130 Z. Nianmei et al. The eq. (16) shows that constant load f and loading frequency ~ have great effect on the conditions that there exists Smale horseshoe in the system. 1) When x/-a-f > 1, the conditions that Melnikov function has simple zero points is: 2o Poo ;to <_ < (20) 2fRa + 22 ,u 2f& + 22 or m R 1 < P-~-~ <R 2 (21) where R 1 = 8a: 4~ 15fl 2 1+ ocosechI ol R 2 = 8a: x/-a 15fl m The eq.(20) means that there exists a limited belted zone in Po-co plane. When R 1 < ~P~ <R2, there is Smale horseshoe in the Poincare map of the system. 1 2) When 0 < f < ~, there exists sole nr*. It should satisfy: 4a - nr cosech zc~ =f 2 a. If 0 _< w _< w*, the critical condition chaos occurs is: Chaotic Belt Phenomena in Nonl&ear Elastic Beam 131 r~ > R 1 (22) /.t The above formula shows that chaotic area is half-infinite. b. If nr > w*, the representative of critical condition that chaos occurs is the same as formula (21). The above analyzation shows that the critical conditions there exists Smale horseshoe in the dynamic system have closed relations with loading way and frequency. ROUTE TO CHAOS There is a set of periodic orbits circling the center (0,0)" ~Pk(Z')=+ -0+k2)flsn l+k2V, k =_ v,k dn v,k l+k 2 l+k 2 l+k 2 (23) The periods of the orbits are: T 4~ l+k2 = K a where K is first type Jocabi elliptic integration. Melnikov functions of subharmonic orbits are: )) + COS ~7(2" + T O o - -Po + 16a 2 __2~ nk3(5 - k____:) (m,.)- Po 15p O+k F fO~%m [ / (KK) n~l~ miseven Pz(m,n)= 2 ct rc ' ~ fll+k2 sh n=l and m is odd here K'= K~/1 - k 2 The threshold that odd order subharmonic bifurcation occurs is: (24) (25) (26) 132 Z. Nianmei et al. ~0> 8k3etS/2 (5-k 2) sh(mZK' 15pz~O+k2) 5n \ 2K ) : Rm When m > oo, that is k > 1, following formula can be obtained: limRm =R o :-~sh zw m ~oo 15/~rW Comparing R1 ,R 2 with Ro, we know: R 0 > R 1 , Ro > R 2 (27) (28) So the system will enter chaos status by limited odd order subharmonic bifurcation. CONCLUSION 1 .Only when the undisturbed differential dynamic systems possess heteroclinic orbits, the chaotic belt phenomena may occur after the system is perturbed. 2.The chaotic areas are affected by not only the ratio of constant load to the amplitude of periodic load but also loading frequency co. If the ratio of constant load to the amplitude of periodic load is greater than 1, the chaotic area is belted at any loading frequency co. If the ratio is smaller than 1, the area in which Smale horseshoe occurs is belted when nr > nr*. But the threshold is lower limit only when 0 _< nr ___ m* 3.If the constant load equals to zero the area that chaos occurs is half infinite. 4.The system may enter chaos status by limited odd order subharmonic bifurcation. REFERENCE Moon F. C. (1988). Experiments on Chaotic Motions of A Forced Nonlinear Oscillator: Stranger Attractors. J. Appl. Mech. 55, 190-196. Panida Dinca Baran (1984). Mathematical Modes Used in Study The Chaotic Vibration of Buckled Beams. Mechanics Research Communications 29:2, 189-196. Zhang Nianmei and Yang G.T. (1996). Dynamic Subharmonic Bifurcation and Chaos of Nonlinear Elastic Beam. J. Nonlinear Dynamic 3:2, 265-274.(in Chiness) Frames and Trusses This Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left Blank INVESTIGATION OF ROTATIONAL CHARACTERISTICS OF COLUMN BASES OF STEEL PORTAL FRAMES T C H Liu and L J Morris Manchester School of Engineering, Oxford Road, Manchester, M13 9PL, UK ABSTRACT Most of the portal frames are designed these days by the application of plastic analysis, with the normal assumption being made that the column bases are pinned. However, the couple produced by the compression action of the inner column flange and the tension in the holding down bolts will inevitably generate some moment resistance and rotational stiffness. Full-scale portal frame tests conducted during a previous research program had suggested that this moment can be as much as 20% of the moment of resistance of the column. The size of this moment of resistance is particularly important for the design of the tensile capacity of the holding down bolts and also the bearing resistance of the foundation. The present research program is aiming at defining this moment of resistance in simple design terms so that it could be included in the design of the frame. The investigation also included the study of the semi-rigid behaviour of the column base/foundation, which, to a certain extent, affects the overall loading capacity and stiffness of the portal frames. A series of column bases with various details were tested and were used to calibrate a finite element model which is able to simulate the action of the holding down bolts, the effect of the concrete foundation and the deformation of the base plate. KEY WORDS