Advances in compositestructures design and simulationTRAINING MODULES
for Researchers
Volume 1
2013
National Aerospace University
“KhAI”
Advances in composite structuresdesign and simulation
Volume 1
TRAINING MODULES
for master and doctoral students
Prepared in the frame of the FP7 KhAI-ERA project activities
2013
National Aerospace University “KhAI”
Published by the National Aerospace University “KhAI” in 2013
Advances in composite structures design and simulation
This training modules collection was jointly prepared by the National Aerospace University “KhAI” andInstitute of Aerospace Engineering, Brno University of Technology in the frame of FP7 KhAI-ERA projecttraining development activities. It is intended for master and doctoral students.
This publication includes training modules elucidate the fundamentals of engineering design methods oftypical constructive elements made of composite materials. Each training module describes problemstatement, list of constraints, as well as analytical or numerical approaches for parameters calculation withexamples of realization.
Reproduction is authorised provided the source is acknowledged. No use of this publication may be madefor resale or for any other commercial purpose.
Available on-line at http://khai-era.khai.edu/en/site/training-modules.html
For enquiries, inputs and feedback on the use of this document please contact:
International S&T Projects Office
National Aerospace University “KhAI”
17Chkalova str., Kharkov, 61070, Ukraine
Phone: +38 (057) 788 40 60
Fax: +38 (057) 719 04 73
e-mail: [email protected]
LEGAL NOTICE
Neither the European Commission nor any person acting on behalf of the Commission is responsible for theuse, which might be made, of the following information.
The views expressed in this report are those of the authors and do not necessarily reflect those of theEuropean Commission.
© KhAI-ERA Consortium, 2013
Foreword
Composite materials are widely used in structural engineering, especially in aviation and rocketry. In theunanimous conviction of the leading research centers in the world the composites represent the mainalternative to traditional metal alloys in the way to improve the efficiency of technical objects. Compositeshave passed a typical way of the any new material introduction – from complete euphoria over their lowdensity and high mechanical properties as compared with metals (but only in certain directions that at firstwas often ignored) to a reasonable rationalism, sometimes bordering skepticism, which is connected withsome disadvantages of these materials (the anisotropy of physical and mechanical properties, fibrous orlayered nature type, a strong properties dependence on the construction manufacturing technology, lack ofproperties stability, the brittle nature of failure, lack of the interlaminar shear strength and bearingstrength, the abnormal values of Poisson's ratio and CTE, etc.). Any unique feature of the compositematerial can be an advantage, but can also become a disadvantage, depends on the skills of the engineer tofind a reasonable balance between the rather original properties of these materials. It is the principal aimof composite structures design. Changing the fiber type and matrix materials, fiber volume fraction andtype of lay-up allows controlling the mechanical properties of the composites in wide range. This is themain and most significant advantage of the composite materials. The ability to change the elastic andstrength properties due to laminate parameters variation (plies orientation, ply thickness fraction andstacking sequence) is an important factor in determining the stress field in any structure, because thedistribution of internal forces is directly dependent on the elastic moduli. That is besides actual regulationof material properties is also possible controlling of the stress-strain state. These are interrelatedprocedures, so the design of structural elements made of composite material unlike to metal alloys,necessarily includes the stage of laminate parameters optimization. Design of any construction begins withthe material selection and ends with calculation of design parameters (dimensions) which provideoperation of the object under entire spectrum of possible loadings (mechanical, thermal, acoustic, etc.).The essential advantage of the classical design schemes (rod, beam, plate and shell, etc.) as compared withmore accurate computational methods (for example the finite element method) is the possibility of cleardemonstration of stress distribution through the volume or section of the structure and rapid analyze ofthe results. Engineer always has to find a compromise between the desire to apply as accurate as possibledesign methods and necessity to analyze the results at all designing stages, which is more efficient byanalytical stress-strain field dependences on external loads magnitude and material properties. That’s whythe most widespread design procedures imply the use of simpler design schemes with subsequentrefinement of the stress-strain state at the stage of strength verification by FEM.
This edition presents the collection of training modules where the fundamentals of engineering designmethods of typical constructive elements are presented. The training modules should not be used as asingle source of information. There are, of course, many other scientific editions where designing problemsof composite elements of aircraft structures are covered in depth with using more complex structuralanalysis.
Professor Yakov KarpovNational Aerospace University “KhAI”
Authors
Yakov KarpovDr. Techn. Sc., Prof.National Aerospace University “KhAI”
Head of Aviation Material Science Department, Deputy Head ofAerospace scientific guidance council of the Ministry of Education andScience of Ukraine, Honored Worker of Ukrainian Science, winner ofState Prize of Ukraine in Science and Technology, UkrainianGovernment award. Author of more then 150 scientific publications(7 textbooks including) and 20 copyright certificates on the inventionin the area of composite structures.
Pavlo GagauzPh.D., Assoc. Prof.National Aerospace University “KhAI”
Associate professor of Aviation Materials Science Department, authorof 10 scientific papers in the field of laminate parameters optimizationand 3 published tutorials. Scientific researches are devoted tocomposite structures optimization and structural mechanics ofaircraft composite structures.
Fedir GagauzPh.D., Assoc. Prof.National Aerospace University “KhAI”
Associate professor of Aviation Materials Science Department, authorof 12 scientific papers in the field of composite structures design andsimulation, has 4 published tutorials, winner of Ukraine PresidentPrize for young scientist. Scientific researches are devoted to designand engineering of composite structures and structural components.
Content
Module 1 Composite Rods Design and Joining 9
Ph.D., Ass. Prof. Fedir Gagauz, National Aerospace University “KhAI”
Module 2 Composite laminated panels. Design and optimization 33
Dr.Sc., Prof. Yakov Karpov, National Aerospace University “KhAI”
Ph.D., Ass. Prof. Pavlo Gagauz, National Aerospace University “KhAI”
Module 3 Composite sandwich panels designand structural-technological solutions 65
Ph.D., Ass. Prof. Pavlo Gagauz, National Aerospace University “KhAI”
Module 4 Composite beams and spars design 85
Dr.Sc., Prof. Yakov Karpov, National Aerospace University “KhAI”
Ph.D., Ass. Prof. Fedir Gagauz, National Aerospace University “KhAI”
Module 5 Designing and strength analysis of the jointsof aircraft composite structures 105
Dr.Sc., Prof. Yakov Karpov, National Aerospace University “KhAI”
Ph.D., Ass. Prof. Fedir Gagauz, National Aerospace University “KhAI”
Ph.D., Ass. Prof. Pavlo Gagauz, National Aerospace University “KhAI”
Module 1 – Composite Rods Design and Joining
Prepared in the frame of the FP7 KhAI-ERA project 9
Training Module 1
Composite Rods Design and Joining
Ph.D., Ass. Prof. Fedir Gagauz
National Aerospace University “KhAI”
2013
Module 1 – Composite Rods Design and Joining
Prepared in the frame of the FP7 KhAI-ERA project 11
Introduction
Rods are widely used in aircraft structures as an individual constructive element in the form of controltubes or pushrods of the mechanization control systems, wing struts, etc., as well as in the form of truss-structures. Considering the typical loading of the rods by tension or compression the composites would bethe most effective for rods manufacturing due to high strength and elastic modulus in main direction.
This training module has two goals of providing enough information for students to design composite rodsand bars as well as providing the simplified approaches to perform strength analyses of the composite rodsin the zone of edge effect.
In issued training module the techniques of designing rods of circular cross-section which made bypultrusion and filament winding are described. The features of strength and buckling analysis of compositerods with open and closed prismatic cross-sections are described in detail. Practical recommendations onthe choice of various design and technological solutions for rod ends are given. The mechanism ofoccurrence of edge effects at the tips of the rods and analytical dependences for evaluating additionaltemperature and Poisson's stresses in the wall of the rod are presented.
This training module should not be used as a single source of information. There are, of course, bookswhere designing problems of composite rods and bars considering more complex structural analysis arecovered in depth.
Training Objectives
become familiar with the principles of loads perception and possible types of load-carrying abilitylosing of compressed rods with tubular and prismatic types of cross-section;
study the engineering design methods of composite rods considering strength and bucklingconstraints;
understand the mechanism of the edge effect occurrence at the tips of the rods due to differenceof Poisson's ratios and coefficient of thermal expansion of the rod and metal fitting
Module components
statement of the problem, list of constraints, numerical design procedure of the cross-section;
simplified analytical formulas to perform stress-strain analyses of the tubular composite rods in thezone of edge effect;
schematic description of the possible methods of realizing the joints of tubular rods with metalfitting and discussion of the advantages and disadvantages
Target audience
- master and doctoral students
Advances in composite structures design and simulation
12 Prepared in the frame of the FP7 KhAI-ERA project
Objective function:G min
Problem Statement
1 2
22 11
12
21
Homogeneous sections:
b 2
G f min l
f 2 R
Heterogeneous sections:
m
ii 1
f b
G f min l
i i ii
f 2 R
m n
ij ij ij 1 i 1
f b
Object of investigation
l
A – AHollow sections:
Open sections:
Homogeneous sections:
Heterogeneous sections:
Module 1 – Composite Rods Design and Joining
Prepared in the frame of the FP7 KhAI-ERA project 13
Constraints
Global buckling:
2glob mincr c2
2 min2
x
k EJN N
k EJ1
K
l
l
k – depends on boundary conditions
minEJ – bending stiffness in principal axis minEJ – bending stiffness in principal axis
xK – shear stiffnessxK – shear stiffness
ConstraintsStrength:
Homogeneous sections:
xc c xt tF f N ; F f N
Homogeneous sections:
xc c xt tF f N ; F f N xc c xt tF f N ; F f N
mxi
i i xii xi i 1
Fmin bE NE
Heterogeneous sections:
from equilibrium equationm m m
i i i i i xi i i xii 1 i 1 i 1
b bE bE
strain compatibility condition1 2 i...
xii xi
Fmin ; i 1,...,mE
Heterogeneous sections:
from equilibrium equationm m m
i i i i i xi i i xii 1 i 1 i 1
b bE bE
strain compatibility condition1 2 i...
xii xi
Fmin ; i 1,...,mE
Advances in composite structures design and simulation
14 Prepared in the frame of the FP7 KhAI-ERA project
Design Variables1. Type of cross-section:
- what is the best?
1. Type of cross-section:
- what is the best?
2. Type of laminate:
- what is the best?
2. Type of laminate:
- what is the best?
3. Cross-sectional dimensions:
1 2
22 11
12
21
- ?i ijR, , a ,
3. Cross-sectional dimensions:
1 2
22 11
12
21
- ?i ijR, , a , 1 2
22 11
12
21
- ?i ijR, , a , - ?i ijR, , a ,
Depends on type of cross-section
– 2 different buckling mode shapes – 1 buckling mode shapes
Constraints
Local buckling:
Axisymmetric mode shape
Nonaxisymmetric mode shape
Module 1 – Composite Rods Design and Joining
Prepared in the frame of the FP7 KhAI-ERA project 15
Composite Rod Design(circular section)
Objective:G 2 R min l
Constraints:
Strength: xc c xt t2 R F N ; 2 R F N3 3
glob xcr c3 3
2 x2
xy
E RN NE R1
G R
l
l
Global buckling:
3f 2 R , J R (for thin-walled section)
Optimal forstrength and
global bucklingis
UD laminate(pultruded rod)
3 3 3 322 globx 1 x
cr2 2 212xy
E R E E RR 1 NGG R
l l lNB: for UD CFRP
3 3 3 322 globx 1 x
cr2 2 212xy
E R E E RR 1 NGG R
l l lNB: for UD CFRP
Cross-section Selection
Circular section Box section
O O 2 Of 2R
or
f 4 aW W W
Circular section Box section
O O 2 Of 2R
or
f 4 aW W W
Stiffness comparison of rods with equal weight : Of fWStiffness comparison of rods with equal weight : Of fW2
2O OO2 2 2 2 42
OO O O2 2 2 42
2 O2
f f8J f83
J 2 f 16f f 212 8
W
W W WW WW
W
2
2O
1.216W
weight
strength
weight
strength
22O O
O O2 2O
f fJ8
22
2f fJ 212 8W W
W WW
?
22O O
O O2 2O
f fJ8
22
2f fJ 212 8W W
W WW
?
