A novel structural modification to eliminate the early coupling between bending and torsional mode shapes in a cable stayed bridge

Abstract

In this paper, a novel structural modification approach has been adopted to eliminate the early coupling between the bending and torsional mode shapes of vibrations for a cable stayed bridge model generated using ABAQUS software. Two lateral steel beams are added to the middle span of the structure. Frequency analysis is dedicated to obtain the natural frequencies of the first eight mode shapes of vibrations before and after the structural modification approach. Numerical simulations of wind excitations are conducted for the 3D model of the cable stayed bridge with duration of 30 s supporting on real data of a strong wind from the literature. Both vertical and torsional displacements are calculated at the mid span of the deck to analyze both the bending and the torsional stiffness of the system before and after the structural modification. The results of the frequency analysis after applying lateral steel beams declared a safer structure against vertical and torsional vibrations and rarely expected flutter wind speed. Furthermore, the coupling between the vertical and torsional mode shapes has been removed to larger natural frequencies magnitudes with a high factor of safety. The novel structural approach manifested great efficiency in increasing vertical and torsional stiffness of the structure.

Keywords

aeroelastic instability / structural damping / flutter wind speed / bending stiffness / torsional stiffness

Introduction

The finite element method (FEM) is a firm and appropriate technique which is dedicated to solve complex engineering problems in different areas: civil engineering, mechanical engineering, biomedical engineering, hydrodynamics, heat conduction, etc. FEM is a robust tool to determine approximate solution of differential equations which are describing different physical operations. Basic FEM procedures are considered the origin of its success. These procedures are: the figuration of the problem in variational form, the finite element discretization of this figuration and the effective solution of the finite element equations which are produced. These elementary stages are the same whatever problem is adopted and in conjunction with the use of the digital computer exhibit a perfect procedure to engineering analysis. Computational methods [–] are well suited to complement experimental testing.

Long span bridges are vulnerable to many types of vibrations when subjected to wind buffeting action due to their slenderness, high flexibility, low structural damping and lightweight. Consequently the aeroelastic interaction between the wind and the long span bridges is generated, where the flutter and the torsional instabilities can take place at certain wind speeds. The vibrational response of long span bridges is affected by many structural characteristics like mass, stiffness and energy dissipation mechanisms [–]. Recently many slender long span bridges have been constructed without taking in consideration the both the bending and torsional stiffness, which resulted in wide displacements especially at the mid span of the deck with probabilities of aeroelastic instability and structural failure. The extra sensitivity of long span cable supported bridges due to wind excitation is related to the very low structural damping in the coupled modes of vibrations widely below 1% and even less than this value in the vibration modes associated with cable vibrations [–]. Flutter and torsional instabilities force limits on the length increase of long span cable supported bridges which can be avoided by better aerodynamic design of the deck or by the use of vibration control methods [–]. Flutter is considered the most dangerous aeroelastic instability in long span bridges, especially after the failure of Tacoma narrow bridge. When flutter occurs, the wind forces change continuously due to structural displacements, while the wind alters the stiffness and structural damping of the system. When the structural damping becomes very low, a small oscillation would be amplified until the structural failure. Furthermore, these oscillations couple to generate what is called the flutter frequency [–].

Since the structural damping of long span bridges is very low, the applications of dashpots and frictional dampers have become an important need to increase the damping of the structure against multiple types of vibrations, such as tuned mass damper (TMD) which are mechanical dampers are modally tuned and applied at critical locations in the long span bridges to increase the structural damping of the structure [].

Sealed tuned liquid column gas damper is another device used to increase the structural damping of long span bridges which is consisted of gas spring effect consideration especially when the vibrations are occurring in low frequency range [].

Electromechanical actuator bearing energy conversion characteristics is another innovative device is used to damp the vibrations in long span bridges. The work of this device is similar to piezoelectric device, where efficient design of this device can lead to optimally damping vibrations of the system and energy harvesting abilities [].

Various devices and mechanical techniques have been undertaken by researchers and designers of long span bridges to accommodate the lateral, vertical and torsional vibrations of the deck which in turn resulted in fair positive outputs. In this study, a novel structural approach is being considered different from the previous methods through increasing the overall bending and torsional stiffness of the structure aiming to remove the danger of coupling between the vertical and torsional mode shapes of vibrations to a safe and rare region. Two lateral steel beams are attached to the middle span of the cable stayed bridge which is the critical region for vibrations. This approach is applied by assuring a little effect of the added weights on the overall deflections of the deck and in such a way to avoid the fatigue of the structural elements.