Column Base, Holding down bolts, Column flexibility, Portal Frame INTRODUCTION Steel portal frames, similar to most other structures, tend to be designed almost independent of the foundation condition, mainly because most practicing engineers cannot readily appreciate or quantify this interaction. While the design of column bases of most of the multi-storey frame structures is govemed by the large axial forces, column bases in portal frames are subjected to a relatively larger lateral shear (Bresler & Lin, 1959). Though there have been some studies recently, the interaction between the soil/foundation block/structural frame is probably the least understood aspect of the whole building. An on-going project was designed to investigate the effect of foundation to the overall 135 136 T.C.H. Liu and L.J. Morris behaviour of steel portal frames following the series of full-scale tests. The research program had been divided into three phases, aiming to quantify the rotational and moment capacities of the column bases in order to check their effects on the overall frame behaviour and to recommend a suitable design for column base details. The first part, which is to be reported in this paper, was to look into the effect of various geometric parameters of the column bases such as the thicknesses of the base plates, column sizes, and size and length of holding down bolts. The study consists of a series of laboratory testing and computational modelling. In the design of a typical portal frame, it is generally assumed that the bases are "pinned" for purposes of analysis, i.e. column does not transfer any moment to the foundation. A typical base consists of a base plate fillet welded to the end of the column member. The base is then attached to the concrete block by means of holding down bolts, anchored within the block. The normal detail for a "pinned" base is to locate two holding down bolts along the neutral axis of the column, one on either side of the web in an attempt to simulate a "pinned" base with the minimum cost. After completion of alignment the plate is grouted into position. In a previous research program, three three-dimensional full-scale pitched-roof portal frames of spans 12m, 12m and 25m respectively were tested. In additional to normal vertical load applied from the roof as in all the three frames, one of the columns in the second frame was also subjected to a horizontal load. In all cases, the columns, designed with "pinned bases", were built as mentioned above except that the concrete blocks were rest on floor. Table 1 shows the bending moment measured in the column just before the frames failed. Only the second frame failed with a plastic hinge formed near to the column head (Engel,1990; Liu, 1988). Frame 1 Frame 2 Frame 3 TABLE 1 COLUMN BENDING MOMENT IN FULL-SCALE FRAMES Column size Height Bending Moment near Bending Moment at (m) to column head (kNm) column base (kNm) 203x133x25UB 3.7 58 13.5 305x165x40UB 2.7 185 35 406xl 78x54UB 3.65 323 64 Though designed and constructed as "pinned", the bases had inevitably attracted some moments. Such moments might be about 20% of the column moment capacity (Liu, 1988) and have to be resisted by the coupled generated by the bearing compression of the base plates against the concrete blocks and the tension developed in the bolts. Since the bases were designed as "pinned", the size of the bolts were determined largely by the applied shear forces (Morris & Plum, 1995). EXPERIMENTAL SET-UP The objective of the isolated column base tests was primarily to calibrate the finite element model. The main feature in the set-up was to ensure that the numerical model was able to reveal a sufficiently accurate interaction between the column base plate and the concrete block. The column in the TABLE 2 SUMMARY OF MATERIAL PROPERTIES Yield stress Modulus of Ultimate strength (N/mm 2) Elasticity (N/mm 2) (N/mm 2) Flange 348.20 187710 500.00 Web 401.00 189365 526.37 HD bolts Concrete 675.00 195200 feu=30N/mm 2 28500 845.00 Rotational Characteristics of Column Bases of Steel Portal Frames 137 Figure 1 Experimental set-up arrangement was laid horizontal for the convenience of load application. It was loaded as a simple cantilever. The whole column base was 'rest' on a 500x1200x1500 concrete block. The whole set-up was geometrically symmetrical about the bottom of the concrete block as shown in Figure 1. A pair of one-metre long M24 holding-down bolts went through the two concrete blocks and held the two sides in position. The type of HD bolts used in the tests was of higher strength Grade 8.8 with an ultimate strength of 845N/mm 2. The material properties were shown in Table 2. A well-established finite element package was previously developed (Liu, 