PSI - Issue 81

Denys Rudavskyi et al. / Procedia Structural Integrity 81 (2026) 151–155

152

Fig. 1. Photos of arch concrete bridges.

The maintenance and strengthening of such structures require cost-effective and minimally invasive technologies. Injection based methods provide a practical solution for restoring load-bearing capacity by filling existing cracks with high-stiffness materials (Panasyuk et al. (2014), Trout (1997), Smith (1992)). This paper focuses on a case study of a nineteenth-century arch bridge modelled using the finite element method (FEM). The study aims to assess the effectiveness of injection reinforcement in reducing stress concentration and restoring the mechanical integrity of a cracked region.

Nomenclature K II

mode II stress intensity factor

angle of crack plane inclination relative to the local tangential direction



E 1 E 2

Young’s module of the base material Young’s module of the filler material

Poisson’s ratio



a , b

semi-axes of the semi-elliptical crack semi-elliptical crack flatt е ning parameter thickness of the local cracked volume applied external uniformly distributed pressure

c

t

p

Tresca stresses

 max

2. Methodology A numerical model of the arch bridge was developed using the open-source FEM package SALOME-MECA with the Code_Aster solver (Fig. 2). The geometric parameters of the bridge segment are defined as follows: length of 12.4 m, width of 5.0 m, height of 3.5 m, and an arch radius of 2.5 m. The global model of the arch bridge was discretised using quadratic solid elements (tetrahedrons TET10) to accurately evaluate the overall stress state under applied load conditions (Fig. 3). Tresca stress distribution was analysed to identify critical regions where the maximum shear stresses occur. The load conditions were modelled as a uniform pressure p applied to the bridge top plate. This value was chosen as the characteristic intensity for the uniformly distributed load component used in rail bridge design codes for global structural force analysis. For the investigated bridge with a circular arch shape, the simulation results revealed that the maximum shear stresses occur along planes inclined at about α = 45°, as shown in Fig. 4. A local model was then extracted from the most stressed area of the arch, representing a semi-elliptical surface crack inclined at various angles α relative to the local normal direction. The crack geometry parameters are defined as follows: a =0.5 m, b =1.0 m, and c =0.0035 m. The crack was modelled both in an unreinforced state and with an injection filler, simulating the stiffened interface region. Boundary conditions were derived from the global model results to ensure consistency between the models. For the numerical simulation, the elastic properties of the base concrete were defined with a Young’s m odulus E 1 =30GPa and a Poisson’s ratio  =0.2. The injection material was modelled as a polyurethane filler with E 2 =0.9 GPa and  =0.35, resulting in a stiffness ratio of E 2 / E 1 = 0.03. This specific filler stiffness was chosen as representative of high-performance resins used for structural restoration, as documented in the injection technology studies by Panasyuk et al. (2014). The shear stress distribution calculated in the finite elements analysis step (Fig. 5) served as the basis for determining the SIF K II . The SIF K II value was then computed using an asymptotic method (Gdoutos (2020)) with an applied external pressure of p = 30 kPa. To accurately capture the singularity at the crack front, a refinement strategy was employed where the element size was reduced to 0.005 m (approximately 1% of the crack semi-axis a ) in the immediate vicinity of the crack. A convergence check for