- Email: [email protected]
Shape optimization of bonded joints
Shape optimization of bonded joints H.L. Groth* and P. Nordlund¢ (*The Royal Institute of Technology and Avesta A B / t T h e Royal Institute of Technology, Sweden)
Shape optimization of bonded joints was performed by use of numerical shape optimization techniques. The aim of this study was to obtain joints that are as strong and light as possible under static loading conditions by changing the profiling of the adherends. Joint types included are the single-lap, the double-lap and the double-strap; a console bonded to a rigid wall was also examined. Shape optimization was found to give a substantial decrease in the stress levels in the adhesive layer and in many cases a much lighter joint was obtained.
Key words: bonded joints; shape optimization; single-lap joint; double-lap joint; doublestrap joint; joint design
Notation E R/-/O S
Young's modulus of elasticity = $1/$2 (RHO = 1 for isotropic materials) yield stress (if only S is given:
SIc, SIT $2c, S2T
SIT = SIC = S2T = $2C = S12 = S )
Sl $2 Sl2
yield stress in x-direction (if only Sl is given: S I x = S i c = $1) yield stress in y-direction (if only $2 is given: S2T = $2C = S2) yield stress in shear
v Crx
O'y r
yield stresses in compression and tension in x-direction yield stresses in compression and tension in y-direction density Poisson's ratio normal stress in the x-direction normal stress in the y-direction shear stress
The stress distribution in adhesive joints has been studied, among many, by Hart-Smith,I A d a m s2, Ojalvo 3 and Cherry and Harrison.4 Theories and ideas have been presented on how one could decrease and smooth out the stress concentrations that appear at the joint ends in lap joints, and how to obtain joints with evenly distributed shear stresses over the total overlap length. Ojalvo 3 and Cherry and Harrison 4 have made proposals on how to shape the adherends in order to obtain a constant shear stress in the adhesive layer. An alternative way, not investigated in the present paper*, of obtaining a constant shear stress and removing stress peaks in the adhesive is to vary the adhesive layer thickness. This has been demonstrated by Adams et al. 5 and Sage.6 Further 'ideas' for decreasing the shear stress peaks may be found in the books by Adams and Wake 7 and Lees 8. Structural shape optimization, using numerical
(finite element) methods, is a recently developed technique in the area of design and analysis that has been applied during the last few years to real structural problems. However, the authors could find no reference in the literature to the application of this technique to adhesive joints. Thus the present paper uses the structural shape optimization program OASIS-ALADDIN 9 to examine theoretically different ways to optimize the geometry of some bonded joints and a bonded structure, in order to achieve joints that are as light and strong as possible. This has been done for static loading conditions only. The joint types treated are the single-lap joint, the double-lap joint and the double-strap joint. One example of a console bonded to a rigid wall was further examined.
*This method is used in the present paper, however, to reduce the stress peaks locally in a double-strap joint model.
The optimization problem can mathematically be formulated for continuous design variables as to:
Analysis
0143-7496/91/040204-09 © 1991 Butterworth-Heinemann Ltd 204
INT.J.ADHESION AND ADHESIVES VOL. 11 NO. 4 OCTOBER 1991
Problem, P:
min w(x)
Subjected to:
gi(x) < 0 xj < x j < ~ j
(1) i = 1,m j = 1,n
The problem, P, is a non-linear optimization problem since in general both the objective function, w, and the constraints, gi, are implicit, non-linear functions of the design variables, xj, and x_j and ~-j represent the lower and upper limits of the design variables, respectively. P may in general have multiple minima. The design variables, x j, are point coordinates describing the shape of the adherends or the console. The optimization problem is solved numerically with one stress analysis module, using finite element analysis and one optimization module in an iterative way l°. A VAXstation 2000 was used for the computations. The constraints used were stresses and linear geometric constraints; the objective functions used were stresses and weight. The aim of the study was to optimize the geometry