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The structural behavior of T-frame bridges is particularly complicated and it is difficult using a general analytical method to directly acquire the internal forces in the structure. This paper presents a spatial grillage model for analysis of such bridges. The proposed model is validated by comparison with results obtained from field testing. It is shown that analysis of T-frame bridges may be conveniently performed using the spatial grillage model.

Rigid frame bridges are appearing in various exotic forms resulting in complex, efficient, and aesthetically pleasing structures, with graceful appearance, compact construction dimension, spacious room under bridge, and a broad eye view, and the plan can be applied to the construction of long span bridges. Because of T-frame bridge special advantages in the rigid frame bridge, more and more the T-frame bridges were used. The safety of the T-frame bridges presents an increasing important concern in design, construction, and service. This special type of flexible, large span T-frame bridges makes the structural analysis more complex and difficult [

Typically, the design of highway bridges in China must conform to the General Code for Design of Highway Bridges and Culverts (JTG D60-2004) specifications. The analysis and design of any highway bridge must consider truck and lane loadings. However, the structural behavior of T-frame bridge is particularly complicated, and many rigorous methods for analysis of T-frame bridges are quite tedious and often difficult.

Pan et al. [

Mabsout et al. [

In a word, the above mentioned methods can not exactly deal with the structural mechanical behavior of the T-frame bridges what’s worse, they may lead to insecurity of structural design. Grillage analysis is probably the most popular computer-aided method for analyzing bridge decks [

This paper presents a spatial grillage model for the analysis of a T-frame bridge. Static and dynamic analysis results of spatial grillage model for the T-frame bridge are compared with results based on results obtained from field testing. The research results shown that analysis of T-frame bridges may be conveniently performed using the spatial grillage model.

Spatial grillage model is a convenient method for analysis of box-girder bridges. In the model, the box-girder slab is represented by an equivalent grid of beams whose longitudinal and transverse stiffnesses are approximately the same as the local plate stiffnesses of the box-girder slab.

In the spatial grillage model analysis, the orientation of the longitudinal members should be always parallel to the free edges while the orientation of transverse members can be either parallel to the supports or orthogonal to the longitudinal beams. According to the grillage model, the output internal force resultants can be used directly. The grillage model involves a plane grillage of discrete interconnected beams. The representation of a bridge as a grillage is ideally suited for carrying out the necessary calculations associated with analysis and design on a digital computer and it gives the designer an idea about the structural behavior of the bridge.

The main challenge in the spatial grillage model is how to obtain an equivalent grillage based on the box-girder deck structure. The spatial grillage model for analyzing box-girder Slab bridge includes figuring out the grillage mesh and grillage member section properties.

Determination of a suitable grillage mesh for a box-girder of rigid frame bridge is, as for a slab deck, best approached from a consideration of the structural behavior of the particular deck rather than from the application of a set of rules. Since the average longitudinal and transverse bending stiffness are comparable, the distribution of load is somewhat similar to that of a torsionally flexible slab, but with forces locally concentrated. The grillage simulates the prototype closely by having its members coincident with the centre lines of the prototypes beams. In addition, there is a diaphragm in the prototype such as over a support, and then a grillage member should be coincident. Based on section shape of the rigid frame bridge and support arrangements, a spatial grillage mesh should be represented by the above mentioned spatial grillage method. At the same time, according to the grillage equivalent theory, the following three important aspects have to be noted: according to mechanical behavior of a rigid frame bridge, one should place the grillage beams along the lines of designed strength; the longitudinal and transverse member spacing should be reasonably similar to permit sensible static distribution of loads; in addition, virtual longitudinal and transverse members are often employed for the sake of convenience in the analysis. The virtual members only offer stiffness, but its weight must be ignored.

