1 ADHESIVE BONDED JOINTS - Altair University2018/06/05 · V.1.2 Version 4.4 Theoretical Background of ESAComp Analyses 3.12.2012 V Joints 1 Adhesive Bonded Joints J2D Second invariant

Version 4.4 V.1.1

3.12.2012 Theoretical Background of ESAComp Analyses

V Joints

1 Adhesive Bonded Joints

1 ADHESIVE BONDED JOINTS

Flemming Mortensen and Ole Thybo Thomsen (Aalborg University, Institute of Mechanical Engineering, Denmark, 2000)

The method used in ESAComp for engineering analysis of adhesive bonded joints of various complexities is

presented. The joints considered are divided in two types: standard and advanced. The standard joints consist of

two or three adherends bonded together with a straight continuous adhesive layer parallel to the in-plane

direction of the adherends. The advanced joints consist of two adherends bonded together with either a single or double-sided scarfed adhesive interface. The adherends are modelled as beams or plates in cylindrical bending.

They are formed from laminates with arbitrary lay-ups using the classical lamination theory (CLT). The adhesive

layer is modelled by a two-parameter elastic foundation model, where the adhesive layer is assumed composed

of a continuous layer of linear tension/compression and shear springs. Since non-linear effects in the form of

adhesive plasticity play an important role in the load transfer, the analysis allows inclusion of non-linear

adhesive properties by an iterative method based upon the linear-elastic approach. The load and boundary

conditions can be chosen arbitrarily. Approaches for predicting the cohesive failure in the adhesive layers and

laminate failure in the joint area are also presented.

SYMBOLS

i

jkA Element of the adherend in-plane stiffness matrix

i

jkB Element of the adherend coupling stiffness matrix

cfi Constant used in system equations

CS Constant used in effective stress formulation

CV Constant used in effective stress formulation

i

jkD Element of the adherend bending stiffness matrix

Ea Adhesive elastic modulus

e Effective strain

eN Effective strain in the N’th iteration step

Ga Adhesive elastic shear modulus

hfi Constant used in system equations

I1 First invariant of the general strain tensor

I2D Second invariant of the deviatoric strain tensor

J1 First invariant of the general stress tensor

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J2D Second invariant of the deviatoric stress tensor

kgi Constant used in system equations

L Length of the overlap zone in the adhesive joint

L1, L2 Length of adherends outside the overlap zone

i

xxM , i

xyM , i

yyM Adherend moment resultants

mfi Constant used in system equations

i

xxN , i

xyN , i

yyN Adherend in-plane stress resultants

i

xxQ , i

xyQ , i

yyQ Adherend shear force resultants

RFadh Reserve factor for cohesive failure of adhesive (linear or non-linear

adhesive model)

RFadh,prop Reserve factor for proportional limit of adhesive (non-linear adhesive

model)

RFFPF Reserve factor for adherend (laminate) first ply failure in the vicinity

of the joint

s Effective stress

*

Ns Calculated stress in the N’th iteration step

sN Experimental stress in the N’th iteration step

sprop Stress proportional limit

DsN Difference between calculated and experimental stress

ti Adherend thickness

ti(x) Adherend thickness as a function of x

ta Adhesive layer thickness

x Adherend in-plane coordinate system in the longitudinal direction

iu0 Longitudinal displacement of the adherend mid-plane (x-direction)

ui Longitudinal displacement of the adherend (x-direction)

iv0 Displacement of the adherend mid-plane in the width direction (y-

direction)

vi Displacement of the adherend in the width direction (y-direction)

wi Transverse displacement of the adherend (z-direction)

a, a1, a2 Transition angles of scarfed adherend

i

xb , i

yb Rotation of mid-plane normal to the adherend

d Weight factor for the change in elastic modulus

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ef Principal strains (f = 1, 2, 3)

l Ratio between compressive and tensile yield stress

sa Adhesive layer out-of-plane normal stress

san Adhesive layer out-of-plane normal stress

sani Adhesive layer out-of-plane normal stress

sc Compressive yield stress

sf Principal stresses (f = 1, 2, 3)

st Tensile yield stress

tax, tay Adhesive layer shear stress

tan, Adhesive layer shear stress

taxi Adhesive layer shear stress

Subscripts

a Adhesive layer

i Adherend (i = 1, 1a, 1b, 2, 2a, 2b, 3)

N Iteration number for non-linear tangent modulus

,x Differentiation with respect to the x-coordinate

,y Differentiation with respect to the y-coordinate

ult Ultimate

Superscripts

i Adherend (i = 1, 1a, 1b, 2, 2a, 2b, 3)

end Adherend end thickness at the overlap zone

end,L Adherend thickness at the left end of the overlap zone

end,R Adherend thickness at the right end of the overlap zone

t Identifier used for non-linear tangent modulus

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1.1 INTRODUCTION

Joining of composite structures can be achieved through use of bolted, riveted or adhesive

bonded joints. The performances of the mentioned joint types are severely influenced by the

characteristics of the layered composite materials, but adhesive bonded joints provide a much

more efficient load transfer than mechanically fastened joints. Accurate analysis of adhesive

bonded joints, for instance by using the finite element method, is an elaborate and

computationally demanding task as described by Crocrombe et al. [3], Harris et al. [11] and

Frostig et al. [5]. Hence, there is an obvious need for analysis and design tools that can

provide accurate results for preliminary design purposes.

This chapter introduces the analysis approach used in ESAComp for determining the stress

and displacement fields in commonly used adhesive bonded joint configurations. The last

sections deal with the handling of plasticity effects in the adhesive layers and failure

prediction of bonded joints.

The bonded joint types considered in ESAComp are:

· Single lap joint (SL)

· Single strap joint (SS)

· Bonded doubler (BD)

· Double lap joint (DL)

· Double strap joint (DS)

· Single sided scarfed lap joint (SSC)

· Double sided scarfed lap joint (DSC)

These joint types are illustrated in Figure 1.1. All the joint configurations can be composed of

similar or dissimilar laminates with an arbitrary lay-up. The joints are subjected to a general

loading condition as shown in Figure 1.2.

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Figure 1.2 Schematic illustration of an adhesive single lap joint subjected to a general loading condition.

According to the complexity of the joints, the lap and strap joints and bonded doublers can be

referred to as standard joints. In these joints, the adhesive layer or layers are parallel to the in-

plane direction of the adherends. Correspondingly, the scarfed joints can be referred to as

advanced joints. The advanced joints are more efficient due to the reduced eccentricity of the

load path, but the advanced joints are also much more expensive to manufacture and they are

therefore only used for high-performance applications.

1.2 STRUCTURAL MODELLING

The structural modelling is carried out by adopting a set of basic restrictive assumptions for

the behaviour of bonded joints. Based on these restrictions, the constitutive and kinematic

relations for the adherends are derived, and the constitutive relations for the adhesive layers

are adopted. Finally, the equilibrium equations for the joints are derived and, by combining all

these equations and relations, the set of governing equations is obtained.

1.2.1 Model dimensions

The adhesive bonded joint configurations were introduced in Section 1.1. The adherend

thicknesses are given by t1 and t2 for all the joints outside the overlap zone. For the double lap

joint, the thickness of the third adherend (lower adherend) is t3. The adherend length outside

the overlap is L1 and L2, and the length of the adhesive layer is L as illustrated in Figure 1.3.

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Figure 1.3 Illustration of adherend lengths and adhesive layer length and thickness.

Inside the overlap zone (0 £ x £ L) the thicknesses are:

· Single lap joint, single strap joint, and bonded doubler:

( ) ( ) 2211 , txttxt == (1.2.1)

· Double lap and double strap joint:

( ) ( ) ( ) 332211 ,, txttxttxt === (1.2.2)

· Single sided scarfed lap joint:

( ) ( ) xL

tttxtx

L

tttxt

endend

end

2222

1111 ,

--=

--= (1.2.3)

Where the superscript end in endt1 and endt2 refers to the thicknesses of the adherends at the free

ends of the overlap, see Figure 1.4.

Figure 1.4 Thicknesses and scarf angle for single sided scarfed lap joint.

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· Double sided scarfed lap joint:

( ) ( )

( ) ( )Lx

xL

tttxtx

L

tttxt

xL

tttxtx

L

tttxt

Rend

b

Lend

bLend

bb

Rend

a

Lend

aLend

aa

Lend

b

Rend

bb

Lend

a

Rend

aa

££

ïïþ

ïïý

ü

--=

--=

--=

--=

0

,

,

,

2

,

2,

22

,

2

,

2,

22

,

2

,

212

11

,

2

,

212

11

(1.2.4)

Where the subscripts a and b and the superscript end in Lend

at ,

2 and Rend

bt ,

2 refer to the thickness

of adherend 2 at the left (L) and right (R) ends of the overlap above and below adherend 1

(Figure 1.5).

Figure 1.5 Thicknesses for double sided scarfed lap joint.

1.2.2 Basic assumptions for the structural modelling

The basic restrictive assumptions for the structural modelling are the following:

Adherends

· The adherends are modelled as beams or plates in cylindrical bending, using ordinary

“Kirchhoff” plate theory (“Love-Kirchhoff” assumptions).

· The constitutive behaviour of the adherends is obtained using the classical lamination

theory (CLT). No restrictions are set on the laminate lay-up, i.e. unsymmetric and

unbalanced laminates can be included in the analysis.

· The laminates are assumed to obey linear-elastic constitutive laws.

· The strains are small and the rotations are very small.

Adhesive layers

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· The adhesive layers are modelled as continuously distributed linear tension/compression

and shear springs.

· Non-linear adhesive properties are considered by using a secant modulus approach for the

non-linear tensile stress-strain relationship in conjunction with a modified Von Mises

yield criterion.

Loads and boundary conditions

· The structural model allows boundary conditions to be chosen arbitrarily as long as the

system is in equilibrium. Sets of prescribed external loads (in-plane and out-of-plane

forces and bending moments) and geometric boundary conditions are defined in

ESAComp to avoid selections of inconsistent boundary conditions, which can lead to

singularity problems in the system of equations.

The system of governing equations is set up for two different cases, i.e. the adherends are

modelled as plates in cylindrical bending or as wide beams. In the following, the case where

the adherends are modelled as plates in cylindrical bending is primarily considered since the

modelling of the adherends as beams is a reduced case of this.

1.2.3 Constitutive relations for adherends modelled as plates

For the purposes of the present investigation, and with references to Figures 1.6 and 1.2,

cylindrical bending can be defined as a wide plate (in the y-direction), where the displacement

field can be described as a function of the longitudinal coordinate only. As a consequence, the

displacement field in the width direction is uniform. Thus, the displacement field can be

described as

( ) ( ) ( )xwwxvvxuu iiiiii === ,, 0000 (1.2.5)

where u0 is the mid-plane displacement in the longitudinal direction (x-direction), v0 is the

mid-plane displacement in the width direction (y-direction), and w is the displacement in the

transverse direction (z-direction). The displacement components u0, v0 and w are all defined

relative to the mid-plane of the laminates, and i = 1, 2, 3 corresponds to the laminates 1, 2 and

3, respectively.

