  0,9  372  435  2,0  0,14 mm²/mm

→ e.g. bars: ϕ8/250 mm = 0,20 mm²/mm

Near support B:

1  (1827,6  250)

Ast 2

101830

s  1827,6  0,9  372  435  2,0  0,15 mm²/mm

(3.22)

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

1  (1827,6  250)

Ast 2

131410

s  1827,6  0,9  372  435  2,0  0,19 mm²/mm

→ e.g. bars: ϕ8/250 mm = 0,20 mm²/mm

Fig. 3.2.17 Beam axis 2 – zones where transverse reinforcement is required

3.2.1.5 Design of slabs supported by beams Figure 3.2.18 shows a design assumption for the load transmission from slabs to beams. Furthermore the static systems for a strip of the slab in both directions are depicted. The loads on the slab will be distributed into areas. It depends on the supporting conditions of the slab boundaries by which angle the loads are carried to the beams and bearing walls. The assumed angles are o 45° for consistent edgings, o 60° for disparate edgings (fixed connection) and o 45° for disparate edgings (freely supported). The dead load G1 for the 180 mm thick slab amounts to 4,5 kN/m². The loads G2 = 3,0 kN/m² and Q = 2,0 kN/m² were discussed in Chapter 1. Assuming the suggested safety factors for G and Q GEd  1,3  (4,5  3,0)  9,75 kN/m² QEd  1,5  2,0  3,0 k N/m² LEd  9,75  3,0  12,75 kN/m²

Longitudinal reinforcement in slabs The minimum reinforcement ratio follows from: ρl,min  0,26 fctm fyk

(3.23)

Limit state design (ULS-SLS) J. Walraven and S. Gmainer For a C25/3 concrete and a B500 steel ρl,min  0,26  2,6 500  0,14%. The effective depth d of the slab on beams has been determined (Chapter 1) as d = 143 mm.

Fig. 3.2.18 Load transfer from slabs to beams and static systems

Longitudinal reinforcement in X-direction at intermediate support - axis 2:

MEd

49,18 106

K 2 

 0,144  0,295 

bd fcd 103 1432  25

1,5

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

 z  0,5(1 1 2K )  0,5  (1 1 2  0,144)  0,922 d

1 49,1106

Asl 

(

)  856 mm²

435 143  0,922

So the reinforcement ratio ρ can be determined (Eqn. 3.24). ρ  As bd

ρ  As  856,36  0,60 % bd 1000 1,43

Longitudinal reinforcement in X-direction at mid span 1-2:

MEd

18,32 106

K 2 

 0,0538  0,295 

bd fcd 103 1432  25

1,5

 z  0,5(1 1 2K )  0,5  (1 1 2  0,0538)  0,972 d

1 18,32 106

Asl 

(

)  303 mm²

435 143  0,972

ρ  As  303  0,21% bd 1000 1,43

Longitudinal reinforcement in X-direction at mid span 2-3 and at support in axis 3:

MEd

36,95 106

K 2 

 0,108  0,295 

bd fcd 103 1432  25

1,5

 z  0,5  (1 1 2  K )  0,5  (1 1 2  0,108)  0,942 d

1 36,95 106

Asl 

(

)  630 mm²

435 143  0,942

ρ  As  630  0,44% b  d 1000 1,43

Longitudinal reinforcement in Y-direction at intermediate support in axis B:

MEd

68,15 106

K 2 

 0,200  0,295 

bd fcd 103 1432  25

1,5

(3.24)

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

 z  0,5(1 1 2K )  0,5  (1 1 2  0,200)  0,887 d

1 68,15 106

Asl 

(

)  1235 mm²

435 143  0,887

ρ  As  1235  0,86 % bd 1000 1,43

Longitudinal reinforcement in Y-direction at mid span:

MEd

52,14 106

K 2 

 0,153  0,295 

bd fcd 103 1432  25

1,5

 z  0,5(1 1 2K )  0,5  (1 1 2  0,153)  0,917 d

1 52,14 106

Asl 

(

)  915 mm²

435 143  0,917

ρ  As  914  0,64 % bd 1000 1,43

In Figure 3.2.19 the theoretical reinforcement ratios of the slab are represented, all greater than the minimum.

Fig. 3.2.19 Ground view of a symmetric part of the slab with reinforcement ratios 67

3.2.2. Flat slab

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

Figure 3.2.20 presents the flat slab with a height of 210 mm. It is favourable to choose an ideal small width for the strips passing over the supporting column, so that locally higher longitudinal reinforcement ratios apply.

Fig. 3.2.20 Flat slab with hidden strong strips 3.2.2.1 Loads and internal forces for the calculation of the flat slab GEd  1,3  (5,25  3,0)  10,73kN/m² QEd  1,5  2,0  3,0kN/m² LEd  GEd  QEd  13,73kN/m² The moments for the flat slab may be obtained simply upgrading with the factor 13,73/12,75 = 1,077 the moments calculated for the slab on beams in the previous analysis. The moments in beam axis B and 2 are shown in Figure 3.2.21.

Fig. 3.2.21 Maximum moments for the calculation of the flat slab 3.2.2.2 Determination of the bending reinforcement Longitudinal reinforcement in the flat slab The effective depth d of the flat slab was assumed (chapter 1) as d = 172 mm. Longitudinal reinforcement in X-direction at intermediate support in axis 2:

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

MEd

286,45 106

K 2 

 0,194  0,295 

bd fcd 3000 1722  25

1,5

 z  0,5(1 1 2K )  0,5  (1 1 2  0,194)  0,891 d

1 286,45 106

Asl 

(

)  4295 mm²

435 172  0,891

ρ  As  4295  0,83 % bd 3000 1,72

Longitudinal reinforcement in X-direction at support in axis 3:

342  1,19; 286

ρ  1,19  0,83%  0,99%

Longitudinal reinforcement in X-direction at support in axis 1:

254  0,89; 286

ρ  0,89  0,83%  0,74%

Longitudinal reinforcement in X-direction at mid-span 1-2:

191  0,67; 286

ρ  0,67  0,83%  0,56%

Longitudinal reinforcement in X-direction at mid-span 2-3:

169  0,59; 286

ρ  0,59  0,83%  0,49%

Longitudinal reinforcement in Y-direction at intermediate support in axis B:

K    MEd 143,12 106 0,194  0,295 

bd 2fcd

1500 1722  25 1,5

 z  0,5  (1 1 2  K )  0,5  (1 1 2  0,194)  0,891 d

A  1  (143,12 106 )  2147 mm² sl 435 172  0,891

ρ    As 2147 0,83% bd 1500 1,72

Longitudinal reinforcement in Y-direction at support in axis A and C:

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

91  0,64; 143

ρ  0,64  0,83%  0,53%

Longitudinal reinforcement in Y-direction at mid-span A-B:

96  0,67; 143

ρ  0,67  0,83%  0,56%

Longitudinal reinforcement in Y-direction at mid-span B-C:

65  0,46; 143

ρ  0,46  0,83%  0,38%

In Figure 3.2.22 the reinforcement ratios of the hidden beams in the flat slab are shown.

Fig.3.2.22 Symmetric part of the flat slab with reinforcement ratios of flat slab “hidden beams” 3.2.2.3 Punching shear - column B2 Figure 3.2.23 shows the punching shear phenomena in general. Fig. 3.2.23 Punching shear cylinder 70

Limit state design (ULS-SLS) J. Walraven and S. Gmainer At the junction column to slab the ULS vertical load from the slab to the column is calculated as VEd =705 kN. To take the eccentricity into account, a factor β can be determined with simplified assumptions according to Chapter 6.4.3 (EC 2). The simplified case may be used only for structures where lateral stability does not depend on frame action and where adjacent spans do not differ by more than 25 %. The approximate values for β can be taken from Figure 3.2.24. For the current example β = 1,15 (interior column) applies.

Upper limit value for design punching shear stress in design At the perimeter of the loaded area the maximum punching shear stress has to satisfy the following criterion (EC 2, chapter 6.4.5):

νEd  βVEd  νRd,max  0,4νfcd u0d

(3.25)

where u0 is the perimeter of the loaded area.

Fig. 3.2.24 Recommended values for β according to EC 2 First, a check of the upper limit value of punching shear capacity is required. Further data Dimensions of column B2: 500/500 mm Effective depths for two-way reinforcement layers dy  210  30  162  172mm dz  210  30  16  162  156 mm Mean effective depth: d  (172  156)  164mm

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

ν  0,6(1 fck )  0,6  (1 25 )  0,54

Maximum allowable punching shear stress

νRd,max  0,4νfcd  0,4  0,54  25 1,5  3,6N/mm²

νEd  βVEd  1,15  705000  2,47N/mm²  νRd,max  3,6N/mm² u0d 4  500 164

The second verification is at perimeter u1 where νEd  βVEd u1d

(3.26)

The basic control perimeter u1 is taken at a distance 2,0d from the loaded area and should be constructed as to minimise its length (see Figure 3.2.25). The definition of control perimeters of different cross sections is in Figure 3.2.26 (Chapter 6.4.2 – EC 2).

Fig. 3.2.25 Punching shear stress at perimeter Fig. 3.2.26 Definition of control perimeters according to EC 2 The length of the control perimeter of the column with 500/500 mm sides is: u1  4  500  2 π  2 164  4061 mm νEd  1,15  705000 4061164  1,22 N/mm² There is no punching shear reinforcement required if: ν Ed  ν Rd,c

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

νRd,c  CRd,ck(100ρl fck )3  νmin where CRd,c  0,18  0,18  0,12 γc 1,5

(3.27)

k  1 200  2,0 d

k  1 200  2,10  2,0 164

 k  2,0

ρl  ρx ρy  0,02

ρl  0,83  0,83  0,83%

νRd,c  0,12  2,0  (100  0,0083  25)3  0,66N/mm²

νRd,c  νmin  0,035k 2fck 2  0,035  2,02  252  0,49N/mm²  ok!

νEd  1,22 N/mm²  νRd,c  0,66N/mm²  Punching shear reinforcement is required.

Capacity with punching shear reinforcement

νRd,s  0,75νRd,c  1,5( d )Asw fywd,ef ( 1 )sinα

sr

u1d

(3.28)

Shear reinforcement within a distance of 1,5d from the column (see Figure 3.2.27) is computed as follows. fywd,ef  250  0,25d  fywd

fywd,ef  250  0,25d  250  0,25 164  291 N/mm²

Fig. 3.2.27 Punching shear reinforcement The steel contribution comes from the shear reinforcement inside a distance 1,5d from the edge of the loaded area, to ensure some anchorage at the upper end. The concrete contribution to resistance is assumed to be 75% of the design strength of a slab without shear reinforcement.

Limit state design (ULS-SLS) J. Walraven and S. Gmainer The distance sr between the punching shear reinforcement perimeters should not be larger than 0,75d as shown in Figure 3.2.28. s  0,75d  0,75164  123mm r νRd,s  νEd  Asw  (νEd  0,75νRd,c )u1sr 1,5fywd,ef

Asw  (1,22  0,75  0,66)  4060,88 123 1,5  291  830 mm²

in each reinforcement perimeter. The length of the outer perimeter, marked as section B in Figure 3.2.28, is: uout  βVEd ν Rd ,c d

(3.29)

uout  1,15  705000 0,66 164  7490 mm

The distance from this perimeter to the edge of the column follows from:

a  (uout  4h)  7490,30  4  500  874 mm  5,33d

2π

2π

The outer punching shear reinforcement, marked as section A in Figure 3.2.28, has to be at a distance of no more than kd from the outer perimeter. The recommended factor k being k =1,5, the outer punching shear reinforcement is at a distance of: 5,33d -1,5d = 3,83d.

Fig. 3.2.28 Punching shear reinforcement perimeters according to EC 2 The distance between the punching shear reinforcement perimeters should not be larger than 0,75d = 0,75·164 =123 mm.

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

Fig. 3.2.29 Punching shear design of slab at column B2

3.2.2.4 Column B2

Second order effects under axial loading General background according to chapter 5.8.2, 5.8.3.1, 5.8.3.2 and 5.8.3.3 (EC 2) o Second order effects may be ignored if they are smaller than 10 % of the corresponding 1st order effects. o “Slenderness” is defined as

λ  l0  l0 i (I ) A

(3.30)

where l0 is the effective height of the column, i is the radius of gyration of the uncracked concrete section, I is the moment of inertia around the axis considered and A is the cross sectional area of the column. For rectangular cross sections the λ value is

λ  3,46 l0 h

and for circular cross sections is

λ  4 l0 . h

Figure 3.2.30 shows the basic cases for l0 according to EC 2.

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

Fig. 3.2.30 Examples of different effective heights of columns according to EC 2

The relative flexibilities of rotation “springs” at the column ends (as in Figure 3.2.30, case f) and g)) may be calculated as

k  θ EI Ml

(3.31)

where θ is the rotation of restraining members for a bending moment M, EI is the bending stiffness of a compression member and l is the height of the column between two rotation-springs.

The effective column height in a frame (see Figure 3.2.31) is different for braced and unbraced frames.

Fig.3.2.31 Determination of the effective column height For braced frames: l0  0,5l (1 k1 )(1 k2 ) 0,45  k1 0,45  k2 For unbraced frames, the largest of:

(3.32)

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

l0  l (1 10 k1  k2 ) and l0  l(1 k1 )(1 k2 )

k1  k2

1 k1 1 k2

(3.33)

Values k1 and k2 are the relative spring stiffness at the ends of columns, I is the clear height of the column between the end restraints. A simplifying assumption according to Chapter 5.8 and 5.8.3.2 (EC 2) is that the contribution of the adjacent “non failing” columns to the spring stiffness is ignored (if this contributes is positive to, i.e. increases the restraint). Furthermore for beams for ( / M) the value (l/2EI) may be assumed taking account of the loss of beam stiffness due to cracking. Assuming the beams are symmetric with regard to the column and their dimensions are the same for the two stories, the following relations are found:

(EI )column

( EI )column

k1  k2  l EI

l 2  2  EI

 0,25 χ

( l )beams ( l )beams

where

(EI )column

χ l

.

EI

( l )beams

The effective column length l0 can for this situation is given in Table 3.2.1 as a function of χ . o Second order effects may be ignored if the slenderness is smaller than the limit value λlim. o In case of biaxial bending the slenderness should be calculated for an Y-direction. Second order effects need only to be considered in the direction(s) in which λlim is exceeded.

χ or k1 = k2 l0 for braced column

Table 3.2.1 Effective heights of columns according to EC 2

0 (fixed

0,25

0,50

1,00

2,00

end)

0,0625

0,125

0,25

0,50

0,50 l

0,56 l

0,61 l

0,68 l

0,76 l

∞ (pinned end) 1,00 1,00 l

l0 for unbraced column: 1,00 l 1,14 l 1,27 l 1,50 l 1,87 l ∞

(the larger of

the values in

the two rows) 1,00 l

1,12 l

1,13 l

1,44 l

1,78 l

∞

A column is qualified as “slender”, which implies that second order effects have to be taken into account, if λ ≥ λlim. The limit value according to chapter 5.8.3.1 (EC 2) is :

λlim  20ABC n

(3.34)

where

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

A 1 , (1 0,2φef )

B  1  2ω,

C  1,7  rm,

n  NEd , Ac fcd

φef ω  As fyd Ac fcd rm  M01 M02

is the effective creep factor. If it is unknown it can be assumed that A = 0,7, is the mech. reinforcement ratio. If it is unknown B = 1,1 can be adopted. is the ratio between end-moments in the column considered with IM02I ≥ IM01I (see Figure 3.2.32). In particular cases it can be supposed that rm is 1,0. Then C = 0,7.

Fig. 3.2.32 Ratio between end-moments in column

Determination of the slenderness λ - column B2 For the current example (see Figure 3.2.33) the first step is the determination of the rotational spring stiffness at the end of the column. The elasticity moduli Ecm are: for columns - concrete class C30/37 Ecm = 33000 MN/m²; for beams concrete class C25/30 Ecm = 31000 MN/m². The moment of inertia for the 4 m high column B2 is:

lcolumn,B2  112  0,54  0,0052m4

The spring stiffnesses are:

Column:

EI  33000  0,0052  43,0MNm

l

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

Beam:

31000   6  0,21

EI  12

 20,15MNm

l

7,125

( EIl )column 43,0

k1  k2  2  2  EI

 2  2  20,15  0,53

( l )beams

l0  0,5l (1 k1 )(1 k2 )  0,5  4  (1 0,53)2  3,1 m

0,45  k1 0,45  k2

0,98

Actual slenderness of the column

λ  3,46l0  3,46  3,1  22,5

h

0,5

Fig. 3.2.33 Configuration of variable load on slab

Limit slenderness of the column In Eqn. (3.35) the default values A = 0,7, B = 1,1 and C = 0,7 are used. The normal force NEd is (Chapter 2) NEd = 4384 kN and MEd = 42 kNm. n  NEd  4384000  0,88 Ac fcd 5002  30 1,5

Therefore:

λlim  20  0,7 1,1 0,7  11,5 0,88

 λ  22,5  λlim  11,5

As the actual slenderness of the column is larger than the limit slenderness, second order effects have to be taken into account.

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

General: Method based on nominal curvature

Mtot  NEd (e0  ei  e2 )

(3.35)

First order eccentricities (Figure 3.2.34) e01 and e02 are different. At the end of the column an equivalent eccentricity e0 may be used, defined as:

e0  0,6e02  0,4e01  0,4e02 .

(3.36)

If e01 and e02 have the same sign (Figure 3.2.34 left), the curvature is increased. Otherwise they would have different signs. Moreover it is assumed that Ie02I ≥ Ie01I.

