The modeling scope of the first level local model spans from the transverse diaphragm of Bd20 to the center line of pier No. 4, as depicted in Fig. 4. Given the numerous variable sections and guiding angles within the model, the beam body, transverse diaphragm, and cable anchor block were modeled independently for the sake of convenient mesh generation. These three components were then assembled together, ensuring they are in contact with each other without any overlap or gaps.

Although tetrahedral elements offer automatic division, they are inherently constant strain elements, thus resulting in lower calculation accuracy compared to hexahedral elements. To achieve more precise calculation outcomes, this model predominantly employs hexahedral meshes for the majority of regions, while reserving tetrahedral meshes solely for complex, variable cross-section areas that are not locally prioritized, as illustrated in Fig. 5. The basic unit size for division is approximately 30 cm in length. In the long side direction of geometric dimensions, the unit size is between 50 to 100 cm, which can reduce the number of units and facilitate structural calculations.

The boundary conditions have a direct and significant impact on the effectiveness of the structure. Ensuring accurate boundary condition definitions is crucial to maintaining consistency in the internal forces and deformations between the local model and the overall model. Given that this local model is a higher-order statically indeterminate structure with asymmetric elastic constraints at its boundaries, directly defining these conditions can be challenging. Consequently, we employ the force method to simplify the structure into a statically indeterminate system. This approach replaces the excessive constraints with generalized forces, resulting in the basic structure depicted in Fig. 6. Upon comparing the global model developed in Midas Civil with the local model created in ABAQUS, it is conclusively demonstrated that the definition of boundary conditions in both instances is accurate, as shown in Fig. 7.

Loads acting on the structure encompass various factors, including gravity, second-phase constant load, prestress, vehicle live load, and temperature variations. These loads have a significant impact on the behavior and performance of the structure, requiring careful consideration and analysis during the design and analysis process.

For linear prestressed bars, the application of pressure to the loading block is utilized to simulate the prestressing effect, as depicted in Fig. 8. However, for curve prestressed bars, a truss model is established specifically to apply the prestress during the cooling process. This approach accounts for the unique geometry and behavior of curved prestressed elements.

The beam section in question adopts a transverse cantilever design, fixed at its center, with four vehicles arranged on the outer edges. The loading position of the Bd22 transverse partition in the longitudinal direction is determined based on the influence line analysis, as illustrated in Figs. 9 and 10. This approach ensures that the loading is applied at the most critical locations, taking into account the specific structural configuration and loading conditions.

As depicted in Figs. 11 and 12, the distribution of the primary tensile stress and compressive stress within the transverse partition at the cable anchor is clearly illustrated. Specifically, the peak tensile stress, recorded at 11.7 MPa, is concentrated in the ① region of the N1 cable anchor’s transverse diaphragm Bd22. Conversely, the maximum compressive stress, amounting to 13.4 MPa, is significantly lower. Fig. 13 provides a visual representation of the N1 transverse diaphragm, highlighting the location of these stress concentrations.

The second-level local model focuses on the Bd22 transverse diaphragm as its center and encompasses a 7 m beam section, as shown in Fig. 14. In this model, the concrete material properties are specifically optimized to incorporate the plastic damage (CDP) model, while the remaining aspects remain in alignment with the first-level local model. The boundary conditions for this model are determined based on the calculations performed in the first-level local model, allowing for a reduction in computational cost while ensuring accurate and reliable results.

Before performing the calculation, the material properties of the second-level local model are initially set to be consistent with the first local model to verify the accuracy and correctness of the newly established local model. By comparing the main tensile stress figures of the two transverse diaphragm in both models, as indicated in Fig. 15, it is determined that the error falls within an acceptable range. This comparison serves as validation that the second-level local model is reliable and can provide accurate results.

The CDP model is an effective tool for simulating the stress-strain behavior of brittle materials, such as concrete, under a wide range of loading conditions, including monotonic, reciprocating, and dynamic loads, especially under moderate confining pressures. This model is a continuum damage calculation approach based on plasticity theory.

Within the CDP model, the tensile and compressive behavior of the material can be independently defined and tailored to specific conditions. This flexibility allows for a more accurate representation of the material’s response to different types of loads. Additionally, the model accounts for stiffness degradation due to damage accumulation, as well as stiffness recovery under reciprocating loading conditions. These features make the CDP model a valuable tool for analyzing the performance of concrete structures under complex loading scenarios.