Based on the earlier assumptions, the following holds also true:

0,,,0,0 ==== i

yy

i

y

i

y

i

y wwvu (1.2.6)

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Figure 1.6 Schematic illustration of adhesive single lap joint “clamped” between two vertical laminates, which

prevent the adherends of the single lap joint from moving and rotating freely in the width direction. This

represents the conceptual interpretation of cylindrical bending as defined in the present formulation.

In the concept of “cylindrical bending”, the boundary conditions at the boundaries in the

width direction are not well defined. However, it is assumed that there are some restrictive

constraints on the boundaries, such that the boundaries are not capable of moving freely. It

should be noted that the concept of “cylindrical bending” is not unique, and that other

definitions than the one used in the present formulation can be adopted, see Whitney [20].

Substitution of the quantities in Eq. (1.2.5) into the constitutive relations for a laminated

composite material gives the constitutive relations for a laminate (i) in cylindrical bending

[20]:

i

xx

ii

x

ii

x

ii

xy

i

xx

ii

x

ii

x

ii

xy

i

xx

ii

x

ii

x

ii

yy

i

xx

ii

x

ii

x

ii

yy

i

xx

ii

x

ii

x

ii

xx

i

xx

ii

x

ii

x

ii

xx

w-DvBuBMw-BvAuAN

w-DvBuBMw-BvAuAN

w-DvBuBMw-BvAuAN

,16,066,016,16,066,016

,12,026,012,12,026,012

,11,016,011,11,016,011

,

,

,

+=+=

+=+=

+=+=

(1.2.7)

where i

jkA , i

jkB and i

jkD (j,k = 1,2,6) are the extensional, coupling and flexural rigidities based

on the classical lamination theory (see Part III, Chapter 2). i

xxN , i

yyN and i

xyN are the in-plane

stress resultants i

xxM , i

yyM and i

xyM are the moment resultants. For the joints with scarfed

adherends the rigidities i

jkA , i

jkB and i

jkD (j,k = 1,2,6) within the overlap zone are changed as a

function of the longitudinal coordinate in accordance with their definition, i.e. i

jkA is changed

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linearly, i

jkB is changed parabolically and i

jkD is changed cubically (j,k = 1,2,6) between their

values at the ends of the overlap zone. This is of course an approximation since the actual

stiffnesses of the laminates are changing by changes within the layers as a function of the

longitudinal direction.

1.2.4 Constitutive relations for adherends modelled as beams

Modelling of the adherends as wide beams can be considered as a special case of cylindrical

bending. When the adherends are modelled as beams, the width direction displacements are

not considered, and only the longitudinal and vertical displacements are included. Thus, the

displacement field in Eq. (1.2.5) is reduced to

( ) ( )xwwxuu iiii == ,00 (1.2.8)

For this case the constitutive relations for a composite beam are reduced to

i

xx

ii

x

ii

xx

i

xx

ii

x

ii

xx w-DuBMw-BuAN ,11,011,11,011 , == (1.2.9)

1.2.5 Kinematic relations

From the “Love-Kirchhoff” assumptions, the following kinematic relations for the laminates

in cylindrical bending are derived:

0,, ,0 =-=+= i

y

i

x

i

x

i

x

ii wzuu bbb (1.2.10)

here ui is the longitudinal displacement, iu0 is the longitudinal displacement of the mid-plane,

and wi is the vertical displacement of the i’th laminate.

The kinematic relations of Eq. (1.2.10) are the same for the beam case as for the cylindrical

bending case except that all the variables associated with the width direction are nil.

1.2.6 Constitutive relations for the adhesive layer

The coupling between the adherends is established through the constitutive relations for the

adhesive layer, which as a first approximation is assumed homogeneous, isotropic and linear

elastic. The constitutive relations for the adhesive layer are established by use of a two-

parameter elastic foundation approach, where the adhesive layer is assumed to be composed

of continuously distributed shear and tension/compression springs. The constitutive relations

of the adhesive layer are suggested in accordance with Thomsen [16–17], Thomsen et al. [18]

and Tong [19]:

Version 4.4 V.1.11


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( ) ( ) ( )( )( ) ( )( )

( )( )ji

ji

ww

vvvv

xtuxtuuu

ji

t

E

a

ji

t

Gji

t

G

ay

j

xj

ji

xi

i

t

Gji

t

G

ax

a

a

a

a

a

a

a

a

a

a

¹

=

ïþ

ïý

ü

-=

-=-=

---=-=,3,2,1,

00

00

st

bbt (1.2.11)

where i and j are the numbers of the adherends, Ga is the shear modulus, and Ea is the elastic

modulus of the adhesive layer.

The consequence of using the simple spring model approach for the modelling of the adhesive

layers is that it is not possible to satisfy the equilibrium conditions at the (free) edges of the

adhesive. However, in real adhesive joints no free edges are present at the ends of the overlap,

since a fillet of surplus adhesive, a so-called spew-fillet, is formed at the ends of the overlap

zone. This spew fillet allows for the transfer of shear stresses at the overlap ends. Modelling

of the adhesive layer by spring models has been compared with other known analysis methods

such as finite element analysis (Crocrombe et al. [3] and Frostig et al. [5]) and a high-order

theory approach including spew fillets (Frostig et al. [5]). The results show that the overall

stress distribution and the predicted values are in very good agreement.

1.3 EQUILIBRIUM EQUATIONS

The equilibrium equations are derived based on equilibrium elements inside and outside the

overlap zone for each of the considered joint types.

1.3.1 Adherends outside the overlap zone

The equilibrium equations are derived for plates in cylindrical bending since the equilibrium

equations for the beam modelling can be considered as a reduced case of this. The equilibrium

equations outside the overlap zone for each of the adherends, i.e. in the regions -L1 £ x £ 0

and L £ x £ L + L2, are all the same (see Figure 1.2) and are derived based on Figure 1.7:

21

,

,

,

,

,

00

0

0

LLxLandxL

QM

QM

Q

N

N

i

y

i

xxy

i

x

i

xxx

i

xx

i

xxy

i

xxx

+££££-

ïïï

þ

ïïï

ý

ü

=

=

=

=

=

(1.3.1)

where i correspond to the adherends i = 1, 2, 3.

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Figure 1.7 Equilibrium elements of adherend outside the overlap zone; -L1 £ x £ 0 and L £ x £ L + L2.

1.3.2 Single lap and single strap joints

The equilibrium equations inside the overlap zone for the single lap joint and the single strap

are derived based on Figure 1.8. For the single lap joint the adherend thickness’ will remain

the same in the entire overlap zone as specified by Eqs. (1.2.1), thus giving the equations:

Lx

ttQM

ttQM

ttQM

ttQM

QQ

NN

NN

aayyxxy

aayyxxy

aaxxxxx

aaxxxxx

axxaxx

ayxxyayxxy

axxxxaxxxx

££

ïïïï

þ

ïïïï

ý

ü

++=

++=

++=

++=

-==

=-=

=-=

0

2,

2

2,

2

,

,

,

222

,

111

,

222

,

111

,

2

,

1

,

2

,

1

,

2

,

1

,

tt

tt

sstttt

(1.3.2)

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Figure 1.8 Equilibrium element of adherends inside the overlap zone for joints with one adhesive layer and

straight adherends; 0 £ x £ L.

1.3.3 Bonded doubler

Inside the overlap zone for bonded doubler joint the equilibrium equations are derived based

on Figure 1.8. and Eqs. (1.2.1) and yields exactly the same equations as for the single lap joint

(see the previous section).

1.3.4 Double lap and double strap joint

The equilibrium equations inside the overlap zone for the double lap joint and the double strap

are derived based on Figure 1.9:

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Figure 1.9 Equilibrium element of adherends inside the overlap zone for joints with two adhesive layers and

straight adherends; 0 £ x £ L.

Version 4.4 V.1.15


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Lx

ttQM

ttQM

Q

N

N

ttQM

ttQM

Q

N

N

ttttQM

ttttQM

Q

N

N

aayyxxy

aaxxxxx

axx

ayxxy

axxxx

aayyxxy

aaxxxxx

axx

ayxxy

axxxx

aay

aayyxxy

aax

aaxxxxx

aaxx

ayayxxy

axaxxxx

££

ïïïïïïïïïïïïïï

þ


ý

ü

+-=

+-=

=

=

=

++=

++=

-=

=

=

+-

++=

+-

++=

-=

--=

--=

0

2

2

2

2

22

22

23

2

33

,

232

33

,

2

3

,

2

3

,

2

3

,

12

1

22

,

12

1

22

,

1

2

,

1

2

,

1

2

,

21

2

11

1

11

,

21

2

11

1

11

,

21

1

,

21

1

,

21

1

,

t

t

s

tt

t

t

st

t

tt

tt

sstt

tt

(1.3.3)

1.3.5 Single sided scarfed lap joint

The equilibrium equations inside the overlap zone for the single sided scarfed lap joint are

derived based on Figure 1.10. They are different from the earlier ones due to the linear change

of the adherend thicknesses and the sloping bond line:

( ) ( )

( ) ( )

Lx

L

ttN

txtQ

M

L

ttN

txtQ

M

L

ttN

txtQ

M

L

ttN

txtQ

M

QQ

NN

NN

end

xy

aayy

xxyend

xy

aayy

xxy

end

xx

aaxx

xxxend

xx

aaxx

xxx

axxaxx

ayxxyayxxy

axxxxaxxxx

££

ïïïïïïï

þ

ïïïïïïï

ý

ü

--

++

=-

-

++

=

--

++

=-

-

++

=

-==

=-=

=-=

0

2

2

,2

2

2

2

,2

2

,

,

,

221

22

2

,

111

11

1

,

221

22

2

,

111

11

1

,

2

,

1

,

2

,

1

,

2

,

1

,

tt

tt

ss

tttt

(1.3.4)

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where the relationship between tax, sa in Eq. (1.3.4) and tan, san shown in Figure 1.10 is

established through equilibrium:

asaatsaasatt 22 coscossin,cossincos aaxanaaxan +=+= (1.3.5)

where a is the scarf angle of the adherends in the overlap zone (see Figure 1.4).

Figure 1.10 Equilibrium elements in the overlap zone for a single sided scarfed lap joint (scarf angles a); 0 £ x £ L.

The adherend thicknesses t1(x), t2(x) vary linearly through the overlap length as specified by

Eq. (1.2.3).

Version 4.4 V.1.17


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1.3.6 Double sided scarfed lap joint

Finally, the equilibrium equations inside the overlap zone for the double-sided scarfed lap

joint are derived based on Figure 1.11:

( ) ( )

( ) ( )

( )

( )

( )

( )

Lx

L

ttN

txtQM

L

ttN

txtQM

Q

N

N

L

ttN

txtQM

L

ttN

txtQM

Q

N

N

txt

txtQM

txt

txtQM

Q

N

N

Lend

b

Rend

bb

xyab

ax

b

y

b

xxy

Lend

b

Rend

bb

xxab

ax

b

x

b

xxx

a

b

xx

ay

b

xxy

ax

b

xxx

Lend

a

Rend

aa

xyaa

ay

a

y

a

xxy

Lend

a

Rend

aa

xxaa

ax

a

x

a

xxx

a

a

xx

ay

a

xxy

ax

a

xxx

abay

aaayyxxy

abax

aaaxxxxx

aaxx

ayayxxy

axaxxxx

££


þ


ý

ü

-+

+-=

-+

+-=

=

=

=

--

++=

--

++=

-=

=

=

÷ø

öçè

æ+-÷

ø

öçè

æ++=

÷ø

öçè

æ+-÷

ø

öçè

æ++=

-=

--=

--=

0

22

22

22

22

22

22

,

2

,

2222

2

22

,

,

2

,

2222

2

22

,

2

2

,

2

2

,

2

2

,

,

2

,

22121

22

,

,

2

,

2212

1

22

,

1

2

,

1

2

,

1

2

,

212

111

11

,

2

12

1

11

11

,

21

1

,

21

1

,

21

1

,

t

t

st

t

t

t

s

tt

tt

tt

sstttt

(1.3.6)

where t1(x), t2a(x) and t2b(x) are the adherend thicknesses, according to Eq. (1.2.4), and ta1 and

ta2 are the adhesive layer thicknesses.

