1. APPLICATIONS USING FINITE ELEMENT METHODS TO REDUCE THE COST OF
COMPOSITE STRUCTURES
M H Armbruster*
, MSc Mech Eng (Wits)
Cape Town, 14 July 2002
1. ABSTRACT
The Finite Element Method (FEM) is a computer based technology for predicting the
stresses and strains in structures. It can also be used to predict deflections, natural
frequencies, heat transfer, dynamic response and buckling and has been popular in the
aerospace and automobile industry to reduce the cost and improve the reliability of load
carrying components. This paper presents examples of composite structures which
have been optimised using FEM. Comparisons are made with components designed
using traditional methods, with the emphasis on the reduction of material mass and
component cost.
2. BRIEF DESCRIPTION OF THE FINITE ELEMENT METHOD
A finite element model is a mathematical representation for the geometry, materials,
loading and constraints (such as supports) of the structure. Modern FEM programs
have simplified the generation of models with a user friendly graphical interface. This
enables even entry level technology students with computer skills and a basic
understanding of the principles involved to build accurate FEM models and generate
useful results.
The old adage of “garbage in, garbage out” does however still apply, but this can be
improved with analyst experience and by following the some quality assurance
guidelines. ANSI/AISC N690 [Ref A1] gives the following guidelines with respect to
using programs for the design of nuclear safety related structures:
- Design assumptions, identified input and output data must be documented.
- The computed results shall be documented together with a comparison of hand
calculations - Hand calculations will often be accurate in regions remote from
discontinuities. Using simple loading conditions such as uniform internal pressure
and/or isotropic material properties can also be used to check a model. These
results are then compared to calculated predictions using “Theory of plates and
shells” [Ref A2] or “Formulas for stress and strain” [Ref A3]. An example of this
approach is shown in Figure 1 and Figure 2 where a FEM model of a glass fibre
reinforced plastic (GRP) road tanker is checked using a simplified internal pressure
case [Ref A4]. Figure 1 compares the FEM and theory results for the axial force
resultant, and Figure 2 for the circumferential force resultant where the effect that
ring stiffeners have on the circumferential stresses is evident. Force resultant is
force per unit width (N/mm) and can be in some cases be estimated by stress
(N/mm2)
multiplied by thickness of the composite (mm).
* Member of M A Technologies cc, (email: matfix@iafrica.com)
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2. - Quality assurance requirements relating to the identification of the program,
documentation and control of its use shall be acceptable: Not all FEM packages or
analysts may be acceptable to the client, regulator or inspection authority.
- Untested computer programs shall not be used for design calculations.
- The computer program shall be uniquely identified by a name and/or number and
revision.
- Each revision of the computer program shall have its own documentation.
The geometry on which the FEM model is based can often be loaded directly from
computer aided design (CAD) programs (See Figure 9,10,11, 25 and 26). Some
programs have modules where the element mesh (consisting of nodes and elements),
loads and constraints can be generated within the CAD environment, and then exported
directly into the FEM program for analysis. This 3D solid model sometimes requires
some simplification for FEM, but means that the main features are used for the
generation of the FEM model and the subsequent 2D general arrangement and detailed
manufacturing drawings. The use of parametric 3D CAD software also allows for easier
updating of the 3D model once the results of the FEM analyses are obtained (and have
been checked).
This paper presents applications where FEM has been used successfully for:
- Concept Design - done early in the design cycle
- Detailed Design – done once the customer is happy with the concept and the
geometry is better defined.
- Laminate Optimisation – cost reduction and reverse engineering.
- Failure Analysis – used by investigation consultants and third party inspection.
- Design Verification – client review of a design, third party inspection.
Each section gives a detailed description of one example and short description of a few
other relevant applications.
3. THE USE OF FEM AT THE CONCEPT DESIGN STAGE
The analysis of composite components using FEM can already be incorporated at a
concept design stage to accurately estimate the amount of material required for
manufacture. This is often done before the manufacturer determines a budget or firm
price. The detailed example presented below explains the methodology typical of this
type of analysis.