for some bonded joints, this also including stress and/or weight optimization. In order to do this it was necessary to investigate suitable constraints and objective functions for the joint optimization process and further to study the influence of the selection of design variables. The finite element analyses carded out were twodimensional and linear with respect to both the geometry and the materials. Eight-noded, plane strain, isoparametric membrane elements with second-order Gauss integration were used. Singularities, such as crack tips and sharp corners, appearing in the finite element models were, if possible, excluded from the optimization process. This was done by assigning the elements close to the singular stress concentrations a high yield stress, which means that they will not be critical in the optimization process. The models consisted of a base contour made of straight lines and Ferguson splines with a number of reference points. The coordinates of the reference points on and at the end points of the lines were either master or slave design variables. They were allowed to move within given limits during the optimization process. The slave variables were linearly dependent on one, or a number of, master design variables. One problem that may appear when using splines is illustrated in Fig. l(a). As can be seen, the spline crosses the lower line, despite the fact that the limits of the design variables are arranged so this should not be the case. This may give degenerated finite elements with poor properties. Sometimes the optimization process was interrupted due to this problem. The problem can be solved by limiting the freedom of the variables with so-called explicit linear constraints. This can be formulated, for instance, so that no master variable obtained a value larger than that for the variable to the left as shown in Fig. l(b). As another example, one may, as illustrated in Fig. l(c), give the next variable a limited movement for both the lower and the upper limit. The Tsai-Hill yield criterion is a general form of the von Mises yield hypothesis and this yield criterion has been used in the present study. Yielding occurs when the Tsai-factor, here denoted TSAI, becomes larger than, or equal to 1. Only stresses below the yield limit are permitted. For two dimensions, the Tsai-factor is written as:
.. ~te~.v.~...abJr.~
....
a
(~.U..p.pe..r.limit for P2 F2
~
A
Uoper limit for P4
~f~....up.~r.Ijmit for PS
P3 P4
b
P5 Upper limit for P2 / e. . . . L. . . . .
=P5"~
r ..........
.............
~
.... ~ r ~ ~
/
I
c
........
.~ . . . . . . . . . . .
Lower limit for P2
Fig. 1 Master variables describing the spline that gives the adherend shape. (a) The spline reaches below what was thought as being the limiting line. (b) Linear explicit constraints limiting the design variable movement in the y-direction. (c) Linear explicit constraints limiting both the lower and upper limit for the design variables
TS.
=
"
"
" ,2,
If one needs a single stress component as an objective function, this may be obtained by giving the yield stresses SIT, Sic . . . . different values. Then the Tsaifactor will become dependent on one, or several, stress components that may be of interest. The different stress components may have different influences, or different 'weights'. When TSAI (r) is written as objective function, this means that the shear stress, r, dominates the Tsai-factor. As an example: S i r = Sic = SET = $2c = 10 000 MPa, Si2 = 20 MPa and RHO = 10 000 gives the Tsai-factor dependent (in practice) only on the shear stress. The adherends were either of steel with E = 210 000 MPa, v = 0.3, S = 300 MPa and ( = 7800 kg m -3, or aluminium with E = 70 000 MPa, v = 0.3 and ( = 2800 kg m -3. The adhesives used were: an epoxy with E = 3300 MPa and v -- 0.3; an epoxy with E = 3300 MPa and v = 0.35; an epoxy with E = 2000 MPa and v = 0.30; and a polyurethane with E = 690 MPa and v = 0.3. The density of all the adhesives was 1000 kg m -3. All materials were assumed to be isotropic. If stresses were used as constraints or an objective function, they were evaluated in the Gauss points of the finite elements. For this reason stresses may appear in figures showing a Tsai-factor higher than one for an -~ optimal design. Finally, the optimization was made for the case of static loading only, and no creep or dynamic effects were accounted for.
