Lecture notes, lectures 4-7 PDF

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Summary

Local Buckling Section Classification Steel as a Structural Material Local Buckling Section Classification Class 4 Section Examples 1 Advantages of Steel as a Structural Material Y The many advantages of steel can be summarized as follows: High Strength Uniformity Elasticity Ductility Toughness Weld...

Description

Local Buckling & Section Classification

 Steel as a Structural Material  Local Buckling  Section Classification  Class 4 Section  Examples

1

Advantages of Steel as a Structural Material Y The many advantages of steel can be summarized as follows: • High Strength • Uniformity • Elasticity • Ductility • Toughness • Weldability

2

Disadvantages of Steel as a Structural Material Y Steel also has many disadvantages that make reinforced concrete as a replacement for construction purposes. Y The disadvantages of steel can be summarized as follows: • Fireproofing Cost • Brittle Fracture • Fatigue • Susceptibility to Buckling • For most structures, the use of steel columns is very economical because of their high strength-to-weight ratios. However, as the length and slenderness of a compressive column is increased, its danger of buckling increases.

3

Disadvantages of Steel as a Structural Material

Lateral-Torsional buckling of a beam

Local buckling of a column 4

Disadvantages of Steel as a Structural Material

Concrete beam section

Steel beam section

5

Shear buckling

Web buckling

Flange buckling

6

Local Buckling Y For efficiency, structural members are generally composed of relatively thin elements. Y Although favourable in terms of overall structural efficiency, the slender nature of these thin elements results in susceptibility to local buckling under compressive stress which must be considered in design. Y Cold-formed sections  Thin-walled sections (plate thickness up to 16 mm) Y Thin-walled sections  Local buckling may occur

7

Hot-Formed vs. Cold-Formed

Hot-formed

Cold-formed 8

9

10

Why Section Classification?

Why Section Classification?

12

Local Buckling Y Local buckling involves the deformation of the component plate elements of a section

: Local buckling of typical cross-sections subjected to pure compression

13

Local Buckling

Local buckling

14

Local Buckling vs Section Classification Y Whether in the elastic or inelastic material range, cross-sectional resistance and rotation capacity are limited by the effect of local buckling. Y Eurocode 3 accounts for the effects of local buckling through section classification. Y The classifications are Class 1, Class 2, Class 3 and Class 4 respectively.

15

Factors affecting Local Buckling Y The factors that affect local buckling (and therefore the cross-section classification) are: • Width /thickness ratios of plate components (cf/tf, cw/tw in EC 3) • Element support conditions • Material strength fy • Fabrication process • Applied stress system (Part subject to bending or compression)

16

Basis of Section Classification  Some are internal element - webs of open beams - flanges of boxes

 Some are outstand element - flanges of I beams - legs of angles and Tees

Outstand Internal

Outstand Internal

Internal Web

Web

Flange Rolled I-section

Web

Flange Hollow section

Internal

Flange Welded box section

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Section Classification  Classification is made by comparing actual width-to-thickness ratios of the plate elements with a set of limiting values, given in Table 5.2 of EN 1993-1-1.  A plate element is Class 4 if it fails to meet the limiting values for a Class 3 element.

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Section Classification  The classification of the overall cross-section is taken as the least favorable of the constituent elements(for example, a cross-section with a class 3 flange and class1 web has an overall classification of Class 3).  However, as stated in clause 6.2.2.4, a crosssection with a Class 3 web and Class 1 or 2 flange may be classified as an effective Class 2 cross- section.

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Definition of the 4 Classes There are four (4) classes of cross-sections: Y Class 1 can develop plastic hinge with the rotation capacity required for plastic analysis without any reduction of resistance Y Class 2 can develop plastic moment resistance but limited rotation capacity due to local buckling Y Class 3 can only develop elastic distribution where extreme fiber stresses can reach yield but local buckling prevents further development of the full plastic moment resistance. Y Class 4 develops local buckling before the attainment of yield

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Definition of the 4 Classes Eurocode 3 defines four classes of the cross-section

