modeling, analysis and optimization of foldable structure using finite element method

Modeling, Analysis and Optimization of Foldable

Structure using Finite Element Method

By: Omar Eladel Mahmoud Mechatronics Engineering Department

Supervised by: Dr. Adel Elsabbagh Dr. Wael Nabil Akl

2

Modeling, Analysis and Optimization of Foldable

Structure using Finite Element Method

By: Omar Eladel Mahmoud Mechatronics Engineering Department

Supervised by: Dr. Adel Elsabbagh Dr. Wael Nabil Akl

3

Abstract

A linear finite element model is developed for a foldable structure (tent) used in regions prone to natural disasters. The developed model is studied and analyzed, and is structurally optimized using genetic algorithms to find the minimum weight for the proposed structure.

Keywords: Foldable Structures, Finite Element Analysis, Genetic Algorithms, Structural

Optimization, Structural Mechanics.

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Acknowledgements

First and foremost, I would like to thank my supervisors, Dr. Adel Elsabbagh and Dr. Wael Akl. Without their knowledge, experience, and support, this thesis would have never been accomplished.

I would also like to thank all my professors and teaching assistants who have helped me and showed a lot of effort and patience to help us be good engineers during the past five years.

I would like to thank Eng. Islam Helaza and Eng. Khaled Youssef who have helped me a lot in using the new software which I learned and used during my work.

I would like to thank Eng. Ahmad Rashied, who is my mentor to engineering, as he taught me a lot in engineering and life. I would also like to thank my friends for their help and support during my work.

Most importantly, none of this could have happened without the support of my parents and my lovely sister. May Allah bless them with happiness, togetherness and love.

Finally, all praise is due to Allah, who made all things happen.

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Contents

Abstract ..................................................................................................................................... 3

Acknowledgements................................................................................................................... 4

Chapter 1: Introduction and Literature Review ........................................................................ 7

Foldable Structures ............................................................................................................... 7

Deployable structures based on pantographs .................................................................. 8

Different Examples of Foldable Structures ..................................................................... 10

Aims and Scope of Work ..................................................................................................... 12

Outline of Thesis ................................................................................................................. 12

Chapter 2: Mechanical Design and Construction ................................................................... 14

Construction of Unit............................................................................................................ 15

Unit Dimensions .............................................................................................................. 17

Construction of Line ............................................................................................................ 17

Construction of Hinges........................................................................................................ 17

Folded Structure ................................................................................................................. 18

Dimensions of folded tent .............................................................................................. 19

Construction Material ......................................................................................................... 19

Chapters 3: Loads and Structural Analysis .............................................................................. 21

Symbols ............................................................................................................................... 22

Dead Load ........................................................................................................................... 22

Wind Load ........................................................................................................................... 22

Summary of Wind Loads ..................................................................................................... 24

Chapter 4: Finite Element Modeling ....................................................................................... 26

Finite Element Method ....................................................................................................... 26

Description of the Model .................................................................................................... 26

Model Summary .................................................................................................................. 27

Hinged Joints ....................................................................................................................... 28

Finite Element Model .......................................................................................................... 28

Load Case 1 ..................................................................................................................... 29

Load Case 2 ..................................................................................................................... 32

Analysis Results ................................................................................................................... 33

Chapter 5: Structural Optimization ......................................................................................... 35

Types of Structural Optimization ........................................................................................ 35

Genetic Optimization Algorithm ......................................................................................... 36

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Advantages of Genetic Algorithms ................................................................................. 36

Disadvantages of Genetic Algorithms ............................................................................. 37

Algorithm Flowchart ........................................................................................................... 37

Optimization Software ........................................................................................................ 38

Optimization Runs ............................................................................................................... 39

Run 1 ............................................................................................................................... 39

Run 2 ............................................................................................................................... 40

Run 3 ............................................................................................................................... 41

Run 4 ............................................................................................................................... 42

Optimization Results ........................................................................................................... 43

Run 1 ............................................................................................................................... 43

Run 2 ............................................................................................................................... 45

Run 3 ............................................................................................................................... 46

Run 4 ............................................................................................................................... 48

Recommendations and Further Work .................................................................................... 51

Unit Construction ................................................................................................................ 51

Optimization ....................................................................................................................... 52

References .............................................................................................................................. 54

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Chapter 1: Introduction and Literature Review

Foldable structures belong to a larger family of structures, Deployable Structures. Deployable structures are classified into: Foldable Structures, Inflatable Structures. Foldable structures are those types of structures which can be folded and unfolded while they are fully assembled.

Deployable structures use has increased tremendously in the last decade, as it is used in space applications (deployable antennas and solar panels), in modern furniture (chairs, tables), and also in regions prone to natural disasters, as it can be used in deploying refugee and emergency camps in few hours, and in building temporary bridges.

This chapter introduces the concepts behind the thesis and gives the reader the knowledge needed about foldable structures and how they are designed and analyzed and then shows some of the previous work done on this topic.

