advances in civil and environmental engineeringpioneer for shell-form design, used various methods...

ACEE – Volume 01 - P 69 - 83

Advances in Civil and Environmental Engineering

Shell Surface with Fibonacci spiral curve fold

H. Soltanzadeh1*, K. K. Choong1, M. Dehestanic

1* School of Civil Engineering, Universiti Sains Malaysia, Penang, Malaysia

2Faculty of Civil Engineering, Babol Noshirvani University of Technology, Babol, Iran

1* Corresponding author,

Email: [email protected]

Abstract

Effect of curve folds in the form of Fibonacci spiral on behaviour of folded shell structures has

been investigated in this study. Two kinds of models with different configuration of the spiral

curves have been generated . Results of analysis show that shell surface with spiral curved fold lines

with variable radius has better performance as compared to that of constant radius. Specifically, in

terms of maximum tensile stress, shell surface with constant radius is 24.95% higher than the model

with variable radius and also maximum compressive stress for shell surface with constant radius is

36.83% higher than the model with variable radius. In terms of deflection, maximum vertical

displacement in model with constant radius and variable radius is approximately the same with

difference of only 4%.

Keywords

Curve folds, Shell Structures, Fibonacci

spiral. Volume 01 – Year 2013

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H. Soltanzadeh, et al / ACEE Volume 01- P 69 - 83

1 Introduction

Shell as one of the most attractive types of

structures has been frequently adopted as a

three-dimensional surface structure which can

be used for large span constructions. The word

or phrase – shell, which consists of two special

features: First the shells are strong and second

being curved surface, smooth and flat, never in

the best case they cannot form a shell

(Farshad, 2009). The load carrying capacity of

a flat and thin surface structure is rather

limited. However, if a surface is folded,

enhancement in stiffness and strength of the

resulting folded surface can be achieved due to

the increase in effective depth (Golabchi,

2010). With the combination of shell and

curve fold lines, very attractive alternative

structural forms can be formed. Heinz Isler,a

pioneer for shell-form design, used various

methods to determine the appropriate shape for

his shells (Isler, 1994). Nature inspired

structures with organic shapes and double

curvature is very important in Isler’s works.

According to Isler’s theory, the critical load in

a shell is a function of the radius of the

element in compression, its thickness and its

elastic modulus with modifying factors that

depend on the form of the shell and an

assessment of how well it is constructed. The

combination of shell surface and curved fold

lines are the main issue in this paper. This

study is carried out to analyse the effect of

combination of shell surface and spiral curved

inspired with Fibonacci sequence on structural

behaviour of shell surface with curved folds.

The idea in this study to draw a spiral curve

originated from Fibonacci equation.

The traditional Japanese art of paper folding

has recently found its way into engineering

applications. Origami techniques can be a

great inspiration source of idea for

architectures and civil engineers. Curved folds

have been studied by (Huffman, 1976) who

created numerous awe-inspiring origami

sculptures. Folded plate structure was first

built as German coal bunker in early 1920s.

Since then many Types of folded plate have

been studied, analysed and built. Huffman

developed a structure consisting of parabolic

and elliptical curved folds. One of the main

features of Huffman’s work is existence of

curved fold lines on surface in contrast to

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traditional folded paper model with straight

folds. Some of Huffman’s ideas are shown in

Figure 1 (Rohim et al., 2008).

Figure 1. 4 parabolic curved fold through centre b) A tower-like paper structure, c) 4 parabolic curved folds meeting at

central square (Rohim et al., 2008)

Many forms in nature, living organisms such

as the skull head are shell forms. The word

shell is an old one and is commonly used to

describe the hard covering of eggs, crustacea,

tortoises, etc (Golabchi, 2010). They appear in

natural forms but also as man-made, load-

bearing components in diverse engineering

systems. Shells and spatial structures are

adopted for construction of large span

structures in which a large space is realized

without columns as the structural components.

In those cases, the structures are expected to

resist various design loads mainly through

their extremely strong capability which can be

acquired through in-plane or membrane stress

resultants and this is just the reason by which

they themselves stand for external loads

without columns as their structural

components in the large span structures

(Ohmori and Yamamoto, 1998).

Without doubt one of the great Spanish

pioneers of Shell designer, engineer and

mathematician is Felix Candela. According to

his words, the strength should come from

form, not mass.