V.1.18 Version 4.4


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Figure 1.11 Equilibrium element of adherends inside the overlap zone for double sided scarfed lap joints; 0 £ x

£ L.

The relationship between tax1, sa1 in Eq. (1.3.6) and tan1, san1 shown in Figure 1.11 as well as

the relationship between tax2, sa2 in Eq. (1.3.6) and tan2, san2 shown in Figure 1.11 is

established through equilibrium:

( )2,1,coscossin,cossincos 22 =+=-= iiaiiiaxianiiiaiiaxiani asaatsaasatt (1.3.7)

Version 4.4 V.1.19


V Joints


where ai (i=1,2) is the scarf angles of the adherends in the overlap zone (see Figure 1.5).

1.4 THE COMPLETE SET OF SYSTEM EQUATIONS

From the equations derived, it is possible to form the complete set of system equations for

each of the bonded joint configurations. Thus, combination of the constitutive and kinematic

relations, i.e. Eqs. (1.2.7) and (1.2.10), together with the constitutive relations for the adhesive

layers, i.e. Eqs. (1.2.11), and the equilibrium equations lead to a set of 8 linear coupled first-

order ordinary differential equations describing the system behaviour of each of the

adherends. The total number of coupled first-order ordinary differential equations within the

overlap zone is therefore 16 for joints with two adherends inside the overlap zone, and 24 for

the joints with three adherends inside the overlap zone. Outside the overlap zones the system

behaviour for all the joints is described by 8 linear coupled first-order ordinary differential

equations, except for the double lap joint which has two adherends in the region L £ x £ L +

L2 and therefore is described by a set of 16 linear coupled first-order ordinary differential

equations in this region.

The set of governing equations for all the considered adhesive bonded joint types are

presented in this section. The governing equations presented are those where the adherends

are modelled as plates in cylindrical bending. The case where the adherends are modelled as

wide beams can be considered as a special case of cylindrical bending, and results in a

reduced set of the governing equations. However, to demonstrate that this is true the

governing equations for the case of adherends modelled as beams are also shown for the

single lap joint.

1.4.1 Single lap and single strap joints

From the equations derived, it is possible to form the complete set of system equations for the

problem. Thus combination of Eqs. (1.2.7), (1.2.10) and (1.3.1) yields for the laminate 1 and 2

in the areas -L1 £ x £ 0 and L £ x £ L + L2 (outside of overlap):

2,1

0

0

0

,

,

,

,

,9,8,7,0

,6,5,4,

,

,3,2,1,0

=

ïïïïï

þ

ïïïïï

ý

ü

=

=

=

=

++=

---=

-=

++=

i

Q

QM

N

N

MkNkNkv

MkNkNk

w

MkNkNku

i

xx

i

x

i

xxx

i

xxy

i

xxx

i

xxxi

i

xxyi

i

xxxi

i

x

i

xxxi

i

xxyi

i

xxxi

i

xx

i

x

i

x

i

xxxi

i

xxyi

i

xxxi

i

x

bb

(1.4.1)

V.1.20 Version 4.4


V Joints


Eqs. (1.4.1) constitute a set of eight linear coupled first-order ordinary differential equations.

The coefficients k1I – k9i (i = 1, 2) contain laminate stiffness parameters and are a result of

isolating i

xu ,0 , i

xv ,0 and i

xxw from i

xxN , i

xyN and i

xxM in Eqs. (1.2.7):

i

i

i

iii

i

ii

i

i

i

iii

iiiiiiiiiii

i

ii

i

ii

i

i

c

c

c

kck

c

kc

ck

c

kck

hkhkhkhkkhk

m

mk

m

mk

mk

2

3

2

319

2

21

2

8

2

117

23163215114

1

23

1

32

1

1

,1

,

,

,,1

--=-=-=

+=+==

-=-==

(1.4.2)

where the coefficients cji, hji and mji (j=1,2,3) are:

i

i

i

i

ii

i

i

i

i

i

i

i

i

i

ii

i

i

i

i

ii

i

i

i

i

i

i

i

i

i

i

i

i

i

i

ii

iii

ii

iii

i

hBc

AmhB

c

cAm

c

cAhBAm

cD

Bh

DcD

cBh

cD

cB

D

Bh

D

Bc

D

BBAc

D

BBAc

311

2

163211

2

3162

2

116111111

211

163

11211

3162

211

116

11

111

11

163

11

1616662

11

1116161

1,

,,

-=--=--=

=-=-=

=-=-=

(1.4.3)

Within the overlap zone, i.e. for 0 £ x £ L, combination of Eqs. (1.2.7), (1.2.10), (1.2.11) and

(1.3.2) yields for the laminate 1 and 2:

Version 4.4 V.1.21


V Joints


( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

212

,

2222

0

21211

0

212

,

2

0

1

0

2

,

222

0

111

0

2

,

2

,92

2

,82

2

,72

2

,0

2

,62

2

,52

2

,42

2

,

22

,

2

,32

2

,22

2

,12

2

,0

211

,

2122

0

11111

0

111

,

2

0

1

0

1

,

222

0

111

0

1

,

1

,91

1

,81

1

,71

1

,0

1

,61

1

,51

1

,41

1

,

11

,

1

,31

1

,21

1

,11

1

,0

4242

22

4242

22

wt

Ew

t

EQ

t

tttGu

t

ttG

t

tttGu

t

ttGQM

vt

Gv

t

GN

t

tGu

t

G

t

tGu

t

GN

MkNkNkv

MkNkNk

w

MkNkNku

wt

Ew

t

EQ

t

tttGu

t

ttG

t

tttGu

t

ttGQM

vt

Gv

t

GN

t

tGu

t

G

t

tGu

t

GN

MkNkNkv

MkNkNk

w

MkNkNku

a

a

a

axx

x

a

aa

a

aax

a

aa

a

aaxxxx

a

a

a

axxy

x

a

a

a

ax

a

a

a

axxx

xxxxxyxxxx

xxxxxyxxxxx

xx

xxxxxyxxxx

a

a

a

axx

x

a

aa

a

aax

a

aa

a

aaxxxx

a

a

a

axxy

x

a

a

a

ax

a

a

a

axxx

xxxxxyxxxx

xxxxxyxxxxx

xx

xxxxxyxxxx

+-=

++

++

++

+-=

+-=

+++-=

++=

---=

-=

++=

-=

++

++

++

+-=

-=

---=

++=

---=

-=

++=

bb

bb

bb

bb

bb

bb

(1.4.4)

Eqs. (1.4.4) constitute a set of 16 linear coupled first-order ordinary differential equations.

1.4.2 Single lap joint with adherends modelled as beam

For the laminates 1 and 2 in the areas -L1 £ x £ 0 and L £ x £ L + L2 (outside of overlap)

combining Eqs. (1.2.10), (1.2.9), (1.2.11) together with the equilibrium equations yields:

2,1

0

0

,

,

,

,6,5,4,

,

,3,2,1,0

=

ïïïï

þ

ïïïï

ý

ü

=

=

=

---=

-=

++=

i

Q

QM

N

MkNkNk

w

MkNkNku

i

xx

i

x

i

xxx

i

xxx

i

xxxi

i

xxyi

i

xxxi

i

xx

i

x

i

x

i

xxxi

i

xxyi

i

xxxi

i

x

bb

(1.4.5)

V.1.22 Version 4.4


V Joints


Eqs. (1.4.5) constitute a set of six linear coupled first-order ordinary differential equations.

The coefficients k1i – k9i (i = 1, 2) contain the laminate stiffness parameters and are

determined by isolation of i

xu ,0 and i

xxw from i

xxN and i

xxM in Eq. (1.2.9):

( )iiiii

ii

iiii

iiii

i

i

D

BBii

BBADD

BB

Dkkk

BBAD

Bk

Ak

i

ii

1111111111

1111

11

423

11111111

112

11

1

1,

,1

11

1111

-+==

--=

-=

(1.4.6)

By comparison with the coefficients for the cylindrical bending case it is seen that the

coefficients for the beam case are strongly reduced and only contain few of the laminate

stiffness parameters.

Within the overlap zone, i.e. for 0 £ x £ L, the governing for laminate 1 and 2 are:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

212

,

2222

0

21211

0

212

,

222

0

111

0

2

,

2

,42

2

,32

2

,

22

,

2

,22

2

,12

2

,0

211

,

2122

0

11111

0

111

,

222

0

111

0

1

,

1

,41

1

,31

1

,

11

,

1

,21

1

,11

1

,0

4242

22

4242

22

wt

Ew

t

EQ

t

tttGu

t

ttG

t

tttGu

t

ttGQM

t

tGu

t

G

t

tGu

t

GN

MkNk

w

MkNku

wt

Ew

t

EQ

t

tttGu

t

ttG

t

tttGu

t

ttGQM

t

tGu

t

G

t

tGu

t

GN

MkNk

w

MkNku

a

a

a

a

xx

x

a

aa

a

aa

x

a

aa

a

aa

xxxx

x

a

a

a

a

x

a

a

a

a

xxx

xxxxxxxx

xx

xxxxxxx

a

a

a

axx

x

a

aa

a

aax

a

aa

a

aaxxxx

x

a

a

a

ax

a

a

a

axxx

xxxxxxxx

xx

xxxxxxx

+-=

++

++

++

+-=

+++-=

--=

-=

+=

-=

++

++

++

+-=

---=

--=

-=

+=

bb

bb

bb

bb

bb

bb

(1.4.7)

Eqs. (1.4.7) constitute a set of 12 linear coupled first-order ordinary differential equations. By

comparison with the equations for the cylindrical bending case it is seen that the equations

Version 4.4 V.1.23


V Joints


display the same overall appearance except that all variables associated with the width

direction are nil in Eqs. (1.4.7).


The governing equations for the bonded doubler joint are exactly the same as for the single

lap joint in the overlap zone and outside the overlap zone in the region L £ x £ L + L2.

