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

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

Geometrically non-linear thermal stress analysis of an adhesively 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 single lap joint subjected to air9ows having di:erent temperature and velocity outside and inside its inner and outer tubes was carried out using the incremental "nite element method. Since the thermal expansion coe
1. Introduction Advancements in adhesive technology have allowed the adhesive joints to be used as a structural element. The deformation and stress states of the adhesively bonded joints have been investigated extensively; 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 to their simple geometry. Generally, the mechanical properties of adhesive and adherends were assumed to be linear elastic, and the adhesive joints have been analysed under structural loads. Attention was paid on the stress and strain concentrations arising around the adhesive free ends, and on parameters a:ecting joint strength, such as overlap length, adhesive thickness, and adhesive=adherend sti:ness ratio. 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 0 1 6 8 - 8 7 4 X ( 0 2 ) 0 0 0 6 2 - 8

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due to di
t2

ft

ft A

157

ØR 2o

ØR 2i

ØR 1i

ØR 1o

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

a l L

r

r0 =

z 0.2

t2

Detail-A A detail

Fig. 1. Dimensions of a TSL joint. Table 1 Mechanical and thermal properties of tubes and adhesive

Modulus of elasticity E, (GPa) Poisson’s ratio  ◦ Coe
Medium carbon steel

Rubber modi"ed epoxy adhesive

209 0.29 11:3 × 10−6 52

3.33 0.34 45 –65 × 10−6 0.19

2. Joint conguration and nite element model The TSL joint consists of an epoxy adhesive layer and two steel tubes. The main dimensions of the TSL joint are shown in Fig. 1. An inner diameter R1i = 10 mm for inner tube, an inner diameter of R2i = 14:4 mm for outer tube, a thickness of 2 mm and a length l = 30 mm for inner and outer tubes, and an adhesive thickness t2 = 0:2 mm were kept constant through the analyses. The overlap length 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 mechanical and thermal properties of the steel and the epoxy adhesive were given in Table 1. In the analysis, the joint members, i.e. tubes and adhesive layer, were assumed to have linear elastic properties, and the material non-linearity was not considered. In production of adhesively bonded joints the adhesive layer is compressed between plates and some amount of adhesive is squeezed out. Since the adhesive accumulations around the adhesive

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free ends, called spew "llet have a considerable e:ect on the peak adhesive stresses arising at the adhesive free ends [1,2], the presence of adhesive "llets were taken into account in the analysis, and the shape of adhesive "llets was idealised to a triangle with a height and a width twice the adhesive thickness (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 of the adherends [1,2]. In case the adherend corners are sharp, the stress singularities occur at these sharp corners due to the geometrical and material discontinuities. In fact, in order to obtain a better bonding surface as possible, the bonding surfaces of the adherends are etched; therefore, these types of sharp corners in the adherends are unusual. The rounded corners cause uniformly varying lower stress levels around them than those in case of the sharp corners. Adams and Harris showed that the 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 as shown in Fig. 1. In the "nite element method, any continuous media under given boundary and loading conditions is divided into elements with "nite size including nodes at its corners and edges. Therefore, the "nite element 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 "nite element, a four-noded quadrilateral axi-symmetric element (called couple "eld element) was used to model the inner and outer tubes, and the adhesive layer. The couple "eld element is capable of the thermal and structural analyses. In cases the results of any step of the thermal analysis may have an important e:ect on the stress and deformation states of a structure, this type of element allows these e:ects to be taken into account without completing the thermal analysis. This capability plays an 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 ends in 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 the adhesive zones around the rounded adherend corners. The "nite element software ANSYS 5.3 was used for the thermal stress analysis [44]. 3. Thermal analysis In this study, the adhesively bonded TSL joint was assumed to experience variable thermal loads along its inner and outer surfaces. Since the mechanical and thermal properties of the adhesive and tubes, such as thermal expansion and conductivity coe
M.K. Apalak et al. / Finite Elements in Analysis and Design 39 (2003) 155 – 174

