(E-book Composite) Airbus Composite Stress Manual Us Mts006 b
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Technical Manual MTS 006 Iss. B Outhouse distribution authorised
Composite stress manual 1 4
5 4
2 1 2
5
Structural Design Manual
Purpose
Scope
To list and homogenise the calculation methods and the allowable values for the composite materials used at the Aerospatiale Design Office. To be used as reference document for all Aerospatiale and subcontractors' stressmen.
Data processing tool supporting this Manual
Summary
Document responsibility
See detailed summary
Dept. code : BTE/CC/SC
Validation
Name : P. CIAVALDINI
Name : JF. IMBERT Function: Deputy Department Group Manager Dept. code : BTE/CC/A Date : 06/05/99 Signature
This document belongs to AEROSPATIALE and cannot be given to third parties and/or be copied without prior authorisation from AEROSPATIALE and its contents cannot be disclosed. © AEROSPATIALE - 1999
Composite stress manual
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© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
Foreword
This issue is incomplete and existing chapters are liable to change. All allowable values and coefficients related to the various materials described in chapter Z are updated with each issue of the manual. This means that different values may be found in the stress dossiers prior to latest issue. The data processing tools are given for information purposes only.
© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
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© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
SUMMARY OF CHAPTERS Iss. DETAILED SUMMARY B INTRODUCTION - COMPOSITE MATERIAL PROPERTIES
A
Date
Editor
A
Jan 98 P. Ciavaldini
B
Apr 99 P. Ciavaldini
COMPOSITE PLATE THEORY
B
*
MONOLITHIC PLATE - MEMBRANE ANALYSIS
C
A
Jan 98 P. Ciavaldini
MONOLITHIC PLATE - BENDING ANALYSIS
D
A
Jan 98 P. Ciavaldini
MONOLITHIC PLATE - MEMBRANE + BENDING ANALYSIS
E
A
Jan 98 P. Ciavaldini
MONOLITHIC PLATE - TRANSVERSAL SHEAR ANALYSIS
F
B
Apr 99 P. Ciavaldini
MONOLITHIC PLATE - FAILURE CRITERIA
G
B
Apr 99 P. Ciavaldini
MONOLITHIC PLATE - FATIGUE ANALYSIS
H
*
MONOLITHIC PLATE - DAMAGE-TOLERANCE
I
**
B
Apr 99 P. Ciavaldini
MONOLITHIC PLATE - BUCKLING
J
*
MONOLITHIC PLATE - HOLE WITHOUT FASTENER ANALYSIS
K
B
Apr 99 P. Ciavaldini
MONOLITHIC PLATE - FASTENER HOLE
L
B
Apr 99 P. Ciavaldini
MONOLITHIC PLATE - SPECIAL ANALYSIS
M
B
Apr 99 P. Ciavaldini
B
B
B SANDWICHIC - MEMBRANE / BENDING / SHEAR ANALYSIS
*
N
SANDWICH - FATIGUE ANALYSIS
O
*
SANDWICH - DAMAGE-TOLERANCE APPROACH
P
*
SANDWICH - BUCKLING ANALYSIS
Q
*
SANDWICH - SPECIFIC DESIGNS
R
*
BONDED JOINTS
S
A
Jan 98 P. Ciavaldini
B BONDED REPAIRS
T
B
Apr 99 P. Ciavaldini
BOLTED REPAIRS
U
A
Jan 98 P. Ciavaldini
B THERMAL CALCULATIONS
V
B
Apr 99 P. Ciavaldini
ENVIRONMENTAL EFFECT
W
*
NEW TECHNOLOGIES
X
*
STATISTICS
Y
*
Z
**
B
Apr 99 P. Ciavaldini
B MATERIAL PROPERTIES
*: chapter not dealt with. **: chapter partially dealt with.
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
HOW TO USE THE COMPOSITE MANUAL? reference of chapter
title(s) of subchapter(s)
reference(s) of subchapter(s)
title of chapter
N 4.2.1
SANDWICH Effect of normal load Ny
1/2
4.2.1 . Effect of normal load Ny Assuming that all layers are in a pure tension or compression condition, a normal load Ny applied at the neutral line results in a constant elongation over the whole cross-section. This elongation may be formulated as follows:
reference of relation
n3
ε =
Ny b (EMi ei + EMc ec + Ems es )
This elongation this unduces: - in the lower skin, a stress σi = Emi ε, - in the core, a stress σc = Emc ε, - in the upper skin, a stress σs = Ems ε. The equivalent membrane modulus of the sandwich beam may be determined by the relationship m14.
Remark: In the case of a sandwich beam in which Emc ec 0
x
z R=
1 2 ∂ w 2 ∂x
Mx > 0 Mxy > 0
tg(β) =
∂w ∂x
w x, y
z
u, v
z
uo, vo
© AEROSPATIALE - 1999
MTS 006 Iss. A
w
wo x, y
Composite stress manual
MONOLITHIC PLATE - BENDING
D
Design method
3 2/4
If the displacements from a point at position Z are defined as u, v and w in the coordinate system (x, y, z), then we may write: u = uo - z
∂w o ∂x
v = vo - z
∂w o ∂y
w = wo where uo, vo et wo represent displacements from the neutral plane in the coordinate system (x, y, z). We deduce (by deriving with respect to coordinates) the corresponding non-zero strains:
d1
εx = εox - z
εy = εoy - z
∂2 w o ∂x 2
∂2 w o ∂y 2
γxy = γoxy - 2 z
∂2wo ∂x ∂y z
εx
z εox
neutral plan
o
tg(α) =
2 ∂ w 2 ∂x
x
where εox, εoy and γoxy rerepresent strains at a point located on the neutral plane and εx, εy and γxy represent strains at any point at position z.
© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
MONOLITHIC PLATE - BENDING
D
Design method
From the general expression for the bending moment: M =
h 2 h − 2
ò
3
σ z dz , we obtain the
relationship between the bending load tensor (M) and the rotation tensor (χ): d2
(M) = (C) x (χ)
Mx
C11
C12
C13
My
= C 21
C22
C 23
M xy
C 31
C32
C 33
∂2wO ∂x 2 ∂2wO ∂y 2 ∂2 w O 2 ∂x ∂y
where
d3
Cij =
å
n k = 1
æ k zk3 − zk3 − 1 ö çç Ε ij ÷÷ 3 è ø
with d4
Ε11(θ) = c4 Εl + s4 Εt + 2 c2 s2 (νtl Εl + 2 Glt) Ε22(θ) = s4 Εl + c4 Εt + 2 c2 s2 (νtl Εl + 2 Glt) Ε33(θ) = c2 s2 (Εl + Εt - 2 νtl Εl) + (c2 - s2)2 Glt Ε12(θ) = Ε21(θ) = c2 s2 (Εl + Εt - 4 Glt) + (c4 + s4) νtl Εl Ε13(θ) = Ε31(θ) = c s {c2 Εl - s2 Εt - (c2 - s2) (νtl Εl + 2 Glt)} Ε23(θ) = Ε32(θ) = c s {s2 Εl - c2 Εt + (c2 - s2) (νtl Εl + 2 Glt)}
c ≡ cos(θ) where θ is the ply direction in the reference coordinate system (o, x, y) s ≡ sin(θ) where θ is the ply direction in the reference coordinate system (o, x, y)
© AEROSPATIALE - 1999
MTS 006 Iss. A
3/4
Composite stress manual
MONOLITHIC PLATE - BENDING
D
Design method
3 4/4
with d5
Εl =
El 1 − ν tl νlt
Εt =
Et 1 − ν tl νlt
If the tensor of angles formed by the strain diagram in each direction is defined by (α): (αx, αy, αxy) we may write in a simplified form the relationship: d6
(χ) = tg (α) By convention, we shall assume that (α) is negative when the upper fibre is in tension. We have:
d7
(ε)z = - (χ) x z z
z
ply No. k
α
zk zk - 1 h
neutral plan
σ
ε
ply No. 1
This relationship makes it possible to determine each ply strain and, therefore, to find (using chapter C) stresses applied to it.
Remark: The terms Cij must be determined with relation to the laminate neutral line (Kirchoff’s assumption). In this case, the neutral plane shall also be used as a reference for the overall load pattern.
© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
MONOLITHIC PLATE - BENDING
D
Equivalent mechanical properties
4
4 . DEFORMATIONS AND EQUIVALENT MECHANICAL PROPERTIES Monolithic plates are microscopically heterogeneous. It is sometimes necessary to find their equivalent bending stiffness properties in order to determine the passing loads and resulting deformations. Equivalent bending elasticity moduli are directly derived from the laminate stiffness matrix (C): 1 E xx bending equi.
d8
(C)-1 =
12 e3
x 1
x
E yy bending equi.
x
x
x x 1 Gxy bending equi.
If reference axes (o, x, y) are coincident with the axes of orthotropy of the laminate, we obtain:
Exxbending equi. = 12
Eyybending equi. = 12
Gxybending equi. = 12
© AEROSPATIALE - 1999
C11 C 22 − (C12 )2 e3 C 22 C 11 C 22 − (C 12 ) 2 e 3 C 11 C 66 e3
MTS 006 Iss. A
Composite stress manual
MONOLITHIC PLATE - BENDING
D
Example
5 1/7
5 . EXAMPLE Let a T300/BSL914 laminate (new) be laid up as follows: 0°: 2 plies 45°: 2 plies 135°: 2 plies 90°: 2 plies Stacking from the external surface being as follows: 0°/45°/135°/90°/90°/135°/ 45°/0°.
k = 8 (0°) k = 7 (45°) k = 6 (135°) k = 5 (90°) k = 4 (90°) k = 3 (135°) k = 2 (45°)
z8 = 0.52 z7 = 0.39 z6 = 0.26 z5 = 0.13 z4 = 0
k = 1 (0°)
Mechanical properties of the unidirectional ply are the following: El = 13000 hb Et = 465 hb νlt = 0.35 νtl = 0.0125 Glt = 465 hb ep = 0.13 mm The purpose of this example is to search for elongations at the laminate external surface, knowing that the laminate is globally subject to the three following moment fluxes in the reference coordinate system (x, y):
© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
MONOLITHIC PLATE - BENDING Exemple
D
5 2/7
Mx = 10 daN My = 0 daN/mm Mxy = - 5 daN/mm z
y
x
Mx = 10 daN
Mxy = - 5 daN
1st step: calculation of stiffness coefficients for the unidirectional ply: {d5} Εl =
13000 = 13057 daN/mm2 1 − 0.35 0.0125
Εt =
465 = 467 daN/mm2 1 − 0.35 0.0125
2nd step: For each ply, stiffness coefficients Εij expressed in daN/mm2 are calculated. {d4} ply at 0° Ε11(0°) = 13057 Ε22(0°) = 467 Ε33(0°) = 465 Ε12(0°) = Ε21(0°) = 0.0125 x 13000 = 163 Ε13(0°) = Ε31(0°) = 0 Ε23(0°) = Ε32(0°) = 0
© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
MONOLITHIC PLATE - BENDING Example
D
5 3/7
ply at 45° Ε11(45°) = 0.7074 13057 + 0.7074 467 + 2 x 0.7072 0.7072 (0.0125 x 13057 + 2 x 465) = 3925 Ε22(45°) = 0.7074 13057 + 0.7074 467 + 2 x 0.7072 0.7072 (0.0125 x 13057 + 2 x 465) = 3925 Ε33(45°) = 0.7072 0.7072 (13057 + 467 - 2 x 0.0125 x 13057) = 3297 Ε12(45°) = Ε21(45°) = 0.7072 0.7072 (13057 + 467 - 4 x 465) + (0.7074 + 0.7074) x 0.0125 x 13057 = 2995 Ε13(45°) = Ε31(45°) = 0.707 x 0.707 {0.7072 13057 - 0.7072 467} = 3146 Ε23(45°) = Ε32(45°) = 0.707 x 0.707 {0.7072 13057 - 0.7072 467} = 3146 ply at 135° Ε11(135°) = 3925 Ε22(135°) = 3925 Ε33(135°) = 3297 Ε12(135°) = Ε21(135°) = 2995 Ε13(135°) = Ε31(135°) = - 3146 Ε23(135°) = Ε32(135°) = - 3146 ply at 90° Ε11(90°) = 467 Ε22(90°) = 13057 Ε33(90°) = 465 Ε12(90°) = Ε21(90°) = 163 Ε13(90°) = Ε31(90°) = 0 Ε23(90°) = Ε32(90°) = 0
© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
MONOLITHIC PLATE - BENDING
D
Example
5 4/7
3rd step: Calculation of laminate inertia matrix (C) coefficients Cij expressed in daN mm. The laminate being provided with the mirror symmetry property, coefficients Cij shall be calculated for the laminate upper half, then they shall be multiplied by 2. {d3} 90° æ . − 0 013 C11 = 2 ç 467 3 è 3
135° 3
+ 3925
0.26
3
− 013 . 3
45° 3
+ 3925
0.39
3
0° − 0.26 3
3
+ 13057
0.52 3 − 0.39 3 ö 3
÷ ø
æ . 3 − 03 . 3 013 0.26 3 − 013 0.39 3 − 0.26 3 0.52 3 − 0.39 3 ö + 2995 + 2995 + 163 C12 = 2 ç163 ÷ 3 3 3 3 ø è æ 0.13 3 − 0 3 0.26 3 − 0.13 3 0.39 3 − 0.26 3 0.52 3 − 0.39 3 ö − 3146 + 3146 + 0 C13 = 2 ç 0 ÷ 3 3 3 3 ø è æ . 3 − 03 013 0.26 3 − 0.13 3 0.39 3 − 0.26 3 0.52 3 − 0.39 3 ö + 2995 + 2995 + 163 C21 = 2 ç163 ÷ 3 3 3 3 ø è æ . 3 − 03 . 3 013 0.26 3 − 013 0.39 3 − 0.26 3 0.52 3 − 0.39 3 ö + 3925 + 3925 + 467 C22 = 2 ç13057 ÷ 3 3 3 3 ø è æ 0.13 3 − 0 3 0.26 3 − 0.13 3 0.39 3 − 0.26 3 0.52 3 − 0.39 3 ö − 3146 + 3146 + 0 C23 = 2 ç 0 ÷ 3 3 3 3 ø è æ 0.13 3 − 0 3 0.26 3 − 0.13 3 0.39 3 − 0.26 3 0.52 3 − 0.39 3 ö − 3146 + 3146 + 0 C31 = 2 ç 0 ÷ 3 3 3 3 ø è æ 0.13 3 − 0 3 0.26 3 − 0.13 3 0.39 3 − 0.26 3 0.52 3 − 0.39 3 ö − 3146 + 3146 + 0 C32 = 2 ç 0 ÷ 3 3 3 3 ø è æ . 3 − 03 . 3 013 0.26 3 − 013 0.39 3 − 0.26 3 0.52 3 − 0.39 3 ö + 3297 + 3297 + 465 C33 = 2 ç 465 ÷ 3 3 3 3 ø è
C11 = 858 C12 = 123 C13 = 55 C21 = 123 C22 = 194 C23 = 55 C31 = 55 C32 = 55 C33 = 151
© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
MONOLITHIC PLATE - BENDING Example
D
Thus, the following matrix is obtained: 858
123
55
(C) = 123
194
55
55
55
151
4th step: Search for the rotation tensor {d2}
Mx
858
123
My
= 123
194
Mxy
55
55
∂ 2 wo ∂x 2 ∂ 2 wo 55 = ∂y 2 ∂2 wo 2 151 ∂x ∂y 55
hence
∂ 2 wo ∂x 2 ∂ 2 wo ∂y 2 ∂2 wo 2 ∂x ∂y
© AEROSPATIALE - 1999
=
1287 . E−3
− 7.617 E − 4
− 1913 . E−4
− 7.617 E − 4
6.199 E − 3
− 198 . E−3
− 1913 . E−4
− 198 . E−3
7.414 E − 3
MTS 006 Iss. A
Mx
=
My Mxy
5 5/7
Composite stress manual
MONOLITHIC PLATE - BENDING Example
∂2wo ∂x 2 ∂2wo
=
∂y 2 ∂2 w o 2 ∂x ∂y
1287 . E−3
− 7.617 E − 4
− 1913 . E−4
− 7.617 E − 4
6.199 E − 3
− 198 . E−3
− 1913 . E−4
− 198 . E−3
7.414 E − 3
D
5 6/7
10
=
0
−5
Thus, we find: ∂2wo
13.82 E − 3
∂x 2 ∂2wo
=
∂y 2 ∂2 w o 2 ∂x ∂y
2.283 E − 3
− 38.98 E − 3
which is the rotation tensor (χ).
5th step: We now propose to calculate strains ε (0°) for the ply at 0° (at the external line of the layer). {d7} εx(0°) = -
∂2 w o h x 2 2 ∂x
εy(0°) = -
∂2 w o h x 2 2 ∂y
γxy(0°) = - 2
© AEROSPATIALE - 1999
∂2w o h x 2 ∂x ∂y
MTS 006 Iss. A
Composite stress manual
MONOLITHIC PLATE - BENDING Example
D
5 7/7
hence: εx(0°) = - 1 x 13.82 E-3 x 0.52 = - 7186 µd εy(0°) = - 1 x 2.283 E-3 x 0.52 = - 1187 µd γxy(0°) = - 1 x - 38.98 E-3 x 0.52 = 20270 µd Stresses in the layer may be determined afterwards. To do this, refer to chapter C.
© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
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© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
MONOLITHIC PLATE - BENDING References
D
BARRAU - LAROZE, Design of composite material structures, 1987 GAY, Composite materials, 1991 VALLAT, Strength of materials M. THOMAS, Analysis of a laminate plate subject to membrane and bending loads, 440.227/79 J.C. SOURISSEAU, 40430.030 J. CHAIX, 436.127/91
© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
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© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
E MONOLITHIC PLATE - MEMBRANE + BENDING ANALYSIS
© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
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© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
MONOLITHIC PLATE - MEMBRANE + BENDING ANALYSIS Notations
1 . NOTATIONS (o, x, y): reference coordinate system (o, l, t): coordinate system specific to the unidirectional fibre εx, εy, γxy: material strains at any point wo: displacement from plate neutral plane (N): normal flux tensor (M): bending moment tensor (ε): membrane type strain tensor (χ): curvature tensor (A): laminate stiffness matrix (membrane) (B): laminate stiffness matrix (membrane/bending coupling) (C): laminate stiffness matrix (bending) θ: fibre orientation k: fibre coordinate system El: longitudinal elasticity modulus of unidirectional fibre Et: transversal elasticity modulus of unidirectional fibre νlt: longitudinal/transversal poisson coefficient νtl: transversal/longitudinal poisson coefficient Glt: shear modulus of unidirectional fibre ep: ply thickness
© AEROSPATIALE - 1999
MTS 006 Iss. A
E
1
Composite stress manual
MONOLITHIC PLATE - MEMBRANE + BENDING ANALYSIS Introduction
E
2
2 . INTRODUCTION We have seen in chapter C that there is a relationship which binds membrane strains and loading of the same type. This relationship may be formulated as follows: (N) = (A) x (ε).
We also saw in chapter D that there is a relationship which binds the curvature tensor and the moment tensor. This relationship may be formulated as follows: (M) = (C) x (χ).
If lay-up has the mirror symmetry property, then both phenomena are dissociated and independent. In other words, the overall relationship which binds the set of strains and the set of loadings may be formulated as follows: Nx
A11
A12
A13
0
0
0
εx
Ny
A 21
A 22
A 23
0
0
0
εy
Nxy
A 31
A 32
A 33
0
0
0
γ xy
= Mx
0
0
0
C11
C12
C13
My
0
0
0
C21
C 22
C23
Mxy
0
0
0
C31
C 32
C33
∂2wo ∂x 2 ∂2wo ∂y 2 ∂2 wo 2 ∂x ∂y
where coefficients Aij and Cij are defined in chapters C and D.
© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
MONOLITHIC PLATE - MEMBRANE + BENDING ANALYSIS Analysis method
E
3 1/2
3 . ANALYSIS METHOD If lay-up is non-symmetrical, then all zero terms of the previous matrix become non-zero and there is a membrane/bending coupling. Both phenomena become dependent. The relationship between loadings and strains is thus: Nx
A11
A12
A13
B11
B12
B13
εx
Ny
A 21
A 22
A 23
B21
B22
B23
εy
Nxy
A 31
A 32
A 33
B31
B32
B33
γ xy
e1
= Mx
B11
B12
B13
C11
C12
C13
My
B21
B22
B23
C21
C 22
C23
Mxy
B31
B32
B33
C31
C 32
C33
where
e2
Bij = -
å
n k = 1
æ k zk2 − zk2 − 1 ö çç Eij ÷÷ 2 è ø
ply No. k zk zk - 1 neutral plane
ply No. 1
© AEROSPATIALE - 1999
MTS 006 Iss. A
∂2wo ∂x 2 ∂2wo ∂y 2 ∂2 wo 2 ∂x ∂y
Composite stress manual
MONOLITHIC PLATE - MEMBRANE + BENDING ANALYSIS Analysis method
E
3 2/2
with e3
Ε11(θ) = c4 Εl + s4 Εt + 2 c2 s2 (νtl Εl + 2 Glt) Ε22(θ) = s4 Εl + c4 Εt + 2 c2 s2 (νtl Εl + 2 Glt) Ε33(θ) = c2 s2 (Εl + Εt - 2 νtl Εl) + (c2 - s2)2 Glt Ε12(θ) = Ε21(θ) = c2 s2 (Εl + Εt - 4 Glt) + (c4 + s4) νtl Εl Ε13(θ) = Ε31(θ) = c s {c2 Εl - s2 Εt - (c2 - s2) (νtl Εl + 2 Glt)} Ε23(θ) = Ε32(θ) = c s {s2 Εl - c2 Εt + (c2 - s2) (νtl Εl + 2 Glt)} where c ≡ cos(θ) where θ is the fibre direction in the reference coordinate system (o, x, y). s ≡ sin(θ) where θ is the fibre direction in the reference coordinate system (o, x, y). with
e4
Εl =
El 1 − ν tl νlt
Εt =
Et 1 − ν tl νlt
Remark: The terms Bij and Cij must be determined with relation to the laminate neutral line (Kirchoff’s assumption). In this case, the neutral plane shall also be used as a reference for the overall load pattern.
© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
MONOLITHIC PLATE - MEMBRANE + BENDING ANALYSIS Example
E
4
4 . EXAMPLE Let a T300/BSL914 laminate (new) be laid up as follows: 0°: 1 ply 45°: 1 ply 135°: 1 ply 90°: 1 ply Stacking from the external surface being as follows: 0°/45°/135°/90°.
k = 4 (0°)
z4 = 0.26 z3 = 0.13
k = 3 (45°) k = 2 (135°)
z2 = 0 z1 = - 0.13
k = 1 (90°) z0 = - 0.26
Mechanical properties of the unidirectional fibre are the following: El = 13000 hb Et = 465 hb νlt = 0.35 νtl = 0.0125 Glt = 465 hb ep = 0.13 mm
© AEROSPATIALE - 1999
MTS 006 Iss. A
neutral plane
1/9
Composite stress manual
MONOLITHIC PLATE - MEMBRANE + BENDING ANALYSIS Example
E
4 2/9
The purpose of this example is to search for strains at the laminate internal and external surfaces, knowing that the laminate is globally subject to the following fluxes in the reference coordinate system (x, y): Nx = 5 daN/mm Ny = 0 daN/mm Nxy = 0 daN/mm Mx = 0 daN æ mm daN ö My = - 0.15 daN ç ÷ è mm ø Mxy = 0 daN
z
y
My = - 0.15 daN
x
Nx = 5 daN/mm
1st step: calculation of stiffness coefficients for the unidirectional fibre: {e4} Εl =
13000 = 13057 daN/mm2 1 − 0.35 0.0125
Εt =
465 = 467 daN/mm2 1 − 0.35 0.0125
© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
MONOLITHIC PLATE - MEMBRANE + BENDING ANALYSIS Example
E
4 3/9
2nd step: For each fibre direction, stiffness coefficients Εij expressed in daN/mm2, are calculated. {e3} fibre at 0° Ε11(0°) = 13057 Ε22(0°) = 467 Ε33(0°) = 465 Ε12(0°) = Ε21(0°) = 0.0125 x 13000 = 163 Ε13(0°) = Ε31(0°) = 0 Ε23(0°) = Ε32(0°) = 0 fibre at 45° Ε11(45°) = 0.7074 13057 + 0.7074 467 + 2 x 0.7072 0.7072 (0.0125 x 13057 + 2 x 465) = 3925 Ε22(45°) = 0.7074 13057 + 0.7074 467 + 2 x 0.7072 0.7072 (0.0125 x 13057 + 2 x 465) = 3925 Ε33(45°) = 0.7072 0.7072 (13057 + 467 - 2 x 0.0125 x 13057) = 3297 Ε12(45°) = Ε21(45°) = 0.7072 0.7072 (13057 + 467 - 4 x 465) (0.7074 + 0.7074) x 0.0125 x 13057 = 2995 Ε13(45°) = Ε31(45°) = 0.707 x 0.707 {0.7072 13057 - 0.7072 467} = 3146 Ε23(45°) = Ε32(45°) = 0.707 x 0.707 {0.7072 13057 - 0.7072 467} = 3146 fibre at 135° Ε11(135°) = 3925 Ε22(135°) = 3925 Ε33(135°) = 3297 Ε12(135°) = Ε21(135°) = 2995 Ε13(135°) = Ε31(135°) = - 3146 Ε23(135°) = Ε32(135°) = - 3146
© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
MONOLITHIC PLATE - MEMBRANE + BENDING ANALYSIS Example
E
4 4/9
fibre at 90° Ε11(90°) = 467 Ε22(90°) = 13057 Ε33(90°) = 465 Ε12(90°) = Ε21(90°) = 163 Ε13(90°) = Ε31(90°) = 0 Ε23(90°) = Ε32(90°) = 0
3rd step: Calculation of laminate (membrane) stiffness matrix (A) coefficients Aij expressed in daN/mm. {c6} 90°
135°
45°
0°
A11 = (467 x 0.13 + 3925 x 0.13 + 3925 x 0.13 + 13057 x 0.13) A12 = (163 x 0.13 + 2995 x 0.13 + 2995 x 0.13 + 163 x 0.13) A13 = (0 x 0.13 - 3146 x 0.13 + 3146 x 0.13 + 0 x 0.13) A21 = (163 x 0.13 + 2995 x 0.13 + 2995 x 0.13 + 163 x 0.13) A22 = (13057 x 0.13 + 3925 x 0.13 + 3925 x 0.13 + 467 x 0.13) A23 = (0 x 0.13 - 3146 x 0.13 + 3146 x 0.13 + 0 x 0.13) A31 = (0 x 0.13 - 3146 x 0.13 + 3146 x 0.13 + 0 x 0.13) A32 = (0 x 0.13 - 3146 x 0.13 + 3146 x 0.13 + 0 x 0.13) A33 = (465 x 0.13 + 3297 x 0.13 + 3297 x 0.13 + 465 x 0.13) hence A11 = 2779 A12 = 821 A13 = 0 A21 = 821 A22 = 2779 A23 = 0 A31 = 0 A32 = 0 A33 = 978
© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
MONOLITHIC PLATE - MEMBRANE + BENDING ANALYSIS Example
E
4 5/9
4th step: Calculation of laminate (bending) inertia matrix (C) coefficients Cij expressed in daN mm. {d3} 90° 135° 45° 0° 3 3 3 3 3 3 3 3 æ ö 0 − (− 013 013 0.26 − 013 (− 013 . ) − (− 0.26) . ) . − 0 . + 3925 + 3925 + 13057 C11 = ç 467 ÷ 3 3 3 3 è ø 3 3 3 3 3 3 3 3 æ 0 − (− 0.13) 013 0.26 − 0.13 ö (− 013 . ) − (− 0.26) . − 0 + 2995 + 2995 + 163 C12 = ç163 ÷ 3 3 3 3 è ø 3 3 3 3 3 3 æ ( − 0.13) 3 − (− 0.26) 3 0 − (− 0.13) 0.13 − 0 0.26 − 0.13 ö − 3146 + 3146 + 0 C13 = ç 0 ÷ 3 3 3 3 è ø 3 3 3 3 3 3 3 3 æ 0 − (− 013 013 0.26 − 0.13 ö (− 013 . ) − (− 0.26) . ) . − 0 + 2995 + 2995 + 163 C21 = ç163 ÷ 3 3 3 3 è ø 3 3 3 3 3 3 3 3 æ ö 0 − (− 013 013 0.26 − 013 (− 013 . ) − (− 0.26) . ) . − 0 . + 3925 + 3925 + 467 C22 = ç13057 ÷ 3 3 3 3 è ø 3 3 3 3 3 3 æ (− 0.13) 3 − (− 0.26) 3 0 − (− 0.13) 0.13 − 0 0.26 − 0.13 ö − 3146 + 3146 + 0 C23 = ç 0 ÷ 3 3 3 3 è ø 3 3 3 3 3 3 æ ( − 0.13) 3 − (− 0.26) 3 0 − (− 0.13) 0.13 − 0 0.26 − 0.13 ö − 3146 + 3146 + 0 C31 = ç 0 ÷ 3 3 3 3 è ø 3 3 3 3 3 3 æ ( − 0.13) 3 − (− 0.26) 3 0 − ( − 0.13) 0.13 − 0 0.26 − 0.13 ö − 3146 + 3146 + 0 C32 = ç 0 ÷ 3 3 3 3 è ø 3 3 3 3 3 3 3 3 æ ö 0 − (− 013 0.13 − 0 0.26 − 013 (− 013 . ) − (− 0.26) . ) . + 3297 + 3297 + 465 C33 = ç 465 ÷ 3 3 3 3 è ø
hence C11 = 75.1 C12 = 6.06 C13 = 0 C21 = 6.06 C22 = 75.1 C23 = 0 C31 = 0 C32 = 0 C33 = 9.59
© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
E
MONOLITHIC PLATE - MEMBRANE + BENDING ANALYSIS Example
4 6/9
5th step: Calculation of membrane - bending coupling coefficients Bij expressed in daN. {e2} 90° æ B11 = - ç 467 è æ B12 = - ç 163 è æ B13 = - ç 0 è
( − 0.13 )
( − 0.13)
æ B32 = - ç 0 è
− ( − 0.26 )
+ 3925
0
2
2
2
− ( − 0.26 )
2 + 2995
0
2
− 3146
0
2
( − 0.13 )
( − 0.13 )
2
( − 0.13 )
2
− ( − 0.26 )
2
+ 2995
0
2
− ( − 0.26)
2 + 3925
0
− 3146
0
2
2 + 3146
− ( − 0.13)
2
− ( − 0.26 )
− 3146
0
2
2 + 2995
− ( − 0.26 )
− 3146
0
2
− ( − 0.13)
2 + 3925
− ( − 0.26 )
2
+ 3297
0
0.26
2
− 0.13
+ 0
0.26
2
− 0.13
+ 3146
0.13
2
− 0
0.13
+ 3146
0.13
2
2 + 3146
0.13
2
2 + 163
− 0
− 0
2
− 0
2
0.26
2
− ( − 0.13)
2
B11 = - 319 B12 = 0 B13 = - 53.2 B21 = 0 B22 = 319 B23 = - 53.2 B31 = - 53.2 B32 = - 53.2 B33 = 0
MTS 006 Iss. A
+ 3297
0.13
− 0
2
2 + 467
0,26
2
2
ö ÷ ø
− 0.13
+ 0
+ 0
0.26
− 0.13
0.26
2
− 0.13
+ 0
0.26
2
− 0.13
2 − 0
2
2 2
2
ö ÷ ø
2
ö ÷ ø
2
ö ÷ ø
2
2
ö ÷ ø
2 + 465
0.26
2
− 0.13
2
ö ÷ ø
ö ÷ ø
− 0.13
2
2
2
2
2
2
2 2
2
2
2
2
− 0.13
2 2
2
− ( − 0.13 )
2
2
2 2
2 2
+ 163
2
− ( − 0.13 )
2
0.26
2
− 0
0.13
2 2
− 0
2
+ 13057
2
hence
© AEROSPATIALE - 1999
0.13
2
2 2
− 0
2
− ( − 0.13 )
2
( − 0.13 )
0.13
2 2
2
æ B33 = - ç 465 è
+ 2995
2
2
− ( − 0.26 )
2
2
2 2
0° 2
2
− ( − 0.13 )
2 2
0.13
2 2
2 ( − 0.13)
+ 3925
2 2
− ( − 0.13)
2 ( − 0.13)
2
2
− ( − 0.26 )
2
45°
− ( − 0.13 )
2
æ B22 = - ç 13057 è
æ B31 = - ç 0 è
135° 2
2
( − 0.13 )
æ B21 = - ç 163 è
æ B23 = - ç 0 è
2
2
ö ÷ ø
ö ÷ ø
Composite stress manual
MONOLITHIC PLATE - MEMBRANE + BENDING ANALYSIS Example
E
6th step: Expression of stiffness overall matrix {e1} Nx
A 11
A 12
A 13
B 11
B 12
B 13
εx
Ny
A 21
A 22
A 23
B 21
B 22
B 23
εy
Nxy
A 31
A 32
A 33
B 31
B 32
B 33
γ xy
= Mx
B 11
B 12
B 13
C 11
C 12
C 13
My
B 21
B 22
B 23
C 21
C 22
C 23
Mxy
B 31
B 32
B 33
C 31
C 32
C 33
∂2wo ∂x 2 ∂2wo ∂y 2 ∂2 wo 2 ∂x ∂y
then Nx
2779
821
0
− 319
0
53.2
εx
Ny
821
2779
0
0
319
53.2
εy
Nxy
0
0
978
53.2
53.2
0
γ xy
= Mx
− 319
0
53.2
75.1
6.06
0
My
0
319
53.2
6.06
75.1
0
Mxy
53.2
53.2
0
0
0
9.59
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MTS 006 Iss. A
∂2wo ∂x 2 ∂2wo ∂y 2 ∂2 wo 2 ∂x ∂y
4 7/9
Composite stress manual
MONOLITHIC PLATE - MEMBRANE + BENDING ANALYSIS Example
E
4 8/9
By reversing the relationship, we find: ε ε γ
x y
xy
∂2w
o
∂x 2 2 ∂ w o
=
1.15 x E − 3
− 5.0 x E − 4
3.80 x E − 4
5.00 x E − 3
1.99 x E − 3
3.61 x E − 3
N
− 5.0 x E − 4
1.15 x E − 3
− 3.8 x E − 4
− 2.0 x E − 3
− 5.0 x E − 3
3.61 x E − 3
N
3.80 x E − 4
− 3.8 x E − 4
1.28 x E − 3
2.33 x E − 3
2.33 x E − 3
0
N
5.00 x E − 3
− 2.0 x E − 3
2.33 x E − 3
3.57 x E − 2
7.23 x E − 3
1.67 x E − 2
M
2.00 x E − 3
− 5.0 x E − 3
2.33 x E − 3
7.23 x E − 3
3.57 x E − 2
− 1.67 x E − 2
M
3.61 x E − 3
3.61 x E − 3
0
1.67 x E − 2
− 1.67 x E − 2
1.44 x E − 1
M
∂y 2 ∂2w
2
o
∂x ∂y
x y
xy
x
y
xy
7th step: Search for the strain tensor By replacing loading by values quoted at the beginning of the example in the previous relationship, we find: εx
5.44 E − 3 mm / mm
(5440 µd)
εy
− 174 . E − 3 mm / mm
(− 1740 µd)
γ xy
154 . E − 3 mm / mm
(1540 µd)
∂2wo ∂x 2 ∂2wo ∂y 2 ∂2 wo 2 ∂x ∂y
© AEROSPATIALE - 1999
= 2.38 E − 2 mm −1 4.57 E − 3 mm −1 2.05 E − 2 mm −1
MTS 006 Iss. A
Composite stress manual
MONOLITHIC PLATE - MEMBRANE + BENDING ANALYSIS Example
E
4 9/9
8th step: Search for strains in the upper fibre at ε (0°) To do this, membrane strains (εx, εy, γxy) are added to strains resulting from the bending 2 æ ∂2wo ∂2wo h ∂ wo h hö x x 2 x ÷ , , effect ç 2 2 2 ∂y 2 ∂x ∂y 2 ø è ∂x {d7} εx(0°) = εx -
∂2 w o h x 2 2 ∂x
εy(0°) = εy -
∂2 w o h x 2 2 ∂y
γxy(0°) = γxy - 2
∂2wo h x 2 ∂x ∂y
hence: εx (0°) = 5.44 E-3 + (-1) x 2.38 E-2 x 0.26 = - 748 µd εy (0°) = - 1.74 E-3 + (-1) x 4.57 E-3 x 0.26 = - 2928 µd γxy (0°) = - 1.54 E-3 + (-1) x 2.05 E-2 x 0.26 = - 3790 µd For the lower fibre, we would find: εx (90°) = 5.44 E-3 + 2.38 E-2 x 0.26 = 11628 µd εy (90°) = - 1.74 E-3 + 4.57 E-3 x 0.26 = - 552 µd γxy (90°) = + 1.54 E-3 + 2.05 E-2 x 0.26 = 6870 µd
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MTS 006 Iss. A
Composite stress manual
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© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
MONOLITHIC PLATE - MEMBRANE + BENDING ANALYSIS References
E
BARRAU - LAROZE, Design of composite material structures, 1987 GAY, Composite materials, 1991 VALLAT, Strength of materials M. THOMAS, Analysis of a laminate plate subjected to membrane and bending loads, 440.227/79 J.C. SOURISSEAU, 40430.030 J. CHAIX, 436.127/91
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MTS 006 Iss. A
Composite stress manual
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© AEROSPATIALE - 1999
MTS 006 Iss. A
Composite stress manual
F MONOLITHIC PLATE - TRANSVERSAL SHEAR ANALYSIS
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
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© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - TRANSVERSAL SHEAR Notations
F
1
1 . NOTATIONS (o, x, y): reference coordinate system (o, l, t): coordinate system specific to the unidirectional fibre El: longitudinal elasticity modulus of unidirectional fibre Et: transversal elasticity modulus of unidirectional fibre νlt: longitudinal/transversal Poisson coefficient νtl: transversal/longitudinal Poisson coefficient Glt: shear modulus of unidirectional fibre ep: ply thickness Ek: longitudinal elasticity modulus with relation to x-axis of ply No. k n: total number of laminate layers θ: fibre orientation El: laminate overall inertia with relation to the (moduli weighted) neutral axis E Wk: static moment with relation to the (moduli weighted) neutral axis of the set of plies k to n τ: shear stress Txy, Tyz, T(β): shear load flux
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - TRANSVERSAL SHEAR
F
Introduction
2
2 . INTRODUCTION The purpose of this chapter is to determine interlaminar shear stresses in a monolithic plate subject to a shear load flux. For simplification purposes, we shall assume that the laminate is made up of n identical fibres but with different directions.
z Tyz > 0 y z
Txz > 0 x
y θ k=n ep
k=1 x
Layer No. k in direction θ has the following longitudinal elasticity modulus with relation to the reference coordinate system (o, x, y): f1
Ek =
1 æ 1 ν ö c s + + c 2 s2 ç − 2 tl ÷ El Et Et ø è Glt 4
4
see chapter C3.
with c ≡ cos(θ) s ≡ sin(θ)
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - TRANSVERSAL SHEAR Introduction - Analysis method
F
2 3
1/5
We shall assume that shear load Txz (direction z load shearing a plane perpendicular to xaxis) creates stress τxz and, based on the reciprocity principle, stress τzx. Similarly, we shall assume that shear load Tyz (direction z load shearing a plane perpendicular to y-axis) creates stress τyz and, based on the reciprocity principle, stress τzy. These shear stresses are called interlaminar stresses.
z
τzy τzx
τyz
τxz
y Tyz
Txz
x
3 . ANALYSIS METHOD To calculate interlaminar stresses τxz (τzx) generated by shear load Txz (Tyz), use the following methodology. We shall only consider the case of a laminate subject to shear load Txz. The analysis principle is the same for Tyz. In this case, inertias (El) and static moments (E W k) are measured with relation to y-axis. Elasticity moduli (Ek) are measured with relation to x-axis.
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - TRANSVERSAL SHEAR Analysis method
F
3 2/5
1st step: The position of the laminate neutral axis is determined. If the laminate lower fibre is used as a reference, then the neutral axis is defined by dimension zg, so that:
å 2å n
f2
zg =
(
E k z k2 − z k2 − 1
k =1 n
(
)
Ek z k − zk − 1
k =1
) z
ply No. n ply No. k zg
zk zk - 1 z1
ply No. 1
y
z0 = 0
2nd step: The (moduli weighted) bending stiffness of laminate El is determined with relation to the lay-up neutral axis
f3
El =
å
n
Ek
k =1
(z
k
− zk − 1 12
)
3
+
å
n
(
Ek z k − z k − 1
k =1
)
æ zk + zk − 1 ö − zg ÷ ç 2 è ø
2
z
ply No. k
zk
zk - 1
zg
y
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - TRANSVERSAL SHEAR Analysis method
F
3 3/5
3rd step: Then the (elasticity moduli weighted) static moment E W k (of the material surface located above the line where interlaminar stress is to be calculated), is determined. This static moment shall be calculated with relation to the plate neutral axis. If the line is a fibre interface surface (z = zk - 1), then we have the following relationship: f4
E Wk =
ö æ zi + z i − 1 Ei z i − z i − 1 ç − zg ÷ i=k 2 ø è
å
n
(
)
z
ply No. k
zk
zk - 1
zg
y
If the line is situated at the centre of a fibre at z =
f5
E Wk =
å
n i = k
(
Ei zi − zi
æ zk + z k Ek ç 2 è
− 1
− 1
) æçè
− zk
z i + zi 2 ö ø
− 1÷
− 1
zk + zk
− 1
2
, the relationship becomes:
ö − zg ÷ − ø
æ zk + z k ç 4 è
− 1
+
zk
− 1
2
ö − zg ÷ ø z
z +z
ply No. k
k
k − 1
2
zk
zg
zk - 1
y
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - TRANSVERSAL SHEAR Analysis method
F
3 4/5
4th step: Shear stress τxzk is determined, so that: f6
τxzk =
Txz . E Wk El
where Txz is the shear load applied to the laminate. By using this analysis method for each ply interface (or at the center of each ply for greater accuracy), it is possible to plot the interlaminar shear stress diagram over the entire plate width. The previous relationship shows that the shear stress is maximum when the static moment is maximum as well, i.e. at the neutral axis (z = zg). z
z
τxzk ply No. k τzxk zg
τxz
y
Remark: The previous analysis is based on a shear load flux Txz applied to a section perpendicular to x-axis. In the case of any section forming an angle β in the coordinate system (o, x, y), the shear load flux in this new section may be expressed as a function of Txz and Tyz.