1.4.4 Double lap joint

The governing equations for the double lap joint are exactly the same as for the single lap

joint in the region L1 £ x £ 0. The governing equations for laminate 1, 2 and 3 within the

overlap zone, i.e. for 0 £ x £ L, are derived by combining Eqs. (1.2.7), (1.2.10), (1.2.11) and

(1.3.3):

V.1.24 Version 4.4


V Joints


( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2

1

11

1

12

,

2

1

21112

0

1

1111

1

11211

0

1

12122

,

2

0

1

11

0

1

12

,

2

1

122

0

1

11

1

111

0

1

12

,

2

,92

2

,82

2

,72

2

,0

2

,62

2

,52

2

,42

2

,

22

,

2

,32

2

,22

2

,12

2

,0

3

2

22

1

11

2

2

1

11

,

2

2

32123

0

2

2122

1

21112

0

1

111

1

2

1212

1

11111

0

2

212

1

1111

1

,

3

0

2

22

0

1

11

0

2

2

1

11

,

3

2

233

0

2

32

1

122

0

1

11

2

21

1

111

0

2

2

1

11

,

1

,91

1

,81

1

,71

1

,0

1

,61

1

,51

1

,41

1

,

11

,

1

,31

1

,21

1

,11

1

,0

4242

22

4242

4422

2222

wt

Ew

t

EQ

t

tttGu

t

ttG

t

tttGu

t

ttGQM

vt

Gv

t

GN

t

Gtu

t

G

t

Gtu

t

GN

MkNkNkv

MkNkNk

w

MkNkNku

wt

Ew

t

Ew

t

E

t

EQ

t

tttGu

t

ttG

t

tttGu

t

ttG

t

tttG

t

tttGu

t

ttG

t

ttGQ

M

vt

Gv

t

Gv

t

G

t

GN

t

Gtu

t

G

t

Gtu

t

G

t

Gt

t

Gtu

t

G

t

GN

MkNkNkv

MkNkNk

w

MkNkNku

a

a

a

axx

x

a

aa

a

aax

a

aa

a

aaxxxx

a

a

a

axxy

x

a

a

a

ax

a

a

a

axxx

xxxxxyxxxx

xxxxxyxxxxx

xx

xxxxxyxxxx

a

a

a

a

a

a

a

axx

x

a

aa

a

aax

a

aa

a

aa

x

a

aa

a

aa

a

aa

a

aax

xxx

a

a

a

a

a

a

a

axxy

x

a

a

a

ax

a

a

a

ax

a

a

a

a

a

a

a

axxx

xxxxxyxxxx

xxxxxyxxxxx

xx

xxxxxyxxxx

+-=

++

++

++

+-=

+-=

+++-=

++=

---=

-=

++=

--÷÷ø

öççè

æ+=

++

+-

++

++

÷÷ø

öççè

æ +-

++÷÷

ø

öççè

æ +-

+-

=

--÷÷ø

öççè

æ+=

+---÷÷ø

öççè

æ--÷÷

ø

öççè

æ+=

++=

---=

-=

++=

bb

bb

bb

bb

b

bbb

bb

(1.4.8)

( ) ( ) ( ) ( )

3

2

21

2

23

,

3

2

32323

0

2

2321

2

12321

0

2

23233

,

3

0

2

21

0

2

23

,

3

2

233

0

2

21

2

211

0

2

23

,

2

,93

2

,83

2

,73

3

,0

2

,63

2

,53

2

,43

3

,

23

,

2

,33

2

,23

2

,13

3

,0

4242

22

wt

Ew

t

EQ

t

tttGu

t

ttG

t

tttGu

t

ttGQM

vt

Gv

t

GN

t

Gtu

t

G

t

Gtu

t

GN

MkNkNkv

MkNkNk

w

MkNkNku

a

a

a

axx

x

a

aa

a

aax

a

aa

a

aaxxxx

a

a

a

axxy

x

a

a

a

ax

a

a

a

axxx

xxxxxyxxxx

xxxxxyxxxxx

xx

xxxxxyxxxx

+-=

++

+-

++

++=

+-=

-+--=

++=

---=

-=

++=

bb

bb

bb

Version 4.4 V.1.25


V Joints


Eqs. (1.4.8) constitute a set of 24 linear coupled first-order ordinary differential equations

within the overlap zone.

The governing equation for the laminates 2 and 3 in the region L £ x £ L + L1 (outside of

overlap) are derived by combining Eqs. (1.2.7), (1.2.10) and (1.3.1):

0

0

0

0

0

0

3

,

33

,

3

,

3

,

3

,93

3

,83

3

,73

3

,0

3

,63

3

,53

3

,43

3

,

33

,

3

,33

3

,23

3

,13

3

,0

2

,

22

,

2

,

2

,

2

,92

2

,82

2

,72

2

,0

2

,62

2

,52

2

,42

2

,

22

,

2

,32

2

,22

2

,12

2

,0

=

=

=

=

++=

---=

-=

++=

=

=

=

=

++=

---=

-=

++=

xx

xxxx

xxy

xxx

xxxxxyxxxx

xxxxxyxxxxx

xx

xxxxxyxxxx

xx

xxxx

xxy

xxx

xxxxxyxxxx

xxxxxyxxxxx

xx

xxxxxyxxxx

Q

QM

N

N

MkNkNkv

MkNkNk

w

MkNkNku

Q

QM

N

N

MkNkNkv

MkNkNk

w

MkNkNku

b

b

b

b

(1.4.9)



The governing equations outside the overlap zone are the same as for the single lap joint, i.e.

Eqs. (1.4.1). Within the overlap zone, i.e. for 0 £ x £ L, the governing equations for the

laminates 1 and 2 are derived by combining of Eqs. (1.2.7), (1.2.10), (1.2.11) and (1.3.4):

V.1.26 Version 4.4


V Joints


( )( ) ( ) ( )( ) ( )( )

( ) ( )( )

( )( ) ( ) ( )( ) ( )( )

( ) ( )( )

212

,

222222

2

021211

022

2

,

2

0

1

0

2

,

222

0

111

0

2

,

2

,92

2

,82

2

,72

2

,0

2

,62

2

,52

2

,42

2

,

22

,

2

,32

2

,22

2

,12

2

,0

211

,

111212

2

011111

011

1

,

2

0

1

0

1

,

222

0

111

0

1

,

1

,91

1

,81

1

,71

1

,0

1

,61

1

,51

1

,41

1

,

11

,

1

,31

1

,21

1

,11

1

,0

24

242

22

24

242

22

wt

Ew

t

EQ

NL

tt

t

txtxtG

ut

txtG

t

txtxtGu

t

txtGQ

M

vt

Gv

t

GN

t

tGu

t

G

t

tGu

t

GN

MkNkNkv

MkNkNk

w

MkNkNku

wt

Ew

t

EQ

NL

tt

t

txtxtG

ut

txtG

t

txtxtGu

t

txtGQ

M

vt

Gv

t

GN

t

tGu

t

G

t

tGu

t

GN

MkNkNkv

MkNkNk

w

MkNkNku

a

a

a

axx

xx

end

x

a

aa

a

aax

a

aa

a

aax

xxx

a

a

a

axxy

x

a

a

a

ax

a

a

a

axxx

xxxxxyxxxx

xxxxxyxxxxx

xx

xxxxxyxxxx

a

a

a

axx

xx

end

x

a

aa

a

aax

a

aa

a

aax

xxx

a

a

a

axxy

x

a

a

a

ax

a

a

a

axxx

xxxxxyxxxx

xxxxxyxxxxx

xx

xxxxxyxxxx

+-=

--

++

++

++

+-

=

+-=

+++-=

++=

---=

-=

++=

-=

--

++

++

++

+-

=

-=

---=

++=

---=

-=

++=

b

b

bb

bb

b

b

bb

bb

(1.4.10)



The governing equations for the double-sided scarfed lap joint are the same as for the single

lap joint outside the overlap zone.

The governing equations for the laminates within the overlap zone, i.e. for 0 £ x £ L, are

derived by combining of Eqs. (1.2.7), (1.2.10), (1.2.11) and (1.3.6):

Version 4.4 V.1.27


V Joints


( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

a

a

a

a

aa

xx

a

xx

end

aaa

x

a

aa

a

a

aax

a

aa

a

aaa

x

a

xxx

a

a

a

a

aa

xxy

a

x

a

aa

a

ax

a

a

a

aa

xxx

a

xxx

a

xxy

a

xxx

a

x

a

xxx

a

xxy

a

xxx

a

xx

a

x

a

x

a

xxx

a

xxy

a

xxx

a

x

a

a

a

a

a

a

a

axx

x

a

aa

a

aax

a

aa

a

aa

x

a

aa

a

aa

a

aa

a

aax

xxx

a

a

a

a

a

a

a

axxy

x

a

a

a

ax

a

a

a

ax

a

a

a

a

a

a

a

axxx

xxxxxyxxxx

xxxxxyxxxxx

xx

xxxxxyxxxx

wt

Ew

t

EQ

NL

tt

t

tttG

ut

ttG

t

tttGu

t

ttGQ

M

vt

Gv

t

GN

t

Gtu

t

G

t

Gtu

t

GN

MkNkNkv

MkNkNk

w

MkNkNku

wt

Ew

t

Ew

t

E

t

EQ

t

tttGu

t

ttG

t

tttGu

t

ttG

t

tttG

t

tttGu

t

ttG

t

ttGQ

M

vt

Gv

t

Gv

t

G

t

GN

t

Gtu

t

G

t

Gtu

t

G

t

Gt

t

Gtu

t

G

t

GN

MkNkNkv

MkNkNk

w

MkNkNku

2

1

11

1

12

,

2222

1

2111

2

0

1

1111

1

11211

0

1

1212

2

,

2

0

1

11

0

1

12

,

2

1

122

0

1

11

1

111

0

1

12

,

2

,92

2

,82

2

,72

2

,0

2

,62

2

,52

2

,42

2

,

22

,

2

,32

2

,22

2

,12

2

,0

3

2

22

1

11

2

2

1

11

,

3

2

32123

0

2

2122

1

21112

0

1

111

1

2

1212

1

11111

0

2

212

1

1111

1

,

3

0

2

22

0

1

11

0

2

2

1

11

,

3

2

233

0

2

32

1

122

0

1

11

2

21

1

111

0

2

2

1

11

,

1

,91

1

,81

1

,71

1

,0

1

,61

1

,51

1

,41

1

,

11

,

1

,31

1

,21

1

,11

1

,0

24

242

22

4242

4422

2222

+-=

--

++

++

++

+-

=

+-=

+++-=

++=

---=

-=

++=

--÷÷ø

öççè

æ+=

++

+-

++

++

÷÷ø

öççè

æ ++

++÷÷

ø

öççè

æ +-

+-

=

--÷÷ø

öççè

æ+=

+---÷÷ø

öççè

æ--÷÷

ø

öççè

æ+=

++=

---=

-=

++=

b

b

bb

bb

bb

b

bbb

bb

(1.4.11)

V.1.28 Version 4.4


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( ) ( ) ( )

( )

b

a

a

a

ab

xx

b

xx

end

bbb

x

a

aa

b

a

aax

a

aa

a

aab

x

b

xxx

b

a

a

a

ab

xxy

b

x

a

ab

a

ax

a

a

a

ab

xxx

b

xxx

b

xxy

b

xxx

b

x

b

xxx

b

xxy

b

xxx

b

xx

b

x

b

x

b

xxx

b

xxy

b

xxx

b

x

wt

Ew

t

EQ

NL

tt

t

tttG

ut

ttG

t

tttGu

t

ttGQ

M

vt

Gv

t

GN

t

Gtu

t

G

t

Gtu

t

GN

MkNkNkv

MkNkNk

w

MkNkNku

2

1

11

1

12

,

2222

1

2111

2

0

1

1111

1

11211

0

1

1212

2

,

2

0

1

11

0

1

12

,

2

1

122

0

1

11

1

111

0

1

12

,

2

,92

2

,82

2

,72

2

,0

2

,62

2

,52

2

,42

2

,

22

,

2

,32

2

,22

2

,12

2

,0

24

242

22

+-=

-+

++

+-

++

++

=

+-=

-+--=

++=

---=

-=

++=

b

b

bb

bb

Eqs. (1.4.11) constitute a set of 24 linear coupled first-order ordinary differential equations

within the overlap zone.

