A Simply-Supported Curved Box-Girder Bridge

FINITE ELEMENT FORMULATION OF THIN-WALLED CURVED BOX-GIRDER BRIDGE

2.5 NUMERICAL EXAMPLES

2.5.2 A Simply-Supported Curved Box-Girder Bridge

varying radius (38.10 m –^∞ ). It is observed that the response parameters corresponding to torsion have increased appreciably with the reduction in radius of curvature. This will thus necessitate increased torsional stiffness requirement as the radius of curvature of the bridge is decreased. While the flexural components have been observed to be almost unchanged, the distortion components reduce slightly with the reduction in the radius of curvature. Thus, the design of bridge remains unaffected except the torsional rigidity, which needs to be augmented as the radius of curvature of the bridge, is decreased.

In order to compare the evaluated values of different response parameters, the thin-walled box-girder bridge has been analyzed using shell elements in the general purpose finite element software ANSYS. The finite element model of the straight box-girder bridge is shown in Fig.2.32. Table 2.3 shows the maximum magnitude of different response parameters as obtained from the analysis using shell elements and have been found to match very well with the results of the straight bridge as shown in the last column of Table 2.1 and 2.2.

Further, all the response parameters have been evaluated for IRC loading (IRC:6,2000) of Class 70R and of Class A loading (Appendix C) and the values of response parameters corresponding to the maximum magnitude for a particular location of the vehicle have been shown in Table 2.4 and 2.5. One lane of Class 70 R loading has been considered, while two lanes of Class A loading is considered for the analysis as the width of the carriageway is more than 5.3 m for the example under consideration. The vehicle loads have been placed symmetrically as well as asymmetrically maintaining a minimum distance from the kerb and also a minimum distance between two rows of vehicle for Class A loading (Table.C.1.2).It is observed that the maximum magnitude of vertical deflection, rotation about z direction and corresponding stress resultant (viz. shear force and bending moment) are high in IRC Class A train of vehicles compared to IRC 70R Tracked as well as Wheeled Vehicle loading class.

The torsional and distortional displacement components and their corresponding stress resultants vary differently for different loading cases as the distance between wheels and their locations significantly influence the values of those response parameters.

E = 2.69 × 10¹⁰ N/m² ν = 0.15

G = 8.995 × 10⁹ N/m²

Iz = 9.41× 10^-1 m⁴ J = 2.01m⁴ JI = 1.87 × 10^-1 m⁶ JII = 4.45 × 10^-1 m⁶ Jd = 2.32 × 10^-2 m²

μt = 0.236 A = 2.4305 m²

0.9144m 0.2032m

3.048m 0.9144m

1.3716m 4.45e3 N

0.2032m

Fig.2.29. Cross-section and loading

Fig 2.30 (a) Vertical deflection (v)

-1.60E-04 -1.20E-04 -8.00E-05 -4.00E-05 0.00E+00

0 5 10 15 20 25 30 35

Span (m)

Vertical deflection (m)

0.609 m 0.609 m

-1.50E-05 -1.00E-05 -5.00E-06 0.00E+00 5.00E-06 1.00E-05 1.50E-05

0 5 10 15 20 25 30 35

Span (m)

Rotation about z (radian)

Fig 2.30 (b) Rotation about x (θx)

Fig 2.30 (c) Rotation about z (θ_z)

Fig 2.30 (d) Rate of rotation about x (_θ_x′)

0.00E+00 2.00E-06 4.00E-06 6.00E-06

0 5 10 15 20 25 30 35

Span (m)

Rotation about x (radian)

-6.00E-07 -3.00E-07 0.00E+00 3.00E-07 6.00E-07

0 5 10 15 20 25 30 35

Span (m)

Rate of rotation about x (radian\m)

Fig 2.30 (e) Distortional angle (γ_d)

Fig 2.30 (f) Rate of distortional angle (γ_d′)

-3.00E+03 -2.00E+03 -1.00E+03 0.00E+00 1.00E+03 2.00E+03 3.00E+03

0 5 10 15 20 25 30 35

Span (m)

Shear force (N)

-3.00E-06 0.00E+00 3.00E-06 6.00E-06

0 5 10 15 20 25 30 35

Span (m)

Distorsional angle (radian)

-1.40E-06 -7.00E-07 0.00E+00 7.00E-07 1.40E-06

0 5 10 15 20 25 30 35

Span (m)

Rate of distorsional angle (radian/m)

Fig 2.30 (g) Shear force (Q_Y)