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - TRANSVERSAL SHEAR Analysis method
F
3 5/5
y
T(β + π/2)
T(β)
ds
-Txz β x -Tyz
As shown in the drawing above, the z equilibrium of the hatched material element leads to the following relationship: T(β) ds - Txz ds cos(β) - Tyz ds sin(β) = 0 hence: T(β) = cos(β) Txz + sin(β) Tyz æ Tyz ö It is easy to show that for β = Arctg ç ÷ , a modulus extremum T(β) (called main shear è Txz ø load flux) is reached that is equal to:
f7
l T(β) l =
Txz 2 + Tyz 2
Example: if shear load fluxes Txz and Tyz are equal, then the maximum shear load flux is located in the plane with a direction β = 45°. Its modulus equals
© AEROSPATIALE - 1999
MTS 006 Iss. B
2 Txy (or
2 Tyz).
Composite stress manual
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© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
B
MONOLITHIC PLATE - TRANSVERSAL SHEAR Example
F
4 1/9
4 . EXAMPLE Let a T300/BSL914 laminate (new) be laid up as follows: 0°: 1 ply 45°: 1 ply 135°: 1 ply 90°: 1 ply Stacking from the external surface being as follows: 0°/45°/135°/90°. 0° 45° 135° 90°
Mechanical properties of the unidirectional fibre are the following: El = 13000 hb (130000 MPa) Et = 465 hb (4650 MPa) νlt = 0.35 νtl = 0.0125 Glt = 465 hb (4650 MPa) ep = 0.13 mm e = 0.52 mm The purpose of this example is to search for interlaminar shear stresses in the laminate, knowing that it is subject to the following shear load flux: Txz = 0.7 daN/mm
z
y
Txz = 0.7 daN/mm x
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
B
MONOLITHIC PLATE - TRANSVERSAL SHEAR Example
F
4 2/9
Knowing the mechanical properties of the unidirectional fibre, elasticity moduli of each fibre should be calculated in direction x. {f1} For the fibre at 90°: k = 1. E1 = Et = 465 hb (4650 MPa) For the fibre at 135: k = 2 E2 =
1 4
0.707 0.707 + 13000 465
4
0.0125 ö æ 1 + 0.707 2 0.707 2 ç − 2 ÷ è 465 13000 ø
E2 = 925 hb (9250 MPa) For the fibre at 45°: k = 3 E3 = 925 hb (9250 MPa) For the fibre at 0°: k = 4 E4 = El = 13000 hb (130000 MPa)
1st step: Analysis of the position of neutral axis zg {f2}
zg =
465 (0.13 2 − 0 2 ) + 925 (0.26 2 − 0.13 2 ) + 925 (0.39 2 − 0.26 2 ) + 13000 (0.522 − 0.39 2 ) 2 ( 465 (0.13 − 0) + 925 (0.26 − 0.13) + 925 (0.39 − 0.26 ) + 13000 (0.52 − 0.39 ))
zg = 0.42 mm
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
B
MONOLITHIC PLATE - TRANSVERSAL SHEAR Example
F
4 3/9
z
z4 = 0.52 z3 = 0.39 z2 = 0.26
zg = 0.42
z1 = 0.13 z0 = 0
2nd step: Analysis of the laminate bending stiffness El with relation to the neutral axis {f3}
El = 465
925
(0.26 − 0.13 ) 3 (0.13 − 0) 3 + 925 + 12 12 (0.52 − 0.39 ) 3 (0.39 − 0.26 ) 3 + 13000 + 12 12 2
ö æ 0.13 + 0 − 0.42 ÷ + 465 (0.13 − 0) ç 2 ø è 2
ö æ 0.26 + 0.13 − 0.42 ÷ + 925 (0.26 − 0.13) ç 2 ø è 2
ö æ 0.39 + 0.26 − 0.42 ÷ + 925 (0.39 − 0.26) ç 2 ø è ö æ 0.52 + 0.39 − 0.42 ÷ 13000 (0.52 − 0.39) ç 2 ø è
2
El = 0.085134 + 0.169352 + 0.169352 + 2.380083 + 7.618211 + 6.087656 + 1.085256 + 2.07025 El = 19.67 daN.mm
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
B
MONOLITHIC PLATE - TRANSVERSAL SHEAR Example
F
4 4/9
3rd step: Analysis of static moments E W k (with relation to the neutral line) at the base and center of each ply. At the top of ply at 0° {f4} E W 4 = 0 daN z
0° 45° 135° 90°
y
At the center of ply at 0° {f5} ö æ 0.52 + 0.39 − 0.42 ÷ − E W 4 = 13000 (0.52 − 0.39 ) ç 2 ø è ö æ 0.52 + 0.39 0.39 ö æ 0.52 + 0.39 − 0.39 ÷ ç + − 0.42 ÷ 13000 ç 4 2 2 øè ø è
E W 4 = 59.15 - 2.11 E W 4 = 57.04 daN z
0°
y
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
B
MONOLITHIC PLATE - TRANSVERSAL SHEAR Example
F
4 5/9
At the base of ply at 0° {f4} ö æ 0.52 + 0.39 − 0.42 ÷ E W 4 = 13000 (0.52 − 0.39 ) ç 2 ø è
E W 4 = 59.15 daN z
0°
y
At the center of ply at 45° {f5} ö ö æ 0.52 + 0.39 æ 0.39 + 0.26 − 0.42 ÷ + 925 (0.39 − 0.26 ) ç − 0.42 ÷ E W 3 = 13000 (0.52 − 0.39) ç 2 2 ø ø è è ö æ 0.39 + 0.26 0.26 ö æ 0.39 + 0.26 − 925 ç − 0.26 ÷ ç + − 0.42 ÷ 2 2 2 øè ø è
E W 3 = 59.15 - 11.42 + 7.67 E W 3 = 55.4 daNp z
45° y
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Composite stress manual
B
MONOLITHIC PLATE - TRANSVERSAL SHEAR Example
F
4 6/9
At the base of ply at 45° {f4} æ 0.52 + 0.39 ö æ 0.39 + 0.26 ö − 0.42 ÷ + 925 (0.39 − 0.26 ) ç − 0.42 ÷ E W 3 = 13000 (0.52 − 0.39) ç 2 2 è ø è ø
E W 3 = 59.15 - 11.42 E W 3 = 47.73 daN z
45° y
At the center of ply at 135° {f5} æ 0.52 + 0.39 ö æ 0.39 + 0.26 ö − 0.42 ÷ + 925 (0.39 − 0.26 ) ç − 0.42 ÷ + E W 2 = 13000 (0.52 − 0.39) ç 2 2 è ø è ø ö æ 0.26 + 0.13 − 0.42 ÷ − 925 (0.26 − 0.13) ç 2 ø è ö æ 0.26 + 0.13 0.13 ö æ 0.26 + 0.13 − 0.13 ÷ ç + − 0.42 ÷ 925 ç 4 2 2 øè ø è
E W 2 = 59.15 - 11.42 - 27.06 + 15.48 E W 2 = 35.35 daN z
135° y
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Composite stress manual
B
MONOLITHIC PLATE - TRANSVERSAL SHEAR Example
F
4 7/9
At the base of ply at 135° {f4} æ 0.52 + 0.39 ö æ 0.39 + 0.26 ö − 0.42 ÷ + 925 (0.39 − 0.26 ) ç − 0.42 ÷ + E W 2 = 13000 (0.52 − 0.39) ç 2 2 è ø è ø æ 0.26 + 0.13 ö − 0.42 ÷ 925 (0.26 − 0.13 ) ç 2 è ø
E W 2 = 59.15 - 11.42 - 27.06 E W 2 = 19.87 daN z
135° y
At the center of ply at 90° {f5} æ 0.52 + 0.39 ö æ 0.39 + 0.26 ö − 0.42 ÷ + 925 (0.39 − 0.26 ) ç − 0.42 ÷ + E W 1 = 13000 (0.52 − 0.39 ) ç 2 2 è ø è ø ö ö æ 0.26 + 0.13 æ 0.13 + 0 − 0.42 ÷ + 465 (0.13 − 0) ç − 0.42 ÷ − 925 (0.26 − 0.13) ç 2 2 ø ø è è ö æ 0.13 + 0 0 ö æ 0.13 + 0 − 0÷ ç + − 0.42 ÷ 465 ç 4 2 2 øè ø è
E W 1 = 59.15 - 11.42 - 27.86 - 21.46 + 11.71 z
E W 1 = 10.12 daN
90°
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Composite stress manual
B
MONOLITHIC PLATE - TRANSVERSAL SHEAR Example
At the base of ply at 90° {f4} E W 1 = 0 daN z
90°
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Composite stress manual
B
MONOLITHIC PLATE - TRANSVERSAL SHEAR Example
F
4 9/9
4th step: calculation of maximum interlaminar shear stress In the example given, it is located at the point where the static moment is maximum, i.e. at the base of the ply at 0°. Its value equals at E W 0 = 59.15 daN, which gives stress τxz0: {f6} τxz0 =
0.7 x 59.15 = 2.1 hb (21 MPa) 19.67
If these interlaminar shear stresses are analysed for each fibre, stresses are distributed along the laminate thickness as follows: τxzk =
0.7 E Wk 19.67 0.52
0.455
0.39
0.325
z (mm)
0.26
0.195
0.13
0.065
0 0
0.5
1
1.5
τ (hb)
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MONOLITHIC PLATE - TRANSVERSAL SHEAR References
F
BARRAU - LAROZE, Design of composite material structures, 1987 GAY, Composite materials, 1991 VALLAT, Strength of materials
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G MONOLITHIC PLATE - FAILURE CRITERIA
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FAILURE CRITERIA Notations
1 . NOTATIONS σl (σl): longitudinal stress in unidirectional fibre σt (σ2): transversal stress in unidirectional fibre τlt (σ6): shear stress in unidirectional fibre εl (εl): longitudinal strain in unidirectional fibre εt (ε2): transversal strain in unidirectional fibre γlt (ε6): shear strain in unidirectional fibre Rl: allowable longitudinal stress Rlt: allowable longitudinal tension stress Rlc: allowable longitudinal compression stress Rt: allowable transversal stress Rtt: allowable transversal tension stress Rtc: allowable transversal compression stress S: allowable shear stress
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1
Composite stress manual
FAILURE CRITERIA Inventory
G
2
2 . INVENTORY OF STATIC FAILURE CRITERIA The purpose of this chapter is to describe various failure criteria of the unidirectional fibre within a laminate. The following criteria shall be presented in chronological order (this is not an exhaustive list): - maximum stress criterion - maximum strain criterion - Norris and Mac Kinnon's criterion - Puck's criterion - Hill's criterion - Norris's criterion - Fischer's criterion - Hoffman's criterion - Tsaï - Wu's criterion For three-dimensional criteria, we shall assume that the composite material is subjected to the following stress tensor and strain tensor: (σ) = (σ1, σ2, σ3, σ4, σ5, σ6) (ε) = (ε1, ε2, ε3, ε4, ε5, ε6)
For two-dimensional criteria, we shall assume that the unidirectional fibre is subjected to the following stress tensor and strain tensor: (σlt) = (σl, σt, τlt) (εlt) = (εl, εt, γlt)
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FAILURE CRITERIA Maximum stress criterion
G
2.1
2.1 . Maximum stress criterion This criterion is applicable for orthotropic materials only. The criterion anticipates failure of the material if: for 1 ≤ i ≤ 6 g1
σi = Xi
for tension stresses
or σi = - X'i
for compression stresses
For the two-dimensional case, the failure envelope may be represented as follows:
σt
σl
τlt
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FAILURE CRITERIA Maximum strain criterion
G
2.2
2.2 . Maximum strain criterion This criterion is applicable for orthotropic materials only. The criterion anticipates failure of the material if: for 1 ≤ i ≤ 6 g2
εi = Yi
for tension strains
or εi = - Y'i
for compression strains
For the two-dimensional case, the failure envelope may be represented as follows:
εt
εl
γlt
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FAILURE CRITERIA Norris and Mac Kinnon's criterion
G
2.3
2.3 . Norris and Mac Kinnon's criterion This criterion is valid for any material. The criterion anticipates failure of the material if:
å
6
1
C i (σ i ) 2 = 1
Coefficients Ci depend on the material used. For the two-dimensional case, the expression becomes: g3
C1 (σl)2 + C2 (σt)2 + C6 (τlt)2 = 1 The failure envelope may be represented as follows: σt
σl
τlt
This is the first criterion which calls for stress dependency.
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Composite stress manual
FAILURE CRITERIA Puck's criterion
G
2.4
2.4 . Puck's criterion This two-dimensional criterion is valid for orthotropic materials only. The criterion anticipates failure of the material if: σ1 = X1
for tension stresses
or σ1 = - X'1
for compression stresses
and 2
g4
2
2
æτ ö æ σ1 ö æσ ö ç ÷ + ç 2 ÷ + ç 12 ÷ = 1 è X1 ø è X2 ø è X6 ø
Coefficients X1, X2 and X6 depend on the material used. σt
σl
τlt
Accuracy close to that of Norris and Mac Kinnon's criterion.
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Composite stress manual
FAILURE CRITERIA Hill's criterion
G
2.5
2.5 . Hill's criterion This criterion is valid for orthotropic materials or for slightly anisotropic materials only. The criterion anticipates failure of the material if: F (σ2 - σ3)2 + G (σ3 - σ1)2 + H (σ1 - σ2)2 + L (σ4)2 + M (σ5)2 + N (σ6)2 = 1 Coefficients F, G, H, L, M and N depend on the material used. For a two-dimensional analysis, the expression becomes: g5
F (σt)2 + G (σl)2 + H (σl - σt)2 + N (τlt)2 = 1
σt
σl
τlt
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FAILURE CRITERIA Norris's criterion
G
2.6
2.6 . Norris's criterion This two-dimensional criterion is valid for orthotropic materials only. The criterion anticipates failure of the material if: F (σ2 - σ3)2 + G (σ3 - σ1)2 + H (σ1 - σ2)2 + L (σ4)2 + M (σ5)2 + N (σ6)2 = 1 and for 1 ≤ i ≤ 6 σi = Xi
for tension stresses
or σi = - X'i
for compression stresses
Coefficients F, G, H, L, M and N depend on the material used. For a two-dimensional analysis, the expression becomes: g6
F (σt)2 + G (σl)2 + H (σl - σt)2 + N (τlt)2 = 1 - X'1 ≤ σl ≤ X1 and - X'2 ≤ σt ≤ X2 and - X'6 ≤ τlt ≤ X6
σt
σl
τlt
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FAILURE CRITERIA Fischer's criterion
G
2.7
2.7 . Fischer's criterion This two-dimensional criterion is valid for orthotropic materials only. The criterion anticipates failure of the material if: 2
g7
2
2
æτ ö æ σl ö æσ ö σ σ ç ÷ + ç t ÷ − K l t + ç lt ÷ = 1 X1 X 2 è X 6 ø è X1 ø è X2 ø
Coefficients X1, X2 and X6 depend on the material used. σt
σl
τlt
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Composite stress manual
FAILURE CRITERIA Hoffman's criterion FAILURE CRITERIA
G
2.8
2.8 . Hoffman's criterion This criterion is valid for orthotropic materials only. The criterion anticipates failure of the material if: C1 (σ2 - σ3)2 + C2 (σ3 - σ1)2 + C3 (σ1 - σ2)2 + C4 (σ4)2 + C6 (σ6)2 + C5 (σ5)2 + C'1 σ1 + C'2 σ2 + C'3 σ3 = 1 Coefficients C1, C2, C3, C4, C5, C6, C'1, C'2 and C'3 depend on the material used. For a two-dimensional analysis, the expression becomes: g8
C1 (σt)2 + C2 (σl)2 + C3 (σl - σt)2 + C6 (τlt)2 + C'1 σl + C'2 σt = 1 σt
σl
τlt
Very good tension accuracy, but very bad compression results.
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FAILURE CRITERIA Tsaï - Wu's criterion
G
2.9
2.9 . Tsaï - Wu's criterion This criterion intends to be as general as possible and then, there is, a priori, no particular hypothesis. This criterion anticipates failure of the material if: For 1 ≤ i ≤ 6 Σ Fi σi + Σ Fij σi σj + Σ Fijk σi σj σk + … = 1 For a two-dimensional analysis, there is: g9
F1 σl + F2 σt + F6 τlt + F11 (σl)2 + F22 (σt)2 + F66 (τlt)2 + 2 F12 σl σt + 2 F26 σt τlt + 2 F16 σl τlt = 1 Coefficient F1, F2, F6, F11, F22, F66, F12, F26 and F16 depend on the material used.
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FAILURE CRITERIA
G
Aerospatiale's criterion
3 1/2
3 . Failure criterion used at Aerospatiale: Hill's criterion As seen previously, Hill's criterion is, in its general form, formulated as follows: F (σt)2 + G (σl)2 + H (σl - σt)2 + N (τlt)2 = 1 This non-interactive criterion is applicable at the elementary ply only. There is a laminate failure when the most highly loaded layer is broken. If the expression is developed, we obtain: (G + H) (σt)2 + (F + H) (σl)2 - 2 H σl σt + N (τlt)2 = 1 By definition, we shall assume that: (G + H) = (1/Rl)2 where Rl is the longitudinal strength of the unidirectional fibre. (F + H) = (1/Rt)2 where Rt is the transversal strength of the unidirectional fibre. 2 H = (1/Rl)2 N = (1/S)2 where S is the shear strength of the unidirectional fibre.
g10
æσ ö There is a failure if h = ç l ÷ è Rl ø
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2
2
æσ ö + ç t÷ è Rt ø
2
æτ ö + ç lt ÷ è Sø
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æσ σ ö − ç l 2 t÷ = 1 è Rl ø
Composite stress manual
FAILURE CRITERIA Aerospatiale's criterion
G
3 2/2
Thus, the following Reserve Factor is deduced: g11
RF =
1 = h
1 2
2
2 æ σl σ t ö æ σt ö æ σl ö æ τ lt ö + + ÷ ç ÷ ç ÷ ç ÷ +ç 2 è Sø è Rl ø è Rt ø è Rl ø
This criterion is the one used by Aerospatiale. In order to avoid having a premature theoretical failure in the resin, the transversal modulus Et was considerably reduced (by a coefficient 2 approximately) with relation to the average values measured. This "design" value is determined so that the transversal strain is greater than the longitudinal one.
B
The allowable plane shear value S of the unidirectional fibre was determined for having, a good test/calculation correlation and significant tension and compression failures of notched or unnotched laminates.
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FAILURE CRITERIA Example
G
4 1/4
4 . EXAMPLE Hill's criterion shall be applied to the example considered in the chapter "plain plate membrane". Stresses applied to fibres are calculated and presented in the corresponding chapter (C.6) and quoted in the following pages. Let a T300/BSL914 laminate (new) be laid up as follows: 0°: 6 plies 45°: 4 plies 135°: 4 plies 90°: 6 plies Mechanical properties of the unidirectional fibre are the following: El = 13000 hb (130000 MPa) Et = 465 hb (4650 MPa) νlt = 0.35 Glt = 465 hb (4650 MPa) Rlt = 120 hb (1200 MPa) Rlc = 100 hb (1000 MPa) Rtt = 5 hb (50 MPa) Rtc = 12 hb (120 MPa) S = 7.5 hb (75 MPa) The laminate is globally subjected to the three following load fluxes in the reference coordinate system (x, y) (see chapter C.6) : Nx = 30.83 daN/mm Ny = - 2.22 daN/mm Nxy = 44.92 daN/mm
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Composite stress manual
FAILURE CRITERIA Example
Reminder of stresses applied to the fibre with a 0° direction σl = 29.42 hb σt = 0.06 hb τlt = 5.03 hb {g10} 2
2
2
æ 29.42 ö æ 0.06 ö æ 5.03 ö æ 29.42 x 0.06 ö h = ç ÷ +ç ÷ +ç ÷ −ç ÷ =1 120 2 è 120 ø è 5 ø è 7 .5 ø è ø 2
h2 = 0.06 + 1.44 E-4 + 0.45 - 1.23 E-4 = 0.51
{g11} Reserve Factor: R.F. =
1 h2
=
1 0.51
= 14 .
Margin = 100 (R.F. - 1) ≈ 40 %
Reminder of stresses applied to the fibre with a 45° direction σl = 80.17 hb σt = - 1.14 hb τlt = - 1.36 hb {f10} æ 80.17 ö h2 = ç ÷ è 120 ø
2
. ö æ − 114 + ç ÷ è 12 ø
2
. ö æ − 136 + ç ÷ è 7.5 ø
2
. )ö æ 80.17 x (− 114 − ç ÷ 2 è ø 120
h2 = 0.45 + 0.009 + 0.033 + 0.006 = 0.498
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4 2/4
Composite stress manual
FAILURE CRITERIA Example
{g11}
Reserve Factor: R.F. =
1 0.498
= 142 .
Margin ≈ 42 %
Reminder of stresses applied to the fibre with a 135° direction σl = - 59.17 hb σt = 2.14 hb τlt = 1.36 hb {g10} æ − 59.17 ö h2 = ç ÷ è 100 ø
2
æ 2.14 ö + ç ÷ è 5 ø
2
. ö æ 136 + ç ÷ è 7.5 ø
2
æ − 59.17 x 2.14 ö − ç ÷ è ø 100 2
h2 = 0.35 + 0.183 + 0.033 + 0.0126 = 0.579
{g11} Reserve Factor: R.F. =
1 0.579
= 131 .
Margin ≈ 31 %
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4 3/4
Composite stress manual
FAILURE CRITERIA Example
Reminder of stresses applied to the fibre with a 90° direction σl = - 8.42 hb σt = 0.95 hb τlt = - 5.03 hb {g10} æ − 8.42 ö h = ç ÷ è 100 ø
2
2
æ 0.95 ö + ç ÷ è 5 ø
2
æ − 5.03 ö + ç ÷ è 7.5 ø
2
æ − 8.42 x 0.95 ö − ç ÷ è ø 100 2
h2 = 0.007 + 0.036 + 0.45 + 8 E-4 = 0.494
{g11} Reserve Factor: R.F. =
1 0.494
= 142 .
Margin ≈ 42 %
Conclusion: the laminate overall margin is therefore 31 %
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4 4/4
Composite stress manual
MONOLITHIC PLATE - FAILURE CRITERIA References
G
BARRAU - LAROZE, Design of composite material structures, 1987 GAY, Composite materials, 1991 VALLAT, Strength of materials Comparative analysis of composite material damaging criteria BOUNIE, Failure criteria of mechanical bonds in composite materials, 1991, 440.180/91
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H MONOLITHIC PLATE - FATIGUE ANALYSIS
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I MONOLITHIC PLATE - DAMAGE TOLERANCE
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MONOLITHIC PLATE - DAMAGE TOLERANCE Notations
1 . NOTATIONS (o, x, y): panel reference frame Nx: x-direction normal flow Ny: y-direction normal flow Nxy: shear flow α i: orientation of fibre “i” εli: longitudinal strain of fibre “i” εti: transverse strain fibre “i” γlti: angular slip of fibre “i” εadm: permissible longitudinal strain of unidirectional fibre γadm: permissible slip of unidirectional fibre σli: longitudinal stress of fibre “i” σti: transverse stress of fibre “i” τlti: shear stress of fibre “i” Rl: permissible longitudinal stress of unidirectional fibre Rt: permissible transverse stress of unidirectional fibre S: permissible shear stress of unidirectional stress κR: reduction coefficient for permissible longitudinal stress κS: reduction coefficient for permissible shear stress
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1
Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE Introduction
I
2
2 . INTRODUCTION The regulatory requirements in terms of structural justification concern, on the one hand, the static strength JAR § 25.305 and, on the other hand, fatigue + damage tolerance JAR § 25.571. For the latter, three cases are to be considered: - § 25.571 (b) Damage tolerance - § 25.571 (c) Safe-life evaluation * § 25.571 (d) Discrete Source For the static strength evaluation, Acceptable Means of Compliance ACJ 25.603 § 5.8 requests resistance to ultimate loads with "realistic" impact damage susceptible to be produced in production and in service. This damage must be at the limit of the detectability threshold defined by the selected inspection procedure. Also, static strength must be demonstrated after application of mechanical fatigue (§ 5.2) and test specimens must have minimum quality level, that is, containing the permissible manufacturing flaws (§ 5.5) and "realistic" impact damage. The static strength range is defined therefore for a detection threshold and by a "realistic" cut-off energy leading to "realistic" impacts. The damage tolerance range is outside the static range. Detection threshold (impact depth in mm) Damage at detectability threshold limit
Damage Tolerance Range
Low thickness
Static strength range
High thickness Impact energy
Static cut-off energy
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Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE Introduction
I
2 3 3.1
Distinction is made between the range above the detectability threshold where all damage will be detectable and the range above the static cut-off energy and below the detectability threshold where the damage will never be detected. In this "Damage tolerance" section, we shall discuss both manufacturing defects and impact damage for the static justification and the fatigue-damage tolerance justification. The basic assumption to be retained is the fatigue damage no-growth concept.