1.5 BOUNDARY CONDITIONS

To solve the adhesive bonded joint problems the boundary conditions and continuity

conditions have to be stated. The continuity conditions must be stated at the ends of the

regions in which the joint is divided as shown in Figure 1.1. In the following the boundary

conditions and continuity conditions are stated for the different joint types.

1.5.1 Single lap joint

The boundary conditions for a single lap joint are

junctionacrossContinuity

QMNN

adherend

adherendLx

QMNN


adherend

adherendx

iNorvMor

QorwNoruprescribedLLLx

xxxxyxx

xxxxyxx

i

xy

i

o

i

xx

i

i

x

ii

xx

i

0

:2

:1:

0:2

:1:0

2,1,

,,::,

1111

2222

0

021

=====

====

=

=ïþ

ïýü+-=

b

(1.5.1)

The boundary conditions for the adherend 2 at x = 0 and for adherend 1 at x = L are derived

from the assumption that the adherend edges are free, see Figure 1.1.

Version 4.4 V.1.29


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1.5.2 Single strap joint

The boundary conditions for a single strap joint are

0:2

:1:

0

0

0

0

:)0(2

:)0(2

:)0(1

:)0(1

::0

,

,,::

2222

2222

0

22

0

22

0

1111

0

1111

2

1

2

1

0

2

11

0

2

11

0

1111

1111

02

====

=

====

====

====

====

+==

+==

+=<

+=>

=

ïþ

ïýü+=

xxxxyxx

xxyxy

xxy

xxxxy

xxxxyxx

xy

xx

xx

xyoxxx

xxx

QMNN


adherend

adherendLx

QNu

Qvu

QMNu

QMNN

LLxatNifadherend

LLxatvifadherend

LLxatNoruifadherend

LLxatNoruifadherend

Symmetryx

NorvMor

QorwNoruprescribedLLx

b

b

b

(1.5.1)

The boundary conditions at x = 0 are derived from the assumptions that there is symmetry

around the centerline of the strap joint. For adherend 2 at x = L, it is assumed that the

adherend edge is free.


Similar assumptions as for single strap joints are used bonded doublers, which yields the

following boundary conditions:

ïþ

ïýü

====+=

=

====+==

====+==

====+==

====+==

=

1111

0

1111

0

2222

2

2222

02

1

22

0

22

02

1

0

1111

02

1

11

0

11

02

1

0

,

,,

:

0

:1

:2

:1

:

:

0:)0(2

0:)0(2

0:)0(1

0:)0(1

::0

xyoxx

xxx

xxxxyxx

xxyxyxy

xxy

xxyxyxy

xxy

NorvMor

QorwNoru

prescribed

QMNN


adherend

adherend

adherend

LLx

Lx

QNuLLxatNifadherend

QvuLLxatvifadherend

QNuLLxatNifadherend

QvuLLxatvifadherend

Symmetryx

b

b

b

b

b

(1.5.2)

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1.5.4 Double lap joint

The boundary conditions for a double lap joint are

0

0

:3

:2

:1:

0

:3

:2

:1:0

3,2,1,

,,::,

3333

2222

1111

0

021

====

====

=

=====

=ïþ

ïýü+-=

xxxxyxx

xxxxyxx

xxxxyxx

i

xy

i

o

i

xx

i

i

x

ii

xx

i

QMNN

QMNN


adherend

adherend

adherendLx



QMNN

adherend

adherend

adherendx

iNorvMor

QorwNoruprescribedLLLx

b

(1.5.3)

For the adherend 2 and 3 at x = 0 and for the adherend 1 at x = L, the boundary conditions are

derived from the assumption that the adherend edges are free.


The boundary and continuity conditions at the ends of the joint and at the ends of the overlap

zone are the same as for the single lap joint in Eq. (1.5.1).


The boundary and continuity conditions are the same as for the double sided stepped lap joint

in Eq. (1.5.3), except that no continuity conditions within the overlap zone is required for the

double sided scarfed lap joint.

1.6 MULTI-SEGMENT METHOD

Each set of governing equations, together with the appropriate boundary conditions for the

particular bonded joint problem considered, constitutes a multiple-point boundary value

problem to which no general closed-form solution is obtainable. Thus, a numerical solution

procedure must be used to solve the bonded joint problems. In general, this can be done by

using methods such as finite difference methods or direct integration methods. The use of a

normal direct integration approach will involve some disadvantages, from which the most

important is that a complete loss of accuracy invariably will occur if the length of the

integration interval is increased beyond a certain value. The loss of accuracy is caused by

subtraction of almost equal and very large numbers in the process of determination of the

unknown boundary values. However, the use of a modified direct integration method, called

the “multi-segment method of integration”, can overcome the loss of accuracy experienced

with the “normal” direct integration methods.

Version 4.4 V.1.31


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The method is based on a transformation of the original “multiple-point” boundary value

problem into a series of initial value problems. The principle behind the method is to divide

the original problem into a finite number of segments where the solution within each segment

can be accomplished by means of direct integration. Fulfilment of the boundary conditions, as

well as fulfilment of continuity requirements across the segment junctions is assured by

formulation and solving a set of linear algebraic equations. As an example, the single lap joint

configuration shown in Figure 1.12 is divided into three regions.

Figure 1.12 Schematic illustration of a single lap joint divided into M1 + M2 + M3 segments.

According to Figure 1.12, the three regions are:

· The region to the left side of the overlap zone, i.e. -L1 £ x £ 0

· The overlap zone, i.e. 0 £ x £ L

· The region to the right side of the overlap zone, i.e. L £ x £ L + L2.

Each of the regions r (r = 1, 2, … nr) are then divided into a finite number of segments Mr, see

Figure 1.12. The segments within a region are denoted by r

jS (j = 1, 2, … Mr) and the j'th

segment extends from r

jx to r

jx 1+ .

The solution procedure adopted in the “multi-segment method of integration” includes four

steps:

· Solution of the governing equation within each segment r

jS in each region r.

· Specification of continuity conditions between each segment within each region r.

· Specification of boundary and continuity conditions at the ends of the regions.

V.1.32 Version 4.4


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· Formulation and solution of a set of linear algebraic equations containing the unknown

variables.

It is beyond the scope of this document to go into further details of the solution method and

the implementation in the ESAComp software (see Mortensen [13] for details). It is essential,

though, to emphasise that the direct integration of the initial value problems is performed by

an embedded Runge-Kutta method with adaptive step size control based on a prescribed

accuracy which enables ESAComp to control the number of segments used.

1.7 PLASTICITY EFFECTS IN THE ADHESIVE LAYER

1.7.1 Introduction

The structural modelling described in Section 1.2 is based on the assumption that the adhesive

layers behave as a linear elastic material. This is a good approximation for most brittle

adhesives, especially at low load levels, and the approach is useful to predict the stress

distribution and the location of peak stress values. However, most polymeric structural

adhesives exhibit inelastic behaviour, in the sense that plastic residual strains are induced

even at low levels of external loading, and plastic yielding will appear in most adhesive

bonded joints as the load is increased to failure, see Hart-Smith [8–10], Pickett [14–15],

Adams [2], Gali [6–7] and Thomsen [16–17]. Thus, the assumption on linear elasticity of the

adhesive is clearly an approximation.

Based on the structural analysis described in Section 1.2, the bonded joint analysis has been

extended to include adhesive plasticity. However, non-linear time and temperature dependent

effects including visco-elasticity, creep and thermal straining are not considered.

1.7.2 Non-linear formulation and solution procedure

The concept of effective stress/strain is one way of approaching the non-linear problem. In

this approach it is assumed that for a ductile material the plastic residual strains are large

compared with the creep strains at normal loading rates. Therefore, a plastic yield hypothesis

can be applied, and the multidirectional state of stress can be related to a simple unidirectional

stress state through a function similar to that of Von Mises.

However, it is widely accepted that the yield behaviour of polymeric structural adhesives is

dependent on both deviatoric and hydrostatic stress components. A consequence of this

phenomenon is a difference between the yield stresses in uniaxial tension and compression,

see Adams et al. [2], [4], Gali et al. [6–7], Harris et al. [11] and Thomsen [16–17]. This

behaviour has been incorporated into the analysis by the application of a modified Von Mises

criterion suggested by Gali et al. [7]:

( )

t

cVSVDS CCJCJCs

ss

ll

lll

=-

=+

=+= ,2

1,

2

13,12 (1.7.1)

Version 4.4 V.1.33


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where s is the effective stress, J2D is the second invariant of the deviatoric stress tensor, J1 is

the first invariant of the general stress tensor and l is the ratio between the compressive and

tensile yield stresses. J2D and J1 are defined by:

( ) ( ) ( )( )

3211

2

13

2

32

2

2161

2

sssssssss

++=

-+-+-=

J

J D (1.7.2)

For l = 1, Eqs. (1.7.1) are reduced to the ordinary Von Mises criterion. At the failure load

level, the first of Eqs. (1.7.1) is transformed into the expression:

( ) ( )ultultVultDultSult JCJCs 1,2,

2

1

+= (1.7.3)

where the subscript ult denotes “ultimate”. Eq. (1.7.1) describes the failure envelope for the

general case of a ductile material. In three-dimensional stress space Eq. (1.7.1) represents a

paraboloid with its axis coincident with the line s1 = s2 = s3.

The effective strain e is given by Gali et al. [7]:

1221

1

1

1ICICe VDS nn -

++

= (1.7.4)

where n is the Poisson's ratio, I2D is the second invariant of the deviatoric strain tensor and I1

is the first invariant of the general strain tensor. I2D and I1 are defined by:

( ) ( ) ( )( )

3211

2

13

2

32

2

2161

2

eeeeeeeee

++=

-+-+-=

I

I D (1.7.5)

The non-linear adhesive properties are included by implementing an effective stress-strain

relationship derived experimentally from tests on adhesive bulk specimens Thomsen [16–17]

and Tong [19]. Thus, it is assumed that the bulk and “in-situ” mechanical properties of the

structural adhesive are closely correlated as discusses by Gali et al. [7] and shown by

Lilleheden [12].

V.1.34 Version 4.4


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Figure 1.13 (a) Effective stress-strain relationship obtained from tensile test on bulk specimen. (b) Illustration of

piece-wise linear approximation to the curve and the solution procedure for the stress analysis in the non-linear

range.

Based on a secant modulus approach for the non-linear effective stress-strain relationship for

the adhesive, as shown in Figure 1.13, the solution procedure for determining the stress

distribution in the adhesive layer can be described by the following steps:

(1) Calculate the effective strains e1 and stresses *

1s (Eqs. (1.7.1) and (1.7.4)) for each point of

the adhesive layer using the linear elastic solution procedure and assuming a uniform

elastic modulus E1 for the adhesive.

