Geometrically non-linear thermal stress analysis of an adhesively bonded tubular single lap joint

Finite Elements in Analysis and Design 39 (2003) 155–174www.elsevier.com/locate/"nel

Geometrically non-linear thermal stress analysis of anadhesively bonded tubular single lap joint

M. Kemal Apalak ∗, Recep G0une1s, Levent F34danci

Department of Mechanical Engineering, Erciyes University, Kayseri 38039, Turkey

Received 4 September 2000; accepted 7 October 2001

Abstract

In this study, the geometrically non-linear thermal stress analysis of an adhesively bonded tubular singlelap joint subjected to air9ows having di:erent temperature and velocity outside and inside its inner and outertubes was carried out using the incremental "nite element method. Since the thermal expansion coe<cientsof tubes and adhesive were di:erent the thermal strains arising along the adhesive–tube interfaces resulted inhigh thermal stresses in the adhesive layer and tubes for di:erent end conditions applied to the ends of itsouter and inner tubes. The joint region was considerably deformed, and serious stress concentrations occurredaround the adhesive free ends. This e:ect became more evident in cases the tube edges were fully restrained.The peak thermal adhesive and tube stresses occurred at the free ends of the adhesive–tube interfaces. Inaddition, increasing the overlap length had an e:ect of reducing the peak adhesive and tube stresses, and anoptimum overlap length=joint length ratio of 0.2 was determined. ? 2002 Elsevier Science B.V. All rightsreserved.

1. Introduction

Advancements in adhesive technology have allowed the adhesive joints to be used as a structuralelement. The deformation and stress states of the adhesively bonded joints have been investigatedextensively; consequently a large number of theoretical and experimental studies are available [1,2].The single lap joints and its derivatives have been used in the stress and deformation analyses due totheir simple geometry. Generally, the mechanical properties of adhesive and adherends were assumedto be linear elastic, and the adhesive joints have been analysed under structural loads. Attention waspaid on the stress and strain concentrations arising around the adhesive free ends, and on parametersa:ecting joint strength, such as overlap length, adhesive thickness, and adhesive=adherend sti:nessratio. Whereas the analytical methods have some limitations in solving the adhesive joint problem

∗ Corresponding author. Tel:. +90-352-437-4901; fax: +90-352-437-5784.E-mail address: [email protected] (M.K. Apalak).

0168-874X/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved.PII: S 0168-874X(02)00062-8

156 M.K. Apalak et al. / Finite Elements in Analysis and Design 39 (2003) 155–174

due to di<culties arising boundary conditions, the "nite element method has been applied suc-cessfully. The elasto-plastic and visco-elastic stress analyses of adhesive joints were also carriedout [1,2].

Another adhesive joint used in load transmission of thin-walled structures is tubular single lap(TSL) joint. Since epoxy adhesives are usually rubber modi"ed in order to obtain higher tough-ness, they present non-linear behaviour under loads. The exact solutions of the stress and deforma-tion problem of TSL joints are di<cult. Adams and Peppiatt [3] carried out elastic stress analysisof a TSL joint, and investigated the torque transmission capabilities of the tubular scarf joints.Thomsen and Kildegaard [4], Chon [5], Graves and Adams [6], and Hipol [7] studied the designand stress analyses of the TSL joints made of composite materials. Lee et al. [8] investigated thee:ects of the adhesive thickness, adherend roughness on the torsional static and fatigue strengths ofthe adhesive TSL joints, and developed a failure model [9–11]. Alwar and Nagaraja [12], and Zhouand Rao [13] considered time-dependent viscoelastic behaviour of the adhesive in the stress analysisof a TSL joint. Thomsen [14] showed the e:ect of non-linear adhesive properties on the adhesivestress distribution in the elasto-plastic stress analysis of the TSL joint made of two orthotropiccircular cylindrical laminated shells. Chen and Cheng [15], Rao and Zhou [16], and Whitcomband Woo [17] presented the studies in which the torsional adhesive stresses, the transverse vibra-tion and damping, and the debond growth of an adhesively bonded tubular joint were investigated,respectively.