3 . DAMAGE SOURCES AND CLASSIFICATION Distinction is made between damage which may occur during manufacture and that which occurs in service.
3.1 . Manufacturing damage of flaws Manufacturing damage or flaws include porosities, microcracks and delaminations resulting from anomalies, during the manufacturing process and also edge cuts, unwanted routing, surface scratches, surface folds, damage attachment holes and impact damage (see § 3.2.3). Damage, outside of the curing process, can occur a detail part or component level during the assembly phases or during transport or on flight line before delivery to the customer. If manufacturing damage/flaws are beyond permissible limits, they must be detected by routine quality inspections. For all composite parts, the acceptance/scrapping criteria must be defined by the Design Office. Acceptable damage/flaws are incorporated into the ultimate load justification by analysis and into the test specimens to demonstrate the tolerance of the structure to this damage throughout the life of the aircraft.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Fatigue damage
I
3.2 3.2.1
3.2 . In-service damage This damage occurs in service in a random manner. Distinction is made between three types of damage: - fatigue, - corrosion and environmental effects, - accidental.
3.2.1 . Fatigue damage Composite materials are said to be insensitive to fatigue; more exactly, their mechanical properties are such that the static dimensioning requirements naturally cover the fatigue dimensioning requirements. This is valid for a laminate submitted to plane loads, less than 60 % of ultimate load. However, complex areas or areas with a sudden variation in rigidity may favour the appearance of delaminations under triaxial loads. Today, it is very difficult to (analytically or numerically) model the growth rate of a possible flaw. This is why a "safe-life" justification philosophy has been adopted. It is based on two principles which must be underpinned by experimental results: - non-creation of fatigue damage (endurance), - no-growth of damage of tolerable size. On account of the dispersion proper to composites and the form of the "Wohler" curves associated with them (relatively flat curve with low gradient), the factor 5 normally used on metallic structures for the number of lives to be simulated during a fatigue test, was replaced by a load factor. All these points will be discussed in detail in section O (MONOLITHIC PLATE FATIGUE).
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MONOLITHIC PLATE - DAMAGE TOLERANCE Corrosion damage - Environmental effects
I
3.2.2
3.2.2 . Corrosion damage and environmental effects a) Corrosion Composites are insensitive to corrosion. Nevertheless, their association with certain metallic materials can cause galvanic coupling liable to damage certain metal alloys. For information purposes, the table below shows various carbon/metal pairs over a scale ranging from A to E. We consider that type A and B couplings are correct and that those of types C, D and E
Coupling with carbon to be avoided
Coupling with carbon correct
are not.
A
Anodised titanium, protected titanium fasteners
A
Titanium and gold, platinium and rhodium alloys
B
Chromiums, chrome-plated parts
B
Passivated austenitic stainless steels
B
Monel, inconel
B
Martensitic stainless steels
C
Ordinary steels, low alloys steels, cast irons
D
Anodic or chemically oxidised aluminium and light alloys
D
Cadmium and cadmium-plated parts
D
Aluminium and aluminium-magnesium alloys
D
Aluminium-copper and aluminium-zinc alloys
b) Environmental effects At high temperatures, aggressions by hydraulic fluids may cause damage such as separation, delamination, translaminar cracks, etc. Rain can cause damage by erosion, etc. All these points will be discussed in detail in section W (INFLUENCE OF THE ENVIRONMENT).
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MONOLITHIC PLATE - DAMAGE TOLERANCE Accidental damage - Inspection of damage
I
3.2.3 4
3.2.3 . Accidental damage This is the most important type of damage and the damage most liable to call into question the structural strength of the part. It can occur during the manufacture of the item (drilling delamination) or in service (drop of a maintenance tool, hail or bird strikes).
4 . INSPECTION OF DAMAGE One of the main preoccupations concerning the damage tolerance of composites is damage detection. This is true both during manufacture and in service. In service, the detectability threshold depends on the type of scheduled in-service inspection. There are four types of inspections: Inspection - Special detailed (ref: Maintenance Program Development: MPD): An intensive examination of a specific location similar to the detailed inspection except for the following differences. The examination requires some special technique such as non-destructive test techniques, dye penetrant, high-powered magnification, etc., and may required disassembly procedures. This type of inspection is mainly conducted during production but can be used exceptionally in service. Inspection - Visual Detailed (ref: Maintenance Program Development: MPD): An intensive visual examination of a specified detail, assembly, or installation. It searches for evidence of irregularity using adequate lighting and, where necessary, inspection aids such as mirrors, hand lens, etc. Surface cleaning and elaborate access procedures may be required. This type of inspection enables BVID (Barely Visible Impact Damage) to be detected.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Inspection of damage
I
4 4.1 4.2
Inspection - General Visual (ref: Maintenance Program Development: MPD): A visual examination that will detect obvious unsatisfactory conditions/discrepancies. This type of inspection may require removal of fillets, fairings, access panels/doors, etc. Workstands, ladders, etc. may be required to gain access. Inspection - Walk Around Check (ref: Maintenance Review Board Document: MRB): A visual check conducted from ground level to detect obvious discrepancies.
In general, the Walk Around check is considered as a general daily visual inspection.
4.1. Minimum damage detectable by a Special Detailed Inspection These inspections are conducted with bulky facilities: ultrasonic, thermographic, X-rays, etc. Minimum detectable sizes are related to the size of the U.S. probes and the accuracy of the tools used, etc.
4.2 . Minimum damage detectable by a Detailed Visual Inspection This type of damage is called BVID (Barely Visible Impact Damage). The geometrical detectability criteria are as follows (cf. ref. 22S 002 10504):
© AEROSPATIALE - 1999
Depth of flaw "δ δ"
Inside box structure (broken fibres)
Outside box structure
Mean
0.1 mm
0.3 mm
"A" value
0.2 mm
0.5 mm
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MONOLITHIC PLATE - DAMAGE TOLERANCE Inspection of damage
I
4.3 4.4
4.3 . Minimum damage detectable by a General Visual Inspection This type of damage is called Minor VID (Minor Visible Impact Damage). The geometrical detectability criteria are as follows (cf. ref. 22S 002 10504):
Depth of flaw "δ δ"
Size of perforation
2 mm or thickness of structure if < 2 mm
20 mm ∅
4.4 . Minimum damage detectable by a Walk Around Check This type of damage is called Large VID (Large Visible Impact Damage). The geometrical detectability criteria are not explicitly defined but the damage must be detectable without ambiguities during a Walk Around Check. We generally use a 50/60 mm ∅ perforation as criterion. The diagram below summarises these four detectability levels according the size of the damage. Special detailed inspection
Detailed visual inspection
General visual inspection
BVID
Depth of indent
δ = 0.3 mm
Minor VID
Large VID
δ = 2 mm 20 mm ∅
diameter
Walk around
50/60 mm ∅
In the remainder of this document, we will consider only visual inspections.
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Size of damage
Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE Classification of damage
I
4.5
4.5 . Classification of accidental damage by detectability ranges Depending on the type of visual inspection considered during the maintenance phases (general or detailed), we will define three clearly separate detectability ranges: a) Damage undetectable by visual means used in service. b) Damage susceptible to be detected during in-service inspections. c) Damage "inevitably" detectable that can be placed into two categories: - Readily detectable damage. - Obvious detectable damage. These ranges are positioned as follows on the previously defined detectability scale: → For Detailed Visual Inspection: Damage susceptible to be detected
Undetectable damage DVI
Inevitably detectable damage WA
BVID
Minor VID
Large VID
→ For General Visual Inspection: Damage susceptible to be detected
Undetectable damage
BVID
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GVI
WA
Minor VID
Large VID
MTS 006 Iss. B
Inevitably detectable damage
Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE Influence of damage - Porosity
I
4.5 5 5.1 5.1.1
Remark: Note that certain authors define the BVID notion according to the type of inspection selected. In this case, for a general inspection: MINOR VID ≡ BVID In our document, we will conserve the initial definition related to the visual detailed inspection.
5 . EFFECT OF FLAWS/DAMAGE ON MECHANICAL CHARACTERISTICS 5.1 . Health flaws 5.1.1 . Porosity → Description
By "porosity", we mean a heterogeneity of the matrix leading, more often than not, to lack of inter- or intra-layer cohesion, generally small in size, but distributed uniformly or almost throughout the complete thickness of the laminate. Note that for unidirectional tapes the porosities have a tendency to be located between the layers whereas, for fabrics, they are more generally located where the weft and warp threads cross. The porosity ratio given is a surface porosity ratio measured by the ultrasonic attenuation method. The permissible absorption level is fixed at 12 dB irrespective of the thickness inspected (cf. note 440.241/90 issue 2 - SIAM curve). All absorption areas above this limit will be considered as a delamination and meet therefore the same criteria as a delamination. However, only T300/N5208, more fluid than T300/BSL914 has a higher tendency to be porous.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Porosity
I
5.1.1
→ Loss of mechanical characteristics due to porosity The test results were interpolated, for the V10F wing, on T300/N5208 with various porosity ratios distributed in all interply areas to determine the influence on the mechanical characteristics for a 3 % ratio considered as the acceptable limit. This ratio combined with the fatigue, ageing and residual test effects at 80° C, led to the following losses in mechanical characteristics:
T300/N5208
3 % porosity Loss of characteristics after F + VC1 + 80° C
Loss of characteristics after F + VC1 + 80° C
BENDING
- 15 %
- 19 %
INTERLAMINAR SHEAR
- 47 %
- 33.5 %
COMPRESSION
- 20 %
- 19 %
TENSILE (high bearing stress) joint not supported
- 20 %
- 19 %
→ Example of porosity acceptance criteria The 3 % acceptance criterion appears therefore as being non-conservative for interlaminar shear. However, let us recall: - that the spar boxes of the wings, movable surfaces or fin are subjected to very low interlaminar stresses, - only T300/N5208 had porosities, - that the 3 % porosity criterion distributed at all interply areas is today no longer applied to primary structures. The permissible porosity ratio depends on the thickness of the laminate.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Delaminations outside stiffener
I
5.1.2 5.1.2.1 1/5
5.1.2 . Delaminations
A delamination is a lack of cohesion between the layers caused by a shear or transverse tensile failure of the resin or, more simply, by forgetting a foreign body.
5 1.2.1 . Delaminations outside stiffener Þ Skin bottom areas → Description A skin bottom delamination is a lack of cohesion between two well-defined plies. Natural delaminations appear during manufacture (surface contamination). A foreign body left in the laminate (separator) will be considered as a delamination.
→ Loss of characteristics due to a delamination For the V10F wing, a lack of interlayer cohesion up to 400 mm2 leads to a loss of compression strength of around 10 % for the two materials (T300/N5208 and T300/BSL914) tested in new condition at θ = 80° C. In aged/fatigue condition the drop in strength is 20 % for T300/N5208 and 13 % for T300/BSL914 in relation to the new state/80° C reference. Fatigue leads to no growth of the flaw.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Delaminations outside stiffener
I
5.1.2.1 2/5
Þ Fastener areas → Description As for the skin bottom delaminations, the lack of cohesion in these areas occurs between two well-defined plies, sometimes at several levels but generally adjacent. These flaws come through to the bore. They are created during the drilling operations. The ultrasonic inspections conducted after each test case showed no evolution of existing flaws. The parameter representing the size of the damage is the number given by: φ =
damage ∅ fastener ∅
damage Ø fastener Ø
Vb Vc where Vb represents the "B value" (see section Y) relevant to all tests characterising the material and where Vc is the calculation value used. Provided that the calculation value is lower than the "B value", the integrity of the item is ensured. For safety reasons, we will impose a minimum margin of 10 % between the calculated value and the "b value".
The parameter representing the drop in characteristics is the number given by: ν =
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MONOLITHIC PLATE - DAMAGE TOLERANCE Delaminations outside stiffener
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5.1.2.1 3/5
Two cases can occur: - if ν ≥ 1.1: no reduction will be made on the initial reserve factor RF, - if ν < 1.1: after reduction, the new reserve factor is equal to RF’ = RF
ν 1.1
The values of ν are given by the graphs in section Z for the prepreg epoxy carbon fibre T300/914. Generally speaking, the graphs gives the values of ν for the flaw (delamination) but also for repairs which may be made on it (injection of resin, NAS cup). They enable you to find therefore: - whether the flaw is acceptable as such, - what type of repair is to be chosen.
→ Examples of acceptance and concession criteria - in standard area, the delamination must be covered by a concession if its surface area is greater than: S mm2
75
120
160
285
440
440
Ø
3.2
4.1
4.8
6.35
7.92
9.52
These permissible delamination values are valid only for isolated delaminations. For delaminated hole concentrations and irrespective of the size of the delaminations, the flaw must be covered by a concession if: - for aligned fasteners, more than 20 % of the holes are delaminated and/or two flaws are less than 5 fastener pitches apart, - for a delamination at a fastener or of another skin bottom area, they are less than 120 mm apart. - in designated area, permissible delamination is defined as follows: S mm2
50
80
110
200
400
400
Ø
3.2
4.1
4.8
6.35
7.92
9.52
These permissible delamination values are valid only for isolated delaminations.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Delaminations outside stiffener
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5.1.2.1 4/5
For delaminated hole concentrations and irrespective of the size of the delaminations, the flaw must be covered by a concession if: - for aligned fastener, more than 10 % of the holes are delaminated and/or two flaws are less than 5 fastener pitches apart, - for a delamination at a fastener or of another skin bottom area, they are less than 120 mm apart. - for areas with several fastener rows: • if the fasteners are on same row: same as above, • if the flaws are located on several rows, they must be covered by a concession if they are less than 175 mm apart.
→ Examples of repairs to be made The table below summarises the repair solutions to be applied when delaminations are detected at fastener holes in materials T300/914, G803/914 and HTA/EH25 depending on the loads and the damage ∅ ratio. fastener ∅
The choice of the solution is governed by the following rules: - for a pure load, the repair or untreated delamination must resist ultimate loads under the most severe environmental conditions, - for a pure bearing stress test, the calculation value Vc is taken as reference. The Vb repair will not be acceptable if is lower than 1. Vc The validation range of the acceptable solutions given in the table below is damage ∅ ≤ 6. fastener ∅
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MONOLITHIC PLATE - DAMAGE TOLERANCE Delaminations outside stiffener
Load
Condition
Untreaded Injection via delamination vent hole
I
5.1.2.1 5/5
Normal injection
NAS cup
New
Acceptable
Unacceptable
-
-
Aged-wet
Acceptable
Unacceptable
-
Acceptable
Acceptable
-
Acceptable
-
Acceptable
Pure tensile Acceptable New
damage ∅
< 4
fastener ∅
Bearing Tensile
Acceptable Aged-wet
damage ∅
< 4.5
fastener ∅
Acceptable damage ∅
< 4.5
fastener ∅
Acceptable New
Unacceptable Unacceptable Unacceptable
damage ∅
< 5
fastener ∅
Pure compression
Acceptable Aged-wet
Unacceptable Acceptable
New
damage ∅
< 4.75
fastener ∅
Bearing compression
damage ∅
< 2.5
fastener ∅
Acceptable
damage ∅
< 5.25
fastener ∅
Acceptable damage ∅
Acceptable
< 2
fastener ∅
Acceptable damage ∅
< 5.25
fastener ∅
Acceptable damage ∅
< 5.25
fastener ∅
Acceptable damage ∅
Unacceptable Unacceptable
Acceptable
Acceptable
Unacceptable
-
Acceptable
Acceptable
Unacceptable
-
Acceptable in "hollow"
Acceptable
Unacceptable
-
Unacceptable
Without bending
Unacceptable Unacceptable
-
Acceptable
Bending 1000 µd
Unacceptable Unacceptable
-
Acceptable in "hollow"
Bending 2500 µd
Unacceptable
-
Acceptable
Aged-wet
< 4
fastener ∅
Without bending JOINT tensile
JOINT compression
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Acceptable
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MONOLITHIC PLATE - DAMAGE TOLERANCE
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Delaminations in stiffener area
5.1.2.2 1/5
5.1.2.2 . Delaminations in stiffener area of an integrally-stiffened panel ❏ Stiffener runouts Stiffener runouts represent a critical point for dimensioning. When these stiffener runouts are made during moulding without later machining operations, these fairly tortured areas may include lacks of cohesion either in the base, or in the stiffener itself.
U-section
Half core Baseplate
U-section
Wedge
Þ Crater → Description This flaw is consecutive to too short a wedge which gives, after machining of the stiffener runout, a crater at the end of the stiffener.
L e
l
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MONOLITHIC PLATE - DAMAGE TOLERANCE Delaminations in stiffener area
I
5.1.2.2 2/5
→ Loss of characteristics due to crater
Size of flaw
ATR 72 T300/914 L = 10 mm l = 4 mm e = 1 mm
ATR 72 HTA/EH25
Test conducted
Conditions
Loss of characteristics due to flaw
Tensile Compression (stiffener runouts not protected)
New θ = 20° C
- 28 %
Tensile Compression (stiffener runouts protected)
New θ = 20° C
0%
Compression (with reinforcement) Compression (without reinforcement)
-4% Aged θ = 70° C
- 12 %
For unprotected stiffener runouts (that is, when it was impossible to thicken the skin to make structure relatively simple to manufacture), this flaw must be covered by a concession. When it is located at protected stiffener runouts (that is with a significant skin overthickness at stiffener runout), this flaw will be covered by a concession only if its size is greater than the following values: L = 10 mm
l = 2 mm
e = 0.5 mm
Þ Punching → Description This flaw is due to an imperfect Mosite cut leading to flaws at stiffener ends.
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MONOLITHIC PLATE - DAMAGE TOLERANCE
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Delaminations in stiffener area
5.1.2.2 3/5
L e
l
e
→ Loss of characteristics due to punching
Size of flaw
Test conducted
Conditions
Loss of characteristics due to flaw
ATR 72 T300/914
Tensile Compression (stiffener runouts not protected)
New θ = 20° C
- 20 %
Tensile Compression (stiffener runouts protected)
New θ = 20° C
0%
L = 10 mm e = 1 mm
Must be covered by a concession when located at unprotected stiffener runouts. When located at protected stiffener runouts, it will be covered by a concession only if it size is greater than the following values: L = 10 mm
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Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE Delaminations in stiffener area
I
5.1.2.2 4/5
Þ Flaws "E", "B", "AB" and "BC" → Description These flaws are located at various levels: FLAW E
FLAW B
FLAW AB
Delamination in radius between U-sections and base
Delamination under wedge
Delamination at skin midthickness
Flaws BC correspond to one or more lacks of cohesion of stiffener wedge as shown on diagram below: Flaw BC A B
C Wedge
U-section Half core
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MONOLITHIC PLATE - DAMAGE TOLERANCE Delaminations in stiffener area
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5.1.2.2 5/5
Þ Loss of characteristics due to flaw
Type of flaw
Test conducted
Conditions
Loss of characteristics due to flaw
V10F T300/N5208 200 mm2 (flaw B)
Tensile (between wedge and base skin)
New θ = 20° C
- 17 %
ATR 72 T300/914 (flaw BC)
Tensile (unprotected stiffener runouts)
Wet ageing θ = 50° C
- 20 %
ATR 72 T300/914 (flaw BC)
Compression (unprotected stiffener runouts)
Wet ageing θ = 50° C
0%
❏ Stiffener top Lack of interlayer cohesion at top of stiffener between the U-section and the wedge does not seem to modify the mechanical characteristics.
❏ Stiffener base Lack of interlayer cohesion in stiffener base hardly modifies the mechanical characteristics. Within the scope of the V10F programme, the greatest drop is less than 10 % in standard stiffener compression case with a type BC flaw.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Delamination in spar radii Delamination on edge of spar flanges
I
5.1.3 5.1.4
5.1.3 . Delamination in spar radii This flaw correspond to lack of cohesion between two well-defined plies in the web/flange blend-in radius.
The maximum permissible surface area for a flaw is 100 mm2. Þ In standard areas: maximum local surface area between 2 ribs for a radius is 250 mm2, including delaminations and foreign bodies. Þ
In designated areas: maximum local surface area between 2 ribs for a radius is
150 mm2, including delaminations and foreign bodies.
5.1.4 . Delamination on spar flange edges l
L
Delamination
Delamination acceptable after repair is defined as follows : - 1 delaminated interface only, - l ≤ 5 mm, - L ≤ 25 mm. An acceptable flaw will however require a Hysol 9321 sealing operation on edge. Any other flaws shall be covered by a concession.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Foreign bodies - Translaminar cracks
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5.1.5 5.1.6
5.1.5 . Foreign bodies Same criteria as given for delaminations (cf. § 5.1.2.1).
5.1.6 . Translaminar cracks Translaminar cracks have been detected on the ATR 72 outer wing spar box, the A340 aileron, the 2000 fin, the A300/A310 (cf. note 494.048/91); however there are none on the flight V10F (cf. note 494.007/91). These are elongated flaws due to the use of a corrosive stripper (MEK, Methyl Ethyl Ketone). Currently, baltane is used. T300/914 and G803/914 have these flaws; the tests conducted on IM7/977-2 and HTA/EH25 showed no translaminar cracks (cf. note 494.056/91). These cracks are detected by ultrasonic inspection in the fastener areas (the back surface echo totally disappears). They concern all ply directions but do not touch between two plies with different orientations. It is in the high crack density area that the ultrasonic signal is totally damped. There a transition zone between this area and the healthy part of the laminate where crack density decreases and the ultrasonic back surface echo reappears. These cracks are parallel to the fibres leaving the holes. They first affect the plies at 0°, then the plies at ± 45°. Some cracks are observed in the central plies at 90°. The axes of these crack networks correspond approximately to the hole diameters. They do not lead to a drop in the mechanical characteristics (cf. note 437.115/91). The existence of flaws at fasteners can be masked by high density translaminar cracks. Therefore, the threshold of the surface areas of the translaminar cracks which must be plotted is coherent with the size of acceptable delaminations.