The CDP model encompasses several crucial plastic parameters such as the flow potential offset value, expansion angle, the ratio of biaxial to uniaxial ultimate compressive strength, the second stress invariant ratio on the meridional plane pertaining to tension and compression, and the viscosity coefficient. Within this model, the flow motive force function is represented by a Drucker-Prager hyperbolic function, as delineated in Eq. (1).(1) G=(fc−m⋅ft⋅tanψ)2+q¯2−p¯⋅tanψ−σ where, the uniaxial compressive strength of concrete is fc , the uniaxial tensile strength of concrete is ft ; p¯=−13σ¯⋅I is the effective static pressure; Mises equivalent effective stress is q¯ , q¯=32S¯2 ; expansion angle is ψ and flow potential offset value is m according to p-q plane values; the partial of the effective stress tensor is S¯ , S¯=P¯⋅I+σ¯ ;

The irrelevant flow rule determines a loading surface, which represents the boundary between elastic and plastic behavior in a material. In the context of a concrete damage model, the loading function is typically used to characterize the conditions under which plastic deformation or damage occurs in the concrete. The specific loading function employed in the model may vary, but it generally depends on the stress state and material properties.(2) F=11−α[q¯−3αp¯+θ(ε~pl)⟨σ¯max⟩−⟨γ−σ¯max⟩] −σ¯c(ε~cpl) where, α is determined by the ratio of biaxial and uniaxial ultimate compressive strength; σ¯max is the largest eigenvalue of σ¯ ; The function ⟨⟩ is defined as ⟨x⟩=12(|x|+x) ; The expression of θ(ε~pl) is shown in Eq. (3) as:(3) θ(ε~pl)=σ¯c(ε~cpl)σ¯t(ε~tpl)(1−α)−(1+α) where, σ¯t represents effective tensile stress, σ¯c represents effective compressive stress. γ represents the ratio of the second stress invariant of the meridian plane of tensile and the second stress invariant of the meridian plane of compression, and its value is selected according to the concrete triaxial compression experiment, γ=3(1−ρ)2ρ+3 , and the coefficient ρ=(J2)TM/(J2)CM , according to the definition of the state quantity p¯ , J2 represents the second stress deflection, and TM represents the tensile yield line ( σ1>σ2=σ3 ). CM represents the compression yield line ( σ1=σ2>σ3 ).

The damage factor serves as an indicator of the nonlinear material properties exhibited by concrete. Given the significant disparity between the tensile and compressive characteristics of concrete, the damage factor d can be further categorized into a compressive damage factor dc and a tensile damage factor dt . The fundamental premise of the CDP model is that the strain experienced by concrete under tension or compression comprises both isotropic elastic strain ( εtel,εcel ) and isotropic plastic strain ( ε~tpl,ε~cpl ). In contrast to elastic materials, this strain can be viewed analogously to the elastic strain ( ε0tel,ε0cel ) of an elastic material, coupled with inelastic strain ( ε~tck,ε~cin )—in the case of tension, specifically, cracking strain. Fig. 16 provides a detailed illustration of these concepts.

The calculation methods of various strain intervals of tension concrete are as follows:

The calculation methods for various strain intervals in tension concrete are outlined as follows:(4) σt=1−dtE0(εt−ε~tpl) (5) σt=E0(εt−ε~tck)

The damage factor dt can be obtained by the following equation:(6) dt=1−σtE0−1σtE0−1+ε~tck(1−1/bt) where, bt=ε~tpl/ε~tck .

Similarly, for the compressed concrete, the damage factor calculation formula is as follows:(7) dc=1−σcE0−1σcE0−1+ε~cin(1−1/bc) where, bc=ε~cpl/ε~cin .

In the computation of the damage factor, the elastic modulus utilized corresponds to the cutting line modulus associated with tensile cracking. In accordance with the findings of the Birtel test, bt=0.1 , it is presumed that the concrete’s tensile rising section is devoid of damage and plastic deformation. Additionally, during the initial stages of concrete compression, it is considered to exhibit purely elastic deformation, with no damage incurred. The simplified stress-strain relationship during this phase is outlined as follows:(8) σt={E0εε≤εt,rρtEcεαt(ε/εt,r−1)1.7+ε/εt,rε > εt,r (9) σc={E0εε≤ε0ρcnEcεn−1+(ε/εc,r)nε0 < ε≤εc,rρcEcεαc(ε/εc,r−1)2+ε/εc,rε > εc,r where, ρt=ft,rEcεt,r , ρc=fc,rEcεc,r , n=Ecεc,rEcεc,r−fc,r , Ec is concrete elastic modulus, ε is concrete strain, ft,r, fc,r are the concrete uniaxial tensile and compressive strength representative value, εt,r , εc,r are the peak tensile and compression strain corresponding to the uniaxial tensile and compressive strength representative concrete, respectively, ε0 is the tensile cracking elastic modulus to simulate the linear concrete compression early corresponding critical strain, αt, αc are the uniaxial tensile stress-strain relationship drop parameter value and compressive stress-strain relationship drop parameter value of concrete.

