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CHAPTER 4
ANALYSIS RESULTS
This chapter is about the stresses the catamaran hull is subjected to, which will be examined in detail.
The first analysis data, concerning the base layers already detailed in the chapter two (Chart. 2.5.1), (Fig. 2.5.2), have been compared with the carbon fiber sandwich panel’s stresses allowed. Once a failure index value has been fixed, in order to guarantee a wide margin of safety, the hull structure has been modified to meet the requirements previously imposed. It has been finally tested with another balance configuration.
4.1 Layer stresses
The first analysis data, show which are the hull highest stresses, the areas and the axis along which they occur and which kind of stress is the main one.
The hull-crossbeam intersection areas, together with the hull-tie rods juncture, result as the most stressed ones. They are primarily subjected to bending along the hull longitudinal direction thus the normal stresses along the x- axis result as the highest. Yet the stresses along the y and z axis as well as the shear ones are a lesser order of magnitude.
In the following pages the stresses along the longitudinal direction are shown layer by layer (Fig. 4.1.1- 4.1.2- 4.1.3- 4.1.4- 4.1.5- 4.1.6)
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Fig. 4.1.1 Longitudinal stress Layer one
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Fig. 4.1.3 Longitudinal stress Layer three
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Fig. 4.1.5 Longitudinal stress Layer five
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Below (Chart 4.1.1) is an outline of the hull panels composition, features and equivalent stresses reached. Layer Thickness (mm) Orient. (°) σ e (MPa) 1-carbon 0,30 0 29,8 2-carbon 0,18 45 41 3-carbon 0,18 -45 26,6 4-nomex 5,00 0 0,6 5-carbon 0,18 -45 40 6-carbon 0,18 45 48
Chart 4.1.1 Stress and features outline
4.2 Tsai Wu failure criteria
As the carbon fiber, the hull is made of, has an inherently orthotropic mechanical behavior, it has been necessary to define a failure criterion which took into account how the layers strength changes with the different fibers orientation.
The starting account has been that the failure occurs as soon as a fiber overtakes its permissible stress. This is a conservative assumption since the fiber fracture in a layer causes a decrease in the panel overall stiffness which does not necessarily brings the panel to collapse. High safety coefficients have been necessary at this stage anyway because of the uncertainty of the loads a dynamic structure as a catamaran is subjected to.
Starting from the Tsai Wu coefficients, calculated as follows, the failure criterion equation (Eqn. 4.1) has been used making the lamina orientation explicit. It determines how, for a panel loaded longitudinally (Fig. 4.2.1), the permissible stress changes with the fiber orientation.
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Fig. 4.2.1 Longitudinal load and fiber orientation
Tsai Wu coefficients:
F
1=
√; F
2=
√ with:
=
longitudinal compressive strength = longitudinal tensile strength = transverse compressive strength = transverse tensile strength47
Failure criterion equation:
σ
1 2[
( )( ) ( ) √
( ) ( )
]
[
( ) √ ( ) √]
Eqn. 4.1Below is the graphic which underlines the variation of the tensile and compressive failure load with the lamina angle θ (Fig. 4.2.2)
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Fig. 4.2.2 displays how much the lamina strength shortens while its orientation angle increases. It is also evident the tensile strength is lower than the compressive one for 45 degrees oriented fibers.
To compare the lamina stresses, resulting from the first analysis, with the permissible ones found above, the Tsai Wu failure index (F.I) has been calculated through the Equation 4.2.
Since the layer collapse occurs when F.I = 1 the safety condition, which must be verified on the hull lamina, has been fixed as F.I 0,4.
F.I = F1*σ1 + F2* σ2 + F11 σ12 + F22 σ22 + F66 τ12 + 2 F12 σ1 σ2 (Eqn 4.2) with: F1*=
; F
2 *=
+
; F
11=
; F
22=
;
F66 =; F
12=
√Charts 4.2.1, 4.2.2 show the failure indexes, correspondent to the highest tensile and compressive stress points on the hull, calculated layer by layer.
Layer σ1 (MPa) σ2 (MPa) τ12 (MPa) F.I Carbon 0° 30 4 1 0,14 Carbon 45° 26 3 2 0,58 Carbon -45° 26 2 2 0,56 Carbon -45° 30 3 3 0,66 Carbon 45° 33 3 3 0,73
Chart 4.2.1 Layers tensile stress
49 Layer σ1 (MPa) σ2 (MPa) τ12 (MPa) F.I Carbon 0° 24 5 2 0,08 Carbon 45° 43 3 2 0,94 Carbon -45° 28 4 2 0,59 Carbon -45° 30 4 2,5 0,63 Carbon 45° 50 4 3 1,1
Chart 4.2.2 Layers compressive stress
4.3 Hull structural reinforcement
The data displayed in Chart 4.2.2 show how the 45 degrees oriented lamina do not meet the safety requirements imposed since their failure index exceeds the limit value 0,4. For the inner layer the compressive failure would also occur.
Thus some changes in the layers composition have been necessary.
As the hull panels, particularly in the hull-crossbeam interface, are subjected to bending, the stresses are normal and their highest value occurs as far as the layer is far from the laminate neutral axis as shown in Fig. 4.3.1 and confirmed by the analysis data in Chart 4.2.1 .
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Thus the layers disposal has been changed. The panel has been made geometrically symmetrical about its neutral axis adding another 0° degrees oriented layer. This reduced the stress level in the 45 degrees oriented ones and made the overall stress symmetrical too (Fig. 4.3.2).
