simplified method of determination of natural-vibration ... · at the present moment, a suspension...

Procedia Engineering 57 ( 2013 ) 343 – 352

1877-7058 © 2013 The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license.

Selection and peer-review under responsibility of the Vilnius Gediminas Technical Universitydoi: 10.1016/j.proeng.2013.04.046

11th International Conference on Modern Building Materials, Structures and Techniques, MBMST 2013

Simplified Method of Determination of Natural-Vibration Frequencies

of Prestressed Suspension Bridge

Vadims Goremikinsa,*, Karlis Rocens

b, Dmitrijs Serdjuks

c, Janis Sliseris

d

a,b,c,dInstitute of Structural Engineering and Reconstruction, Riga Technical University, Azenes Str. 16, LV-1048, Riga, Latvia

Abstract

A suspension bridge is the most suitable type for a long-span bridge. Increased kinematic displacements are the major disadvantage of

suspension bridges. This problem can be solved by application of prestressed cable truss.

Dynamic approach is one of regulated bridge design parts. Simplified determination method of natural-vibration frequencies of

prestressed suspension structure and its experimental validation is presented in this paper.

Natural-vibration frequencies and mode shapes of the model depending on the prestressing level were determined. It was experimentally

proved, that mode shape with one half-wave does not appear for the model.

The difference between results, which were calculated by the developed simplified determination method of natural-vibration frequencies

of prestressed suspension structure and experimentally achieved by the model testing, does not exceed 20%. Therefore the method is

applicable for preliminary dynamic analyses of structures.

© 2013 The Authors. Published by Elsevier Ltd.

Selection and peer-review under responsibility of the Vilnius Gediminas Technical University.

Keywords: mode shapes; cable truss; prestressing; natural frequencies.

1. Introduction

Suspension bridges are structures where the deck is continuously supported by the stretched catenary cable [1].

Suspension bridges are the most important and attractive structures possessing a number of technical, economical and

aesthetic advantages [2].

At the present moment, a suspension bridge is the most suitable type of structure for very long-span bridges. Suspension

bridges represent 20 or more of all the longest span bridges in the world. The bridge with the longest centre span of 1991 m

is the Akashi Kaikyo Bridge [3]. So long spans can be achieved because main load carrying cables are subjected to tension

and distribution of normal stresses are close to uniform [4].

Increased deformability is one of the basic disadvantages of suspension bridges [5]. Increased deformability is

conditioned by appearance of elastic and kinematic displacements. The elastic displacements are caused by large tensile

inner forces. The elastic displacements are maximal at the centre of span in case of symmetrical load application. The

kinematic displacements are caused by initial parabolic shape change, resulting from non-symmetrical or local loads [6-7],

see Fig. 1. These displacements are not connected with cable elastic characteristics. Serviceability limit state is dominating

for suspension cable structures.

* Corresponding author.

E-mail address: [email protected]

Available online at www.sciencedirect.com

© 2013 The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license.

Selection and peer-review under responsibility of the Vilnius Gediminas Technical University

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344 Vadims Goremikins et al. / Procedia Engineering 57 ( 2013 ) 343 – 352

The elastic displacements can be reduced by applying low strength steel structural profiles, elastic modulus increase,

reinforced concrete application and cable camber increase [8].

Fig. 1. Initial shape change under the action of non-symmetrical load Fig. 2. Suspension bridge stabilization by the prestressing

The problem of increased kinematic displacements can be solved by increasing the relation of dead weight to imposed

load, which is achieved by adding of cantledge [9]. However, this method causes the increase of material consumption.

Stiffness of suspension structure can be increased also by increasing of girder stiffness, increasing of main cable camber,

connecting of main cable and girder at the centre of span, application of diagonal suspenders or inclined additional cables,

application of two chain systems, stiff chains and stress ribbons [10-11]. Nevertheless, these systems are characterized also

with material consumption increase, and system stiffness is not sufficient in many cases [12-13].

Usage of prestressed cable trusses is another method of fixing the problem of increased kinematic displacements under

the action of unsymmetrical load [14-15]. Different types of cable trusses are known, such as convex cable trusses, convex-

concave cable trusses, cable trusses with centre compression strut or parallel cable truss [16]. But one of the most efficient

and convenient for application for bridges is concave cable truss [15], see Fig. 2. Cable truss usage allows the development

of bridges with reduced requirements for girder stiffness, but overall bridge rigidity will be ensured by prestressing of

stabilization cable [8]. The deck can be made of light composite materials in this case [17-18].

Dynamic approach is one of the regulated bridge design parts after accident with the Tacoma Narrows Bridge [19-20].