Advances in composite structures design and simulation
16 Prepared in the frame of the FP7 KhAI-ERA project
m= 5;n=6
m=9;n=8
m=3 ;n=4
m=2;n=4
m= 7;n=8
m=15 ;n=1 0
Nonaxisymmetric mode shape
Local buckling:
Local buckling:
2axcr x y c
2N E E N3
23nax mcr m,n c2m,n m,nm
2RN min L NRQ6
4 2 2 4m,n x m x yx xy m n y nL E 2 E 2G E
4 4xy 2 2m n
m,n m ny xy x x
21QE G E E
m n
m n;Rl
Axisymmetric mode shape
Nonaxisymmetric mode shape
Module 1 – Composite Rods Design and Joining
Prepared in the frame of the FP7 KhAI-ERA project 17
Advances in composite structures design and simulation
18 Prepared in the frame of the FP7 KhAI-ERA project
Global buckling:
2glob mincr c2
k EJN N
l
1
2xy xz
cr crN N
22yz
2 2
4 EJ1 EJ
l l
xy-plane: simply supported; kxy=1xz-plane: clamped; kxz=4
x
y
z
x
y
z
xy xzz yk EJ k EJadditional constraint xy xzz yk EJ k EJ
additional constraint
m 1
ib
x1ExiE mb
m
i ii 1
G b minl
mxti
i i xi ti xi i 1
Fmin bE NE
Composite Rod Design(heterogeneous section)
Strength:
homogeneous section:
xiE const; i 1,...,m
heterogeneous section:
mxci
i i xi ci xi i 1
Fmin bE NE
from strain compatibility condition and equilibrium equation:
Objective:
xt t xc cF f N ; F f N
Module 1 – Composite Rods Design and Joining
Prepared in the frame of the FP7 KhAI-ERA project 19
Local buckling:
i xi
i c m
i i xii 1
EN N
b E
strain compatibility condition
m
i i ci 1
1 2 m
1 x1 2 x2 m xm
N b N
N N N...E E E
equilibrium equation
22i xi yi i xi
c mixyi yxi
i i xii 1
k E E ENb12 1 bE
22i xi yi i xi
c mixyi yxi
i i xii 1
k E E ENb12 1 bE
cri iN N
m 1
ib
x1ExiE mb
for simply supported panel with free edge:
2
i xi yi 3cri ii2
i xyi yxi
k E EN N
12b 1
Local buckling:
criN – critical distributed load of i-facecriN – critical distributed load of i-face
xi yxi xyi xyi yxii
xi yi
0,3E G 1k 0,4
E E
for simply supported panel:
xi yxi xyi xyi yxii
xi yi
E 2G 1k 2 1
E E
Advances in composite structures design and simulation
20 Prepared in the frame of the FP7 KhAI-ERA project
Local buckling:
22yii i
cr i xi ixyi yxi
EkminE b12 1
m mloccr cri i cr xi i i
i 1 i 1N N b E b
from strain compatibility condition:
22 m
yii ixi i i ci xi i i 1xyi yxi
Ekmin E b NE b12 1
22 myii i
xi i i ci xi i i 1xyi yxi
Ekmin E b NE b12 1
cr1 cr2 cri crmN N ... N ... Nadditional constraint: cr1 cr2 cri crmN N ... N ... Nadditional constraint:
Module 1 – Composite Rods Design and Joining
Prepared in the frame of the FP7 KhAI-ERA project 21
Rod-to-Bushing Jointsextra-windingextra-winding
Advantages: simple construction and low-technology machining is unnecessaryDisadvantages: difficult to guarantee the uniform thickness of glue surface glue bonding requires the pressure unmanageable matching of glue curing temperature with limitingtemperature of bonded elements is needed
Fork-Joint
spherical planebearing
alignment errors or angular misalignments compensation operational deformations have no effect
Advances in composite structures design and simulation
22 Prepared in the frame of the FP7 KhAI-ERA project
Joints With Cutting Edges
partially polymerized rod
extra-winding
Advantages: uniform glue thickness the concurrent glue curingand rod polymerization
Disadvantages: the extra-winding requires can be applied for thethin-walled rods only
Tapered Joints
Advantages: uniform thickness of glue coating - more strength elements can be pressed more uniform shear stress distributionDisadvantages: the machining is required rod tension provokes the glue tearing realizability for the thick-walled rods only
Module 1 – Composite Rods Design and Joining
Prepared in the frame of the FP7 KhAI-ERA project 23
Glue-Mechanical Joint
Advantages: essential increase of load carrying ability glue-mechanical joint ensures more reliabilityDisadvantages: high cost
Advantages: the concurrent glue curing and rod polymerization glue quality guaranteed application pressure is ensured with yarn (or tape) tension
Disadvantages: demountable (expensive) mandrel is required
Joints of the Rods Manufactured byFilament Winding
Advances in composite structures design and simulation
24 Prepared in the frame of the FP7 KhAI-ERA project
Joints With Special Threaded Bushing
Basic technological actions: coating by the separating layer the winding and the rod polymerization the disassembling and extraction of the winding mandrel the bushing unscrewing the glue coating of the bushing twisting-in of the bushing and the glue polymerization
Module 1 – Composite Rods Design and Joining
Prepared in the frame of the FP7 KhAI-ERA project 25
b
b
N
N
NN
NN
RR
R
yxx yx
x y
yxy xy
x y
E E
E E
Hook’s law:
NN2 R
y x2
0
l xy xy
xE
xy
x
NRRE
y x 00 rigid bush
y
y xyx
ENE
Poisson’s Edge Effect
Advances in composite structures design and simulation
26 Prepared in the frame of the FP7 KhAI-ERA project
– Young modulus,length and thickness of bush
b b bE , , l
y2 21 23
xx
2 2 2 2 2 22 1 2 1
3E3N 2k ; k ;R ER E
1 1r k k ; t k k2 2
k b b b y1B E E2 l
– Young modulus,length and thickness of bush
b b bE , , l
y2 21 23
xx
2 2 2 2 2 22 1 2 1
3E3N 2k ; k ;R ER E
1 1r k k ; t k k2 2
k b b b y1B E E2 l
x y2
xy yx
E EN 2
3 1
rx
2 32 2x
k
re cos tx sin txtw R 1
E R1 r r t3B
Edge Loads Estimation:Analytical approach to stress calculation
of frame-stiffened cylindrical shell can be used
3 2x
2E d wM12 dx
3 3
x3
EdM d wQdx 12 x
Distributed bending moment: Distributed shear force:3 2
x2
E d wM12 dx
3 3
x3
EdM d wQdx 12 x
Distributed bending moment: Distributed shear force:
y
x.bxz
y
N
N
M
Q
N
Edge Loads:
x.b – bending stressx.b – bending stress xz – interlaminar shear stressxz – interlaminar shear stress
Module 1 – Composite Rods Design and Joining
Prepared in the frame of the FP7 KhAI-ERA project 27
Example of Edge Stress Analysis:Rod R=15mm; δ=2mmUD CFRP: Еx=100GPа; Еy=10GPа; μxy=0,35; F1t=1500MPaBush ℓb=30mm; δb=4mmSteel: Еb=210GPа
Maximal Tensile Load: t 1tN 2 R F 282.7 kN Maximal Tensile Load: t 1tN 2 R F 282.7 kN
x y2
xy yx
E E2 461.7 kN
3 1
5xy
x
NRR 7.875 10 mE
1 1 1 11 2k 149.1m ; k 190.5 m ; r 171.0 m ; t 83.9 m
7kB 1.29 10 N
5xy
x
NRR 7.875 10 mE
1 1 1 11 2k 149.1m ; k 190.5 m ; r 171.0 m ; t 83.9 m
7kB 1.29 10 N
rx
2 32 2x
k
re cos tx sin txtw R 1
E R1 r r t3B
rx
2 32 2x
k
re cos tx sin txtw R 1
E R1 r r t3B
x y2
xy yx
E EN 2
3 1
1 2r x r x1
22 3
2 21 x1 2 1
2 k
re erw R 1
r E R1 r r rr 6B
2 2 4 4 2 2 4 41 1 1 2 2 1 1 2r k k k ; r k k k
Edge Loads Estimation:
Stress analysis:
max 26M
bending stress:max
max
max3Q2
interlaminar shear stress:
yQ
hoop stress:yyQ
Stress analysis:
max 26M
bending stress: max 26M
bending stress:max
max
max3Q2
interlaminar shear stress: max3Q2
interlaminar shear stress:
yQ
hoop stress: yQ
hoop stress:yyQ
Advances in composite structures design and simulation
28 Prepared in the frame of the FP7 KhAI-ERA project
Module 1 – Composite Rods Design and Joining
Prepared in the frame of the FP7 KhAI-ERA project 29
free rod extension
restricted strain
free bush extension
А С
B
х
хrхb
r
b
хb
xr x x r x х х xb b x b b х bE E Т E E Т
Temperature Edge Effect
b b b x x r x bxr x хb b
x r b b х r b b
Е f Е fЕ T Е Т
Е f Е f Е f Е f
Equilibrium equation:r xr b xbf f 0 x r x b b b
хx r b b
Е f Е fTE f E f
x r x b b bх
x r b b
Е f Е fTE f E f
Axial Stresses:
NB:
simplified approach to hoop stress analysis can be used
bush stiffness >> rod stiffness
yy xy
x
ENE
bush is treated as rigid body yy xy
x
ENE
bush is treated as rigid body
maxy yx 0
1032.1MPa 0.35 1500 52.5MPa100
Example:
if N>0 R 0 Q 0 rod wall press to bushif N<0 R 0 Q 0 rod wall tear out the bushif N>0 R 0 Q 0 rod wall press to bushif N<0 R 0 Q 0 rod wall tear out the bush
FEM simulation provide: more accurate stress analysis reliability of estimate through-the-thickness stress distribution
Advances in composite structures design and simulation
30 Prepared in the frame of the FP7 KhAI-ERA project
Conclusions:Methods for edge stresses relaxation:
tapered thickness of the rod wall and the bushing in the joint additional reinforcement for required Young’s modules,Posson’s ratios and coefficients of thermal expansion decreasing of the anisotropy grade special bushing configuration
yr
byb
b
Rp
Rp
Hoop Stresses:
y
b b b
R R 1 T
R R 1 T
Free:
y
b b b
R R 1 T
R R 1 T
Free:
yr yb bb b b
y y b b y
pRpR R 1 R 1 R R 1 R 1E E E E
Restricted:
yr yb bb b b
y y b b y
pRpR R 1 R 1 R R 1 R 1E E E E
Restricted:
b b1R R2
Strain compatibility condition: b b1R R2
Strain compatibility condition:
y b b y b b2 22 2
b b y b y b
E E R Rp T
R E 1 T R E 1 T
Module 1 – Composite Rods Design and Joining
Prepared in the frame of the FP7 KhAI-ERA project 31
References
1. Timoshenko, S.P., Gere, J.M. Mechanics of Materials, 4-th ed., PWS Publishing Co., Boston,1997.
2. Pilkey, W.D. Formulas for stress, strain, and structural matrices, 2-nd ed.,John Wiley & Sons, Inc., New Jersey, 2005.
3. Jones, R.M. Mechanics of Composite Materials, 2-nd ed., Taylor&Francis, Inc., Philadelphia,1999.
4. Vasiliev, V.V. Mechanics of Composite Structures, Taylor&Francis, Inc., Washington, 1993.
5. Karpov, Ya.S. Composite items and structural components design, [in Russian], Kharkiv,KhAI, 2010. – 768 p.
Module 2 – Composite laminated panels. Design and optimization
Prepared in the frame of the FP7 KhAI-ERA project 33
Training Module 2
Composite laminated panels.Design and optimization
Dr.Sc., Prof. Yakov Karpov
National Aerospace University “KhAI”
Ph.D., Ass. Prof. Pavlo Gagauz
National Aerospace University “KhAI”
2013
Module 2 – Composite laminated panels. Design and optimization
Prepared in the frame of the FP7 KhAI-ERA project 35
Introduction
Most structural elements of the airframe such as the skin of the fuselage, wings and tail, the floor of thepassenger and cargo cabin, the elements of interior etc., represent flat and curved panels and plates. Inconjunction with the rods, bars and beams they can form a high-efficient thin-walled airframe structureshaving a small mass and required strength and stiffness. The panels provide perception of the distributednormal and shear forces applied along the edges of the panel, as well as the normal pressure distributedalong the surface. In general case, bending stresses which are caused by aerodynamic pressure arenegligible in comparison with stresses due to in-plane loads, thats why for airframe weight decreasing it isadvisable to use composite material in panels manufacturing.
The most significant advantage of the composite materials is the possibility of controlling the elastic andstrength properties due to changing the type of lay-up. Thus, the designing of composite structures shouldinclude the stage of the lay-up optimization, where the optimal thickness ratio of plies and angles of theirorientation are determined.
This training module provides general information on aircraft composite panels design. Typical loading ofaircraft wing panel is considered and basic assumptions are involved. Proposed design approach is based onclassical mechanics of laminated plates which allows to formulate goal function (total laminate thickness)as function of laminate parameters (plies orientations and thickness ratios) and design constraints.
Training Objectives
studying the fundamentals of the laminate stacking sequence optimization according to strength,buckling and bending stiffness constraints;
learning the general strategy of ordinary laminated composite panels optimization according tostrength, buckling and bending stiffness constraints.
Module components
lay-up optimization under strength constraints in terms of the maximum stress failure criterion andTsai-Wu failure criterion;
composite panel optimization under the buckling constraints;
results of the eigenvalue problems solution for critical shear loads calculation of simply supportedcomposite panels;
composite panel optimization under the stiffness constraints.
Target audience
- master and doctoral students.
Advances in composite structures design and simulation
36 Prepared in the frame of the FP7 KhAI-ERA project
loading
gene
ric p
erfo
rman
ce c
riter
ia usual ordinary(unstiffened)
panels
sandwichpanels
stiffenedpanels
Efficiency of Basic Design Models
Typical Loading of Aircraft Wing Panel
z
x
y
xN
yN
xyq
p
a
b
Torsion
Cross Bending
Module 2 – Composite laminated panels. Design and optimization
Prepared in the frame of the FP7 KhAI-ERA project 37
Basic relations of the theory of plates x x y y xy xyN N ; N N ; q q ;
2 2
11 122 22 2
22 122 22
332
x
y
xy
w wM D D ;x y
w wM D D ;y x
wM D ;x y
Distributed edge load (in-plane forces):
Distributed bending and torsional moments:
Distributed shear forces:
2 2
11 12 32 2
2 2
22 12 32 2
2
2
xyxx
y xyy
MM w wQ D D D ;x y x x y
M M w wQ D D D .y x y y x
Unstiffened Panel Design.General Problem Statement
Minimize objective function (for aircraft structures panel weight)considering predefined set of design constraints:
panel strength;
buckling loads;
bending stiffness (panel deflection);
structural and technological (manufacturing) requirements.
Main design variables are:
ply orientation angles and thicknesses;
laminate stacking sequence (LSS).
Advances in composite structures design and simulation
38 Prepared in the frame of the FP7 KhAI-ERA project
Basic Assumptions only non-hybrid laminates are under consideration (means
one objective function);
laminate stacking sequence are assumed to be balanced(orthotropic) and symmetric (substantially reduces designspace and simplifies design equations);
bending stresses which are induced by aerodynamic pressureare negligible in comparison with stresses due to in-planeloading (gives conservative designs);
local strength approach individual ply failure means thefailure of laminate at all;
external loads and laminate thickness/stiffness are uniformalong the panel.