Fig. 3.2.34 Effects of the first order eccentricities e01 and e02

The eccentricity ei by imperfection follows from Chapter 5.2 (7) – EC 2.

ei  θi l02

(3.37)

where l0 θi  θ0αhαm

is the effective column height around the axis regarded, according to Chapter 5.2 (5) – EC 2,

θ0  1 200 [rad]

is the basic value,

αh  2l ; 23  αh  1 is the reduction value for the height,

αm  m

0,5(1 1 ) m

is the reduction value for the number of building elements and is the number of the vertical elements which are required for the total impact.

The second order eccentricity e2 follows from l02 εyd e2  KφKr 2 π 0,45d

(3.38)

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

where

Kφ  1 (0,35  fck  λ )φef  1,0 and Kr  nud  nEd  1,0.

200 150

nud  nbal

and

φef  ( M0Eqp )φ,t . M0Ed

where φ,t  2 or 3 final creep ratio

M 0Eqp M 0Ed

is the ratio between permanent load to design load.

Bending moment including second order effects e0  MEd  42  0,010m  10mm NEd 4384 At least the maximum of {l0 / 20, b / 20 or 20 mm} should be taken for e0. So the maximum e0 value is: e0  b20  500 20  25mm.

θ0  1 200 ; αh  24  1; αm  0,5  (1 11)  1  θ0  1 200 11  0,005 ei  0,005  4000 2  10mm φef  0,3  2 1,5  2  2  0, 4

Kφ  1 (0,35  30 200  22,1 150 )  0,4  1,14

nud  1 ρfyd  1 0,03  435  1,65

fcd

The reinforcing ratio estimated value is ρ = 0,03.

nEd  NEd  2 4384000  0,88 Acfcd 500  20

nbal = 0,4 for concrete classes up to C50/60

Kr  1,65  0,88 1,65  0,4  0,62

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

500 εyd  fyd  1,15  2,17 103 Es 200000

32002 2,17 103

e2  1,14  0,62  2 

 14mm

π 0,25  454

Mtot  4384  (25  10  14) 103  214,82 kNm

Column reinforcement For the reinforcement of the column, the interaction diagram for a symmetric reinforced rectangular cross section is used. The diagram is valid for reinforcing steel with fck = 500 N/mm².

μEd  2 MEd bh fcd

(3.39)

νEd  NEd bhfcd

(3.40)

μEd ; νEd  ωtot  As,tot  As1  As2  ωtot bh fyd fcd

(3.41)

214,82 106

μEd 

 0,086;

5003  30

1,5

4384 103

νEd 

 0,877

5002  30

1,5

The "-" is conventionally used for the compression force NEd. d1 h  46 500  0,092  0,10  ωtot  0,20

So the diagram (Figure 3.2.35) with the ratio d1 / h = 0,10 can be used.

As,tot  As1  As2  0,20 

 2299 mm² → e.g. 8ϕ20 = 2513 mm²

The maximum and minimum areas are:

As,min  0,10NEd  0,10  4384000  1008 mm²

fyd

1,15

As,max  0,04Ac  0,04  5002  10000mm²

The calculation of the column reinforcement has been carried out for the most unfavourable direction. The bending moment in the other direction is only slightly smaller (Figure 3.2.32). Therefore, without further calculation, in the other direction 8ϕ20 are also used, leading to the reinforcement configuration in Figure 3.2.36.

Limit state design (ULS-SLS) J. Walraven and S. Gmainer Fig. 3.2.35 Interaction diagram – double symmetric reinforced rectangular cross section (Zilch and Zehetmaier, 2010) Figure 3.2.36 shows the cross section of the column with the reinforcement. 83

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

Fig. 3.2.36 Layout of the reinforced column B2 3.2.2.5 Design of shear walls The stability of the building is ensured by two shear walls (one at each end of the building in axis B1 and B6) and a central core between the axis B7 and B8 (see Figure 3.2.1). The dimensions of the shear walls are in Figure 3.2.37.

Fig. 3.2.37 Dimensions of the shear walls and the core

The moments of inertia around a centroid axis parallel to the X-global axis are:

Ix = (0,25·2,03)/12=0,167 m4

for shear walls 1 and 2

I = 0,497 m4

for the central core

The contribution of shear wall 1 to the total is:

0,167

 0,20

(2  0,167  0,497)

(20%)

Second order effects If second order effects are smaller than 10% of the first order moments, they can be neglected. (Chapter 5.8.2(6) EC2). Alternatively, according to Chapter 5.8.3.3, for bracing systems without significant shear deformations second order effects may be ignored if:

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

FV,Ed  k1 ns

EcdIc

ns  1,6 L

(3.42)

where

FV,Ed

is the total vertical load (both on braced and unbraced elements),

ns

is the number of storeys,

L

is the total height of the building above its fixed foundation,

Ecd

is the design E-modulus of the concrete, Ecd = Ecm / λcE, λcE = 1,2 and

Ic

is the moment of inertia of stabilizing elements.

The suggested value of the factor k1 is k1 =0,31 if the section is cracked, the double if uncracked. For the shear wall the following actions are applied (see Chapter 2): the maximum moment My = 66,59 kNm with the corresponding normal force N = -2392,6 kN.

σN  NA  2,392 (2  0,25)  4,78MN/m²

σM  MW  0,06659 0,1667  0,40  4,78 MN/m² where W=bh2/6=(0,25·2,02)/6=0,1667 m3. The shear wall remains indeed uncracked therefore factor k1 may be taken k1 = 2·0,31=0,62 according to clause 5.8.3.3 (2) – EC 2.

Whole building global second order effects Assuming a six-storey building the shear walls’ total inertia is Ic=(2·0,167+0,497)=0,83 m4. Applying clause 5.8.3.3 – EC 2

33  106  0,83

FV ,Ed  0, 62  6  2 1, 2

 30961 kN

6  1,6

Assuming that 30% of the variable load is permanent, the unit load per story is qEd = 9,75+0,3·2·1,5=10,65 kN/m². The total area for the load per story is 30·14,25=427,5 m² so the total load per story can be estimated as 10,65·427,5=4553 kN. For a six-storey building FV,Ed can be roughly calculated as:

FV ,Ed  6553  6  27318 kN  30961 kN

The condition being fulfilled second order effects may be ignored. For overall buckling in the X-direction the external shear walls contribution may be neglected and the central C core is considered. The C section of the central core has a moment of inertia around the Yaxis Iy =(1,8·3,63 - 1,55·3,103)/12 = 3,15 m4.. Assuming the most severe value k1 = 0,31 the result is (0,31·3,15)=0,98 > (0,62·0,83)=0,51: the condition is fulfilled in the X-direction also.

Verification by the moment magnification factor Another possibility to check if second order effects should be considered is to determine the moment magnification factor (clause 5.8.7.3 EC 2). The simplified (5.30) formula can be written as:

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

MEd 

M0Ed  f M0Ed NEd

1

NB

where

(3.43)

f 1 NEd 1 NB

is the moment magnification factor

π 2EI

NB 

2 is the Euler’s buckling axial force

(1,12l )

NEd is the total axial force FV,Ed = 27318 kN. In order f ≤ 1,1 it results NEd  0,091 NB Substituting in the relations above it results:

(3.44)

l FV ,Ed  0,85 EI

103 19 

27318 33  0,66  0,85 1,2  0,83

(3.45)

The assumption that the cross section is uncracked is correct because the condition in Eqn. (3.44) is fulfilled. Minimum reinforcement AS,v,min should be provided.

3.2.3. Slab with embedded elements The top view of the slab with embedded lighting elements is represented in Figure 3.2.38. The bearing beams lie on axis A, B and C, the slab spans in transverse direction in between. A section of the lighting clay elements is in Figure 3.2.39. The clay elements are used as permanent formwork . The upper concrete slab in the cross-section is 50 mm thick. Light reinforcement is applied in transverse direction at mid-depth. On top of the concrete slab mostly foamed or polystyrene concrete is applied, in which heating and electricity pipes and tubes are embedded, and at the bottom a clay finishing layer (see also Figure 3.2.39) is projected. Because of those protecting layers the concrete cover can be small (mostly governed by the bond criteria) and environmental classes X2-X3, elsewhere used as a general basis for the design, do not apply.

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

Fig. 3.2.38 Ground view of the slab with the embedded elements

Fig. 3.2.39 Cross section of the slab with lighting elements

3.2.3.1 Bending reinforcement - beam axis B

Mid-span The design bending moments at mid-span and at the internal support (intersection point of axes B and 2) are MEd = 177,2 kNm and MEd = 266 kNm respectively. At mid-span the effective width beff is (clause 5.3.2.1 EC2):

beff = beff,i + bw where beff,i = 0,2bi + 0,1l0

With l0=0,85l1=0,856000=5100 mm, bw=250 mm and bi =7125/2=3562 mm it results beff = 2695 mm. For the maximum bending moment at mid-span MEd = 177,2 kNm

MEd 

177,2 106

 0,0287

bd 2fcd 2695  375 16,7

From the diagram in Figure 3.2.7 z = 0,98d = 367 mm. The area of the tensile reinforcement is

Limit state design (ULS-SLS) J. Walraven and S. Gmainer Asl MEd  177,2 106  1108mm² zfyd 0,98  375  435 4 bars ϕ20 = 1256 > 1108 mm² are adequate. A mid-span cross-section showing the ribbed slab with embedded elements is in Figure 3.2.40.

Internal support At the intermediate support B-2 the effective width is again calculated with

Here

beff = beff,i + bw with beff,i = 0,2bi + 0,1l0.

l0 = 0,15(l1 + l2) = 0,15(6000 + 6000) = 1800 mm so that

beff = 0,23562 + 0,11800 = 892 mm. This results in an effective width

beff = 2892 + 250 = 2035 mm. At first it is verified whether compressive reinforcement is required. The value K is

K  2 MEd 

266 106

 0,45  0,295

bd fcd 250  375 16,7

So compression reinforcement is indeed necessary. The parameter (d-d’) is the distance between the compression reinforcement and the tension reinforcement. The reinforcement Asc follows from:

(K  Kbal )fcd bd 2

Asc 

'

fyd (d  d )

(3.46)

0,45  0,295 16,7  250  3752

Asc 

 605mm²

435  (375  30)

The tensile reinforcement is then:

Kbal fcd bd 2

Ast 

 Acs

fyd zbal

(3.47)

0,295 16,7  250  3752

Ast 

 605  1899mm

435  0,82  375

This reinforcement can be spread over the effective width beff = 2035 mm. So, 10 bars ϕ16 = 10·201 = 2010 mm² are adequate. The bars can be spread over the effective width.

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

Fig. 3.2.40 Cross sections of bearing beam, axis B, at mid-span 2 (left) and of slab with embedded elements adjacent to bearing beam B ( right)

3.2.3.2 Bending reinforcement of the slab with embedded elements A cross section of the floor is in Figure 3.2.39. The uniformly distributed load on the slab consists of the dead load components G1 = 2,33 kN/m² and G2 = 3,0 kN/m². The variable load is 2,0 kN/m². The design load is: QEd = 1,3·(2,33 + 3,0) + 1,5·2,0 = 9,93 kN/m². For a unit slab width the bending moments at the intermediate support (B) is MEd = 63,0 kNm/m and at mid-span MEd = 39,2 kNm/m (Figure 3.2.41).

Mid-span

K  2 MEd 

39,2 106

 0,065  0,295

bd fcd 1000 197 16,7

The inner lever arm is 0,97d = 0,97197=191 mm, the area of the bending tensile reinforcement is:

MEd 39,2 106

Asl 



 472mm / m

zfyd 191 435

Per rib this means 472/2= 235 mm² or 2 bars ϕ12 mm = 226 mm². (A 4% lower value is acceptable since no moment redistribution was applied).

Fig.3.2.41 Design bending moment at internal support and mid-span - beam with embedded lighting elements uniformly distributed loaded

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

Internal support As the support may be considered to provide limited restraint to rotation according to clause 5.3.2.2(4) – EC2 the theoretical moment may be reduced to take into account the reaction distribution over the width of the support. The design support reaction is FEd,sup = 1,25x9,93x7,125 = 88,5 kN/m The moment reduction is MEd = FEd,supt /8 = 88,5·0,25/8 = 2,76 kN/m therefore the moment is MEd = 63,0 – 2,76 = 60,2 kNm/m. First it is controlled whether compression reinforcement is necessary:

K  2 MEd 

60,2 106

 0,417  0,295

bd fcd 240 190 16.7

So compression reinforcement is required:

(K  Kbal )fcd bd 2 (0,417  0,295) 16.7  240 1902

Asc 



 253 mm

fyd (d  d ')

435  (190  30)

Per rib this is 253/2 = 127 mm² or 2 bars ϕ10mm = 156 mm². Tensile reinforcement is then

Kbal fcd bd 2

0,295 16,7  240 1902

Ast 

 Asc 

 253  882 mm

fyd zbal

435  0,82 190

Per rib this is 882/2 = 441 mm² or 2 bars ϕ18 = 508 mm². The application of this reinforcement is limited to the width of the rib, as it cannot be placed in the thin upper slab (50 mm) because of the presence of the transverse reinforcement (see below).

Bending reinforcement of slab with embedded elements In Figure 3.2.39 the net span of the top 50 mm slab between the ribs is 380 mm. The design load per square metre is

QEd = 1,3·(1,2 + 3) + 1,52=8,5 kN/m². The design bending moment at the fixed-end supports is

MEd  (1 / 12)l 2QEd  (1 / 12)  0,382  8,5  0,10 kNm/m

K  2 MEd 

0,10 106

 0,01

bd fcd 1000  25 16,7

So the inner lever arm is z = 0,99d = 0,9925 = 24,7 mm and the required longitudinal reinforcement is

MEd 0,1106

Asl 



 9mm /m

zfyd 24,7  435

A practical reinforcement made of an orthogonal mesh bars ϕ6-200 mm applied at mid-height of the slab, is by far sufficient.

3.2.3.3 Shear capacity

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

Bearing beam in axis B In the bearing beam axis, adjacent to the intermediate support (axis 2) the maximum shear force is VEd = 270,73 kN (see Chapter 2). In the governing section at distance d from the support this value is reduced to VEd,red = 240 kN. If this is larger than VRd,c (see Eqn. 3.18) calculated shear reinforcement is required. Here

k  1 200  1 200  1,74

d

ρl  Asl bd  2010 250  360  0,022  0,02

VRd,c  0,12 1,74  (100  0,02 16,7)1/3 250  360 103  69kN

is smaller than the design shear force of 270,73 kN: shear reinforcement has to be provided. Assuming an inclination angle with cot = 2.5 the required shear reinforcement follows from Eqn. 3.19. VRd,s  VEd  Asw  VEd s zfywd cot

asw  Asw s  240 1000 0,82  375  435  2,5  0,72mm²/mm

Stirrups ϕ10/175 mm (0,89 mm² / mm) are adequate.

Bearing ribs with embedded elements The maximum shear force in the ribs follows from Figure 3.2.42. VEd,red at a distance d (=190 mm) from the edge of the supporting beam (see also Figure 3.2.42) is

44,2 

VEd ,red

VEd,red  41,1 kN.

4,45 (4,45  0,19  0,125)

Fig. 3.2.42 Distribution of shear force along rib of slab with embedded lighting elements 91

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

VRd,c  [CRd,c k(100ρl fck )3 ]bw d where CRd,c  0,18  0,18  0,12 γc 1,5

k  1 200  2,0 d

k  1 200  2,02  2,0  k  2,0 190

VRd,c  [0,12  2,0  (100  0,02  25)3 ]  240 190 103  40,2kN

VRd,c  40,2 kN  VEd,red  41,1kN 

This 2% lower than the design shear value can, reasonably, be ignored. In order that that the slab with embedded lighting elements is formally regarded as a slab, the rule in clause 5.3.1(6) (“Transverse ribs are provided at a clear spacing not exceeding 10 times the overall depth of the slab”) gives a distance of 10230 = 2300 mm. One transverse rib is actually provided at distance 7,125/2 = 3,56 > 2,30 m so two ribs at a distance 7,125/3 = 2,38  2,30 m should preferably be provided.

Slab with embedded elements The design shear stress at the fixed ends is vEd = (0,1908460)/(100025)=0,06 N/mm², far below vRd,c.

3.3 Serviceability limit states

3.3.1. SLS deflection - general

The control of deflection can be done o by calculation or o by tabulated values

3.3.1.1 Deflection control by calculation

For span-depth ratios below 7,5 m the following limits according to chapter 7.4.2 (EC 2) no further checks are needed.

l

ρ0  ρ0 3/2 

 K 11 1,5 fck  3,2 fck   1 

d 

ρ

 ρ  

if ρ  ρ0

(3.48)

l

ρ0

ρ 

 K 11 1,5 fck



fck

 if ρ  ρ0

d 

ρ  ρ 12

ρ0 

where

l/d is the limit span/depth,

K is the factor to take into account the different structural systems,

(3.49)

Limit state design (ULS-SLS) J. Walraven and S. Gmainer ρ0 is the reference reinforcement ratio  fck 103, ρ is the required tension reinforcement ratio at mid span to resist the moment due to the design loads (at support for cantilevers) and ρ’ is the required compression reinforcement ratio at mid span to resist the moment due to design loads (at support for cantilevers). Figure 3.3.2 shows the previous expressions in a graphical form assuming K = 1.

Fig. 3.3.1 Eqn. 7.16a/b EC 2 Eqn.(3.49) and Eqn.(3.50) are based on many different assumptions (age of loading, time or formwork removal, temperature, humidity, creep effects) and represent a conservative approach. The coefficient K depends on the static system of the structure - Figure 3.3.1.

