Ultimate limit states
Clause 5.5.2 Cross-section classification (clause 5.5.2)
6.2.5. Bending moment
Clause 6.2.5 Clause 6.3.2 Cross-section resistance in bending is covered in clause 6.2.5, and represents the in-plane
flexural strength of a beam with no account for lateral torsional buckling. Lateral torsional buckling checks are described in clause 6.3.2. Situations where lateral torsional buckling may be ignored are not, however, uncommon. In the cases listed below, member strength may be assessed on the basis of the in-plane cross-sectional strength, and no lateral torsional buckling checks need be made:
• where sufficient lateral restraint is applied to the compression flange of the beam
• where bending is about the minor axis
• where cross-sections with high lateral and torsional stiffness are employed, for example square or circular hollow sections
Clause 6.3.2.3
• or generally where the non-dimensional lateral torsional slenderness, λLT £ 0.2 (or in some cases where λLT£ 0.4 (see clause 6.3.2.3)).
The design bending moment is denoted by MEd (bending moment design effect). The design resistance of a cross-section in bending about one principal axis, Mc, Rdis determined in a similar manner to BS 5950: Part 1.
Eurocode 3 adopts the symbol W for all section moduli. Subscripts are used to differentiate between the plastic, elastic or effective section modulus (Wpl, Welor Weff, respectively). The partial factor γM0is applied to all cross-section bending resistances. As in BS 5950: Part 1, the resistance of Class 1 and 2 cross-sections is based upon the full plastic section modulus, the resistance of Class 3 cross-sections is based upon the elastic section modulus, and the resistance of Class 4 cross-sections utilizes the effective section modulus. The design expressions are given below:
for Class 1 or 2 cross-sections (6.13)
for Class 3 cross-sections (6.14)
for Class 4 cross-sections (6.15)
CHAPTER 6. ULTIMATE LIMIT STATES
pl y
where the subscript ‘min’ indicates that the minimum value of Welor Weffshould be used; i.e.
the elastic or effective modulus should be based on the extreme fibre that reaches yield first.
Example 6.3: cross-section resistance in bending
A welded I section is to be designed in bending. The proportions of the section have been selected such that it may be classified as an effective Class 2 cross-section, as described in Section 6.2.2 of this guide. The chosen section is of grade S275 steel, and has two 200 × 20 mm flanges and a 600 × 6 mm web. The weld size (leg length) s is 6.0 mm.
Assuming full lateral restraint, calculate the bending moment resistance.
Section properties
The cross-sectional dimensions are shown in Fig. 6.9.
For a nominal material thickness (tf= 20.0 mm and tw= 6.0 mm) of less than or equal to 40 mm the nominal values of yield strength fyfor grade S275 steel (to EN 10025-2) is found from Table 3.1 to be 275 N/mm2. Note that reference should be made to the UK National Annex for the nominal material strength (see Section 3.2 of this guide).
Clause 3.2.6 From clause 3.2.6:
E = 210 000 N/mm2
Clause 5.5.2 Cross-section classification(clause 5.5.2)
Outstand flanges (Table 5.2, sheet 2):
c = (b – tw– 2s)/2 = 91.0 mm c/tf= 91.0/20.0 = 4.55
Limit for Class 1 flange = 9ε = 8.32 8.32 > 4.55 \ flange is Class 1
Web – internal part in bending (Table 5.2, sheet 1):
c = h – 2tf– 2s = 548.0 mm c/tw= 548.0/6.0 = 91.3
Limit for Class 3 web = 124ε = 114.6 114.6 > 91.3 \ web is Class 3
235/fy 235/275 0.92
ε = = =
b
h d
tw
tf s
y y
z
z hw
b = 200.0 mm tf = 20.0 mm hw = 600.0 mm tw = 6.0 mm s = 6.0 mm
Wel, y = 2536249 mm3
Fig. 6.9. Dimensions for a welded I section with 200 × 20 mm flanges and a 600 × 6 mm web
Overall cross-section classification is therefore Class 3.
Clause 6.2.2.4 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.
Clause 6.2.2.4 Effective Class 2 cross-section properties(clause 6.2.2.4)
Plastic neutral axis of effective section
The depth to the plastic neutral axis of the effective section, as indicated by z in Fig. 6.10, may be shown (based on equal areas above and below the plastic neutral axis) to be
z = h – tf– s – (2 × 20εtw) (D6.4)
= 600.0 – 20.0 – 6.0 – (2 × 20 × 0.92 × 6.0)
= 352.1 mm
Plastic modulus of effective section
Wpl, y, eff= btf(h – tf) + tw{(20εtw+ s)[z – tf– (20εtw+ s)/2]}
+ tw(20εtw× 20εtw/2) + tw[(h – tf– z)(h – tf– z)/2]
= 2 704 682 mm3
Clause 6.2.5 Bending resistance of cross-section(clause 6.2.5)
Based on the elastic section modulus Wel, y= 2 536 249 mm3, the bending resistance of the cross-section would have been 697.5 kN m. Therefore, for the chosen section, use of the effective Class 2 plastic properties results in an increase in bending moment resistance of approximately 7%.
