142977150 Structural Analysis Report

UNIVERSITI TEKNOLOGI MARA

STATIC ANALYSIS OF AN AIRCRAFT

WING STRUCTURE USING SUPERELEMENT

ABDUL MALIK HUSSEIN BIN ABDUL JALIL

Thesis submitted in fulfillment of the requirements for the degree of

Masters of Science

Faculty of Mechanical Engineering

July 2006

Abstract

This study describes the use of superelement for the stress and deflection

analysis of a typical modern fighter wing structure. Three methods of analyses were

carried out and compared. One is the theoretical analysis, the second is the finite

element analysis with the conventional finite element modeling approach and the

third is the finite element analysis with the superelement approach. The theoretical

analysis was divided into stress and deflection calculations. The stress analysis was

carried out using the simple beam theory. The deflection analysis was carried out

using the integration and energy method. For the finite element analysis, the finite

element models of the wing were developed. Both the Finite Element Analysis and

Superelement Analyses were performed using the NASTRAN finite element

software. CQUAD4 and BAR2 elements were used to represent the individual

components of the wing such as the skin and stringers. For the finite element

analyses using the superelement approach, the wing was divided to five (5) and

seven (7) substructures respectively known as superelements. Analyses were also

carried out by reducing the typical wing structure into the center and outer wing

without using superelements and the behavior observed. Partial Reanalysis was also

carried on one superelement that was modified. Wing loading at 1-g flight condition

was assumed. For all these methods, the direct stress and deflection are sought and to

be compared. The finite element analysis using the conventional approach produced

the same results as the finite element analysis using the superelement approach.

Running the partial re-analysis on one superelement reduced the analysis time greatly

as compared to running the analysis with the conventional approach, from 12.4

seconds reduced to 1.03 seconds.

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Candidate’s Declaration

I declare that the work in this thesis was carried out in accordance with the

regulations of Universiti Teknologi MARA. It is original and is the result of my own

work, unless otherwise indicated or acknowledged as referenced work. This topic has

not been submitted to any other academic institution or non-academic institution for

any other degree or qualification.

In the event that my thesis be found to violate the conditions mentioned above, I

voluntarily waive the right of conferment of my degree and agree to be subjected to

the disciplinary rules and regulations of University Teknologi MARA.

Name of Candidate: ABDUL MALIK HUSSEIN BIN ABDUL JALIL

Candidate’s ID No.: 2003307829

Programme: EM780

Faculty: MECHANICAL ENGINEERING

Thesis Title: STATIC ANALYSIS OF AN AIRCRAFT WING

STRUCTURE USING SUPERELEMENT

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Acknowledgements I would like to thank Allah s.w.t for his graciousness in giving me the strength

and will to fulfill the requirements of this thesis, my supervisor, Dr. Wahyu

Kuntjoro, if not for his kind patience and guidance, I will be definitely lost in my

quest for answers. I would like to extend my gratitude to Dr Assanah bin Mohd

Mydin, Managing Director of Caidmark Sdn. Bhd. for sponsoring and allowing me

time occasionally from work to pursue my studies, not forgetting my wife and son

for their sacrifice and understanding in order for me to complete my studies. I would

like to thank also staff from Msc Software Malaysia for their kind support and also

all at Uitm Shah Alam as without any of this support, I will be lost. May Allah s.w.t

bless all your kindness.

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Table of Contents Title Page i

Abstract ii

Candidate’s Declaration

Acknowledgements iii

Table of Contents iv

List of Tables vi

List of Figures vii

List of Abbreviation x

1. Introduction 1 1.1 Background 1 1.2 Problem Identification 2 1.3 Objective 4

2. Theoretical Considerations 5 2.1 Superelement Concept 5 2.1.1 Finite Element Analysis 5 2.1.2 Superelement Analysis 9

3. Literature Review 16

4. Methodology 30 4.1 Wing Loading at 1-g Symmetrical Level Flight Condition 35 4.2 Performing Theoretical Analysis on the Wing 36 4.2.1 Theoretical Stress Analysis 37 4.2.1.1 Theoretical Stress Analysis for Point 1 37 4.2.1.2 Theoretical Stress Analysis for Sections A, B, C and Wing Tip 38 4.2.2 Theoretical Deflection Analysis 41 4.3 Performing Finite Element Analysis Using Conventional Approach 44 4.3.1 Creating the Wing Model (Finite Element Model of the Wing) 44 4.4 Superelement Analysis 45 4.4.1 Performing Finite Element Analysis Using Superelement Approach 45 4.4.2 Creating Five (5) Superelements 45

