CONNECTION DESIGN: STATIC LOADING · Lecture 11.2.3 : Welded Connections ... 3. Eurocode 3, Part 1.3: ... 3. PHYSICAL CHARACTERISTICS Thermal insulation is often required for walls

RAKENNUSTEKNIIKKA Olli Ilveskoski 30.08.2006 rev2 10.01.2007

213

Opiskelija perehtyy liitosten suunnitteluun.ks ESDEP – oppimisympäristö: http://www.terasrakenneyhdistys.fisuomenkielinen versio

Course Contents

CONNECTION DESIGN: STATIC LOADING

Lecture 11.1.1 : Connections in Buildings

Lecture 11.1.2 : Introduction to Connection Design

Lecture 11.2.1 : Generalities on Welded Connections

Lecture 11.2.2 : Welded Connections - Basis for Weld Calculation

Lecture 11.2.3 : Welded Connections - Applications of Fillet Weld Calculation

Lecture 11.3.1 : Connections with Non-Preloaded Bolts

Lecture 11.3.2 : Connections with Preloaded Bolts

Lecture 11.3.3 : Particular Aspects in Bolted Connections

Lecture 11.4.1 : Analysis of Connections I: Basic Determination of Forces

Lecture 11.4.2 : Analysis of Connections: Distribution of Forces in Groups of Bolts and Welds

Lecture 11.4.3 : Analysis of Connections: Transfer of Direct Tension or Compression and Shear

Lecture 11.4.4 : Analysis of Connections: Resistance to Moment by Combined Tension and Compression

Lecture 11.5 : Simple Connections for Buildings

Lecture 11.6 : Moment Connections for Continuous Framing

Lecture 11.7 : Partial Strength Connections for Semi-Continuous Framing

Lecture 11.8 : Splices in Buildings


212

Lecture 10.1 : Composite Construction - General

Top

1. INTRODUCTION

2. COMPOSITE ACTION IN BEAMS

3. COMPOSITE MEMBERS

3.1 Composite Beams

3.1.1 Propped construction

3.1.2 Resistance of section

3.1.3 Continuous beams and slabs

3.2 Shear Connection

3.3 Beam-to-Column Connection

3.4 Composite Columns

3.5 Partially Encased Steel Sections

3.6 Composite Slabs

4. COMPOSITE FLOOR CONSTRUCTION

5. COMPOSITE BRIDGES

6. CONCLUDING SUMMARY

7. REFERENCES

8. ADDITIONAL READING


211


• Cold-formed products provide the designer with a very wide range of items, capable of being used in a variety of different ways.

• Very efficient structural elements may be produced by combining different types of cold-formed element or by using such elements in association with other materials.

• Use of precoated sheeting, together with the incorporation of as many features as possible in the factory production process drastically reduces the need for expensive site operations.

• Corrosion protection, thermal and acoustic insulation and good visual appearance are all readily achievable.

• The use of this material permits several simple techniques of site jointing to be employed.


1. European Convention for Constructional Steelwork: "European Recommendations for the Design of Light Gauge Steel Members", Publication 49, ECCS, 1987.

2. European Convention for Constructional Steelwork: "European Recommendations for the Design of Profiled Sheeting", Publication 40, ECCS, 1983.

3. Eurocode 3, Part 1.3: "Cold-formed Thin-gauge Members and Sheeting" (in preparation).

Previous | Next | Contents


210


209

Two types are available:

• A skeleton completely assembled by bolting on site, Figure 16.• A skeleton based on the use of steel panels preassembled in the works.

Using the second principle, it is possible with quite large panels, e.g. 3mx12m to introduce a large degree of pre-fabrication into the structure. All sections are pregalvanised and contain all the holes necessary for wiring, plumbing, etc.

5.3 Temporary Accommodation

Modular units for houses, offices, construction site accommodation, etc., may conveniently be produced using cold-formed sections and flat products.

5.4 Storage

Storage racking systems, Figure 17, may conveniently be made from cold-formed components, with the forming process being used to produce not simply the most appropriate shapes but also to introduce slots and holes that facilitate rapid assembly and demounting.


208


207

Trussed rafters are also used in greenhouse construction.

5.2 Housing

Light gauge sections are appropriate for use as the steel frame of housing, Figure15.


206

filling of columns to provide fire protection is possible, whilst powder painting eliminates the need for final site painting. Members may have all the holes required for fixings automatically produced during the roll-forming process.

One of the largest uses of cold-formed material is in purlins Figure 13, spanning between 4 and 15 metres. Sections of heights between 100 and 300 mm are used and a number of proprietary systems are available. These systems include concepts such as purlin sleeves and overlapping, Figure 14, in order to obtain optimum structural performance. Because manufacturers have often undertaken extensive testing and other development work themselves, design information in support of particular systems is normally available. This information substantially reduces both the labour and complexity of the design process.


205

A parallel development is the use of lattice portal frames, with both the rafters and the columns being lattice members. Spans of up to 60 metres are possible.

One of the more traditional applications has been in purlins spanning between the heavier main frames in portal frame buildings. For smaller frames, the columns and rafters themselves may be suitable cold-formed sections, as indicated in Figure 12.

The insulation of mezzanine floors for storage or as a means of creating office space is a particularly suitable application for cold-rolled sections. Their use as columns and beams permits a lightweight construction using pre-galvanised sections and straightforward fastening.

The availability of cold-formed sections up to 500 mm in depth permits their use as main framing members, either in multi-storey braced frame construction or in portal frames. For the latter, spans up to 25 m are possible. In office buildings concrete


204

Cold-forming products are normally so light in weight that easy handling is possible. Since most can be handled manually, fast and easy erection is the norm. For transportation, sections and sheeting can normally be nested, thus requiring little space and permitting unloading in quantity. Use of precoated material ensures corrosion protection during transport and erection.

5. TYPES OF LIGHT-WEIGHT STRUCTURES

Early applications of cold-formed thin-walled steel sections were restricted to situations where weight saving was important. With the advance in the raw material itself and the manufacturing processes, the range of actual and potential use is virtually unlimited.

5.1 Industrial Buildings

Trusses of the types shown in Figure 11 may be found in industrial and storage buildings. The main chords are usually channel sections joined back to back. The web members are normally single channels. A high degree of lateral stability may be provided by using suitably wide chord members. Pre-galvanised high yield steel is normally used, with joints being made by simple bolting. Clear spans of up to 50 metres are possible.


203

4. USE IN SERVICE

Advances in manufacturing technology enable the range of products and the features that may be included directly in the forming process to increase continually. Clearly clever utilisation of this facility will lead to a progressive reduction in expensive site operations. Figure 10 shows how the termination of the section may now be integrated into the roll-forming process.


202

Since insulated panels are proprietary products, relevant manufacturers literature should be consulted as a way of ascertaining specific properties and recommendations on usage. Of particular concern are dimensional stability, voluminal mass, heat conductivity coefficient, resistance to compression, resistance to bending, reaction to fire, resistance to water vapour, dew point, ability to act as a heat bridge, and fixing arrangements.

3.1 Acoustics

The need for adequate sound insulation is an increasingly stringent requirement for buildings. The two characteristics required for limiting the transmission of noise are absorption and insulation. It is often possible to combine the dual functions of thermal and acoustic insulation in the one type of composite panel. Particular ways to improve acoustic insulation are to increase the thickness of the profile and/or the density of the insulation or to add an additional sheet between the two skins.

3.2 Fire Resistance

Because light gauge sheets are so thin, they do not posses much inherent fire resistance if exposed directly to elevated temperatures. However, properly insulated double-wall cladding can attain a fire resistance of at least 90 minutes. Support roofing and composite flooring may well reach at least 60 minutes without additional protection.

3.3 Condensation

A steel wall, especially when not insulated, may be exposed to condensation since its coefficient of thermal conductivity is rather high. However, several anti-condensation coatings have been developed. These coatings are applied using an industrial process to the inside of the wall. They largely prevent subsequent condensation. One such example is "Grafo Therm". It is a water based mixture including porous components with large specific surfaces suitable for absorbing water. It appears in the form of a coating containing very fine granules of a light grey shade which have a pleasing appearance.

3.4 Durability

Profiled sheeting is always pre-coated and is obtainable in thicknesses between 0,3mm and 4 mm and widths of up to 1.500 mm. Typically steel strengths are 320N/mm2 to 350N/mm2, with values up to 550 N/mm2 presently available. Galvanising is the normal form of pre-coating. This coating provides cathodic protection to surfaces where the steel is uncoated, e.g. either accidentally or at edges formed by shearing or drilling, Figure 9. The zinc film may be painted in order to improve further the corrosion resistance, as well as to enhance appearance. Specially produced galvanised and pre-painted systems provide very significant corrosion resistance due to the synergy that exists between the two processes. For members it is also possible to apply paint after roll forming, such as polyester powders by the electrostatic process. In all cases coatings are available in a large range of colours.


201

2.3 Fasteners

Various proprietary types of fasteners, including spot welding, may be used with light gauge products. Figure 8, which illustrates the main types of mechanical fasteners, shows:

• Bolts with nuts.• Shot fired pins.• Screws:

- self tapping.- self drilling.- special types for translucent sheeting.

• Blind rivets.• Crimping of the seams to provide a mechanical interlock between adjacent

sheets.• Welding may be either:

- Arc welding.- Resistance welding.- Weld-brazing.

This last process is relatively new and is specifically designed to avoid damage to galvanised or pre-painted coatings. It works by introducing another metal that melts more easily than those to be joined and, using a blowpipe flame, results in a quickly made, tight and resistant joint.

3. PHYSICAL CHARACTERISTICS

Thermal insulation is often required for walls and may be provided by any of rockwool, glass-wool, chipboard, polystyrene, cellular glass or expanded perlite.


200

contribution is made by the cladding. Such an arrangement can substantially reduce the need for bracings.

2.2 Profiled Sheeting and Concrete

In the construction of multi-storey steel framed buildings, the use of composite metal deck flooring has increased significantly in recent years. Metal decking, typically spanning about 3 metres between secondary beams, may be laid rapidly and, when secured to the top flanges of the beams using shot fired steel pins, provides both a working platform and a protection to the operatives further down in the building. Concrete may be pumped or supplied by skips using a crane to the floors, which are normally reinforced with a light prefabricated mesh to control any cracking that might occur as a result of shrinkage during hardening of the concrete. Composite action with the metal decking is ensured through the use either of the decking profile itself or through indentations on part of the sheeting that provides a shear key arrangement preventing relative movement between the hardened concrete and the metal decking. Shear studs may be welded through the decking onto the top flanges of the beams so as to provide composite action with the primary beams spanning between columns in the frame.