Fig. 3.3.2 Coefficient K depending on the static system

The expressions have been derived for an assumed stress in the reinforcing steel at mid span stress σs = 310 N/mm². σs has to be evaluated under the quasi permanent load combination. If another stress level is applied or if more reinforcement than minimum required is provided, the values obtained by Eqn.(3.49) and (3.50) can be multiplied with the factor

310  500 σs fyk As,req As,prov

(3.50)

There are rules for spans larger than 7,5 m in Chapter 7.4.2 (EC 2):

Limit state design (ULS-SLS) J. Walraven and S. Gmainer o For beams and slabs (no flat slabs) with spans larger than 7 m, which support partitions liable to damage by excessive deflections, the values l/d given by Eqn. (7.16) in the EC 2 should be multiplied by 7/leff (leff in meters). o For flat slabs where the greater span exceeds 8,5 m, and which support partitions to be damaged by excessive deflections, the values l/d given by expression (7.16) in the EC 2 should be multiplied by 8,5/leff.

3.3.1.2 Tabulated K values and basic ratios (l/d) Table 3.3.1 gives the K values (Eqn. 7.16 – EC 2), corresponding to the different structural system and the slenderness’ limits (l/d) values for relatively high (ρ=1,5%) and low (ρ=0,5%) longitudinal reinforcement ratios at mid-span. These values, calculated for concrete quality C30/37 and σs = 310 N/mm², and satisfy the deflection limits given in chapter 7.4.1 (4) and (5) in EC 2.

Table 3.3.1 Tabulated values for l / d

Structural system Simply supported slab/beam End span Interior span Flat slab Cantilever

Factor K 1,0 1,3 1,5 1,2 0,4

l / d

ρ = 1,5 % ρ = 0,5 %

3.3.1.3 Slab on beams For the ULS design the transmission of loads to the bearing beams has been assumed in the approximate way shown in Figure 3.2.18. This leads to a distribution of the bending reinforcement as in Figure 3.2.19. For the control of the deflection the strip is considered spanning between the mean beams at axes 1 and 2, with a required reinforcement of 0,44% (in the left of Figure 3.2.19 spanning from left to right). In the case of a two-way spanning slab, the check has to be carried on the basis of the shorter span (l = 6,00 m) and the related reinforcement. According to EC 2, Chapter 7.4.2, the span to depth ratio l/d should satisfy Eqn.7.16a (see also Eqn.3.49 in this report). Assuming As,req = As,prov, fck = 25 MPa, ρ0  103 fck  0,005 , ρ = 0,0044 and K = 1,3 (end-span):

l  1,3  [11 1,5  25  0,5  3,2  25  ( 0,5  1)3/2 ]  1,3  [11 8,5  0,80]  26,4

d

0, 44

0, 44

This relation is valid for a default steel stress σs = 310 MPa. Deflection control is made for the quasipermanent load condition when the total load is (G1 + G2 + ψ2Qk ) = 4,5 + 3,0 + 0,3·2 = 8,1 kN/m2. Differently from ULS, in SLS conditions the variable load is present on both spans, so the bending moment MEk = 8,1·6,02/14,2 = 20,5 kN/m while MEd = 36,95 kNm. As the depth of the neutral axis does not differ that much in ULS and SLS conditions, the steel stress under the quasi-permanent load) may be assumed as:

σs  MEk fyd  20,5  435  241 MPa

MEd

36,95

Therefore the allowable l/d ratio may be increased to:

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

l  310  26,4  33,9 d 241

The actual span-depth ratio is

l  6000  100  125  40,1  33,9.

d

The effective reinforcement area should therefore be at least (40,1/33,9) -1=18% higher than the theoretical 0,44%, one, i.e. 44·1,18 = 0,52% and the coefficient As,req /As,prov =1,18 should be considered. Increasing the quantity of steel in turns reduces the steel stress and guarantees the deflection control is satisfied. As an alternative, the refined calculation method could be used (Chapter 7.4.3 EC2).

3.3.1.4 Flat slab For the ULS the reinforcement has been calculated assuming a total slab thickness of 210 mm, 30 mm higher than in the previous case. For a flat slab with a relatively low reinforcement ratio (ρ = 0,5%) Table 3.3.1 gives a span-depth ratio l/d = 24. In Chapter 3.2.2 the ULS calculated bending reinforcement in the Y-direction at mid-span A-B was 0,56%. This means that the mentioned value l/d = 24 should be almost OK. It may be assumed that, even if the models are different (slabs on beams vs flat slab), the steel stress ration under the quasi-permanent load condition does not differ from the previous one, so the allowable l/d ratio can be assumed to be l  310  24  30,9 d 241 For flat slab the longer span ly = 7125 – 125 – 100 = 6900 mm has to be considered, so the actual ratio of l/d is: l  6900  40,1 d 172 That means that, as in the previous case, a (40,1/30,9) -1=30% increase of the longitudinal reinforcement is required, ,i.e. the effective reinforcement should be 0,5·1,30 = 0,65 > 0,56%. As an alternative, a refined calculation could to be considered.

3.3.1.5 Slab with embedded elements

The reinforcement ratio at mid-span is

ρ  Asl  226  0,23 102 bd 500 197

The ribbed slab has a T section with b = 500 mm and bw = 120 mm then b/bw = 4,16 > 3 which introduces a reduction factor 0,80 in the formulae. According to chapter 7.4.2 (2) – EC 2 no detailed calculation is necessary if the l /d ratio of the slab is smaller than the limit value:

l  0,5  0,5 3/2 

 0,80 1,3  11 1,5· 25·  3,2· 25·  1   0,80 1,3  11 16,3  20,4 49,6

d



0,23

 0,23  

Limit state design (ULS-SLS) J. Walraven and S. Gmainer In this case it not necessary to consider neither the steel stress/310 nor the As,req /As,prov coefficients as the actual value of l/d is : l  7125  36,2  49,6 d 197

3.3.2. SLS crack width - general

The crack width is the difference between the steel and concrete elongations over the length (2lt), where lt is the “transmission length” necessary to increase the concrete strength from 0 to the tensile strength fctm (Figure 3.3.3): the maximum distance between two cracks has to be 2lt otherwise a new crack could occur in-between (see Figure 3.3.3) It can be demonstrated that the transmission length lt is equal to:

lt  1 fctm  4 τbm ρ

(3.51)

Fig. 3.3.3 Definition of the crack width

For the calculation of the maximum (or characteristic) crack width, the difference between steel and concrete elongations has to be calculated for the largest crack distance sr,max = 2lt. The formula for crack width control according to clause 7.3.4 Eqn. (7.8) – EC 2 is:

w k  sr ,max (εsm  εcm )

(3.52)

where

sr,max

is the maximum crack distance

(εsm  εcm )

is the difference in deformation between steel and concrete over the maximum crack distance.

Formulations for sr,max and (εsm  εcm ) are given in the formula in clause 7.3.4 Eqn. (7.9) – EC 2:

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

σs  kt fct,eff (1  αe ρp,eff )

ρp,eff

σs

(εsm  εcm ) 

 0,6

Es

Es

(3.53)

where

σs is the stress in the steel assuming a cracked section,

αe is the moduli ratio Es / Ecm,

ρp,eff  (As  ξAp ) Ap

is the effective reinforcement ratio (including eventual prestressing steel Ap)

ξ is the moduli ratio Es / Ecm

kt is a factor depending on the duration of load (0,6 for short and 0,4 for long term loads). For the crack spacing sr,max a modified expression has been derived, including the concrete cover. From experimental observations the crack at the outer concrete surface is wider than at the reinforcing steel level. Moreover, cracks are always measured at the surface of the structure (see Figure 3.3.4).

Fig. 3.3.4 Measured crack width w

The maximum final crack spacing sr,max according to clause 7.3.4 Eqn. (7.11) – EC 2 is given by:

sr ,max  k3c  k1k2k4   p,eff

(3.54)

where c ϕ k1 k2 k3 k4

is the concrete cover, is the bar diameter, is the bond factor (0,8 for high bars, 1,6 for bars with an effectively plain surface, e.g. prestressing tendons), is the strain distribution coefficient (1,0 for tension and 0,5 for bending: intermediate values can be used), is recommended to be 3,4 and is recommended to be 0,425.

In order to apply the crack width formulae, basically established for a concrete bar in tension, to a structure under bending, a definition of the “effective tensile bar height” is necessary. This effective height hc,ef is the minimum of:

2,5 (h-d) or

(h-x)/3

or

h/2.

Limit state design (ULS-SLS) J. Walraven and S. Gmainer

Fig. 3.3.5 Effective tensile bar height according to EC 2

Table 3.3.2 shows the EC 2 requirements for the crack width control (recommended values).

Table 3.3.2 Recommended values for crack width control

Exposure class

Prestressed members with bonded tendons

Quasi-permanent load

Frequent load

X0,XC1

0,3

XC2, XC3, XC4 0,3 XD1, XD2, XS1, XS2, XS3

0,2 Decompression

3.3.2.1 Crack width control – slab with embedded elements Internal support Assuming concentric tension in the upper slab of 50 mm depth (see Figure 3.3.7) leads to the following calculation.

Fig. 3.3.7 Geometry and bending moments 98

Limit state design (ULS-SLS) J. Walraven and S. Gmainer The steel stress σs,qp under the quasi-permanent load and the reinforcement ratio are: : σs,qp  Qqp As,req fyd  0,597  0,73  435  190,2N/mm² QEd As,prov

ρs,eff  Asl bd  1256 1000  50  2,51% The crack distance (according to Eqn.(3.55)) is

sr,max  3,4 19  0,8 1,0  0,425  20 0,0251  335,3 mm

The average strain (according to Eqn.(3.54)) amounts to

190,2  0,4  2,6  (1 7  0,0251)

εsm  εcm 

0,0251 200000

 0,71103  0,57 103

The characteristic crack width (according to Eqn. (3.53)) is

wk  sr,max εsm  εcm   335,3  0,71103  0,24  0,30 mm

Mid-span The steel stress σs,qp under the quasi-permanent load and the reinforcement ratio are: σs,qp  Qqp As,req fyd  0,597 1,04  435  270N/mm² QEd As,prov

s,eff  Asl . bheff For heff, the smallest of 2,5(h-d), (h-x) / 3 or h / 2 should be chosen. The critical value for heff is: 2,5(h-d)=2,5·33=82,5 mm.

ρs,eff  226 120  82,5  2,28%

The crack distance according to Eqn.(3.55) is

sr,max  3,4  29  0,8  0,5  0,425  12 0,0228  188,1 mm

The average strain (according to Eqn.(3.54)) amounts to

270  0,4  2,6  (1 7  0,0228)

εsm  εcm 

0,0228 200000

 1,09 103  0,81103  ok!

The characteristic crack width according to Eqn. (3.53) is

Limit state design (ULS-SLS) J. Walraven and S. Gmainer wk  sr,max εsm  εcm   188,11,09 103  0,20mm  0,30mm. References Zilch, K., and G. Zehetmaier. 2010. Bemessung im konstruktiven Betonbau, nach DIN 1045-1 (Fassung 2008) und EN 1992-1-1 (Eurocode 2). Springer Verlag.

CHAPTER 4 DETAILING OF THE REINFORCEMENT Jose ARRIETA PROES S.A., Spain

Detailing of reinforcement J. Arrieta

Detailing of reinforcement J. Arrieta 4.1 Detailing - general

EN 1992-1-1 Section 8 is dedicated to detailing rules for ribbed reinforcement, meshes and prestressing tendons subjected predominantly to static loading. Different rules than those in Eurocode 2 apply to reinforcement of buildings in seismic zones. Section 8.2 defines the minimum bar spacing required to guarantee a good placing and compaction of concrete, to grant adequate bond. Table reports the minimum spacing smin = (dg + 5) mm for each bar diameter, assuming the maximum aggregate size dg = 20 mm. To bend a bar avoiding cracks in the bar and/or failure of the concrete inside the bend a minimum diameter of the mandrel ϕm,min is required in Section 8.3. For each bar diameter in Table 4.1.1 the mandrel diameter ϕm,min is given, assuming that provisions in Section 8.3 (3) are fulfilled.

Table 4.1.1. Minimum spacing and mandrel diameter

ϕ (mm) 8 10 12 14 16 20 25

smin (mm) 25 25 25 25 25 25 25

ϕmand,min (mm) 32 40 48 56 64 140 175

4.1.1. Anchorage length To avoid longitudinal cracks and spalling of concrete the anchorage of longitudinal reinforcement has to satisfy the conditions given in Section 8.4. Applying these provisions are obtained the anchorage lengths in Table 4.1.2, Table 4.1.3 and Table 4.1.4 for structural elements of the building subjected to several conditions: o Tension or compression o Good or poor bond conditions (related to concreting) o Straight anchorage or standard bend, hook or loop The design anchorage length lbd has been calculated for straight bars; for standard bends, hooks or loops, the simplified procedure described in clause 8.4.4 (2), based on the equivalent anchorage length lb,eq has been used. For standard bends or hooks, table data are based on the assumption smin  2cnom. Confinement by transverse reinforcement or transverse pressure was neglected.

Detailing of reinforcement J. Arrieta

ϕ (mm) 8 10 12 14 16 20 25

Table 4.1.2. Anchorage lengths for FOOTINGS (C25/30 cnom = 40 mm)

lb,d straight anchorage (mm)

Tension

Compression

Good Poor Good Poor

lb,eq standard bend, hook or loop (mm)

Tension

Compression

Good Poor Good Poor

Table 4.1.3. Anchorage lengths for BEAMS AND SLABS (C25/30 cnom = 30 mm)

ϕ (mm) 8 10 12 14 16 20 25

lb,d straight anchorage (mm)

Tension

Compression

Good Poor Good Poor

lb,eq standard bend, hook or loop (mm)

Tension

Compression

Good Poor Good Poor

ϕ (mm) 8 10 12 14 16 20 25

Table 4.1.4. Anchorage lengths for COLUMNS (C30/37 cnom = 30 mm)

lb,d straight anchorage (mm)

Tension

Compression

Good Poor Good Poor

lb,eq standard bend, hook or loop (mm)

Tension

Compression

Good Poor Good Poor

Regarding the anchorage of links and of shear reinforcement using bends and hooks, the provisions in clause 8.5 result in the length after the curved part llink in Table 4.1.5.

Detailing of reinforcement J. Arrieta

Table 4.1.5. Length after the curved part for links

ϕ (mm) 6 8 10 12

llink (mm) Bend Hook

4.1.2. Lap length

The transmission of forces between the bars may be obtained using laps. To avoid the spalling of concrete and/or large cracks, the laps should be staggered far from high moments/forces zones and symmetrically arranged in any section. Using provisions in clause 8.7 the lap lengths l0 for beams, slabs and columns (Tables 4.1.6, 4.1.7, 4.1.8 and 4.1.9) have been determined for the following conditions: o Tension or compression o Good or poor bond conditions related to concreting o Different percentages 1 of lapped bars in a zone around the selected bar lap

Table 4.1.6 Lap lengths for BEAMS AND SLABS (C25/30 cnom = 30 mm) - TENSION

ϕ (mm) 8 10 12 14 16 20 25

Lap length lo (mm)

Good bond conditions

Poor bond conditions

ρ1<25 ρ1=33 ρ1=50 ρ1>50 ρ1<25 ρ1=33 ρ1=50 ρ1>50

226 260 316 339 323 371 452 484

283 325 396 424 404 464 565 605

375 432 525 563 536 617 751 804

468 538 655 702 669 769 936 1003

561 645 785 841 801 922 1122 1202

747 859 1045 1120 1067 1227 1493 1600

979 1126 1370 1468 1398 1608 1957 2097

Detailing of reinforcement J. Arrieta

Table 4.1.7 Lap lengths for BEAMS AND SLABS (C25/30 cnom = 30mm) - COMPRESSION

ϕ (mm) 8 10 12 14 16 20 25

Lap length lo (mm)

Good bond conditions

Poor bond conditions

ρ1<25 ρ1=33 ρ1=50 ρ1>50 ρ1<25 ρ1=33 ρ1=50 ρ1>50

323 371 452 484 461 530 646 692

404 464 565 605 577 663 807 865

484 557 678 726 692 796 969 1038

565 650 791 848 807 928 1130 1211

646 743 904 969 922 1061 1291 1384

807 928 1130 1211 1153 1326 1614 1730

1009 1160 1413 1513 1441 1658 2018 2162

Table 4.1.8 Lap lengths for COLUMNS (C30/37 cnom = 30mm) – TENSION

ϕ (mm) 8 10 12 14 16 20 25

Lap length lo (mm)

Good bond conditions

Poor bond conditions

ρ1<25 ρ1=33 ρ1=50 ρ1>50 ρ1<25 ρ1=33 ρ1=50 ρ1>50

200 230 280 300 286 329 400 429

250 288 350 375 357 411 500 536

332 382 465 499 475 546 665 712

415 477 580 622 592 681 829 888

497 571 695 745 710 816 994 1065

661 760 926 992 945 1086 1322 1417

867 997 1213 1300 1238 1424 1733 1857

Table 4.1.9 Lap lengths for COLUMNS (C30/37 cnom = 30mm) – COMPRESSION

ϕ (mm) 8 10 12 14 16 20 25

Lap length lo (mm)

Good bond conditions

Poor bond conditions

ρ1<25 ρ1=33 ρ1=50 ρ1>50 ρ1<25 ρ1=33 ρ1=50 ρ1>50

286 329 400 429 408 470 572 613

357 411 500 536 511 587 715 766

429 493 600 643 613 705 858 919

500 575 701 751 715 822 1001 1072

572 658 801 858 817 939 1144 1225

715 822 1001 1072 1021 1174 1430 1532

893 1028 1251 1340 1276 1468 1787 1915

If the diameter of the lapped bars is  20 mm and the percentage ρ1 of lapped bars is  25%, transverse reinforcement is required (clause 8.7.4). Otherwise, any transverse reinforcement or links necessary for other reasons may be considered sufficient for the transverse tensile forces without further justification.

