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Paper Number 004

Investigation of Free In-plane Vibration of Curved bridges

Navid Heidarzadeh

Former graduate student, Sharif University of Technology, Tehran, Iran.

Shervin Maleki

Associated Professor, Sharif University of Technology, Tehran, Iran.

ABSTRACT:

In this paper the free in-plane vibration of curved bridges is investigated. The exact

closed form solution for free vibration of arches, proposed by Tufekci and Arpachi

(1996), which takes into account the bending effects, shear effects, axial tension,translational and rotational inertia is adopted for two possible cases of boundary

conditions of curved bridges. The results obtained by this method are compared to those

obtained by the finite element method (FEM) using shell elements. The effect ofgeometry, including span length and opening angle, on the natural frequencies and mode

shapes is parametrically studied. Results show that the difference between the natural

frequencies obtained by closed form solution and those obtained by FEM is less than 3percent. Results also reveal that the first mode is more sensitive to boundary conditions

than other modes. Increasing the central angle subtended by the arch in the case ofconstant chord length, decreases the natural frequency of the first mode, but increases the

frequency of the second mode.

Keywords: Curved bridges, Free in-plane vibration, Closed form solution, Finite element

method.

1 INTRODUCTIONAs a result of complicated geometry and limited rights of way, horizontally curved bridges arebecoming the norm of highway interchanges and urban expressways. This type of superstructure hasgained popularity since the early 1960s because it addresses the needs of transportation engineering

(Chen and Duan 1999).

The out-of-plane vibration of curved bridges due to traffic movement is of strong engineering interestand has received considerable attention in recent years. A good literature survey about the out-of-plane

vibration of curved bridges has been presented by K. Senneh et al, 2001. In contrast, the in-planevibration of curved bridges has not adequately been studied yet. The in-plane vibration of curved

bridges provides a good understanding of the seismic behavior and responses of such bridges. Thenatural frequencies and mode shapes can directly be used for calculating the seismic responses using

an acceleration response spectrum.

Curved bridges can have asymmetrical responses similar to those of skewed bridges (Chen and Duan

1999). Coupling effects in the horizontal directions due to curved alignment of the bridge, makes thein-plane behavior of curved bridges very complicated. It seems that the irregularities in the dynamic

behavior of curved bridges can be highly attributed to curve alignment of the deck rather than thesubstructure.

For in-plane vibration analysis, classical beam theory is usually used to approximate the actual deck

behavior (Abdel-Salam et al. 1988, Singh 1996 and Sextos et al. 2004). It is well-known that forbeams having large cross-sectional dimensions in comparison to their length, and for beams in which

high frequency modes of vibration are critical, the elementary Euler-Bernouli beam, which is derived

based on the assumption that the deflection of beams are due to bending only, may yield unsatisfactory


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results. In contrast the Timoshenko beam theory which takes into account the rotary inertia, sheareffect and axial extension, gives a better approximation to the actual beam behavior (Tufekci and

Arpachi 1998).

In recent years many finite element formulations have been developed for curved beam elements withlumped masses (Wolf 1971, Palaninathan et al. 1985, Lin 1998, Patel et al. 1999, Raveendranath et al.

2000, Przemyslaw et al. 2001 and Wu et al. 2003). However, a lumped mass idealization, although

applicable, is not the natural approach for certain types of structures such as curved bridges (Chopra2001). Tufekci and Arpachi (1998), proposed an exact solution to the governing equation of freevibration of circular curved beams which accounts for bending effect, shear effect, axial extension,

translational and rotational inertia and the distributed mass. A similar procedure does exist for beamswith variable cross section (Tong et al. 1998).

In this paper, to identify the effects of deck curvature on characteristics of free in-plane vibration of

curved bridges, the simple case of a single span curved bridge is considered. In order to achieve exactresults, the mass has been considered distributed over the length of the curve. The exact closed form

solution for free vibration of arches, proposed by Tufekci and Arpachi (1998), which takes into

account the bending effect, shear effect, axial extension, translational and rotational inertia is adaptedand solved using two possible cases of boundary conditions for curved bridges.

2 BRIDGE GEOMETRYFigure 1 shows typical cross sections of curved bridges. In this study, as the cross-sectional

dimensions of the deck are usually considerable in comparison to their length, the Timoshenko beamtheory is used to approximate the actual deck behavior.

At the bridge supports, the elastomeric bearings are assumed to be present. As the shear stiffness of

elastomeric bearings is insignificant, concrete shear keys are used to prevent large displacements and

falling of the superstructure due to earthquake induced motions.

