Composite Beams Columns to Eurocode 4

EUROPEAN CONVENTION FOR CONSTRUCTIONAL STEELWORK CONVENTION EUROPEENNE DE LA CONSTRUCTION METALLIQUE EUROPAISCHE KONVENTION FUR STAHLBAU

ECCS - Technical Committee 11 Composite Structures

Composite Beams and Columns to Eurocode 4

FIRST EDITION

1993

No72

All rights reserved. No part of this publicationmay be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the Copyright owner :

ECCS CECM EKS

General Secretariat Avenue des Ombrages, 32136 bte 20 8-1200 BRUSSEL (Belgium) Tel. 382-762 04 29 Fax 382-762 09 35

ECCS assumes no liability with respect to the use for any application of the material and information contained in this publication.

FOREWORD The Eurocodes are being prepared to harmonize design procedures between countries which are members of CEN (European Committee for Standardization) and have been published initially as ENV documents (European pre-standards - prospective European Standards for provisional application). The Eurocode for composite construction (referred to in this publication as EC4) is: ENV 1994-1-1: Eurocode 4 Design of composite steel and concrete structures Part 1.1: General rules and rules for buildings The national authorities of the member states have issued National Application Documents (NAD) to make the Eurocodes operative whilst they have ENV-status. This publication "Composite Beams and Columns to Eurocode 4" has been prepared by the ECCS-Technical Committee 11 to provide simplified guidance on composite beams and columns in supplement to EC4 and to facilitate the use of EC4 for the design of composite buildings during the ENV-period. "Composite Beams and Columns to Eurocode 4" contains those rules from EC4 that are likely to be needed for daily practical design work. It is a self-standing document and contains additional information as simplified guidance, design tables and examples. References to EC4 are given in [ 1. Any other text, tables or figures not quoted from EC4 are deemed to satisfy the rules specified in EC4. In case of doubt, when rules are missing (e.g. for the design of composite slabs, etc.) or when more detailed rules are required, EC4 should be consulted in conjunction with the National Application Document for the country in which the building project is situated.

The working group of ECCS-TC 11, responsible for this publication is:

The other members of ECCS-TC 1 1 are:

Anderson, D. Beguin, P. Bode, H. Brekelmans, J . Falke, J. Janss, J. Lawson, R. M. Mutignani, F.

Arda, T.S. Aribert, J.M. Axhag, F. Bossart, R. Cederwall, K. Lebet, J.P. Leskela, M. Schleich, J.B. Stark, J.W.B. Tschemmernegg, F.

United Kingdom France Germany (Chairman of TC11) Netherlands Germany Belgium United Kingdom Italy

Turkey France Sweden Switzerland Sweden Switzerland Finland Luxembourg Netherlands Austria

Particular thanks are given to those organisations who supported the work. Besides ECCS itself and its members, specific contributions were made by: Bauberatung Stahl, Bundesvereinigung der Priifingenieure fur Baustatik, The Department of Trade and Industry British Steel (Sections, Plates & Commercial Steels)

Germany Germany UK UK

The text was prepared for publication by the Steel Construction Institute, UK. Page 1

~

This publication presents useful information and worked examples on the design of composite beams and columns to Eurocode 4 ‘Design of composite steel and concrete structures’ (ENV 1994-1-1). The information is given in the form of a concise guide on the relevant aspects of Eurocode 4 that affect the design of composite beams and columns normally encountered in general building construction. Each section of the publication reviews the design principles, gives design formulae and makes cross-reference to the clauses of Eurocode 4. Information on the design of composite slabs is also given, although the publication concentrates on the influence of the slab on the design of the composite beam. Pesign aids are also presented to assist in selecting the size of steel beams to be used in certain applications. Worked examples cover the design of composite beams with full and partial shear connection, continuous beams, and composite columns.

Page 2

I

COMPOSITE BEAMS AND COLUMNS TO EUROCODE 4 CONTENTS

Page SUMMARY NOTATION

2 7

PART 1: DESIGN GUIDE

1.

INTRODUCTION 1.1 1.2 1.3

Scope of Publication Cross-referencing Partial Safety Factors

2.

INITIAL DESIGN

3.

ACTIONS AND COMBINATION RULES FOR DESIGN 3.1 3.2 3.3

3.4 4.

9 9 10 10 11 14

Fundamental Requirements Definitions and Classifications Design Requirements

14 14 15

3.3.1 General ‘3.3.2 Ultimate limit state 3.3.3 Serviceability limit state

15 15 16

Design of Steel Beams

17

MATERIALS AND CONSTRUCTION

18

4.1

Description of Forms of Construction

18

4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.1.6

18 18 19 19 20 20

4.2

Types of columns Types of beams Types of slabs Types of Shear connectors Types of erection Types of connection

Properties of Materials

22

4.2.1 Concrete 4.2.2 Reinforcing steel 4.2.3 Structural steel

22 23 23 Page 3

4.3 5.

24

Partial Safety Factors for Resistance and Material Properties

24

COMPOSITE OR CONCRETE SLABS 5.1 5.2

5.3

5.4

5.5 6.

4.2.4 Profiled steel decking for composite slabs

25

Introduction Initial Slab Design

25 26

5.2.1 5.2.2 5.2.3 5.2.4

26 27 27 28

Proportions of composite slabs Construction condition Composite action Deflections

Influence of Decking on the Design of Composite Beams

28

5.3.1 Ribs transverse to beams 5.3.2 Ribs parallel to beam

29 29

Detailing Rules for Shear Connectors Welded Through Profiled Steel Decking

30

5.4.1 Welding and spacing of studs 5.4.2 Additional requirements for steel decking

30 31

Minimum Transverse Reinforcement

31

ULTIMATE LIMIT STATE: COMPOSITE BEAMS

33

Basis of Design of Composite Beams

33

6.1.1 General 6.1.2 Verification of composite beams 6.1.3 Effective width of the concrete flange 6.1.4 Classification of cross-sections 6.1.5 Distribution of internal forces and moments in continuous beams

33 33 34 35 40

Resistance of Cross Sections

42

6.2.1 6.2.2 6.2.3 6.2.4 6.2.5

42 42 44 45 45

6.1

6.2

General Positive moment resistance Negative moment resistance Vertical shear Momen t-shear interaction

Page 4

6.3

7.

Shear Connection

46

6.3.1 6.3.2 6.3.3 6.3.4 6.3.5

46 46 48 49 53

6.4

Partially Encased Beams

56

6.5

Lateral Torsional Buckling of Continuous Beams

57

SERVICEABILITY LIMIT STATE: COMPOSITE BEAMS

59

7.1

General Criteria

59

7.2

Calculation of Deflections

59

7.2.1 7.2.2 7.2.3 7.2.4 7.2.5

8.

General Resistance of shear connectors Spacing of shear connectors Longitudinal shear force Transverse reinforcement

Second moment of area Modular ratio Influence of partial shear connection Shrinkage-induced deflections Continuous beams

59 61 63 63 64

7.3

Vibration Checks

65

7.4

Crack Control

65

ULTIMATE LIMIT STATE: COMPOSITE COLUMNS

67

8.1

Introduction

67

8.2

Design Method

68

8.2.1 General

8.2.2 Design assumptions 8.2.3 Local buckling 8.2.4 Shear between the steel and concrete components

68 68 68 69

Simplified Method of Design of Composite Columns

70

8.3.1 Resistance of cross-sections to axial load 8.3.2 Resistance of members to axial load 8.3.3 Resistance of cross-sections to combined compression and uniaxial bending 8.3.4 Analysis for moments applied to columns 8.3.5 Resistance of members to combined compression and uniaxial bending

70 71

8.3

74 76 76 Page 5

~

~

~

8.3.6 Limits of applicability of the simplified design method

78

9.

FIRE RESISTANCE

80

10.

CONSTRUCTION AND WORKMANSHIP

82

10.1 10.2 10.3 10.4 10.5 10.6

General Sequence of Construction Stability Accuracy during Construction and Quality Control Loads during Construction Stud Connectors Welded through Profiled Decking

82 82 82 82 83 83

11.

REFERENCES

85

12.

DESIGN TABLES AND GRAPHS FOR COMPOSITE BEAMS

86

12.1

Moment Resistance of Composite Beam Relative to Steel Beam

87

12.2 Second Moment of Area of Composite Beam Relative to Steel Beam

88

12.3 Design Tables for Composite Beams Subject to Uniform Loading

89

ANNEX 1

DESIGN FORMULAE FOR COMPOSITE COLUMNS

94

PART 2:

WORKED EXAMPLES

97

1.

Simply Supported Composite Beam with Solid Slab and Full Shear Connection

2.

Simply Supported Composite Beam with Composite Slab and Partial Shear Connection

3.

Continuous Composite Beam with Solid Slab

4.

Composite Column with End Moments

Page 6

NOTATION Notation is not presented in detail here and reference should be made to Eurocode 4 Part 1.1. However, the use of the following common symbols and subscripts is given to help understanding of this publication.

Symbols: A beff d

E fck *

fY

F G h I L M N

Q t

V W YF

Y

x

E

X

cross-sectional area effective width of slab diameter of shear connector; depth of web considered in shear area modulus of elasticity of steel characteristic compressive (cylinder) strength of concrete yield strength of steel force in element of cross-section; load (action) permanent loads (actions) depth of element second moment of area length or span moment (with subscripts as below) axial force variable loads (actions) thickness of element of cross-section shear force plastic section modulus partial safety factor for loads partial safety factor for materials (with subscripts as below) slenderness d ( f y /235) reduction factor on axial resistance due to imperfection

Subscripts to symbols: a C

k P S

PP R

S Rd Sd W

structural steel concrete characteristic value profiled steel decking (sheeting) reinforcement plastic resistance (in bending, shear or compression) resistance (of member) internal force or moment design value of resistance design value of internal force or moment web of steel section

Page 7

Member axes: X

Y 2

along the axis of the member major axis bending minor axis bending

Terminology: This publication adopts the terminology used in Eurocode 4 Part 1.1. However, there are some important terms which may be defined to assist in understanding this document. These are: Hogging moment

Negative moment causing compression in the bottom flange of the beam.

Sagging moment

Positive moment causing tension in the bottom flange of the beam.

Moment resistance

Resistance of the steel or composite cross-section to bending actions.

Stud connector

A particular form of shear connector comprising a steel bar and flat

head that is welded automatically to the beam.

Decking

Profiled steel sheet which may be embossed for composite action with the concrete slab.

Transverse reinforcement

Reinforcement placed in the slab transversely (across) the steel beam.

Page 8

1.

INTRODUCTION Eurocode 4 Part 1.1 deals with the ‘design of composite steel and concrete structures’. The publication ‘Composite beams and columns to Eurocode 4’ presents simplified guidance in accordance with the main Eurocode, but concentrates on the common forms of structure that are encountered in building construction. Although the publication retains the principles and application rules of the Eurocode, it is not written in a code format because of the need to offer further explanation on the design principles. It is intended that each section is read as a design guide with cross-reference to the relevant clauses in EC4 (or EC3 or EC2, as appropriate). Because of this less formal presentation it is possible to introduce additional information and design aids in the form of tables and graphs. Part 1 of the document covers the design methods for composite beams and composite columns. Also given are some design tables for composite beams using standard steel sections. Part 2 presents a number of fully worked examples for simply supported and continuous composite beams, and composite columns.

1.1

Scope of Publication A decision was made to limit the scope of the publication to the information that

’90% of designers will need 90% of the time’. In this sense, simply supported or continuous beams in braced construction are most typical of modern buildings. Similarly, composite beams are increasingly associated with composite slabs, rather than solid slabs. Composite columns are also increasingly popular. In summary, the document covers the following aspects in detail: Composite beams with composite or solid slabs Braced frames (non-sway) Simply supported (simple) connections Continuous beams (or with connections equivalent to the moment resistance of the beam) Welded stud shear connectors Full or partial shear connection Class 1 or 2 sections (class 3 webs are permitted for continuous beams) Composite columns (encased I sections or concrete filled sections) under axial load Composite columns with moments using simplified interactions Partially encased sections

The document makes only general reference (and does not include detailed information) on: Global analysis of composite frames Design of connections Behaviour of composite slabs Cracking in concrete Other forms of shear connector Use of precast concrete slabs Lightweight concrete Lateral-torsional buckling Fire resistance aspects General analysis of composite columns Specifically excluded is the use of: Non-uniform cross-sections Class 3 or 4 sections Sway frames Partial strength connections 1.2

Cross-referencing This publication is to be read as a self standing document and cross-refers to other sections within the text. To aid cross-referencing to Eurocode 4 (EC4) or other Eurocodes, the source clauses in these Eurocodes are presented in brackets at the start of each section, or adjacent to the relevant part of the text. All references to Eurocodes or EN standards or other important publications are listed in full at the back of the publication.

1.3

Partial safety factors National authorities are able to select partial safety factors on loads and materials which are given as ‘boxed values’ in the Eurocodes. Because this document is intended to be read throughout Europe the recommended boxed values have been used in the text, Worked Examples and Design Tables. Further information on partial safety factors is given in Sections 3 and 4.3.

Page 10

INITIAL DESIGN Composite beams comprise I or H section steel beams attached to a ’solid’ or ’composite’floor slab by use of shear connectors. Composite slabs comprise profiled steel decking which supports the self weight of the wet concrete during construction and acts as ’reinforcement’ to the slab during in-service conditions. Composite beams behave as a series of T beams in which the concrete is in compression when subject to positive moment and the steel is mainly in tension. The beams may be designed as simply-supported, or as continuous over a number of supports. The relative economy of ’simple’ or ’continuous’ construction depends on the benefits of reduced section size and depth in relation to the increased complexity of the design and the connections in continuous construction. Composite beams may be designed to be unpropped for reasons of speed of construction. Propped construction may be appropriate where it is necessary to control deflections of the steel beam during construction. The sizing of the composite beam is independent of the form of construction provided the steel beam is able to support the loads developed during concreting. The following recommendations are made for initial sizing of composite beams. It is important to recognise the difference between secondary beams which directly support the decking and composite slab and primary beams which support the secondary beams as point loads. Primary beams usually receive greater loads than secondary beams and therefore are usually designed to span a shorter distance for the same beam size. Alternatively, long span primary beams, such as composite trusses, can be designed efficiently with short span secondary beams. These cases are illustrated in Figure 2.1. General features: Slab depth

-

typically 120mm to 180mm depending on fire resistance, structural and other requirements

Slab span

-

2.5m to 3.5m unpropped 3.5m to 5.5m propped subject to maximum span: depth ratio of 35 for a slab with continuity at one end (see Section 5 for further guidance).

Grid sizes

-

primary and secondary beams can be designed for approximately the same depth when grid dimensions are in proportion of 1 : lV2 respectively

Beam design The following beam proportions should give acceptable deflections when the section size is determined for moment resistance. a)

b)

Simply supported Secondary beam

-

span: depth ratio of 18 to 20 (depth = total beam and slab depth)

Primary beam

-

span: depth ratio of 15 to 18

Secondary beam

-

span: depth ratio of 22 to 25 (end bays)

Primary beam

-

span: depth ratio of 18 to 22

Continuous

Steel grade

-

higher grade steel (Fe 510) usually leads to smaller beam sizes than lower grade steel (Fe 360 or Fe 430)

Concrete grade

-

C 25/30 for composite beams.

Shear connectors

-

19mm diameter welded stud connectors are placed typically at 150mm spacing. These studs can be welded through the steel decking up to 1.25mm thick. 22mm diameter welded stud connectors where throughdeck welding is not used.

Page 12

column

T

L

span of slab

primary beam

-

L i

span of slab

-primary beam

&- -m21 - 8 - - - -&

L-I :

P4

12 -18m

Figure 2.1

Framing plans for medium and long span beams Page 13

3.

ACTIONS AND COMBINATION RULES FOR DESIGN

3.1

Fundamental Requirements [2.1] A structure shall be designed and constructed in such a way that: 0

with acceptable probability, it will remain fit for the use for which it is required, having due regard to its intended life and its cost, and

0

with appropriate degrees of reliability, it will sustain all actions and influences likely to occur during execution (ie. construction period) and subsequent use, and have adequate durability in relation to maintenance costs.

A structure shall also be designed in such a way that it will not be damaged by events

like explosions, or impact or consequences of human error to an extent disproportionate to the original cause.

3.2

Definitions and Classifications [2.2] Limit States Limit states are states beyond which the structure no longer satisfies the design performance requirements. Limit states are classified into: a 0

ultimate limit states serviceability limit states.

Ultimate limit states are those associated with collapse, or with other forms of structural failure which may endanger the safety of people. Serviceability limit states correspond to states beyond which specified in-service criteria are no longer met by the structure. Actions Definitions and principal classification*)' An action (F) is:

*)l

0

a force (load) applied to the structure (direct action), or

0

an imposed deformation (indirect action); for example, temperature effects or differential settlement.

Fuller definitions of the classification of actions will be found in the Eurocode for Actions. Page 14

Actions are classified as: 0

permanent actions (G), eg. self-weight of structures, fittings, and fixed equipment.

0

variable actions (Q), eg. imposed loads, wind loads or snow loads.

e

accidental actions (A), eg. explosions or impact from vehicles.

Characteristic values of actions F, are specified a

in the Eurocode for Actions or other relevant loading codes, or

0

by the client, or the designer in consultation with the client, provided that the minimum provisions specified in the relevant loading codes or by the competent authority are observed.

The design value F, of an action is expressed in general terms as:

where yF Fk

3.3

= =

partial safety factor for actions characteristic value of the action

Design Requirements [2.3]

3.3.1 General It shalI be verified that no relevant limit state is exceeded. All relevant design situations and load cases shall be considered, including those at the construction phase. Possible deviations from the assumed directions or positions of actions shall be considered. Calculations shall be performed using appropriate design models (supplemented, if necessary, by tests) involving all relevant variables. The models shall be sufficiently precise to predict the structural behaviour, commensurate with the standard of workmanship likely to be achieved, and with the reliability of the information on which the design is based.

3.3.2 Ultimate limit state Verification conditions When considering a limit state of failure of a section, member or connection (fatigue excluded), it shall be verified that:

Page 15

r

where s d is the design value of an internal force or moment (Or of a respective vector of several internal forces or moments) and Rd is the corresponding design resistance, associating all structural properties with the respective design values. Combination of actions For each load case, design values for the effects of actions shall be determined from combination rules involving design values of actions, as identified by Table 3.1. The most unfavourable combinations are considered at each critical location of the structure, for example, at the points of maximum negative or positive moment. In Table 3.1 a combination factor of 0.9 is taken into account. Eurocodes permit the use of other combination factors, if reliable load data are is available. Load combinations to be considered:

permanent actions, eg. self weight variable actions, eg. imposed loads on floors, snow loads, wind loads the variable action which causes the largest effect at a given location

* If the dead load G counteracts the variable action Q:

partial safety factor for permanent actions partial safety factor for variable actions

**

If a variable load Q counteracts the dominant loading:

Yo = 0

Table 3.1

Combinations of actions for the ultimate limit state

3.3.3 Serviceability Limit State For each load case, design values for the effects of actions shall be determined from combination rules involving design values of actions as identified by Table 3.2.

Page 16

Load combinations to be considered:

’.

I

Gk + Qk.max

2.

Table 3.2

3.4

Parameters defined in Table 3.1.

Combinations of actions for the serviceabilitv limit state

Design of Steel Beams The steel beam is to be designed in accordance with Eurocode 3. The loads to be considered shall include the self weight of the beam and slab and an additional load taking account of the construction operation. Although no information is given in EC4 on these additional construction loads to be used in the design of the steel beams, it is consistent with the design of slabs to assume a construction load of 0.75 kN/m’ in the design of the beams.

Page 17

4.

MATERIALS AND CONSTRUCTION

4.1

Description of Forms of Construction

4.1.1 Types of column Composite columns may be of the form shown in Figure 4.1. There are two main types; concrete encased (totally or partially), and concrete-filled columns.

Figure 4.1

Types of column

4.1.2 Types of beam Composite beams may be of the form shown in Figure 4.2. Beams are usually of IPE or HE section (or UB or UC section). Partial encasement of the steel section provides increased fire resistance and resistance to buckling.

Figure 4.2

Types of beam

Shear connectors between the slab and beam provide the necessary longitudinal shear transfer for composite action. The shear connection of the steel beam to a concrete slab can either be by full or partial shear connection. This action is considered in Section 6. Page 18

4.1.3 Types of slab Slabs are either: e e

concrete slabs: composite slabs:

Prefabricated, or cast in situ, or Profiled steel decking and concrete (see Section 5).

Slabs are generally continuous but are often designed as a series of simply supported elements spanning between the beams.

h

Figure 4.3

Types of composite and concrete slabs

4.1.4 Types of shear connector

7-d-L

h 23d generally h 14d ductile

Page 19

4.1.5 Types of erection Beams and/or profiled steel sheets may be either propped or unpropped during concreting of the slab. The most economic method of construction is generally to avoid the use of temporary propping. Propping is needed where the steel beam is not able to support the weight of a thick concrete slab during construction, or where deflection of the steel beam would otherwise be unacceptable. 4.1.6 Types of connection There are many types of connection. Some examples are given in Figure 4.6 for beam-to-column and beam-to-beam connections. In design to EC4, the two forms of connection generally envisaged are (i) nominally pinned or (ii) rigid and full strength. No application rules are given for partial strength connections, as defined in EC4 [4.10 5.31.

anti-crack reinforcement

1

secondary beamreinforcement

1

a. Examples of “nominally pinned” connections both in the construction and cornposite stages Figure 4.6

Examples of connections in composite frames Page 20

reinforcement

1

I L e x t e n d e d end plate

b. Example of “rigid” and full strength connection

tensile reinforcement

c. Example of connections that are pinned in the construction stage and ’partial strength’ in the composite stage Figure 4.6

(Continued) Examples of connections in composite frames

In Figures 4.6(b) and (c), the connections may be considered to be rigid, but may or may not develop the full strength of the composite section. In the case of Figure 4.6(c) the connection is pinned in the construction stage, but is made moment resisting by the slab reinforcement and fitting pieces which transfer the necessary tension and compression forces.

Page 21

4.2

Properties of Materials The material properties given in this Section are those required for design purposes.

4.2.1 Concrete [3.1J Normal and lightweight concrete may be used. In this Section, data for normal weight concrete are given. For lightweight, concrete see EC4 [3.1.4.1(3)]. Strength Class of Concrete

C20/25

C25/30

C30/37

C35/45

C40/50

C45/55

C50/60

20

25

30

35

40

45

50

2.2

2.6

2.9

3.2

3.5

3.8

4.1

fck (compressive strength) f,,, (tensile strength)

The strength class (ie. C20) refers to cylinder strength of concrete, fck. The cube strength is given as the second figure (ie. /25). Shrinkage (long-term free shrinkage strain ecs) for normal weight concrete:

in dry environment (filled members excluded)

325 x 10-6

in other environments and for filled members

200 x 10-6

The secant modulus of elasticity for short term loading is given in Table 4.2 below. Strength Class C

E,,,

(kN/mm2)

Table 4.2

C20/25

C25/30

C30/37

C35/45

C40/50

C45/55

C50/60

29

30.5

32

33.5

35

36

37

Secant modulus of elasticity for concrete Ecm for short-term loading

Modular ratio, n

=

EJE,,

using E, as in Table 4.4.

For long term (permanent) loads, the modulus of elasticity for concrete is reduced due to creep and is taken as Ec,,/3, leading to an increase in n by a factor of 3. In most cases of imposed loading the representative value of modulus of elasticity is taken as Ec,/2 [3.1.4.2(4)]. Although not generally required for general design: Coefficient of linear thermal expansion, aT

-

10 x 10-6/ "C

Page 22

4.2.2 Reinforcing steel [3.2] Refer to EN 10 080, which is the product standard for reinforcement. Types of Steel according to ductility characteristics: high (class H) or normal (class N)

e

according to surface characteristics: plain smooth or ribbed bars

0

Steel grades B 500:

characteristic yield strength fsk = 500N/mm2

The modulus of elasticity of reinforcing steel is taken as for structural steel.

