MECHANICS. DIFFERENTIAL EQUATION SYSTEM OF ELASTICITY AND ITS BOUNDARY ELEMENTS PROBLEM
6. ANALYSIS OF TWO-DIMENSIONAL BENT BARS USING FI- FI-NITE ELEMENT METHOD BASED PROGRAM SYSTEM FI-NITE ELEMENT METHOD BASED PROGRAM SYSTEM
6.2. The used finite elements in modeling
The chapter 4 clarified that program system based on the finite element method use two types of element for modeling beam structures. The TRUSS element for modeling structure loaded axial forces only and the BEAM element for modeling loaded axial and shear forces, bending and torque moments. In both cases, the finite elements are planar, so that is characterized by a single straight line.
The properties of the TRUSS elements have already described in the previously chapters.
6.2.1. Properties of the BEAM element
The BEAM elements can be divided into two groups. The BEAM2D elements for models characterized by planar-stressed state, such as generally the planar structures, with symme-trical cross-section bars, loaded the plane of the structure only. The BEAM3D elements are used for three-dimensional modeling. These is usually the three-dimensional constructions, or
two-dimensional constructions loaded perpendicular to own plane, or two-dimensional con-struction consisting of asymmetrical cross-section bars.
The chapters 7-8. will deal with BEAM3D elements.
The BEAM2D elements are two-node uniaxial elements, have three degree of freedom in both nodes (two displacement and a rotational degrees of freedom). The local coordinate sys-tem of the element is shown in figure 6.1. The coordinate syssys-tem x-axis pointing from the first to the second node, the y-axis parallel to the global coordinate system XY plane and perpen-dicular to x-axis, the z axis is perpenperpen-dicular to x and y axis and create a right-handed Carte-sian coordinate system.
Figure 6.1 The BEAMD2D element
The linear static analysis are required the real constants of BEAM2D elements. In this case it means cross-sectional area, the moment of inertia, depth of the section and shear factor. The value of shear factor depends on the shape of the section.
We will also need the material properties of the bar. In this case, the elastic modulus and the Poison’s ratio determination is sufficient because there are planar-stressed state in all points of the BEAM2D elements. If the tare weight of the structure must be considered as a load, the material density determination is needed.
The BEAM2D elements are suitable buckling and thermodynamic analysis. This requires additional real constants and material properties.
6.2.2. The shear deformation
The shear deformation is usually neglected. It is possible simply to take this into account us-ing finite element model for more accurate results.
The shear deformation is deduced from work of internal forces. The work of shear forces of the two-dimensional beam:
dxdydz b
GI S
W V2 2
2 2 int
Sort of the equation the shear (shape) factor is:
b dA S I f A
A 2 2 s 2
The work of the shear forces in constant cross-section beam:
GAdx f V W
2 s int
l
This is the shape factor, and this inverse using in the finite element solution as shear fac-tor. The shear factor values of some often used cross section shown in figure 6.2.
The cross section fs The Shear factor
Rectangle
6/5=1,2 5/6=0,833
Circle
10/9=1,11 9/10=0,9
Thin walled pipe
2 0,5
Figure 6.2 Shear factor of sections
In the technical practice we often use section where the tensioned chords and the sheared web are separable (see Figure 6.3). In this case, the approximate value of shear factor:
The cross section fs The Shear factor
A/AWeb AWeb/A
Figure 6.3 The simplified definition of shear factor 6.3. The study solution
The solution of the finite element studies we follow the following procedure:
Analysis of the problem,
Creation of the geometry model,
Define the properties of finite elements (element types, real constants, material properties),
Define the boundary conditions and loads,
Run the analysis,
Evaluation of the results.
The open frame is shown in figure 6.4, loaded 10 kN on marked point. The force lies in plane of the structure. The bars are 100x100x4 cold bended box sections.
We have to determine the reaction forces, the stress generated in the bars, the deflections, and the bending-, torque moments and shear force diagrams.
Figure 6.4 The cross section
The finite element programs usually contain built-in 3D geometric modeling, graphics pre- and postprocessor. Thus, we can prepare the geometric model in its (see Figure 6.5)
Figure 6.5 Geometric modeler in the finite element program
These built-in geometric modelers do not always offer you the convenience of modern CAD systems. Often we have to analyze existing models. In this case, the data exchange procedure can be convenient and efficient with other CAD systems by any available standard file format such as SAT, IGS, DXF, etc.. (see figure 6.6).
Figure 6.6 Import geometric model from another geometric modeler
Do not forget, in this case the geometric model only helps to create a finite element mesh. It does not comply with the rules of technical drawing, and has no relevance to real shape of the structure. It is true in this exercise, because the 100 mm box sections appears only lines (see Figure 6.7). Thus, we have to transform (simplify and extend) the technical documentation before finite element analysis. This is shown in Figure 6.7, which shows the imported geome-tric model.
Figure 6.7 The imported geometric model It is also shown that the elements lie in the XY plane.