In the rest state with no horizontal forces being applied to the superstructure, shear keys are not incontact with the elastomeric bearings and there is a small gap between the two. These gaps prevents

the deck and shear keys from engaging as a result of unexpected loads during construction or service

loads. Two possible positions of shear keys are shown in Figure 2. In the case of Figure 2a, the shear

keys only prevent transverse displacements. Longitudinal displacement and rotation around thevertical axis are unrestrained. This is a common detail at expansion joints, which are normally

provided for thermal effects. The boundary condition, corresponding to these bearings, is consideredas a roller in the analytical model.

In the case of Figure 2b, both the longitudinal and transverse displacements are restrained, but the

rotational behavior of structure is complicated due to the gap. If the gap width and the distance

between the centerline of the two nearby bearings are expressed in terms ofx andy, respectively (Fig.2b), the maximum rotation will be equal to 2x/y. For practical range ofx, between 1 to 2 cm, and y,

between 4 to 8 m, the maximum rotation is in the range of 0.0025 radians to 0.01 radians. Also theLRFD Bridge Design Specification (AASHTO 1998) recommends providing a safety margin againstapplicable unfactored loads and uncertainties. It suggests the allowable rotation to be taken as 0.005

radians, unless an approved quality control plan justifies a smaller value.

b

h

tw

ft

(c)(b)(a)

Figure 1: Typical cross section of curved bridges, (a) Steel I girder, (b) Steel box girder, (c) Concrete box girder


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Therefore, if the rotation of the superstructure at supports is more than the mentioned range, theboundary condition at supports should be chosen as clamped, otherwise, a hinged condition may be

assumed. As the rotation due to free vibration is a variable quantity over time, the boundary conditionsmust be frequently changed in the analytical model. Considering the difficulties in modeling time

varying boundary conditions, which needs the rigid body motion and contact mechanics to be taken

into account; two cases of boundary condition (roller-hinged and roller-clamped) are considered in thispaper. It is anticipated that the actual behavior of curved bridges is somewhat between these two cases.

It seems that this assumption is satisfactory for most of the engineering practice.

3 METHODOLOGYAs mentioned in the introduction, an exact closed form solution presented by Tufekci and Arpachi(1998) is used to calculate circular natural frequencies and mode shapes. The theory behind this

method is explained here briefly.

A schematic of general circular arch with an arbitrary geometry, distributed mass and boundaryconditions subjected to externally distributed forces is shown in Figure 3a. In this figure, is the

radius of curvature, is the central angle subtended by the arch, P is the distributed tangential force, Qis the distributed transverse force and Tis the distributed moment. Note that, the tangential and radialaxes of the local coordinate system, which is rotated degrees from the centerline of the curve, are

shown as and , respectively. A free body diagram of a differential length of the curve and the acting

forces is shown in Figure 3b, where Nis the internal axial force, Vis the internal shear and Mis theinternal moment. The dynamic equilibrium of this differential element can be explained by three

equilibrium equations as indicated below:

(b)(a)

Figure 3: (a) A schematic of general circular arch, (b) Free body diagram of a differential length of thecurve

dx

y

(b)(a)

Figure 2: Two possible positioning of shear keys (a) transverse direction only (b) both transverse andlongitudinal directions


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- Dynamic equilibrium of internal forces, external forces and inertial forces in the tangential direction:

dN cosd

2

2 V sind

2

d P+ d2

t

wd

d

2

(1)

- Dynamic equilibrium of internal forces, external forces and inertial forces in the transverse direction:

2N sind

2

dN sind

2

+ dV cosd

2

+ Q d+ d2

t

ud

d

2

(2)

- Dynamic equilibrium of internal moments, external moments and inertial forces about the vertical

axis:

dM d V+ d T+ I

A d

2t

d

d

2

(3)

In these equations, = mass per length of the curve; u = tangential displacement; w = transverse

displacement; = angle of rotation about the vertical axis; I = moment of inertia with respect tovertical axis and A = cross-sectional. Neglecting the second order terms in Equations 1-3 and

considering the external forces as inertial forces for free vibration gives:

N

d

dV

2 w ;

V

d

dN

2 u

;M

d

d V

I

A

2

(4)

The in-plane behavior of elastic curved beams, with axial and shear deformations taken into account,can be represented by three equations below:

w

d

du

E AN+ ; u

d

dw +

G AV+

;

d

d

E IM

(5)