3.1 Mass reduction of a helicopter undercarriage cross tube axle
An analysis was done to evaluate the mass reduction that could be achieved by
using a carbon fibre composite to replace the high strength steel used in the cross
tube axle of an military helicopter. The most severe loading is a crash load case,
and these were defined by the client.
This analysis ensured that the loading conditions were modelled correctly before
the component was optimised. The original steel axle was first analysed and
compared to results provided by the client. Close correlation with the client’s safety
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3. factors was obtained, and this confirmed that the FEM model represented the
loading accurately.
A different material was then defined for each circumferential band of the FEM
model, with the laminate composition varying from band to band (Figure 3). The
outer band represented the outer tip of the axle. The inner band represented the
middle of the axle at the centreline of the helicopter. The benefit of this technique is
that the material composition of the axle can be varied from band to band. This
enables both the orientation and the thickness of each orientation to be
continuously varied along the length of the axle, in response to the variation of the
axial, circumferential, shear and radial stresses. The layup consisted of a 0°, ±45°
and 90° layers. A nomex honeycomb was used for the portion between the
fuselage supports in order to increase the buckling strength. (See Figure 5 and 6).
The output was compared using the Tsai-Wu failure criteria. Each layer in the 30
different bands was evaluated, and the thickness of this layer was increased (or
decreased) in multiples of the per layer thickness in order to increase (or decrease)
the load carrying capacity of that layer.
A spreadsheet was used to convert the output of the FEM prediction into a number
of layers for the next FEM iteration. After this process was repeated a number of
times, the optimum was reached and no further reduction of mass was possible for
a given Tsai Wu safety factor. This method ensured that the entire component had
the same safety factor. The mass of the axle was reduced from 32kg for the
original steel component to a predicted mass of 6 kg for the carbon fibre
composite, with no sacrifice of the margin of safety.
This same technique was used for the optimum design of composite aircraft
seating. A economy class seat and business class seats are shown in Figures 31
and 16 respectively. Both seats were analysed for a range of different loading
conditions which include load and deflection limiting cases.
3.2Other Examples
Figure 6 shows a concept analysis of a carbon fibre archery bow. Figure 8 shows
a hollow carbon fibre windsurfer which was analysed for various load cases as
specified by the client. All of these components were successfully placed in
production based on laminates very similar to those specified during the concept
stage.
4. DETAILED DESIGN USING FEM
The use of FEM can be carried through from the concept design stage to the detailed
design. It is especially useful for evaluating different loading conditions, as specified by
code or client requirements. Traditional methods require complex calculations when
determining the natural frequency, buckling, shrinkage, thermal distortion, redundant
supports or stress concentrations. The use of FEM greatly simplifies the calculation of
these details.
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4. 4.1 Design of Pressure Vessels to ASME X
The adequacy of vessel designs to ASME X [Ref A9] shall be qualified to one of
two basic methods:
- Class I design through the destructive test of a prototype
- Class II design by mandatory design rules and acceptance testing by non-
destructive methods
FEM methods can be used in both cases to ensure a successful outcome for the
qualification of the vessels. For Class I vessels, the detailed FEM model can be
used to represent the geometry and construction of the proposed vessel. This can
then be ‘tested’ to the qualification test pressure of six times the design pressure.
Regions of high stress, discontinuity or stress concentration can be identified and
suitably reinforced before the vessel is constructed. The FEM model can also be
used to evaluate the first ply and last ply failure, using the failure criteria most
applicable to the materials specified.
Class II vessel designs are based on mandatory design rules and acceptance
testing. These methods rely on the determination of the engineering constants for
the laminate which are then used in the calculations and can easily be compared
to results from the FEM model. Rules are also given with respect to the maximum
strain. This can also be easily verified by FEM.
