A SIMPLIFIED METHOD FOR ESTIMATING THE NATURAL …

A SIMPLIFIED METHOD FOR ESTIMATING THE NATURAL FREQUENCY OF BRIDGE SUPERSTRUCTURES

T. MEMORY B.Eng (Hons) Postgraduate Student School of Civil Engineering, Queensland University of Technology

G.H. BRAMELD B.E (Hons), M.Eng Sc, B.Com, PhD Associate Professor School of Civil Engineering, Queensland University of Technology

and

D. THAMBIRATNAM B.Sc Engin (Hons), M.Sc, PhD Senior Lecturer School of Civil Engineering, Queensland University of Technology

SUMMARY

The draft edition of the limit state NAASRA bridge design specifications expresses the Dynamic Load Allowance as a function of the first flexural frequency of the bridge superstructure. In this paper, a simplified, accurate method for estimating the fundamental frequency of bridges is proposed and the higher vibration modes excited by vehicular traffic are examined. The significance of dynamic modulus of elasticity for concrete in vibration analysis is quantified. The effect of asphaltic concrete surfacing is also examined.

KEYWORDS: Bridge, Dynamics, Modulus, Natural, Frequency

ACKNOWLEDGEMENTS: Physical Infrastructure Centre, Queensland University of Technology

RI Heywood (Editor) Bridges - Part of the Transport System !'ages 539-550

Terry Memory is currently undertaking postgraduate research at QUT after being awarded a QUT Physical Infrastructure Centre scholarship. During the period of 1986 to 1989 he worked with a Brisbane based consultancy and completed his Bachelor Degree with 1st Class Honours in 1990. His major interests included dynamic analysis of bridge superstructures and highrise buildings.

Gerald Brameld joined the Department of Civil Engineering at QIT in 1974, following over eight years of bridge design with Cameron McNamara. Since then he has continued his interest in bridge engineering through experimental and theoretical research. His major interests are in post cracking behaviour of superstructures, structural reliability, and the response of superstructures to load. He is presently the Director of the Physical Infrastructure Centre at QUT.

David Thambiratnam is a senior lecturer in the School of Civil Engineering at QUT. Prior to joining QUT in June 1990, he was a senior lecturer in the Department of Civil Engineering at the National University of Singapore. David has several years of industrial, teaching and research experience in structural engineering and has over 70 publications to his credit.

540 AUSTROADS Conference Brisbane 1991

INTRODUCTION

From as early as 1910 bridge designers have factored up the live load to account for the dynamic loads induced by vehicles. This factor has usually been referred to as the "impact factor" and was specified in design codes as a function of the span length. The Ontario Highway Bridge Design Code (1983) and the draft NAASRA bridge design specifications (1987) refer to this entity as the "dynamic load allowance" (DLA). Furthermore, both of these codes express the DLA as a function of the first flexural frequency of the bridge superstructure. The DLA provision from the draft NAASRA specifications has been reproduced in figure 1.

0.40

0.25 - 020

1.0 2.5 45 6.0 10.0

First Flexural Frequency, Hz

Fig. 1 - NAASRA dynamic load allowance provision.

The draft NAASRA specification (1987) also designates a grillage analysis as the lowest tiered analysis method. The objective of this paper is to provide a simplified and accurate method for calculating the fundamental frequency of a grillage using static analysis software. The method proposed may therefore be used in conjunction with the premise illustrated in figure 1 to obtain a Dynamic Load Allowance._

CALCULATING THE FIRST FLEXURAL FREQUENCY

BEAM ANALOGY

A common practice is to idealise a bridge structure as a single beam. The assumptions made when beam theory is applied to bridge superstructure behaviour are:

(a) The bridge must be idealised as a prismatic section of uniform stiffness and mass along the full length of the span.

(b) The bridge must be straight, non-skewed and symmetric about the centreline.

(c) The beam analogy has no capacity to model transverse energy distribution.

(d) With respect to dynamic response a beam analogy can only simulate vehicles travelling along the centreline of the bridge.