Grillage member section properties include longitudinal grillage member section properties and transverse grillage member section properties. Based on rigid frame bridge structural features, the amount of each deck for prototype bridge is represented by the appropriate grillage member. The flexural inertia of each grillage member is calculated about the centroid of the section it represents. The section properties of a transverse grillage member, which solely represents slab, are calculated as for slab. For this

The Quhai rigid frame bridge is over the Qu river in Dongguang, Guangdong province, China. A photo of the bridge just before its opening is shown in Figure

A photo of Quhai T-frame bridge.

General view of Quhai T-frame bridge: (a) plan view; (b) elevation view; (c) typical cross-section of box-girder deck.

The Quhai Bridge is a rigid frame design for a single box with double chamber bridge, with two main palaces with hang holes, T-frame of 2

Three-dimensional linear elastic finite element models of the spatial grillage model of the Quhai Bridge have been constructed using SAP2000 finite element analysis software. In the finite-element model, 3D beam4 elements were adopted to create the grillage model that will be used to determine the internal stress resultants, natural frequencies, and corresponding mode shapes. The spatial grillage model was shown in Figure

Equivalent grillage model.

Grillage model of main bridge for Quhai Bridge.

According to the influence line method of the control section, internal forces of longitudinal and transverse grillage members were presented. Internal force values of the control section for the grillage model were shown in Tables ^{6} ^{6} ^{7}

Moment under design live load (control sections of 1# grillage/N·m).

Section location | Trailer 120 max/min | Truck 20 max/min | Control moment |
---|---|---|---|

Pier (14#) | 0/0 | 0/0 | 0/0 |

0.5 L | 3.9 |
2.9 |
3.9 |

Corbel | 0/0 | 0/0 | 0/0 |

Pier (15#) | 0/−3.5 |
0/−4.1 |
0/−4.1 |

Corbel | 0/−5.6 |
0/−1.1 |
0/−1.1 |

0.5 L | 4.2 |
3.2 |
4.3 |

Corbel | 0/0 | 0/0 | 0/0 |

Pier (16#) | 0/−3.5 |
0/−4.1 |
0/−4.1 |

Shear force under design live load (control sections of 1# grillage/N).

Section location | Trailer 120 max/min | Truck 20 max/min | Control moment |
---|---|---|---|

Pier (14#) | 5.3 |
2.9 |
9.1 |

0.5 L | 2.7 |
1.8 |
5.7 |

Corbel | 9.1 |
7.9 |
9.1 |

Corbel | 8.6 |
7.4 |
8.6 |

Pier (15#) | 1.1 |
1.7 |
1.7 |

Pier (15#) | 0/−1.1 |
0/−1.7 |
1.7 |

Corbel | 0/−8.6 |
0/−7.5 |
8.6 |

Corbel | 6.2 |
3.8 |
9.7 |

0.5 L | 3.1 |
2.1 |
5.5 |

Corbel | 9.1 |
7.9 |
9.1 |

Corbel | 8.6 |
7.5 |
8.6 |

Pier (16#) | 1.1 |
1.7 |
1.7 |

Pier (16#) | 0/−1.1 |
0/−1.7 |
1.7 |

Envelope of internal force of grillage model for side hang beam (1# grillage).

Envelope of bending moment for side hang beam (1# grillage axis)

Envelope of shear force for side hang beam (1# grillage axis)

Envelope of internal force of grillage model for T-frame (1# grillage).

Envelope of bending moment for T-frame (1# grillage axis)

Envelope of shear force for T-frame (1# grillage axis)

Envelope of internal force of grillage model for middle hang beam (1# grillage).

Envelope of bending moment for middle hang beam (1# grillage axis)

Envelope of shear force for middle hang beam (1# grillage axis)

From the dynamic analysis using the spatial grillage model, the first natural frequency of the hang beam (14-15#) and T-frame is 3.6 Hz and 1.31 Hz, respectively, and vibration mode is symmetric vertical bending; the first natural frequency of the hang beam (16-17#) and T-frame is 3.6 Hz and 1.31 Hz, respectively, and vibration mode is symmetric vertical bending; the first natural frequency of the middle-span hang beam is 3.6 Hz, and vibration mode is symmetric vertical bending. The first mode shapes of the bridge are shown in Figure

the first vibration mode of grillage model for hang beam and T-frame.

