# 3D-finite Element Analysis of Beam Design

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### Abstract:

Any design and development activities involves in huge amount of time and money in bringing out the final product to the market, whilst functionality of the product being crucial under all scenarios without fail or malfunctioning over a period of time.

Earlier design was carried out by the conventional methods from planning to final manufacturing of a components and the behavior of the product was understood only when it was not meeting its functionality.

Recent developments in the above said area is vast, as this enables an engineer to study the behavior of a component/assembly, whist suggesting precautionary measures or a possible solution in validating the member thereby saves an organization time and effort.

Thanks to the recent developments in the field of Stress analysis, along with the CAD packages, which actually enable us to visualize the component in 3D and analysis and design, validate it before it is actually released for manufacturing.

Furthermore the robustness of CAE packages enables us to visualize the behavior of the component/assembly when it is actually put to work defining constrains under which it has to perform.

Industries strongly rely on these packages to reduce the time and money involvement of a company and it is important for an Engineer to adapt the methods presented in this paper in the right approach so as to meet the design criteria which should be practical in nature.

### Introduction:

This paper demonstrates the Design of a beam which has to be validated under several constrains/operating conditions, and understanding its behavior under these real time situations.

Application of Stress methods using "Solid Works Simulation" package is demonstrated to understand the behavior of the beam.

3D Finite element analysis is one of the approaches in understanding the behavior of the load paths under different situations and with different boundary conditions.

Several beam sections are validated to design the best beam under the given load conditions and the best beam based on several criteria are made, by demonstrating several plots.

Hand/Theoretical calculations and results from Simulation are interpreted in order to study the behavior of the beam.

Methods of this Stress Simulation and relevant steps are explained by plotting various plots like the Stress, Displacement and Factor of Safety by relevant comments at certain stages are done for the company to understand the process and design validation.

Further it is important for the safety engineer to understand the usage of 3D finite element method so as to interpret the results and to make design changes before the component being put it function.

### Beam analysis: [Part 1]

The figure below shows the beam on which the loads are acting at points P1, P2 and P3 of magnitude 18KN, 26KN and 20KN respectively.

Beam 1 and 2 are bolted with pins through the two beams and the beam is supported at two locations. Analyzing the above situation, several considerations are needed in order to apply and analyze the situation. The above situation is a case of "simply supported beams at either ends and loaded at the center".

### Design phase:

The given sections are designed using Solid works package as per the dimensions provided.

The cross-section of beams designed is plotted below.

Consider the cross-section 1 for analysis.

Below shows the cross-section 1 with dimensions being A= 0.3m, B=0.3m respectively.

### 3D element solid element type analysis using finite element method:

### Cross-section 1: [Beam with circular hole]

As shown above the assembly is created using solid works as "Solidworks.asm" format and is meshed and analysis is carried out. Several steps are carried out like constrains, load conditions, assigning material are done in order to study the behavior of the assembly.

Load points are defined at three locations as shown; either of the beams is connected by means of metal pads of 3mm thick with pins to support them. As we apply the loads at points P1, P2 and P3, simulation is carried out and a report on the desired results is obtained and are plotted below.

Further to the design of the beam with relevant dimensions, simulation of the assembly is carried out using Solid works simulation. Several boundary conditions are implied, like the loads at the given locations, applying material, bolts at four locations and finally meshing the assembly to perform the analysis.

Repeating the above procedure for rest of the cross-sections for design of beam, following plots will account for the values of Von-misses stress, displacement and factor of safety.

### Deflection Calculations:

From the bending moment diagram, we observe that the maximum deflection occurs at the centre of the beam. The maximum load due to all the three loads can be found out. By using the Principle of Superposition, the deflection due to each load can be interpolated to the centre.

Consider a load 'P' acting on a beam AB at a distance of 'a' from end 'A' as shown in figure. The bending moment plot shown in figure above, shows a discontinuity at the point x=a.

Solving for each of the lengths of the beam

For length AD,

(d2y /dx2) = (M/EI) = (Pbx/EIL) --------1

Integrating equation 1, we get,

y = (Pbx3/6EIL)

For length DB,

y = (Pax3/2EI) - (Pax3/6EIL) + B1x + B2

To determine the four constants A1 and A2, two boundary conditions and two continuity conditions are used.