4.2 Design of a GRP underground storage tank (UST)
The following example covers the use of FEM in the design of glass reinforced
plastic (GRP) underground storage tanks. An 3D CAD rendering of the proposed
design is shown in Figure 9. Although this vessel is not required to be designed to
comply to any specific pressure vessel code such as BS 4994 [Ref A5] or ASME X
[Ref A9], there are performance requirements specified by the relevant SABS 1668
[Ref A11] specification. The introduction of this product into export markets would
also require compliance to European [Ref A12 to A16] specifications.
Figure 11 shows a section through the UST which was modelled in this analysis.
The cylindrical shell consists of a sandwich with internal CSM corrosion barrier,
helical and hoop filament winding, and a layer of parabeam as the core to give a
double wall construction. This has the benefit of leak detection and redundancy
with respect to protecting the environment from the contents of the tank. A
schematic of the construction is shown in Figure 10 with the corrosion barrier in
brown, filament winding in green and the parabeam core in blue. The ring stiffener
is shown in purple and this is further evaluated in the next paragraph.
Figure 12 to 15 evaluates the effect that different ring stiffener constructions, and
the use of a cross bracing inside the vessel, has on the stresses. The loading
modelled represents a concentrated 10 000 kg vertical load as specified by SABS
1668 [Ref A11]. Figure 12 has GRP ring stiffener with two GRP pipes as cross
bracing. Figure 13 is the same construction with the bracing removed. Figure 14
has the GRP ring stiffener replaced with a rolled steel section. Figure 15 shows a
section of a GRP pipe with no ring stiffeners. Comparing these FEM models to
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5. each other with respect to the stresses, deflection and masses quickly allows an
optimum design to be chosen. The other loading conditions such as impact,
internal pressure, external pressure and burial loads can also be evaluated. By
doing a matrix of FEM ‘testing’ of various constructions vs loading conditions, the
optimised design can quickly be determined. The tests required for qualification of
the vessel can also be modelled to ensure that the vessel is strong enough to pass
all of the qualification and acceptance tests.
4.3 Other Examples
Figure 16 shows the detailed design using FEM for a carbon fibre business class
seatback. Figure 17 and 18 show how FEM was used in the detailed design of a
carbon fibre wing for a supersonic aircraft. Figure 18 show how a FEM analysis is
used to predict the natural frequency of the first mode of vibration.
5. LAMINATE OPTIMISATION – COST REDUCTION AND REVERSE ENGINEERING
The following example shows how FEM can be used to optimise a pressure vessel
originally designed by a GRP pressure vessel design code such as ASME X [Ref A9] or
BS 4994 [Ref A5]. BS 4994 limits the design strain of the GRP vessel laminate such
that the minimum safety factor is greater than 8. The allowable strain in the laminate is
limited to ensure that breakdown of the resin-reinforcement bond does not occur in any
part of the structure.
The use of helical filament winding in the construction of the shell is also handled very
conservatively when using the standard approach recommended by BS 4994. Provision
is however made to perform a rigorous anisotropic elastic analysis, whereby the
contribution of each layer in the laminate is allowed for, and account is taken of the
interaction between the normal and shear strains. In this case, the strain transverse to
the fibre direction shall be less than 0.1%. FEM codes that support this type of analysis
include ABAQUS [Ref A6], MSc Nastran [Ref A7] EMRC Nisa [Ref A8] and ANSYS.
5.1 Laminate Optimisation of a GRP pressure vessel originally designed to BS
4994
The example presented in Figure 19 represents the fibre stress in the corrosion
barrier of a conical bottom of a 4 m diameter, 40 m3
silo which contains liquid with
a relative density of 1.5. The conical bottom was originally designed according to
BS 4994, and this construction was the first one analysed. None of the
constructions analysed in this exercise were allowed to exceed the design unit
loading for each layer through the laminate. The original laminate shown in Fig 19
had no circumferential or radial ribs, and the cone, skirt and overlay laminate had a
mass of 1900 kg. Figure 19 to 21 show ¼ symmetry used for the analysis, while
the mass is calculated for the entire skirt and cone assembly.