Fig. 4.3.2 Panel layers bending stress
with:
a = unidirectional layer 0 degrees oriented b = bidirectional layer +/- 45 degrees oriented c = core
This decreased the highest stress in the layers, as in Chart 4.3.1 for the tensile stresses and in Chart 4.3.2 for the compressive ones
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Layer σx (MPa) σy (MPa) τxy (MPa) F.I
Carbon 0° 20 3 0,7 0,1 Carbon 45° 26 2 1,6 0,56 Carbon -45° 23 2 1,3 0,49 Carbon -45° 25 2 1,7 0,53 Carbon 45° 25 2 1,8 0,53 Carbon 0° 16 3,5 1,8 0,1
Chart 4.3.1 Layers tensile stresses
Layer σx (MPa) σy (MPa) τxy (MPa) F.I
Carbon 0° 21 4 1,2 0,1 Carbon 45° 43 3 1,3 0,94 Carbon -45° 30 3 1,6 0,64 Carbon -45° 26 2 1,7 0,54 Carbon 45° 27 2 1,8 0,57 Carbon 0° 19 3 1 0,1
Chart 4.3.2 Layers compressive stresses
Since the failure index was still beyond the fixed safety level and no failure would have occurred with this configuration anyway, it has been necessary to thicken every layer. This procedure involved all the hull surfaces. The thickness enhancement has been different surface by surface, due to the different stress level the hull panels are subjected to. This has been done not to reach too high thicknesses and weight values where not necessary.
In Appendix one are shown the details of every panel layer thickness. For more details about all the stresses refer to Appendix two.
Below are (Chart 4.3.2, 4.3.3) the highest stresses (tensile and compressive) results concerning the catamaran final configuration.
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Layer σx (MPa) σy (MPa) τxy (MPa) F.I
Carbon 0° 17 1,7 0,4 0,06 Carbon 45° 12 1,3 1,2 0,25 Carbon -45° 13 1,1 1 0,27 Carbon -45° 13 0,8 1 0,26 Carbon 45° 12 1 1 0,24 Carbon 0° 14 2 0,8 0,06
Chart 4.3.2 Final configuration tensile stresses
Layer σx (MPa) σy (MPa) τxy (MPa) F.I
Carbon 0° 14 2,1 0,6 0,07 Carbon 45° 18 1,5 1 0,38 Carbon -45° 12 1,4 1 0,25 Carbon -45° 15 1 1 0,31 Carbon 45° 13 1,4 1,2 0,27 Carbon 0° 14 2 0,5 0,07
Chart 4.3.3 Final configuration compressive stresses
The hull panels thickness enhancement caused an increase of the overall hull weight whose value changed from 12 Kg to 18 Kg for each hull. This final value is in accordance with the weight limit imposed by the project requirements fixed at the beginning.
The weight increase however changed the catamaran roll configuration as well as the aerodynamic force of equilibrium. Thus it has been necessary to repeat the procedure, described above, to balance the catamaran roll moment under the newly calculated force.
The balance force found for the final configuration is Fa = 685 N. As the model had been updated with its real weight forces another analysis has been run. The resultant stresses differ from the previous ones by less than 2% and all of them meet the failure index safety requirements.
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4.4 Overboard skipper configuration test
The hull structure has been finally checked as the skipper enhances the weights momentum, sailing with his barycenter totally overboard. This causes a higher aerodynamic load on the sail, with Φ =10°, and it changes the skipper weight distribution on the hull.
Indeed, as shown in Fig. 4.4.1, the skipper downloads its weight only partially on the hull because he is kept attached to the catamaran by a rod constrained at the top of the mast.
Fig. 4.4.1 Skipper overboard configuration roll balance
Fcn
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with:
R2 = rod reaction
β = rod-catamaran transverse axis angle Lr = metacenter- rod reaction distance
Fig. 4.4.2 shows the detail of the skipper weight balanced with the forces exchanged with the catamaran and the rod
Fig. 4.4.2 Skipper-hull forces
with:
R1 = hull-skipper reaction Fa = hull-skipper friction force
The forces magnitude has been calculated with the equilibrium along the x and y axis: R1cosΦ = R2cos(β-Φ) + FasenΦ
Ws = FacosΦ + R2sen(β-Φ) + R1senΦ with:
Φ= 10 degrees β= 60 degrees
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The roll balance is defined as follows referring to Fig. 5.4.1
Mfa = Mfw
Mfa = Fa a + (Fd+Fw)b + Fcnt e
Mfw = (Wh+Ws) Lcb cosΦ + Wcb
cosΦ + Wm Lb cos(γ+Φ) + Wh c + R2Lr
The aerodynamic force value which satisfies the balance condition is Fa= 940 N
Once the new catamaran geometry model (detailed in Appendix 3) has been defined, modifying the original in order to insert the new skipper-catamaran interaction, a new analysis has been run.
Below are shown the results for the tensile and compressive stresses are shown layer by layer (Chart 4.4.1, 4.4.2).
The failure index threshold level has been fixed to 0,5 for this configuration since this analysis has been carried on overestimating the skipper force in order to have a more conservative result. All the layer stresses meet the requirements.
Layer σx (MPa) σy (MPa) τxy (MPa) F.I
Carbon 0° 16 4 1 0,12 Carbon 45° 24 2 1 0,49 Carbon -45° 19 2 2 0,41 Carbon -45° 23 3 3 0,49 Carbon 45° 22 3 3 0,48 Carbon 0° 48 2 1 0,12
56 Layer σx σy τxy F.I Carbon 0° 46 2 1 0,11 Carbon 45° 23 2 2 0,47 Carbon -45° 23 2 2 0,47 Carbon -45° 22 1 1 0,46 Carbon 45° 26 1 1 0,48 Carbon 0° 16 2 1 0,07
Chart 4.4.2 Layers compressive stresses