-4.00E+04 -3.00E+04 -2.00E+04 -1.00E+04 0.00E+00

0 5 10 15 20 25 30 35

Span (m)

Bending moment (N-m)

Fig 2.30 (i) Torsional moment ( )Mx -1.20E+04

-8.00E+03 -4.00E+03 0.00E+00 4.00E+03 8.00E+03 1.20E+04

0 5 10 15 20 25 30 35

Span (m)

Torsional moment (N-m)

-4.00E+02 0.00E+00 4.00E+02 8.00E+02 1.20E+03 1.60E+03

0 5 10 15 20 25 30 35

Span (m) Torsioanl bi moment ( N-m2 )

Fig 2.30 (h) Bending moment (M_Z)

-3.00E+03 -2.00E+03 -1.00E+03 0.00E+00 1.00E+03 2.00E+03 3.00E+03

0 5 10 15 20 25 30 35

Span (m)

Distorsional moment (N-m)

-2.00E+03 0.00E+00 2.00E+03 4.00E+03 6.00E+03 8.00E+03

0 5 10 15 20 25 30 35

Span (m) Distorsional Bi moment (N-M2)

Fig 2.30 (k) Distortional moment ( )MD

Fig 2.30 (l) Distortional bi moment ( )B_II

12000

7990

7990 12000

20010.015

6 D.0.f (with out shear lag)

15589.3532 20010.015 27266.9

24432.4954 31097.685

33096.4328 17709.0904

17709.0904

55801.298

44447.222 33096.4328

31097.685 7 24097.548

24097.548

6 D.0.f (with shear lag) 9 D.o.f

27266.9

54533.8 54533.8

54533.8

24097.548

44447.222 44447.222

44447.222

Fig 2.31 (a) Longitudinal normal stresses at mid-span (N/m²)

55801.298

Fig 2.31 (b) Transverse distortional bending stress at mid-span (N/m²)

Radius of box-girder

( )

R m Response

parameter

38.10 76.2 152.4 304.8 ∝

v

(m) -1.46e-04 -1.40e-04 -1.38e-04 -1.37e-04 -1.36e-04 θz(radian) 1.36e-05 1.33e-05 1.33e-05 1.33e-05 1.33e-05 θx(radian) 6.03e-06 4.81e-06 4.26e-06 4.01e-06 3.75e-06

θx(radian/m) 6.00e-07 4.43e-07 3.54e-07 3.09e-07 2.63e-07 γd(radian) 2.08e-06 5.59e-06 7.40e-06 8.28e-06 8.63e-06

γd(radian/m) 7.68e-07 9.21e-07 1.01e-06 1.05e-06 1.07e-06

Radius of box-girder

( )

R m

Response parameter

38.10 76.2 152.4 304.8 ∝

QY (N) 2.22e3 2.22e3 2.22e3 2.22e3 2.22e3

Mz(N-m) -3.39e4 -3.39e4 -3.39e4 -3.39e4 -3.39e4

Mx(N-m) -1.18e4 -8.18e3 -6.44e3 -5.50e3 -4.75e3

BI(N-m²) 1.090e3 1.085e3 1.083e3 1.080e3 1.078e3

MD(N-m) 2.28e3 2.33e3 2.35e3 2.36e3 2.37e3

BII(N-m²) 5.21e3 5.69e3 5.95e3 6.07e3 6.12e3

Table 2.2 Effect of radius of box-girder bridge on stress resultants

Table 2.1 Effect of radius of curvature of box-girder bridge on responses

Fig. 2.32 Finite element model of box-girder bridge using shell element

Table 2.3 Different response parameters corresponding to a straight box-girder bridge modeled using shell elements

v

(m) θ_z(radian) θ_x(radian) Q_Y(N) M_z(N-m) M_x(N-m) Response

parameter -1.37e-4 1.33e-5 3.87e-6 2.22e3 -3.39e4 -4.72e3

Table 2.4 Displacement response due toIRC Loadings in two lane box-girder bridge IRC Loadings

IRC Loading

Case

IRC 70R Tracked Vehicle IRC 70R Wheeled Vehicle IRC A Train of Vehicle Response

parameter

Symmetrical Loading

Non- Symmetrical

Loading

Symmetrical Loading

Non- Symmetrical

Loading

Symmetrical Loading

Non- Symmetrical

Loading v(m) -2.14e-02 -2.14e-02 -2.77e-02 -2.77e-02 -2.79e-02 -2.79e-02 θz(radian) 2.09e-03 2.09e-03 2.75e-03 2.75e-03 2.81e-03 2.81e-03

