Abstract:

This report is set to
compare finite
element analysis with a theoretical model to verify its validity. Dassault
Systèmes’ Abaqus is utilized in this report to analyse a truss frame model. The
findings of both models are viable and the differences can be ignored due to accuracy;
theoretical model yields values with low number of significant figures.

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Abbreviations: FEA, Finite element analysis.

1.0  Introduction:

Engineering components can be of diverse shapes, in
two1
or three2
dimensions. In this report, we confined our discussion to linear
elastic analysis. The success rate of the numerical analysis is dependent on
the comparison with the theoretical method. FEA aids in understanding the
behaviour of constrained models, to extrapolate the performance; by to
calculating the safety margin. To amend model to optimal design. The precision
of numerical model highly depends on the approximation of all initial parameters.

2.0  Theoretical
Model:

This model is to be used in comparison with the
numerical model. This model fails to take variables like young modulus, E
or passion ratio into account. This is predicted to show a minor significance in
less accurate notations. However, higher accuracy yields differences in lower
prefixes.

2.1 Support Reactions:

Calculations for the fixed
pin at node 1:

Calculations for roller
pin at node 2:

Where: A is the reaction
force at the pin and all angles are in degrees.

2.2 Forces in all elements:

At node 4:

 

 

At node 3:

3.0 Numerical Model:

Finite element analysis,
or FEA, is an approximate method of solving engineering problems. Generally
used to solve problems with no exact solution. It’s rather a
numerical method than analytical. The need to FEA arises as analytical methods
cannot handle complex problems like engineering problems. For instance, we can
calculate, analytically, the stresses and strains in a bent beam using
engineering mechanics of materials or the mathematical theory of elasticity.

But neither will be very effective in determining what is happening in part of
a car suspension system in cornering. The first appearance of FEA originated to
find stresses and strains in engineering models under concentrated loads. FEA
demands a large amount of computation power.

“The method is now applied to problems involving a
wide range of phenomena, including vibrations, heat conduction, fluid mechanics
and electrostatics, and a wide range of material properties, such as
linear-elastic (Hookean) behaviour and behaviour involving deviation from
Hooke’s law (for example, plasticity or rubber-elasticity).” says Modlen, G.

(2008).

 

 

 

 

 

 

 

 

Figure 3.0.1. Initial Model, in white. Model, rainbow,
under applied force at node 4

Shown in figure 3.0.1 is the resultant concentrated
force applied to node 4. The white line represents the frame when no forces
applied. The frame has a Young’s Modulus, E of 210 GPa. Poisson’s ratio3,
at
0.31. Geometrically, the frame has 4 nodes and 5 elements. An element length of
1m and a circular profile diameter of

15 mm. It’s a frame; since it’s equilateral since
all elements are equal. This also means angles are 60º degrees. Node 1 is a
fixed pin and node 2 is a roller. 

The magnitude of  equals to 2kN as shown in the figure 3.0.1. As
a result of the applied force, shown in the contoured graph, the frame is seen
to deform and change in mechanical and geometrical values.

3.1 Nodal Displacements

The displacement of a respective point within
an element is fixed by the displacements of nodes of an element, that is a
function of the nodal displacements. The system matrix is simply a
superposition of the individual element stiffness matrices with proper
assignment of element nodal displacements and associated stiffness coefficients
to system nodal displacements.

Figure 3.1.1 presents
the magnitudes of displacement, U in nodes

3.2.1 Nodal Forces

Table 3.2.1.1 presents
the active nodal forces

     Node Label    RF.Magnitude          RF.RF1          RF.RF2            U.U2         S.Mises

                          @Loc 1          @Loc 1          @Loc 1          @Loc 1          @Loc 1

————————————————————————————————-

               1          3.E+03    -113.687E-15          3.E+03    -3.00000E-33     7.62394E+06

               2     1.00000E+03              0.    -1.00000E+03     1.00000E-33     4.90111E+06

               3              0.              0.              0.     4.49152E-06     6.53481E+06

               4              0.              0.              0.    -130.254E-06     9.80221E+06

 

 

  Minimum                     0.    -113.687E-15    -1.00000E+03    -130.254E-06     4.90111E+06

         At Node               4               1               2               4               2

 

  Maximum                 3.E+03              0.          3.E+03     4.49152E-06     9.80221E+06

         At Node               1               4               1               3               4

 

           Total     4.00000E+03    -113.687E-15     2.00000E+03    -125.762E-06     28.8621E+06

3.2.2 Support
Reactions

Table 3.2.2.1 presents
the acting forces on elements

  Part Instance     Node ID 
Attached elements   RF, Magnitude

—————————————————————

     TRUSS-1           1               1          3.E+03

     TRUSS-1           1               4          3.E+03

     TRUSS-1           1               5          3.E+03

     TRUSS-1           2               1          1.E+03

     TRUSS-1           2               2          1.E+03

     TRUSS-1 
         4               3              0.

     TRUSS-1           4               4              0.

3.3 Forces in all
elements

Output sorted by
column “Element Label”.