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4.4.3 Creating Seven (7) Superelements 51 4.5 Performing Analysis on Two Separate Wing Structures 58 4.5.1 Theoretical Stress Analysis for Outer Wing 59 4.5.2 Theoretical Deflection Analysis for Outer Wing 60 4.5.2.1 Deflection at Section A Due to Bending Moment Using Energy Method 61 4.5.2.2 Deflection at Section A Due to Shear Force Using Energy Method 61 4.5.3 Theoretical Stress Analysis for Center Wing 62 4.5.4 Theoretical Deflection Analysis for Center Wing 62 4.6 Partial Re-Analysis on Typical Wing Structure 62 5. Results and Discussion 63 5.1 Typical Wing Structure 63 5.1.1 Stress Analysis Results for A Typical Wing Structure 63 5.1.2 Deflection Analysis Results for A Typical Wing Structure 65 5.2 Two Separate Wing Structures 67 5.2.1 Outer Wing Structure 67 5.2.1.1 Stress Analysis Results for Outer Wing Structure 67 5.2.1.2 Deflection Analysis Results for Outer Wing Structure 69 5.2.2 Center Wing Structure 72 5.2.2.1 Stress Analysis Results for Center Wing Structure 72 5.2.2.2 Deflection Analysis Results for Center Wing Structure 72 5.3 Partial Re-Analysis 73

6. Conclusion 75

7. Bibliography 78

8. Appendices 83 8.1 Appendix A 83 8.2 Appendix B 87 8.3 Appendix C 96 8.4 Appendix D 98 8.5 Appendix E 127 8.6 Appendix F 144 8.7 Appendix G 147

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List of Tables

1. Table 5.1: Forces Acting Along the Wing

2. Table 5.2: Theoretical Direct Stress Acting Along the Wing

3. Table 5.3: FEA (conventional approach) Direct Stress Acting along the Wing

4. Table 5.4: FEA (superelement approach) Direct Stress Acting along the Wing

with Five (5) Superelements

5. Table 5.5: FEA (superelement approach) Direct Stress Acting along the Wing

with Seven (7) Superelements

6. Table 5.6: Theoretical Displacement Acting Along the Wing

7. Table 5.7: FEA (conventional approach) Displacement Acting along the Wing

8. Table 5.8: FEA (superelement approach) Displacement Acting Along the Wing

with Five (5) Superelements

9. Table 5.9: FEA (superelement approach) Displacement Acting Along the Wing

With Seven (7) Superelements

10. Table 5.10: Theoretical Direct Stress Acting Along the Wing

11. Table 5.11: FEA (conventional approach) Direct Stress Acting Along the Wing

12. Table 5.12: Theoretical Displacement Acting along the Wing

13. Table 5.13: FEA (conventional approach) Displacement Acting along the Wing

14. Table 5.14: Results of Reaction Forces of Partial Outer Wing Structure

15. Table 5.15: Theoretical Stress Acting at Point 1 16. Table 5.16: FEA (conventional approach) Stress Acting at Point 1

17. Table 5.17: Theoretical Deflection Acting at Section c

18. Table 5.18: FEA (conventional approach) Deflection Acting at Section c

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List of Figures 1. Figure 1: Sequence in Finite Element Software Application

2. Figure 2: Example of A Complete Airplane

3. Figure 3: Airplane Broken Down to Six Levels of Substructures or

Superelements

4. Figure 4: Layout of Axial Bar Arrangement

5. Figure 5: Superelement 1 and 2 (Residual Structure)

6. Figure 6: Finite Element Model of the Axial Bar

7. Figure 7: Results of Forces acting at Point 1 and 2

8. Figure 8: Results of Displacement at Point 1, 2 and 3

9. Figure 9: Cessna’s Finite Element Model of Total Wing

10. Figure 10: Example of Superelements

11. Figure 11: Substructure or Superelement [22]

12. Figure 12: Rigid Body Dynamic Model of Rear Suspension [33]

13. Figure 13: Coupled Rigid Body and Flexible Body Dynamic Model of Rear

Suspension [33]