In practice, the adhesive joints also experience thermal loads as well as structural loads. Sincethe thermal loads cause thermal strains in solids, in case the degrees of freedom (translation androtation) of structures are restrained the thermal strains would cause thermal stresses in the structures.Since the adhesive and adherends (plate or cylindrical shell) have di:erent thermal and mechanicalproperties they present di:erent deformation states. However, the strains should be compatible alongthe adhesive and adherend interfaces, therefore the thermal stresses occur. First, Kukovyakin andSkory [18] considered the e:ect of the adhesive and adherend thermal expansion coe<cients inthe thermal elastic stress analysis of a bonded cylindrical joint. The following studies concentratedon the thermal residual stresses arising in the adhesive butt, single lap and tubular joints made ofmetals and composites [19–36]. However, the thermal stress analyses were carried for a uniformtemperature distribution, i.e. a constant temperature distribution in all joint regions, or along thegeometrical boundaries. Due to variable thermal conditions the heat transfer by conduction andconvection across the adherend and adhesive thickness was not considered. The variable thermalboundary conditions result in non-uniform temperature distributions in all joint members. As a resultof the non-uniform temperature distribution and di:erent material properties the thermal strains atthe di:erent locations of the adhesive joint, even in zones with similar material properties becomeat di:erent levels. Consequently, the stress distributions in the adhesive layer and the adherends ofthe adhesive joint would be considerably a:ected.

In this study, the thermal stress analysis of a TSL joint was carried out for variable thermalboundary conditions allowing the conductive and convective heat transfers in order to determine thee:ect of the thermal strains on its stress states. First, the thermal analysis of the TSL joint wascarried out in order to compute the thermal strain distributions. Later, based on the thermal straindistributions the geometrical non-linear stress analysis of the TSL joint was carried out using theincremental "nite element method in which the small strain–large displacement theory was applied.More details can be found in Refs. [37–42].

M.K. Apalak et al. / Finite Elements in Analysis and Design 39 (2003) 155–174 157

L

l

r0= 0.2 t2

A detail

ØR

1o

a

ØR

1i

Aft

ftt2

ØR

2o

ØR

2i

r

z

Detail-A

Fig. 1. Dimensions of a TSL joint.

Table 1Mechanical and thermal properties of tubes and adhesive

Medium carbon Rubber modi"edsteel epoxy adhesive

Modulus of elasticity E, (GPa) 209 3.33Poisson’s ratio � 0.29 0.34Coe<cient of thermal expansion �; (1=

◦C) 11:3× 10−6 45–65× 10−6

Thermal conductivity k; (W=m K) 52 0.19

2. Joint con�guration and �nite element model

The TSL joint consists of an epoxy adhesive layer and two steel tubes. The main dimensions ofthe TSL joint are shown in Fig. 1. An inner diameter R1i = 10 mm for inner tube, an inner diameterof R2i = 14:4 mm for outer tube, a thickness of 2 mm and a length l = 30 mm for inner and outertubes, and an adhesive thickness t2 = 0:2 mm were kept constant through the analyses. The overlaplength a was taken as 5 mm for the "rst analysis.

The inner and outer tubes made of steel were bonded using an epoxy-based adhesive. The me-chanical and thermal properties of the steel and the epoxy adhesive were given in Table 1. Inthe analysis, the joint members, i.e. tubes and adhesive layer, were assumed to have linear elasticproperties, and the material non-linearity was not considered.

In production of adhesively bonded joints the adhesive layer is compressed between plates andsome amount of adhesive is squeezed out. Since the adhesive accumulations around the adhesive


free ends, called spew "llet have a considerable e:ect on the peak adhesive stresses arising at theadhesive free ends [1,2], the presence of adhesive "llets were taken into account in the analysis, andthe shape of adhesive "llets was idealised to a triangle with a height and a width twice the adhesivethickness (ft = 0:4 mm).The previous studies on the stress and deformation states of the single lap joints have shown

that the peak adhesive stress and strains occurred at the adhesive vicinities around the corners ofthe adherends [1,2]. In case the adherend corners are sharp, the stress singularities occur at thesesharp corners due to the geometrical and material discontinuities. In fact, in order to obtain a betterbonding surface as possible, the bonding surfaces of the adherends are etched; therefore, these typesof sharp corners in the adherends are unusual. The rounded corners cause uniformly varying lowerstress levels around them than those in case of the sharp corners. Adams and Harris showed thatthe local geometry of the adherend edges had a considerable e:ect on the adhesive stresses [43].For this reason, the corners of both adherends were rounded with a radius r0 = 0:2; t2 = 0:04 mm asshown in Fig. 1.

