Advances in Steel Structures - part 17 pptx

140 T.C.H. Liu and L.J. Morris Figure 6 Effect of base plate thickness on rotational characteristics membrance action of the base plate is able to support a further increase in the bolt force until the behaviour comes to a final stage where the bolts eventually fail. The diameter of the holding down bolts affects directly the initial elastic rotational stiffness as shown in Figure 7. The moment carrying capacity at the second stage and the in-plane membrane stiffness in the third stage are basically not affected as they depend largely on the thickness of the base plate. Figure 8 summaries the effect on the elastic rotational stiffness due to the HD bolt diameter and the base plate thickness. In the range of consideration, the variations seem to be fairly linear. However, when the bolt diameter becomes very large, the stiffness of the column base with 12mm plate should approach a magnitude of about 12000kNrn/rad. When the base plate becomes very thick, the stiffness due to the pair of 24mm HD bolts would be 17000kNm/rad. Obviously, if similar analysis is extended Figure 7 Effect of diameter of HD bolts on rotational characteristics Rotational Characteristics of Column Bases of Steel Portal Frames 141 Figure 8 Factors affecting Rotational stiffness of column bases to some smaller or weaker HD bolts, e.g. Grade 4.6, the ultimate moment capacity would be substantially reduced, as shown in Figure 7, by shortening the third stage. FURTHER PARAMETRIC STUDY ON ROTATIONAL STIFFNESS Apart from the study on various geometric factors, a series of parametric analyses was also carried out. The objective of this was to establish and quantify the effect of each of the components in the column base on its rotational flexibility of the column base. The components of interest include the stiffness of the concrete block, HD bolt and the base plate. Five different cases were considered: 9 Case 1: Rigid concrete block with infinitely rigid HD bolts; 9 Case 2: No concrete block, but the base rotates about the toe with infinitely rigid HD bolts; 9 Case 3: Rigid concrete block with normal HD bolts; 9 Case 4: No concrete block, but the base rotates about the toe with normal HD rigid bolts; 9 Case 5: Normal concrete block and HD bolts The F.E. analyses were carried out until the base plates had yielded extensively. For the cases where the thicknesses of base plates were 20mm, the average rotational stiffness upto 30kNm were noted; whereas in the cases of 12mm, the stiffnesses upto 15kNm were recorded. The results are tabulated in Tables 3 and 4. Comparing cases 4 and 5, the flexibility due to compression of the concrete block is 23.27• .6 rad/kNm for the 20mm case and 16.87x 10 -6 rad/kNm when the thickness if 12ram. This is because the bearing area for the thinner plate is much larger than the thicker one. The difference between cases 2 and 3 shows that the flexibility due to extension of bolts are about 55x10 "6 rad/kNm from the two thickness cases. This agrees very well with the elastic flexibility obtained by simple calculation (58x 10 -6 rad/kNm) assuming all other components rigid. The flexibility due to base plate deformation is expected to dominate the difference between the two cases. The flexibility due to the bending of the 12mm plate together with the end-portion of the column is found to be 110.71x10 6 rad/kNm and that of the 20mm plate is only 34.68x 10 .6 rad/kNm. 142 T.C.H. Liu and L.J. Morris TABLE 3 ROTATIONAL STIFFNESS FOR 20MM BASE PLATE Rotation (x 10 "3 radian) Case 1 0.805 Case 2 1.040 Case 3 2.110 Case 4 2.683 Case 5 3.381 (Test result = 7800 kNrn/rad I Single Bolt equivalent force (kN) 98.1 eccentricity (mm) 152.9 Stiffness (kNm/rad) 35104.0 68.2 220.0 28837.6 34.68 74.7 201.0 14221.4 70.32 68.2 220.0 11181.6 89.43 69.0 217.3 8873.3 112.70 Flexibility (~trad/kNm) 28.49 TABLE 4 ROTATIONAL STIFFNESS FOR 12MM BASE PLATE Rotation (x 10 "3 radian) Case 1 0.960 Case 2 1.660 Case 3 1.820 Case 4 2.458 Case 5 2.711 (Test result = 5300 kNm/rad' Single Bolt force (kN) 59.8 equivalent eccentricity (mm) 125.4 Stiffness (kNrn/rad) 15624.1 Flexibility (~trad/kNm) 64.00 34.1 220.0 9032.6 110.71 38.2 196.5 8248.8 121.23 33.9 221.4 6102.6 163.86 36.8 204.0 5533.2 180.73 CONCLUSION In this paper, a few design parameters have been considered and their effects on the rotational stiffness been examined. They include the thickness of the base plate, bolt size and the stiffness of the concrete block. The contribution of flexibility by the concrete block is about 20% on the 20mm thick base plate whereas that on the 12mm thick base plate is only 8%. The stress distribution within the toe region is very complex. It requires further investigation. A factor, which is not considered here, is the reduced effective column section. The tensile stress transmitted from the bolts would diffuse gradually into the column. The effective stiffness of the column at the plate-column junction could probably be halved the normal value