Foldable Structures

A large group of structures have the ability to transform themselves from a small, closed or stowed configuration to a much larger, open or deployed configuration. These are generally referred to as deployable structures though they might also be known as erectable, expandable, extendible, developable structures [Jensen, 2003].

The main reason for recent interest in deployable structures has been their potential application in space. Launch vehicles which put satellites into space are limited in size. The largest presently in use is NASA's space shuttle; it has a cargo bay 4.6 m in diameter, and 18.3m long. Many present and proposed space missions require structures larger than this. One way of making a structure fit into the limited space available for launch is to fold it, and then to automatically deploy it once in space. Another option is to use erectable structures that are taken to space in pieces, and put together in orbit, either by man, or by robots. However, it has been realized that using men in space is not only dangerous, but also expensive, because extra safety precautions are required. Robots are not sufficiently advanced to be able to autonomously erect a complex structure in space. This leaves deployable structures as the only viable option for almost all large space structures.

Another reason for using deployable (or erectable) structures in space is the high loads due to vibration experienced by a structure during launch. By using a deployable structure, a lightweight structure can be safely packaged and protected during this critical time. There are a large number of possible applications which require large structures in space. These structures may be required for astronomy, earth observation, communications, or to provide solar power.

A common use of deployable structures is as deployable solar panels to provide the power requirements of modern satellites. The solar panels of the Hubble Space Telescope are a well-known example (Cawsey, 1982).

These types of structures have many advantages. Some important benefits are: Speed of erection; ease of erection and prefabrication; ease of transportation and storage; reusability; Minimal skill requirements for erection and relocation; reasonable cost; simplicity of connections; changeability of geometry of structure; possibility to map (match) the structure to any shape; lightweight and packed in the deployed configuration.

Generally, foldable structures with scissor-like elements are classified into two groups as Compatible and Incompatible structures as in Shan. In compatible structures, there is no stress and residual strain in folded state, during deployment and deployed state. These types of structures behave as mechanisms in all states, so it is essential to add other elements for

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stabling the structure. In incompatible structures there is no stress in the folded state, but during deployment and in the deployed configuration, residual stress and curved members are developed. Therefore, there is no need to add other elements for stabling the structures. Also snap-through phenomenon is occurred during deployment process. This type is investigated by Gantes et al. In this thesis compatible foldable structures with scissor-like elements (SLE) are investigated.

Deployable structures based on pantographs

Scissor units, otherwise called scissor-like elements (SLEs) or pantographic elements, consist of two straight bars connected through a revolute joint, called the intermediate hinge, allowing the bars to pivot about an axis perpendicular to their common plane. The upper and lower end nodes of a scissor unit are connected by unit lines. For a translational unit, these unit lines are parallel and remain so during deployment. In figure 1.1 and 1.2 a plane and a curved translational unit are shown, the plane unit being the simplest translational unit having identical bars. When these units are linked, a well-known transformable single-degree-of-freedom mechanism is formed, called a lazy-tong, shown in Figure 1.2.

Fig. 1.1: Plane and curved unit

Fig. 1.2: Planar translational frame (lazy tong)

Curved frames are shown in figure 1.3; it is constructed from the curved unit shown in figure 1.1.

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Fig. 1.3: Curved frame

Another example of the design of units is the polar frame, which consists of polar units, that is formed when the two unequal semi-bars a and b are connected together with the hinge that is moved away from the center of the bars.

Fig 1.4: Polar unit

Fig 1.5: Polar frame in its deployed and undeployed states

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Different Examples of Foldable Structures

Fig 1.6: Positive curvature structure with translational units in two deployment stages

Fig 1.7: Negative curvature structure with translational units in two deployment stages

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Fig. 1.8: The three different folded states of a model of a pantograph structure made of

plastic straws

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Aims and Scope of Work

Although many different deployable systems have been proposed, few have successfully found their way into the field of temporary constructions. A cause for this limited use can be found in the complexity of the design process. This entails detailed design of the connections which ensure the expansion of the structure during the deployment process. Therefore, not only the final deployed configuration is to be designed, but an insight is required in the mobility of the mechanism, as a means to achieve that final erected state. Also, designing deployable structures requires a thorough understanding of the specific configurations which will give rise to a fully deployable geometry.

The aim of the work presented in this thesis is to design a foldable structure used as tent which can be used in regions prone to natural disasters. These tent shall be deployed fast to construct camps in less time than traditional methods. Then the structure is analyzed using finite element method the optimization of the structure is done using genetic algorithms so that one can find the minimum weight with the optimal size of the structure members.

Outline of Thesis

In chapter 1 (this chapter), introduction about foldable structures is given, followed by some of the examples and literature review about the previous work done on this topic.

In chapter 2, the mechanical design and construction of the structure is studied, showing the hinges and connections of the structure and its topology, size, how it is folded and unfolded.

