The simplified methods described above do not allow for the reduction in frame stiffness due to compressive forces, nor for the effects of axial shortening and unsymmetrical loading on sway.
The reduction in stiffness can easily be included by using additional horizontal shears (Vogel, 1983). If Pu denotes the total vertical loading carried by the columns at storey level u, then the total shear at this level should be increased by 1·2Pu. .
Moy (1977) has given an estimate of the sway due to differential axial shortening.
This was derived for frames subject to uniform horizontal loading, with the floors assumed to be rigid and column cross-sectional areas varying linearly from the top level to the bottom. Anderson and Islam (1979b) used this approach on a 15-storey frame and found the accuracy to be reasonable. However, when sway due to differential axial shortening or unsymmetrical vertical loading is likely to be significant, then simplified methods become appropriate only to the initial design stage. The final design should be obtained using a standard elastic analysis program to examine trial sections, or from specialised programs, such as those of Anderson and Salter (1975) or Majid and Okdeh (1982).
3.5.1 Use of Linear Programming
A flow diagram for the procedure due to Anderson and Salter is shown in Fig. 3.10. Due to the non-linear relationship between deflection and stiffness, a number of iterations will usually be necessary before the
FIG. 3.10. Flow diagram (Anderson
and Salter, 1975).
deflection limits are satisfied. The method is based on standard routines for elastic analysis and linear programming, and therefore can be developed easily. The procedure is applicable to a wide variety of frames, and secondary effects are readily included.
3.5.2 Minimum Cost Design
The previous method aims to minimise cost by generating a light design, but no account is taken of the variations in price per tonne that exist in the supplier’s section catalogue, nor of the restricted number of sections available. The program developed by Majid and Okdeh (1982) overcomes these limitations by costing alternative selections of available sections using a supplier’s price list. This method also adopts an iterative procedure to avoid direct solution of the structure’s stiffness equations. The general procedure is explained with reference to the single storey fixed base frame of Fig. 3.11. For simplicity, axial deformations will be ignored, and the shear in each column will therefore be F/2, with equal rotations θ at B and C.
FIG. 3.11. Single-storey frame.
For equilibrium of AB
(3.42) where MAB and MBC are the clockwise moments acting on AB at A and B, respectively.
Slope-deflection enables these moments to be expressed in terms of θ and the specified sway ∆. Substituting for the moments in eqn (3.42) leads to
(3.43) For equilibrium at B, the sum of the column and beam end moments must be zero. Using the slope-deflection equations for these moments enables the following expression to be obtained
(3.44)
A value of Ib is selected to correspond to a beam section, and an initial value assigned to θ, say zero. Ic is then calculated from eqn (3.43), and θ from eqn (3.44). This enables Ic to
be recalculated, using the value for θ just determined. Iteration continues until the required degree of accuracy is attained. A section is then selected for the columns. Other designs are initiated with different beam sections, and the various designs priced.
For multi-storey multi-bay frames, expressions are derived for column inertias and vertical and rotational displacements by applying slope-deflection and axial stiffness equations at each joint in turn. To treat the sway at each storey as a known quantity, the horizontal deflection x is given by
(3.45) with the vertical distance Y measured from the roof downwards. The constants a1−a3 are determined from the following conditions:
(i) at the top of the frame (Y=0), dx/dY= , (ii) at ground level (Y=H), x=0,
(iii) at ground level, dx/dY=0 due to fixity at the bases.
Hence
(3.46) As n increases the frame becomes more flexible and the profile approaches a linear profile of slope . The value of n used in the calculations is therefore the highest integer value that does not give exponential overflow in the computer.
It can be assumed that the column inertias satisfy eqn (3.1), but the economy of other relative values can also be investigated. Initial beam sections can be generated by rigid-plastic design under vertical loading, or by use of the design equations due to Anderson and Islam. Other beam sections can also be specified in the search for further economy.
Secondary effects are included in the method, which has been demonstrated on very large frames, up to 32 storeys in height with unequal bay widths. The results are generally satisfactory, although the form of the deflection profile may impose unnecessarily severe restrictions on sway of the bottom storeys.
