This paper will consider the response of a generic four-span, five support bridge subject to real time-history accelerations taken from the SMART-1 array, Taiwan ...
Proceedings of the 8th U.S. National Conference on Earthquake Engineering April 18-22, 2006, San Francisco, California, USA Paper No. 875
EFFECTS OF ORIENTATION TO THE EPICENTRE ON THE RESPONSE OF LONG SPAN BRIDGES SUBJECT TO MULTIPLE SUPPORT EXCITATION USING SMART-1 ARRAY DATA AND CORROBORATIVE EXPERIMENTAL RESULTS N. A. Alexander1, J. Norman2, D. Virden3, A. Crewe4 , D. Wagg5 and A.A. Chanerley6
ABSTRACT This paper will consider the response of a generic four-span, five support bridge subject to real time-history accelerations taken from the SMART-1 array, Taiwan. It considers the effects of (a) support pier to deck stiffness ratio and (b) bridge horizontal alignment to the epicentre on the bridges response. The SMART-1 accelerogram data is corrected using a wavelet de-noising algorithm, bidirectional filtering and a second order instrument deconvolution. The wavelet de-noising algorithm has not been applied previously to this data set and is designed to mitigate the quantisation noise present. The numerical modelling of the bridge is achieved by employing a vectorial Euler-Lagrange approach. Numerical solutions are obtained by a Runge-Kutta integration scheme. This paper will include corroborative experimental data using the multiple support excitation rig. The multiple support excitation rig is a series of five single-axis shaking tables each at a spacing of one meter; this enables the response of bridge structures to multiple support excitation to be modelled. Comparisons between the physical and numerical models shall be made.
Introduction There have many studies on the influence of spatially heterogeneous ground motions on the dynamical behavior of long multispan bridges. Many researchers, Harichandran et al. (1996) etc., tend to use a SVEGM type model to characterize this spatial-heterogeneity between two arbitrary stations i and j. These models tend to assume some exponential approximation for the absolute coherency and a linear function for the phase variation between these two stations. However these models fail to capture precisely the complicated and random variations present in real spatially heterogeneous ground motions. In the paper the responses of multispan bridge is evaluated numerical when subject to real accelograms recorded at the SMART-1 array. Nontrivial experimental validation of the numerical modeling is performed. The experimental test rig 1 2 3 4 5 6
Lecturer, Dept. of Civil Engineering, University of Bristol, Uk. Ph.D. Student, Dept. of Civil Engineering, University of Bristol, Uk. Ph.D. Student, Dept. of Mechanical Engineering, University of Bristol, Uk. Senior Lecturer, Dept. of Civil Engineering, University of Bristol, Uk. Lecturer, Dept. of Mechanical Engineering, University of Bristol, Uk. Senior Lecturer, Dept. of Electrical Engineering, University of East London Uk.
models the entire bridge and independently excites each support with real timehistories. A general multi-span bridge with multi-support excitation During a seismic event the bridge beam and ground deform as shown in Figure 1. This shows only the lateral horizontal motion of the bridge and the ground. Bridge piers connecting the bridge and ground are modelled as simple springs. This problem can be thought of as the dynamics of a beam on discrete elastic foundations. The kinetic energy of this system is given by T and the potential energy by U in equation (1). The ground deformation is given by y g (x, t ) and
the beam deformation by y(x, t ) whose mass per unit length is given by mb ( x ) and its flexural
rigidity by EI ( x ) . The beam is supported by p pier springs, of stiffness ki , that are located at positions xi . These springs encapsulate pier geometry, end fixity and local ground conditions. The ground reactions at these columns are given by Ri . Support pier i deck beam
y ( x, t )
ki
y g ( x, t )
Ri
ground x
xi Figure 1, generalized beam on discrete moving elastic foundation
T = ∫ mb y& 2 dx , U = 1 2
1 2
∫
2 EI y ′′ 2 dx + 12 ∑ k i ( y (xi ) − y g ( xi )) + ∑ Ri y ( xi ) p
p
i =1
i =1
(1)
A Rayleigh-Ritz spatial-temporal formulation is employed. The beam displacements are approximated by an n term shape function and the ground displacement a p term shape function described in (2). Hence, by substitution of (2) in (1) equations (3) and (4) are obtained. By applying the vectorial form of Euler-Lagrange principle the equations of motion for this system can be derived.