199

1.1.3 Sandwich panels

In recent years the use of prefabricated sandwich panels, (Figure 6) formed from two metal sheets with a foam core, has increased considerably. Such an arrangement, which may be installed very rapidly, is particularly suitable as a means of providing thermal insulation at the same time as the basic weather shield. It consists of two metal faces bonded to an internal layer of rigid foam. Such panels may be installed very quickly thus saving time on site.

2. COMPOSITE CONSTRUCTION2.1 Cold-Formed Sections and Sheeting

It is commonplace to use the opportunity offered by the flexibility inherent in the forming processes to arrange for cold-formed products to act in conjunction with other components. One of the most common examples is the use of Z purlins, Figure 7, acting in association with roof sheeting to provide horizontal diaphragm action which helps to stabilise the building. In addition the sheeting acts to provide torsional restraint that improves the load-carrying resistance of the purlins themselves. The concept of deliberately relying on diaphragm action of sheeting has led to the concept of "stressed skin action" in buildings in which a significant


198

Because precoated sheeting is used for these flooring arrangements, the lower face may be used to provide a sufficiently visually attractive ceiling in some cases.

Since many steel cladding and roofing profiles are also available in equivalent geometrical shapes manufactured from translucent material, it is normally quite straightforward to create areas of natural lighting in buildings using this arrangement.

In order to properly fix cladding and roofing panels it is necessary to employ a range of smaller components at the various junctions. These products are termed flashings and will frequently need to be provided in a range of shapes and sizes. The products may sometimes also fulfil additional functions, e.g. to act as gutters. The folding operation is ideally suited to the production of this multiplicity of different items.

1.1.2 Members

Cold-formed members are available in a very wide range of shapes and sizes. Since they are normally formed from a single sheet, the cross-section should be such as not to require additional jointing, e.g. an I-section can only be formed by joining two channels back to back at the web. Figure 5 provides some idea of the range of products currently available.


197

In such arrangements it is important to ensure that the insulation can safely span the troughs between adjacent upper flanges of the corrugations.

For vertical walls, the outer skin of either a single or double wall-cladding, Figure 3, has to ensure air tightness and water tightness of the vertical walls, whilst at the same time providing an attractive visual aspect to the building. The internal skin is then effectively a structural tray or cassette, spanning horizontally between columns that provides a flat internal wall and supports the insulation.

Trapezoidal sheeting may also be used for flooring, e.g. in pre-fabricated houses.

Sheeting may also be used as permanent shuttering to concrete, in which case it may be designed merely to support the weight of the wet concrete and any construction loading. A more efficient arrangement is, however, to use the steel sheeting in the final condition also to produce composite action in the floor as shown in Figure 4.


196

For flat roofs, Figure 2b, the steel profile has, in addition to supporting its own dead weight, the function of carrying the insulation and the maintenance and climatic (snow and wind) loads.


195

Whilst the first of these methods is suitable for large volume production, e.g. sheeting for cladding, the second is more appropriate for low volume production of items that need to be tailored for specific applications, e.g. flashings used to seal the joins between components.

1.1 Available Products

1.1.1 Profiled sheeting

Profiled sheeting is available in a wide range of geometries and may be adapted for several different uses.

In single or double skin roof sheeting, Figure 2, the steel sheet is normally used as the external profile. Its main function is to ensure the security of the roofing. The main function of the internal sheet in a double skin roof is to support the insulation, although by incorporating suitable perforations it may assist in improving the acoustic properties of the building. Double skins may use the same profile for both sheets - in which case the roofing is generally called "parallel texture", or it can use a different profile internally, thereby replacing the purlins. In this latter case the arrangement is termed "crossed texture".


194

1. INTRODUCTION

The general trend in recent years towards lighter and more economic forms of steel construction has led to increases in the use of cold-formed, light-gauge elements. These elements may be either members or flat products.

Members such as purlins, sheeting rails and main framing components may be used in place of heavier and more costly hot-rolled products or substituted for alternative materials such as timber. Flat products such as wall cladding, roof sheeting and floor decking are often used most effectively by combining them with other materials.

The main advantages in using cold-formed products are:

i. A very wide range of components are available through the flexibility provided by the forming process.

ii. Good corrosion resistance through the use of previously coated material.

iii. Good quality surface finish, available in a variety of different types.

iv. Ability to provide high levels of thermal and acoustic insulation when used in conjunction with suitable insulating materials.

v. Availability of relatively simple fastening methods, several of which may be used on site.

vi. High ratios of load-bearing resistance to structural weight.

vii. Permits the use of considerable off-site prefabrication.

Cold-formed products are usually produced from the original coils of thin sheet material by either:

• Roll-forming in which the coil is progressively unwound and various shapes made in a continuous process by passing the sheet through a series of rolls that progressively develop the required shape, Figure 1, or

• Folding in which pre-cut lengths are bent to form a suitable structural element.


193




ESDEP WG 9

THIN-WALLED CONSTRUCTION

Lecture 9.7: Application of Thin-Walled Construction

OBJECTIVE/SCOPE

To present the range of available cold-formed, light gauge products and to illustrate their range of application.

PREREQUISITES

Lecture 4A.1: General Corrosion

Lecture 5.1: Introduction to Computer Aided Design and Manufacture

Lecture 9.1: Thin-Walled Members and Sheeting

RELATED LECTURES

Lecture 9.2: Design Procedures for Columns

Lecture 9.3: Design Procedures for Beams

Lecture 9.4: Design Procedures for Sheeting

Lecture 9.5: Stressed Skin Design

Lecture 9.6: Connections in Thin-Walled Construction

SUMMARY

The range of cold-formed products currently available is described and the particular features offered by products produced in this way are explained. Current uses of such products, often acting to provide several functions, are discussed.


192

Lecture 9.7 : Application of Thin-Walled Construction

Top

1. INTRODUCTION

1.1 Available Products

1.1.1 Profiled sheeting

1.1.2 Members

1.1.3 Sandwich panels

2. COMPOSITE CONSTRUCTION

2.1 Cold-Formed Sections and Sheeting

2.2 Profiled Sheeting and Concrete

2.3 Fasteners

3. PHYSICAL CHARACTERISTICS

3.1 Acoustics

3.2 Fire Resistance

3.3 Condensation

3.4 Durability

4. USE IN SERVICE

5. TYPES OF LIGHT-WEIGHT STRUCTURES

5.1 Industrial Buildings

5.2 Housing

5.3 Temporary Accommodation

5.4 Storage


191

LIITTOLAATAN JA -PILARIN MITOITUS

18. Tehtävänä on omakotitalon kaksiaukkoisen välipohjarakenteen mitoitus liittorakenteisena Rautaruukin mitoitustaulukoita käyttäen. Laataston jännevälit ovat 6.0 m ja välipohjarakenteen leveys on 5m. Rakenteessa käytetään betonia K25-2 ja terästä A500 HW. Välipohjarakenteen kuormitus on OLI, palovaatimusta rakenteelle ei aseteta. Piirrä laatan raudoitus.

Valitaan raudoitus taulukosta: 6.1 T4, mikä tarkoittaa, että 6.1metrin jännevälillä laatan paksuuden ollessa 140mm ja liittolevyn paksuuden 0.7mm riittää, kun tuelle asennetaan raudoite T4, mikä vastaa T10k150- raudoitusta.

19. Koulun aulan pilari on nivelisesti tuettu molemmista päistään, pilarin kuormitus Nd=750 kN, pituus L=4m. Suunnittele pilari liittorakenteisena taulukoita käyttäen, kun palonkestovaatimus on 60 min.

Valitaan neliöpilari 250x250x12.5, teräs S355, betoni K40-2, teräsA500HW.Taulukon mukaan pilarin B+4T25 A60 normaalivoimakapasiteettion 4m nurjahduspituudella noin 800 kN. Pilari siis varustetaan palotilanteen varalta 4 kpl T25 pääteräksillä ja haoilla ja betonoidaan K40-2 betonilla.


190

α 0.34:=

Muunnettu hoikkuus:

λkLcix π⋅

fyE

⋅:= λk 1.109=

Apusuure β:

β1 α λk 0.2−( )⋅+ λk

2+

2 λk2

⋅

:= β 1.032=

Reduktiokerroin:

χ β β2 1

λk2

−

−

:=χ 0.53=

Nrcx χ fd⋅ A⋅:= Nrcx 1.637 106×= Mrx fd Wx⋅:= Mrx 3.243 108×=

NR fd A⋅:= NR 3.088 106×= Nelx

NR

λk2

:= Nelx 2.512 106×=

C 1.0:=

NdNrcx

CMdMrx

⋅

1

1Nd Nrcx⋅

NR Nelx⋅−

⋅+ 0.766= Valittu poikkileikkaus OK.

KAKSOISSYMMETRISEN PURISTETUN JA TAIVUTETUNSAUVAN KESTÄVYYS B7Mastojäykistetyn teollisuushallin pääpilarin , L=6000mmm, laskentakuormat ovat Nd=200 kN ja Md=200 kNm. Valitse HE- profiili, kun teräs on S235.

Alkuarvot: Kokeillaan HE280B L 6 103⋅:=

Md 200 106⋅:= Nd 200 103⋅:= ix 121:= fy 235:=

A 131.4102⋅:= Wx 1380 103⋅:= fd 235:= E 2.1 105⋅:=

Alustava mitoitus:

Ndfd A⋅

Mdfd Wx⋅

+ 0.681= <1 jatketaan HE280B-profiililla

Tarkempi mitoitus yhteisvaikutusyhtälöllä:

fy 235=Nd 2 105×= Md 2 108×=

Lc 2.1 6000⋅:= ix 121=


189

KANTAVAN PROFIILIPELLIN ALUSTAVA VALINTATAULUKOIDEN AVULLA / B6

Valitse vesikaton kantava profiilipelti taulukoidenavulla seuraaviin rakenteisiin, lumikuorma qk=1.8 kN/m2.Ks. osoite www.rautaruukki.fi

a) kylmä autokatos, kantava profiilipelti yksiaukkoinen, aukonpituus 5m

b) lämpöisen teollisuushallin vesikatteen kantava, 2-aukkoinen profiilipelti, orsiväli 3000 mm, päällä lämpöeriste ja vedeneristys, paino 0.3 kN/m2.

c) lämpöisen teollisuushallin ulkoseinän verhouspelti, orret k/k 1200, tuulen paine 0.5 kN/m2.