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Delaminations consecutive to a shock
5.1.7 1/4
5.1.7 . Delaminations consecutive to a shock (during production and in service) → Description An impact causes lack of interlayer cohesion at several levels depending on the energy of the impact. Damaged area
Delaminated area Impactor indent
→ Loss of characteristics due to a delamination Generally speaking, a composite material with a non-through delamination is much more sensitive from a structural strength viewpoint to compression or shear loads (resin) than to tensile loads (fibre). The drops in characteristics within the scope of the V10F programme are: - 18 % in tensile strength for a maximum invisible impact, - 36 % in compression strength for a maximum invisible impact. All points of the tests conducted on the V10F test specimens were plotted on the graph below (the points of the static and fatigue test specimens are combined on this curve as it has been demonstrated that the ageing effect is not significant for damage tolerance). The curve used at Aerospatiale for the new states/residual test at ambient temperature and aged/fatigue states/residual test at ambient temperature is shown on the curve below by comparison at static test specimen and fatigue test specimen points.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Delaminations consecutive to a shock
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5.1.7 2/4
Behaviour to impact damage V10F Static test specimen (CES) Fatigue test specimen (CEF) 0
500
1000
1500
2000
2500
- 1000
- 1500
Allongement de compression (µd)
- 2000
- 2500
i32 - (- 2800 µd) Arrêt CEF
- 3000
i22 - (- 3108 µd) Rupture CES
- 3500
- 4000
COURBE ACTUELLE VALEURS DE CALCUL Etat neuf/température ambiante ou Etat vieilli/fatigue à 20° C/température ambiante
- 4500
CES CEF
- 5000
Delaminated surface area (mm2) → Ultimate strength of a delaminated laminate The problem is generally posed as follows: we take a laminate consisting of a set of tapes (or fabrics) that we will assume to be made of the same material, each one of them having a specific orientation in relation to the reference frame (o, x, y). The laminate is submitted to shear forces (of membrane type) Nx, Ny and Nxy. In the presence of a delamination (without ply failure) in surface area Sd, what is the strength of the plain composite plate? Today, there are three methods for evaluating the residual strength in the presence of a delamination (established from experimental results) which call on the stresses and/or strains of the unidirectional fibre and not those of the laminate considered as a homogeneous plate. Each fibre direction must therefore be justified.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Delaminations consecutive to a shock
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5.1.7 3/4
We will describe here these three methods in chronological order. 1st method: This first method consists in calculating a failure criterion determined from the strains of each fibre in relation to their intrinsic frame (o, l, t). By referring to the "plain plate - calculation method" section, it is possible to calculate the strains in the various layers of the composite from the global flows Nx, Ny and Nxy applied to the laminate and from the characteristics of the material used. For layer "i" defined by its orientation α i, the strains of the fibre "i" in its own frame are defined by the following strain tensor: (εli, εti, γlti). We can define the following failure criterion C1 for each layer "i": 2
i1
C1 =
æ γ ö æ εl i ö ÷ + ç lt i ÷ ç ÷ çε çγ ÷ è adm ø è adm ø
2
where εadm and γadm are the permissible strains (longitudinal and shear) of the unidirectional fibre (equivalent). These values (obtained from the test results) depend on the material and the surface area Sd of the delamination considered and the types of loads. They are given in section Z (sheets giving calculation values and coefficients used). This criterion was used for the dimensioning of the ATR 72 wing panels (dossier 22S00210460). 2nd method: This second method consists in calculating a failure criterion C2 (Hill type criterion in which the permissible stresses are reduced by coefficients κR and κS) calculated from the stresses in each fibre in relation to their intrinsic frames (o, l, t).
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MONOLITHIC PLATE - DAMAGE TOLERANCE Delaminations consecutive to a shock
I
5.1.7 4/4
By referring to the "plain plate - calculation method" section, it is possible to calculate the strains in the various layers of the composite from the global flows Nx, Ny and Nxy applied to the laminate and from the characteristics of the material used. For layer "i" defined by its orientation α1, the stresses of the fibre "i" in its own frame are defined by the following stress tensor: (σli, σti, τlti). We can define the following failure criterion C2 for each layer "i":
i2
C2 =
æ σ li ç çκ R è R l
2
2
2
σ li σ t i æ τ ö æσ ö ö ÷ + ç t i ÷ + ç lt i ÷ − ç κ S÷ çR ÷ ÷ (κ R R l )2 è s ø è tø ø
where Rl, Rt and S are the permissible longitudinal, transverse and shear stresses of the unidirectional fibre respectively (equivalent) and where κR and κs are the reduction coefficients for these permissible stresses. These coefficients depend on the material used and the surface area of the delamination considered and are determined from the test results. They are given in section V (sheets giving calculation values and coefficients used). This criterion was used for the sizing of the A330/340 inboard and outboard aileron panels. 3rd method: This method consists in calculating a failure criterion C3 (similar to the one of method 1) calculated from the strains of each fibre in relation to their intrinsic farmes (o, l, t). For layer "i" defined by its orientation αi, the strains of the fibre "i" in its own frame are defined by the following strain tensor: (εli, εti, γlti). We can define the following failure criterion C3 for each layer "i" :
i3
C3 =
æ ε li ç çε è a
2
æ γ ö ÷ + ç lt i çγ ÷ è adm ø
2
ε ε ö ÷ + li t i ÷ (ε ab )2 ø
where: if 2 ôεadmô ≤ ôγadmô
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MONOLITHIC PLATE - DAMAGE TOLERANCE Delaminations consecutive to a shock Visual flaws - Sharp scratches
i4
1
εa =
3
−
2 (ε adm )
2
i5
I
5.1.7 5.2 5.2.1
2
(γ adm )2
1
εab =
3
2 (ε adm )
2
−
6
(γ adm )2
else εa = εadm εab = + ∞ The particularity of this method is that it takes into account (in a significant manner) the load transverse to the fibre. Tests have shown that presence of a tensile force perpendicular to the fibre direction compression increases the ultimate strength of the laminate. Criterion C3 takes this phenomenon into account. Indeed, if εti is of tensile type and εli of compression type, the third term of the criterion C3 becomes negative and tends to increase the reserve factor and therefore the margin (RF = 1/C3). Today, it is recommended to use this third finer method based on a high number of experimental results.
5.2 . Visual flaws
5.2.1. Sharp scratches → Description Sharp scratches are made by scalpels or cutting tools. Sharp scratches lead to drops in tensile characteristics of around 15 %; for compression, we assume that there is no drop in characteristics.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Sharp scratches
I
5.2.1
→ Examples of acceptance criteria for sharp scratches A long anomaly is acceptable without concession within following limits: Þ On the ATR 72 outer wing carbon box structure, → In standard areas: permissible scratches are defined as follows: - maximum length: 100 mm, - maximum depth: 1 ply irrespective of the thickness. → in designated areas: the acceptance criteria are as follows: - maximum length: 100 mm, - maximum depth: 1 ply irrespective of the thickness, - all scratches though to a hole, an hedge or stopping less than 5 mm away must be covered by a concession. Any scratch concentrations must be covered by a concessions if the flaws are less than 20 mm apart. Þ On A330/A340 inboard and outboard ailerons, if length of scratch is less than 100 mm and if its depth is less than 0.15 mm for tapes and 0.3 mm for fabrics, sealing with Hysol 9321 will be performed. Þ On A330/A340, A320, A319, A321 nose landing gear doors (carbon fabrics G803/914), → at fittings, the permissible scratches are defined as follows: - maximum length: 10 mm, - maximum depth: 1 ply irrespective of the thickness. → outside fittings: the acceptance criteria are as follows : - maximum length: 250 mm, - maximum depth: 1 ply irrespective of the thickness.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Indents - Scaling
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5.2.1 5.2.2 5.2.3
1/3
Þ On A330/A340, A320, A319, A321 main landing gear doors (carbon fabrics G803/914), → at fittings, the acceptance criteria as follows: - maximum length: 10 mm, - maximum depth: 1 ply irrespective of the thickness. → outside fittings: the acceptance criteria area as follows : - maximum length: 100 mm, - maximum depth: 1 ply irrespective of the thickness.
5.2.2 . Indents "Indent" type flaws due, for instance, to abrasion of skin by a rototest are permissible if: - surface area of indent is ≤ 20 mm2 (∅ 5), - only the 1st ply is totally damaged, that is 2nd ply visible. Any flaw concentrations must be covered by a concession if two indents are less than 100 mm apart.
5.2.3 . Scaling → Description By "scaling", we mean separation or removal of several fibres (locally) altering only the first surface ply on monolith edge or on outgoing side of drilled holes. → Examples of scaling acceptance criteria Þ On ATR 72 outer wing carbon box structure, → in standard areas: the permissible scaling flaws are defined as follows: Maximum surface area = 30 mm2 Maximum depth: 1 ply for th < 20 plies 2 plies for th ≥ 20 plies
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MONOLITHIC PLATE - DAMAGE TOLERANCE Scaling
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5.2.3 2/3
For scaled hole concentrations, this flaw must be covered by a concession if, for aligned fasteners, more than 20 % of the holes are scaled and/or two flaws are less than 5 fastener pitches apart.
Flaw 1
Flaw 2
→ in designated area: permissible scaling flaws are defined as follows: Maximum surface area = 20 mm2, Maximum depth: 1 ply irrespective of the thickness. For scaled hole concentrations, this flaw must be covered by a concession if: - for aligned fasteners, more than 10 % of the holes are scaled and/or two flaws are less than 5 fastener pitches apart,
Flaw 1
Flaw 2
- for areas with several fasteners rows (e.g. piano area)
175 mm Flaw 1
For flaws 1 and 3: to be covered by a concession Flaw 2 For flaws 1 and 2: if S1 and S2 ≤ permissible surface area permissible
Flaw 3
• for fasteners on same row: same as above, • for flaws on several rows; must be covered by a concession if they are less than 175 mm apart.
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Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE Scaling
I
5.2.3 3/3
All scaled areas will be sealed with Hysol 9321 to restore flat surface and avoid scaling developing during later operations. Þ On A330/A340 inboard and outboard ailerons, scaling on 1 ply of skin will be sealed with Hysol 9321. Permissible scaling flaws are defined as follows: → panels (delaminations at fasteners) Maximum surface area = 30 mm2 For flaw concentrations at fasteners, two flaws on same row must be separated by 9 fasteners. Areas with several fastener rows: - on same row: see above, - between different fastener rows minimum distance = 175 mm → panels (leading edge joints), ribs, spar Maximum surface area = 30 mm2 Maximum depth: 0.2 mm For flaw concentrations, 5 flaws maximum on 10 consecutive fasteners. → panels (other areas) (scaling at fasteners) Maximum surface area = 30 mm2 Maximum depth: 0.2 mm For flaw concentrations, two flaws on a given row must be separated by 9 fasteners.
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Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE Steps
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5.2.4 1/2
5.2.4 . Steps → Description This is a fold of one or more skin plies which may occur between two (spar support) blocks or on a sandwich skin during co-curing or in spar webs.
50 mm
Filleralu
→ Examples of acceptance criteria Þ On ATR 72 outer wing carbon box structure → On bearing surfaces (spar, rib passage) - Standard areas: steps on spar and rib passage areas are acceptable within a limit of 0.3 mm. This type of flaw will be compensated for by Filleralu over a width of 50 mm on either side of the step. - Designated areas: this flaw must be covered by a concession irrespective of its geometry. → On stiffeners - standard areas: steps on stiffener flanges are acceptable within a height limit of 0.3 mm provided that: • there are no flaws in stiffener radius, • two flaws are at least 400 mm apart in Y-direction (wing frame), • two adjacent stiffeners are not affected in the same section, • an ultrasonic inspection demonstrates absence of "delamination" type flaws. - designated areas: steps on stiffener flanges must be covered by a concession.
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Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE Steps - Justification of permissible manufacturing flaws
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5.2.4 2/2
Þ On A330/A340 inboard and outboard ailerons under spar and rib bearing surfaces, steps lower than or equal to 0.2 mm and with a width lower than or equal to 3 mm will be accepted, but: - they must never be trimmed, - they will be compensated for by Filleralu, - in other areas, acceptable height is 0.4 mm. Þ On A330 Pratt et Whitney thrust reverser sandwich skins mainly in areas with high curvatures, steps with a height less than 0.5 mm are accepted in production. Steps greater than 0.5 mm will be examined case by case.
6 . JUSTIFICATION OF PERMISSIBLE MANUFACTURING FLAWS
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Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE Justification of in-service damage
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7 7.1 7.1.1
7 . JUSTIFICATION OF IN-SERVICE DAMAGE 7.1. Justification philosophy A justification philosophy in agreement with European regulations (JAR) is associated with each damage detectability range § 4.5 (undetectable damage; damage susceptible to be detected [during inspection]; readily and obvious detectable damage).
7.1.1. Justification philosophy for undetectable damage ACJ 25.603 § 5.1 : The static strength of the composite design should be demonstrated through a programme of component ultimate load tests in the appropriate environment, unless experience with similar design, material systems and loadings is available to demonstrate the adequacy of the analysis supported by subcomponent tests, or component tests to agreed lower levels. ACJ 25.603 § 5.2 : The effect of repeated loading and environmental exposure which may result in material property degradation should be addressed in the static strength evaluation… ACJ 25.603 § 5.5 : The static test articles should be fabricated and assembled in accordance with production specifications and processes so that the test articles are representative of production structure. ACJ 25.603 § 5.8 : It should be shown that impact damage that can be realistically expected from manufacturing and service, but not more than established threshold of detectability for the selected inspection procedure, will not reduce the structural strength below ultimate load capability. This can be shown by analysis supported by test evidence, or by test at the coupon, element or subcomponent level.
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Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE Justification of in-service damage
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7.1.2 7.1.3 1/3
Undetectable damage, whether due to accidental impacts (in-service damage undetectable by a detailed visual inspection and therefore corresponding to BVID) or manufacturing flaws must be covered by a static justification at ultimate load under the most severe environmental conditions (humidity and temperature) and at end of aircraft life. During the certification tests, this damage will be introduced into minimum margin areas
7.1.2 . Justification philosophy for readily and obvious detectable damage As laid down in the regulations, any damage which cannot withstand the limit loads must be readily detectable during any general visual inspection (50 flights) or obvious. Þ Damage readily detectable within an interval of 50 flights must withstand 0.85 LL. Þ Obvious damage (engine burst) which occurs in flight with crew being aware of it must withstand 0.7 LL (get-home loads capability).
7.1.3 . Justification philosophy for damage susceptible to be detected during scheduled in-service inspections Þ Regulatory aspects ACJ 25.603 § 6.2.1 : Structural details, elements, and subcomponents of critical structural areas should be tested under repeated loads to define the sensitivity of the structure to damage growth. This testing can form the basis for validating a no-growth approach to the damage tolerance requirements… ACJ 25.603 § 6.2.3 : ...The evaluation should demonstrate that the residual strength of the structure is equal to or greater than the strength required for the design loads (considered as ultimate)... ACJ 25.603 § 6.2.4 : ...For the case of no-growth design concept, inspection intervals should be established as part of the maintenance programme. In selecting such intervals the residual strength level associated with the assumed damage should be considered.
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Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE Justification of in-service damage
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7.1.3 2/3
Þ General For metallic structures, the two fundamental damage tolerance parameters are the initiation of the damage and its growth before detection. Many tests have been conducted therefore to evaluate the growth speed of the damage and the time required to reach its critical size and therefore its residual strength (limit load). The critical loading mode is mainly tensile loading. εresidual
Repair
εU.L. εL.L. METALLIC Growth Initiation threshold
Time
Inspection intervals
In contrast, impact damage to the composite structure of perforation/delamination type cause, when it occurs, a very substantial drop in the mechanical strength but it does not grow under the fatigue load levels on civil aircraft. The critical loading mode is mainly compression (and shear) loading εresidual
εU.L. εL.L. COMPOSITE Time At time of impact
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Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE Justification of in-service damage
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7.1.3 3/3
Several methods are used for demonstrating conformity with regulations: a) Semi-probabilistic methods If the no-growth concept of the flaw is demonstrated (by fatigue test), the size of the damage no longer depends on an evolving phenomenon but on a random event (accidental). For the damage range between BVID and VID, the aim of the (analytical) justification will be to determine an inspection interval so that the probability (Re) of simultaneously having a flaw and a load greater than its residual load will be a highly improbable event (probability per flight hour less than 10-9). This probabilistic damage occurrence versus time aspect therefore replaces the deterministic concept for metallic materials where the occurrence of a flaw depends either on fatigue initiation, or, for certain areas, on an accidental impact; the effect of the latter being a modification in the threshold. The complete philosophy can be summarised by the curve below. It expresses the load level to be demonstrated and the type of justification versus the damage range considered. The portion of the curve between the BVID and the VID depends on the results of the probabilistic analysis.
Probabilistic analysis TOLDOM
εresidual
≥ L.L.
Re = E - 9
≥ U.L.
εBVID εU.L. εVID
εL.L.
0.85 εL.L. 0.7 εL.L. ULTIMATE LOADS BVID
These methods are used by AS and CEAT.
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VID OBVIOUS READILY DETECTABLE
Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE Aerospatiale semi-probabilistic method Determining inspection intervals
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7.1.3 7.1.3.1 7.1.3.1.1
1/6
b) deterministic method (Boeing) This method is based on two analysis and test configurations: - demonstrating positive margins at ultimate load with BVID, - demonstrating positive margins at limit load with extensive damage. The non-growth aspect of the fatigue damage must be demonstrated.
7.1.3.1 . AEROSPATIALE semi-probabilistic method (cf. note 432.0162/96) 7.1.3.1.1 . Process for determining inspection intervals As stated above and in § 4.5, certain damage is susceptible to be detected during inspections which implies that the aircraft may possibly fly between two inspections with damage in a structure the residual strength of which may be lower than the ultimate loads. In order not to design composite structures less reliable than metallic ones, an inspection programme has been defined so that the probability of simultaneously having a flaw and a load greater than its residual strength will be a highly improbable event (probability per flight hour less than 10-9). In mathematical form, this requirement can be written: probability of occurrence of an impact with given energy (Pat) x probability of not detecting the resulting flaw (1- Pdat) x probability of occurrence of loads greater than the residual strength of the damage (Prat) ≤ 10-9/fh or again: i6
Pat x (1 - Pdat) x Prat ≤ 10-9/fh This condition involves several notions that we will specify in the following sections.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Determining inspection intervals
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7.1.3.1.1 2/6
❑ Pdat: probability of detecting the damage. We defined, in § 4.2 to 4.4, the visual detection criteria for "A" value and "B" value damage and the mean value for various types of in-service inspections. Knowing that the "A" values correspond to a detection probability of 99 %, the "B" values to a probability of 90 % and the mean values to a probability of 50 %, we can deduce the curve below which shows the probability of detection versus the depth of the indent and the type of inspection. Depth of indent (mm) General visual inspection: * Detailed visual inspection: ** 5 * 2 0,5 **
0,3
0
0.5
0.99 1
Pdat Detection probability
❑ Pat: probability of occurrence of an impact with given energy. Several sources of impacts can be considered (this list is not restrictive): - projection of gravel, - removal of the item, - dropping of tools or removable items, - shock with maintenance vehicle. Each impact source will be defined by its incident energy. As for detection, we will define an impact source by a statistical distribution (in this case, the Log-normal distribution). We will therefore speak of the impact probability (or, more precisely, the impact energy range) that we will call (Pat) and which will be characterised by mean energy Em and a standard deviation (according to Sikorsky, the standard deviation σ has a constant value equal to 0.217). The probability of having an impact energy between E et E+2 Joules is equal to E+2
Pat =
ò
f (E) x dE
E
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MONOLITHIC PLATE - DAMAGE TOLERANCE
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Determining inspection intervals
7.1.3.1.1 3/6
∞
We also obtain
ò
f (E) x dE = 1
0
f(E)
Pat
Em
E 2J
The impact energy will generally be limited to 50 J (cut-off energy), except for THS root: 140 J corresponding to the energy of a tool box failing from the top of the fin. Now that the impact has been defined, we must find the relation between the incident energy (E), the size of the damage (Sd) and its indentation (f). Generally, we have : Sd = Ksd æEö f = Kf ç ÷ èeø
E e 3.3
Test campaigns are however necessary to determine the coefficients Ksd and Kf which depends on the types of materials, their thickness and the item bearing conditions. ❑ Prat: probability of having a loading case greater than the residual strength of the impacted laminate. As we saw in paragraph § 5.1.7, the residual strength of a laminate with a delamination defined by its surface area Sd can be determined by the numbers C1, C2 and C3 that we will call more generally C in the remainder of this section. The need to have three variables to characterise the number C (εl, εt, γlt ou σl, σt, τlt) makes all theoretical exploitations of the item loads (or deformations) difficult.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Determining inspection intervals
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7.1.3.1.1 4/6
ε admissible which represents the permissible C strain of damage of size Sd under a single compression load.
We will therefore define a number εresidual =
This residual deformation depends of course on the size of the damage Sd. The general form of this relation can be represented by the following curve: Sd
εnominal
εresidual
It is therefore possible to determine, for each point on the item studied, the probability of occurrence of the load leading to the failure of the laminate with a delamination of size Sd Knowing that the following gust occurrence probabilities are generally admitted: - 2 x 10-5 for limit loads, - 1 x 10-9 for ultimate loads, We can plot the curve below associating a probability of occurrence Prat with all residual strength levels (εresidual = k x εL.L.) such that: i7
Prat = 10- 8.6 k + 3.9 Prat
2 x 10
-5
10
-9
εL.L.
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εU.L.
εresidual
Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE Determining inspection intervals
I
7.1.3.1.1 5/6
This curve will in fact be compared to a log-normal type occurrence law (or a first approximation linear law) for a larger deformation range. PROBABILITY DETERMINATION LOGIC DIAGRAM - to have an impact in a given energy range, - to detect damage, - to encounter a load greater than the residual strength of the laminate.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Determining inspection intervals f(E)
I
7.1.3.1.1 6/6
Known impact source
Impact energy range
Pat
Energy
Em mean
Detection probability f
f Depth of indent
General visual inspection: * Detailed visual inspection: **
*
æEö Kf ç ÷ èeø
3.3
** Energy
Pdat Pdat
1
Sd
Sd Delaminated surface
Ksd
εresidual
Prat
2 x 10
-5
Prat
10
-9
εL.L.