y ( x, t ) = a(t )Τφ ( x ) ,
y g (x, t ) = b(t )Τψ ( x )
(2)
Τ Τ T = 12 a& ∫ mb φφ dx a&
U = 12 a
Τ
∫
(3) p
(
)
EI φ ′′φ ′′ dx a + 12 ∑ k i a φ ( xi ) − b ψ ( xi ) +b Τ
i =1
Τ
Τ
2
Τ
p
∑ R ψ (x ) i =1
i
i
(4)
Τ
Beam deformation y = a φ a1 = b1
a2
k2
a5 = b5
a4
a3 k4
k3
b2 b4
b3 h
h
Ground Τ yg = b ψ
h
h
Figure 2, a four-span continuous-beam on three flexible supports
Four-span bridge problem
The degrees of freedom chosen for the problem shown in Figure 2 are the displacements at the top and bottom of the column support springs. The degrees of freedom on the beam a are initial unknown and require the numerical solution of the system equations, while the ground degrees of freedom b are essentially known and will form part of the excitation of the system. Lagrangian polynomials are employed for the shape functions φ = ψ as they do not require any inversion of a boundary condition matrix. However, there is one extra set of boundary conditions for this bridge that is different from the general case. This is there are no column support piers at the beginning and end of the bridge: thus a1 = b1 and a5 = b5 , therefore degrees of freedom a1 and a5 are known. In this problem the system can be reduced to just three degrees of freedom a2 a3 and a4 which represent the absolute displacement of the top of the support columns. ⎡m11 m12 ⎤ ⎡b&&1 ⎤ ⎡c11 c12 ⎤ ⎡b&1 ⎤ ⎡k 11 k 12 ⎤ ⎡b1 ⎤ ⎢m T m ⎥ ⎢ ⎥ + ⎢cT c ⎥ ⎢ ⎥ + ⎢k T k ⎥ ⎢ u ⎥ && ⎦ ⎣ 12 22 ⎦ ⎣ u& ⎦ ⎣ 12 22 ⎦ ⎣ u 22 ⎦ ⎣ ⎦ ⎣ 12 ⎡k = ⎢ 33 ⎣ 0
0 ⎤ ⎡ b1 ⎤ ⎡q11 0 ⎤ ⎡ b&1 ⎤ − ⎢ ⎥ k 44 ⎥⎦ ⎢⎣b 2 ⎥⎦ ⎢⎣ 0 q 22 ⎥⎦ ⎣b&2 ⎦
(5)
The equations of motion can be stated as equations (5) and partitioned to achieve the reduced and final system (6) in terms of the degree of freedom vector u = [a2 , a3 , a4 ]Τ . Classical Rayleigh viscous damping model is assumed. The matrices in equation (6) are defined in equations (7) and (8). The pier to deck stiffness ratio parameters κ i and frequency parameter ϖ are introduced in (9). Ground displacement vectors are b1 = [b1,b5 ]Τ and b 2 = [b2 , b3 , b4 ]Τ .
