RESULT AND DISCUSSION- PART-I: DYNAMIC BEHAVIOUR OF THEORETICAL MODEL
6.7 Response Statistics for Model-3
6.7.1 Vehicle Response
The mean and standard deviation of displacement, velocity and acceleration at the centroid of vehicle body has been shown in Fig. 6.38 to Fig. 6.40. The vehicle body vertical displacement is influenced by both bending and twisting modes. The mean displacement shows interaction of various modes of vibration with appearance of higher frequency components as compared to that seen in other two models. More rapid fluctuation of peaks is observed in mean displacement, velocity and acceleration time history of vehicle centroid.
Standard deviation of mean displacement at centroid of vehicle body displays a high peak when vehicle is nearer the mid span of the bridge. It may be noted that vehicle is excited by two sources-one is pavement unevenness and another is bridge deflection. Twist of the vehicle body is non uniform as the elastic torsional vibration takes place and varies along the length of the vehicle. The mean and standard deviation of torsional rotation, torsional velocity and torsional acceleration are depicted in Fig.6.41 to Fig.6.43.
(a) Mean (b) Standard deviation Fig.6.38. Vertical displacement of vehicle at C.G
The response mean shows that as long as vehicle is within the bridge, the output frequency is affected by the interaction of various modes in the coupled system showing high fluctuation of magnitudes of peak. In the analysis, no wheel input from bridge pavement is considered as soon as the vehicle completely crosses the bridge and therefore free vibration takes place at different frequency. This is more clearly visible in mean acceleration time history. In the standard deviation of transverse rotation, rolling velocity and acceleration, multiple peaks are
seen. The response magnitudes in translatory and rotational motion increase with the increase in vehicle forward velocity.
(a) Mean (b) Standard deviation Fig.6.39. Vertical velocity of vehicle at C.G
(a) Mean (b) Standard deviation Fig.6.40. Vertical acceleration of vehicle at C.G
(a) Mean (b) Standard deviation Fig.6.41. Torsional rotation of vehicle at C.G
(a) Mean (b) Standard deviation Fig.6.42. Torsional velocity of vehicle at C.G
(a) Mean (b) Standard deviation Fig.6.43. Torsional acceleration of vehicle at C.G
The mean and standard deviation of displacement, velocity and acceleration time history for each of the four unsprung masses have been shown in Fig.6.44 to Fig.6.55. It may be noted that as rear wheel enters into the bridge later, response time histories of rear unsprung masses have been shown with time delay. No remarkable influence of velocity on the magnitudes of wheel displacement is clearly reflected in the entire segment of the time history. However, it is seen that each wheel represents somewhat different features in terms of the location of peak and magnitude for each of vehicle speed considered. This may be due to rolling of vehicle as the wheel experiences different input from longitudinal and transverse bridge surface profile. The mean and standard deviation of wheel responses are found to be lower compared to the model where rolling of the vehicle body has been ignored.
(a) Mean (b) Standard deviation
Fig.6.44. Displacement of front wheel in right side of the vehicle
(a) Mean (b) Standard deviation Fig.6.45. Velocity of front wheel in right side of the vehicle
(a) Mean (b) Standard deviation
Fig.6.46. Acceleration of front wheel in right side of the vehicle
(a) Mean (b) Standard deviation Fig.6.47. Displacement of front wheel in left side of the vehicle
(a) Mean (b) Standard deviation Fig.6.48. Velocity of front wheel in left side of the vehicle
(a) Mean (b) Standard deviation Fig.6.49. Acceleration of front wheel in left side of the vehicle
(a) Mean (b) Standard deviation Fig.6.50. Displacement of rear wheel in right side of the vehicle
(a) Mean (b) Standard deviation Fig.6.51. Velocity of rear wheel right in side of the vehicle
(a) Mean (b) Standard deviation Fig.6.52. Acceleration of rear wheel in right side of the vehicle
(a) Mean (b) Standard deviation Fig.6.53. Displacement of rear wheel in left side of the vehicle
(a) Mean (b) Standard deviation Fig.6.54. Velocity of rear wheel in left side of the vehicle
(a) Mean (b) Standard deviation Fig.6.55. Acceleration of rear wheel in left side of the vehicle 6.7.2 Bridge Response
In this section, we present the effect of important parameters on the mean and standard deviation of mid span deflection, velocity and acceleration of the bridge due to passage of a
flexible full car body. Dynamic Amplification factors have been presented and its changes due to simultaneous variation of bridge-vehicle parameters have been discussed.
