Simplified verification for buildings without direct calculation

Clause 6.4.3(1) As calculations for the U-frame model are quite extensive, a simplified method has been

developed from it. Clause 6.4.3(1) defines continuous beams and cantilevers that may be designed without lateral bracing to the bottom flange, except at supports. Its Table 6.1 gives limits to the steel grade and overall depth of the steel member, provided that it is an IPE or HE rolled section. The contribution from partial encasement is allowed for in paragraph (h) of this clause.

These results are derived from equation (D6.11) for the elastic critical buckling moment, making assumptions that further reduce the scope of the method. Accounts of its origin are available in both English⁶¹and German.⁶²It is outlined in Johnson and Fan,⁵⁶and is similar to that used in the treatment of lateral–torsional buckling of haunched beams.⁶³

The basis is that there shall be no reduction, due to lateral buckling, in the resistance of the beam to hogging bending. It is assumed that this is achieved when £ 0.4. This value is given in a Note to clause 6.3.2.3 of EN 1993-1-1, which can be modified by a National Annex.

Any National Annex that defines a lower limit should therefore also state if the method is permitted.

λLT

The slenderness is a function of the variation of bending moment along the span. This was studied using various loadings on continuous beams of the types shown in Fig. 6.9(c), and on beams with cantilevers. The limitations in paragraphs (a) and (b) of clause 6.4.3(1) on spans and loading result from this work.

Simplified expression for , and use of British UB rolled sections

Table 6.1 applies only to IPE and HE rolled sections. Criteria for other rolled I- and H-sections are deduced in Appendix A. The basis for the method is as follows.

For uncased beams that satisfy the conditions that apply to equation (D6.11) for Mcr, have a double symmetrical steel section, and are not concrete encased, the slenderness ratio for a Class 1 or Class 2 cross-section may conservatively be taken as

(D6.14) The derivation from equation (D6.11) is given in Appendix A. Most of the terms in equation (D6.14) define properties of the steel I-section; bfis the breadth of the bottom flange, and other symbols are as in Fig. 6.9(a).

To check if a particular section qualifies for ‘simplified verification’, a section parameter F is calculated. From equation (D6.14), it is

(D6.15) Limiting values of F, Flimsay, are given in Appendix A (see Fig. A.5) for the nominal steel grades listed in Table 6.1. The horizontal ‘S’ line at or next above the plotted point F gives the highest grade of steel for which the method of clause 6.3.3 can be used for that section.

Some examples are given in Table 6.1, with the values of F_limin the column headings. Many of the heavier wide-flange sections in S275 steel qualify for verification without direct calculation, but few UB sections in S355 steel do so.

In ENV 1994-1-1, verification without direct calculation was permitted for hot-rolled sections of ‘similar shape’ to IPE and HE sections, that conformed to Table 6.1 and a geometrical condition similar to the limit on F. In EN 1994-1-1 this has been replaced by a reference to National Annexes.

Use of UB rolled sections with encased webs to clause 5.5.3(2)

It is shown in Appendix A (equation (DA.4)) that the effect of web encasement is to increase F_limby at least 29%. All the sections shown in Table 6.1 now qualify for S275 steel, and all

λ_LT

Table 6.1. Qualification of some UB rolled steel sections for verification of lateral–torsional stability, in a composite beam, without direct calculation

Section

except one do so for S355 steel. Web encasement is thus an effective option for improving the lateral stability of a rolled steel section in a continuous composite beam.

The method of clause 6.4.2 should be used where the steel section does not qualify.

Use of intermediate lateral bracing

Where the buckling resistance moment M_{b, Rd}as found by one of the preceding methods is significantly less than the design resistance moment MRdof the cross-sections concerned, it may be cost-effective to provide discrete lateral restraint to the steel bottom flange. Where the slab is composite, a steel cross-member may be needed (Fig. 6.10(a)), but for solid slabs, other solutions are possible (e.g. Fig. 6.10(b)).

Clause 6.3.2.1(2) of EN 1993-1-1 refers to ‘beams with sufficient restraint to the compression flange’, but does not define ‘sufficient’. A Note to clause 6.3.2.4(3) of EN 1993-1-1 refers to Annex BB.3 in that EN standard for buckling of components of building structures ‘with restraints’. Clause BB.3.2(1) gives the minimum ‘stable length between lateral restraints’, but this is intended to apply to lateral–torsional buckling, and may not be appropriate for distortional buckling. Further provisions for steel structures are given in EN 1993-2.⁶⁴

There is no guidance in EN 1994-1-1 on the minimum strength or stiffness that a lateral restraint must have. There is guidance in Lawson and Rackham,⁶³based on BS 5950: Part 1, clause 4.3.2.⁶⁵This states that a discrete restraint should be designed for 2% of the maximum compressive force in the flange. It is suggested⁶³that where discrete and continuous restraints act together, as in a composite beam, the design force can be reduced to 1% of the force in the flange. Provision of discrete bracing to make up a deficiency in continuous restraint is attractive in principle, but the relative stiffness of the two types of restraint must be such that they are effective in parallel.

Another proposal is to relate the restraining force to the total compression in the flange and the web at the cross-section where the bracing is provided, to take advantage of the steep moment gradient in a region of hogging bending.