KATTO- ja SEINÄORREN ALUSTAVA VALINTATAULUKOIDEN AVULLA B6

Valitse teollisuushallin vesikatto-orsi. Orsi on 2-aukkoinen,aukon pituus on 6m, k/k = 2400 mm.Ks osoite www.rautaruukki.fi , Teräsrakenne-oppimisympäristö

Valitse teollisuushallin seinäorsi. Orsi on 2-aukkoinen,aukon pituus on 6m, k/k = 1200 mm.Ks osoite www.rautaruukki.fi, Teräsrakenne-oppimisympäristö


188

Wx 252000:=

VR 0.6 fd⋅ tw⋅ b⋅:= VR 2.378 105×= > V2d 5.375 104

×=

>MR fd Wx⋅:= MR 5.922 107×= M2d 5.375 107×=

Yhteisvaikutusyhtälö:

Md M2d:= Vd V2d:=

MdMR

0.63VdVR

⋅+

1.05= 1.38< OK

Käyttörajatilamitoitus:

Ix 27700000:= E 2.1 105⋅:=

ymaxgk qk+( ) L4

⋅187 E⋅ Ix⋅

:= ymax 6.32=L

40012.5= OK

ymax2 0.521 gk qk+( )⋅L4 10 2−⋅

E Ix⋅⋅:= ymax2 6.158=

SIVUTTAISTUETTU KAKSIAUKKOINEN PALKKIALUSTAVAN MITOITUKSEN MUKAAN / B7

Mitoita sivuttaistuettu kaksiaukkoinen valssattu teräspalkki alustavan mitoituksen kaavoilla, kun aukkomitta L= 5000 mm, kuormituksena ja tasainen kuorma qk=10 kN/m.

Alkuarvot: Teräs S235 IPE L 5000:= gk 1:= qk 10:=

fy 235:= γ 1.0:= fdfyγ

:=

Laskentakuormat:

gd 1.2 1⋅:= qd 1.6 qk⋅:=

M2d gd qd+( )L2

8⋅

:= M2d 5.375 107×=

V2d 0.625 gd qd+( )⋅ L⋅:= V2d 5.375 104×=

Kimmoteorian mukainen mitoitus poikkileikkausluokan 3 perusteella:

WtarpM2d

fd:= Wtarp 2.287 105

×= IPE 220 Wx=252 000 mm3

IPE 220- profiilin kimmoteorian perusteella määritetyt kestävyydet ovat:

fd 235= b 220 9.2−( ):= tw 8:= Aw b tw⋅:=


187

LASKUESIMERKKEJÄ B7 MUKAAN

Osiossa esitetään laskuesimerkkejä RakMK B7 mukaan. Tehtävänä on ratkaista tehtävät EC3 mukaan.


186

Esimerkki 4.5

Määritettävä kuvan 4.57 pienahitsin vaadittu a-mitta, kun teräs on S 275, t = 10 mm, Lw= 200 mm ja FSd = 300 kN.

Kaavasta 4.35 saadaan a ≥⋅ ⋅ ⋅

⋅300 0 85 1 25 3

200 0 430, ,

,mm = 6,4 mm ⇒ valitaan a = 7 mm


185

Esimerkki 4.4

Laskettava kuvan 4.56 liitoksessa tarvittava pienahitsin a-mitta, kun teräs on S235, Lw = 80 mm, t = 15 mm ja FSd = 150 kN.

Sijoittamalla annetut arvot kaavaan 4.36 saadaan

akN

mm kN mm≥

⋅ ⋅ ⋅

⋅

150 0 8 1 25 280 0 360 2

, ,, /

= 7,4 mm ⇒ valitaan a = 8 mm


184

Esimerkki 3.6. Keskisesti kuormitettu pilari.Tarkista kuvan 3.40 mukaisen pilarin normaalivoimakestävyys, kun pilari pääsee nurjahtamaan poikkileikkauksen heikompaan suuntaan. Pysyvän kuorman osuus normaalivoimasta Ngk = 200 kN.

1. LähtöarvotTeräs S235 ⇒ fy = 235 N/mm2

Profiili HE200A ⇒ A = 5383 mm2

iz = 49,8 mmpoikkileikkausluokka 1, koskauuman d/tw = (190 - 2 ⋅ 10 - 2 ⋅ 18)/6,5 = 20,6 < 33ε = 33

laipan c/tf = 2002 10⋅

= 10 = 10ε

Nurjahduspituus Leff = γ ⋅ L = 0,8 ⋅ 4000 mm = 3200 mm

Koska profiilin h/b <1,2 ⇒ nurjahduskäyrä c⇒ α = 0,49

⇒ λπ

_

,,

=⋅

⋅⋅3200

49 81 0 235210000

= 0,6842

⇒ φ = 0,5 ⋅ [1 + 0,49 ⋅ (0,6842 - 0,2) + 0,68422] = 0,8527

⇒ χ =+ −

10 8527 0 8527 0 68422 2, , ,

= 0,7344

Nb.Rd = 0 7344 1 0 53800 23511

22, ,

,,

⋅ ⋅ ⋅mmkN

mm= 844 kN

NSd = (1,2 ⋅ 200 + 1,5 ⋅ 380) kN = 810 kN < Nb.Rd ⇒ profiili on sopiva.


183

Yhteisvaikutusta ei tarvitse tarkistaa, koska normaalivoimaa ei esiinny ja taivutusmomentin sekä leikkausvoiman suurimmat arvot vaikuttavat eri poikkileikkauksissa. IPE 550 on siten sopiva profiili.

>MSdMbRd 555.714kNm=MbRd χLT βw⋅ Wply⋅fy

γM⋅:=

χLT 0.933=χLT 1

φLT φLT2λLTmuunn2−+

:=

φLT 0.659=φLT 0.5 1 αLT λLTmuunn 0.2−( )⋅+ λLTmuunn2+ ⋅:=

αLT 0.21:=λLTmuunn 0.47=λLTmuunn( ) λLTλl

βw⋅:=

βw 1:=λl 93.913=λl πEfy

⋅:=

λLT 44.138=λLT

L1iz

C1( ) 1120

L1izhtf

2

⋅+⋅

:=

C1 1:=

> VSd

VplRd 276.599kN=VplRd Avfy

γM 3⋅

⋅:=

Av2 6349.2mm2=Av2 1.04 h⋅ tw⋅:=

Av 2242.52mm2=Av A 2 b⋅ tf⋅− tw 2 r⋅+( ) tf⋅+:=

Leikkaus


182

L1 2m=

b 210 mm⋅:=h 550 mm⋅:=

r 24 mm⋅:=tf 17.2 mm⋅:=tw 11.1 mm⋅:=iz 43.1 mm⋅:=

Wply 2787000mm3⋅:=

A 8450 mm2⋅:=

Vaaditun taivutusvastuksen ja jäyhyysmomentin perusteella valitaanIPE 550:

Poikkileikkausarvot:A = 8450 mm2Iy = 671200000 mm4Wpl.y= 1310 103 mm3 (Wel = 1160 103 mm3)iz = 43.1 mmtw = 11,1 mmtf = 17,2 mmhs = (400 - 13,5) mm = 387 mmr = 24 mm (pyöristyssäde)gk = 0,663 kN/m < 0,7 kN/m (oletus)h =550 mmb = 210 mm

Itarv 57977.431cm4=Itarv k δf⋅MkE

⋅ L⋅:=

Mk maxMk:=k 250:=δf 548

:=

Taipuman rajatila on L/250 (vrt. taulukko 2.4), sillä kokonaistaipuman rajatila antaa pienemmän tuloksen jäyhyysmomentille. k = 250

Taipuma

Wpltarv 2239.787cm3=WpltarvMsdfyd

:=Msd maxMsd:=PL1

Taivutus

E 2.1 105⋅N

mm2⋅:=

fyd 213.636N

mm2=fyd

fyγM

:=γM 1.1:=fy 235N

mm2⋅:=S235

LujuudetMITOITUS


181

maxMsd 478.5kNm=

maxMsd 3 Fgd1 Fqd1+( )L2

⋅

Fgd1 Fqd1+( ) 5 m⋅ 3 m⋅+ 1 m⋅+( )⋅[ ]− pgd2L2

8⋅+

:=

maxVsd 120.36kN=maxVsd 3 Fgd1 Fqd1+( ) Lpgd2

2⋅+:=

Muuttuvien ja pysyvien kuormien aiheuttamat voimasuureet käyttötilassa:

pgd2 0.84kNm

=pgd2 γG pgk2⋅:=Palkin op:n metrikuorma:

Fqd1 32.4kN=Fqd1 γQ kk⋅ L1⋅ qk1⋅:=Orren pistekuorma lumesta: Fgd1 5.76kN=Fgd1 γG kk⋅ L1⋅ gk1⋅:=Orren pistekuorma rakenteista:

γQ 1.5:=γG 1.2:=

MURTORAJATILA

maxMk 333.95kNm=

maxMk 3 Fgk1 Fqk1+( )L2

⋅

Fgk1 Fqk1+( ) 5 m⋅ 3 m⋅+ 1 m⋅+( )⋅[ ]− pgk2L2

8⋅+

:=

maxVk 84.1kN=maxVk 3 Fgk1 Fqk1+( ) Lpgk2

2⋅+:=

Muuttuvien ja pysyvien kuormien aiheuttamat voimasuureet käyttötilassa:

maxMq1 259.2kNm=maxMq1 3 Fqk1⋅L2

⋅ Fqk1( ) 5 m⋅ 3 m⋅+ 1 m⋅+( )⋅−

:=

maxVqk1 64.8kN=maxVqk1 Fqk1 3⋅:=

Muuttuvan kuorman= lumikuorman aiheuttama voimasuureet käyttötilassa:


180

pgk2 0.70kNm

⋅:=Palkin op:n metrikuorma:

Fqk1 21.6kN=Fqk1 kk L1⋅ qk1⋅:=Orren pistekuorma lumesta: Fgk1 4.8kN=Fgk1 kk L1⋅ gk1⋅:=Orren pistekuorma rakenteista:

Voimasuureet:

KÄYTTÖRAJATILA:

kk 6 m⋅:=L1 2 m⋅:=L 14 m⋅:=

gk2 0.70kNm

⋅:=

qk1 1.8kN

m2⋅:=

gk1 0.40kN

m2⋅:=

Palkin kuormitus:kate + eriste 0,30 kN/m2orret 0,10 “ pysyvä kuorma yhteensä 0,40 kN/m2

lumikuorma 1,80 kN/m2

palkin omapaino 0,70 kN/m2

Palkin geometria:jänneväli L = 10 morsiväli L1 = 2 mkehäväli kk = 6 m

Kuva: Vapaasti tuettu vesikaton pääkannattaja. kNm 1000N m⋅:=

kN 1000N:=

Esimerkki : Vapaasti tuettu vesikaton pääkannattajaMitoitettava vapaasti tuettu vesikaton pääkannattaja, jonka jänneväli L = 14 m. Kehäväli on 6,0 m ja orsiväli L1 = 2,0 m. Teräs S235J2G3


179

max MSd = (80,5 ⋅ 5 -12

⋅ 0,84 ⋅ 52 - 38,16 ⋅ 4) kNm = 239 kNm

S235 ⇒ fy = 235 N/mm2 (t < 16 mm)Poikkileikkausluokka 1 ⇒ γM0 = 1,1

⇒ Wpl ≥11 239 10

235

6, ⋅ ⋅mm3 = 1119 ⋅ 103 mm3

Taipuman rajatila on L/250 (vrt. taulukko 2.4), sillä kokonaistaipuman rajatila antaa pienemmän tuloksen jäyhyysmomentille. ⇒ k = 250

δf = 548

⇒ I ≥ 250548

130 102 1 10

100006

5⋅ ⋅⋅⋅

⋅,

mm4 = 16120 ⋅ 104 mm4 (kaava 3.24)

Vaaditun taivutusvastuksen ja jäyhyysmomentin perusteella valitaan IPE 400.