Analyses of natural-vibration frequencies of prestressed suspension structures are realized with labour-intensive discrete

methods at present [21-23]. There is a lack of simple natural-vibration frequency determination methods, which can be used

for approximate design.

The new natural-vibration frequency determination method and its experimental validation are described in this paper.

2. Simplified Method of Determination of Natural-Vibration Frequencies of Prestressed Suspension Structure

For simple suspension bridge simplified equations Eq. (1) and Eq. (2) can be used for mode shapes with odd and even

half-wave number, respectively [10].

6 4 24 4 2 2

, 4 2 2 3

82 c c

v i

c

E A fi EI iH

ml l m i L l m

ππ π ⎛ ⎞ω = + + ⎜ ⎟π⎝ ⎠

, (1)

4 4 2 2

, 4 2v i

i EI iH

ml l m

π πω = +

, (2)

where: i - half-wave number of mode shape;

ωv,i - angular frequency of vertical vibration;

EI - bending stiffness of girder;

l - span;

m - mass of 1 meter of span;

H - horizontal support reaction of main cable; Ec - modulus of elasticity of main cable;

Ac - cross-section area of main cable;

f - camber of main cable;

Lc - length of main cable.

Radicand of the equations Eq. (1) and Eq. (2) can be divided to components: component dependent on girder Eq. (3),

component dependent on support reaction of main cable Eq. (4), component dependent on main cable characteristics

Eq. (5).


4 4

1 4

i EI

ml

π

α = , (3)

2 2

2 2

iH

l m

π

α = , (4)

6 4 2

3 2 3

82c c

c

E A f

i L l m

π⎛ ⎞α = ⎜ ⎟π⎝ ⎠

. (5)

In comparison with simple suspension bridge, prestressed suspension structure (Fig. 3 and Fig. 4) has additional

stabilization cable, so we can add two additional components for stabilization cable: component dependent on support

reaction of the stabilization cable Eq. (6) and component dependent on the stabilization cable characteristics Eq. (7).

2 2

4 2s

iH

l m

π

α = , (6)

6 4 2, ,

5 2 3,

82 c s c s s

c s

E A f

i L l m

π⎛ ⎞α = ⎜ ⎟π⎝ ⎠

, (7)

where: Hs - horizontal support reaction of stabilization cable;

Ec,s - modulus of elasticity of stabilization cable;

Ac,s - cross-section area of stabilization cable; fs - camber of stabilization cable;

Lc,s - length of stabilization cable.

Fig. 3. Prestressed suspension structure Fig. 4. Cross-section of prestressed suspension structure

Equation of calculation of natural-vibration frequencies of prestressed suspension structure was composed assuming

main and stabilization cables are situated in rows, but left and right cables situated parallel, see Fig. 5.

Fig. 5. Situation of components of prestressed suspension structure

Equations of determination of natural-vibration frequencies for mode shapes with odd half-wave number Eq. (8) and

even half-wave number Eq. (9) were created. These equations allow to calculate natural-vibration frequencies of prestressed

suspension structure.

3 52 4, 1

2 2v i

α +αα +αω = α + + , (8)

2 4, 1

2v i

α +αω = α + . (9)


3. Experimental Determination of Natural-Vibration Frequencies of Physical Model

3.1. Description of Experimental Model

Physical model was developed to determine natural-vibration frequencies of prestressed suspension structure, see Fig. 6,

Fig. 7). The span of the physical model of the prestressed suspension structure is equal to 2.1 m. Camber of the main cable

is equal to 275 centimetres. The deck is connected to the main cable by suspensions in 15 points. Width of the model is

equal to 0.4 m [24-25].

Fig. 6. Scheme of the physical model of the prestressed suspension

structure

Fig. 7. The physical model of the prestressed suspension bridge

The elements of the model of the prestressed suspension structure are made of steel cables with modulus of elasticity

60000 MPa. Tensile strength of wires of the cables is equal to 1770 MPa. The diameters and cable types of the elements are

shown in Table 1.

Table 1. Characteristics of the Cable Elements of the Physical Model

Elements Cable type Diameter Breaking force

Main cable 6x19+WSC 10.0 mm 63.0 kN

Stabilization

cable 6x19+WSC 8.0 mm 40.3 kN

The deck of the model of prestressed suspension structure is made from oriented strand board (OSB). It does not have

significant load bearing capacity. It only distributes load among suspensions due to deformability of OSB.

The prestressing is organized in the stabilization cables and is developed by rotating of a screw and moving of a bar. The

stabilization cables are supported by the block, see Fig. 8. The tensile force in the stabilization cable was measured by the

electronic dynamometer Scaime IPB50, see Fig. 9. The electronic dynamometer work principle is based on changes of

electrical bridge resistance. The precision of measurements for the electronic dynamometer is 2.5 N [26].