Detailing of reinforcement J. Arrieta 4.2 Detailing of structural members Section 9 of EN 1992-1-1 establishes rules to satisfy the safety, serviceability and durability requirements. Minimum and maximum reinforcement areas are defined to avoid concrete brittle failure and/or formation of wide cracks, and to resist forces coming for restrained actions. 4.2.1. Detailing of footing B-2 The columns of the building have direct concrete footings with total depth 800 mm which geometry is shown on Figure 4.2.1.

Fig.4.2.1 Definition of the footing

4.2.1.1 Design of the footing

To design and verify footing B-2, the soil pressures for ULS determined in the geotechnical part for the verification of the bearing capacity of the soil, was used. These pressures were obtained from the analytical method described in EN 1997-2 Annex 1. Figure 4.2.2 and the following equations summarize the mentioned model and allow to evaluate the soil pressure at the base of the foundation for the ULS action effects NEd, MEd,y and MEd,z.

σEd  NEd B ' L '



MEd,z

L '  L  2eL and eL  



NEd





MEd ,y

B '  L  2eB and eB 

 

NEd

Detailing of reinforcement J. Arrieta

Fig.4.2.2 Model for the bearing resistance calculation

Table 4.2.1 includes the mentioned internal forces and the eccentricities in each direction. As it can be seen the eccentricities are very low (less than 1 mm) and we will consider them as zero.

Table 4.2.1 Internal forces and eccentricities at ULS

Combination 1 2 3 4 5 6 7 8 9 10

NEd (kN) -4554,80 -4837,96 -4990,35 -4985,91 -4491,62 -5435,54 -5359,70 -5359,70 -4502,78 -5780,18

MEd,y (kNm) 3,82 -3,11 -2,71 -3,26 -2,73 -2,98 1,44 1,44 4,08 -2,36

MEd,z (kNm) -3,78 0,60 0,66 -2,31 -1,38 -3,46 0,71 0,71 -3,50 -4,49

eL (mm) 0,8 -0,1 -0,1 0,5 0,3 0,6 -0,1 -0,1 0,8 0,8

eB (mm) 0,8 -0,6 -0,5 -0,7 -0,6 -0,5 0,3 0,3 0,9 -0,4

Detailing of reinforcement J. Arrieta The highest soil pressure is: σEd  NEd B ' L '  5780,18 2,00  2,00  1445 kN/m2 In order to design the bottom reinforcement area in the footing, the approach in clause 9.8.2.2 of EN 1992-1-1 for the anchorage of the bars provides the maximum force in the reinforcement. The effective pressure ’Ed has to be considered to calculate the reinforcement of the footing subjected to the soil pressure and to its self-weight (Figure 4.2.3). σ 'Ed  σEd  qsw,d  1445  1,35  25  0,80  1418 kN/m2

Fig.4.2.3 Effective pressure

Figure 4.2.4 and the following equations describe how to define the anchorage of the bars: assuming x = b/2 – 0,35a the maximum tensile force on the reinforcement is obtained.

Fs  x   Rd  x  ze  x  zi

being



b

x

ze  x    0,35a 



Rd  x   σ 'Ed b ' x

b



Fs,max  Fs   0,35a 

2



Fig.4.2.4 Model for tensile forces in cracks 109

Detailing of reinforcement J. Arrieta

The main characteristics of the footing are:

Concrete: C25/30

Steel: B500

a = 500 mm

e = 0,15a = 75 mm [9.8.2.2 (3)]

b = b’ = 2000 mm

h = 800 mm

cnom = 40 mm

d = h – cnom – 1,5ϕ = 736 mm (assuming ϕ = 16 mm)

zi = 0,90d = 662 mm (assuming ϕ = 16 mm) [9.8.2.2 (3)]

Substituting the footing characteristics, the expression for the tensile force in the reinforcement Fs  x   2836  x  0,825  0,5x 0,662 The maximum value of the tensile force Fs,max and the needed area of the reinforcement As are: Fs,max  Fs 0,825  1457,9 mm2  As  Fs,max  3353 mm2  1716 fyk γs

4.2.1.2 Arrangement of the reinforcement The minimum bar diameter to be used in a footing is 8 mm [9.8.2.1 (1)], so the provided reinforcement is correct. The clear distance between bars is 98 mm (Figure 4.2.5) greater than smin = 25 mm (see Table 4.1.1). To verify the conditions of the bar straight anchorage, it has to be verified that lb + cnom < xmin where xmin is the distance of the first crack located [9.8.2.2 (5)] at distance xmin = h/2. From Table 4.1.2 the design anchorage length in case of tensile force and good bond conditions is lbd = 500 mm.

Fs  xmin   Fs 0,40  1071,0 kN 

lb  Fs  xmin  lbd  1071,0  500  360 mm

As fyd

1485,7

lb  cnom  400 mm  xmin  400 mm

Detailing of reinforcement J. Arrieta Fig.4.2.5 Reinforcement of the footing 111

Detailing of reinforcement J. Arrieta Figure 4.2.6 shows the actual force in the reinforcement Fs,Ed (action) as a function of x and the capacity of the reinforcement Fs,Rd (resistance) taking into account the anchorage. From the distance xmin of the first crack is always Fs,Rd > Fs,Ed.

1600,0

1400,0

1200,0

Fs

1000,0 800,0 600,0

Fs,Ed Fs,Rd

400,0

200,0

0,0 0,40

0,50

0,60

0,70

0,80

0,90

x

Fig.4.2.6 Forces in the reinforcement ϕ16

For bars with diameter ϕ = 20 mm: d = h – cnom – 1,5ϕ = 800 – 40 – 1,5·20 = 730 mm zi = 0,90d = 657 mm [9.8.2.2 (3)] Fs,max  Fs 0,825  1469,0 kN 

As  Fs,max  33,79 cm2 fyk γs

 1120

In this case the clear distance between bars should be 162 mm > smin = 25 mm (see Table 4.2.11). For the straight anchorage of the bars, taking into account that for a ϕ=20mm bar lbd = 686 mm (Table 4.2.2):

Fs  xmin   Fs 0,40  1079,1kN 

lb  Fs  xmin  lbd  1079,1  686  493 mm

As fyd

1501,7

lb  cnom  533 mm  xmin  400 mm In this case a straight anchorage cannot be used, and a bend as in Figure 4.2.7, with la = 195 mm is needed.

Detailing of reinforcement J. Arrieta Fig.4.2.7 Bend anchorage for bars ϕ20 Fig.4.2.8 Forces in the reinforcement ϕ20 Figure 4.2.8 shows the actual force Fs,Ed and the resistance forces Fs,Rd1 and Fs,Rd1 corresponding respectively to straight and bend anchorages. For punching verification [6.4], in this specific case the basic control perimeter, which may be assumed at a distance 2,0d from the loaded area, is outside the footing.

4.2.2. Detailing of beams

Detailing of reinforcement J. Arrieta

Clause 9.2 contains the detailing rules for the beams. The material properties for the beams of the building are the following:

o Concrete: fck = 25 N/mm2;

γc = 1,50;

fctm=0,30fck2/3 = 2,56 N/mm2

o Steel:

fyk = 500 N/mm2;

γs = 1,15

The minimum area As,min of longitudinal tension reinforcement [9.2.1.1 (1)] is:

As,min  0,26 fctm bt d  0,0013bt d  As,min  0,00133bt d fyk

bt being the mean width of the tension zone. The maximum area As,max of longitudinal tension or compression reinforcement [9.2.1.1 (3)] is: As,max  0,04 Ac

For the curtailment of the longitudinal tension reinforcement the “shift rule” [9.2.1.3 (2)], gives the following shift length al: al  z cot θ  cot α  2 Assuming z = 0,9d and vertical links as shear reinforcement (α = 90º), the shift length is: al  0,45d cot θ cotθ has the same value as for the design of shear reinforcement. For shear reinforcement, the minimum ratio is [9.2.2 (5)]:

ρw,min  Asw  0,08 fck

sbw sinα

fyk

 ρw,min  0,0008

As α = 90º

 Asw     0,0008bw  s min The maximum longitudinal spacing between links sl,max is given by: sl,max  0,75d 1 cot α   0,75d

The maximum value for the transverse spacing of the legs of shear links st,max is defined by: st,max  0,75d  600 mm

Detailing of reinforcement J. Arrieta 4.2.2.1 Beam A2 – B2 – C2 for the case 1 This beam corresponds to case 1: two way slab on beams. The geometry is defined in Figure 4.2.9: Assuming ϕw = 8 mm for links and ϕ = 16 mm for longitudinal reinforcement, we obtain:

Fig.4.2.9 Geometric definition of beam A2 – B2 – C2

d  h  cnom  w  2  400  30  8  8  354 mm

250 mm bt   1100 mm

for positive moments for negative moments

bw  250 mm Substituting these values in the previous expressions we obtain:

118 mm2

As,min  0,00133btd  

518 mm

for positive moments for negative moments

As,max  0,04Ac  6520 mm2

al  0,45d cot θ  400 mm for cot θ  2,5

 Asw 

mm2

   0,0008bw  0,20

 s min

mm

sl,max  0,75d  266 mm

st,max  0,75d  266 mm

Detailing of reinforcement J. Arrieta From calculations we obtain the envelope of the internal forces: positive and negative moments and shear, and then the envelope of the required longitudinal and transverse reinforcement forces. In Figure 4.2.10 we can see the envelope of the required force Fs,Ed for the top and the bottom longitudinal reinforcement. Due to inclined cracks these envelopes are shifted of length al obtaining F*s,Edd that represents the capacity of the reinforcement required. For the selected number of longitudinal bars the resisted force Fs,Rd taking into account the anchorage lengths are calculated and drawn. At any section it must be : Fs,Rd ≥ F*s,Ed. Fig.4.2.10 Curtailment of longitudinal reinforcement - Beam A2 – B2 – C2 For the shear reinforcement, we obtain the required force in the transverse reinforcement Fsw,Ed from the envelope of the shear and we dispose links with a capacity of Fsw,Rd so that: Fsw,Rd ≥ Fsw,Ed. (Figure 4.2.11). Fig.4.2.11 Transverse reinforcement envelope - Beam A2 – B2 – C2 Figure 4.2.12 shows the final arrangement of the longitudinal and the transverse reinforcement for this beam.

Detailing of reinforcement J. Arrieta Fig.4.2.12 Reinforcement - Beam A2 – B2 – C2 117

Detailing of reinforcement J. Arrieta 4.2.2.2 Beam B1 – B2 – B3 for the case 3 This is the case of monodirectional slab with embedded lighting elements. The geometric dimensions of this beam are in Figure 4.2.13.

Fig.4.2.13 Geometric definition of beam B1 – B2 – B3

Assuming as in the precious case ϕw = 8 mm for the links and ϕ = 16 mm for the longitudinal reinforcement, we obtain:

d  h  cnom  w  2  400  30  8  8  354 mm

250 mm bt   600 mm

for positive moments for negative moments

bw  250 mm

118 mm2

As,min  0,00133btd  

282 mm

for positive moments for negative moments

As,max  0,04Ac  722 mm2

al  0,45d cot θ  400 mm for cot θ  2,5

 Asw 

mm2

   0,0008bw  0,20

 s min

mm

sl,max  0,75d  266 mm

Detailing of reinforcement J. Arrieta st,max  0,75d  266 mm As for beam 2, we obtain the envelopes of the internal forces and the required reinforcement. Figure 4.2.14 shows the envelope of the required force Fs,Ed for the top and the bottom longitudinal reinforcement, the shifted envelope F*s,Ed and the resisted force Fs,Rd. At any section Fs,Rd ≥ F*s,Ed. Fig.4.2.14 Curtailment of the longitudinal reinforcement of beam B1 – B2 – B3 For the shear reinforcement, we obtain the required force in the transverse reinforcement from the envelop of shear and we dispose links with a capacity of Fsw,Rd so that: Fsw,Rd ≥ Fsww,Ed (Figure 4.2.15). Fig.4.2.15 Transverse reinforcement envelop - Beam B1 – B2 – B3 In Figure 4.2.16 is the final arrangement of the longitudinal and the transverse reinforcement for this beam.

Detailing of reinforcement J. Arrieta Fig.4.2.16 Reinforcement - Beam B1 – B2 – B3 120

Detailing of reinforcement J. Arrieta 4.2.3. Detailing of slabs The material properties of slabs are the same as those of the beams. 4.2.3.1 Slab AB12 for case 1 In this case (two way slab on beams) the slab has a total height of 18 cm. In Figure 4.2.17 is shown the geometry of the slab. Fig.4.2.17 Geometric definition of slab AB12 for case 1 Provisions for the detailing of the reinforcement for this type of slabs are exposed in Clause 9.3. The minimum and maximum values for the reinforcement area, As,min and As,max, are the same as for the beams [9.3.1.1 (1)]. For a slab unit width (b = 1 mm): d  h  cnom    2  180  30  12  6  132 mm (  12 mm) bt  1mm

Detailing of reinforcement J. Arrieta mm2 As,min  0,00133btd  0,176 mm mm2 As,max  0,04Ac  7,200 mm The maximum spacing of the bars smax,slabs is [9.3.1.1 (3)]: smax,slabs  2,0  h  250 mm  smax,slabs  250 mm And the shift length for the curtailment of the reinforcement [9.3.1.1 (4)] is: al  d  132 mm The design of the slab gives the required reinforcement in each (X,Y) direction, for each strip (Figure 4.2.18). Fig.4.2.18 Required reinforcement of slab AB12 for case 1 122

Detailing of reinforcement J. Arrieta Finally we dispose the real reinforcement, taking into account the previous provisions about limit values for areas, diameters and distances (Figure 4.2.19).

Fig.4.2.19 Arrangement of reinforcement of slab AB12 for case 1

4.2.4. Detailing of columns

The characteristics of a column, as a structural member, are defined in §5.3.1 and the detailing rules are in section 9.5. For the longitudinal reinforcement (9.5.2), the minimum diameter of the bars is ϕmin=8 mm and the total amount of its area is limited by a minimum and a maximum value:

 0,10NEd



As,min  max 

;0,002Ac 

 fyd



As,max  0,04 Ac

where Ac is the sectional area of the column.

Detailing of reinforcement J. Arrieta

For transverse reinforcement (9.5.3) the minimum diameter ϕt,min is defined and the maximum spacing st,max are :



1

t,min  max 6 mm ;  long 



4

st,max  min 20long ;bmin;400 mm This maximum spacing is reduced by a factor 0,60, in zones near a beam or a slab, and in lapped joints if the diameter of the bars is greater than 14 mm. In this case, a minimum of 3 bars must be present. No longitudinal compression bar can be at a clear distance greater than 150 mm from a restrained bar; restraining is done through transverse reinforcement or splices. When there is a change of direction in a longitudinal bar, the lateral forces may be ignored if the slope of the change is less or equal to 1 in 12, otherwise pushing forces have to be considered.

4.2.4.1 Column B2 for the case 2

The column analysed corresponds to case 2: flat solid slab. The geometrical data are in Figure 4.2.20.

The materials used to make the columns have the following properties:

o Concrete: fck = 30 N/mm2;

γc = 1,50;

fctm  0,30fck 2/3  2,90 N/mm2

o Steel:

fyk = 500 N/mm2;

γs = 1,15

Applying the actual values of the column to the previous expressions we obtain:

o Longitudinal reinforcement

min  8 mm

As,min



max 0,23NEd ;500

mm2  

As,max  10000 mm2

o Transverse reinforcement

6 mm  t,min   long  4

if long  24 m m if long  24 m m

st,max  min 20long ;400 mm The characteristics of a column, as a structural member, are defined in Cause 5.3.1 and the detailing rules in Section 9.5. For the longitudinal reinforcement (9.5.2), the bar’s minimum diameter is ϕmin=8 mm and the total amount is limited by a minimum and a maximum value:

 0,10  NEd



As,min  max 

;0,002Ac 

 fyd



Detailing of reinforcement J. Arrieta

As,max  0,04 Ac

where Ac is the transverse area of the column. For transverse reinforcement (9.5.3) the minimum diameter ϕt,min and the maximum spacing st,max are:



1

t,min  max 6 mm ; long 



4

st,max  m in 20long ; bmin ; 400 m m  The maximum spacing is reduced by a factor 0,60 in zones near a beam or a slab, and in lapped joints if the diameter of bars is greater than 14 mm. In this case a minimum of 3 bars must be present. No longitudinal compression bar can be at a clear distance greater than 150 mm from a restrained bar; restraining is obtained using transverse reinforcement or splices When there is a change of direction in a longitudinal bar, the lateral forces may be ignored if the slope of the change is less or equal to 1:12, otherwise pushing forces have to be considered.

4.2.4.2 Column B2 - case 2

The column considered corresponds to case 2 - flat solid slab. Geometrical data are in figure 20.

The materials used have the following properties:

o Concrete: fck = 30 N/mm2;

γc = 1,50;

fctm  0,30fck 2/3  2,90 N/mm2

o Steel:

fyk = 500 N/mm2;

γs = 1,15

Applying the actual values of the column to the previous expressions we obtain:

o Longitudinal reinforcement

min  8 mm

As,min



max 0,23NEd ;500

mm2  

As,max  10000 mm2

o Transverse reinforcement

6 mm  t,min   long  4

if long  24 mm if long  24 mm

st,max  min 20long ;400 mm

Detailing of reinforcement J. Arrieta

Fig.4.2.20 Geometric definition column B2 – Case 2

Fig.4.2.21 Transverse reinforcement column B2 - Case 2

Detailing of reinforcement J. Arrieta

An uniform distributed reinforcement is provided along the perimeter of the column. If the total area of reinforcement required is As,rqd, applying the mentioned rules, the area of reinforcement really disposed is As,dsp (Table 4.2.11).