T T & T && m 22 u&& + c 22 u& + k 22 u = −k 12 b1 + k 44 b 2 − c12 b1 − q11b&2 − m12 b1
m 22
56 ⎤ ⎡ 1792 − 384 256 ⎤ ⎡ 296 2 ⎢ 2 ⎢ ⎥ T = − 384 1872 − 384⎥ , m 12 = − 174 − 174⎥⎥ 2835 ⎢ 2835 ⎢ ⎢⎣ 256 − 384 1792 ⎥⎦ ⎢⎣ 56 296 ⎥⎦
k 22
⎡ 176 5 ⎢ = ϖ 2 ⎢− 632 15 368 ⎢⎣ 15
κi =
− 632 15 301 5 632 15
−
368 15 632 15 176 5
−
⎤ ⎡κ 2 ⎥ 2⎢ ⎥ + k 44 , k 44 = ϖ ⎢ 0 ⎥⎦ ⎢⎣ 0
0
κ3 0
0⎤ ⎡− 172 15 ⎢ T 0 ⎥⎥ , k 12 = ϖ 2 ⎢ 361 30 92 ⎢⎣ − 15 κ 4 ⎥⎦
EI ki h 3 , ϖ2 = EI mb h 4
(6)
(7) 92 − 15 ⎤ 361 ⎥ 30 ⎥ ⎥ − 172 15 ⎦
(8)
(9) Correction of SMART-1 data
The SMART-1 Array was the first large array of digital accelerometers specially designed to investigate the near-field properties of earthquake ground motion, Abrahamson et al. (1987) . It was located in the northeast corner of Taiwan near the city of Lotung on the Lanyang plain. The near-surface geology under the array is predominantly recently laid alluvial deposits; the water table was at or near the surface. The topographic of the surface was very flat. In this paper data from the largest event recorded, that had an epicentre within close proximity to the array (event 43), was used. The records were processed, Abrahamson et al. (1987), to remove glitches, replace data drop-outs and remove the DC baseline. The digital recorder applied a low pass five-pole butterworth filter, with a cut-off frequency of 25Hz, was employed to mitigate the effects of aliasing. This will impose a nonlinear phase change on the records, which is far from ideal, however at least the same change is imposed on all records. Deconvolution of the instrument characteristic, SA-3000, was not performed. In this paper a further correction process is performed on the data in order to mitigate three potential sources of noise. (i) The records are wavelet denoised using a db8 wavelet decomposition down to 5 levels, Chanerly et al. (2002). This approach applies a power threshold to the data, i.e. all components of the signal below this power threshold are filtered out. This process is not dependant on stationary assumptions or frequency content of the noise. The purpose of this correction scheme is to mitigate the noticeable quantisation noise present in the data. This quantisation noise is generally low in power but broad in spectrum. This method has not been applied to the SMART-1 data previously. (ii) A standard single degree of freedom instrument deconvolution is employed. This used the natural frequency of the SA-3000 and its ratio of critical damping; 140Hz and 0.7 respectively. (iii) A low pass 4th order Butterworth filter is applied, bidirectionally in time so as not to corrupt the phase content, with a cut-off frequency of 0.25Hz. In order to produce ground displacement the corrected ground acceleration are numerically integrated twice by a Simpson’s 1/3rd rule. Low-cut 4th order Butterworth filter of 0.25Hz is bidirectionally applied on the velocity and displacement to remove the drift error normally associated with the integration of
timeseries.
Figure 3, Left, the bridge model skeleton and right, the bridge model with additional masses added to allow for scaling effects.
The experimental bridge model
The bridge model is a 1:50 scale model of a 200m long bridge. The bridge has 4 spans 50m long, and is supported by two abutments and three piers. In this study the piers are of an equal length of 21m. A 200m long bridge was chosen because similar bridges have been used in studies by Lupoi et al (2005) and Zapico et al (2003) and these studies have shown that bridges of this size may be susceptible to a significant increase in response when considering MSE. The length was also chosen because a large proportion of bridges on major road projects are of a similar length. For example, on the new Egnatia Motorway in Greece, 103 of the 612 bridges were between 100m and 300m long, whilst only 4 were longer than 600m, Ahmadi-Kashani (2003). The bridge model is constructed from a 4m, 60 × 60 × 3.2 mm box section, see Figure 3. The piers are 420mm long, and have a 20 × 25 mm solid section. The abutments are pinned in plan, but fixed in elevation. This is achieved by having a smooth and greased 20mm diameter rod which passes through a thickened 20mm hole at each end of the deck, this can be seen clearly in the left hand side of Figure 3. The rods are fixed to a 200 × 100 × 10 mm box section, orientated so that it is shaken in the stiff direction. The model and box section have a combined natural frequency of 70Hz, well