3. Moving Force Problem
3.1. A Beam Traversed by a Moving Force
3.1.3. A Two-Span Continuous Beam Subjected to a Moving Constant Force
(a) Analytical solution
Substituting the approximate mode shapes of the continuous beam Eq. (3.38) into the equation of motion of a beam subjected to a moving constant force in modal coordinates Eq. (3.11), one can obtain
X 5 % # : Y Z[[∑ A ”sin 3p\` _ 6 $J pm$ (3.39)
where is the th natural frequency of the two-span continuous beam. Similarly to Eq. (3.17), the solution of Eq. (3.39) can be written as
3 6 # ∑ hi[ $b•j[¦‚*Ap§sin ( p\` _ - : psin3 6¨ $J pm$ (3.40) where p #p\` _L[ , # : Y Z[[L[* , P # & 2/! 3 6! 3 6d _ and ! 3 6 is the th
mode shape of the continuous beam.
By using the expressions for ! 3 6 in Eq. (3.30) to Eq. (3.37), P is calculated as following:
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P$$#\%2/], P%%#$%¢ ^2/], PHH#\%2/], PJJ#$%¡ ^2/]
P££#\%2/], P¤¤#%£¤ ^2/], P¡¡#\%2/], P¥¥#$$£ ^2/] (3.41)
The displacement of the continuous beam is the sum of mode shapes of the beam times corresponding modal coordinates
3 , 6 # ∑ ! 3 6 3 6l
m$ (3.42)
It can be seen from Eq. (3.42) that the contribution of the th frequency to the modal displacement of the continuous beam response is | ∑14S#1Ajsin 3S^] 6∑ ª
1:3 o62Ao o|
$J
pm$ .
The acceleration of the continuous beam can be written as
«3 , 6 # ∑ ! 3 6lm$ aX 3 6 (3.43)
where aX is the second derivative of with respect to time , which is
a X 3 6 # ∑ hi[ $b•j[¦‚*Ap9: ( p\` _ - % sin (p\`_ - 5 pω y %sin 3 6< $J pm$ (3.44)
It can be seen from Eq. (3.43) and Eq. (3.44) that the contribution of the th frequency to the modal acceleration of the continuous beam response is
/ # | ∑$J A-sin 3_\ 6 m$ ∑ hi[L[ * $b•j[¦‚*Ap p| $J pm$ (3.45) (b) Validation
To verify the above analytical solutions for the two-span continuous beam subjected to a moving force by using the approximate mode shapes of the beam, a numerical example is given here for comparing the analytical solutions with numerical solutions. The material and geometric properties of a two-span continuous beam are from (Nguyen et al., 2009): 2 # 5400 kg/mH, / # 7.73 m% , ' # 28.25 GPa , # 7.84 mJ and # 4.255 × 10J kg. The example bridge is a simply supported
bridge with the length of # 30 m (Nguyen et al., 2009). The length of the continuous beam is assumed to be ] # 2 # 60 m. The vehicle loading is treated as a constant force of # ° moving at a speed of 360 km/h. The above properties are different from the properties used in section 3.1.2 for deriving the approximate mode
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shapes of the continuous beam, which is to verify the universality of the mode shapes derived in section 3.1.2. The numerical method of solving this problem is described below.
The equation of motion of the continuous beam after applying MS method can be described in modal coordinates as
aX 3 6 5 % 3 6 # :Y@Z[3`;6[[ (3.46)
The first eight analytical frequencies and mode shapes of the continuous beam, which are given in Eq. (3.27) to Eq. (3.29), respectively, are adopted in the Newmark integration which can give directly the acceleration of the beam (Yang et al., 2004). Eight beam modes are used in the calculations.
To study the influence of the vehicle speed on the structural dynamic response, a notion of critical speed is introduced here (Ouyang, 2011)
— #1L\= (3.47)
where $ is the first natural frequency of the beam in the unit of rad/s. — # 868.388 km/h is calculated in this example.
A speed ratio of the vehicle speed to the critical speed is introduced here
` `±²#
\`
1L= (3.48)
which matches the frequency ratio $ defined in Eq. (3.16), and $ # 0.415 is calculated in this example.