14. Figure 14: Finite Element Model of the Front Link and Knuckle [33]

15. Figure 15: Overall Finite Element Model of the Wing [34]

16. Figure 16: Finite Element Model of Wing Lower Skin [34]

17. Figure 17: Finite Element Model of the Crack Zone [34]

18. Figure 18: Configuration of A typical –Aero Engine [42]

19. Figure 19: Finite Element Model of Engine Casing [42]

20. Figure 20: The Global Model of the Space Shuttle Redesigned Solid Rocket

Motor (RSRM) [50]

21. Figure 21: Finite Element Model of an Anti-Ship Missile [51]

22. Figure 22: Finite Element Model of A Heat Exchanger [58]

23. Figure 23: End Plate Structure of the Heat Exchanger [58]

24. Figure 24: The Modified Structure of the End Plate [58]

25. Figure 25: Layout of the Aircraft [59]

26. Figure 26: Superelement of the Fuselage [59]

27. Figure 27: Superelement of the Right Wing [59]

28. Figure 28: Layout of F/A-18 Aircraft

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29. Figure 29: Structure of Wing

30. Figure 30: Methodology for Comparison of Analysis for a Typical Wing

Structure

31. Figure 31: Wing Model Showing Ribs Section 1 to Section 9

32. Figure 32: Sections of Analysis for Typical Wing Structure

33. Figure 33: Deriving the Bending Moment

34. Figure 34: Reaction Forces Acting at the Wing-Lug Joints

35. Figure 35: Cross Section of Stringer

36. Figure 36: Cross Section of Section 1

37. Figure 37: Theoretical Deflection Diagram

38. Figure 38: Finite Element Model of the Typical Wing Structure

39. Figure 39: Superelement 1: Aft Center Beam

40. Figure 40: Superelement 2: Middle Center Beam

41. Figure 41: Superelement 3: Forward Center Beam

42. Figure 42: Superelement 4: Inner Half Wing

43. Figure 43: Superelement 5: Outer Half Wing




47. Figure 47: Superelement 4: Inner Quarter Wing

48. Figure 48: Superelement 5: Center Quarter Wing 1


50. Figure 50: Superelement 7: Outer Quarter Wing

51. Figure 51: Finite Element Model of Outer Wing Structure

52. Figure 52: Finite Element Model of Center Wing

53. Figure 53: Sections of Analysis for Outer Wing

54. Figure 54: Sections of Analysis for Deflection of Outer Wing


56. Figure 56: Direct Stress Distribution of A Typical Wing Structure

57. Figure 57: Comparison of Direct Stress Distribution of A Typical Wing

Structure

58. Figure 58: Displacement Distribution of A Typical Wing Structure

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59. Figure 59: Comparison of Displacement of A Typical Wing Structure

60. Figure 60: Direct Stress Distribution for Outer Partial Wing

61. Figure 61: Comparison of Stress Results of Outer Wing Structure

62. Figure 62: Displacement Distribution of Outer Wing

63. Figure 63: Comparison of Deflection Results of Outer Wing Structure

64. Figure 64: Reaction Force in the x-Direction

65. Figure 65: Reaction Force in the y-Direction

66. Figure 66: Reaction Force in the z-Direction

67. Figure 67: Stress Distribution of A typical Wing Structure after Partial Re-

Analysis

Appendices

68. Figure 68: Stress Analysis at Point 1

69. Figure 69: Reaction Forces at Fuselage Support

70. Figure 70: Reference for Deflection Analysis

71. Figure 71: Description of Sections

72. Figure 72: Creating a New Group

73. Figure 73: To View New Group Created

74. Figure 74: To Associate the Nodes to the Elements in the Group ‘s1’

75. Figure 75: Creating Superelement ‘sp1’




79. Figure 79: Superelement 4: Inner Half Wing

80. Figure 80: Superelement 5: Outer Half Wing




84. Figure 84: Superelement 4: Inner Quarter Wing




88. Figure 88: Selecting Superelement Subcase

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1.5 List of Abbreviation

1. VPD – Virtual Product Development

2. FEA – Finite Element Analysis

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1. Introduction 1.1 Background The Finite Element method synthesizes complicated structural systems as a