Sign Convention
Laminate Stacking Sequence Ply Orientation Angles
1
2
3
1ply 1,
(3)1
(3)2
(2)1(2)2
12
x
y
22
2k
12
n
2n
2ply 2,
ply k, k
1ply n 1, n
ply n, n
(1)1(1)2
Module 2 – Composite laminated panels. Design and optimization
Prepared in the frame of the FP7 KhAI-ERA project 39
Main Difficulties problem complexity is unknown plies orientation count n is
one of design variables;
principal constraints are slack inequalities;
objective function and certain constraints are implicit (as wellas nonlinear) functions of design variables;
nonconvex (and so multimodal) design space;
ply and total laminate thicknesses (may be also ply orientationangles) are discrete design variables which usually meansmultiple optimum solutions with comparable performance;
laminate stacking sequence optimization is combinatorialproblem.
1,
nk
kG min
1 2 12; ; 1, 1,.., ;k k kf k n
2
0 0 01;
y xyx
x y xy
N qNN N q
0;maxw w
Mathematical Formulation of PanelOptimization Problem
Minimize panel thickness
which have to meet requirements of
- structural stability
- laminate strength(any lamina failure criterion)
- bending stiffness
- lay-up validity 0.k km
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40 Prepared in the frame of the FP7 KhAI-ERA project
0
2
4
6
8
10
0 15 30 45 60 75 90ply orientation, deg.
lam
inat
e th
ickn
ess,
mm
.
no 0-plies25% of 0-plies50% of 0-plies75% of 0-plies
Typical Dependence of [0/±] Lay-Up Thickness onPly Orientation Angle . Strength Constraints (AS4/3501-6)
Module 2 – Composite laminated panels. Design and optimization
Prepared in the frame of the FP7 KhAI-ERA project 41
Lay-up Optimization Under StrengthConstraints
Vector of design variables (LSS has no effect)
1 2 1 2 1 2 1 2 1, ,..., , , ,..., , ,..., , , ,..., ,
, 1,..., , .
n n n n
k s
U Uk,s n k s
or
Problem statement
1 2 12
,
; ; 1, 1,..,strk k kk
min
f k n.
1; 1.
nk k k
k
Ply thickness ratio
1
2
3
4
5
6
0 45 90ply orientation, deg.
lam
inat
e th
ickn
ess,
mm
.
Effect of Ply Thicknesses Discontinuity. Multiplicity ofOptimal Solutions. Strength Constraints (AS4/3501-6)
015
020
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42 Prepared in the frame of the FP7 KhAI-ERA project
11 22 33 1 2 12
11 22 12 1 2 1 21
2 2 ;
2 2 .
B B B E E G
B B B E E E
So, there are only two independent membrane stiffness terms
2 2 ;
.
xy x y
k k
sin cos
For the purpose of certain simplification of ply-by-ply stress stateanalysis a new lamina parameter was specified as function ofply orientation angle
laminate volumetric strain
maximal possible ply shear strain.
Such approach requires also two additional parameters to becalculated (being constants for all plies are evaluated just once)
x y
22o xy x y
0 ;z
22 12 11 122 2 3311 22 11 2212 12
; ; .x y y x xyx y xy
N B N B N B N B qBB B B B B B
1
nij k ij kk
B Q
Basic Fundamental ResultsLaminate strains
determine laminate stiffness under in-planeloading (membrane stiffness terms)
2 2 211 11 1
2 2 222 22 2
2 233 33 12
2 212 12 1 21
1 2 1 2 1 21 12
;
;
;
;
; 2 4 .
k k k kk
k k k kk
k k kk
k k kk
Q Q E Rsin Ssin cos
Q Q E Rsin Ssin cos
Q Q G Ssin cos
Q Q E Ssin cos
R E E S E E E G
Module 2 – Composite laminated panels. Design and optimization
Prepared in the frame of the FP7 KhAI-ERA project 43
Functional dependence represents geometrical locus of k-points. Thus solution of equation
122 2 2 2 0xy x ycos sin
gives extreme values of -parameter and corresponding laminaprincipal axes directions (orientations of plies with zero shearstress state)
1 2 11
; .2 2
xyo o o
x yarctg
For any laminate with arbitrary lay-up there are identically true,axiomatic conditions
2 2 212 12 ; ; .o k o o k o ok k
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44 Prepared in the frame of the FP7 KhAI-ERA project
Maximum stress failure criterion
21 21 12
1 21 21 2 12 12
1 21 2 12
1 21 21 2 12 12
22 1 1; ;
1 1 1 1
2 1 2 1; .
1 1 1 1
pcmin
p cmax
FFminE E
F FminE E
1 1 1 2 2 2 12 12 12
12
12
2 212 212 12
2 212 12
; ; ;
; , ;
; ; ,
c k p c k p k
min max o
k
min o o max o
F F F F F F
FG
F F FG G
1444444444444444444444444442 444444444444444444444444443
U
2
12,
G
2 21 2 12
1 1 21 21
2 2 12 12
2 212 12
; ; ;2 2
11 1 ;
21
1 1 ;2
.
k kk k k o k
k k
k k
k o k
E
E
G
Ply strains and stresses (relative to ply local coordinate system)
Strength allowable range of k-values (have to meet laminatefeasible limits)
; ; ; ; . or k min max k min max k o oI
1 2 12, , 1 1 , rootsstr str min max .
Module 2 – Composite laminated panels. Design and optimization
Prepared in the frame of the FP7 KhAI-ERA project 45
Tsai-Wu failure criterion
2 2 21 1 2 2 11 12 1 2 22 661 2 12
2
2 1
1 11
4 2
; ; , 0;
; , 0,
k k k kk k k
kk
min maxk
min max
p p p p p p
A B C ,
A
A
5 621 4 2 3 4
2
24 ; ; ;
4
4 1; ; ; ; .
o
min a b max a b a,b
g gA g g B g g C g
B B A Cmin max
A
Advances in composite structures design and simulation
46 Prepared in the frame of the FP7 KhAI-ERA project
1 11 2
2 1
2 2
, ;1,
2 , .
xy x y
o o xy, o o
o o o xy
sin cos .
sign signarccos
sign sign
Each value of -parameter corresponds to at least two valuesof ply orientation angle
Main conclusions:
regardless of loading conditions for any arbitrary laminatethere could be no more than four plies with equal strength;
allowable by strength and feasible range of -parameterdefines allowable range for ply orientation angles.
Module 2 – Composite laminated panels. Design and optimization
Prepared in the frame of the FP7 KhAI-ERA project 47
Reformulation of problem statement to unconstraintoptimizational problem with continuous design variables
General Approach to Laminate Design
,kstr str
kU max min
kstr kf U total laminate thickness which is required by
strength condition for individual k-th ply.
Design vector + basic mechanics of composite materials
str f U minimal laminate thickness which guaranteesstrength of corresponding lay-up U ;
KuhnTucker problem and conditions
Basic results of resolving this equations set:
function str() has no more than four extrema;
optimal solutions are always among [0/90/±] or [±1/±2]lay-ups.
1 1 1
1
1 0, 1,2,..., ;
1 1 0, 1,2,..., ;
0, 1,2,..., .
strk k
strn n nstr tk k k t
kk k tstr
n ttkt
k n
k n
k n
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48 Prepared in the frame of the FP7 KhAI-ERA project
2 21
2 22
12
1 1 1 21 2
2 2 2 12 1
12 12 12
, ;
, ;
, 2 2 ;
, ;
, ;
, .
x y xy
x y xy
y x xy
U cos sin sin cos
U sin cos sin cos
U sin cos
U E
U E
U G
Reduced strains and stresses for individual ply with orientationangle ( could be equal to 0, 90, or )
1 1 2 21 2
1 1 2 2
, 0; , 0;, ,
, 0; , 0.t t
c c
F FF U F U
F F
Strength limits for individual ply
1 2 1 2 30 90 1 , 11,22,33,12.ij ij ij ijB U u Q u Q u u Q u ij
Design vector
1 2 3 1 2, , , , ,U u u u v v
1 2,v v thickness ratios of all plies with orientation angles of0 and 90 accordingly.
Reduced (by laminate thickness) membrane stiffness terms
22 12 11 122 2 3311 22 11 2212 12
; ; .x y y x xyx y xy
N B N B N B N B qU U U
BB B B B B B
Optimization Strategy for [0/90/±] Lay-Up
Reduced laminate strains
Module 2 – Composite laminated panels. Design and optimization
Prepared in the frame of the FP7 KhAI-ERA project 49
Strength constraint for laminate
1 2 3; ; ,str str str strU max
1 11
1 1
2 22
2 2
1 2 1 23
3 3 1 2 1 2
0, 0;
, 0 , 0.
0, 0;
, 90 , 0.
0, 1;
, ; , , 1.
str
str
str
u vU
U u v
u vU
U u v
u u v vU
max U u U u u u v v
Example. Assuming function willreturn minimal allowable value of thickness for quasi-isotropiclaminate [0/90/±45].
0.25, 0.25, 45U , str U
1 2 12
1 2 12, ; ; ;U max
F F F
2 2 2
1 1 2 2 12
1 1 2 2 12, ;U
F F F F F
Strength constraint for individual ply
- maximum stress criterion
- TsaiHill criterion
21 21
1 1 1 2 2
2 2 22 11 12 1 2 22 661 2 12
, ,
1, ;
2
, 2 .
U a a a
a U p p
a U p p p p
- TsaiWu criterion
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Correlation of Laminate Thickness for Different Lay-Ups.Unidirectional Carbon/Epoxy System
MR50/LTM25,Carbon/Epoxy UD
1
2
3
4
5
-4 -3 -2 -1 0 1 2 3 4loads ratio, Nx/qxy
[0/90][0/90/±45][0/90/±x], "10%"quasi-isotropic
Ultimate failure load
; ; .u u ux str x y str y xy str xyN N N N q q
Strength margin of safety or load factor (relative to design loads)
0 .str
str str
mU U
Example. Design vector for laminate [04/902/(±60)2]s
0 4, 0 2, 60U . . .
0 020
1.strstr str
mU U
Strength of laminate is ensured if load factor
Module 2 – Composite laminated panels. Design and optimization
Prepared in the frame of the FP7 KhAI-ERA project 51
Correlation of Laminate Thickness for Different Lay-Ups.Fabric Glass/Epoxy System
M10E/3783,Glass/Epoxy Fabric
1
1,5
2
-4 -3 -2 -1 0 1 2 3 4loads ratio, Nx/qxy
[0/90][0/90/±45][0/90/±x], "10%"quasi-isotropic
Correlation of Laminate Thickness for Different Lay-Ups.Unidirectional Carbon/Epoxy System
AS4/3501-6,Carbon/Epoxy UD
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4loads ratio, Nx/qxy
[0/90][0/90/±45][0/90/±x], "10%"quasi-isotropic
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Buckling forces for simple supported rectangular plate
Basic Fundamental Results
2 2 211 22 11 22 11 22
0 0 02 2; ; ,x x y y xy xy
D D D D D DN K N K q Kabb a
2
22 , 1 1 ;x
mK m m m mm
22
1 12 , 1 1 ;yK n n n n n
n
xyK f ,
222 12 33
11 11 22
2; .
a D D Db D D D
requires partial eigenvalue problem solution.
Dimensionless factors
Composite Panel OptimizationUnder Buckling Constraints
Problem formulation
2
0 0 0
,
1
y xyxbuc
x y xy
min
N qN .N N q
Additional assumptions:
panel is simple supported along all edges (conservativedesign);
transverse (interlaminar) shear strains no affect bucklingloads (nonconservative design).
Module 2 – Composite laminated panels. Design and optimization
Prepared in the frame of the FP7 KhAI-ERA project 53
Bending stiffness terms
3 3 3
2
1 1 1
1.
12 48
n n n n kij ij s s ij k kk kk s k s k kD Q Q z
For heterogeneous, “cluster” LSS 1 2 3 ... n
3 33 3
1 1.
12 12
n n nij ij ij s skk s k s k
D D Q
For homogeneous LSS with large amount of plies
3 3
1.
12 12
nij ij ij ij kkk
D D B Q
24m
Buckling mode shapes
11 1 1 1 m m m m n n n n
m 1; n 1 m 2; n 1
m 3; n 1 m 2; n 2
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Typical Dependence of Buckling Constraint Function buc
on Ply Orientation Angle
cross-ply laminate [±] specially orthotropiclaminate [0], [90] or [0/90]
Lagrange's method of multipliers
Main results of resolving this system of equations:
function buc() has no more than three extrema;
optimal results are always among specially orthotropic [0/90]or cross-ply [±] laminates.
1
1 0, 1 2 ;
1 0, 1 2 ;
1 0.
buc
k kbucn buc
kk kk
buc
k , ,...,n
k , ,...,n
Module 2 – Composite laminated panels. Design and optimization
Prepared in the frame of the FP7 KhAI-ERA project 55
Reduced bending stiffness terms
31 1 2 1
31 2 3 2
1
1 , 11,22,33,12.
ij ij ij ij
ij ij
D U Q Q Q
Q Q ij
222 12 33
11 11 22
22
2 2
2; ;
12 ; 2 ; ;
1; .
2
x y xy
y xyxn q
x y xy
a D D DU Ub D D D
mK U K U n K U f ,m n
N qN b aK U K UK a K b K
Necessary for calculations design and buckling loads functions
For definition see supplementary materialsxyK
Optimization Strategy for [0/90/±] Lay-Up
Design vector have to deal with LSS permutations
1 2 3 4 5 1 2 3 1 2, , , , , , , , .U u u u u u
1 1 20, , 90, , 1U v v v 1 2 1, 0, 90, 1 ,U v v v
1 2,v v thickness ratios of all plies with orientation angles of0 and 90 accordingly.