4.2.3 Structural steel [3.3] Nominal values of material strength are as given below. The nominal values may be adapted as characteristic values in calculations. Thickness t mm*) t 5 40mm

Nominal steel grade Fe 360 Fe 430 Fe 510

fr fU

Table 4.3

40mm

<

t 5 lOOmm

fy

f"

fy

fU

235 275 355

360 430 5 10

215 255 335

340 410 490

-

yield strength

-

ultimate tensile strength

Nominal values of strength of structural steels to EN 10 025 (in N/mm')

No values of material strength are given for high-strength steel. For this steel, clause 3.2.1(2) of EC3 is applicable. modulus of elasticity shear modulus coefficient of linear thermal expansion density

Table 4.4

Ell Ga (YT P

-

2 1 m 81000 10 x 10-6 7850

[N/rnrn2] [N/mm2] [/"Cl [kg/m31

Design values of other properties of steel Page 23

4.2.4 Profiled steel decking for composite slabs Composite slabs are dealt with in this publication only as far as they affect the design of the composite beam. Reference should be made to EC4 for further information on the design of composite slabs, with EN 10 147 as the product standard for steel sheeting. 4.3

Partial Safety Factors for Resistance and Material Properties [2.3.3.2]

In general, resistance is determined by using design values of strength of the different materials or components as given in the individual chapters of EC4 or in this publication. Recommended values for fundamental and accidental combinations are given in Table 4.5. These values may be modified by the various National Authorities and are given as ‘boxed values’ in EC4. Combination Structural Concrete Steel

Y O

Profiled Steel Reinforcement Steel Decking

Yc

Ys

Shear COMectOrS (studs, angles, friction grip bolts) and Longitudinal Shear in Slabs

5 YW Yvs

Fundamental

1.10

1S O

1.15

1.10

1.25

Accidental

1 .oo

1.30

1 .oo

1 .oo

1 .oo

Table 4.5

Partial safety factors for resistance and material properties

Values for bolts, rivets, pins, welds, and slip resistance of bolted connections are as given in EC3 clause 6.1.1(2). Where the member resistance is influenced by the buckling of the structural steel section, a specific safety factor YRd = [ l . 101 is recommended [2.2.3.2(2)], [4.6.3], [4.8.3.2]. When the design value Rd is determined by tests, refer to Eurocode 4.

Page 24

5.

COMPOSITE OR CONCRETE SLABS

5.1

Introduction This section reviews the different forms of concrete slab that may be used in conjunction with composite beams, and the factors that influence the design of the beams. The detailed design of composite slabs, which is covered in chapter 7 of EC4, is not treated here. Three types of concrete slab are often used in combination with composite beams. These three types are listed as follows: 0

*

0

Solid slab: This is a slab with no internal voids or rib openings, normally cast-in place using traditional wooden formwork. Composite slab: This is a slab which is cast-in-place using decking (coldformed profiled steel sheeting) as permanent formwork to the concrete slab. When ribs of the decking have a re-entrant shape and/or are provided with embossments that can transmit longitudinal forces between the decking arid the concrete, the resulting slab acts as a composite slab in the direction of the decking ribs. Precast concrete slab: This is a slab consisting of prefabricated concrete units and cast-in-place concrete. There are two forms that may be used: Thin precast concrete plate elements of approximately 50mm thickness are used as a formwork for solid slabs or alternatively, deep precast concrete elements are used for longer spans with a thin layer of cast-in-place concrete as a wearing surface. Deep precast concrete units often have hollow cores which serve to reduce their dead weight. The units may be designed to act compositely with ‘the steel beams, but this aspect is outside the scope of this document.

No further information is given on solid or pre-cast concrete slabs in this section. In the design of composite slabs the following aspects have to be considered:

*

The cross-sectional geometry of the slab: In some cases the full crosssectional area of the slab cannot be used for composite beam calculations. A reduced or “effective”cross-sectional area must be calculated. Formulae for determining effective slab widths are given in Section 6.1.3.

0

The influence of the slab on the shear connection between the slab and the beam: Stud behaviour and maximum strength may be modified due to the shape of the ribs in the slab (see Section 6.3.2.2). The correct placement of studs relative to ribs is of great importance.

e

The quantity and placement of transverse reinforcement: Transverse reinforcement is used to ensure that longitudinal shear failure or splitting of the concrete does not occur before failure of the composite beam itself.

Page 25

r

Figure 5.1

5.2

Typical coniposite slab with re-entrant deck profile

Initial Slab Design

5.2.1 Proportions of composite slab A typical composite slab is shown in Figure 5.1. In general such slabs consist of:

decking (cold formed profiled steel sheeting), concrete and light mesh reinforcement. There are many types of decking currently marketed in Europe. These can be, however, broadly classified into two groups: 0

Re-entrant rib geometries. An example of such a profile is shown in Figure 5. I. Note that embossments are often placed on the the top flange of the deck.

0

Open or trapezoidal rib geometries. An example of such a profile is shown in Figure 5.2. Note that embossments are often placed on the webs of the deck.

Slab depths range from 100 to 200mm; 120 to 180mm being the most common depending on the fire resistance requirements. Decking rib geometries may vary considerably in form, width and depth. Typical rib heights, h,, are between 40mm and 85mm. Centre-line distances between ribs generally vary between 150mm and 300mm. Embossment shapes and sheet overlaps also vary between decking manufacturers. Page 26

generally vary between 150mm and 300mm. Embossment shapes and sheet overlaps also vary between decking manufacturers. In general, the sheet steel is hot-dipped galvanised with 0.02mm of zinc coating on each side. The base material is cold-formed steel with thicknesses between 0.75mm and 1.5mm. The yield strength of the steel is in the range of 220 to 350N/mm2. Deeper decks permit longer spans to be concreted without the need for propping. Ribs deeper than 85mm, however, are not treated in this document. For such ribs composite action with the steel beam may be significantly reduced, thus requiring special attention.

5.2.2 Construction condition Normally, decking is first used as a construction platform. This means that it supports construction operatives, their tools and other material commonly found on construction sites. Good construction practice requires that the decking sheets be attached to each other and to all permanent supports using screws or shot-fired nails. Next, the decking is used as formwork so that it supports the weight of the wet concrete, reinforcement and the concreting gang. The maximum span length of the decking without propping can be calculated according to the rules given in Part 1.3 of EC3. Characteristic loads for the construction phase are 1.5 kN/m2 on any 3 metres by 3 metres area and 0.75 kN/m2 on the remaining area, in addition to the self weight of the slab. Typically, decking with a steel thickness of 1.2mm, and a rib height of 60mm, can span between 3m and 3.5m without propping.

50 Figure 5.2

Typical composite slab using a trapezoidal deck profile, showing the main geometrical parameters

5.2.3 Composite action After the concrete has hardened, composite action is achieved by the combination of chemical bond and mechanical interlock between the steel decking and the concrete. The chemical bond is unreliable and is not taken into account in design. Composite slab design is generally based on information provided by the decking manufacturer, Page 27

in the form of allowable imposed load tables. These values are determined from test results and their interpretation as required in EC4 clause 10.3. In most catalogues the resistance to imposed load is given as a function of decking type and steel sheet thickness, slab thickness, span length and the number of temporary supports. Generally, these resistances are well in excess of the applied loads, indicating that composite action is satisfactory or that the design is controlled by other limitations. However, care should be taken to read the catalogue for any limitations or restrictions due to dynamic loads, and concentrated point and line loads. 5.2.4 Deflections [7.6.2.2] Deflection calculations in reinforced concrete are notoriously inaccurate, and therefore some approximations are justified to obtain an estimate for the deflections of a composite slab. The stiffness of a composite slab may be calculated from the cracked section properties of a reinforced concrete slab, by treating the cross-sectional area of decking as an equivalent reinforcing bar. However, if the maximum ratio of span length to slab depth is within the limits of Table 5.1 no deflection check is needed. The end span should be considered as the general case for design. In this case it is assumed that minimum anti-crack reinforcement exists at the supports. Experience shows that imposed load deflections do not exceed span/350 when using the span to depth ratios shown in Table 5.1. More refined deflection calculations will lead to greater span to depth ratios than those given in Table 5.1. Maximum Span: Depth ratios Normal weight concrete Light weight concrete

End span

Internal span

Single span

35 30

38 33

32 27

Table 5.1: Maximum span to depth ratios of composite slabs

5.3

Influence of Decking on the Design of Composite Beams Profiled steel change decking performs a number of important roles, and influences the design of the composite beam in a number of ways. It: 0

may provide lateral restraint to the steel beams during construction;

0

causes a possible reduction in the design resistance of the shear connectors;

0

acts as transverse reinforcement leading to a reduction in the amount of bar reinforcement needed.

These factors are addressed more fully in Section 6. The orientation of the sheeting is important. Decking ribs may be oriented in two ways with respect to the composite beam:

Page 28

Decking ribs transverse to the steel beam, as shown in Figure 5.3. The decking may be discontinuous (Figure 5.3a), or continuous (Figure 5.3b) over the top flange of the beam. e

Decking ribs parallel to the steel beam, as shown in Figure 5.4.

The shear connectors may be welded through the decking, or placed in holes formed in the troughs of the decking. In the latter case the shear connectors can also be welded to the steel beam off-site. When the through welding procedure is used on site, studs may not be welded through more than one sheet and overlapping of sheets is not permitted.

5.3.1 Ribs transverse to the beam The concrete slab in the direction of the beam is not a homogeneous (solid) slab. This has important consequences for the design of the composite beam, as only the depth of concrete over the ribs acts in compression. Additionally, there is often a significant influence on the resistance of the shear connectors due to the shape of the deck profile.

Figure 5.3

Decking ribs transverse to the beam

5.3.2 Ribs parallel to the beam In the construction phase, decking with this orientation is not considered effective in resisting lateral torsional buckling of the steel beam. In this case, the complete cross-section of the slab may be used in calculating the moment resistance of the beam. The orientation of the ribs also implies that there will be little reduction in the resistance of the studs due to the ribs in the concrete slab.

Page 29

Figure 5.4 5.4

Decking ribs parallel to the beam

Detailing Rules for Shear Connectors Welded Through Profiled Steel Decking [6.4.3.1]

5.4.1 Welding and spacing of studs When the decking is continuous and transverse to the beam (Figure 5.3a), the correct placement of studs in relation to the decking rib is of great importance. The most important rules for welded headed studs are repeated here: Welded headed studs are normally between 19mm and 22mm in diameter. Stud diameters up to 19mm are generally used for through deck welding only. For welded studs the upper flange of the steel beam should be clean, dry and unpainted. For satisfactory welding, the deck thickness should not exceed 1.25mm if galvanized, or 1S m m if ungalvanized. In all cases, welding trials shall be performed. The following limitations should also be observed: 0

The flange thickness of the supporting beams should not be less than 0.4 times the diameter of the studs, unless the studs are located directly over the web.

0

After welding, the top of the stud should extend at least 2 times diameter of the stud above the top of the decking ribs and should have a concrete cover of at least 20mm.

0

The minimum distance between the edge of the stud and the edge of the steel flange is 20mm.

0

The transverse spacing between studs should not be less than 4 times the diameter of the stud.

0

The longitudinal spacing between studs should not be less than 5 times the stud diameter and not greater than six times the overall slab depth nor 800mm [6.4.3(3)]. Page 30

5.4.2 Additional requirements for steel decking Studs must be properly placed in decking ribs. A summary of these rules are shown in Figure 5 . 5 , and listed below: e

Studs are usually attached in every decking rib, in alternate ribs, or in some cases, in pairs in every rib. If more studs are needed than are given by a standard pattern these additional studs should be positioned in equal numbers near the two ends of the span.

e

Some modem decks have a central stiffener in the rib which means that it is impossible to attach the stud centrally. In such cases it is recommended that studs are attached to the side of each stiffener closest to the end of the beam shown as the favourable side in Figure 5.5. This means that a change in location at midspan is needed.

e

Alternatively, studs can be ‘staggered’ so that they are attached on each side of the stiffener in adjacent ribs.

0

At discontinuities in the decking, studs should be attached in such a way that both edges of the decking at the discontinuity are properly ‘anchored’. If the decking is considered to act as transverse reinforcement this may mean placing studs in a zigzag pattern along the beam, as shown in Figure 5.5.

The minimum distance of the centre of the stud to the edge of the decking is defined in EC4 7.6.1.4(3) as 2.2 times the stud diameter. Similar rules may be established for other forms of shear connectors such as shotfired cold-formed angles.

5.5

Minimum Transverse Reinforcement Transverse reinforcement must be provided in the slab to ensure that longitudinal shearing failure or splitting does not occur before the failure of the composite beam itself (see Section 6.3.5). The decking is not allowed to participate as transverse reinforcement unless there is an effective means of transferring tension into the slab, such as by through-deck welding of the shear connectors. Where the decking is continuous, the decking is effective in transferring tension and can act as transverse reinforcement. This is not necessarily the case if the ribs are parallel to the beam because of overlaps in the sheeting.

Minimum amounts of transverse reinforcement are required. The reinforcement should be distributed uniformly. The minimum amount is 0.002 times the concrete section above the ribs.

Page 31 I

unfavourable side

i

stiffener

beam

22

favourable side

rshear

end of s p a n 1

connector -

0 2

2 2.2d

,

beam

=butt joint

t

stiffener

,-shear connector

O I

IFigure 5.5

Detailing of shear connectors in decks with a central stiffener

Page 32

6.

ULTIMATE LIMIT STATE OF COMPOSITE BEAMS

6.1

Basis of Design of Composite Beams

6.1.1 General [4.1] The following clauses outline the design rules for composite beams. The treatment is largely restricted to Class 1 and Class 2 sections which are capable of developing their plastic moment of resistance without local buckling problems. Partially encased beams are also included. The majority of composite beams encountered in practice are thereby covered. Composite structures and members should be so proportioned as to satisfy the basic design requirements for the ultimate limit state using the appropriate partial safety factors and load combinations. Continuous composite beams may be analysed in all cases by elastic global analysis, and Class 1 beams by plastic hinge analysis.

transverse reinforcement

-headed studs -

L

partially encasedA

steel sections: either rolled or welded

Figure 6.1

Typical cross-sections of composite beams

Figure 6.1 shows typical cross-sections. Other combinations between steel sections and slabs are also used, but are not covered in this document.

6.1.2 Verification of composite beams [4.1.2] Composite beams shall be checked for: 0

resistance of critical cross sections [4.4]

e

resistance to longitudinal shear [6]

0

resistance to lateral-torsional buckling [4.6] in the case of continuous span beams or cantilevers (see Section 6.5)

0

resistance to shear buckling [4.4.4]and web crippling [4.7]. Page 33

1

The possible critical sections to be checked, are summarised below:

II!

I!

rn! rn!

I

-

I

Figure 6.2

Critical sections for design calculation and related action effects

Critical cross-sections: 1-1 11-11 111-111

bending resistance vertical shear resistance bending moment - vertical shear interaction

Regions: IV-IV

KLI VI1

}

longitudinal shear resistance of the shear connectors longitudinal shear resistance of the slab and transverse reinforcement lateral torsional buckling of bottom flange.

Critical cross-sections are for example the sections I, I1 and I11 shown in Figure 6.2, and also sections subjected to heavy concentrated loads or reactions. In case of single span beams, subject to uniform load, no bending moment - vertical shear interaction has to be considered.

6.1.3 Effective width of the concrete flange [4.2.2] The effective width be, for elastic global analvsis may be assumed to be constant over the whole of each span. It may be taken as the value at midspan (beam supported at both ends), or as the value at the support (cantilever). The effectivebreadth for verification of cross-sections should be taken as the midspan value (for sections in positive bending), or as the value at the support (for sections in negative bending).

Page 34

Figure 6.3

Effective width of concrete slab, be,

The effective width on each side of the steel web should be taken as PO /8, but not greater than half the distance to the next adjacent beam web (see Figure 6.3). The length PO is: 0

equal to the span of simply supported beams

0

the approximate distance between points of zero bending moment in case of continuous composite beams (see Figure 6.4).

//

,'/

Figure 6.4

/

.'/ , / / / /

/'/

/ /'/

,'/ //' /

/'/ /

,'// ' / /

/'/ /

/

Length 4, for continuous beams

6.1.4 Classification of cross-sections [4.3] 6.1.4.1

General Composite beams are classified into 4 Classes depending on the local buckling behaviour of the steel flange and/or the steel web in compression.

Page 35 ~

The classification system of cross-sections of composite beams is as follows: Class 1 (plastic) cross-sections are those which can form a plastic hinge with sufficient rotation capacity for plastic hinge analysis. Class 2 (compact) cross-sections are those which can develop their plastic moment resistance, but have limited rotation capacity. 0

Class 3 cross-sections are those in which the calculated stress in the extreme compression fibre of the steel member can reach its yield strength, but local buckling is liable to prevent development of the plastic moment resistance.

e

Class 4 cross-sections are those in which it is necessary to make explicit allowances for the effects of local buckling when determining the moment resistance or compression resistance of the section.

Class 3 and 4 cross-sections are not further considered in this document. A cross-section is classified according to the least favourable class of its steel elements in compression, according to the following Tables 6.1 to 6.4. Steel webs and flanges in compression are classified according to their width to thickness ratios and stress distributions. The positions of the plastic neutral axes of composite sections should be calculated for the effective cross-section using design values of strengths of materials.

Cross-sections under positive bending, where the plastic neutral axis lies in the concrete or in the steel flange, belong to Class 1 independent of the width to thickness ratios of the web and the flanges. Under certain circumstances the classification can be upgraded (refer to Section 6.4 and to EC4).

6.1.4.2

Flanges

I

Flanges in compression

I

rolled

welded

I I

I

Steel

I

E

I

I

0.81

I

Fe 360 Fe 430

I I

1 1 10E

I

I

I

10E

I

9E

I

Fe 510

9E

Table 6.1 Maximum width-to-thickness ratios, c/t, for steel outstand flanges in compression Page 36

The following observations may be made concerning rolled sections: 0

The steel compression flange, if properly attached to the concrete flange, may be assumed to be of Class 1. All IPE, HEB and HEM sections belong to Class 1 (with regard to their flanges).

0

To classify steel flanges of HEA sections, see Table 6.2, which is based on the requirements of Table 6.1.

0

Other restrictions are given in EC4 [6.4.1 SI. HEA Sections

Fe 360

Fe 430

Fe 510

160 180 200

1 1 1

1 2

2

2 3 3

240 260 280

1 2 2

2 3 3

3 3 3

300 320 340

2 1 1

3 2 1

3 3 3

360 400 450

1 1 1

1 1 1

2 1 1

Table 6.2 Classification of HEA Sections (based on flange proportions) 0

HEA sections deeper than 450 mm belong to Class 1.

0

HEA sections of Class 3 belong to Class 2, if they are partially encased (see Section 6.4)

6.1.4.3

Webs Webs: (internal elements perpendicular to axis of bending)

Web subject to bending 01

= 0.5

Web subject to bending and compression 0 I O1 I1.0

B

Stress distribution

Class

1

Web subject to compression Q = 1.0

d/t I 72

E

d/t I33

E

when 01 > 0.5: d/t I396 ~l(1301- 1)

when

01

I

d/t I36 ~

2

0.5: E/CY

~

d/t I83 E

dlt I38

E

when 01 > 0.5: d/t I456 d(1301- 1) when 01 S 0.5: d/t I41.5 €/a

Table 6.3 Maximum width-to-thickness ratios for steel webs

Webs of all IPE and HE sections subject to bending, or bending and compression with a neutral axis characterized by Q I0.5, belong to Class 1. In case of single span beams under positive bending, local instability of the steel web is not critical for any IPE or HE profiles. If the steel web is stressed fully in compression, Tables 6.4 a - d can be used for the classification based on the requirements in Table 6.3. A Class 3 web that is encased in concrete in accordance with Section 6.4 [4.3.1 (6) to (9)] may be assumed to be in Class 2 [4.3.3.1(2)]. An uncased Class 3 web may be represented by an effective depth of web equivalent to a Class 2 web. The cross-section may then be analysed plastically and the section treated as Class 2 [4.3.3.1(3)], provided that the compression flange is Class 1 or 2.

Page 38

Tables 6.4

Classification of steel webs fully in compression (a= l), based on Table 6.3 ~

IPE Sections

~

~

~~

Fe 360

Fe 430

Fe 510

2

3

4

140 160 180

200 220 240 270 300 330 360

Sections smaller than IPE 140 are in Class 1 ~

HEA Sections

Fe 360

Fe 430

Fe 510

~

340 360 400 450

500

550 a00

Sections smaller than HE 340A are in Class 1

Sections smaller than HE 450B are in Class 1

HEM Sections

Fe 360

600 650 700

800 900

1 2

Fe 430

I Fe 510

1 1 1

1 1 2

2

3 4

3

Sections smaller than HE 600M are in Class 1 6.1.5 Distribution of internal forces and moments in continuous beams [4.5] 6.1.5.1

General Bending moments in composite beams at ultimate limit state (ULS) may be determined by elastic or rigid-plastic global analysis, using factored loads. The design bending moments shall not exceed the resistance of the composite beam. The verification shall be done at critical cross-sections (see section 6.1.2).

6.1 S . 2

Plastic global analysis [4.5.2]

Plastic global analysis (or plastic hinge analysis) may be used for all continuous beams, provided that the following requirements are met [4.5.2.2(2)] [4.2.1(3)]: e e e e

the steel cross-section is symmetrical about the plane of its web, lateral torsional buckling does not occur, the steel compression flange at a plastic hinge location is laterally restrained, sufficient rotation capacity is available.

Rotation capacity is sufficient when the following requirements are met: e

the effective cross-sections at plastic hinge locations are in Class 1 and, elsewhere, all others are in Class 1 or 2

e

adjacent spans do not differ in length by more than 50% of the shorter span: 0.66 ILk/Lk+l 5 1.5

e

end spans do not exceed 115% of the length of the adjacent span: Le I 1.15 L;

e

the reinforcement in concrete sections under tension fulfils the requirements of high ductility (see Section 4.2.2 or EC2)

In case of heavy concentrated loads, refer to EC4 [4.5.2.2(2)(d)]. Class 2 sections are only permitted where no rotation capacity is required.

Page 40

6.1.5.3

Elastic global analysis [4.5.3]

Elastic global analysis is based on a linear stress-strain relationship [4.5.3.1].

No account need be taken of bending moments due to shrinkage.

Loss of stiffness, due to cracking of concrete in negative moment regions and yielding of steel, influences the distribution of bending moments in continuous composite beams. Two methods of elastic global analysis are permitted by EC4 at the ultimate limit state to determine the bending moment distribution: e

uncracked section method, based on midspan effective width ignoring any longitudinal reinforcement (method 1);

0

cracked section analysis, based on a section in the region of the internal support comprising the steel member together with the effectively anchored reinforcement located within the effective width at the support (method 2).

,

Method 1

uncracked Figure 6.5

Method 2

cracked

Definition of “uncracked” and “cracked” sections for elastic global analysis

Method 2 is more suitable for computer analysis. However, this method may be also used at the serviceability limit state to accurately determine the moments in cases of crack control in the slab. This method assumes that, for a length of 15%of the span on each side of the support, the section properties are those of the cracked section under negative moments (see Figure 6.5). I2 is the cracked second moment of area which is less than the uncracked value, I,. Refer to Section 7.2 for the calculation of these properties. The elastic bending moments for a continuous composite beam of uniform depth within each span may be modified by reducing maximum negative moments by amounts not exceeding the percentage of Table 6.5. The resulting positive bending moments are then found by static equilibrium. Page 41

Class of cross-section in negative moment region

1

2

For “uncracked” elastic analysis - method 1

40

30

For “cracked” elastic analysis - method 2

25

15

Table 6.5 Limits to redistribution of negative moments at supports, in terms of the maximum percentage of the initial bending moment to be reduced 6.2

Resistance of Cross-Sections

6.2.1 General The design bending resistance may be determined by plastic theory, but only where the effective composite section is in Class 1 or Class 2. 6.2.2 Positive moment resistance [4.4.1.2(2)] The following assumptions shall be made in the calculation of MR, = Mpl,Rd(see Figures 6.6 to 6.8). In all cases Msd IMpf,Rd for adequate design. The effective areas of longitudinal reinforcement in tension and in compression are stressed to their design yield strength f& /ysin tension or compression. Alternatively, reinforcement in compression in a concrete slab may be neglected. Profiled steel decking in compression shall be neglected. The presence of profiled steel decking, when running transverse to the main span, reduces the area of concrete that may resist compression forces. Hence, the maximum possible depth of concrete in compression is h,, which is the depth of concrete flange above upper flange of profiled decking of depth, $. The calculation method for MpI,Rd depends on the location of the plastic neutral axis. Three cases to be considered for doubly symmetric sections are as follows:

Neutral axis in the concrete flange

Case 1

0,SS

fck IT

hc

Figure 6.6 Fa ZC

Neutral axis in the concrete-flange: plastic stress distribution

= A, . f,, /ya = Fa /(b,ff . 0.85 fck /yc ) Ih,

Mp,,Rd = Fa (ha/2 Case 2

+ h,

h, -

Z,

/2)

Neutral axis in the steel flange

1 b e

f

f

l

h,

U

Figure 6.7

'a2

Neutral axis in the steel flange: plastic stress distribution

= h, be, . 0.85 fck /yc For the neutral axis to lie in the flange: Fa > F, > F, where F, = d t,,, . fy /ya F C

Taking moments about the top flange, it follows that the moment resistance is: M,,,,,,

2:

Case 3

Figure 6.8

Fa ha /2

+ F,

(2h,

+ h,

)/2

Neutral axis in the web

Neutral axis in the web: plastic stress distribution

For the neutral axis to lie in the web: F, < F, Hence, the depth of web in compression: = 0.5 ha - Fc/(2 t,,, fy/ya) zcw Neutral axis depth: = h, 3- h, ZC

+ z,

Page 43

where

is the plastic moment resistance of the steel section alone.