Poikkileikkausarvot: Iy = 23130 ⋅ 104 mm4

Wpl.y = 1310 ⋅ 103 mm3 (Wel = 1160 ⋅ 103 mm3)iz = 39,5 mmtw = 8,6 mmhs = (400 - 13,5) mm = 387 mmr = 21 mm (pyöristyssäde)gk = 0,663 kN/m < 0,7 kN/m (oletus)

Av = 8450mm2 - ( )2 180 13 5 8 6 2 21 13 5⋅ ⋅ + + ⋅ ⋅mm mm mm mm, , , = 4273 mm2

Leikkauskestävyys Vpl.Rd = 42730 235

11 32

2

mmkN mm

⋅⋅

, /,

= 527 kN > VSd (kaava 3.7)

λLT =

⋅ + ⋅

200039 5

1 1120

2000 39 5400 13 5

2

,

/ ,/ ,

= 47,3 (kaava 3.30)

λ1 = 93,9 ⇒ λ_

LT =47 393 9

,,

= 0,504 ( kaava 3.26 kun βw = 1)

⇒ χLT = 0,9229 (taulukosta 3.4 interpoloimalla)

⇒ Mb.Rd = 0 9229 1 1310 100 235

113 3

2

,, /

,⋅ ⋅ ⋅ ⋅mm

kN mm= 258 kNm > MSd

Yhteisvaikutusta ei tarvitse tarkistaa, koska normaalivoimaa ei esiinny ja taivutusmomentin sekä leikkausvoiman suurimmat arvot vaikuttavat eri poikkileikkauksissa. IPE 400 on siten sopiva profiili. Jos teräkseksi valitaan S355J2G3, voidaan profiiliksi valita IPE 360.


178

Kuva 3.8. Vapaasti tuettu vesikaton pääkannattaja

Esimerkki 3.2Mitoitettava vapaasti tuettu vesikaton pääkannattaja (kuva 3.8), jonka jänneväli L = 10 m. Kehäväli on 6,0 m ja orsiväli L1 = 2,0 m. Teräs S235J2G3.

.

Palkin kuormitus:kate + eriste 0,30 kN/m2

orret 0,10 “ pysyvä kuorma yhteensä 0,40 kN/m2

lumikuorma 1,80 kN/m2

Ominaislumikuorma orrelta palkille Fqk = 6,0 m ⋅ 2,0 m ⋅ 1,8 KN/m2 = 21,6 kN⇒ max Vqk = 2 ⋅ 21,6 kN = 43,2 kN

max Mqk = (43,2 kN ⋅ 5,0 m - 21,6 kN ⋅ (3,0 + 1,0)m) = 130 kNm

Arvioidaan palkin omaksipainoksi gk ≈ 0,7 kN/m

⇒ max Mk = 18

0 7 10 1302⋅ ⋅ +

, kNm = 139 kNm

Laskentakuormat: gd = 1,2 ⋅ 0,7 kN/m = 0,84 kN/mFd = 1,2 ⋅ 6,0 m ⋅ 2,0 m ⋅ 0,4 kN/m2 + 1,5 ⋅ 21,6 kN = 38,16

kN

max VSd = 12

⋅ 0,84 kN/m ⋅ 10 m + 2 ⋅ 38,16 kN = 80,5 kN


177

LASKUESIMERKKEJÄ:Saarinen, Eero. Teräsrakenteiden suunnittelu, Eurocode. Sähköinen versio. Teräsrakenneyhdistys r.y. 2002. Julkaisematon

Saarinen, Eero. Teräsrakenteiden suunnittelu, Eurocode. Sähköinen versio. Teräsrakenneyhdistys r.y. 2002. Julkaisematon

Esimerkki 3.1Määritä kuvan 3.1 vedetyn sauvan kestävyys, kun teräs on S355 ja levyn paksuus t = 10 mm.On tutkittava leikkaukset I - I ja II - II:

AI = 10 mm ⋅ (120 - 18) mm = 1020 mm2

AII = 10 mm ⋅ (120 - 2 ⋅ 18 + 502/(4 ⋅ 60)) mm = 944 mm2

Tässä tapauksessa Anet = AII .γM0 = 1,1 (poikkileikkausluokka 1)γM2 = 1,25A = 10 mm ⋅ 120 mm = 1200 mm2

Npl.Rd = 1200 mm2 ⋅ 0,355 kN/mm2/1,1 = 387 kNNu.Rd = 0,9 ⋅ 944 mm2 ⋅ 0,510 kN/mm2/1,25 = 347 kN

⇒ Vetokestävyys Nt.Rd = 347 Kn


176

5.4 Design Steps for Compression Members

To design a simple compression member it is first necessary to evaluate its two effective lengths, in relation to the two principal axes, bearing in mind the expected connections at its ends. Secondly, the required second moment of area to resist the Euler critical loads should be calculated to give an idea of the minimum cross-section necessary. The verification procedure should then proceed as follows:

• the geometric characteristics of the shape, and its yield strength give the reference slenderness (Equation (9)).

• χ is calculated, taking into account the forming process and the shape thickness, using one of the buckling curves and (Equations 10 and 11).

Buckling resistance of a compression member is then taken as:

Nb.Rd = χ A fy/γM1 (13)

the plastic resistance as:

Npl.Rd = A fy/γM0 (14)

and the local buckling resistance as:

No.Rd = Aeff fy/γM1 (15)

If these are higher than the design axial load, the column is acceptable; if not, another larger cross-section must be chosen and checked.

In addition, torsional or flexural-torsional buckling of the column must be prevented.


o Many different kinds of cross-sections are used as compression members; these include simple members, built-up, tapered and stepped columns.

o A stub column (with ≤ 0,2) can achieve the full plastic resistance of the cross-section and buckling does not need to be checked.

o If > 0,2, reduction of the load resistance must be considered because of buckling. Columns with medium slenderness fail by inelastic buckling and slender columns by elastic buckling.

o European buckling curves give the reduction factor for the relevant buckling mode depending on the shape of the cross-section, the forming-process, the reference slenderness and the axis about which buckling can occur. They take into account experimental and theoretical approaches and give reliable results.


175

pinned. The top chord is designed for axial load and bending. Computer analysis is mentioned below.

3. Moments resulting from rigid joints and truss deflection

Stresses resulting from secondary moments are important in trusses with short thick members. Approximate rules specify when such an analysis should be made. Secondary stresses will be insignificant if the slenderness of the chord members in the plane of the truss is greater than 50 and that of most of the web members is greater than 100. In trusses in buildings, the loads are predominantly static and it is not necessary to calculate these stresses. The maximum stresses from secondary moments occur at the ends of members and are not likely to cause collapse. However, where fatigue effects are significant, these secondary stresses have to be considered. The method of analysis for secondary moments is set out below.

4.3 Rigorous Elastic Analysis

Rigid jointed, redundant or continuous trusses or trusses with loads applied between the nodes can be analysed using a plane frame program based on the matrix stiffness method of frame analysis. The truss can also be modelled taking account of joint eccentricity. Member sizes must be determined in advance using a manual analysis. All information required for design is output including joint deflections.

It is important that a consistent approach is adopted for analysis and design. This means that if secondary moments are to be ignored then the primary axial forces to be used in design must be obtained from the simple analysis of the truss as a pin jointed frame. The axial forces obtained from a rigid frame computer analysis may be modified considerably by the joint moments.


174

j is the number of joints.

However compliance with this formula for the truss as a whole does not preclude the possibility of a local mechanism in part of the truss.

Manual methods of analysis for trusses, where the loads are applied at the nodes, are joint resolution, the method of sections and the force diagram. Joint resolution is the quickest method for analysing parallel chord lattice girders when all the forces are required. The method of sections is useful where the values of the forces in only a few critical members are required. The force diagram is the best general manual method. Computer programs are also available for truss analysis.

4.2 Secondary Stresses in Trusses

In many cases in the design of trusses and lattice girders, it is not necessary to consider secondary stresses. These stresses should, however, be calculated for heavy trusses used in industrial buildings and bridges.

Secondary stresses are caused by:

• Eccentricity at connections• Loads applied between the truss nodes• Moments resulting from rigid joints and truss deflection.

They are discussed in detail below:

1. Eccentricity at connections

Trusses should be detailed so that either the centroidal axes of the members or the bolt gauge lines meet at a point at the nodes. Otherwise, members and connections should be designed to resist the moments due to eccentricity. These moments should be divided between members meeting at joints in proportion to their rotational stiffnesses. Stresses due to small eccentricities are often neglected.

2. Loads applied between the truss nodes

Moments due to these loads must be calculated and the stresses arising combined with those due to primary axial loads; that is the members concerned must be designed as beam-columns. This situation often occurs in roof trusses where the loads are applied to the top chord through purlins which may not be located at the nodes, as shown in Figure 7. The manual method of calculation is first to analyse the truss for the loads applied at the nodes which gives the axial forces in the members. Then a separate analysis is made for bending in the top chord which is considered as a continuous beam. The ridge joint E is fixed because of symmetry, but the eaves joint A should be taken as pinned; otherwise, moment will be transferred into the bottom chord if the joint between the truss and column is assumed to be


173

carefully estimated from material weights given in handbooks and manufacturers' literature.

2. Imposed loads. These are given in Eurocode 1 [1] for floors in various types of building and for roofs with or without access. The imposed load may cover the whole or part of the member and should be applied in such a way as to cause the most severe effect.

3. Wind loads. These are given in Eurocode 1 [1] and can be estimated from the location of the building, its dimensions and the sizes of openings on its faces. Wind generally causes uplift on roofs and this can lead to reversal of load in truss members in light construction. In multi-storey buildings, wind gives rise to horizontal loads that must be resisted by the bracing.

In special cases, trusses resist dynamic, seismic and wave loads. A careful watch should be kept for unusual loads applied during erection. Failures may occur at this stage when the final lateral support system is not fully installed.

For bridges, in addition to the dead loads and the vertical effects of live loads due to highway or railway loading, horizontal effects of live load have to be considered. These include braking and traction effects, centrifugal loads and accidental skidding loads. Temperature effects are significant in some bridges.

4. ANALYSIS OF TRUSSES4.1 General

Trusses may be single span, statically determinate or indeterminate, or may be continuous over two or more spans, as shown in Figure 6. Only single span, statically determinate, trusses are considered in this section.