θx(radian) ** 1.86e-04 ** 3.21e-04 ** 2.72e-04

θx(radian/m) ** 3.10e-05 ** 4.86e-05 ** 4.88e-05

γd(radian) ** -4.59e-04 ** -3.37e-04 ** -5.10e-04

γd(radian/m) ** 4.70e-05 ** 5.10e-05 ** 5.27e-05

** Not applicable due to symmetric nature of the load

IRC Loadings IRC

Loading Case

IRC 70R Tracked Vehicle IRC 70R Wheeled Vehicle IRC A Train of Vehicle Response

parameter

Symmetrical Loading

Non- Symmetrical

Loading

Symmetrical Loading

Non- Symmetrical

Loading

Symmetrical Loading

Non- Symmetrical

Loading QY(N) -3.43e05 -3.43e05 -4.91e05 -4.91e05 -5.389e05 -5.389e05

Mz(N-m) -5.17e06 -5.17e06 -5.90e06 -5.90e06 -5.93e06 -5.93e06

Mx(N-m) ** 5.72e05 ** 8.97e05 ** 9.02e05

BI(N-m²) ** 5.56e03 ** 1.05e04 ** 8.81e03

MD(N-m) ** -2.73e04 ** -3.33e04 ** -3.91e03

BII(N-m²) ** -5.33e04 ** -6.48e04 ** -6.33e04

** Not applicable due to symmetric nature of the load

Table 2.5 Stress resultant due to IRC Loadings in two lane box-girder bridge

2.5.3 A Two-Span Double-Cell Box- Girder Subjected to Eccentric Load

A two span continuous double-celled straight box-girder model subjected to two-eccentric point loads has been analyzed. The span of each box-girder is 3.5 m. The longitudinal elevation and cross-section are shown in Fig. 2.33 (a) and (b). Diaphragms are located at the supports preventing in-plane distortion of the cross-section. Sixteen thin-walled box-girder elements were used for the present analysis. Fig.2.34 (a), (b), (c) and (d) show the distribution of bending moment, torsional bi moment distortional moment and distortional bi-moment along the span. Fig.2.35 (a) and (b) show the distribution of longitudinal stresses and transverse bending moments across the cross-section at mid-span. The results have been observed to match with Zhang and Lyons (1984).

E = 2.90 × 10¹⁰ N/m² ν = 0.18

G = 1.229 × 10¹⁰ N/m²

Iz = 5.0309 × 10^-4 m⁴ J = 1.314 × 10^-3 m⁴ JI = 6.6403 × 10^-6 m⁶ JII = 5.3408 × 10^-6 m⁶ Jd = 0.116 × 10^-3 m²

μt = 0.311 A = 0.05025 m²

3500 3500 200

100 100

1500 1500 500

Diaphragm

Fig. 2.33 (b) Cross-section (mm) and member properties

25 25

30 30

690

250

Fig. 2.33 (a) Longitudinal elevation (mm)

-1.40E+04 -7.00E+03 0.00E+00 7.00E+03 1.40E+04

0 1 2 3 4 5 6 7

Span (m)

Bending moment (N-m)

-1.20E+02 -8.00E+01 -4.00E+01 0.00E+00 4.00E+01 8.00E+01

0 1 2 3 4 5 6 7

Span (m) Torsional bimoment ( N-m2 )

-8.00E+02 -4.00E+02 0.00E+00 4.00E+02 8.00E+02

0 1 2 3 4 5 6 7

Span (m)

Distorsional moment (N-m)

Fig 2.34 (a) Bending moment (M_Z)

Fig 2.34 (b) Torsional bi-moment ( )BI

Fig 2.34 (c) Distortional moment ( )M_D

-1.25E+02 0.00E+00 1.25E+02 2.50E+02

0 1 2 3 4 5 6 7

Span (m) Distorsional bimoment ( N-m2 )

Fig. 2.35 (a) longitudinal normal stresses at mid-span section (N/m²)

0.0 3519

0.02639

0.01759 0.0 2639

0.03519 0.01759

1.95e-5 1.874e-5 7.524e-4

1.95e-5 5.223e-4

5.20e-4 7.53e-4

1.873e-5

2..3 e-5

2.39e-5

Fig.2.35 (b) Transverse distortional bending stress at mid-span section (N/m²) Fig.2.34 (d) Distortional bi-moment ( )B_II

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