Flutter wind speed

The flutter and buffeting response analysis of long span bridges exposed to turbulent wind excitation is generally performed in the frequency domain based on the approach originally. The primary effect of the self-excited forces is to change the structural damping through the aerodynamic damping [,]. The torsional-to-vertical frequency ratio is decisive for the critical flutter wind speed. In traditional bridge design, this means that the required torsional stiffness of the bridge deck is increased when the span goes up, which means that the mass of the bridge deck per unit length is increased. The principle of decreasing the torsional frequency to the level of the vertical frequency or below has been labeled the non-flutter design principle [91,92]. Dyrbye and Hansen [91] explained that for very long span suspension bridges, the torsional rigidity of a closed box girder is too small to resist flutter. To obtain large critical flutter wind speeds, they proposed a design in which the torsional and vertical frequencies deliberately were made identical. This concept was validated experimentally by Andersen et al. [].

Flutter analysis

Identifying the flutter wind speed for the design of long span bridges, needs to perform wind tunnel analysis so that to calculate the flutter derivatives in addition to using frequency domain. A vibrating bridge deck under wind flow creates self-induced forces that depend on displacement vector y=(v,w,{\phi }_x)and its derivative, where v is horizontal, w is vertical and {\phi }_x is the rotational degrees of freedom of the deck (see Fig. 1).

The relationship between aeroelastic force f_z and the displacement vector can be written employing flutter derivatives as formulated by Simiu and Scanlan, 1986 [94]

(1)\begin{array}{l}{f}_z=\left\{\begin{array}{l}{D}_z\\ L_z\\ M_z\end{array}\right\}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho VKB\left(\begin{array}{l}\hspace{0.17em}\hspace{0.17em}{P}_{\mathrm{1}}^{*}\hspace{1em}-P_{\mathrm{5}}^{*}\hspace{1em}\hspace{0.17em}-B{P}_{\mathrm{2}}^{*}\\ -H_{\mathrm{5}}^{*}\hspace{1em}\hspace{0.17em}{H}_{\mathrm{1}}^{*}\hspace{1em}\hspace{0.17em}\hspace{0.17em}B{H}_{\mathrm{2}}^{*}\\ -B{A}_{\mathrm{5}}^{*}\hspace{1em}B{A}_{\mathrm{1}}^{*}\hspace{1em}{B}^{\mathrm{2}}{A}_{\mathrm{2}}^{*}\end{array}\right)\left\{\begin{array}{l}\dot{v}\\ \dot{w}\\ {\dot{\phi }}_x\end{array}\right\}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho V^{\mathrm{2}}{K}^{\mathrm{2}}\left(\begin{array}{l}\hspace{0.17em}\hspace{0.17em}{P}_{\mathrm{4}}^{*}\hspace{1em}-P_{\mathrm{6}}^{*}\hspace{1em}\hspace{0.17em}-B{P}_{\mathrm{3}}^{*}\\ -H_{\mathrm{6}}^{*}\hspace{1em}\hspace{0.17em}{H}_{\mathrm{4}}^{*}\hspace{1em}\hspace{0.17em}\hspace{0.17em}B{H}_{\mathrm{3}}^{*}\\ -B{A}_{\mathrm{6}}^{*}\hspace{1em}B{A}_{\mathrm{4}}^{*}\hspace{1em}{B}^{\mathrm{2}}{A}_{\mathrm{3}}^{*}\end{array}\right)\left\{\begin{array}{l}v\\ w\\ {\phi }_x\end{array}\right\},\end{array}

where B is the deck width, \rho is the air density, V is the acting wind speed, K= Bw/V is the reduced frequency with w as the response frequency, A_{\mathrm{1}}^{*}, H_{\mathrm{1}}^{*}and P_{\mathrm{1}}^{*}(i = 1,2,…,6) are the flutter derivatives obtained experimentally. The multimodal flutter analysis is used to solve this problem. Eq. (1) can be expressed in a matrix form as:

(2)f_z=K_{z}y+C_z\dot{y},

where K_z and C_z are aeroelastic stiffness and damping matrix, while y represents a displacement vector of a node along the deck. The matricesK_z and C_z for the entire bridge, can be obtained by assembling the matrix of each bar element of the deck. The dimension of the matrices coincides with the total number of degree of freedom of the bridge deck. The system of equations that governs the dynamic behavior of the deck under aeroelastic forces is expressed as:

(3)M\ddot{y}+C\dot{y}+Ky=f_z,

where M, C, and K are mass, damping, and stiffness matrices. By combining Eq. (2) and Eq. (3), we get:

(4)M\ddot{y}+(C-C_z)\dot{y}+(K-K_z)y=\mathrm{0.}

To solve the problem of Eq. (4), a modal analysis is performed. The displacement vector can be written as a function of the most relevant m mode shapes grouped by a modal matrix, \Phi.