The results from the first iteration indicated that the cone and overlay laminates
between the cylinder, skirt and cone were over-designed. These were then
successively reduced, while still staying within the strength requirements. This
laminate is indicated by Fig 20, and the mass of the construction was 1608 kg.
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6. The third iteration shown in Figure 21 shows a construction with circumferential
and radial stiffeners, which are connected onto the skirt. The thickness of the
conical section and overlay are reduced up to the design unit loading as the ribs
carry some of the load. The laminate mass is reduced to 1516 kg. The mass of the
overlays required to bond the ribs to each other and to the skirt and cone would
have to be added to this estimate.
The final laminate chosen was the second iteration with a predicted mass of 1608
kg. This gave a 292 kg mass saving, without the complication of constructing the
radial and circumferential ribs of the third iteration. The cost reduction resulting
from this new optimised design exceeded the cost of the FEM analysis, even
though only one vessel was manufactured. This approach would be very cost
effective when a series of identical tanks needs to be manufactured.
5.2 Other Examples
Other examples where laminate optimisation was successfully applied are shown
in Figures 22 to 24. Figure 22 shows how a detailed design using FEM was used
to optimise a glass reinforced sandwich satellite TV antenna. Figure 23 and 24
show how an ice scraper designed to BS 4994 was modified after analysis by FEM
to allow the structure to support a torsion load induced by a ice scraper startup.
6. FAILURE ANALYSIS USING FEM
The use of FEM is well suited to failure analysis of structures. The structure can be
modelled to represent the failed product based on measurement or as built drawings.
The design or failure load can then be applied to the structure. Some examples are
shown in Figure 31 to 32. Figure 31 shows a carbon fibre economy class seatback
which was loaded with a 360 lbs ultimate load. The red regions indicate areas of high
stress, and corresponded with regions where failure was observed in practice.
Figure 32 shows the model of the bottom corner of a flat-bottomed pressure vessel.
Each layer of material was given it’s axisymmetrical orthotropic properties. This model
was generated in response to the client observing cracks in the corner of the cylinder
to flat bottom junction.
Figure 33 shows a axisymmtrical FEM model of a section through a glass wrapped
PVC pipe with a steel coupling. The steps in the metal portion of the pipe are there to
transfer the axial load from the steel component to the composite. This analysis was
done in response to the original design not being able to attain the test pressure.
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7. 7. DESIGN VERIFICATION – CLIENT REVIEW OF DESIGN AND 3rd
PARTY
INSPECTION
The use of FEM to independently check a design allows it to be used as a quality
assurance tool. Both the design and manufacture of composite products must comply
with the client requirements in order to be acceptable. Traditional methods often
involved extensive testing of the component, but this would occur relatively late in the
development cycle. Problems experienced during these tests, could result in delays as
the design was changed and new prototypes were built and tested. The alternative
was building products with large safety factors, but this may become too costly in
today’s competitive market.
7.1 Design verification of a yacht keel ad keel support
The example presented in Fig 25 to 28 is the design verification of a yacht keel and
supporting structure. The design is done by naval architects using the guidelines of
ABS 1986 [Ref A10], and is often based on experience from previous yachts. Fig
25 shows a 3D rendering of the yacht. The relevant portion to be analysed is
shown in Figure 26. This is a ½ symmetrical solid model of the hull, keel, keel
supporting structure and bolts which was loaded into the FEM package for mesh
generation and load definition. Size limitations of the FEM package resulted in the
two models being run separately.
Figure 27 shows the effect from a grounding load (as defined by Ref A10) applied
to the tip of the keel. The deflection of the keel is exaggerated to show the
rearwards movement under load. This shows the fibre stress in the outer
longitudinal fibre which runs along the length of the keel. These can then be
compared to the allowable stresses to determine of the design is acceptable.