Early work on non-classical structural

configuration was carried out by (Suresh

Panda and Natarajan, 1981) who described a

doubly curved quadratic shell element as a

model for the analysis of practical, non-

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classical structure constructed of fiber

reinforced laminated composites. They

reported high accuracy for the model for a

practical case.

A comprehensive study on numerical analysis

of regular and irregular folded shell structures

was conducted by (Chroscielewski et al,

1997). They formulated the shell theory for the

problem and developed shell finite elements.

Eventually admissible efficiency and general

applicability for the methodology were

reported. The numerical examples were

representative for a wide class of practical

engineering problems.

The structural characteristics of folding

structures depend on the shape of the folding

(longitudinal or pyramidal), on their

geometrical basic shape (plane, cupola, free-

from), on its material (concrete, timber, metal,

synthetics), on the connection of the different

folding planes and on the design of the

bearings (Trautz and Herkrath, 2009).

In an interesting nature inspired field, an

alternative structural concept for roof

structures that combines both structural

efficiency and architectural beauty inspired by

nature was studied by (Razzack et al, 2006).

This study is carried out with the following

two objectives:

1. To propose a procedure to generate shell

surface with spiral curve folds.

2. To investigate the structural behavior of

shell surface with spiral curved folds.

The outcome of the research study could also

serve as initial wok in the development of

novel form of shell surface with curved fold

lines for possible consideration as structural

form.

2 Methodology

Fibonacci equation is attractive and has

reasonable and significant relationship for

drawing the spiral curve. The sequence formed

from the ratio of adjacent numbers of the

Fibonacci sequence converges to a constant

value of 1.6180339887…, called “phi”, whose

symbol is ɸ. The ratio ɸ= 1.6180339887… is

called the “golden ratio”. A rectangular that

has sides in this proportion is called the

“golden rectangular” (Figure 2).

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Figure 2. Golden rectangular

A rectangle with longer side and shorter side

b, when placed adjacent to a square with sides

of length a, will produce a similar rectangle

with longer side a+b and shorter side a. This

illustrates the relationship

≡ ɸ. This

fraction,

, is called the golden ratio. This

equation has one positive solution in the set of

algebraic irrational numbers.

ɸ= √

339887…

The rectangular is the basis for generating a

curve known as the “golden spiral”.

Some of the Fibonacci sequence numbers are

shown as below:

(1)

The Fibonacci sequence obeys the recursion relation:

( ) ( ) ( ) ( ) ( ) (2)

This sequence is constructed by

choosing the first two numbers which is then

followed assigning the rest by the rule that

each number be the sum of the two preceding

numbers. This simple rule generates a

sequence of numbers having many surprising

properties, of which few are listed below:

- Take any three adjacent numbers in the

sequence, square the middle number;

multiply the first and third numbers. The

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difference between these two results is always

1.

- The sum of any ten adjacent numbers

equals 11 times the seventh one of the ten.

The Fibonacci spiral as shown in Figure

3 Squares following the dimensions of the

Fibonacci code are used to construct such a

spiral.

Figure 3. Fibonacci spiral curve

The Fibonacci sequence starts with 0 and 1

and each following number is the sum of the

two previous ones. Therefore the sequence

goes, 1, 2, 3, 5, 8, 13, 21 and on. This is how

the squares with the correct unit sizes are made

can be known.

The central first square is drawn first.

Afterward the curving quarter circles through

the corners of the squares starting with the

center going counter clockwise and work its

way out to the biggest square.

For drawing the model, an internal square is

needed. If the dimensions of this internal

square are set as 2x2 (To better perception the

small number was opted, however the

procedure will be the same for other numbers)

the external square would be the sum of the

two digits after 2, which means 3 and 5, equal

to 8.

The dimension of the radius of the curve is

chosen to be 3. Furthermore one of the corners

of the internal square is chosen (A1) as shown

in Figure 4 and drawn towards the external

square edge with a quarter of arch in counter

clockwise with the radius of 3.

2 1

3

5

8 13

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After the first curve has been drawn, the

remaining three curves are obtained through

corresponding translation and rotation of 90˚

(second curve) 180˚ (third curve) and 270˚

(fourth curve) respectively around the origin

(0, 0) (Figure 4).

