510 C.K. Lau et al. Structural Performance Measurement for Kap Shui Mun Bridge 511 512 C.K. Lau et al. FREE AND FORCED VIBRATION OF LARGE-DIAMETER SAGGED CABLES TAKING INTO ACCOUNT BENDING STIFFNESS Y. Q. Ni, J. M. Ko and G. Zheng Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong ABSTRACT In this paper, a finite element formulation for vibration analysis of large-diameter structural cables taking into account bending stiffness is developed. This formulation is suitable for suspended and inclined cables with sag-to-span ratio not limited to being small. The proposed method provides a tool for accurate evaluation of natural frequencies and mode shapes of structural cables. A numerical verification is made by analyzing with the proposed method the modal properties of a set of cables with different degrees of bending stiffness and sag extensibility and comparing them with the results available in the literature. The modal behaviour and dynamic response of the main cables of the Tsing Ma Bridge in free cable stage are also predicted and compared with the measurement results. KEYWORDS Sagged cable, bending stiffness, finite element method, modal property, transient dynamic response, three-dimensional analysis. INTRODUCTION Advance in modem construction technology has resulted in increasing application of large-diameter structural cables in long-span cable-supported bridges. The Tsing Ma Bridge, a suspension bridge with the main span of 1377m, has been built recently in Hong Kong. As a result of carrying both road and rail traffic, the Tsing Ma Bridge has the most heavily loaded cables in the world. The cable section of the Tsing Ma Bridge is about 1.1m in diameter after compacting. The Akashi Kaikyo Bridge in Japan, which is the world's longest suspension bridge with the main span of 1990m, also has the main cables of about 1.1m diameter. It is well known that the existing theory for cable analysis is developed on the assumption that the cable is perfectly flexible and only capable of developing uniform normal stress over the cross-section. For the large-diameter structural cables, however, the effect of the bending stiffness should be not negligible in performing accurate dynamic analyses. 513 514 Y.Q. Ni et al. Stay cables are the most crucial elements in cable-stayed bridges. Changes in cable forces through degradation or other factors affect internal force distributions in the deck and towers and influence bridge alignment, and are therefore important in assessing the structural condition. The dynamic method has been applied to quantitative measurement of cable tension forces (Takahashi et al. 1983; Okamura 1986; Kroneberger-Stanton and Hartsough 1992; Casas 1994). In most of these applications, the cables were idealized as taut strings by ignoring the sag and bending stiffness effects. This idealization simplifies the analysis but may introduce unacceptable errors in tension force evaluation (Casas 1994; Mehrabi and Tabatabai 1998). Recent research efforts in this subject have been devoted to the development of accurate analytical models to relate the modal properties to cable tension by considering bending stiffness and/or sag extensibility (Zui et al. 1996; Yen et al. 1997; Mehrabi and Tabatabai 1998; Russell and Lardner 1998). Zui et al. (1996) derived empiric formulas for estimating cable tension from measured frequencies, which took into account the effects of bending stiffness and cable sag. Yen et al. (1997) proposed a bridge cable force measurement scheme that considered sag extensibility, bending stiffness, end conditions, and intermediate springs and/or dampers. Mehrabi and Tabatabai (1998) developed a general finite difference formulation for free vibration of a fiat-sag horizontal cable accounting for sag extensibility and bending stiffness. Russell and Lardner (1998) related the tension to natural frequencies of an inclined cable through formulating a curvature equation which took into account the sag extensibility. This paper presents a new finite element formulation for free and forced vibration of cables. This formulation takes into account the combined effects of all important parameters involved, such as sag extensibility, bending stiffness, end conditions, cable inclination, and lumped stiffness and mass. The formulation