connected collection of objects, called finite elements that embody local physical

laws [1]. The use of finite element analysis has made its way to a stage where they

are widely used in various engineering applications and are improving steadily over

the past decade. Engineers are able to predict the behaviour of these elements as it

would be in the form of mathematical models which will then be solved, resulting in

a set of linear algebraic equations. There are many references that can be found to

better understand the concept of using finite element as an analysis tool [2,3]. It is a

form of numerical analysis which can be used for stress prediction and also to

perform structure optimization [4, 5]. Although the name ‘finite element method’

was a recent invention, the application was put to use much earlier [6]. There are

many finite element softwares that can be found in the market today, such as,

LUSAS, ANSYS and NASTRAN [7-10]. These softwares has capabilities from low

to sophisticated usage combined with excellent graphics capabilities. This thesis

utilizes the NASTRAN finite element software.

One of the key studies that contributed to the finite element method used

today is that carried out by Turner and colleagues in 1956 [11]. This involves the use

of simple finite elements (pin jointed bars and triangular plates with in-plane-loads)

to analyze aircraft structures. From then on, the development of finite element has

expanded to carryout stress analysis [12], all kinds of field problems that can be

formulated into variational form [13] and also in fluid mechanics [14]. In general,

there are three approaches that can be used to solve various finite element problems.

They are the direct equilibrium method, work or energy methods and weighted

residual methods.

Three terms are often used in application of the finite element software which

are pre-processor, solution process and post-processor, as in Figure 1. Pre-processor

is the process of preparing the geometry, selecting the elements, discretization of the

domain, selection of materials, applications of loadings and the specifications of

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boundary conditions. Based on these inputs, the software will set up the equations

which will be solved through the solution process. Post-processor is where the user

can evaluate the stress distribution, structural displacements, pressure distribution or

heat flux distribution.

2

Pre-Processor

Solution

Post-Processor

User Input

Output Presentation

Figure 1: Sequence in Finite Element Software Application 1.2 Problem Identification Superelements are defined as grouping of finite elements that, upon

assembly, maybe considered as an individual element for computational purposes. It

is an analysis procedure that supports collaborative analysis and is very useful for

large models that are developed by different organizations.

Figure 2: Example of A Complete Airplane

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Figure 2 shows an example of a complete airplane. To analyze this model

using the Finite Element Analysis with the conventional approach will take a very

long time as it has a very large number of degrees of freedom.

Figure 3: Airplane Broken Down to Six Levels of Substructures or

Superelements

Figure 3 shows the same airplane model being divided into six different

substructures or superelements to ease the analysis. Here, each superelement can be

analyzed individually, hence saving analysis time should there be a modification

made.

These reduce matrices for the individual superelements are combined to form

an assembly solution. The results of the assembly are then used to perform data

recovery (calculations of stresses, displacement etc) for the superelements [15].

Superelements allow a big, complex structure to be analyzed, by dividing this

structure to individual components. These individual components will be analyzed

and then assembled together to produce a complete analysis results.

Superelements can consist of physical data (elements and grid points) or can

be defined as an image of another superelement or as an external superelement (a set

of matrices from an external source to be attached to the model). The image

superelement can save processing time in that they are able to use the stiffness, mass

and damping from their primary superelement, which reduces the amount of

calculations needed. Full data recovery is available for image superelements. An

image superelement can be an identical image or a mirror image copy of the primary.

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The other type of superelement is the external superelement where a part of a model

is represented by using matrices of an outside source. For these matrices, no internal

geometry information is available only the grid points to which the matrices are

attached are known.

