EFFECTS ON NUMBER OF CABLES FOR MODAL ANALYSIS OF CABLE-STAYED BRIDGES

Transcription

1 EFFECTS ON NUMBER OF CABLES FOR MODAL ANALYSIS OF CABLE-STAYED BRIDGES Yang-Cheng Wang Associate Professor & Chairman Department of Civil Engineering Chinese Military Academy Feng-Shan 83000,Taiwan Republic of China Chun-Ho Hua Lecturer Department of Civil Engineering Chinese Military Academy Feng-Shan 83000,Taiwan Republic of China ABSTRACT. Due to the use of computer technique and the use of high strength material. The span lengths of cable-stayed bridges have been increased. Compared with continuous and suspension bridges, cable-stayed bridge deck has subjected to much stronger axial force caused by the horizontal component of cable reactions. The axial forces make the geometric nonlinearity for the bridges. Cable-stayed bridge is supported by cables instead of internal piers. Therefore, the prestress of the cable, inclined angle between the cable and the bridge's deck as well as the cross section areas of the cables are the most important features for this type of structure. With various number of cables, the cablestayed bridges may have different prestress, inclined angles and the cable cross section areas. The stiffness of the bridge deck may be changed due to the axial forces. In this paper, the three dimensional finite element model of the bridge having the similar geometry with Quincy Bayview Bridge has been built. The modal analysis has been carried out by using different number of cables. The natural frequencies and their corresponding mode shapes are found and compared with those obtained from ambient test. The important effects on the number of cables have been drawn. The numerical results have been presented in tabular and graphical forms. Keywords: Cable-Stayed Bridge, Cable, Modal Analysis NOMENCLATURE M C K U U U F Mass Matrix Damping Matrix Stiffness Matrix Nodal Displacement Vector Nodal Velocity Vector Nodal Acceleration Vector External Force Vector 1. INTRODUCTION Cable-stayed bridges have been more interesting in the recent years due to increasing span length and their aesthetics. Cable-stayed bridges have subjected to strong axial force due to thecable reactions compared with those of suspensions and conventional continuous bridges. In order to reduce the cable tension and the axial force acting on the bridge deck, Agrawal 1 proposed that the cable tension can be reduced rapidly by using more number of cables. In the field of dynamic analysis such as aerodynamic analysis due to wind loading, transient dynamic analysis due to traffic loading and the bridge subjected to seismic loading, these analyses are getting important for this type of bridge. No matter what kind of dynamic analysis is ap- 1374

2 plied, the natural frequency of the bridge is necessary to be studied. In this paper, cable-stayed bridges having the geometry similar to a realistic cable-stayed bridge supported by different number of cables are analyzed. This type of cablestayed bridges has been investigated 2 ' 3 ' 4 ' 5 by finite element method and ambient test 6. The natural frequencies and their corresponding mode shapes are found and presented in tabular and graphical forms. 2. PROBLEM IDEALIZATION In order to effectively model and solve the problem, the following idealizations are made: Members were initially straight and piecewise prismatic. The material behavior was linearly elastic and the moduli of elasticity E in tension and compression are equal. The effects of residual stresses was negligibly small. Bayview Bridge crossing Mississippi River is studied in this paper. Quincy Bayview Bridge is designed 7 in 1983 and completed in The bridge has three spans. It consists of a main span of 900 ft and two equal side spans of 440 ft for total span length of 1780 ft. The structure is described as the following sections; 3.1 Cables The Quincy Bayview Bridge has 25 different cross section areas of 56 cables ranging from in 2 to in 2 constructed of 0.25 in diameter wires with an ultimate strength of 240 psi. The inclined angles between the deck and the cables are different ranging from 38 to The cables are 7-wire cables. Twenty-eight of them support the main span and 14 of them support each side span. The cables are connected at the bottom flange of the main girder. The first interval from the supports is 62 ft. The other intervals on the side span are 63 ft and the interval on main span is 60 ft. The cables are attached to the pylons at 9 ft intervals beginning at 6 ft from the top of the pylon. The cables have the The cable element is a straight, tension-only same cross section areas which are symmetric in element with uniform properties from end to the longitudinal direction. end. The modus of elasticity of the steel and cable is 30 x 10 6 psi, and the poisson's ratio, v, is 0.3 and the unit weight is 490 lb/ft 3. The modulus of elasticity of concrete is equal to 4.47 x 10 6 psi, Poisson's ratio, v, is 0.25 and the unite weight is 150 lb/ft 3. The end supports are attached to the ground with a pinned connection. The pylon and the deck are also connected with a pinned connection and the bridge deck remains continuous. 3.2 Bridge Deck Figure 1 represents the typical cross section of the deck. The deck consists of a nine inch precast slab with two precast traffic barriers, five longitudinal steel stringers with equal spacing of 7.25 ft and floor beams transverse the main girder with spacing from 17 ft to 24 ft which transfers stringer loads to the main girder. 3. STRUCTURAL DESCRIPTION In order to investigate the realistic bridge, a bridge having similar geometry to that of Quincy Figure 1 The Typical Cross Section of the Bridge Deck 1375

