This paper introduces the design of simply supported PC track beam in the second phase of Chongqing Light Rail Jiaoxin Line, compares it with the first phase, and discusses the design idea of track beam.
1 overview
Track beam has dual functions of bearing vehicle load and forming train running track. The track beam structure is mainly simply supported PC track beam. The design service life of track beam is 100a.
The second phase of Chongqing Light Rail Jiaoxin Line has a total length of 4.708km, with a minimum curve radius of 100m and a maximum superelevation of 12%. In the design of track beam in the second phase project, according to the test of 25m straight track beam and considering the existing formwork of beam factory, the standard span of track beam in the straight section is set at 24 m, and all track beams are 0.85m wide and 1.5m high, which is consistent with the first phase project. Simply supported PC track beam bearings are all cast steel bearings.
2. Establishment of calculation model
The spatial stress characteristics of track beam are obvious, mainly in the following two aspects: (1) minimum curve radius 100m, superelevation12%; (2) The clearance requires that the transverse width of the track beam is only 0.85m m. Due to the complexity of the monorail beam, a special program RTV is compiled for calculation and analysis. The program adopts three-dimensional straight beam element. There are the following difficulties in establishing track beam model: (1) track beam support simulation; (2) Simulation of prestress effect of track beam; (3) Simulation of construction process; (4) Running load simulation.
2. 1 simulation of track beam support
For curved beams, the supporting direction of the structural boundary will not appear parallel to the overall coordinate axis direction of the structure. In order to describe the constraint conditions of oblique bearing nodes, the node coordinate system can be established, so as to introduce the constraint conditions along the supporting direction of oblique bearing.
(1) Establish the node coordinate system.
△i=L△*i，Pi=LP*i
Where: L is the transformation matrix from node coordinate system to global coordinate system;
△I and P*i are the displacement vectors of nodes in the node coordinate system;
△i and Pi are the displacement vectors of nodes in the global coordinate system;
φ is the included angle between the nodal coordinate system and the global coordinate system.
(2) modify the global stiffness equation
The element stiffness matrix is modified before it is merged into the total stiffness, and the element stiffness matrix in the overall coordinate system is modified to the element stiffness matrix in the node coordinate system in the diagonal support direction. The method of modifying the element stiffness matrix is as follows:
Where: ke is the element stiffness matrix in the global coordinate system;
K*e is the element stiffness matrix in the nodal coordinate system.
Similarly, before the unit node force vector is incorporated into the total node force vector, the unit node force vector is modified according to the relationship between the node coordinate system and the total coordinate system.
(3) Solving the internal force of the element
Firstly, by solving the global stiffness equation, the node displacement in the obliquely supported node coordinate system is obtained. Under the condition of obliquely supported nodes, the node displacement is multiplied by the element stiffness matrix to obtain the element internal force in the obliquely supported node coordinate system. Then, the internal force in the global coordinate system can be obtained by the relationship between the coordinate system of the diagonal bracing node and the global coordinate system. Finally, according to the relationship between the global coordinate system and the local coordinate system, the internal force of the unit in the local coordinate system can be obtained.
2.2 Simulation of prestress effect of track beam
Prestressed concrete structure is a system in which prestress and concrete interact to achieve internal force balance. In order to analyze the interaction between them, prestressed tendons and concrete can be regarded as independent separated bodies, and the equivalent load of prestressed tendons on the structure can be obtained by analyzing the internal force balance of prestressed tendons separated bodies. The equivalent load of prestress on the structure includes node load and non-node load. The resulting structural internal force can be calculated by structural mechanics.
At present, there are many literatures about the prestress simulation of curved beam under curved beam element, but there are few literatures about the prestress simulation of straight beam element. This paper deduces this and applies RTV programming well.
2.3 Construction process simulation
See table 1 for the main construction steps of track beam.
The internal force in the final stage of the structure can be obtained by superimposing the internal force generated by the load or fulcrum movement in each construction stage.
