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.

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.

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Modern Materials’. Nature Materials 10 (11), 823–837

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Mann, N., and Whitby, M. (n.d.) The Photographic Periodic Table of the

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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.