WhereE= modulus of elasticity; G = shear modulus and = shear factor. Equations 4 and 5 can bewritten in matrix form as:

y ( )d

dC y ( ) (6)

Where:

y

w

u

M

N

V

; C

0

1

0

0

2

0

1

0

0

0

0

2

0

0

2

I

A

0

0

0

0

E I

0

0

0

E A

0

0

0

0

1

0

G A

0

1

0

(7)

An exact solution exists for the particular case of a constant coefficient matrix in the form of:

y ( ) e C

y 0( ) (8)


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In Equation 8, e.C

is a matrix exponential term and y(0) is the initial value vector at =0. e.C

can beexpressed exactly using Caylay-Hamilton theory. Nowadays powerful mathematical software are

available which have exponential matrix functions, such as MATLAB (Mathworks Inc. 2002). Thissoftware uses the Potzer method for calculating the exponential matrix term with double precision.

The y(0) can be obtained according to the boundary conditions. Equation 8 at both ends of the curve

suggests that:

y

2

e

2y 0( )

;y

2

e

2y 0( )

(9)

For three possible boundary conditions at each support of a curved bridge, i.e., roller, hinged andclamped, three components of the solution vector, y, is equal to zero (roller: u=0;M=0;N=0, hinged:

w=0; u=0;M=0, clamped: w=0; u=0; =0).

Equation 9 actually contains 12 linear equations in which the 6 components of the initial value vector,the 3 components of the solution vector at =-/2 and the 3 components of the solution vector at =/2

are redundant. Choosing the 6 equations corresponding to zero value components of solution vectors

at =-/2 and =/2, gives 6 equations in terms of 6 components of the initial value vector:

H y 0( ) O (10)Where H is the 66 coefficient matrix and O is the zero vector. For non-trivial solution, setting the

determinant ofH to zero will give the circular natural frequencies. The initial value vector, y(0), for

each mode can also be calculated from Equation 10. Mode shapes are specified by substituting thenormalized initial values into Equation 8.

4 VALIDATION OF SIMPLIFIED MODELThe results of the closed form solution for a typical box girder curved bridge of 40 m span with anopening angle of 60 degrees (case A8 in Table 1) were verified against the finite element model using

the program SAP2000 (Computers and Structures, Inc. 2005). Shell elements were used to model the

deck. Figure 1c shows the cross-section of the deck. The mathematical program, MATLAB

(Mathworks Inc. 2002) was used for mathematical calculations of the closed form solution.Figure 4 shows the first five circular natural frequencies and mode shapes obtained by the twomethods. The difference between the circular natural frequencies is less than 3 percent. Mode shapes

are also in good agreement with the FEM model.

5 PARAMETRIC STUDYIn this section, the effect of geometrical parameters in the free vibration of curved bridges isinvestigated. The curved box girder bridge with spans of 20, 40 and 60 m are considered with opening

angles of 20, 40 and 60 degrees. Boundary conditions were hinged-roller (A) and clamped-roller (B).

Figure 1c shows the cross section of the deck. Geometrical properties and section properties are given

in Table 1. The circular natural frequencies, considering the hinged-roller and clamped-roller

boundary conditions are given in Tables 2 and 3, respectively. The mode shapes of A4-A6 and B4-B6cases are shown in Figure 5 and Figure 6, respectively.

Generally, results indicate that single span curved bridges with common dimensions and boundary

conditions can be categorized as high frequency structures. The lowest circular natural frequencythrough the 18 cases studied, is 12.6 rad/s.


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Closed Form Solution Finite Element Model

Mode 1, = 39.3 rad/s Mode 1, =38.2 rad/s

Mode 2, = 105.5 rad/s Mode 2, = 105.6 rad/s




(a) (b)

Figure 4: The first five circular natural frequencies and mode shapes obtained by (a) the proposed theory and(b) the FEM model, heavier line is the undeformed shape of the arch centreline

Table 1: Geometrical and section properties

M(ton)I(m4)Av(m

2)A(m

2)twtfbh(m)R(m)L(m)(degree)Case

1.513.824.33.24.280.20.28.21.0057.62020A1, B1

1.516.030.33.25.180.20.28.21.50115.24020A2, B2

1.518.136.33.26.080.20.28.22.00172.86020A3, B3

1.513.824.33.24.280.20.28.21.0029.22040A4, B4

1.516.030.33.25.180.20.28.21.5058.54040A5, B5

1.518.136.33.26.080.20.28.22.0087.76040A6, B6

1.513.824.33.24.280.20.28.21.0020.02060A7, B7

1.516.030.33.25.180.20.28.21.5040.04060A8, B8

1.518.136.33.26.080.20.28.22.0060.06060A9, B9


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Table 2: Circular natural frequencies considering the hinged-roller boundary condition