RJ Heywood (Editor) Bridges - Part of the Transport System 541

Dyn

amic

Loa

d A

llo w

ance

BEAM ANALOGY COMPARED TO 2-D MODELS

When a bridge is idealised as a single beam the transverse stiffness is effectively infinite, therefore at maximum amplitude all the energy in the system must be in the form of longitudinal (bending) strain energy. In reality, a proportion of the energy in the systems is distributed as transverse strain. If the fundamental mode of vibration is longitudinal, the 2-D idealisation must yield less longitudinal potential energy compared with the single beam analogy. As the masses of the two idealisations remain constant, the calculated fundamental frequency of the 2-D model must therefore be less than that of the beam analogy. This conclusion is only valid for straight, non-skewed bridges, or bridges idealised with support fully released with respect to transverse rotation.

PROPOSED APPLICATION OF THE RAYLEIGH METHOD

Using the Rayleigh energy principles, Clough and Penzien (1975), eqn (1) can be derived assuming the fundamental mode shape is taken as the deflected shape due to selfweight.

ft 1

i-n

g E Yi

2n

E 2 y;

(1)

where: f, = fundamental frequency, Hz g = gravitational acceleration, 9.81 m/s' y, = displacement at the i th node due to self weight

m = mass of the i th node

This relationship was applied to the static deflections calculated from a grillage analysis. Obviously if the grillage is symmetric the energies need only be calculated for half the grillage. Furthermore it was found that applying the method to only one half of a central girder yielded results of surprising precision. This means that in the design office the method can ,be quickly performed on only a handful of deflections calculated from a static analysis.

Transverse Support Rotations Fixed Transverse Support Rotations Released

15 ,,. 14 ,

13

15 ,„; 14 t., 13

: ,t, 12 :6: 12 cr 11 t 11

'- 10 "- 10 9 9

0 10 20 30 0 10 20 30

Degree of Skew Degree of Skew

Eigenvalue Soin. Rayliegh 1/2 Beam — - — - — Beam Analogy

Fig. 2 - Comparison of theoretical results.


To validate the Rayleigh half beam method for single span bridges a parametric study was undertaken. The study involved the progressive skewing of a grillage. Two support conditions were also considered. They were the supports fully fixed against transverse rotation and the supports fully released. The fully fixed case was representative of post-tensioned concrete deck unit structures, while the released case represented girder-slab deck type structures.The results from the Rayleigh half beam method were compared with the beam analogy and the eigenvalue solution from SuperSap, a dynamic analysis package. The results are presented in figure 2.

THE BEAM ANALOGY COMPARED TO TEST DATA

Lee, Ho and Chung (1987) observed a fundamental frequency of 12.9 Hz for a simply supported, straight, non-skewed concrete superstructure. Using the beam analogy the calculated fundamental frequency was 12.6 Hz. The interesting point about this investigation was that the dynamic modulus of elasticity was used for the theoretical calculation. Equation 2, from BS-The Structural Use of Concrete (1972), was used to relate the static modulus of elasticity to the dynamic.

Es - 1.25 E, - 19 (2)

Alternatively the Ontario Code (1983) specifies the use of the static modulus for calculating the first flexural frequency.

Nine simply supported, straight, non skewed bridges were investigated by Wills (1977). After changing the modular ratio to approximately 5, it was found that the difference between calculated and observed frequency was reduced to approx. 5%. The point to note is that using the dynamic modulus of elasticity for concrete would result in a modular ratio of approximately 5.

A trend for the observed fundamental frequency, fogs, to be greater than the calculated frequency was noticed in the literature. Equation 3, after Green (1977) was developed from the extensive data base held by the Ontario Ministry of Transportation.

foils = 0.95 fcme + 0.72

for 2 < fogs < 7 Hz (3)

It should be noted that the observed frequency was recorded while vehicles were on the bridge. However, these values were later corrected to allow for the mass of vehicles and the forcing function of the vehicle.

This brief literature review suggests the following:

(a) For straight, non-skewed bridges of approximately uniform mass and stiffness, the beam analogy underestimates the fundamental frequency by approximately 5% when compared with field results. For design purposes this error is insignificant.

(b) When using the beam analogy to estimate the fundamental frequency, no allowance is made for longitudinal skew. Consequently full scale testing of skewed bridges has been avoided.

RJ Ileywood (Editor) Bridges - Part of the Transport System 54.3

(c) There is a conflict of opinion in the literature regarding the use of the dynamic modulus of elasticity for concrete.