Single-lap joint A model of a single-lap joint, shown in Fig. 2, was first made, since this may be regarded as a 'reference case'
INT.J.ADHESION AND ADHESIVES OCTOBER 1991
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Objective function: Constraints: Design variables:
10.0 1.61
[
Load
I
77.0
Geometry:
1.6 ~
141.3
Fig. 2 The single-lap joint, geometry and boundary conditions. All dimensions in mm
'A
~Master
Boundary conditions: Load:
vadables 1
Number of elements: Material data: Slave variables Fig. 3
Design variables in the single-lap joint model
and a start for all types of stress analysis of adhesive joints. The objective function was to minimize the Tsai-factor in the adhesive layer. The design variables were the y-coordinates for the master variables on the spline marked A in Fig. 3. An initial profiling was made by assigning the master variables initial coordinates, corresponding to a linearly decreasing adherend thickness as shown in Fig. 4(a). This pre-profiling also made the optimization execute faster. Every slave variable was connected directly to the antisymmetrically located master variable (cf., Fig. 3). The master variables had explicit linear constraints that implied that every master variable in the upper part of the model must have a smaller y-coordinate than the preceding point to the left. The explicit linear constraints must be present to ensure that the finite element mesh did not fail during the optimization process. This was found to be a proper assumption after some initial investigations. Otherwise problems may occur, partly because of the geometrical non-linearity of the joint that is not accounted for in the analysis.
minimize the maximum Tsai-factor, TSAI, in the adhesive layer. explicit geometric constraints. eight master variables on the adherend that may vary in the y-direction. see Fig. 2. The joint width is 25.4 ram. Pre-shaped adherends according to Fig. 4(a). according to Fig. 2. the loading is a distributed load along the adherend edge in the positive xdirection (see Fig, 2) with a magnitude of 322,58 N. This correspondes to a nominal shear stress in the adhesive layer of 1 MPa. 48. adherends -- aluminium. adhesive -- epoxy, E = 2000 MPa, v = 0.3. $1 - - S 2 = 2 M P a . Si2 = 1.01 MPa, RHO = 10000.
Optimization results The optimal profiling obtained is shown in Fig. 4(b). The maximum Tsai-factor in the joint decreased by 25%, from 4.61 to 3.4 after 15 iterations. This gives an increase in the load-carrying capacity of 33%. It should be noted that the geometry of the singlelap joint is geometrically non-linear due to the eccentricity of the load path. This is not accounted for in the stress analysis. Hence this may not be the ideal example of joint optimization, using a linear elastic analysis.
Double-lap joint The double-lap joint was simpler to optimize. One reason for this is that the joint has not the same geometric non-linearity as the single-lap joint. The model is shown in Fig. 5.
• ---I
I
I a
•
[
1 ~y,.~,>,%,,, ~,J,,,,J,),,,,,; .....................
Fig. 4
206
Geometry of the single-lap joint: (a) initially; (b) after 15 iterations
INT.J.ADHESION AND ADHESIVES OCTOBER 1991
Fig. 5
The double-lap joint and the equivalent model
for a joint without profiling 7. The final adherend geometry is weakest in the ends of the adhesive layer. This reduces the shear stresses there and increases the stresses in the middle part of the joint. The objective function converged after six iterations. The maximum shear stress in the adhesive layer decreased by 23%. This gives a corresponding increase in the loadcarrying capacity of 30%. Note that stresses in Fig. 6 for the optimal shape may be larger than 1.0 in the nodes. In the Gauss-points they are always < 1.0.
S h e a r stress as objective f u n c t i o n
The double-lap joint model appeared as the model for the single-lap joint, but without slave variables and with other boundary conditions, as shown in Fig. 5. The explicit linear constraints present within the single-lap joint model were removed as the rotation of the double-lap joint is reduced. Also, only one adherend in the model is pre-shaped (the other side is locked with boundary conditons due to symmetry). Objective function:
minimize the maximum shear stress, TSAI(r), in the adhesive layer. none. see Figs 5 and 6(a). The joint width is 25.4 ram. as for the single-lap joint in Fig. 2. In addition the symmetry line is locked in the y-direction. 322.58 N (Fig. 2), this corresponding to a nominal shear stress in the adhesive layer of 1 MPa. 90. adherends - - aluminium. adhesive - - epoxy, E = 2000 MPa, v = 0.3. Sl = $2 = 10 000 MPa. S12 = 1.01 MPa, RHO = 10000.