above the first three natural frequencies of the model. All the bridge components are made from S275 grade steel. The right hand side of Figure 3 shows the bridge model with added mass. 160 added masses were attached to the bridge deck with threaded bar, increasing the mass of the bridge model to almost 500kg. The masses are attached in groups of four and are isolated from each other so as not to change the properties of the bridge deck. The added masses are used to significantly reduce the natural frequencies of the bridge model so that they are realistic, therefore allowing for scaling of the model. The ground excitations were simulated by use of a five single axis, actuators, see Figure 4. Each actuator is mounted on a pair
of bearings, which move over a single steel shaft. The shaft is attached at each end to a stiff frame. For further performance specification see Norman et al. (2005)
Figure 4; the five, single axis, actuators with the bridge model mounted
Numerical model updating
In order to obtain a calibration of the numerical parameters,ϖ and κ = κ 2 = κ 3 = κ 4 , an attempt was made to match the first three mode natural frequencies, fi , of the experimental bridge model with the three natural frequencies, ni , of the numerical bridge model. The error in modal frequencies squared, S , is given by (10). The least square error S min can be obtained by a process of numerical minimisation, using a Nelder-Mead Simplex method. This process almost exactly matches both the first and second mode frequencies with numerical system parametersϖ = 18.23 Hz, κ = 3.336 . However it was unable to match the third mode frequencies. After model updating, the numerical modal frequencies are ni = {5.571,10.43, 24.23}Hz compared with the estimated experimental natural frequencies f i = {5.55,10.43,17.09}Hz. ⎛ f − ni (ϖ , κ , α ) ⎞ ⎟⎟ S = ∑ ⎜⎜ i fi i =1 ⎝ ⎠ 3
2
(10)
Figure 5 shows a comparison of the calibrated numerical solution to (6) with the experimentally recorded results. This figure compares the displacement timehistories at the top of the first column, a2 (t ) , and central column, a3 (t ) . The numerical and experimental models match extremely well; this confirms that the low order system (6) is a reasonable model a complex multi-span bridge. In these figures the first and second mode damping ratio are 0.0117 and 0.0087 respectively.
3
experimental numerical
displacement, top of central column [mm]
displacement, top of first column [mm]
2 1.5 1 0.5 0
-0.5 -1
-1.5 -2
-2.5 -3 21
21.5
22
22.5
23
t [secs]
23.5
24
24.5
25
experimental numerical
2
1
0
-1
-2
-3
-4 21
21.5
22
22.5
23
t [sec]
23.5
24
24.5
25
Figure 5, Comparison between experimental and numerical timehistories for first and central column supports ( for stations C00 to I01 ground motions) I01
I12
SMART-1 Inner ring Station
I02
I11
I10
bridge I03
C00 0
200m
I09
I08
I04
I05 I07
I06
Figure 6, Positioning of bridge in SMART-1 array
Influence of bridge alignment with SMART-1 data.
The multispan bridge is placed in the geographical location of the SMART-1 array. The 13 accelerometer of the inner array are used to determine the five input ground motions required for the bridge. This is achieved by using a cubic interpolation algorithm. Figure 6 displays the bridge located on one of the 12 radial lines. The system (6) is solved to produce the bridge responses for all 12 radial lines. A parallel experimental study is performed. Bridge alignment response spectra are displayed in Figure 7. The horizontal axes gives information about the bridge alignment, for example a value of 5 would indicate the bridge located between the centre station C00 and inner station I05. The central pier peak displacement, max a3 (t ) , show a good correlation between the experimental results and the numerical simulations. For the first pier peak displacement, max a2 (t ) , there are differences; these are most probably due to a lack of precision in estimating the damping ratios. Vertical dotted lines on this diagram indicated the estimated epicentral direction. For this value of pier to deck stiffness ratio κ = 3.336 it is clear that the maximum lateral displacement response of the bridge is not aligned with the epicentral
direction nor normal to epicentral direction. 6
max | a (t) |
5.5
3
Experimental 5
max | a (t) | 3
Numerical
Displacement [mm]
4.5
4
3.5
max | a (t) | 2
Numerical
3
2.5
max | a (t) | 2
Experimental
2
1.5
EP
EP 1
0
2
4
6
8
10
12
Bridge alignment, by SMART-1 Station No.
