90 W.P. Howson and F.W. Williams obtained by superposing N (which need not be integer) such frames, in the sense implied by flames (b) and (c), must also share the critical load W c and the deflected shape of frame (a), even if the frames are all clamped together. Hence putting N=2 and N=4 gives the required proofs for frames (b) and (c), respectively. Moreover, frame (d) can be obtained by fastening together two frame (a)'s and a frame (b), which are situated side by side in the appropriate way. Since frames (a) and (b) share the same critical load Wc and sway with an anti-symmetric deflection pattem, the process of fastening them together to form frame (d) leaves the critical load We and the deflections unaltered. The proof that frames (a)-(d) share the same deflections under lateral loading is essentially identical to the proof for buckling given above, if it is noted that the lateral loads of Figures (a)-(c) can be replaced by an anti-symmetric load pattern, which must cause an anti-symmetric deflection pattem, because the beams are assumed to be inextensible. For example, the F of Figure (a) can be replaced by F/2 at the left hand end of the top beam and F/2 at the right hand end of the top beam. Similarly, the 4F at the top left hand joint of Figure (d) can be replaced by loads of F/2, F, 3F/2 and F at the four top storey joints, etc. Most multi-bay frames do not obey the Principle of Multiples, Home and Merchant (1965), Lightfoot (1956). However, a well established method exists for reducing multi-bay multi-storey frames to single bay multi-storey 'substitute' frames which can then be used to obtain approximate lateral loading or critical buckling results for the multi-bay case. The substitute frame has the same number of storeys and the same storey heights as the actual frame, but differs in that it has only one bay, is symmetric and carries symmetrical vertical loads. The required details of the substitute frame are found from the actual frame as follows: the substitute column k is equal to half the sum of the k's for all actual columns at the same storey level; the substitute beam k is equal to the sum of the k's for all beams at the same storey level; the horizontal loads at the nodes at both ends of a beam are equal to half of the sum of the horizontal loads at all actual nodes at that storey level; and the values of p for the substitute columns are equal to the sum of the axial forces in all actual columns at the same storey level divided by the sum of the n2EI / g2 for all actual columns at the storey level. Hence p is equal to the axial force in a substitute column divided by the value of its Euler load, n2EI / g2. Applying the above rules to the flame of Figure l(d) gives the flame of Figure l(c), on which the forces 4F and 8F can be replaced by anti-symmetrical pairs of forces. Hence it can be deduced that when a frame obeys the Principle of Multiples the rules yield a substitute frame which gives exactly correct results for the actual frame, remembering that inextensible member theory is assumed. The Grinter frame of Figure 1 (e) has been advocated as a means of obtaining approximate results for use in codes and has been known for a very long time, Grinter (1936), Wood (1974). Rules for obtaining it from the actual frame are identical to those given for the substitute frame above, except that, as can be seen by comparing Figures 1 (c) and (e), the vertical loads and column k's are twice as large and the beam k's are three times as large. Note that the rolling supports at the right-hand ends of the beams of the Grinter frame prevent rotation while leaving horizontal motion unrestrained. The Grinter frame has been favoured due to its computational simplicity, because computation only involves one node per storey and they are connected to form a chain, so that the stiffness matrix has the minimum possible bandwidth, i.e. it is tri-diagonal. Because of the symmetry, the substitute frame of Figure l(c) can be analysed by considering only half of the frame, which looks like the Grinter frame of Figure l(e) except that the beams are now of half length and are pinned to the rollers. Therefore, for buckling problems there is no horizontal force in the columns and so the stability functions n and o, Home and Merchant (1965), can be used to obtain an overall stiffness matrix with only one degree of freedom, the joint rotation, at each storey level. If the refinement of the n and o Unified Pr&ciple of Multiples for Lateral Deflection, Buckl&g and Vibration 91 functions is not introduced, the problem is still very simple because it has only two degrees of freedom, rotation and horizontal deflection, at each storey level. Similarly, the lateral load calculations require only two degrees of freedom at each joint, or one degree of freedom if the Principle of Superposition is used to first apply the lateral loads with rotation prevented at the joints and then to calculate the clamping moments at the joints and apply them with reversed sign, so that the ensuing calculations are for a problem with no horizontal force in the columns and hence can again use the n and o functions. Thus the nodes form a chain and there is only one degree of freedom (rotation) at each storey level if the n and o functions are used (so that the overall stiffness matrix is tri-diagonal) and otherwise there are two degrees of freedom per node, i.e. rotation and horizontal displacement. Therefore, the substitute frame and the Grinter frame give identical results with identical computational