is first developed for the pure cable without considering bending stiffness. Then the additional contribution of the flexural, torsional and shear rigidities to stiffness matrix is derived by reference to a curvilinear coordinate system. Analytical results using the proposed formulation are verified with available results in the literature and compared with experimentally measured modal data of bridge cables. FINITE ELEMENT FORMULATION Three-Node Curved Element of Pure Cable Without losing generality, the cable static equilibrium profile is assumed in the x-y plane as shown in Figure 1. This initial (static) configuration is defined by x(s) and y(s), here s denotes the arc length coordinate. Let L, E, A and m be the cable length, modulus of elasticity, cross-sectional area, and mass per unit length respectively. In static equilibrium state, the cable is subjected to dead loads (cable self weight and lumped masses) and the cable tension is H(s). The cable is then subjected to the action of dynamic external forces px(s, t), py(s, t), and pz(s, t). The dynamic configuration of the cable is described by the displacement responses u(s, t), v(s, t), and w(s, t) measured from the position of static equilibrium in the x-, y- and z-directions respectively. Let U = {u(s, t) v(s, t) w(s, t)} r and P = {px(s, t) p~(s, 0 pz(s, O} T. By using the Lagrangian strain measure, the extensional strain in the cable due to dynamic loads, ignoring bending stiffness, can be expressed as ~ eO+el (dx Ou dY Ov 1 ~s 2 -~s z ~s ds Os+ 'ds Os)+2"[( ) +( ) +( )2] (1) The finite element formulation is derived from the Hamilton's principle Free and Forced Vibration of Large-Diameter Sagged Cables 515 OU 51 = 5~ I~ OUrc3t OUc3t V,. EA2 ~2 H(s) e]dsdt + ~ ~SU r (q + P c Ot )dsdt = 0 (2) where V,. is the elastic strain energy stored in the initial state; q is the dead load vector existent in the initial state; c = diag[cx Cy ez] is the viscous damping coefficient matrix. An isoparametric curved element with three-nodes is introduced to describe the cable. As shown in Figure 1, the shape functions in the natural coordinate system are given by 1(1_~2 ~2 _1 1 2 N 1 = 89 - ~- ) , N 2 = 1- ' N3 -2 (1 + ~)- 5 (1-~, ) (3) and the coordinates and the displacement functions are expressed as x = Z Nixi, Y = Z NiYi (4) U = Z Niui, v = Z Nivi, w = Z Niwi (5) By defining nodal displacement vector r}~ }r {5}={{8}[ {8} r {8}3 ={uljvljwlju2jv2jw2ju3jv3jWaj (6) Eqn. 5 can be expressed as T T U={u v w} r =[N,I N2I N3I ] {{5} r {5}2 r {8}3 } =[N]{8} (7) Substituting Eqns. 4 to 7 into Eqn. 1 yields e o =[Bo]{d}=[{Bol } {B02 } {B03}]{6}, e t =[Bt]{d}=[{BI1 } {Bt2 } {B13}]{6 } (8a, b) , , 1 {u'N; v'N; w'N;} (9a, b) {B0i } = -55-1{x'N[ y N i 0}, {Bli } = 2j 2 where J = ds/d~ and the prime denotes the derivative with respect to ~. After substituting Eqns. 7 to 9 into Eqn. 2, integrating Eqn. 2 by parts, and considering the static equilibrium equation in the initial state, the governing motion equation for the elementj is derived as [M/]{~jI+[Cs]{fJSI+[Koj + Kls({Uj})+ K2j({UjI{ujIT)]{Uj} = {Pj.} (10) in which, [Mj] =mJ~+][N]r[N]d~, , [Cj] =JI+([N]r[c][N]d~ (lla, b) {Pj} =J~+~[N] r{PId~ , [K0j] = EAJ~+][Bo]r[Bo]d{ +I~+~H[N']r[N']d~ (11c, d) [Klj] = EAJ~+~([Bt]T[Bo]+ 2[Bo][Bt])d~ , , [K2j] = 2EAJ~+~[Bt]T[Bt]d~ (lle, f) (a) 1 (xlj, Yv) y tj _= x ~ 3 (x3j, y3j) ,r 7 z (b) ~=-1 ~=0 ~=+1 1 2 ~' 3 Figure 1. Three-Node Curved Cable Element: (a) Physical Coordinate; (b) Natural Coordinate 516 Y.Q. Ni et al. The global equation of the cable is then obtained through assembling the element mass matrix, damping matrix, stiffness matrix, and nodal load vector by the standard assembly procedure. It is noted that in Eqn. 10 the stiffness matrix includes linear stiffness [K0], cubic stiffness [K1] and quadratic stiffness [/(2]. The present study only addresses linear problem of cable dynamics by discarding the nonlinear stiffness terms. The nonlinear dynamic analyses of cables refer to Ni et al. (1999a, b). Cable Taking into Account Bending Stiffness The additional stiffness matrix stemming from flexural rigidities of the cable is derived with respect to the same curved element as shown in Figure 1. However, a new local coordinate system in terms of tangential and normal axes is introduced to relate displacements with