Advances in Steel Structures - part 15 ppt

120 H. Unterweger The first term considers the base section (A 0, v~ 0 for slenderness ratio 7~) under preload and the second term considers the strengthen section (A, 1< for slenderness ratio ~). The acting stresses are limited by the yield stress. For short columns (~:-> 1,0) the verification procedure is equal to the elastic cross resistance (Eqn. 1). Therefore this procedure also seems uneconomic with increasing preload, because the capacity of the strengthening plates cannot be taken into account. This statement will be confirmed in the following. Suggested solution The extent of preload and strengthening is characterized by the preload ratio o~ - referred to the load capacity NR, 0 of the base section (Eqn. 4), and the strengthening ratio ~ (Eqn. 5, AA is the area of strengthening plates). Nv Nv (~ - - (4) NR, o Ao" too" fy AA (5) The simple analytic calculation model with the essential assumptions is shown in Figure 3. For determi- nation of deformations and moments second order theory is used, including equivalent initial geometric bow imperfections (sine curve). This leads to increasing factors including the ideal elastic buckling load Nki,0 (base section) and Nki (strengthened section) respectively. The individual calculation steps are: - calibration of the simple model in form of determining the initial bow imperfections e 0 to fulfil the ultimate load capacity of the base section according to the code buckling curves. Using the European buckling curves leads to Eqn. 6 for e 0, where a* is a constant depending on the relevant buckling curve (a* = 0,21 § 0,76 for curves a § d) and W 0 is the section modulus. W o e 0 = a*. (~0-0,2) 9 A o (6) - determination of deformation w v under preload N v of the base section. - unloading of the strengthed section leads to the deformation w 0 - neglecting the residual stress distribution in the section - Determination of the ultimate load N R of the strengthed section, with elastic cross resistance of the cross section. Regarding a practical design procedure the resulting load capacity N R is expressed in form of a modi- fied buckling reduction factor •* referred to the base section (Equ. 7 - 9). Doing this, the efficiency of the strengthening plates can be seen immediately. N R = tc*.Ao.fy (7) (8) 1 / wo < K* = ~. 1+~-• o~';Co) x(l_{x.;Co.~o2).~2+ (9) Ultimate Load Capacity of Columns Strengthened under Preload The resulting load capacity N R depends on the following parameters: - cross section parameters, slenderness and buckling reduction factor of the base section (Ao, WO, I0-> ~0-> r'O) - cross section parameters, slenderness and buckling reduction factor of the strengthed section (A,W, I->~) - preload ratio o~, strengthening ratio 121 Figure 3 : Calculation procedure for N R of the strengthed column under preload. To determine the extent of the strengthening plates AA an iteration process is necessary, because W and depends on AA. Reduction of load capacity due to welded strengthen plates Due to the welding process of the strengthening plates residual stresses are introduced, which lead to a decrease of the buckling load capacity. Their quantities and distribution are in general hardly predic- table, due to the high scatter of influence factors. Therefore the influence of the welding process on the load capacity is estimated in an engineering manner. Following the Europian buckling curves the effect of welding on the bearing capacity can be estimated in form of an additional geometric imperfection ev = 0,5. e 0 . Considering this effect in the analytic model (working with Wv* = w v + ev) leads to a ma- ximum decrease of the load capacity of about 12 % for medium slenderness ratios, shown in Figure 4. 122 H. Unterweger Figure 4 : Reduction fweld of the calculated ultimate load N R due to welding of the strengthening plates for careful execution. Comparison of suggested and engineering solution To evaluate the suggested load capacity (Equ. 7 - 9 -> NR, ex) with the engineering procedure (Equ. 3 -> NR, ing ) a comparison for practical columns is useful. In Figure 5 the increase of the load capacity using the suggested solution referred to the engineering procedure is shown for the two border line cases. On the one hand type 1- buckling about y- axis, which is the most effective case for the strength- ening plates; and on the other hand type 2 - buckling about z- axis. In the first case the differencies in load capacity are given for all slenderness ratios, whereas in the latter case with increasing slenderness ratio the results