Averaged at nodes

Table 3.3.1 presents
the acting forces on elements

Node Label    RF.Magnitude          RF.RF1          RF.RF2            U.U2         S.Mises

@Loc 1          @Loc 1          @Loc 1          @Loc 1          @Loc 1

————————————————————————————————-

1          3.E+03    -113.687E-15          3.E+03    -3.00000E-33     7.62394E+06

2     1.00000E+03              0.    -1.00000E+03     1.00000E-33     4.90111E+06

3              0.              0.              0.     4.49152E-06     6.53481E+06

4              0.              0.              0.    -130.254E-06     9.80221E+06

Minimum                    
0.    -113.687E-15    -1.00000E+03    -130.254E-06     4.90111E+06

At Node               4               1               2               4               2

Maximum                
3.E+03              0.          3.E+03     4.49152E-06     9.80221E+06

At Node               1               4               1               3               4

Total     4.00000E+03    -113.687E-15     2.00000E+03    -125.762E-06     28.8621E+06

3.4 Stress in all
elements

Stress analysis for trusses, beams, and other simple structures are
carried out based on dramatic simplification and idealization: – mass
concentrated at the center of gravity – beam simplified as a line segment (same
cross-section) • Design is based on the calculation results of the idealized
structure & a large safety factor (1.5-3) given by experience.

Table 3.4.1 presents
the stress in elements

  Part
Instance  Element ID        Type        Int. Pt.        S, Mises

——————————————————————————–

    
TRUSS-1           4        T2D2               1     13.0696E+06

    
TRUSS-1           3        T2D2               1     6.53481E+06

    
TRUSS-1           5        T2D2               1     6.53481E+06

    
TRUSS-1           1        T2D2               1      3.2674E+06

    
TRUSS-1           2        T2D2               1     6.53481E+06

4.0 Analysis & Discussion:

4.1 Material discernment: Poisson’s’s Ratio,
Elastic Moduli & Mohs Scale

Poisson’s’s ratio, ? is the ratio of transverse
contraction strain to longitudinal extension strain in the direction of
stretching force. In comparing a material’s
resistance to distort under mechanical load rather than to alter in volume, Poisson’s’s
ratio offers the fundamental metric by which to compare the performance of any
material when strained elastically. 

Molybdenum, as shown in figures, has a toughness of 5.5—by
passing iron. Allowing it to be
used in many industrial applications. A young modulus of 4.7700E4 Pa, making it
elastically suitable for a truss frame. Has an atom around twice as heavy as an
iron atom, which allows for better crystallization processes.

4.2 Discussion

FEA model values are instant values of steps; the
higher the number, the more the model can be manipulated. The calculated theoretical
model failed to include the elastic moduli, as well as Poisson’s’s ratio. It is
shown in some results, i.e. nodal force number 1.

5.0 Conclusions:

In finite element
analysis, solution accuracy is judged in terms of convergence as the element
“mesh” is refined. FEA models, when defined correctly, are more representative
of real life emulation. In FEA the thickness of the elements was 1.767E4.

Whereas the theoretical model failed to include the thickness. Both models are viable
and the differences can be ignored due to accuracy; theoretical model yields
values with a low number of significant figures. For  purposes of comparison, analytical model values
are to be rounded, resulting in matching values.

Bibliography:

Widas, P.

(1997) ‘Introduction to Finite Element Analysis’. Virginia Tech Material
Science and Engineering online (January 2008), 87. available from

Greaves,
G.N., Greer, A.L., Lakes, R.S., and Rouxel, T. (2011) ‘Poisson’s’s Ratio and
Modern Materials’. Nature Materials 10 (11), 823–837

Gray, T.,
Mann, N., and Whitby, M. (n.d.) The Photographic Periodic Table of the
Elements online available from 13
December 2017

Tura, A.,
Dong, Z., Trivedi, S., Shibata, Y., Tanimoto, Y., Maruyama, N., Nagakura, M.,
Roylance, D., Pérez-Pevida, E., Brizuela-Velasco, A., Chávarri-Prado, D.,
Jiménez-Garrudo, A., Sánchez-Lasheras, F., Solaberrieta-Méndez, E.,
Diéguez-Pereira, M., Fernández-González, F.J., Dehesa-Ibarra, B., Monticelli,
F., Parashar, S.K., Sharma, J.K., Mohammed, S., Desai, H., Marci, P., Florian,
Z., Bart, S., Luan, Y., Guan, Z., Cheng, G., Li, W., Rungsiyakull, C., Field,
C., Lin, D., Zhang, L., Li, Q., Swain, M., Gröning, F., Fagan, M.,
O'Higgins, P., Evans, S.P., Parr, W.C.H., Clausen, P.D., Jones, A.,
Wroe, S., Du, J., Lee, J.H., Jang, A.T., Gu, A., Hossaini-Zadeh, M., Prevost,
R., Curtis, D.A., Ho, S.P., Dalla Pria Bankoff, A., Choi, A.H., Conway, R.C.,
Taraschi, V., and Ben-Nissan, B. (2012) ‘A Review of Improved Fixation Methods
for Dental Implants. Part II: Biomechanical Integrity at Bone-Implant
Interface’. Journal of Biomechanics online 2016 (5), 2702–2705.

available from

1 Membrane, plate, shell, etc…

2 3D fields can be: temperature, displacement, stress,
flow velocity, etc…

1 This report was submitted as for the completion
requirements of 208MAE/CWK1—practical finite element analysis.

3 Refer to chapter 4.1
for a point of reference.