Design of a Prosthesis for Running in Composite Materials

7/27/2019 Design of a Prosthesis for Running in Composite Materials

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2. Bidimensional static analysis of a beam-based model: restraining

reactions and internal actions.

The prosthesis has been represented as a beam with three elbows, to reproduce

approximately the profile of the real model. The Fig. 2.1 shows the proportions of lengths and

angles, that have been chosen in order to adjust the schematic representation of a beam-based

model to real forms and dimensions. Length L corresponds to 10 cm, while angle is of 120.The model has been submitted to a static analysis: in Fig. 2.2 are shown the vertical load P,

applied by the runner in the connection point of the prosthesis, and the three degrees of

constraint, represented by the reactions of a sleeve and the ground reaction force. The sleeve

is placed in the same connection point and allows only vertical displacement, so it acts with an

horizontal force and a momentum. The ground reaction force acts normally to the horizontal

part of the beam, that is oriented parallel to the ground, and it is conceived as a distributed

load (q is a force per unit length). The equality between degrees of freedom and of constraint

and the configuration of the constraints make the system isostatic. The Fig. 2.2 shows also the

explicit values of the constraints, found with a simple static analysis.

Fig. 2.1: Form and proportions of a C-shaped Fig. 2.2: Representation of the users load and the

prosthesis modeled as a beam with three elbows. constraints acting on the prosthesis.

Internal actions has been also evaluated; results are showed in Figg. 2.3, 2.4, 2.5. We can

obtain the order of magnitude of forces and momenta acting on the model using a suitable

value for the force P. We have chosen 2850 N, that is the mean value transmitted on the

prosthesis by an amputated leg during the athletic gesture [1]. We can notice that the maxima

shear forces involve the two horizontal pieces of beam: in the upper the value is constant,

while in the lower the value has a linear trend with the highest value present on the left end.

The highest value coincides with P so it is equal to 2850 N. Normal forces act only on the twooblique parts, and have a constant value of about 2470 N. The momenta reach their maximum

value, about 520 Nm, on the elbow formed by the two oblique parts.


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Fig. 2.3: Representation of the internal shear Fig. 2.4: Representation of the internal axial actions; Cactions; words C and CC indicate clockwise and indicates compressive direction.

counterclockwise direction respectively.

Fig. 2.5: Representation of the bending momenta oriented

towards the stretched fibers zone.

3. Creation of the model in Abaqus

The prosthesis has been devised as an unidirectional laminate in epoxy matrix reinforced

with carbon fibers (volumetric fraction of 0,6). The Tab. 3.1 reports the properties of a single

lamina.

P P

P

0

0

C

CC


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(GPa) (GPa) (GPa) (GPa) (GPa)148 9,66 0,3 4,55 4,55 2,95 (MPa) (MPa) (MPa) (MPa) (MPa)

1314 1220 43 168 48

Tab. 3.1: Mechanical properties of an unidirectional lamina T300/934 Carbon/Epoxy; E and G are the normal

and the shear modulus respectively; is the Poissons coefficient; F is the ultimate strength.

The object has been modeled as a 3D deformable shell with an height of 35 cm and a width of

8 cm as showed in Figg. 3.1 and 3.2. A reference point, used to apply the vertical force, has

been created within the upper horizontal portion of the profile; the Fig. 2.2 shows this point.

Fig. 3.1: Profile view of the model with its dimensions. Fig. 3.2: View with the width dimension;in the red circle is highlighted the

application point of the force.

An additional rectangular element has been created with the function of reference plate for

the application of ground constraints. Therefore an interaction has been imposed between the

lower extremity of the prosthesis and the plate, neglecting the friction between the two

elements; a vertex of the plate has been restrained as an encastre. Also the final edge of the

prosthesis has been constrained allowing only vertical displacement and rotation respect to

the axis in the direction of the width; Fig. 3.3 clarifies configuration and constraints. A last

constraint, that permits only vertical displacement, has been applied to the reference point for

the application of the force. For each analyzed case the direction of force is vertical.The profile of the prosthesis and the direction of the width generate the principal directions, 1

and 2 respectively, the orientation of plies are referred to.

