The Finite Element Method

FEM for Trusses (Finite Element Method) Part 3

High Order One-Dimensional Elements For truss members that are free of body forces, there is no need to use higher order elements, as the linear element can already give the exact solution, as shown in Example 4.1. However, for truss members subjected to body forces arbitrarily distributed in the truss elements along its axial direction, […]

FEM for Beams (Finite Element Method) Part 1

Introduction A beam is another simple but commonly used structural component. It is also geometrically a straight bar of an arbitrary cross-section, but it deforms only in directions perpendicular to its axis. Note that the main difference between the beam and the truss is the type of load they carry. Beams are subjected to transverse […]

FEM for Beams (Finite Element Method) Part 2

Worked Examples Example 5.1: A uniform cantilever beam subjected to a downward force Consider the cantilever beam as shown in Figure 5.2. The beam is fixed at one end, and it has a uniform cross-sectional area as shown. The beam undergoes static deflection by a downward load of P = 1000 N applied at the […]

FEM for Beams (Finite Element Method) Part 3

Solution Process Let us now try to relate the information provided in the input file with what is formulated in this topic. The first part of the ABAQUS input normally describes the nodes and their coordinates (position). These lines are often called ‘nodal cards1’. The second part of the input file are the so-called ‘element […]

FEM for Frames (Finite Element Method) Part 1

Introduction A frame element is formulated to model a straight bar of an arbitrary cross-section, which can deform not only in the axial direction but also in the directions perpendicular to the axis of the bar. The bar is capable of carrying both axial and transverse forces, as well as moments. Therefore, a frame element […]

FEM for Frames (Finite Element Method) Part 2

Equations in Global Coordinate System Having known the element matrices in the local coordinate system, the next thing to do is to transform the element matrices into the global coordinate system to account for the differences in orientation of all the local coordinate systems that are attached on individual frame members. Assume that the local […]

FEM for Frames (Finite Element Method) Part 3

Case Study: Finite Element Analysis of a Bicycle Frame In the design of many modern devices and equipment, the finite element method has become an indispensable tool for the many successful products that we have come to use daily. In this case study, the analysis of a bicycle frame is carried out. Historically, intuition and […]

FEM for Two-Dimensional Solids (Finite Element Method) Part 1

Introduction In this topic, we develop, in an easy to understand manner, finite element equations for the stress analysis of two-dimensional (2D) solids subjected to external loads.The element developed is called a 2D solid element that is used for structural problems where the loading-and hence the deformation-occur within a plane. Though no real life structure […]

FEM for Two-Dimensional Solids (Finite Element Method) Part 2

Linear Rectangular Elements Triangular elements are usually not preferred by many analysts nowadays, unless there are difficulties with the meshing and re-meshing of models of complex geometry. The main reason is that the triangular elements are usually less accurate than rectangular or quadrilateral elements. As shown in the previous section, the strain matrix of the […]

FEM for Two-Dimensional Solids (Finite Element Method) Part 3

Strain Matrix After mapping is performed for the coordinates, we can now evaluate the strain matrix B. To do so in this case, it is necessary to express the differentials in terms of the natural coordinates, since the relationship between the x and y coordinates and the natural coordinates is no longer as straightforward as […]

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