(5)y=\Phi w{e}^{\mu t},

where \mu and w are complex values. By plugging Eq. (5) into Eq. (4), the system is transformed to:

(6)({\mu }^{\mathrm{2}}IW+\mu C_{R}W+K_{R}W)e^{\mu t}=\mathrm{0},

where {\Phi }^{\mathrm{T}}M\Phi =I,\hspace{1em}{\Phi }^{\mathrm{T}}(C-C_Z)\Phi =C_R,\hspace{1em}{\Phi }^{\mathrm{T}}(K-K_Z)\Phi =K_R.

Equation (6) becomes a nonlinear eigenvalue problem as:

(7)(A-\mu I)W_{\mu }{e}^{\mu t}=\mathrm{0},

where A=\left(\begin{array}{l}-C_R\hspace{1em}-K_R\\ \hspace{0.17em}I\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\mathrm{0}\end{array}\right)and W_{\mu }=\left(\begin{array}{l}\mu w\\ \hspace{0.17em}w\end{array}\right).

To solve this problem, we need mode shapes and natural frequencies of the bridge under study, which can be obtained by a finite element model. The solution to Eq. (7) is expressed as {\mu }_j={\theta }_j+i{\gamma }_j(j=\mathrm{1},\mathrm{...},\mathrm{2}m) where \theta is related to structural damping and \gamma is the damping frequency. Flutter is produced when \theta becomes null with increasing wind speed. This eigenvalue problem is solved to obtain flutter speed.

Finite element model

A cable stayed bridge model is created in ABAQUS with 324 m length and 22 m width, the main parts of the bridge is the deck which consists of connected reinforced concrete deck segments with 2.6 m height. Four reinforced concrete pylons with square shapes 4 m × 4 m dimensions and 103 m height, and 80 stay cables are connecting the deck to the pylons in a fan shape arrangement, each cable with cross section area 0.00785 m² (see Fig. 2).

Two lateral steel beams with length 146 m, thickness= 0.5 m and width 10 m at the pylons and 5 m at the middle part are added to the middle span tied with the deck and the pylons (see Fig. 3). Main steel bar diameter is 0.06 m and diameter of the temperature steel bars in addition to the stirrups is 0.04 m. The boundary condition of the deck is fixed in one side and free for longitudinal translation in the other side. The pylons are fixed at the bottom and each two pylons are connected by six reinforced concrete ties with 4 m × 4 m dimensions and 22 m length. The stay cables equivalent Young’s modulus of elasticity has been used to approximate the sagging occurrence in the cables because it was modeled as truss elements (see Table 1).

The deck is modeled as (C3D10: A10-quadratic tetrahedron) elements, pylons and ties are modeled as (C3D8R: An 8 node linear brick reduced integration hourglass control) elements, the reinforcing steel bars and the lateral steel beams are modeled as (B31: A 2- node linear beam in space) elements and the stay cables are modeled as (T3D2: A 2- node linear 3D truss) elements.

Mesh convergence

Adequate refined mesh is important to insure that the calculated results are efficient. Coarse meshes can yield inaccurate results in analyses using implicit or explicit methods. The numerical solutions of the models will tend toward a unique value as the mesh is refined. The mesh will be converged when additional mesh refinement produces a negligible change in the results. Mesh convergence was performed supporting on the results of the natural frequencies for eight mode shapes of vibrations. A uniform mesh refinement was considered with equal element size for each element type, where the results are calculated 14 times (see Fig. 4).

The mesh convergence apparently starts with a constant path approximately from 300000 elements. To create the model of the cable stayed Bridge with the structural modification, the total of 393413 elements were used in the convergence region by considering finer mesh size consisting of (10872-C3D8R elements, 108125-C3D10 elements, 6088-T3D2 elements and 268328-B31elements).

Wind load

A strong wind with design speed of 54 m/sec has been dedicated in the numerical simulations, and the frequency of the excitation is arranged to produce a frequency falls in the frequency ranges of first eight mode shapes of vibrations of the cable stayed bridge model so that to include the vertical and torsional modes. The wind pressure assigned in the simulations is with duration of 30 s (see Fig. 5). The wind fluctuation data has been prepared depending on exact data from the literature in addition to modification of the excitation frequency [95]

The cable stayed bridge model is divided into five regions over its height by taking in account the wind pressure variation due to elevation from the ground. The wind pressure is designed to excite the cable stayed bridge model perpendicular to the longitudinal axis without skewedness. The wind forces (lift and drag) and pitching moment are assigned to the cable stayed bridge regions as pressure values supporting on the level of each region along the height. It is worthy to mention that after adding the two lateral steel beams, the level of wind pressure assignation would be modified because the wind will act on the new added structural member.