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Class 1 Cross-Section Moment

fy

Plastic moment on gross section

Mpl Local Buckling

fy

Cross-section model in design

Φ

M

M pl

Sufficient

1 Real behaviour of cross-section 1



pl

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Class 2 Cross-Section Moment

fy

Plastic moment on gross section

Mpl Local Buckling

fy

Cross-section model in design

Φ

M

M pl

Limited 1

Real behaviour of cross-section 1



pl

23

Class 3 Cross-Section Moment

fy

Elastic moment on gross section

Mpl Mel Local Buckling

Cross-section model in design

fy

Φ

M

M pl

1

M 1 M el

Real behaviour of cross-section 1



pl

24

Class 4 Cross-Section Moment

fy

Elastic moment on effective section

Mpl Mel Local Buckling

Cross-section model in design

fy

Φ

M

M pl

1

M 1 M el

Real behaviour of cross-section 1



pl

25

Cross-Section Resistance – Compression Y Cross-section resistance in compression Nc,Rd: Class 1, 2 and 3:

Class 4:

N c,Rd 

Af y

 M0

Nc,Rd 

A f eff

y

M0

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Cross-Section Resistance - Bending Class 1 & 2 cross-sections:

M c,Rd  M pl 

W f pl

y

M0

Class 3 cross-sections:

Class 4 cross-sections:

M c,Rd  M el 

M c,Rd 

Wel f y

M0

Weff f y

 M0

27

Compressed Width C Y Definition of compressed widths - flat widths: r r

s s (a) Outstand flanges

(b) Internal compression parts

Section

Outstand flanges

Internal web

Rolled

c = (b – tw – 2r)/2

c = h – 2tf – 2r

Welded

c = (b – tw – 2s*)/2

c = h – 2tf – 2s

*s is the weld width 28

Compressed Width C  Definitions of α and ψ for classification of cross-sections under combined bending and compression. (a) Class 1 and class 2 cross-sections. (b) Class 3 cross-sections Ψ is the ratio of end stress (fy)

(a)

(b) 29

Internal Compression Parts

Table 5.2 (sheet 1 of 3): Maximum width-to-thickness ratios for compression parts

30

Outstand Flanges

Table 5.2 (sheet 2 of 3): Maximum width-to-thickness ratios for compression parts

31

Angles and Tubular Sections

Table 5.2 (sheet 3 of 3): Maximum width-to-thickness ratios for compression parts

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Class 4 Section Y Cross-sections with class 4 elements may be replaced by an effective cross-section Y Designed in a similar manner to class 3 sections using elastic cross-sectional resistance limited by yielding in the extreme fibers. Y Effective width formulae for individual elements are provided in Table 4.1 & 4.2 in Eurocode 3 Part 1.5.

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Effective Width Formulae

EN1993‐1‐5Table4.1 34

Y The reduction factor ρ may be taken as following

Where  p 

fy

 cr



bt 28.4 k

 & k are given in Table 4.1/4.2 35

Effective Section Method Y Class 4 cross-sections under axial compression

Centroidal axis of effective section

Centroidal axis

Non-effective zone

Centroidal axis

Centroidal axis of effective section

36

Effective Section Method Y Class 4 cross-sections under bending moment

Centroidal axis

Centroidal axis of effective section Non-effective zone

Centroidal axis

Centroidal axis of effective section

37

Effective Class 2 cross-section Y Generally, Class 3 cross-section would assume an elastic distribution of stresses, and its bending resistance would be calculated using the elastic modulus Wel. Y However, Eurocode 3 (clauses 5.5.2(11) and 6.2.2.4) makes special allowance for cross-sections with Class 3 web and Class 1 or 2 flange by permitting the cross-section to be classified as effective Class 2 cross-section.

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Effective Class 2 cross-section Y Accordingly, part of the compressed portion of the web is neglected, and plastic section properties for the remainder of the cross-section are determined. Y the compressed portion of the web be replaced by a part of 20εtw adjacent to the compression flange, another part of 20 εtw adjacent to the plastic neutral axis of the effective crosssection.

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Effective Class 2 cross-section

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Example 1 – Section Classification Cross-section resistance of beam under in-plane bending •

•

A welded I section beam is under pure bending. The chosen section is of grade S275 steel, and has two 200 × 20 mm flanges and a 560 × 6 mm web. The weld size (leg length) s is 6.0 mm. Assuming full lateral restraint, determined the cross-section resistance.