and thereby increases the rotational flexibility. It is also not included in this part of the research the behaviour of the underlying soil. Any moment reversal could produce differential settlement causing possible rotation of the foundation block. This might lead to a reduction of the column base moment. While it is essential to quantify the possible stiffness and the moment capacity of the column base for their detail design, it is not recommendable to take this into account when designing the portal frame. REFERENCES: Bresler, B. & Lin, Y.Y. (1959), Design of steel structures, John Wiley & sons, N.Y. Engel, P. (1990) The Testing and Analysis of Pitched Roof Portal Frames, Ph D Thesis, University of Salford. Rotational Characteristics of Column Bases of Steel Portal Frames 143 Liu, T.C.H. (1988), Theoretical Modelling of Steel Portal Frame Behaviour, Ph D Thesis, University of Manchester. Liu, T.C.H. & Morris, L.J., The development of a shear hinge and the effect on connection flexibility, Proc. Of the Asian-Pacific Conf on Computational Mechanics, Hong Kong, Dec 1991 Liu, T.C.H, & Morris, L.J., The effect of connection flexibility on portal frame behaviour, Int Workshop on connections in steel structures, AISC/Eurcom, Pittsburgh, April 1991 Morris, L.J. and Plum, D.R. (1995) Structural Steelwork Design to BS5950, Longman Scientific & Technical, 2 no Edition, U.K. 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 ULTIMATE STRENGTH OF SEMI-RIGID FRAMES UNDER NON-PROPORTIONAL LOADS B.H.M. Chan, L.X.Fang and S.L. Chan The Hong Kong Polytechnic University, Hunghom, Hong Kong SAR, PR China. ABSTRACT This paper presents a numerical procedure for practical design and elasto-plastic large deflection analysis of semi-rigid steel frames under non-proportional loads. Most structures are first under a set of vertical loads such as self-weight and live load before the application of the lateral loads due to wind or seismic forces. The response of a structure under this load sequence cannot be obtained by the principle of super-imposition of these two loading cases due to the non-linear structural behaviour. However, it is often treated in a non-linear analysis as proportional loads for simplicity, which contains a certain degree of uncertainty in accuracy. In this paper, the effects of load sequence are studied and a comparison is made between the case for a structure under proportional and non-proportional loads. It was found that the two results are considerably different that an accurate analysis should allow for this effect. KEYWORDS Steel frames, Semi-rigid Frames, Elasto-plastic analysis, Second-order inelastic analysis, Ultimate strength, Proportional and Non-proportional Loads INTRODUCTION Currently most of the second order inelastic analyses of steel-flamed structures are performed under the assumption of proportional extemal loads. However, real structures are often subjected to non-proportional loads. The objective of this paper is to study the load-deflection behaviour of steel flames under proportional and non-proportional loads. A numerical example with a portal flame of steel I-sections is analysed for this purpose using a geometric and material nonlinear finite element computer program, GM-NAF (Geometric & Material Non-linear Analysis of Frames). 145 146 PLASTIC HINGE MODEL B.H.M. Chan et al. In the second-order inelastic analysis of steel frames, the first-yield moment, Mer, accounting for the residual stress, (Yres, Can be determined as, Me r ._(O.y__O.res FIZ (1) where cry is the specified yield strength, F is the axial load of the member, A is the cross-sectional area and Z is the elastic sectional modulus. In this paper, the Section Assemblage Concept (Chan& Chui 1997) is adopted to determine the yielded zone, 2~, which is shown in Figure 1, for a section as, F for ~ < d . 2Cryt 2 ~: (F-crytd) d for d d ~+ <~< +T 2Bcry 2 2 2 (2) in which B is the overall breadth of the flanged section, d is the depth of the web, T and t are the thickness of flange and web, respectively. (Yy ' i loy Stress Block Figure 1" Stress distribution for wide-flange section under combined axial force and moment Once the extent of the plastic region 2~ is known, the reduced moment capacity Mpr of the section under combined axial force and bending moment are obtained as follows, Mp r =[BT(D_T)+I(d)2 _~//2/t]o.y for Up r = I(O/2 _ ~br ~O.y for d" ~<_ , 2 d d (3) <V< +T 2 2 A spring is employed for simulation of yielding and the formation of plastic hinge. When no yielding occurs, the spring stiffness is infinite and, when a plastic hinge is formed, the spring stiffness is zero. The spring stiffness, ks, of sections between the first-yield and the fully plastic moments is then taken as, Ultimate Strength of Semi-Rigid Frames under Non-Proportional Loads 6EI Mpr -M for Mer < M < Mpr ks= Z ]M-Mer] (4) 147 where E is the elastic modulus of elasticity, I is the second moment of inertia and L is the member length and the strain hardening effect is