# Bending Performance of a Prestressed Concrete Composite Girder Bridge with Steel Truss Webs
## 3 Nonlinear Finite Element Model
### Material Constitutive Relationship  
The tension and compression constitutive relations of the steel truss webs and longitudinal steel reinforcement were based on a three-fold model in Figure 6a, with the yield strengths and elastic moduli determined from material property tests; the yield strength of the steel truss webs \(\sigma_{y}\) is 375.6 MPa, the elastic modulus \(E_{\mathrm{s}}\) is 2.08 \(\times\) 10\({}^{5}\) MPa, the Poisson ratio \(\mu_{\mathrm{s}}\) is 0.3, and the yield strength of the longitudinal steel reinforcement \(\sigma_{y}\) is 369.8 MPa. The elastic modulus \(E_{\mathrm{s}}\) = 2.04 \(\times\) 10\({}^{5}\) MPa, the Poisson's ratio \(\mu_{\mathrm{s}}\) = 0.3, and the strain limit \(\varepsilon_{\mathrm{u}}\) = 0.01. In Figure 6a, \(\varepsilon_{y}\) is the steel yield strain and \(E_{\mathrm{s}}^{*}\) = 0.3\(E_{\mathrm{s}}\) is the secant modulus of the steel strengthening stage [23]. The constitutive relationship of prestressed reinforcement is based on the double-broken line model in Figure 6b. The standard tensile strength \(\sigma_{\mathrm{pk}}\) is 1860 MPa and the elastic modulus \(E_{\mathrm{s}}\) is 1.95 \(\times\) 105 MPa. [PERSON]'s ratio \(\mu\) is 0.3, and the strain limit \(\varepsilon_{\mathrm{u}}\) is 0.015.  
The compression and tension constitutive relationship of concrete is shown in Figure 6c. The Hongnestad model of reference [24] is adopted for the stress-strain relationship under uniaxial compression. The ascending stage is a parabola and the descending stage is an oblique straight line. The uniaxial tensile strength of concrete \(\sigma_{\mathrm{t}}\) is 0.26\(\sigma_{\mathrm{c}}\)2\({}^{/3}\)[25].  
The values of relevant parameters are determined according to the results of material property tests. The cube compressive strength of the bottom plate \(\sigma_{\mathrm{c}}\) is 59.4 MPa, and the elastic modulus \(E_{\mathrm{c}}\) is 3.19 \(\times\) 10\({}^{4}\) MPa, while they are 56.0 MPa and 3.23 \(\times\) 10\({}^{4}\) MPa, respectively, for the top plate. The Poisson ratio of concrete \(\mu\) is 0.2, and other parameters are specified in the reference [26], in which \(\sigma_{0}\) is 0.68\(\sigma\)c, \(\varepsilon_{0}\) is 0.002, and \(\varepsilon_{\mathrm{u}}\) is 0.003. In ABAQUS software, the concrete damaged plasticity model was adopted to simulate the tensile fracture failure of concrete [27]. For the concrete in the tension area, when the fracture energy of the concrete reaches the limit value, the concrete cracks and goes out of work. Using the concept of brittle fracture, the energy \(G_{\mathrm{f}}^{\mathrm{I}}\) required per unit opening area of the crack is taken as the material parameter, and the material strength is assumed to be changed linearly to zero after cracking. The value of \(G_{\mathrm{f}}^{\mathrm{I}}\) is between 0.04 N/mm and 0.12 N/mm, and for concrete with a compressive strength of approximately 20 MPa and 50 MPa, the lower limit and upper limit values are taken, respectively [28].  
Figure 6: Material constitutive relationships: (**a**) Steel; (**b**) Prestressed steel strand; (**c**) Concrete.