Table 4.2.11 Longitudinal reinforcement for Column B2 – Case 2

Floor L-2/L-1 L-1/L0 L0/L1 L1/L2 L2/L3 L3/L4 L4/L5 L5/Roof

As,rqd [mm2] 5581 3551 1082 0 0 0 0 0

As,min1 [mm2] 1305 1177 1012 838 670 504 344 216

As,min2 [mm2] 500 500 500 500 500 500 500 500

As,disp [mm2] 5892 3768 1232 904 904 628 628 628

12ϕ25 12ϕ20 8ϕ14 8ϕ12 8ϕ12 8ϕ10 8ϕ10 8ϕ10

Closed links are used for transverse reinforcement as in Figure 4.2.21. Following the prescriptions of the code the links spacing of the links is reduced in the zones close to the slab and in the laps.

Table 4.2.12 shows the extension of the zones and the links’ diameter and spacing in each zone.

Table 4.2.12 Transverse reinforcement for column B2 – Case 2

Floor L-2/L-1 L-1/L0 L0/L1 L1/L2 L2/L3 L3/L4 L4/L5 L5/Roof

ϕt,min [mm] 8 6 6 6 6 6 6 6

st,max [mm] 400 400 280 240 240 200 200 200

L [mm] 1340 1340 1072 751 643 643 536 536

Links

At1

At2

ϕ8 - 240

ϕ8 - 400

ϕ6 - 240

ϕ6 - 400

2ϕ6 - 160

2ϕ6 - 280

2ϕ6 - 140

2ϕ6 - 240

2ϕ6 - 140

2ϕ6 - 240

2ϕ6 - 120

2ϕ6 - 200

2ϕ6 - 120

2ϕ6 - 200

2ϕ6 - 120

2ϕ6 - 200

Finally in Figure 4.2.22 the arrangement of the column reinforcement is represented.

Detailing of reinforcement J. Arrieta Fig.4.2.22 Reinforcement for column B2 - Case 2 128

CHAPTER 5 SOME GEOTECHNICAL ASPECTS OF BUILDING DESIGN (EN 1997) Roger FRANK Université Paris-Est, Ecole des Ponts ParisTech, Laboratoire Navier Geotechnical team (CERMES), France

Some geotechnical aspects of building design (EN 1997) R. Frank

Some geotechnical aspects of building design (EN 1997) R. Frank 5.1 Introduction Eurocode 7 deals with all the geotechnical aspects of the design of structures (buildings, bridges and civil engineering works) and should be used for all the problems of interaction of structures with the ground (soils and rocks), through foundations or retaining structures Eurocode 7 allows the calculation of the geotechnical actions on the structures, as well the resistances of the ground submitted to the actions from the structures. It also gives all the prescriptions and rules for good practice required for properly conducting the geotechnical aspects of a structural project or, more generally speaking, a purely geotechnical project. Eurocode 7 consists of two parts: o EN 1997-1 Geotechnical design - Part 1: General rules (CEN, 2004) o EN 1997-2 Geotechnical design - Part 2: Ground investigation and testing (CEN, 2007). In the following, it is applied to the geotechnical design of a reinforced concrete building, designed by applying the principles of Eurocode 2. The example building is a six-storey building, with two storeys below the ground level. It is designed as a reinforced concrete skeleton construction and has total dimensions of 30,25 m in length, 14,25 m in width and a height of 19 m above ground level – details are in Chapter 1. The inner columns are founded on square spread foundations of dimensions B = 2 m and L = 2 m; the outer columns and the shear walls are supported by a peripheral diaphragm retaining wall of width 0,6 m, of height 9 m (embedded 3 m below the 2 levels of parking) – see Figure 5.2.1. After some considerations about the geotechnical data, the following calculations will be presented: o for column B2: bearing capacity and sliding resistance of the spread foundations (ULS verifications); o comments on the settlement of the columns (SLS verification). 5.2 Geotechnical data The soil investigation can consist of core sampling, laboratory tests (e.g. identification and tri-axial compression tests), field tests (e.g. pressuremeter tests MPM and cone penetration CPT tests), etc. see EN 1997-2 (CEN, 2007) for the use of these tests in geotechnical design). The selection of appropriate values of soil properties for foundations (or other geotechnical structures) is probably the most difficult and challenging phase of the whole geotechnical design process and cannot be extensively described here. In the Eurocodes approach, in particular the Eurocode 7 one, characteristic values of materials properties should be determined before applying any partial factor of safety. Figure 5.2.2 shows the link between the two parts of Eurocode 7 and, more important, gives the path leading to characteristic values. The present ‘philosophy’ with regard to the definition of characteristic values of geotechnical parameters is contained in the following clauses of Eurocode 7 – Part 1 (clause 2.4.5.2 in EN1997-1): ‘(2)P The characteristic value of a geotechnical parameter shall be selected as a cautious estimate of the value affecting the occurrence of the limit state.’ ‘(7) […]the governing parameter is often the mean of a range of values covering a large surface or volume of the ground. The characteristic value should be a cautious estimate of this mean value.’

Some geotechnical aspects of building design (EN 1997) R. Frank These paragraphs in Eurocode 7 – Part 1 reflect the concern that one should be able to keep using the values of the geotechnical parameters that were traditionally used (the determination of which is not standardised, i.e. they often depend on the individual judgment of the geotechnical engineer,). However two remarks should be made at this point. o on the one hand, the concept of 'derived value' of a geotechnical parameter (preceding the determination of the characteristic value) has been introduced (see Fig.5.2.1); o on the other hand, there is now a clear reference to the limit state involved (which may look evident, but is, in any case, a way of linking traditional geotechnical engineering and the new limit state approach) and to the assessment of the mean value (and not to a local value; this might appear to be a specific feature of geotechnical design which, indeed, involves 'large' areas or 'large' ground masses). Statistical methods are mentioned only as a possibility: ‘(10) If statistical methods are employed […], such methods should differentiate between local and regional sampling […].’ ‘(11) If statistical methods are used, the characteristic value should be derived such that the calculated probability of a worse value governing the occurrence of the limit state under consideration is not greater than 5%. NOTE In this respect, a cautious estimate of the mean value is a selection of the mean value of the limited set of geotechnical parameter values, with a confidence level of 95%; where local failure is concerned, a cautious estimate of the low value is a 5% fractile.’ For the sake of simplicity, in the present study, it is assumed that the whole building is founded on a very stiff clay with the following characteristics: o undrained shear strength (in terms of total stresses): cu = 300 kPa o total unit weight γk = 20 kN/m3 The water-table is assumed to be at natural ground level.

Type of test F= field L= laboratory Correlations Test results and derived values EN 1997-2 EN 1997-1

F1 F2

L 1

L 2

C1 C2

Cautious selection

Information from other sources on the site, the soils and rocks and the project

Geotechnical model and characteristic value s of geotechnical properties Application of partial factors

Design values of geotechnical properties Fig.5.2.1 General framework for the selection of derived values, characteristic values and design values of geotechnical properties (CEN, 2007)

Some geotechnical aspects of building design (EN 1997) R. Frank 5.3 Actions on the foundations 5.3.1. Structural and geotechnical actions The ‘structural’ actions to be considered on the foundations are taken from the structural analysis in Chapter 2. The combinations of forces and moments for column B2 are given at the sole of the foundation (0,8 m thick) both at ULS (for permanent and transient design situations, i.e. fundamental combinations) and at SLS – see Tables 5.3.1. For the diaphragm retaining wall, active and passive pressures from the ground and the groundwater (geotechnical actions) must also be considered. Table 5.3.1. Forces and moments on the foundation of column B2 - ULS and SLS

Some geotechnical aspects of building design (EN 1997) R. Frank

5.3.2. General: the three design approaches of Eurocode 7

When checking STR/GEO Ultimate Limit States for permanent and transient design situations (fundamental combinations), 3 Design Approaches (DA) are offered by Eurocode EN 1990 and Eurocode EN 1997-1 (Eurocode 7 – Part 1; CEN, 2004). The choice, for each geotechnical structure, is left to the National Application Document. For the bearing capacity of spread foundations and for retaining structures, these approaches can be summarised as follows.

5.3.2.1 Design approach 1 (DA1)

Two combinations (DA1-1 and DA1-2) should be used. It should be checked that an ULS is not reached for either of the two combinations. Combination 1 (DA1-1) is called the ‘structural combination’ because safety is applied on actions (i.e. partial load factors γF  1,0 are applied to action effects) while the design value of the geotechnical resistance Rd is based on the materials characteristic resistance. With the recommended values given in Note 2 of Table A2.4 (B) of EN 1990, for Eqn.6.10):

Ed {γFFrep} ≤ Rd {Xk}

(5.1)

where γF means γG,sup = 1,35; γG,inf = 1,00; γG,set = 1,35; 1,20 or 0; and γQ = 1,20 to 1,50 or 0. Combination 2 (DA1-2) is called the ‘geotechnical combination’ because the safety is applied on the geotechnical resistance Rd, through partial material factors γM > 1,0, applied at the ‘source’ to the ground parameters themselves. No safety factor is applied on unfavourable permanent (‘structural’ or ‘geotechnical’) actions. Note that for the resistance of piles and anchors resistances factors γR are used instead of material factors γM.

Some geotechnical aspects of building design (EN 1997) R. Frank

With the recommended values given in the Note of Table A2.4 (C) of EN 1990 for Eqn.6.10):

Ed {γFFrep} ≤ Rd {Xk /γM}

(5.2)

where γF means γG,sup = 1,00; γG,inf = 1,00; γG,set = 1,00 or 0; and γQ = 1,15 to 1,30 or 0. Table 5.3.3 summarises the recommended values of load factors used for DA1-1 (set A1) and DA1-2 (set A2).

Table 5.3.3 Partial factors on actions (γF) or the effects of actions (γE) (Table A.3 in EN 1997-1)

Action

Symbol

Set

A1

A2

Unfavourable

Permanent

γG

Favourable

Unfavourable

Variable

γQ

Favourable

1,35

1,0

1,0

1,0

1,5

1,3

For DA1-2, the recommended values for the partial factors γM both for ‘geotechnical’ actions and resistances are those of set M2 given in Table 5.3.4 (except for resistances of piles and anchors).

Some geotechnical aspects of building design (EN 1997) R. Frank

Table 5.3.4 Partial factors for soil parameters (γM) (Table A.4 in EN 1997-1)

Soil parameter

Symbol

Set

M1

M2

Angle of shearing resistance*

γ’

Effective cohesion

γc

Undrained shear strength

γcu

Unconfined strength

γqu

Weight density

γγ

* This factor is applied to tan φ'

1,0

1,25

1,0

1,25

1,0

1,4

1,0

1,4

1,0

1,0

5.3.2.2 Design approach 2 (DA2 and DA2*)

Only one combination should be used to check that the ULS is not reached. Safety is applied on both actions and resistances. On the action side, the factors can be applied either on the actions themselves (DA2, factors γF) or on the actions’ effects (DA2*, factors γE). Thus, o for DA2:

Ed {γFFrep} ≤ Rd {Xk}/γR

(5.3)

o for DA2 *:

γEEd {Frep} ≤ Rd {Xk}/γR

(5.4)

The recommended values for γF or γE are given in Note 2 of Table A2.4 (B) of EN 1990, for Eqn.6.10:

γG,sup = 1,35; γG,inf = 1,00; γG,set = 1,35; 1,20 or 0; and γQ = 1,20 to 1,50 or 0

The recommended values of the resistance factors for spread foundations and retaining structures are those for set R2 given in Table 5.3.5 and 5.3.6, respectively.

Table 5.3.5 Partial resistance factors (γR) for spread foundations (Table A.5 in EN 1997-1)

Resistance

Symbol

Set

R1

R2

R3

Bearing Sliding

γR;v

1,0

1,4

1,0

γR;h

1,0

1,1

1,0

Table 5.3.6 Partial resistance factors (γR) for retaining structures (Table A.13 in EN 1997-1)

Resistance

Symbol

Set

R1

R2

R3

Bearing capacity

γR;v

1,0

1,4

1,0

Sliding resistance

γR;h

1,0

1,1

1,0

Some geotechnical aspects of building design (EN 1997) R. Frank

Earth resistance

γR;e

1,0

1,4

1,0

5.3.2.3 Design approach 3 (DA3)

Only one combination should be used to check that the ULS is not reached. Safety is applied on both actions (factors γF) and on the geotechnical resistance Rd, through partial material factors γM > 1,0, applied at the ‘source’ to the ground parameters themselves. This writes:

Ed {γFFrep; Xk /γM } ≤ Rd {Xk /γM}

(5.5)

The recommended values for the actions are given: o for ‘structural’ actions, in Note 2 of Table A2.4 (B) of EN 1990, for Eqn.6.10:

γG,sup = 1,35; γG,inf = 1,00 and γQ = 1,20 to 1,50 or 0

and o

for ‘geotechnical’ actions, in the Note of Table A2.4 (C) of EN 1990 for Eqn.6.10:

γG,sup = 1,00; γG,inf = 1,00; γG,set = 1,00 or 0; and γQ = 1,15 to 1,30 or 0.

The recommended values of partial material factors γM for ground parameters are those of set M2 of Table 5.3.4.

5.3.2.4 Summary for DA1, DA2 and DA3 (for “fundamental” combinations) For spread foundations and retaining structures, the three Design Approaches, for ULS in permanent and transient design situations, can be summarised in a symbolic manner with sets A, M and R of Tables 5.3.3, 5.3.4, 5.3.5 and 5.3.6, as follows (“+” means “to be combined with”): 1. Design Approach 1 (DA1) Combination 1: A1 “+” M1 “+” R1 Combination 2: A2 “+” M2 “+” R1 2. Design Approach 2 (DA2) Combination: A1 “+” M1 “+” R2 3. Design Approach 3 (DA3) Combination: (A17 or A28) “+” M2 “+” R3 For the design of axially loaded piles and anchors, see EN 1997-1 (CEN, 2004).

7 on structural actions 8 on geotechnical actions

Some geotechnical aspects of building design (EN 1997) R. Frank 5.4 Column B2 - design of foundation

5.4.1. Bearing capacity (ULS)

The ULS condition is (Eqn.6.1 in EN 1997-1):

Nd ≤ Rd

(5.6)

where Nd is the design value of the axial component acting on the base of the foundation, due to structural and geotechnical actions; Rd is the design value of the resistance of the ground (ground bearing capacity) below the base of the foundation.

Geotechnical resistance (bearing capacity) The resistance R (bearing capacity) is calculated with the sample method of Annex D of EN 1997-1 (CEN, 2004) – see Annex 1 below. I For undrained conditions (with α = 0 and q = 0) as in the present case, R may be expressed as:

R = A’ (π+2) cu sc ic

(5.7)

with A’ = B’ L’ = (B-2eB)(L-2eL)

sc = 1+0,2 B’/L’

1

H

ic  2 1 1   A 'cu 

H being the resultant horizontal force (resultant of Vy and Vz)

Eccentricity is calculated by:

o in the transversal (B) direction:

eB = My/N

o in the longitudinal (L) direction:

eL = Mz/N

For the calculations of eB, eL, sc and ic, the design values Nd, Vyd and Vzd , as well as Mzd and Myd, which depend on the Design Approach under consideration, are needed. Thus the resistance depends on the actions, which is quite common in geotechnical engineering (because of this, it is sometimes necessary to check the calculations both with unfavourable and favourable values of a number of actions…).

Partial factors γM on cu and γR;v on the bearing capacity R are taken from the recommended values in Tables 5.3.4 and 5.3.5 respectively, for each Design Approach.

For DA1-1, DA2 and DA3, Nd, Vyd and Vzd, Mzd and Myd are given in Table 5.3.1 (derived with set A1 on actions – see Table 5.3.3). The governing combination of actions is taken as the one with the largest value of Nd:

Nd = 5,81 MN

Vyd = -4,53·10-3 MN

Vzd = -1,54·10-3 MN

Myd = -2,36·10-3 MNm

Mzd = -4,49·10-3 MNm

Hd = 4,78·10-3 MNm

Hd is the resultant of Vyd and Vzd. Horizontal loads and moments on this foundation seem negligible.

Some geotechnical aspects of building design (EN 1997) R. Frank

For DA1-2, these loads have to be divided by a factor somewhere between 1,11 and 1,35, depending on the ratio of permanent G to variable loads Q. For DA1-1, DA2 and DA3

eB = 4,1·10-4 m (!), eL = 7,8·10-4 m (!), B’ ≈ B and L’ ≈ L and sc = 1,2

Correction factor ic, and the total resistance R also depend on the Design Approach through γM (on cu) and γR;v (see Tables 5.3.4 and 5.3.5). Design Approach 1 Combination DA1-1: γM = 1,0 ; γR;v = 1,0

thus

cud = 300 kPa ; sc ≈ 1,2 , ic ≈ 1

and Rd = 4·5,14·1,2·1·300·10-3 /1,0 = 7,4/1,0 = 7,4 MN and Nd ≤ Rd is verified.

Combination DA1-2: γM = 1,4 ; γR;v = 1,0

thus: and

cud = 300/1,4= 214 kPa ; sc ≈ 1,2 , ic ≈ 1 Rd = 4·5,14·1,2·1·214·10-3 /1,0 = 5,28/1,0 = 5,28 MN

Let us assume that Nd is equal to Nd for A1-1 divided by 1,11, thus Nd = 5,23 MN and Nd ≤ Rd is verified.

According to DA1, the foundation is safe with regard to bearing capacity of the ground.

Design Approaches 2 and 3

Design Approaches 2 and 3 yield the same level of safety, because one of the values for the factors

γM and γR;v is equal to 1,4 and the other one is equal to 1,0. thus Rd = 4·5,14·1,2·1·300·10-3 /1,4 = 5,28 MN.

thus

Nd ≤ Rd is not verified.

It can be seen easily that the size of the footing should be around:

A’ = 1,4 x Nd /(π+2) cu sc ic ≈ 4,39 , that is, say: B = L = 2,10 m. The difference with the assumed B = 2,0 m is small.