159

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 joint has 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’s ◦ applications. The initial temperature of all the joint members was assumed to be uniform at 20 C before the variable thermal boundary conditions were applied. The TSL joint exposes to air9ows in di: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 of the inner and outer tubes pointed by 1–2–3– 4 –5 lines in Fig. 3. Whereas the air 9ows horizontally along 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 surfaces of both tubes, the air 9ow is horizontal to the tube surfaces 6 –7 and 9 –10, but is perpendicular to the tube surfaces 7–8 and 8–9. The heat transfer takes places by convection from air to tubes or adhesive, and by conduction through tubes and adhesive layer of the TSL joint. The heat transfer by convection requires the computation of the heat transfer coe
160

M.K. Apalak et al. / Finite Elements in Analysis and Design 39 (2003) 155 – 174 U2 = 1 m/s

T2 = 20°C

T0 = 20°C T0 = 20°C

T1 = 120°C 2 3

1 A

T0 = 20°C 9

10

CL

U1 = 1 m/s

8

5

4 C

T0 = 20°C 6

7 B

r

T0 = 20°C

T2 = 20°C

U2 = 1 m/s z

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 transfer coe
Nu  ; L

(2) ◦

where  is thermal conductivity of the air (kcal=m h C), L is plate length (m) and Nusselt number is de"ned as Nu = 0:836 Re1=2 Pr 1=3 ;

(3)

where Reynolds and Prandtl numbers, respectively, U∞ L ; Re =  cp  ; Pr =  ◦

(4) (5)

where cp is speci"c heat (kcal=kg C) and  is dynamic viscosity (kg=ms). The thermal properties of the air are computed based on an average temperature as follows: T∞ + T ; (6) Tf = 2

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161

Table 2 ◦ Thermal properties of the air and heat transfer coe
Tf ( C)  (m2 =s) ◦  (W=m C) ◦  (kcal=m h C) ◦ cp (kcal=kg C)  (kg=ms) ◦ hm (W=m2 C)

Horizontal air 9ow

7–8, 8–9

2–3, 3– 4

6 –7

9 –10

20 15:11 × 10−6 0.0257 — — — 112.887

120 23:23 × 10−6 0.0328 — — — 104.799

(20 + 20)=2 15:11 × 10−6 — 0.0221 0.24 1:82 × 10−5 28.489 35.2462

1–2

4 –5

(120 + 20)=2 19:92 × 10−6 — 0.0251 0.241 2:05 × 10−5 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 air and the heat transfer coe
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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 right free 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 joint was carried out based on the small strain–large displacement theory. The incremental "nite element method was used to solve the stress problem. The thermal strains computed based on the temperature 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 in Fig. 5. In the "rst boundary condition (BC-I), one corner at the free edges of both inner and outer tubes was "xed in all (r; ; z) directions (Fig. 5a), and in case of the BC-II the free edges of the tubes were "xed (Fig. 5b). The BC-III assumes that the free edge of the inner tube was "xed whereas the uppermost 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 (not scaled) of the TSL joint were compared in Fig. 6 for each boundary condition. In cases of the BCs-I and -II, large translations and rotations occurred in the joint region due to the applied restraints at the 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, -II and -III, respectively. For all boundary conditions, the inner and outer tubes experience higher

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

163

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 adhesive layer is subjected to moderate stresses along the middle overlap region whereas the serious stress concentrations occur around the adhesive free ends. Similarly, the inner and outer tubes also experience stress concentrations in the tube regions corresponding to the adhesive free ends. In cases 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 nearly 10 times lower than those in the BCs-I and -II (Fig. 9). This is because the free edge of the inner tube is free in the z-direction. Therefore, it is evident that the TSL joint is subjected to considerably high stresses in the joint region in cases the tube edges are partly or completely restrained. Since the high stress concentrations are observed around the adhesive free ends, the distributions of the radial rr , normal zz and shear r stresses in the left and right adhesive free ends were plotted in Figs. 10 –12, respectively. In cases of the BCs-I and -II, the stress components concentrate 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 adhesive "llet are critical adhesive regions in which the stress concentrations occur. However, the stress levels around the rounded tube corners are 10 times higher than those along the outer surface of the adhesive "llets. As a result, the "rst crack initiation can be expected at the adhesive zone around the rounded tube corner and propagates to the free surface of the adhesive "llet. In both cases, the stress components reach high levels, thus, the normal stress zz is dominant and is higher 1.8–2.6 times higher than the radial stress rr and shear r stress, respectively. However,