stress resultants. As shown in Figures 2 and 3, the nodal displacements at any node i are expressed as {t~}i {U i V i W i ~si ~ti Ozi }T (12) where ui is the in-plane displacement in tangential direction; v; is the displacement in transverse direction; wi is the displacement in z-direction; Osi is the total rotation in tangential direction; 0ti is the angle of twist; and Ozi is the total rotation of transverse bending. Similar to Eqn. 7, the displacement vector is expressed with isoparametric interpolation functions as {U}={u v w O~ 0 t Oz}r=[NlI N2I N3I]{{b'}~ {b'}~ {8}~}r=[N]{b "} (13) The strain vector is written as {s}- {x" z K" t a Yv, Yw~ } (14) where Ir and x~ are the in-plane and out-of-plane curvature changes respectively; a is the cross- sectional torsion change; Yvs and Yws are the shear strains. They are expressed as aO z 1 ~u a~s ~t K" z - d- , K" t = (15a, b) as R as as R ao, Os ~ aw ct as R ' Yvs = -~s - O z, Yws +0s (15c, d,e) in which R is the curvature radius of the element. It is noted that R is not a constant in the case of sagged cables. It is calculated using the formulae R [1 (dd_~Yx) ]3/2 dx "d2y) = + 2 /( _ z w- (16) The strain-displacement relation can be obtained from Eqns. 13 and 15 as {s} =[B]{J'} =[[B1] [B:z] [B3]]{J'} (17) i~si S \ Oz~ /5/ o,, I ~-X Iz Msi s Mzi R rds I", ~X Figure 2. Displacements at Node i Figure 3. Stress-Resultants at Node i in which, Free and Forced Vibration of Large-Diameter Sagged Cables I N; 0 0 0 0 1 0 0 -R.N' -J.N i ,0 o 0 [Bi ] "- ~ R-; 0 0 0 L0 o R.N~ RJ.N i 0 and the stress-strain relationship is given by R.N; 0 0 -RJ.N i 0 517 (18) {o-} = {M z m t T V z V s }T =diag[Eiz E1 s GJ flGA flGA]{g} = [D]{e'} (19) The additional element stiffness matrix due to flexural rigidities is obtained from Eqns. 17 and 19 as tXa I : JI _ [Sl [OISld (20) The additional stiffness matrix given in Eqn. 20 is derived by referring to the local coordinate system. It should be transformed into the element stiffness relation in the global x-y-z coordinate system before performing assembly to obtain overall stiffness matrix. Similarly, the element stiffness matrix given in Eqn. 11, with 9• dimension, should be expanded as an 18x 18 matrix to cater for the rotation degrees of freedom. NUMERICAL VERIFICATION The proposed formulation has been encoded into a versatile finite element program. In this section, a numerical verification is conducted through comparing the computed results by the present method with the analytical results available in Mehrabi and Tabatabai (1998). They used the string equation and a finite difference formulation to calculate the frequencies of the first two in-plane modes of four suspended cables with a same length (100m) but different sag-extensibility (~2) and bending-stiffness (~) parameters. Table 1 shows the parameters of the four cables (the definition of the parameters refers to the reference). Cable 1 (~2 = 0.79, ~ = 605.5) has a moderate sag and a low bending stiffness; Cable 2 (L2 = 50.70, ~ = 302.7) has a large sag and an average bending stiffness; Cable 3 (~2 _ 1.41, ~ = 50.5) has a moderate sag and a high bending stiffness; Cable 4 (~2 = 50.70, ~ = 50.5) has a large sag and a high bending stiffness. Modal properties of the four cables are analysed by using the proposed finite element formulation. The static profiles of the cables are assumed as parabolas. Sixty equi-length Cable No. ~2 1 0.79 605.5 2 50.70 302.7 3 1.41 50.5 4 50.70 50.5 TABLE 1 Cable Parameters m (kg/m) g (N/kg) L (m) H (106N) E (Pa) A (m 2) J (m 4) 400.0 9.8 100.0 2.90360 1.5988e+10 7.8507e-03 4.9535e-06 400.0 9.8 100.0 0.72590 1.7186e+10 7.6110e-03 4.6097e-06 400.0 9.8 100.0 26.13254 2.0826e+13 7.8633e-03 4.9204e-06 400.0 9.8 100.0 0.72590 4.7834e+08 2.7345e-01 5.9506e-03 TABLE 2 Comparison of Computed Frequencies of In-Plane Modes (Hz) Cable String equation No. ~2 ~ 1st mode 2nd mode 1 0.79 605.5 0.426 0.852 2 50.70 302.7 0.213 0.426 3 1.41 50.5 1.278 2.556 4 50.70 50.5 0.213 0.426 Finite difference method 1 st mode 2nd mode 0.440 0.428 1.399 0.447 0.853 0.464 2.679 0.464 Present method 1 st mode 2nd mode 0.441 0.854 0.421 0.460 1.400 2.682 0.438 0.461 518 Y.Q. Ni et al. elements are used in the computation. Table 2 presents a comparison of the computed natural frequencies obtained by the string equation, the