3 The deck is supported by anchor piers at each end nite element model used in this paper is referred with pinned connections. The piers are concrete to Hua's model. The number of supporting of columns whose bases are fixed at the bed rock cables are modified based on the model. under the water level. 4. SOLUTION PROCEDURES 3.3 Pylons There are two pylons of Quincy Bayview Bridge. Each of the two pylons consists of two concrete columns and two struts. The upper strut connects with the two columns at the level of 78 ft from the top of the pylon and the lower strut supports the bridge deck. The modal analysis is used to extract the natural frequencies and mode shapes of a linear elastic structure. In the modal analysis, free, undamped vibrations are assumed, i.e. F=O and C=O. The governing equation is expressed as; MU+GU+KU=F(x,t) (1) 3.4 Boundary Conditions The boundary of the bases of the pylons are constrained in all of the translation and rotation directions. The both end of the side spans are con::;trained in the vertical direction. It is considered by pinned connection. The bridge deck and the lower strut::; of the pylons are coupling together in longitudinal and vertical direction. All of the boundary condition::; considered are made in the finite element model of the bridge. 3.5 Finite Element Model The numerical analysis i::; feasible to solve this type of structure. The structure is modeled as a three dimensional finite element model. The cables are discretized a::; a three dimensional tension-only truss element. Each node of the element has three degree of freedom, i.e. translation in x-, y- and z-direction. As the truss element subjected to compre::>sive forces, its modulus of elasticity, E, will be considered as it is approaching to zero. The pylons and the stringers are considered as three dimensional beam elements. Each node of the beam element has six degrees of freedom, i.e., translation in x-, y- and z-direction as well as the rotation about x-, y and z-direction. The bridge deck and the composite girder were modeled as a plate element. Each node of the plate element also has six degrees of freedom which are the same as those of beam element. Hua 2 et. al. provided the global model of this cable-stayed bridge. This fi- Substituting F=O and C=O, the Equation (1) becomes; MU +KU = 0 (2) For linear system, free vibrations will be harmonic of the form, U = U 0 coswt (3) Substituting U and U in the governing equation gives, (4) For non-trial solution, [K - w 2 M] must be zero; the determinant of IK-.\MI = 0 (5) where >. = w 2. If n is the order of the matrices, then the equation results in a polynomial of order n, which should have n roots; wi, w,, w. This is an eigenvalue problem, whose solution are the eigenvalues, >.i, and the corresponding eigenvectors Ui. The eigenvalues represent the natural frequencies of the system ( wi = v:\) and the eigenvectors represent their corresponding mode shapes. 5. NUMERICAL RESULTS Based on the finite element model and the solution procedure, the numerical results were found. Figure 2 represents the axial force due to the cable reactions acting on the bridge deck. The horizontal axis of the figure represents the location along the bridge deck. The vertical axis of 1376

4 the figure represents the normalized axial force. The unity is considered as the horizontal component of the anchor cable reaction of the bridge supported by 56 cables (the original bridge). The center-dash line represents the axial forces acting on the original bridge supported by 56 cables. The solid line represents the axial force acting on the bridge deck supported by 168 cables. The dash line represents the axial forces acting on the bridge supported by 280 cables. Figure 2 indicates that the original bridge has the strongest axial forces acting on the main span bridge deck < " 6-5 " 0. "' " 3 E "'2 " c z' Cable Nurnb<"r bb-, Bridge Deck Figure 2 The Axial force Acting on the Bridge Deck for Various Number of Cables In side spans, as the number of cable increased, the axial force acting on the bridge deck is increasing. As the cable number of 280, the axial force in side span is greater than that of main span created by the original bridge's supporting cables. The bridge deck around the pylon is always subjected to stronger compressive axial forces. On the other hand the midpoint of main span is always subjected to tensile axial forces. It is hard to conclude that the bridge is dominated by the compressive axial force of by the tensile axial forces. Even though the axial force changes the natural frequency of the structural system, it is not easy to find the location dominating the bridge behavior. Table 1 represents the frequencies of the bridges supported by various number of cables. As the number of cables increased the natural frequency is also increased. It means that the structural stiffness is enhanced. Based on Figure 2, as the number of cables increased, the bridge deck around the pylon is subjected to stronger axial force than that of the original bridge. On the other hand, the bridge deck at midpoint of the main span is subjected to stronger tensile axial force than those of the original bridge. Table 1 The Frequencies of the Cable-Stayed Bridges Having Various Numbers of Supporting Cables Number of Cables Mode Number Freq. Mode Freq. Mode Freq. Mode (Hz) Shape (Hz) Shape (Hz) Shape F L F L F L F F F F T p T T p F T T F: Flexural Mode L: Lateral Mode T: Torsional Mode P: Pylon Buckling Mode 1377