Constraints: Geometry: Boundary conditions:
Load:
Number of elements: Material data:
Tsai-factor as objective f u n c t i o n
Objective function:
Constraints: Geometry:
Boundary conditions:
Load: Number of elements: Material data:
Optimization results The final geometry, shown in Fig. 6(b), seems logical if one recalls how the shear stress distribution appears
minimize the maximum Tsai-factor, TSAI, in the adhesive. Tsai-factor, TSAI, in the adherends. see Figs 5 and 7(a). The initial adherend thickness was increased from 1.6 mm to 2.15 mm (compared with Fig. 2). The joint width is 25.4 mm. as for the single-lap joint. In addition the symmetry line is locked in the y-direction. 3225 N. 148. adherends - - aluminium, S = 170 MPa. adhesive - - epoxy, E = 2000 MPa, v = 0.35. Sl = 40 MPa. Szr = 20 MPa, $2c = 60 MPa. S 12 = 40 MPa, RHO = 1. Levels Min 0.02 A 0.02 B 0.15 C 0.29 D 0.42 E 0.56 F 0.69 G 0.82 H 0.96 I 1.09 J 1.23 Max 1.50
a Levels Min 0.01 A 0.01
L_ ~-a BDN •
rl i*=
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f'i,,,,
I
-
-------_,_~
-
C
0.23
D E
0.34 0.45
F
0.55
H
0.77
I J Max
0.88 099 1.21
b Fig. 6 levels
The double-lap joint: (a)
initial g e o m e t r y
of the adherend profiling; (b) final geometry
of the adherend
profiling a f t e r
15 iterations. Both with TSAI(c)
INT.J.ADHESION A N D A D H E S I V E S OCTOBER 1 9 9 1
207
3
Levels Min 0.01 A 0.01 B 0.07 C 0.13 D 0.19 E 0.25 F 0.31 G 0.37 H 0.43 I 0.49 J 0.55 Max 0.67
a
Levels
/ b..
a
il-/I
~
--,,
f
f
\
\
\
Min
0.00
A B C D E F G H I J Max
0.00 0.06 0.11 0.16 0.22 O.27 0.32 0.38 0.43 0.48 0.59
b Fig. 7
The double-lap joint: (a) initial geometry with Tsai-factor; (b) final geometry with Tsai4actor after 15 iterations
Optimization results First the optimization gave that the fight corner, see Fig. 7(b), in the adhesive was critical. Thus the adherend was shaped (slightly) at that end. When this was done the left corner became critical. At this end the design variables could not do much to decrease the stresses, as the stresses have to be equal to the nominal adherend thickness outside the overlap in this area. The optimization stopped as the objective function was to minimize the maximum Tsai-factor which was located in the left corner. The objective function decreased by 11% and a very minor change of the shape was obtained, as can be seen from Fig. 7(b). The corresponding increase in the load-carrying capacity is 12%.
the rest of the adhesive layer. This would make the optimization stop (compare the double-lap joint with the Tsai-factor as objective function). The initial finite element model is shown in Fig. 10(a). Shear stress as objective function
Objective function:
minimize the maximum shear stress, TSAI(r), in the adhesive layer.