Both prosthesis and plate have been meshed by using quad structured elements of dimension

7 mm.

35 cm 15 cm

8 cm


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Fig. 3.3: Configuration of prosthesis and plate; the red arrows highlight the boundary conditions

regarding the lower part of the system.

4. Choice of the lamination thickness

The prosthesis has been initially designed as a superposition of thirty unidirectional laminas,

covering the entire object along its profile and all oriented in direction 1. The thickness of

each lamina has been set in order to obtain a vertical displacement (of the point in which the

force is applied) included in the range of the values considered optimal during running, i.e.

between 35 and 55 mm [2]. The force value used for this aim is 2850 N, as explained in thesecond paragraph.

The Fig. 4.1 shows how the displacement values vary respect to the thickness of a single

lamina.

0,17 0,18 0,19 0,20 0,21 0,2230

35

40

45

50

55

60

65

Verticaldisplacement

(mm)

Thickness (mm)

Fig. 4.1: Trend of the vertical displacement respect to the thickness of a single lamina

Encastre on a

vertex of the plateConstraints for

displacement and

rotation on the

final edge of the

prosthesis


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We have chosen a thickness value of 0,21 mm, that corresponds to a displacement of 37,6 mm.

In this way the total thickness of the laminate is of 6,3 mm.

5. Resistance verification

Fixed the thickness of the laminate, we have checked that the choice guaranteed an adequate

degree of safety during working. The value of force applied for this verification is 5000 N, that

results from increasing by 75% the working value of 2850 N (for precision the resulting value

is 4987,5 N, that we have round up). The Tsai-Hill failure criterion (Eq. 5.1) has been used,

considering only the stresses in the plane, since the laminate is very thin.

( ) (

) ()

To have a first indication about a possible fracture we have looked at the maximum mean

values of tensile, compressive and shear stresses reached by the prosthesis; we haveconsidered the first ply since the highest degree of stress is localized in the outermost plies, in

a symmetric way (the maximum tensile value in the ply 1 is roughly equal to the maximum

compressive value in the ply 30 and vice versa). The data are presented in the Tab. 5.1.

Minimum mean value Type of stress Maximum mean value

-1330 MPa 750 MPa-26 MPa 14 MPa-6 MPa 6 MPa

Tab. 5.1: Maximum mean values of stress obtained on ply 1, simulating the application of 5000 N on the

prosthesis with thirty laminas, each of them with thickness of 0,21 mm.

is the most problematical type of stress; in fact on the ply 1 it reaches a compressive valueof 1330 MPa, that overcomes the material strength = 1220 MPa; analogous questionregards the ply 30 on which is the tensile value higher than the strength.

The more stressed zone, in which reaches the highest value, is the blue region indicated inFig. 5.1.

Fig. 5.1: Red arrows highlighted the blue zone, in which reaches the maximum compressive value forthe ply 1.


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At first we have examined this zone; as for the other zones considered later, we have applied

the equation 5.1 in the whole thickness, taking into account the variation of the stress values

through it. Data in Tab. 5.2 show that, for the examined zone, the outermost plies break

according to the Tsai-Hill criterion.

(MPa) (MPa) Thickness(mm)