A truss is usually statically determinate when:

m = 2j - 3,

where m is the number of members in a truss


172

3. LOADS ON TRUSSES AND LATTICE GIRDERS

The main types of loads on buildings are shown in Figure 5, namely:

1. Dead loads. These are caused by self-weight, sheeting, decking, floor or roof slabs, purlins, beams, insulation, ceilings, services and finishes. Dead loads for the construction to be used in any particular case must be


171

The selection of members depends on the location, use, span, type of connection and the appearance required. Hollow sections are more expensive than open sections but are cheaper to maintain and have a better appearance. However, in exposed trusses corrosion can occur at the crevices which are formed at gusset positions. Angles are the sections traditionally used for small span truss construction.


170

single member. Loads applied in such a way as to cause bending, shear, or torsion will usually result in inefficient use of material.

Trusses and lattice girders are classified in accordance with the overall form and internal member arrangement. Pitched trusses are used for roofs. Parallel chordlattice girders are used to support roofs and floors and for bridges, although in continuous bridges, additional depth is often required at the piers. In the past, proper names were given to the various types of trusses such as the Fink truss, Warren girder, etc.. The most commonly used truss is single span, simply supported and statically determinate with joints assumed to act as pins.

The Vierendeel girder should also be mentioned. It consists of rigid jointed rectangular panels as shown in Figure 1d. This truss is statically indeterminate and will not be further considered in this lecture, although it has a pleasing appearance and is often used in foot-bridges.

The saving over a plate girder is clear when the webs are considered. In a truss the webs are mainly fresh air - hence less weight and less wind force.

A truss can be assembled from small easily handled and transported pieces, and the site connections can all be bolted. Trusses can have a particular advantage for bridges in countries where access to the site is difficult or supply of skilled labour is limited.

2. TYPICAL MEMBERS

Truss, lattice girder and bracing members for buildings are selected from:

• open sections, primarily angles, channels, tees and joists.• compound sections, i.e. double angle and channels.• closed sections, in practice structural hollow sections.

For bridges, members are selected from:

• rolled sections.• compound sections.• built-up H, top hat and box sections.

Typical sections are shown in Figure 4.


169

The principle of a truss is simple. The structure is composed of top and bottom chords triangulated with diagonals in the webs so that each member carries purely axial load. Additional effects do exist but in a well designed truss these will be of a secondary nature.

A global moment on a truss is carried as compression and tension in the chords. A global shear is carried as tension or compression in the diagonal members. In the simplified case, where joints are considered as pinned, and the loads are applied at the panel points, the loading creates no bending moment, shear, or torsion in any


168

1. INTRODUCTION

The truss or lattice girder is a triangulated framework of members where loads in the plane of the truss or girder are resisted by axial forces in the individual members. The terms are generally applied to the planar truss. A 'space frame' is formed when the members lie in three dimensions.

The main uses are:

• in buildings, to support roofs and floors, to span large distances and carry relatively light loads, see Figure 1.

• in road and rail bridges, for short and intermediate spans and in footbridges, as shown in Figure 2.

• as bracing in buildings and bridges, to provide stability where the bracing members form a truss with other structural members such as the columns in a building. Examples are shown in Figure 3.


167

Lecture 7.12: Trusses and Lattice Girders

OBJECTIVE/SCOPE

To introduce two-dimensional trusses: types, uses and principal design considerations.

PREREQUISITES

None.

RELATED LECTURES

None.

SUMMARY

This lecture presents the types and uses of trusses and lattice girders and indicates the members that are most often used in their construction. A discussion of overall truss design considers primary analysis, secondary stresses, rigorous elastic analysis, cross- braced trusses and truss deflections. The practical design of truss members is discussed.


166

First-order or second-order analysis may be used. If the analysis is first-order, second-order effects may be allowed for in an appropriate way when designing the columns by using the results of a first-order analysis and either:

- using amplified sway moments, or

- using the sway-mode buckling lengths.

• When second-order elastic global analysis is used, in-plane buckling lengths for the non-sway mode may be used for member design.

• In the amplified sway moments method, the sway moments found by a first-order elastic analysis should be increased by multiplying them by the ratio:

where VSd is the design value of the total vertical load.

Vcr is its elastic critical value for failure in a sway mode.

• The amplified sway moments method should not be used when the elastic critical load ratio VSd/Vcr is more than 0,25.

• Sway moments are those associated with the horizontal translation of the top of a storey relative to the bottom of that storey. They arise from horizontal loading and may also arise from vertical loading if either the structure or the loading is asymmetrical.

• As an alternative to determining VSd/Vcr directly, the following approximation may be used in beam-and-column type frames:

where δ, h, H and V are as defined previously.

• When the amplified sway moments method is used, in-plane buckling lengths for the non-sway mode may be used for member design.

• When first-order elastic analysis with sway-mode in-plane buckling lengths is used for column design, the sway moments in the beams and the beam-to-column connections should be amplified by at least 1,2 unless a smaller value is shown by analysis to be adequate.

Rules for the application of plastic analysis procedures to sway frames are given in Clause 5.2.6.3 of Eurocode 3 [1].


165

ns is the number of storeys.

• Columns which carry a vertical load NSd of less than 50% of the mean value of the vertical load per column in the plane considered, shall not be included in nc.

• Columns which do not extend through all the storeys included in ns shall not be included in nc. Those floor levels and roof levels which are not connected to all the columns included in nc shall not be included when determining ns.

• These initial sway imperfections apply in all horizontal directions, but need only be considered in one direction at a time.

• The possible torsional effects on the structure of anti-symmetric sways, on two opposite faces, shall also be considered.

• If more convenient, the initial sway imperfection may be replaced by a closed system of equivalent horizontal forces, see Figure 7.

• In beam-and-column building frames, these equivalent horizontal forces should be applied at each floor and roof level and should be proportionate to the vertical loads applied to the structure at that level, see Figure 8.

• The horizontal reactions at each support should be determined using the initial sway imperfection and not the equivalent horizontal forces. In the absence of actual horizontal loads, the net horizontal reaction is zero.


164

5. DEFLECTIONS

Excessive deflections, whilst not normally leading to a structural failure, may, nonetheless, impair the serviceability of the structure.

Deflection might, for example, cause:

• cracking of plaster ceilings.• misalignment of crane rails.• difficulty in opening large doors.

Since these affect the performance of the structure in its working conditions, it is usual to conduct this type of check at service load levels. Eurocode 3 suggests a limit of span/300 for the maximum deflection of beams supporting floors under service live load [1]. Such limits should, however, be regarded as advisory only, and smaller or larger values can be more appropriate in any given situation. Wherever possible, specialist advice should be sought; crane manufacturers, for example, should be able to give detailed guidelines regarding permissable deflection for gantry beams.

7.3 Sway Frames

Sway frames shall be analysed under those arrangements of the variable loads which are critical for failure in a sway mode. In addition, sway frames shall also be analysed for the non-sway mode.

The initial sway imperfections, and member imperfections where necessary, shall be included in the global analysis of all frames.

The allowance for imperfections in the analysis of sway frames is intended to cover effects such as lack of verticality, lack of straightness, residual stresses, etc. It is expressed in Eurocode 3 by means of a set of equivalent geometrical imperfections [1]. These imperfections are not actual construction tolerances but, because they are intended to represent the effect of a number of factors, are likely to be larger than such tolerances. The form specified in Eurocode 3 is:

• The effects of imperfections shall be allowed for in frame analysis by means of an equivalent geometric imperfection in the form of an initial sway imperfection φ determined from:

φ = kc ks φo

with φo = 1/200

kc = [0,5 + 1/nc]0,5 but kc ≤ 1,0

and ks = [0,2 + 1/ns]0,5 but ks ≤ 1,0

where nc is the number of columns per plane.


163

4. DESIGN OF BEAMS FOR SHEAR

Although bending will govern the design of most steel beams, situations will arise, e.g. short beams carrying heavy concentrated loads, in which shear forces are sufficiently high for them to be the controlling factor.

Figure 6 illustrates the pattern of shear stress found in a rectangular section and in an I-section assuming elastic behaviour. In both cases shear stress varies parabolically with depth, with the maximum value occurring at the neutral axis. However, for the I-section, the difference between maximum and minimum values for the web, which carries virtually the whole of the vertical shear force, is sufficiently small for design to be simplified by working with the average shear stress, i.e. total shear force/web area. Since the shear yield stress of steel is approximately 1/√3 of its tensile yield stress, a suitable value for "permissible" shear stress when using elastic design is 1/√3 of the permissible tensile stress.

The ultimate shear resistance (based on plastic principles) used in Eurocode 3 [1] is fyd/√3; this is used in conjunction with a shear area Av, examples of which are:

rolled I-section, load parallel to web Av = A-2btf + (tw+2r)tf

plates and solid bars Av = A

circular hollow sections Av = 2A/π

In cases where high shear and high moment coexist, e.g. the internal support of a continuous beam, it may sometimes be necessary to allow for interaction effects. However, since the full shear capacity may be developed in the presence of quite large moments and vice versa (Eurocode 3 only requires reductions in moment resistance when the applied shear exceeds 50% of the ultimate shear resistance [1]) this will not often be required.


162

will enable the structure to withstand loads beyond that load which produces the first plastic hinge, until eventually collapse will occur when sufficient hinges have formed to convert the structure into a mechanism, as shown in Figure 5. Utilisation of this property of redistribution is termed "plastic design". It can only be used for continuous structures and then only when certain restrictions on cross-sectional geometry, member slenderness, etc.., are observed. The topic is covered in detail in Lecture 7.8.2 in the context of beams and in Lecture 7.11 in the context of frames.


161

Utilisation of the plastic part of the stress-strain curve for steel enables moments in excess of those which just cause yield to be carried. At full plasticity the distribution of bending stress in a doubly-symmetrical section will be as illustrated in Figure 4, with half the section yielding in compression and half in tension.

The corresponding moment is termed the fully plastic moment Mpl. It may be calculated by taking moments of the stress diagram about the neutral axis to give:

Mpl = fy Wpl (3)

where fy is the material yield stress (assumed identical in tension and compression).

Wpl is the plastic section modulus.

Basing design on Equation (3) means that the full strength of the cross-section in bending is now being used, with the design condition being given by:

Wpl ≥ M/fyd (4)

where M is the moment at cross-section under consideration.

fyd is the design strength (material yield strength divided by a suitable material factor).

When M is due to the factored loads Equation (4) represents the design condition of ultimate bending strength used in Eurocode 3 for beams whose cross-sections meet at least the Class 2 limits [1]. It is usual in codes such as Eurocode 3 for the value of fyd to be taken as the material yield strength, reduced slightly so as to cover possible variations from the expected value.