Structural modification and mode shapes

The first 8 mode shapes of vibrations have been obtained, the types of vibrations and the related frequency were determined for each mode shape. The first 4 mode shapes are vertical vibrations modes for the model before structural modification and their frequencies are 0.242, 0.347, 0.509 and 0.613 Hz respectively. While for the model after modification, the first four mode shapes are vertical vibrations too, and their frequencies are 0.390, 0.450, 0.579 and 0.752 Hz simultaneously (see Table 2). This means that no changes took place in the type of vibrations for the mentioned mode shapes but the magnitudes of frequencies have been increased to higher critical wind speeds with a difference range of (0.07-0.148) Hz between the two cases. The fifth and sixth mode shapes for the model before modification are coupled lateral torsional and vertical torsional vibrations with frequencies 0.655 and 0.776 Hz respectively, and the same mode shapes for the model after modification are coupled lateral torsional vibrations with frequencies 0.820 Hz and 0.856 Hz respectively. While the seventh and eighth mode shapes for the model before modification are torsional and vertical vibrations respectively with frequencies 0.789 and 0.813 Hz respectively. In the other hand the same mode shapes for the model after modification are lateral and torsional vibrations with frequencies 0.960 and 0.986 Hz respectively. This indicates that after structural modification the danger of coupling between the vertical and torsional vibrations has been removed and secured to remote region having natural frequency of 1.610 Hz and safety against flutter instability has been increased widely.

Simulation of frequency analysis for the first eight mode shapes of vibrations of the cable stayed bridge model before and after structural modification are shown in (Fig. 6). The first 4 mode shapes of the model after structural modification have manifested appreciable decrease in the displacement for the same scale factor as shown by the amplitude of the displacement of the deck and the magnitude of the displacement represented by color. Wile for the mode shape 5, the lateral torsional vibration showed larger torsional displacement after modification. Furthermore, the mode shapes 6, 7 and 8 has been altered after the modification totally, where the types of vibrations have been changed and commonly the frequencies of all the 8 mode shapes have been increased indicating an increase in the bending and torsional stiffness.

Results of vertical displacements

The magnitudes of vertical displacements at the center of the mid span during 30 s of wind excitation after modifying the model of the cable stayed bridge have decreased widely compared to the same situation before modification. It is worthy to mention that the amplitudes of vibrations are smaller and the frequency of deck vibration has increased (see Fig. 7).

The maximum displacement reached is 0.0175 m compared to the maximum displacement before modification is much smaller where it was 0.0325 m. These results indicate high suppression of vertical vibrations as a result, stiffer and safer structure.

Results of torsional displacements

The torsional vibrations after the structural modification have decreased in wide range too. First the maximum torsional displacements reached 0.02 m and the frequency of fluttering has decreased very obviously till the end of the wind excitation. While the maximum torsional displacements was 0.035 m before the modification, as well as the fluttering rate was quicker (see Fig. 8). This is a firm fact that the structural modification increased the torsional stiffness of the structure.

Results of control efficiency

Due to wind excitation for the duration of 30 s, the deck exhibited significant vertical and torsional vibrations. To control these vibrations, lateral steel beams to increase the structural stiffness of the system were adopted. The deck vibrations of the cable stayed bridge model after structural modification have decreased to a large limit. For vertical displacement at the mid span of the deck center, the displacement decreased in a wide range in such a way for the overall response the control efficiency reached 64.73% compared to the same response of the deck at the same point without structural modification. While the torsional displacement of the deck at the mid span point decreased too much, and the control efficiency for this case was 88.69% in comparison with the deck torsional response before structural modification. This means that this novel method have an efficient action in suppressing both the vertical and torsional vibrations of the deck.

Conclusions

A summary of the conclusion points is listed below:

1) The novel structural modification of the cable stayed bridge model succeeded to increase the bending and torsional stiffness of the structure to a wide range in such a way that it decreased 64.73% and 88.69% of the vertical and torsional vibrations of the deck simultaneously.

2) The natural frequencies of the system were modified positively, where the flutter wind speed has been changed to larger magnitudes. The maximum gap between the first eight mode shapes frequencies before and after the structural modification reached 0.173 Hz, which is an indication that the safety of the structure against vertical and torsional vibrations has been guaranteed to a large factor of safety.

3) This novel approach was successful to insure the safety of the structure against a dangerous coupling between the vertical and torsional mode shapes of vibrations, by removing it from the nearest mode shape 6 with a natural frequency of 0.776 Hz before the structural modification to the mode shape 16 with a natural frequency of 1.610 Hz after the structural modification.

4) Comparing to the previous approaches that have been adopted for vibrations control in long span bridges such as tuned mass dampers, Sealed tuned liquid column gas dampers and electromechanical actuator, this novel approach showed larger efficiency as vibrations suppressor.

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