In tests at the University of Warwick,⁶⁶ bracings that could resist 1% of the total compression, defined in this way, were found to be effective. The calculation of this compressive force involves an elastic analysis of the section, not otherwise needed, and the rule is unsafe near points of contraflexure. There is at present no simple design method better than the 2% rule quoted above, which can be over-conservative. An elastic analysis can be avoided by taking the stress in the flange as the yield stress.

The design methods based on equation (D6.11) for M_crwork well for complete spans, but become unsatisfactory (over-conservative) for short lengths of beam between lateral bracings. This is because the correct values of the factors C4are functions of the length between lateral restraints. For simplicity, only the minimum values of C4are given in the figures in Appendix B of this guide. They are applicable where the half-wavelength of a buckle is less than the length L in equation (D6.11). This is always so where L is a complete span, but where L is part of a span, the value given may be over-conservative. This is

(a) (b)

Fig. 6.10. Laterally restrained bottom flanges

illustrated at the end of Example 6.7, where the effect of providing lateral bracing is examined.

Flow charts for continuous beam

The flow charts in Fig. 6.11 cover some aspects of design of an internal span of a continuous composite beam. Their scope is limited to cross-sections in Class 1, 2 or 3, an uncased web, a uniform steel section, and no flexural interaction with supporting members. It is assumed that for lateral–torsional buckling, the simplified method of clause 6.4.3 is not applicable.

Figure 6.11(a) refers to the following notes:

Classify sections at A and C, to clause 5.5

Is section Class 1 or 2?

Class 1 or 2 Class 3 Use effective section in Class 2, to clause 5.5.2(3)?

Find lLT at A and C, to clause 6.4. Is steel element a ‘rolled or equivalent welded’ section?

Yes

Use elastic global analysis, with relevant modular ratio, for actions on steel and composite members and relevant load arrangements to find M_Ed (= M_{a, Ed} + M_{c, Ed}) at A, B and C.

Redistribute hogging bending moments M_{c, Ed} to clause 5.4.4, to find new M_Ed at sections A, B and C. See Fig. 5.4

Fig. 6.11. (a) Flow chart for design for ultimate limit state of an internal span of a continuous beam in a building, with uniform steel section and no cross-sections in Class 4

Are all critical sections in Class 1 or 2, to clause 6.6.1.1(14)? See comments on clause 6.6.2

No

Are all critical cross-sections in Class 1 or 2?

Sagging region. Find n_s = hN_{c, f} /P_Rd for region each side of B

Hogging region. Find n_h = N_{c, f} /P_Rd for regions adjacent to A and C

Find P_Rd for stud connectors to clause 6.6.3.1 or 6.6.4. For other connectors, find P_Rd that satisfies clause 6.6.1.1 and determine if ‘ductile’

Find n = n_s + n_h for critical lengths AB and BC. Are connectors ‘ductile’?

Space n connectors uniformly over lengths AB (and CD) to clause6.6.1.3(3)

Space n connectors over AB (and CD) in accordance with longitudinal shear the stability of any part of the member?

Additional checks required to clause6.6.1.3(4) Yes

No

Check spacing to clause6.6.5.5(2) and revise if necessary Yes

Fig. 6.11. (Contd) (b) Flow chart for design of an internal span of a continuous beam in a building – shear connection

(1) The elastic global analysis is simpler if the ‘uncracked’ model is used. The limits to redistribution of hogging moments include allowance for the effects of cracking, but redistribution is not permitted where allowance for lateral–torsional buckling is required (clause 5.4.4(4)). ‘Cracked’ analysis may then be preferred, as it gives lower hogging moments at internal supports.

(2) If fully propped construction is used, M_{a, Ed}may be zero at all cross-sections.

Example 6.3: lateral–torsional buckling of two-span beam

The design method of clause 6.4.2, with application of the theory in Appendix A, is illustrated in Example 6.7, which will be found after the comments on clause 6.6. The length of the method is such that the ‘simplified verification’ of clause 6.4.3 is used wherever possible.

Limitations of the simplified method are now illustrated, with reference to the two-span beam treated in Example 6.7. This uses an IPE 450 steel section in grade S355 steel, and is continuous over two 12 m spans. Details are shown in Figs 6.23–6.25.

Relevant results for hogging bending of the composite section at support B in Fig.

6.23(c) are as follows:

Mpl, Rd= 781 kN m = 0.43 Mb, Rd= 687 kN m

The design ultimate loads per unit length of beam, from Table 6.2, are permanent: 7.80 + 1.62 = 9.42 kN/m

variable: 26.25 kN/m

The steel section fails the condition in paragraph (g) of clause 6.4.3(1), which limits its depth to 400 mm. The beam satisfies all the other conditions except that in paragraph (b), for its ratio of permanent to total load is only 9.42/35.67 = 0.26, far below the specified minimum of 0.4.

The simplest way to satisfy paragraph (g) would be to encase the web in concrete, which increases the depth limit to 600 mm, and the permanent load to 11.9 kN/m.

The condition in paragraph (b) is quite severe. In this case, it is 11.9 ≥ 0.4(11.9 + qd)

whence

q_d£ 17.8 kN/m

This corresponds to a characteristic floor loading of 17.8/(1.5 × 2.5) = 4.75 kN/m²

which is a big reduction from the 7 kN/m²specified, even though the required reduction in (to £ 0.4) is less than 10%.

This result illustrates a common feature of ‘simplified’ methods. They have to cover so wide a variety of situations that they are over-conservative for some of them.