Other examples of cross-section bending resistance checks are also included in Examples 6.5, 6.8, 6.9 and 6.10.
Clause 6.2.9 For combined bending and axial force, of which bi-axial bending is a special case (with the
applied axial force NEd= 0), the designer should refer to clause 6.2.9.
Clause 6.2.5(4) Clause 6.2.5(5) In the compression zone of cross-sections in bending, (as for cross-sections under uniform
compression), no allowance need be made for fastener holes (where fasteners are present) except for oversize or slotted holes. Fastener holes in the tension flange and the tensile zone of the web need not be allowed for, provided clause 6.2.5(4) and 6.2.5(5) are satisfied.
CHAPTER 6. ULTIMATE LIMIT STATES
pl, , eff y c, , Rd
M0
for effective class 2 sections
y
Fig. 6.10. Effective Class 2 properties for a welded I section. (a) Cross-section. (b) Stress distribution
6.2.6. Shear
Clause 6.2.6 The resistance of cross-sections to shear is covered in clause 6.2.6. The design shear force is denoted by VEd(shear force design effect). 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. The shear stress distribution in a rectangular section and in an I section, based on purely elastic behaviour, is shown in Fig. 6.11.
In both cases in Fig. 6.11, the shear stress varies parabolically with depth, with the maximum value occurring at the neutral axis. However, for the I section (and similarly for the majority of conventional structural steel cross-sections), the difference between maximum and minimum values for the web, which carries almost all the vertical shear force, is relatively small. Consequently, by allowing a degree of plastic redistribution of shear stress, design can be simplified to working with average shear stress, defined as the total shear force VEddivided by the area of the web (or equivalent shear area Av).
Clause 6.2.6(2) Since the yield stress of steel in shear is approximately 1/÷3 of its yield stress in tension, clause 6.2.6(2) therefore defines the plastic shear resistance as
(6.18) and it is the plastic shear resistance that will be used in the vast majority of practical design situations.
Clause 6.2.6(3)
The shear area Avis in effect the area of the cross-section that can be mobilized to resist the applied shear force with a moderate allowance for plastic redistribution, and, for sections where the load is applied parallel to the web, this is essentially the area of the web (with some allowance for the root radii in rolled sections). Expressions for the determination of shear area Avfor general structural cross-sections are given in clause 6.2.6(3). The most common ones are repeated below:
Fig. 6.11. Distribution of shear stresses in beams subjected to a shear force VEd. (a) Cross-section.
(b) Shear stress distribution
• Rolled I and H sections, load parallel to the web:
Av= A – 2btf+ (tw+ 2r)tf but ≥ ηhwtw
• Rolled channel sections, load parallel to the web:
Av= A – 2btf+ (tw+ r)tf
• Welded I, H and box sections, load parallel to the web:
Av= η
Â
hwtw• Welded I, H, channel and box sections, load parallel to the flanges:
Av= A –
Â
hwtw• Rolled rectangular hollow section of uniform thickness, load parallel to the depth:
Av= Ah/(b + h)
• Rolled rectangular hollow section of uniform thickness, load parallel to the width:
Av= Ab/(b + h)
• Circular hollow section and tubes of uniform thickness:
Av= 2A/π where
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 the 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)
η is the shear area factor – see clause 5.1 of EN 1993-1-5 (and the discussion on p. 50 of this guide).
Clause 6.2.6(4)
Clause 6.2.6(2) The code also provides expressions in clause 6.2.6(4) for checking the elastic shear
resistance of a cross-section, where the distribution of shear stresses is calculated assuming elastic material behaviour (see Fig. 6.11). This check need only be applied to unusual sections that are not addressed in clause 6.2.6(2), or in cases where plasticity is to be avoided, such as where repeated load reversal occurs.
The resistance of the web to shear buckling should also be checked, though this is unlikely to affect cross-sections of standard hot-rolled proportions. Shear buckling need not be considered provided:
for unstiffened webs (D6.5)
for webs with intermediate stiffeners (D6.6)
where
kτ is a shear buckling coefficient defined in Annex A.3 of EN 1993-1-5.
CHAPTER 6. ULTIMATE LIMIT STATES
w
It is recommended in clause 5.1 of EN 1993-1-5 that η be taken as 1.20 (except for steel grades higher than S460, where η = 1.00 is recommended). However, numerical values for η are given in the National Annexes, and designers should refer to these for specific guidance, though a conservative value of 1.00 may be used in all cases.
Clause 6.2.7(9)
For cross-sections that fail to meet the criteria of equations (D6.5) and (D6.6), reference should be made to clause 5.2 of EN 1993-1-5, to determine shear buckling resistance. Rules for combined shear force and torsion are provided in clause 6.2.7(9).
Example 6.4: shear resistance
Determine the shear resistance of a 229 × 89 rolled channel section in grade S275 steel loaded parallel to the web.
Section properties
The section properties are given in Fig. 6.12.
For a nominal material thickness (tf= 13.3 mm and tw= 8.6 mm) of less than or equal to 40 mm the nominal values of yield strength fyfor grade S275 steel (to EN 10025-2) is found from Table 3.1 to be 275 N/mm2.