are more and more equal. The differencies increase with growing preload ratio t~ and strengthening ratio ~. For the theoretical case of a preload ratio of t~ = 1,0 and small slenderness we get the highest difference AN = ~. NR,ing, which is equal to the difference between elastic and plastic section resistance (Eqn. 1, 2). This example shows the economic advantages of the suggested solution. Figure 5 : Increase of the ultimate load N R using the proposed solution compared to the engineering approach for two border line cases; a.) type 1, y - axis, b.) type 2, z- axis. Ultimate Load Capacity of Columns Strengthened under Preload Overestimation of load capacity due to plastification of the base section 123 The analytical model neglects the effect of plastification of the base section, which grows with increa- sing preload ratio o~ and also with higher material strength, because of increased plastic zones. To find out the extent of reduction of the load capacity comprehensive finite element calculations with ABAQUS (1996) were made. The web was modelled with shell elements and the flanges and strength- ening plates with special beam elements, including progressive plastifications in thickness direction. The highest reduction of load capacity due to plastification are obtained for type 1 - buckling about y- axis. Fortunately the decrease of load capacity is very small, e.g. 2 - 5 % for a preload ratio ct = 0,5. From the results a simplified conservative procedure for practical use in form of a reduction factor fNR, plast (Figure 6) can be given. Figure 6 : Reduction factor fNR,plast of the calculated ultimate load N R due to plastification of the base section. Figure 7 : Load bearing capacity N R of strengthen columns using European rolled sections, expressed in form of a modified buckling factor •* a.) type 1 - y ; b.) type 2 - z. 124 Improved solution for practical design H. Unterweger For practical design a direct determination of the extent of strengthening plates AA, expressed by the strengthening ratio ~, is desirable. A comprehensive study shows that for every type of strengthened base section the ratios W / W 0 and ~ / 7~ 0 can be expressed in form of a linear relationship of ~. Intro- duction of these information in Eqn. 8 and 9 leads to equivalent buckling factors K:* depending on the slenderness of the base section, the preload ratio t~ and the strengthening ratio ~. In Figure 7 for type 1- buckling about y- axis and type 2 - buckling about z- axis, using European rolled sections, the load capacities in form of •* are given. The effectiveness of the strengthening plates is easy to survey. In Figure 8 the suggested simple design procedure for direct determination of the extent of the strengthening plates, based on design charts for the global buckling reduction factors K:*, is shown. In Unterweger (1996, 1998) the design charts for rolled European universal columns for type 1 and 2 are presented. Figure 8 : Starting position and procedure of a practical design of column strengthening (partial factors omitted). References DIN 18800, Teil 1 und 2 (1990). Stahlbauten - Bemessung und Konstruktion. Deutsches Institut f'dr Normung. Eurocode 3 (1993). Design of steel structures; Part 1.1: General rules. CEN. Rebrov and Raboldt (1981). Zur Berechnung von Druckst~iben, die unter Belastung verst~kt werden. Informationen des VEB MLK 20. Unterweger H. (1996). Berechnung von unter Belastung verstarkten stahlernen Druckstaben, unpublished. Unterweger H. (1998). Druckbeanspruchbarkeit von unter Vorbelastung verst~kten Stiitzen. Stahlbau 68: 3, 196- 203. CHAOTIC BELT PHENOMENA IN NONLINEAR ELASTIC BEAM" Zhang Nianmei 1 Yang Guitong 2 Xu Bingye ~ 1 Department of Engineering Mechanics, Tsinghua University, Beijing, China 2 Institute of Applied Mechanics, Taiyuan University of Technology, Taiyuan, China ABSTRACT The chaotic motions of axial compressed nonlinear elastic beam subjected to transverse load P 8P 0 (f + coscot)sin rex = are studied in this paper. The constitutive equation of the beam is 1 threefold multinomial. The damping force in the system is nonlinear. Considering material and geometric nonlinearity, nonlinear governing equation of the system is derived. By use of nonlinear Galerkin method, differential dynamic system is set up. Melnikov method is used to analyze the characters of the system. The results show that chaos may occur in the system when the load parameters P0 and f satisfy some conditions. The zone of chaotic motion is belted. The route from subharmonic bifurcation to chaos is analyzed in the paper. The critical conditions that chaos occurs are determined. 