In chapter 3, the loads that acts on the structure, as it is subjected to its own weight and wind loads. The wind loads are estimated and calculated according to the Egyptian Code for Loads on Structures.

In chapter 4, a finite element model is developed and solved using commercial finite element analysis package (ANSYS), and the analysis results are shown on a sample of domain of the solution, as the extended solution of the optimization problem is studied later.

In chapter 5, the optimization problem is studied. First, theres an introduction about structural optimization, and then about genetic algorithms and how to use them in a brief and how genetic algorithms are applied to optimization problems. Then the optimization model of the tent structure is developed and the solver used (modeFrontier) is shown. After that, several runs have been held showing all the parameters of the run. Finally, the results of the optimization runs are shown and briefly explained.

In chapter 6, Recommendations and further work are suggested so as to be the start point for further research projects and theses.

Finally, the references and bibliography are at the end of the thesis.

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Chapter 2: Mechanical Design and Construction

There are a lot of design methodologies used in the design of foldable structures, based mainly on some constraints that are considered during design, as geometric and kinematic constraints.

The tent studied in this project is designed to be on the shape of arch, 6.8 m diameter and 4.9 m length, consisting of scissor-like units.

Fig 2.1: Geometric design of foldable structure

Fig 2.2: General assembly of tent

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Fig 2.3: Elevation and Plan views of tent

Construction of Unit

A scissor-like element, called duplet, consists of two elements, named uniplets. In general, there are two types of duplets, regular and irregular duplets as illustrated in figure 2.4 in the same plane. The duplet is capable of rotation about its intermediate pivot.

Regular duplets are rectangular and the irregular ones are trapezoidal. Using trapezoidal duplets in two directions results in a dome, and the rectangular ones in two directions results in a flat structure. A barrel vault consists of regular, in Y direction, and irregular, in X direction.

Fig 2.4: Regular and irregular units

Each unit consists of two intersecting lines, each line consists of pipes connected together using plates that are hinged to each other.

As shown in figure 2.5, the unit looks like isosceles trapezoid; the dimensions of the trapezoid are shown on the figure.

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Fig 2.5: Geometry of unit

Fig 2.6: Construction of single unit, unfolded state

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Fig 2.7: Construction of single unit, folded state

Unit Dimensions

Length of link = 760 mm

Construction of Line

Fig 2.8: Construction of line

Construction of Hinges

The hinge between the two lines is shown in figure 2.9, it consists of two plates joined using bolt or rivet to transmit forces only between members and allows relative rotation between members.

Fig 2.9: Hinge between lines

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Folded Structure

According to the design and constraints considered, to be able to fold the structure; the angles between links are:

Connection Angle (Unfolded state) Angle (Folded state)

Unit 83 145

Transverse Unit 35 145

Special End Transverse Unit 35 145

Special End Transverse Unit (Mid angle)

17.5 72.5

Table 2.1: Angles of folded and unfolded state of tent

Fig 2.10: Folded tent

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Fig 2.11: Side view of folded tent

Dimensions of folded tent

Width 3.3 m

Length 2.4 m

Table 2.2: Dimensions of folded tent

Construction Material

The used material of construction is steel due to its availability and low cost; table 2.3 shows the mechanical properties of the used material.

Material S355JR

Yield Strength (MPa) 355

Ultimate Tensile Strength (MPa) 450

Maximum Allowable Stress 230

Factor of Safety 1.54

Table 2.3: Properties of S355JR

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Chapters 3: Loads and Structural Analysis

The aim of this chapter is to study the loads that act on the structure of the tent. The calculation of the loads is based on the Egyptian Code of Loads on Structures.

The designed tent shall be used in regions prone to natural disasters, as it can be used in deploying refugee and emergency camps in few hours, and in building temporary bridges so it is subjected to wind loads due to its presence in open areas.

The loads on the tent are: 1. Dead load (own weight of tent) 2. Wind load

For wind load, as per the code, there are two load cases which are acting on the tent:

1. Load Case 1: Transverse Wind 2. Load Case 2: Longitudinal Wind

Fig 3.1: Load case 1, transverse wind

Fig 3.2: Load case 2, longitudinal wind (Plan view of tent)

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Symbols

Air density, kg/m3 Vw Wind velocity, m/s Ct Topography factor Cs Structure shape factor q Wind pressure, Pa zo Land steepness factor, m z Height, m h Height, m Ls Height, m Ce External wind pressure factor Pe External wind pressure, Pa w Width of tent, m A Projected area subjected to

wind, m2 F Wind force, N n Number of nodes subjected

to given force A,B,C,1,2 Subscript given to section

which wind load is calculated at (Fig 3.1, 3.2)

Dead Load

The weight of the tent is calculated from ANSYS and it is the main objective function of the optimization process (minimizing the weight of the weight, constrained by given value of maximum stress and maximum deflection).