In the "nite element method, any continuous media under given boundary and loading conditionsis divided into elements with "nite size including nodes at its corners and edges. Therefore, the "niteelement type is important in modelling the structure and to achieve a reasonably accurate solution.Since the thermal stress analysis of the TSL joint requires a two-dimensional axi-symmetric "niteelement, a four-noded quadrilateral axi-symmetric element (called couple "eld element) was used tomodel the inner and outer tubes, and the adhesive layer. The couple "eld element is capable of thethermal and structural analyses. In cases the results of any step of the thermal analysis may havean important e:ect on the stress and deformation states of a structure, this type of element allowsthese e:ects to be taken into account without completing the thermal analysis. This capability playsan important role on the geometrical non-linear stress analysis.

A series of analyses have shown that the mesh re"nement is essential around the adhesive free endsin which high stress and strain gradients occurred in order to achieve reasonable results. Therefore,the "nal mesh details are shown in Fig. 2. The "nite element size was taken as 0:02 mm in theadhesive zones around the rounded adherend corners. The "nite element software ANSYS 5.3 wasused for the thermal stress analysis [44].

3. Thermal analysis

In this study, the adhesively bonded TSL joint was assumed to experience variable thermal loadsalong its inner and outer surfaces. Since the mechanical and thermal properties of the adhesiveand tubes, such as thermal expansion and conductivity coe<cients, are di:erent the thermal strainswould be expected to cause high stress distributions in the joint members as a result of a non-uniformtemperature distribution through the joint members. However, the temperature distribution in the jointcorresponding to each step of the geometrical non-linear stress analysis should be determined forspeci"ed variable thermal boundary conditions.

In the previous studies, a uniform temperature distribution along the boundaries of the adhe-sive joints or in the whole adhesive joint was prescribed and then the thermal strains were com-puted using the temperature di:erences relative to the initial uniform joint temperature[19–36].


Fig. 2. Mesh details of the "nite element model of the TSL joint.

The thermal stress analysis of the TSL joint is three-dimensional problem. Since the TSL jointhas an axi-symmetrical geometry, the problem can be reduced to two-dimensional (axi-symmetrical)case. The thermal boundary conditions applied to the TSL joint were taken from the real world’sapplications. The initial temperature of all the joint members was assumed to be uniform at 20◦Cbefore the variable thermal boundary conditions were applied. The TSL joint exposes to air9ows indi:erent temperature and velocity along inner and outer surfaces of its both tubes as shown in Fig.3. Thus, the air 9ows in a temperature of 120◦C and a velocity of 1 m=s along inner surfaces ofthe inner and outer tubes pointed by 1–2–3–4–5 lines in Fig. 3. Whereas the air 9ows horizontallyalong surfaces 1–2 and 4–5, it is perpendicular to the inner tube edge along surfaces 2–3 and 3–4.Similarly, the air 9ows in a temperature of 20◦C, and a velocity of 1 m=s along the outer surfacesof both tubes, the air 9ow is horizontal to the tube surfaces 6–7 and 9–10, but is perpendicular tothe tube surfaces 7–8 and 8–9.

The heat transfer takes places by convection from air to tubes or adhesive, and by conductionthrough tubes and adhesive layer of the TSL joint. The heat transfer by convection requires thecomputation of the heat transfer coe<cients between the air and the tube or the adhesive surfaces.The heat transfer coe<cient is dependent on the 9ow direction of the air with respect to the adherendand adhesive surfaces (vertically or horizontally), surface (wall) temperature, the air properties,


r

z

B T0 = 20°C

3

1

10 9

T0 = 20°C

2

4 5

6 7

8

A

C T0 = 20°C

T0 = 20°C

T0 = 20°C

T1 = 120°C U1 = 1 m/s

T2 = 20°C U2 = 1 m/s

T2 = 20°C U2 = 1 m/s

L C

Fig. 3. Thermal boundary conditions of a TSL joint (A: inner tube, B: adhesive layer, C: outer tube, T0: initial temperature).

etc. Each of surfaces has di:erent geometry and is subjected to 9ow conditions, the heat transfercoe<cients should be computed for each surface separately.