A1 A2 A3 A4 A5 A6 A7 A8 A9

Mode 1: 116.0 36.1 17.3 101.7 31.9 15.3 83.8 26.3 12.6

Mode 2: 219.6 110.0 61.3 227.5 107.1 58.0 232.6 100.6 53.4

Mode 3: 325.3 121.5 76.4 316.7 124.3 80.0 305.8 129.4 84.1

Mode 4: 526.2 215.4 118.2 519.7 209.6 115.1 497.3 200.4 110.2

Mode 5: 533.9 313.1 178.6 527.2 305.3 174.2 528.7 293.1 167.0

Mode 6: 644.9 330.6 223.8 637.5 328.7 221.8 625.5 324.7 217.9

Mode 7: 698.0 410.9 240.2 698.2 402.6 235.3 694.8 389.0 227.8

Mode 8: 740.4 481.7 301.3 725.9 482.0 295.3 704.6 481.3 285.3

Mode 9: 943.0 507.2 362.1 924.8 497.4 355.0 895.0 482.4 343.5

Mode 10: 1028.9 535.8 372.6 1015.0 533.7 368.2 992.9 527.6 360.6

Table 3: Circular natural frequencies considering the clamped-roller boundary condition

B1 B2 B3 B4 B5 B6 B7 B8 B9

Mode 1: 140.5 49.8 25.1 126.4 45.3 22.9 109.3 39.3 19.9

Mode 2: 219.9 110.6 68.7 227.8 109.7 64.7 232.6 105.5 59.9

Mode 3: 334.4 132.0 77.4 326.8 132.7 81.2 316.8 135.0 84.8

Mode 4: 533.5 221.8 124.7 520.5 216.4 121.8 499.7 207.8 117.3

Mode 5: 578.1 316.3 182.8 577.6 308.6 178.5 577.1 296.7 171.5

Mode 6: 645.8 330.7 223.8 640.7 328.8 221.9 631.0 324.8 218.1

Mode 7: 739.2 412.2 242.8 724.4 404.1 238.0 700.7 390.7 230.5

Mode 8: 853.7 495.5 302.9 847.6 494.0 296.9 837.0 481.5 287.1

Mode 9: 944.9 508.0 362.9 927.4 499.7 355.9 899.3 496.3 344.5

Mode 10: 1074.5 549.6 372.6 1061.0 542.9 368.2 1038.1 531.5 360.6

For the clamped-roller boundary condition, the natural frequency of the first mode is 40 percent

greater than the hinged-roller boundary condition. The difference is less than 4 percent for othermodes.

The frequencies of the first three modes decrease up to 1/3 and 1/2, when the span length increases

from 20 m to 40 m and from 40 m to 60 m, respectively. The dynamic characteristics show

complicated trend when the opening angle changes. For example, the frequency of the first mode

decreases up to 16 percent when the opening angle increases from 20 degrees to 40 degrees and from40 degrees to 60 degrees. In contrast the frequency of second mode increases up to 5 percent.

Mode shapes are approximately similar for each mode in all the cases of A1-A9, and B1-B9. Thisindicates that mode shapes are not sensitive to geometrical parameters. Mode shapes in the case of

hinged-roller boundary condition are not very different from those with clamped-roller boundary

condition. In all cases, the first mode shape activates the tangential displacement at the roller end ofthe bridge. The second mode shape has an inflection point near the centerline of the curved bridge.

The third mode also has an inflection point near the centerline. Modes 4 and 5 have three inflection

points at various locations along the span.


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-10 -8 -6 -4 -2 0 2 4 6 8 10 12

x(m)

-20 -15 -10 -5 0 5 10 15 20 25

x(m)

(a) (b)

-3 0 -2 5 -2 0 -1 5 -1 0 -5 0 5 1 0 1 5 2 0 2 5 3 0 3 5

x(m)

(c)

Base Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

Figure 5: (a) case A4, (b) case A5, (c) case A6

-10 -8 -6 -4 -2 0 2 4 6 8 10 12

x(m)

-20 -15 -1 0 -5 0 5 10 15 2 0 2 5

x(m)

(a) (b)

-3 0 -2 5 -2 0 -1 5 -1 0 -5 0 5 1 0 1 5 2 0 2 5 3 0 3 5

x(m)

