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©SSEDTA 2001

Structural Steelwork Eurocodes Development of

A Trans-national Approach Course: Eurocode 3 Module 7 : Worked Examples Lecture 20 : Simple braced frame

Contents: 1. Simple Braced Frame

1.1 Characteristic Loads 1.2 Design Loads Fd = γF Fk 1.3 Partial Safety Factors for Strength

2. Floor Beam - Fully Restrained 2.1 Classification of Cross-section

2.1.1Flange buckling 2.1.2 Web buckling

2.2 Shear on Web 2.3 Deflection Check 2.4 Additional Checks if Section is on Seating Cleats 2.5 Crushing Resistance 2.6 Crippling Resistance 2.7 Buckling Resistance 2.8 Summary

3. Roof Beam – Restrained at Load Points 3.1 Initial Section Selection 3.2 Classification of Cross Section

3.2.1 Flange buckling 3.2.2 Web buckling

3.3 Design Buckling Resistance Moment 3.4 Shear on Web 3.5 Deflection Check 3.6 Crushing, Crippling and Buckling 3.7 Summary

4. Internal Column 4.1 Loadings 4.2 Section properties

Structural Steelwork Eurocodes – Development of a Trans-National Approach Worked examples Design of a 3-storey unbraced frame

15/02/07 2

4.3 Classification of Cross-Section 4.3.1 Flange (subject to compression) 4.3.2 Web (subject to compression)

4.4 Resistance of Cross-Section 4.5 Buckling Resistance of Member 4.6 Determination of Reduction Factor χy 4.7 Determination of Reduction Factor χz

5. External Column 5.1 Loadings 5.2 Section properties 5.3 Classifcation of Cross-Section

5.3.1 Flange (subject to compression) 5.3.2 Web (subject to compression)

5.4 Resistance of Cross-Section 5.5 Buckling Resistance of Member 5.6 Determination of Reduction factor χy 5.7 Determination of Reduction factor χz

6. Design of Cross-Bracing 6.1 Section Properties 6.2 Classification of Cross-Section 6.3 Design of Compression Member

6.3.1 Resistance of Cross-section 6.3.2 Design Buckling Resistance 6.3.3 Determination of Reduction Factor χ?

6.4 Design of Tension Member 6.4.1Resistance of Cross-Section

7. Concluding Summary

Structural Steelwork Eurocodes – Development of a Trans-National Approach Worked examples Design of a 3-storey unbraced frame

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1. Simple Braced Frame

The frame consists of two storeys and two bays. The frames are at 5 m spacing. The beam span is 7,2 m. The height from column foot to the beam at floor level is 4,5 m and the height from floor to roof is 4,2 m. It is assumed that the column foot is pinned at the foundation.

7,2 m

4,2 m

4,5 m

7,2 m

Roof Beam

Floor Beam

Internal Column

External Column

Figure 1 Typical Cross Section of Frame

It is assumed that resistance to lateral wind loads is provided by a system of localised cross-bracing, and that the main steel frame is designed to support gravity loads only. The connections are designed to transmit vertical shear, and it is also assumed that the connections offer little, if any, resistance to free rotation of the beam ends. With these assumptions, the frame is classified as ‘simple’, and the internal forces and moments are determined using a global analysis which assumes the members to be effectively pin-connected.

6.4.2.1(2)

5.2.2.2

1.1 Characteristic Loads Floor: Variable load, Qk = 3,5 kN/m2 Permanent load, Gk = 8,11 kN/m2 Roof: Variable load, Qk = 0,75 kN/m2 Permanent load, Gk = 7,17 kN/m2

1.2 Design Loads Fd = γF Fk 2.2.2.4(1) Floor: Gd = γG Gk. At ultimate limit state γG = 1,35 (unfavourable) Gd = 1,35 x 8,11 = 10,95 kN/m2

Qd = γQ Qk. At ultimate limit state γQ = 1,5 (unfavourable) Qd = 1,5 x 3,5 = 5,25 kN/m2

2.2.2.4(2) Table 2.2 2.2.2.4(2) Table 2.2

Structural Steelwork Eurocodes – Development of a Trans-National Approach Worked examples Design of a 3-storey unbraced frame

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Roof: Gd = γG Gk. At ultimate limit state γG = 1,35 (unfavourable) Gd = 1,35 x 7,17 = 9,68 kN/m2

Qd = γQ Qk. At ultimate limit state γQ = 1,5 (unfavourable) Qd = 1,5 x 0,75 = 1,125 kN/m2

2.2.2.4(2) Table 2.2 2.2.2.4(2) Table 2.2

The steel grade selected for beams, columns and joints is Fe360. (fy = 235 N/mm2)

Table 3.1

1.3 Partial Safety Factors for Strength The following partial safety factors for strength have been adopted during the design: • Resistance of Class 1,2 or 3 cross-section, γM0 = 1,1 • Resistance of member to buckling, γM1 = 1,1 • Resistance of bolted connections, γMb = 1,25

2.3.3.2(1)

5.1.1(2) 5.1.1(2) 6.1.1(2)

The following load case, corresponding to permanent and variable actions (no horizontal loads) is found to be critical.