Wind Load

The following calculations are based on the Egyptian Code of Loads on Structures, Chapter 7: Wind loads on buildings and structures.

Table 7-1:

Table 7-2:

Annex 7-A:

C_s: is assumed to be 1000 the value given in the code due to changing units to SI units, as q

and P are given in the code in kN/m2.

1.25kg

m3

Vw 33m

s

Ct 1.2

Cs 1 1000

q 0.5 Vw2

Ct Cs1000

1 106

816.75Pa

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Section 7-5:

Region := A k 1

Transverse Wind: Section 7-6-7: h 3m

Table 7-8:

Longitudinal Wind: Section 7-6-2: Assuming: rectangular structure (plan view of tent)

External Pressure: Note: Positive pressures (forces) indicate direction into surface, while negative pressures

(forces) indicate direction out of surface.

Area exposed to pressure: Note: All areas are projected areas not real areas

zo 0.05m

z 0 10

Ls 6m

h

Ls

0.5

CeA 0.75

CeB 1.2

CeC 0.5

Ce1 0.8

Ce2 0.5

PeA CeA k q 612.563Pa

PeB CeB k q 980.1 Pa

PeC CeC k q 408.375 Pa

Pe1 Ce1 k q 653.4Pa

Pe2 Ce2 k q 408.375 Pa

w 4m

AA h sin45

180

w 8.485m2

AC AA 8.485m2

LB 2 h2

4.243m

AB LB h 12.728m2

A1 0.5 h2

14.137m2

A2 A1

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Total Forces:

Forces on each node for FE model:

Summary of Wind Loads

Section Number of nodes Force on node (N)

A 16 324.86

B 36 -346.518

C 16 -216.574

D 68 135.842

E 68 -84.901

Table 3.1: Summary of wind loads

FA PeA AA 5.198 103

N

FB PeB AB 1.247 104

N

FC PeC AC 3.465 103

N

F1 Pe1 A1 9.237 103

N

F2 Pe2 A2 5.773 103

N

nA 16

FnA

FA

nA

324.86N

nB 36

FnB

FB

nB

346.518 N

nC 16

FnC

FC

nC

216.574 N

n1 68

Fn1

F1

n1

135.842N

n2 68

Fn2

F2

n2

84.901 N

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Chapter 4: Finite Element Modeling

The studied structure is a complex structure, because it consists of a lot of elements connected with each other with hinged connections, which increases the complexity of the structure. So classical methods wont be efficient and effective in the analysis of the structure, so a finite element model has to be developed to effectively analyze and solve this structure.

ANSYS is used as the finite element modeling and analysis software. The model is created and analyzed in ANSYS Mechanical APDL (ANSYS Parametric Design Language), although APDL is more difficult in development of finite element models but it guarantees flexibility and full control over the finite element model.

Finite Element Method

Finite Element Analysis (FEA) was first developed in 1943 by R. Courant, who utilized the Ritz method of numerical analysis and minimization of variational calculus to obtain approximate solutions to vibration systems. Shortly thereafter, a paper published in 1956 by M. J. Turner, R. W. Clough, H. C. Martin, and L. J. Topp established a broader definition of numerical analysis. The paper centered on the "stiffness and deflection of complex structures".

FEA consists of a computer model of a material or design that is stressed and analyzed for specific results. It is used in new product design, and existing product refinement. A company is able to verify a proposed design will be able to perform to the client's specifications prior to manufacturing or construction. Modifying an existing product or structure is utilized to qualify the product or structure for a new service condition. In case of structural failure, FEA may be used to help determine the design modifications to meet the new condition.

There are generally two types of analysis that are used in industry: 2-D modeling, and 3-D modeling. While 2-D modeling conserves simplicity and allows the analysis to be run on a relatively normal computer, it tends to yield less accurate results. 3-D modeling, however, produces more accurate results while sacrificing the ability to run on all but the fastest computers effectively. Within each of these modeling schemes, the engineer can insert numerous algorithms (functions) which may make the system behave linearly or non-linearly. Linear systems are far less complex and generally do not take into account plastic deformation. Non-linear systems do account for plastic deformation, and many also are capable of testing a material all the way to fracture.

The analysis approach in this thesis is 2D analysis using BEAM188 elements in ANSYS library. The BEAM188 element has the capability of transforming 2D analysis results to 3D plots and results due to its methodology of modeling cross sections of beam elements, as the cross section of the beam element is defined not using its geometrical properties only (Ixx, Iyy, etc) but the shape itself is given to ANSYS. And using integration points all over the cross section, the analysis results can be shown with all its variation (not the average) all over the cross section in a 3D way.

Description of the Model

The finite element model developed is constructed using 3D beam elements (line elements), connected with each other with hinges. The boundary conditions are assumed to be fixed supports (UX=0, UY=0, UZ=0, ROTX=0, ROTY=0, ROTZ=0).