Figure 28 shows the section of the hull and keel support structure. The loads are
transferred onto this structure via the two anchor bolts. The stresses determined
from this plot can be compared to the allowable stresses to check the design. In
both of these cases, the design was found to be acceptable, and the manufacturer
was happy to commence with mass production of the yachts with the peace of
mind that the design has been independently verified.
7.2 Other Examples
Figure 29 shows the FEM model for a large silo designed using BS 4994 [Ref A5].
The designed was checked using FEM for design loading of 250mm vacuum, 150
km/hr wind, 8 ton vertical top load and full contents. The design was found to be
suitable, and was also checked using a bucking analysis.
Figure 30 shows the FEM verification of a modification to a torpedo hull subjected
to external pressure (buckling analysis). The shell was analysed for the before
modification and after modification configuration. No reduction in the buckling
strength was predicted. The actual structure was modified and successfully passed
all tests.
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8. 8. CONCLUSION
Finite Element Methods are a powerful tool to enable today’s engineers to
successfully design, build and analyse composite structures. The ability of these
techniques to accurately predict the load distribution in the structure and determine
the effect of stress concentrations enables the designer to compensate by adding
extra reinforcing material in these regions. Systematic use of this technique can result
in the design of highly optimised structures, which complies with all the customer
requirements, and does not weigh or cost more than it needs to.
APPENDIX A : REFERENCES
A1. American Standards Institute, ANSI/AISC N690-1994, “Specification for the design
fabrication, and erection of safety related structures of safety-related structures for
nuclear facilites”
A2. Timoschenko, S., Woinowski, S. and Krieger, “Theory of plates and shells , 2nd
Edition”, McGraw Hill.
A3. Roark, R.J. and Young, W.C., “Formulas for stress and strain, 5th
Edition”, McGraw
Hill.
A4. Armbruster, M.H , “MSc Thesis Dissertation: The effect of liquids on the stress
distribution in a glass fibre reinforced plastic road tanker”, University of the
Witwatersrand, 1992
A5 British Standards Institute BS 4994: 1987, “Design and construction of vessels and
tanks in reinforced plastics”
A6 Abaqus Finite Element Program
A7 MSc Nastran Finite Element Program
A8 EMRC Nisa Finite Element Program
A9 American Society of Mechanical Engineers: ASME X, “1989 ASME boiler and
pressure vessel code: Fibreglass-reinforced plastic pressure vessels”
A10 American Bureau of Shipping, “ABS 1986-1993: Guide for building and classing
offshore racing yachts”
A11 South African Standard Specification: SABS 1668:1997, “Fibre-Reinforced Plastic
(FRP) tanks for buried (underground) storage of petroleum products”.
A12 European Standard: CEN/TC 210/WG6 N 53, “Filament wound FRP pressure
vessels – materials, design, calculation, manufacturing and testing”.
A13 European Standard: CEN/TC 210/SC1 N 74, “Requirements and test methods for
double wall tanks”
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9. A14 European Standard: Draft prEN 976-1, “Underground tanks of glass-reinforced
plastics (GRP) – Horizontal cylindrical tanks for the non-pressure storage of liquid
petroleum based fuels – Part 1: Requirements and test methods”.
A15 European Standard: Draft prEN 976-2, “Underground tanks of glass-reinforced
plastics (GRP) – Horizontal cylindrical tanks for the non-pressure storage of liquid
petroleum based fuels – Part 2: Transport, handling storage and installation”
A16 European Standard: Draft prEN 976-3, “Underground tanks of glass-reinforced
plastics (GRP) – Horizontal cylindrical tanks for the non-pressure storage of liquid
petroleum based fuels – Part 3: Requirements and test methods for double wall
tanks”
APPENDIX B : PICTURES
Figure 1. Comparison between theory and FEM predictions for the longitudinal stress
resultant in a GRP road tanker subjected to uniform internal pressure.
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10. Figure 2. Comparison between theory and FEM predictions for the circumferential stress
resultant in a GRP road tanker subjected to uniform internal pressure.