Mapl,Rd

6.2.3 Negative moment resistance The composite cross-section consists of the steel section together with the effectively anchored reinforcement located within the effective breadth of the concrete flange at the support (see Figures 6.9 and 6.10). The reinforcement is located at height, a, above the top flange of the steel beam. In the calculation of

Case 4

Mpl,Rd

two cases have to be considered:

Neutral axis in steel flange for negative bending

+!

Figure 6.9

Fs

Neutral axis in the steel flange: plastic stress distribution (high degree of reinforcement)

For the neutral axis to lie in the top flange: Fa > F, > F,

Case 5

Neutral axis in steel web for negative bending

1 beff 7 P 7 - ----

2qf$/la I

I

I

Figure 6.10 Neutral axis in the web: plastic stress distribution For the neutral axis to lie in web: F, < F, Page 44

Depth of web in tension: = ha/2 - F, /(2t,,, . fy /ya ) ZCW Neutral axis depth: = hc h, zc, ZC

+ +

Mp],Rd = Mapl,Rd

+ F, (ha/2 + a) -

F,2 /(4 t,,, fy

/Ya

)

6.2.4 Vertical shear [4.4.2.2(2)] The contribution of the concrete slab to the resistance to vertical shear is normally neglected. Therefore, the shear force resisted by the structural steel section should satisfy: 'Sd

where

vpl,Rd

V,],Rd

is the plastic shear resistance given by:

vpj,Rd

=

(fy

h/3)/ra

The shear area A, may be taken as follows (see EC3, 5.4.6): Rolled I, H and C sections loaded parallel to the web, A, = 1,04 ha t,,, Built-up I sections, A, = d t,,, In addition, the shear buckling resistance of a steel web shall be checked in the following cases [4.4.2.2(3)]: e

for a vertically unstiffened and uncased web, where d / t , > 69 .(for all IPE and HE profiles, d/t,,, < 69 E)

e

for an unstiffened web encased in concrete in accordance with Section 6.4, where d/t, > 124 E ;

E;

If necessary, the shear buckling resistance can be calculated with the rules given in EC4 r4.4.41. 6.2.5 Moment-shear interaction [4.4.3] Where the vertical shear V S d exceeds half the plastic shear resistance, Vp1,,d, allowance shall be made for its effect on the resistance moment at the same cross section. The plastic resistance moment should then be calculated using a reduced design yield strength fy,rd for the shear area A,, according to:

Page 45

Figure 6.11 Normal stress distribution for M-V interaction in negative bending

The following interaction criterion, based on Figure 6.1 1, should be satisfied:

where:

6.3

MRd

is the design moment resistance as given in Sections 6.2.2 and 6.2.3.

Mf,Rd

is the design plastic moment resistance of the cross-section consisting of the flanges only.

Shear Connection [6]

6.3.1 General Shear connectors and transverse reinforcement shall be provided throughout the length of the beam to transmit the longitudinal shear force between the concrete slab and the steel beam at the ultimate limit state, ignoring the effect of natural bond between the materials. Ductile connectors are those with sufficient deformation capacity to justify the assumption of ideal plastic behaviour of the shear connection in the structure. Headed studs may be considered as ductile i f height after welding, h 2 4d 16 mm 5 d 5 22 mm and N/N,2 the minimum degree of shear connection (see Figure 6.14).

6.3.2 Resistance of shear connectors [6.3.2.1] 6.3.2.1

Solid slabs The design shear resistance of a welded headed stud (as shown in Figure 4.5) with a normal weld collar should be determined from the smaller of

P,, = 0.8 f,, (7rd2/4)lyv

Equation (1)

or P,, = 0.29 a d 2

/E,,)/?,

Equation (2)

whichever is smaller, where d and h are the diameter and height of the stud respectively (see Figure 4.5) ~

is the specified ultimate tensile strength of the material of the stud but not greater than 500 N/mm2. The commonly specified strength is 450 N/mm2.

fU

fck

is the characteristic cylinder strength of the concrete at the age considered;

Em

is the mean value of the secant modulus of the concrete in accordance with Table 4.2;

0.2 [(h/d)

Q

=

Q

= 1

+ 11

for 3 I h/d I4; for h/d > 4,

The partial safety factor yv is given as 1.25 in Table 4.5. The resulting design resistances of stud connectors obtained from equations (1) and (2) are presented in Table 6.6 below. Equation (1) f,, = Diameter d [mml 22 19 16

450

500

109.4 81.6 57.9

121.6 90.7 64.3

Equation (2)

(01

= 1.0)

C 25/30 C 30/37 C 35/45 C 40/50 C 45/55

98.1 73.1 51.9

110.0 82.1 58.2

121.6 90.7 64.3

(132.9) (99.1) (70.3)

(142.9) (106.6) (75.6)

Table 6.6 Design resistance PRd[kNl of stud connectors with h/d > 4 The value in brackets denote values which exceed those given by Equation (1); the lower value should therefore be used in design.

Influence of shape of profiled steel decking [6.3.3]

6.3.2.2

Where profiled steel decking with ribs parallel to the supporting beam is used, the studs are located within a region of concrete that has the shape of a haunch. The design shear resistance should be taken as their resistance in a solid slab, multiplied by the reduction factor k, given by the following expression:

kp

=

0.6 . b, /h,

. [(h/% -

l)] 5 1.0

where h Ihp

+ 75

Figure 6.12 Beams with steel decking ribs parallel to the beam Where studs of diameter not exceeding 20 mm are placed in ribs transverse to the supporting beam with a height h, not exceeding 85 mm and a width bo not less than h,, the design shear resistance should be taken as their resistance in a solid slab (calculated as given above except that fu should not be taken as greater than 450 Nlmm’) multiplied by the reduction factor given by the following expression:

where N, is the number of stud connectors in one rib at a beam intersection, not to exceed 2 in computations. For studs welded through the steel decking, k, should not be taken greater than 1.0 when N, = 1, and not greater than 0.8 when N, 1 2. 6.3.3 Spacing of shear connectors [6.1.3] [6.4] Stud connectors may be spaced uniformly over a length L,, between adjacent critical cross-sections (see Section 6.1.2) provided that: 8

all critical sections in the span considered are in Class 1 or Class 2

8

N/N, satisfies the limits for partial shear connection given in Section 6.3.4.2, when L is replaced by L,.,, and

8

the plastic resistance moment of the composite section does not exceed 2.5 times the plastic resistance moment of the steel member alone.

Detailing rules for placement of studs are given in Section 5.4.

6.3.4 6.3.4.1

Longitudinal shear force Full shear connection r6.2.1.13

a. Single span beams [6.2.1.1(1)] For full shear connection, the total design longitudinal shear V, to be resisted by shear connectors between the point of maximum positive bending moment and the end support should be: V, = F,, where or

FCf= A, f bya F,f = o*8zfck

hc /Yc

whichever is the smaller.

b. Continuous span beams [6.2.1.1(2)] For full shear connection, the total design longitudinal shear V, to be resisted by shear connectors between the point of maximum positive bending moment and an intermediate support shall be calculated according to Figure 6.13 as follows: V, = F,,

+ (A,

fsk )/ys

where A, is the effective area of longitudinal slab reinforcement.

This calculation is illustrated in Figure 6.13 for a particular case of the positions of the plastic neutral axes in negative and positive bending.

Figure 6.13(a)

Moments in continuous beam Page 49

(b)

Internal force distribution

Figure 6.13

Calculation of the longitudinal shear force in continuous beam

The number of shear connectors for full shear connection shall be at least equal to the design longitudinal shear force V,, divided by the design resistance of a connector, PRd. Therefore, the number of shear connectors in the zone under consideration is:

PRd takes into account the influence of the shape of the profiled sheeting, as given in Section 6.3.2.2. 6.3.4.2 Partial shear connection with ductile shear connectors [6.2.1.2] Partial shear connection may be used if all cross-sections are in Class 1 or 2. Ideal plastic behaviour of the shear connectors may be assumed if a minimum degree of shear connection is provided (see Figure 6.14, where L is the beam span (metres)).

Page 50

9y//

0 '0

/

'

I

/

1

I

I I

I I

I

I

I

I & I

-

I

I

I

5

0

I ' I I

I 10

I

15

20

25

-

Liml

Figure 6.14 Minimum degree of shear connection The minimum degree of shear connection is defined by the following equations: (a) (b) (c)

+

0.03 L N/Nf I0.4 N/Nf 2 0.25 4- 0.03 L N/N, 1 0.04 L

where A, = top flange area and

where 3At 2 Ab where A, = Ab where A, = A, A,,

=

(see further restrictions)

bottom flange area

In all cases, N/Nf 2 0.4, where N Nf

= number of shear connectors = number of shear connectors for full shear connection.

Equation (c) (line c in Figure 6.14) may be used when the following conditions are satisfied: e e e

e e

a single, centrally placed 19 mm diameter stud per trough, with a height after welding of not less than 76 mm rolled I or H sections has equal flanges the concrete slab is composite with sheeting that spans perpendicular to the beams and is continuous across it. deck profile with b, /h, 2 2 and h, 5 60 mm linear interaction method is used (see following)

The general method for other cases is line b in Figure 6.14.

Page 51

The moment resistance of a composite beam designed for partial shear connection may be determined by either of the following methods:

1. 2.

Stress block (or equilibrium) method Linear interaction method.

In Method 1 the force transferred to the concrete is determined by the longitudinal resistance of the shear connectors. Equilibrium equations can be established explicitly, in a similar manner to Section 6.2.2. The relationship is defined by the curve ABC in Figure 6.15. Method 2 is more conservative, but is often preferred because it is a simple method of determining the moment resistance, knowing the moment resistances of the steel beam and the composite section for full shear connection. The relationship is defined by the line AC in Figure 6.15. Because it is a more conservative treatment of partial shear connection, the linear interaction method may be used with a less severe restriction on the minimum degree of shear connection.

%

I.o

Mpl,Rd -

- -

P--Fcf

-a? f

-d

B

equilibrium method

linear interaction method

/ /

M p l , Rd

I I.O

N FC or Fcf

Nf

Figure 6.15 Relation between F, and MSd for partial shear connection The force transferred by the shear connectors, Fc, in the linear interaction method is: FC

-

- Mapl,Rd Fcf Mpl,Rd - Mapl,Rd MSd

where Mapl,Rdis the design moment resistance of the structural steel section. Fc, is the longitudinal shear force required for full shear connection, and Msd I Mpl,Rd, as determined for full shear connection. Page 52

The design longitudinal shear force, V, can be determined from Figure 6.13 with FCf replaced by the force transferred by the shear connectors, F,. I f F, is determined for a known distribution of shear connectors, then the maximum value of the reduced moment resistance may be calculated using the same linear interaction equation. This reduced moment resistance, MRd, should be not less than Msd, which is the applied moment.

Figure 6.16 Reduced bending moment resistance of cross-sections A good approximation to the equilibrium method can be obtained by considering the

moment-axial force interaction for an I section, as illustrated in Figure 6.16. It follows that:

where F, = force transferred by shear connectors = C PRd and N S Nf Fa = axial resistance of steel section = A, fy /ya The interaction between moment and shear is covered in Section 6.2.5. 6.3.5 Transverse reinforcement 6.3.5.1 Longitudinal shear in the slab [6.6.1] The design longitudinal shear per unit length for any potential surface of longitudinal shear failure within the slab, VSd, shall not exceed the design resistance to longitudinal shear, VRd, of the shear surface considered. VSd should be determined in accordance with Section 6.3.4 and be consistent with the design of the shear connectors at the ultimate limit state.

Page 53

Potential failure surfaces are shown in Figure 6.17. Top and bottom reinforcement in the slab may be considered to be effective. Where steel decking is continuous over the beam, or is effectively anchored by shear connectors, it may also be considered to act as transverse reinforcement. Failure surface a-a controls in these cases. Failure surface b-b is not considered critical in EC4 in cases where steel decking is used. In determining VSd, account may be taken of the variation of longitudinal shear across the width of the concrete flange. Longitudinal shear is considered to be transferred uniformly by the shear connectors.

Figure 6.17 Typical potential surfaces of shear failure (examples)

6.3.5.2 Design resistance to longitudinal shear [6.6.2(1) to (5)] The design resistance of the concrete flange (shear planes a-a illustrated in Figure 6.17) shall be determined in accordance with the principles in Clause 4.3.2.5 of EC2. Profiled steel sheeting with ribs transverse to the steel beam may be assumed to contribute to resistance to longitudinal shear, provided it is continuous across the top flange of the steel beam or if it is welded to the steel beam by stud shear connectors. In the absence of a more accurate calculation the design resistance of any surface of potential shear failure in the flange or a haunch should be determined from:

whichever is smaller, where 7Rd

is the basic shear strength to be taken as 0.25 fctk0.05 /yc, (see Table 6.7)

1c1c1c1

Concrete strength rRd

J* c

20125

25/30

30/37

35/45

0.25

0.30

0.33

0.37

4050

45/55

50/60

Table 6.7 Basic shear strength 7Rd (in N / I I U ~ ~ ) rl rl

= 1 for normal-weight concrete, = 0.3 0.7 (p/24) for lightweight-aggregate concrete of unit weight p in

+

w/m3, A,,

is the mean cross-sectional area per unit length of beam of the concrete shear surface under consideration,

A,

is the sum of the cross-sectional areas of transverse reinforcement (assumed to the perpendicular to the beam) per unit length of beam crossing the shear surface under consideration (Figure 6.17) including any reinforcement provided for bending of the slab,

Vpd

is the contribution of the steel decking, if applicable, as given in 6.3.5.3.

For a ribbed slab the area of concrete shear surface A,, should be determined taking into account of the effect of the ribs. Where the ribs run transverse to the span of the beam, the concrete within the depth of the ribs may be included in the value of A,, but for parallel ribs it should not be included in A,,. These cases are illustrated in Figure 6.17. Transverse reinforcement considered to resist longitudinal shear shall be anchored so as to develop its yield strength in accordance with EC2. At edge beams, anchorage may be provided by means of U-bars looped around the shear connectors.

6.3.5.3 Contribution o,f profiled steel decking as transverse reinforcement [6.6.3(1) to (2)] Where the profiled steel sheets are continuous across the top flange of the steel beam, the contribution of the decking with ribs transverse to the beam may be taken as:

'pd

-

fYP YPP

where

Vpd

is per unit length of the beam for each intersection of the shear surface by the sheeting,

A, is the cross-sectional area of the profiled steel decking per unit length of the beam, and fyp is its yield strength, given in N/mm2.

Page 55

Where the decking with ribs transverse to the beam is discontinuous across the top flange of the steel beam, and stud shear connectors are welded to the steel beam directly through the decking, the contribution of the decking should be taken as:

VPd

where

S

Ppb,Rd

s

6.4

'pb,Rd -

but

fYP -

I *P

Yap

is the design resistance of a headed stud against tearing through the steel sheet [7.6.1.4]) and

is the spacing centre-to-centre of the studs along the beam.

Partially Encased Beams A typical partially encased composite beam is shown in Figure 6.18. The partial

encasement provides fire protection to the steel beam.

Figure 6.18

Cross-section of partially encased beam

Reauiremen ts: Concrete encasement the webs of composite beams shall be [4.3.1(6) to (S)]: 0

reinforced by longitudinal bars and stirrups, and/or welded mesh [4.3.1(7)]

0

mechanically connected to the web by stud connectors, welded bars, or bars through holes

0

capable of preventing buckling of the web and of any part of the compression flange towards the web.

Influence of encasement: 0

Substantial increase in fire resistance is possible (reference should be made to EC4 Part 1.2);

*

A web in Class 3 may be represented by an effective web of the same cross-

section in Class 2 [4.3.3.1(2)]; Web encasement may be assumed to contribute to resistance against local buckling [4.3.2,4.3.3], shear buckling [4.4.2.2] and lateral-torsional buckling [4.6.2]. However, the shear buckling resistance for an unstiffened and encased web shall be verified (by testing), if d/t, > 124 E [4.4.2.2(3)]; e

No application rules are given for the contribution of concrete encasement of a steel web to resistance in bending and vertical shear.

Where the depth h of a partially encased steel member does not exceed the limit given in Section 6.5 [4.6.2(m)] by more than 200 mm [4.6.2], verification of lateral torsional buckling is not necessary provided the conditions in EC4, [4.6.2 (a) to (k)] are satisfied, (see Section 6.5). 6.5

Lateral-Torsional Buckling of Continuous Beams [4.6] The concrete slab may usually be assumed to prevent the upper flange of the steel section (connected to the concrete part) from moving laterally. In negative moment regions of continuous composite beams the lower flange is subject to compression. The tendency of the lower flange to buckle laterally is restrained by the distortional stiffness of the cross-section (inverted U-frame action).

Web encasement in accordance with Section 6.4 [4.3.1] may be assumed to contribute to resistance to lateral-torsional buckling [4.6.2], see Table 6.8.

No direct calculation for the lateral stability of a composite beam is necessary when the following conditions are satisfied: adjacent spans do not differ in length by more than 20% of the shorter span or where there is a cantilever, its length does not exceed 15% of the adjacent span. the loading on each span is uniformly distributed and the design permanent load exceeds 40% of the total load. the shear connection in the steel-concrete interface satisfies the requirements of Sections 5.4 and 6.3. e

the slab proportions are typical of those in general building design (see Section 5.2). ha Imaximum depth of steel member is as given in Table 6.8.

Page 57

I t b -

l t b - t

Profile

Fe 360

Fe 430

Fe 510

Fe 360

Fe 430

Fe 510

IPE HEA

600 800

550 700

400 650

800 1o00

750 900

600 850

All HEB sections satisfy the requirements for lateral torsional buckling. The data in Table 6.8 for IPE sections is also applicable for UB and other equivalent rolled sections, provided the depth/width of the steel section does not exceed 2.75. In other cases a check for lateral torsional buckling according to EC4 [4.6] and [Annex B] is required. If necessary, additional discrete lateral restraints may be provided to the compression flange, for example, by bracing or transverse members. For edge beams, fully anchored top reinforcement is required in the slab [4.6.2(h)].

Page 58

7.

SERVICEABILITY LIMIT STATES [ 5 ] , [2.3.4]

7.1

General Criteria The serviceability requirements for composite beams concern the control of deflections, cracking of concrete and, in some cases, vibration response. Deflections are important in order to prevent cracking or deformation of the partitions and cladding, or to avoid noticeable deviations of floors or ceilings. Floor vibrations may be important in long span applications, but these calculations are outside the scope of the Code (see Section 7.3). Loads to be used at the serviceability limit state are presented in Section 3.3.3 [2.3.4]. Normally unfactored loads are used. In order to calculate serviceability effects, account should be taken of any non-linear effects due to cracking, steel yielding etc. Most designers base assessments at the serviceability limit state on elastic behaviour (with certain modifications for creep and cracking etc). To avoid consideration of post-elastic effects, limits are often placed on the stresses existing in beams at the serviceability limit state.

No stress limitations are made in Eurocode 4 because: 0

post-elastic effects in the mid span region are likely to be small and have little influence on deflections;

0

the influence of the connections on the deflection of simply supported beams has been neglected;

0

account is taken of plasticity in the support region of continuous beams.

Deflection limits are not specified in EC4. Reference is made to Eurocode 3 (see Tables 7.1 and 7.2) for limits on deflections due to permanent and variable loads.

7.2

Calciilation of deflections [5.2.2(2)J

7.2.1 Second moment of area Deflections are calculated knowing the second moment of area of the composite section based on elastic properties (see Figure 7.1). Under positive moment the concrete may be assumed to be uncracked, and the second moment of area of the composite section (expressed as a transformed steel section) is:

I

, , ,6

= sagging in the final state relative to the

straight line joining the supports.

state 0

Table 7.1

6,

= pre-camber (hogging) of the beam in the unloaded state (state 0)

6,

= dueto G (variation of the deflection of the beam due to the permanent loads) (state 1)

6,

= due to Q (variation of the deflection of the beam due to the variable loading) (state 2)

Vertical deflections to be considered

Limits

Conditions

*

~mllx

62

roofs generally

1/200 1/250

roofs frequently carrying personnel other than for maintenance

1/250 1/300

floors generally

1/250 1/300

floors and roofs supporting brittle finish or non-flexible partitions

1/250 1/350

floors supporting columns (unless the deflection has been included in the global analysis for the ultimate limit state)

1/400 1/500

* where,,,a,

1/250

can impair the appearance of the building

For cantilever: L = twice cantilever span

Table 7.2

Recommended limiting values for vertical deflections

where:

n

is the ratio of the elastic moduli of steel to concrete (see Section 7.2.2) taking into account creep

r

is the ratio of the cross-sectional area of the steel section, A,, relative to the area of the concrete section (beffhc).

I,

is the second moment of area of the steel section

Note: other terms are as defined previously. The effective slab width, be, is the same as used at the ultimate limit state see Section 6.1.3 [5.2.2(3)]. The ratio Ic!Ia therefore defines the improvement in the stiffness of the composite section relative to the steel section. This ratio is presented in Section 11.2 for all IPE and HE sections (up to 600mm deep) for typical slab depth. Typically, Ic/Iais in the range of 2.5 to 4.0, indicating that one of the main benefits of composite action is in reduction of deflections. It is not usually necessary to calculate the ‘cracked’ second moment of area under positive moment as the elastic neutral axis will normally lie in the steel beam, or near the base of the slab. Where it is necessary to know the ‘cracked’ second moment of area under negative moment, a simple formula may be derived from Figure 7.2. Assuming that the reinforcement is placed at mid-height of the slab above the sheeting, this formula is:

4,”

-

A, (hc+2h,

+

4 (1 +rs)

ha)? +

‘a

where:

r,

is the ratio of the cross-sectional area of reinforcement, A,, within the effective’breadthof the slab to the cross-sectional area of the beam, A,.

This formula may be used in establishing the moments in elastic global analysis (method 2 in Section 6.1.5.3), or in crack control calculations (see Section 7.4). 7.2.2 Modular ratio r3.1.4.21 The values of elastic modulus of concrete under short term loads are given in Table 4.2. The elastic modulus under long term loads is affected by creep, which causes a reduction in the stiffness of the concrete. The modular ratio, n, is the ratio of the elastic modulus of steel to the time-dependent elastic modulus of concrete (see Section 4.2.1). Typically, the modular ratio for normal weight concrete is 6.5 for short term (variable) loading. The elastic modulus of concrete for long term (permanent) loads is taken as one-third of the short term value, leading to a modulus ratio of approximately 20 for long term (permanent) loading in an internal environment.

b-eEr

-

I,

filed steel shee --

I-

-

Ea

strain

stress ELASTIC STRESSES

CROSS-SECTION

Figure 7.1: Elastic analysis of composite beam under positive moment

I

ES

t-

- - - - --

/

II I,

L

-7

\

\

\

t

-I-T

OS

-

Ze

- -

-

-1-1

---I

Ea

strain

oa

stress

Figure 7.2: Elastic analysis of composite beam under negative moment

Page 62

For building of normal usage, surveys have shown that the proportions of variable and permanent imposed loads rarely exceed 3: 1. Although separate deflection calculations may be needed for the variable and permanent deflections, a representative modular ratio is usually appropriate for calculation of imposed load deflection. This value may be taken as twice the short term modular ratio (ie. approximately 13) for buildings of normal usage.

7.2.3 Influence of partial shear connection [5.2.2(6)] Deflections increase due to the effects of slip in the shear connectors. These effects are ignored in composite beams designed for full shear connection. For cases of partial shear connection using headed stud shear connectors, the deflection, 6, is increased according to:

-a -- 1 + c

1--

6,

N Nf

-6, - 1 6,

where: N/N,

is the degree of shear connection at the ultimate limit state

6,

is the deflection of the composite beam with full shear connection

6,

is the deflection of the steel beam under the same loads

C

is a coefficient, taken as 0.3 for unpropped construction and 0.5 for propped construction.

The difference between these coefficients arises from the higher force in the shear connectors at serviceability in propped construction. An additional simplification is that slip effects can usually be ignored when N/Nf1 0.5 in unpropped construction [5.2.2(5)]. This is because of beneficial effects that are ignored in calculating deflections making the above equation too conservative in many cases.