5.4.2. Sliding (ULS)

The basic equation (Eqn. 6.2 in EN 1997-1) is:

Hd ≤ Rd + Rp;d

(5.8)

where Hd is the design value of the horizontal component of the load acting on the base of the foundation; Rp;d is the passive earth force in front of the spread foundation, which, for simplicity is not considered here as in order to take into account any passive force the earth has to be properly in contact in front of the footing and well compacted, which is not always the case. Rd is the sliding resistance, which for undrained conditions is (Eqns. 6.4a and 6.4b in EN 1997-1):

Rd = { A’cu /γM} / γR;h

(5.9)

where γM and γR;h are taken from the recommended values in Tables 5.3.4 and 5.3.6 respectively, for each Design Approach in persistent transient design situations.

Some geotechnical aspects of building design (EN 1997) R. Frank

As for the bearing capacity, the sliding resistance in undrained conditions also depends on the values of the actions (through A’ = B’ L’, which itself depends on the eccentricities); in drained conditions the sliding resistance is moreover directly proportional to the vertical load, which is thus a favourable action - and its design value should be obtained using the factors for favourable actions. The largest value of Hd for persistent transient design situations (fundamental combinations) is – see Table 5.3.1:

Hd = 5,03 kN (which is quite small) with Myd ≈ - 2,09·10-3 MNm, Mzd ≈ -4,70·10-3 MNm, Nd ≈ 5,5 MN (for DA1-1, DA2 and DA3).

The eccentricities eB and eL remain negligible and B’ ≈ B and L’ ≈ L. Thus A’≈ BL≈ 4m².

Design Approach 1

Combination DA1-1: γM = 1,0 ; γR;h = 1,0

thus

cud = 300 kPa and Rd = 4·0,300 /1,0 = 1,2 MN and Hd ≤ Rd is largely verified.

Combination DA1-2: γM = 1,4 ; γR;h = 1,0

thus

cud = 300/1,4= 214 kPa and Rd = 4·0,214/1,0 = 0,86 MN, with Hd < 5,03 kN

and

Hd ≤ Rd is largely verified.

According to DA1, the foundation is safe with regard to sliding.

Design Approach 2 γM = 1,0 ; γR;h = 1,1

thus

cud = 300 kPa and Rd = 4·0,300 /1,1 = 1,09 MN

and

Hd ≤ Rd is largely verified.

Design Approach 3 γM = 1,4 ; γR;h = 1,0

thus

cud = 300/1,4= 214 kPa and Rd = 4·0,214/1,0 = 0,86 MN

and

Hd ≤ Rd is largely verified.

5.5 Comments on settlements (SLS) 5.5.1. Compensated foundation The total weight of the present building is less than the weight of the ground removed to build the underlying parking. Assuming for the weight density of the ground γ = 20 kN/m3, the initial pressure at the base level of the excavation is around (3·20) kN/m3 = 60 kPa. Thus, settlements will be limited as the construction of the building simply consists in partly “putting back” something similar to the ground in its initial position. For this type of foundation, called “compensated foundation”, in practice settlements are ignored. Nevertheless, for the sake of illustration, various assumptions are made in order to estimate the maximum possible settlement. Settlements are usually checked under the vertical load Q obtained with quasi-permanent SLS combinations. For column B2, from Table 5.3.1: Q = 4,2 MN which corresponds to the applied pressure on the ground:

Some geotechnical aspects of building design (EN 1997) R. Frank q = Q / (BL) = 4,2/(2·2) = 1,05 MPa.

5.5.2. Calculation based on the results of Mènard pressuremeter method

Eurocode 7 – Part 2 (EN1997-2) provides, in informative Annexes, several methods for determining the settlement of spread foundations. In the following the Ménard pressuremeter (MPM) method is used, described in Annex E.2 of EN 1997-2 (CEN, 2002). The settlement is expressed as:

 2B0

a  λd B 

αλcB 

s  q  σv0      



 9Ed 

 B0 

9Ec  

(5.10)

In the current case: q = 1,05 MPa σvo = 0 assumed for simplicity, as if the soil is simply loaded from its initial natural level (without the unloading due to the excavation) B = 2 m B0 = 0,6 m For a square foundation: thus λd = 1,12 and λc = 1,1 For an overconsolidated clay in Eqn. (5.10) the exponent α may be assumed α = 1,0 The Ménard pressuremeter modulus is assumed to be constant over depth and equal, as a minimum, for highly overconsolidated clays to 16 times the limit pressure, his one being equal to 9cu on average (cu is the undrained shear strength of the clay - see e.g., Frank, 1999 and Baguelin, et al. ,1978) EM ≈ (9·16) cu = 144·300·10-3 = 43 MPa Ed = Ec = 43 MPa sB2 = (1,05 – 0,00)·[(2 0,6 1,12·2)/(9·43 0,6) + (1·1,1·2,0) / (9·43)] = 1,05·[0,0116+0,0057] = 0,017 m = 17 mm. The span between two columns is L = 6 m: assuming that the differential settlement δs = sB2/2, the relative rotation is: b = sB2 / 2L = 8,5/6000= 1,4·10-3 Annex H of EN 1997-1 (informative) states that for buildings in the most of cases a relative rotation β = 1/500 = 2·10-3 is acceptable. If this building was just built to “rest” on the stiff clay loaded from its natural level (with no excavation at all), the differential settlement would be clearly acceptable. Furthermore, some conservative assumptions have been made: for instance, it is most likely that the stiffness of the clay would increase with depth.

5.5.3. Adjusted elastic method

Eurocode 7 - Part 1 (CEN, 2004) allows the use of pseudo-elastic methods written under the form:

s = qBf / Em

(5.11)

Some geotechnical aspects of building design (EN 1997) R. Frank where Em is the design value of the modulus of elasticity. This method assumes that an estimate of an equivalent and unique Young’s modulus value is possible to represent the ground affected by the load and for the correct level(s) of deformation… All the difficulty lies in the assessment of Em. For the Fort Canning Tunnel in Singapore, Mair (2011) quotes a value of the undrained modulus Eu for the hard clay matrix Eu ≈ 500 MPa, from back-analysis of settlements of buildings on rafts. Note that the undrained shear strength of the Singapore clay matrix is cu > 150 kPa. The value for Eu is consistent with unload-reload moduli from pressuremeter tests and moduli from plate loading tests in the same clay. Thus, it would be 10 times the “first loading” modulus EM obtained with the Ménard pressuremeter quoted above, which is not surprising. The elastic calculation (Eqn.5.11) leads to: sB2 = 1,05·2,0·0,66/500 = 0,0023 = 2,8 mm. This settlement, largely less to the one calculated using the first loading Ménard modulus together with the Ménard empirical formula (Eqn. 5.10), is probably more realistic, as an elastic approach is more appropriate for the reloading phase of the clay (after excavation). The result confirms that the settlement of such a “compensated” foundation can be ignored in practice. References Baguelin, F., Jézéquel, J.F., Shields, D.H. (1978). The Pressuremeter and Foundation Engineering. Trans Tech Publications, Clausthal, Germany. CEN 2002. Eurocode: Basis of structural design. EN 1990: 2002. European Committee for Standardization (CEN): Brussels. CEN 2004. Eurocode 7: Geotechnical design - Part 1: General rules. EN 1997-1:2004 (E), November 2004, European Committee for Standardization: Brussels. CEN 2007. Eurocode 7: Geotechnical design - Part 2: Ground investigation and testing. EN19972:2007 (E), March 2007, European Committee for Standardization: Brussels. Frank R. 1999. Calcul des fondations superficielles et profondes, Presses de l’Ecole des ponts et Techniques de l’ingénieur, Paris. MELT-Ministère de l'Equipement, du logement et des transports 1993. Règles Techniques de Conception et de Calcul des Fondations des Ouvrages de Génie Civil (in French: Technical Rules for the Design of Foundations of Civil Engineering Structures). Cahier des clauses techniques générales applicables aux marchés publics de travaux, FASCICULE N°62 -Titre V, Textes Officiels N° 93-3 T.O., 182 pages.

Some geotechnical aspects of building design (EN 1997) R. Frank

ANNEX 1

From EN 1997-2 (CEN, 2004): Annex D (informative): A sample analytical method for bearing resistance calculation.

Annex D.3: Undrained conditions

The following symbols are used in Annex D.3

A' = B'  L'

the design effective foundation area

b

the design values of the factor for the inclination of the

base, with subscript c (with subscripts q and γ, only used for drained

conditions)

B

the foundation width

B'

the effective foundation width

D

the embedment depth

e

the eccentricity of the resultant action, with subscripts B

and L

i

the inclination factor of the load, with subscript c (with

subscripts q and γ for drained conditions only)

L

the foundation length

L'

the effective foundation length

q

overburden or surcharge pressure at the level of the

foundation base

s

the shape factor of the foundation base, with subscript c

(with subscripts q and γ for drained conditions only)

V

the vertical load

α

the inclination of the foundation base to the horizontal

γ

weight density of the soil below the foundation level

The notations used are given in Figure D.1.

Figure D.1 — Notations

D.3 Undrained conditions The design bearing resistance may be calculated from:

R/A' = (+2) cu bc sc ic + q

with the dimensionless factors for: o the inclination of the foundation base: bc = 1 – 2α / (π + 2); o the shape of the foundation:

sc = 1 + 0,2(B'/L')

for a rectangular shape;

sc = 1,2

for a square or circular shape.

(D.1)

Some geotechnical aspects of building design (EN 1997) R. Frank

ANNEX 2

From EN 1997-2 (CEN, 2004): Annex E (informative): E.2 Example of a method to calculate the settlements for spread foundations The following is an example of a method to calculate the settlement s, of spread foundations using a semi-empirical method developed for MPM tests. The example was published by the French Ministère de l‘Equipement du Logement et des Transport (1993). For additional information and examples, see EN1997-2, Annex X, §X.3.2.

 2B0

a  λd B 

αλcB 

s  q  σv0      



 9Ed 

 B0 

9Ec  

where

Bo B d, c  Ec Ed v0 q

is a reference width of 0,6 m; is the width of the foundation; are shape factors given in Table E.2; is a rheological factor given in Table E.3; is the weighted value of EM immediately below the foundation; is the harmonic mean of EM in all layers up to 8B below the foundation; is the total (initial) vertical stress at the level of the foundation base; is the design normal pressure applied on the foundation.

Table E.2 — The shape coefficients, c, d, for settlement of spread foundations

L/B

Circle

Square

d

1,12

1,53 1,78 2,14 2.65

c

1,1

1,2

1,3

1.4

1,5

Table E.3 — Correlations for deriving the coefficient  for spread foundations

Type of ground Peat

Description

EM/pLM



Clay

Over-consolidated Normally consolidated Remoulded

16

9–16

0,67

7–9

0,5

Silt Over-consolidated >14 0,67

Normally consolidated

5–14

0,5

Sand

>12

0,5

5–12

0,33

Sand and gravel Rock

Extensively fractured Unaltered Weathered

>10

0,33

6–10

0,25

0,33 0,5 0,67

CHAPTER 6 FIRE RESISTANCE ACCORDING TO EN 1992-1-2 Caroline MORIN and Fabienne ROBERT CERIB, France

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert 6.1 Introduction Fire is a definite danger to any construction and needs to be prevented and fought by all possible means. The fire may occur anywhere, in any session and in any phase in the lifetime of a building (construction, service, refurbishment or demolition). The aim of this chapter is to give a general overview of the fire design according to Eurocodes (EN 1990, EN 1991-1-2 and EN 1992-1-2) through the example out of the concrete building. The fire loadbearing capacity of three concrete members (a column, a beam and a slab) will be determined. The global analysis of the overall structure is not covered. EN 1990 concerns the basis of structural design. EN 1991-1-2 describes the thermal and mechanical actions for the structural design of building exposed to fire. EN 1992-1-2 describes the principles, requirements and rules for the structural design of concrete buildings for the accidental situation of fire exposure, including the safety requirements, design procedure and design aids. EN 1991-1-2 and EN 1992-1-2 are intended to be used in conjunction with EN 1991-1-1 and EN 1992-1-1. In this chapter, the prescriptive approach is adopted (in opposite to the performance-based code), i.e. it uses nominal fires to generate thermal actions like the standard temperature-time curve (EN 19911-2, Sec-3). Fire resistance is defined as “..the ability of a structure, a part of a structure or a member to fulfil its required functions (load bearing function and/or fire spreading function) for a specified load level, for a specified fire exposure and for a specified period of time…”. The methods given in EN 1992-1-2 are applicable since concrete materials used in the building are normal weight concrete materials with strength class lass then the limit strength class C90/105. In this chapter the different methods given in EN 1992-1-2, Sec-4 are used: o tabulated data (EN 1992-1-2, Sec 5); o simplified calculation methods (EN 1992-1-2, Sec-4); EN 1992-1-2 gives alternative procedures, values and recommendations for classes with notes indicating where national choices have to be made. Therefore the National Standard implementing EN 1992-1-2 should have a National Annex containing the Eurocode all Nationally Determined Parameters to be used for the design of buildings, and where required and applicable, for civil engineering works to be built in the relevant country. For this example, the French National Annex has been selected. 6.2 Data concerning building 6.2.1. Description of the building The building is described in 1.2.1, Chapter 1. The plan view and the main sections of the building are given in Figure 6.2.1 to 6.2.4. Focus is on: o The beam in axis 2 which consists of a continuous beam. Its cross section is a T-beam where its effective width has been calculated in chapter concerning Limit State Design (ULS-SLS). The effective width beff at mid-span is equal to 2,6 m and at intermediate support is equal to 1,83 m. The length Lbeam of the continuous beam is equal to 7,125 m. The width bw of the web is 0,25 m. The height of the slab hslab is 0,18 m. The height of the beam hbeam is 0,40 m; o The 4 m high column B2 is the one in the second basement. Its effective length l0,column has been calculated in the previous chapter concerning ULS-SLS and is equal to 3,1 m. The slenderness λcolumn of the column at normal temperatures is equal to 22,5. The cross-section is a square of 0,50 m. Its section Ac,column is equal to 0,25 m²;

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert o The slab on the beams (A1B2) is a two-way slab of uniform thickness (hslab = 0,18 m). The width of the slab in X-direction lx is equal to 6 m and the width of the slab in y-direction ly is equal to 7,125 m. Fig. 6.2.1 Plan view of the slab on beams Fig. 6.2.2 Section 1 of the building 148

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert Fig. 6.2.3 Section 2 of the building with the 8 floors Fig. 6.2.4 Elements verified under fire (dimensions) – column, beam and slab 149

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert 6.2.2. Mechanical material properties

6.2.2.1 General

The values of material properties shall be considered as characteristic values to be used with simplified and advanced calculation methods. The mechanical properties of concrete and reinforcing steel at normal temperature are given in EN 1992-1-1 for normal temperature design. Design values of mechanical (strength and deformation) material properties Xd,fi are defined as follows (Eqn 6.1):

X d ,fi  kθ X k γ M,fi

(6.1)

Xk is the characteristic value of strength or deformation property for normal temperature as described in EN 1992-1-1, kθ is the reduction factor for a strength or deformation property dependent on the material temperature (Xk,θ/Xk) and γM,fi is the partial safety factor for the relevant material property for the fire situation. For thermal and mechanical properties of concrete and reinforcing steel, γM,fi is taken equal to 1,0. Table 6.2.1 gives, for each member, the classes of concrete and reinforcement steel (see Ch.1).

Table 6.2.1 Concrete class and steel class of members

Slab C25/30 Grade 500 class B

Beam C25/30 Grade 500 class B

Column C30/37 Grade 500 class B

The exposure class considered is XC2-XC1. The nominal cover cnom (see Ch.1) due to non-uniformity of EU National choices, to avoid country specific conditions it was fixed to 30 mm.

6.2.2.2 Concrete

The concrete used is assumed to be made of siliceous aggregates. In EN 1992-1-2, Sec-3 strength and deformation properties of uniaxially stressed concreted at elevated temperatures are given in terms of stress-strain relationship. This relationship is described by two parameters: the compressive strength fc,θ and the strain εc1,θ corresponding to fc,θ. Values are given in Table 6.2.2, as a function of concrete temperatures. The reduction factor for concrete strength dependent on the material temperature is in Figure 6.2.5. Mathematical model for stress-strain relationships of concrete under compression at elevated temperatures is as in Eqn 6.2 for ε < εc1,θ:

σ θ 

3  ε  fc,θ

  ε 

εc1,θ   2  

 

  εc1,θ  





(6.2)

For εc1,θ < ε < εcu1,θ and numerical purposes, a descending branch should be adopted.

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert

Table 6.2.2 Values of the main parameters of the stress-strain relationships - normal concrete with siliceous aggregates (from EN 1992-1-2, Sec-3, Table 3.1)

Temperature (°C) 20 100 200 300 400 500 600 700 800 900 1000 1100 1200

fc,θ/fck 1.00 1.00 0,95 0.85 0.75 0.60 0.45 0.30 0.15 0.08 0.04 0.01 0.00

εc,θ 0.0025 0.0040 0.0055 0.0070 0.0100 0.0150 0.0250 0.0250 0.0250 0.0250 0.0250 0.0250 -

εc1,θ 0.0200 0.0225 0.0250 0.0275 0.0300 0.0325 0.0350 0.0375 0.0400 0.0425 0.0450 0.0475 -

Fig. 6.2.5 Coefficient kc(θ) for decrease of concrete strength fck (1: siliceous aggregates, 2: calcareous aggregates) 6.2.2.3 Reinforcing bars The reinforcing steel used is cold worked steel. The strength and deformation properties of reinforcing steel at elevated temperatures is obtained from stress-strain relationships described in EN 1992-1-2, Sec-3, defined by three parameters: the slope of the linear elastic range Es,θ, the proportional limit fsp,θ and the maximum stress level fsy,θ. Values of these parameters are given in Table 6.2.3, as a function of steel temperature. The mathematical model for stress-strain relationships of reinforcing steel at elevated temperatures is presented in Figure 6.2.6.