For continuous (statically indeterminate) structures, attainment of Mpl at the point of maximum moment will not normally imply collapse. Providing nothing triggers unloading at this point, e.g. local buckling does not occur, then the local rotational stiffness will virtually disappear, i.e. the cross-section will behave as if it were a hinge, and the pattern of moments within the structure will alter from the original elastic distribution as successive plastic hinges form. This redistribution of moments


160


159

it follows that limiting this to a fraction of the material yield stress gives a design condition of the form:

W ≥ M/fd (2)

where W =

fd is the limiting normal bending stress.

When M is taken as the moment produced by the working loads, this approach todesign is termed 'elastic' or 'permissible stress' design and is the method traditionally used in many existing codes of practice. In more modern Limit States codes fd is taken as the material strength fy possibly divided by a suitable material factor γM and M is taken as the moment due to the factored loads i.e. the working loads suitably increased so as to provide a margin of safety in the design. Equation (2), in this case, represents the condition of first yield. Values of W for the standard range of sections are available in tables of section properties.

Selection of a suitable beam therefore comes down to:

1. determination of the maximum moment in the beam,2. extraction of the appropriate value of fd from a suitable code,3. selection of a section with an adequate value of W subject to

considerations of minimum weight, depth of section, rationalisation of sizes throughout the structure, etc...

Clearly, sections for which the majority of the material is located as far away as possible from the neutral axis will tend to be the most efficient in elastic bending. Figure 2 gives some quantitative idea of this for some of the more common structural shapes. I-sections are most often chosen for beams because of their structural efficiency; being open sections they can also be connected to adjacent parts of the structure without undue difficulty. Figure 3 gives some typical examples of beam-to-column connections.


158

If it is designed with an elastic-plastic analysis (elasto-plastic or elastic-perfectly plastic), the values of eo are functions of the buckling length L, and are given in Table 3.

3. DESIGN OF BEAMS FOR SIMPLE BENDING

For a doubly-symmetrical section or a singly-symmetrical section bent about the axis of symmetry, the basic theory of bending, assuming elastic behaviour, gives the distribution of bending stress shown in Figure 1.

Since the maximum stress σmax is given by:

σmax = (1)

where M is the moment at cross-section under consideration.

d is the overall depth of section.

I is the second moment of area about the neutral axis (line of zero strain).


157

Curve a represents quasi perfect shapes: hot-rolled I-sections (h/b > 1,2) with thin flanges (tf ≤ 40mm) if buckling is perpendicular to the major axis; it also represents hot-rolled hollow sections.

Curve b represents shapes with medium imperfections: it defines the behaviour of most welded box-sections; of hot-rolled I-sections buckling about the minor axis; of welded I-sections with thin flanges (tf ≤ 40mm) and of the rolled I-sections with medium flanges (40 < tf ≤ 100mm) if buckling is about the major axis; it also concerns cold-formed hollow sections where the average strength of the member after forming is used.

Curve c represents shapes with a lot of imperfections: U, L, and T shaped sections are in this category as are thick welded box-sections; cold-formed hollow sections designed to the yield strength of the original sheet; hot-rolled H-sections (h/b ≤ 1,2 and tf ≤ 100mm) buckling about the minor axis; and some welded I-sections (tf ≤40mm buckling about the minor axis and tf > 40mm buckling about the major axis).

Curve d represents shapes with maximum imperfections: it is to be used for hot-rolled I-sections with very thick flanges (tf > 100mm) and thick welded I-sections (tf> 40mm), if buckling occurs in the minor axis.

Table 4 helps the selection of the appropriate buckling curve as a function of the type of cross-section, of its dimensional limits and of the axis about which buckling can occur. For cold-formed hollow sections, fyb is the tensile yield strength and fya is the average yield strength. If the cross-section in question is not one of those described, it must be classified analogously.

It is important to note that the buckling curves are established for a pin-ended, end loaded member; it is necessary carefully to evaluate the buckling lengths if the boundary conditions are different, see Lecture 7.7.

5.3 Equivalent Initial Bow Imperfection

To study a column using second order theory, it is necessary to choose geometrical imperfections (initial out-of-straightness and eccentricities of loading) and mechanical imperfections (residual stresses and variations of the yield stress). Eurocode 3 proposes values for a bow imperfection, eo, whose effect is equivalent to a combination of the two previous kinds of imperfections [1].

If the column is designed using elastic analysis, eo is as follows:

eo = α ( - 0,2) Wpl/A for plastic design of cross-sections

or,

eo = α ( - 0,2) Wel/A for elastic design of cross-sections (12)


156

3.

5.2 Basis of the ECCS Buckling Curves

From 1960 onwards, an international experimental programme was carried out by the ECCS to study the behaviour of standard columns [2]. More than 1000 buckling tests, on various types of members (I, H, T, U, circular and square hollow sections), with different values of slenderness (between 55 and 160) were studied. A probabilistic approach, using the experimental strength, associated with a theoretical analysis, showed that it was possible to draw some curves describing column strength as a function of the reference slenderness. The imperfections which have been taken into account are: a half sine-wave geometric imperfection of magnitude equal to 1/1000 of the length of the column; and the effect of residual stresses relative to each kind of cross-section.

The European buckling curves (a, b, c or d) are shown in Figure 14. These give the value for the reduction factor χ of the resistance of the column as a function of the reference slenderness for different kinds of cross-sections (referred to different values of the imperfection factor α).

The mathematical expression for χ is:

χ = 1/ {φ + [φ2 −€ 2]1/2} ≤ 1 (10)

where: φ = 0,5 [1 + α ( - 0,2) + 2] (11)

Table 1 gives values of the reduction factor χ as a function of the reference

slenderness .

The imperfection factor α depends on the shape of the column cross-section considered, the direction in which buckling can occur (y axis or z axis) and the fabrication process used on the compression member (hot-rolled, welded or cold-formed); values for α, which increase with the imperfections, are given in Table 2.


155

• In the case of columns with non-doubly symmetric cross-sections, the formation of "effective holes" may cause a shift in the neutral axis position resulting in eccentric compression and hence a beam-column problem.

• Beam-columns (compression and biaxial bending) are verified by using an interaction formula where the checks for the column in compression only, for the beam with bending about the y-axis only, and for the beam with bending about the z-axis only, are combined.

5. REFERENCES

[1] "Eurocode 3: "Design of steel structures" ENV 1993-1-1: Part 1.1, General rules and rules for buildings, CEN, 1992.


1. "Eurocode 3: Part 1.3: "Cold formed thin gauge members and sheeting", CEN, (in preparation).

2. Eurocode 3: Background Document 5.5. (Justification of the design resistances, for buckling verifications)


154

The lateral-torsional buckling check for beams is analogous to that for columns. The

non-dimensional slenderness LT = , is calculated with Mu=Weff.fy for the effective cross-section, and with Mcr calculated for the gross cross-sectional values. The design lateral-torsional buckling resistance is then given by:

Mb,Rd = χLT . Mu /γM1

where χLT is the reduction factor for the relevant buckling curve.

3.3 Beam-Columns

In the case of members that are subject to compression and monoaxial or biaxial bending (e.g. in the case of columns with monosymmetrical or unsymmetrical cross-sections) the design check is carried out using an interaction formula in which the checks for a centrally compressed column, for a beam with bending about the y-axis only, and for a beam with bending about the z-axis only, are combined.

If lateral-torsional bucking is prevented the interaction formula is as follows:

If lateral-torsional bucking can occur:

where eNy or eNz are the eccentricities due to the shift of the neutral axis for compression only. The resistances NbRd and Nbz.Rd are related to the case of central compression; Moy.Rd and Mby.Rd are related to bending about the y-axis only; and Moz.Rd is related to bending about the z-axis (weak axis) only.


• The design of members with Class 4 sections is carried out as for members with Class 3 sections (elastic analysis, elastic cross-sectional resistance limited by yielding in the extreme fibres) except that an effective cross-section (derived from gross sections with "effective holes", where buckles may occur) is used.

• The buckling checks for columns and the lateral-torsional buckling check for beams, requires the critical values, Ncr and Mcr, to be calculated using the gross cross-sectional data without considering "effective holes".


153

3. DESIGN OF MEMBERS3.1 Columns in Compression

As shown by the effective cross-sections 1 and 2, in Figure 4, the neutral axes of doubly-symmetrical cross-sections will not change with the formation of effective holes. Hence the compression load NSd that is central to the gross cross-sections will also be central to the effective cross-section.

The column buckling design check, therefore, is based on the non-dimensional

slenderness = , where Ncr is calculated on the basis of the gross cross-section and Npl is calculated using the effective cross-sectional area Aeff (Npl = Aeff . fy).

The design buckling resistance is given by:

NbRd = χ . Npl /γM

where χ is the reduction factor for the relevant buckling curve.

• For singly symmetrical cross-sections - type 3 in Figure 4 - or unsymmetrical cross-sections, the formation of effective holes may lead to a shift, eN, in the neutral axes position. The compression load, NSd, that is central to the gross cross-section will, therefore, be eccentric to the effective cross-section and hence will cause an additional bending moment M=NSd.eN. The member is now a beam-column and must be checked in accordance with Section 3.3.

3.2 Beams in Bending

Beams must be checked using the section modulus determined for the effective cross-sections, as given in Figure 5. In general the attainment of the yield strength at the compression face will limit the design bending resistance of effective cross-sections:

Mo,Rd = Weff . fy /γM1

For cross-sections similar to type 3 of Figure 5, Weff will be determined for the actual edge (e):

Weff =

and not the edge of the effective hole.


152

The various compression elements in a cross-section (such as a web or a flange) can, in general, be in different classes and a cross-section is normally classified by quoting the least favourable (highest) class of its compression elements.

It is important, particularly in plastic design, that the sections selected for various members should be, in all cases, appropriate for the assumed mode of behaviour.

When any of the compression elements of a cross-section fail to satisfy the limits given in Table 5 for Class 3, the section is classified as "slender" and local buckling shall be taken into account in the design. This may be done by means of the effective cross-section method which is discussed in detail in Lecture 7.3.


• The methods of analysis used are influenced by the geometry of the cross-sections and, more particularly, by the width to thickness ratios of the plate elements in compression.

• It must be guaranteed that no local instability can occur before a complete mechanism is achieved or before the plastic or elastic moment can be reached.

• Four cross-sectional classes are identified, each corresponding to a different performance requirement: plastic, compact, semi-compact and slender cross-sections.

• Limiting proportions for the elements of a cross-section, which enable the appropriate classifications to be made, are given in the lecture.

• When any of the compression elements of a cross-section fail to satisfy the limiting proportions for Class 3 (semi-compact), local buckling shall be taken into account in the design.


