Advances in composite structures design and simulation · PDF filePublished by the National Aerospace University “KhAI” in 2013 Advances in composite structures design and simulation

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


17Chkalova str., Kharkov, 61070, Ukraine

Phone: +38 (057) 788 40 60

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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


Module 4 Composite beams and spars design 85



Module 5 Designing and strength analysis of the jointsof aircraft composite structures 105




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


2013



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


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:





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:


xc c xt tF f N ; F f N


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


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


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



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



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

?



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


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






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



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



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:



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



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



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



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



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



– 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



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


E R1 r r t3B

rx

2 32 2x

k


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





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



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



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


Training Module 2

Composite laminated panels.Design and optimization

Dr.Sc., Prof. Yakov Karpov


Ph.D., Ass. Prof. Pavlo Gagauz


2013



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.



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



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).



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



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



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)



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



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



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



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 .



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



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.



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



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


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



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



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



Correlation of Laminate Thickness for Different Lay-Ups.Fabric Glass/Epoxy System

M10E/3783,Glass/Epoxy Fabric

1

1,5

2



Correlation of Laminate Thickness for Different Lay-Ups.Unidirectional Carbon/Epoxy System

AS4/3501-6,Carbon/Epoxy UD

1

2

3

4





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).



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



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



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


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




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



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.




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



2

4

6

8

10

12

14

0 45 90ply orientation angle, deg.

4

6

8

10

12

14


0

2

4

6

8

10


0

2

4

6

8

10

12


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



Estimating of Design Space for Laminates Optimizationby Optimal Design of Cross-Ply Lay-Up



References


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.




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

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17,1

0417

,818

2,8

29,1

3322

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19,9

1518

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17,9

4917

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17,2

7617

,133

17,0

6217

,042

17,1

0317

,250

17,4

5117

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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



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

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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

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16,5

0817

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19,0

0320

,272

21,4

7022

,393

0,8

10,1

0810

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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

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23,9

5225

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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

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1,4

12,7

0513

,427

14,1

6914

,902

15,5

7616

,199

17,1

4818

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19,0

6219

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20,9

0622

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24,3

3825

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27,1

5728

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1,6

13,5

6214

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15,0

3215

,774

16,4

6517

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18,2

6119

,246

20,2

2721

,160

22,1

1323

,991

25,7

3527

,296

28,6

9530

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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

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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

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31,7

8733

,226

2,2

16,1

2116

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17,6

1018

,376

19,1

3619

,843

21,2

7822

,594

23,6

4924

,690

25,6

6027

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29,5

9831

,398

33,1

2934

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2,4

16,9

7017

,699

18,4

6719

,239

20,0

0720

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22,1

9023

,593

24,7

6725

,830

26,8

1928

,848

30,8

2732

,667

34,5

3336

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2,6

17,8

1818

,549

19,3

2220

,100

20,8

7621

,647

23,0

9524

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25,8

3826

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27,9

6830

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32,0

4433

,921

35,8

1837

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2,8

18,6

6519

,398

20,1

7520

,959

21,7

4322

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23,9

9425

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26,8

6528

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29,1

0731

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33,2

5135

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37,0

8938

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3,0

19,5

1020

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21,0

2721

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22,6

0723

,394

24,8

8826

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27,8

1329

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30,2

3632

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34,4

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Module 3 – Composite sandwich panels design and structural-technological solutions


Training Module 3

Composite sandwich panels designand structural-technological solutions



2013



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



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



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.



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)



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


Reduced membrane stiffness terms

22 12 11 122 2 3311 22 11 2212 12


N B N B N B N B qU U U

BB B B B B B


Reduced laminate strains



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



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



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

.



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



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



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



Structural-Technological Solutionsfor Composite Sandwich Panels

Basic methods for sandwich panel manufacturing



Handling edges



Joining of Composite Sandwich Panels





Joining with Frame Elements



References



3. Kollar, L.P., Springer, G.S. Mechanics of Composite Structures, Cambridge University Press,2003. – 480 p.


Module 4 – Composite beams and spars design


Training Module 4

Composite beams and spars design





2013



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




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:



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,



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



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



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



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



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;



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





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



Structural-Technological Solutions





Edge Effects

zf f zsh sh

хf f хsh sh

0;0;

zf zsh хf хsh;

Mechanism of temperature edge effect:

Equilibrium equation:




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:



– 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;



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.




Module 5 – Designing and strength analysis of the joints of aircraft composite structures


Training Module 5

Designing and strength analysis of the jointsof aircraft composite structures







2013



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




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



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



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



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)



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]



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



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:



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]



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



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



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



' ' ' ' ' ''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:



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:



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



Use of special washers


Use of metallic washers and hybridizing of the laminatewith metallic foil




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



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


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



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



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


1x1 b

2x2 b

2b2x 1x

b22x 1x

d 0dx

d 0dx

dd d1 1 ПE dx E dx dx



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


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



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



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



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



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 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 П chkNkП b shk

lllll

Maximum peak of shearstress is on the end of lessflexible (more stiff) plate



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



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



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



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



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



Application in aircraft structures.Concentrated load transfer

Special fasteners:

Application in aircraft structures.Distributed load transfer



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



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



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

Documents

Advances in composite structures design and simulation · PDF filePublished by the National Aerospace University “KhAI” in 2013 Advances in composite structures design and simulation