Values in

compression

Values in

traction

-1338,69 -26,79 -0,29 0,00 1,21

-1293,78 -25,89 -0,28 0,10 1,13

-1203,97 -24,09 -0,26 0,32 0,98

-1114,16 -22,30 -0,24 0,53 0,83

-1024,34 -20,50 -0,22 0,73 0,71

-934,53 -18,70 -0,20 0,95 0,59

-844,71 -16,90 -0,18 1,16 0,48

-754,90 -15,11 -0,16 1,37 0,38

-665,09 -13,31 -0,14 1,58 0,30

-575,27 -11,51 -0,12 1,79 0,22

-485,46 -9,71 -0,11 2,00 0,16

-395,64 -7,92 -0,09 2,21 0,11

-305,83 -6,12 -0,07 2,42 0,06

-216,02 -4,32 -0,05 2,63 0,03

-126,20 -2,52 -0,03 2,84 0,01

-36,39 -0,73 -0,01 3,05 0,00

53,43 1,07 0,01 3,26 0,00143,24 2,87 0,03 3,47 0,02

233,06 4,67 0,05 3,68 0,04

322,87 6,46 0,07 3,89 0,08

412,68 8,26 0,09 4,10 0,13

502,50 10,06 0,11 4,31 0,20

592,31 11,86 0,13 4,52 0,28

682,13 13,65 0,15 4,73 0,36

771,94 15,45 0,17 4,94 0,47

861,76 17,25 0,19 5,15 0,58

951,57 19,04 0,21 5,36 0,711041,38 20,84 0,23 5,57 0,85

1131,20 22,64 0,24 5,78 1,00

1221,01 24,44 0,26 5,99 1,17

1310,83 26,23 0,28 6,20 1,35

1355,73 27,13 0,29 6,30 1,44Tab. 5.2: Application of the Tsai-Hill equation (last two columns) on a square of the blue zone showed inFig. 5.1. Red color highlights the exceeding (or the equaling) of the breaking threshold.

As consequence of these results, we have tried to increase the thickness of each lamina, going

from 0,21 mm to 0,23 mm. Also in this case the object breaks in the ply 30, even if the limitvalue 1 in the equation 5.1 is not much exceeded.


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Therefore we have decided to change strategy, in order to avoid an excessive increasing of

thickness (and so of weight) of the prosthesis. Since we had already checked with thickness

0,21 mm that the only critical region is the showed one in Fig. 5.1, we have tried to use 30

plies, 0,21 mm thick, covering the entire profile of the proshtesis (as in the first choice) and to

add four further plies, 0,28 mm thick, just in that region. To be more precise this zone is

indicated in Fig. 5.2.

Fig. 5.2: With red color is showed the zone in which four

plies 0,28 mm thick has been added.

This solution works well. At first we notice that the maximum mean values of stress are quite

lower than the strength ones, as reported in Tab. 5.3.

Minimum mean value Type of stress Maximum mean value

-813 MPa 466 MPa-16 MPa 8 MPa-5 MPa 5 MPa

Tab. 5.3: Maximum mean values of stress obtained on the ply 1, simulating the application of 5000 N on the

prosthesis with thirty laminas, 0,21 mm thick, covering all the profile and four laminas, 0,28 mm thick, added

only in the region of Fig. 5.2 .

In the most critical zone the maximum value reached by using the equation 5.1 is 0,57. The

Tsai-Hill criterion has been successfully satisfied also in the red region showed in Fig. 5.3, in

which reaches the maximum tensile value in the ply 1; applying here the equation 5.1 thevalue 0,16 is not exceed.

To do a complete check of the safety condition, we have also considered the more stressedregion for and Confronting the Tabb. 3.1 and 5.3 an important indication is that themaximum mean values, for these two types of stress, are lower than the strength values of an

order of magnitude. With regard to the more stressed regions are indicated in Fig. 5.4.They are the same zones (blue in compression and red in traction, for the ply 1) already

checked since critical for . With regard to Fig. 5.5 shows that the maximum values areon the boundary of the critical zone, positive on one side and negative on the other side.

Applying the equation 5.1 a value of 0,27 results, well lower than 1.


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6. Energetic considerations

Storage and restitution of energy by materials used in sports application play, in general, a

fundamental role with regard to the performance and the safety of the athletes. This fact

posed a serious question about the possible disparity of performance condition between the

Fig. 5.3: Red color indicates the zone in which reachesthe maximum tensile values for the ply 1.

Fig. 5.4: Blue color indicates the zone in which reaches its maximum compressivevalues, while red color is associated to its maximum tensile values, for the ply 1.


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runners competing with a prosthesis and the able-bodied ones.

We have evaluated the energy restituted by the prosthesis, in static condition, under a

working vertical force of 2850 N. For this and the following analysis we have obviously used

the lamination configuration that satisfies the Tsai-Hill criterion. Since the materials forming

the prosthesis are modeled as elastic, the restituted energy equals the stored energy, without

any dissipation. The obtained value is about 47 J, that is consistent with the values obtained ina more accurate study in which, for a carbon/epoxy prosthesis, the different phases of

running has been considered [3].

