Finite Element Analysis of Structures

Abstract

The use of computers in engineering started in 1960. Before 1960, engineers used physical models to check the results of complex structures. Recently, these physical models have been used for education purposes, namely, in the minor Bend and Break. Conversion from small-scale structures to real-sized structures was taught in this minor (P. C. J. Hoogenboom, n.d.). Therefore, This research aims to find a relation between small-scale and real-sized structures. Namely, the conversion of internal forces and displacements from a small-scale structure to a real-sized structure in both linear and non-linear material behaviour. A linear calculation yields a linear relation between the results and the applied force on the structure, unlike a non-linear calculation. The conversion rules apply to linear material behaviour. However, whether they also apply to physical and geometrical non-linear material behaviour is uncertain. This uncertainty yields the following research question: “There are conversion rules for deflection and stresses from small-scale structures to real-sized structures. Do these apply to non-linear behaviour too?” Multiple structures were analysed. A small-scale and real-sized model of these structures was modelled in the finite element program SCIA Engineer (SCIA, 2024). The geometrical properties of the smallscale structures were 10 times smaller than the geometrical properties of the real-sized structures. The type of structures is itemized below: • A steel truss (with and without bending stiffness) • A steel beam (statically determinate and statically indeterminate) • A steel Vierendeel girder • A concrete two-way slab • A concrete shell roof Geometrical and physical non-linear calculations were done. The geometrical non-linearity focused on local and global buckling, which yields large displacements while physical non-linearity focused on the plasticity of hinges and elastoplastic stress-strain diagrams. Relevant forces in the structures’ members and nodes were computed with linear and non-linear calculations. Additionally, displacement and deformations of relevant members or nodes were analysed. Furthermore, the applied loads are point loads, line loads or distributed loads, which had their conversion rules from small-scale to real-sized structures. Namely, the real-sized point load is 100 times larger than the small-scale point load. The real-sized line load is 10 times larger than the small-scale line load and distributed loads are identical in both structures. This yields identical stresses in the small-scale and real-sized structures. Displacements differ by a factor of 10 and internal forces by a factor of 100 or 1000, depending on the type of applied load. The steel trusses, the statically determinate beam, the Vierendeel girder and the concrete shell roof were all analysed with a geometrical non-linear calculation. All linear analyses followed the conversion rules perfectly. The relevant displacements in the small-scale structures differed by a factor of 10 from the displacements in the real-sized structures. The internal forces differed by a factor of 100 or 1000. Similarly, in the geometrical non-linear calculation, the displacements differed by a factor of 10 and the internal forces by a factor of 100 or 1000. However, there were small numerical errors in the nonlinear calculation. This happens when SCIA Engineer makes a non-linear calculation due to its iterative calculation method. The statically indeterminate beam and the concrete two-way slab were analysed with a physical non-linear calculation in addition to geometrical non-linearity. The plasticity of hinges was tested in the statically indeterminate beam and the elastic stress-strain curve of the two-way slab model was changed to an elastoplastic stress-strain curve which yields large displacements. Again, the internal forces differed by a factor of 100 or 1000, depending on the type of applied load. The displacements differed by a factor of 10, which is in line with the existing conversion rules. In conclusion, for the analysed structures, the conversion rules apply to both geometrical and physical non-linear material behaviour.