(c)

Base Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

Figure 6: (a) case B4, (b) case B5, (c) case B6

6 CONCLUSIONThis paper investigated the effect of deck curvature on characteristics of free in-plane vibration ofsingle-span curved bridges. The exact closed form solution for free vibration of arches, proposed by

Tufekci and Arpachi (1996) is adopted for two possible cases of boundary conditions of curved

bridges. The Timoshenko theory which takes into account the bending effect, shear effect, axialextension, translational and rotary inertia was used to approximate the superstructures behavior. The

natural frequency and mode shapes were obtained in exact closed form and compared with the finiteelement method. Further, the effect of geometrical parameters on the free vibration of curved bridges

was investigated in a parametric study. The significant results are as follows:


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The difference between the circular natural frequencies obtained by closed form solution withthose obtained by FEM is less than 3 percent even in the range of 300 rad/s and higher mode

shapes.

In the case of clamped-roller boundary condition, the natural frequency of the first mode is 40percent greater than the hinged-roller boundary condition. The difference is less than 4 percentfor other modes.

Increasing the central angle of the arc and keeping a constant chord length, decreases the naturalfrequency of the first mode, but increases the frequency of the second mode.

Mode shapes are not very sensitive to geometrical parameters and boundary conditions. Modeshapes in the case of hinged-roller boundary condition are similar to those with clamped-roller

boundary condition.

REFERENCES:

Abdel-Salam M. N. and Heins C. P., Seismic response of curved steel box girder bridges.Journal of structuralengineering, ASCE. Vol. 114, No. 12 (1988)

Chen W. F. and Duan L., Bridge Engineering Handbook, CRC press, (1999)

Chopra A. K., Dynamics of structures, theory and application to earthquake engineering, Prentice Hall, 2001

Lin S. M., Exact solution for extensible circular curved Timoshenko beams with nonhemogeneous elasticboundary condition,Acta Mechanica, Vol. 130, pp. 67-79 (1998)

Litewka P. and Rakowski J., Free vibration of shear-flexible and comperessible arches by FEM, InternationJournal of Numerical Methods in Engineering, Vol. 52, pp. 273-286 (2001)

LRFD Bridge Design Specification, 2nd Ed. American Association of State Highway and TransportationOfficials,AASHTO, Washington, D.C. (1998).

MATLAB, version 7.1,Mathworks Inc.

Palaninathan R., Chandrasekharan P. S., Curved beam stiffness matrix formulation, Computers and Structures,Vol. 21, No. 4, pp. 663-669 (1985)

Patel B. P., Ganapath M. and Saravanan J., Shear flexible field-consistent curved spline beam element forvibration analysis,Internation Journal of Numerical Methods in Engineering, Vol. 46, pp. 387-407 (1999)

Raveendranath P., Singh G., Pradhan B., Free vibration of arches using a curved beam element based oncoupled polynomial displacemebt field, Computers and Structures, Vol. 78, pp. 583-590 (2000)

SAP2000, version 9.03 (2005). Integrated structural analysis and design software. Computers and Structures,Inc., Berkeley, CA.

Senneh K. M., Kennedy J. B., State-of-the-art in design of curved box girder bridges, Journal of BridgeEngineering/may/june. pp 159-167 (2001)

Sextos A., Kappos A. J, Mergos P., Effect of soil-structure interaction and spatial variability of ground motionon irregular bridges: the case of Kristallpigi bridge, 13

thworld conference on Earthquake Engineering,

Vancouver, B.C., Canada, 2004

Singh R., Seismic response analysis of curved bridge, Journal of Individual Studies by Participants in theInternational Institute of Seismology and Earthquake Engineering, Tokyo, Japan, 1996

Tong X., Mrad N., Tabarrok B., In-plane vibration of circular arches with variable cross-sections, Journal ofSound and Vibration, Vol. 212(1), pp. 121-140 (1998)

Tufekci E. and Arpachi A., Exact solution of in-plane vibration of circular arches with account taken of axialextension, transverse shear and rotatory inertia effects,Journal of Sound and Vibration, Vol. 209(5), pp. 845-856 (1998)

Wolf J. A., Natural frequencies of circular arches,Journal of Structural Division, Proceedings of the AmericanSociety of Civil Engineers, Sep. 1971, pp. 2337-2350

Wu J. S., Chiang L. K., Free vibration analysis of arches using curved beam elements, Internation Journal ofNumerical Methods in Engineering, Vol. 58, pp. 1907-1936 (2003)


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