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 Total Downloads : 4211
 Authors : Srirama Satish Kumar, A. Swarna Kumari
 Paper ID : IJERTV1IS7089
 Volume & Issue : Volume 01, Issue 07 (September 2012)
 Published (First Online): 25092012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Design And Failure Analysis Of Geodesic Dome Of A Composite Pressure Vessel
Srirama Satish Kumar 1, A. Swarna Kumari2
1PG student, Department of mechanical engineering, JNTU college of engineering, Kakinada, A.P, INDIA.
2Assistant professor, Department of mechanical engineering, JNTU college of engineering, Kakinada, A.P, INDIA.
Abstract
Composite pressure vessels are low weight structures mainly used in defence, aviation applications. A composite pressure vessel along with optimized dome ends avoids critical stresses that are incorporate with the structure when the structure is internally pressurized. Design of the Geodesic dome portion which leads to the optimization of the dome structure is calculated by using geodesic path equation. The thicknesses of laminate structure at various portions of the dome region and cylindrical region are calculated by using netting analysis. Carbon fibre is reinforcement material and epoxy resin is the matrix material selected for the present study. The operating and failure conditions of the Geodesic Dome portion are predicted by using classical laminate theory of composites. Finite element analysis of Geodesic dome end and cylindrical portion is done by using Ansys
12.0. The failure sequence of the fibres associated with different fibre orientations is identified. Mat lab code is developed to study the failure sequence of the fibres. Based on TsaiWu theory and maximum stress theory the operating and failure pressures of the composite structure are predicted.
Keywords: Geodesicdome profile, ANSYS, Tsai Wutheory, composite pressure vessel.

Introduction
The structures with parameters like high strength, high stiffness, with optimized mass are most effective and advantageous structures. These Composite structures are most widely used in aerospace and chemical engineering applications, composite rocket motor cases and CNG tanks. Failure prediction of the composite pressure vessel is very important. There are only two ASME standards are available for Glass fibre reinforced composite pressure vessels (water storage tanks).Composite pressure vessel fabricated by filament winding technique. Composite pressure vessel consists of filament wound laminated structure, along with polar
fittings and thin non metallic low elastic modulus layer which does not share pressure load, prevents leakage of pressurized fluid, and efficiently transfers the loads to the composite structure. Metallic Polar fittings are provided at the dome region to avoid the failure at the ends of the dome surface (vicinity of the polar opening). In filament winding technique, filaments immersed in slightly heated resin, wound around collapsible foam mandrel with some tension, and then after completion of winding process, set the mandrel for curing in an air circulated oven for the specified time and temperature. By digging the foam from the cured structure we get the required composite pressure vessel. Lining process is carried out by using low modulus metallic or non metallic material, generally aluminium or polyurethane rubber layer can be used for lining. The tension of the fibres directly affects the volume fraction of the fibre and matrix material. Winding of the resin impregnated fibres around the mandrel with more tension increases volume fraction of the fibres and decreases the volume fraction of resin content. The design of the mandrel should be carried out in such a way that the structure should equally stressed at all regions when the pressure load is applied.

Design of the collapsible mandrel
The outer surface of the collapsible mandrel becomes inner surface of the composite pressure vessel after curing. Collapsible mandrel consists of cylindrical portion and two dome regions. Domes surface is highly stressed region under circumferential and longitudinal loads. Geodesic dome profile is most commonly used for pressure vessels. The material for the fabrication of the mandrel is foam or sand with polyvinyl alcohol
binder (water soluble mandrel). The aluminium polar fittings are preassembled with collapsible mandrel before winding. Mandrel should not have voids or air gaps in the case of foam mandrel. The length of the cylindrical portion, the radius of the cylindrical portion, pole opening radius is taken as 1000mm, 206mm, 100mm for the present study. Fig .2 represents the collapsible mandrel along with Geodesic dome ends.