In case of an air9ow perpendicular to tube surface, the heat transfer coe<cient is given as [45]

hm = 0:205(U∞Deqv

�

)0:731 �′airDeqv

; (1)

where U∞ is air velocity (m=s), Deqv is equivalent diameter (m), � is kinematic viscosity (m2=s)and �′air is thermal conductivity of the air (W=m ◦C).

In case of the horizontal 9ow, the heat transfer coe<cient is given as [45]

hm =Nu�L

; (2)

where � is thermal conductivity of the air (kcal=m h ◦C), L is plate length (m) and Nusselt numberis de"ned as

Nu= 0:836Re1=2 Pr1=3; (3)

where Reynolds and Prandtl numbers, respectively,

Re =U∞L�

; (4)

Pr =cp��

; (5)

where cp is speci"c heat (kcal=kg ◦C) and � is dynamic viscosity (kg=ms). The thermal propertiesof the air are computed based on an average temperature as follows:

Tf =T∞ + T

2; (6)


Table 2Thermal properties of the air and heat transfer coe<cients hm (W=m2 ◦

C) along tube inner and outer surfaces and adhesiveouter surfaces

Vertical air 9ow Horizontal air 9ow

Surface No→ 7–8, 8–9 2–3, 3–4 6–7 9–10 1–2 4–5

Tf (◦C) 20 120 (20 + 20)=2 (120 + 20)=2

� (m2=s) 15:11× 10−6 23:23× 10−6 15:11× 10−6 19:92× 10−6

�′(W=m◦C) 0.0257 0.0328 — —

� (kcal=m h◦C) — — 0.0221 0.0251

cp (kcal=kg◦C) — — 0.24 0.241

� (kg=ms) — — 1:82× 10−5 2:05× 10−5

hm (W=m2 ◦C) 112.887 104.799 28.489 35.2462 28.152 34.829

Fig. 4. The temperature distributions in the enlarged joint region of the TSL joint (in◦C).

where T∞ and T are the air and plate temperatures, respectively. The thermal properties of the airand the heat transfer coe<cients are given in Table 2.

In the thermal analysis, after the heat transfer reached a steady state, the temperature distributionin the adhesive joint was considered. This temperature distribution in the joint region is shown inFig. 4. The temperature levels in the joint members decrease from 69◦C to 55◦C. The temperaturedi:erence between the highest and lowest temperature levels is 14◦C because the tubes are thin.However, the temperature di:erence causing the thermal strains reaches a maximum value of 49◦C(between the highest and initial temperatures). The high temperatures arise along the inner adhesive"llet and decrease through the joint.

Finally, it is obvious that the adhesive TSL joint has a non-uniform temperature distribution.Consequently, the thermal strain distributions are expected to become non-uniform in comparisonwith those in case of a uniform temperature distribution.


Fig. 5. Boundary conditions of the tubular single lap joint: (a) (BC-I), Only one corner of two tubes fully "xed,(b) (BC-II), The free ends of two tubes fully "xed and (c) (BC-III), Left edge of the inner tube fully "xed and the rightfree edge of the outer tube free only in the z-direction.

4. Geometrically non-linear stress analysis

In this section, the geometrical non-linear stress analysis of the adhesively bonded TSL jointwas carried out based on the small strain–large displacement theory. The incremental "nite elementmethod was used to solve the stress problem. The thermal strains computed based on the temper-ature distributions obtained from the thermal analysis were applied to the TSL joint incrementally,and the thermal stress distributions were computed for three tube edge conditions as shown inFig. 5.

In the "rst boundary condition (BC-I), one corner at the free edges of both inner and outer tubeswas "xed in all (r; �; z) directions (Fig. 5a), and in case of the BC-II the free edges of the tubeswere "xed (Fig. 5b). The BC-III assumes that the free edge of the inner tube was "xed whereas theuppermost and lowermost nodes of the free edges of the outer tube were "xed only in the r-direction(Fig. 5c).