Figure 4. Internal square parameters

Structural Modelling

In this research, two models have been studied

and analysed. In model I (Figure 5), the curve

with a radius of 3 resumes from one of the

corners of internal square with dimensions of

2×2 and after touching the intersection point of

the outer square with dimension of 8x8

continues with the same radius.

Figure 5. Spiral curve for model I

In model II (Figure 6), the curve with a radius

of 3 resumes from one of the corners of

internal square with dimensions 2 ×2 and after

touching the collision point to the outer

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dimensions of the square radius of 8x8 continues with the radius 5.

Figure 6. Spiral curve for model II

In other words two models have been studied

in this work; model I with constant radius and

the model II with variable radius. Both

models are generated based on Fibonacci

sequence.

The grid lines consist of three sets of points,

one set for each axis: x, y and z. At each point

is a plane. The grid lines attributes of the

models were assigned with details as

following:

The number of elements is 17 elements in x

direction, 16 elements in y directions and 5

elements in z direction. In this study, the

support conditions used are pin-pin and

intersection point of curve with outer square is

point of supported.

The material used for the models or in the

other word the type of shell is considered as

aluminium, due to the lightness and flexibility.

The properties of the material such as

thickness, Young’s modulus, ultimate tensile

stress ( ), yield tensile stress (

), yield

compressive stress (

), available ultimate

stress ( ) and available yield stress (

) are

listed in Tables 1(a, b, c).

00


Table 1-a. The properties of the material used for models

Table 1-b. Description of the Aluminium profile

Thickness

M

Unit Mass

kN-s2/m4

Unit Weight

kN/m3

E

kN/m2

Poisson’s Ratio

m/m

Total Weight

kN

0.01 2.71E+00 2.66E+01 69637052.49 0.33 20.575

Table 1-c. Characteristics and mechanical properties of the Aluminium profile

kN/m2 kN/m

2 kN/m

2 kN/m

2 kN/m

2

262000.80 241316.53 241316.53 165474.19 137895.16

The models were analysed under self- weight

(body force) of aluminium regarding the

properties given in Table 1 in order to

investigate the structural behaviour in terms of

stress and deflection.

The syncline and anticline are terms used to

describe folds (Figure 7). A syncline is the

opposite of an anticline. A syncline is a

downward fold or dip and on the other hand

the anticline is the upward fold in a plane

which the quantity of those in any two models

is the same and equal to 0.75meter.

Figure 7. A plan view of the anticline and syncline of model

Material Area Type Alum Type Alloy

ALUM Shell Wrought 6061-T6

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3 Results and discussion

Regarding the displacement of the support

joints the results of the analysis given in

Tables 2 and 3, it is found that rotation in

model I in the location of supports is higher

than rotation in model II.

Table 2. Displacement of the support joints (boundary condition) in first model

Joints of support U1 (m) U2 (m) U3 (m) R1

(Radians) R2

(Radians) R3

(Radians)

1 0.000000 0.000000 0.000000 -0.025474 0.016941 0.008071

2 0.000000 0.000000 0.000000 0.025474 -0.016941 0.008071

3 0.000000 0.000000 0.000000 0.016948 0.025489 -0.008064

4 0.000000 0.000000 0.000000 -0.016948 -0.025489 -0.008064

In tables Ui and Ri denote the displacement and rotation in i- direction, respectively.

Table 3. Displacement of the supports joints (boundary condition) in second model

Joints of support U1 (m) U2 (m) U3 (m) R1

(Radians) R2

(Radians) R3

(Radians)

1 0.000000 0.000000 0.000000 -0.019523 0.014592 0.004663

2 0.000000 0.000000 0.000000 -0.019563 0.014902 0.004725

3 0.000000 0.000000 0.000000 -0.019884 0.014160 -0.004900

4 0.000000 0.000000 0.000000 -0.020456 0.014215 -0.004975

Results of maximum displacement due to the

surface parameter for two models in x

direction, y direction and z direction are

summarized in Table 4.

Table 4. Maximum displacements in models

Maximum

Displacement U1 (m) U2 (m) U3 (m)

R1

(Radians)

R2

(Radians)

R3

(Radians)

Model I 0.010388 0.010364 0.034519 0.030393 0.025967 0.015519

Model II 0.010624 0.010676 0.035933 0.032406 0.02752 0.015984

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Table 5 and Table 6 show, maximum values of

stresses resultant in model I are more than

those in model II.