εU.L.
εresidual
The inspection interval must be such that risk of failure in the interval: Pat x (1 - Pdat) x Prat ≤ 10-9/flight hours
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E e
Energy
Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE Determining and calculating inspection intervals
I
7.1.3.1.1 7.1.3.1.2 1/4
Prat Linear law 1
Log-normal law 0.5 2 x 10
-5
10
-9
εmean
εL.L.
εU.L.
εresidual
This curve, like all statistical distribution curves, is characterised by a mean value and a standard deviation. A simple calculation enables us to obtain the following expressions: εmean = 10(Log (εU.L.) - 0.5554) σ = 0.0928 To sum up, it is clear that by choosing a given impact energy range, the values of Pat, f, Pdat, Sd, εresidual and Prat are implicitly determined. The drawing above shows the links between these various quantities.
7.1.3.1.2 . Inspection interval calculation software The calculation tool is based on the fundamental principle described above: all damage susceptible to be detected during an inspection must have a probability of encountering a load greater than its residual strength lower than 10-9 per flight hour (maximum value at end of aircraft life or before last inspection). This principle involves three probabilities: ❑ Pat: probability of occurrence of an impact with a given energy. ❑ Pdat: probability of detecting the damage. ❑ Prat: probability of occurrence of a loading case greater than the residual strength of the impacted laminate.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Calculating inspection intervals
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7.1.3.1.2 2/4
This principle can be stated in a more useable form: The probability of having damage susceptible to encounter a load greater than its residual strength is equivalent to the sum of the probabilities of having: - damage relevant to an incident energy between 0 and 2 J susceptible to encounter a load greater than its residual strength and - damage relevant to an incident energy between 2 and 4 J susceptible to encounter a load greater than its residual strength and - damage relevant to an incident energy between 48 and 50 J susceptible to encounter a load greater than its residual strength. By discretizing the incident energy and therefore the type of the damage, each flaw range can be dealt with independently of the others. f(E)
E
We can therefore apply the fundamental principle to each energy interval then add the results. First of all we will consider an incident energy range between E and E+2 Joules. The trickiest bit is to determine the probability of existence of damage of a well-defined size versus time knowing that its probability of occurrence is equal to Pat (per flight hour) and its probability of non-detection during inspections is equal to (1 - Pdat).
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MONOLITHIC PLATE - DAMAGE TOLERANCE Calculating inspection intervals
I
7.1.3.1.2 3/4
If Pat is the probability of occurrence of the flaw per flight hour at time t1 (before first inspection for instance) the probability of existence of the flaw is equal to: 1 - (1 - Pat)t1. After the first inspection, the probability of occurrence of the flaw is therefore reduced to: [1 - (1 - Pat)t1] (1 - Pdat) then increases according to same curve as before but with a time shift as initial probability is no longer zero. We repeat this operation up until the last inspection. The form of the function makes the calculations difficult; it is for this reason that we compare the curve to its tangent: 1 - (1 - Pat)t ≈ t x Pat. This approximation remains valid as long as the term t x Pat is small in comparison with 1. This therefore gives the following configuration: Probability of occurrence of a flaw
1 1 - (1 - Pat) ^ t1 [1 - (1 - Pat) ^ t1] (1 - Pdat)
IT1
IT2
t1
t2
IT3
t3
IT4
t4
t
The curve [1 - (1 - Pat) ^ t] will be compared to its limited development: t x Pat Probability of occurrence of a flaw
1
IT x Pat x (1 - Pdat)
IT
IT x Pat x (1 - Pdat) ^ 2
IT x Pat x (1 - Pdat) ^ 3
IT
IT ERL = n x IT
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IT x Pat
IT x Pat
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IT x Pat
IT
t
Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE Calculating inspection intervals
I
7.1.3.1.2 4/4
For constant inspection intervals, the mean probability of occurrence of the flaw is equal to: IT x Pat n − 1 n − i x IT x Pat x (1 − Pdat )i + 2 n i =1
å
The maximum probability of occurrence of the flaw (Rd) is equal to: n −1
i8
Rd = IT x Pat +
å IT x Pat x (1 − Pdat )
i
i=1
The mean probability of failure (Rr) of the flaw is therefore equal to: ìïIT x Pat n − 1 n − i üï Prat x í x IT x Pat x (1 − Pdat) i ý + 2 n ï ï i =1 î þ
å
The maximum probability of failure (Rr) of the flaw is therefore equal to: n −1 ìï üï Rr = Prat x íIT x Pat + IT x Pat x (1 − Pdat) i ý ï ï i =1 î þ
å
i9
To find the mean overall risk per flight hour, all we need to do is to integrate this result into all possible incident energy ranges. ìïIT x Pat n − 1 n − i üï Prat x í + x IT x Pat x (1 − Pdat) i ý 2 n ï ï E=0J i=1 î þ
E = 50 J
å
å
The overall maximum risk per flight hour (Re) is equal to: Re =
i10
ìï Prat x íIT x Pat + ï E=0J î
E = 50 J
å
ü
n −1
iï
i =1
þ
å IT x Pat x (1 − Pdat) ýï
This risk must be lower than 10 E-9. The table below summarises (by giving the mathematical links between the various variables) the method used to determine Re.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Load level K
I
7.1.3.1.2 7.1.3.1.3 1/5
IT & ERL
E
f
Sd
εresidual
Prat
Pdat
Pat
Rd
0-2J 2-4J 4-6J . . . . . . . . . . 44 - 46 J 46 - 48 J 48 - 50 J
Rr
i8
i9
i10 Re 7.1.3.1.3 . Load level K to be demonstrated in the presence of Large VID The previous analysis can be substantiated by a static test with VID and a load level k.CL (1 ≤ k ≤ 1.5). ❑ First method: This method consists in initially evaluating the reduction coefficient α on the permissible strengths of the material so that the final calculated risk Re is equal to 10-9 per flight hour (this determination can only be done by successive approximations). This means that we can suppose that the damage tolerance behaviour of the material is degraded in relation to that really used, that is a material whose strength (under compression loading after impact) will be equal to a certain percentage, called α, of that of the real material.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Load level K
I
7.1.3.1.3 2/5
In this case, the (εresidual; Sd) is submitted to a homothety in relation to the x-axis.
Sd Basic curve → Re Reduced curve -9 → Re = 10 /fh
xα εresidual
The number
1 can therefore legitimately be compared to a reserve factor. α
We will thus define a static test with VID (Visible Impact Damage) such that the margin in relation to the residual strain ε (VID) of the flaw is the one defined above. We obtain: 1 ε ( VID) = α K x εL.L.
hence: K=α
i11
ε ( VID) = α x k ( VID) ε L.L.
value representing the load level K to be demonstrated with VID. ❑ Second method: Another method would consist in directly considering the probability and load notions.
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MONOLITHIC PLATE - DAMAGE TOLERANCE Load level K
I
7.1.3.1.3 3/5
It is clear that for a static test, we can consider that the probabilities of occurrence of the flaw (Pat) and the probabilities of detecting (Pat) and not detecting (1 - Pdat) the flaw are equal to 1 as we are sure that it is present in the item. If we write the equivalence between the test and the maximum risk per flight hour from a probabilistic viewpoint, we obtain: Re = Prat x PatVID x (1 - PdatVID) = Prat. The method consists therefore in determining a fictive ultimate load level such that the probability of the flaw residual load level is equal to Re. The drawing below shows that we must randomly subject the curve (strain level ε; Prat) to 1 a homothety with a factor so as to move point A to point B level. In this case, it appears η that the permissible load level of the VID has a probability of occurrence Re. We see that this transformation also moves point A' to point B' which corresponds to the fictive ultimate load level that must be applied to the structure. Prat
1
Permissible deformation of VID /η
2 x 10
-5
10
-9
A'
B'
A
B
Re εU.L. fictive εL.L.
εU.L.
εresidual
ε VID
By zooming in onto the part of the graph which concerns us and imposing a logarithmic scale on the y-axis, we obtain the following representation:
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Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE
I
Load level K
7.1.3.1.3 4/5
- Log(Prat)
B
- Log(Re)
B'
9
/η
8.6 x K - 3.9 A
A'
4.7
1 K(VID)
k =
1.5
12.9 x K ( VID )
− Log(Re ) + 3.9
− Log(Re ) + 3.9
8 .6
ε residual ε L.L.
We obtain: η=
axis( A ) − Log(Re) + 3.9 = axis(B ) 8.6 x k( VID)
We can deduce the fictive ultimate load level to be demonstrated in the presence of VID K =
i12
12 .9 x k ( VID ) − Log(Re ) + 3.9
Load level K must always be between 1 and 1.5. The graph below represents the previous relation (the maximum risk Re per flight hour on the x-axis and the load level K to be demonstrated on the y-axis). Each curve is relevant to a residual load level of the flaw K.
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Composite stress manual
MONOLITHIC PLATE - DAMAGE TOLERANCE Load level K
I
7.1.3.1.3 5/5
K Safety factor to be applied to limit loads
ULTIMATE LOAD 1.5
1.4 K = 1.17
1.3
K = 1.7
k = 20 k = 1.9
Re = E-15
k = 1.8
k = damage residual load level
k = 1.7
1.2 k = 1.6 k = 1.5 k = 1.4
1.1
k = 1.3 k = 1.2 k = 1.1
LIMIT LOAD 1
Risk of failure per flight hour in Log Log(Re)
0.9 9
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J MONOLITHIC PLATE - BUCKLING
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K MONOLITHIC PLATE - HOLE WITHOUT FASTENER ANALYSIS
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MONOLITHIC PLATE - HOLE WITHOUT FASTENER Notations
K
1
1 . NOTATIONS (o, x, y): reference coordinate system of panel (o, 1, 2): orthotropic axis of laminate φ: angle formed by loading with the orthotropic axis α: angular position of point to be calculated with the orthotropic coordinate system Ex: longitudinal modulus of laminate in the reference coordinate system Ey: transversal modulus of laminate in the reference coordinate system Gxy: shear modulus of laminate in the reference coordinate system νxy: Poisson coefficient of laminate in the reference coordinate system E1: longitudinal modulus of laminate in the orthotropic coordinate system E2: transversal modulus of laminate in the orthotropic coordinate system G12: shear modulus of laminate in the orthotropic coordinate system ν12: Poisson's ratio of the laminate in the orthotropic coordinate system σ ∞x : stress to infinity σx (y): stress along y-axis σt (α): tangential stress around circular hole K ∞T : hole coefficient for an infinite plate width K LT : hole coefficient for a finite plate width β: "finite plate width" coefficient L: plate width ∅: hole diameter R: hole radius
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Composite stress manual
MONOLITHIC PLATE - HOLE WITHOUT FASTENER Introduction - Theory - First method
K
2 3.1 1/3
2 . INTRODUCTION The purpose of this chapter is to assess stresses at the edge of a hole without fastener on an axially loaded composite plate and to anticipate failure of a notched laminate.
3 . GENERAL THEORY 3.1 . First method (Withney and Nuismer) From a theoretical point of view, the problem is formulated as follows: let an infinite plate be subjected to stress flux σ ∞x and with the diameter hole: ∅. The method is valid only if the x-axis is the laminate orthotropic axis. What is the stress σx (y) distribution along the y-axis?
y
σx (y) σ∞ x σx (y = R) x ∅ = 2R
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Composite stress manual
MONOLITHIC PLATE - HOLE WITHOUT FASTENER Theory - First method
K
3.1 2/3
First, the following number needs to be considered: k1
K ∞T =
σ x ( y = R) σ ∞x
=
hole edge stress inf inite stress
This coefficient expresses hole edge stress concentration for the case of an infinitely large plate. This is the hole coefficient. For a composite plate, this term may be formulated as a function of the mechanical properties of the laminate as follows:
k2
æ E ö E x K ∞T = 1 + 2 ç − ν xy ÷ + x ç E ÷ G y xy è ø
Stress σx (y) evolution along the y-axis may be expressed as follows:
k3
σx (y) =
σ ∞x 2
2 4 æ æ ç 2 + æç R ö÷ + 3 æç R ö÷ − K ∞ − 3 ç 5 T ç ç è yø è yø è è
(
)
æ Rö ç ÷ è yø
6
If y = R, then this function is reduced to expression k1. If the material is near-isotropic, then:
k4
σx (R + do) ≈ σ ∞x
k5
with: ξ =
© AEROSPATIALE - 1999
2 + ξ2 + 3 ξ4 2
R R + do
MTS 006 Iss. B
æ Rö −7ç ÷ è yø
8
öö ÷÷ ÷÷ øø
Composite stress manual
MONOLITHIC PLATE - HOLE WITHOUT FASTENER Theory - First method
K
3.1
If the plate is not infinitely large and has a length L, then:
k6
σ∞ σx (y) = β x 2
2 4 6 8 ö æ æ ö ç 2 + æç R ö÷ + 3 æç R ö÷ − K ∞ − 3 ç 5 æç R ö÷ − 7 æç R ö÷ ÷ ÷ T ç è yø ç è yø è yø è y ø ÷ø ÷ø è è
(
)
with: 3
k7
∅ö æ 2 + ç1 − ÷ è Lø β= as a first approximation ∅ö æ 3 ç1 − ÷ è Lø or ∅ö æ 3 ç1 − ÷ 6 æ è 1 1 æ ∅ Mö Lø ∞ = + ç ÷ K T − 3 çç 1 − 3 2è L ø β ∅ö è æ 2 + ç1 − ÷ è ø L
(
in which: M2 =
)
æ ∅ Mö ç ÷ è L ø
2
ö ÷÷ as a second approximation ø
æ ö ∅ ç 3 æç 1 − ö÷ ÷ è Lø ÷ −1 1− 8 ç 1 − 3 ç ÷ ∅ ç 2 + æç 1 − ö÷ ÷ è è ø Lø æ ∅ö 2ç ÷ èLø
2
y
σx (y) σ∞ x σx (y = R) L
x ∅ = 2R
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3/3
Composite stress manual
MONOLITHIC PLATE - HOLE WITHOUT FASTENER Theory - Second method
K
3.2 1/3
3.2 . Second method (NASA) This method is based on a NASA study. For an infinite plate, it expresses the ratio K ∞T between the loading stress to infinity σ ∞x and the tangential normal stress at the edge σt (α) around the hole. The position of the point is defined by the angle α with relation to the orthotropic It shall be assumed that loading is uniaxial.
y
2
σ∞ x
σt (α) α ∅ = 2R
x φ
1
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Composite stress manual
MONOLITHIC PLATE - HOLE WITHOUT FASTENER Theory - Second method
K
3.2 2/3
The first step consists in searching for the orthotropic axes (o, 1, 2) of the material. Angles φ and α are thus determined (α being the angular coordinate of the point to be considered with relation to the orthotropic coordinate system). The hole coefficient expression is the following: k8
K ∞T =
{
σt(α ) Eα = (− cos 2 φ + (k + n) sin 2 φ) k cos 2 α + (1 + n) cos 2 φ − k sin 2 φ sin 2 α − ∞ E σx 1
[
]
n (1 + k + n) sin φ cos φ sin α cos α}
with
k9
k=
k10
Eα = E1
K11
n=
E1 E2
1 sin 4 α +
ö E1 1æ E cos 4 α + ç 1 − 2 ν12 ÷ sin 2 2α E2 4 è G12 ø
ö æE E 2 çç 1 − ν 12 ÷÷ + 1 ø G12 è E2
where E1, E2, G12 and ν12 are the mechanical properties of the laminate in the orthotropic coordinate system (o, 1, 2).
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MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - HOLE WITHOUT FASTENER Theory - Second method
K
3.2 3/3
If the laminate lay-up is equilibrated, the expression is simplified and becomes: K ∞T =
σ t (α ) σ ∞x
=
Eα − k cos 2 α + (1 + n) sin 2 α E1
{
}
If the laminate lay-up is nearly-isotropic, the expression is reduced to: K ∞T =
σ t (α ) σ ∞x
= − cos 2 α + 3 sin 2 α
For a nearly-isotropic lay-up and uniaxial loading, hole coefficients for 0°, 45°, 135° and 90° fibre directions are thus: 3, 1, 1 and - 1.
y
1 1 1 1
3
K∞ T
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σ∞ x
1 -1
x
Composite stress manual
MONOLITHIC PLATE - HOLE WITHOUT FASTENER Theory - Third method
K
3.3
3.3 . Third method (isotropic plate theory) If the material is isotropic (or nearly-isotropic) and if the plate is infinitely large, then the stress tensor may be formulated for any point P (identified by its coordinates r and α) on the plate as follows:
k12
σr =
σ ∞x 2
σt =
σ ∞x æ 3 R4 ö R2 ö σ ∞ æ ç 1 + 2 ÷ − x ç 1 + 4 ÷ cos 2α 2 è 2 è r ø r ø
τrt = −
æ 3 R4 R 2 ö σ ∞x æ R2 ö 1 1 4 − + + − ÷ cos 2α ç ÷ ç 2 è r2 ø r4 r2 ø è
σ ∞x æ 3 R4 2 R2 ö ç 1 − 4 + 2 ÷ sin 2α 2 è r r ø
y
t r P
r
σ∞ x
α x ∅ = 2R
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Composite stress manual
MONOLITHIC PLATE - HOLE WITHOUT FASTENER Theory - Fourth method
K
3.4 1/2
3.4 . Fourth method (empirical) This method is simple, fast but conservative. For more details, refer to chapter L (MONOLITHIC PLATE - FASTENER HOLE) by considering the bearing load as zero. Let a plate of length L be subjected to stress triplet σ ∞x , σ ∞y , and τ ∞xy . σ∞ y
y τ∞ xy
L
x
σ∞ x
∅
The first step consists in calculating the principal stresses σ p∞ and σ p'∞ and in weighting them with the net cross-section coefficient
L . L−∅
Thus, the main net stresses σ Np and σ Np' are obtained. Both stresses are then divided by coefficients Kt (K ct or K tt for direction p) and K't (K' ct or K' tt for direction p').
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Composite stress manual
MONOLITHIC PLATE - HOLE WITHOUT FASTENER Theory - Fourth method
K
3.4 2/2
These coefficients (smaller than 1) are a function of the material, the elasticity moduli in the direction considered (p or p'), the hole diameter (∅) and the type of load (tension "t" or compression "c"). They are found in the form of graphs (for carbon T300/914 layer in particular) in chapter Z (sheets 3 and 4 T300/914). The two following final stresses are obtained : σ Fp =
σ Fp' =
σ Np Kt σ Np' K t'
Both stresses are expressed in the main coordinate system (o, p, p').
σN p'
p'
y σN /Kt p σN /K't p'
L
p ∅ x
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Composite stress manual
MONOLITHIC PLATE - HOLE WITHOUT FASTENER "Point stress" - Failure criterion
K
4.1 1/2
4 . ASSOCIATED FAILURE CRITERIA 4.1 . Failure criterion associated with the "point stress" method (Whitney and Nuismer) To determine the failure of a notched laminate, it is generally allowed (for composite materials) to search for edge stresses at a certain distance do from the hole edge. Indeed, edge distance stress release through microdamages causes them to be analyzed at the edge distance do in practice. This distance depends on the type of load of the fibre considered (compression or tension), on the hole diameter and on the material (see chapter Z sheets 9 and 10 for T300/914). At the composite material stress office of the Aerospatiale Design Office, one considers ("point stress" method) that there is a failure in the laminate when the longitudinal stress of the most highly loaded fibre (located at the edge distance do) tangent to the hole is greater than the longitudinal stress allowable for the fibre. k13
There is a failure if: σl (y = R + do) > Rl σl: longitudinal stress of the fibre tangent to the hole
fibre at 0°
fibre at 90° fi b
re
at
45
°
Rl: longitudinal stress allowable for the fibre
fib re at 13 5°
do
σl
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σl
Composite stress manual
MONOLITHIC PLATE - HOLE WITHOUT FASTENER "Point stress" - Failure criterion
K
4.1 2/2
For complex loads, there is a software (PSH2 on mx4) which automatically models a finite element mesh and finds loads in fibres that are tangent to the hole. Longitudinal stress analysis is performed in a circle of elements, its center of gravity being located at the hole edge distance do.
2 do
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Composite stress manual
MONOLITHIC PLATE - HOLE WITHOUT FASTENER "Average stress" - Failure criterion
K
4.2
4.2 . Failure criterion associated with the "average stress" method (Whitney and Nuismer) This method consists in determining the average stress average σx average (ao) between coordinate points (0, R) and (0, R + ao). It is assumed that the plate is infinitely large and the loading uniaxial. y σx average (ao)
(ao)
σ∞ x
x ∅ = 2R
Based on the previous theory (see K 3.1), the following may be formulated as: σx average (ao) =
1 ao
ò
R + ao
R
σ x (y) dy
After development, we obtain:
k14
σx average (ao) ≈ σ ∞x
k15
with: ξ =
© AEROSPATIALE - 1999
2 − ξ2 − ξ4 2 (1 − ξ)
R R + ao
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - HOLE WITHOUT FASTENER Empirical method - Failure criterion
K
4.3
It is possible to choose ao so that: σx average (ao) ≈ σx (R + do) This condition allows the "point stress" and the "average stress" method to become equivalent. The "average stress" method is rarely used at Aerospatiale, the same failure criterion as for the "point stress" method may be applied: one considers that there is a failure in the laminate when the longitudinal stress of the most highly loaded fibre tangent to the hole is greater than the longitudinal stress allowable for the fibre.
4.3 . Failure criterion associated with the empirical method After determining stresses σ Fp and σ Fp' , a smooth calculation must be performed (see chapter C) in order to assess longitudinal stresses in fibres tangent to the hole. B
The Hill's failure criterion shall be used to each single ply (see chapter G3). It may be noted that this method is relatively conservative because both coefficients Kt and K't are assessed for different points, each one being the most critical with relation to directions p and p'. On the other hand, coefficient Kt and K't values were determined only for diameters between ∅ 3.2 to ∅ 11.1. It is, therefore, necessary to use the theory for large diameters.
