2019 - CIV 2225 Steel Beams PDF

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⇒ CIV2225 Steel Beams 1. Section Classification 2. Beam Section Capacity 3. Full Lateral Restraint (FLR) 1 1 Classification Local Buckling Section Classification in Different Standards Slenderness Limits or Width-to-Thickness Ratio Limits Examples 2 Some beams may fail before reaching the yield ...

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CIV2225 Steel Beams

1. Section Classification 2. Beam Section Capacity 3. Full Lateral Restraint (FLR)

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1.Section Classification

Local Buckling Section Classification in Different Standards Slenderness Limits or Width-to-Thickness Ratio Limits Examples

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1.1 Local Buckling A steel beam cannot sustain infinite curvature, and at some curvature failure occurs. A common mode of failure is local instability (buckling) of the plate elements in the section, although material fracture is another possible failure mode.

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Some beams may fail before reaching the yield moment or the plastic moment. If the beam can reach the plastic moment, the rotation capacity (R) is a measure of how much the plastic hinge can rotate before failure occurs.

Zhao, X.L, Wilkinson, T. and Hancock, G.J. (2005), Cold-formed tubular members and connections, Elsevier Science Pty Ltd, Oxford, UK

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Compact

Required rotation capacity (Rreq) = 4 in AS4100 for plastic design

Non-compact

Slender

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1.2 Section Classification in Different Standards Specification Eurocode 3 AS4100 (Australia) AISC LRFD (USA)

Class 1 Compact

Section classification Class 2 Class 3 Non-compact

Class 4 Slender

Compact

Non-compact

Slender

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1.3 Slenderness Limits or Width-to-Thickness Ratio Limits Definition AS4100 EC3 Part 1-1

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The definition of slenderness or width-to-thickness ratio varies slightly between different standards taking into account either the clear width between the flanges or webs, or the flat width considering the curved corner radii. In AS4100 clear width is used to define element slenderness, whereas in EC3 Part 1.1 flat width is used to define width-to-thickness ratio. clear width (not considering corner radius): e.g. b-2tw, d-2tf

flat width (considering corner radius): e.g. b-2rext, d-2rext

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For example, the element slenderness (λe) in AS4100 or width-tothickness ratio for flanges and webs in a cold-formed RHS or I-section (dimensions shown in Figure 4) or CHS (circular hollow section) is defined as follows, where fyf and fyw are yield stress of the flange and web respectively.

yield stress is involved in the definition, but not in the limiting value

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yield stress is not involved in the definition, but in the limiting values

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The slenderness or width-to-thickness ratios are compared with some limiting values to determine the cross-section classes.

The origin of slenderness limits was based on the elastic local buckling behaviour of perfect plates. Material non-linearity (particularly for cold-formed steels), geometric imperfections and residual stresses all affect the local buckling behaviour. Different slenderness limits are also specified for flanges and webs for the same cross section.

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Notes: 1. Different symbols may be used in different standards or books. 2. In AS4100 the yield stress fy is included in the definition of element slenderness. The slenderness limit is expressed as a constant (i.e. independent of the yield stress). 3. In EC3 the yield stress fy is not included in the definition of width-tothickness ratio. However the yield stress is included in the limiting width-tothickness ratio. 4. In AS4100 the cross-section classification (compact, non-compact or slender) only applies to sections under pure bending. For sections under pure compression the concept of fully effective or not is used rather than cross-section classification. 5. In EC3 the same cross-section could have different classes (1, 2, 3 or 4) depending on if it is under pure bending, pure compression or combined bending and compression, i.e. different limiting width-to-thickness ratios are specified for pure bending, pure compression or combined bending and compression. 12

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AS4100 Here are the procedures to determine cross-section classes: 1. Calculate the element slenderness (λe) for each element in flange and web 2. Choose the one with the largest (λe/λey) ratio as the critical section slenderness (λs) The class is

The values of λey and λep are given in Table 5.2 of AS4100 – which is reproduced here as Table 2.

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Zhao, X. L. and Hancock, G. J. (1991), Tests to determine plate slenderness limits for cold-formed rectangular hollow sections. Journal of Australian Institute of Steel Construction, 25(4), 2-16

See AISC 1991 pap15

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1.4 Examples Example 1 Question: What is the definition of rotation capacity?

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Answer: The rotation capacity (R) is a measure of how much the plastic hinge can rotate before failure occurs. It can be estimated from a dimensionless moment versus curvature diagram, as R = κ1/κp - 1, where κp = Mp/EI.

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Example 2 Determine the class for a light-welded I-section subject to pure bending with the following dimensions and properties: Overall flange width b = 200 mm Overall depth d = 600 mm Flange thickness tf = 16mm Web thickness tw = 6 mm Weld leg length s = 6 mm Yield stress of flange fyf = 275 MPa Yield stress of web fyw = 275 MPa

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Solution using AS4100 The I-section is light-welded (LW). Flange

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Web

The web is more critical.

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Section slenderness λs = 99.29 Plasticity slenderness limit λsp = 82 This I-section is a Non-compact section since λsp ≤ λs ≤ λsy

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Example 3 Question: What is the characteristic of Class 4 section? What is the equivalent class in AS4100?

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Answer: Class 4 sections cannot reach the yield moment due to local buckling. They are also known as slender sections in AS4100.