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56 Prepared in the frame of the FP7 KhAI-ERA project
Panel Thickness Increment Induced by Partial Swapping ofPlies in Optimal Cross-Ply Laminate (AS4/3501-6)
1 2
0/
1 2
90/
1 3
2 22
11 22
12.buc n q n
abU K K KD D
Buckling constraint for composite panel
Example. Assuming function willreturn minimal value of allowable thickness for quasi-isotropic lay-up [0/±60] with outer cross plies and inner longitudinal plies.
60, 0, 90, 2/3, 1/3U , buc U
Ultimate buckling load
; ; .u u ux buc x y buc y xy buc xyN N N N q q
Buckling margin of safety or buckling load factor
0
3 3 .bucbuc buc
mU U
Module 2 – Composite laminated panels. Design and optimization
Prepared in the frame of the FP7 KhAI-ERA project 57
Central deflection of simple supported rectangular plate
Basic Fundamental Results
2 2
11 22,max w
pa bw KD D
with, being the same dimensionless factors as specified forbuckling problem above.
42 2 4
61 1
16 1, 2 .
2 2w mn
mnm n
m n mK sin sin L m n nmnL
20.01664
1 2wK
For initial design purpose it could be assumed that
Composite Panel OptimizationUnder Stiffness Constraints
Problem formulation
00
,
1, 0,002 0,01 .maxdef
min
w w ...w
l
Additional assumptions:
same as for buckling optimization problem (simple supportededges, laminates possess absolute interlaminar stiffness);
transverse load p is uniformly distributed along panelsurface.
Advances in composite structures design and simulation
58 Prepared in the frame of the FP7 KhAI-ERA project
Optimization Strategy for [0/90/±] Lay-Up
Same as previous design vector with LSS consideration
1 2 3 1 2, , , , .U
1 32 2
01 2
12.w
defpa b KU
wD D
Deflection constraint for composite panel
Transverse load safety factor for accounting of design variablediscontinuity
0
3 3 .defdef def
mU U
Since deflection constraint function def depends on laminatebending stiffness terms as well, main results remain the same asfor buckling optimization problem:
function def() has no more than three extrema on all plyorientation angle range;
optimal results are always among specially orthotropic [0/90]or cross-ply [±] laminates.
System of equations for Lagrange's method of multipliers
0, 1 2 ; 0, 1 2 ; 0k k
k , ,...,n k , ,...,n
1
1 .n def
kk
with Lagrangian function
Module 2 – Composite laminated panels. Design and optimization
Prepared in the frame of the FP7 KhAI-ERA project 59
2
4
6
8
10
12
14
0 45 90ply orientation angle, deg.
4
6
8
10
12
14
0 45 90ply orientation angle, deg.
0
2
4
6
8
10
0 45 90ply orientation angle, deg.
0
2
4
6
8
10
12
0 45 90ply orientation angle, deg.
strbuc
str
buc
strbuc
str
buc
Correlation of Strength and Buckling Design Spaces forCross-Ply Laminate (AS4/3501-6)
Composite Panel Optimization.General Strategy for [0/90/±] Lay-UpProblem formulation
,
1, 1,...,4; 1; 1str buc defk
min
k .
Constraints set in terms of laminate thickness
, , .str buc defU max
Constraints set in terms of safety margins
, , 1.str buc defmin
Advances in composite structures design and simulation
60 Prepared in the frame of the FP7 KhAI-ERA project
Estimating of Design Space for Laminates Optimizationby Optimal Design of Cross-Ply Lay-Up
Module 2 – Composite laminated panels. Design and optimization
Prepared in the frame of the FP7 KhAI-ERA project 61
References
1. Jones, R.M. Mechanics of Composite Materials, 2-nd ed., Taylor&Francis, Inc., Philadelphia,1999.
2. Barbero, E.J. Introduction to Composite Materials Design, 2-nd ed., CRC Press, Boca Raton,2010. – 336 p.
3. Kollar, L.P., Springer, G.S. Mechanics of Composite Structures, Cambridge University Press,2003. – 480 p.
4. Karpov, Ya.S. Composite items and structural components design, [in Russian], Kharkiv,KhAI, 2010. – 768 p.
Advances in composite structures design and simulation
62 Prepared in the frame of the FP7 KhAI-ERA project
Supplementary materialsco
effic
ient
s fo
r crit
ical
she
ar lo
ads
calc
ulat
ion
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
1,2
1,4
1,6
1,8
2,0
2,5
0,07
511
,693
8,61
27,
589
6,95
56,
207
5,74
25,
458
5,29
45,
212
5,18
85,
259
5,42
85,
657
5,92
16,
207
6,95
5
0,1
11,8
948,
765
7,73
47,
064
6,31
95,
854
5,57
15,
408
5,32
65,
303
5,37
35,
542
5,77
06,
033
6,31
97,
064
0,15
12,2
959,
070
8,02
27,
283
6,54
16,
079
5,79
85,
636
5,55
45,
531
5,60
15,
769
5,99
56,
257
6,54
17,
283
0,2
12,6
939,
373
8,30
87,
502
6,76
36,
304
6,02
45,
863
5,78
25,
759
5,82
85,
995
6,22
06,
481
6,76
37,
502
0,25
13,0
879,
674
8,59
27,
720
6,98
46,
528
6,25
06,
089
6,00
95,
986
6,05
56,
221
6,44
46,
703
6,98
47,
720
0,3
13,4
799,
974
8,87
47,
938
7,20
66,
751
6,47
56,
315
6,23
56,
213
6,28
16,
446
6,66
86,
926
7,20
67,
938
0,4
14,2
5410
,567
9,43
28,
374
7,64
77,
196
6,92
36,
765
6,68
66,
664
6,73
16,
894
7,11
47,
369
7,64
78,
374
0,5
15,0
1711
,155
9,98
28,
808
8,08
67,
639
7,36
97,
213
7,13
57,
112
7,17
97,
340
7,55
87,
811
8,08
68,
808
0,6
15,7
6911
,736
10,4
609,
242
8,52
48,
081
7,81
37,
658
7,58
17,
559
7,62
57,
785
8,00
18,
251
8,52
49,
242
0,7
16,5
0812
,312
10,8
949,
675
8,96
18,
521
8,25
58,
102
8,02
58,
004
8,06
98,
227
8,44
18,
690
8,96
19,
675
0,8
17,2
1412
,882
11,3
2810
,108
9,39
78,
960
8,69
68,
544
8,46
88,
446
8,51
18,
668
8,88
09,
128
9,39
710
,108
0,9
17,8
4213
,448
11,7
6210
,539
9,83
29,
397
9,13
58,
984
8,90
98,
887
8,95
29,
107
9,31
89,
564
9,83
210
,539
1,0
18,4
6514
,009
12,1
9510
,985
10,2
659,
833
9,57
29,
423
9,34
89,
327
9,39
19,
545
9,75
59,
999
10,2
6510
,985
1,2
19,7
5115
,117
13,0
6011
,846
11,1
3010
,701
10,4
4410
,295
10,2
2210
,201
10,2
6410
,417
10,6
2410
,866
11,1
3011
,846
1,4
20,9
6716
,208
13,9
2212
,705
11,9
9111
,565
11,3
1011
,163
11,0
9011
,070
11,1
3211
,283
11,4
8911
,729
11,9
9112
,705
1,6
22,1
6817
,161
14,7
8313
,562
12,8
5012
,426
12,1
7212
,027
11,9
5411
,934
11,9
9612
,146
12,3
4912
,588
12,8
5013
,562
1,8
23,3
5718
,061
15,6
4214
,417
13,7
0513
,283
13,0
3012
,886
12,8
1412
,793
12,8
5513
,004
13,2
0713
,444
13,7
0514
,417
2,0
24,5
3418
,955
16,5
0015
,269
14,5
5814
,137
13,8
8513
,741
13,6
7013
,649
13,7
1013
,859
14,0
6114
,298
14,5
5815
,269
2,2
25,7
0019
,843
17,3
5516
,121
15,4
0914
,988
14,7
3714
,593
14,5
2214
,502
14,5
6314
,711
14,9
1215
,149
15,4
0916
,121
2,4
26,8
5420
,769
18,2
1016
,970
16,2
5715
,836
15,5
8615
,442
15,3
7115
,351
15,4
1215
,560
15,7
6115
,997
16,2
5716
,970
2,6
27,9
9921
,647
19,0
6317
,818
17,1
0416
,683
16,4
3216
,289
16,2
1816
,198
16,2
5916
,406
16,6
0716
,844
17,1
0417
,818
2,8
29,1
3322
,522
19,9
1518
,665
17,9
4917
,527
17,2
7617
,133
17,0
6217
,042
17,1
0317
,250
17,4
5117
,688
17,9
4918
,665
3,0
30,2
5823
,394
20,7
6519
,510
18,7
9218
,369
18,1
1817
,975
17,9
0417
,883
17,9
4418
,092
18,2
9418
,531
18,7
9219
,510
coef
ficie
nts
for c
ritic
al s
hear
load
s ca
lcul
atio
n
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
1,2
1,4
1,6
1,8
2,0
2,5
0,07
511
,693
8,61
27,
589
6,95
56,
207
5,74
25,
458
5,29
45,
212
5,18
85,
259
5,42
85,
657
5,92
16,
207
6,95
5
0,1
11,8
948,
765
7,73
47,
064
6,31
95,
854
5,57
15,
408
5,32
65,
303
5,37
35,
542
5,77
06,
033
6,31
97,
064
0,15
12,2
959,
070
8,02
27,
283
6,54
16,
079
5,79
85,
636
5,55
45,
531
5,60
15,
769
5,99
56,
257
6,54
17,
283
0,2
12,6
939,
373
8,30
87,
502
6,76
36,
304
6,02
45,
863
5,78
25,
759
5,82
85,
995
6,22
06,
481
6,76
37,
502
0,25
13,0
879,
674
8,59
27,
720
6,98
46,
528
6,25
06,
089
6,00
95,
986
6,05
56,
221
6,44
46,
703
6,98
47,
720
0,3
13,4
799,
974
8,87
47,
938
7,20
66,
751
6,47
56,
315
6,23
56,
213
6,28
16,
446
6,66
86,
926
7,20
67,
938
0,4
14,2
5410
,567
9,43
28,
374
7,64
77,
196
6,92
36,
765
6,68
66,
664
6,73
16,
894
7,11
47,
369
7,64
78,
374
0,5
15,0
1711
,155
9,98
28,
808
8,08
67,
639
7,36
97,
213
7,13
57,
112
7,17
97,
340
7,55
87,
811
8,08
68,
808
0,6
15,7
6911
,736
10,4
609,
242
8,52
48,
081
7,81
37,
658
7,58
17,
559
7,62
57,
785
8,00
18,
251
8,52
49,
242
0,7
16,5
0812
,312
10,8
949,
675
8,96
18,
521
8,25
58,
102
8,02
58,
004
8,06
98,
227
8,44
18,
690
8,96
19,
675
0,8
17,2
1412
,882
11,3
2810
,108
9,39
78,
960
8,69
68,
544
8,46
88,
446
8,51
18,
668
8,88
09,
128
9,39
710
,108
0,9
17,8
4213
,448
11,7
6210
,539
9,83
29,
397
9,13
58,
984
8,90
98,
887
8,95
29,
107
9,31
89,
564
9,83
210
,539
1,0
18,4
6514
,009
12,1
9510
,985
10,2
659,
833
9,57
29,
423
9,34
89,
327
9,39
19,
545
9,75
59,
999
10,2
6510
,985
1,2
19,7
5115
,117
13,0
6011
,846
11,1
3010
,701
10,4
4410
,295
10,2
2210
,201
10,2
6410
,417
10,6
2410
,866
11,1
3011
,846
1,4
20,9
6716
,208
13,9
2212
,705
11,9
9111
,565
11,3
1011
,163
11,0
9011
,070
11,1
3211
,283
11,4
8911
,729
11,9
9112
,705
1,6
22,1
6817
,161
14,7
8313
,562
12,8
5012
,426
12,1
7212
,027
11,9
5411
,934
11,9
9612
,146
12,3
4912
,588
12,8
5013
,562
1,8
23,3
5718
,061
15,6
4214
,417
13,7
0513
,283
13,0
3012
,886
12,8
1412
,793
12,8
5513
,004
13,2
0713
,444
13,7
0514
,417
2,0
24,5
3418
,955
16,5
0015
,269
14,5
5814
,137
13,8
8513
,741
13,6
7013
,649
13,7
1013
,859
14,0
6114
,298
14,5
5815
,269
2,2
25,7
0019
,843
17,3
5516
,121
15,4
0914
,988
14,7
3714
,593
14,5
2214
,502
14,5
6314
,711
14,9
1215
,149
15,4
0916
,121
2,4
26,8
5420
,769
18,2
1016
,970
16,2
5715
,836
15,5
8615
,442
15,3
7115
,351
15,4
1215
,560
15,7
6115
,997
16,2
5716
,970
2,6
27,9
9921
,647
19,0
6317
,818
17,1
0416
,683
16,4
3216
,289
16,2
1816
,198
16,2
5916
,406
16,6
0716
,844
17,1
0417
,818
2,8
29,1
3322
,522
19,9
1518
,665
17,9
4917
,527
17,2
7617
,133
17,0
6217
,042
17,1
0317
,250
17,4
5117
,688
17,9
4918
,665
3,0
30,2
5823
,394
20,7
6519
,510
18,7
9218
,369
18,1
1817
,975
17,9
0417
,883
17,9
4418
,092
18,2
9418
,531
18,7
9219
,510
Module 2 – Composite laminated panels. Design and optimization
Prepared in the frame of the FP7 KhAI-ERA project 63
2,5
3,0
3,5
4,0
4,5
5,0
6,0
7,0
8,0
9,0
10,0
12,0
14,0
16,0
18,0
20,0
0,07
56,
955
7,46
07,
664
7,93
48,
250
8,60
09,
367
10,1
5910
,750
11,2
0111
,693
12,7
4713
,820
14,5
3315
,291
16,0
66
0,1
7,06
47,
602
7,81
08,
083
8,40
28,
754
9,51
210
,316