Figure 3. FEM model of the one half of the helicopter composite axle, with different materials
assigned to each band.
Figure 4. FEM plot showing the stress in the outer longitudinal layer as a result of the crash
load case
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11. HELICOPTER COMPOSITE AXLE
LOADCASE CRASH 9
0
2
4
6
8
10
12
14
16
18
20
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
SPANWISE STATION (-Y)
TOTALTHICKNESS
(NOMEX+CARBON)
TOTAL THICK CARBON THICK NOMEX THICK
Figure 5. Variation of thickness along the length of the helicopter composite axle (centreline
on the left)
0
20
40
60
80
100
120
140
NOOFLAYERS
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
MATERIAL TYPE (1=TIP)
HELICOPTER COMPOSITE AXLE
LOADCASE CRASH 9
0 +45 +90 -45
Figure 6. Variation of the laminate construction along the length of a helicopter
composite axle (centreline on the right)
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12. Figure 7. Analysis of a carbon archery bow using FEM from the concept stage.
Figure 8. FEM buckling analysis of a hollow carbon fibre windsurfer board used to
determine initial laminate details
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13. Figure 9. 3D rendered model of a 23m3
GRP underground storage tank (UST).
Figure 10. 3D rendered model showing a section through the wall of the GRP
underground storage tank.
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14. Figure 11. A section through the GRP UST which is exported from the 3D CAD package
and imported to the FEM program for mesh generation. The rib stiffener and
overlay laminate are clearly visible.
Figure 12. Stresses in the corrosion barrier for a section of the GRP UST supported by
a cross brace and GRP ring stiffener.
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15. Figure 13. Stresses in the corrosion barrier for a section of the GRP UST supported by
a GRP ring stiffener.
Figure 14. Stresses in the corrosion barrier for a section of the GRP UST supported by
a Steel ring stiffener overlaid with GRP.
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16. Figure 15. Stresses in the corrosion barrier for a section of the GRP UST consisting of a
GRP shell section only (no ribs stiffeners or cross bracing)
Figure 16. Detailed design using FEM for a carbon fibre business class seatback.
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17. Figure 17. Detailed design of a carbon fibre wing for a supersonic aircraft at ultimate
load using FEM .
Figure 18. FEM analysis to predict the natural frequency of the first mode of vibration .
This was determined to be at 46 Hz
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18. Figure 19. FEM of a section of the conical bottom of a silo as calculated by BS 4994
Figure 20. FEM of a section of the conical bottom of a silo after optimisation
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19. Figure 21. FEM of a section of the conical bottom of a silo with proposed
circumferential and radial stiffeners.
Figure 22. Detailed design using FEM for a glass reinforced sandwich satellite TV
antenna.
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20. Figure 23. Top dome of an ice scraper vessel designed to BS 4994 optimised by FEM
to carry the torsion loads.
Figure 24. Shell of an ice scraper vessel designed to BS 4994 optimised by FEM to
carry the torsion loads
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21. Figure 25. 3D rendering of the yacht which was analysed
Figure 26. 3D solid model of the keel, keel support, keel bolts and hull.
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22. Figure 27. The effect that the specified grounding load has on the fibres along the axis
of the keel.
Figure 28. The effect that the specified grounding load has on the keel support structure
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23. Figure 29. Buckling design verification of a silo designed for 250mm vacuum, 150 km/hr
wind, full contents and a 8 ton vertical top load.
Figure 30. Design verification of a modification to a torpedo hull subjected to external
pressure (buckling analysis)
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24. Figure 31. Model of an economy class carbon fibre seatback used to determine the
regions responsible for seat structural failure
Figure 32. Model of the corner of a flat bottomed pressure vessel. Each layer of material
was given it’s axisymmetrical orthotropic properties.
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25. Figure 32. Axisymmtrical FEM model of a section through a glass/polyester wrapped
PVC pipe with a steel coupling. The steps in the pipe are there to transfer the
axial load from the steel component to the composite.
25