(2) If the calculated effective stresses *

1s are above the proportional limit denoted by sprop,

determine the effective stresses s1 for each point of the adhesive layer according to the

corresponding effective strains e1 (using the experimental relationship given by Figure

1.13) calculated in step (1).

(3) Calculate the difference Ds1 = *

1s -s1 between the “calculated” and the “experimental”

effective stress, and determine the specific secant-modulus tE2 defined by:

1

1

12 }1{ E

s

sE t

÷÷ø

öççè

æ D-= d (1.7.6)

where d is a weight-factor, which determines the change of the modulus in each iteration.

(4) Rerun the procedure (steps (1)-(2)) with the elastic modulus E1 for each adhesive point

modified as per step (3).

(5) Compare the “calculated” effective stresses s* for each adhesive point with the

“experimental” values s obtained from the effective stress-strain curve (Figure 1.12).

(6) Repeat steps (4)-(5) until the difference between the “calculated” and “experimental”

stresses (Ds) drops below a specified fraction (2%) of the “experimental” stress value.

Convergence is usually achieved within a few iterations. The non-linear stress-strain

relationship obtained from tensile test on bulk specimen as illustrated in Figure 1.12a is in

Version 4.4 V.1.35


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ESAComp defined by a piece-wise linear approximation to the curve as illustrated in Figure

1.12b.

The procedure described above has previously been used for the analysis of non-linear

adhesive behaviour in tubular lap joints by Thomsen [16–17].

The maximum effective stress and strain criteria have been investigated by [2] and [3] by

incorporating the two criteria in a finite element analysis of double and single lap joints. Their

investigations showed that for brittle adhesives there was a very close correlation with

experimental results by using the maximum effective stress criterion. For toughened ductile

adhesives they found that the maximum effective strain criterion gave the best prediction of

the joint strength. From the finite element analyses it was also possible to predict the failure

mode fairly accurately.

1.8 VALIDATION OF THE MODELS

The validation of the adhesive layer model by comparison with a high-order theory approach

and finite element models has been presented by Mortensen in reference [13].

1.9 FAILURE ANALYSIS

Failure in adhesive bonded joints can be divided into the following four types:

1. The adhesive may fail due to high shear and transverse normal stresses (cohesive failure).

2. The adhesive/adherend interfaces may fail due to high shear and transverse normal

stresses (interface failure).

3. The adherends may fail due to the external loads coupled with the large bending moment

concentrations induced in the regions near to the ends of the overlap.

4. If the adherends are made of composite material they may fail due to ply-failure caused by

high interlaminar shear stresses.

The failure types 1 and 3 are considered in the ESAComp implementation as described in the

following subsections. The failure type 2 usually appears due to insufficient bonding or

surface preparation and is therefore primarily a question of proper manufacturing. The failure

type 4 is not predicted in the current ESAComp implementation, but the adhesive shear

stresses and adherent resultant shear forces obtained form the joint analysis can be used as the

basis for assessing the criticality of this mode.

1.9.1 Cohesive failure analysis

The cohesive failure analyses are divided into to types of analysis – linear and non-linear

adhesive failure analysis. The procedure for determine the to failure levels are described in

this section.

V.1.36 Version 4.4


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Linear cohesive failure analysis

The linear cohesive failure level is reached when the effective adhesive stresses *

1s are equal

to the proportional limit (sprop) for the adhesive bulk data.

The procedure for determining the reserve factor (RFadh) is as follows:

1. Calculate the adhesive layer effective stresses corresponding to the applied load vector

{F} and determine the maximum value ( *

maxs ).

2. Calculate the reserve factor as RFadh = sprop /*

maxs .

3. If the reserve factor RFadh ³ 1.0, the results from load response analysis equal to the applied

load vector are shown.

4. If the reserve factor RFadh < 1.0, the load vector is multiplied with the reserve factor, i.e.

RFadh{F}, and the load response analysis is performed again with the reduced load vector.

The results from the analysis are shown, i.e. the results at failure load level.

5. If the joint poses two adhesive layers, step 3 and 4 are performed for each of the adhesive

layers. The lowest reserve factor of the two adhesive layers is displayed as the reserve factor

for the joint (RFadh).

Non-linear cohesive failure analysis

In the non-linear cohesive failure analysis two, reserve factors are displayed – the reserve

factor at which the proportional limit is reached, i.e. where plasticity starts (RFadh,prop) and the

reserve factor to failure level (RFadh). The reserve factor to the proportional limit (RFadh,prop) is

determined as for the linear cohesive failure analysis described above. The reserve factor to

failure level or the ultimate load-bearing capability of the bonded joints are determined by an

iterative use of the non-linear solution procedure described in Subsection 1.7.2, where the

external loads are modified between each iteration. The iteration scheme is repeated until the

calculated maximum effective strain reaches the ultimate value eult.

The procedure for determining the proportional adhesive reserve factor (RFadh,prop) and the

failure reserve factor (RFadh) is as follows:

1. Calculate the adhesive layer effective stresses corresponding to the applied load vector

{F} and determine the maximum value ( *

maxs ).

2. Calculate the reserve factor as RFadh,prop = sprop / *

maxs .

3. Call the non-linear solution procedure described in Subsection 1.7.2.

4. Increase or decrease the load vector by multiplying it with the fraction of the ultimate

effective adhesive strain and the maximum effective adhesive layer strain (eult / *

maxe ), i.e.

{F}i+1

= {F}i* eult /

*

maxe , where i is the iteration number starting from 1 to nfailure.

5. Call the non-linear solution procedure described in Subsection 1.7.2, with the modified load

vector input {F}i+1

.

6. Repeat step 5 and 6 until the calculated maximum effective strain reaches the ultimate

value eult, i.e. until emax = eult.

Version 4.4 V.1.37


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7. Calculate the reserve factor to failure RFadh = {F}failure

/{F}applied

.

8. If the reserve factor RFadh ³ 1.0, call the non-linear solution procedure with the applied load

vector {F}applied

to display output from the load response equal to the applied load vector

together with the reserve factors.

Due to the simple way of modelling the adhesive layer (the adhesive is not modelled as a

continuum), it is not possible to predict the failure mode with this approach. However, it

should be possible to predict the joint strength with reasonable accuracy by applying the

maximum effective stress or strain criteria, since equally simple models of the adhesive layer

have been used successfully for the prediction of the joint strength by [8–10]. However, the

predictions should be used for comparative purposes only. For a realistic evaluation of the

predicted results they should be compared with experimental results.

1.9.2 Laminate failure

The failure of the adherends due to external loads combined with joint induced bending

moments is predicted using the laminate first ply failure (FPF) analysis of ESAComp (Part III,

Chapter 5). Potentially critical locations in the vicinity of the joint are considered as illustrated

in Figure X. The in-plane forces and bending moments acting at these locations are obtained

from the joint analysis. For comparison, the FPF reserve factors at the end supports are also

computed. As a result, the reserve factor for adherend (laminate) FPF are

V.1.38 Version 4.4


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REFERENCES

1. Adams, Robert D. and Wake, William C., Structural Adhesive Joints in

Engineering, Elsevier Applied Science Publishers, 1984, 1. ed.

2. Adams, R. D. and Coppendale, J. and Peppiatt, N. A., Failure Analysis of

aluminium-aluminium bonded joints, Adhesion, 1978, vol. 2, pp. 105–119.

3. Crocombe, A. D. and Adams, R. D., An effective stress/strain concept in

mechanical characterization of structural adhesive bonding, Journal of Adhesion,

1981, vol. 13, 2, pp. 141–155.

4. Adams, R. D., Stress analysis: a finite element analysis approach, Developments in

Adhesives, Applied Science Publishers, London, 1981, 2 ed.

5. Frostig, Y. and Thomsen, O. T. and Mortensen, F., Analysis of Adhesive Bonded

Joints, Square-end and Spew-Fillet: Closed-Form Higher-Order Theory Approach,

Report No. 81, Institute of Mechanical Engineering, Aalborg University, Denmark,

1997, submitted.

6. Gali, S. and Ishai, O., Interlaminar stress distribution within an adhesive layer in

the nonlinear range, Journal of Adhesion, 1978, vol. 9, pp. 253–266.

7. Gali, S. and Dolev, G. and Ishai, O., An effective stress/strain concept in

mechanical characterization of structural adhesive bonding, International Journal of

Adhesion and Adhesives, 1981, vol. 1, pp. 135–140.

8. Hart-Smith, L. J., Adhesive bonded single lap joints, Technical report NASA CR

112236, Douglas Aircraft Company, McDonnell Douglas Corporation, USA, 1973.

9. Hart-Smith, L. J., Adhesive bonded double lap joints, Technical report NASA CR

112237, Douglas Aircraft Company, McDonnell Douglas Corporation, USA, 1973.

10. Hart-Smith, L. J., Adhesive bonded scarf and stepped-lap joints, Technical report

NASA CR 112235, Douglas Aircraft Company, McDonnell Douglas Corporation,

USA, 1973.

11. Harris, J. A. and Adams, R. D., Strength prediction of bonded single lap joints by

non-linear finite element methods, International Journal of Adhesion and

Adhesives, 1984, vol. 4, pp. 65–78.

12. Lilleheden, L., Properties of adhesive in situ and in bulk, International Journal of

Adhesion and Adhesive, 1994, vol. 14, 1, pp. 31–37.

13. Mortensen, F., Development of Tools for Engineering Analysis and Design of

High-Performance FRP-Composite Structural Elements, Ph.D.-thesis, Institute of

Mechanical Engineering, Aalborg University, Denmark, 1998, Special Report No.

37. (http://www.aub.auc.dk/phd/)

14. Pickett, A. K., Stress analysis of adhesive bonded lap joints, Ph.D. thesis,

University of Surrey, 1983.

Version 4.4 V.1.39


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15. Pickett, A. K. and Hollaway, L., The analysis of elasto-plastic adhesive stress in

adhesive bonded lap joints, Composite Structures, 1985, vol. 4, pp. 135–160.

16. Thomsen, O. T., Analysis of adhesive bonded generally orthotropic circular

cylindrical shells, Ph.D.-thesis, Institute of Mechanical Engineering, Aalborg

University, Denmark, 1989, Special Report No. 4.

17. Thomsen, O. T., Elasto-static and elasto-plastic stress analysis of adhesive bonded

tubular lap joints, Composite Structures, 1992, vol. 21, pp. 249–259.

18. Thomsen, O. T. and Rits, W. and Eaton, D. C. G. and Brown, S., Ply Drop-off

effects in sandwich panels - theory, Composites Science and Technology, 1996,

vol. 56, pp. 407–422.

19. Tong, L., Bond strength for adhesive-bonded single-lap joints., Acta Mechanic,

Springer-Verlag, 1996, vol. 117, pp. 101–113.

20. Whitney, J. M., Structural Analysis of Laminated Anisotropic Plates, Technomic

Publishing Company. Inc., Lancaster, 1987.

Version 4.4 V.2.1


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2 Mechanical Joints

2 MECHANICAL JOINTS

Timo Brander (HUT/LLS, 2002)

The procedure for analyzing uniaxial in-plane load induced stresses in mechanically fastened single lap and

double lap joints is presented. The external load can be either tensile or compressive uniaxial load in the joint

length direction. First, the fastener loads are calculated. The stresses of an infinite adherend on the fastener hole

are calculated from the fastener load and from the by-pass load. The joint failure load and the failure mode are

calculated. The procedure is primarily intended for analyzing bolted joints, but also riveted joints with solid

rivets can be analyzed using this procedure providing that proper values for certain parameters describing

fastener flexibility are defined.