10,9
4211
,399
11,8
9412
,952
14,0
4714
,791
15,5
5616
,333
0,15
7,28
37,
885
8,10
28,
380
8,70
49,
059
9,82
210
,630
11,3
2311
,791
12,2
9513
,362
14,4
9015
,280
16,0
5416
,842
0,2
7,50
28,
166
8,39
18,
676
9,00
49,
363
10,1
3010
,942
11,7
0012
,180
12,6
9313
,770
14,9
4515
,769
16,5
8117
,369
0,25
7,72
08,
445
8,67
88,
969
9,30
29,
664
10,4
3611
,253
12,0
5712
,566
13,0
8714
,175
15,3
4816
,268
17,0
7017
,903
0,3
7,93
88,
678
8,96
39,
260
9,59
89,
965
10,7
4111
,563
12,3
7212
,948
13,4
7914
,578
15,7
5216
,714
17,5
5918
,399
0,4
8,37
49,
112
9,52
89,
838
10,1
8510
,560
11,3
4712
,179
12,9
9813
,702
14,2
5415
,378
16,5
6017
,725
18,5
3619
,423
0,5
8,80
89,
545
10,0
8510
,408
10,7
6711
,150
11,9
4712
,790
13,6
4714
,444
15,0
1716
,168
17,4
1618
,618
19,5
2220
,452
0,6
9,24
29,
979
10,6
3610
,973
11,3
4211
,720
12,5
4313
,396
14,2
6415
,086
15,7
6916
,949
18,2
1219
,412
20,5
0621
,382
0,7
9,67
510
,411
11,1
3111
,518
11,8
9712
,296
13,1
3313
,999
14,8
7715
,711
16,5
0817
,721
19,0
0320
,272
21,4
7022
,393
0,8
10,1
0810
,844
11,5
6712
,069
12,4
5912
,867
13,7
1914
,597
15,4
8516
,332
17,1
9718
,483
19,7
8721
,063
22,3
0423
,385
0,9
10,5
3911
,275
12,0
0212
,614
13,0
1513
,433
14,3
0115
,191
16,0
9016
,948
17,8
2519
,236
20,5
6521
,853
23,1
6524
,330
1,0
10,9
8511
,706
12,4
3713
,115
13,5
6713
,995
14,8
7815
,781
16,6
9217
,561
18,4
4919
,980
21,3
3522
,639
23,9
5225
,288
1,2
11,8
4612
,567
13,3
0414
,026
14,6
5715
,105
16,0
2116
,950
17,8
8418
,774
19,6
8521
,410
22,8
5324
,264
25,6
0126
,952
1,4
12,7
0513
,427
14,1
6914
,902
15,5
7616
,199
17,1
4818
,105
19,0
6219
,974
20,9
0622
,682
24,3
3825
,793
27,1
5728
,514
1,6
13,5
6214
,284
15,0
3215
,774
16,4
6517
,161
18,2
6119
,246
20,2
2721
,160
22,1
1323
,991
25,7
3527
,296
28,6
9530
,162
1,8
14,4
1715
,140
15,8
9316
,644
17,3
8418
,061
19,3
6120
,374
21,3
7922
,376
23,3
0825
,224
27,0
1528
,725
30,2
9831
,702
2,0
15,2
6915
,995
16,7
5317
,511
18,2
6218
,955
20,3
5721
,490
22,5
2023
,538
24,4
9026
,444
28,2
7930
,114
31,7
8733
,226
2,2
16,1
2116
,848
17,6
1018
,376
19,1
3619
,843
21,2
7822
,594
23,6
4924
,690
25,6
6027
,652
29,5
9831
,398
33,1
2934
,835
2,4
16,9
7017
,699
18,4
6719
,239
20,0
0720
,769
22,1
9023
,593
24,7
6725
,830
26,8
1928
,848
30,8
2732
,667
34,5
3336
,235
2,6
17,8
1818
,549
19,3
2220
,100
20,8
7621
,647
23,0
9524
,530
25,8
3826
,960
27,9
6830
,033
32,0
4433
,921
35,8
1837
,564
2,8
18,6
6519
,398
20,1
7520
,959
21,7
4322
,522
23,9
9425
,456
26,8
6528
,080
29,1
0731
,208
33,2
5135
,249
37,0
8938
,980
3,0
19,5
1020
,246
21,0
2721
,817
22,6
0723
,394
24,8
8826
,375
27,8
1329
,118
30,2
3632
,373
34,4
4736
,472
38,3
4740
,264
coef
ficie
nts
for c
ritic
al s
hear
load
s ca
lcul
atio
n
Module 3 – Composite sandwich panels design and structural-technological solutions
Prepared in the frame of the FP7 KhAI-ERA project 65
Training Module 3
Composite sandwich panels designand structural-technological solutions
Ph.D., Ass. Prof. Pavlo Gagauz
National Aerospace University “KhAI”
2013
Module 3 – Composite sandwich panels design and structural-technological solutions
67 Prepared in the frame of the FP7 KhAI-ERA project
Introduction
According to statistics the exhaustion of the load-bearing capacity of thin-walled constructions is due to thebuckling. This fact led to widespread using in aircraft structures the sandwich panels and skins, which aremore efficient than conventional laminated panels. The efficiency of sandwich panels explains by increasingthe moment of inertia that caused by the separation of the base material on two face sheets. Their jointbehavior under bending loads is guaranteed by core, which can be made of foam, honeycombs, tubularfiller, etc.
Usually the task of the sandwich panels designing can be divided conventionally into two distinct phases:the designing of face sheets considering strength constraints and the calculation of minimally requiredthickness of core from buckling and deflection constraints.
This training module describes the engineering method for the sandwich panels designing. The semi-analytical dependences for determination of the critical distributed forces and deflection of the panel aregiven.
Training Objectives
studying the engineering approaches and design methods of composite sandwich panel understrength, buckling and stiffness constraints;
learning the semi-analytical dependences for determination of the critical distributed forces anddeflection of the sandwich panel considering the out-of-plane shear stiffness of the core;
studying the typical constructive modifications of the sandwich panel edges.
Module components
statement of the problem, list of constraints, basic assumptions and optimization strategy forcomposite sandwich panel designing;
analytical dependences for the shear stresses calculations in the core of honeycomb-type andtubular core;
schematic description of the possible alternatives of the joints of sandwich panels with the frameelements.
Target audience
- master and doctoral students
Advances in composite structures design and simulation
68 Prepared in the frame of the FP7 KhAI-ERA project
panel strength;
buckling loads;
transverse stiffness (panel deflection);
structural or/and technological (manufacturing) constraints.
Minimize objective function (basically panel weight)
which is subjected to the set of design constraints:
s c gG ab h min
General Problem Statement
Sign Convention
Module 3 – Composite sandwich panels design and structural-technological solutions
Prepared in the frame of the FP7 KhAI-ERA project 69
22 211
4 3 4
ij ij ijh hD B B ,
h h
Bending stiffness terms
0 1 10 0 05 5err errh . % ; h . %.
11
2 2x yc cxz yzxz yz
h hh ; h .G GG G
Shear stiffness terms for sandwich panel
Transverse shear modulus of composite sheets
12 5 10 GPaxz yzG G G .. .
For foam core 20 100 MPa.c cxz xz c cG G G A ..
Basic Assumptions for Initial Design
skins are identical with regard to LSS and thickness;
laminate stacking sequence are assumed to be orthotropic;
regardless of specific laminate stacking sequence skinsnegligibly affect overall bending stiffness (acting more likemembranes);
task sharing: sheets are intended for ensure panel strength,while core is intended to provide panel stability and bendingstiffness.
external loads and laminate thickness/stiffness are uniformalong the panel.
Advances in composite structures design and simulation
70 Prepared in the frame of the FP7 KhAI-ERA project
Strength Constraints
1 2 12; ; 1, 1,.., ;k k kf k n
Two basic approaches for laminate strength validation:
1) ply-by-ply analysis by chosen lamina failure criterion:
2) laminate in-the-large strength analysis.
Example for maximum stress failure criterion
1 1 1 2 2 2 12 12; ; , 1,.., ;
; ;
c k t c k t k
xyyxxc xt yc yt xy
F F F F F k n
qNNF F F F F .
Honeycomb core
So, it could be assumed c cx xz y yzhG ; hG .
0 866 0 577c cc cxz c yz c
c cG . G ; G . G .
R R
1 17 0 78c cxz yzG . GPa; G . GPa.
0 1mm 2mmc c. ; R : Aluminum core with regular hexagon cell,
For scheme (a)
Module 3 – Composite sandwich panels design and structural-technological solutions
Prepared in the frame of the FP7 KhAI-ERA project 71
2 21
2 22
12
1 1 1 21 2
2 2 2 12 1
12 12 12
, ;
, ;
, 2 2 ;
, ;
, ;
, .
x y xy
x y xy
y x xy
U cos sin sin cos
U sin cos sin cos
U sin cos
U E
U E
U G
Reduced strains and stresses for individual ply with orientationangle ( could be equal to 0, 90, or )
1 1 2 21 2
1 1 2 2
, 0; , 0;, ,
, 0; , 0.t t
c c
F FF U F U
F F
Strength limits for individual ply
1 2 1 2 30 90 1 , 11,22,33,12.ij ij ij ijB U u Q u Q u u Q u ij
Design vector
1 2 3 1 2, , , , ,U u u u v v
1 2,v v thickness ratios of all plies with orientation angles of0 and 90 accordingly.
Reduced membrane stiffness terms
22 12 11 122 2 3311 22 11 2212 12
; ; .x y y x xyx y xy
N B N B N B N B qU U U
BB B B B B B
Optimization Strategy for [0/90/±] Lay-Up
Reduced laminate strains
Advances in composite structures design and simulation
72 Prepared in the frame of the FP7 KhAI-ERA project
Strength constraint for laminate
1 2 3; ; ,str str str strU max
1 11
1 1
2 22
2 2
1 2 1 23
3 3 1 2 1 2
0, 0;
, 0 , 0.
0, 0;
, 90 , 0.
0, 1;
, ; , , 1.
str
str
str
u vU
U u v
u vU
U u v
u u v vU
max U u U u u u v v
Example. Assuming function will returnminimal necessary value of total thickness for sheets with cross-ply lay-up [±45].
0, 0, 45U , str U
1 2 12
1 2 12, ; ; ;U max
F F F
2 2 2
1 1 2 2 12
1 1 2 2 12, ;U
F F F F F
Strength constraint for individual ply- maximum stress criterion
- TsaiHill criterion
21 21
1 1 1 2 2
2 2 22 11 12 1 2 22 661 2 12
, ,
1, ;
2
, 2 .
U a a a
a U p p
a U p p p p
- TsaiWu criterion
Module 3 – Composite sandwich panels design and structural-technological solutions
Prepared in the frame of the FP7 KhAI-ERA project 73
2 2 2 2 2 2
3
2 216 x yx xy y xy xystr
E b a G a E a b G bph max , ,a bd
2 2
2 2x y xy xy yb ad E E G E .a b
For initial design
where
2 2 2 211 3 22 3
2 2 2 216 16w w
xmax ymaxpA D b D a pA D a D bQ ; Q ,
a ba b a b
Maximal shear forces (on edges)
2 211 11 33 22 33
2 2
11211 2 2
1x y x y y x
wyx
x y
T D D D Da bA .
TLa b
where
Core shear failure constraint
yxxz yz
QQ ; ,h h
3 3 3 3
11 3 22 3 3 12 333 2 3 2 2x y
w w w wQ D D ; Q D D ; D D D .x x y y x y
where
21 1
16 1m n
mnm n
pw sin xsin y,mnA
Transverse deflection of panel
m nm n; .a a
where
Advances in composite structures design and simulation
74 Prepared in the frame of the FP7 KhAI-ERA project
Buckling and Deflection Constraints
2
0 0 01y xyx
x y xy
N qN .N N q
Global buckling criterion
Buckling forces (assuming core is absolutely rigid in shear)
0 22 2 2 211 22 11 22
0
04 4
x x x xx y
y y y y
xy xy xy xy
N k k kE ED D B B h hN k k k .
ab ab abq k k k
First, rough value of core thickness
2 2
22
4y xy yx xbuc
x y xy x yx y
N q Nab N Nh .k k k k kE E
3
2
2
2
xmax cхz h
c
ymax ymax cyz h
c c
Q R;
hQ R Q R
;h h
2
2
xmaxхz c
p
ymaxyz c
p
Q t;
hQ
.