164

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Fig. 7. The variations of (a) radial rr , (b) normal zz and (c) shear r stress components in the joint region for the BC-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 of the BC-I. In case of the BC-III, the TSL joint has also di:erent stress concentration regions inside the adhesive "llets (Fig. 12); thus, the adhesive cap–inner and outer edge interfaces, the tube–adhesive interfaces, and the free ends of the tube-adhesive interfaces. The stress components in the BC-III are 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 boundary conditions as a result of the thermal strains arising due to a non-uniform temperature distribution throughout the joint members. The adhesive layer and particularly adhesive "llets are critical joint regions. Therefore, the thermal loads as well as the structural loads should be taken into account in the design and strength analysis of the adhesively bonded joints.

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

165

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

5. E ect of overlap length Due to the advancement in the adhesive technology rubber toughened epoxy adhesives can withstand high plastic strains. It means that as the adhesive layer yields locally the adherends may also yield under the structural and thermal loads. Therefore, not only adhesive layer but also the adherends should be analysed and their stress concentration regions should be determined. A detailed analysis of tube and adhesive stresses for all boundary conditions showed that the peak tube and adhesive stresses occurred at P; Q; R; S and T points as shown in Fig. 13. These critical locations correspond 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 peak adhesive stresses, such as tapering the adherend edges, or increasing the overlap length. The previous

166

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

studies concerning the stress analysis of the adhesive TSL joint subjected to tensile=torsional loads showed that the overlap length had a considerable e:ect of reducing the adhesive stress concentration and 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, the ratio of the overlap length to the joint length a=L was changed from 0.09 to 0.33 for all boundary conditions. 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 and T ) were plotted in Fig. 14 for the boundary conditions I, II and III, respectively. In cases of the BCs-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

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

167

Fig. 10. The variations of (a) radial rr , (b) normal zz and (c) shear r stress components inside the left and right adhesive "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 stresses after 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, and von 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 stress levels occurred in the BCs-I and -II. Thus, the peak stresses are higher by 550 – 650% than those in the BC-III. Whereas the overlap length=joint length ratio a=L between 0.09 and 0.33 resulted in a 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 case of the BC-III. The most critical stresses in the inner tube occurred for the BCs-I and -II. Thus, the peak stresses were higher by 600 –700% than those in the BC-III. Increasing the overlap length resulted in decreases of 30 – 60% in the peak tube stresses as shown in Fig. 16. Increasing the overlap length gives rise to a minor e:ect of reducing the peak tube stresses after the overlap length=joint length ratio a=L of 0.2.

168

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Fig. 11. The variations of (a) radial rr , (b) normal zz and (c) shear r stress components inside the left and right adhesive "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 to the variable thermal loads along the outer and inner surfaces of the tubes and various tube edge conditions was carried out using the small strain–large displacement theory. The thermal analysis showed that the TSL joint is subjected to a non-uniform temperature distribution. In addition, the high thermal stress concentrations were observed around the adhesive free ends, and the stress levels propagated along the adhesive layer and throughout both inner and outer tubes by loosing their magnitudes. The peak adhesive and tube stresses occurred at the free ends of the adhesive–inner=outer tube interfaces, and the lowest stresses at the adhesive caps neighbouring the tube edges. In addition, the adhesive stress distributions around the rounded tube corners were smooth. In cases the free edges of the inner and outer tubes were partly or fully restrained the thermal strains resulted in large thermal stresses in the adhesive layer and the tubes. Increasing the overlap length had an important reducing e:ect of peak adhesive and tube stresses for the present thermal and tube edge boundary conditions.