effort. The authors have always considered the substitute frame to be preferable to the Grinter frame because it gives much greater physical insight. In particular, as can now be seen, the substitute frame relates in an obvious way to the Principle of Multiples whereas the Grinter frame does not. For example, when an actual frame does not obey the Principle of Multiples, a 'feel' for the probable accuracy of results given by a substitute frame can be obtained by a quick estimate of how close an approximation to the actual frame can be obtained by applying the Principle of Multiples to the substitute frame. The substitute frame has many fewer design variables than the actual frame. Therefore, parametric studies undertaken with the substitute frame can give the designer insights into the behaviour of the full range of possible actual frames, i.e. the full range of multi-storey multi-bays frames, with very small computational effort and without the designer overload referred to in the Introduction occurring. S EI "37" Figure 2: Simple system used to represent cladding for a Grinter frame, where the dashed line represents one storey (with flexural rigidity EI) of its column A simple established way of allowing for cladding when using Grinter frames, Wood (1974), which can also be used for substitute frames, Williams (1979), is shown in Figure 2. The effect of the system shown is to resist relative horizontal movement of adjacent storeys with stiffness S, which is usually expressed in the dimensionless form = Sg 3 / EI (1) 92 W.P. Howson and F.W. Williams APPLICATION OF THE PRINCIPLE OF MULTIPLES TO VIBRATION PROBLEMS The preceding sections have established a case for using substitute frames for wind load and buckling calculations, despite the availability of massive and cheap computer power, both for frames which do or do not obey the Principle of Multiples. Therefore, the focus is now changed to examine the extent and usefulness of the corresponding applications to vibration problems, to which very little attention has been given, Bolton (1978), Roberts and Wood (1981), Williams et. al. (1983). It is assumed that 'exact' member theory is used in the sense that the distributed mass of the members and the attached floors are incorporated when calculating the member stiffness matrices, which are therefore transcendental functions of both frequency and load per unit length. Hence, there are two restrictions to the application of the Principle of Multiples to vibration problems which were not there for the lateral load and buckling problems. These are that members sharing the same subscript on Figure 1 must have the same mass per unit length as well as the same value of k and that all bays must have identical spans. The second requirement occurs because, whereas a beam (because it is in contraflexure) contributes 6k to the overall stiffness matrix of the half substitute frame analysed for the lateral loading and buckling cases, in vibration problems the stiffness contributed depends both on k and on a dynamic stability function which is a transcendental function of both the beam span and the mass per unit length. Therefore rules must be adopted to establish the values of g and la for the beams of the substitute frame. The rules adopted herein are that L is taken as the average value of the bay widths of the actual frame, so that EI can be calculated from the substitute beam k yielded by the rules given in the previous section, while l.t for the substitute beam is obtained by dividing its g into the total mass of all beams at the same storey level of the actual frame. Arguments essentially identical to those in the previous section then show that frames (a)-(d) of Figure 1 have identical sway natural frequencies. It is impossible to devise a Grinter frame of the type shown in Figure l(e) which will share exactly the natural frequencies of the actual frame even when the actual frame obeys the Principle of Multiples. This is because the dynamic stability functions for a member built-in at its far end do not behave identically to those of a member which crosses an axis of anti-symmetry of the mode. Of course, this could be overcome by modifying the Grinter frame of Figure 1 (e) such that the right-hand ends of the beams, as well as being on rollers, are free to rotate. However, as well as this modified frame no longer strictly being a Grinter frame, it is essentially half of the substitute frame used previously with all flexural rigidities and loads doubled. Hence it shares exactly the computational advantages of the substitute frame without giving the insight advantages. Therefore, the authors consider that their preference for the substitute frame as opposed to the Grinter frame is additionally vindicated when vibration, as opposed to just lateral load and critical buckling problems, is considered. RESULTS To keep the description of the results concise, they were all obtained for variants of the four bay, eight storey frame with built-in foundations shown in Figure 3. This frame has sensible properties, as follows. Young's modulus (E) = 200 GN/m 2. The beams are all identical, with length 7.2m, second moment of area (I) 6,000cm 4, cross-sectional area (A) = 52.5cm 2 and mass per unit length (~t) = 3,500 kg/m, which includes an allowance for floor mass and the mass of live floor loading. The columns are all of length 4.0m and their other properties were identical for any chosen storey i (i = 1,2 8), such Unified Principle of Multiples for Lateral Deflection, Buckling and