Module 3 – Composite sandwich panels design and structural-technological solutions
Prepared in the frame of the FP7 KhAI-ERA project 75
Comparison of different theories for calculation of buckling loads
, град.a/b=0.5 a/b=1.0 a/b=2.0
Nx0,Н/мм
Nx0*,Н/мм , % Nx0,
Н/ммNx0*,Н/мм , % Nx0,
Н/ммNx0*,Н/мм , %
0 254.7 172.7 47.5 158.9 129.1 23.1 198.5 179.2 10.8
15 246.4 172.2 43.1 183.8 149.4 23.0 275.3 240.2 14.6
30 219.8 162.4 35.3 233.6 187.5 24.6 467.3 313.4 49.1
45 174.0 134.8 29.0 258.6 205.4 25.9 517.1 316.6 63.3
60 117.3 95.79 22.4 233.6 162.3 44.0 432.4 249.5 73.3
75 68.82 60.05 14.6 137.6 106.9 28.8 275.3 176.2 56.3
90 49.63 44.79 10.8 99.26 81.87 21.2 198.5 140.6 41.2
, град.a/b=0.5 a/b=1.0 a/b=2.0
Nx0,Н/мм
Nx0*,Н/мм , % Nx0,
Н/ммNx0*,Н/мм , % Nx0,
Н/ммNx0*,Н/мм , %
0 254.7 172.7 47.5 158.9 129.1 23.1 198.5 179.2 10.8
15 246.4 172.2 43.1 183.8 149.4 23.0 275.3 240.2 14.6
30 219.8 162.4 35.3 233.6 187.5 24.6 467.3 313.4 49.1
45 174.0 134.8 29.0 258.6 205.4 25.9 517.1 316.6 63.3
60 117.3 95.79 22.4 233.6 162.3 44.0 432.4 249.5 73.3
75 68.82 60.05 14.6 137.6 106.9 28.8 275.3 176.2 56.3
90 49.63 44.79 10.8 99.26 81.87 21.2 198.5 140.6 41.2
, град.a/b=0.5 a/b=1.0 a/b=2.0
qxy0,Н/мм
qxy0*,Н/мм , % qxy0,
Н/ммqxy0*,Н/мм , % qxy0,
Н/ммqxy0*,Н/мм , %
0 577.6 261.5 121 333.2 201.1 65.7 235.7 173.1 36.2
15 686.7 301.8 128 402.0 233.8 71.9 320.2 216.0 48.2
30 811.9 359.2 126 524.1 288.7 81.6 525.7 300.7 74.8
45 739.3 365.5 102 585.2 313.8 86.5 739.3 364.9 103
60 525.7 300.7 74.8 524.1 288.7 81.6 812.4 358.5 127
75 320.2 216.0 48.2 402.0 233.8 71.9 686.7 301.8 128
90 235.7 173.1 36.2 333.4 201.1 65.7 577.6 261.5 121
, град.a/b=0.5 a/b=1.0 a/b=2.0
qxy0,Н/мм
qxy0*,Н/мм , % qxy0,
Н/ммqxy0*,Н/мм , % qxy0,
Н/ммqxy0*,Н/мм , %
0 577.6 261.5 121 333.2 201.1 65.7 235.7 173.1 36.2
15 686.7 301.8 128 402.0 233.8 71.9 320.2 216.0 48.2
30 811.9 359.2 126 524.1 288.7 81.6 525.7 300.7 74.8
45 739.3 365.5 102 585.2 313.8 86.5 739.3 364.9 103
60 525.7 300.7 74.8 524.1 288.7 81.6 812.4 358.5 127
75 320.2 216.0 48.2 402.0 233.8 71.9 686.7 301.8 128
90 235.7 173.1 36.2 333.4 201.1 65.7 577.6 261.5 121
2 mm; 50 MPaG
where
2 2
4 2 2 411 12 33 22
22 233 11 22 12 33
2 2 2 211 33 22 33
2 2
2
1 11
mn mn m y n xmn
mn
mn m m n n
mn mn m n
mnmn m n n m
x y x y
L TA ;
S
L D D D D ;
T D L D D D D ;
TS D D D D .
0 0 02 232mn mn
x y xym,n m,nm n
A AN min ; N min ; A w q B w,ab
More accurate expressions for buckling forces according FSDT
Advances in composite structures design and simulation
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Local buckling modes
maxw w .Transverse stiffness constraint
2 211 11 33 22 33
2 2
2 11211 2 2
116 x y x y y x
maxyx
x y
T D D D Dp a bw .
TLa b
Initial approximation for core thickness by stiffness constraint
38
defab ph .
d w
For set of constraints
str buc defh max h , h , h .
Maximal panel deflection
Module 3 – Composite sandwich panels design and structural-technological solutions
Prepared in the frame of the FP7 KhAI-ERA project 77
Shear buckling of honeycomb sell wall
2 33
2 12c c c
c xmax ymax cc
R Eq Q Q k .h hR
Shear buckling of tubular core (tubes are laid along x-axis)
32 2
12
p x yxmaxp p
E EQ tq k .
h ha
Assumptions for face-sheet wrinkling prediction
Advances in composite structures design and simulation
78 Prepared in the frame of the FP7 KhAI-ERA project
Structural-Technological Solutionsfor Composite Sandwich Panels
Basic methods for sandwich panel manufacturing
Module 3 – Composite sandwich panels design and structural-technological solutions
Prepared in the frame of the FP7 KhAI-ERA project 79
Handling edges
Advances in composite structures design and simulation
80 Prepared in the frame of the FP7 KhAI-ERA project
Joining of Composite Sandwich Panels
Module 3 – Composite sandwich panels design and structural-technological solutions
Prepared in the frame of the FP7 KhAI-ERA project 81
Advances in composite structures design and simulation
82 Prepared in the frame of the FP7 KhAI-ERA project
Joining with Frame Elements
Module 3 – Composite sandwich panels design and structural-technological solutions
Prepared in the frame of the FP7 KhAI-ERA project 83
References
1. Jones, R.M. Mechanics of Composite Materials, 2-nd ed., Taylor&Francis, Inc., Philadelphia,1999.
2. Barbero, E.J. Introduction to Composite Materials Design, 2-nd ed., CRC Press, Boca Raton,2010. – 336 p.
3. Kollar, L.P., Springer, G.S. Mechanics of Composite Structures, Cambridge University Press,2003. – 480 p.
4. Karpov, Ya.S. Composite items and structural components design, [in Russian], Kharkiv,KhAI, 2010. – 768 p.
Module 4 – Composite beams and spars design
Prepared in the frame of the FP7 KhAI-ERA project 85
Training Module 4
Composite beams and spars design
Dr.Sc., Prof. Yakov Karpov
National Aerospace University “KhAI”
Ph.D., Ass. Prof. Fedir Gagauz
National Aerospace University “KhAI”
2013
Module 4 – Composite beams and spars design
Prepared in the frame of the FP7 KhAI-ERA project 87
Introduction
It is impossible now to identify any area of technique where the beams were not used. Among theelements of aircraft structures the beams represent the spars and the ribs of wing and tail, cross-beams ofthe cabin floor, landing gear struts etc.
Using of composites in the designing of beams is very promising due to possibility of specifying the optimallay-up of individual elements to ensure the best possible perception of loads. According to features of load
perception the optimal lay-up for flanges would be [0] and most efficient lay-up for web is [45]. Such
character of lay-up for individual elements of beams allows using the differential design approach theflanges are calculated according to tensile or compressive forces initiated by bending moment and axialforce, and thickness of web is defined from strength restriction under shear force only.
In this training module the problems of designing and manufacturing of composite beams and spars arepresented. The methodology of individual elements designing in the cross-section of composite beamunder several load cases is described that is the most relevant since the exploitation of real aircraftstructures occurs under significantly changing flight conditions. A lot of attention paid to the problem of theweb buckling and to the analysis of additional stresses in the elements due to edge effect.
Training Objectives
studying the design methods of flanges and web in the beam cross-section under strengthconstraints;
learning the general constructive approaches to ensure web stability;
understanding the mechanism of the edge effect occurrence in the flanges due to difference ofoptimal lay-up of the individual structural elements.
Module components
statement of the problem, list of constraints, basic assumptions and optimization strategy for thecomposite beam design;
engineering methods of web designing under strength and buckling constraints;
simplified analytical formulas to perform stress-strain analyses of the flanges and web in the zoneof edge effect;
schematic description of the composite beams manufacturing.
Target audience
- master and doctoral students.
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Rational Cross-section
Bending and shear stresses distribution for heterogeneoussection of the beam with optimal lay-up of the flanges and web:
Typical diagrams of tensile strength and shear strengthfor different laminates:
Object of InvestigationCross-sectional loads under general type of loading:
M – bending moment
Q – shear force
N – axial force
Design model hypothesis – linear strain distribution:
Module 4 – Composite beams and spars design
Prepared in the frame of the FP7 KhAI-ERA project 89
RestrictionsStrength of the flanges under bending stresses:
bz х N u u 1t 1c
ef u u
uz х N b b 1t 1c
ef b b
1 M N y F ; F F FH b 2
1 M N Н y F ; F F FH b 2
Problem Statement
Design variables:
u u u b b b w u b u b w wG b b Н b b 2 min
– thickness of the flanges;
w – thickness of the web;
u bb , b – thickness of the web
Objective function:
u b,
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Strength of the flanges under shear stresses:
Beam design model:
yu
ef u
yb
ef b
QILSS ;
H bQ
ILSS ;H b
y 1212
ef u,b 12 w 45
Q Ge FH G e G
Thin-walled bar model:
e=1 for C – section;e=0.5 for I - section
Module 4 – Composite beams and spars design
Prepared in the frame of the FP7 KhAI-ERA project 91
Strength of the web:
y
45ef w
QF
H
2 3xy x y w y
xy yx ef ef
k E E Q12(1 )H L H
Buckling stability of the web under shear loading:
x y xy xy wE , E , , G ,– depends on boundary conditions,See supplementary materials to training module 2.
xyk
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Design Procedure. Several LC
j j j b1c z x 1tN
ef u u
j j j u1c z x 1tN
ef b b
1F M N y FH b 2
1F M N Н y FH b 2
Strength conditions of the flanges under bending stresses:
j 1, ,m; m number of LC
Beam optimization with strength restrictionsThe designing procedure is based on previous algorithm by varyingvalues of the flanges width.
ymin
ef
Qb
H ILSS
Module 4 – Composite beams and spars design
Prepared in the frame of the FP7 KhAI-ERA project 93
ExampleBeam with strut under 2 LC:
0 k
0 k
LC1: q 12 kN m; q 8 kN m;LC2 : q 8 kN m; q 4 kN m.
j 1, ,8
4242420-84-84-840-2.2402.528.444.490-5.04-16.89
87654321Load Case
4242420-84-84-840-2.2402.528.444.490-5.04-16.89
87654321Load Case( j)zM , kNm( j)xN , kN
Advances in composite structures design and simulation
94 Prepared in the frame of the FP7 KhAI-ERA project
BzM 1Nm
-0.244
0.244
3
-0.854
0.142
2
-0.204-0.149
-0.395-0.646-2.260-0.135
0.6710.102
1.7690.6460.3770.270
876541LC
-0.244
0.244
3
-0.854
0.142
2
-0.204-0.149
-0.395-0.646-2.260-0.135
0.6710.102
1.7690.6460.3770.270
876541LC
Definition of the stress-limits for flanges:Example
u1F , MPa
b1F , MPau2F , MPa
b2F , MPa
Bu1 u1tu B B B
u u1 u2Bu2 u2cu
F min F 0.270MPaF min F , F 0.102MPa
F min F 0.102MPa
B Bz z
u bB Bef u u ef b b
M M0.87mm; 0.66mmH b F H b F
Bb1 u1tb B B B
b b1 b2Bb2 u2cb
F max F 0.135MPaF max F , F 0.135MPa
F max F 0.149MPa
ExampleDefinition of the flanges loading:
1t 1c u b efF 2280MPa; F 1725MPa; b b 60mm; H 0.9H 189mm.
+
-
+
-
3
+
-
+
-
2
---++-
+++--+
----++
++++--
876541Load Case
+
-
+
-
3
+
-
+
-
2
---++-
+++--+
----++
++++--
876541Load Case z х N ef
ef u 1t х
M N y H Н2H b F N
z х N ef
ef u 1c х
M N y H Н2H b F N
z х ef N
ef b 1t х
M N H y2H b F N
z х ef N
ef b 1c х
M N H y2H b F N
Numbers of load cases with tension of the upper flange: tu=5,6,7,8;Numbers of load cases with compression of the upper flange: cu=1,2,3,4;Numbers of load cases with tension of the bottom flange: tb=1,6,7,8;Numbers of load cases with compression of the bottom flange: cb=2,3,4,5;
Module 4 – Composite beams and spars design
Prepared in the frame of the FP7 KhAI-ERA project 95
Bending stiffness:
3 3ws wb ws wb
1 str buck ws wb
3 3ws wb ws wb
2 str buck ws wb
3 3ws wb ws wb
3 str str buck buck ws wb
D b11 b1112 12 4
D b22 b2212 12 4
D b12 2b33 b12 2b3312 12 4
Homogeneous web:
2xy 1 2 y
ef ef
k D D QH L H
Buckling Stability Providing
Buckling stability restriction:
1 2D D increasing
Plies lay-up optimization Constructive transformation
Advances in composite structures design and simulation
96 Prepared in the frame of the FP7 KhAI-ERA project
Module 4 – Composite beams and spars design
Prepared in the frame of the FP7 KhAI-ERA project 97
3 3yw2 w wywrib min
rib
512
t E tKf E 5E f12 KE
Web with stiffening rib:Necessary condition of the stiffening rib’s efficiency:
rib 1E EribE
yw strE b22
– elastic modulus of the rib in y-dir of the beam. Generally,
f – cross-sectional area of the stiffening rib
K – cross-sectional shape factor
Example:
2yw w rib w r
0x w rib
3w
2w yw w 0 0 w
32rib r r
rib r r r w 0
E t E f 2 b1z2 E t E f
D E z z3
E bD b b zt 12
2r
r rbf b ; K12
2 2w ribD D D
Advances in composite structures design and simulation
98 Prepared in the frame of the FP7 KhAI-ERA project
Structural-Technological Solutions
Module 4 – Composite beams and spars design
Prepared in the frame of the FP7 KhAI-ERA project 99
Advances in composite structures design and simulation
100 Prepared in the frame of the FP7 KhAI-ERA project
Edge Effects
zf f zsh sh
хf f хsh sh
0;0;
zf zsh хf хsh;
Mechanism of temperature edge effect:
Equilibrium equation:
Strain compatibility condition:
Module 4 – Composite beams and spars design
Prepared in the frame of the FP7 KhAI-ERA project 101
11 23 21 13 22 13 12 23xf zf2 2
11 22 12 11 22 12
a a a a a a a a;a a a a a a
хzf хzff f11 12 21 13 zsh zf
zf zsh sh хf sh хsh
f22 23 хsh хf
хf хsh sh
1 1a ; a a ; a Т ;E E E E
1 1a ; a ТE E
f fxsh xf zsh zf
sh sh;
zsh хshzf хfzf хzf zf zsh хzsh zsh
zf хf zsh хsh
хsh zshхf zfхf zхf хf хsh zхsh xsh
хf zf хsh zsh
Т; ТЕ Е Е Е
Т; ТЕ Е Е Е
According to Duhamel – Neyman hypothesis:
Relations of the temperature edge effect :
Additional stresses in the flanges determine from strains compatibility conditionand equilibrium equations:
Stresses in the web-shoulder:
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– active axial stresses in the flange and web-shoulder, respectively(from bending moment and axial force)
хzshf f хzf f11 22 12 21
zf sh zsh хf sh хsh хf sh хsh
shf13 хzf хzsh 23
хf хsh
1 1 1 1a ; a ; a a ;E E E E E E
a ; a 0.Е Е
11 23 21 13 22 13 12 23xf zf2 2
11 22 12 11 22 12
a a a a a a a a;a a a a a a
*хf zf zf хf f
хf zхf zf хzf хzfхf zf zf хf хf
*хsh zsh zsh хsh sh
хsh zхsh zsh хzsh хzshхsh zsh zsh хsh хsh
;Е Е Е Е Е
;Е Е Е Е Е
* *f sh,
Relations of the Poisson’s edge effect :According to Hook's law:
Additional stresses in the flanges determine from strains compatibility conditionand equilibrium equations:
Stresses in the web-shoulder: f fxsh xf zsh zf
sh sh;