125 126 KEYWORDS Z. Nianmei et al. chaos, bifurcation, heteroclinic orbit, periodic orbit, dynamic system, saddle INTRODUCTION The chaotic phenomena in solid mechanics fields bring more and more interesting. In 1988, F.C.Moon analyzed the chaotic behaviors of beams experimentally first. Then he studied the dynamics response of linear elastic beam subjected transverse periodic load. The chaotic motions of linear damping beams have been studied by many scholars at home and abroad in resent years. The dynamic behaviors of nonlinear damping beams subjected to transverse load P=SPo(f + coscot)sin ~rx l m are studied in this paper. The critic conditions that chaos occurs in the system are determined by use of Melnikov method. The results show that the chaotic areas may be limited ribbon zones. BASIC EQUATIONS The dynamic behavior of a simply supported nonlinearly elastic beam is studied. Two constant compressive loads N are applied at its two ends. The length of the beam is l. The constitutive relation of beam material satisfies: o- =Eo~ +E16 2 ) (1) where, E and E 1 are material constants. We assume that deformation of the beam is still small deformation after buckling. The buckling critical load of the beam is: ~2EI1A1 Here A I = 1 + 3E1602 . 11 stands for inertia moment, 11 = ~y2dA . A is the cross section area of the A beam. 60 is the strain at neutral surface, it satisfies: Chaotic Belt Phenomena in Nonlinear Elastic Beam 127 ( 3~2El/a I ~'211 E1Eo 4 + E18o 3 + 1-~ 602 + 60 =0 (3) AI 2 AI 2 The beam is subjected to transverse load P =6Po( f +coscot)sin ~r___x_x after buckling. Then the 1 governing equation of the system is: c32M c32w 02w c3w Ow + N~ + m = 8P o (f + coscot)sin ~rx _ 6/.t~~ ax ~ Ox ~ ~ T at at (4) where 8/~ is damping coefficient, m is the mass of unit length of the beam. The boundary conditions of the system are: w(o): wq)-o (5) w"(0) = w"(l)=O (6) Following formula can stand for the strain at the point which distancement from neutral plain is y: O0 c = 60 - y c3 x (7) where 0 is the rotating angle of cross section of the beam at x. It satisfies: 1 Ow sin 0 = (8) 1+~" 00x Submitting geometric relations and physical relations into eq.(4) and omitting the higher order items than three, follow formula can be obtained: c, 4w_+c212 2 3 +6 w 4wl 128 Z. Nianmei et al. V6(t~3W~ 2 {~2W q_,3~2W~ 2 04W 1 ,,[ ,,sin"" =EI &(f + cosco / ~, N m a2w + ~~ (9) E11 E11 O t 2 X m W Eq.(9) is turned into dimensionless form using dimensionless amounts x=- w = r =COot t' 7' ' , cOO = ' coo ml4 " According to boundary conditions (5) and (6), we suppose follow displacement mode: n m w = c,o(v)sin rcx (10) Applying Galerkin method to the dimensionless governing equation, differential dynamic system can be obtained: =-,~q,- p~' +,~o (," +,:os-,)- ~ ~] (11) where ~-~(-~+~,~), p-,.,.,,(c,_- ~) C 1 = A1 , Cz = A1 , C4= 3ELI2 l+eo 2(1+ eo) 3 212(1+6o)312 P = ~ ISc~176 -~o - P~ , -N = N 12 Ell Ell EI1 If system (11) is not perturbed, g = 0. Then eq. (11) is integrable Hamilton system: 'r = v' (12) Hamilton function means the total energy of kinetic energy and potential energy. The energy keeps constant on the same orbit: h = ~ + + = const (13) 2 2 4 Chaotic Belt Phenomena in Nonlinear Elastic Beam The phase trajectory of undisturbed system may be determined by following formula: 129 d(p ~ ~(p4 (14) =+- 2h - aq92 2 The formula (14) shows that the phase trajectory has closed relation with the value of a, ft. The dynamic response of the system in the case of stable post-buckling path (a > 0) and fl <0 are studied in this paper. DYNAMIC ANALYZATION The unperturbed system have three balance states in real space. (0,0) is a center. (- ~]a/- fl,O) and (~/a/- fl,O) are hyperbolic saddles. The heteroclinic orbits passing though two saddles are: (15) The Melnikov function ofheteroclinic orbits is as follows: oo -oo :-; ~0 + ~(,~ + ~ cos~,~0) (16) here ~o : f~d~:.; - seth ~ ~- a~-:-; _0o 2 15r r 2 a - ~(odr- 2 - -oo (17) (18) (19) . obtained: =-, ~q ,- p~' +,~o (," +,:os-, )- ~ ~] (11) where ~-~ (-~ +~,~), p-,.,.,,(c, _- ~) C 1 = A1 , Cz = A1 , C4= 3ELI2 l+eo 2(1+ eo) 3 212(1+6o)312 P = ~ ISc~176 -~ o - P~ , -N. 19 6- 203. CHAOTIC BELT PHENOMENA IN NONLINEAR ELASTIC BEAM" Zhang Nianmei 1 Yang Guitong 2 Xu Bingye ~ 1 Department of Engineering Mechanics, Tsinghua University, Beijing, China 2 Institute. border line cases. On the one hand type 1- buckling about y- axis, which is the most effective case for the strength- ening plates; and on the other hand type 2 - buckling about z- axis. In the