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert

Fig. 6.2.6 Mathematical model of the reinforcing steel stress-strain relationships at elevated temperatures according to EN 1992-2, Sec-3

Table 6.2.3 Values of the main parameters of the cold worked reinforcing steel stress-strain relationships at elevated temperatures (from EN 1992-1-2, Sec-3, Table 3.2a)

Temperature (°C) 20 100 200 300 400 500 600 700 800 900 1000 1100 1200

fsy,θ /fyk 1,00 1,00 1,00 1,00 0,94 0,67 0,40 0,12 0,11 0,08 0,05 0,03 0,00

Es,θ /Es 1,00 1,00 0,87 0,72 0,56 0,40 0,24 0,08 0,06 0,05 0,03 0,02 0,00

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert 6.2.3. Materials’ physical and thermal properties

Concrete thermal and physical properties are described in EN 1992-1-2, Sec-3 as a function of temperature θ and other variables. Differently from the thermal conductivity, thermal strain εc(θ), specific heat cp(θ) and density ρ(θ) are not nationally Determined Parameters,

6.2.3.1 Thermal strain of concrete and steel

o Concrete - the strain εc(θ) at temperature θ of a siliceous concrete is (Fig. 6.2.7):

εc θ   1,8 104  9 106  θ  2,3 1011  θ3

for 20C  θ  700C

εc θ   14 103

for700C  θ  1200C

(6.3) (6.4)

Fig. 6.2.7 Concrete strain f(θ) (1: siliceous aggregates, 2: calcareous aggregates)

o Steel - the strain εs(θ) at temperature θ of reinforcing steel is (Fig. 6.2.8):

εsc θ   2,416 104  1,2 105  θ  0,4 108  θ2 for 20C  θ  750C

εc θ   11103

for 750C  θ  860C

εsc θ   6,2 103  2 105  θ

for 860C  θ  1200C

(6.5) (6.6) (6.7)

Fig.6.2.8 Steel strain f(θ) (1: reinforcing steel, 2: prestressing steel) 153

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert 6.2.3.2 Concrete specific heat Calculation in the following are based on a moisture content u=1,5 % of concrete weight. Figure 6.2.9 illustrates the variation of the specific heat as a function of concrete temperature θ: for u=1,5 % the value of cp,peak is 1470 J/kgºK. Fig. 6.2.9 Specific heat cp, f(θ) and moisture contents by weight - siliceous concrete 6.2.3.3 Concrete thermal conductivity The variation of concrete thermal conductivity λc with temperature is set by the National annex within a range defined by lower and upper limits. For the following calculations, Eqns. 6.8, 6.9 and 6.10 were adopted, which correspond to the curves in Figure 6.2.10 described in the French National Annex.

Fig. 6.2.10 Thermal conductivity of concrete (French National Annex of EN 1992-1-2)

λc  2  0,2451θ 100  0,0107θ 1002 W/mK λc  0,02604θ  5,324 W/mK λc  1,36  0,136θ 100  0,0057θ 1002 W/mK

θ  140C 140  θ  160C θ  160C

(6.8) (6.9) (6.10)

6.2.3.4 Density of concrete and reinforcing bars The variation of density with temperature influenced by water loss is described in EN 1992-1-2, Sec-3.

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert 6.2.4. Reinforced members’ sections (column, beam and slab) 6.2.4.1 Column B2 According to Chapter 3, the column reinforcement is made of 12ϕ20 (37,69 cm²) in a symmetric manner with stirrups ϕ12/200 mm (Figure 6.2.11 and Table 6.2.4).

Fig. 6.2.11 Layout of the reinforced column B2

Table 6.2.4 Steel reinforcement of column B2

Longitudinal 12ϕ20

Transversal ϕ12/200 mm

The axis distance of the longitudinal steel bars from the concrete surface is: acolumn: (30 + 12 + 20/2) mm = 52 mm.

6.2.4.2 Beam in axis 2 The beam in axis 2 is a continuous beam with spans 7,125 m long, reinforced as in Table 6.2.5.

Table 6.2.5 Steel longitudinal (lower/upper) and transversal reinforcement of beam in axis 2

Title 1 Upper Lower Stirrups

Perimeter support 7ϕ12 3ϕ16 ϕ6/175

Middle span 2ϕ10 3ϕ16 ϕ6/175

Intermediate support 9ϕ12 3ϕ16 ϕ6/175

At middle span, the axis distance amid-span,beam of steel reinforcement from the nearest exposed surface is 44 mm. At support, the axis distance asupport,beam of steel reinforcement from the nearest nonexposed surface is 42 mm.

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert 6.2.4.3 Slab on beams For this study only one type of horizontal slabs, i.e. slab on beams, is considered. The slab thickness hslab is 0,18 m. The slab reinforcement is in Figure 6.2.12 and in Table 6.2.6 and Table 6.2.7.

Fig. 6.2.12 Layout of the reinforced slab

Table 6.2.6 Longitudinal reinforcement of the slab in x-direction

Title 1 Upper Lower

Perimeter beam strip (1,75 m) ϕ14/250 mm ϕ12/250 mm

Middle strip (3,5 m) ϕ14/125 mm ϕ12/125 mm

Intermediate beam strip (1,75 m) ϕ14/250 mm ϕ12/250 mm

Table 6.2.7 Longitudinal reinforcement of the slab in y-direction

Title 1 Upper Lower

Perimeter beam strip (1,5 m) ϕ16/250 mm ϕ12/500 mm ϕ14/500 mm

Middle strip (3 m) ϕ16/125 mm ϕ12/250 mm ϕ14/250 mm

Intermediate beam strip (1,5 m) ϕ16/250 mm ϕ12/500 mm ϕ14/500 mm

The axis distance ax,slab of reinforcing steel in x-direction from the nearest exposed surface is :

ax,slab  cnom  φx 2  30  122  36 mm

(6.11)

The axis distance ay,slab of reinforcing steel in Y-direction from the nearest exposed surface is :

ay,slab  cnom  φx  φy  30  12  14  49 mm

(6.12)

6.2.5.

Actions

The thermal and mechanical actions are taken from EN 1991-1-2. These actions, considered at normal temperature, shall be applied because they are likely to act in fire situation. The emissivity

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert

related to the concrete surface is taken as 0,7 (EN 1992-1-2, Sec-2). The following actions are considered: o Dead weight Gslab based on reinforced concrete unit weight of 25 kN/m3 and on the geometry of the slab; o Imposed permanent actions GImp (finishing, pavement, embedded services and partitions); o Variable actions Q1. To obtain the relevant effects of actions Efi,d,t during fire exposure, the mechanical actions are combined in accordance with EN 1990 for accidental design situations. In France, the representative value of the variable action Q1 is the frequent value ψ1,1Q1. (ψ1,1 = 0,5 for dwellings - EN 1990, Annex A, Table A1.1). The design loads are LEd = 13,125 kN/m² at normal temperatures and LEd,fi = 8,5 kN/m² under fire see Table 6.2.8. The ratio LEd,fi /LEd is equal to 0,65.

Gslab 4,5 kN/m²

Table 6.2.8 External actions on slabs

GImp 3 kN/m²

Q1 2 kN/m²

LEd,fi 8,5 kN/m²

The ratio lx/ly is equal to 0,84. The design isostatic moment in X-direction M0Ed,fi,x-slab is μx = 0,052. The design isostatic moment in y-direction M0Ed,fi,y-slab is μy = 0,671 and is equals to μyM0Ed,fi,x-slab. For beam in axis 2, the maximum bending moments in the fire situation can be obtained multiplying by 0,65 the design moments in chapter 3 . The resulting isostatic moment M0Ed,fi,beam is 128,7 kN.m. For column, the normal force NEd,fi under fire is 2849 kN (the design normal force NEd is 4384 kN) and the bending moment at the ends MEd,fi,column under fire is 14 kN.m. Actions acting on slab, beam and column are summed up in Table 6.2.9.

Table 6.2.9 Exterior actions acting on beam (axis 2) and on the column B2

M0Ed,fi,x-slab 15,9 kN.m

M0Ed,fi,y-slab 10,7 kN.m

M0Ed,fi,beam 128,7 kN.m

NEd,fi,column 2849 kN

MEd,fi,column 14 kN.m

6.3 Tabulated data 6.3.1. Scope Where simple calculation models are not available, the Eurocode fire parts give design solutions in terms of tabulated data (based on tests or advanced calculation methods), to be used within the specified limits of validity. In this case the member is assumed as isolated. Indirect fire actions are not considered, except those resulting from thermal gradients. Tabulated data are verified design solutions for a standard fire exposure up to 240 minutes. The values, given in tables in terms of minimal cross-sectional dimensions and of minimum nominal axis distance, apply to normal weight concrete made with siliceous aggregates (Figure 6.3.1). Using tabulated data, according to EN 1992-1-2, Sec-5 no further checks are required concerning shear and torsion capacity and spalling. Tabulated data are based on a reference load level ηfi = 0,7. Linear interpolation between values in the tables may be carried out.

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert

Fig. 6.3.1 Sections through structural members showing nominal axis and minimum dimensions (EN 1992-1-2, Sec-5)

6.3.2. Column 500/52

Tabulated data are given for braced structures. To assess the fire resistance of columns, two methods (A and B) are provided in EN 1992-1-2, Sec-5. For this study, method B is used. Validity of the method is first checked before using tabulated data: o Load level, ncolumn, at normal temperature conditions is :

ncolumn 

N0Ed,fi

0,7   Acfcd  Asfyd 

(6.13)

o First order eccentricity under fire conditions, e, is :

e  M0Ed,fi N0Ed,fi

(6.14)

Under the condition e/b ≤ 0,25 (e = 0,004). o Slenderness of the column under fire conditions fi as mentioned in EN 1992-1-2, Sec-5, Note 2, is assumed to be equal to λ at normal temperature in all cases. o The mechanical reinforcement ratio at normal temperature is:

  As fyd Ac fcd All of those parameters are sum up in Table 6.3.1.

(6.15)

Table 6.3.1 Parameters for method B (Tabulated data) – column B2

n

e

fi

ω

0,61

0,004

22,5

0,33

According to Figure 6.3.2, by linear interpolation between the different column tables the minimum dimensions required for ω = 0,33 and n=0,61 are 500/46. Therefore the column fire resistance is R90.

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert Fig. 6.3.2 Fire resistance, minimum column dimensions and axis distances for reinforced concrete columns with rectangular or circular section (EN 1992-1-2, Sec-5) 6.3.3. Beam 250/44 Tabulated data in En 1992-1-2, Sec-5 apply to beams exposed to fire on three sides. In this building, the upper side is insulated by slabs during the whole fire duration. The beam has a constant width bw,beam (equals to 0,25 m). Figure 6.3.3 provides minimum values of axis distance to the soffit and sides of continuous beams and minimum widths of the beam, for standard fire resistance from R30 to R240. The beam has only one layer of reinforcement. Interpolation between columns 2 and 3 gives a width of 250 mm and an axis distance of 40 mm. However, as indicated under the table, for of width values less than values in column 3, an increase of a is required. Then, for R120, asd is equal to 45 mm (35 + 10 mm). The continuous beam can ensure its load-bearing capacity up to 90 minutes. For R90 and above, the area of top reinforcement over the intermediate support, for up to a distance of 0,3leff from the center line of the support is sufficient (see EN 1992-1-2, Sec-5). Note: it is assumed that redistribution of bending moment under normal temperature design does not exceed 15 %. Otherwise, the continuous beam should be considered as simply supported.

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert Fig. 6.3.3 Minimum dimensions and axis distances - continuous reinforced concrete beam (EN 1992-1-2, Sec-5) 6.3.4. Slab 180/36/49 Fire resistance of reinforced concrete slabs is considered adequate if the values of Figure 6.3.4 are applied. The minimum slab thickness hs ensures adequate separating function (criteria E and I). In this building, continuous solid two-way-slabs are supported at all four edges. The values given in Figure 6.3.4 (column 2 and 4) apply to one-way or two-way continuous slab. The ratio of the lengths in y and x-directions ly /lx is equal to 1,32<1,5. Columns 2 and 4 of Figure 6.3.4 apply. Note: moment redistribution is assumed not to exceed 15 % for ambient temperature design.

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert

Fig. 6.3.4 Minimum dimensions and axis distance for reinforced concrete simply supported one-way and two-way solid slabs (EN 1992-1-2, Sec-5)

The French National Annex gives additional rules on rotation capacity on supports. In case of continuous slab, if the condition related to the slab thickness is verified (see Eqn 6.16), the calculation under fire may be avoided provided the axis distance of column 5 of Figure 6.3.4 is used. The continuous slab can maintain its load-bearing capacity up to 180 minutes. For 240 minutes, the minimum axis distance of the reinforcement in X-direction (36 mm) is less than the one in the table (40 mm). In y-direction, if we use the French National Annex, column 5 may be used leading to an axis distance of 50 mm (> 49 mm). The French National Annex requires on supports, under ambient temperature, reinforcement at least equal to 50 % of the isostatic bending moment, over a length at least equal to 1/3 of the longest contiguous span. The condition for the thickness of the slab is:

h  h0  b0 100ΩR L  a0

(6.16)

Limiting values for angle Ω of the yield hinge (ΩR) are based on reinforcement properties: o ΩR = 0,25 for class A (bars and rods) o ΩR = 0,25 for class B (bars and rods) o ΩR = 0,08 for Wire Fabrics L is the half the sum of the two ideal spans located west and east of the support. In Y-direction, L equals to 7,125 m and in X-direction, L equals to 5,40 m. Coefficients a0, b0 and h0 are in Table 6.3.2.

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert

Table 6.3.2 Coefficients a0, b0 et h0

REI

a0

b0

h0

-1,81

0,882

0,0564

-2,67

1,289

0,0715

-3,64

1,868

0,1082

-5,28

3,097

0,1860

-40,20

105,740

2,2240

The numerical application leads to the height for different duration of fire in Table 6.3.3 (ΩR = 0,25).

Table 6.3.3 Minimum height h of the slab (ΩR = 0,25)

ΩR = 0,25 30 min 60 min 90 min 120 min 180 min

L=7,125 m (in Y-direction) 0,109 m 0,137 m 0,153 m 0,166 m 0,195 m

L=5,40 m (in X-direction) 0,081 m 0,105 m 0,118 m 0,127 m 0,135 m

From table 6.3.3 the slab maintains its load-bearing capacity up to 120 minutes, for 180 minutes the condition is not verified (0,195 m > hslab).

6.3.5. Summary

Resistance to fire R calculated with tabulated data are in Table 6.3.4.

Table 6.3.4 Duration of load bearing capacity of members with tabulated data

Method Tabulated data

Column R90

Beam R90

Slab R120

6.4 Simplified calculation methods 6.4.1. Methodology In this part, the member is considered as isolated. Indirect fire actions are not considered, except those resulting from thermal gradients. Simplified calculation methods are used to determine the ultimate load-bearing capacity of a heated cross-section under the relevant combination of actions. In the fire situation It has to be verified that the design effect of actions Ed,fi is less than or equal to the corresponding design resistance Rd,t,fi. Temperatures profiles in concrete cross-sections subjected to a fire standard exposure are calculated using software Cim’Feu EC2 developed in France and concrete thermal properties (see 6.2.3).

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert In EN 1992-1-2, Sec-4 and in EN 1992-1-2, Annex B, three simplified methods are described: o ‘500°C isotherm method’: this method is applicable to a standard fire exposure and any other heat regimes which cause similar temperature fields in the fire exposed member. The method is valid for minimum width of cross-section depending on the fire resistance or on the fire load density (see EN 1992-1-2, Annex B, Table B1). The thickness of the damaged concrete a500 is assumed equal to the average depth of the 500°C isotherm in the compression zone of the cross-section. Concrete with temperatures in excess of 500°C is assumed not to contribute to the member load bearing capacity, while the residual concrete cross-section retains its initial values of strength and modulus of elasticity. o ‘Zone method’: this method provides more accurate results that the previous one, especially for columns but is applicable to the standard temperature-time curve only. The fire damaged cross-section is represented by a reduced cross-section ignoring a damaged zone of thickness az at the fire exposed sides. o The method based on estimation of curvature deals with columns where second order effects under fire are significant, assessing a reinforced concrete cross-section exposed to bending moment and axial load. This method is based on the estimation of the curvature (EN 1992-1-1, Sec-5). 6.4.2. Column In this study, the structural behaviour of the column is significantly influenced by second order effects under fire conditions. The damage of the outer layers of the column due to high temperatures, combined with the drop of the elasticity modulus at the inner layers, results in a decrease of the stiffness. Therefore according to EN 1992-1-2, Annex B, a procedure is presented to calculate the load-bearing capacity of a reinforced concrete cross-section subjected to bending moment and axial load using the method based on estimation of curvature, considered as an isolated member under fire conditions. The estimation of curvature is described in EN 1992-1-2, Sec-5. As a safe simplification the effective length under fire conditions, l0,fi,column, may be assumed equal to l0,column at normal temperature (l0,fi,column is equal to 3,1 m, see Chapter 3). In the first step, using software (CIM’feu EC2) temperatures profiles in the concrete cross-section subjected to a standard 180 minutes fire exposure are determined omitting the presence of reinforcement - see Figure 6.4.1. Concrete thermal properties are defined in section 6.2.3. Fig. 6.4.1 Temperature profiles in cross-section of the column In the second step, the temperatures in the center of the reinforcing bars are determined, see Table 6.4.1. For each reinforcing bar, the reduction coefficient factor ks(θ) is calculated.