2. Floor Beam - Fully Restrained The beam shown in Figure 2 is simply supported at both ends and is fully restrained along its length. For the loading shown, design the beam in grade Fe360, assuming that it is carrying plaster, or a similar brittle finish. Fd = γG Gk + γQ Qk Design load, Fd = (5 x 1,35 x 8,11) + (5 x 1,5 x 3,5) = 81 kN/m

Table 2.1

7,2 m

81 kN/m

Figure 2 Loading on Fully Restrained Floor Beam

Design moment, M F L 8Sd

d 2

=

Where MSd is the design moment in beam span, Fd is the design load = 81 kN/m, and L is the beam span = 7,2m.

M 81x7,2

8 525 kNmSd

2

= =

Design shear force, V F L 2

81x7,2 2

292 kNSd d= = =

To determine the section size it is assumed that the flange thickness is less than 40 mm so that the design strength is 235 N/mm2, and that the section is class 1 or 2.

Table 3.1

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The design bending moment, MSd, must be less than or equal to the design moment resistance of the cross section, Mc.Rd: MSd ≤ Mc.Rd

Mc.Rd = Mpl.y.Rd = W fpl y

M0γ Where Wpl is the plastic section modulus (to be determined), fy is the yield strength = 235 N/mm2, and γM0 is the partial material safety factor = 1,1.

5.4.5.1(1)

Table 3.1 5.1.1(2)

Therefore, rearranging:

W M f

525x10 x1,1 235

2457 cmpl.required sd M0 y

3 3= = =

γ

Try IPE 550 Section properties: Depth, h = 550 mm, Width, b = 210 mm Web thickness, tw = 11,1 mm Flange thickness, tf = 17,2 mm Plastic modulus, Wpl = 2787 cm3

5.4.5.1

This notation conforms with Figure 1.1 in Eurocode 3: Part1.1. 2.1 Classification of Cross-section

As a simply supported beam is not required to have any plastic rotation capacity (only one hinge required), it is sufficient to ensure that the section is at least class 2 to develop the plastic moment resistance.

5.3 5.3.2 and Table 5.3.1

Figure 3 shows a typical cross-section for an IPE. IPE sections have been used in this example to reflect the European nature of the training pack.

tw

t

d

c

f

Figure 3 A Typical Cross-Section

2.1.1 Flange buckling

Class 1 limiting value of c/tf for an outstand of a rolled section is 10ε. ε = 235 / fy and fy = 235 N/mm

2, therefore ε =1.

Calculate the ratio c t f

, where c is half

the width of the flange = 105 mm, and tf is the flange thickness = 17,2 mm (if the flange is tapered, tf should be taken as the average thickness). c t

105 17,2

6,10 f

= =

Table 5.3.1 (Sheet 3)

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2.1.2 Web buckling

Class 1 limiting value of d/tw for a web subject to bending is 72ε. ε = 235 / fy and fy = 235 N/mm

2, therefore ε =1.

Calculate the ratio d tw

, where d is the depth between root radii = 467,6

mm and tw is the web thickness = 11,1 mm. d t

467,6 11,1

42,1 w

= =

c t

10 f

< ε and d t

72 w

< ε

∴ Section is Class 1 and is capable of developing plastic moment.

Table 5.3.1 (Sheet 1)

Table 5.3.1 (Sheets 1 and 3)

2.2 Shear on Web The shear resistance of the web must be checked. The design shear force, VSd, must be less than or equal to the design plastic shear resistance, Vpl.Rd:

VSd ≤ Vpl.Rd

Where Vpl.Rd is given by A f / 3

v y

M0γ

5.4.6

For rolled I and H sections loaded parallel to the web, Shear area, Av = 1,04 h tw, fy is the yield strength = 235 N/mm2, and γM0 is the partial material safety factor = 1,1.

5.4.6(4)

Table 3.1 5.1.1(2)

∴ =V 1,04ht f

3xpl.Rd w y

M0γ = =

1,04 x 550 x 11,1 x 235 3 x 1,1x10

783 kN3

This is greater than the shear on the section (292 kN). The shear on the beam web is OK. If the beam has partial depth end-plates, a local shear check is required on the web of the beam where it is welded to the end-plate.

V A f / 3

pl.Rd v y

M0

= γ

where Av = twd, and d is the depth of end-plate = (for example) 300 mm.

V 11,1 x 300 x 235 3 x 1,1 x 10

411 kNpl.Rd 3= =

This is greater than the shear on the section (292 kN). The local shear on the beam web is OK.

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