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Composite stress manual
HOLE WITHOUT FASTENER First example
K
5.1 1/4
5 . Example 5.1 . First example Let a T300/BSL914 (new) square laminate plate of width L = 120 mm be laid up as follows: 0°: 6 plies 45°: 4 plies 135°: 4 plies 90°: 2 plies In the coordinate system (o, x, y), it is subjected to the following loading: N ∞x = 10 daN/mm N ∞y = 0 daN/mm N ∞xy = 0 daN/mm The plate has a diameter hole ∅ = 40 mm. y
2 4 6 4
L = 120
x ∅ = 40
L = 120
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Nx = 10
Composite stress manual
HOLE WITHOUT FASTENER First example
K
5.1 2/4
Let's analyse, along the y-axis, the evolution of stress flux Nx (y). The mechanical properties of the laminate in the reference coordinate system are the following: Ex = E1 = 6256 daN/mm2 (62560 MPa) Ey = E2 = 3410 daN/mm2 (34100 MPa) Gxy= Gv12 = 1882 daN/mm2 (18820 MPa) νxy = ν12 = 0.4191 νyx = ν21 = 0.2285 The value of K ∞T is deduced as follows: {k2}
K ∞T = 1 +
æ 6256 ö 6256 − 0.4191÷ + 2ç = 3.28 ç 3410 ÷ 1882 è ø
This number represents the hole edge coefficient for the case of a plate of infinite width. Since the plate does not have an infinite width L = 120 mm, we are led to calculate the following number : {k3} 40 ö æ 2 + ç1 − ÷ 120 ø è β= 40 ö æ 3 ç1 − ÷ 120 ø è
© AEROSPATIALE - 1999
3
= 1.148
MTS 006 Iss. B
Composite stress manual
HOLE WITHOUT FASTENER
K
First example
5.1 3/4
We thus obtain the evolution of normal stress fluxes along the y-axis:
Nx (y) = 1.148
10 2
2 4 6 8 ö æ ö æ ç 2 + æç 20 ö÷ + 3 æç 20 ö÷ − (3.28 − 3) ç 5 æç 20 ö÷ − 7 æç 20 ö÷ ÷ ÷ ç ç ç ÷ ÷ ÷ ç ÷ ÷ ç ç è y ø è y ø è y ø è y ø ÷ø ø è è
2 4 8 æ æ æ 20 ö 6 æ 20 ö æ 20 ö æ 20 ö ö÷ ö÷ ç ç Nx (y) = 5.74 ç 2 + çç ÷÷ + 3 çç ÷÷ − 0.28 5 çç ÷÷ − 7 çç ÷÷ ÷ ç è y ø è y ø è y ø è y ø ÷ø ø è è
40 37.65 35
30
25
Nx (y)
20
15 12.32 10
10
5
0 0
10
20
30 y
40
50
60
And we obtain, at the plate edge (y = 60) a flux of 12.32 daN/mm and at the hole edge (y = 20) a flux of 37.65 daN/mm.
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Composite stress manual
HOLE WITHOUT FASTENER First example
K
5.1 4/4
If one determines the flux at a hole edge distance do = 1 mm (see do in tension for the T300/914), one gets: Nx (y = 20 + 1) = 32.47 daN/mm. A smooth plate calculation (chapter C) with this flux makes it possible to determine the longitudinal stress of the most highly loaded fibre (fibre at 0°): σl = 32.41 hb. On the other hand, as the allowable longitudinal tension stress of the same fibre is equal to Rl = 120 hb, based on the "point stress" failure criterion, we obtain: æ 120 ö − 1÷ 100 = 270 % Margin: ç è 32.41 ø
At a hole edge distance do = 1 mm (see tension do for fibre T300/914 in chapter Z), flux Nx is now only 32.47 daN/mm. A smooth plate calculation makes it possible to find that fibres with a 0° direction are subjected to a 32.41 hb longitudinal stress at this particular hole edge distance. The longitudinal tensile strength of fibre T300/914 being 120 hb, the targeted margin is thus: æ 120 ö Margin = 100 ç − 1÷ = 270 % 32 . 41 è ø
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
HOLE WITHOUT FASTENER
K
Second example
5.2 . Second example Let a T300/BSL914 (infinitely large) laminate plate be laid up as follows: 0°: 6 plies 45°: 4 plies 135°: 4 plies 90°: 2 plies In the coordinate system (o, x, y), it is subject to the following loading: N ∞x = 2.8 daN/mm N ∞y = - 7.8 daN/mm N ∞xy = 5.3 daN/mm The plate has a diameter hole ∅ = 40 mm.
Ny = - 7.8
y Nxy = 5.3
2 4 6 4 Nx = 2.8 x ∅ = 40
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5.2 1/5
Composite stress manual
HOLE WITHOUT FASTENER
K
Second example
5.2 2/5
Let's determine the normal stress fluxes of the hole edge at point P (fibre at 0° tangent to hole). To do this, we shall use the second method First of all, (in order to eliminate the shear flux), let's be positioned in the main coordinate system (o, p, p') which forms a 22.5° angle with the reference coordinate system (o, x, y). Stress fluxes then become N p∞ = 5 daN/mm, N p'∞ = - 10 daN/mm. Orthotropic axes (o, 1, 2) are coincident with the reference coordinate system (o, x, y). The plate and its loading may then be described as follows: Np' = - 10
p'
4
2 4 6
2
φ' = 112.5°
P
y
α = 90°
Np = 5 p
φ = 22.5° ∅ = 40 1 x
In the coordinate system (o, p, p'), the mechanical properties of the laminate are the following: Ep = 5800 daN/mm2 (58000 MPa) Ep' = 3749 daN/mm2 (37490 MPa) Gpp' = 1788 daN/mm2 (17880 MPa) νpp' = 0.3481 νp'p = 0.225
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MTS 006 Iss. B
Composite stress manual
HOLE WITHOUT FASTENER Second example
K
5.2 3/5
In the reference coordinate system (o, x, y) and in the orthotropic coordinate system (o, 1, 2), the laminate properties are the following: Ex = E1 = 6256 daN/mm2 (62560 MPa) Ey = E2 = 3410 daN/mm2 (34100 MPa) Gxy= G12 = 1882 daN/mm2 (18820 MPa) νxy = ν12 = 0.4191 νyx = ν21 = 0.2285 A first step shall consist in calculating the effect of the main flux N p∞ at point P as follows: We have: {k9}
k=
6256 = 1.354 3410
{k10} E 90° = E1
1 6256 1 æ 6256 ö sin 4 90° + cos 4 90° + ç − 2 x 0.4191÷sin 2 2 x 90° 3410 4 è 1882 ø
E90° 1 = =1 E1 1
{k11}
n=
© AEROSPATIALE - 1999
æ 6256 ö 6256 − 0.4191÷ + 2ç = 2.48 ç 3410 ÷ 1882 è ø
MTS 006 Iss. B
Composite stress manual
HOLE WITHOUT FASTENER
K
Second example
5.2 4/5
{k1} K ∞T =
σ t ( α = 90°) σ p∞
=
{
6256 ( − cos 2 22.5° + (1.354 + 2.48 ) sin 2 45°) 1.354 cos 2 90° + 6256
((1 + 2.48 ) cos 2 22.5° − 1.354 sin 2 22.5°) sin 2 90° − 2.48 (1 + 1.354 + 2.48) sin 22.5° cos 22.5°
sin 90° cos 90°}
K ∞T =
σ t (α = 90°) σ p∞
= 2.773 p'
4
2
2
4 6
y
2.773 Np = 5
P p
1 x
A second step shall consist in calculating the effect of the main flux N p'∞ at point P. {k1} K' ∞T =
σ t ( α = 90°) σ p∞'
=
{
6256 ( − cos 2 112.5° + (1.354 + 2.48 ) sin 2 45°) 1.354 cos 2 90° + 6256
((1 + 2.48 ) cos 2 112.5° − 1.354 sin 2 112.5°) sin 2 90° − 2.48 (1 + 1.354 + 2.48 ) sin 112.5° cos 112.5°
sin 90° cos 90°}
K' ∞T =
© AEROSPATIALE - 1999
σ t (α = 90°) σ p∞'
= − 0.646
MTS 006 Iss. B
Composite stress manual
HOLE WITHOUT FASTENER
K
Second example
5.2 5/5
Np' = - 10
p'
4
2
2
4 6
y
- 0.646 P p
1 x
The deduction is that the normal stress flux tangent to the hole crossing point P is equal to: Nt (P) = 2.773 N p∞ + (- 0.646) N p'∞ = 2.773 x 5 + (- 0.646) x (- 10) = 20.31 daN/mm
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MONOLITHIC PLATE - HOLE WITHOUT FASTENER References
K
BARRAU - LAROZE, Design of composite material structures, 1987 GAY, Composite materials, 1991 VALLAT, Résistance des matériaux S.C. TAN, Finite width correction factors for anisotropic plate containing a central opening, 1988
B
J. Rocker, Composite material parts: Design methods at fastener holes 3 ≤ φ ≤ 100 mm. Extrapolation to damage tolerance evaluation, 1998, 581.0162/98 W.L. KO, Stress concentration around a small circular hole in a composite plate, 1985, NSA TM 86038 WHITNEY - NUISMER, Uniaxial failure of composite laminates containing stress concentration, American Society for testing materials STP 593, 1975 ERICKSON - DURELLI, Stress distribution around a circular hole in square plate, loaded uniformly in the plane, on two opposite sides of the square, Journal of applied mechanics, vol. 48, 1981
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L MONOLITHIC PLATE - FASTENER HOLE
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MONOLITHIC PLATE - FASTENER HOLE Notations
L
1
1 . NOTATIONS (o, x, y): initial coordinate system (o, M, M'): coordinate system specific to the bearing load (o, P, P'): stress main coordinate system F: bearing load ∅: fastener diameter Sf: countersink surface of fastener e: actual thickness of laminate e*: thickness taken into account in bearing calculations p: fastener pitch σ Nt : net cross-section stress at the hole σm: bearing stress σR: allowable stress of material (general designation) σxa: allowable normal stress of material in direction x σya: allowable normal stress of material in direction y τxya: allowable shear stress of material τvisa: allowable shear stress of screw N Bx N By N
gross fluxes in panel
B xy
N Nx N Ny N
net cross-section fluxes
N xy
N NM N NM'
net cross-section fluxes in the coordinate system specific to the bearing load
N NMM' Nm M
additional flux due to the bearing load
β: bearing load angle with relation to the initial coordinate system
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Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE Notations
L
NPN
net cross-section global fluxes in the main coordinate system NPN'
α: main coordinate system angle with relation to the bearing load N Fx N Fy N
corrected final fluxes
F xy
K mc : compression bearing coefficient K mt : tension bearing coefficient Km : bearing coefficient in the broad meaning of the term K ct : compression hole coefficient K tt : tension hole coefficient Kt: hole coefficient in the broad meaning of the term Kf: bending hole coefficient
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1 2/2
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE
L
General - Failure modes
2.1 2.2
2 . GENERAL/FAILURE MODES The purpose of this chapter is to assess the structural strength of a notched and loaded laminate fitted with fastener. Depending on the loading level and the type of geometry, such a system may fail as per several failure modes.
2.1 . Bearing failure F ≥ σRm ∅e
e
∅
F
∅
F
2.2 . Net cross-section failure F ≥ σxa (b − ∅) e
b
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MTS 006 Iss. B
∅
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE
2.3 2.4 2.5
L
General - Failure modes
2.3 . Plane shear failure F ≥ τxya 2 (L − 0,35 ∅) e
e
L
45°
F
2.4 . Cleavage failure F ≥ σya ∅ö æ çL − ÷ e è 2ø
e
L
∅
∅
F
2.5 . Cleavage: net cross-section failure σxa (b - ∅) + τxya L ≤
2F e e
L
∅
b
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MTS 006 Iss. B
F
∅
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE
L
Failure modes - Pitch definition
2.6 3.1
2.6 . Fastener shear failure 4F ≥ τvisa π ∅2 e
∅
F
∅
where: σxa is the allowable normal stress of the notched material in direction x σya is the allowable normal stress of the notched material in direction y τxya is the allowable shear stress of the notched material σRm is the allowable bearing stress of the material τvisa is the allowable shear stress of the screw
3 . SINGLE HOLE WITH FASTENER The purpose of this sub-chapter is to outline the justification method of a hole with a fastener to which is applied a bearing load in any direction, the laminate being subjected to membrane type surrounding load fluxes and/or bending moment fluxes. The failure mode associated with this method is a combined net cross-section failure mode in the presence of bearing (see 2.1 and 2.2).
3.1 . Pitch p definition If the main loading is in the F1 direction, the pitch taken into account in the calculations p1 + p2 . shall be equal to: p = 2
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Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE Membrane analysis - Short cut method - Theory
L
3.2.1 1/8
If the main loading is direction F2, the pitch (which is more commonly called edge distance) taken into account in the designs shall be equal to: p = 2 p3. For complex loading (or for simplification purposes), the following pitch value may be used: p = mini (p1; p2; 2 p3). It should be noted that for membrane or membrane and bending loading, pitch p is limited to k ∅ where k depends on the material used. The value of k is generally between 4.5 and 5. For pure bending loading, this limitation does not apply.
F1
p1
p2
p3 F2
p=
p1 + p2 2
3.2 . Membrane analysis - Short cut method 3.2.1 . Theory Generally speaking, a failure is reached at a fastener hole when: l1
σ Nt + Km σm ≥ Kt σR In the case of a membrane loaded single hole with fastener, the various justification (broadly summed up by relationship I1) steps must be followed:
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Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE
L
Membrane analysis - Short cut method - Theory
3.2.1 2/8
1st step: For load introduction zones (fittings, splices), the membrane gross flux NB to be taken into account at fasteners is deduced from the constant flux to infinity N∞ by the following relationship: l2
NB =
p N∞ if p > 5 ∅ 5∅
NB = N∞ if p ≤ 5 ∅
If the zone to be justified is a typical zone (ribs, spars), then: NB = N∞
N∞ B
N
5Ø Flux
The drawing above shows the difference between the flux to infinity and the actual flux at fasteners for a load introduction zone and highlights the existence of a working strip at each fastener of a width equivalent to 5 Ø. This phenomenon is comparable to the one described in chapter M.1.
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MONOLITHIC PLATE - FASTENER HOLE Membrane analysis - Short cut method - Theory
L
3.2.1 3/8
2nd step: It consists in transforming pitch corrected gross fluxes (see previous step) into net cross-section flux in the initial coordinate system:
y F β0 I I I I I I I
Note the bearing load direction (β = - 30°).
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Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE
L
Bending analysis
3.3 1/5
3.3 . Bending analysis - Short cut method If the notched plate is subjected to bending moment fluxes Mx, My and Mxy, follow the additional steps described hereafter:
1st step: Determine stresses on the external and internal surfaces corresponding to bending loads only. As a first approximation, these stresses may be assessed by the general relationship Mv 6M ≈ 2 . In that case, the material shall be considered as homogeneous. σ≈ l e external surface
internal surface
σ Be
σB l
It is nevertheless recommended to determine these stresses with the computing software PSD48 (stacking homogenizing and analysis) which takes into account stiffness variations within the laminate or to refer to chapter D. external surface
internal surface
σ Be
σB l
Thus, for each design direction (x, y and xy), the following stresses are obtained: σe Bx , σe By , τe Bxy : gross stresses on external surface. σi Bx , σi By , τi Bxy : gross stresses on internal surface.
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE Bending analysis
L
3.3 2/5
2nd step: From these stresses, "equivalent" membrane gross fluxes are evaluated. ∆neBx
σ eBx
∆neBy
σ eBy
∆neBxy
τ eBxy
l10
for external surface
=e ∆niBx
σiBx
∆niBy
σiBy
∆niBxy
τ iBxy
for int ernal surface
B
∆n e
external surface e
B
∆n i
internal surface
3rd step: On the contrary of membrane analysis, no majoration between fluxes to infinity N∞ and gross fluxes NB will be taken into account at load introduction areas. NB = N∞ 4th step: "Equivalent" membrane net fluxes are evaluated from "equivalent" membrane gross fluxes. l11
∆ne Nx = ∆ne Bx
∆ne Ny = ∆ne By
∆ne Nxy
© AEROSPATIALE - 1999
p p−∅−
Sf e
p
Sf p−∅− e p = ∆ne Bxy Sf p−∅− e
for external surface (with countersunk fastener head)
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE Bending analysis
p p−∅ p ∆ni Ny = ∆ni By p−∅ p ∆ni Nxy = ∆ni Bxy p−∅
L
3.3 3/5
∆ni Nx = ∆ni Bx
for internal surface (no countersunk fastener head)
Confer to sub-chapter L.3.1 to determine fastener pitch. N
∆n e
external surface
N
internal surface
∆n i
5th step: "Equivalent" membrane net fluxes are divided by the coefficient Kf (bending hole coefficient) which depends on the material (in general Kf = 0.9). Hence, we get the (majorated) "equivalent" membrane net fluxes:
l12
∆ne Fx = ∆ne Fy
=
∆ne Fxy =
∆ni Fx = ∆ni Fy ∆ni Fxy
© AEROSPATIALE - 1999
= =
∆n e Nx Kf ∆n e Ny Kf
for external surface
∆n e Nxy Kf ∆ni Nx Kf ∆niNy Kf
for internal surface
∆n i Nxy Kf
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE Bending analysis
L
3.3 4/5
6th step: Final membrane fluxes from relation I9 are, then, added to fluxes calculated from relation I12. l13
N Fx + ∆ne Fx N Fy + ∆ne Fy
for external surface (without bearing)
N Fxy + ∆ne Fxy
N Fx + ∆ni Fx N Fy + ∆ni Fy
for internal surface (with bearing)
N Fxy + ∆ni Fxy
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MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE
L
Membrane + bending analysis - Summary table
3.3 5/5
The overall method for the membrane and bending analysis is summarized in the figure here below. External surface Membrane Bending Data in the initial coordinate system
Nx
B
∆n x
B
∆n y
B
∆n yx
B
p
B
Nx =Nx
p−∅−
Net crosssection analysis
N
Ny =Ny
p−∅−
N
N
Sf
Sf
N xy = N xy p−∅−
p p−∅−
∆n y = ∆n y
p−∅−
N
N
Sf
Sf
p−∅−
e
p−∅−
Sf
Sf
N
p−∅−
p−∅−
e
N
N
N M'
Sf
α F
N
NP
p
−
N
∅
Hole coefficient maximizing
Kt N
NP'
Kt
Kt
α
∆n y
N
F
∆n xy =
∆n xy
↓
N
β N
m
N M ± Km N M N
↓
N M' N
N
NP
↓
N
N P'
N P'
α-β
α-β N
F
N
NP
NP
Kt
Kt
N
N
NP'
NP'
Kt
Kt
α-β
α-β
∆n x =
∆n x Kt
N
F
∆n y =
∆n y Kt
N
F
∆n xy =
∆n xy
Rotation in the initial coordinate system
Nx
Nx
↓
F
Ny
F
F
N xy
Nx
↓
F
Ny
F
F
N xy
F
F
N x + ∆n x
F
F
N y + ∆n y
F
N xy + ∆n xy
N y + ∆n y F
N xy + ∆n xy
© AEROSPATIALE - 1999
F
Ny
N xy N x + ∆n x
Addition of fluxes
Kt F
MTS 006 Iss. B
∅
p p
N MM'
Kt F
−
B
N
β
N
N
F
∆n y =
p
∅
p
∆n xy = ∆n xy
N M'
∆n x Kt
−
B
∆n y = ∆n y
N
∆n x =
p
e
β
N
N
∅
p
N MM'
NP
↓
p
N MM' N
N
−
p
B
∆n x = ∆n x
N
m
N
N P'
p
N
∅
NM
N M ± Km N M
NP
−
B
N
N
p p
N xy = N xy
β
Rotation in the main coordinate system
B
e
N
↓
∆n yx
B
B
N MM'
↓
B
Ny =Ny
N M'
N
Addition of bearing
∆n y
Sf
p
B
N xy = N xy
↓
B
B
NM
↓
B
Nx =Nx
N
Rotation in the load coordinate system
Ny
e
N
Ny =Ny
N
∆n x
p
B
e
p
B
∆n xy = ∆n xy
B
p
B
N
Nx
Ny
Nx =Nx
e
p
B
e
Sf
B
Ny
B
N
p
B
B
Ny
∆n x = ∆n x
e
p
B
B
B
Ny
Internal surface Membrane Bending
Nx
B
Ny
N
Neutral line Membrane
F
F
F
F
F
F
−
∅
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE Justifications - Nominal deviations
L
3.4 3.5.1
3.4 . Justifications Whatever the type of load (membrane or membrane + bending), make sure that: - the plain monolithic plate subject to "equivalent" membrane load fluxes (NF + ∆neF) or (NF + ∆niF) is acceptable from a structural strength point of view (refer to chapter C), - the allowable bearing stress of material σm (which depends on the material, the fastener diameter and the thickness to be clamped - see chapter Z) is greater or equal to the bearing stress applied corresponding to a laminate thickness that is smaller or equal to 1.3 ∅ for single shear or 2.6 ∅ for double shear (see sub-chapter L.2.1 - 4th step):
3.5 . Nominal deviations on a single hole This sub-chapter is directly related to concession processing. Here, simple rules are outlined, that shall allow the stressman to assess the effect of a geometrical deviation, such as a fastener diameter, its pitch or edge distance, on an initial margin. The following paragraphs are valid only for a hole with fastener subject to membrane fluxes.
B
However, for greater accuracy, it is recommended to redo the calculation or use the software psg33.
3.5.1 . Changing to a larger diameter Following a drilling fault, it is sometimes necessary to change to a repair size or to oversize the fastener.
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE
L
Nominal deviations - Pitch decrease
3.5.2
Based on the theory we have just presented, any diameter change (∅ changes to ∅') shall have an effect on: - the net cross-section coefficient: the resulting reduction shall be equal to: Sf' e k= Sf p−∅− e p − ∅' −
l14
- the bearing stress: we shall assume that there is no effect on the bearing stress, even if it tends to decrease (this assumption is conservative), - the hole coefficient: if we assume that the hole coefficient value is in the most unfavorable case Kt = 0.003684 ∅2 - 0.08806 ∅ + 0.886 (see corresponding curve in chapter Z - material T300/914), the resulting reduction shall be equal to:
k' =
0.0037 ∅' 2 − 0.088 ∅' + 0.89 0.0037 ∅ − 0.088 ∅ + 0.89 2
≈
∅ ∅'
Thus, the general relationship may be given as follows:
l15
RF' ≈ RF k k' ≈ RF
Sf' p − ∅' − ∅ e ∅' p − ∅ − Sf e
3.5.2 . Pitch decrease If loads are parallel to the free edge, no reduction is necessary on the reserve factor: RF' ≈ RF F2
p' p © AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE Nominal deviations - Edge distance decrease
L
3.5.3 1/2
If loads are perpendicular to the free edge, the reduction on the reserve factor is equal to: Sf e RF' ≈ RF Sf p−∅− e p' − ∅ −
l16
F1
p' p
3.5.3 . Edge distance decrease If loads are parallel to the free edge, the reduction on the reserve factor is equal to: Sf e RF' ≈ RF Sf p−∅− e p' − ∅ −
p
p' F2
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE Nominal deviations - Edge distance decrease
L
3.5.3 2/2
If loads are perpendicular to the free edge, the reduction on the reserve factor is equal to:
l17
æ p' ö RF' ≈ RF çç ÷÷ èpø
0.54
æ p' ö RF' ≈ RF çç ÷÷ èpø
0.73
æ p' ö RF' ≈ RF çç ÷÷ èpø
1.65
→ (100 % ± 45°)
→ (50 % 0; 50 % ± 45°)
→ isotrope
F1 p
p'
Important remarks: - These empirical relationships are valid only for low edge distance variations (2 ∅ ≤ p' ≤ 2.5 ∅). - For low edge distances, the fact that the failure mode described in sub-chapter L.2.3 is not critical shall have to be demonstrated.