Design of Geodesic Dome Profile
The geodesic profile is the elliptical curve connecting the two points taking the consideration of shortest
distance. We need to define Geodesic profile between
in the Figure.1 along the line AFB. At a distance of
1.225 times of the pole opening radius the axial coordinate becomes infinity at that particular radius (i.e., point of inflection). In previous papers a tangent line drawn to connect the point of inflection and pole opening radius. Another geodesic profile equation is provided which can avoid the tangent line usage for connecting pole opening and inflection point. Second geodesic profile equation used for the calculation of dome coordinates. From the point of inflection the aluminium polar fitting is introduced to avoid the mode of failure at the vicinity of the pole opening [3], [1].
pole opening radius and cylindrical portion radius [8],[3]. Generally friction is required to keep the path
(2)
stationary. The fibre wound on the geodesic dome profile according to the claurits principle [4] does not slip and it directly follows the Geodesic path, this type of winding is called Geodesic winding, which does not require any friction to keep the fibre stationary. Geodesic dome profile is created by using Geodesic path equation [8].
(1)
Axial coordinate Polar opening radius
r = radial coordinate, R=radius of the cylindrical portion
Figure 1. Geodesic dome profile pole opening
Geodesic curve which is obtained from the above Geodesic equation has the point of inflection at which the radius of curvature reverses its direction as shown
Where
=radius at point of inflection=1.225 times of pole opening.
The thickness of the metallic polar fitting can be calculated by using equation
t
Yield stress of the material t= thickness of the polar fitting P=design pressure
2.3. Design of the laminate structure
The fibre material is carbon, the matrix material is epoxy resin selected for the present study. Circumferential layer and helical layers are required for constructing the laminate structure on cylindrical portion. Circumferential winding is not possible on the dome region of the mandrel. At the dome region slipping of the fibres occurs during winding. Doilies are group of layers in two mutually perpendicular directions are needed to cover the surface of the dome. The geodesic dome profile is calculated by using Geodesic path equations 1, 2, which are specified in the previous section. The below Tables. 1 , 2 represents the Geodesic dome coordinates for the required pole opening radius and cylindrical portion radius. Point of inflection (at which radius of curvature reverses its direction) is also considered. Laminate structure for the composite pressure vessels is clearly explained in the Table 3.
Table 1.Geodesic dome profile up to the point of inflection
z(mm)
r(mm)
0
206
14.612
205
21.05
204
25.93
203
30.074
202
33.67
201
36.91
200
50.42
195
59.428
190
68.432
185
75.63
180
82.83
175
88.24
170
108.05
145
126.05
122.5


Failure analysis of the composite structure
Classical laminate theory is used for analysis in which the behaviour of composite laminate structure is treated as an orthotropic material. Maximum stress, TsaiWu theories [9], [10] followed to predict the failure, Steps involved in the analysis:

Find the reduced stiffness matrix ( from the orthotropic material properties of the carbon/epoxy structure.

Calculate reduced transformed stiffness matrix by using fibre angle orientation. Find coupling stiffness matrix, flexural stiffness matrix, and bending stiffness matrix (A, B, D matrices)
Table 2. Geodesic dome profile from the point of inflection to the up to pole opening
z(mm)
r(mm)
126.4963
120
126.826
118
127.1306
116
127.3961
114
127.6668
112
127.8946
110
128.0958
108
128.2689
106
128.411
104
128.5189
102
128.5763
100
Previously the angle of doilies is taken as the winding angle of the dome portion at that particular radius, winding of the dollies is done by using filament winding machine. Now for the present study the dollies with 0Â° and 90Â° fibre layer orientations are considered. The process of covering the dome region doilies in the direction of 0Â° and 90Â° is carried out by manually. After completion of the circumferential winding with required thickness on arbitrary cylindrical mandrel, cut the layers according to the required dimensions using the template and cover the surface of the dome portion, by keeping the circumferential layers along 0Â° and 90Â° direction manually. Winding angle is calculated from the claurits principle. Thickness of hoop, helical, doilies is taken as 0.2mm, 0.45mm, 0.5mm. Thickness calculations are done by using netting analysis [4].
,k=number of layers
By using thickness coordinates from the midplane throughout the laminate for each layer.

Calculate mechanical loading i.e., forces, (external moments [M] =0) of composite pressure vessel
.
Where P= applied pressure, d=diameter at particular portion, t=laminate thickness.

From the normal loading and moments find mid plane strains and curvatures

Apply classical laminate theory to find local stresses, local strains, global stresses and global strains. Mat lab code is developed for the above algorithm to study the failure sequence of the laminate structure.