Section properties For a nominal material thickness (tf = 20.0 mm and tw = 6.0 mm) of large than 16 mm the nominal values of yield strength fy for grade S275 steel (to EN 10025-2) is 265 N/mm2. E = 210 000 N/mm2

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b = 200.0 mm tf = 20.0 mm hw = 560.0 mm h = 600.0 mm tw = 6.0 mm s = 6.0 mm Wel,y = 2536 248 mm3

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Cross-section classification

  235/ f y  235/ 265  0.94 Outstand flanges : c = (b – tw – 2s)/2 = 91.0 mm c/tf = 91.0/20.0 = 4.55 Limit for Class 1 flange = 9ε = 8.46 8.46 > 4.55 So, flange is Class 1

43

Web – internal part in bending: c = h – 2tf – 2s = 548.0 mm c/tw = 548.0/6.0 = 91.3 Limit for Class 3 web = 124ε = 116.56 116.56 > 91.3 web is Class 3 Overall cross-section classification is therefore Class 3. However, as stated in clause 6.2.2.4, a cross-section with a Class 3 web and Class 1 or 2 flanges may be classified as an effective Class 2 cross-section.

44

Example 2 – Section Classification under combined Bending and Compression •

A member is to be designed to carry combined bending and axial load. In the presence of a major axis (y-y) bending moment and an axial force of 300kN, determine the crosssection classification of a 406×178UB54 in grade S275 steel.

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Section properties For a nominal material thickness (tf = 10.9 mm and tw = 7.7 mm) of less than 16 mm the nominal values of yield strength fy for grade S275 steel (to EN 10025-2) is 275 N/mm2. E = 210 000 N/mm2 Cross-section classification under pure compression

  235/ f y  235/ 275  0.92 Outstand flanges : cf = (b – tw – 2s)/2 = 74.8 mm cf/tf = 74.8/10.9 = 6.86 Limit for Class 1 flange = 9ε = 8.32 8.32 > 6.86

So, flange is Class 1 46

Web – internal part in compression: cw = h – 2tf – 2r = 360.4 mm cw/tw = 360.4/7.7 = 46.81 Limit for Class 3 web = 42ε = 38.8 46.81 > 38.8 web is Class 4 Under pure compression, the overall cross-section classification is therefore Class 4.

47

Cross-section classification under combined loading Flange classification remains as Class 1. Web – internal part in bending and compression : From Table 5.2 sheet 1, for a Class 2 cross-section:

c 456  t 13 -1 c 41.5  when  0.5: t  when  0.5:

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Where  may be determined from equation for an I- and Hsection where the neutral axis lies within the web.

overall cross-section classification under the combined bending and compression is therefore Class 2.

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Thank You!

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Fully Restrained Beam  Background  In-plane Bending  Shear  Bending and Shear

1

Background  Steel beams can often be designed on the basis of under pure in-plane bending by ensuring the design shear and moment resistances of the selected cross-section are not exceeded by the design maximum shear and bending moment values.  Check for stiffness by ensuring that the beam does not deflect so much that it affects serviceability considerations.

2

What is “Fully Restraint” ? Beams which are unable to move laterally are termed “fully restrained beams"

Unaffected by out-of-plane buckling (lateral--torsional instability)

3

Fully Restrained Beam Beams may be considered fully restrained if: Y full lateral restraint is provided by positive attachment of a floor system to the top flange of a simply-supported beam Y adequate torsional restraint of the compression flange is provided, for example by purlins and profiled roof sheets Y closely spaced cross beams are provided such that the minor axis slenderness is very low Y Sections with very high torsional and lateral stiffness, for example rectangular hollow section beams

4

Fully Restrained Beam

composite slab restrained beam

5

Fully Restrained Beam Friction force

tw

(h – tf)

L

Rectangular hollow section 6

Bending Moment Resistance Y Eurocode 3 states that both cross-sectional (in-plane bending) and member buckling resistance must be checked, i.e. M Ed  Mc,Rd

Cross-section check (In-plane bending)

M Ed  M b,Rd

Member-buckling check

MEd = design bending moment Mc,Rd = design bending resistance about one principal axis of a cross-section Mb,Rd = design buckling resistance moment