(d) The beam analogy for non skewed simply supported bridges (theoretical) overestimates the fundamental frequency of vibration. In contrast field observations suggest the opposite. This suggests that present idealisation principles are inadequate as a detailed 2-D analysis will yield results further removed from reality compared to the beam analogy.

DYNAMIC MODULUS OF ELASTICITY FOR CONCRETE

In the case of vehicle induced vibration it could be argued that the response is primarily static with a superimposed dynamic response. With this point of view, one would be inclined to use the static modulus. However, this static-dynamic component concept is applied to a discrete period of time. It was therefore suggested that a truer representation of a vehicle-bridge system is to consider the response as a combination of a fast dynamic response and a slower dynamic response, the latter being primarily a function of the vehicle speed and span length. Consider a typical example. Let a truck traverse a 15 m span bridge at 16.7 m/s (60 km/hr). The so called static response actually occurs in 0.9 second. This is by no means slow or static loading.

SIX MILE CREEK BRIDGE

To consolidate the previous discussion and for the purpose of demonstrating natural frequency calculation procedures, a case study was undertaken. The study was of Six Mile Creek Bridge, which is a short span, steel and concrete highway bridge located on the Cunningham Highway west of Brisbane, Australia. The dynamic amplification factors were measured by O'Connor and Pritchard (1984). The dynamic amplification factor used in the design was 0.3 from the ASSHTO code, however values as high as 1.32 were recorded in the field. Stress records also gave a natural frequency of 10 - 12.3 Hz with a mean of 11.4 Hz. It is important to note that particular attention was given to the mechanical bridge bearings to ensure that they acted as close as possible to frictionless pins.

RAYLEIGH HALF BEAM CALCULATION EXAMPLE

To demonstrate the calculation procedure of the Rayleigh half beam method, the fundamental frequency of Six Mile Creek bridge, based on the static modulus of elasticity for concrete, is presented. The calculations were performed on the static displacements and mass of the girder C, which was the girder closest to the centreline. Girder C has been diagrammatically represented below in figure 3.


NODE DISPLACE. SHAPE (S) MASS (M) M.S M.S2 -0.000832 -0.262

821 -214.97 I 56.29

-0.001603 j -0.504 I 821 -413.90 j 208.66

-0.002258 -0.710 821 -583.25 414.35

C2

C3

C4

C5 -0,002758 3 -0.868 R71 -712.34 # 618.06

C6 -0.003071 -0.966 821 -793.26 766.46

C7 -0.003179 I -1.000 1587 -1587.00 1587.00

Repeat values for nodes C2 to C6 incl. -2717.72 f 2063.82

y.,„x = 0.003179 I

TOTALS -7022.44 5714.64

Fig. 3 - Girder C as per the grillage analysis

TABLE 1

RAYLEIGH HALF BEAM METHOD CALCULATIONS

j - -9.81 x -7022.44 - 61.58 radls

0.003179 x 5714.64 (3)

fi - 9.80 Hz 2 n

COMPARISON OF CASE STUDY NATURAL FREQUENCIES

Three methods for calculating the fundamental frequency were used for the case study. These were the single beam analogy, the Rayleigh method, applied to all the nodes of the grillage


and half of one of the central girders, and a dynamic analysis which employed eigenvalue/vector theory.

As a comparison between the static and dynamic moduli of elasticity for concrete was important, both cases were considered. The results are presented in table 2.

TABLE 2

SUMMARY OF NATURAL FREQUENCIES

Frequencies (Hz) I Natural

Using E = Es Using E = ED % Difference

Beam Analogy

Mode 1 10.89 11.37 4.4

Rayleigh 1/2 Beam Method

Mode 1 9.80 10.24 4.5

Eigenvalue Solution

Mode 1 9.95 10.40 4.5

Mode 2 11.08 11.83 5.2

Mode 3 19.09 21.92 2.8

The natural frequencies and respective mode shapes from the dynamic analysis have been reproduced below.

Fig. 3 - Mode 1, f,= 9.95 Hz

546

Fig. 4 - Mode 2, f2 = 11.08 Hz

AUSTROADS Conference Brisbane 1991

Fig. 5 - Mode 3, f3 = 19.09 Hz Fig. 6 - Mode 4, f, = 40.43 Hz

The fifth mode of vibration was the second longitudinal mode. The torsional and two transverse modes, mode numbers 2, 3, and 4 respectively, are non-existent in the single beam analogy. This highlights how analytical vehicle single beam simulations will never yield results representative of reality.