Two uniplets which form a duplet are in fact beam elements with three nodes acting as pin-joints having only translational degrees of freedom. No torsion is produced in the members,

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however, axial forces and bending moments can be developed. A uniplet together with its degrees of freedom and the corresponding nodal forces are illustrated in figure 4.1.

Fig. 4.1: Uniplet beam element and its nodal forces

The stiffness matrix of a uniplet can be obtained by assembling first the stiffness matrix of two beam type elements, and then condensing and removing the rotational degrees of freedom of three nodes. The result is a 9 x 9 stiffness matrix for the uniplet element. Once such a matrix is obtained, the analysis of foldable structures follows a standard stiffness method.

Model Summary

Element type BEAM188

No. of cells (for Pipe cross section) 8

No. of cells (for Plate cross section) 4

Element size 10 mm

No. of elements 27368

Modulus of Elasticity 210 GPa

Poisson Ratio 0.3

Density 7600 kg/m3

Table 4.1: Model Summary

Warping degree of freedom Unrestrained

Cross section scaling Function of stretch

Element behavior Linear form

Shear stress output Torsional only

Section force/strain output At integration points

Stress/Strain (sect points) NONE

Stress/Strain (elmt/sect nds) NONE

Section integration Automatic

Taper section interpretation Linear

Results file format Average (corner nodes)

Table 4.2: BEAM188 Key Options

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Hinged Joints

ANSYS has two methods to model the hinged joints between beam elements; one can use Coupling and Constraints Equations or Multi Point Constraint Elements (MPC).

Coupling and constraints equations method depends on developing extra equations used to solve the stiffness equations. These extra equations couple the joint degrees of freedom with each other.

As an example, if its required to couple node a and node b in X,Y, ROTX, ROTZ directions, then 4 equations will be developed as follows:

These four extra equations are used simultaneously with the stiffness matrix to solve for

those coupled DOFs. By developing simple verification models to study the difference between both methods, it

was concluded finally that both methods yielded the same results for the developed cases, so Coupling equations method is used because its simpler to implement in APDL code.

The tent has 456 hinged joints, and each joint has only 1 free DOF (axis of rotation), so the tent model has 2280 coupling set.

Finite Element Model

Fig 4.2: Isometric view of the finite element model of the tent (2D view)

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Fig 4.3: Finite element model of the joints (shown in 3D)

Load Case 1

Load case 1 is the case where wind acts in transverse direction to the tent, as shown before in chapter 3.

Fig 4.4: Loads on finite element model for load case 1 (Elevation)

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Fig 4.5: Loads on finite element model for load case 1 (Isometric view)

Fig. 4.6: Displacement of model after applying loads (Elevation)

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Fig. 4.7: Displacement of model after applying loads (Isometric)

The stress distributions across the members are shown in figure 4.8. The majority of load is

concentrated in the mid-plate which joins both links together through the pivot.

Fig 4.8: Maximum stress at the lowest end of the support pipes

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Load Case 2

Load case 2 is the case where wind acts in longitudinal direction to the tent, as shown before in chapter 3.

Fig 4.9: Loads on finite element model for load case 2 (Isometric view)

Fig 4.10: Loads on finite element model for load case 2 (Plan)

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Fig 4.11: Displacement of model after applying loads (Left: Side View, Right: Plan)

Analysis Results

Because it is intended to do optimization all over the structure, the results given here are for a specific domain of dimensions only.

Load Case Input Parameters Output Results

Do (mm) tp (mm) wpl (mm) tpl (mm) (MPa) umax (mm) Mtotal (kg)

Load Case 1 60 5 100 10

55.1 8.9 1838.7

Load Case 2 229.1 44

Table 4.3: Summary of a sample of results The maximum stress is calculated based on von-Mises Theory, which is shown next in

the case of principal stresses only (shear stresses = 0).

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Chapter 5: Structural Optimization

Structural optimization process is vital process for complex structures as the studied tent, as it aims at optimizing the structure from the point of view of shape, size or topology. Structural optimization is the subject of making an assemblage of materials sustains loads in the best way. We want to find the structure that performs this task in the best possible way. However, to make any sense out of that objective we need to specify the term best. The first such specification that comes to mind may be to make the structure as light as possible, i.e., to minimize weight. Another idea of best could be to make the structure as stiff as possible, and yet another one could be to make it as insensitive to buckling or instability as possible. Clearly such maximizations or minimizations cannot be performed without any constraints. For instance, if there is no limitation on the amount of material that can be used, the structure can be made stiff without limit and we have an optimization problem without a well defined solution. Quantities that are usually constrained in structural optimization problems are stresses, displacements and/or the geometry. Note that most quantities that one can think of as constraints could also be used as measures of best, i.e., as objective functions. Thus, one can put down a number of measures on structural performanceweight, stiffness, critical load, stress, displacement and geometryand a structural optimization problem is formulated by picking one of these as an objective function that should be maximized or minimized and using some of the other measures as constraints.