Cardona [16] states that the main advantage of sub structuring techniques is

to allow the detailed modeling of components with complex geometry and structural

functions while keeping a relatively simple global dynamic model with a number of

degrees of freedom as small as possible. Other advantages of using superelements

includes the ability to solve problems using components that exceeds computer

resources for a single large analysis, partial redesign only requires a partial

reanalysis, supports local/global analysis allowing the analyst to refine the model in

important regions of the structure allows multiple level of sub structuring for

dynamic analysis.

The application of finite element analysis within the aircraft industry has

mainly concentrated on providing an inside into both detail and structural behavior.

The testing of structures still forms a large part of the design and qualification

process, with analysis providing additional information to support these activities

[17]. The next era would be to provide detailed simulation of a structure where such

testing programmes can be significantly reduced.

1.3 Objective The objective of this study is to carry out the stress and deflection

analysis on a typical fighter aircraft wing structure using superelement. In order to

achieve the objective, three methods of analyses were carried out and compared. One

is the theoretical analysis, the second is the finite element analysis with the

conventional approach and the third is the finite element analysis using the

superelement approach.

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2. Theoretical Considerations 2.1 Superelement Concept

The superelement concept is described in detail using the axial bar

arrangement as described below. Finite Element Analysis is first conducted, followed

by the superelement analysis and compared.

5

2000 N

2m 2m 2m 2m

5 4

3000 N

1 2 31 2 3 4

1000 N

Figure 4: Layout of Axial Bar Arrangement Figure 4 shows the layout of the axial bar arrangement. The finite element

analysis of the axial bar using the conventional approach was carried out. For the

finite element analysis using the superelement approach, the axial bar was divided

into 2 superelements. The analysis using the MSC.FEA 2003 software was also

carried out.

2.1.1 Finite Element Analysis For the conventional finite element analysis, the elements are arranged in the

form of : f = k.u

E = 72e3 N/m2 µ = 0.3

A = 1 m2

LAE =

210^3721 ×× = 36 × 103 .. 2.1

k = 36 × 103

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Element 1

⎭⎬⎫

⎩⎨⎧

x

x

ff

2

1 = 36 × 103 .. 2.2 ⎥

⎦

⎤⎢⎣

⎡−

−1111

⎥⎦

⎤⎢⎣

⎡2

1

uu

Expanding the equation:

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

x

x

x

x

x

fffff

5

4

3

2

1

= 36 × 103 .. 2.3

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡−

−

0000000000000000001100011

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

5

4

3

2

1

uuuuu

Element 2

⎭⎬⎫

⎩⎨⎧

x

x

ff

3

2 = 36 × 103 .. 2.4 ⎥

⎦

⎤⎢⎣

⎡−

−1111

⎥⎦

⎤⎢⎣

⎡3

2

uu


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

x

x

x

x

x

fffff

5

4

3

2

1

= 36 × 103 .. 2.5

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

0000000000001100011000000

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

5

4

3

2

1

uuuuu

Element 3

⎭⎬⎫

⎩⎨⎧

x

x

ff

4

3 = 36 × 103 .. 2.6 ⎥

⎦

⎤⎢⎣

⎡−

−1111

⎥⎦

⎤⎢⎣

⎡4

3

uu


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

x

x

x

x

x

fffff

5

4

3

2

1

= 36 × 103 .. 2.7

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

0000001100011000000000000

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

5

4

3

2

1

uuuuu

6

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Element 4

⎭⎬⎫

⎩⎨⎧

x

x

ff

5

4 = 36 × 103 .. 2.8 ⎥

⎦

⎤⎢⎣

⎡−

−1111

⎥⎦

⎤⎢⎣

⎡5

4

uu


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

x

x

x

x

x

fffff

5

4

3

2

1

= 36 × 103 .. 2.9

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−1100011000

000000000000000

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

5

4

3

2

1

uuuuu

Combining the Matrix for the Structural Equation

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

x

x

x

x

x

fffff

5

4

3

2

1

= 36 × 103 .. 2.10

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−

−−−

−

1100012100

012000012100011

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

5

4

3

2

1

uuuuu

These matrices equations are then combined to form the structural equation.

The boundary conditions are then applied, u1 = u5 = 0. Then, the forces are applied to

the equations. The equations are then solved to calculate the displacement that occurs

at the respective elements. The values of these displacements are then applied to the

individual element equations to calculate the reaction forces at each node.