ignored. SEMI-RIGID CONNECTION MODEL The exponential model as follows and proposed by Lui and Chen (1988) is used in this paper to demonstrate the moment-rotation behaviour of semi-rigid connections and is given by, g c =nj~.lCj{1-expl-lOr]ll+kcf]Orl+g 0 .: 2ja)J (5) In equation (5), Mc is the moment applied at the connection; Or is the relative rotation corresponding to the moment Mc; Mo is the initial moment; kcf is the connection stiffness at the strain-hardening stage; a is the scaling factor; and Cj are the curve-fitting constants given in Lui and Chen (1988). NUMERICAL PROCEDURE For non-proportional loading, the Newton-Raphson method is used for the vertical loads in the first load sequence whilst the Minimum Residual Displacement method (Chan 1988) is used for lateral loads in the second load sequence. This arrangement is needed since the vertical loads can only be confined to the designed level by the Newton-Raphson method. To detect the hysteretic behaviour of the plastic hinges and semi-rigid connections during loading stages, one can determine the sign of the incremental moment, AM, and then compared with that of the total moment, M. For the virgin loading path it can be sensed by AM.M>0 (6) while for the unloading path, aM.M_<0. (7) NUMERICAL EXAMPLE The double-bay pitched-roof portal frame illustrated in Figure 2 is adopted for this example. Three types of semi-rigid beam-to-column connections are assumed. These connections are of the extended end plate (EEP), flush end plate (FEP) and top-and-seat angle (TSA) types. The out-of-plumbness for each column is taken as 1/200 (EC3 1993) and the residual stress pattern is considered according to E.C.C.S. (1983). To illustrate the effects of proportional loads (PL) and 148 B.H.M. Chan et al. non-proportional loads (NPL) in the semi-rigid portal frame, the applications of the loading in PL and NPL cases for each type of semi-rigid connections are as follows: 1. PL: The vertical loads P and the lateral load H, which are described in Figure 2, are applied to the portal frame simultaneously with the same load factor (i.e. ~,v=~,h) until the frame collapses; and 2. NPL: The vertical loads P are firstly applied to the corresponding value of ~.v at which the frame collapses in the PL case (see Figure 3). Then, the lateral load H is applied until the frame collapses. This procedure is to ensure that the same vertical load level is maintained as a basis for comparison of the PL and NPL cases. ~,hH ~,vP LvP .~ ~ 20 ~ 20 ~ ~ [. 4mL" 4m I. 4m I 4m I E = 200 kNm -2 (Yy = 275 Nmm 2 P = 200 kN; H = 40 kN Column: 203x203x46UC Rat~er : 254x 146x31UB Figure 2: Double-bay pitched-roof portal frame. The load deflection curves for the semi-rigid frames are shown in Figures 3 and 4, and the maximum values of P, H with corresponding values of u are also given in parentheses in the figures. Note that, for the NPL cases, curves in Figure 3 are with respects to the first load sequences while Figure 4 for the second load sequences. Therefore, it is reminded that the curves for the NPL cases in Figure 4 do not start from the origin. In Figure 4 it can be found that the ultimate lateral loads for the portal frame with EEL connections under PL and NPL are very close and the differences in ultimate lateral loads and displacements are within 5%. On the other hand, for the frame incorporated with TSA connections, the ultimate lateral load in the NPL case is about 14.6% greater than that in the PL case. However, for the frame with FEL connections, the behaviour in the PL and NPL cases is totally different. The ultimate lateral load in the NPL case is only 7.9% of the value in the PL case. It is because as observed in Figure 3 the stiffness of the portal frame with FEP commences to decrease from its elastic value, at which the prescribed vertical load level has been applied. Ultimate Strength of Semi-Rigid Frames under Non-Proportional Loads 120 80 A z 60 Ik 40 20 24 22 2O 18 16 14 8 6 4 2 0 i ( a EEP-PL(P=107.27kN; u=5.724cm) _- EEP-N PL(P 107.27kN; u=-3.063cm) FEP-PL(P=100.52kN; u=8.177cm) ~, FEP-N PL(P=100.52kN; u=-4.112cm) : TSA-PL(P=74.227kN; u=l 1.23cm) - e- - - TSA-NPL(P=74.227kN; u=-3.419cm) l l i , | , , ! , -6 -4 -2 0 2 4 6 8 10 12 14 16 18 u (cm) 100 Figure 3" Vertical loads P versus lateral displacement u. 149 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 u (cm) Figure 4" Lateral load H versus lateral displacement u. . u =-3 .063cm) FEP-PL(P=100.52kN; u=8 .177 cm) ~, FEP-N PL(P=100.52kN; u =-4 .112cm) : TSA-PL(P=74.227kN; u=l 1.23cm) - e- - - TSA-NPL(P=74.227kN; u =-3 .419cm) l l i , | , , ! , -6 -4 -2 0 2 4 6 8. of yielding and the formation of plastic hinge. When no yielding occurs, the spring stiffness is infinite and, when a plastic hinge is formed, the spring stiffness is zero. The spring stiffness,. corresponding to the moment Mc; Mo is the initial moment; kcf is the connection stiffness at the strain-hardening stage; a is the scaling factor; and Cj are the curve-fitting constants given in Lui