The first vibration mode of T-frame

The first vibration mode of hang beam (14-15# and 16-17#)

The first vibration mode of hang beam (15-16#)

The field static or dynamic tests on bridges have been of great interest not only in investigating bridge fundamental behavior but also for calibrating finite-element models. Several results of field tests and correlated finite-element analyses have been presented for the Quhai Bridge. The field load tests on the Quhai Bridge employed the corresponding design load, to simulate the design live loads of the bridge. The field load tests on the Quhai Bridge employed the heavily loaded dump trucks, each with approximately 30t weight, to simulate the design live loads of the bridge. Due to the difficulty to hire the same type of such heavy dump trucks in the area, there were, in total, 5 trucks employed during the static load tests. The individual axle load and spacing of each dump truck were carefully measured at the nearby weight station before it was moved to the bridge.

In addition, the applied test loads should be identical to the design live loads of the bridge. The applied test loads are normally designated by the static test load efficiency:

Photos of three critical load cases: (a) field loading on the main span (16-17#); (b) field loading on the main span (14-15#); (c) field loading on the main span (15-16#).

Deflection values of the prototype bridge for the corresponding grillage under working condition I and working condition II are shown in Figure

experimental deformations under working condition I and working condition II (14-15#).

Deflection curve of from loading position I to III

Deflection curve of from loading position IV to unloading

Experimental deformations under working condition III (15-16#).

According to the comparative analysis of the strains between the field test and theoretical analysis, the ratio

Dynamic properties can be obtained by measurement of vibrations produced by ambient loads and vehicle bump. The experimental program includes dynamic characterization of the structure in normal conditions and when a half of the bridge is covered by traffic. The response of the structure was measured at 7 selected points using accelerometers. Preliminary results obtained from an FE dynamic analysis were used to determine the optimum location of the sensors. The first mode shapes of the bridge according to field dynamic test and theoretical value are presented in Table

Experimental and theoretical base frequency (Hz).

Experiment location | Experiment | Theory | Vibration mode |
---|---|---|---|

15# T-frame | 2.24 | 1.31 | Symmetric vertical bending |

Middle-span hang beam | 2.70 | 3.60 | Symmetric vertical bending |

16# T-frame | 2.22 | 1.30 | Symmetric vertical bending |

The static and dynamic behaviors of a rigid T-frame bridge were investigated analytically and experimentally. Based on the comparison study on analysis results obtained from the conventional and proposed analysis methods, one may obtain more economical designs using the spatial grillage model. Main contents of the grillage model include the grillage mesh and the grillage member section properties. The precision of the grillage model mainly lies on the simulation of the equivalent grillage stiffness and the element property. According to the comparative analysis, the bridge possesses a relatively small stiffness to resist the deformation. As a result, field test results have indicated that the bridge works in the elastic stage, but the bridge has a relatively smaller load-carrying capacity under the design load conditions. Therefore, some suggests of reinforcement or maintenance for the bridge were presented to increase bearing capacity, and prestressed outside were employed to solve deflection of beam, relatively smaller effective prestress and web shear crack. In addition, bearings of right and left hang beam were replaced to recover mechanical behavior of the original design.

The authors wish to thank the referees and Sheng-yong Chen for some helpful comments. Moreover, they are grateful for the financial support provided by the Science Foundation of China Postdoctor (Grant no. 20110490183), the Science Foundation of Ministry of Housing and Urban-Rural Development of the People’s Republic of China (Grant no. 2012-K2-6), the Education Department Science Foundation of Zhejiang (Grant no. Y201122051), the Science Foundation of Zhejiang University of Technology (Grant no. 2011XY022).