For segment AD, y (0) = 0 = A2

For segment DB,

y (L) = 0 = (PaL2/3EI) + B1L + B2

Equating the deflections and slope on both segments at x=a, and solving the four equations, we get,

A1 = - (Pb/6EIL) (L2 - b2)

A1 = 0

B1 = - (Pa/6EIL) (2L2 + a2)

B2 = - (Pa3/6EI)

Hence we get the following equation, for length "AD"

y = (Pbx/6EIL) (x2 - L2 + b2) ....... (2)

Considering the load P1 = 18KN, the deflection at midpoint, we have,

P = 18000N, x = 1.4m, b = 1.9m, L = 2.8m, E = 220 X 109N/m2. Substituting these values in equation (2), we get

y = (2.9407 X 10-8) / I m

Hence, below are the values

- For cross section 1: y1circle = 0.04523mm
- For cross section 2: y1oct = 0.0454mm
- For cross section 3: y1sqr = 0.0465mm
- For cross section 4: y1isect = 0.06022mm

For segment AD, using the expressions obtained for B1 and B2 in the deflection equation, we get,

y = - (Pa/6EIL) [(x3/2) - (x3/6L) - {x (2L2 + a2)/6L} + (a2/6)] --------------2

Considering the load P2 = 20KN, deflection at mid point can be calculated using,

P = 20000N, x = 1.4m, a = 1.7m, L = 2.8m, E = 220 X 109 N/m2.

Substituting the above values in equation (2), the deflection at mid point D is found to be:

y = (2.2074 X 10-8)/I m

Hence,

- For cross section 1: y2circle = 0.03395mm
- For cross section 2: y2oct = 0.0341mm
- For cross section 3: y2sqr = 0.0349mm
- For cross section 4: y2isect = 0.0452mm

Similarly, considering the load P3 = 26000N, deflection at mid point is,

y = (54.0484 X 10-9)/I m

Hence,

- For cross section 1: y3circle = 0.0831mm
- For cross section 2: y3oct = 0.0835mm
- For cross section 3: y3sqr = 0.0854mm
- For cross section 4: y3isect = 0.1107mm

Total deflection is given by:

y = y1 + y2 + y3

Hence,

- For cross section 1: y = 0.1622mm
- For cross section 2: y = 0.1630mm
- For cross section 3: y = 0.1668mm
- For cross section 4: y = 0.2161mm

### Factor of safety for the beams.

Factor of safety is given by the formula:

FOS = ?yield / ?max

Given, yield stress of the material, ?yield = 650N/mm2

Using the above data, we get,

- For cross section 1: FOS = (650/8.6) = 75.58
- For cross section 2: FOS = (650/8.64) = 75.23
- For cross section 3: FOS = (650/8.84) = 73.53
- For cross section 4: FOS = (650/11.46) = 56.72

By the above results, the cross section with the highest FOS can be chosen for designing the beam. Hence it can be recommended to choose the cross section with circular hole for final design.

### Part 2

The zone is red color is critical, means it has high stress and displacement. Hence clamping used will play a major role.

From the plot, the maximum displacement at this location is 0.6511 mm, which is less than the customer's expectations and hence the design is safe.

As this displacement is almost 3.8 times of the specified value [2.5mm], no design changes or precautions would be needed. Therefore,

Maximum displacement < Specified value.

### Part 3

Finite element method is one of the methods widely used and applied among the industries in the recent years and is used to study the behavior of the part by assigning various properties on to it.

### Method of simulation:

Static studies in Solid works simulation calculate displacements, reaction forces, strains, stresses, failure criterion, factor of safety, and error estimates. Available loading conditions include point, line, surface, acceleration (volume) and thermal loads are available.

Below criteria are important and are followed in this document so as to obtain values which are realistic in nature;

The approach is done in three phases and are,

- Bottom up assembly-Phase 1
- Defining load points-Phase 2
- Simulation-Phase 3

### Phase 1.

Assembly of beams with relevant dimensions was done with fully defining the sketch geometry.

Generating bosses with desired lengths and creating the profile as needed.

Mates being defined between each parts using mate options in assembly mode.

### Phase 2.

Split of 10mm was done at the top surface of the beam was done in order to imply point loads.

### Phase 3.

Solid works simulation tool was used to access the simulation options.

Steel was applied from the material database for all the components in the assembly.

Connections were defined so as to make the assembly a rigid structure by defining the locations and this creates an effect of holding both the beams by means of bolts.