5 Therefore, the natural frequency of this type of bridge will be increased as the number of cable increased. Regarding the mode shape corresponding to the natural frequency, the first flexural mode of the bridge supporting by 168 cables is missing. The first mode of the original bridge is flexural but this mode is missing and the lateral mode is instead for the bridge supported by 168 cables. The corresponding frequencies of these two modes have an excellent agreement. Compared with the second flexural mode of the original bridge and the first flexural mode of the bridge supporting by 280 cables, the frequencies have also a good agreement. The difference between these two modes is less than 5%. If the number of cable increases, the cables take more load. Not only the bridge deck takes care of more loads but also the pylons take care of more loads. The pylons are only subjected to compression. The stiffness of the pylon will be reduced by compressive cable reactions. Table 1 indicates that the fifth and sixth modes are pylon flexural even though the moment of inertia of the pylon is as 94 times as much as that of the bridge deck. In order to present the mode shape of the natural vibration, Figure 3 shows the lateral mode of the bridge supported by 168 cables. The free vibration modes of the original bridge are referred to Hua's et. al. paper. and 5, respectively. The fourth and sixth modes, i.e. the first and the second torsional modes, are presented in Figures 6, 8, respectively. The fifth mode, i.e. the first pylon buckling mode, is represented in Figure 7. Figure 4. The First Flexural Mode of the Top View Figure 3. The First Lateral Mode of the Bridge Supported by 168 Cables. The second and third modes, i.e. the first and second flexural modes, are presented in Figures 4 Figure 5. The Second Flexural Mode of the If it is disregarding the agreement of mode shape, the natural frequencies is only considered. Figure 9 represents the comparison of the first three frequencies for the bridges supported by 56, 168 and 280 cables. The horizontal axis of Figure 9 represents the mode number of the bridge. The vertical axis represents the corresponding frequencies of the first three modes in Hz. 1378

6 0.9 r-.-56al-.1-ce-s cab: I L , I 0.8 Figure 6. The First Torsional Mode of the Figure 7. The First Pylon Flexural Mode of the,.-... N '-" u z a first mode second mode third mode MODE NUMBER Figure 9. The Comparison of Natural Frequencies of the First Three Modes of the Bridge Having Various umber of Cables. As discussed in the previous, if the number of cables is increased. the frequencies are also m creased but the increments are not linear. 6. CONCLUSIONS Based on the previous work, the important conclusions can be drawn as the following: Figure 8. The Second Torsional Mode of the For realistic bridges. as the supporting cables increased. the bridge deck around the pylon subjected to stronger compressive axial force than that of the bridge supported by less number of cables. The axial forces will reduce the stiffness of the bridge deck. On the other hand, the midpoint of the main span subjected to stronger tensile force which enhances the stiffness of bridge deck. In global analysis, it is hard to determine that the location subjected to compressive axial force or the location subjected to tensile forces will dominate the structural behavior. 1379

7 As tht> number of cables increased, the pylon will subjected to stronger compressive axial force. It will also reduce the stiffness of the pylon. For the global analysis, as the number of cables increased, the free vibration frequency will increase for this type of bridge. 8. REFERENCES [1] T.P. Agrawal," Cable-Stayed Bridges- Parametric Study", Journal of Bridge Engineering, Vol.2, No.2, 1997, pp [2] Chun-Ho Hua and Yang-Cheng Wang, "Modeling of a Cable-Stayed Bridge for Dynamic Analysis", Proceeding of 12nd International Modal Analysis Conference, Vol.II, pp [3] John C. Wilson and Wayne Gravelle," Modelling of a Cable-Stayed Bridge for Dynamic Analysis", Earthquake Engineering and Structural Dynamics, Vol. 20, pp , [4] Ermopoulos, J. CH., Vlahinos, A.S., and Wang, Yang-Cheng "Stability Analysis of Cable-Stayed Bridges", International Journal of Computers & Structures, Vol. 22 No. 12, pp June [5] Vlahinos, A.S. and Wang, Yang-Cheng "Nonlinear Dynamic Behavior of Cable Stayed Bridges", in the Proceedings of the 12th International Modal Analysis Conference, Vol.II, pp.l , [6] John C. Wilson and Tao Lin," Ambient. Vibration Measurements on a Cable-St.ayeG. Bridge", Earthquake Engineering and Structural Dynamics, Vol. 20, pp , [7] Mojeski and Master, "Structural Drawings of Quincy Bayview Bridge", Mojeski and Master Consulting Engineering, Marrisburg, Pennsylvania,