A general structural optimization problem now takes the form:

Fig. 5.1: Structural optimization problem formulation

Types of Structural Optimization

Sizing optimization: This is when x is some type of structural thickness, i.e., cross-sectional areas of truss members, or the thickness distribution of a sheet. A sizing optimization problem for a truss structure is shown in figure 5.2.

Fig. 5.2: Size optimization of truss links cross sections

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Shape optimization: In this case x represents the form or contour of some part of the

boundary of the structural domain. Think of a solid body, the state of which is described by a set of partial differential equations. The optimization consists in choosing the integration domain for the differential equations in an optimal way. Note that the connectivity of the structure is not changed by shape optimization: new boundaries are not formed. A two-dimensional shape optimization problem is seen in figure 5.3.

Fig. 5.3: Shape optimization of beam; find the function (x) that describes the shape of the

beam-like structure

Topology optimization: This is the most general form of structural optimization. In a discrete case, such as for a truss, it is achieved by taking cross-sectional areas of truss members as design variables, and then allowing these variables to take the value zero, i.e., bars are removed from the truss. In this way the connectivity of nodes is variable so we may say that the topology of the truss changes as shown in figure 5.4.

Fig. 5.4: Topology optimization of a truss. Bars are removed by letting cross-sectional areas

take the value zero

Genetic Optimization Algorithm

Genetic algorithms are the selected optimization algorithm to do the structural optimization process, because it has many advantages for optimization processes with huge

number of design variables. Genetic algorithms are a part of evolutionary computing, which is a rapidly growing area of artificial intelligence. They are inspired by Darwins theory of evolution as the solution of the problem is genetically evolved.

Advantages of Genetic Algorithms

1. It can solve every optimization problem which can be described with the chromosome encoding.

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2. It solves problems with multiple solutions. 3. Structural genetic algorithm gives us the possibility to solve the solution

structure and solution parameter problems at the same time by means of genetic algorithm.

4. Genetic algorithm is a method which is very easy to understand and it practically does not demand the knowledge of mathematics.

5. Genetic algorithms are easily transferred to existing simulations and models.

Disadvantages of Genetic Algorithms

1. Certain optimization problems (they are called variant problems) cannot be solved by means of genetic algorithms. This occurs due to poorly known

fitness functions which generate bad chromosome blocks in spite of the fact that only good chromosome blocks cross-over.

2. There is no absolute assurance that a genetic algorithm will find a global optimum. It happens very often when the populations have a lot of subjects.

Algorithm Flowchart

The following figure shows the logical flow of the algorithm. The optimization software starts with initial population (designs) and evaluates all these designs to find which of them has the most fitness (optimal designs), then starts the process of crossover, selection and mutation between the most fit designs and evaluates the results and so till finding the optimal solution at the end of the optimization run.

Fig. 5.5: Algorithm flowchart

1. [Start] Generate random population of n chromosomes (suitable solutions for

the problem) 2. [Fitness] Evaluate the fitness f(x) of each chromosome x in the population 3. [New population] Create a new population by repeating following steps until

the new population is complete 4. [Selection] Select two parent chromosomes from a population according to

their fitness (the better fitness, the bigger chance to be selected) 5. [Crossover] With a crossover probability cross over the parents to form a new

offspring (children). If no crossover was performed, offspring is an exact copy of parents.

6. [Mutation] With a mutation probability mutate new offspring at each locus (position in chromosome).

7. [Accepting] Place new offspring in a new population

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8. [Replace] Use new generated population for a further run of algorithm 9. [Test] If the end condition is satisfied, stop, and return the best solution in

current population 10. [Loop] Go to step 2

Optimization Software

As shown in figure 5.1, ANSYS is used as the finite element solver, and it needs external software to run the optimization algorithm and evaluate the designs, so modeFrontier software is used as the genetic algorithm solver. The following figure shows the logical flow and interface of the software used (modeFrontier).

Fig 5.6: modeFrontier logical flow diagram

Figure 5.2 shows different group of elements, these groups are the main components of the flow of the genetic algorithm software:

1. Input variables: they are the optimization problem input variables that are changed to find the optimal solution.

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2. Input file: it is macro file written in ANSYS APDL which contains the finite element model code, and the input variables are written in it.

3. ANSYS batch file: it is DOS command used to run ANSYS software using the input file.

4. Output file: it is the file which contains the results of the finite element analysis done by ANSYS.

5. Output variables: they are the variables that are read from the output file so that modeFrontier could evaluate them according to the design criteria.

6. Design objectives and constraints: the design criteria is decided in this part of the model, it is divided to objectives and constraints; objectives are usually the variables that have to be maximized or minimized and constraints are the limits that must be considered while optimizing.

Optimization Runs

Several runs have been made to find the optimal solution of the tent structure. The main objective function is to minimize the weight and the constraints are maximums stress and maximum deflection not to exceed definite values.