Based on the geometrical non-linear stress analysis, the deformed and undeformed geometries (notscaled) of the TSL joint were compared in Fig. 6 for each boundary condition. In cases of the BCs-Iand -II, large translations and rotations occurred in the joint region due to the applied restraints atthe free edges of the tubes (Figs. 6a and b). However, this e:ect was not observed in the BC-III(Fig. 6c) because one edge of the inner tube is free in the z-direction.

Since the high stress and strain gradients occur in the joint region, the radial �rr , normal �zz,and shear �r� stress distributions in the joint region are shown in Figs. 7–9 for the BCs-I, -IIand -III, respectively. For all boundary conditions, the inner and outer tubes experience higher


Fig. 6. Deformed and undeformed geometries of the TSL joint for (a) BC-I, (b) BC-II and (c) BC-III.

stresses along the overlap region in comparison with those in the other tube regions. The adhe-sive layer is subjected to moderate stresses along the middle overlap region whereas the seriousstress concentrations occur around the adhesive free ends. Similarly, the inner and outer tubes alsoexperience stress concentrations in the tube regions corresponding to the adhesive free ends. Incases of the BCs-I and -II, whilst the normal stress �zz is dominant the radial �rr and shear �r�stresses present similar, but still serious levels (Figs. 7–8). In case of the BC-III, the radial �rr ,normal �zz and shear �r� stress levels are almost similar, but the stress distributions are nearly10 times lower than those in the BCs-I and -II (Fig. 9). This is because the free edge of theinner tube is free in the z-direction. Therefore, it is evident that the TSL joint is subjected toconsiderably high stresses in the joint region in cases the tube edges are partly or completelyrestrained.

Since the high stress concentrations are observed around the adhesive free ends, the distribu-tions of the radial �rr , normal �zz and shear �r� stresses in the left and right adhesive free endswere plotted in Figs. 10–12, respectively. In cases of the BCs-I and -II, the stress componentsconcentrate on di:erent adhesive locations in the left and right adhesive "llets (Figs. 10 and 11).Thus, the adhesive zone around the rounded tube corners and the outer free surface of the ad-hesive "llet are critical adhesive regions in which the stress concentrations occur. However, thestress levels around the rounded tube corners are 10 times higher than those along the outer sur-face of the adhesive "llets. As a result, the "rst crack initiation can be expected at the adhesivezone around the rounded tube corner and propagates to the free surface of the adhesive "llet. Inboth cases, the stress components reach high levels, thus, the normal stress �zz is dominant and ishigher 1.8–2.6 times higher than the radial stress �rr and shear �r� stress, respectively. However,


Fig. 7. The variations of (a) radial �rr , (b) normal �zz and (c) shear �r� stress components in the joint region for theBC-I (stresses in Pa).

in case of the BC-II, the stress levels are slightly higher (1.13–1.23 times) than those in case ofthe BC-I.

In case of the BC-III, the TSL joint has also di:erent stress concentration regions inside theadhesive "llets (Fig. 12); thus, the adhesive cap–inner and outer edge interfaces, the tube–adhesiveinterfaces, and the free ends of the tube-adhesive interfaces. The stress components in the BC-IIIare 2.1–2.6 times lower than those in the BCs-I and -II (Figs. 10 and 11).

Finally, considerably high stress distributions occur in the TSL joint having structural boundaryconditions as a result of the thermal strains arising due to a non-uniform temperature distributionthroughout the joint members. The adhesive layer and particularly adhesive "llets are critical jointregions. Therefore, the thermal loads as well as the structural loads should be taken into account inthe design and strength analysis of the adhesively bonded joints.


Fig. 8. The variations of (a) radial �rr , (b) normal �zz and (c) shear �r� stress components in the joint region for theBC-II (stresses in Pa).

5. E ect of overlap length

Due to the advancement in the adhesive technology rubber toughened epoxy adhesives can with-stand high plastic strains. It means that as the adhesive layer yields locally the adherends may alsoyield under the structural and thermal loads. Therefore, not only adhesive layer but also the ad-herends should be analysed and their stress concentration regions should be determined. A detailedanalysis of tube and adhesive stresses for all boundary conditions showed that the peak tube andadhesive stresses occurred at P; Q; R; S and T points as shown in Fig. 13. These critical locationscorrespond to the free ends of the adhesive layer–tube interfaces, and the rounded tube corners.

