*6.1. Computational Model*

In order to design the final acceptance test programme, the bridge response needs to be estimated. Therefore, numerical simulations of its behavior under static loads and natural frequency extraction were performed by means of FEM (refer to [57]) using Abaqus 6.14-2 code. The computational model was created according to the bridge technical drawings and it is a particular one. It is created to describe the global response of the soil-steel bridge and to check whether the assumptions and approach to design were appropriate. Thus, the computational domain needs to remain in correspondence with the precision required by the design process. From this reason, we employ basic modelling techniques and some simplifications. The model should not be overdetailed, but it still has to allow for obtaining reliable results.

Its size (distance from the bridge to the outer boundaries of the domain) was selected in such a way that it did not affect the results in the area of the steel shell. A structured mesh of finite elements was created. First-order solid C3D8I elements (refer to [58]) were used to model the soil continuum. These elements are enhanced by incompatible modes. This formulation enables very strong reduction of the locking effect, as stiffening in bending, being a consequence of the parasitic shear stresses or Poisson's effect, is almost eliminated. The steel parts were represented by shell S4 elements (see for example papers [34,59–61] for other applications of shell elements) with linear shape functions and full 2 × 2 in-plane Gauss-Legendre integration scheme and some additional procedures preventing the locking effect. In the central part of the structure, namely in the region where the in situ measurements were taken, the corrugated sheets were modelled in detail having regard to the real geometry. A simplification was made close to the structure ends, where the steel sheets were treated as a shell endowed with equivalent orthotropic properties and thickness. It is worth mentioning that, often, the whole steel shell is treated as the one having some equivalent properties and uniform thickness (see for example papers [42,62]). Nevertheless, such a simplification cannot be done here, because we are interested in accurate estimation of stresses in the shell-corrugated sheets. The whole computational domain of the soil-steel composite structure is shown in Figure 20 and the detail depicting modelling approach of steel shell in Figure 21.

**Figure 20.** Finite element method (FEM) computational domain of the bridge.

**Figure 21.** Steel shell in the numerical model: corrugated sheets in the center, equivalent shell close to the bridge ends.

The soil and steel were treated in the analyses as isotropic homogenous materials. The elastic constants of the soil were determined on the basis of the requirement of backfill compaction—minimal relative density of the material should be not less than 98% of the maximum density of the soil achieved in the laboratory. This corresponds, according to the polish design standards still used by site engineers [63], to the elastic modulus being 170 MPa. However, often, at the site, the properties of the backfill are better than minimums. So it was also in this case. Aggregate with very good parameters was used to construct the backfill. The reports on the compaction acceptance of the backfill layers showed that the compaction was ranging from 100% to 103%. Therefore, other possible soil properties are also considered, with elastic modulus being 200 MPa (corresponding to 100% compaction) or 230 MPa (corresponding to 103% compaction). Thus, we try to calibrate the numerical model, based on the obtained measurements and assign soil properties enabling best fitting of the FEM estimations. The material properties of steel and soil used in the computational analyses are collected in Table 2. An evident simplification of the material law for the backfill is done. Nevertheless, it is justified by the following reasons. A global model is created and we focus mainly on the appropriate estimation of displacements and stresses in the steel shell. Therefore, detailed description of soil material law is not so important. The backfill main properties are calculated based on well-established engineering procedures, which should enable reasonable description of its response. Second, behavior of the bridge under design loads is recreated and by assumption this loading condition should not lead to any failure. What is more, if failure occur in the backfill, the real response of the bridge would be much different from the FEM estimation, which would also clearly indicate that the simplified approach is not a proper one. This will be verified in the next chapter. It is also worth mentioning that the backfill is treated in a similar way by others and good results are obtained, as presented for example in [64].


**Table 2.** Yielding of steel rafter in all the considered local models.

It also needs to be noted that the bridge was assembled in summer and its temperature during the tests was similar to the one at which it was mounted. There was only a couple of degrees difference. What is more, soil elastic modulus is only slightly affected by the temperature changes, as reported in [65]. Therefore, the material properties of the backfill does not need to be adjusted.

The following boundary conditions were assigned to themodel. The steel shellis fixedin the foundations. Vertical displacements are restrained at the lateral lower surface of the embankment. The displacements perpendicular to the vertical side surfaces of the embankment are constrained as well. A contact was defined between steel shell and the surrounding backfill.

The loads were applied to the model in accordance with trucks positions, described in the Section 3 and their axle loads. In consequence, uniformly distributed loads were defined at truck tires-road contact area. The shapes of tire contact patches were simplified and treated as rectangular. No additional application of temperature loading was necessary to compensate eventual change of the bridge geometry due to the change of temperature during different testing configurations. This is due to the fact, described earlier as well, that the weather during the whole testing day (cloudy or partly cloudy) was basically the same, the bridge was not exposed to the sun, therefore variation of its temperature was negligible.

Finally, static analyses were executed to estimate bridge response during the in situ tests.