Fig. 8. Prestressing mechanism of the stabilization cable


Fig. 9. Measurement of prestressing by electronic dynamometer

To connect the deck with the load bearing cable, adjustable suspensions are used, that allow levelling of the deck. The

suspensions are connected to the cables using U-bolt clips [26], see Fig. 10.

Fig. 10. Connection of the deck and main cable by the suspensions

Acceleration sensors were used to obtain natural-vibration frequencies of the model, see Fig. 11.

Fig. 11. Acceleration sensor

The acceleration sensors were situated on the points with numbers 3, 5, 7, 9, 11 and 13 of the model, see Fig. 12 and

Fig. 13. The acceleration sensors measure acceleration at defined time interval with step equal to 0.006255 s and save it to

internal memory. After experiment data can be loaded to PC.


Fig. 12. Situation of the acceleration sensors on the model

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

GS1 GS2 GS3 GS4 GS5 GS6

20 kg

Fig. 13. Scheme of situation of the acceleration sensors on the model

Vibration excitation in vertical direction was realized by cutting of suspended weight (20 kg), see Fig. 13. The weight

was connected to the point with number 13 to avoid some mode shape loose. Experiments were done for 6 prestressing

levels: 0, 2, 4, 6, 8 and 10 kN, that correspond to stresses in the stabilization cable equal to 0, 69.7, 139.4, 209.1, 278.7 and

348.4 MPa, respectively.

3.2. Experimental Data Handling

Program ME’scopeVES was used for data handling from the acceleration sensors. The program transforms acceleration-

time dependence into frequency-response function using the Fourier transformation algorithm. Frequency-response function

consists of real and imaginary parts, or magnitude and phase. Natural-vibration frequencies were calculated from

magnitude-frequency dependence. Mode shapes were obtained by connecting imaginary part peaks of each point for each

natural-vibration frequency [27], see Fig. 7.

3.3. Results of the Experiment

Natural-vibration frequencies of the physical model of the prestressed suspension structure were determined depending

on the prestressing level. First three experimentally calculated natural-vibration frequencies in vertical direction are

generalized in Table 2, Table 3 and Table 4, respectively. The first vertical natural-vibration frequency changed from 7.24

to 21.69 Hz, the second frequency changed from 14.55 to 32.61 Hz and the third frequency changed from 21.78 to 40.53

Hz, while the prestressing level changed from 0 to 1000 kg, respectively.

Table 2. 1st Natural-Vibration Frequency in Vertical Direction, Hz

Prestres-sing level P, kN GS 1 GS 2 GS 3 GS 4 GS 5 GS 6 Ave-rage

0 7.18 7.27 7.18 7.33 7.23 7.26 7.24

2 12.37 12.32 12.36 12.35 12.46 12.39 12.38

4 15.26 15.33 15.28 15.40 15.35 15.42 15.34

6 18.18 18.23 18.11 18.11 18.00 18.10 18.12

8 19.95 19.97 20.11 19.92 19.99 20.06 20.00

10 21.71 21.69 21.77 21.56 21.80 21.62 21.69


Table 3. 2nd Natural-Vibration Frequency in Vertical Direction, Hz


0 14.56 14.44 14.75 14.87 13.96 14.71 14.55

2 21.14 20.66 21.24 21.12 20.79 21.18 21.02

4 24.63 24.55 24.59 24.85 24.58 24.9 24.68

6 28.19 27.8 28.17 28.17 27.45 28.15 27.99

8 30.42 30.22 30.83 30.38 30.49 30.59 30.49

10 32.61 32.54 32.75 32.44 32.75 32.58 32.61

Table 4. 3rd Natural-Vibration Frequency in Vertical Direction, Hz


0 21.84 21.41 21.93 22.31 22 21.17 21.78

2 28.62 26.72 26.62 26.79 26.82 27.07 27.11

4 31.61 30.66 30.56 30.79 30.51 30.95 30.85

6 35.75 34.45 34.41 34.51 34.12 34.28 34.59

8 38.4 37.57 37.78 37.72 36.18 37.72 37.56

10 40.59 40.75 40.99 40.6 39.38 40.86 40.53

The dependences between the prestressing level and natural-vibration frequencies, as well as approximation curves are

shown on Fig. 14. The dependence between the natural-vibration frequency and prestression level was evaluated in the form

of second order polynomial.

Fig. 14. Dependence between prestressing level and natural-vibration frequencies

Mode shapes of the first two natural-vibration frequencies in vertical direction are shown on Fig. 15 and Fig 16,

respectively. It was not possible to detect mode shape for third natural-vibration frequency precisely.