7.2.4 Shrinkage induced deflections [3.1.3]

~

EC4 states that shrinkage deflections need only be calculated for simply supported beams when the span: depth ratio of the beam exceeds 20, and when the free shrinkage strain of the concrete exceeds 400 x 10-6 [5.2.2(9)]. In practice, these deflections will only be significant for spans greater than 12m in exceptionally warm dry atmospheres. The curvature, K,, due to a free shrinkage strain, es, is: ~

K,

E,

=

(h

2hJ A, 2(1 +nr) I, +

h,

+

Page 63

n is the modular ratio appropriate for shrinkage calculations (n = 20). The deflection due to this curvature is:

6, = 0.125 K, L2 This deflection formula ignores continuity effects at the supports and probably overestimates shrinkage deflections by a considerable margin.

7.2.5 Continuous beam [5.2.2(7)] The deflection of a continuous beam is modified by the influence of cracking in the negative moment region. This may be taken into account by calculating the second moment of area of the cracked section under negative moment (ignoring the concrete). The ne ative moment at the supports is assumed to vary as a reduction factor of (I,II$0.3Ftimes the negative moment at the serviceability limit state based on an analysis of the uncracked section, where I, is the uncracked, and I, is the cracked second moment of area. The lower limit to this reduction factor on negative moment is 0.6, which is applicable where there is minimum reinforcement in the slab. This method may be used where the difference in adjacent spans is less than 25 %. Alternatively, a more precise method is to use the ‘cracked section’ analysis model of elastic global analysis described in Section 6.1.5 in order to determine the negative moments directly. No further reduction in negative moments should be made in this case. In continuous beams, there is a possibility of yielding in the negative moment region. To take account of this effect the negative moments may be further reduced. In reality, this reduction is a function of the moment resistance of the composite section under negative and positive moment. A conservative way of taking this into account is to multiply the ‘elastic’ negative moments at the supports by a further reduction factor. This factor is given in EC4 [5.2.2(8)]as 0.7, where load sufficient to cause yielding is applied to the hardened section, which is the normal design case. Together with the minimum factor of 0.6 due to concrete cracking, the final negative moment may be conservatively taken as 0.42 times the elastic moment based on uncracked analysis. The mid-span deflection of a beam, as influenced by the support moments, may be calculated from:

where:

C

=

0.6 for uniform load 0.5 for central point load

M Oand 6, are the mid-span moment and deflection of the equivalent simply-supported beam and M,and M, are the negative moments at the supports (for the same loading condition), reduced Tor cracking and yielding as noted above.

As an approximation, a deflection coefficient of 3/384 is usually appropriate for determining the deflection of a continuous composite beam subject to uniform loading on equal adjacent spans. This may be reduced to 4/384 for end spans. The second moment of area of the section is based on the uncracked value.

7.3

Vibration Checks [5.1(1)] This section is included because a check on the potential vibration response may be necessary for long span beams designed for light imposed loads. A simple measure of the natural frequency of a beam is:

f = 18/@sw where as, is the instantaneous deflection (mm) caused by reapplication of the self weight of the floor and beam to the composite member. A minimum limit on natural frequency, f, is proposed as 4 cycledsec for most building applications except where there is vibrating machinery, or 3 cycles/sec for car parks. The limit may be raised to 5 cycles/sec for special buildings such as sports halls.

7.4

Crack Control [5.3] It is necessary to control cracking of concrete only in cases where the proper functioning of the structure or its appearance would be impaired. Internally within buildings, durability is not affected by cracking. Similarly when raised floors are used, cracking is not visually important. Where it is necessary to control cracking, the amount of reinforcement should exceed a minimum value in order to avoid the presence of large cracks in the negative moment region. This minimum percentage, p, is given by: p = -AS - -k,.k.A C

fCt

r5.3.21

US

where: k,

is a coefficient due to the bending stress distribution in the section: (k, = 0.9)

k

is a coefficient accounting for the decrease in tensile strength of concrete (k = 0.8)

fct

is the effective tensile strength of concrete. minimum adopted.

us

is the maximum permitted stress in the reinforcement (see Table 5.2 of EC4).

A value of 3N/mm2 is the

Page 65

A typical value of p is 0.4 to 0.6% which is well in excess of the minimum of 0.2% necessary in unpropped construction for shrinkage control and transverse load distribution. However, these bars need only be placed in the negative moment region of the beams or slabs. This reinforcement may also act as fire reinforcement or transverse reinforcement.

An additional criterion is that the bars should be of small diameter and should be spaced relatively closely together in order to be more effective in crack control. Maximum bar diameters are given in Table 7.3 as a function of the reinforcement stress, U,, and maximum crack width, w. If crack control is necessary, more information is given in EC4,clause 5.3.4.

I

crack widths in concrete wk =

us

0.3 m m

N/mm2

I

320

I

12

22

I

360

II

10

18

400 450

Table 7.3

8

II

6

I

14

-17

Maximum bar diameters (mm) for different reinforcement stresses and crack widths at the serviceability limit state

Page 66

8.

COMPOSITE COLUMNS

8.1

Introduction

Composite columns are of two main types: totally or partially concrete encased steel sections (Figures 8.1.a and b) or concrete filled steel sections (Figures 8. lc and d). The encased composite column is encountered frequently in building construction, often because the concrete encasement also provides sufficient fire protection. Partially encased columns (Figure 8. lb) are easier to construct and provide free structural steel surfaces for later welding or attachments. For both types of columns fire resistance can be enhanced by additional reinforcement. Where a hollow steel section is used with an internal filling of concrete, the profile provides formwork for the concrete. For short circular columns, the effect of confinement leads to an additional increase in the load bearing resistance.

tb = b c - T

I4. Y

a.

7

?-

ez

-

h hc

ez

A-

-

Y

h=hc

b.

I

i 7 Figure 8.1

Types of cross-sections of composite columns

Page 67

8.2

Design Method

8.2.1 General In EC4 two methods of design are given. The first one, which is general, takes account of second order effects including imperfections. It ensures that for the most unfavourable combinations of actions at the ultimate limit state, instability does not occur and that the resistance of individual cross-sections subjected to bending and longitudinal forces is not exceeded. Numerical methods of analysis are necessary for verification using this method. The second method - presented in the following pages - is a simplified one, permitting design without the aid of a computer. It is based on assumptions given below and it adopts the European buckling curves for steel columns as the basic design curves for composite columns. The limits of applicability of the simplified method are given in Section 8.3.5. When the limits are not fulfilled, the general method in EC4 [4.8.2] has to be applied. 8.2.2 Design assumptions E4.8.3.13 Both approaches for the design of composite columns are based on the following main assumptions. 0

full interaction between concrete and steel up to the point of collapse;

0

allowances must be made for imperfections which are consistent with those adopted for assessing the strength of bare steel columns;

0

proper account must be taken of the steel and concrete stress-strain curves;

a

plane sections remain plane.

8.2.3 Local buckling [4.8.2.4] It is also assumed that all the material strengths are attained without any local buckling of the steel parts of the cross section.

To prevent premature buckling, the maximum width-to-thickness ratios for the steel parts in compression have to satisfy the following values (for notations, see Figure 8.1). a 0

0

concrete filled circular hollow sections concrete filled rectangular hollow sections partly encased I-sections

where e

-

1235

d/t I90 e2 b/t I52 E b/$ I44 E

(f,, in N/mm2)

f"

Table 8.1 gives the maximum width-to-thickness ratios for steel sections. Page 68

Type of cross section (Figure 8.1)

Nominal steel grade Fe 360

Fe 430

Fe 510

Concrete filled circular hollow section (d/t)

90

77

60

Concrete filled rectangular hollow section

52

48

42

44

41

36

(W Partly encased I-section (b/tf)

Table 8.1

Limiting width-to-thickness ratios to avoid local buckling

If the maximum width-to-thickness ratios are exceeded, special methods of analyses should be applied.

No verification for local buckling is needed for totally encased sections but a sufficient cover must be provided in order to prevent premature spalling of the concrete. Therefore the minimum concrete cover c, may not be less than 40mm or 116 of the width of the flange of the steel column. 8.2.4 Shear between the steel and concrete components [4.8.2.6] Internal forces and moments applied from members connected to the ends of a column length have to be distributed between the steel and concrete components of the column, by considering the shear resistance at the interface between steel and concrete. The shkar resistance is provided by bond stresses and friction at the interface, or by mechanical shear connection, so that no significant slip occurs. As the natural bond between steel profile and concrete is uncertain, the design shear

resistance due to bond may not exceed the following values r4.8.2.71. e

for totally concrete encased sections

0.6 N/mm2

e

for concrete filled sections

0.4 N/mm2

a

for flanges in partially concrete encased sections according to Figure 8. lb

0.2 N/mm2

for the webs in partially concrete encased sections according to Figure 8.lb

0.0 N/mm2

0

An exact determination of bond stresses between structural steel and concrete requires extensive calculation. Stresses may be determined in a simplified way either according to elastic theory or from the plastic resistance of the cross section. The variation of stresses in the concrete member between two critical sections can be used for the determination of bond stresses.

If loads are introduced into the column, it must be ensured that after a certain introduction length the individual components of the cross section are loaded according to their resistance, so that no significant slip occurs between these parts. The introduction length for the shear force should not be assumed to exceed twice the smaller of the two cross section dimensions. 8.3

Simplified Method of Design

8.3.1 Resistance of cross-sections to axial loads [4.8.3.3] In dealing with axially loaded columns it is important to differentiate between these columns whose strength is augmented by triaxial containment of concrete (concrete filled circular hollow steel sections) and those where no such action occurs (totally or partially encased rolled sections, concrete filled rectangular hollow steel sections). 8.3.1.1 Encased steel sections and concrete filled rectangular hollow steel sections The plastic resistance of the cross section is given simply by the sum of the components, as follows:

e

A,, A, and A,

are the cross-sectional areas of the structural steel, the concrete and the reinforcement, respectively

e

fy, fck

and f&

are the characteristic strengths of the above components

and

are partial safety factors at the ultimate state (ya = 1.10, yc = 1.5 and ys = 1.15)

’?a,

e

ar

YC

7s

= 1 = 0.85

for concrete filled sections for all other cases.

6 is the steel contribution factor= (A, fy/ya)/Npl,Rd and 0.2 I6 I0.9 8.3.1.2 Concrete filled circular hollow sections After a certain stage of loading of a concrete-filled circular hollow steel section, the Poisson’s expansion of the concrete exceeds that of the steel and from that point the concrete is triaxially contained by the radial forces associated with the development of hoop tension in the steel section. The development of these hoop tensile forces in the tube combined with the compressive axial forces in the steel shell lowers the, effective plastic resistance of the steel section in accordance with the von Mises failure criteria. On the other hand the increase in concrete strength over the normal unconfined cylinder strength more than offsets any reduction in the resistance of the steel with the result that such columns show an enhanced strength. It is important to note, however, that the effects of triaxial containment tend to diminish as the column length increases. Consequently this effect may only be Page 70

-

considergd up to a relative slenderness X 5 0.5. For most practical columns the value of X = 0.5 corresponds to a length to diameter ratio (P/d) of approximately 12. In addition the eccentricity of the normal force (e) may not exceed the value d/10, d being the outer diameter of the circular hollow steel section. The eccentricity, e, is defined by:

-

e

Mrnax,Sd NSd

where

is the maximum design moment is the design axial force.

Mrnax,Sd NSd

The plastic resistance of the cross section of a concrete-filled circular hollol section is given by: I

e

t is the wall thickness of the steel tube

e

92 = 920

e

'vl0 = 4.9 -

e

qz0 = 0.25

1Oe

+ (1 - 920) d 18.5 (3

x + 17 x'

+ 2x)

and qlo 2 0 and

920 I

1.0

When the relative slenderness h exceeds the value 0.5 or the eccentricity e exceeds d/10, then q l = 0 and q2 = 1.0 are used (ie. no account is taken of the benefits of confinement). 8.3.2 Resistance of members to axial loads [4.8.3.8] A composite column has a sufficient compression resistance, if for both axes of bending:

where

NpI,Rd

is the cross section resistance in accordance with Sections 8.3.1.1 or 8.3.1.2.

Page 71

X

where Q!

is the reduction factor for the relevznt buckling mode given in terms of the relevant relative slenderness h and the relevant buckling curve. Values of the reduction factor for the appropriate slenderness may be obtained from Table 8.3. Otherwise, it may be determined from:-

+ = 0.5 [l +

Q!

(i - 0.2) + i2]

is an imperfection factor corresponding to the appropriate buckling curve and is given in Table 8.2.

I Imperfection factor cx Table 8.2

I

0.21

I

Values of imperfection factor

0.34

I

I

0.49

(Y

h

is the relative slenderness for the plane of bending considered, and is:

(EI),

is the effective elastic flexural stiffness of the cross section given by: (EI), = E, I,

+ 0.8 E,,

I,

+ E, I,

I,, I, and I,

are the second moments of the area of structural steel, concrete and reinforcement for the buckling axis considered

Ea, Es

are the moduli of elasticity of structural steel and reinforcement

Ecd = E,,/y, and yc = 1.35, where E,, is the secant modulus of elasticity of the concrete given in Table 4.2.

P

is the buckling length of the column.

Page 72

Relative slenderness

Concrete-tilled hollow-section

x

a

b

C

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.o 1.1 1.2 1.3 I .4 1.5 1.6 1.7 1.8 1.9 2.0

1.oooo 0.9775 0.9528 0.9243 0.8900 0.8477 0.7957 0.7339 0.6656 0.5960 0.5300 0.4703 0.4179 0.3724 0.3332 0.2994 0.2702 0.2449 0.2229

1.oooo 0.9641 0.9261 0.8842 0.8371 0.7837 0.7245 0.6612 0.5970 0.5352 0.4781 0.4269 0.3817 0.3422 0.3079 0.278 1 0.252 1 0.2294 0.2095

1 0.9491 0.8973 0.8430 0.7854 0.7247 0.6622 0.5998 0.5399 0.4842 0.4338 0.3888 0.3492 0.3 145 0.2842 0.2577 0.2345 0.2141 0.1962

-

Encased sections Encased sections (strong bending axis)* (weak bending axis)*

.oooo

* of steel section -

Table 8.3

Reduction factor x = f(X)

For slender columns, the influence of long-term behaviour of the concrete (creep and shrinkage) on the resistance should be considered. This influence can be taken into account by a simple modification of the modulus of elasticity of the concrete from Ecd to E,:

where N,,

is the design normal force, and

NG.Sdis the permanently acting part of the normal force.

x

Creep and shrinkage should be considered when the relative slenderness lies outside the limiting value of Table 8.4.

Page 73

Braced systems Concrete encased sections

0.8

Concrete filled sections

0.8 1-6

where, 6 =

*a

fy'ya

Npl,Rd

Table 8.4

Limiting values of effects

x in order to neglect of creep and shrinkage

8.3.3 Resistance of cross sections to combined compression and uniaxial bending[4.8.3.11] Unlike the interaction diagram for a bare steel section where the moment resistance undergoes a continuous reduction with increase in axial load, for very short composite columns the moment resistance may be increased by the presence of axial load. This is because the prestressing effect of an axial load may in certain circumstances prevent cracking and so make the concrete more effective in resisting moments. Figure 8.2 represents the non-dimensional interaction curve for compression and uniaxial bending for a composite cross section.

Figure 8.2

Interaction curve for compression and uniaxial bending

Such a curve for short composite columns can be determined by considering different positions of the neutral axis over the whole cross section and determining the internal action effects from the resulting stress blocks. This approach can only be carried out by a computer analysis.

With the simplified method of EC4, it is possible to calculate by hand four or five points (ACDB and E) of the interaction curve. The interaction curve may be replaced by the polygon ACDB(E) through these points (Figure 8.3). The method is applicable to the design of columns with cross-sections that are symmetrical about both principal axes.

Mpl,Rd

Figure 8.3

i

IMrnax. Rd

Interaction curve with polygonal approximation

For point A (compression resistance)

For point B (moment resistance) NB = 0 and MB =

M,I,Rd

For point C (moment resistance =

(plastic resistance moment, see Annex 1) Mpl,Rdfor

N > 0)

Point D is less important than point C, and can be safely neglected as it leads to an increase of moment resistance with axial load. Point E need not be determined for major axis bending of encased I sections or if the design axial force does not exceed Npm,Rd. In other cases, refer to Eurocode 4.

8.3.4 Analysis for bending moments applied to columns [4.8.3.10] Analysis for bending moments according to second order theory may be neglected for non-sway systems i f 1.

The relative slenderness

x does not exceed 0.2 (2 - r)

where r is the ratio of the smaller to larger end moment (-1 S r I1). For transverse loading within the columns length, r = 1. 2.

N,, /Nc,

I0.1,

where N,, is defined in Section 8.3.2.

The flexural stiffness, (EIa, which is necessary for the analysis using second order theory, can be determined following Section 8.3.2. As a simplification, the moment according to second order theory can be calculated by multiplying the maximum first order bending moment by a factor, k, such that:

where:

N,,

is the design normal force

N,,

is the critical load according to 8.3.2

.is the moment factor given below. For colu'mns with transverse loading within the column length the value for P must be taken as 1.0. For pure end moments, p can be determined from:

P

=

0.66

+ 0.44 r

but

1 0.44

8.3.5 Resistance of members to combined compression and uniaxial bending [4.8.3.13] When there are moments in addition to the axial load, the buckling loads as obtained by the method described in Section 8.3.2 have to be reduced. The simplified method uses the previous buckling load as a starting point and refers to the cross section interaction curve described above. Figure 8.4 illustrates the procedure for cases in which point E need not be determined. In this case, lines AC and BC may be defined easily. The principle of calculation is shown in the same figure.

Page 76

Figure 8.4

Design procedure for compression and uniaxial bending

The axial resistance of a column in the absence of moment is given by x.Np,Rd (see Section 8.3.2). Therefore, at the level x = NRd/Npl,Rd, no additional bending moment can be applied to the column. The corresponding value for bending pk of the cross section is therefore the “moment for imperfection” of the column and the influence of this imperfection is assumed to decrease linearly to the value xn. If the design axial force of the column is N,,, then xd =

-.NSd Npl.Rd

The distance p defines the ultimate moment resistance that is still available, having taken account of the influence of second order effects in the column.

xn = x(l-r)/4 IXd, where r the ratio of the lesser to the greater end moment of the column. For simplification it is allowed to consider r = 1 (equal end moments) and xn = 0. This simplification leads to results which are conservative for other values of r. Where xn > 0, refer to EC4 r4.8.3.1.31 for typical values of xn. In any case, if transverse loads occur within the column length, xn = 0. For the verification of the column in combined compression and uniaxial bending, the maximum design bending moment within the column length (Mmax,Sd)shall meet the following condition:

Mmax,Sdis the maximum applied factored moment taking account of second order effects (see EC4 [4.8.3.10].

In Annex 1 of this publication, formulae are given for the determination of the plastic moment resistance of the section, Mpl,Rd. The distance p can be calculated as follows when equivalent reduction factor at point C in Figure 8.4.

Therefore, it must be verified that, for Xd >

Xd

2

xPm,where xPmis

the

xpm:

These formulae may be re-arranged in terms of the axial load that may be resisted for a given applied moment. For the above cases:

8.3.6 Limits of applicability of the simplified design method The composite columns are of double symmetrical and uniform cross section over the column length. The design clauses were based on numerical studies covering a range of steel contribution, 6, varying from 0.2 up to 0.9. As the method has only been checked over this range, designs using the simplified method should have 6 values lying within this range:

0.2 I 6 I 0.9 If 6 is less than 0.2 the column shall be designed according to Eurocode 2 and if 6 exceeds 0.9 according to Eurocode 3. 0

The non-dimensional slenderness

x shall not exceed 2.0. Page 78

e

If the longitudinal reinforcement is considered in design, a minimum percentage of 0.3% of the concrete area must be provided. The maximum percentage of reinforcement at the concrete cross section which can be applied in the analysis is 4%. For reasons of fire protection, greater percentages of the reinforcement can be included, but shall not be taken into account for the "normal design". As 0.3% I I 4.0%

A, e

Concrete filled sections may be fabricated without any reinforcement. For concrete encased sections the minimum reinforcement is given in EC4 [4.8.3.1(3)].

e

Generally for the concrete cover of completely encased profiles a minimum cover of 40mm has to be provided. The maximum cover that may be used in calculations is also restricted, as follows: 40mm I c, I 0.3h

and

b c, 2 (see 8.2.3) 6

40mm I cy I 0.4b

Page 79

9.

FIRE RESISTANCE The behaviour of composite elements in case of fire depends largely on the thickness of steel flanges, position of concrete and reinforcement etc. In order to achieve an optimal design, the requirements from fire design have to be taken into account when carrying out the cold design. For detailed information on fire design, refer to EC4, Part 1.2.

To enable the designer to estimate whether a chosen type of column or beam will have sufficient fire resistance and how it can be improved, the following overview is given. Columns

* ,’

The load resistance of concrete-filled hollow sections in the case of fire is mainly determined by the concrete and the reinforcement. The steel section loses its resistance quite early, but the concrete and reinforcement continue to resist loads. The concrete insulates the reinforcement in the concrete.

In the case of high load levels, additional longitudinal reinforcment is required. Columns of concrete-filled hollow sections reach fire resistance class R 90 when designed for full load utilisation at normal temperatures and up to R 180 when designed for partial load utilisation at normal temperatures. The load resistance of encased sections is determined mainly by the hot-rolled steel section, both in case of normal temperature and fire design. Concrete insulates the steel section and the reinforcement is used largely to prevent spalling of the concrete in fire. Columns of encased sections can reach a fire resistance class R240 even when designed for full load utilisation at normal temperatures. The load resistance of partially encased sections is determined by the reinforcement and the web of the steel section in case of fire. The steel flanges are heated quickly and hence lose load resistance. Columns of partially encased sections reach fire resistance class R60 and up to R120 depending on the detailing and the load utilisation.

Page 80

~

Beams If composite beams are required to have fire resistance, this can be achieved by conventional protection of the steel profile. Fire resistance classes R 30 to R240 are reached by conventional board or spray fire protection. The critical temperature of a composite beam is the same as for an equivalent steel beam. Alternatively, the steel section may be partially encased in concrete, and fire resistance classes of R30 to R120 can be achieved. Connections also have to be fire protected. Connections which are surrounded by concrete reach the same fire resistance class as the adjacent columns and beams.

Slabs

m;

For fire resistance of concrete slabs see EC2, Part 1.2. Fire resistance class R 90 is usually achieved. For the fire resistance of a composite slab see EC4 Part 1.2, European Technical Approvals or Testing Certificates. The fire resistance depends largely on the shape of the profiled decking. Composite slabs using trapezoidal decking in general need additional reinforcement to reach R 90. Composite slabs using Holorib-type decking in general exceed R90 without additional reinforcement in the slab other than that required for distribution of loads and for crack control.

Page 81

10.

CONSTRUCTION AND WORKMANSHIP

10.1 General [9.1] This section gives specific recommendations related to the design of composite structures. However, EC4 does not give a full treatment of all aspects of construction and workmanship. In addition, the relevant clauses of appropriate Parts of Eurocode 2 and Eurocode 3 are applicable to composite structures.

10.2 Sequence of Construction [9.2] The sequence of construction shall be compatible with the design. All information necessary to ensure this compatibility shall be clearly indicated and described on the final drawings and specifications. These shall include instructions for control measurements in the different phases of construction, if appropriate. The speed and sequence of concreting should be required to be such that partly matured concrete is not damaged as a result of limited composite action occurring from removal of props etc. 10.3 Stability [9.3] The stability of the steelwork shall be ensured during construction, particularly before the development of composite action. It shall not be assumed that permanent or temporary formwork (including decking) provides restraint to steel members susceptible to buckling unless it has been demonstrated that the formwork and its fixings are capable of transferring sufficient restraining forces from the supports to the steel member.

10.4

Accuracy during Construction and Quality Control [9.4] 0

Static deflection during and after concreting The edge supports for the slab during concreting should be such that they can follow the deflections of the steel beams during concreting. When unpropped construction is used, measures should be taken to limit the additional thickness of the floor slabs resulting from deflections of the steel beam, unless the extra thickness of concrete is taken into account in the final design.

0

Compaction of concrete Special attention should be paid to the achievement of satisfactory compaction around shear connectors and in concrete-filled steel tubes.

Page 82

0

Shear connection in beams and columns For headed studs the quality of the stud welding shall be checked by visual inspection. Special attention shall be given to the weld collar and the length of the stud. Any studs with defective welding shall be replaced. In addition, a number of studs shall be bent until the head of each stud is displaced laterally from its original position a distance of approximately one quarter of the height of the stud. The stud weld shall not show any signs of cracking. The satisfactory studs shall be left in the bent position.

0

Corrosion protection in the interface Steel parts of composite beams in buildings in general need not be protected against corrosion unless particular corrosion action has to be taken into account. If the steel parts must be protected against corrosion by painting, the painting may also be applied in the interface and to the shear connectors. Where protection from corrosion is required without the interface and shear connectors being fully painted, the protection should extend at least 30mm into the interface.

