# The Finite Element Method Linear Static And Dyn...

Originally developed to address specific areas of structural mechanics and elasticity, the finite element method is applicable to problems throughout applied mathematics, continuum mechanics, engineering, and physics. This text elucidates the method's broader scope, bridging the gap between mathematical foundations and practical applications. Intended for students as well as professionals, it is an excellent companion for independent study, with numerous illustrative examples and problems.The authors trace the method's development and explain the technique in clearly understandable stages. Topics include solving problems involving partial differential equations, with a thorough finite element analysis of Poisson's equation; a step-by-step assembly of the master matrix; various numerical techniques for solving large systems of equations; and applications to problems in elasticity and the bending of beams and plates. Additional subjects include general interpolation functions, numerical integrations, and higher-order elements; applications to second- and fourth-order partial differential equations; and a variety of issues involving elastic vibrations, heat transfer, and fluid flow. The displacement model is fully developed, in addition to the hybrid model, of which Dr. Tong was an originator. The text concludes with numerous helpful appendixes.

## The finite element method Linear static and dyn...

Partial Differential Equation Toolbox lets you import 2D and 3D geometries from STL or mesh data. You can automatically generate meshes with triangular and tetrahedral elements. You can solve PDEs by using the finite element method, and postprocess results to explore and analyze them.

Grid shells supporting transparent or opaque panels are largely used to cover long-spanned spaces because of their lightness, the easy setup, and economy. This paper presents the results of experimental static and dynamic investigations carried out on a large-scale free-form grid shell mock-up, whose geometry descended from an innovative Voronoi polygonal pattern. Accompanying finite-element method (FEM) simulations followed. To these purposes, a four-step procedure was adopted: (1) a perfect FEM model was analyzed; (2) using the modal shapes scaled by measuring the mock-up, a deformed unloaded geometry was built, which took into account the defects caused by the assembly phase; (3) experimental static tests were executed by affixing weights to the mock-up, and a simplified representative FEM model was calibrated, choosing the nodes stiffness and the material properties as parameters; and (4) modal identification was performed through operational modal analysis and impulsive tests, and then, a simplified FEM dynamical model was calibrated. Due to the high deformability of the mock-up, only a symmetric load case configuration was adopted.

Grid shells are afflicted by some static problems as deformability, buckling, and imperfection sensitivity (Schlaich and Schober 1996, 1997) and by constructional problems such as nodes assembly, feasibility of the panels geometry and curvature, connectivity of the elements, etc. because of their three-dimensional spatial complexity. Therefore, a large displacement analysis typically applies to a grid shell due to the intrinsic deformability caused by its lightness. As they carry loads mainly by compressive forces, buckling failures (local, global, snap-through, or even worse combinations of previous) should be avoided. Stability analysis is carried out considering both second-order effects and imperfections, and a strong abatement of the buckling multiplier with respect to the ideally perfect structure is usually detected (Bacco and Borri 1993; Bulenda and Knippers 2001; Cai et al. 2013; Dini et al. 2013; Gioncu 1995).

The second goal of this study was to execute non-destructive tests and consequently simulate the observed mock-up behavior with simplified FEM models. We performed two sets of experimental campaigns. The first set intended to investigate the statics of the mock-up, submitted to symmetrical loading patterns. The descending representative FEM static model did not consider the presence of the PET panels. Moreover, to approach the experimental behavior, we varied the beams rotational end-release stiffness and reduced the characteristic material properties, simulating the non-rigid connectivity, the local non-linearities, and structural imperfections.

The present study relied on three base assumptions regarding non-linearity, size of the models, and symmetry. In the non-linear static analyses, only geometrical non-linearity was included. Because the field of proportionality between stress and consequent displacement was indeed not exceeded, material non-linearity was omitted. Moreover, other non-linearities were omitted such as contact and friction at the nodes. Concerning the size, FEM models conveniently represent the real size of the prototype (2.40 m side), avoiding to extend the structural test results to a larger scale model and to introduce further uncertainty. Among multiple loading configurations, we considered only symmetrical load cases in accordance with the symmetry of the vault. Other possible and probably more dangerous loading configurations may be asymmetrical loads. However, the Voronoi Static-Aware algorithm is still limited to the optimization of only symmetrically loaded surfaces (i.e., under dead loads, Pietroni et al. 2015; Tonelli et al. 2016a; Tonelli 2015); consequently, the high deformability of the mock-up excluded to extend the study to other loading scenarios.