Fig. 5.5: Blue color indicates the zone in which reaches the maximum negativevalues, while red color is associated to the maximum positive values, for the ply 1.

7. Relation between force and vertical displacement

We have done some simulations with different values of force, in order to evaluate its

influence on the vertical displacement of the point in which the force is applied. The results

are presented in Fig. 7.1.

The force values applied go from 1000 N to 5000 N, with increases of 500 N. The slope of the

curve is obviously positive and tend to decrease slowly, but not in a regular way. The slope of

a linear fitting on the whole interval is about 11 mm/kN.

8. Influence of the material properties

We have done simulations with the working force value of 2850 N, changing one by one the

values of the elastic moduli and of the Poissons coefficient of 10% respect to the ones present

in the Tab. 3.1. In this way we have estimated the effect of each property on the vertical

displacement and on the energy storage of the prosthesis. The Tab. 8.1 shows that is theonly property the variation of which has a significant effect on the prosthesis behavior.

Positive

Negative


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0 1000 2000 3000 4000 5000

0

10

20

30

40

50

60

Verticaldisplacement(mm)

Force (N)

Fig. 7.1: Trend of the vertical displacement respect to the vertical force

Property Property value Displacement (mm) Displacement

variation (%)

Energy

(J)

Energy

variation (%)

125,8 GPa 42,57 +15 55,07 +16,1148 GPa 37,03 47,42

170,2 GPa 33,77 -8,8 44,54 -6,1

8,21 GPa 37,04 +0,03 47,43 +0,02

9,66 GPa 37,03 47,4211,11 GPa 37,025 -0,01 47,42 0

0,255 37,05 +0,05 47,43 +0,020,3 37,03 47,42

0,345 37,01 -0,05 47,41 -0,02 3,87 GPa 37,11 +0,2 47,52 +0,24,55 GPa 37,03 47,42

5,23 GPa 36,98 -0,1 47,35 -0,1

Fig. 8.1: Effect of each of the four principal mechanical properties on the prosthesis response in terms of

displacement and energy storage.


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9. Production technology

A suitable production technology for the prosthesis designed in this work is the hand layup

and the autoclave molding. This technology allows a level of freedom, in the choice of

geometry, quite higher respect to other technologies as pultrusion and filament winding. It

permits also high volumetric fraction of fibers (that in this case is 0,6) and accurate control oftheir direction. However the working temperature doesnt overcome 40-50 C, so the

temperature reached by the autoclave should not be much higher.

10. Conclusions

In this work we have designed a C-shaped prosthesis for running in carbon fibers and epoxy

resin, modeled as an unidirectional laminate. A static analysis performed (with FEM method)

applying a vertical force in the connection point with the human body has revealed that themost critical zone, in terms of stresses, is the first horizontal portion of the profile. In this zone

we have inserted four further plies, 0,28 thick, in addition to the thirty others, 0,21 thick,

covering the whole profile. The prosthesis has passed the resistance verification with a load

increased by 75% the working one. The quantity of energy stored and restituted by the object

in static conditions, under the typical working load, is about 47 J. The axial modulus in

direction 1 has turned out to be the only mechanical property having significant effect on the

displacement and the energetic responses. This suggests that it is the most important

mechanical property that must be taken into account to obtain an optimization of

performances.

Bibliography

[1] D. I. Miller,Biomechanical considerations in lower extremity amputee running and sportsperformance, Aust J Sports Med, 13, 55-67, 1981.[2] D. Pailler, P. Sautreuil, J.-B. Piera, M. Genty, H. Goujon,volution des prothses des sprintersamputs de membre infrieur, Annales de radaptation et de mdecine physique, 47, 374-381,2004.

[3] C. Colombo, A. Curti, Modellazione e ottimizzazione di una protesi transtibiale per attivit

sportiva, XXXVIII Convegno Nazionale AIAS, 2009.

Documents

Design of a Prosthesis for Running in Composite Materials