## 6 In Situ Investigation of Full-Scale Construction of Arch-Shaped Steel Sheets
### FEM Model of Full-Scale Construction  
A model being the representation of the arch in the central part of the hall was used for calculations. The model consists of a single ABM 240 profile with the geometry adopted on the basis of the previous analyses. Material constants (modulus of elasticity, Poisson's modulus, shear modulus and coefficient of linear thermal expansion) have been adopted for structural analysis in accordance with EN 1993-1-1 standard [28]. The computation adopted an elastic-plastic multi-linear material model, determined according to the tests. The strength characteristics of steel were identified through a series of laboratory tests on 10 samples of steel sheet with a nominal thickness of 1.40 mm. The obtained mean yield strength amounted to f\({}_{\text{y}}\) = 340 MPa, and ultimate strength was f\({}_{\text{u}}\) = 390 MPa. A typical course of the stress-strain relationship in a single test, and elastic-plastic multi-linear material model is presented in Figure 10.  
The test data were implemented via transformation functions in the \(\sigma_{\text{true}}\)-\(\epsilon_{\text{In}}\) system (elastic-plastic multi-linear material model) [3] according to the Equations (1), (2) presented below:  
\[\epsilon_{In} = ln\big{(}1+\epsilon_{eng}\big{)} \tag{1}\] \[\sigma_{true} = \sigma_{eng}\big{(}1+\epsilon_{eng}\big{)} \tag{2}\]  
where:  
\(\epsilon_{\text{In}}\)--relative logarithmic strain,  
\(\sigma_{\text{true}}\)--true stress,  
\(\sigma_{\text{eng}}\)--engineering stress (test result),  
\(\epsilon_{\text{eng}}\)--engineering strain (test result). Figure 10: Stress–strain curves. The free ends of the model are propped up in joints (a possibility to rotate towards X axis). The remaining degrees of freedom are blocked. On the side edges, boundary conditions are assumed that map the cooperation of the adjacent profiles (remote point). In the remote point system, displacement towards Z and Y are released, the remaining degrees of freedom are blocked. Due to lack of snow during the measurement period in winter, and the negligible influence of the pressure of wind on construction, only the thermal load was considered. The purpose of the test was to determine the displacements and stress of the characteristic points located on the surface of the test object exposed to the action of thermal loads. For numerical calculations, the external and internal temperature of arch structure from three areas (corresponding to measured AOIs) was adopted. The temperature was measured precisely with the utilization of thermoelectrics. Utilization of 3D DIC Measurements for Validation of FEM Model of Full-Scale Construction of Arch-Shaped Steel Sheets  
Displacement measurements were taken in AOI1, AOI2 and AOI3 areas and 3 reference points were selected for each area (Figure 11) to validate the numerical model, 9 points in total. The tests lasted for several months, however validation of the numerical model was limited to a much shorter period. Below (Figures 12 and 13) we present exemplary results of measurements, which were performed in 12 h and 25 min. during a cold night and sunny day in April (largest external as well as internal temperature gradient occurred). The obtained results allowed to validate the numerical model, that can be used to calculate load capacity and stability of the coverings of the thin-walled arch-shaped sheet metals. Figure 13 presents the results of the displacements measurement and those of numerical calculations for the chosen reference points from the analysed measurement areas. The results were juxtaposed in relation to the change of temperature depending on the time of measurement. Displacements measured and determined on the basis of calculations are compatible in terms of increment directions and have similar values. Additionally, Figure 14 shows the analysis of the discrepancies of the calculation results and the test results depending on the value of temperature gradient. A certain regularity may be noticed in the distribution of discrepancies of the results.