DISCUSSION OF RESULTS

From table 2 it was noted that the second mode of vibration had a calculated frequency range of 11.08 - 11.83 Hz, depending on which modulus was used. These values were not only very close to the observed mean frequency but also the forcing frequency associated with the wheel hop phenomena, after Cantieni (1983). This immediately cast doubt on the precision of the field measured frequencies. The first two natural frequencies had theoretical ranges of 9.95 - 11.37 Hz and 11.08 - 11.83 Hz respectively. Given that the second natural frequency also falls within the range of field observation, 10 - 12.3 Hz, it is quite possible that the field measuring device at times recorded the second mode of vibration. This is more clearly appreciated when the measuring procedure is considered.

The frequency range of 10 - 12.3 Hz was estimated from stress records, that is from strain gauges placed in the longitudinal direction only. Given that the fundamental mode of vibration for the bridge was longitudinal bending, all subsequent modes of free vibration must have a longitudinal component. This means that the strain gauges placed in the longitudinal direction will record the longitudinal component of -any mode of vibration. The observed frequency range was obtained from selected stress records where the traffic appeared to resonate the bridge. It is therefore suggested that the higher observed natural frequencies, of the order of 12 Hz, were an underestimate of the true second mode of vibration. It is an underestimate because vehicles on the bridge during the test add mass to the system without increasing the stiffness. The effect of support stiffness in this case was thought to be insignificant as the mechanical bearings were thoroughly cleaned and had teflon inserts installed to reduce friction. The observed frequency of 10 Hz, by the same argument, must represent an underestimate of the fundamental frequency. This conclusion is a matter of fact because the lowest observed frequency was 10 Hz. Having concluded that the bearings for this bridge behave very much like ideal pin supports, it is also true to say that the eigenvalue solution is the most accurate. Referring again to table 2 it is seen that calculated fundamental frequency based on Young's modulus of elasticity yields a value less than 10 Hz. This violates the previous argument based on the field observations. Alternatively, when the dynamic modulus of elasticity for concrete was used, the calculated frequency complied very well with field result interpretations. It was therefore concluded that the use of the dynamic

RI Heywood (Editor) Bridges - Part of the Transport System 547

modulus of elasticity for concrete is essential for realistic modelling of time dependant bridge-vehicle systems. This conclusion also verifies the argument previously presented.

As it was often the case that the length of the vehicle exceeded that of the bridge span, it was very difficult to quantify the effect of vehicle mass on the test results. Consequently it can not be verified conclusively that the fundamental frequency was 10.40 Hz; however as it is believed that the bridge bearings accurately portray the characteristics of an ideal pin, it is strongly suggested that the fundamental frequency of Six Mile Creek bridge at the time of testing was 10.40 Hz. In this case both the beam analogy and the Rayleigh half girder method for the Young's modulus case yield results which are less than 6% in -error. Using the dynamit modulus, the Rayleigh 1/2 Beam method yields a very accurate result of 10.24 Hz while the beam analogy yields an error of 9.3 %.

A problem arises however for designers of steel/concrete composite structures. That is, for the bulk of the design the static modulus of concrete is used and the corresponding transformed sections are calculated for the grillage analysis. For the sole purpose of establishing a dynamic load allowance it is unlikely that designers will re-calculate section properties, re-analyse the grillage and calculate the natural frequency and then apply an error margin of ± 10%. Given this, it is highlighted that the use of a static modulus of elasticity for concrete when calculating the fundamental frequency of a composite bridge superstructure will yield a lower bound solution. The extent to which the fundamental frequency is underestimated depends on the sensitivity of the section properties to a change in the modular ratio. For each design this sensitivity should at least be crudely appraised. Knowing that the natural frequencies of a system are a function of the square root of stiffness the significance of the dynamic modulus can be appreciated if nothing else.

For concrete bridge superstructures it is essential that the dynamic modulus is accounted for. As the structure is entirely concrete a simple adjustment factor, A, can be legitimately applied to the calculated frequency based on the static modulus.