0

Profiled steel decking as permanent formwork - Fixing of sheets The steel sheets shall be fixed: 0

during laying to keep them in position and to provide a safe working platform; to ensure adequate connection between adjacent sheets and between the sheets and supporting beams;

0

to transmit horizontal forces and shear, where necessary.

10.5 Loads During Construction The values of the construction and storage loads assumed in design of the decking shall be clearly shown on the relevant site plans. Those responsible for controlling work on site shall ensure that these loads are not exceeded.

10.6 Stud Connectors Welded through Profiled Decking Stud connectors may be welded through the steel sheet to the supporting beams under the following conditions: 0

Any paint on the steel beam near the weld should be removed.

0

When the sheet is ungalvanised, the gross thickness should not exceed 1S m m and any corrosion should be minimal.

The overall thickness of a galvanised steel sheet should not exceed 1.25mm. 0

Wet conditions at the time of welding should be avoided. Before welding, the sheet should be in close contact with the steel.

0

Stud connectors should not be welded through more than one thickness of sheet.

0

The diameter, d, of stud connector should not exceed 19mm.

0

Welding trials should be carried out.

Other detailing requirements are given in Section 5.4.1.

Page 84

I

11.

REFERENCES The following references are presented in this publication:

1.

ENV 1992-1-1: Eurocode 2 Design of concrete structures Part 1.1: General rules and rules for building, 1992

2.

ENV 1993-1-1: Eurocode 3 Design of steel structures Part 1.1: General rules and rules for building, 1992

3.

ENV 1994-1-1: Eurocode 4 Design of composite steel and concrete structures Part 1.1: General rules and rules for building, 1992

4.

EN 10 025: Hot rolled products of non-alloy structural steels: Technical delivery conditions, 1990

5.

EN 10 142: Specification for continuously hot-dip zinc coated low carbon steel sheet and strip for cold forming: Technical delivery conditions, 1991

6.

pr EN 10 080: Steel for the reinforcement of concrete - weldable ribbed reinforcing steel BS500: Technical delivery conditions for bars, cribs and welded fabric, 1991

7.

European Convention for Constructional Steelwork Good practice in composite slabs (to be published based on a document prepared by ECCS TC 7.6 in 1993). Other references to the design and construction of composite beams and slabs are not included in this publication. Consult your national steel advisory centre for more information.

Page 85

12.

DESIGN TABLES AND GRAPHS FOR COMPOSITE BEAMS The following Tables and Figures present information to assist designers in selecting the size of steel section to be used as a composite beam for different spans and loads. Figures 12.1 and 12.2 show the benefits of composite action in terms of moment resistance, and second moment of area (stiffness). This information is presented for IPE and HE sections in grade Fe 360 or 510 steel. Certain assumptions have been made regarding the slab depths and width. The Design Tables are presented for certain standard cases using Holorib decking and slab depths typical of current practice. The information is given in terms of the maximum span that can be achieved for a specified imposed load and beam size. Additional information is also presented in the Tables, which are as follows: Table Table Table Table

12.1: 12.2: 12.3: 12.4:

Fe 360 steel, Fe 360 steel, Fe 510 steel, Fe 510 steel,

Holorib deck, 19 mm Holorib deck, 22 mm Holorib deck, 19 mm Holorib deck, 22 mm

diameter diameter diameter diameter

shear connectors shear connectors shear connectors shear connectors

Page 86

0 0 (D

0 0

m

C

0

. I

n

0 0

d-

E E

U

c

0

. I

CI

0

a, CT) 0 0 rn a, a,

-

iij

-

m W I

Y-

0

0 0

c CI

nl Q

5 VI

I

Q, Y 0

VI

0

Y

C

0

. I

5 VI Q)

Y

. I

8

k 8

cr

0 Q,

z

cp

Y

.Y

ML

Y

! E

(cr

4 I

0 0

. I

Y

0 0 F

d

7

tl 2 a

W

-

d

e

0

ci

m

ni

0

ni

0 0 7

0 0

CD

P n

k

Q,

U

E m

r

nQ,

+

5n 0)

U

0 0 U3

0 0

e

0 0

i

rn

cr

0

cr

0

m W

0 0

I

cv

4 I

0 0

-

w

-

L

0

0

6

cd

0 Irj

+

0

0 ccj

0 c)

0

0 0

c\i

7

I

cr

0 0

. I c)

2

~

DESIGN TABLES FOR COMPOSITE BEAMS IN FE 360 STEEL Table 12.1 Holorib deck; 19 nun diameter shear connectors BEAU DATA

SLAB DATA

Internal beam Uniform load Beam spacing Steel strength Shear connectors diameter

Fire resistance Slab depth concrete Strength (cylinder/cube)