The failure of the laminates is predicted by using maximum stress theory and Tsaiwu theory.
Figure 2. Collapsible mandrel with geodesic dome end (All dimensions are in mm)
Table 3. Winding layup details around the collapsible mandrel
Starting station 
Ending station 
radius 
Winding angle 
Sequence of winding pattern Around the mandrel 
2.5263 
20.523 
122.5 
54.7 
(54.7/54.7/54.7/doily/54.7/doily/54.7/54.7/doily/54.7/54.7/doily/54.7 
20.523 
40.3363 
145 
43.602 
(43.6/43.6/43.6/doily/43.6/doily/43.6/43.6/doily/43.6//doily/43.6/43.6 
40.3363 
52.9463 
170 
36.03 
(36.03/doily/36.03/doily/36.03/36.03/doily/36.03/36.03 
52.9463 
69.1483 
180 
33.748 
(33.7/doily/33.7/doily/33.7/33.7/doily/33.7/33.7 
69.1483 
91.663 
190 
31.7 
[31.7/doily/31.7/doily/31.7/31.7/doily/31.7/31.7]s 
91.663 
113.9543 
200 
30 
[30/doily/30/doily/30/30/doily/30]s 
113.9543 
128.5763 
205 
29.19 
(29.1/doily/29.1/doily/29.1/29.1/doily/29.1)s 
128.5763 
1128.576 
206 
29.04 
(90/90/29.04/90/90/29.04/90/90/29.04/90/90/29.04/90/90/29.04/90 
1128.576 
1143.188 
122.5 
29.19 
(29.1/doily/29.1/doily/29.1/29.1/doily/29.1)s 
1143.188 
1165.486 
145 
30 
[30/doily/30/doily/30/30/doily/30]s 
1165.486 
1188.004 
170 
31.7 
[31.7/doily/31.7/doily/31.7/31.7/doily/31.7/31.7]s 
1188.004 
1204.206 
180 
33.748 
(33.7/doily/33.7/doily/33.7/33.7/doily/33.7/33.7 
1204.206 
1216.816 
190 
36.03 
(36.03/doily/36.03/doily/36.03/36.03/doily/36.03/36.03 
1216.816 
1236.626 
200 
43.602 
(43.6/43.6/43.6/doily/43.6/doily/43.6/43.6/doily/43.6//doily/43.6/43.6 
1236.626 
1257.1526 
205 
54.7 
(54.7/54.7/54.7/doily/54.7/doily/54.7/54.7/doily/54.7/54.7/doily/54.7 

Maximum stress theory
Ultimate strength of the carbon fibre epoxy layer is around 1260Mpa. Finding the strains and stresses in material coordinates which is illustrated in section3 involves number of iterative calculations, stresses induced in the material and global coordinates are Using Mat lab program. The stress induced in the material coordinates should not exceed ultimate strength of the carbon/epoxy composite. The layer which exceeds the ultimate strength that layer fails first. (The stress induced in the layers should not exceed 1260Mpa).

TsaiWu theory
In this theory interaction between longitudinal, transverse, shear stress is considered. Using TsaiWu theory [9], [10] to find the strength ratio for all layers
+++ ,
Strength , , ,
=Ultimate longitudinal tensile strength = Ultimate transverse tensile strength =Ultimate inplane shear strength
, , are the local stresses induced in the lamina
Identify the minimum value of strength ratio of all the layers. Strength ratio gives the maximum value of the normal load. Strength ratio divided by the thickness of corresponding layer gives the allowable normal stress of the laminate.

Operating and failure conditions of cylindrical and Geodesic dome
Tsai Wu theory is the best theory to predict the failure of composite structure. The present structure consists of epoxy impregnated carbon fibres which are oriented with different fibre angles. At the Cylindrical portion 22 hoop layers and 10 helical layers are required. The total thickness provided at the cylindrical 8.9mm. Symmetric laminate is selected for the pesent study. The helical winding angle is 29.04 and hoop winding angle is 90 both are constant throughout the cylindrical portion. Form the failure analysis (discussed in section3), we can say that the operating pressure of the cylindrical portion is 25MPa and the ultimate failure pressure of the cylindrical portion is 32Mpa. Dome portion consists of different winding angles at different portions. The cross section of the dome consists of increase of thickness from the cylindrical portion to the polar opening. Slope of the contour is different at each point on the dome. This reason causes the more thickness of the composite laminates at the vicinity of the polar opening. The details of the winding angle at different portions of the dome are given in the table.3. The dome contour consists of different laminate structures with different winding angles, with different thicknesses. We need to find the laminate which is going to fail first under the given pressure. The dome portion the failure of first ply occurs at 21.5Mpa at radius 200mm. At this portion
10 helical, 6doilies, total thickness 10.5mm are provided. The Operating pressure of the dome portion is 20Mpa. The complete failure of the dome portion occurs at a pressure of 25MPa.