3.6 CONCLUDING REMARKS
This chapter has described two approximate methods to design medium-rise unbraced rectangular frames in which sway deflections control the choice of sections. The method of Anderson and Islam has the advantage that it actually generates a design without the need for trial analyses. Provided secondary effects, particularly differential axial shortening, are not significant, the design will be satisfactory. The choice of sections can be refined by using the analysis due to Wood and Roberts. Both methods can be used by hand, or programmed for a micro-computer. The second method enables account to be taken of cladding stiffness.
If sway due to differential axial shortening is significant, as in very slender frames, then specialised computer methods are more appropriate. Two such methods have been described.
REFERENCES
ANDERSON, D. and ISLAM, M.A. (1979a) Design of multi-storey frames to sway deflection limitations. Structural Engineer, 57B(1), 11–17.
ANDERSON, D. and ISLAM, M.A. (1979b) Design equations for multi-storey frames subject to sway deflection limitations. University of Warwick, Research Report CE3.
ANDERSON, D. and LOK, T.S. (1983) Design studies on unbraced, multistorey steel frames.
Structural Engineer, 61B(2), 29–34.
ANDERSON, D. and SALTER, J.B. (1975) Design of structural frames to deflexion limitations.
Structural Engineer, 53(8), 327–33.
BATE, S.C.C. (1973) Design philosophy and basic assumptions. Concrete, 7(8), 43–4.
BRITISH STANDARDS INSTITUTION (1969) Specification for the use of structural steel in building, BS 449, Part 2.
BRITISH STANDARDS INSTITUTION (1977) Draft standard specification for the structural use of steelwork in building, 13908DC, Part 1, Simple construction and continuous construction.
COUNCIL ON TALL BUILDINGS, COMMITTEE 17 (1979) Stiffness. Monograph on Planning and Design of Tall Buildings, ASCE, New York, Vol. SB, Chapter SB-5, pp. 345–400.
ECCS (1978) European Recommendations for Steel Construction, The Construction Press, London.
JOINT COMMITTEE OF THE INSTITUTION OF STRUCTURAL ENGINEERS AND THE WELDING INSTITUTE (1971) Fully-rigid multi-storey welded steel frames, Second Report.
MAJID, K.I. and ANDERSON, D. (1968) The computer analysis of large multi-storey framed structures. Structural Engineer, 46(11), 357–65.
MAJID, K.I. and OKDEH, S. (1982) Limit state design of sway frames. Structural Engineer, 60B(4), 76–82.
MOY, F.C.S. (1974) Control of deflexions in unbraced steel frames. Proc. ICE, Pt. 2, 57, 619–34.
MOY, F.C.S. (1977) Consideration of secondary effects in frame design. Journal of the Structural Division, ASCE, 103, ST10, 2005–19.
VOGEL, U. (1983) Simplified second-order elastic and elastic-plastic analysis of sway frames.
Proceedings, Third International Colloquium, Stability of Meta Structures, Toronto, Structural Stability Research Council, Bethlehem, USA, pp. 377–87.
WOOD, R.H. (1974) Effective lengths of columns in multistorey buildings. Structural Engineer, 52(7), 235–44; (8), 295–302; (9), 341–6.
WOOD R.H. and ROBERTS, E.H. (1975) A graphical method of predicting side-sway in the design of multi-storey buildings. Proc. ICE, Pt 2, 59, 353–72.
APPENDIX
Major axis moments of inertia for Universal sections in cm4 for design of six-storey two-bay frame.
Universal beams Universal columns 254×146×31 4439 203×203×46 4564 305×127×37 7162 203×203×52 5263 356×171×45 12091 203×203×60 6088
406×178×54 18626 203×203×71 7647
254×254×73 11360
254×254×89 14307
254×254×107 17510
The complete range of Universal sections is given in the Structural Steelwork Handbook published by BCSA/Constrado, London, 1978.