L. LIN

Copyright © 2013 SciRes. OJCE

Ec(t) = Ec(28){exp[s(1 - (28/t)1/2)]}1/2 (1)

in which Ec(t) is the elastic modulus at age t, Ec(28) =

38.7 GPa is the average elastic modulus at age t = 28

days, and s = 0.25 is a coefficient for normal hardening

cement assumed for the concrete used in the bridge.

4.5. Calibration of the Model

The calibration was conducted using the model shown in

Figure 3. Since the calibration was based on the meas-

ured data during the pull tests, it was important that the

masses of the bridge model correspond to those of the

bridge during the pull tests. As reported by [6], the barri-

ers and the pavement had not yet been in place at the

time of the pull tests. Therefore these masses were not

included in the calibration of the model. Given the avai-

lable data described in Sections 4.1 to 4.4, the following

parameters were used as reference parameters in the ca-

libration of the model:

• Tilts at locations 3 and 4 of pier P31 recorded during

the static pull test,

• Acceleration time history of the transverse vibrations

at location 9 (Figure 5) recorded during the dynamic

pull test,

• Predominant frequencies of the recorded vibrations at

locati on 9,

• Damping ratios of the predominant modes of the re-

corded vi br a t ions, and

• Modulus of elasticity of the concrete of 40,000 MPa

corresponding to the age of the concrete at the time

when the pull tests were conducted. It was determined

by using Equation 1.

The parameter that was varied in the calibration of the

model was the foundation stiffness. While the soil under

the foundations of the piers is designated “rock” in the

design drawings, it is believed that they may have certain

soil-structure interaction effects, which is appropriate to

consider them in analyzing the dynamic behaviour of the

bridge. Rotational springs (at the bases of the piers) in

the longitudinal and transverse directions were intro-

duced in the model to represent the foundation stiffness.

The calibration consisted of iterative performing the fol-

lowing sequence of tasks:

1) The selection (in the first iteration), or the adjust-

ment (in the subsequent iterations) of the rotation al stiff-

ness of the springs;

2) Static analysis of the model by applying a horizon-

tal force of 1 MN to pier P31, 6 m above the mean sea

level, in the transverse direction of the model;

3) Using the results from 2), the tilts at locations 3 and

4 of pier P31 were computed, and compared with the

values measured during the static pull test;

4) Time-history analysis of the model by applying a

loading representative of that used in the dynamic pull

test. The loading was applied to pier P31, 6 m above the

mean sea level, in the transverse direction of the model;

5) Fourier analysis of the response time history of the

model at location 9 to compute the Fourier amplitude

spectrum. This spectrum was compared with that of the

vibrations recorded at location 9 during the dynamic pull

test (Figure 6).

It should be mentioned herein that in case of differ-

ences between the computed and measured tilts in task 3)

of a given iteration were larger than those in the previous

iteration, then tasks 4) and 5) were not proceeded, and a

new itera t ion was undertake n.

4.5.1. C om parison of Tilts

Results for the tilts of selected iterations of the calibra-

tion process are given in Table 1. The table shows the

rotational stiffness used in the iterations and the corre-

sponding tilts at locations 3 and 4 resulted from the static

analysis of applying a horizontal force of 1 MN to pier

P31, 6 m above the mean sea level. For comparison, the

tilts measured during the static pull test are also given in

the table. It can be noted that the stiffness of the founda-

tion between 3.35 MN·m/µR and 4.45 MN·m/µR pro-

vides tilts that are quite close to the measured tilts. The

differences between the computed and the measured tilts

were less than 3%. Tabl e 1 also sho ws t hat the computed

tilts corresponding to fixed-base conditions are signifi-

cantly smaller than the measured values. This was ex-

pected since the bridge structure with rigid bases is stiffer

than that with flexible bases.

4.5.2. Time-History Analysis of the Model

For each of the foundation stiffness considered in the

calibration process, a time-history analysis was con-

ducted to compute the response time histories of the mo-

del. The load was assumed to increase linearly from 0 to

1.4 MN in 15 s, and then it was kept constant for 10 s,

Table 1. Foundation stiffness used in selected iterations and

computed tilts at locations 3 and 4.

Foundation

stiffness

(MN·m/μR)

Computed tilts (in μR) and differences (in %)

between computed and measured values

Location 3 Difference Location 4 Difference

∞(Fixed-bases) 34.78 −19.1(1) 38.15 −18.1

5.05 40.71 −5.3 44.72 −4.0

4.45 41.70 −3.0 45.54 −2.0

4.05 42.29 −1.6 46.36 -0.5

3.35 43.77 +1.8 48.00 +3.0

2.65 46.34 +7.8 50.67 +8.8

Measured values 42.99 46.59

(1)Difference wi th negat ive sig n indicates t hat the co mputed v alue is small er

than the measured value.