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Example 4 Determine the class for a cold-formed RHS subject to pure bending with the following dimensions: Overall flange width b = 50 mm Overall depth d = 75 mm Flange thickness tf = 2.5 mm Web thickness tw = 2.5 mm Yield stress fyf = fyw = fy = 350 MPa

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Solution using AS4100

Flange

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Web

The flange is more critical. Section slenderness λs = 21.24 Plasticity slenderness limit λsp = 30 This cold-formed RHS is a compact section since λs < λsp 27

2. Section Capacity 2.1 Behaviour 2.2 Section Capacity 2.3 Example

AS4100 only

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2.1 Behaviour As mentioned in Figure 2, steel sections can be classified as compact, non-compact or slender in AS4100 or classified as Class 1 to Class 4 in EC3 Part 1-1. The strength of short beams is influenced by the local buckling of its plate elements.

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 For the whole section, λs = λe, λsp = λep , λsy = λey from the critical element with the greatest value of λe/λey .

e = critical element s = whole section

 A section is: compact (region 1), non-compact (region 2), or slender (region 3) Moment Capacity

Region 1

2

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Sfy Zfy Typical point Zefy λ ey λ i Yield Limit

Plasticity Limit

λ ep

Plate Slenderness b λ=t

fy 250 30

2.2 Section Capacity

AS4100 only

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AS4100 The nominal section moment capacity is defined as Ms = f y Z e The design section moment capacity is defined as φMs = φfyZe where Ze is the effective section modulus, fy is the yield stress of the section, the capacity factor φ is taken as 0.9.

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The formulae for the effective section modulus (Ze) depend on the classification of the section.

where Z is the elastic section modulus, S is the plastic section modulus, Zc is the effective section modulus (Ze) for a compact section, λs is the section slenderness, λsy is the section yield slenderness limit and λep is the section plasticity slenderness limit. 33

The element slenderness λe of both the flange and the web are compared with the yield and plastic slenderness limits λey and λep, to determine the classification of the section. The section slenderness λs is determined from the element (flange or web) which is more critical, having the higher value of λe/ λey.

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Try to derive the condition (do/t >?) when the second limit governs. Given fy = 250MPa and λsy=120.

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2.3 Example A hot-rolled I-section beam (6 m span) is simply supported with a design UDL of 24 kN/m. The beam is fully restrained so that it can achieve its section capacity. The dimensions and properties of the I-section are: Overall flange width b = 146 mm Overall depth d = 256 mm Is the I-section adequate if full lateral Flange thickness tf = 10.9 mm restraint is provided? Web thickness tw = 6.4 mm Root radius r = 8.9 mm Radius of gyration ry = 34.5 Plastic section modulus Sx = Wpl = 486 × 103 mm3 Elastic section modulus Zx = Wel = 435 × 103 mm3 Yield stress of flange fyf = 320 MPa Yield stress of web fyw = 320 MPa

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Solution using AS4100 (1). Cross-section classification Flange

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Web

The flange is more critical. Section slenderness λs = 7.25 Plasticity slenderness limit λsp = 9 This I-section is a Compact section since λs < λsp 38

(2). Section capacity

Action M* = wL2/8 = 24*62/8 = 108 kNm

The I-section is adequate if full lateral restraint is provided.

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3. Full Lateral Restraint (FLR) 3.1 Behaviour 3.2 FLR length

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3.1 Behaviour When a beam is being bent about its major axis, flexural torsional buckling may occur. As shown in Figure 8, the beam deflects downwards, but at some stage buckling occurs over the length of the member, in which the cross section moves laterally (out of the plane of bending) and twists. The buckling deformations create bending about the minor axis and occur over the entire length of the beam, and hence this is sometimes called a member buckle, and the associated strength is sometimes called a member strength.

41 Trahair, N.S., Flexural–Torsional Buckling of Structures, E & FN Spon, London, 1993

Flexural-torsional buckling is also called lateral buckling, lateral-torsional buckling, or out-of-plane buckling. Figure 9 shows the flexural-torsional buckling of an RHS under experimental conditions.

Member capacity will be covered in CIV3221

Zhao, X.L., Hancock, G.J. and Trahair, N.S. (1995), Lateral buckling te 42 of cold-formed RHS beams. Journal of Structural Engineering, ASCE, 12 ), 1565-1573

3.2 FLR length

If full lateral restraint (FLR) is provided to a beam the member capacity of the beam is the same as the section capacity. The length below which the section capacity can be achieved is called FLR (Full Lateral Restraint) length in AS4100.

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where ry is the radius of gyration about the minor principal axis

ry =

Iy A 44

the ratio βm shall be taken as one of the following as appropriate: -1.0 -0.8 for segments with transverse loads; or the ratio of the smaller to the larger end moments in the length L, (positive when the segment is bent in reverse curvature and negative when bent in single curvature) for segments without traverse loads.

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1.3 Example A welded I-section beam (6 m span) is simply supported with a design UDL of 24 kN/m. The dimensions and properties of the I-section are: Overall flange width b = 146 mm Overall depth d = 256 mm Flange thickness tf = 10.9 mm Web thickness tw = 6.4 mm Iy ≈ 5.66 × 106 mm4 Yield stress of flange fyf = 320 MPa Yield stress of web fyw = 320 MPa What is the FLR length? 46

Solution using AS4100 Area A ≈ 4882 mm2 Radius of gyration ry ≈ 34.8 mm βm = - 0.8 FLR length LFLR = ry (80+50βm ) √(250/fy) = 34.8 x (80 + 50 (-0.8)) √(250/320) = 1230.4 mm

Choose spacing = 1200 mm 47

Summary

1. Section Classification 2. Beam Section Capacity 3. Full Lateral Restraint (FLR)

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