//
Comparison
w i t h O j a l v o ' s results
An adherend shape similar to the one obtained by Ojalvo 3 in an optimization study was made to see if a similar optimal shape was obtained using the numerical approach. Ojalvo 3 wanted to obtain a constant shear stress in the outerpart of the adhesive layer of a double-lap joint. It should be noted that he did not try to optimize the stresses in the whole adhesive layer. The model has spew-fillet elements and the adhesive layer is made thicker to the left end, cf., Fig. 8. The two elements to the right (of., Fig. 9) in the adhesive layer have been assigned a higher yield stress, due to the stress peak there that could not be removed by the program if these elements had the same yield stress as
208
INT.J.ADHESION A N D ADHESIVES OCTOBER 1 991
aIOI
I
Jr
i
c
/J /
or
Fig. 8 Geometry of the double-lap joint used by Ovaljo 3 and the doublestrap joint, a = 4 0 . 2 mm, b = 2 0 ram, c = 0 . 7 4 ram, d = 2 0 0 mm, e = 1 0 0 ram, f = 4 1 0 mm, g = 1.2 mm, h = 0.2 ram, i = 20 mm
Fig. 9 The double-lap joint used by Ojalvo 3 and the double-strap joint, showing shaded elements having a higher yield stress
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I
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I
I
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I
I
I
I
I
b Fig. 10 The double-lapjointused by Ojalvo3:(a) initialgeometry; (b}finalgeometryafter 20 iterstions ()b() () tJ (I z/i/i/////ill//////////////////////////e////////////////
Constraints:
Variables:
Geometry: Boundary conditions:
Load:
Number of elements: Material data:
(no active) explicit geometric constraints, implying that no master variable may have a larger value on the y-coordinate than the variable next to the left. eight master variables for the adherend profiling that may vary in the y-direction. see Fig. 8. The width of the joint is 20 mm. line C in Fig. 8 is locked in the y-direction, the lines A and B are locked in the xdirection. 31 800 N. The loading is a distributed load along line D (Fig. 8) in the positive xdirection. This correspondes to a nominal shear stress in the adhesive layer of 7.95 MPa. 96. adherends -- aluminium. adhesive -- polyurethane, E = 690 MPa, v = 0.3. The two elements to the right in Fig. 9: S = 10 000 MPa, RHO = I0000. The rest of the adhesive layer: Sl = Sz = 10000 MPa. S]1 = 7 MPa, RHO = 10 000.
Optimization results
The profiling changed from the smooth initial shape to a shape that had a bump as shown in Fig. 10(b). The stress levels decreased 1.9% only. This is because a near-optimum solution was used as the ~pre-profiling 3. From Fig. 10(b) it is evident that Ojalvo'gives essentially the same result as the present approach, except for a small bump that appeared on the adherend.
Fig. 11
Double-strap
joint and the equivalent
model
The two elements to the right in the adhesive layer have been assigned a higher yield stress (Fig. 9), due to the same reasons as for the Ojalvo model. Weight as objective function
A series of optimizations was made with a similar model, shown in Fig. 12(a). Four cases, denoted case 1 to 4, were examined. The only parameter that was changed was the load level. Also, the overlap length was introduced as a design variable. The shaded elements in Fig. 9 have obtained the higher yield stress. When optimizing for weight, the stresses in the joint (including the adhesive) are kept below the yield limit. Objective function: Constraints:
minimize the total weight. 1) Tsai-factor in the whole structure. 2) (no active) explicit geometric constraints, implying that no master variable may have a larger value on the y-coordinate than the variable next to the left. nine master variables for the adherend profiling that may vary in the y-direction. In addition to this, the overlap length may be
Variables:
a
~ i 1 1 1 [ 1 Ill(llll l l l I I I
,,
1 I
b
¢
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I
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Double-strap joint
d
The basic geometry of the double-strap joint, shown in Fig. I 1, is similar to the model of the double-lap joint (cf., Fig. 5) with some additional boundary conditions introduced. The model hasspew-fillet elements and the adhesive layer is made thicker to the left end in order to the reduce the adhesive stresses locallys (cf., Fig. 8).
e
I I l [ l l l l / l l l l III]1111/111
I
,
,
LJ
F1
tlltllllllltmEtl
I I 1 1
t
I
I
tl
I
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I
I
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Fig. 12 Double-strap joint: (a) initial geometry for cases 1 to 4; (b) final geometry for case 1 (80%); (c) final geometry for case 2 (82,5%); (d) final geometry for case 3 (85%); (e) final geometry for case 4 (87.5%}
INT.J.ADHESION AND ADHESIVES OCTOBER 1991
209
Geometry:
Boundary conditions:
Load:
N u m b e r of elements: Material data:
varied ('d' in Fig. 8). Note that for variables describing 'd' and 'e' in Fig. 8, the size of e is a "slave', moving proportional to "d'. see Figs 8 and 12(a). Actual sizes are as follows: a = 4.2 mm, b = 2 mm, c = 0.2 ram, d = 20 mm, e = 4 mm, f = 41 mm, g = 0.12 mm, h = 0.2 mm, i = 2 mm. The width of the model is 20 mm. line C is locked in the ydirection, the lines A and B are locked in the x-direction (Fig. 8). the loading is a distributed load along line D in the positive x-direction. The load is expressed in % of the tension fracture load of 84 000 N for the inner adherend, marked 'b' in Fig. 8. This correspondes to a nominal shear stress in the adhesive for the 200 m m overlap length of 21 MPa. In case 1 the load level was 80%; in case 2, 82.5%; in case 3, 85%; and in case 4, 87.5%. 175. the shaded elements in Fig. 9: S = 10 000 MPa, RHO = 10 000. adherends - - aluminium, S = 210 MPa. adhesive - - epoxy, E = 3300 MPa, v = 0.3. Si = $2 = 40 MPa. Si2 = 20 MPa, RHO = 1.
