Study solution

The finite element study procedure:

1. problem analysis,

2. create a geometry for generate a finite element mesh,

3. define properties of finite elements (element type, real constant, material properties), 4. determine boundary conditions, and loads,

5. solve the model,

6. evaluation of the results

At both ends supported trusses are loaded at two nodes. The forces are 120-120 kN each.

(see Figure 4.3). The bars are steel pipes 100x10.

To be determined:

 deflection of the structure,

 stresses generated in the bars,

3 x 4 = 12 m

3 m

120 kN

120 kN

Figure 4.3 The tested trusses

The finite element programs usually contain built-in 3D geometric modeler, graphics pre- and postprocessor. Thus, we can prepare the geometric model in its (see Figure 4.4)

Figure 4.4. 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 with other CAD systems can be convenient and efficient by any available standard file format such as SAT, IGS, DXF, etc.. (see Figure 4.5).

Figure 4.5. Import geometric model

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 a technical drawing, and has no relevance to the real shape of the structure. It is true in this exercise, because the 100 mm diameter pipes appear only lines (see Figure 4.3). Thus, we have to transform (simplify and extend) the technical docu-mentation before the finite element analysis. This is shown in Figure 4.6, which shows the imported geometric model. The one piece chord bars are divided at nodes, because it helps the finite element mesh generation.

It should be remembered, that we have to choose a unit system for finite element model-ing. If the SI is selected, one drawing unit will be a meter during the data exchange of geome-tric models.

Figure 4.6. The imported geometric model

It is also shown that the elements lie in the XY plane.

In the next step we determine the element group (see Figure 4.7).

We have clarified that we use linear behavior, TRUSS2D elements.

Figure 4.7. Determination of element group

It is also necessary to determine the material properties of finite elements. It is sufficient to specify the value of the modulus of elasticity for the truss element (see Figure 4.8). Making sure to use the selected unit system. In this case, it is the SI system, where the dimensions are determined in meters, and the modulus of elasticity in Pa, (N/m2).

Figure 4.8. Determination of material properties Next task is determine the real constants of the elements. (see Figure4.9).

A complex finite element models contain various types of elements, so we have to also de-termine the associated element group.

As previously described, the real constant is only the cross-sectional area for TRUSS2D elements. Do not forget, we have to use the selected unit system in this case too.

Figure 4.9. Real constants determination

After defining the the mesh properties, may follow the finite element mesh generation. The FEM programs offer several methods for this (see Figure 4.10).

Because the bar forces do not change along the length of the bars, it is sufficient to be placed one element in each objects.

Figure 4.10. Parametric mesh generation

Because, the finite element mesh created each geometry object separately, it is necessary to merge the nodes in each end of the bars (see Figure 4.11). The redundant nodes are removed from the finite element model.

Figure 4.11. Merge of the end of bars

In the next step the boundary conditions should be given. In this case, these are two, 0 dis-placement constrains on the ends of the trusses.

The left side two degrees of freedom are fixed x and y directions and the other end only the y direction is fixed (see Figure 4.12).

Figure 4.12. Specify displacement constraints

Finally, it should be given the loads (which shown in Figure 4.3), two 120 kN concentrated force (see Figure 4.13). The direction of forces must be given in the global coordinate system, so the downward forces are negative sign.

Figure 4.13 Defining the concentrated forces

By the finite element model is built. The calculation follows (see Figure 4.14).

Figure 4.14 Run a linear static analysis

After the successfully solving, the display and evaluation of the results follows.

The displaying stresses generated in bars (see Figure 4.15) can be done in several ways.

The stresses are interpreted on the element and in the element local coordinate system.

There is a possibility that the results display on deformed shape. The deformation is not real of course, the program generates a specific scale factor, so that data can be evaluated.

Figure 4.15 Display stresses

The results (see Figure 4.16) must be evaluated. The negative sign indicate compressive stress.

Notice, that the bars were straight, can be interpreted no bending moment generated in them.

Figure 4.16. Stresses on deformed shape Our aim was to examine the deflection (see Figure 4.17).

Figure 4.17 Deformed shape

The deformations can be bi-directional displacement of nodes. The deflection is the y dis-placement in the global coordinate system (see Figure 4.18). The negative sign of results represent a downward displacement. The value of the scale according to SI unit system.

Figure 4.18 Displacement in y direction

It is possible to display the exact numerical results at nodes, forces generated in bars and dis-placement components (Figure 4:19 to 4:20).

Because the truss elements are loaded only by tension-compression stresses, so the table include only these stresses, interpreted in the element local coordinate system

Figure 4.19 Fig. Stress component list

The displacements of nodes are interpreted in the global coordinate system (see Figure 4.20).

4.20. Fig. The displacements of nodes

4.4. Remarks

During the solution, we have not dealt with buckling of the compressed bars. If this is a real problem, it should have to verify with solution a finite element problem, or with any analytic method.

During the solutions the tare weight (~81.59 kN) was neglected because this order of magnitude smaller than the external load.

Both problems are explained in later chapters which deal with BEAM elements.

Furthermore, the structural joint was not examined. The other specialized areas of struc-tural design deal with this problems.

5. TWO-DIMENSIONAL BENT BARS VARIATION PROBLEM,