W S E D DAD )rN/rna

3 m Fe 360 Welded 19 m

3.5

90 mine 130 mm NW 25/30

6.0

4.5

7.5

~~~

Le

)ESIGNATION ~

:PE

IEA

IEB

LE LA DE DA DS

~

DE D S

LA

DA

LE

DE DS

LA

DA

LE 4.3 4.9 5.4 5.8 6.4 7.0 7.8 8.5 9.3 10.4 11.6 14.4 15.5

a d

DE D S

LA

DA

LE

DB D S

LA

DA

~~

160 180 200 220 240 270 300 330 360 400 450 SUO 550 600

4.3 5.0 5.6 6.3 7.2 8.1 9.0 10.0 11.1 12.5 13.9 14.8 15.8

100 120 140 160 180 200 220 240 260 280 300 320 340 360 400 450 500 550 600

4.2 4.9 5.5 6.2 7.0 7.8 8.6 9.3 10.2 10.9 11.6 12.2 13.4 14.3 15.2 15.9 16.7

g g g g g g g g g g g g

100 120 140 160 180 200 220 240 260 280 300 320 340 360 400 450 500 550 600

4.0 4.7 5.5 6.3 7.1 7.9 8.7 9.6 10.3 11.2 11.9 12.6 13.2 13.9 14.8 15.7 16.5 17.2

g g g g g g g g g g g g g

=

= = = =

a a

a g g g g g g g d

i i

3 15 4 5 6 8 9 11 12 14 16 19 19 18

18 21 24 27 30 33 36 40 44 48 45 42

3 17

4.3 5.0 5.6 6.3 7.2 8.1

9.0 10.0 11.1

a

3

a

4 5 6 8 9 11 12 14 16 19 6 5

a g g g g g g g

12.5 13.9 d 10.6 e 10.6 e

4 20 5 22 6 24 8 26 9 29 10 31 11 34 1 3 36 14 39 15 4 1 16 43 i 1 8 45 i 1 8 42 i 18 40 i 1 8 38 i 1 8 37

4.2 4.9 5.5 6.2 7.0 7.8 8.6 9.3 9.9 10.6 11.3 10.6 10.6 10.6 10.6 10.6 10.6

g g g g g g g g g g g e e

3 4 5 6 8 9 10 12 13 14 16 17 18 18 18 17 17 17

4.0 4.7 5.5 6.1 6.9 7.6 8.4 9.2 10.0 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6

g g g g g g g g g e e e e e e e e e

i i i i

17 19 22 24 27 30 32 34 37 40 42 44 45 43 40 38 37 35

3 4 5

6

8 9 10 12 15 17 18 12 9 e 7 e 5 e 4 e 3 3 4 5 8 10 11 13 14 15 15 13 11 10 8 6 4 4

3

4.3 5.0 5.6 6.2 7.0 7.9 8.7 9.4 10.5 11.6 12.9 14.7 15.7

a a

4 5 7 8 9

g g g d 11 d 12 d 13 d 14 d 16 d 18 i 23 i 23

4.1 4.8 5.4 6.1 6.9 7.6 8.4 9.1 10.0 10.7 11.3 11.9 13.1 14.2 15.1 15.9 16.6

g g g g g g g g g g g g g

3.9 4.6 5.4 6.2 7.0 7.7 8.6 9.3 10.1 10.9 11.6 12.3 12.9 13.8 14.7 15.6 16.4 17.1

g g g g g g g g g g g g g

i i i i

4 5 6 7 9 11 12 13 15 17 18 19 22 23 23 22 22

15 18 21 22 25 27 29 28 32 33 35 44 41

4.3 5.0 5.6 6.2 7.0 7.9 8.6 9.4 10.4 11.6 12.9 10.6 10.6

a

d d d d d d e e

4 5 7 8 9 11 12 12 14 16 18 8 6

4.1 4.8 5.4 6.1 6.9 7.6 8.4 8.8 9.6 10.3 10.9 10.6 10.6 10.6 39 1 0 . 6 37 1 0 . 6 36 1 0 . 6

g g g g g g g g g g g e e e e e e

4 5 6 8 9 11 12 16 18 19 20 16 12 9 7 5 4

4.1 4.7 5.2 5.8 6.4 7.2 8.1 8.8 9.6 10.3 10.9 11.5 12.6 14.0 15.0 15.7 16.5

5

4.5 5.3 6.0 6.8 7.5 8.3 9.0 9.8 10.6 11.2 11.8 12.4 13.5 14.6 15.5 16.2 17.0

16 18 20 22 24 26 26 31 33 35 37 38 41 41

3 16 5 18 6 20 7 23 9 25 1 1 27 12 29 14 3 1 15 3 3 17 36 1 8 38 20 40 21 41 i 22 42 i 22 39 i 22 3 7 i 22 36 i 22 35

.

4.6 5.4 5.9 6.7 7.4 8.2 9.0 9.7 10.6 10.6 10.6 10.6 10.6 10.6 10.6 1O.t 10.6

a g

g g

g g g g g g g g e e

e e e

e e e

e

6 10 11 13 15 16 18 20 17 14 12 10 7 6 5 4

5 15

i

7 8 7 8 9 10 11 12 14 16 29 30

17 18 17 17 17 19 19 19 21 23 40 40

g g d d d

5 6 7 8 9

15 17

d d d d d d d d d c

i i i

g g g g g g g g g g g g g

i i i i

a

6

d

7 8 8 8 9 10 11 12 14 16 11 8

d d d d d d d d d e

e

14 16 18 20 21 22 25 29 29 29 29

25 27 29 30 32 34 36 39 38 36 35

g g d d 6.3 d 7.1 d 7.7 d 8.3 d 9.0 d 9.7 d 10.2 d 10.6 e 10.6 e 10.6 e 10.6 e 10.6 e 10.6 e

6 7 9 11 13 14 16 18 20 22 23 24 27 29 28 28 28

17 19 20 22 24 26 27 29 31 33 34 36 38 38 36 35 34

g g g g g g g g 10.1 g 10.6 e 10.6 e 10.6 e 10.6 e 10.6 e 10.6 e 10.6 e 10.6 e

17 18 It)

d 11 2 1 g g g g g g g g

4.3 4.9 5.4 5.7 6.3 6.9 7.7 8.4 9.3 10.4 11.6 10.6 10.6

4.1 4.7 5.2 5.7

4.5 5.3 5.7 6.4 7.1 7.9 8.6 9.3

5 6 7

4.0 4.4 4.9 5.2 5.7 6.3 7.0 7.6 8.5 9.6 12.2 13.4 14.8

d d

d d d d d d d d c c c

5 5 6 6 7 8 9 10

11 12 24 27

30

4.0 11 4.4 1 2 4.8 10 5 . 1 11 5.7 11 6.3 12 6.9 12 7.6 14 8.5 1 5 9.5 28 10.6 30 10.6 3 3 10.6

d

d d d

11

8 9 11 16 17 19 20 21 21 16 12 9 7 6

4.4 4.8 5.2 6.5 7.2 7.9 8.5 9.3 10.0 10.6 11.2 12.3 13.6 14.8 15.6 16.3

d d d g g g g g g g

6 7 7 12 14 16 18 20 22 23 g 25 g 28 g 32 i 35 i 35 i 34

13 12 11 20 21 22 24 25 27 28 29 32 34 36 35 34

6 8 11 13 15 17 19 20 22 22 19 17 13 10 8 6 5

4.3 5.2 5.9 6.6 7.3 8.1 8.8 9.5 10.2 10.9 11.5 12.1 13.1 14.4 15.3 16.1 16.9

d g g g g g g g

13 17 18 20 22 23 25 26 21 29 30 32 33 36 35 34

g g g g g g

i i

i

6 8 10 12 14 16 18 20 22 24 25 27 30 34 34 34 34

maximum span (m) for shear connectors in every rib maximum span (m) for shear connectors in alternate ribs imposed load deflection (mm) for span LE imposed load deflection (mm) for span LA deflection (mm) of unpropped beam of span LE, due to self weight of slab and beam Page 89

4.3 4.8 5.1 5.7

d

d d d d d d d d d e e

d

6.3 d 6.9 d 7.5 d 8.3 d 8.9 9.4 9.9 10.6 10.6 10.6 10.6 10.6

d d d e e e e e

5 5

6 6 7 8 9

9 11 13 17 13 10

6 6 7 11 12 14 15 17 19 19 20 20 15 11 9 7

4.3 d 6 4.8 d 11 5.4 d 12 6.0 d 1 3 6.7 d 15 7 . 4 d 16 8.0 d 18 8.6 d 19 9.5 d 22 10.1 d 23 10.6 d 24 10.6 e 2 1 10.6 e 16 10.6 e 12 10.6 e 9 10.6 e 8 33 10.6 e 6

Table 12.2 Holorib deck; 22 m m diameter shear connectors BEAM DATA

SLAB DATA

Internal beam Uniform load Beam spacing Steel strength Shear connectors diameter

Fire resistance Slab depth Concrete Strength (cylinder/cube)

Welded

90 mina

130 mm

Nw 25/30

I!

I(

?!POSED

PE

LEA

BB

LE LA DE DA DS

LE

DE DS

160 380 200 220 240 270 300 330 360 400 450 500 550 600

4.3 5.0 5.6 6.3 7.2 8.1 9.0 10.0 11.1 12.5 13.9 14.8 15.8

a a a g g g g g g g

100 120 140 160 180 200 220 240 260 280 300 320 340 360 400 450 500 550 600

4.2 4.9 5.5 6.2 7.0 7.8 8.6 9.3 10.2 10.9 11.6 12.2 13.4 1 4 ~ 15.2 15.9 16.7

g g g g g g g g g g g g

100 120 140 160 180 200 220 240 260 280 300 320 340 360 400 450 500 550 600

4.0 4.7 5.5 6.3 7.1 7.9 8.8 9.6 10.3 11.2 11.9 12.6 13.2 13.9 14.8 15.7 16.5 17.2

= = = = =

LA

DA

3 4 5

12 14 16 i 19 i 19 i 18

15 18 21 24 27 30 33 36 40 44 48 45 42

4.3 5.0 5.6 6.3 7.2 8.1 9.0 10.0 11.1 12.5 13.9 10.5 10.5

3 4 5 6 8 9 10 11 13 14 15 16 18 18 18 18 18

17 20 22 24 26 29 31 34 36 39 41 43 45 42 40 38 37

4.2 4.9 5.5 6.2 7.0 7.8 8.6 9.3 10.2 10.9 10.5 10.5 10.5 10.5 10.5 10.5 10.5

g g g g g g g g g g e e e e e e e

3 4 5 6 8 9 10 12 13 14

3 17 4 19 5 22 6 24 8 27 9 30 10 32 1 2 34 1 3 37 14 40 16 42 17 44 18 45 i 1 8 43 i 18 40 i 17 3 8 i 17 37 i 17 35

4.0 4.7 5.5 6.3 7.1 7.9 8.8 9.6 10.3 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5

g g g g g

3 4 5

i 3i i i i g g g g g g g g g g g g g

3 4 5 6 8 9 11

6.0

4.5

3.5

kN/ma

NES IGNAT ION

a

a a g g g g g g g

6

8 9 11 12 14 16 i 19 e 5 e 4

11

9 7 6 4 3 3

6

8 g 9 g 10 g 12 g 13 e 11 e 10 e 9 e 8 e

6

e

5 4 3 2

e e

e

LE 4.3 5.0 5.6 6.2 7.0 7.9 8.8 9.7 10.8 12.2 13.6 14.7 15.7

DE DS

LA

4 5 7 8 9 11 13 14 16 19 22 23 23

15 18 21 22 25 27 30 32 36 40 43 44 41

4.3 5.0 5.6 6.2 7.0 7.9 8.8 9.7 10.8 12.2 13.6 10.5 10.5

4 5 6 7 9 11 12 13 15 17 18 19 22 i 23 i 23 i 22 i 22

16 18 20 22 24 26 28 31 33 35 37 38 41 41 39 37 36

4.1 4.8 5.4 6.1 6.9 7.6 8.4 9.1 10.0 10.7 10.5 10.5 10.5 10.5 10.5 10.5 10.5

3 16 5 18 6 20 7 23 9 25 11 27 12 29 14 3 1 15 3 3 17 36 18 38 20 4 0 21 41 i 22 42 i 22 39 i 22 37 i 22 36 i 22 35

4.6 5.4 6.2 7.0 7.7 8.6 9.3 10.1 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5

a a

g g g g g g g g g

i i

4.1 4.8 5.4 6.1 6.9 7.6 8.4 9.1 10.0 10.7 11.3 11.9 13.1 14.2 15.1 15.9 16.6

g g g g g g g g g g g g g

3.9 4.6 5.4 6.2 7.0 7.7 8.6 9.3 10.1 10.9 11.6 12.3 12.9 13.8 14.7 15.6 16.4 17.1

g g g g g g g g g g g g g

.

a

a g

g g g g

g g g g e e

g g g g g g g g g g e e e e e e e

g g g g g g g g

DA

LE

4 5 7 8 9 11 13 15 17 20 22 6 5

4.3 4.9 5.5 6.1 6.8 7.7 8.5 9.2 10.0 11.2 12.4 14.4 15.5

a g g g g g d d d d d c

4 5 6 8 9

4.1 4.7 5.3 5.9 6.7 7.4 8.1 8.8 9.6 10.3 10.9 11.5 12.6 14.0 15.0 15.7 16.5

g g g

11

12 14 15 17 14 12 9 7 6

4 4

5 6 7 9 11

12 14 15 e 15 e 13 e 11 e 10 e 8 e 6 e 5 e 4 e 3

4.5 5.3 6.0 6.8 7.5 8.3 9.0 9.8 10.6 11.2 11.8 12.4 13.5 14.6 15.5 16.2 17.0

DE DS

i

5 7 8 9

15 17 19 20 22 24 26 26 26 28 30 40 40

4.3 4.9 5.5 6.1 6.8 7.7 8.5 9.1 10.0 11.1 12.3 10.5 10.5

a g g. g g g d

15 17 18 20 22 23 25 27 29 30 32 34 36 39 38 36 35

4.1 4.7 5.3 5.9 6.7 7.4 8.1 8.8 9.6 10.2 10.5 10.5 10.5 10.5 10.5 10.5 10.5

g g g g g g d d d d e e

6 17

4.5 5.3 6.0 6.8 7.5 8.3 9.0 9.8 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5

g g g g g g g g

11

13 15 15 16 18 20 29 30

5 6 8 9

g g 11 g 13 d 14 d 16 d 18 d 19 g 21 g 22 g 25 g 29 i 29 i 29 i 29

g g g g g g g g g g g

g g

i i i i

7.5

LA

7 9 11 13 14 16 18 20 22 23 24 27 29 28 28 28

19 20 22 24 26 27 29 31 33 34 36 38 38 36 35 34

d d d d e e

e e e e

e

e e e

e e e

e e e

DA

LE

6 7 8 9

4.2 4.7 5.2 5.7 6.4 7.0 7.6 8.2 9.2 10.2 11.2 13.4 14.8

11

13 15 15 16 18 20 8 6

5 6

8 9 11 13 15 16 18 19 18 16 12 9 7 6 5

6

8 9 11

13 15 17 18 20 17 15 13 10 8 6 5 4

4.0 4.6 5.1 5.8 6.3 6.9 7.5 8.0 8.7 9.4 9.9 10.4 12.3 13.6 14.8 15.6 16.3

4.4 5.2 5.8 6.6 7.2 7.9 8.5 9.2 9.9 10.5 11.5 12.1 13.1 14.4 15.3 16.1 16.9

DK DS c

c c c d d d

d d d d c E

g g

6 7 8 10 10 11 12 12 14 16 17 27 30

4.2 4.7 5.2 5.7 6.3 6.9 7.5 8.2 9.1 10.2 11.2 10.5 10.5

c c c c d d d d d d d e

7 8 9 10 10 11 12 12 14 16 17 10

e

8

6 14

4.0 4.6 5.1 5.7 6.2 6.8 7.4 8.0 8.7 9.3 9.8 10.4 10.5 10.5 10.5 10.5 10.5

g g

6 7 9 10 11 12 13 14 15 17 18 19 16 12 9 7 6

15 16 18 17 17 18 18 19 21 22 22 32 34 36 35 34

7 8 10 12 13 14 16 18 20 21 25 27 30 34 34 34 34

15 17

d g g g g

i

L i

DA

14 14 15 16 17 17 16 17 18 20 20 30 33

7 d 8 d 10 d 11 d.12 d 13 d 14 d 15 d 17 d 18 d 19 g 28 g 32 i 35 i 35 i 34

g g d d d d d d d

LA

maximum span (m) for shear connectors in every rib maximum span (m) for shear connectors in alternate ribs imposed load deflection (mm) for span LE imposed load deflection (mm) for span LA deflection (mni) of unpropped beam of span LE, due to self weight of slab and beam

Page 90

18

20 20 21 22 23 24 25 30 32 33 36 35 34 33

4.4 5.2 5.8 6.5 7.1 7.8 8.5 9.1 9.9 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5

d d d d d d d d d d e

e e e e

g g d d d d d d d d

e e

e e e e e

7 8 10 12 13 15 16 18 20 21 18 16 13 10 8 6 5

DESIGN TABLES FOR COMPOSlTE BEAMS IN FE 510 STEEL Table 12.3 Holorib deck; 19 mm diameter shear connectors SLAB DATA

Internal beam Uniform load Beam spacing Steel strength Shear connectors diameter

:HWSED DAD kN/ma bESIGNATION :PE

--LEA

1EB

LE LA DE DA DS

Welded

DE DS

g 3 16 g 4 18 g 5 20 g 6 22 g 6 24 g 8 27 g 9 30 g 11 33 g 12 36 a 14 40 g 16 44 i 19 48 i- 19 45 i 18 42

4.0 4.5 5.1 5.7 6.3 7.2 8.1 9.0 9.6 10.8 10.6 10.6 10.6 10.6

g g g g g g g g g o e e

100 120 140 160 180 200 220 240 260 280 300 320 340 360 400 450 500 550 600

4.2 4.9 5.5 6.2 7.0 7.8 8.6 9.3 10.2 10.9 11.6 12.2 13.4 14.3 15.2 15.9 16.7

g g g

3 4 5 6 8 9 10 11 13 14 15 16 18 18 18 18 18

4.0 4.6 5.3 6.0 6.7 7.5 8.3 9.0 9.9 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6

100 120 140 160 100 200 220 240 260 280 300 320 340 360 400 450 500 550 600

4.0 4.7 5.5 6.3 7.1 7.9 8.7 9.6 10.3 11.2 11.9 12.6 13.2 13.9 14.8 15.7 16.5 17.2

g g

= =

=

g

g g

g g

g g g g

i i i i

g

g g g g g g g g g g

i i i i i

17 20 22 24 26 29 31 34 36 39 41 43 45 42 40 38 37

3 17 4 5 6 8 9 10 12 13 14 16 17 18 18 18 17 17 17

19 22 24 27 30 32 34 37 40 42 44 45 43 40 38 37 35

6.0

4.5 DA

LA

4.0 4.5 5.1 5.7 6.3 7.2 8.1 9.0 10.0 11.1 12.5 13.9 14.8 15.8

=

I1

3.5

Le

90 mins 130 mm NW 25/30

I'

160 180 200 220 240 270 300 330 360 400 450 500 550 600

=

Fire resistance Slab depth Concrete Strength (cylinder/cube)

.

4.5 5.3 6.0 6.8 7.6 8.4 9.2 10.0 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6

3 4 5

DE DS

LE

15 17 11 8 e 6 e 5

3.9 4.4 5.0 5.6 6.2 7.0 7.9 8.8 9.7 10.8 12.2 13.6 14.7 15.7

g 4 g 5 g 6 g 7 g 8 g 9 g 11 g 13 g 14 a 16 g 19 g 22 i 23 i 23

g 5 g 6 g 7 g 8 g 10 g 11 g 12 g 14 g 15 e 16 e 14 e 12 e 9 e 7 e 5 e 4 e 3

4.1 4.8 5.4 6.1 6.9 7.6 8.4 9.1 10.0 10.7 11.3 11.9 13.1 14.2 15.1 15.9 16.6

g

3.9 4.6 5.4 6.2 7.0 7.7 8.6 9.3 10.1 10.9 11.6 12.3 12.9 13.8 14.7 15.6 16.4 17.1

g 3 g 5 g 6 y 7 g 9 g 11 g 12 g 14 g 15 g 17 g 18 g 20 g 21 i 22 i 22 i 22 i 22 i 22

6 6 8 9

16 17 19 21 22 25 27 30 32 36 40 43 44 41

4.4 5.0 5.6 6.2 7.0 7.9 8.4 9.4 0.4 10.6 10.6 10.6 10.6

4 16 5 18 20 22 24 26 28 31 33 35 37 38 41 41 39 37 36

4.5 5.1 5.8 6.5 7.3 8.0 0.7 9.6 10.3 10.6 10.6 10.6 10.6 10.6 10.6 10.6

-1-

g g g

g g g g g

e e e e

e e e e e

11

.

6 7 8 10 11 13 14 15 15 13 11 10 8 6 4 4 3

g

LA

g 6 g 7 g 9 g 11 g 12 g 13 g 15 g 17 g 18 g 19 g 22 i 23 i 23 i 22 i 22

16 18 20 23 25 21 29 31 33 36 38 40 41 42 39 37 36 35

.

.

1.4 5.1 5.9 6.7 7.4 8.2 9.0 9.7 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6

DA

g g g g g

5

6 7 8 9

g ii

g g g e

15 17 19 15 e 11 e 8 e 6

g g

7 8

g g g g 9 g g e e e

10 11 13 14 16 18 19 18 16 12 9 7 5 4

e e e e

g g g g g g g g

e e

e e

7 8 10 12 13 15 16 18 20 17 14 12

e 10 e 7 e 6 e 5 e 4

LE

4.3 4.9 5.5 6.1 6.8 7.7 8.5 9.4 10.5 11.8 13.1 14.5 15.5

DE DS

g g g g g g g g g g g g

6 16 7 17

8 19

9 11 13 15 17 19 22 26 29 i 30

4.1 4.7 5.3 5.9 6.7 7.4 8.1

20 22 24 27 29 31 35 38 41 40

7.5

LA 4.3 4.9 5.5 6.1 6.8 7.1 8.1 9.0 10.0 10.6 10.6 10.6 10.6

DA

g g g g g g g g g e e e e

.

g g g g g g g 8.8 g 9.6 g 10.3 g 10.9 g 11.5 g 12.6 g 14.0 g 15.0 i 15.7 i 16.5 i

5 6 8 9 11 13 14 16 18 20 21 22 25 29 29 29 29

15 17 4.4 g 18 5.0 g 20 5.6 g 22 6.3 g 23 7.1 g 25 7.7 g 27 8.4 g 29 9.2 g 30 9.9 g 32 10.5 g 34 10.6 e 36 10.6 e 39 10.6 e 38 10.6 e 36 10.6 e 35 10.6 e

g g q g g g g g g g g g g

6 7 9 11 13 14 16 18 20 22 23 24 27 29 28 28 28

17 19 20 22 24 26 27 29 31 33 34 36 38

4.6 5.3 6.0 6.8 7.5 8.3 9.0 9.8 10.6 11.2 11.8 12.4 13.5 14.6 15.5 16.2 17.0

i i I

i

30 36 35 34

4.3 5.0 5.7 6.4 7.1 7.9 0.6 9.3 10.1 10.6 10.6 10.6 10.6 10.6 10.6 10.6 10.6

g g g g g g g g g e

e e e

e e e e

6 7 8

9 11 13 17 19 22 19 14 11 8

8 10 11 13 15 17 18 20 22 23 21 16 12 9 7

6

8 10 11 13 15 17 19 20 22 22 19 17 13 10

8 6 5

LE

DE DS

4.3 4.0 5.3 5.9 6.7 7.4 8.3 9.1 10.2 11.4 12.7 14.0 15.4

g 7 15 g 8 16 g 9 17 d 11 18 g 13 20 g 15 22 g 17 24 g 19 25 g 22 28 g 25 31 g 28 34 g 32 36 g 36 39

4.0 4.6 5.2 5.0 6.5 7.2 7.9 8.5 9.3

g 6 14 g 7 15 g 9 17 g 10 16 g 12 20 g 14 21 g 16 22 g 18 24 g 20 25 g 22 27 g 23 28 g 25 29 g 28 32 g 32 34 35 36 i 35 35 i 34 34

10.0 10.6 11.2

12.3 13.6 14.8 15.6 16.3

-

4.4 5.2 5.9 6.6 7.3 8.1 8.8 9.5 10.2 10.9 11.5 12.1 13.1 14.4 15.3 16.1 16.9

7 15 8 17 10 18 12 20 14 22 16 23 18 25 20 26 22 27 24 29 25 30 27 32 30 33 34 36 i 34 35 i 34 34 i 34 33

g g g g g g g g g g g g g g

maximum span (m) for shear connectors in every rib maximum span (m) for shear connectors in alternate ribs imposed load deflection (mm) for span LE imposed load deflection (mm) for span LA deflection (mm) of unpropped beam of span LE, due to self weight of slab and beam Page 91

LA

4.3 4.8 5.3 5.9 6.3 7.1 7.8 8.7 9.7 10.6 10.6 10.6 10.6

.

4.3 4.8 5.4 6.1 6.8 7.5 8.1 8.9 9.5 10.1 10.6 10.6

10.6 10.6 10.6 10.6

4.1 4.8 5.5 6.2 6.9 7.6 8.3 9.0 9.8 10.4 10.6 10.6 10.6 10.6 10.6 10.6 10.6

DA

g

7

g 8 g 9 d 11 g g g g g

e e e e

15 17 19 21 24 24 18 13 10

-

g 9 g 11 I] 13 g 14 9 16 g 18 g 20 g 22 g 24 g 25 e 26 e 20 e 15 e 11 e 9 e 7

g 9 g 11 g 13 g 14 g 17 g 18 g 20 g 22 I] 24 g 26 e 24 e 21 e 16 e 12 e 9 e 8

e

6

Table 12.4 Holorib deck; 22 mm diameter shear connectors SLAB DATA

BEAM DATA 11

Internal beam Uniform load Beam spacing Steel strength Shear connectors diameter

Fire resistance 90 mins Slab depth 130 mm Concrete Nw Strength 25/30 (cylinder/cube)

3 m Fe 510 Welded 22 mm

IMPOSED

LOAD kN/m' DES IGNATIOE IPE

HEA

[EB

LE LA DE DA

DS

3.5 LE

DE DS

LA

DA

160 180 200 220 240 270 300 330 360 400 450 500 550 600

4.0 4.5 5.1 5.7 6.3 7.2 8.1 9.0 10.0 11.1 12.5 13.9 14.8 15.8

g 3 16 4.0 g 3 g 4 18 4.5 g 4 g 5 20 5.1 g 5 g 6 22 5.7 g 6 g 6 24 6.3 g 6 g 8 27 7.2 g 8 g 9 30 8.1 g 9 g 11 33 9.0 g 11 g 12 36 10.0 g 12 g 14 40 11.1 g 14 g 16 44 10.5 e 8 i 19 48 10.5 e 6 i 19 45 10.5 e 5 i 18 42 10.5 e 4

100 120 140 160 180 200 220 240 260 280 300 320 340 360 400 450 500 550 600

4.2 4.9 5.5 6.2 7.0 7.8 8.6 9.3 10.2 10.9 11.6 12.2 13.4 14:3 15.2 15.9 16.7

g g g g g

100 120 140 160 180 200 220 240 260 280 300 320 340 360 400 450 500 . 550 600

4.0 4.7 5.5 6.3 7.1 7.9 8.7 9.6 10.3 11.2 11.9 12.6 13.2 13.9 14.8 15.7 16.5 17.2

g g g g g g g g g g g g g

= = =

= =

g g g g g g g

i i i i i

i i i

i i

I

LE

DE DS

DA

LE

5 6 7 8 9 11 13 15 17 11 8

g

5

4.3 4.9 5.5 6.1 6.8 7.7 8.5 9.4 10.5 11.8 13.1 14.5 15.5

g 4 g 5 g 6 g 8 g 9 g 11 g 12 g 14 g 15 e 16 e 14 e 12 e 9 e 7 e 6 e 4 e 4

4.1 4.7 5.3 5.9 6.7 7.4 8.1 8.8 9.6 10.3 10.9 11.5 12.6 14.0 15.0 15.7 16.5

g g g g g g g g g

5 6 7 9 11 12 14 15 15 13 11 10 8 6 5 4 3

4.5 5.3 6.0 6.8 7.5 8.3 9.0 9.8 10.6 11.2 11.8 12.4 13.5 14.6 15.5 16.2 17.0

g g g g g g g g g g g g g

LA

3.9 g 4.4 g 5.0 g 5.6 g 6.2 g 7.0 g 7.9 g 8.8 g 9.7 g 10.8 g 12.2 g 13.6 g 14.7 i 15.7 i

4 5 6 7 8 9 11 13 14 16 19 22 23 23

16 17 19 21 22 25 27 30 32 36 40 43 44 41

4.4 5.0 5.6 6.2 7.0 7.9 8.8 9.7 10.8 10.5 10.5 10.5 10.5

g

4 5 6 7 9 11 12 13 15 17 18 19 22 23 23 22 22

16 18 20 22 24 26 28 31 33 35 37 38 41 41 39 37 36

4.1 4.8 5.4 6.1 6.9 7.6 8.4 9.1 10.0 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5

3 5 6 7 9 11 12 14 15 17 18 20 21 22 22 22 22 22

16 18 20 23 25 27 29 31 33 36 38 40 41 42 39 37 36 35

4.6 5.4 6.2 7.0 7.7 8.6 9.3 10.1 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5

3 4 5 6 8 9 10 11 13 14 15 16 18 18 18 18 18

17 20 22 24 26 29 31 34 36 39 41 43 45 42 40 38 37

4.2 4.9 5.5 6.2 7.0 7.8 8.6 9.3 10.2 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5

g g g g g g g g g e e e e e

3 4 5 6 8 9 10 12 13 12 11 9 7 6 e 4 e 3 e 3

4.1 4.8 5.4 6.1 6.9 7.6 8.4 9.1 10.0 10.7 11.3 11.9 13.1 14.2 15.1 15.9 16.6

g g g g g g g g g g g g g

3 4 5 6 8 9 10 12 13 14 16 17 18 18 18 17 17 17

17 19 22 24 27 30 32 34 37 40 42 44 45 43 40 38 37 35

4.0 4.7 5.5 6.3 7.1 7.9 8.7 9.6 10.3 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5

g 3 g 4 g 5 g 6 g 8 g 9 g 10 g 12 g 13 e 11 e 10 e 9 e 8 e 6 e 5 e 4 e 3 e 2

3.9 4.6 5.4 6.2 7.0 7.7 8.6 9.3 10.1 10.9 11.6 12.3 12.9 13.8 14.7 15.6 16.4 17.1

g g g

i i i i

g

g g g g g g g g g

i i i i i

7.5

6.0

4.5

.

g g g g g g g g

e e e e

g g g g g g g g

e e e e e e e

e e

6

g i

g g g g g

i i i

i i i i

LB

g g g g g g g g g e e e e

6 7 8 9 11 13 15 17 19 14 11 8 6

4.3 4.8 5.3 5.9 6.7 7.4 8.3 9.1 10.2 11.4 12.7 14.0 15.4

g

4.1 4.7 5.3 5.9 6.7 7.4 8.1 8.8 9.6 10.3 10.5 10.5 10.5 10.5 10.5 10.5 10.5

g g g g g g g g g g e e e e e e e

5 6 8 9 11 13 15 16 18 20 18 16 12 9 7 6 5

4.0 4.6 5.2 5.8 6.5 7.2 7.9 8.5 9.3 10.0 10.6 11.2 12.3 13.6 14.8 15.6 16.3

4.5 5.3 6.0 6.8 7.5 8.3 9.0 9.8 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5

g g g g g g g g e e e e e e e e

6 8 9 11 13 15 17 18

4.4 5.2 5.9 6.6 7.3 8.1 8.8 9.5 10.2 10.9 11.5 12.1 13.1 14.4 15.3 16.1 16.9

g g g g g g g g g g g g g g

LA

6 16 17 19 20 22 13 24 15 27 17 29 19 31 22 35 26 38 29 41 30 40

4.3 4.9 5.5 6.1 6.8 7.7 8.5 9.4 10.5 10.5 10.5 10.5 10.5

5 6 8 9 11 13 14 16 18 20 21 22 25 29 29 29 29

15 17 18 20 22 23 25 27 29 30 32 34 36 39 38 36 35

6 7 9 11 13 14 16 18 20 22 23 24 27 29 28 28 28

17 19 20 22 24 26 27 29 31 33 34 36 38 38 36 35 34

g 7 g 8 g 9 g 11 g g g g g g

DA

DE DS

e

20 17 15 13 10

8 6 5 4

DE DS

7 15

LA

DA

4.3 4.8 5.3 5.9 6.7 7.4 8.3 9.1 10.2 10.5 10.5 10.5 10.5

g

g g g g g g g g g g g g g g

6 7 9 10 12 14 16 18 20 22 23 25 28 32 i 35 i 35 i 34

14 4.0 .I5 4.6 17 5.2 18 5.8 20 6.5 21 7.2 22 7.9 24 8.5 25 9.3 27 10.0 28 10.5 29 10.5 32 10.5 34 10.5 36 10.5 35 10.5 34 10.5

g g g g g g g g g g

7 8 10 12 14 16 18 20 22 24 25 27 30 34 i 34 i 34 i 34

15 17 18 20 22 23 25 26 27 29 30 32 33 36 35 34 33

g 8 16 g 9 17 g 11 18 g 13 20 g g g g g g g g

22 24 25 28 31 34 36 39

15 17 19 22 25 28 32 36

maximum span (in) for shear connectors in every rib maximum span (in) for shear connectors in alternate ribs imposed load deflection (mm) for span LE imposed load deflection (mm) for span LA deflection (mm) of unpropped beam of span LE, due to self weight of slab and beam

Page 92

4.4 5.2 5.9 6.6 7.3 8.1 8.8 9.5 10.2 10.5 10.5 10.5 10.5 10.5 10.5 10.5 10.5

7

g 8 g 9 g 11 13 15 17 19 22 18 e 14 e 10 e 8 g g g g g e

e e

e e e e e

g 9

g g g g g g g e e e e e e e e

6 7 9 11 13 15 17 18 20 22 23 20 16 12 9 7 6

7 8 10 13 15 16 18 20 23 21 18 16 13 10 8 6 5

ADDITIONAL NOTES ON TABLES Internal Beam The composite beam supports a composite slab with equally spaced adjacent beams.

Deck The steel deck is the re-entrant type, Holorib, which is 50mm deep and 150 mm rib spacing.

Imposed Load The designer should include an allowance for partitions in the imposed load.

Shear Connectors Welded stud (19 mm dia. x 95 mm or 22 mm dia. x 95 mm) are included. Studs are placed in every rib (at 150 mm spacing) or in alternate ribs (at 300 mm spacing).

Fire Resistance The slab depth is selected to achieve a certain fire resistance. Check with EC4 Part 1.2. The results are conservative for deeper slabs. Failure Criteria a

moment resistance of the beam exceeded in the construction stage

b

shear stress in composite beam exceeds 0.5 V,,,,, - the design continues by reducing the moment resistance of the section

C

moment resistance of the composite beam exceeded

d

moment resistance of the composite beam with partial shear connection exceeded

e

limit on degree of shear connection not satisfied (see Section 6.3.4.2)

f

serviceability stress in steel exceeded (not used in EC4)

g

total deflection exceeded (taken as L/200 for floors with suspended ceilings)

h

imposed load deflection exceeded (L/350 for floors supporting brittle partitions)

1

natural frequency < 4 Hz

NB: For some cases, the beam span corresponding to criterion e is output. This occurs in cases where R, < R, and the minimum degree of shear connection is not achieved.

Page 93

nu II n

E E

L

5

0

U W

1 E

-

0

U

P L 0 .I

0

e

&

LL

0 U

a

W U aa & aa 0 U 0

W E

ro U E

aa aa s

U

0 w .4 E

..

r(

-

aa

Y

E M

=a

4

aa

I

a

a U

0

0

U 4

aJ U v)

.4 U

U

cn

ro

4

0

a

va aJ

Q)

k

0 0

0

U

ro

a

E

aa

a 0

U

4

a

U

aa U

a

W 0

-4

k U 0 U 0

m

4 4

m E 0

aa

U U

sry 0

a n

U

c (

& U 0

I

-va 4

--

& U

E

ro

s

e

-

a

M m

m 4

-

U 0

W aa

U aa

v)

3

aa n

a U

U

U ro

aa U &

-

ry 0

E

0

-4 U U m a l

aa

U

w

0

U

ro &

a

U 0 & U 0

Ln 0

m 4

r

a

2% w

II

a

a

U

a

-

3

m

a

s

* ro

a

s

m

a

3

0 V

n

.. 1

zl U

N \ U

*

VI

0

E

3

n

0

m

r”

a

3

al

a

I

I

U

Q

a

0 -4

3 I

-m 4

I

Q w

N

N O

4

a

U

n

-

II U

a

3

Q

0

a

II

4

N

a

II

II

0

a

n

a

3

rp

-

0

a

I

N O

U

0 U

m

a

.U

U .E

0

3

a .

4

II II

N O

N O

3

U

n

% 3

W

a n a CI

D9

m v)

5 s E

I

t

a

m

a 2

a U 0 -4 U .4

c al

a

w

*

n

w

U I

N \

a VI

0

m

Q al

Q

6

&

c

9 al

al

S a Q

a

0

n

II U

a

3r

m c1 U a

0

c

a

al .a

N

0)

*

0

zs 0

n

r-2

4 -

n

4

U

II

.4

II

al

0

m

a

n

3

c

alas

ad

UP

n

.Z

m a ual -4a -4

* U

*

0

4 +

3 I

w

N O

n

a

=I

N

U

W N

m

n

-4

-4

m I

n

oi

n 3 -

w

-N 4

a Q

I

4

4

U

w

m

4) w

N

e

-U 0

n

U

-m 4

-N 4

al

-4

a

I

U

a

L

0 -4

aa a

a# c

w

fl, a

3

W

U

0)

I

-4

a

c

0

m

I 0 w &

m c

a I

P U U

4 U

N

n 11

U

a

3

-.

c

cn

rn

a

z

Q

a en 0

0 -M 4

0 a3

0

a

0

-4

-4

&i U

M a3

0

A

ql

a3

a U

F

-4

a# a c)

El er, Q

n

0 M

V

P4

ra

PI

a3

I

0 a

0 Q

-4

>

z

3 " I

w

W

4

O

M

a

N E

a

n

0)

re)

a

n

II

I1

0 Q

3 -

a

U

r

* Q

R

0

U1 -4

U

a

w

3

Q

-c. 0

Q

a

*

P

g s - N

U I N

\

a CI

V 0 Q

a 9

E

.n

0

U 4

U

0 . 4

M a3

U 4

0

a

4

U .4 U

a3 a3

c) M

v)

n

a

a3

M

* a

U a

a3 P 4

E 0

M-

4

U N

.r(

a\ o v a

iz

.4

v)

P4

z

- 4 VI

w

0 4

0

a Q a

a

- V I

x

U 8-4 N

1-

0

a a n

er,

z-

I1

n

0

U

0

U

ra

U

0 Q

Q

ra

a .r( 0

>

+tt L2t"7I

0 -0 4 U

z n en

5 3

pz 0

-

a

a3

A

* Q

s

U

W

Z w m

zcn

a3 M

4 0

n

4

0

n

L

h

a3

A .p

z:

1 c. U -4

'

I

U

E i

aa a

COMPOSITE BEAMS AND COLUMNS TO EUROCODE 4

PART 2: WORKED EXAMPLES The following Worked Examples are intended to assist in understanding the requirements of Eurocode 4 Part 1.1. Worked Examples 1 and 2 refer to simply supported composite beams, and Worked Example 3 to a continuous composite beam of the same span. These Examples follow the same format which is given in the Index before each Example. Pages are numbered separately for each example. Worked Example 4 refers to an encased I section column subject to combined bending and axial force. Worked Example No. 1:

Simply supported composite beam with solid slab and full shear connection.

(11 pages)

Worked Example No. 2:

Simply supported composite beam with composite slab and partial shear connection. ( 14 pages)

Worked Example No. 3:

Continuous composite beam with solid slab

(2 1 pages)

Worked Example No. 4:

Composite column with end moments

(10 pages)

Page 97

WORKED EXAMPLE no.1

This worked example refers to a simply supported composite beam considered as a floor beam in a building. The span of the beam is equal to 12m with a spacing of 3.33m. The variable load has been considered equal to 3.0 l W m 2 and the load for interior finishings equal to 1 .O kN/mZ. The beam is verified as propped until the concrete has hardened and full shear connection has been assumed between steel and concrete. For simplicity the contribution of the longitudinal reinforcement has been neglected. The effect of creep has been taken into account for deflection checks (serviceability limit states), while those due to temperature and shrinkage have been ignored. The structural solution uses solid slab, rolled steel beam and headed stud connectors.

12000

I

1

Wwdmd atud connectors dbm.19 mm h-100 mm

concrete C25/30

IPE 450 Fe 360

3333

I

3333

INDEX 1.

COMPOSITE BEAM CHARACTERISTICS

........... Page 3

2.

VALUES OF ACTIONS

...........

3

3.

LOAD COMBINATIONS

...........

3

3.1 Ultimate limit states 3.2 Serviceability limit states

........... ...........

3 3

MATERIALS

...........

4

4.1 Concrete 4.2 Reinforcing steel 4.3 Structural steel 4.4 Shear connectors 4.5 Partial safety factors

........... ........... ........... ........... ...........

4 4 4 4

5.

DESIGN MOMENT AND SHEAR

...........

5

6.

ULTIMATE LJMIT STATE VERIFICATION

...........

5

6.1 Cross-section properties ........... 6.2 Plastic resistance moment of a section with full shear connection ........... 6.3 Vertical shear ........... 6.4 Bending and vertical shear ...........

5 6 7 7

SHEAR CONNECTORS

...........

7

7.1 Longitudinal shear force 7.2 Design resistance of shear connectors 7.3 Number of shear connectors

...........

7 8 8

4.

7.

........... ...........

5

8.

TRANSVERSE REINFORCEMENT

9.

SERVICEABILITY LIMIT STATE VERIFICATION

10

9.1 Calculation of maximum deflection

10

esaniple 1

8

...........

Page 2

1.

COMPOSITE BEAM CHARACTERISTICS Span: L = 12000mm Structural scheme : simply supported beam Beam spacing : b = 3333 mm Construction type : propped

2.

VALUES OF ACTIONS Characteristic values

permanent loads

variable loads

accidental loads

g'k

= 3.00.3.33

= 10.00 kN/m normal weight concrete

gI1k

-

= 0.80 kN/m steel beam

g"'k

=

qlk

= 2.00.3.33

qllk

=

ak

=O

3.

LOAD COMBINATIONS

3.1

Ultimate limit states

1.00.3.33

1.00.3.33

= 3.33 kN/m interior finishings

= 6.67 kN/m imposed load =

3.33 kN/m partitions

[Table 3.1J

YG'~Gk,jfyQ'Qk,1

considering only the most unfavourable action

yG .CGk,j +O. 9*yQ*CQk j

considering all unfavourable variable actions

whichever is larger. Partial safety factors: 1.35 yy 7 1.50

yG =

3.2

Serviceability limit states CGk,j

esample 1

+ 1.0 Qk.1

[Table 3.21

considering only the most unfavourable variable action

Page 3

ZGk,j + 0.9CQk,i

considering all unfavorable variable actions

4.

MATERIALS

~4.21

4.1

Concrete

[4.2.1]

Concrete strength class : C25/30 Characteristic strength : fck= 25 N/mm2 6 Shrinkage : E,, = 325.10Secant modulus of elasticity of short-term ,dading :

E,,

= 30.5 kN/mm2

[7.2.21

modular ratios :

n =E,&,

= 210/30.5 = 6.88

n = E,/(Ec,/3) = 20.65 4.2

(short term effects) (long term effects) [4.2.2]

Reinforcing steel Type of steel : S 420 Steel grade : fsk= 420 N / m z Modulus of elasticity : E, = 210 kN/mm2

4.3

[4.2.3]

Structural steel Nominal steel grade

: Fe 360

Nominal yield strength : f, = 235 N / m 2

(t 5 40 mm)

Modulus of elasticity : E, = 2 10 kN/mm2 4.4

Shear connectors Type : headed studs 19 mm diameter Ultimate tensile strength : f, = 450 N / m 2

esample 1

Page 4

4.5

Partial safety factors yM

M.31

4.5.1 Ultimate limit states Structural steel

ya = 1.10

Concrete

yc = 1.50

Steel reinforcement

ys = 1.15

Shear connectors

yv = 1.25

4.5.2 Serviceability limit states yM

5.

= 1.0

DESIGN MOMENT AND SHEAR

-A-@& 1

qd 12000

= 34.08 kN/m

1

imposed load qlk+qtIk= qk

yG.&+yQ*(q'k+q''k) = 1.35~(10.00+0.80+3.33)+1.50~(6.67+3.33) = = 34.08 kN/m At mid5pan section (maximum bending moment)

MSd = qd.L2/8= 34.08.12?/8= 613.4 kNin At supports VSd

= qd.L/2 = 34.08.12/2 = 204.4 kN

6.

ULTIMATE LIMIT STATE VERIFICATION

6.1

Cross-section properties Effective width of concrete flange: beff= 2.12000/8 = 3000 inin

example 1

(< 3333tnm)

l6.1

[6.1.3]

[6.1.4]

Classification of cross-section

A cross section is classified according to the least favourable class of its steel elements in compression.

IPE 450

1, = 450.0 intn d = 378.8m1n b = 190.0 tntn tf = 14.6 tntn t, - 9.4 rntn r = 21.0 tntn A, = 9880min2 I%, = 33740.104mtn4 We,,, = 1500.103tnm3 W,,,, = 1702.103tnm3 i, = 185 inin i, = 41.2 Inin

Upper flange Class 1 when acting compositely because tlie flange is restrained from buckling by attachment of sliear connectors. (see EC4, 16.4.1S(2)J 6.2

Plastic resistance moment of a section with full shear connection [6.2.1-21

From equ il i briuiii of longitudinal forces: 0.85.f,, *bef f ~ h , / ~ ,

= (0.85~25~3000~120/1.5)~10'3 = = 5100 kN (concrete slab)

f,*A,/y, = (235*9880/1.10)-10'3 = 2111 kN (steel beam) Tlie plastic neutral axis lies in the slab; in tlus case tlie depth (z,) of the neutral axis is given by:

z, = (fy.A,/y,)/(0.85.fC,.b,,,/y,)= 49.7 tnin h

M example I

Page 6

The plastic moment of resistance of the composite beam Mpl,Rdis given by: = (f ,,*Aa/ya) (h/2+1ic-2,/2) =

Mpl.Rd

= (235~9880/1.10)~(0.45/2+0.12-0.0497/2)~10” = = 675.8 kNrn > 613.4 kNm and the verification is satisfied.

6.3

[6.2.4]

Vertical shear Vpl.Rd

= A;(f,/

d3)/y, = [4399.2-(235/d3)/1.101-10”= 542.6 kN

where

A, = 1.04.ha.t,\,= 4399.2 inm2 v,l,Rd

= 542.6 k N > V S d = 204.4 kN

Shear buckling d/t,,, = 40.3 < 6 9 =~ 69 and therefore the steel web is verified. 6.4

Bending and vertical shear

L6.2.51

As the beam is subject to unifonn load it is not necessary to check for coincident moment and shear. 7.

SHEAR CONNECTORS

7.1

Longitudinal shear force

[6.3.41

Full shear connection

v, = Fcf where

F,f = A;fS/Ya = (9880.235/1.1)*105= 2111 kN or, neglecting the tenn related to longitudinal reinforcement, as usual

Fcf= 0.85.A;fC~/yc = (0.85*3000-120~25/1.5)-103= 5100 kN wh icliever is stna 11er.

example 1

Page 7

7.2

Design resistance of shear connectors

i6.31

The design resistance of a stud is given by: a) PRd = 0.8.f,,.(n.d2/4)

/v,

or b) P,d = 0.29.a.ds.~(f,,.E,,~) /y,

with a = 1 for h/d > 4

whichever is smaller.

Try 19 mm diam. studs 100 mm high a) PRd = [0.8*450.(~.192/4)/1.25]*10~3 = 81.7 kN b) P,,

= [0.29.1.19”..\1(25.30500)/1.25]*10”= 73.1 k N c 81.7 kN

tf = 14.6 > 0.4-d = 7.6 inin 7.3

[5.4.1]

Number of shear connectors

required : 11= 2*FCf/PRd = 2-2111/73.1 = 57.7 coiuiectors per total length provided : n = 2.29 = 58 headed studs o 19 mm Mapl.Rd

Mpl,Rd

(steel beam) = (235~17O2~1O3/1.1)~l0” = 363.6 kNin = 675.8 kNin

Mpl.Rd~apl.Kd=

2.5

and therefore the studs inay be spaced utufonnly over the entire beam length. spacing = 12000/58 = 200 inin > 5.d = 5-19 = 95 tntn

c 6-h, = 6.120 = 720 tnm 8.

TRANSVERSE REINFORCEMENT

[6.3.51

Minimum transverse reinforcement for solid slab

A,> 0.002.A, = 0.002~120~1000 = 240 intn’/tn reinf. bar 4 8 mm / 200 mm - 2 layers A, = A,+A, = 2.251.3 mm’/tn

example I

Page 8

Longitudinal shear in the slab 0

I

At

I

I

Section a-a

where TRd

7 0.30 N/IYUII'

q=l

(C25/30) (normal weight concrete)

4"= 120.1000 = 120-10; Vpd = 0

mm'/m

(contribution of the steel sheeting)

and therefore =273.6 kN/m a) VRd = (2.5~120~10'*1~0.30+2*251.3~420/1.15)~10~3

b) vRJ = (0.2~120~10'~1~~25/1.5)~10~~= 400 kN/m The longitudinal design shear is given by

vsJ = 73.1.1000/207 = 353 kN/m For each shear plane a-a : vSd/2 = 353.1/2 = 177 273.6 k b h and the verification is satisfied. e.mple 1

1 Page Y

Section b-b A,, = (2.95+30)-1000= 220-103tnm2/m VRd

= (2.5~220~1O3~1~0.3O+2~251.3~420/1.15)~1O3 = 348.6 kN/m

VRd

= 348.6 kN/tn

2 VSd

= 353 kN/tn

and the verification is satisfied. 9.

SERVICEABILITY LIMIT STATE VERIFICATION

P.1

9.1

Calculation of maximum deflection

P.21

In buildings it is nonnally satisfactory to consider the deflection wider the frequent combination of loading. hi this example it is assumed that all load is of long duration. n = 20.65

be& = 3000/20.65 =145.2 intn

b e f f / n =145.2 mm

E

E

0

2 nu

c

A inin?

1 2

Z

inn

17433.4 285 9880.0 27313.4

example 1

AT

inin3

4968.5.103

ez - z 1'" A.(e, - z)* I" tntn tnm4 tnm4 tnn4 -103.0 2092.0.104 18528.2*104 20620.2*104 181.9 33740.0-104 32693.3~10~ 66433.3.104 87O53.6*1OJ

Page 10

e, = 4968.5.103/27313.4= 181.9 mm qtot= 10.00+0.80+3.33+6.67+3.33= 24.13 kN/m 4nax

= 5.qt0,*L4/(384-Ea*I) =

= 5-24.13*1o3~12O0O4/(384~2 10.87053.6.10')= = 35.6 mm = L/337 < L/250 recommended limit value

for floors

[Table 7.21

Calculation of deflection due to shrinkage is not required as E,, does not exceed 400.1 04.The following calculation of the additional deflection due to shrinkage is given only to demonstrate the calculation method if required.

For simplification the long term value of elastic modulus of concrete is used for shnkage deflection calculation

N, = (E,,/3).b.h,*~,, = = (30.5/3)-3333*120*0.325-10-' = 1321.5 kN

e,

= z-e, = 285-181.9 =

103.1 mm

6, = N.e,-L2/(8.Ea.I) = = 13.4 IIMI = 132 1.5.103.1.120002/(8210*87053.6*10')

WORKED EXAMPLE no.2

This worked example refers to a simply supported composite beam considered as a floor beam in a building. The span of the beam is equal to 12 m with a spacing of 3.33 m. The variable load has been considered equal to 3 . 0 W/m2 and the load for interior finishings equal to 1.0 kN/m2. The beam is unpropped during construction and is designed for partial shear connection between the steel and concrete. For simplicity the contribution of the longitudinal reinforcement has been neglected. The effect of creep has been taken into account for deflection checks (serviceability limit states), while those due to temperature and shrinkage have been ignored. The structural solution uses a composite slab (sheeting with ribs transverse to the supporting beam), rolled steel beam and headed stud connectors. The steel beam is also checked for its ability to support the loads during construction.

Headed itud connector8 diam.19 mm h=lW mm (95 mm) concrete CzS/lO

esampli. 2

w

‘3

P

Page 1

INDEX 1.

COMPOSITE BEAM CHARACTERISTICS

...........

Page 3

2.

DESIGN LOADS

...........

3

3.

LOAD COMBINATIONS

...........

3

3 . 1 Ultimate limit states 3.2 Serviceability limit states

........... ...........

3 3

MATERIALS

...........

4

...........

...........

4 4 4 4 4 5

DESIGN MOMENT AND SHEAR

...........

5

5 . 1 Construction stage 5.2 Composite stage

........... ...........

5 6

ULTIMATE LIMIT STATE VERIFICATION

...........

6

6.1 Construction stage 6.2 Composite stage

........... ...........

6 8

S H E A R CONNECTORS

...........

9

7.1 Longitudinal shear force 7.2 Design resistance of shear connectors 7.3 Number of shear connectors

...........

9

........... ...........

10 11

4.

4.1 Concrete 4.2 Reinforcing steel 4.3 Structural steel 4.4 Profiled steel sheeting 4.5 Connecting devices 4.6 Partial safety factors 5.

6.

7.

........... ........... ...........

...........

8.

TRANSVERSE RENORCEMENT

12

9.

SERVICEAEHLITY LIMIT STATE VERIFICATION

13

9.1 Calculation of maximum deflection

13

esample 2

...........

Page 2

1

1.

COMPOSITE BEAM CHARACTERISTICS Span : L = 12000 mm Structural scheme : simply supported beam Beam spacing : b =3333mm Construction type : unpropped

2.

DESIGN LOADS Characteristic values

permanent loads

g'k

= 2.40.3.33

= 8.00 kN/m normal weight concrete

gllk

= 0.15.3.33

= 0.50 kN/m profiled steel sheeting

gtllk

-

= 0.80 kN/m steel beam

g1I1lk

=

1.00.3.333 = 3.33 kN/m interior finishings

qlk

=

1.00.3.333

qltk

= 2.00.3.333 = 6.67 kN/m

q"lk

= 0.75.3.333 = 2.50 kN/m

ak

=O

variable loads

accidental loads

3.

LOAD COMBINATIONS

3.1

Ultimate limit states

YG 'CGk,j +rQ.Qk,

= 3.33 kN/m

partitions imposed load construction load

[Table 3.13 considering only the most unfavourable action

1

,

yG.ZGk,j +0.9yq.CQk

considering all unfavourable variable actions

whichever is larger. Partial safety factors: yG = 1.35

yy =. 1.50

3.2

Serviceability limit states

[Table 3.21

CGk,j + 1.0 Qk.1

considering only the most unfavourable variable action

ZGk,j + 0.9 ZQk,i

considering all unfavorable variable actions

esampie 2

Page 3

4.

MATERIALS

14.21

4.1

Concrete

[4.2.1]

Concrete strength class : C25/30 Characteristic strength : fck

= 25

N/mZ

Shrinkage : E,, = 3 2 5 ~ 1 0 ~ Secant modulus of elasticity of short-term loading :

E,,

= 30.5 kN/m2

[7.2.21

modular ratios :

4.2

n = E,/Ecm = 210/30.5 = 6.88

(short term effects)

n = E,/(Ec,/3) = 20.65

(long term effects) [4.2.2]

Reinforcing steel

Type of steel : S 420 Steel grade : fsk= 420 N/mm’ Modulus of elasticity : E, = 2 10 kN/mm2 4.3

[4.2.3]

Structural steel

Nominal steel grade : Fe 360 Nominal yield strength : fy = 235 N/mm’ Modulus of elasticity : E, 4.4

= 210

(t 5 40 mm)

kN/mm2

Profiled steei sheeting

Nominal steel grade : Fe 360 Characteristic strength : fyb= fyp= 235 N / m ’ Modulus of elasticity : E, = 2 10 kN/mm2 4.5

Shear connectors

Type : headed studs 19 mm diameter Ultimate tensile strength : f,, = 450 N/mm2

[4.2.4]

Yield strength : fy = 350 N / m Z 4.6

Partial safety factors yM

4.6.1

Ultimate limit states

Structural steel

ya = 1.10

Profiled steel sheetingyap = 1.10 Concrete

yc = 1.50

Steel reinforcement ys = 1.15 Shear connectors

yv = 1.25

Serviceability limit states

4.6.2

yM = 1.0

5.

DESIGN MOMENT AND SHEAR

5.1

Construction stage

Loading

At Iliidspan section (masimum bending moment)

M s ~= 16.3.12V8 = 293.5 kNm At supports

Vsd = 16.3*12/2= 97.8 kN e.uample 2

Page 5

5.2

Composite stage

yG.zgk+Yq'(q'k+q"k) = 1.3 5'( 8.00+0.50+0.80+3.3 3)+ 1.5043.33+6.67)= = 32.05kN/m

At midspan section (maximum bending moment) MSd = 32.05*122/8 = 576.9WlIl

At supports v,d = 32.05.12/2 = 192.3 kN

6.

UL"l3lATE LIMIT STATE VERIFICATION

6.1

Construction stage

6.1.1

Cross-section properties Cross section classification

L6.1

[6.1.4]

A cross section is classified according to the least favourable class of its steel elements in compression. z

*%=

= 450.0mm

d =378.8mm b = 190.0mm tf = 14.6mm &= 9.4m r = 21.Omm

A, = 9880mm2 1a.y

L-l IPE 450

e.uample 2

= 33740.10' mJ = 1500.10' mm'

Wpl.y= 1702.10' mm3 $ = 185mm 4 = 41.2mm

Page 6

Umer flange (b/2)/tf = 95114.6 = 6.5 1 E = 1 (Fe360) 6.51 < 1 0 =~ 10 and therefore the flange is in Class 1

d/t,

= 378.819.4 = 40.3 1 (Fe 360) 40.3 < 7 2 =~ 72 E=

and the requirements for Class 1 cross-section are statisfied. 6.1.2 Plastic resistance moment of the section = f,,*Wpl/y, = (23 5.1702.1 03/ 1.10). 1Oa =

1. R d

= 363.6 kNm > Ms, = 293.5 kNm

and the verification is satisfied. 6.1.3 Vertical shear

where A, = 1 .04*ha.t,,,= 4399.2 mm’

Shear buckling d/t,

= 40.3

< 6 9 =~ 69

and therefore the steel web is verified. 6.1.4 Bending and vertical shear

As the beam is subject to uniform load it is not necessary to check for coincident moment and shear.

example 2

Page 7

6.1.5 Lateral-torsional buckling of the steel beam

It is assumed that the steel beam is laterally restrained by the steel sheeting during construction. In order to provide restraint, the sheeting is fixed to the beam either by the action of through-deck welding, or by shot-fired pins.

6.2

Composite stage [6.1.2]

6.2.1 Cross-section properties Effective width of concrete flange:

[6.1.3]

Cross section classification Upoer flange Class 1 from calculation in 6.1.1

6.2.2 Plastic resistance moment of the section with full shear connection C6.2.1-21 From equilibrium of longitudinal forces: 0.85 fc b, f f ' h,/y, *

*

= (0.85.25*~000.65/1.5)*10"= = 2762.5 kN

(concrete slab)

f,.AJ-ya = 235.9880/1.10 = 21 11 kN (steel beam)

1"8" h

P-7

3 I

f

The plastic neutral axis lies in the slab; in this case the depth (z,) of the neutral axis is gven by:

esaniple 2

Page 8

The plastic moment of resistance of the composite beam M,,.,,

is given by:

= (f,,*Aa/ya)(11/2+h, -2,/2) = = = (235~9880/1.10)~(0.45/2+0.12-0.0497/2)*10” = 675.8 kNin > 576.9 kNm

Mpl.Rd

and the verification is satisfied.

6.2.3 Vertical shear

l6.2.41

V p l . ~= d Av.(fy/J3)/ya= [4399.2.(235/.\/3)/1.10].10” = 542.6 k N

where

A, = 1.04.ha-t,,,= 4399.2 inin? Vpl.Rd

= 542.6 k N > VSd = 192.3 k N

Shear buck1 ing d/t,v = 40.3 < 6 9 =~ 69 and therefore the steel web is verified.

6.2.4 Bending and vertical shear

L6.2.51

As the beam is subject to unifonii load it is not necessary to check for coincideiit moment and shear.

7.

SHEAR CONNECTORS

7.1

Longitudinal shear force

[6.3.41

Full shear connection Vl = Fcf

where

Fcf = Aa.fy/ya - (9880*235/1.1).10-3 = 21 11 k N or, neglecting the tenn relating to longitudinal reinforcement Fc, = 0.85*Ac.fc,/yc= (0.85.3000.65*25/1.5).10-3= 2762.5 k N whichever is sina 11er.

example 2

Page 9

Partial shear connection Linear interaction method

v! =

=

dMS

I , Rd

1, R d-Map 1, Rd)

where

MSd = 576.9 kNm M p l , R d = 675.8 kNm

Mapl,Rd= 363.6 kNm

538 (point A)

and then

VI

= 21 1 ls(576.9-363.6)/(675.8-363.6)= = 21 11.0.683 = 1442.3 kru'

Fc

7.2

= 1442.3 kN

Design resistance of shear connectors The design resistance of a stud is given by:

Try 19 mm diam. studs 100 mm (95 m m after welding) high a) PR, = [0.8~450~(r~~19')/1.25]~10~= 81.7 kN

esample 2

Page 10

b) PR, = [0.29~1~19'.\1(25~30500)/1.25]~10~3 = 73.1 kN < 81.7 Taking into account the reduction factor for profiled sheeting ki

= reduction factor for one stud per trough (valid for 11, S 85 inm and b, 2 11, ) = (0.7/dNr)* (bo/hp (Iflip)-11 = = 0.7. (75/55)*[(95/55)-11 = 0.69 )e[

Q

57 It is possible to reduce the number of studs in the middle of the span to give N = 57. Check the minirnum value of N/N,. N/Nf 2 0.25+0.03.L = 0.25+0.03*12= 0.61

N/N, = 57/84 = 0.67 > 0.61 Idd = (100-51/19 = 5.0 > 4 and the studs may be considered as ductile. Mapl,Rd Mpl,Rd

exainplc 2

(steel beam) = (235~1702~103/1.1)~10-6 = 363.6 kNin

= 675.8 k N m

Page 11

ql.Rd/Mapl,Rd =

1.86 < 2.5

and therefore the studs may be spaced uniformly.

8.

TRANSVERSE REINFORCEMENT

[6.3.5]

Minimum transverse reinforcement for comoosite slab 4 2 O.OOZ.& = 0.002.65.1000 = 130 m ’ / m

reinf. bar 4 8 m m / 300 m m

4 = 167.6 mm’/m

Longitudinal shear

{*[

............................................. .........................................

I a

Section a-a a) VR d = 2 . 5 - 4 * ~ - TdR+4:.fsk/yS+Vpd

or b) VRd = 0 . 2 . k q*f,k I*/,+vP d l d 3 whichever is smaller, where fRJ = 0.30 N / W ‘

q=l

4 e.umple 2

= 65-1000 = 65-10’

m ’ / m ( without concrete rib contribution )

Page 12

for profiled steel sheeting continuous over the beam. For lmm thick steel;

Ap = 1575 mm2/m = Ap.f,,p/y,p = (1575.235/1.1)-10”=336.5 kN/m

vpd

and therefore for each shear plane a-a :

a) VRd = (2.5~65*103~1~0.30+167.6~420/1.15)~10‘3+ 336.5 = 446.3 kN/m b) VRd = (O.2.65*1O3*1*25/1.5)*1OJ+ 336.5/43 = 410.9 kN/m The longitudinal design shear is given by

= (1/2)~50.5~1000/150 = 168.3 kN/m < 410.9 kN/m

VSd

and the verification is satisfied.

9.

SERVICEABILITY LIMIT STATE VERIFICATION

9.1

Calculation of maximum deflections of beams

9.1.1 Construction stage = 5-q.L4/(384-E;I) = = 5~(8.00+0.50+0.80)~10-3~120004/(384~210~33740.0~104)= = 35.4 min

6‘

9.1.2 Composite stage It is assumed hi this example that all the variable load is of long-term duration. Take the same value of 11 as for Example 1. b e r f / n -145.2 mrn

E E 0

c

example 2

Page 13

~