finite difference formulation and the present method. It is observed that for all the four cases the results by the present method match well with those by the finite difference formulation (both the methods take into account sag extensibility and bending stiffness). It is also seen that the computed natural frequencies using the taut string equation (ignoring sag extensibility and bending stiffness) are quite different from those calculated with the present method and finite difference formulation, indicating a considerable influence of the sag extensibility and bending stiffness. CASE STUDY: TSING MA BRIDGE CABLES The modal properties of the Tsing Ma Bridge (Figure 4) in different construction stages have been measured through ambient vibration survey (Ko and Ni 1998). One stage under measurement is the free cable stage. In this stage, only the tower-cable system was erected but none of deck segments has been hoisted into position. The modal parameters of the main span cable and the Tsing Yi side span cable in the free cable stage are calculated and compared with the measurement results. The cable length and sag are 1397.8m and 112.5m for the suspended main span cable and 329.1m and 5.7m for the inclined Tsing Yi side cable. The horizontal component of the tension force is 122642 kN for both the cables. The main span cable is partitioned into 77 elements and the Tsing Yi side span cable is partitioned into 17 elements in the computation. The analyses are conducted by assuming the cable supports to be pinned ends and fixed ends respectively. It is seen in Tables 3 and 4 that the computed natural frequencies of both in-plane and out-of-plane modes of the two cables agree favorably with the measurement results. The dynamic responses of the two cables under in-plane and out-of-plane excitations are then analyzed using the present method. For the damped response analysis, the damping is assumed in the Rayleigh damping form [C] = cz[M] + [3[K] with the coefficients cz = 0.05 and 1~ = 0.01. Figure 5 shows the predicted lateral dynamic response of the damped Tsing Yi side span cable at the cable midspan when an out-of-plane pulse excitation is laterally exerted at the same position. The pulse excitation is F(t) = 500 kN for 0 < t < tcr = 0.5s. Figure 6 illustrates the predicted vertical dynamic response of the damped Tsing Yi side span cable at the cable midspan when an in-plane harmonic excitation is vertically applied at the same position. The harmonic excitation is F(t) = F0.cos2nfi with F0 = 500 kN and f= 0.03569 Hz. It is observed that the damped dynamic response rapidly attains the steady state after several cycles, having the response frequency identical to the exciting frequency. Figure 7 shows the Figure 4. Elevation of Tsing Ma Bridge Free and Forced Vibration of Large-Diameter Sagged Cables 519 TABLE 3 Natural Frequencies of Main Span Cable in Free Cable Stage (Hz) Mode Out-of-plane modes In'plane modes No. 1 st 2nd 3rd 1 st 2nd 3rd Computed: pinned ends 0.0522 0.1040 0.1557 0.1008 0.1471 0.2081 Computed: fixed ends 0.0528 0.1052 0.1578 0.1020 0.1488 0.2091 Measured 0.0530 0.1050 0.1560 0.1020 0.1430 0.2070 TABLE 4 Natural Frequencies of Tsing Yi Side Span Cable in Free Cable Stage (Hz) Mode No. Computed: pinned ends Computed: fixed ends Measured Out-of-plane modes 1st 2nd 3rd 0.2352 0.4696 0.7154 0.2450 0.4946 0.7534 0.2360 0.4770 0.7400 In-plane modes 1st 2nd 3rd 0.3527 0.4693 0.7216 0.3569 0.4943 0.7593 0.3430 0.4780 0.7310 Figure 5. Damped Response of Cable Midspan under Lateral Pulse Excitation (tcr = 0.5s) Figure 6. Damped Response of Cable Midspan under Vertical Harmonic Excitation Figure 7. Undamped Response of Cable Midspan under Vertical Harmonic Excitation . response, three-dimensional analysis. INTRODUCTION Advance in modem construction technology has resulted in increasing application of large-diameter structural cables in long-span cable-supported. introduced to describe the cable. As shown in Figure 1, the shape functions in the natural coordinate system are given by 1(1_~2 ~2 _1 1 2 N 1 = 89 - ~- ) , N 2 = 1- ' N3 -2 (1 + ~ )-. respect to ~. After substituting Eqns. 7 to 9 into Eqn. 2, integrating Eqn. 2 by parts, and considering the static equilibrium equation in the initial state, the governing motion equation for the