Figure 7, Comparison of Spectral displacements for 2π variation in bridge alignment
As a further parametric study consider the polar response spectral plot, Figure 8. This diagram has polar coordinates κ∠θ ; where radius κ is the pier to deck lateral stiffness ratio and bearing angle θ (clockwise from North) is the alignment of the bridge. The contour levels indicate the value of the central pier peak displacement max a3 (t ) . Different levels of κ change the fundamental mode shape of the bridge as indicated in this figure. Thus this figure is a response spectrum that describes the influence of pier to deck stiffness ratio and bridge alignment. The worst response occurs at location 5∠1350 and this is a bridge with relatively flexible piers and aligned approximately in the epicentral direction. However generally, there is no correlation between bridge alignment to an epicentral direction (even approximately) and the worst bridge response. Consider the bridges in location a 50∠900 this is near the maximum for κ = 50 and this is neither in an epicentral or normal to epicentral direction. Figure 9 is a zoom up of Figure 8 for the region of low κ . It clear shows the worst bridge alignment is constantly changing with increasing κ and it is not correlated to epicentral or normal to epicentral directions. Conclusions
This paper considers the influence of bridge alignment on the lateral responses of bridge when subject to real spatially heterogeneous ground motions. Rather than employing a SVEGM type model real ground motion records were extracted from the SMART-1 array database. This paper is based on one event, and 26 spatially heterogeneous accelerograms. The numerical, three degree of freedom, model compares very well with the experimental test results from the multispan bridge. This corroborative evidence validates the theoretical and numerical formulation of the problem. The use of model updating was valuable in calibrating the numerical and experimental models. However, care was required to avoid fitting a set of parameters to the
numerical model that resulted in a physically unrealizable system. Bridge mode shapes κ=1
12
N
1
6.5
mm
0 0
h
2h
3h
κ = 100
11
4h
6
2
5.5
0
5
0
h
2h
3h
4h
10
3
20
40
60
80
100
4.5
4
κ
3.5
9
4
3
2.5
8
2
5
1.5
7
EP
6
Figure 8, Polar response spectra, influence of κ vs. alignment N 6.5
6
5.5
5
4.5
5
7.5
10
12.5
15
17.5
20
κ
4
3.5
3
2.5
2
1.5
EP
Figure 9, Zoom up of detail in polar response spectra, influence of
κ
vs. alignment
The parametric study into the influence of bridge alignment did not validate any
directional hypothesis; e.g. the worst lateral responses should occurred when the bridge alignment is a normal to or epicentral direction. In a polar response spectral plots the critical bridge alignments were shown to be uncorrelated with any predetermined or physically meaningful directions. They also showed that there was significant variation in bridge responses with various horizontal alignments on an identical bridge; this was as much as 50% variation in the worst case. A set of 12 independent tests on the bridge model that was excited by five independent actuators validated the numerical results. Further studies need to be performed on the frequency parameter ϖ and consider the influence of dissimilar support piers. In addition the effect of other earthquake events needs to be explored to extend the scope of this study. Acknowledgments
The authors wish to thank Engineering and Physical Sciences Research Council (EPSRC) for funding this research under grant GR/R99539/01, DJW supported by an Advanced Research Fellowship. The authors also wish to thank the Institute of Earth Science, SMART-1 Array data repository, Tawain. http://www.earth.sinica.edu.tw/~smdmc/ for the uses of the SMART-1 data set. References
Abrahamson N.A., Bolt B.A., Darragh R.B. Penzien J., Tsai Y.B., 1987, The Smart-1 accelerograph array (1990-1987): A review, Earthquake Spectra, 3(2) , Ahmadi-Kashani, K., 2004. Seismic design of Egnatia motorway bridges, Greece, Bridge Engineering 157, 83-91. Alexander N.A. 2005, Incoherent ground motion in multi-support dynamics of bridges. Proceedings of the Tenth International Conference on Civil, Structural and Environmental Engineering Computing, B.H.V. Topping, (Editor), Civil-Comp Press, Stirling, United Kingdom, paper 236. Chanerley A.A., Alexander N.A., 2002, An approach to seismic correction that includes wavelet de-noising. Proceedings of 6th InternationalConference on Computational Structures Technology, Prague. 10(44) Harichandran R.S., Hawwari A., Sweldan B.N., “Response of Long-Span Bridges to Spatially Varying Ground Motion” J. Struc. Eng. 122(5), pp 476-484, 1996. Lupoi, A., Franchin, P., Pinto, P. E. and Monti, G., 2005. Seismic design of bridges accounting for spatial variability of ground motion, Earthquake Engineering and Structural Dynamics 34, 327-348. Norman, J., Virden, D., Crewe, A. and Wagg, D., 2005. Physical Modelling of Bridges subject to multiple support excitation, Proceedings of the 8th National Conference on Earthquake Engineering, San Francisco, California, USA. Zapico, J. L., Gonzalez, M. P., Friswell, M. I., Taylor, C. A. and Crewe, A. J., 2003. Finite element updating of a small scale bridge, Journal of Sound and Vibration, 993-1012.
