Analysis of curved composite structures through · PDF fileaccuracy of the beam theory" ->...

Introduction CUF Mapped models Composite applications Conclusion

Analysis of curved composite structuresthrough refined 1D finite elements with

aerospace applicationsErasmo Carrera, Alberto García de Miguel, Alfonso Pagani,

Marco Petrolo

Mul2 Team, Department of Mechanical and Aerospace Engineering,Politecnico di Torino

Full

ASME International Mechanical Engineering Congress and ExpositionNovember 16th , 2016, Phoenix (AZ) USA

Erasmo Carrera, Alberto García de Miguel, Alfonso Pagani, Marco Petrolo

CARRERA UNIFIED FORMULATION - 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.FULLCOMP.NET


International Mechanical Engineering Congress and Exposition IMECE2016, November 16th , Phoenix (USA)

The FULLCOMP projectFULLy integrated analysis, design, manufacturing and health-monitoring of COMPosite structures

The FULLCOMP project is funded by the European Commission under aMarie Sklodowska-Curie Innovative Training Networks grant for European Training Networks(ETN).

The FULLCOMP partners are:1 Politecnico di Torino (Italy) - Coordinator2 University of Bristol (UK)3 ENSMA (Bordeaux, France)4 Leibniz Universitaet Hannover (Germany)5 University of Porto (Portugal)6 University of Washington (USA)7 RMIT (Melbourne, Australia)8 Luxembourg Institute of Technology9 Elan-Ausy (Hamburg,Germany)

12 PhD studentsFull spectrum of design of composite structures: manufacturing, health-monitoring, failure,modeling, multiscale approaches, testing, prognosis and prognostic.

www.fullcomp.net





Overview

1 From classical beam theories to the unified formulation

2 Non-local hierarchical theories of structure

3 Cross-sectional mapping

4 Component-wise analysis for composite structures

5 Conclusions and future developments





From classic to refined kinematicsEuler-Bernoulli beam theory

ux (x, y, z) = ux1 (y)

uy (x, y, z) = uy1 (y) − x∂ux1 (y)

∂y − z∂uz1 (y)

∂yuz(x, y, z) = uz1 (y)

Timoshenko beam theoryux (x, y, z) = ux1 (y)uy (x, y, z) = uy1 (y) + x φz(y) − z φx (y)uz(x, y, z) = uz1 (y)

Saint Venant beam theoryux (x, y, z) = ux1 (y) − z θ(y)

uy (x, y, z) = uy1 (y) + x φz(y) − z φx (y) + ψ(x, z)∂θ(y)∂y

uz(x, y, z) = uz1 (y) + x θ(y)

Washizu’s statement:"For a complete removal of the inconsistency and an improvement of theaccuracy of the beam theory" -> enrich beam kinematics with higher-order terms





Beam kinematics through the unified formulation

u(x, y, x) = Fτ(x, z) uτ(y)

ux (x, y, z) = F1(x, z) ux1 (y) + F2(x, z) ux2 (y) + F3(x, z) ux3 (y) + ... + FM(x, z) uxM (y)

uy (x, y, z) = F1(x, z) uy1 (y) + F2(x, z) uy2 (y) + F3(x, z) uy3 (y) + ... + FM(x, z) uyM (y)

uz(x, y, z) = F1(x, z) uz1 (y) + F2(x, z) uz2 (y) + F3(x, z) uz3 (y) + ... + FM(x, z) uzM (y)

τ = 1, ...,M (number of expansion terms)

N(y)i

Classical1D FE

F (x,z)τ

Cross-SectionFunctions

+

transverse stresses

z,x

y

o(x3,z3)o(x,z) o(x5,z5)

F =

Classical models, such as Euler or Timoshenko, can be obtained as particular cases of thelinear models.





CUF stiffness matrixCUF displacement field

u = Fτ(x, z)Ni(y) uτi (1D)

PVD - Static

δLint = δuTτiK

ijτsusj

δLext = PδuT

fundamental nucleus

K ijτsxx = C22

∫Ω

Fτ,x Fs,x dΩ

∫lNiNjdy + ...

i, j - Shape function indexes.

τ, s - Expansion function indexes

Assembly Techniqueτ-s block i-j block global stiffness matrix

node element structure





Hierarchical Legendre Expansions, HLEVertex polynomials

Fτ =14

(1 − rτr)(1 − sτs)

Side polynomials

Fτ =12

(1 − s)φp(r)

Internal polynomials

Fτ = φpr (r)φps (s) pr + ps = p

Cross-Section

nodes

Disconnected

nodes

1

2

3

4

5

6

7

Po

lyn

om

ial d

eg

ree

τ=1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

18 19 20 21

24 25 26 27

31 32 33 34

17

22 23

28 29 30

35 36 37 38

HL1 HL2 HL3 HL4vertex expansions

side expansions

internal expansions

e.g. HL4 = 4th order expansion, 17×3 DOFs

generalized displacements unknowns, hierarchicalkinematicscross-section discretization





Cross-sectional mapping

HLE models: coarse section discretizations, large domains

Need of representing the exact geometry of curved domains

Blending function method→ non isoparametric expansions (i.e. themapping functions, Q, do not correspond with the expansionfunctions, Fτ)

z

x

(X1,Z1)

(X2,Z2)

(X3,Z3)

(X4,Z4)

z2( )

x2( )Q

(-1,1) (1,1)

(1,-1)(-1,-1)

x2(η) = ax + bxη + cxη2 + dxη

3

z2(η) = az + bzη + czη2 + dzη

3

⇓ Mapping functions

Qx = Fτ(ξ, η)Xτ +(x2(η) −

( 1 − η2

X2 +1 + η

2X3

)) 1 + ξ

2

Qz = Fτ(ξ, η)Zτ︸︷︷︸ +(z2(η) −

( 1 − η2

Z2 +1 + η

2Z3

)) 1 + ξ

2︸︷︷︸isoparametric blending function

term term





3D-like modelling using CUF beams

y

x x

zz

Geometrically exact curvedsections 3D-like boundary conditions

Thin-walled structuresComposites





Wing structure

Airfoil NACA 2415

X

Z

Y

1 m

6 m

L = 1766 N/m

Aluminum: E = 75 Gpa, ν = 0.33

HLE mapped model

1

2

5

3

4

+5 HLE section domains

10 beam elements

Shell-like displacement solutions

Model uz m σyy MPa σyz MPa DOFsLoad point, y = L Load point, y = L/2 Leading edge, y = L/2 m

HL3 0.214 30.044 9.686 3720HL5 0.220 31.646 8.609 7905HL8 0.223 32.000 8.839 17670MSC Nastran 0.225 32.301 8.761 395280





3D-like stress solutions

σxx

σxy

σyy

σxz

σzz

σyz





The Component-Wise Approach

τ

s

τ

s

τ

s

Layer 1 Layer 2 Layer 3

Fundamental nucleus expansions

3x3 Fundamental nucleus

τ

s

Assembled LW matrix

Layer-Wise approach

for laminate structures

laminae

1D elements

Only 1D beam elements are used to model all components.

No need of reference surfaces. This might be useful in a CAD-FEM interface scenario.

Straightfoward refining of the model on particular zones of interest.





Curved sandwhich

p=10kN/m2

L = 1 m, h = w = 40 mm,tf = 1 mm, tc = 5 mm

B

A

9 HLE domains10 beam elements

Faces: E = 75 GPa, ν = 0.33

Core: E = 0.1063 GPa, ν = 0.32

model uz × 102 m σyy × 10−7 Pa σyz × 10−6 Pa DOFsPoint A point B Point B

MSC Nastran solutionsSolid -3.715 3.353 4.324 166617Shell -3.900 3.496 2.721 71000

HLE model solutionsHL2 -3.032 3.421 -6.551 3720HL3 -3.711 3.307 4.484 952HL4 -3.712 3.293 4.466 9021HL5 -3.712 3.288 4.276 12927HL6 -3.712 3.290 4.256 17670HL7 -3.712 3.298 4.308 23250HL8 -3.712 3.297 4.321 29667

0

0.5

1

1.5

2

2.5

3

3.5

-3 -2 -1 0 1 2 3

σ yy

x 10

-7 [P

a]

z [mm]

SolidShellHL3HL5HL7

-6

-4

-2

0

2

4

-3 -2 -1 0 1 2 3

σ yz

x 10

-6 [P

a]

z [mm]

SolidShellHL3HL5HL7





Fiber-matrix cell

F = 0.1 N

L = 4 mm

0.1 mm

0.0

8 m

m

A

B

C

D

E [GPa] ν

Fiber 202.038 0.2128Matrix 3.252 0.355

HLE mapped model 1D elements

+

model uz × 102 [mm] σyy × 10−8 [Pa] σyy × 10−8 [Pa] σyz × 10−8 [Pa] DOFsPoint A, y = L Point B, y= L/2 Point C, y= L/2 Point D, y= L/2

MSC Nastran -7.818 9.492 7.094 -2.383 268215TE (N=5) -7.795 9.327 7.090 -2.375 7623LE (12L9+8L6) -7.933 9.450 7.046 -2.500 7533HLE (5HL5) -7.773 9.383 7.070 -2.746 6603

-1x109

-8x108

-6x108

-4x108

-2x108

0

2x108

4x108

6x108

8x108

1x109

σyy

-3x107

-2.5x107

-2x107

-1.5x107

-1x107

-5x106

0

5x106

σyz





Cross-ply laminateComponent EL ET Ez GLT GLz GTz νLT νLz νTz

Fiber (ortho) 202.038 12.134 12.134 8.358 8.358 47.756 0.2128 0.2128 0.2704Layer (ortho) 103.173 5.145 5.145 2.107 2.107 2.353 0.2835 0.2835 0.3124Matrix (iso) 3.252 1.200 0.355

2706 DOFs 35433 DOFs 14601 DOFs

-5x108

-4x108

-3x108

-2x108

-1x108

0

1x108

2x108

3x108

4x108

5x108

-1x109

-8x108

-6x108

-4x108

-2x108

0

2x108

4x108

6x108

8x108

1x109

-8x108

-6x108

-4x108

-2x108

0

2x108

4x108

6x108





Main Conclusions and Perspectives

1 Hierarchical theories of structure with mapping capabilities extend thebeam modelling to the accurate analysis of curved thin-walled structures.

2 The accuracy is mainly controlled by the polynomial order and refinement ofthe expansions over the cross-section.

3 The Component-Wise approach provides a tool for the efficient analysis ofmulti-component structures, such as fiber-reinforced composites.

4 Reduction of DOFs by one order of magnitude, or more, in comparison withcommercial solid models for the same level of accuracy.

Ongoing developments and future extensionsMulti-scale analysis of fiber-reinforced components.Extension of CUF to non-linear problems: progressive damage, plasticity,etc.Introduction of curved beams: extension of CUF beam models to a newrange of structural geometries.





Thank you for the kind attention,

any questions?

Full





Analysis of curved composite structuresthrough refined 1D finite elements with

aerospace applicationsErasmo Carrera, Alberto García de Miguel, Alfonso Pagani,

Marco Petrolo

Mul2 Team, Department of Mechanical and Aerospace Engineering,Politecnico di Torino

Full

ASME International Mechanical Engineering Congress and ExpositionNovember 16th , 2016, Phoenix (AZ) USA



Analysis of curved composite structures through · PDF fileaccuracy of the beam theory" ->...

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