The geometrical modi"cations of the joint members can have an e:ect of reducing the peakadhesive stresses, such as tapering the adherend edges, or increasing the overlap length. The previous


Fig. 9. The variations of (a) radial �rr , (b) normal �zz and (c) shear �r� stress components in the joint region for theBC-III (stresses in Pa).

studies concerning the stress analysis of the adhesive TSL joint subjected to tensile=torsional loadsshowed that the overlap length had a considerable e:ect of reducing the adhesive stress concentrationand that an optimum overlap length exists [1,2].

In order to determine the e:ect of the overlap length on the peak adhesive and tube stresses, theratio of the overlap length to the joint length a=L was changed from 0.09 to 0.33 for all boundaryconditions. The variations of the normalized radial �rr , normal �zz and shear �r� stress components,and von Mises stress evaluated at the critical adhesive locations (inside right adhesive "llet, R andT ) were plotted in Fig. 14 for the boundary conditions I, II and III, respectively. In cases of theBCs-I and -II the TSL joint experiences peak stresses 3–3.5 times higher than those in the BC-III.In cases of the BCs-I and -II, increasing overlap length resulted in a decrease of about 35–59%in the peak adhesive stresses (Figs. 14a and b) whereas the overlap length has a minor e:ect (an


Fig. 10. The variations of (a) radial �rr , (b) normal �zz and (c) shear �r� stress components inside the left and rightadhesive "llets for the BC-I (stresses in Pa).

increase of about 3.5–7%) on the adhesive stresses as shown in Fig. 14c for the BC-III. In addition,increasing the overlap length does not result in considerable e:ect of reducing peak adhesive stressesafter an overlap length=joint length ratio a=L of 0.2 for all boundary conditions.

The variations of the normalized radial �rr , normal �zz and shear �r� stress components, andvon Mises stress evaluated at the peak stress locations of the inner (P; S) and outer tubes (Q; T )were plotted in Figs. 15 and 16 for the BCs-I, -II and -III, respectively. The most critical stresslevels occurred in the BCs-I and -II. Thus, the peak stresses are higher by 550–650% than thosein the BC-III. Whereas the overlap length=joint length ratio a=L between 0.09 and 0.33 resulted ina decrease of about 25–30% in the critical locations in the outer tube for the BCs-I and -II (Figs.15a and b), a small decrease of 8–10% in the peak tube stresses was observed (Fig. 15c) in caseof the BC-III.

The most critical stresses in the inner tube occurred for the BCs-I and -II. Thus, the peak stresseswere higher by 600–700% than those in the BC-III. Increasing the overlap length resulted in de-creases of 30–60% in the peak tube stresses as shown in Fig. 16. Increasing the overlap lengthgives rise to a minor e:ect of reducing the peak tube stresses after the overlap length=joint lengthratio a=L of 0.2.


Fig. 11. The variations of (a) radial �rr , (b) normal �zz and (c) shear �r� stress components inside the left and rightadhesive "llets for the BC-II (stresses in Pa).

6. Conclusions

In this study, the geometrically non-linear thermal stress analysis of a TSL joint subjected tothe variable thermal loads along the outer and inner surfaces of the tubes and various tube edgeconditions was carried out using the small strain–large displacement theory. The thermal analysisshowed that the TSL joint is subjected to a non-uniform temperature distribution. In addition, thehigh thermal stress concentrations were observed around the adhesive free ends, and the stress levelspropagated along the adhesive layer and throughout both inner and outer tubes by loosing their mag-nitudes. The peak adhesive and tube stresses occurred at the free ends of the adhesive–inner=outertube interfaces, and the lowest stresses at the adhesive caps neighbouring the tube edges. In addi-tion, the adhesive stress distributions around the rounded tube corners were smooth. In cases the freeedges of the inner and outer tubes were partly or fully restrained the thermal strains resulted in largethermal stresses in the adhesive layer and the tubes. Increasing the overlap length had an importantreducing e:ect of peak adhesive and tube stresses for the present thermal and tube edge boundaryconditions.


Fig. 12. The variations of (a) radial �rr , (b) normal �zz and (c) shear �r� stress components inside the left and rightadhesive "llets for the BC-III (stresses in Pa).

Fig. 13. Critical stress points of the TSL joint.