E given fDYN fSTAT -D (5) Es

where En = Dynamic moudulus of elasticity of concrete. Es = Static modulus of elasticity of concrete. f„, = fundamental frequency based on the dynamic modulus of elasticity of concrete.

fSTAT = fundamental frequency based on the static modulus of elasticity of concrete.

Sub eqn(2) into eqn(5)

fDYN AE ' fSTAT where AE - \1( 0.8 +

15.2

(6) Es

To appreciate the significance the dynamic modulus of concrete for superstructures constructed entirely of concrete consider a bridge with an Es = 25 MPa. From eqn(6) the difference between fpy, and fsTAT is 18 %.


THE EFFECT OF ASPHALTIC CONCRETE

The addition of asphaltic concrete to a bridge deck effectively adds uniformly distributed mass without changing the stiffness of the system. This means that a bridge may have different natural frequencies throughout its life as a result of deck re-surfacing. Nowhere in the code is this long term consideration mentioned. As the fundamental frequency is inversely proportional to the square root of mass, the following adjustment factor can be derived for the purpose of quantifying the effect of AC surfacing and re-surfacing.

fA AAc f B where

A AC ".

M B (7)

A

where MB = mass of the superstructure before the addition of AC. MA = mass of the superstructure after the addition of AC. f, = the fundamental frequency before the addition of AC. f, = the fundamental frequency after the addition of AC.

To verify this, the case where 50 mm of AC was added to Six Mile Creek bridge, which has actually occurred, was considered. From the dynamic analysis the results were:

f, = 9.95 Hz

fA = 9.33 difference = -6.2 %

using eqn(7):

fA = 9.32 that is: AAc = -6.3 %

As a point of interest, if instead of 50 mm of AC, 80 mm were added to Six Mile Creek bridge the fundamental frequency would have changed by 10 %.

CONCLUSIONS

1. Applying the Rayleigh method to one half of a central girder in a grillage analysis yields theoretically accurate results for simply supported bridges.

2. The use of the dynamic modulus of elasticity for concrete results in a more realistic evaluation of bridge superstructure vibration characteristics. For completely concrete structures it is essential that the dynamic properties be accounted for. This can be achieved simply by using the adjustment factor AE presented in the text.

3. The Ontario Code (1983) specifies the use of the static modulus of elasticity for materials when calculating the first flexural frequency. The reason for calculating the first flexural frequency is to establish a dynamic load allowance from the graph developed from full scale testing. The first flexural frequencies used in the development of this graph were back calculated from test data to eliminate the effects of vehicle mass and forcing function. It is therefore suggested that during this procedure the static modulus was used and that the necessity to maintain consistency is the reason why the OHBDC specifies the use of the static modulus.


4. Vehicles excite various longitudinal, torsional and transverse modes of vibration, therefore single beam idealisations must be rendered unrealistic with respect to vehicle-bridge interaction.

5. Transverse support conditions have a significant effect on the vibrational behaviour of skewed or curved bridges.

6. Present support idealisations, that is, pin supports, render analytical models of complex superstructures unrealistic as the model fails to represent support stiffness.

REFERENCES

British Standards Institution (1972). The Structural Use of Concrete - CP110, Part 1, London, England.

Cantieni, R (1983). Dynamic testing of highway bridges in Switzerland - 60 years experience of EMPA. Swiss federal laboratories of materials testing and research, report 211.

Clough, R.W., Penzien, J. (1975). Dynamics of Structures. Mc Graw Hill, New York

Draft NAASRA Bridge Design Specifications (1987)

Green, R (1977). Dynamic response of Bridge superstructures - Ontario observations. Bridges, TRRL Report SR 275 UC Crowthorne, pp 40-47.

Lee,Ho and Chung (1987). Static and dynamic tests of concrete bridge. Journal of Structural Engineering, ASCE, Vol 113, No.1, Jan.,pp 61-74.

O'Connor, C and Pritchard, W. (1985). Impact studies on small composite girder bridge. Journal of the Structural Engineering, ASCE, Vol 111, No.3, Mar.,pp 641-653.

Ontario Highway Bridge Design Code, Second Edition (1983), Ministry of Transportation and Communication, Ontario.

Wills, J (1977). Correlatioli of calculated and measured dynamic behaviour of bridges. Bridges, TRRL Report SR 275 UC Crowthorne, pp 70-89.


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A SIMPLIFIED METHOD FOR ESTIMATING THE NATURAL …