The maximum allowable stress is 180 MPa for runs 1, 2, 3 as the used material is S275JR, so the design constraint defined in the optimization code is 180 MPa.

For run 4, the material is changed to S355JR with yield strength 355 MPa, so the

allowable stress increased to 230 MPa to help decrease the weight.

Run 1

These run is made on load case 1 (transverse wind) only.

Optimization Algorithm Parameters

No. of initial population 100

No. of generations 15

Initial population generation method RANDOM

Cross-over probability 0.6

Selection probability 0.1

Mutation probability 0.15

Elitism probability Enabled

Table 5.1: Optimization algorithm parameters for run #1

Constraints and Objectives

Constraints

Objectives

Table 5.2: Constraints and objectives of run #1

40

Design Parameters

The optimization parameters are: Outer diameter of each pipe, mid-plate connecting pipes to each other, keeping thicknesses of pipes and plates constant.

Parameter From To Step

Dp 40 100 10

tp 6 6 -

wpl 50 110 10

tpl 5 5 -

Table 5.3: Design parameters of run #1

Run 2

These run is made on load case 2 (longitudinal wind) only.











Constraints

Objectives

Table 5.5: Constraints and objectives of run #2

41

Design Parameters



Dp 60 100 10

tp 6 6 -

wpl 170 200 10

tpl 8 8 -


Run 3

These run is made on load case 2 (longitudinal wind) only.











Constraints

Objectives Table 5.8: Constraints and objectives of run #3

42

Design Parameters



Dp 50 80 6

tp 6 6 -

wpl 90 120 10

tpl 15 15 -


Run 4

Itll be shown in the next section that the 3 runs made yielded a large weight, so some changes are made to the design of the structure so that the weight could be reduced more. The modifications made are changing the longitudinal links to plates hinged together instead of the regular shape of the unit (pipes hinged together using plates).

These optimization run is made on load case 2.

Fig. 5.7: Modified shape of longitudinal links










43


Constraints

Objectives Table 5.11: Constraints and objectives of run #3

Design Parameters

The optimization parameters are: Outer diameter of each pipe, mid-plate connecting pipes to each other, keeping thicknesses of pipes and plates of pipes and plates of pipes constant.


Dp 40 90 6

tp 8 - -

wpl 60 110 10

tpl 15 - -

wl 20 - -

tl 2 - -


Optimization Results

Run 1

Analysis Results

It is obvious that run #1 is the optimal solution for load case 1 only, so another optimization must be done on load case 2 to check if it is safe for both load cases or not.

Load Case v,max (MPa) umax (mm) Mtotal (kg)

Load Case 1 172.22 25.415 1870

Load Case 2 842.724 211.342

Table 5.13: Results of Run 1

Design Parameters

Parameter Dimension (mm)

tp 6

tpl 5

wpl 80

Dp,average 65

Table 5.14: tp, tpl, wpl for run #1

44

Fig 5.8: Frequencies of occurrence of cross sections Dp for run #1

The following figure (as well as fig 5.6, 5.8) shows the variation of total volume (mass) of

structure vs. the design ID. The plot shows great randomness in the first several hundreds of runs, because GA starts with initial random population and then explores the domain of the most fit solutions in this random population, so the solution starts to decrease and saturate near the optimal (minimum) value of volume. The horizontal red line shows the optimal (minimum) solution with its ID.

Fig 5.9: Total volume vs. Design ID for run #1

0

20

40

60

80

100

120

140

40 50 60 70 80 90 100

Fre

qu

en

cy o

f O

ccu

ren

ce

Cross Sections, mm

Frequencies of Cross Sections

45

Run 2

Analysis Results

The second run satisfies all the constraints on the design from point of view of stress and deflection, but the problem in it is the width of the plate (wpl) which is 170 mm, which isnt practical in the construction of the structure, so the need to make another run appears.


Load Case 1 67.733 4.483 2730

Load Case 2 174.641 35.061

Table 5.15: Results of run #2

Design Parameters


tp 6

tpl 8

wpl 170

Dp,average 76.76



0

50

100

150

200

250

60 70 80 90 100

Fre

qu

en

cy o

f O

ccu

ren

ce

Cross Sections, mm

Frequency of Cross Sections

46


Run 3

Analysis Results

The third run is the optimal run, as it satisfies all the constraints and the objectives of the design in an optimal manner.


Load Case 1 66.6 3.33 2330

Load Case 2 179.365 32.552


Design Parameters


tp 6

tpl 15

wpl 90

Dp,average 62.33


47



0

20

40

60

80

100

120

140

160

180

50 56 62 68 74 80

Fre

qu

en

cy o

f O

ccu

ren

ce

Cross Sections, mm


48

Run 4

Analysis Results

The third run is the optimal run, as it satisfies all the constraints and the objectives of the design in an optimal manner.


