# Fatigue Performance Test and Numerical Analysis of Composite Girders with CSW-CFST Truss Chords
## 4 Non-Linear Numerical Analysis  
### Non-Linear Numerical Analysis Model  
A three-linear model was adopted for the tensile and compressive constitutive relationship of steel, and the typical damage plastic model considering material non-linearity was adopted for the constitutive relationship of concrete. The relevant parameters are listed in Tables 2 and 3, respectively. A finite element model of the test specimen was constructed using MSC.MARC. The concrete bridge deck slab, the concrete in the chord, the chord, the corrugated steel webs, the cross-bracing, the K-bracing and the horizontal-bracing were all simulated with solid elements, and the rebar in the concrete bridge deck slab was simulated with truss elements. Relative slippage between the rebar and the concrete bridge deck slab and between the concrete bridge deck slab and the corrugated steel webs were ignored, and embedded and tie options were used to simulate the contact between the rebar and the concrete bridge deck slab and that between the concrete bridge deck slab and the corrugated steel web, respectively. The interfaces between the concrete and steel were assumed to be perfectly connected without relative slip except for the interface between the chord and concrete. It was necessary to ensure a one-to-one correspondence of finite element nodes on the outer surface of concrete in the chord to those on the inner surface of the chord. These were forbidden from sharing the same nodes during the process of meshing of concrete-filling the chord to simulate the non-linear contact between the inner wall of the steel tube and the outer surface of the concrete in the chord, and to prevent occurrence of spurious sudden stress variations produced by numerical effects rather than by actual loads. The mesh of the numerical model is shown in Figure 16.  
Figure 16: Mesh of numerical model: (**a**) entire model; and (**b**) concrete in chord. Figure 15: Fatigue failure mode. The simulation of contact between the chord and concrete-filling the chord must ensure continuity of deformation and load transfer. Two main methods for non-linear simulation of contact between the chord and concrete were tried, namely, glue and touch [26]. The glue method considers that the value of the relative slippage among the components that contacted each other is zero. The touch method considers the actual relative slippage among the components that will contact each other. Comparison of results obtained with different non-linear contact simulation methods and the experimental results is shown in Figure 17. The hot-spot stress obtained with the glue contact method was markedly less than the test results. At the location with the maximum hot-spot stress, the deviation was 35.5%, indicating that the glue contact method overestimated the constraining effect of concrete in the chord, resulting in unconservative estimations. The hot-spot stress obtained by the touch contact method was in good agreement with the test results. At the position with the maximum hot-spot stress, the deviation was only 5.7%. Therefore, it is advisable to use the touch contact method to simulate the non-linear contact between the chord and concrete in the chord, and the results obtained may have a certain safety margin. The boundary conditions of the non-linear numerical model simulated those of the fatigue performance test. The degrees of freedom of one end of the finite element model in the X and Z direction were constrainted, and that of the other end in the Z direction were constrainted. The degrees of freedom of the finite element model in the Y direction were not constrained--the finite element model could rotate freely around the X, Y and Z directions. Vertical loads symmetrical about the mid-span section were applied to the concrete deck slab. The boundary conditions of the non-linear numerical analysis model are shown in Figure 3.  
Calculation accuracy is affected by the size of the mesh; thus, mesh convergence should be checked.