As the span of the model was equal to 2100 mm and ends of the girder were not connected to supports, mode shape of

first natural-vibration frequency consists of two half-waves, mode shapes of second and third natural-vibration frequencies

consist of three and four half-waves, respectively. Mode shapes are shown on Fig. 17.


Fig. 15. Mode shape of 1st natural-vibration frequency

Fig. 16. Mode shape of 2nd natural-vibration frequency

Fig. 17. Mode shapes of natural vibrations of the physical model

Mode shape with one half-wave did not appear for the model. This effect is for the prestressed suspension structure with

small stiffness girder.

4. Validation of Analytical Model

The geometrical and physical parameters of the model of the prestressed suspension structure are generalized in Table 5.

Support reactions of the main and stabilization cables for different prestressing levels were calculated by FEM program

ANSYS and are generalized in Table 6.

Table 5. Geometrical and Physical Parameters of the Model

Parameter of the Physical Model Value

Modulus of elasticity of girder E = 2.3 GPa

Modulus of elasticity of main cable Ec = 60 GPa

Modulus of elasticity of stabilization cable Ec,s = 60 GPa

Girder height h = 0.012 m

Girder width b = 0.4 m

Moment of inertia of girder I = 5.76 · 10-8 m4

Model span l = 2.1 m

Cross-section area of main cable Ac = 4.49 · 10-5 m2

Cross-section area of stabilization cable Ac,s = 2.87 · 10-5 m2

Camber of main cable f = 0.275 m

Camber of stabilization cable fs = 0.174 m

Length of main cable Lc = 2.196 m

Length of stabilization cable Lc,s = 2.138 m

Dead weight of model G = 45.3 kg

Mass of 1 meter of span m = 21.47 kg / m

Natural-vibration frequencies of the model were determined by the developed analytical model according to the

Equations (8) and (9) and are generalized in Table 7. As it was experimentally proved, mode shape of first natural-vibration

frequency of the prestressed suspension structure with small stiffness girder consists of two half-waves.

The differences between analytically and experimentally obtained results are shown in Table 8.


Table 6. Horizontal Support Reaction Depending on Prestressing Level

Prestressing Level P,

kN Prestressing Level, MPa

Support reaction from

main cable H, kg m/s2

Support reaction from

stabilization cable Hs, kg m/s2

2 69.7 200 112.5

4 139.4 400 205.4

6 209.1 600 295.6

8 278.7 800 383.5

10 348.4 1000 464.9

Table 7. Analytically Determined Natural-Vibration Frequencies of the

Physical Model of the Prestressed Suspension Structure

Prestressing

level P, kN

Number of natural-vibration frequency

1 2 3

Natural frequency ωi, Hz

2 10.7 23.8 32.4

4 13.2 26.5 35.9

6 15.3 28.9 39.1

8 17.2 31.2 42.1

10 18.8 33.2 44.8

Table 8. Comparison of Experimentally and Analytically Determined

Natural-Vibration Frequencies

Prestressing

level P, kN

Number of natural-vibration frequency

1 2 3

Difference between experimentally ant analytically

determined natural-vibrations frequencies, %

2 -15.9 11.5 16.2

4 -16.0 6.8 14.1

6 -18.2 3.2 11.6

8 -16.5 2.1 10.7

10 -15.4 1.8 9.4

The maximum difference between experimentally and analytically determined natural-vibration frequencies does not

exceed 20%, which is enough for design process of structures alternatives.

5. Conclusions

The first vertical natural-vibration frequency of the physical model of the prestressed suspension structure with span

2100 mm changed from 7.24 to 21.69 Hz, the second frequency changed from 14.55 to 32.61 Hz and the third frequency

changed from 21.78 to 40.53 Hz, while prestressing level changed from 0 to 10 kN, respectively.

Mode shape of the first natural-vibration frequency of the model consists of two half-waves, mode shapes of the second

and third natural vibration frequencies consist of three and four half-waves, respectively. Mode shape with one half- wave

does not appear.

Changing of the prestressing level allow to adjust dynamic characteristics of prestressed suspension bridge. Increasing of

prestressing level increases natural-vibration frequency and contra wise. This advantage of prestressed suspension structure

allows improving dynamic characteristics of the bridge and excluding possibility of resonance appearance.

The difference between results, which were calculated by the developed simplified determination method of natural-

vibration frequencies of the prestressed suspension structure and experimentally achieved by the model testing, does not

exceed 20%. Therefore the method is applicable for preliminary dynamic analyses of structures.

Acknowledgements

This work has been supported by the European Social Fund within the project “Support for the implementation of

doctoral studies at Riga Technical University”.

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