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

169

Fig. 12. The variations of (a) radial rr , (b) normal zz and (c) shear r stress components inside the left and right adhesive "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 behaviours are more complicated. For this respect, the analysis results reported in this study are informative 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 more complex geometry.

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

Normalized stress components, (σij/σmax)

170

1.00 0.90 0.80 0.70

R

0.60 0.50 0.40 0.30

(a)

(σeqv)max = 63.29 (σrr )max = -15.17 (σzz)max = -31.32 (σrθ)max = -35.64 0.09

MPa MPa MPa MPa

0.15

0.20

0.28

0.33

0.20

0.28

0.33

Normalized stress components, (σij/σmax)

1.00

(b)

0.90 0.80 0.70

R

0.60 0.50 0.40 0.30

(σeqv)max = 74.36 (σrr )max = -16.75 (σzz)max = -34.54 (σrθ)max = -41.99 0.09

MPa MPa MPa MPa

0.15

Normalized stress components, (σij/σmax)

1.00 0.99 0.98 0.97 0.96

T

0.95

(σeqv)max = 19.10 (σrr )max = -8.87 (σzz)max = -14.81 (σr θ)max = -10.51

0.94 0.93 0.92

(c)

0.09

0.15

0.20

0.28

MPa MPa MPa MPa

0.33

Overlap length / joint length ratio, a/L

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 boundary conditions: (a) I, (b) II and (c) III.

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

171

Normalized stress components, (σij/σmax)

1.00

(a)

0.95

T

0.90

(σeqv)max = 162.30 MPa (σrr )max = -24.92 MPa (σzz)max = -168.70 MPa (σrθ)max = -19.67 MPa

0.85 0.80 0.75 0.70 0.65 0.09

0.15

0.20

0.28

0.33

Normalized stress components, (σij/σ max)

1.00

(b)

0.95

T

0.90

(σeqv)max = 187.43 MPa (σrr )max = -28.39 MPa (σzz)max = -194.94 MPa (σrθ)max = -22.38 MPa

0.85 0.80 0.75 0.70 0.65 0.09

0.15

0.20

0.28

0.33

0.20

0.28

0.33

Normalized stress components, (σij/σ max)

1.00 0.98 0.96 0.94 0.92

0.88 0.86 0.84

(c)

Q

0.90

(σeqv)max = 28.66 (σrr )max = -15.23 (σzz)max = -4.35 (σrθ)max = -2.85 0.09

MPa MPa MPa MPa

0.15

Overlap length / joint length ratio, a/L

Fig. 15. 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 outer tube of the TSL joint for the boundary conditions: (a) I, (b) II and (c) III.

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M.K. Apalak et al. / Finite Elements in Analysis and Design 39 (2003) 155 – 174 Normalized stress components, (σij/σ max)

1.00

P

0.95 0.90

(σeqv)max = 150.79 MPa (σrr )max = -24.27 MPa (σzz)max = -185.67 MPa (σrθ)max = -20.53 MPa

0.85 0.80 0.75 0.70

(a)

0.09

0.15

0.20

0.28

0.33

0.20

0.28

0.33

0.20

0.28

0.33

Normalized stress components, (σij/σmax)

1.00 0.95 0.90 0.85 0.80 0.75 0.70

(b)

P

(σeqv)max = 175.95 MPa (σrr )max = -28.51 MPa (σzz)max = -216.03 MPa (σrθ)max = -23.94 MPa 0.09

0.15

Normalized stress components, (σij/σmax)

1.00

(c)

0.90 0.80 0.70 0.60

S

0.50 0.40 0.30 0.20

(σeqv)max = (σrr )max = (σzz)max = (σrθ)max = 0.09

24.60 -1.47 -0.60 -0.15

MPa MPa MPa MPa

0.15

Overlap length / joint length ratio, a/L

Fig. 16. 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 inner tube of the TSL joint for the boundary conditions: (a) I, (b) II and (c) III.

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