The next step is to determine element group (see Figure 6.8).
Figure 6.8 Determination of element group
We have clarified that we use linear behavior, BAEM2D elements (see Figure 6.9).
Figure 6.9 Select the BEAM2D elements and determination of these properties Next task is to determine the real constants of elements (see Figure 6.10).
Figure 6.10 Real constants definition
As previously described we have to define real constants of BEAM2D elements, the cross-sectional area of the bars, the inertial moment (Iz), deep of the section, and the shear factor (see Figure 6.11). Making sure use the selected unit system what is in this case the SI system.
Figure 6.11 Real constants definition
Needs to be explained in the fourth and fifth real constants (End-release code). The end-release code for each and is specified by a six digit number with combinations of 0 and 1. The six digit code corresponds in order to the six degrees of freedom at each end of the beam ele-ments. For example, end release code 000001 for a BEAM2D element represent a condition in which the moment about z axis is zero and forces in x- and y direction are to be calculated.
The degree of freedom refers to the element local coordinate system (see Figure 6.1).
The seventh and eighth real constants use only in thermal analysis, so we do not deal with them now.
Still, the definition of material properties (see Figure 6.12).
Figure 6.12 Definition of material properties
It is sufficient to specify the value of the modulus of elasticity and Poisson’s coefficient for the beam elements, as shown in figure 6.13 and figure 6.14.
Figure 6.13 Definition of the elastic modulus
Figure 6.14 Definition of the Poison’s coefficient If necessary, we can define more material properties.
After defining properties of the finite element mesh, may follow the finite element mesh generation. The FEM programs offer several methods for this, now we select the automatic mesh (see figure 6.15).
The size of the elements is determined by required precision of the results, the available capabilities of the computer and the available time. Now we choose 0,1 m average element size.
Figure 6.15 Automatic mesh generation
The finite element mesh and the numbered nodes shown in the Figure 6.16.
Figure 6.16 The finite element mesh
It visible in the figure that created an independent node at each three endpoint of the beam elements. Because, the finite element mesh created each geometry object separately, it is ne-cessary to merge the nodes in each end of the bars (see Figure 6.17).
Figure 6.17 Merge of the end of bars
In the next step the boundary conditions should be given. In this case, these are two size 0 displacements on the supports.
We fix two degrees of freedom of the structure, in x and y directions at the both support (see figure 6.18).
Figure 6.18 Displacement constraints
Finally, it should be given the loads the 5 kN concentrated force (see Figure 6.19). The direc-tion of forces must be given in the global coordinate system, so the downward forces are neg-ative sign.
Figure 6.19 Defining the load The completed finite element model is presented in Figure 6.20.
Figure 6.20 The completed finite element model Follows, the running linear static analysis (see Figure 6.21).
Figure 6.21 Run linear static analyses
After the successful solving, follows the display and evaluation of results.
The displaying stresses generated in bars (see Figure 6.22) can be done in several ways.
The stresses are interpreted on the element and in the element coordinate system like case of the TRUSS elements.
Figure 6.22 Display stresses
The results are shown in Figure 6.23. The deformation is not real, of course, the program ge-nerates a specific scale factor, so that data can be evaluated.
Notice, that the bars are bent, due to bending moments.
Figure 6.23 Stresses on deformed shape
It is possible to display stress components (see Figure 6.24). The negative sign of the stress indicate compressive stress.
Figure 6.24 Display stress components
We examine the deflections i.e., y direction displacements in next step, (see Figure 6.25).
Figure 6.25 Display the deflection
The Figure 6.26 shows the results. The negative signs indicate downward displacements.
Figure 6.26 The deflections
We can display the moment and shear force diagrams in beam elements (see Figure 6.27).
Figure 6.27 Display the bending moment diagram
The bending moment diagram shown in figure 6.28. There is not numerical value in diagram, even so useful because it helps to determine the minimal stressed locations.
Figure 6.28 Bending moment diagram
It is possible to display the reaction forces and moments generated in supports (see Figure 6.29).
Figure 6.29 Display the reactions forces
It is possible to list the force and moments components generated in elements (see Figure 6.30).
Figure 6.30 List the force and moments components The listing of the nodal forces and moments are shown in the figure 6 31.
Figure 6.31 The nodal forces and moments The listing of the stress component shows figure 6.32.
Figure 6.32 The stress component list
The numerical results tables can be appear incomplete, some component is 0. As explained by the BEAM2D elements. The shear forces perpendicular to plane of structure, bending mo-ments in this plane and torque does not exist in this case.
6.4. Remarks
During the solutions we do not deal with buckling of the compressed bars. If this is a real problem, one should be to verify with solution a finite element problem, or with any analytic method.
During the solutions the tare weight was neglected.
Both problems are explained in later chapters.
Furthermore, the structural joint was not tested. The other specialized areas of structural design deal with this problems.
7. APPLICATION THE PRINCIPLE OF THE MINIMUM