Applying the Boundary Conditions, u1 = u5 = 0

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

x

x

x

fff

4

3

2

= 36 × 103 .. 2.11 ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−

210121

012

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

5

4

3

uuu

Applying the Forces

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

NNN

300020001000

= .. 2.12 ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−

3723360336372336

0336372

EEEEE

EE

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

4

3

2

uuu

Solving the Equations

1000 = (72E3×u2) – (36E3×u3) .. 2.13 7

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2000 = (-36E3×u2) + (72E3 × u3) .. 2.14

3000 = (-36E3×u3) + (72E3 × u4) .. 2.15

From 2.13,

u2 = 372

)336(1000 3

eue ×+ = 0.014 + 0.5u3 .. 2.16

Insert 2.16 into 2.14,

2000 = -36e3(0.014 + 0.5u3) + (72E3×u3) – (36E3×u4) 2000 = -504 – 18000u3 + 72000u3 – 36000u4

= 54000u3 – 36000u4 -504

u3 = 54000

504200036000 4 ++u = 0.67u4 + 0.046 .. 2.17

Insert 2.17 into 2.15,

3000 = -36E3(0.67u4 + 0.046) + 72E3u4 3000 = -24120u4 – 1656 + 72000u4

= 47880u4 – 1656

u4 = 47880

16563000 + = 0.097 m = 97 mm

Insert value of u4 into 2.17,

u3 = 0.67(0.097) + 0.046 = 0.111m Insert u3 into 2.16,

u2 = 0.014 + 0.5(0.111) = 0.014 + 0.056 = 0.0695 = 69.5mm Calculating Reaction Forces

f1x = 36 × 103 [u1 – u2] = 36 × 103 [0 – 0.0695] = -2502 N ( - sign indicating direction)

f5x = 36 × 103 [-u4 + u5] = 36× 103 [-0.097 + 0] = 3492 N

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Total Reaction Forces:

f1x + f5x = 3492 + 2502 = 5994N, which is about 6000N, initial forces applied. 2.1.2 Superelement Analysis

9

Superelement 1 Superelement 2

Figure 5: Superelement 1 and 2 (Residual Structure) Figure 5 shows the division of the axial bar into two (2) superelements: Structural Matrix for Superelement 1 (Node 1 and Node 3)

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

321

fff

= L

AE .. 2.18 ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−

110121

011

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

3

2

1

uuu

Applying boundary conditions, u1 = 0, the boundary of the elements of the

equations are transformed. These transformed matrices will then be reduced.

Boundary conditions are applied to the forces equations and the stiffness matrices are

reduced to the boundary to form superelement 1.

When applied boundary conditions, u1 = 0, the stiffness matrices will be reduced:

⎭⎬⎫

⎩⎨⎧

3

2

ff

= L

AE⎥⎦

⎤⎢⎣

⎡−

−1112

.. 2.19 ⎥⎦

⎤⎢⎣

⎡3

2

uu

This reduced stiffness matrix will be in the form of:

LAE

⎥⎦

⎤⎢⎣

⎡−

−1112

= 36 × 103 .. 2.20 ⎥⎦

⎤⎢⎣

⎡=−=−==1112

ttto

otoo

KKKK

2000 N

2m 2m 2m 2m

1000 N 3000 N

1 2 3 4 1 2 3 4 5

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These values will then be used in the calculation based on the formulas

provided.

Boundary Transformation This boundary transformation is required to reduce the stiffness matrix from

the finite element to the superelement form. The formula to obtain the boundary

transformation [Got] for superelement 1 is as follows:

[Got] 1 = 36 × 103 (-Koo-1 × Kot) .. 2.21

= 36 × 103 (-21 × -1 )

= 3^1072

1×

× 36 × 103

= 0.5

Once the boundary transformation value is obtained, the stiffness matrix is reduced

further based on the given formula.

Reduce Stiffness Matrix to Boundary

[Ktt] 1 = 36 × 103 [ + KotT . Got] .. 2.22

−

ttK = 36 × 103 [1 + (-1 . 0.5)] = 18 × 103 Resulting Stiffness as seen in grid point 3 = 18 × 103 Besides the stiffness matrix, the forces applied to the finite element structure

too need to be reduced from the finite element form to the superelement form using

the formula provided.