151

Table 5 Maximum slenderness ratios for the elements of a rolled section in compression or in bending

Class 3 cross-sectionElement Class 1

cross-section

Class 2

cross-section Formula kσ b*/t

or d/tw

Flange (1)

(b*/t)

9ε 10ε21ε

0,43 14ε

(1)

Web in compression d/tw

33ε 38ε21

1,0 42ε

Web in pure bending d/tw

72ε 83ε25,4ε

23,9 124ε

fy 235 275 355

ε = ε 1,0 0,92 0,81

(1) In practice, b, the half-width of the flange is considered instead of b*. For this reason, the values given in the "Essentials of Eurocode 3" is b = 15 ε > b*.

The most important limiting proportions of the elements of a cross-section, which enable the appropriate classifications to be made, are specified in Eurocode 3 [1]. Appendix 1 gives the limiting proportions for compression elements of Class 1 to 3.

The limiting values of the width-to-thickness ratio (b/t) of the plate elements of sections apply to members in steel of a specific yield strength. In order to cover all grades of steel, Eurocode 3 presents local buckling data non-dimensionally, in terms

of a reduction factor ε = , where 235 represents the yield stress of mild steel and fy that of the steel considered.


150

0,5 to 0,9 approximately. Eurocode 3 [1,2] has adopted p3 = 0,74 as the limit plate

slenderness of Class 3 compression elements and p3= 0,9 for elements in bending where the yield strength may be reached in the extreme fibre of the cross-section.

For plate elements for which p < p3, no plate buckling can occur before the maximum compressive strength reaches the yield strength.

A Class 1 section must develop a resistance moment equal to the plastic capacity of the section and must maintain this resistance through relatively large inelastic deformations. In order to fulfil these conditions without buckling, the entire plate element must be yielded and the material must be strained in the strain-hardening region (see Table 2); this is only possible for elements with low reference

slendernesses ( p < p1), see Figure 2.

On the basis of certain theoretical approaches [3, 4, 5] values of p1 between 0,46 and 0,6, are proposed in various standards. The difference can be explained in the

choice of the amount of necessary rotation capacity. A value of p1= 0,6 corresponds to a limited rotation capacity which is estimated to be sufficient for usual plastic design (continuous beams, non-sway frames, etc.). In Eurocode 3 [1], the proposed value is:

p1 = 0,5

A Class 2 (or compact section) is one which can just reach its plastic moment resistance but has a rapid drop-off in resistances at that point (Table 2). The plate element is yielded and the material strained in the plastic range; it occurs for

elements with medium reference slendernesses p2 where:

p1 < p2 < p3

In Eurocode 3 [1], the proposed value is p1 = 0,6.

Using formula (2), and the appropriate values of p and kσ , the limiting b/t ratios can be calculated. Table 5 gives some limiting value of b/t for the elements of the cross-section of a rolled I-profile in compression or bending.


149

When plated elements of sections are submitted to any kind of direct stress, other than uniform compression (e.g. webs of a girder in bending), the buckling factor kσ

has to be modified to take account of the stress gradient, given by the stress ratio, ψ.

Table 4 gives the buckling factors for different stress ratios ψ , for internal or outstand elements. In the latter case a distinction is made for elements with tip in compression or in tension.

Table 4 Buckling factors and stress distribution

I II III

σ1

σ1 1= maximum compressive stress and s is positive

σ1

σ2σ2σ2

σ1

ψ σ σ= /2 1 +1 1 > > 0ψ 0 0 > > -1ψ -1

σ2

σ1ψ =

Bucklingfactorkσ

Case Iinternalelement

Case IIoutstandelement

Case IIIoutstandelement

4,0

0,43

0,43

8,41,1 + ψ

0,578ψ + 0,34

0,57 - 0,21 +0,07

ψψ2

7,64

1,70

0,57

7,64 - 6,26 +10

ψψ2

1,7 - 5 +17,1

ψψ2

0,57 - 0,21 +0,07

ψψ2

23,9

23,8

0,85

c. Limit plate element slendernesses

The actual behaviour is somewhat different from the ideal elastic-plastic behaviour represented in Figure 2 because of:

i. initial geometrical and material imperfections,

ii. strain-hardening of the material,

iii. the postbuckling behaviour.

Initial imperfections result in premature plate buckling, which occurs for p < 1.

The corresponding limit plate slenderness p3, for Class 3 sections, may differ substantially from country to country because of statistical variations in imperfections and in material properties which are not sufficiently well known to be quantified accurately; a review of the main national codes shows that it varies from


148

For p < 1, = 1 which means that the plate element can develop its squash load

σu = fy. For p > 1, decreases as the plate slenderness increases, σu being equal to σcr. Substituting the Equation (1) value for σcr into the above and taking υ = 0,3 gives:

p = (2)

This expression is quite general as loading, boundary conditions and aspect ratio all influence the value of the buckling factor kσ .

The factor kσ is a dimensional elastic buckling coefficient, depending on edge support conditions, on type of stress and on the ratio of length to width (a/b), aspect ratio, of the plated element.

In general, the plated elements of a section have an aspect ratio much larger than unity and most of them are submitted to uniform compression. For such cases, Table 3 gives buckling factors for plated elements having various long edge conditions.

Table 3 Elastic buckling factor kσ

Support conditions at long edges

Clamped + clamped

Clamped + simply supported

Simply supported + simply supported

Clamped + free

Simply supported + free

Free + free

Buckling factor kσ

Various supportconditions

6,97

5,40

4,00

1,28

0,43

(b/a)2

a

b

a/b >> 1


147

For a plate element with an aspect ratio, α = a/b (length-to-width), greater than about 0,8, the elastic critical buckling stress (Euler buckling stress) is given by:

σcr = kσ (1)

where kσ is the plate buckling factor (see below),

υ Poisson's coefficient,

E Young's modulus.

The critical buckling stress is proportional to (t/b)2 and, therefore, is inversely proportional to (b/t)2. The plate slenderness, or width-to-thickness ratio (b/t), thus plays a similar role to the slenderness ratio (L/i) for column buckling.

In accordance with the definition of Class 3 sections, the proportions of the plate element, represented by the b/t ratio, must be such that σcr would exceed the material yield strength fy so that yielding occurs before the plate element buckles. The ideal elastic-plastic behaviour of a perfect plate element subject to uniform compression may be represented by a normalised load-slenderness diagram, where the normalised ultimate load:

= σu/fy

and the normalised plate slenderness:

p =

are plotted as ordinates and abscissae respectively (Figure 2).


146

3. CRITERIA FOR CROSS-SECTION CLASSIFICATION

The classification of a specific cross-section depends on the width-to-thickness ratio, b/t, of each of its compression elements. Compression elements include any component plate which is either totally or partially in compression, due to axial force and/or bending moment resulting from the load combination considered; the class to which a specified cross-section belongs, therefore, partly depends on the type of loading this section is experiencing.

a. Components of cross-section

A cross-section is composed of different plate elements, such as web and flanges; most of these elements, if in compression, can be separated into two categories:

• internal or stiffened elements: these elements are considered to be simply supported along two edges parallel to the direction of compressive stress.

• outstand or unstiffened elements; these elements are considered to be simply supported along one edge and free on the other edge parallel to the direction of compressive stress.

These cases correspond respectively to the webs of I-sections (or the webs and flanges of box sections) and to flange outstands (Figure 1).

b. Behaviour of plate elements in compression


145

Class 4 Slender cross-sections: those in which yield in the extreme fibres cannot be attained because of premature local buckling.

Table 2 Cross-section requirements and classification

M

M

M

Mpl

Mpl

Mpl

Mel

Mel

Localbuckling

Localbuckling

Localbuckling

φ

φ

φ

Model ofBehaviour

MomentResistance

Rotation Capacity Class

Plastic momenton full section

Plastic momenton full section

fy

fy

Elastic momenton full section

Elastic moment oneffective section

fy

fy

M

Mpl

1

1

1

1

Localbuckling

φφ

φ

φ

φ

φrot

M

M

M

M

φpl

φpl

φpl

φpl

φpl

Mpl

Mpl

Mpl

Mpl

Mal

Mpl

Sufficient

Limited

None

None

1

1

1

1

1

2

3

4

The moment resistances for the four classes defined above are:

for Classes 1 and 2: the plastic moment (Mpl = Wpl . fy)

for Class 3: the elastic moment (Mel = Wel . fy)

for Class 4: the local buckling moment (Mo < Mel).

The response of the different classes of cross-sections, when subject to bending, is usefully represented by dimensionless moment-rotation curves.

The four classes given above are recognised for beam sections in bending. For struts loaded in axial compression, Classes 1, 2 and 3 become one, and, in the absence of overall buckling are referred to as "compact"; in this case Class 4 is referred to as "slender".


144

limit state, according to Model III, is achieved when yielding occurs at the most stressed fibre. Sometimes yielding in the extreme fibre cannot even be attained because of premature plate buckling of one component of the cross-section; in such cases, the above ultimate limit state should apply only to effective cross-sections (Model IV).

It is obviously not possible to have a model where a plastic method of analysis is combined with an elastic cross-section verification. Indeed, the moment redistribution which is required by the plastic analysis cannot take place without some cross-sections being fully yielded.

2. REQUIREMENTS FOR CROSS-SECTION CLASSIFICATION

In the previous section, the models are defined in terms of structural design criteria; these are actually governed by conditions related to stability problems. Plastic redistribution between cross-sections and/or within cross-sections can take place provided that no premature local buckling occurs, as this would cause a drop-off in load carrying capacity.

It must be guaranteed that no local instability can occur before either the elastic (Model III), or the plastic (Model II), bending resistance of the cross-section, or the formation of a complete plastic mechanism (Model I), is achieved.

Such a mechanism, as envisaged by Model I, can occur provided that the plastic hinge, once formed, has the rotational capacity required for the formation of a plastic mechanism.

To ensure sufficient rotation capacity, the extreme fibres must be able to sustain very large strains without any drop-off in resistance. In tension, the usual steel grades have sufficient ductility to allow for the desired amount of tensile strains; in addition, no drop-off is to be feared before the ultimate tensile strength is reached. With compressive stresses, however, it is not so much a question of material ductility, as of ability to sustain these stresses without instability occurring.

Table 2 gives a summary of the requirements for cross-sections in terms of behaviour, moment capacity and rotational capacity. As can be seen from this table, the limits are referred to cross-section classes, according to Eurocode 3 [1], each corresponding to a different performance requirement:

Class 1 Plastic cross-sections: those which can develop a plastic hinge with sufficient rotation capacity to allow redistribution of bending moments in the structure.

Class 2 Compact cross-sections: those which can develop the plastic moment resistance of the section but where local buckling prevents rotation at constant moment in the structure.

Class 3 Semi-compact cross-sections: those in which the stress in the extreme fibres should be limited to yield because local buckling would prevent development of the plastic moment resistance of the section.


143

1. INTRODUCTION

When designing a structure and its components, the designer must decide on an appropriate structural model. The choice of model effects:

• the analysis of the structure, which is aimed at the determination of the stress resultants (internal forces and moments), and

• the calculation of the cross-section resistance.