Table 5. Maximum tensile stress of the models

Case Sx (kN/m2) Sy (kN/m2) Sxy (kN/m2)

Max tensile stress in model I 426457.44 251149.1 126169.79

Max tensile stress in model II 338350.44 123442.2 121007.94

Table 6. Maximum compressive stress of the models

Case Sx (kN/m2) Sy (kN/m2) Sxy (kN/m2)

Max compressive stress in model I -379710.4 -182784.9 -121367.6

Max compressive stress in model II -248763 -127216 -66034.92

The results of analysis including stress distribution (Figure 8 and 9) were evaluated in

this study.

Figure 8. Stress resultant in local xy plane (Sxy) in model

(Compressive

stress)

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Figure 9. Stress resultant in local xy plane (Sxy) in model II

The obtained results demonstrated the situation of the bending moment as shown in

figures 10 and 11.

Figure 10. Contour of moment (Mmax) in model I

(Tensile

stress)

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Fi gure 11. Contour of moment (Mmax) in model II

From the result of this study which has been

carried out to investigate the effect of folds in

the form of Fibonacci spiral on structural

behaviour of shell structures, the following

conclusions can be made:

Shell surface with spiral curved fold

lines with variable radius shows better

performance as compared to fix radius.

Specifically,

(a) The supports rotation in the model

II found to be 23.36% less than model I.

(b) The difference between maximum

displacement values was found around 0.001

m, where the maximum displacement of the

model II has shown the increase around 4%

compared with model I.

(c) The values of maximum tensile and

compressive stress of model I in x and y

directions as well as xy plane are higher than

the model II. In other words, all stresses in the

model II are distributed evenly compare with

model I. The percentage of the maximum

tensile stress for the model II in x direction is

20%, in y direction is 50.84% and in xy plane

is 4% decreased from model I. On the other

hand the percentage of the maximum

compressive stress for the model II in x

direction is 34.48%, in y direction is 30.4%

and in xy plane is 45.6% decreased from first

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one. The reduction of the stresses is due to the

distribution of stresses evenly, from a

concentrated location through the surface of

the model.

4 Concluding Remarks

From the result of this study to investigate the

effect of folds in the shell structures, the

following conclusions can be made:

Radius changing will cause a smooth surface

that will have better performance compared

with sharp surface. It can be observed the

effect of smoothening of curved fold lines

could provide the best performance in term of

deflection and stresses.

Furthermore, smoothened surface can provide

better way for stress to distribute more evenly

throughout the surface. The advantages of

utilizing shell surface with spiral curved fold

lines are providing an excellence appearance

from the geometry and behaviour.

References

Chroscielewski, J., Makowski, J., Stumpf, H.,

1997. Finite element analysis of smooth

folded and multi-shell structures,

Compute. Methods Appl. Mech. Eng.

141, 1-46.

David A. Huffman, 1976. Curvature and

Creases: A Primer on paper. IEEE

Transactions on Computers. C-25:10,

1010-1019.

Farshad, M., 1992. Design and Analysis of

Shell Structures. Published: Dordrecht,

Boston, Kluwer Academic.

Golabchi, M., 2002. Shell and Folded Plate

Structures, Tehran University Academy

Press, Iran.

Isler, H., 1994. Concrete Shells Today, Spatial,

Lattice and Tension Structures.

Proceedings of the LASS-ASCE

International Symposium, Atlanta,

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Natarajan, R., Suresh Panda, 1981. Analysis of

laminated composite shell structures by

finite element method. Computers and

Structures. 14(3-4), 225-230.

Razzack, S.A., Choong K. Keong, Taksiah A.

Majid, 2006. A Study on Shell Surface

with Folds, A Nature Inspired Idea From

Leaves of Johannesteijsmannia Altifron,

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Journal of the International Association

for Shell and Spatial Structures 47(1),

63-73.

Rohim, R., J.Y. Kim, Choong K. Keong, 2008.

Shell Surface with Curved Fold Lines

Inspired by Paper Folding Art.

Proceedings of the 6th International

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Spatial Structures 18-21.

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Application of Folded Plate Principles

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