Vibration 4 @ 7.2m I I I e I r I f I f I f- I ,,J I I ,a I I @ 93 Figure 3" The datum frame. The diagonals shown dashed are absent except for case 6 of Table 1 that Ii = 15,000 Yi cm4, Ai = 131.25~f~ cm2 and ~i = 100~]-kg/m, which is for columns with no allowance for cladding, where Yi = 1 + 0.35 (8- i) (2) Table 1 gives the first three natural frequencies of this structure as case 1, i.e. as the datum problem. The beams and columns were represented by Bernoulli-Euler member theory, with distributed mass allowed for exactly, Howson et. al. (1983). The results given by the substitute frame are compared with those given by the actual frame using inextensible and extensible member theory. Table 1 additionally contains the corresponding natural frequency results for all the remaining cases, each of which is a variant of the datum problem or of the substitute frame used to represent it, as briefly defined in the second column of the Table and more fully described as follows. Cases 2 and 3 give altemative substitute frames for case 1, for instructional reasons. For case 2 the beams were analysed as massless and hmaped masses equal to half to the mass of the beam were added at each end of the beam. For case 3, the Grinter frame of Figure 1 (e) was used. Case 4 used 'exact' theory, Howson et. al. (1983), to allow for the effect of axial force, as well as of distributed mass, on the flexure of the columns. The axial forces were obtained as if half of the mass of each beam had been lumped at its ends, both for the substitute and actual frames, but the beam 94 Case 10 W.P. Howson and F.W. Williams TABLE 1 RESULTS FOR ALL PROBLEMS (HZ) Description of Problem Datum problem Datum with the masses of beams of substitute frame lumped at their ends Grinter frame results for datum Datum and allow for effect of axial forces on column flexure Datum with cladding added (~ = 5) Datum with both central bay spans doubled Datum with EI of second column from the left doubled Datum with stiff cladding (g = 15) for substitute frame, to represent structural bracing of one bay of actual frame As case 5, except that g = 5 is represented exactly as in Figure 2, not by an equivalent diagonal As case 8, except that g = 15 is represented exactly as in Figure 2, not by an equivalent diagonal Substitute 0.234 0.756 1.437 0.234 0.758 1.446 0.234 0.754 1.428 0.208 0.708 1.374 0.557 1.504 2.518 0.172 0.565 1.095 0.240 0.783 1.510 0.901 2.363 3.054 0.557 1.504 2.519 0.901 2.364 3.054 Frame Actual (EA >oo) 0.234 0.753 1.432 as case 1 as case 1 0.207 0.705 1.369 0.557 1.503 2.515 0.172 0.563 1.086 0.239 0.779 1.505 0.901 2.362 3.004 as case 5 as case 8 Actual (True EA) 0.233 0.752 1.431 as case 1 as case 1 0.206 0.698 1.350 0.543 1.479 2.486 0.171 0.562 1.084 0.238 0.779 1.503 0.633 1.893 2.991 as case 5 as case 8 stiffnesses were still calculated using distributed mass. It should be noted that the axial forces were 22.6% of those which would have caused buckling of the substitute frame, i.e. the critical load factor for the substitute frame was 1/0.226 = 4.42. Case 5 gives results when cladding (the mass of which was neglected) represented by bracing equivalent to ~ = 5 at every storey was added to the actual (i.e. multi-bay) datum problem. The Unified Principle of Multiples for Lateral Deflection, Buckling and Vibration 95 authors deliberately chose software which does not have coding to represent the spring and rigid cranked beam system of Figure 2 when representing the substitute frame because such a feature is unlikely to be available to a designer seeking to use the substitute frame to undertake a parametric study. Instead the authors used approximately equivalent massless diagonal bracing in each of its bays. By assuming that the beams and columns were inextensible (which is reasonable because the cross-sectional area A of the beams and columns far exceeds that of the bracing) the bracing members for storey i were readily shown to have A i = 1.263 yi ~ cm 2 (3) for the substitute frame and one quarter of this value for each bay of the actual frame, for which all the diagonals were parallel to each other, so that the structure was not symmetric. Cases 6 and 7 were included to show the effects of further deviation from the requirements of the Principle of Multiples. In case 6 the span of the two central bays was doubled, with the substitute beam length being taken as the average of the sum of the actual beam lengths. In case 7 the EI of the second column from the left was doubled at every storey level. Case 8 was solved in order to see to what extent ~ (again modelled by diagonal bracing) could be used in the substitute frame to represent an actual frame which was braced only in the one bay indicated by the dashed lines on Figure 3. These diagonals and those of the substitute frame all have the value of A given by Eqn. 3. Cases 9 and 10 are identical to cases 5 and 8 respectively, except that ~ for the substitute frames was modelled as shown in Figure 2, instead of by the equivalent diagonals of Eqn. 3. SOME CONCLUSIONS FROM THE RESULTS OF TABLE 1 All cases of Table 1 (except cases 8 and 10 which are discussed later) demonstrate good agreement between the substitute frame results and those obtained for the actual frame when using extensible member theory, i.e. the true EA' s. This strongly suggests that the first three modes of the actual frame were sway dominated anti-symmetric ones, since these are the only modes which the substitute frame can find. The correctness of this conclusion was verified by