Load Case 1 100.9 11 1536

Load Case 2 222.9 91.8


Design Parameters


tp 8

tpl 15

wpl 90

tl 2

wl 20

Dp,average 64.6



0

10

20

30

40

50

60

70

80

90

40 50 60 70 80 90

Fre

qu

en

cy o

f O

ccu

ren

ce

Cross Sections, mm


49


51

Recommendations and Further Work

This chapter shows the recommendations based on the experience gained during this work, so as to not repeat the same mistakes again and make further development on this research topic. As well as the further work that can be extended based on this work.

Unit Construction

There are some other designs for the foldable unit other than the design used in this thesis. One of the best and easy designs to design and manufacture and also assemble is to use double angles assembled back to back as shown in figure 6.1.

Figure 6.1: Isometric view of the recommended unit

Figure 6.2: Elevation view of the unit in folded and unfolded states

As shown in figure 6.2, this unit has an advantage over the used unit, that it can be folded

with angles obviously larger than the units constructed using pipes, which shall make the structure in the folded state much more compact.

52

Optimization

In chapter 5, the optimization of the structure using genetic algorithms is proposed. The optimization work can be extended to use other algorithms as Ant Colony algorithm, as some papers used it in optimization of traditional structures, but it can be extended to foldable structures.

It is recommended also to increase the number of initial population of the genetic algorithms so as to explore more regions of the domain of the input parameters so that the optimization algorithm can find the optimal solution and not get stuck in local minima (this is one of the disadvantages of genetic algorithms).

Objectives and constraints of the optimization problem shall be extended to include buckling limit and stability of structure. As this is an important criterion in the design of steel structures.

Parallel computing shall be considered in running the genetic algorithms on such complex structures so as to reduce time of the analysis and optimization.

54

References

[1] Simon D. Guest, 1994, Deployable Structures: Concepts and Analysis [2] Niels D. Temmerman, 2007, Design and Analysis of Deployable Bar Structures for Mobile

Architectural Applications [3] A. Kaveh, A. Davaran, 1996, Analysis of Pantograph Foldable Structures, Computers and

Structures, Vol. 59, No. 1, pp. 131-140 [4] W. Shan, 1992, Computer Analysis of Foldable Structures, Computers and Structures, Vol.

42, No. 6, pp. 903-912 [5] G. Tibert, 2002, Deployable Tensegrity Structures for Space Applications [6] M. Babei, E. Sanaei, 2009, Geometric and Structural Design of Foldable Structures,

Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium, Valencia, Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures, Universidad Politecnica de Valencia, Spain

[7] M. Galante, 1996, Genetic Algorithms as an Approach to Optimize Real-World Trusses, International Journal for Numerical Methods in Engineering, Vol. 39, pp. 361-382

[8] A. Kaveh, S. Shojaee, 2007, Optimal Design of Scissor-Link Foldable Structures Using Ant Colony Optimization Algorithm, Computer-Aided Civil and Infrastructure Engineering, 22, pp. 56-64

[9]C. J. Gantes, J. J. Connor, R. D. Logcher, Y. Rosenfeld, 1989, Structural Analysis and Design of Deployable Structures, Computers and Structures, Vol. 32, No. , pp. 661-669

[10] J. S. Zhao, F. Chu, Z. J. Feng, 2009, The mechanism theory and application of deployable structures based on SLE, Mechanism and Machine Theory, Vol. 44, pp. 324-335

[11] J. S. Zhao, J. Y. Wang, F. Chu, Z. J. Feng, J. S. Dai, 2011, Structure synthesis and statics analysis of a foldable stair, Mechanism and Machine Theory, Vol. 46, pp. 998-1015

[12] M. Hultman, 2010, Weight optimization of steel trusses by a genetic algorithm [13] C. Kane, M. Schoenauer, 1996, Topological Optimum Design using Genetic Algorithms,

Control and Cybernetics, Vol .25, No. 5 [14] M. P. Bendsoe, O. Sigmund, 2003, Topology Optimization: Theory, Methods and

Applications, Springer [15] P. W. Christensen, A. Klarbring, 2009, An Introduction to Structural Optimization,

Springer [16] G. R. Liu, S. S. Quek, 2003, The Finite Element Method: A practical course, Butterworh-

Heinemann [17] J. N. Reddy, 1993, An Introduction to the Finite Element Method, McGraw-Hill [18] D. V. Hutton, 2004, Fundamentals of Finite Element Analysis, McGraw-Hill [19] F. P. Beer, E. R. Johnston, J. T. DeWolf, D. F. Mazurek, 2009, Mechanics of Materials,

McGraw-Hill [20] ANSYS Structural Analysis Guide [21] ANSYS Theory Reference [22] ANSYS Element Reference [23] ANSYS Parametric Design Language Guide [24] ANSYS Verification Manual [25] modeFrontier User Manual [26] Egyptian Code of Loads on Structures, Ministry of Housing, Utilities and Urban

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modeling, analysis and optimization of foldable structure using finite element method

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