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE "Point stress" finite element method - Description
L
3.6.1
3.6 . "Point stress" finite element method (membrane analysis) 3.6.1 . Description of the method Procedure PSH2 allows the calculation of stresses in fibres around a circular hole with fastener in a multilayer composite plate subjected to membrane type surrounding fluxes. It is based on a finite element display of a drilled plate. Mapping calls for two separate parts: - the bolt (rivet/screw/bolt), - the drilled plate. The drilled hole is modeled by 8-junction quadrangular elements and 6-junction triangular elements. The area adjacent to the hole is modeled by two rings of elements. The ring nearest to the hole is thin and is not utilized directly on issued sheets. Issues are presented on the second ring, the center of gravity of elements being at a design distance from the hole corresponding to the point stress theory (do). 2 do
Contact elements between the plate and the bolt (which also simulate clearance between the fastener and the edge distance) are of the variable stiffness type. Their stiffness is very low when there is no contact with the plate, their stiffness is very high if there is a contact. Loading is achieved by (normal and shear) fluxes on plate edges.
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE "Point stress" finite element method - Justifications
L
3.6.2
3.6.2 . Justifications Make sure that: - longitudinal stresses in fibres tangent to the hole edge distance (and located at a
Fibre at 0°
Fibre at 90° Fib re
at 45 °
distance do) are smaller than the longitudinal stress allowable for fibre Rl,
Fi br e
at 13 5°
do
σl
σl
- The allowable bearing stress of the material σm is greater or equal to the bearing stress applied corresponding to a laminate thickness that is smaller or equal to 1.3 ∅ for single shear or 2.6 ∅ for double shear (see sub-chapter L.2.1 - 3rd step).
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE Multiple holes - Independent holes - Interfering holes
L
4.1 4.2
4 . MULTIPLE HOLES The previous study allowed us to find the structural effect of a single hole with fastener (or distant enough from others) on a monolithic plate subject to membrane or bending type loads. We shall now study the effect of several lined up holes. We shall assume that the plate is subjected to a membrane type uniaxial load flux that is perpendicular to the row of fasteners. If loading is parallel to the row of fasteners, refer to chapter L.3.4.2 calculation.
4.1 . Independent holes If each fastener pitch is greater of equal to 5 ∅, each fastener may be considered as a single hole. Refer to sub-chapter L.3.
5∅
pas = 5 ∅
5∅
5∅
4.2 . Interfering holes (0 < d < 3.5 ∅) If the distance between two holes is smaller than 5 ∅, the net cross-section coefficient to be used changes to: 5∅−d
l18
5∅−d−∅−
© AEROSPATIALE - 1999
Sf e
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE Multiple holes - Very close holes
L
4.3 1/2
On the other hand, the hole coefficient in tension must also be modified. It changes to:
l19 B
2 æ ö æ 5 ∅ − dö æ5 ∅ − dö η Kt ≈ ç 0.065 ç ÷ − 0.65 ç ÷ + 2.625 ÷ k t (see values of η on next page) ç ÷ è ∅ ø è ∅ ø è ø
The hole coefficient in compression is unchanged (cf. note 440.197/84), but the connection of the holes is ignored for the net section calculation. These new values are to be taken into account in relationships l3 and l8. pas = 5 ∅ - d
d
4.3 . Very close holes (d = 3.5 ∅ soit p = 1.5 ∅) When holes are very close to each other, the diameter ∅' envelope hole shall be considered. The net cross-section coefficient then changes to: pitch
l20
pitch − ∅' −
Sf e
The hole coefficient is not modified by the number η but applies to diameter ∅'. pitch = 4.25 ∅ ∅' 1.5 ∅ d
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE
L
Multiple holes - Kt correction coefficient
4.3 2/2
Kt correction coefficient
E
2
P
P
1.9
O
1.8
L
1.7
V
E
1.6
1.5 N
η
E
1.4
U
1.3
R
O
1.2
T
1.1
1 1
1.5
2
2.5
3
3.5
4
4.5
5
5∅−d ∅ pitch = 5 ∅ - d
pitch ∅' 1.5 ∅
d
d
© AEROSPATIALE - 1999
MTS 006 Iss. B
5∅
pitch = 5∅
5∅
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE
L
First example
5.1 1/7
5.1 . First example Let a T300/BSL914 (new) laminate be laid up as follows: 0°: 6 plies 45°: 4 plies 135°: 4 plies 90°: 6 plies Total thickness: e = 20 x 0.13 = 2.6 mm It is subjected to the three following fluxes in the initial coordinate system (o; x; y): N Bx = 8 daN/mm N By = - 6 daN/mm N Bxy = 20 daN/mm and to the bearing load: F = 185 daN β = - 30° The fastener is a ∅ 4.8 mm countersunk head one (100° countersink angle, which corresponds to a 4.91 mm2). The fastener pitch is 21.6 mm. y 6 4
F = 185 daN 6 β = - 30°
4
x
The purpose of the example is to determine the three final fluxes that shall be used for the equivalent smooth plate design, which shall provide the hole margin looked for (this calculation shall be covered in chapter C).
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE First example
L
5.1 2/7
Design of net cross-section fluxes in the initial coordinate system: {l3} NNx = 8 x
21.6 4.91 21.6 − 4.8 − 2.6
NNy = ( − 6) x
NNxy = 20 x
= 11.59 daN / mm
21.6 21.6 − 4.8 −
4.91 2.6
21.6 4.91 21.6 − 4.8 − 2.6
= − 8.69 daN / mm
= 28.97 daN / mm
Flux transfer in the bearing coordinate system (o, M, M'): {l4} N NM = 31.61 daN/mm N NM' = - 28.71 daN/mm N NMM' = 5.7 daN/mm β = - 30° Bearing flux addition: 3 cases shall be considered The bearing stress is equal to: σm =
185 = 14.82 hb 4. 8 x 2. 6
{l5} Nm M = 14.82 x 2.6 = 38.54 daN/mm The flux in the load direction being a tension flux (+ 31.61 daN/mm), the value of K mt is thus equal to 0.135 (see chapter Z - material T300/914 - Sheet 2).
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE First example
L
5.1 3/7
If the bearing flux minimized by coefficient K mt , is added to previously determined fluxes, the three configurations K mt > 0; K mt < 0 and K mt = 0 are obtained: {l6} K mt = 0.135 N NM = 31.61 + 0.135 x 38.54 = 36.81 daN/mm N NM' = - 28.71 daN/mm N NMM' = 5.7 daN/mm β = - 30° {l6} K mt = - 0.135 N NM = 31.61 - 0.135 x 38.54 = 26.4 daN/mm N NM' = - 28.71 daN/mm N NMM' = 5.7 daN/mm β = - 30° {l6} K mt = 0 N NM = 31.61 + 0 = 31.61 daN/mm N NM' = - 28.71 daN/mm N NMM' = 5.7 daN/mm β = - 30°
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE
L
First example
5.1 4/7
Rotation in the main coordinate system (o ; P ; P'): {l7} N NP = 37.3 daN/mm N PN' = - 29.2 daN/mm α = 4.9° P
y
M α = 4.9°
β = - 30° x N
N P = 37.3 daN/mm N
N P' = - 29.2 daN/mm
{l7} N NP = 26.98 daN/mm N PN' = - 29.29 daN/mm α = 5.8° {l7} N NP = 32.14 daN/mm N PN' = - 29.24 daN/mm α = 5.4° Angle α is the angle formed by the main coordinate system and the bearing coordinate system.
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE
L
First example
5.1 5/7
Application of hole coefficients: Monolithic lay-up under study gives the following elasticity and shear moduli in the main axes: α + β = - 34.9°
E = 4470
G = 2078
α + β = - 35.8°
E = 4455
G = 2092
α + β = - 35.4°
E = 4461
G = 2086
E = 2.151 G E = 2.13 G E = 2.139 G
K tt ≈ 0.6
K ct ≈ 0.87
K tt ≈ 0.6
K ct ≈ 0.87
K tt ≈ 0.6
K ct ≈ 0.87
The values are derived from chapter Z (T300/914 sheets 3 and 4). Which gives the following new values for corrected main fluxes : {l8} 37.3 = 62.17 daN/mm 0.6 − 29.2 N PN' = = - 33.56 daN/mm 0.87 α - β = 34.9°
N NP =
{l8} 26.98 = 44.97 daN/mm 0.6 − 29.29 N PN' = = - 33.67 daN/mm 0.87 α - β = 35.8°
N NP =
{l8} 32.14 = 53.57 daN/mm 0 .6 − 29.24 = - 33.61 daN/mm N PN' = 0.87 α - β = 35.4°
N NP =
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE First example
L
5.1 6/7
Rotation in the initial coordinate system (o; x; y): A rotation of angle (β - α) is achieved: {l9} N Fx = 30.83 daN/mm N Fy = - 2.22 daN/mm N Fxy = 44.92 daN/mm y
N Fxy = 44.92 daN/mm
N Fx = 30.83 daN/mm
x
N Fy = - 2.22 daN/mm
{l9} N Fx = 18.06 daN/mm N Fy = - 6.76 daN/mm N Fxy = 37.31 daN/mm {l9} N Fx = 24.32 daN/mm N Fy = - 4.36 daN/mm N Fxy = 41.17 daN/mm These fluxes are then used in a smooth plate design. Calculation shall be continued in chapter C.6. A 31 % (RF = 1.31) margin shall be found.
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE First example
L
5.1 7/7
New, let's assume that, as a result of a defective drilling operation, the fastener diameter had to be changed to a ∅ 6.35 mm with a 8.62 mm2 countersunk surface. What would be the new margin? {l15}
RF' = 1.31 x
B
4.8 x 6.35
8.62 2.6 = 0.92 4.91 21.6 − 4.8 − 2.6
21.6 − 6.35 −
Which corresponds to a - 8 % margin, thus non allowable. However, a full manual analysis (or using software PSG33) would have made it possible to find a 0 % margin. If the calculation is conservative, it is due to the fact that the decrease of the bearing stress corresponding to fastener oversizing was not taken into account (see chapter L.3.5.1). The preceding example shall also be fully covered in the composite material manual part
B
"Calculation programs" (PSG33 instructions).
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE Second example
L
5.2 1/5
5.2 . Second example Let's assume that three moment fluxes are superposed on membrane fluxes: Mt Bx = - 4 daN mm/mm B y B Mt xy
Mt
= 3 daN mm/mm = 5 daN mm/mm z
y β = - 30°
F = 185 daN Mt By = 3 daN
x
Mt Bx = - 4 daN
Mt Bxy = 5 daN
If the material is considered (as a first approximation) as homogeneous, a strength l 2 .6 2 moment per unit of length equal to: = = 1.127 mm2 is found. v 6 Assuming that a positive moment flux creates compression stresses on the external surface, we obtain: for the external surface: σe Bx =
4 = 3.55 hb (35.5 MPa) 1.127
σe By =
−3 = - 2.66 hb (- 26.6 MPa) 1.127
τe Bxy =
−5 = - 4.44 hb (- 44.4 MPa) 1.127
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE Second example
L
5.2 2/5
For the internal surface: σi Bx =
−4 = - 3.55 hb (- 35.5 MPa) 1.127
σi By =
3 = 2.66 hb (26.6 MPa) 1.127
τi Bxy =
4 = 4.44 hb (44.4 MPa) 1.127 EXTERNAL SURFACE - 4.44 hb 3.55 hb
y x
- 2.66 hb 4.44 hb - 3.55 hb
2.66 hb INTERNAL SURFACE
The purpose of this example is to determine which bending type fluxes must be added to membrane type fluxes for the fastener hole calculation. The "equivalent" gross bending type fluxes necessary for the calculations thus have the following value: {l10} for the external skin: ∆ne Bx = 3.55 x 2.6 = 9.23 daN/mm ∆ne By = - 2.66 x 2.6 = - 6.92 daN/mm ∆ne Bxy = - 4.44 x 2.6 = - 11.54 daN/mm
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE Second example
L
for the internal skin: ∆ni Bx = - 3.55 x 2.6 = - 9.23 daN/mm ∆ni By = 2.66 x 2.6 = 6.92 daN/mm ∆ni Bxy = 4.44 x 2.6 = 11.54 daN/mm The "equivalent" net bending type fluxes thus have the following value: {l11} for the external skin: 21.6
∆ne Nx = 9.23
21.6 − 4.8 −
∆ne Ny = - 6.92
4.91 2.6
= 13.37 daN/mm
21.6 4.91 21.6 − 4.8 − 2.6
∆ne Nxy = - 11.54
= - 10.02 daN/mm
21.6 4.91 21.6 −4.8 − 2.6
= - 16.72 daN/mm
for the internal skin: ∆ni Nx = - 9.23
∆ni Ny = 6.92
21.6 = 8.9 daN/mm 21.6 − 4.8
∆ni Nxy = 11.54
© AEROSPATIALE - 1999
21.6 = - 11.87 daN/mm 21.6 − 4.8
21.6 = 14.84 daN/mm 21.6 − 4.8
MTS 006 Iss. B
5.2 3/5
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE Second example
L
5.2 4/5
Hole coefficient weighting {l12} for the external skin: ∆ne Fx =
13.37 = 14.86 daN/mm 0. 9
∆ne Fy =
− 10.02 = - 11.13 daN/mm 0 .9
∆ne Fxy =
− 16.72 = - 18.58 daN/mm 0 .9
for the internal skin: ∆ni Fx =
− 11.87 = - 13.19 daN/mm 0 .9
∆ni Fy =
8. 9 = 9.89 daN/mm 0. 9
∆ni Fxy =
14.84 = 16.49 daN/mm 0. 9
All prior calculations were made in the initial coordinate system (o; x; y). These "equivalent" bending type fluxes are thus to be added to the membrane type fluxes found in the first example (see summary table on next page).
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE
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Second example
5.2 5/5
This table summarizes the various steps of "equivalent" membrane flux calculation of the previous example. External surface Membrane Bending
Neutral line Membrane
Internal surface Membrane Bending
Data in the initial coordinate system
8 -6 20
9.23 - 6.92 - 11.54
8 -6 20
8 -6 20
- 9.23 6.92 11.54
Net crosssection design
11.59 - 8.69 28.97
13.37 - 10.02 - 16.72
11.59 - 8.69 28.97
10.29 - 7.71 25.71
- 11.87 8.9 14.84
↓
31.61 - 28.71 5.7 - 30°
28.06 - 25.48 5.06 - 30°
↓
Rotation in the bearing load coordinate system
↓
↓
+ Km 36.81 - 28.71 5.7 - 30°
- Km 26.4 - 28.71 5.7 - 30°
Km = 0 31.61 - 28.71 5.7 - 30°
+ Km 33.26 - 25.48 5.06 - 30°
- Km 22.86 - 25.48 5.06 - 30°
Km = 0 28.06 - 25.48 5.06 - 30°
↓
32.14 - 29.24 35.4°
↓
37.3 - 29.2 34.9°
26.98 - 29.29 35.8°
32.14 -29.24 35.4°
33.69 - 25.91 34.9°
22.59 - 26.01 36°
28.53 - 25.95 35.4°
↓
Hole coefficient maximizing
53.57 - 33.61 35.4°
14.86 - 11.13 - 18.58
62.17 - 33.56 34.9°
44.97 -33.67 35.8°
53.57 - 33.61 35.4°
56.15 - 29.78 34.9°
37.65 - 29.90 36°
47.55 -29.83 35.4°
- 13.19 9.89 16.49
Rotation in the initial coordinate system
24.32 - 4.36 41.17
↓
30.83 - 2.22 44.92
18.06 - 6.76 37.31
24.32 - 4.36 41.17
27.7 - 0.87 38.82
14.35 - 5.68 30.81
21.43 - 3.03 35.12
↓
Addition of final fluxes
39.18 - 15.49 22.59
161 % marging
14.51 9.02 55.31
1.16 4.21 47.3
8.24 6.86 51.61
8 % marging
Addition of bearing flux
↓
Rotation in the main coordinate system
31 % marging
The minimum margin is the only one considered, i.e.: 31 % for membrane design 8 % for membrane + bending design
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
MONOLITHIC PLATE - FASTENER HOLE References
L
BARRAU - LAROZE, Design of composite material structures, 1987 GAY, Composite materials, 1991 VALLAT, Strength of materials LAFON, Carbon fibre structures: simplified rules for sizing at fastener holes, 1983, PL No. 139/83 BOUNIE, Failure criteria of mechanical bonds in composite materials, 1991, 440.181/91 LAFON, Justification of design methods used for carbon fibre structures - thin sheet subject area, 1983, 440.156/83
B
J. ROCKER, Composite material parts: design methods at fastener holes, 3 ≤ ∅ ≤ 100 mm. Extrapolation to damage tolerance evaluation, 1998, 581.0162/98 LAFON, TROPIS, Structural strength of outer wing - justification of design values, 1989, 440.233/89
B
LAFON - LACOSTE, Synthesis of drilled carbon specimen tests, 1984, 440.197/84 issue 2.
© AEROSPATIALE - 1999
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M MONOLITHIC PLATE - SPECIAL ANALYSIS
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Composite stress manual
N SANDWICH - MEMBRANE / BENDING / SHEAR ANALYSIS
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© AEROSPATIALE - 1999
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Composite stress manual
SANDWICH - MEMBRANE / BENDING / SHEAR Notations
1 . NOTATIONS Ny: normal load flux Mx: moment flux Mz: moment flux Tx: shear load flux Tz: shear load flux Emi: membrane elasticity modulus of lower skin Efi: bending elasticity modulus of lower skin Gi: shear modulus of lower skin ei: thickness of lower skin Emc: membrane elasticity modulus of core material Efc: bending elasticity modulus of core material Gc: shear modulus of core material ec: thickness of core material Ems: membrane elasticity modulus of upper skin Efs: bending elasticity modulus of upper skin Gs: shear modulus of upper skin es: thickness of upper skin zg: neutral axis position with respect to the lower skin Σ El: overall inertia of elasticity moduli weighted plate EW: elasticity moduli weighted static moment B
µd: microstrain (10-6 mm/mm)
© AEROSPATIALE - 1999
MTS 006 Iss. B
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1
Composite stress manual
SANDWICH - MEMBRANE / BENDING / SHEAR Specificity - Construction principle - Design principle
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2 3 4
2 . SPECIFICITY A sandwich is a three-phase structure consisting of a core generally made out of honeycomb or foam with a low elasticity modulus and two thin and stiff face sheets. Sandwich structures have a very high specific bending stiffness.
external face sheet adhesive bonding interface
core (honeycomb) internal face sheet
3 . CONSTRUCTION PRINCIPLE The face sheets and core are assembled by bonding with synthetic adhesives. There are several alternative manufacturing processes: - multiple phase process: face sheets are cured separately, then bonding of face sheets to the honeycomb is performed as a second operation, - semi-cocuring process: the external face sheet is cured separately, the honeycomb and the internal face sheet are then cocured on the external face sheet, - single phase or "cocuring" process: face sheets and the honeycomb are cured in one single operation.
4 . DESIGN PRINCIPLE The design rules that shall be developed are derived from the classical elasticity (refer to "distribution of load among several closely bound structural elements" in chapter A.7).
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
SANDWICH - MEMBRANE / BENDING / SHEAR Sandwich plates - Sandwich beams
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4.1 4.2 1/3
First of all, we shall consider that the three materials together are completely ordinary. Then, we shall simplify the relationships obtained by considering that face sheets are thin and stiff and that the sandwich core is thick and flexible.
4.1 . Sandwich plates Like monolithic metal or composite plates, sandwich plates are under the general plate equation (see § A.7.4). The determination of matrices (Aij), (Bij) and (Cij) which connect the strain tensor to the load tensor is described in chapters C, D and E.
4.2 . Short cut theory - "Sandwich" beams Here, we shall outline a short cut method applicable to sandwich beams. This method does not take into account transversal loading, transversal effects so-called "Poisson" effects and membrane-bending coupling. This simplification may lead to an error of approximately 10 % on results obtained in cases of complex loading. From the overall deformation point of view, sandwich plates obey the conventional equations of classical elasticity theory. Stiffness equivalences (with iso-cross-section) with homogeneous beams are described by relationships n14 to n18. Let a sandwich beam be made up of: - an upper skin of thickness es, of membrane elasticity modulus Ems and of equivalent bending elasticity modulus Efs, - a core thickness ec, of membrane elasticity modulus Emc and of equivalent bending elasticity modulus Efc, - a lower skin of thickness ei, of membrane elasticity modulus Emi and of equivalent bending elasticity modulus Efi.
© AEROSPATIALE - 1999
MTS 006 Iss. B
Composite stress manual
SANDWICH - MEMBRANE / BENDING / SHEAR
N
Sandwich beams
4.2 2/3
The bending modulus concept comes from the fact that lower and upper skins are generally (in the case of honeycomb sandwiches) laminates with different membrane and bending moduli (see chapters C and D). Its value depends on ply stacking. This concept was extended to all three materials. First of all, we shall develop the full sandwich beam theory while taking into account face sheet thickness and bending stiffnesses, then we shall outline at the end of each subchapter, the simplified relationships in which face sheets shall supposedly be thin and subject to membrane stress only. The neutral line of the sandwich beam is defined by dimension zg to that: Emi
n1
zg =
ei 2 e ö e ö æ æ + Emc ec ç ei + c ÷ + Ems e s ç ei + ec + s ÷ è è 2 2ø 2ø Emi ei + Emc ec + Ems es
Remark: In the case of a beam in which Emc ec
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