Fixtures create an effect of holding the beam as required and are done at the either ends.

Loads in terms of Newton were applied on to the points which were defined at phase 2.

Mesh size was defined for the entire assembly and this inturn divides the geometry and several nodes are created for analysis.

Finally the meshed model will provide us the study report, Von-misses stress, Factor of safety and Displacement of all the four cross-sections are obtained.

### Possible mistakes in simulation:

It is up to the safety engineer in order to take extreme care before the analysis is performed so as to avoid the failure or inaccurate results during or before the simulation is actually performed. Mistakes should be avoided to the maximum extent while conducting simulation, as this might deviate the results and are not practical in nature and hence lead to misinterpretation.

Some of them are listed below.

- Applying the material: This result in wrong stress and strain plots, displacement plots, Factor of safety, this inturn results in wrong load path distribution.
- Defining boundary conditions: Defining boundary conditions is crucial in terms of accurate results. Loadings should be done as per the real situation and unwanted assumptions have to be avoided.
- Generating mesh: Applying mesh is one of the important criteria as this procedure being the base on which the elements of the member or the beam is divided into several millions of individual pieces and are analyzed by applying degrees of freedom.
- Mesh size: Mesh size is important in order the material/component to take the load conditions. Larger mesh could result in small deflection and results may not be practical in nature.
- Clamping face: Wrong clamping face in simulation would completely alter the end result and this leads in wrong interpretation of the results obtained from the stress plot, displacement plots and Factor of safety.

### Mesh sizes and types:

Solid works simulation currently includes solid continuum elements, curved surface shell elements (thin and thick) and truss and frame line elements. The shells are triangular with three vertex nodes or three vertex and three mid-edge nodes. Solids are tetrahedral with four vertex nodes or four vertex and six mid-edge nodes. They use linear and quadratic interpolation for the solution based on whether they have two or three nodes on an edge. The linear elements are also called simplex elements because their number of vertices is one more than the dimension of the space. The size of each element indicates a region where the solution is approximated by a spatial polynomial. Most finite element systems, including SW Simulation, use linear or quadratic complete polynomials in each element. You can tell by inspection which is being used by looking at an element edge. If that line has two nodes the polynomial is linear.

If it has three nodes then the polynomial is quadratic.

When the model is set for simulation, the program sub-divides the model into many tetrahedral small elements, these small points share a common point called as "NODE". Below shows the small element where a common node is shared by curves, lines and edges.

### Difference between hand calculation and simulation:

Few difference do exists between theoretical and hand calculations.

Hand calculations: Hand calculations are often called as theoretical calculations, because of the fact that it does not take into consideration of several constrains could not be defined as we could do it in simulations.

- Material cannot be assigned in hand calculations.
- Mesh cannot be created for better and accurate result.
- Deflection, stress plot, displacement plots could not be visualized in hand calculations.
- Several assumptions might be required and thereby accounts in the deviation of the result from that of simulation.
- Hand calculations are based on the available formulae like from the design data hand book and are not different compared to simulation results.
- Result analysis like the animation of the result and high stress regions could not be obtained from hand calculations.
- Changes in boundary conditions would require repeating the procedure in hand calculations and time consuming process.

### Conclusion:

Study of 3D-Finite element analysis of beam design assembly, address the capabilities of simulation. The idea of using the presented methods and techniques helps in optimizing the product before manufactured.

This helps an industry in being changing their design at this stage based on the results obtained from simulation. Simple to complex parts/assemblies are simulated by this method, by defining several boundary conditions.

The advancement in FEA area is vast, and has the capabilities of creating an environment of real time engineering situation and much finer results could also be obtained, as it provides options for finer mesh and hence more accurate the results.

Finally this method of optimizing or validating the product at the initial level before design is done, has its own advantages, whilst it is worth understanding the customer's requirement along with understanding the basic concepts of FEA makes a worth effort towards any engineering problem.

Hence I strongly suggest for any organization to follow the process of FEA and get the full benefit of the same, as they could save time in the process of optimization of the product.

### References:

- Class tutorial.
- S Timo shenko and D H young. 5th Edition. Elements of strength of materials.
- Strength of materials by Bela I. Sandor.
- Solid works study material.
- Strength of materials by Ferdinand L. Singer and Andrew Pytel, 3rd Edition
- Strength of materials by Surya Patnaik and Dale Hopkins, Title: A new unified theory for the 21st century.