Finite element analysis of Geodesic dome portion
Finite element analysis of the Geodesic dome portion is carried out by considering the structure as orthotropic laminate structure. Shell 99 is selected as element type for the present study [8], [7], [2]. The layer sequence at the dome portion, along with orientation, thickness is completely specified in the Table 3. The below Table 4 represents the material properties of carbon epoxy laminate.
Table 4. Properties of Carbon/epoxy
Properties of carbon/Epoxy 
Values 
139Gpa 

9.8Gpa 

4.818Gpa 

4.818Gpa 

0.3 

0.021 
Where = volume fraction of fibres.
= volume fraction of matix.
Figure 3. Finite element model of Geodesic dome portion
After specifying the layer configurations select the proper boundary conditions and apply the load in cyclic pattern. The failure of the Geodesic dome portions is predicted by using maximum stress theory, which is illustrated in section 3.1. Tsai Wu theory and maximum stress theory gives approximately equal results. From the maximum stress theory the burst pressure of the structure is identified as 22Mpa.
Figure 4. The stress distribution of the Geodesic dome at its failure pressure
Figure 5. Finite element model of half section (axissymmetric) of Geodesic dome portion
Figure 6. The stress distribution of the half section (axissymmetric) of Geodesic dome at its failure pressure
Figure 7. Finite element model of cylindrical portion of composite pressure vessel
Figure 8. The stress distribution of cylindrical portion at failure pressure
All the above Figures represent finite element model and the stress distribution of the cylindrical and Geodesic dome portion at failure pressure.
Results and Discussions
The failure pressure of Geodesic dome portion is predicted by using finite element analysis software (Ansys 12.0), and classical laminate theory. The failure of the structure starts from the pressure of 21.5Mpa and the complete failure of the structure occurs at 25Mpa. The operating pressure of the structure should not exceed 21Mpa.Cylindrical portion consists of hoop layers. The layers with fibre orientation 90 participate more actively in load bearing; the fibres which are oriented in the angle other than 90 also participate in load bearing but lower than the hoop layers. This gives us a clear idea about predicting the failure of the layers. Increasing the pressure load beyond the limit causes the failure of hoop or circumferential layers first; this leads the total failure of the structure. The dome portion also consists of doilies oriented in 0, and 90, and helical layers. Here also hoop layers bear maximum load, and the next maximum load bearing layers are 0
layers. For increasing the load bearing capacity of the composite structure we must increase the circumferential layers at the cylindrical portion, and doilies at the dome portion. The failure sequence of the layers associated with different fibre orientation is as follows: the fibres with 90 orientation fails first, next the failure of the 0 layers takes place, lastly the failure of the helical layers takes place which leads the complete failure of the structure.
Table 5. Operating and failure pressures of composite structure
2000
stress(Mpa)
1500
1000
500
0
30 30 30 90 90 30 30 30
layer oreinatation
(Mpa)
Pressure 
Operating pressure 
Failure pressure 
Cylindrical portion 
25Mpa 
Failure Starts from 25Mpa, 
complete 

failure occurs at 

32Mpa 

Dome portion 
21Mpa 
Starts from 21.5Mpa and 
complete 

failure occurs at 

25Mpa(at 

radius 200mm) 

Composite structure 
20Mpa 
25Mpa 
Figure 9. Failure stresses for each layer at dome portion
1500
stress(Mpa)
1000
500
90
29.04
90
90
90
29.04
90
0
layer orienatation
(Mpa)
The above Table. 5 represent the operating and failure conditions of Geodesic dome as well as cylindrical portion.
The variation of the induced stresses in the corresponding layers is shown in the below Figures. 9, 10. The failure stresses induced in the layers which are oriented with different winding angles are calculated by using Mat lab code (which involves iterative calculations discussed in section3) are represented in the following Figures.
From the graphs indicated in the following Figures. 9,10 we can clearly estimate the failure sequence of layers. The stress induce in the hoop layers is exceeding the ultimate strength of the layers (1260Mpa), from the maximum stress theory it is clear that the failure of the hoop layers will takes place first.
Figure 10. Failure stresses for each layer at cylindrical portion
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