The PET panel is point supported at its vertices, which resulted clamped between the printed plastic node and a metallic washer screwed on the node itself. So that, given a non-linear constraint condition, the panel can detach and slide in case of tension stress. No direct contact is between the panels and the beams. After the attempt to include the panels into the model, we considered only their dead load: gravitational masses weighing 50 g were loaded at each node as simulating the printed joint and the panel influence. Since the aim was to provide a global simplified representation of the mock-up, requiring an in-depth modelling the complexity of the panels, joinery system would not have been compatible with a less refined global modelling scale. Finally, the model consisted of 465 nodes and 697 beam elements.

By minimizing this indicator I, we selected and scaled the modal shapes 10, 12, and 13, for the areas identified, respectively, by apices N326, N312 and N333, N340 (Fig. 2a). In addition, boundary nodes of neighboring areas were scaled according to the two adjacent eigenforms, and the nodal coordinates obtained were mediated to avoid singularities and guarantee smoothness of the surface. The application of such displacement field to the GS0 nodal coordinates returned the imperfect geometry. Finally, linear static analyses tested the acceptability of the imperfect model that manifested about the same stress field of GS0. The similarity between physical and numerical models is shown in Fig. 6.

We studied the static behavior by means of tests performed under time-invariant load configurations, where the ith load increment is maintained until the stabilization of the measure. The aim was to simulate the serviceability loading of the mock-up, transforming a distributed load into equivalent nodal forces. Due to the feasibility of the test, 16 nodes of the vault were loaded. Organized in groups of four, the 16 nodes were symmetrically located with respect to the axes of the prototype. Hollow metal discs, each weighing 125 g, constituted the applied load and were hanged to each node by means of a ligature and a double-hooked-end self-balancing steel element. Such metallic extremity had the advantage of reducing vibrations transmitted to the structure.

The parameters derived from the dynamic experimentation are natural frequencies, damping ratios, and modal shapes (Ewins 2000; Maia and Silva 1997). The high deformability of the prototype suggested the use of an output-only method, such as the operational modal analysis (OMA) (Zhang et al. 2005). In addition, we used impulsive excitation tests to give feedback to the OMA data. Fundamental assumptions for the dynamic tests are linearity, stationarity, and observability, and the test programming complied with these assumptions.

Figure 9 presents the main results of the static tests, performed as mentioned at paragraph 2.3. The numbering of the sensitive control nodes identifies the non-linear curves in the graph. We decided to exclude the first set from the results, because it was not deemed representative. Typically, measurements N326 and N333 had similar displacements in the Z-positive direction. The node N340 was less disposed to motion, and the node N312 had intermediate behavior and manifested a homothetic curve with respect to the displacements. The symmetric nodes N455 and N456 described very similar paths with significant increases in the gravity direction, about three times higher than those at the vertices. In addition, all the nodes displayed a significant irreversible deformation after the unloading, so the final curvature of the shell changed. Two main sources of non-linearities influenced the static response: high deformability of the prototype (namely geometrical non-linearity) and the joints restraints. Indeed, because of the full contact in compression and the unpredictable friction in traction, the contact between the wooden beams and the nodes was non-linear. A similar but less important (in this loading condition due to the relative lower resistance) non-linearity was in the clamping system of the PET panel (full contact in compression, possible sliding in traction).

The method used within this work concerns the Voronoi static-aware vault, but is extendable to other Voronoi static-aware case studies. Simulating the size and the state of the mock-up, a disadvantage is that the findings are strictly related to the specific shape, internal tessellation, building technology, materials, and local errors of assembly. 041b061a72