1988) particularly for the analysis of the full-scale portal frame tests. It was also proved to be very successful for the analysis of various types of connections (Liu & Morris, 1991, 1991). In the finite element model, the steel columns were descretised into 8-noded shell elements and the concrete blocks were refined into 8- noded brick elements. The part of the concrete blocks beyond the tension flange of the columns was Figure 2 F.E. mesh of the column base + concrete foundation block 138 T.C.H. Liu and L.J. Morris Figure 3 Moment-rotation curves of the column not modelled in order to reduce the problem size. Due to symmetry about the web plate, only half of the assembly was modelled. Link interface elements were placed in between the two components in order to determine whether or not they were in contact. The holding down bolts were modelled by line elements following the stress-strain characteristic which was obtained from a separated tension test. The bonding between the HD bolts and the concrete would quickly vanish after once or twice of loading and unloading. Therefore, it was assumed that the bolts were free to extend in tension from the beginning of the loading. Also, the pre-loads in the bolts, about 25kN, were ignored in the model. The base of the concrete block was assumed to be fixed. A point load was applied at a distance of 2m from the base plate. A typical mesh showing the deformation is shown in Figure 2. One of the crucial factors that can determine the accuracy the model is the effect of the base plate. Two thicknesses were used in the test, 12mm and 20mm representing two possible stiffnesses of the same column base. Figure 4 Bolt forces vs. Applied Bending Moment Rotational Characteristics of Column Bases of Steel Portal Frames 139 Figure 5 Effect of base plate thickness on bolt force The moment-rotation characteristic and the bolt force vs. applied bending moment curves obtained from the F.E. models and the tests were plotted in Figures 3 and 4 respectively for the two different thicknesses of base plates. The comparison was excellent except that the F.E. models depicted a stiffer behaviour. This is mainly due to the in-accurate assessment of the compression stiffness of the concrete block. However, it is interested to note that, though there is a large difference in the stiffnesses between the two cases, the bolt forces do not differ a lot. The column base with a thicker base plate rotated about the toe of the base plate, i.e. about 220mm from the centroid. The bolt force would therefore be, 1 Mapp = 2.27M Pbolt - 2 0.22m app where Mapp is the applied bending moment. This agrees very well with the results obtained for the 20mm case from the tests and F.E. modelling. For a more flexible base plate of thickness 12mm, the plate was able to bend and part of it was in contact with the concrete block. The prying action increased the forces in the bolts. However, after the bolts extended further, the prying action faded away and hence the bolt forces came back to a similar level as found with thicker plates. Since the centroids of the couple formed by the tension force in the bolts and the compression force by the reaction should normally be very close to the compression toe of the column, the bolt forces were fairly independent of the base plate thickness and bolt size. Figure 5 shows that the prying action increased the bolt forces by about 20% for thinner base plate. MOMENT-ROTATION CHARACTERISTICS Further computational analysis were carried out to examine the effect of various geometric parameters. The computational models were analysed upto a complete collapse, mainly due to bolt failure. Figure 6 shows the effect due to a variation of the base plate thickness. In general, a full range moment- rotation curve consists of four parts. The first part is the elastic regions where every component remains elastic. However, the behaviour is not linear, as a result of the moving centroid of the reaction from the concrete block. With high strength HD bolt, the elastic portion is followed by a static growth in moment of resistance due to an extensive flexural yielding in the base plate. Thereafter, the tensile . 1991, 1991). In the finite element model, the steel columns were descretised into 8-noded shell elements and the concrete blocks were refined into 8- noded brick elements. The part of the concrete. were placed in between the two components in order to determine whether or not they were in contact. The holding down bolts were modelled by line elements following the stress-strain characteristic. Chaos of Nonlinear Elastic Beam. J. Nonlinear Dynamic 3:2, 26 5-2 74. (in Chiness) Frames and Trusses This Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally