MSc thesis presentation - Aerospace Structures - July 2015

Static and free-vibration analysis of damaged
metallic and composite aerospace structures
through advanced beam models
Alessandro Rosati
POLITECNICO DI TORINO
I Facoltà di Ingegneria
Tesi di laurea magistrale in Ingegneria Aerospaziale
Prof. Erasmo Carrera
Dott. Marco Petrolo
Dott. Alfonso Pagani
Relatori:
22 luglio 2015

1D Beam models
Politecnico di Torino
Tesi di laurea magistrale in Ingegneria Aerospaziale 2
Introduction
 Low computational cost
 Reliable results for slender, compact and
homogeneous structures in bending
⊗ Strong assumptions/simplifications needed
⊗ Difficulties in handling with complex problems
(composites, short and thin-walled structures)
Real wing 1D beam model

Carrera Unified Formulation (CUF)
Carrera Unified Formulation
i, j: shape function indexes (depending on the FE discretization)
τ, s: expansion function indexes (depending on the model order)
Fundamental Nucleus:

Modelli CUF 1D Taylor e Lagrange
TE models:
LE models:
Timoshenko classical model, obtained as a particular case of N = 1
L9 isoparametric polynomial:

Component–Wise (CW)
Each component of the structure is modelled via beams only!

Present work outlines
Present work outlines
• development of a tool able to simulate, via TE and LE CUF models, the
behavior of aeronautical multi-component structures subjected to
increasing loads
Purpose: to understand the evolution of the stresses concerning
to the main parts of the studied structures and to calculate the
values of final loads
• evaluate the variation of natural frequencies of some structures,
treated in previous works, as a result of the introduction of damages
Purpose: to understand the free-vibration behavior of structures
in response to the introduction of damages and test CUF
capabilities to deal with non-destructive testing of structures

Damage Modelling
Damage Modelling
• modelled as percentage reductions or modifications of the elastic/shear
modules of constitutive materials
• a proper factor has been introduced: it is defined as d* = 1 − d, in which
d is the entity of damage (the damage percentage is given by 100 × d)
Ed*, m = d* × Em
Gd*, mn = d* × Gmn
Deteriorations in the mechanical characteristics of the structural components
(m, n are the involved directions in the adopted reference frame, m ≠ n)

Progressive Failure Analysis
Failure Analysis
Objectives:
to evaluate the ultimate load of multi-component structures
 the structure is subjected only to
bending by means of a increasing
transverse point load Fz
 the evolution of stresses on each
component of the structure has to be
studied
 failure criteria have to be defined
The ultimate load is intended as the maximum external load at which the structure
collapse and it is no more able to perform its duties

Failure Criteria
Failure Analysis
The total failure of the multi-component structure is modelled as the loss of all
constrained stringers.
Single failures on stringers:
• a ultimate stress value has to be chosen, as
function of the material employed
• when a general stringer reaches its ultimate
stress level, it is considered lost and a null elastic
modulus has to be assigned to it (d* = 0)
Traction:
Yield stress σ02
Compression:
Euler buckling stress

Procedure and Assumptions
Failure Analysis
• there is a linear correspondence between
external load Fz and internal tensions σ
• there is a linear correspondence between
stresses σ and strains ε (elastic
behaviour)
• isotropic materials are adopted
Assessment Phase:
Comparison between results obtained by means of
EBBT CUF model and iterative analysis based on
ideal-monocoque method

Isotropic material definition
Failure Analysis
An isotropic aluminium alloy material has been adopted for every component
of the studied structures:
E is the elastic modulus of the material,
ν the Poisson ratio,
ρ the density,
Yield stress
E = 75 GPa
ν = 0.33
ρ = 2700 kg/m3
σ02 = 0.3 GPa
A rib has been located at the free-tip of every structure, where
the external load is applied, and modelled with a elastic modulus
ten times higher than the one adopted for other components

Ultimate load evaluation: 4-stringers wing-box
Failure Analysis: 4-stringers wing-box
b = 1 m
h = 0.5 m
t = 2 × 10−3 m
L = 3 m
As = 1.6 × 10−3 m2
• 10 B4 element mesh along the y−axis
• Clamped in y = 0
• Fz applied at [b/2, L, h/2]
• Resistant panels

Ultimate load evaluation: 6-stringers wing-box
b = 1 m
h = 0.5 m
t = 2 × 10−3 m
L = 3 m
As = 1.6 × 10−3 m2

Initial damages: 6-stringers wing-box
Ultimate Load × 10-5 N

Shear-instabilities: 6-stringers wing-box
• onset of instability phenomena on panels
• the instability condition modelled as a damage
Critical shear tension:

Multi wing-boxes
Failure Analysis: multi wing-boxes
b = 1 m
h = 0.5 m
t = 2 × 10−3 m
L = 9 m
ℓ = 3 m
As = 1.6 × 10−3 m2

Free-vibration analysis
• Isotropic, orthotropic and composite
materials have been investigated
• The Modal Assurance Criterion (MAC)
has been adopted in order to yield a
degree of consistency between mode
shapes of undamaged and damaged
structure
Objectives:
to understand the dynamic behaviour of structures
in response to the introduction of damages

Sandwich beam
• clamped-clamped
• face sheets (f) bonded to a core (c)
• isotropic material
L = 1 m
hc = 0.16 m
hf = 0.02 m
Ef = 68.9 GPa
Ec = 179.1 MPa
ρ f = 2687.3 kg/m3
ρ c = 119.69 kg/m3
ν f = ν c = 0.3

Sandwich beam
Component-Wise model is able to detect mode shapes involving the external layers only,
because of the remarkable difference in the elastic modulus values
between the materials of faces and core respectively
Mode 5 - 796.021 Hz Mode 6 - 1070.623 Hz Mode 7 - 1081.442 Hz

Damages : bottom face-sheet
d* = 0.5

MAC evaluation
Undamaged - d* = 0.1Undamaged - d* = 0.5Undamaged - d* = 0.9
CW solutions, comparisons in mode shapes
of undamaged and damaged structure

MAC evaluation
CW solutions, comparisons in mode shapes considring diferrent damage locations:
the entire layer and a 20% of the total length zone at the free-tip and middle-span
Undamaged - d* = 0.1 Undamaged - d* = 0.1 Undamaged - d* = 0.1

Composite-type Longeron
• clamped-free
• 16 B4 along the y−axis
• total length: 1 m
UD and -45/45 layers:
ORTHOTROPIC
EL = 40 GPa
ET = 4 GPa
GLT = GTT = 1 GPa
ν LT = ν TT = 0.25
Foam:
ISOTROPIC
E = 50 MPa
ν = 0.25

Damages: bottom UD layer
d* = 0.5

MAC evaluation, UD layer

MAC evaluation, UD layer
CW solutions, comparisons in mode shapes considring diferrent damage locations, on the
bottom UD layer for 20% of the total length
Undamaged - d* = 0.5 Undamaged - d* = 0.5 Undamaged - d* = 0.5
Free-tip Middle-span Clamped-end

Damages: internal -45/45 thin layer
d* = 0.5

MAC evaluation, thin layer

Conclusions
Conclusions
• reliable ultimate loads in bending have been evaluated and significant
structural behaviours have been pointed out at a low computational cost
• mode shapes and frequencies of undamaged and damaged structures
calculated through solid FEM analyses have been almost duplicated by
LE and high-order TE models
• MAC number evaluation has been in many cases a useful tool in order to
point out discrepancies in the mode shapes of undamaged and damaged
structures

Conclusions
Conclusions
Overcome the limitations imposed by the fundamental assumptions of classical
beam theories
Some limitations in the approximation of some geometries can be found
TE models:
Is able to furnish reliable solutions and is not affected by limitations due to
geometry/complexity of the structure to be analyzed.
Is able to correctly predict local shapes typical of sandwich or laminated structures
CW approach:

THANKS
FOR YOUR ATTENTION!
Alessandro Rosati 22 luglio 2015
Any suggestions or questions are welcome

MSc thesis presentation - Aerospace Structures - July 2015

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