~~~~~

beff /ii = 3000/20.65 = 145.2 inin

1 2

A min2 9443.1 9880.0 19323.1

Z

inm 312.5

A*(e, - z ) ~ I" inin3 inrn rnm4 mm4 m 4 2951.0*103 -159.8 332.4010~ 24108.7010~ 2 W l . 2 0 1 0 ~ 152.7 33740.0010~ 23042.6010~ 56782.6010~ 8 1223.9010' Aoz

e, - z

1'"

e, = 2951.0.103/19323.1= 152.7 inin = 5.q.L4/(384-Ea*1) = = 5~(3.33+3.33+6.67)~10'3~120004/(384~210~81223.9~104) = = 21.1 inin

Calculation of deflection due to shrinkage is not required as E,, does not exceed 400.10"'. 6'+6"-60 = 35.4+21.1-15 = 41.5 inin = L/289 < L/250

[Table 7.11

where

a0 = pre-camber of the steel beain = 15inm in this case The effects of iiicoinplete interaction have been neglected because: N/Nf = 0.61 > 0.50 height of the rib 1, = 55 iiiin < 80 inin.

example 2

Page 14

WORKED EXAMPLE no.3

This worked example refers to a two equal span continuous composite beam considered as a floor beam in a building. Tlie spans of the beam are equal to 12m with a spacing of 3.33m. The variable load has been considered equal to 4.50 kN/m2 and the load for interior finislzings equal to 1.0 kN/m2. The beam is unpropped during construction. The effects of shrinkage and temperature have not been taken into account. Tlie structural solution considers a solid slab, rolled steel beam and headed stud connectors. Additional reinforcing bars are placed in the slab in the region of the central support.

Headed stud connectors diarn.19 rnm h-100 rnrn concrete C25/30

IPE 450

Fe 360

3333

example 3

3333

Page 1

INDEX 1. COMPOSITE BEAM CHARACTERISTICS

. ... . .. .. .page 3

2.VALUES OF ACTIONS

...........

3

3. LOAD COMBINATIONS

...........

3

........... ...........

3 4

...........

4

........... ........... .. . . . . . . . . . ........... ...........

4 4 4 4 5

...........

5

5.1 Construction stage 5.2Composite stage

........... ...........

5 9

6.SHEAR CONNECTORS

...........

13

........... ...........

13 14 15

...........

17

8. SERVICEABILITY LIMIT STATE VERIFICATION ...........

18

3.1 Ultimate limit states 3.2Serviceability limit states 4.MATERIALS 4.1Concrete 4.2Reinforcing steel 4.3 Structural steel 4.4Shear connectors 4.5Partial safety factors 5. ULTIMATE LIMIT STATE

6.1 Longitudinal shear force 6.2Design resistance of shear connectors 6.3Number of shear connectors

7. TRANSVERSE REINFORCEMENT

8.1 Calculation of maximum deflection 8.2Cracking of concrete

esnniple 3

...........

........... ...........

18 20

1.

COMPOSITE BEAM CHARACTERISTICS Span : L = 12000 + 12000 mm Static scheme : continuous beam Beam spacing : b =3333 mm Construction type : unpropped

*

2.

VALUES OF ACTIONS Characteristic values

permanent loads

g'k

= 3.00.3.33

g'lk

-

g"lk

=

qk

= 3.50-3.33

9"k

=

1.00.3.33

=

3.33 kN/m partitions

q"Ik

= 0.50-3.33

=

1.67 kN/m construction load

= 0.80 kN/m steel beam

I

variable loads

= 10.00 kN/m normal weight concrete

1.00.3.33

= 3.33 kN/m interior finishings = 11.67 kN/m imposed load

( reduced value to take account of lower

probability of loading in continuous beam )

accidental loads

ak

3.

LOAD COMBINATIONS

3.1

Ultimate limit states YG.zGk,js"/Q'Qk,

1

YG *CGk+ j +O. 9*yQ*ZQk,j

=O

[Table 3. I] considering only the most unfavourable action considering all unfavourable variable actions

whichever is larger Partial safety factors:

example 3

Page 3

3.2

[Table 3.21

Serviceability limit states CGk,j + 1.0 Qk,,

considering only the most unfavourable variable action

EGk.j + 0.9CQk,i

considering all unfavorable variable action

4.