6.7.2.1 Effect of vehicle speed
In a Bridge-Full car Vehicle interaction problem, longitudinal and transverse surface roughness will play important role in transmitting dynamic force on the pavement. This has been investigated considering three different vehicle forward velocities. In this illustration, poor category of surface unevenness as per ISO classification has been considered. The mean and standard deviation of mid span deflection, velocity and acceleration have been shown in Fig.6.56- Fig.6.58. There is a systematic increase of peak magnitude of mean displacement with speed with a notable shifting of peak towards the left. This once again confirms the modification of driving frequency with increase in vehicle speed. However, in this case standard deviation does not show any definite trend. In derivatives of displacement, more number of waves appears and dependence of excitation frequency on the vehicle speed is not clearly observed. High frequency component of spatial disturbance in bridge deck has been found to appear in acceleration response. Magnitudes of standard deviation are not very significant. The coefficient of variation of peak displacement, velocity and accelerations are found to be 0.17, 0.214 and 0.113 respectively.
(a)Mean (b) Standard deviation Fig.6.56. Bridge displacement at mid span
(a)Mean (b) Standard deviation Fig.6.57. Bridge velocity at mid span
(a)Mean (b) Standard deviation Fig.6.58. Bridge acceleration at mid span
6.7.2.2 Effect of eccentricity of vehicle path
In a Full Car Model, a distance between vehicle body longitudinal axis and the centre line of the bridge has been considered as eccentricity (ex). In generating numerical results under this section vehicle velocity 60 km/h has been considered. The vehicle flexural and torsional stiffness are taken uniform (EvIv=5.3×106 N.m2, GvJv=9.02x102 N-m2). Fig. 6.59 presents mean and standard deviation of bridge mid span vertical deflection for different eccentric position of vehicle. An increase of 6.8% to 9.1% in the bridge dynamic responses has been observed as the eccentricity varies from 0.5 m to 1.5m. It may be mentioned that bridge torsional mode is activated when load eccentricity exists. Moreover rolling of vehicle alters the pattern of vehicle loading.
(a) Mean (b) Standard deviation
Fig.6.59. Bridge mid-span displacement due to different eccentricity of vehicle path (ex), (A) ex=0.5 m, (B) ex =1 m, (C) ex =1.5 m
6.7.2.3 Effect of approach road settlement
Bridge mid span displacements for different level of approach settlement has been shown in Fig. 6.60, by considering a full car vehicle model with 12 m vehicle axle spacing and 2 m tread width. Investigation has been performed for constant speed of 60 km/h. Result shows that approach settlement up to 15 mm has negligible effect on the bridge dynamic response.
However, significant increment in the range of 13.2% to 36.2% with high fluctuation has been observed in the bridge response as settlement increases from 25 mm to 40 mm. It is apprehended that high impact effect due to sudden change of the road profile of significant amount might have triggered higher vibrational mode of bridge.
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(a) Mean (b) Standard deviation Fig.6.60. Bridge mid span deflection for different settlement
6.7.2.4 Effect of vehicle structural modes
The present vehicle model incorporates rigid as well as elastic mode in both bending and torsion. In order to investigate contribution of vehicle body structural mode on bridge response, only rigid mode has been considered first and then elastic modes are included in the analysis. The contribution of significant number of structural mode of vehicle on bridge response has been examined in a bar diagram presented in Fig 6.61 by comparing maximum mean and standard deviation of mid span deflection. Four cases have been considered. These are (A) Only rigid modes (heave, pitch and roll) of vehicle body (B) Three rigid modes along with first elastic mode in both bending and torsion (C) Three rigid modes along with first three structural modes in bending and torsion (D) Three rigid modes along with first five structural modes in bending and torsion. Bridge response magnitude has been found to be higher when only rigid body motion of the vehicle is considered in the analysis. The mid span mean displacement has been found to be 35% less when flexible vehicle loading has been considered on the bridge. From the same figure it is also seen that when flexible modes of the vehicle are considered, summation of first five modes for finding the response is adequate.
(a) Mean (b) Standard deviation
Fig.6.61. Maximum displacement of bridge at mid span (A) Only rigid modes of vehicle body, (B) Three rigid modes along with first elastic mode in both bending and torsion (C) Three rigid modes along with first three structural modes in bending and torsion (D) Three rigid modes along with first five structural modes bending and torsion