According to the internal forces generated in each construction stage and the cross-sectional characteristics of the structure in this construction stage, the new cross-sectional stress in this stage can be calculated. The section stress of the structure in the final stage can be obtained by adding the section stress increments in each construction stage.
The elastic displacement of the structure in each construction stage is multiplied by the corresponding creep coefficient of each stage at the time of displacement occurrence, and the creep deformation of the structure in each construction stage corresponding to the elastic displacement can be obtained.
Where: φij is the creep deformation coefficient of the I-th construction stage corresponding to elastic deformation at the beginning of the J-th construction stage; XEj is the elastic deformation at the beginning of the I-th construction stage.
2.4 Running load simulation
There are two methods to calculate the internal force and displacement of the structure under working load:
(1) influence line method
Firstly, the influence line of internal force and displacement of the calculation point is obtained, and then the most unfavorable internal force or displacement of the point structure is calculated according to the most unfavorable loading position.
(2) Train operation simulation method
Simulate the actual operation of the train, record the change of internal force or displacement at a certain point of the structure with time under the load condition of each step from the first axle to the last axle, and then find out the most unfavorable internal force or displacement at that point.
The above two calculation methods have their own advantages and disadvantages. In the design of track beam in the second phase project, the above two calculation methods are selected to calculate and proofread the program respectively.
3 track beam calculation
The calculation of track beam adopts two sets of programs at the same time, one is self-compiled program RTV, and the other is SOFISTIK program (corresponding secondary development is carried out according to needs). Taking a track beam with a span of 20m (radius R= 100m) as an example, the calculation results of the two programs are introduced. Unless otherwise specified, the selected points are points around the span 1/4.
(1) Calculation of internal force under dead load in each stage (see Table 2)
(2) Displacement calculation in each stage (see Table 3)
(3) Calculation of live load internal force during operation period (see Table 4)
(4) Calculation of normal stress at corner under working load combination (see Table 5).
According to the above calculation, it can be seen that the calculation results of the two programs are consistent.
4 track beam design
4. 1 track beam classification
Compared with the first phase project, the classification of track beams has been reduced. According to the stress analysis, the internal force difference between R=4000m track beam and R=∞ track beam is very small, so the two kinds of track beams can be classified and merged. See Table 8 for details.
4.2 Prestressed steel bar design
(1) When the number of required steel strands exceeds 40, 15-5 steel strands shall be adopted, and the number of anchorage devices at the beam end shall be up to 5 rows. Although the anchorage ratio of 15-5 is slightly larger than that of 15-4, the collision probability of embedded parts such as ordinary steel bars and prestressed steel bars will generally decrease.
(2) Because the axle of the live load is greater than the dead load of the track beam, the stress of the upper and lower edges of the track beam may control the design under the most unfavorable working conditions. In order to make the upper and lower edges of beams have a certain compressive stress reserve, some types of beams should be moved up across the center of prestressed steel bars as needed to make them closer to the axial distribution of beams.
4.3 ordinary steel bar design
Compared with the first phase project, according to the calculation results of internal force and reinforcement, the configuration of ordinary reinforcement of track beam in the second phase project is optimized, as shown in Table 9.
The optimal design of ordinary steel bars is embodied in the following aspects:
(1) Cancel diagonal stirrups.
(2) On the premise of meeting the requirements of shear stirrups first, considering that the internal stirrups have less torsional effect on the track beam, except for the track beam with r = 100 m, the diameter of the internal hoop is changed from Ф16 to Ф12. ..
(3) Because the torque of R= 100m track beam is much greater than that of other track beams, and the normal stress at the corner is more unfavorable, the diameter of longitudinal reinforcement is increased from Ф16 in the first phase of the project to Ф18.
5 conclusion
This paper is some design experience of straddle-type monorail beam, aiming at communicating with design colleagues, so as to achieve the purpose of designing track beam economically and safely and better serve monorail traffic.
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