Module 4 – Composite beams and spars design
Prepared in the frame of the FP7 KhAI-ERA project 103
References
1. Timoshenko, S.P., Gere, J.M. Mechanics of Materials, 4-th ed., PWS Publishing Co., Boston,1997.
2. Pilkey, W.D. Formulas for stress, strain, and structural matrices, 2-nd ed.,John Wiley & Sons, Inc., New Jersey, 2005.
3. Jones, R.M. Mechanics of Composite Materials, 2-nd ed., Taylor&Francis, Inc., Philadelphia,1999.
4. Barbero, E.J. Introduction to Composite Materials Design, 2-nd ed., CRC Press, Boca Raton,2010. – 336 p.
5. Karpov, Ya.S. Composite items and structural components design, [in Russian], Kharkiv,KhAI, 2010. – 768 p.
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 105
Training Module 5
Designing and strength analysis of the jointsof aircraft composite structures
Dr.Sc., Prof. Yakov Karpov
National Aerospace University “KhAI”
Ph.D., Ass. Prof. Fedir Gagauz
National Aerospace University “KhAI”
Ph.D., Ass. Prof. Pavlo Gagauz
National Aerospace University “KhAI”
2013
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 107
Introduction
Joints are the basis for the operation of any technical object. Aircraft structures differ the great quantity offunctional, operational and technological joints and connections. An essential feature of all types ofconnections is that they are the source of irregularity of the stress field from one hand and from another -require some special properties of the joining materials (hardness, wear resistance, adherence etc.).
In general composite structures can be joined by any type of joints traditional for mechanical engineeringexcept welded and brazed joints. Since composite material exists due to adhesive joints of individual plies,then adhesive joints of composite parts is more natural than any other type of joint. The other reason isthat adhesive joints are structurally more efficient than mechanically fastened joints because they canprovide more opportunities for uniform stress distribution and will lead to increasing of the load-carryingability. Due to high sensitiveness to manufacturing deficiencies the assurance of adhesive joints quality isthe one of significant problem. Thats why mechanical fastening has preference over adhesive bondingespecially in the highly stressed critical aircraft structures. However the mechanically fastened joints ofcomposite structures require solving some significant problems related to providing necessary shearingstrength and bearing strength and to decreasing of sensitivity to stress concentrators.
This training module describes design procedures and stress analysis methods in the structural joints forcomposite structures.
Training Objectives
studying the analysis technique for stressess prediction in the adhesive and fastened joints ofcomposite structures;
learning the possibility of load-carrying ability increasing of the adhesive and mechanically fastenedjoints and current tendencies in development of advanced types of joints.
Module components
methods of stress analysis in the adhesive and mechanically fastened joints;
test equipment and testing procedures for bearing strength determination of composite materials;
analytical dependences for the stress concentration coefficients calculations;
schematic description of possible constructive solutions for the increasing of load-carrying ability ofmechanically fastened joints.
Target audience
- master and doctoral students.
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108 Prepared in the frame of the FP7 KhAI-ERA project
Classification based on joints functionality:
- movable (sliding) and fixed joints
Classification based on joints maintenance:
- separable (detachable) and permanent joints
Classification based on type of design scheme:
- discrete and continuous joints
Classification based on physical-chemical process of joints
realization:
- mechanically fastened, welded, adhesive and
brazed (soldered) joints
Joints classification
Some facts and information in the areaof joints designing
Joints add 20% to airframe weight
Responsible for 80% of aircraft failures
Are the sources of irregularities (both structural andstress distribution)
Require additional special features of the joining parts(machinability, wear resistance, microhardness, etc.)
Load-carrying ability of the mechanical joints of thecomposite structures is 2-3 times less compared withthe metallic ones
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 109
The generality of the coverage of existing and potentialtypes of joints
Continuity or the possibility of monotonous orquasimonotonous transition of one joints class to another
The ability to predict trends in the development of jointsand improving of their operability
Possibility of a comparative evaluation of the effectivenessand quality of the classes and individual types of joints
Figurativeness and informative description of the types ofjoints
The possibility of predetermination of design schemes andmodels
Advantages of joints classification based onthe geometry of connective points dislocation
Classification based on the geometry ofconnective points dislocation
discrete joints – load transfer through discretelylocated points (bolts, screws, rivets etc.)
linear joints – load transfer through line of force
surface joints – load transfer through contact surface
volume joints – load transfer through whole cross-section of joining members
Advances in composite structures design and simulation
110 Prepared in the frame of the FP7 KhAI-ERA project
Problems of stress analysis of thecomposite plates being joined:
ss.scr24N ;d
Shear-out strength constraint:
i iss.i
N2 c
Bearing strength constraint:
bs.i bs.scri
N min ,d
Net section strength constraint:
x.i
xt.ii
k N Fb d
Stress analysis of discrete joints
ss bs x, , ?k ??
Comparative assessment of joints efficiencyEfficient factor = workability (operability) factor
ultimate failure load of jointworkability factorultimate failure load of members
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 111
There are no analytical dependences for bearingstrength of composite materials prediction
Value of the bearing strength doesn’t correlate withultimate tensile strength of material
Bearing strength significantly depends on diameterof fastener [Ref.1]
Anisotropy of bearing strength (influence of the forceresultant direction on bearing strength) can be neglected[Ref.1]
Accurate design of the mechanically fastened jointsrequires pretesting of composite materials for allpossible diameters of fasteners and applicable lay-up
Problems of the bearing strength predictionfor composite materials
Problems of the shear-out strength predictionfor composite materials
90,0 ss 0,90
General approach:
0,90 90,0 ss ?For UD composites
ss experiment
For other composites with different lay-up
or ss ILSS (conservative design approach)
Advances in composite structures design and simulation
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Testing tool for bearing strengthdetermination
Proposed testing methodfor bearing strength determination
bs
bsNd
Testing technique based on tensile tests of compositespecimens and determination of bearing strength by point ofdeviation from linearity [Ref. 2]
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 113
Results of the bearing strength tests [Ref.3]
Test specimen for bearing strengthdetermination
xx
yy
xx
yy
Testing directions (studying anisotropy of bearing strength):0º; 22.5º; 45º; 67.5º; 90º
Test specimen thickness:1.74; 3.48; 5.22; 6.96; 8.7; 10.44
Studying lay-up:
4 4 s
3 3 s
s
s
[0 / 45 / 45 / 0 ][0 / 45 / 0 / 90 / 45 / 0 ][ 45 / 45 / 0 / 45 / 0 / 90 / 45 / 0 / 45 / 45][0 / 45 / 0 / 45 / 0 / 90 / 45 / 0 / 45 / 0]
Studying fastener diameter:3; 5; 6; 8; 10
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114 Prepared in the frame of the FP7 KhAI-ERA project
Stress concentration of the tensile plate
4 4xy 2 2
x xy x y
21 sin 1 cossin cosE E G E E
2 2x0
x
x x xxy
y xy y
E 1 sin cosE
E E E2 ;E G E
Distribution of tangential stresses [Ref.4]:
Elastic modulus in tangential direction:
Inconsistency: not depend on hole diameter
max2
x xx xy
y xy
E Ek 1 2E G
max2
x xx xy
y xy
E Ek 1 2E G
Problems of the stress concentrationprediction for composite materials
The problem was studied by Lekhnitskii [Ref.4] fororthotropic plates with cylindrical and elliptical hole undervarious type of loading (tension, compression, shear).Accepted assumptions: Hole diameter much lesscomparing the plate dimensions Hole is located far fromplates edges
r
rr
r
r
0
maxk
Coefficient of stressconcentration:
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 115
Efficiency improvement ≡ Load-carrying ability increasing
221
2 21
dd n4 4
d nd
Effectiveness increase of discrete joints
Approaches for load-carrying ability increase: Change of the jointing parameters to minimize the stress Improvement of strength properties of joining members in
the area of connection
The most elementary method – increase of fasteners quantityPossible criterion which can be used – equal shearingstrength of the fasteners:
Minimization of stress concentration
For UD CFRP (AS4/3501-6):
x147 147k 1 2 0.27 810.3 7
x xx xy
y xy
E Ek 1 2E G
The optimal lay-up of composite material in the joints can bedetermined according to inverse dependence on the elasticmodulus in transverse direction and in-plane shear modulus:
Reinforcement by plies with lay-up [90] and [45]
Advances in composite structures design and simulation
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Stress analysis in the multi-rowmechanically fastened joints
x.i
x.ii
k N Ft nd
Influence of the fastener quantityon load-carrying ability of discrete joints
Shear-out strength:i i
ss.iN
2 nc
Bearing strength: bs.i1 i
Nnd
Net section strength:
– rise n times
– rise timesn
– drop (if not depend on dia, but it is)xk
Possible way to keep orincrease strength in net-section – using ofstaggered jointsor multi-row joints
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 117
Main difficultiesThe equilibrium equations for the joined members(according to method of decomposition) and for joint at whole:
n n
1xn 1x0 xi 2xn 2x0 xii 1 i 1
1x0 2x0 1xn 2xn
N N Q 0; N N Q 0;
N N N N
The problem (n-1) times statically indeterminate.Hence, strain compatibility conditions should be involved
Uniform stress distributionthrough the thicknessof the parts being joined
Hypothesis of the Duamel – Neyman (summation of themechanical and thermal deformations)
No friction (forces and stresses in the joining membersbetween two adjacent rows of the fasteners are constant)
Assumptions of the design scheme
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Flexibility of the joining members
1xi 2xi1i 1xi 2i 2xi
1 1П ; Пb E b E
For constant width, thickness and elastic propertiesof the plates between two adjacent rows of fasteners:
For distributed thickness and elastic properties of thejoining parts between two adjacent rows of fasteners:
i 1 i 1
i i
x x
1хi 2хixi 1i 1xi xi 2i 2xix x
1 dx 1 dxП ; П ;t b (x)E (x) t b (x)E (x)
i i 1x , x
i 1 i 1
i i
x x
1xi 1xi 2xi 2xixi xix x
1 1(x)dx; (x)dx,t t
– boundary coordinates of the area between two
adjacent rows of the fasteners
Strain state of the jointbetween two adjacent rows of fastener
'xi зхi
' 'xi 2xi
'xi 1xi
' ''x,i 1 3х,i 1
BB Q П
B C t
AD t
DD Q П
' ' 2xi 2xixi 2xi xi 2xi xi 2xi
2xi 2i 2xi
' 1xixi 1xi xi 1xi
1i 1xi
NB C t t T t TE b ENAD t t T
b E
зхП – fastener flexibility
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 119
' ' ' ' ' ''BB BC AD DD
Strain compatibility conditionsof the multi-row joints
xi зхi x,i 1 зх,i 1 xi 2xi 2хi 1xi 1хi xi 1xi 2xiQ П Q П t N П N П t T
According to strain state of the joint between two adjacentrows of fasteners:
Flexibility of the fastener
iзхi
xiП
Q
Flexibility of the fastener depends ondiameter of the fastener, thicknessand elastic properties of theconnected parts, diameterclearance or tightness etc.