Thus a model implies the use of a method of analysis combined with a method of cross-section resistance calculation.

There are several possible combinations of methods of analysis and methods of cross-section calculation, for the ultimate limit state, involving either an elastic or plastic design approach; the possible combinations are listed in Table 1.

Table 1 Ultimate Limit State Design - Definition of Design Models

Model Method of Global Analysis (Calculation of internal forces and moments)

Calculation of Member Cross-Section Resistance

I

II

III

IV

Plastic

Elastic

Elastic

Elastic

Plastic

Plastic

Elastic

Elastic Plate Buckling

Model I is related to plastic design of structures. Full plasticity may be developed within cross-sections, i.e. the stress distribution corresponds to a fully rectangular block, so that plastic hinges can form. These have suitable moment rotation characteristics giving sufficient rotation capacity for the formation of a plastic mechanism, as the result of moment redistribution in the structure.

For a structure composed of sections which can achieve their plastic resistance, but have not sufficient rotation capacity to allow for a plastic mechanism in the structure, the ultimate limit state must refer to the onset of the first plastic hinge. Thus, in Model II, the internal forces are determined using an elastic analysis and are compared to the plastic capacities of the corresponding cross-sections. For statically determinate systems, the onset of the first plastic hinge produces a plastic mechanism; both methods I and II should thus give the same result. For statically indeterminate structures, Model II, in contrast to Model I, does not allow moment redistribution.

When the cross-sections of a structure cannot achieve their plastic capacity, both analysis and verification of cross-sections must be conducted elastically. The ultimate


142


ESDEP WG 7

ELEMENTS

Lecture 7.2: Cross-Section Classification

OBJECTIVE

To describe the classification of cross-sections and explain how this controls the application of the methods of analysis given in Eurocode 3 [1].

PREREQUISITES

Lecture 7.1: Methods of Analysis of Steel Structures

RELATED LECTURES

Lecture 7.3: Local Buckling

Lectures 7.5.1 & Lecture 7.5.2: Columns

Lectures 7.8: Restrained Beams

Lectures 7.9: Unrestrained Beams

Lectures 7.10: Beam Columns

Lecture 7.11: Frames

Lecture 14.10: Simple Braced Non-Sway Multi-Storey Buildings

RELATED WORKED EXAMPLES

Worked Example 7.1: Cross-Section Classification

SUMMARY

The analysis methods used are primarily dependent upon the geometry of the cross-section and especially on the width-thickness ratio of the elements which make it up.

The lecture describes how sections are classified as plastic, compact or semi-compact and gives the limiting proportions of the elements by which these classifications are made.


141

Workgroup Contents

Lecture 7.2 : Cross-Section Classification

Top

1. INTRODUCTION

2. REQUIREMENTS FOR CROSS-SECTION CLASSIFICATION

3. CRITERIA FOR CROSS-SECTION CLASSIFICATION


5. REFERENCES



140

Lecture 7.10.3 : Beam Columns III

Lecture 7.11 : Frames

Lecture 7.12 : Trusses and Lattice Girders


139

Opiskelija perehtyy teräsrakenteiden mitoitukseen.ks ESDEP – oppimisympäristö: http://www.terasrakenneyhdistys.fisuomenkielinen versio

DESIGN OF ELEMENTS

Course Contents

ELEMENTS

Lecture 7.1 : Methods of Analysis of Steel Structures

Lecture 7.2 : Cross-Section Classification

Lecture 7.3 : Local Buckling

Lecture 7.4.1 : Tension Members I

Lecture 7.4.2 : Tension Members II

Lecture 7.5.1 : Columns I

Lecture 7.5.2 : Columns II

Lecture 7.6 : Built-up Columns

Lecture 7.7 : Buckling Lengths

Lecture 7.8.1 : Restrained Beams I

Lecture 7.8.2 : Restrained Beams II

Lecture 7.9.1 : Unrestrained Beams I

Lecture 7.9.2 : Unrestrained Beams II

Lecture 7.10.1 : Beam Columns I

Lecture 7.10.2 : Beam Columns II


138

Kuva: MitoitustaTRY ry:n / Risto Liljan Oppimisympäristö

Kuva: Palkin mitoitustaTRY ry:n / Risto Liljan Oppimisympäristö


137




136

Kuva: PoikkileikkausluokatTRY ry:n / Risto Liljan Oppimisympäristö



135

Kuva: KuormatTRY ry:n / Risto Liljan Oppimisympäristö

Kuva: MateriaaliominaisuudetTRY ry:n / Risto Liljan Oppimisympäristö


134




133

Kuva: Määräykset ja ohjeetTRY ry:n / Risto Liljan Oppimisympäristö

Kuva: RajatilamitoitusTRY ry:n / Risto Liljan Oppimisympäristö


132

MITOITUS https://www.virtuaaliamk.fi/opintojaksot/030501/1132142124407/1133882367215/1136369180381/1136369611545.html.stx

Kuva: TRY ry:n / Risto Liljan Oppimisympäristön mitoitusperusteet

- Yleistä- Runkojärjestelmät- Palkit- Profiilipellit- Katto- ja seinäorret- Ohutuumapalkit- Ristikot- Pilarit- Jäykistys- Liitokset- Palosuojaus- Liittorakenteet- Tuotemallinnus ja suunnitelmat


131


130

The second type, namely the steel section with concrete between the flanges, can support considerable central loads and high bending moments. The amount of shuttering is significantly reduced. Other advantages are a good resistance to mechanical damage without the need for corner reinforcement and the ability to use conventional steel connections between the columns and steel beams similarly concreted and reinforced between the flanges, as shown in Figure 8b. Such composite sections may reach any desired fire resistance level.

Composite Steel Deck Floors

Composite floors utilising profiled steel decks are very frequently used in building as shown in Figure 9. These floors can have a fire resistance of up to four hours without any fire protection applied to the soffit. Floors with minimal reinforcement have at least 30 minutes fire resistance and a single layer of reinforcement can give up to two hours fire resistance. For longer periods of fire resistance for floors with high loading and long spans, additional reinforcement may be necessary.


129

The cross-section of this type of column is either rectangular or circular as shown in Figure 7(a, b, c). The performance in fire depends mainly on the member size and the tensile and flexural properties of the concrete. If non-reinforced concrete is used the fire resistance is normally 30 minutes (Figure 7a). However, a rating of 120 minutes can be achieved by the inclusion of reinforcing bars or steel-fibre reinforcement (Figure 7b).

The steel core column (Figure 7c) is a further development of the concrete filled hollow section but with the main part of the load-carrying steel cross-section protected against fire by layers of concrete. The fire resistance of this type of column varies from 60 minutes to higher values depending upon the thickness of the concrete layer. These columns are used as centrally loaded members with small load eccentricities.

Rolled Steel Sections Encased in Concrete

Different types of composite construction utilising steel sections are manufactured into fire resistant columns and beams.

One of the great advantages of composite columns is their uniform outside dimensions in multi-storey buildings. By varying the thickness of the steel section, the material qualities of both steel and concrete, and the percentage of reinforcement, the cross-section of the column may be adapted to support an increased load without significant changes in the outer dimensions. Each type of composite column has specific advantages and ranges of application.

The oldest type of composite column, Figure 8a, is the steel section encased in concrete. Its advantages are a high allowable load level in fire conditions and a high load-carrying resistance not only for centrally applied loads, but also for bending moments. The fire resistance is normally 90 minutes or more.


128

such circumstances the time to attain a certain steel temperature is governed by the factor

λi Am / di A

The required thickness of insulation for a structural steel member may be determined by using a nomogram which relates critical temperature, applied load, section factor and fire resistance. For example, consider an IPE 500 beam supporting an actual load/collapse load, η = 0,625 requiring a fire resistance of 120 minutes and exposed to heat on 3 sides. The nomogram for protected steel is reproduced in Figure 6 for a simply supported beam supporting a concrete slab (k factor of 0,7). The product of η x k = 0,625 x 0,7 = 0,438 and the section factor, Am /A = 132 m-1, for a beam exposed to heat on three sides.

From the nomogram the value of the factor {λi Am / di A} is 690 W/(m3.K) to satisfy 120 min fire resistance with η x κ = 0,438

(1)

Values of thermal conductivity, λi, may be obtained from manufacturers' data and examples are given in the nomogram. For instance, when λi = 0,1 W/m°C (typical of many protection materials), the required thickness of insulation, di ≥ 0,1 x 0,19 = 19 mm.

4. COMPOSITE CONSTRUCTION

The use of composite steel/concrete components in buildings is becoming increasingly important in fire resistant design because they offer several choices for influencing the rise in temperature of the steel [4, 5]. One is the position and mass of the concrete and a second option is the possibility of redistributing the internal stresses to protected and cooler parts of the section.

Concrete-Filled Hollow Steel Columns


127

Tests have shown the need to evaluate the performance of intumescent coatings over a variety of shapes and orientations of the substrate. In the long term it is anticipated that this form of fire protection may be installed by the steel fabricator.

Although these materials have resistance to impact and abrasion, mechanical damage can occur, particularly on columns, requiring maintenance of the paint system to be carried out.

Specification of Fire Protection Thickness

In Eurocode 3: Part 1.2 [3] an equation is given to calculate the rise in temperature of protected steelwork. The thermal conductivity of the insulation material, λi, and its thickness di are taken into account as λi /di. The heat capacity of the insulation is also included. The thermal conductivity of insulating materials changes with their mean temperature. This change can be taken into account in more precise calculations.

However, if no detailed information is available and if only an approximate answer is required, the analysis may be based on average values of λi, which are assumed to be valid for the whole temperature range during a fire. It may be shown that under


126

Typical Intumescent Coating Thicknesses Used on I-Sections

Range of basecoat thickness for different fire resistance periods* (mm)

Coating type

30 mins 60 mins 90 mins 120 mins

Solvent based thin coats

0,25 - 1,0 0,75 - 2,5 1,50 - 2,50 -

Epoxy resin-based thick coats

4,0 - 5,0 4,0 - 11,0 6,0 - 16,5 6,0 - 16,5

* Data from UK practice

The thin film coatings or mastics foam and swell under the influence of heat to produce an insulating char layer up to 50 times thicker than the original film thickness. These products can be applied by spray, brush or roller. In order to apply thicker coats multiple treatments are necessary. Control measurements on thickness are required using proprietary measuring equipment which has been developed for assessing paint thickness. Only a limited amount of investigation of durability has been conducted on the ability of certain products to be used externally. Most of the products have good resistance to impact and abrasion.

A simplified graph showing fire resistance periods provided by a single coat of intumescent paint is shown in Figure 5.

Documents

CONNECTION DESIGN: STATIC LOADING · Lecture 11.2.3 : Welded Connections ... 3. Eurocode 3, Part 1.3: ... 3. PHYSICAL CHARACTERISTICS Thermal insulation is often required for walls