The aforementioned method relies on the fundamental assumption of strain equivalence, which posits that the strain experienced by a damaged unit under the application of stress is identical to the strain response exhibited by an undamaged unit under the influence of effective stress σ¯ . In the context of external force application, the constitutive relationship of the damaged material can be modeled using the constitutive behavior of the undamaged material, with the sole modification being the substitution of the stress with effective stress. Additionally, there exists the energy equivalence hypothesis, which states that the elastic energy of damaged material, when compared to the undamaged state, can be equivalent if the stress is replaced with an equal-effect force or the elastic modulus is substituted with an equivalent elastic modulus.

Elastic residual energy of undamaged material:(10) W0e=σ22E0

Elastic residual energy of damaged material:(11) Wde=σ22Ed=σ¯22E0

The effective stress is:(12) σ¯=σ1−d

From the above conditions:(13) d=1−σE0ε

Fig. 17 presents a comparative analysis of the tensile and compression damage factors for C40 concrete, derived separately using strain equivalence and energy equivalence principles. The results indicate that damage factors calculated based on the energy equivalence method yield lower values compared to those obtained through strain equivalence. Furthermore, the disparity between the tensile damage factors is more significant than that observed in the compressive damage factors. This suggests that the energy equivalence approach may provide a more conservative estimate of damage, particularly in tensile conditions, while still maintaining a relatively close approximation for compressive damage.

Under the application of unidirectional axial force, concrete exhibits a unique unilateral stiffness recovery phenomenon. Specifically, when the external force is removed from tension and the load is reversed to compression, the tensile cracks tend to close, allowing for a partial restoration of compressive stiffness. Conversely, when the external force transitions from compression to tension, the cracks formed during compression remain open, rendering the tensile stiffness recovery largely ineffective.

In the ABAQUS concrete damage plasticity model, this behavior is accounted for by assigning separate recovery factors for the tensile and compressive directions. Under tensile and compressive cyclic loading, the tensile direction is associated with a tensile recovery factor ωt=0 , while the compressive direction corresponds to a compressive recovery factor ωc=1 .

After thorough refinement, the outcomes of the analysis for the transverse partition and the beam body are presented in Figs. 18 and 19. Through the assessment of the yield strain, we can observe the crack morphology in the transverse partition. However, it is noteworthy that the concrete within the beam body remains uncracked. Furthermore, the maximum compressive stress has undergone a significant increase, reaching 28.9 MPa (as opposed to the 13.4 MPa recorded under the linear elastic model).

After determining the cracking location and general shape of the crack, this paper continues to explore the geometric characteristics of the crack to verify the safety of the structure.

The finite element method divides the structure into a finite number of elements with small volume and clear boundary, and the deformation can be transferred between the elements with stiffness according to the established shape function, and the internal force can be obtained by multiplying the deformation and stiffness. The finite element calculation is convenient and fast, but it over-relies on the grid. When simulating cracking, the crack can only be developed along the grid line, rather than through the inside of the unit. And the connection between the elements cannot be broken, that is, the element is always continuous, which limits the scope of application of finite element, so researchers developed infinite elements, which makes the structure get rid of the bondage of the grid, any node can have a relationship with each other, can simulate discontinuous and uneven materials, but the shape function of infinite elements is very difficult to define, and the calculation cost is large. Therefore, some researchers have developed the extended finite element method, which inherits the advantages of convenient and fast finite element calculation. At the same time, by supplementing the shape function with discontinuity, the model gets rid of the shackles of the grid, and the element is no longer continuous under certain conditions, which can better simulate the problem of fracture failure, as shown in Fig. 20.

Compared with the finite element method, the pre-processing and calculation of the extended finite element method are much more difficult. The maximum crack width is concerned in this paper, so only the most serious crack part is taken as the local model based on the previous calculation results. After cracking, the pin effect of the rebar cannot be ignored, but it also greatly increases the difficulty of convergence. In order to speed up the calculation, two rows of rebar etc., are replaced into one row, and a transition unit is established between the rebar and concrete to simulate the bond slip of the two, as shown in Fig. 21.

The treatment method of the third local model is the same as the second one, so the model validation is not carried out.

The calculation results of the third local model are shown in Figs. 22 and 23. From Fig. 22, the fracture surface can be seen significantly. Meanwhile, Fig. 23 reflects the development of the maximum crack width with the loading, and the maximum crack width is 0.19 mm, which meets the specification requirements.