It can be seen from Figure 3.6 that the present results of the continuous beam at first mid-span agree well with corresponding numerical results, which validates the approximate mode shapes of the two-span continuous beam and the analytical solutions of the beam by using the approximate mode shapes. The dynamic displacement ratio is defined as the ratio between the dynamic displacement at first mid-span and the static deflection at the same point. The maximum dynamic displacement ratio is the Dynamic Amplification Factor (DAF) which is 1.425 in this example.
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(a) (b)
Figure 3.6. Comparison between present results and numerical results at first mid-
span for the vibration of a two-span beam subjected to a moving force: (a) displacement, (b) acceleration
The acceleration spectrum of the bridge vibrating freely at location 3L/16 after the passage of the moving force by FFT of the data calculated analytically using approximated modes is shown in Figure 3.7. The first two bridge frequencies are excited into large amplitudes, which is different from that for a simply supported bridge whose first mode is largely excited only (Yang et al., 2004).
Figure 3.7. Excited bridge frequencies of the bridge acceleration response at 3L/16
A m p li tu d e (m / s 2 )
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(c) DAF of simply supported beam and two-span continuous beam
Yang et al. (2004) defined the Impact Factor for a simply supported beam bridge as #³´(µ*-b³i(µ*-
³i3µ*6 (3.49)
where ¶·(_%- and ¶ (_%- denote the maximum dynamic and static response of the bridge at the beam mid-span subjected to a moving load. Comparing the definition for the DAF in this thesis and the Impact Factor I by Yang et al. (2004), it is found that DAF=I+1. Yang et al. (2004) proposed a formula to estimate I for the displacement of a simply supported bridge traversed by a vehicle with a speed ratio of $ as follows
¸
# {
$.£Jj.¡¡ º»¼ j= º»¼ j==½ .£¾ .£ (3.50) Therefore, ¸ # 1.54 × 0.415 # 0.639 for a simply supported bridge with the same properties and span length as the two-span continuous bridge in this example. DAF would be 1.639 for the simple bridge, which is larger than the DAF for the two-span continuous bridge (1.425).It is found that the impact factor ¸ basically only changes with the speed ratio $ (Yang et al., 2004). The DAF of the simply supported beam and DAF of the two- span continuous beam with the same span length against the speed ratio are shown in Figure 3.8. It can be seen in the figure that the DAF at the first mid-span of the two- span continuous beam is smaller than that of the simply supported beam in the whole speed ratio range from 0.05 to 1. However, the DAF at the second mid-span of the continuous beam increases from 1 at speed ratio of 0.45 to about 3.25 at speed ratio of 0.85. The reason of the trend of DAF at the second mid-span is discussed below.
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Figure 3.8. Comparison between DAF of simply supported beam and DAF of two-
span continuous beam against speed ratio
The first two mode shapes of the two-span continuous beam are depicted in Figure 3.9. There is one cycle of the first mode in the whole length of the beam, whereas there exists two cycles of the second mode in the whole length of the beam. The critical speed where the beam is excited into resonance at the first frequency is
¿¼# $] # L%\=_# L\=1 which is same as Eq. (3.47). The vehicle speed where the
beam is excited into resonance at the second frequency is
¿¼% #Y*%_ #L%\*1 (3.51)
The ratio between ¿¼% and ¿¼ is
`ÀÁ* `ÀÁ # L* %L= # E** %E=* # H.¢%¤¤* %×H.$J$¤*# 0.7811 (3.52)
which is a constant number. The reason why the DAF of the two-span beam at the second mid-span is maximum and very high (DAF=3.21) at the speed ratio of around 0.85 is that the first and second modes of the beam are both almost excited into high amplitudes at this speed ratio, which is also found in Figure 3.7. The reason why DAF is not maximum at resonance speed ratio of 1 is that the beam displacement amplitude increases with time and there is not enough time for the beam response excited into largest amplitude (the period of the moving force on the beam is limited).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Speed ratio S 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5
Simply supported mid-span Two-span first mid-span Two-span second mid-span
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(a) (b)
Figure 3.9. (a) First, (b) second normalized mode shapes of a two-span continuous
beam