SYMBOLS

Ab Fastener cross-sectional area

Aij In-plane stiffness matrix of a laminate

Ai1, Ai2 Cross-sectional area of adherend (= W h)

As Effective area of plate over which the fastener shear acts

[A] Adherend extension matrix

[B] Adherend coupling stiffness matrix

{B} Fastener/hole extension vector

C Fastener head rotational stiffness

D Fastener diameter

d0 Characteristic distance

E Young's modulus

f Fastener/hole flexibility

G Shear modulus

h Adherend thickness

I Stiffness moment of inertia

JF Joint flexibility

JS Joint stiffness

k Effective stiffness, per unit thickness, of adherend supporting

fastener

V.2.2 Version 4.4


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L Overall joint length

el,, er Length of adherend before fastener 1 (left) and fastener N (right),

respectively

M Internal moment in a fastener

N Number of fasteners along joint

P Shear load in a fastener

{P} Fastener load vector

Q Shear load

q Reaction of plate supporting fastener per unit length

r Radius of a fastener hole or fastener

S Stiffness of fastener in a single adherend

W Width of a single line of fasteners

X Total in-plane longitudinal load of a single line of fasteners of

width W

b Rotation of fastener axis due to shear, b = Q/(lGbAb)

d In-plane longitudinal extension; Total deformation of a fastener and

adherend

e Strain

l Shape factor for circular beam, l = 6(1+nb)/(7+6nb )

m Coefficient of friction

n Poisson's ratio

q Direction angle from the x-axis

s Stress

t Shear stress

yb Rotation of a fastener axis due to bending

Subscripts

A,B Single lap joints composing a double-lap joint, bearing

b Fastener

bp By-pass

c Compression, characteristic

i Index, ith fastener or pitch along joint

f Failure

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p Adherend (plate), pitch

r Radial direction

t Tension

xy Orthotropic in-plane coordinate system

q Tangential direction on the hole boundary

1,2 Adherends in a single lap joint

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2.1 INTRODUCTION

Practically all real life structures consist of several sub-structures, or are connected to other

structures. Thus, structures almost inevitably contain joints. The two most commonly used

joining methods of composite structures are mechanical joining and bonding.

Highly loaded mechanical joints use rivet or bolt fasteners. The ESAComp analysis

procedures are basically similar in joints using either fastener. When the fastener loads are

calculated it is assumed that the fasteners are bolts. It is also assumed, by default, that the

bolts are tightened to torque that gives adequate rotational stiffness to the bolt head but does

not damage the adherends by through-the-thickness loads. The friction between adherends is

not considered. The fastener load induced stresses are calculated assuming a pin type fastener.

This means that no clamping in the laminate thickness direction is considered at this stage of

analysis. Thus, in this respect, the procedure gives conservative joint failure loads with

respect to bolted joints where at least some clamping or constraint is present.

2.2 ANALYSIS APPROACH

The analysis of mechanical joints is based on the following assumptions:

1. Adherend thicknesses are constant

2. Effects of adherend bending are neglected

3. Adherend strains are assumed constant through the thickness of the adherend

4. Load from a fastener to the adherend is transferred purely by bearing (pin joint).

The analysis procedure can be outlined as follows:

1. The fastener loads are solved according to the theory presented in ESDU 85035 [1] and

85034 [2] in which the flexibility of the components (adherends) and fasteners is included.

A new ESDU should replace the aforementioned ESDUs. A draft version of that, ESDU

S681D [3], is also used. For unsymmetric laminates zero-curvature moduli are used.

2. The stress field at the fastener hole is solved using the theory of anisotropic plates.

However the present solution is limited to orthotropic adherends where A16 = A26 = 0 and

[B] = 0. The solution includes the effects of the fastener load and by-pass load. The

applied theory applies for infinite adherends only. The procedure is as follows:

a) The stresses induced by the fastener are solved according to presentation of Zhang and

Ueng [4]. The solution is based on the theory of anisotropic plates by Lekhnitskii [5].

b) The by-pass load induced stresses are solved using the circular open hole solution for

infinite plate (see Part IV, Chapter 3).

c) In both cases the solution is based on linear elastic behavior of materials. Thus, the

stress fields can be summed.

3. The failure modes and margins of safety are evaluated at various points around the

fastener hole to assess the load carrying capability and the potential failure mode of the

Version 4.4 V.2.5


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2 Mechanical Joints

joint.

4. It is also possible to analyze the case of finite width adherends. However, this procedure

can be applied only to tangential stress at q = ± p/2, i.e. at locations deviating ±90° from

the fastener load direction.

2.3 MECHANICAL JOINT LOAD RESPONSE

2.3.1 Fastener and by-pass loads

The fastener loads are determined according to principles presented in references [1–3]. The

approach is valid only for single row, single lap joints subjected to in-plane tension loads.

However, the same analysis can also be applied to double lap joints and multiple row joints

provided the rows are identical. The analysis provides the loads carried by each fastener, by-

pass loads and the overall in-plane flexibility of the joint.

The approach is in principle valid only for isotropic materials. However, isotropy has a

significant effect only on the fastener/hole flexibility. Thus, it is believed that this approach

can be applied also to orthotropic materials with adequate confidence.

The analysis approach is based on small displacement elastic theory and it does not consider

the moment effects due to the eccentricity of the loading. The joint geometry and notation is

shown in Figure 2.1.

el er

L

p1 pi pN-1

X

X X-P1 X-åPk PN

P1 åPk X-PN

Faste

ner

1

Faste

ner

2

Faste

ner

i

Faste

ner

i+1

Faste

ner

N-1

Faste

ner

N

I 1 I 2

Figure 2.1 Single lap joint geometry and notation

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Each pitch between the fasteners is considered separately, and the effects are summed. For

compatibility, the pitch extensions for adherends 1 and 2 at fastener positions i are equal

dd 21 = ii (2.3.1)

The pitch extensions are made up of two components a) adherend extensions (dip1, dip2) and b)

fastener/hole deformations (di).

The adherend extensions for the adherend 1 and for the adherend 2 are

úû

ùêë

é-å P X

E A

p =

i

1 = k1x1

1p1 k

ii

iid (2.3.2)

úû

ùêë

éå P

E A

p =

i

1 = k2x2

2p2 k

ii

iid (2.3.3)

where Pk is the fastener load of kth fastener and X the total load of a single line of fasteners.

Adherend stiffnesses are zero curvature stiffnesses. The fastener flexibility, f, is related to

both adherend 1 and adherend 2. The pitch extension due to the fastener and fastener hole

flexibility is

i i i i id = P f P f+1 +1 - (2.3.4)

Combining Equations (2.3.1 - 4) the compatibility equation for each pitch becomes

0iiiik

ii

i

k

ii

i =-+úû

ùêë

é

÷÷÷

ø

ö

ççç

è

æ-ú

û

ùêë

é-÷÷

ø

öççè

æ åå fPfPP

EA

p

P XE A

p

1+1+

i

1 = k2x2

2i

1 = k1x1

1 (2.3.5)

where fi is the sum of the individual fastener/hole flexibility in each adherend as explained in

the next subsection.

Overall load compatibility yields

X = Pk=1

N

å k (2.3.6)

Equations (2.3.5) and (2.3.6) represent N simultaneous equations which may be written in the

matrix form

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2 Mechanical Joints

[ ]{ } { }B = PA (2.3.7)

The fastener loads Pi can be solved from this equation.

The by-pass load for each pitch is

å=

-=i

k

kbpi PXP1

(2.3.8)

The joint extension is found by adding (a) the adherend extensions of the top adherend, (b) the

extensions due to the flexibility of the end fastener N, and (c) the free adherend extensions in

adherends 1 and 2 at each end, between fasteners 1 and N and points I1 and I2, respectively:

EA

eX +

EA

eX + fPP X

EA

p =

x22

r

x11

li

=1k1x1

1

1N

=1i

FF

NNk

ii

i +úû

ùêë

éúû

ùêë

é-åå

-

d (2.3.9)

The joint flexibility is defined as

X

JFd

= (2.3.10)

and the corresponding joint stiffness is

JSJF

=1

(2.3.11)

The flexibility of a double lap joint is obtained by analyzing two single lap joints, which are

partitioned from the double lap joint about the midplane of the center adherend as shown in

Figure 2.2. The flexibility of the double lap joint is

J F = J F J F

J F + J FBA

A B

A B

(2.3.12)

where A and B refer to single lap joints composing the double lap joint.

The corresponding stiffness is

JS JS JSAB A B= + (2.3.13)

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2 Mechanical Joints

Center line of center plate

h1

h2 X

Single lap joint A

Single lap joint B

Adherend 1 Adherend 2

X/2

X/2

Figure 2.2 Double lap joint modeled as two single lap joints

The bearing stress for each fastener is

hD

PiB =s (2.3.14)

2.3.2 Fastener flexibility

In Equation (2.3.5) the only unknown parameter is the fastener/hole flexibility, f. It can be

solved by considering a single fastener in shear in a single flat plate (adherend) as shown in

Figure 2.3. The fastener is restrained against rotation. The local deformations of the plate are

included in the analysis. The fastener flexibility is defined as

2

,2,1,h

zjP

= f j -==d

(2.3.15)

If two unequal adherends are connected as a single lap joint, the fastener flexibility is

obtained by summing the flexibility of the fastener in the individual adherends

f + f = f21 (2.3.16)

and the stiffness is

f + f

1 = S

21

(2.3.17)

Version 4.4 V.2.9


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2 Mechanical Joints

z

x

Load P

Center line ofunloaded bolt

Center line ofloaded bolt

Unloaded bolt

d

h

Elastic support

Figure 2.3 Fastener/hole flexibility

The basic governing differential equation for the fastener acting as a beam on an isotropic

elastic foundation is

0 =y I E

k +

zd

xd

I E

A G +

A G

k

zd

xd

A G

A G + 1

bb2

2

bb

szx

bb4

4

bb

szx

úû

ùêë

é-ú

û

ùêë

é

ll (2.3.18)

where l is the shape factor for circular beam, As is the effective area of plate over which the

bolt shear acts, k is the effective stiffness of the plate supporting the bolt and Ib is the stiffness

moment of inertia of the bolt.

The reaction of the plate foundation is

2

2

dz

xdAGxkq szxpz +-= (2.3.19)

The linear small displacement assumption is made which gives

V.2.10 Version 4.4


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2 Mechanical Joints

dz

d I E = M b

bbbz

y (2.3.20)

The following four boundary conditions are applied:

2

,h

zdz

dxAGPQ szxbbz -=-= (2.3.21a)

2

,0h

zM zb -== (2.3.21b)

2

,h

zdz

dxAGQ szxzb =-= (2.3.21c)

2

,h

zCM bzb =Y-= (2.3.21d)

Equation (2.3.18) can be written as

02

''

1

'''' =++ xaxax (2.3.22)

where the primes refer to the differentiation with respect to z. The equation is solved using the

boundary conditions of (2.3.21a - d).