The adhesive joints in the real world experience harder environmental conditions, and their be-haviours are more complicated. For this respect, the analysis results reported in this study are informa-tive about the adhesive TSL joints subjected to variable thermal loads and di:erent edge conditions,and give knowledge for predicting the stress and deformation states of the adhesive joints with morecomplex geometry.


Overlap length / joint length ratio, a/L

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.09 0.15 0.20 0.28 0.33

N

orm

aliz

ed

str

ess c

om

po

ne

nts

, (

σ ij/σ

ma

x)

( σ eqv ) max = 63.29 MPa

( σ rr )max = -15.17 MPa

(σ zz )max = -31.32 MPa

( σ rθ ) max = -35.64 MPa

R

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.09 0.15 0.20 0.28 0.33

N

orm

aliz

ed

str

ess c

om

po

ne

nts

, (

σ ij/σ

ma

x)

( σσσσ

eqv ) max = 74.36 MPa

( rr )max = -16.75 MPa

( zz )max = -34.54 MPa

( rθ ) max = -41.99 MPa

R

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1.00

0.09 0.15 0.20 0.28 0.33

N

orm

aliz

ed

str

ess c

om

po

ne

nts

, (

σ ij/σ

ma

x)


( σ rr )max = -8.87 MPa

( σ zz )max = -14.81 MPa

( σ r θ ) max = -10.51 MPa

T

(a)

(b)

(c)

Fig. 14. The e:ect of the overlap length on the normalized radial stress �rr , normal stress �zz and shear stress �r�

components and von Mises �eqv stress at the critical locations in the right adhesive "llet of the TSL joint for the boundaryconditions: (a) I, (b) II and (c) III.



0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

0.09 0.15 0.20 0.28 0.33

N

orm

aliz

ed

str

ess c

om

po

ne

nts

, (

σ ij/ σ

ma

x)


( σ rr ) max = -15.23 MPa

( σ zz ) max = -4.35 MPa

( σ rθ ) max = -2.85 MPa

Q

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0.09 0.15 0.20 0.28 0.33

N

orm

aliz

ed

str

ess c

om

po

ne

nts

, (

σ ij/σ

ma

x)

( σ eqv ) max = 162.30 MPa

( σ rr )max = -24.92 MPa

( σ zz )max = -168.70 MPa

( σ rθ ) max = -19.67 MPa T

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0.09 0.15 0.20 0.28 0.33

N

orm

aliz

ed

str

ess c

om

po

ne

nts

, (

ij/ σ

σm

ax)

( σ eqv ) max = 187.43 MPa

( σ rr )max = -28.39 MPa

( σ zz )max = -194.94 MPa

( σ rθ ) max = -22.38 MPa T

(a)

(b)

(c)


components and von Mises �eqv stress at the critical locations in the outer tube of the TSL joint for the boundaryconditions: (a) I, (b) II and (c) III.



0.70

0.75

0.80

0.85

0.90

0.95

1.00

0.09 0.15 0.20 0.28 0.33

N

orm

aliz

ed

str

ess c

om

po

ne

nts

, (

σ ij/ σ

ma

x)

( σ eqv ) max = 150.79 MPa

( σ rr ) max = -24.27 MPa

( σ zz ) max = -185.67 MPa

( σ rθ ) max = -20.53 MPa

P

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0.09 0.15 0.20 0.28 0.33

N

orm

aliz

ed

str

ess c

om

po

ne

nts

, (

σ ij/σ

ma

x)

P

( σ eqv ) max = 175.95 MPa

( σ rr )max = -28.51 MPa

( σ zz )max = -216.03 MPa

( σ rθ ) max = -23.94 MPa

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.09 0.15 0.20 0.28 0.33

N

orm

aliz

ed

str

ess c

om

po

ne

nts

, (

σ ij/σ

ma

x)


( σ rr ) max = -1.47 MPa

( σ zz ) max = -0.60 MPa

( σ rθ ) max = -0.15 MPa

S

(a)

(b)

(c)


components and von Mises �eqv stress at the critical locations in the inner tube of the TSL joint for the boundaryconditions: (a) I, (b) II and (c) III.


References

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Geometrically non-linear thermal stress analysis of an adhesively bonded tubular single lap joint