The formula for applying boundary conditions to the forces is as follows:

Pf1 = [

1

2

PP ] = [

01000 ] .. 2.23

This is reduced to the boundary in the superelement form using the following formula:

10

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= [Pt] 1 = [ + GotT . Po] .. 2.24

−

tP

11

= P31 = [ 1 + (0.5).(1000)] = 500N

−

3P

0

Structural Matrix for Superelement 2 (Node 3 and Node 5)

[Kgg] 2 = L

AE

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−+−

−

4545

45453434

3434

0

0

KKKKKK

KK .. 2.25

= 36 × 103 .. 2.26 ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−

110121

011

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

5

4

3

uuu

Applying Boundary Condition, u5 = 0, the boundary of the elements of the

equations are transformed. These transformed matrices will then be reduced.

Boundary conditions are applied to the forces in the stiffness equations and the

stiffness matrices are reduced to the boundary to form superelement 2.

Applying Boundary Condition, u5 = 0

[Kgg] 2 = 36 × 103 .. 2.27 ⎥⎦

⎤⎢⎣

⎡−

−2111

⎥⎦

⎤⎢⎣

⎡4

3

uu

The next step is to apply the boundary conditions to the forces equations for

superelement 1 and 2. The procedure for this is similarly applied as superelement 1.

Boundary Transformation

[Got] 2 = )( 4534

34

KKK+−

− = 3^10723^1036

×−×− = 0.5 .. 2.28

Reduce Stiffness Matrix to Boundary

[Ktt] 2 = 4534

4534.KK

KK+

= 3^10363^10363^10363^1036

×+×××× = 18×103 .. 2.29

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Resulting Stiffness as seen in grid point 3 = 18 × 103 Applying boundary conditions to the forces:

Pf1 = [

5

4

PP ] = [

03000 ] .. 2.30

[P3]2 = [ + . Po] .. 2.31 −

tP−

otG

12

= P3

2 + (0.5).(3000) = 1500N

Once the reduce matrices for the superelement 1 and 2 have been formed, the

model is now treated as a residual structure with the following equation;

For Remaining Grid Point 3 Residual Structure (Figure 5)

[Kgg] = [Kaa] = [Kaa1 + Kaa

2 + Kggo] .. 2.32

Kaa

1 & Kaa2 = [Ktt] 1 & [Ktt] 2, reduced matrix for superelement 1 and 2. Kgg

o

represents the stiffness matrix of any element in the structure. Since there are no

elements, Kggo = 0. The force is also created for the boundary condition.

K = Kaa1 + Kaa2 = 18×103 + 18×103 = 36×103 .. 2.33

For Force:

Pt = F1 + F2 + F3 .. 2.34 = 500 + 1500 + 2000 = 4000N The following formula is applied when retrieving the Solution Retrieving Solution Residual Structure (Figure 5)

ua = Ktt-1. Pt .. 2.35

0

0

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u3o =

3^10364000×

= 0.11 m .. 2.36

Superelement 1

Fixed Boundary Solution = uo = [Koo]-1 {P2} .. 2.37

u20 =

3^10721×

× 1000 = 0.014m

Solution for Boundary Motion = u2 = [Got] 1 . u3 .. 2.38

= 0.5 × 0.11 = 0.055m

u2 = 0.055 + 0.014 = 0.069m Superelement 2

Fixed Boundary Solution = u4o = [Koo]-2 {P4} .. 2.39

=3^1072

1×

× 3000 = 0.0417m

Solution for Boundary Motion = u4

a = [Got] 2 . u3 .. 2.40 = 0.5 × 0.11 = 0.055m

u4 = 0.055 + 0.014 = 0.0967m Finite Element Analysis Using MSC.FEA 2003 (Nastran)_Software

Point 1 Point 2 Point 3

Figure 6: Finite Element Model of the Axial Bar

Figure 6 shows the finite element model of the axial bar as calculated in

section 1. This model was also divided into two superelements. The finite element

13

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Documents

142977150 Structural Analysis Report