The equation for the rotation of fastener axis due to bending, y, required in Equation

(2.3.21d) is

bb

zb

b

b

AG

Q

dz

dx =

dz

dx =

+ = dz

dx

lby

yb

--

(2.3.23a,b)

From Equations (2.3.20) and (2.3.23) follows

Version 4.4 V.2.11


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2 Mechanical Joints

÷÷ø

öççè

æ

bb

zp

2

2

bbzbAG

q +

zd

xd IE = M

l (2.3.24)

to which the foundation reaction, Equation (2.3.19), is placed. The shear force, Q, is obtained

by derivation from Equation (2.3.24)

÷÷ø

öççè

æ--

dz

dq

AG

1 +

zd

xd IE =

dz

dM = Q

pz

bb

3

3

bbbz

bz l (2.3.25)

Equation (2.3.18) includes three quantities As, C, and k, which are typically not available. To

provide the highest possible accuracy the values for these parameters should be determined

through tests. The following estimates can mainly be used for metallic adherends. If the

fastener head is effectively restrained against rotation, the fastener head rotational stiffness, C,

may be considered as infinite. The effective area of adherend over which shear acts, As, may

be estimated to be As = 0.1 D2. The effective adherend stiffness, k, should be determined

experimentally according to the method described in ESDU 85034 Appendix B [2]. The value

k = 0.18 Ep given in [2] should only be used when titanium alloy fasteners are used in an

aluminum alloy adherend.

2.3.3 Fastener load induced stresses at the pin-loaded hole

The fastener load induced stresses at the fastener hole are determined according to the

presentation of Zhang and Ueng [4]. The analytical solution is based on the theory of

anisotropic plates, but in the present solution, it has been restricted to orthotropic plates to

obtain compact analytical solutions.

The expression of radial stress is

qnqs 3cos)1(2

1)33()1(5cos)1(

)1( 00 úû

ùêë

é --

+--++-+

= knc

nkkcrgc

uknu

rgc

cxyr

qnnp

cos)()1(

)22(2

)1( 00

úû

ùêë

é+-

++++-

-+- nk

rgc

ucnknk

rgc

uc

r

Pxyxy (2.3.26)

and for shear stress

qnqt q 3sin)1(2

1)2()1(5sin)1(

)1( 00 úû

ùêë

é --

+++-+--+

-= knc

nknkcrgc

uknu

rgc

cxyr

V.2.12 Version 4.4


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2 Mechanical Joints

qnnp

sin)()1(

)22(2

)1( 00

úû

ùêë

é+-

+-++-

--+ nk

rgc

ucnknk

rgc

uc

r

Pxyxy

(2.3.27)

When the effect of friction is considered, the following condition applies

qsmqtpp

q drdr rr òò -=2/

0

2/

0

(2.3.28)

In Equations (2.3.29) and (2.3.30) following parameters are used

2/1

÷÷ø

öççè

æ=

y

x

E

Ek (2.3.31)

2/1

)(2úúû

ù

êêë

é+-=

xy

xxy

G

Ekn n (2.3.32)

xyy

yxxy

G

k

Eg +

-=

nn1 (2.3.33)

1

11

A

ABc

-= (2.3.34)

( ) ( )knkBknkA

ABPgu

xyxy +----

-=

nnp 11

110

2 (2.3.35)

where

( ) ( )xyxy knknknknA nmn 1515611101011191 -+-+-++= (2.3.36)

( ) ( )nnkkknB xy ++-+-= 233101101 nm (2.3.37)

The tangential stress sq can be expressed as

54321 qqqqqq ssssss ++++= (2.3.38)

Version 4.4 V.2.13


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2 Mechanical Joints

where

[ qqnqnp

s qq

2224

1 sincos)21(cos xyxy

x

kr

P

E

E+---=

] qqn cossin)22( 42nk xy --++ (2.3.39)

[îíì

+-+úû

ùêë

é ---

= qnqqqs qq

22202 cos)()21()cossin(2cos

2

)1(nkkk

n

rgc

uc

E

Exy

x

] úû

ùêë

é +-+--+- qqqqqqqnn 2sin2

)cossin(2cos2sinsinsin)( 222222 nknkknk xyxy

[ ] }qqnqnn cossin)()2(cos)( 222

xyxyxy kknknkn -++++-+- (2.3.40)

[îíì

+úû

ùêë

é --+

= qqqqs qq

2220

3 cos)21()cossin(2cos2

)1(2knk

n

rgc

uc

E

E

x

] [ ( )qqqqqqn 222 cossin2cos2sinsinsin kn xy -++

][ ] }qqnqnq cossin)2()(cos)(2sin2

2222 kkknn

xyxy +-+-+-+ (2.3.41)

[îíì

+-+úû

ùêë

é --+

= qnqqqqs qq

2220

4 cos)()21()cossin(2cos24cos2

)1(2nkkk

n

rgc

uc

E

Exy

x

] úû

ùêë

é +-+--+- qqqqqqqqnn 4sin2sin2

)cossin(4cos2sinsinsin)( 22222 nknkknk xyxy

[ ] }qqnqnn cossin)()2(cos)( 222

xyxyxy kknknkn -++++-+- (2.3.42)

qs qq2

05 sin2

1uE

rc

c -= (2.3.43)

Eq in the previous equations is the adherend Young’s modulus in the q-direction.

2.3.4 By-pass load induced stresses

The by-pass load induced stresses is solved according to Part IV, Chapter 3 of this document.

The stresses are calculated either on the fastener hole or on a characteristic curve (Point Stress

Criterion), where the point of calculation is determined as

V.2.14 Version 4.4


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2 Mechanical Joints

22

,0

pq

p££-+= drrc (2.3.44)

where d0 is the characteristic distance.

The stresses calculated above apply to a hole in an infinite plate.

2.3.5 Failure of the joint

The failure of the joint is determined primarily on the fastener hole according to the following

procedure based on reference [6]:

1. The fastener load induced laminate stresses sr, sq and trq are calculated.

2. The by-pass load induced stresses are added to the fastener load induced stresses.

3. The FPF analysis approach (Part III, Chapter 5) with the selected failure criterion is

applied to predict the failure load in terms of reserve factor or margin of safety. In tension

loading the failure mode is determined as follows

-15° £ qf £ 15° bearing failure mode

30° < qf < 60° shear-out failure mode

75° < qf < 90° tension failure mode

qf is the angle of the point where the combined stress reaches the critical value. At

intermediate values of qf failure may be caused by a combination of the modes.

In compression loading the failure mode is

-15° £ qf £ 15° bearing failure mode

75° < qf < 90° compression failure mode

At intermediate angles, the failure mode is undefined.

Alternatively, if the characteristic distances are known, the failure can be calculated as

follows:

1. The fastener load induced laminate stresses sr, sq and trq are calculated along the

characteristic curve

rc r

c

c

c

r c r

c

= r

r, =

r

r, =

r

r s s s s t tq q q q (2.3.45)

2. The by-pass load induced stresses are added to the fastener load induced stresses along the

characteristic curve.

The final step is identical to the step 3 above.

Version 4.4 V.2.15


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2 Mechanical Joints

Some comments concerning the approach presented above are:

1. Characteristic distances should be determined experimentally for the laminate if relevant

values are not found in literature.

2. The approach does not take into account the stresses induced by the other closely situated

fasteners/holes.

3. The approach, where the stresses are calculated on the hole boundary, gives conservative

failure loads for the joint compared to the case, where the values are calculated on the

characteristic curve.

2.3.6 Finite width joint

The theory presented above for fastener induced stresses and by-pass load induced stresses is

valid for infinite plates only. However, in typical laminates with realistic end and side

distances and pitches the theory is applicable with reasonable accuracy.

The finite width correction is applied only to tangential normal stress at q = ± p/2. No other

stress components are corrected nor included in the failure analysis. The corrected stress for

the fastener load and for the by-pass load is

2

,p

qsss qqq ±=÷÷ø

öççè

æ+÷÷

ø

öççè

æ= ¥

¥¥

¥ pb

bpT

Tb

bT

T

K

K

K

K (2.3.46)

where sq¥ is the tangential stress in an infinite adherend and subscripts b and bp refer to

fastener and by-pass load, respectively. KT¥ and KT denote the stress concentration on the hole

boundary on the axis normal to the applied load for infinite plate and finite plate, respectively

[7]. The ration of KT and KT¥ is called Finite Width Correction (FWC) factor [7].

The FWC for the fastener load is [8]

32

167.29196.23820.82880.0 ÷ø

öçè

æ+÷ø

öçè

æ-÷ø

öçè

æ+=÷÷ø

öççè

æ¥ W

D

W

D

W

D

K

K

bT

T (2.3.47)

where W is the width of a plate containing a central opening or fastener. The formula is valid

for isotropic materials but it is used here due to the lack of a corresponding formula for

orthotropic materials. In addition, typical laminate structures for mechanical joints have close

to quasi-isotropic properties.

V.2.16 Version 4.4


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2 Mechanical Joints

The FWC for the by-pass load is [7]

( )( )

( )úúû

ù

êêë

é÷ø

öçè

æ--÷ø

öçè

æ+-+

-=÷÷

ø

öççè

æ ¥¥

26

313

2

1

/12

/131 M

W

DKM

W

D

WD

WD

K

KT

bpT

T (2.3.48)

where

÷÷ø

öççè

æ -+-+=¥

66

2

122211122211

66 2

21

A

AAAAAA

AKT (2.3.49)

and

( )( )( )2

3

2

/2

11/12

/1381

WD

WD

WD

M

-úû

ùêë

é-

-+

--

= (2.3.50)

Equations 2.3.46–48 apply for orthotropic plates (laminates).

Version 4.4 V.2.17


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REFERENCES

1. "Computer program for the flexibility of single and double lap thin plate joints

loaded in tension". ESDU 85035. Engineering Sciences Data Unit, London, 1985.

2. "Flexibility of a single bolt shear joint". ESDU 85034. Engineering Sciences Data

Unit, London, 1985.

3. "Flexibility of, and load distribution in, multi bolt lap joints subjected to in-plane

axial loads". Draft data item S681D. ESDU International Plc., Fifth Draft,

September 1996.

4. Zhang, K. and Ueng, C.E.S. "Stresses Around a Pin-loaded Hole in Orthotropic

Plates". Journal of Composite Materials, Vol. 18, September 1984, pp. 432–446.

5. Lekhnitskii, S.G., Anisotropic Plates, English edition (Translated by S.W. Tsai and

T. Cheron), Gordon and Breach, London, 1968.

6. Ueng, C.E.S. and Zhang, K. "Strength Prediction of a Mechanically Fastened Joint

in Laminated Composites". AIAA Journal, Vol. 23, No. 11, November 1985, pp.

1832–1834.

7. Seng C. Tan, Stress Concentrations in Laminated Composites, Technomic

Publishing Company, Inc., USA, 1994

8. Walter D. Pilkey, Peterson’s Stress Concentration Factors, John Wiley & Sons,

Inc., USA, 1997

Documents

1 ADHESIVE BONDED JOINTS - Altair University2018/06/05 · V.1.2 Version 4.4 Theoretical Background of ESAComp Analyses 3.12.2012 V Joints 1 Adhesive Bonded Joints J2D Second invariant