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert

Table 6.4.1 Temperatures and reduced strength of steel reinforcement

As

x

y

ϕ(mm) (mm²) (mm) (mm)

314 52 52

314 202 52

314 298 52

314 448 52

314 52 202

314 448 202

314 52 298

314 448 298

314 52 448

314 202 448

314 298 448

314 448 448

T (°C) 741,4 488,6 488,6 741,4 488,6 488,6 488,6 488,6 741,4 488,6 488,6 741,4

ks(θ) 0,116 0,701 0,701 0,116 0,701 0,701 0,701 0,701 0,116 0,701 0,701 0,116 Σ

Asks(θ) (mm²) 36 220 220 36 220 220 220 220 36 220 220 36 1908

ks(θ)fyd(θ) (MPa) 57.93 350.39 350.39 57.93 350.39 350.39 350.39 350.39 57.93 350.39 350.39 57.93

ks(θ)fyd(θ)As (MN) 0,018 0,110 0,110 0,018 0,110 0,110 0,110 0,110 0,018 0,110 0,110 0,018 0,952

Strength properties of concrete and steel reinforcement under fire conditions are:

o Concrete:

fcd,fi,column(20°C) = 30/1=30 MPa

o Steel reinforcement:

fyd,fi,column(20°C) = 500/1=500 MPa

Table 6.4.1 indicates that Asfyd,fi(θ) is equal to 0,952 MN. According to EN 1992-1-2, Annex B, the thickness of the damaged concrete, a500,column, is equal to 60 mm for an exposure of 180 minutes and a width of 500 mm. So, Acfcd,fi(θ) is equal to 4,33 MN (Table 6.4.2).

Table 6.4.2 Damaged zone az according to EN 1992-1-2, Annex B

Time of exposure R90 R120 R180

az (mm) 40,6 49,8 60

Ac,fi (cm²) 1754 1603 1444

Ac,fifcd,fi (MN) 5,26 4,81 4,33

As mentioned in section 2.5, the axial normal force NEd,fi,column = 2,849 MN and the bending moment MEd,fi,column = 14 kNm. The first order eccentricity e0,column is equal to 0,033 m taken into account additional eccentricity (effect of imperfections, see EN 1992-1-2, Sec-5 and chapter 3). The first order bending moment for fire conditions M0Ed,fi,column is108 kNm. The moment-curvature diagram for the axial normal force NEd,fi,column is determined for each reinforcing bar and for each concrete zone using the relevant stress-strain diagram (see Figure 6.4.2). The resistant moment is calculated for different curvatures and the ultimate moment MRd,fi,column is determined. Calculations lead to an ultimate moment MRd,fi,column = 246 kNm with a curvature (1/r)fi,column equals to 0,035 m-1. The nominal second order moment M2,fi,column for the curvature corresponding to the maximum moment MRd,fi,column,max is determined, as follow (see EN 19992-1-1, Sec-5):

²

M2,fi  NEd,fi 1 r  l0 c  2849  0,035  3,1 / π  96 kNm

(6.17)

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert

Where c is a factor depending on the curvature distribution (c equals to π²). For (1/r)fi,column equals to 0,035 m-1 and NEd,fi,column equals to 2849 kN, M2,fi,column is equal to 96 kNm. The remaining ultimate first order moment capacity M0Rd,fi,column,max is then calculated as the difference between the ultimate moment capacity and the nominal second order moment (Figure 6.4.2), as follow:

M0Rd,fi,column  MRd,fi,column  NEd,fi 1 r  l02 c  246  96  150 kNm

(6.18)

M0Rd,fi,column is equal to 150 kNm. Comparing in the final step the remaining ultimate first order moment capacity M0Rd,fi,column with the first order bending moment for fire conditions M0Ed,fi,column:

M0Rd,fi,column  150 kNm  M0Ed,fi,column  108 kNm

(6.19)

All results are summed up in Table 6.4.3.

Fig. 6.4.2 Ultimate moment capacity, second order moment and ultimate first order moment capacity as a function of the curvature (1/r) (EN 1992-1-2, Annex B)

Table 6.4.3 Calculation results for column under fire conditions at 180 minutes

MRd,fi,column 246 kNm

M0Rd,fi,column 150 kNm

M2,fi,column 96 kNm

M0Ed,fi,column 108 kNm

N0Ed,fi,column 2849 kN

(1/r)fi,column 0,035 m-1

The column load-bearing capacity under fire is R = 180 minutes.

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert 6.4.3. Continuous beam Calculations are made at mid-span, internal and end supports on the basis of the design moments (section 6.2.5), geometry of the beam (section 6.2.4) and the reinforcing bars sections and cover (section 6.2.5). 6.4.3.1 Mid-span In a first step, temperatures profiles in the concrete cross-section subjected to a 120 minutes standard fire exposure are determined omitting the presence of reinforcement, with software (CIM’feu EC2), see Figure 6.4.3. Concrete thermal properties are presented in section 6.2.3.

Fig. 6.4.3 Temperatures profiles in the beam

Temperatures in steel reinforcement are presented with the corresponding strength reduction factor, are in Table 6.4.4, for 120 minutes of fire exposure.

Table 6.4.4 Temperatures reduced sections and tension under 120 minutes of fire exposure mid-span of the continuous beam in axis 2

Layer 1 1 ∑

Steel 1ϕ16 2ϕ16 3ϕ16

As (cm²) 2,01 4,02 6,03

T (°C) 507 677 -

ks 0,65 0,18 -

ks As (cm²) 1,31 0,72 2,03

Fs (kN) 65,5 36 101,5

The total tension force in steel reinforcement under fire at 120 minutes is Fs,fi,mid-span,beam = 101,5 kN. The effective depth dmid-span,beam is 356 mm. Equilibrium of forces leads to an effective height of the compressive zone equal to 8,82 mm. The lever arm zmid-span,beam of the internal forces is 352 mm. The resistant moment of the section at mid-span under fire is MRd,fi,mid-span,beam = 36 kNm. 6.4.3.2 Internal support The total tension force in steel reinforcement under fire at 120 minutes is Fs,fi,intermediate,beam = 508,5 kN. As mentioned before, EN 1992-1-2, Sec-4 allows simplified calculation methods to determine the ultimate load-bearing capacity of a heated cross section and to compare the capacity with the relevant combination of actions. In this case, the ‘500°C isotherm method may be used because the fire exposure is a standard one and the cross-section has a width bw = 250 mm greater than 160 mm at 120 minutes. This method gives a reduction of the cross-section size assuming a heat damaged zone at the concrete surfaces. Damaged concrete, i.e. concrete with temperatures in excess of 500°C, is

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert assumed not to contribute to the load bearing capacity of the member, while the residual concrete cross-section retains its initial values of strength and modulus of elasticity. The procedure for calculating the resistance of the cross-section at intermediate support of the beam in the fire condition is as follows: o The isotherm of 500°C for the standard fire exposure is determined according to temperature profiles in the cross-section; o The resulting width bw,fi and an effective height dfi of the cross-section are determined excluding the concrete outside the 500 °C isotherm (Figure 6.4.4): bw,fi = 160 mm and dfi =278 mm;

Fig. 6.4.4 Fire exposure on three sides, reduced cross-section of a reinforced concrete beam at support (EN 1992-1-2, Annex B)

o The temperature of reinforcing bars in tension and their reduced strength are determined (see table 6.4.5).

Table 6.4.5 Temperatures, reduced sections and tension under 120 minutes of fire exposure at intermediate support of the continuous beam in axis 2

Layer 1 1 1 1 1 ∑

Steel 1ϕ12 2ϕ12 2ϕ12 2ϕ12 2ϕ12 9ϕ12

As (cm²) 1,13 2,26 2,26 2,26 2,26 10,17

T (°C) 50 93 99,5 99,5 99,5 -

ks

ks As (cm²)

Fs (kN)

1,13

56,5

2,26

2,26

2,26

2,26

-

10,17

508,5

The total tension force in reinforcement under fire at 120 minutes is Fs,fi,intermediate support,beam = 508,5 kN. o The ultimate load-bearing capacity is then calculated The lever arm zintermediate,beam between the tension reinforcement and concrete is 256 mm. The resisting moment of the section at intermediate support under fire MRd,fi,intermediate,beam = 130 kNm.

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert 6.4.3.3 End support The procedure for calculating the resistance of the cross-section at end support of the beam in the fire condition is the same as for the intermediate support: o The isotherm of 500°C for the standard fire exposure is first determined; o The new width bw,fi and a new effective height dfi of the cross-section is determined excluding the concrete outside the 500 °C isotherm: bw,fi = 160 mm and dfi = 278 mm; o The temperature of reinforcing bars in tension and their reduced strength are determined (see Table 6.4.6).

Table 6.4.6 Temperatures, reduced sections and tension under 120 minutes of fire exposure end support of the continuous beam in axis 2

Layer 1 1 1 1 ∑

Steel 1ϕ12 2ϕ12 2ϕ12 2ϕ12 7ϕ12

As (cm²) 1,13 2,26 2,26 2,26 7,92

T (°C) 50 73,5 82 82 -

ks

ks As (cm²)

Fs (kN)

1,13

0,057

2,26

0,113

2,26

0,113

2,26

0,113

-

0,396

The total tension force in reinforcement under fire at 120 minutes is Fs,fi,intermediate support,beam = 396 kN. The lever arm zend,beam between the tension reinforcement and concrete is 256 mm. The resisting moment of the section at end support under fire is MRd,fi,end,beam = 101 kNm.

6.4.3.4 Summary The resisting moments MRd,fi,beam of the continuous beam are presented in Table 6.4.7.

Table 6.4.7 Resisting and design moments for the beam in axis 2 (in kNm)

MRd,fi,mid-span,beam 36

MRd,fi,intermediate- support,beam

MRd,fi,end-support,beam 101

MRd,fi,beam 151,50

M0Ed,fi,beam 128,7

The total resistant moment MRd,fi,beam is then compared with the isostatic moment of the corresponding beam to verify if M0Ed,fi,beam: MRd,fi,beam > M0Ed,fi,beam. The total resistant moment of the beam is MRd,fi,beam = 151,50 kNm > M0Ed,fi,beam 128,7 kNm. The beam load-bearing capacity under fire is R = 120 minutes.

6.4.4. Two-way slab Calculations have been made in the two directions of the slab (x and y-), according to design moments (see Table 6.2.8 and Table 6.2.9), geometry of the slab (see section 6.2) and the reinforcing bars (sections and cover, see Table 6.2.6 and Table 6.2.7). In a first step, temperatures profiles in the concrete cross-section subjected to a fire exposure at 180 minutes have been determined omitting the presence of reinforcement with software (CIM’feu EC2), see Figure 6.4.5. Thermal properties of the concrete are in section 6.2. Tables 6.4.8 and 6.4.9 present temperatures in the center of the reinforcing bars with the associated strength reduction factor ks(θ) for the two directions.

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert

Fig. 6.4.5 Temperatures profiles in slab

Table 6.4.8 Temperatures, reduced sections and tension under 180 minutes of fire exposure mid-span and support of the continuous slab (X-direction)

At mid-span At support

Steel 8ϕ12 4ϕ14

As (cm²) 9,05 6,16

T (°C) 619 140,4

ks(θ) 0,3468

ks As (cm²) 3,14 6,16

Fs (kN) 157 308

Table 6.4.9 Temperatures, reduced sections and tension under 180 minutes of fire exposure mid-span and support of the continuous slab (Y-direction)

At mid-span At support

Steel 4ϕ12+4ϕ14 4ϕ16

As (cm²) 10,68 8,04

T (°C) 497 164

ks 0,68

ks As (cm²) 7,24 8,04

Fs (kN) 362 402

Calculations are done successively for the two directions: o In X-direction At mid-span, the total tension force in reinforcement after 180 minutes fire is Fs,fi,mid-span,x-slab = 157 kN. The effective depth dmid-span,x-slab is 144 mm. the effective height of the compressive zone is 1,3 mm. The lever arm zmid-span,slab of the internal forces is 143,3 mm. The resisting moment of the section at mid-span under fire is MRd,fi,mid-span,x-slab = 22,50 kNm/m. At support, calculations give a resisting moment under fire MRd,fi,support,x-slab = 36 kNm/m. The total resistant moment is MRd,fi,x-slab= 36 + 22,5 = 58,50 kNm/m. The design moment in X-direction for the two-way slab is MEd,fi,x-slab = 15,9 kN.m/m. The load-bearing capacity of the slab in the Xdirection is verified at 180 minutes. o In Y-direction At mid-span, the total tension force in reinforcement after 180 minutes fire is Fs,fi,mid-span,y-slab =362 kN. The effective depth dmid-span,y-slab is 131 mm, the effective height of the compressive zone is 14,6 mm. The lever arm zmid-span,y-slab of the internal forces is 122 mm. The resisting moment of the section at

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert mid-span under fire is MRd,fi,mid-span,y-slab = 44,14 kNm/m. At support, calculations lead to a resisting moment under fire MRd,fi,support,y-slab = 39,70 kN.m/m. The total resistant moment is MRd,fi,y-slab = 44,14+39,70 = 83,84 kNm/m. The design moment in Y-direction for the two-way slab is MEd,fi,y-slab = 10,7 kN.m/m. The load-bearing capacity of the slab in the Y-direction is verified at 180 minutes. All results ( X and Y direction) are presented in Table 6.4.10.

Table 6.4.10 Two-way slab - resisting and design moments (kNm/m)

MRd,fi,x-slab 58,50

MEd,fi,x-slab 15,9

MRd,fi,y-slab 83,84

MEd,fi,y-slab 10,7

The load-bearing capacity of the two-way slab under fire is R = 180 minutes.

6.5 Advanced calculation methods Advanced calculation methods, based on fundamental physical behaviour, are based on a “global” structural analysis (analysis of the entire structure) for the fire situation and provide a realistic analysis of structures exposed to fire. Indirect fire actions are considered throughout the structure. In global structural analysis the relevant failure mode in fire exposure, the temperature-dependent material properties and member stiffnesses, effects of thermal expansions and deformations (indirect fire actions) have to be taken into account (EN 1992-1-2, Sec-2). Advanced calculation methods include (EN 1992-1-2, Sec-4): o a thermal response model based on the theory of heat transfer and the thermal actions presented in EN 1991-1-2. Any heating curve can be used, provided that the material (concrete and steel) properties are known for the relevant temperature range. o a mechanical response model taking into account the changes of mechanical properties with temperature. The effects of thermally induced strains and stresses due to temperature and temperature differentials have to be considered. Compatibility must be ensured and maintained between all parts of the structure (limitation of deformations). Geometrical nonlinear effects are taken into account. The partition of deformation may be assumed. Special attention is given to boundary conditions.

6.6 Conclusions

For determining the load-bearing capacity of the column, beam and slab, tabulated data and simplified methods have been used. Calculations were based on member analysis. The use of simplified methods results in longer fire resistances R, see Table 6.6.1. In future work, advanced calculation methods should provide a more accurate estimation of the duration of the load-bearing capacity of the entire or of a part of the structure.

Table 6.6.1 Duration of load bearing capacity of members with tabulated data and simplified calculation methods

Tabulated data Simplified method

Column R90 R180

Beam R90 R120

Slab R120 R180

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert References EN 1990:2003. Eurocode 0: Basis of Structural Design. CEN. EN 1991-1-1:2003. Eurocode 1 : Actions on structures. Part 1-1: General actions – Densities, self- weight and imposed loads for buildings. CEN. EN 1991-1-2:2003. Eurocode 1: Actions on structures. Part 1-2: General actions, actions on structures exposed to fire. CEN. EN 1992-1-1:2005. Eurocode 2: Design of concrete structures. Part 1-1: General rules and rules for buildings. CEN. EN 1992-1-2:2005. Eurocode 2: Design of concrete structures. Part 1-2: General rules - Structural fire design. CEN.

Fire resistance according to EN 1992-1-2 C. Morin and F .Robert

European Commission EUR 26566 EN – Joint Research Centre – Institute for the Protection and Security of the Citizen Title: Eurocode 2: DESIGN OF CONCRETE BUILDINGS Author(s): Francesco Biasioli, Giuseppe Mancini, Martin Just, Manfred Curbach, Just Walraven, Sausanne Gmainer, Jose Arrieta, Roger Frank, Caroline Morin, Fabienne Robert Editors: Francesco Biasioli, Martin Poljanšek, Borislava Nikolova, Silvia Dimova, Artur V. Pinto Luxembourg: Publications Office of the European Union 2014 – 182 pp. – 21.0 x 29.7 cm EUR – Scientific and Technical Research series – ISSN 1831-9424 ISBN 978-92-79-36548-5 doi:10.2788/35386 Abstract This document is a report with worked examples presenting step-by-step the design of a reinforced concrete cast on site building following Eurocode 2. The design process has been divided between different authors, some of whom were involved in the preparation and/or assessment of Eurocode 2. Each chapter of the report focuses on a different step in the design process: conceptual design, structural analyses, limit states design and verification, detailing of the reinforcement as well as some geotechnical aspects of building design. Last chapter gives general overview of the fire design according to the Eurocodes. The materials were prepared and presented at the workshop “Eurocode 2: Design of Concrete Buildings” held on 20-21 October 2011 in Brussels, Belgium. The workshop was organized by JRC with the support of DG ENTR and CEN, and in collaboration with CEN/TC250/Sub-Committee 2. The document is part of the Report Series ‘Support to the implementation, harmonization and further development of the Eurocodes’ prepared by JRC in collaboration with DG ENTR and CEN/TC250 “Structural Eurocodes”.

LB-NA-26566-EN-N

z As the Commission's in-house science service, the Joint Research Centre's mission is to provide EU policies with independent, evidence-based scientific and technical support throughout the whole policy cycle. Working in close cooperation with policy Directorates-General, the JRC addresses key societal challenges while stimulating innovation through developing new methods, tools and standards, and sharing its know-how with the Member States, the scientific community and international partners. doi:10.2788/35386 ISBN 978-92-79-36548-5