MATERIALS

~4.21

4.1

Concrete

[4.2.1]

Concrete strength class : C25/30 Characteristic strength : fck= 25 N/mm' Shnkage : E,,

= 325.10-6

Secant modulus of elasticity of short-term loading :

E,,

= 30.5 kN/mm2

[7.2.21

modular ratios :

n = E,&,

n = E,/(E,,/3) 4.2

(short term effects)

= 210/30.5 = 6.88

(long term effects)

= 20.65

[4.2.2]

Reinforcing steel

Type of steel : S 420 Steel grade : fsk = 420 N/mm2 Modulus of elasticity : E, = 2 10 kN/mm2 4.3

[4.2.3]

Structural steel

Nominal steel grade

: Fe 360

Nominal yield strength : f,

= 235

N/mmz

(t 5 40 mm)

Modulus of elasticity : E, = 2 10 kN/mm2 4.4

Shear connectors

Type : headed studs 22 mm diameter Ultimate tensile strength : f, = 450 N/mm2 esaniple 3

P;ge 4

4.5

Partial safety factors yM

4.5.1 Ultimate limit states Structural steel

ya = 1.10

Concrete

yc = 1.50

Steel reinforcement ys = 1.15 Shear connectors

yv = 1.25

4.5.2 Serviceability limit states

5.

ULTIMATE LIMIT STATE

5.1

Construction stage

E6.1

5.1.1 Cross-section properties Cross section classification A cross section is classified accordins to the least favourable class of its steel elements in compression.

71

*%

ha = 450.0 mm d =378.8mm b = 190.0 mm

Flange (b/2)/tf = 9Y14.6 = 6.51 ~ = (Fe360) l 6.51< 1 0 =~ 10 and therefore the flange is in Class 1 example 3

Page 5

d t = 378.819.4= 40.3 E = 1 (Fe 360) 40.3 < 72s = 72 and the requirements for Class 1 cross-section are satisfied. 5.1.2 Global analysis

Moments and forces are determined by elestic global analysis in this case, and the restrictions of the cross-section are determined by plastic section analysis. a) Design loads on both spans

yG .(gtk+gttk)+yQ .q"'k = 1.35.10.8+1.5.1.67 = 1 7.08ki'i'/m At internal support (maximum negative bendingmoment) MSd= 17.08-12218 = 307.5lcNm

At internal support (shear force)

Vs(i = 0.625*17.08*12 = 128.2ld'i Masimum positive bending moment Msd = (9/128).17.08.122 = 172.9kNm

b) Concrete and coiistnictioii design loads on one span only yG

+g"k)+yQ

q " l k

=I 7.08 kN/m

YG*g"k = 1.35.0.8=1.08 kN/m

17.08 kN/m

"1

12000

9

12000

1L

1.08 kN/m

163.4 kNm

At internal support (maximum negative bending moment)

MSd = 17.08~122/16+1.08~122/16 = 163.4 kNm Maximum positive bending moment

MSd = 231.6 kNm 5.1.3

Plastic resistance moment of the steel section Mapl, R d

= f;W,,/y,

= (235~1702~10~/1.10~~10-~ =

= 363.6 kNin MSd = 307.5 < M a p l S K=d 363.6 kNin 5.1.4

Design plastic shear resistance

A, = 1.04.ha.t,,, = 1.04.450.0.9.4 = 4399.2 inin?

V S =~ 128.2 < Vpl,Rd = 542.6 k N 5.1.5

Bending and vertical shear

L6.2.51

Vsd < 0.5*Vp,,,d = 271.3 kN therefore, vertical shear lias no effect 011 the plastic resistance moment.

I

exatnplc 3

5.1.6

Lateral-torsional buckling of the steel beam

[ EC3 - 5 . 5 . 2 Annex F 3

The design buckling resistance moment of a laterally unrestrained beam is given by = XLT‘P w*wpl,y*fyhM 1 =

Mb,Rd

1).106= = (0.45-1.00*1702*103*235/1. = 164.4 kNm

where

-

XLT = 1/[0LT+(0’LT’hZLT)O5]=

= l/[l.51+(l.512-1.332)O’]= 0.45

-

-

0LT = 0.5.[ l+aLT-(ALT-0.2)+hLT2] = = 0.5*[1+O.

~ L T = 0.21

2 1 1.33-0.2)+ 1.33’1 = 1.51 e(

(rolled section)

hLT

= (hLT/h~)*13w’” = 125193.9= 1.33

ALT

= 0 9-(L/i,)/{C,Os[ 1+( 1/20).(L/i,)2/(h/tf)2]O’3}= = 0.9-(12000/41.2)/{1.8790.5[1+( 1/20).(12000/41.2)’/450/14.6)’]0~’~} =

= 125.0 ( for shape of bending moment diagram )

C1

= 1.879

AI

= ~*(E/f,)”.5 =93.9.~ = 93.9.(235/fV)O.’ = 93.9

kv

= 1.0

YM 1

=

(class 1 section)

1.1

Check of stability for the bottom flange of B-C span (load combination b) Mb,Rd =

164.4 kNm> Msd = 163.4 kNm

Therefore, the beam is stable without hrther restraint.

esariiple 3

Page X

5.2

ComDosite stage

5.2.1

Cross-section properties

Effective width of concrete flange:

[6.1.3]

for sections in negative bending bey = 2*0.5*L/S= 0.125.L = 1500 IIUTI for sections in Dositive bendinu, beff= 2.0.8*L/S= 0.2.L = 2400 mm At internal support, negative moment resistance is obtained by considering the tensile resistance of the reinforcement.

- -

bar d i m . 12 mm/150 mm

4 = 1 130 mm'

in the effective width of slab

depth of the web in compression z,.\~ = ha/2-F,/(2t,v.f,,b/,) = 225-412700/(2*9.4*235/1.1) = 225-102.7 = 122.3 IIUII esample .3

Page 9

Mpl,Rd

- Map Rd +F, .(ha/2+a)-F, 2/(4tw-fy/ya)= = 1702000-235/1.1+412700*(450/2+90)-4127002/(4~9.4~235/1. 1)= = 472.4 kNm

where

a = distance of reinforcement from top of steel flange = 90 mm concrete cover = 30 mm Check of cross-section class Web in bending and compression d = 378.8 mm a.d = h,-~,,~-(h,-d)/2 = 450-122.3-(450-378.8)/2 = = 292.2 mm

a = 292.2/378.8 = 0.77 According to TabIe 6.3 d/t,,. = 378.W9.4 = 40.2 < 3 9 6 . ~ 14 3 . ~ 1=) 43.8 web in Class 1 FIange in compression (Table 6.1) C/tf = 95/14.6 = 6.5 < 10 flange in Class 1 and therefore the section is in Class 1.

In midspan region compression in concrete: O.85*fck.b,.yhc/~,=(0.85.25.2400. 20/

080 kN

Tension in steel

f,*&/ya = 235*9880/1.10= 21 1 1 kN

Page 10

csaniple 3

~~

~

Therefore plastic neutral axis lies in the slab. In this case the depth (z,) of the neutral axis is given by: Z, = (f,.~/y~)/(O.SS.f,,.b,*.,/./,) = 62.1 IIUII < 120 IIUII

Jc The plastic moment of resistance of the composite beam b$l,Rd

.Rd

is given by:

= (f,..~/:/,)(h,/2+h,-z,/2) = = (235~9880/1.10)~(0.451'2+0.12-0.0621/2)~10-' = = 662.7 kNm

5.2.2 Global analysis

Bending moments Neeative moment Taking into account a negative moment redistribution equal to 40%, for Class 1 section and "uncracked" elastic analysis , the des& moment at internal support results:

Desip load

= yG.(elk+g'lk)+l/Q.(qI q I1k) = I

= 1.35.(10.8+3.33)+1.5.(11.67-3.33)

=41.6 kN/m

Moment before redistribution = 4 1.6.12YS = 748.4 h i m Moment after redistribution:

M,, = 0.6.748.4 = 449.1 lcNm Msd = 449.1 C M,l,~,j= 472.4 kNm

I

!20@0

1

12000

449.1 KNrn

541 KNm

/n

It should also be noted that plastic global analysis could be used for Class 1 sections. Positive moment From equilibrium the reaction at external support after redistribution is:

R = qd*L/2-Msd/L= 41.6.12/2-449.1/12 = 212.1 kN MSd = R2//(2.qd)= 212.12/(2.41.6) = 541 kNin

and the verification is satisfied. hi this example patteni load for positive bending moment is not critical. But there is the need to consider such a load combination generally.

5.2.3 Vertical sliear At iiitenial support

Vpl,Rd = 542.6 k N > Vsd = 286.9 k N Shear buckling d/t," = 40.3 < 6 9 . = ~ 69 and therefore the steel web is verified. 5.2.4

Bending and vertical shear at internal support

449.1 < 471.7 kNin

example 3

Page 12

where, for the steel section, ignoring the reinforcement:

5.2.5

Lateral-torsional buckling of composite section

Checks have been camed out on the adequacy of the resistance to lateral-torsional buckling of the composite beam as in EC4 clause 4.6.2. "lie design is shown to be satisfactory as is normally the case.

*

6.

SHEAR CONNECTORS

6.1

Longitudinal shear force

i6.3.41

Full shear connection a) between simple end support and point of maximum positive moment

L 12000

length = 0.421.12000 = 5052 inin

where force transfer is 21 11 kN from the slab to the beam. b) between point of max. positive moment and internal support

I

6948

12000

i

length = 12000-5052 = 6948 min Vi = F,f+A,.f,,/y,

= 2111+(1130*420/1.15)*10-3 = = 2523.7 kN

example 3

Page 13

1

Partial shear connection

where

MSd = 541 lcNm 1Mpl,Rd = 662.7 lcNm M a p l , R d = 363.6 kNm

and then

VI ,

= 21 11*(541-363.6)/(662.7-363.6) = = 21 11-0.593= 1252 kN

Therefore, degree of shear connection required so that Msd is achieved is 0.593. The minimum degree of shear connection that is required is obtained as follows.

b) Shear connection between max. sagging bending and the internal support VI.

6.2

=

1252+(1130*420/1.15).103= 1665 kN

Design resistance of shear connectors The design resistance of a stud shear connector is given by: a) P R d = 0.8.fu-(~.d2/4)/y,

or b) P R d = 0.29.a.d2~(f,kE,,)/y~. whichever is smaller. Try 22 mm diam. studs 100 mm high a) P R d = 0.8.450.(~.22Y4)/1.25= 109.4 kN

b) P i d = 0.29.1.222d(25-30500)/1.25 = 98.0 kN

esample 3

Page 14

6.3

fliimber of shear connecton

a) Full shear coiiiiection

FCf/PRd = 2111/98.0 = 21.5

no.22 headed studs o 22 m m

Partial shear connection - Minimum degree of shear connection N/Nf 2 0.25+0.03 L = 0.61 > 0.593 Therefore provide no.14 headed studs U 22 mm to achieve the minimum degree of shear connection. N/Nf = 14/22 = 0.64 > 0.61

h/d = (100-5)/22 = 4.32 > 4 and the studs are considered as ductile.

Spacui g

= 5052/14 = 360 inm

Mapl,Rd

(Steel beam) = 363.6 kNin

( inay be

300 inin in practice )

Actual moment of resistance is: Mpl. Rd(red)=Mapl, K d f ( M p l , K d - M a p l . R d ) ' ( N ~ f )

=363.6+0.64*(662.7-363.6) =555.0

So the plastic moment of resistance of the beam with 64% shear connection exceeds the design moment by 2.5% Check: Mpl,Rd(red)/Mapl,~d = 1.53 < 2.5 and therefore the studs may be spaced unifonnly over the entire beam length.

example 3

Page 15

4

Full shear connection - Miniinuin degree of shear connection Fcf/P,d = 2523.7/98.0 = 25.7

110.26 headed studs U 22 rnm Partial shear connection N

=14+F,/P,d

where:

F,

=tensile force in rei~iforceinent.

N

=14+ 1130.42Q 1.15.9800

~18.2

Use 19

and the studs are considered as ductile.

Tlierefore. spacing

= 6948/19 = 365 m m ( inay be 300 in111in practice)

Use 300 m m spacing of shear con~iectorsover the entire span, CN = 1200/30 = 40.

* rL'L'L full connection would require : XN=22+26=48

A-

5052

6948

partial connection

: m = 1 4 + 1 9 =33

TRANSVERSE REINFORCEMENT

7.

C6.3.51

Minimum transverse reinforcement for solid slab 4 2 0.002.4= 0.002~120~1000 = 240 m ' / m

reinf. bar 4 8 m m / 200 m m - 2 layers

4 = &+Ab = 2-251.3 m ' / m Longitudinal shear in the slab

Section a-a a) VR d

= 2.5..\ v . q-TR d +&.fskh,ls+vPd

or b) VR d = 0 . 2 . 4 7 - fck /yc +Vp d /d3 *

s

whichever is smaller, where ' I R ~= 0.30N / m '

(C?5/30)

q=l

(normal weight concrete)

4,,= ~ ~ O - ~ O=O120.10' O mm:/m Vpd

=0

(contribution of the steel sheeting)

and therefore 1.3.420/1.15)~10~' = 273.6 kN/m a) v R ~= (2.5.120.10'.1~0.~0+2~25 b) vRJ = (0.2.120*10'*1*25/1.5).10-3= 400 kN/m esample .3

[Table 6.71

The longitudinal design shear is given by VSd

= 98.0.1000/300 = 326.7 kN/m

For each shear plane a-a vSd/2 = 326.7/2 = 163.3 < 273.6 kN/m and the verification is satisfied. Section b-b Acv = (2-95+35).1000= 225.103 mtn’/m VRd

= (2.5~225~1O3~1~0.3O+2~251.3~420/1.15)~1O3 = 352.3 kN/m

VRd

= 352.3 kN/m >vsd = 326.7 kN/in

and the verification is satisfied.

8.

SERVICEABILITY LIMIT STATE VERIFICATION

8.1

Calculation of maxiniuni deflection

8.1.1 Construction stage

p1 = 10.80 kN/m 61

= 0.00542*p1*L4/(E.I,) = = 0.00542-10.80*10-3~1 200O1/(210*3374O*1O4) =

= 17.1 inin 8.1.2 Composite stage

example 3

Page 18

mm' 1 2

mm'

rnm

13946.7 285 9880.0

mm

3974.8-1Oj

mm'

- 1 18.1 1673.6.10' 166.8 33740.0.10'

rnm'

mmJ

19478.1-10.' 21 151.7-10' 27495.6-10-1 6 1235.6.10-1

beff /n = 2400/20.65 = 116.2 mm

e, = 3974.8.10'/23826.7 = 166.8 mm For simply supported beam and assuming full shear connection: Load considered for serviceability deflection = interior finishing, imposed load partition :

and

p2= 3.33+11.67+3.33= 18.33 kN/m

= 22.5 mm

Correction factor for support moment Mh C

[7.2.5]

= 0.6.0.7= 0.42

This correction factor c = 0.42 reduces the hogging bending moment in order to take account of the effects of cracking and yielding. The value seems to be conservative. Detailed analysis of stresses may show that cracking and/or yielding does not occur.

,

18.33 kN/m

13.33 C

6

A

12000

I

12000

196.5 kNrn

blh = 0.063.(18.33+3.33).122= 196.5 kNm

Puse 19

1-0.42*M,/(p2.L2/8)=1-0.42*196.5/( 18.33.12*/8)= 0.75

6, = 0.75.22.5 = 16.9 tntn

,a,

=

= 17.1+16.9 = 34.0 inin equal to W353 < L/250

on:itcarenti-

Minitnuin degree of shear connection N/Nf = 0.61 > 0.5 and the effects of incomplete interaction are ignored.

8.2

CrackinP- of concrete

Minimum reinforcement (for no control of crack widths): A, 2 0.002.3333.3.120 = 800 1nm2 Use bar diatn. 12 mm/150 mm

A, = 113.10 = 1130 mn2> 800 min2 The tnlliiinuin area of reinforcement required to ensure that the reinforcement remains elastic when cracking first occurs is given by:

A, >k~k;f,,;A,/a,,

= 0.8~0.7~3~180000/320 = 945 inin?

where: z, = zg-h,/2 = 138.1-60 = 78.1 tntn

zg = [b, ff.h’,/2+11.A, (11,/2+h~)]/(b, ff.h,+~i.A,) =

= [ 1500~120’/2+6.88~9880(450/2+120)]/(1500~120+6.88~9880) = = 138.1 tntn

k, = I/{1+[hC/(2z,,)1 1= = 1/{1+[120/(2.78.1)]}= 0.565 < 0.7 and then k, = 0.7

k = 0.8 A, = beff.lic= 1500.120 = 180000 Inin‘

fctc = 3 N/mln’

us,= 320 N/mm2 (for crack width wk = 0.3 tnrn )

example 3

Page 20

A, = 113.10 = 1130 mm2 > 945 mm2 where A, is referred to the effective width.

esimple 3

Page 2 1

.

WORKED EXAMPLE no.4

The example refers to a concrete encased composite column subject to compression load and uniaxial bending. The length of the column is 4 m. The verification has been carried out according to the simplified method given in this publication. No influence of shear forces has

been considered.

300

e.wple 4

Page 1

INDEX

1.

COMPOSITE COLUMN CHARACTERISTICS

. . . . . . . . . age .p 3

2.

DESIGN VALUES OF ACTIONS

...........

3

3

MATERIALS

...........

3

3.1 Concrete 3.2 Reinforcing steel 3 . 3 Structural steel 3.4 Partial safety factors

........... ........... ........... ...........

3 3 3 4

CROSS SECTION GEOMETRIC AND STATIC CHARACTERISTICS

...........

4

ULTIMATE LIMIT STATE CHECKS

...........

4

5.1 Resistance of cross-section to axial compression 5.2 Effective elastic flexural stiffness of cross-section 5.3 Non-dimensional slenderness 5.4 Simplified method

........... ...........

4 4 5 6

4.

5.

esample 4

...........

...........

1.

COMPOSITE COLUMN CHARACTERISTICS

Columnlength : L = 4000mm

2.

Type of construction

=

Steel profile

= HEA200

concrete encased section

DESIGN VALUES OF ACTION Design axial load for the column length

-

Maximum second-order design bending moment about axis y-y

- Mvmas,Sd = 140 kNm

Maximum design bending moment about axis z-z

= M,max,Sd =

3.

MATERIALS

3.1

Concrete

N,,

=850kN

0 kNm

[4.2.1]

Concrete strength class: 20/25 Characteristic strength: fCk= 20 N/mm2 Secant modulus of elasticity of short-term loading:

E,,

= 29.0 kN/mm2

Modular ratios: 3.2

Reinforcing steel

[4.2.2]

type of steel : S 420 steel grade : fsk = 420 N / m 2 modulus of elasticity : E, = 210 kN/mm2 3.3

Structural steel

[4.2.3]

Nominal steel grade : Fe 510 nominal yield strength : fv= 3 5 5 N / m 2 modulus of elasticity : E,

esamplr. -1

(t I 40 mm)

= 2 10 kN/mm2

Page 3

3.4

Partial safety factors YM

3.4.1 Ultimate limit states

4.

structural steel

Ya

= 1.10

concrete

Y C

=

steel reinforcement

y

,

1.50

= 1.15

CROSS SECTION GEOMETRIC AND STATIC CHARACTERISTICS

c, CZ

= (300-200)/2 = 50 IYKII = (300-190)/2 = 55

Reinforcement

mm

: 4412 bars

A,

= 452.4m.m'

5.

ULTIMATE LIMIT STATE CHECKS

5.1

Resistance of cross-section to axial compression

18.3.1J

The plastic resistance to axial compression is given by Np1, Rd = & *fy/ya+&.(0.85*f,k/y, )+A,'fsk lys

5.2

Effective elastic flexural stiffness of cross-section

Short term loading The flexural stiffness of cross-section about the main axes is (E.I), example 4

= E,.I,+O.S.E,d.I,+E,.I,

Page J

about y-y (E*I), = E, *I, +O. 8.E,d .I,,+E, *Is

=

= E,~I,y+0.8~(E,m/y,)~I,y+E,~I,!, = = 2 10*3692*10+0.8.(29/1.3 5).(300'/12-3692*1P)+2 10.2.226.2.1152 =

= 1.998.10"

kN*mm2

about z-z (E.I),

= E,

*Iaz+O.8*E,d .ICZ+Es*Isz=

= E,~I,Z+0.8~(E,m/y,)I,,+Es*Isz= = 2 10*1336.102+0.8*(29/1 .35).(3OOJ/12-1336.10')+2

10.2.226.2.115 2 =

1.543.10'O kN.m.m2

long term loading The eccentricity e of the normal force is defined by e = Mm

as ,S dmS d

in this case e,.

=

140/850 = 0.165

e, = 0 e/d = 0.165/0.300 = 0.55 < 2 where d is the overall depth of the cross section in the plane of bending considered. The non-dimensional slenderness results

h < 0.8(see 4.3) and therefore the influence of creep and shrinkage on the ultimate load need not to be considered. 5.3

Non-dimensional slenderness

The slenderness for the determination of the load bearing capacity of the column is given by

e s m p i e -I

Page 5

where Npl,R = ~.f,+A,.(O.85.f,k)+As.f,L

=

= [5380.355+(3002-5380).(0.85.20)+452.4.420)]103 = = 3538 kN

N,, 1,

= (E*I)e.~2/12

= 4000 mm

= 1,

buckling length of the column

N,.,!

= (E*I)ey*7t2/Iy2 = 1 .998*10'o*~2/4000' = 12320 kN

Ncrz

= (E.I),,.Ic~/~,~ = 1.54~.10'"*~'/4000' =

9520 kN

and then -

&. = 0.536

-

A, = 0.610

5.4

Simplified method

C8.3.61

The conditions to be satisfied for the applicability of the method are: a) cross-section Izeometry The cross section of composite column (under examination) is double-symmetrical and uniform over the entire length.

b) steel contribution ratio

6

= (b'fy/ya)/NpI , Rd = (5380~355.103/1.1)/2860= 0.607

0.2 < 6 < 0.9 c) non-dimensional slenderness h ?,,,,,as

=

h, = 0.610 < 2

d) concrete cover cy = 50 mm 40 mm < C, < 0.4.b = 0.4-200= 80 IIMII esample 4

Page G

cz = 55 mm 40mm < c, < 0.3ha = 57 mm

e) longitudinal reinforcement

4 4 12 bars

A, = 452.4 m 2

Minimum reinforcement = 0.003.4= 0.003.(3002-5380) = 253.9< 452.4mm2 Maximum reinforcement that may be considered in calculations

0.04.4= 0.04*(300’-5380) = 3384.4> 452.4 mm2

5.4.1 Check of column axial compression resistance

[8.3.2]

The check is satisfied if for both axes

where

N,,,,,

= 2860 k.N

X

=

Q

= 0.5.[l+a(h-0.2)+h2] = 0.786

a

=0.49

I/[D+(Q’-~’)~~] = 0.780 b

d on buckling about z-z

curvec

and then

850 < 0.780.2860= 2230 k.N

5.4.2

Resistance of cross-section in com bined compression and and uniaxial bending

[8.3.3]

Compressive resistance of the whole area of concrete:

esample 4

Page 7

i

Plastic bendine resistance MpI Rd plastic section modulus of reinforcement W,, =4*(122.~/4).112= 50.7.10jTI~P II

Asn = 4-(12'-~/4)= 452.4 W' plastic modulus of structural steel

plastic modulus of concrete part of section Wpc = b,~h,'/4-Wp,-W,, = 6269.1Oj W'

position of neutral axis (in the web)

h, = [ ~ ~ f c ~ - ~ , ~ ( ~ ~ f s ~ - f c ~ ) ] / [ ~ ~ b , ~ f c ~ ~ ~ ~ ~ , ( 2 ~ ) ] where

h, = [959000-452.4.(2.365.2-11.3)]/[2.300~ 11.3+2.6.j.(2.322.7-11.3)] (h, < h/2-t, = 19012-10 = 85 ITII~)

= 42.1 mm

W,,, = fw*h,'= 6.5.42.1' = 11537.0 W' Wpcn= bc.h,.,z-Wpan-Wpsn= 520944.0 mm3

- fyd.(Wpa-Wpan)+0.5.f~d.(Wpc-Wpcn)'fsJ.(Wps-W %l.Rd

)= Psn

= 322.7~(429000-11537)+0.5~11 .3.(6269000-520944)+ +365.2.(50668-0)= 185.8 kXm

M (kNm)

example 4

0

lZo

Mpl,Rd 155.8

Puze 8

5.4.3 Resistance of member in combined compression and and uniaxial bending

[8.3.4]

where

1

x=

0.780

Xd