In general case:
3хf 1 1х 2 2х
5 1 1П 0.8dE Е Е
Methods for the fastener flexibility determination:
1 2
d d
3х1 1х f 2 2х f
1.25 1 3 1.25 1 3ПЕ 8Е Е 8Е
Approach #1:
Approach #2:
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Example
x1 х x 3х x2 3х 1х x
x1 х x x2 х x 3х x3 3х 1х x
x1 x2 х x x3 х x 3х x4 3х 1х x
x1 x2 x3 x4
Q П t П Q П NП t
Q П t Q П t П Q П NП t
Q Q П t Q П t П Q П NП t
Q Q Q Q N
1x0 2xn
1xn 2x0
N N NN N 0
Load transfer(no thermal loading):
Thickness and elasticproperties of the members are constant, i.e. 1хi 2хiП П const
Resolving system of equations:
х 1х 2хП П П
Resolving system of equations
Rewriting of the strain compatibility conditions of the joint:
According to equilibrium of “disassembled” parts:i
1xi 1x0 xkk 1
N N Q
i
2xi 2x0 xkk 1
N N Q
i 13х,i 13хi
1хi 2хi xk xi 1хi 2хi x,i 1xi xik 1
1x0 1хi 2x0 2хi 1xi 2xi
ППП П Q Q П П Qt t
N П N П T ; i 1,...,n 1
n
xk 1x0 1xn 2xn 2x0k 1
Q N N N N
Equilibrium equation:
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 121
Structural and technological solutionsfor bearing strength increase
Use of metallic bushing for the hole quality improvement
Mechanically fastened multi-row jointwith uniform loading of the fasteners
xiNQnCriterion:
1хi 2хi 3хi 3х,i 1 1хixi
i NП П N П П NП ; i 1,...,n 1n nt
Strain compatibility conditions of the joint (no thermal loading):
1хi 2хi 2хi 1 1x
1хi 1хi 2 2x
П П П En n xor 1, i.e.П i П i E x
lIf the fastener flexibilities are equal:
Scarf joint
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Use of special washers
Structural and technological solutionsfor bearing strength increase
Use of metallic washers and hybridizing of the laminatewith metallic foil
Structural and technological solutionsfor bearing strength increase
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 123
Basic assumptions of design scheme
Materials of adherends are orthotropic in х-у axes
Normal stresses in adherends and shear stresses in thebond layer are distributed uniform through thethickness
Bond layer percept only shear stress
Geometry and elastic properties of adherend and bondlayer are constant along surface of the joint
Applied forces are distributed uniform along theadherends edges
One-dimensional design model can be used
Рисунок 36.4
Design scheme of adhesive joint (parameters designation):
Mechanism of the shear stress appearance:
Adhesive (surface) joints
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Equilibrium and deformed stateof adhesive joint
Equilibrium equationsof the adherends:
1x1 b
2x2 b
db dx bdx 0dx
db dx bdx 0dx
1x1 b
2x2 b
d 0dx
d 0dx
bb xz 2 1 b xz b b b
btg u u П
G
Strain compatibility condition:
Models of adhesive joint [Ref.5]Actual distribution of the shear stressin the adhesive jointsAccepted assumptions: shear deformations concentrate
in adhesive only – classic approach shear deformations concentrate in adhesive and
in half-thickness of adherends – Volkersen modification
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 125
Volkersen modification:
b
b b
bb
b
tgG
G
1 g 2
b b b1 1 2 21 xz1 2 xz2 g g g g
xz1 xz2 g; ;
2 G 2 2 G 2 G
1 g 2 g1 2b
b xz1 xz2 gП
2G 2G G
Flexibility of bond layer. Approach 2
2 1 b b bb
u u П П
From strain compatibility condition:
Classical model of adhesive joint (rigid adherends):
gbb g
b b g g
1ПG G
gbb
b g gtg
G G
Flexibility of bond layer. Approach 1
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1x1 b
2x2 b
2b2x 1x
b22x 1x
d 0dx
d 0dx
dd d1 1 ПE dx E dx dx
b b1x 2x
1 2
d d;dx dx
Strain compatibility condition:2
bc b2
2x 2 1x 1
d1 1 ПE E dx
2b
b 1x 2x b2dП П Пdx
22 b
b 2dk 0dx
2 1x 2x
1x 2xb 1x 1 2x 2
П П 1 1k ; П ; ПП E E
Derivation of the shear stress
Resolving system of equations1x
1 b
2x2 b
2 1 b b
d 0dx
d 0dx
u u П
xux
b2x 1x b
d Пdx
1x 2x1x 1x 2x 2x
1x 2xT; T
E E
b2x 1x2x 1x b
2x 1x
dT ПE E dx
Strain compatibility condition:
1x1 b
2x2 b
2b2x 1x
b22x 1x
d 0dx
d 0dx
dd d1 1 ПE dx E dx dx
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 127
1x 2x1 2
d d 0dx dx
Equilibrium equations of the adhesive joint:
1 1x 2 2x 3C
10 20 1n 2n3
N N N NCb b
b2x 1x2x 1x b
2x 1x
dT ПE E dx
2x 2x 2 1x 1x 1 2x 1x b 1 2П П T П C chkx C shkx
1 1x 2 2x 3
2x 2x 2 1x 1x 1 2x 1x b 1 2
CП П T П C chkx C shkx
Derivation of the normal stresses
Strain compatibility condition:
Рисунок 36.4
22 c
c 2dk 0dx
c 1 2C shkx C chkx
С1, С2 – integration constant
Boundary conditions:
10 201x 2x
1 2
N Nx 0 : ;b b
1n 2n1x 2x
1 2
N Nx : ;b b
l
Boundary conditions
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10 2n
1n 20
N N NN N 0
T 0
Example. Load transfer
NN
1 2х хb
b
П chk x П chkxN
kП b shk
ll
1 2
1 2
х хb
b
х хb
b
П chk П0 NkП b shk
П П chkNkП b shk
lllll
b
1 1x 3 2х b 1 2 2x 1x1х 2х
2 2x 3 1х b 1 2 2x 1x1х 2х
b 1 2
1 C П kП C chkx C shkx TП П
1 C П kП C chkx C shkx TП П
C shkx C chkx
20 2х 10 1х1 2x 1x
b
10 1х 20 2х2n 2х 1n 1х2 2x 1x
b b b
10 20 1n 2n3
N П N П1C TkП b
N П N ПN П N П chk 1 chkC TkП b shk shk kП b kП shk
N N N NCb b
l ll l l
Basic equations for stress analysisin the adhesive joints under tension
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 129
N 1 N
10 1n
20 2n
N N; N NN 0; N 1 N
T 0
N
2 1х хb
b
П 1 chkх П chk х chkхN
kП b shk
ll
2 1
2 1
х хb
b
х хb
b
П 1 П chk0 N
kП b shk
П 1 chk П 1 chkN
kП b shk
ll
l ll
l
Example. Load sharing
N N10 1n
20 2n
N N NN N 0
T 0
1хb
b
П chk x chkхN
kП b shk
ll
b
1
1
хb
b
хb
b
П chk 10 NkП b shk
П chk 1NkП b shk
lllll
Example. Reinforcement by strap
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T 10 20
1n 2n
2 1
N N 0N N 0
T T T
2х 1хb
b
chkх chk xТ
kП shk
ll
b 2х 1хb
b 2х 1хb
1 chk0 ТkП shkchk 1ТkП shk
ll
lll
b
Example. Thermal loading
NN 10 20
1n 2n
N N NN N 0
T 0
1 2х хb
b
П chk х П chk chkхN
kП b shk
l ll
1 2
b
2х х
bb
Nk chk0b shkП П chkN
kП b shk
ll
lll
b
Example. Load reversing
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 131
NN 1 2х х
bb
П chk x П chkxN
kП b shk
ll
1 2
1 2
х хb
b
х хb
b
П chk П0 NkП b shk
П П chkNkП b shk
lllll
2 1x 2x
g
g
П Пk
G
Classic model: Model of Volkersen:2 1x 2x
g1 2
xz1 xz2 g
П Пk
2G 2G G
Influence of design model
Influence of the members flexibility on shearstress distribution
NN 1 2х х
bb
П chk x П chkxN
kП b shk
ll
1 2
1 2
х хb
b
х хb
b
П chk П0 NkП b shk
П П chkNkП b shk
lllll
Maximum peak of shearstress is on the end of lessflexible (more stiff) plate
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1х 2хП П maxNk kcth2b 2
l
max2b kN th
k 2
l
l kth 12
l
maxmax2bN
k
g1х
g
Gk 2П
Classic model (rigid adherends):
Model of Volkersen:
1х
g
xz g
2Пk
G G
gmaxmax
1х g
2N b ;П G
gmaxmax
1х xz g
2N bП G G
k kG
Influence of the bond layer flexibilityon the load transfer capability
more ductile and flexible adhesive is preferable
Influence of the length of joint on the loadtransfer capability
b
maxmin
b
maxmin
1х 2х
b b
П ПNk k0 cth2b 2
ll
l kcth 12
l
maxminNk2b
max41k
lnl
bNk
k2 2b sh2
ll cr
2 40k lnl max cr l l
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 133
Рисунок 40.2
bb const
b var
Рисунок 40.2
bb const
b var
2b kN b thk 2
ll
max41k
lnl l 1,71N b
k
maxmax
N 1,71 85%N 2
2 kN b thk 2
ll
Structural and technological solutionsfor adhesive joints effectiveness increasing
Scarf (tapered) joints
Potential margin of transferring load:
Preliminary estimationof adhesive joints workability
Classic model (rigid adherends):
Model of Volkersen:
gx
хg
2EG
gx
хxz g
2EG G
maxmax xN b
x, actual stresses and thickness of joining members
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Advanced structural solutions of jointsfor highly-stressed composite structures
The prevalent principle of the load transfer capability increasefor mechanically fastened joints – minimization of fastenerdiameter and conversion to multi-row joint
One of the possible approach of improvement for adhesivejoints – increasing of out-of-plane shear modulus andinterlaminar shear strength of the laminate
According to unbiased analyses of different types of joints –the most effective is the volume joint with load transferthrough whole cross-section of joining members
Idea of the joint with transversal micro-fasteners
Stress distribution in the scarfand stepped-lap joints
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 135
Patented by KhAI researchers in 1980s
1. Method of fibre-reinforcedcomposite structures joining,Gaydachuk V.E., Karpov Ya.S., etal (USSR Inventor’s CertificateNo.1121867 МКИ4 В 64 С №1/12 ,Publ.10/01/1983)
2. Assembly unit for heterogeneousstructures joining, Gaydachuk V.E.,Karpov Ya.S., et al(USSR Inventor’s CertificateNo.1110071 МКИ4 В 64 С №1/12,Publ. 07/01/1983)
Basic idea of the joint with transversalmicro-fasteners
Uncuredlaminate
Metalfittings
Transversalmicro-fasteners
Time, pressure, temperature
Cured composite
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136 Prepared in the frame of the FP7 KhAI-ERA project
Pressing-in Welding Milling
Manufacturing of the metal fitting withtransversal micro-fasteners
Micro-fasteners arrangement scheme fabricated by milling
Advantages of the joint with transversalmicro-fasteners
Uniform transfer of load with minimal path distortion
Undamaged reinforced fibers (filaments, tows, etc.)
Elimination of the composites machining and fastenersinstallation procedures
Reduce of the manufacturing risks due to human factor
Automated manufacturing of metal fittings withtransversal micro-fasteners
Using of traditional and typical metal fittings forassembling of the aircraft primary structures andcomponents
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 137
Application in aircraft structures.Concentrated load transfer
Special fasteners:
Application in aircraft structures.Distributed load transfer
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138 Prepared in the frame of the FP7 KhAI-ERA project
Determination of the bond layer parameters
Strain compatibility condition: g f
Equilibrium equation: f f g x y f x ys t t s t t
fs – cross-sectional area of micro-fastener
b f f g fG G G 1 – rule of mixture
ff
x y
st t
– areal density of the micro-fasteners
Basic assumptions for stress analysis ofthe joint with transversal micro-fasteners
Analysis technique is based on design scheme of theadhesive joints with some additions to bond layer parametersdetermination
Basic assumptions: Ideal connection between adhesive (glue), adherends and
micro-fasteners Uniform arrangement pattern of the micro-fasteners Changing of material orientation
in the zone of fastenerpenetration can be neglected
Module 5 – Designing and strength analysis of the joints of aircraft composite structures
Prepared in the frame of the FP7 KhAI-ERA project 139
Stress analysis of the joint withtransversal micro-fasteners
According to model of Volkersen: g1 2b
1 2xz gП
2G 2G G
General approach to the stress analysis of the adhesive join
Stress distribution between adhesive and micro-fasteners:
g fg f
g g
G G;G G
Determination of the laminate parametersUnit cell of the laminate
(half-thickness)
xz
2tgG
The average shear strain:
2
xz02
f f xz f0
z dzG z
dzG z G 1 z
Displacement of the fastener:
2
f f xz fxz 0
dzG z G 1 z2G
Advances in composite structures design and simulation
140 Prepared in the frame of the FP7 KhAI-ERA project
References
1. Karpov, Ya.S. Joints of composite items and structural components, [in Russian], Kharkiv,KhAI, 2006, 359 p.
2. MIL-HDBK-17/1F (Vol. 1 of 5), Department of deffence handbook: Composite materialshandbook – Polymer matrix composites. Guidelines for characterization of structuralmaterials, 2002, 586 p.
3. Dveyrin, A.Z., Krivenda, S.P. The bearing strength test of the laminates [in Russian],Problems of design and manufacture of aircraft structures: sc. ed. of Zhukovsky NationalAerospace University "KhAI", Issue 1(65), Kharkiv, KhAI, 2011. - pp. 20 – 28.
4. Lekhnitskii, S.G. Anisotropic Plates, Gordon and Breach Science Publishers, New York, 1968.
5. MIL-HDBK-17/3F (Vol. 3 of 5), Department of defense handbook: Composite materialshandbook – Polymer matrix composites materials usage, design, and analysis, 2002, 693 p.
6. Karpov, Ya.S. Jointing of high-loaded composite structural components. Part 1: Design andengineering solutions and performance assessment, Strength of materials, Vol. 38, No.3,2006, pp. 234 – 240.
7. Karpov, Ya.S. Jointing of high-loaded composite structural components. Part 2: Modeling ofstress-strain state, Strength of materials, Vol. 38, No.5, 2006, pp. 481 – 491.
8. Karpov, Ya.S. Jointing of high-loaded composite structural components. Part 3:An experimental study of strength of joints with transverse fastening microelements,Strength of materials, Vol. 38, No.6, 2006, pp. 575 – 585.
Prepared under the aegis of KhAI-ERA projectfunded by the European Commission’s Directorate-Generalfor Research & Innovations under the FP7 CapacitiesSpecific Programme on International CooperationGrant Agreement no 294311National Aerospace University
“KhAI”17 Chkalova str., Kharkiv
61070 Ukrainewww.khai.edu