7

Bending Resistance Mc,Rd Y

Class 1 & 2 cross-sections:

Y

Class 3 cross-sections:

M c,Rd  M pl,Rd 

Wplf y  M0

Wel,min f y M c,Rd  M El,Rd 

 M0

Weff ,min f y

Y Class 4 cross-sections:

M c,Rd 

 M0

8

Bending Resistance Mc,Rd – Clause 6.2.5 Y Subscripts are used to differentiate between the plastic, elastic or effective section modulus. • Plastic modulus Wpl • Elastic modulus Wel • Effective modulus Weff

Y

The partial factor γM0 is applied to all cross-section bending resistances , and equal 1.0.

9

Elastic Modulus, Wel For rectangular section

Moment Resistance = Force × Lever Arm

10

Elastic Modulus, Wel For rectangular section

where Ac is the area of compression block At is the area of tension block r is the lever arm

11

Elastic Modulus, Wel For I-section

For example, 457 x 191 UB 67: Wel,y (formula) = 1279 cm3 Wel,y (blue handbook) = 1300 cm3 12

Plastic Modulus, Wpl For rectangular section

13

Plastic Modulus, Wpl of a Beam Section For UB &UC

14

Plastic Modulus, Wpl of a Beam Section

For example, 457 x 191 UB 67: Wpl

(formula)

= 1452 cm3

Wpl (blue handbook) = 1470 cm3 15

Shear Resistance Y

The design shear force is denoted by VEd (shear force design effect).

Y The design shear resistance of a cross-section is denoted by Vc,Rd and may be calculated based on a plastic (Vpl,Rd) or an elastic distribution of shear stress. Y The design shear force, VEd, should satisfy:

V Ed

Vc, Rd

1.0

16

Plastic Shear Resistance Vpl,Rd Y

The usual approach is to use the plastic shear resistance Vpl,Rd in practice.

Y

The plastic shear resistance is essentially defined as the yield strength in shear multiplied by a shear area Av.

Y

The yield strength in shear is related the yield strength in tension using the von Mises yield criterion.

17

Shear Area Av Y

The shear area Av is in effect the area of the crosssection that can be mobilised to resist the design shear force with a moderate allowance for plastic redistribution.

Y

For sections where the action is applied parallel to the web, this is essentially the area of the web (with some allowance for the root radii in rolled sections).

18

Shear Area Av Shear areas Av are given in Clause 6.2.6(3). a) Rolled I and H sections, load parallel to web:

Av = A – 2btf + (tw + 2r)tf but ≥ hwtw   1.0 (refer to NA to SS)

b) Rolled channel parallel to web:

sections,

load

Av = A – 2btf + (tw + r)tf

19

Shear Area Av c) Rolled T-section, parallel to web

load

Av = 0,9 (A - btf) d) Welded I, H and box sections, load parallel to web Av = η∑(hwtw)   1.0 (refer to NA to SS)

e) Welded I, H, channel and box sections, load parallel to flanges Av = A - ∑(hwtw) 20

Shear Area Av f)

Rolled RHS of uniform thickness, •

load parallel to depth: Av = Ah/(b+h)

•

Load parallel to width: Av = Ab/(b+h)

g) CHS and tubes of uniform thickness:

Av = 2A/

21

Definitions A is the cross-sectional area b is the overall section breadth h is the overall section depth hw is the overall web depth (measured between flanges) r is the root radius tf is the flange thickness tw is the web thickness (taken as the minimum value if the web is not of constant thickness)  = 1.0 (from UK National Annex)

22

Shear Buckling Resistance Y

The resistance of the web to shear buckling should be checked especially for welded sections, though this is unlikely to affect standard hot-rolled sections.

Y

Shear buckling need not be considered provided: h w

tw where   235 fy

 72

 

for unstiffened webs

;  = 1.0 (from UK National Annex)

23

Shear Buckling of Thin Web (hw/tw ≥72)

24

Combined Bending and Shear

25

Combined Bending and Shear

high shear case (maybe)

low shear case

26

Combined Bending and Shear 

Clause 6.2.8(2) states that provided the design shear force is less than half the plastic shear resist...