calculating the natural frequency for the lowest non-sway (i.e. symmetric) mode and, in case 6, eliminating anti-symmetric modes between 0.562 Hz and 1.084 Hz for which the mode could be seen upon inspection to be a 'local' mode, i.e. one dominated by flexure of individual members with very little sway occurring. By comparison with case 1, it can be seen from cases 2-6, respectively, that : the horizontal beam inertias are important but their transverse inertias have negligible effect; the Grinter frame results are very close to the substitute frame ones, so that the use of Grinter frames for structures which obey the Principle of Multiples may only cause very small errors; allowing for the flexural magnification due to axial forces of practical magnitudes causes significant reductions of the fundamental (12% in this case) and higher natural frequencies and these reductions can be calculated very accurately from the substitute frame; allowing for the stiffening effect of cladding can greatly increase the fundamental (by 133% in this case) and higher natural frequencies and again the substitute frame can be used to calculate these increases very accurately. 96 W.P. Howson and F.W. Williams Note that none of the cases 1-5 of Table 1 obey the Principle of Multiples because the outer two of the five columns have twice the required properties, but that nevertheless the excellent agreement of the final two columns of results confirms that the inextensible assumption of the Principle of Multiples is extremely accurate. Cases 6 and 7 show that this agreement remains good for frames which depart more radically from obeying the Principle of Multiples. The reason for the substitute frame results for cases 8 and 9 differing so much (by up to 42%) must principally be the extensibility of the beams and columns, because the actual frame with EA >oo gave results almost identical to those of the substitute frame. Physical reasoning suggests that, because only one bay of the actual frame is braced, the extensions and contractions of beams and columns caused by the forces in the diagonal bracing will be largely confined to the beams in the braced bay and the two columns bounding the bay. This further suggests that the beams and columns of the substitute frame should not be treated as inextensible but should instead be given the EA values of an individual beam and column of the braced bay of the actual frame. When this was done the values of 0.901, 2.363 and 3.054 in Table 1 were replaced by 0.607, 1.861 and 2.777, i.e. the maximum difference of +42% from the 'full frame with actual EA' results was reduced to -7% for the third natural frequency and the fundamental was in error by only -4%. (Note that if the bracing of the actual frame is evenly distributed between the four bays, so that each bay has one quarter of the A, the substitute frame is unaltered but the actual frame results of 0.835, 2.248 and 2.992 are much closer to them, as would be expected because the columns of the actual frame will then change length very little.) Hence, the results of cases 8 and 10 lead to the tentative but important new result that the substitute frame method, with an appropriate value of ~ (modelled either via Figure 2 or the diagonals of Eqn. 3) and with appropriate values of EA, gives a useful indication of the natural frequencies for the sway modes of frames which have bracing in a minority of their bays. Finally, comparison of the results of cases 9 and 10 with those of cases 5 and 8 justifies the use of the equivalent diagonal of Eqn. 3 when software incorporating the model of Figure 2 is not available. FURTHER THOUGHTS An unbraced frame is usually one of a set of similar frames which are parallel to it and are connected to it by beams perpendicular to it, e.g. to form a building of rectangular planform. The first author, together with a Master's student, have made a very promising preliminary investigation of predicting the sway modes of such structures which sway parallel to the frames, by applying the rules given earlier to obtain a substitute frame, but with the modification that all the frames are used when applying the rules, e.g. the substitute column k is equal to half of the sum of the k's for all actual columns at the same storey level, regardless of which frame the column lies in, etc. This concept is derived from the fact that floors can reasonably be regarded as being rigid in their own planes, so that all frames share the same horizontal displacements. Of course, such substitute frame results will be exact if the frames are identical to one another, are identically loaded and individually obey the Principle of Multiples. This is clearly true, because the frames would then deflect identically to one another even in the absence of floors and the substitute frame would share exactly the behaviour (i.e. lateral displacements, buckling load factor or natural frequencies) of the substitute frame yielded by one frame alone, since all the stiffnesses and loading of the latter substitute frame would be multiplied by the number of frames to give the former one. Unified Principle of Multiples for Lateral Deflection, Buckl&g and Vibration REFERENCES 97 Bolton A. (1976). A simple understanding of elastic critical loads. Struct. Engr. 54:6, 213-218. See also correspondence 54:11,457-462. Bolton A. (1978). Natural frequencies of structures for designers. Struct. Engr. 56A:9, 245-253. See also correspondence 59A:3, 109-111. Grinter L.E. (1936). 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