DESIGN OF MULTI-STOREY STEEL FRAMES TO SWAY DEFLECTION

LIMITATIONS

D.ANDERSON

Department of Engineering, University of Warwick, Coventry, UK

SUMMARY

Methods are described for the design of multi-storey steel frames to specified limits on horizontal sway deflection. Approximate methods for rectangular frames require only simple calculations, and their use is illustrated by a worked example. More general approaches are also given. These necessitate iterative calculation and take the form of specialised computer programs. Accurate allowance can then be made for secondary effects which are of particular significance in the design of very slender unbraced structures.

NOTATION

a Constant

B Width of frame

E Young’s modulus of elasticity

F Shear in columns due to wind

h Storey height

I, I′ Moment of inertia of cross-section

K Member stiffness

k Distribution factors

L Bay width

M Parameter M [with suffix] Bending moment at end of member

m Number of bays

N Parameter n Power O Parameter

P Axial forces in columns

py Design strength

r Ratio of bay width to storey height

S Cladding stiffness

Failure of a structure has been defined by Bate (1973) as ‘unfitness for use’, one possible cause being excessive deformation. This results in damage to the cladding or finishes of a building, hinders operations within, and may cause alarm or unpleasant sensations to the occupants. It follows that, when considering serviceability, the designer should calculate deflections under the working (unfactored) loads expected in normal use of the structure.

In multi-storey steel building frames, beam deflections can readily be determined by analysis of a limited frame consisting of the member under consideration and the adjacent beams and columns (Joint Committee, 1971). The main problem relating to deflection concerns horizontal sway in unbraced frames. This form of deflection arises mainly from wind, and its control may govern the member sections.

In the past, there has been no firmly established limit for the sway: height ratio to be used in design. One survey reported by the Council on Tall Buildings (1979) showed that the limiting value had varied from 1/1000 to 1/200, whilst the British code BS449 (BSI, 1969) gave no recommendation for multi-storey structures. More recently, however, both British and European guides have settled on a value of 1/300 for calculations based on bare frames (BSI, 1977; ECCS, 1978). If the complete structure including cladding is considered, then the more restrictive 1/500 has been proposed (ECCS, 1978).

For adequate safety, ultimate strength is checked under enhanced values of loading, obtained by factoring the working loads. Under combined loading the factors range typically from 1·2 to 1·4. These values are sufficiently high to prevent significant plasticity at working load, and therefore deflection calculations are based usually on elastic behaviour.

A number of approximate methods are available for the calculation of sway, some enabling direct design to specified limits. These methods are suitable for hand calculation, and are sufficiently accurate for medium-rise frames. If a given frame is to be analysed for sway, the charts produced by Wood and Roberts (1975) are most convenient. An alternative procedure is due to Moy (1974), which has the advantage that it also provides guidance on what changes will be required in section properties if deflections in a trial design are found to be excessive. However, if control of sway is likely to govern member sizes then equations due to Anderson and Islam (1979a) enable a suitable design to be obtained directly, without the need for a trial set of sections.

For slender high-rise structures, it is most economical to provide a stiff core. However, if an unbraced frame is preferred for architectural or functional reasons, then secondary effects, particularly loss of stiffness due to compressive axial forces and sway due to differential axial shortening, should be considered. The former effect can be included in approximate methods without difficulty, but if axial shortening is significant it will be preferable to use a more accurate computer-based approach. If only analysis is required, many standard programs are available. Direct design by computer is also possible, as demonstrated by, amongst others, Anderson and Salter (1975) and Majid and Okdeh (1982).

In order to choose the most convenient procedure in design, it is helpful to know at any early stage whether ultimate strength or the serviceability limit on sway will dominate the choice of sections. Guidance on this has been given recently by Anderson and Lok (1983), following a parametric study on medium-rise frames. Their work is described below, before proceeding to the design methods referred to above.

3.2 GOVERNING DESIGN CRITERION UNDER COMBINED LOADING

3.2.1 Design Studies

The frames examined were rectangular in elevation, of four, seven and ten storeys in height, and from two to four or five bays in width. Two ratios of bay width to storey height r were considered, namely 1·33 and 2·0, although within a particular frame these two dimensions were constant. All bases were fixed.

The unfactored loads are given in Table 3.1 together with the maximum and minimum basic wind speeds. For simplicity, the resulting horizontal wind pressure was taken as uniform over the height of the frame, although designers often use a reduced pressure on the lower storeys. On the other hand, no allowance was made for eccentricity of vertical loading arising from fabrication and erection tolerances, and no account was taken of any reduction in live loading permitted for the design of columns. In the studies, the

maximum value of floor loading was combined with minimum values of wind loading, and vice-versa. A number of other load combinations were examined also.

The design strength of structural steel, py, was taken as 240 N/mm², corresponding to the grade commonly used in medium-rise unbraced frames. Sway due to unfactored horizontal wind load was to be restricted to 1/300 of each storey height for the bare frame, in accordance with recent recommendations (BSI, 1977; ECCS, 1978).

TABLE 3.1

Minimum sections were determined by designing against failure by beam-type plastic hinge mechanisms or by squashing, using partial safety factors γf of 1·4 and 1·6 on dead and imposed load, respectively (BSI, 1977). These sections were then increased, as appropriate, to satisfy the restriction on sway at working load. The method of Anderson and Islam (1979a) described below was used. Column sections were made continuous over at least two storeys, but beam sections were changed at each floor level if required.

The designs were then subjected to a second-order elasto-plastic computer analysis (Majid and Anderson, 1968), with γf values of 1·4, 1·2 and 1·2 applied to dead, imposed and wind loads, respectively (BSI, 1977). If the factored load level was achieved before collapse occurred, then ultimate strength under combined loading was not the governing criterion for that particular frame.

3.2.2 Results

The results are summarised in Table 3.2. These are applicable to frames whose steel design strength is in the region of 240 N/mm², such as British Grade 43 and European Fe 360 material. The designer needs to determine the ratio of the sum of the column axial forces P to

TABLE 3.2

LIMITING VERTICAL: HORIZONTAL LOAD RATIO, P/F

Frame Bay width: storey height Average P/F

Four-storey 2·0 40

Seven-storey 2·0 40

Ten-storey 2·0 40

Four-storey 1·33 75

Seven-storey 1·33 65