,/
/ / / / /
Stiff wall
/ / /
560
10oI
/
/ / /
¢
/ / /
I
250
~o. Fig. 13
Geometry of console. All dimensions in mm
bonded to a stiff wall with an epoxy adhesive. Master variables were located at the ends of the flanges. The master variables to the splines describing the console flanges, shown in Fig. 14, had explicit linear constraints, implying that two adjacent variables may not vary too much in the x-direction. This was done in order to avoid an oscillating shape of the flanges. There was also one explicit linear constraint limiting the flange length to a m a x i m u m of 610 ram. Tsai-factor as objective function
Objective function:
Constraints:
Optimization results One may notice from the results shown in Figs 12(b) to 12(e) that the overlap length increases with increased loading. This seems to be logical. In these cases a b u m p appears on the adherend profiling. Large reductions in weight were made, except for case 4 where the nominal shear stress approaches the yield stress in shear of 20 MPa. The weight reductions obtained were: case 1, 55%; case 2, 51%; case 3, 43%; and case 4, an increase of 21%. Doing 15 iterations with nine design variables, one case and 175 elements for this model took about 40 hours for a optimization on the VAXstation 2000 that was used. It could be worth noting this rather long execution time.
80 000 N
Variables:
Geometry:
minimize the m a x i m u m Tsai-factor, TSAI, in the adhesive layer. 1) Tsai-factor in the whole structure. 2) explicit geometric constraints so that two adjacent variables may not vary too much in the xdirection. variable flange length and thickness of the flanges, see Fig. 14. see Fig. 13. The width of the model was 100 mm. The adhesive layer thickness was 1 m m with spew-fillet elements.
m
Console
Finally a model of a console made of steel was analysed as an application example. The geometry of the model is shown in Fig. 13. The same model was used for both weight and stress optimization. Only the objective function was changed. The console was
210
INT.J.ADHESION AND ADHESIVES OCTOBER 1991
2
b
Fig. 14 Master variables in the console model: (a} in the y-direction; (b) in the x-direction
according to Fig. 13. Console bonded to a stiff wall. the console was loaded with a distributed load, 80 000 N, in the negative y-direction according to Fig. 13. 106. adherend -- steel. adhesive -- epoxy, E = 3300 MPa, v = 0.3. S iT = 20 MPa, Sic = 60 MPa. $2 = 40 MPa, Sl2 = 40 MPa, RHO = 1.
Boundary conditions:
Load:
Number of elements: Material data:
Optimization results According to the T s a i - H i l l criterion the stress were too high, leading to fracture in both the adhesive and the
console in the first iteration. After a few iterations the stresses became permitted in the whole structure, cf., Figs 15(a) and 15(b).
7
Levels Min 0.00 A 0.12 B 0.23 C 0.35 D 0.46 E 0.58 F 0.69 G 0.81 H 0.93
~
I 1.04 J 1.16 K 1.27 L 1.39 M 1.50 N 1.62 O 1.73 Max 1.85
Levels MIn 0.00 A 0.08 B 0.16 C 0.24 D 0.31 E 0,39 F 0.47 G 0.55 H 0.63 I 0.71 J 0,79 K 0.86 L 0.94 M 1.02 N 1.10 O 1.18 Max 1.26 Fig, 1 6 Final geometry of console, after 1 5 iterations with Tsai-factor. Weight as objective function
The oscillations on the bottom flange are due to the fact that there are too many design variables within a very short distance, when the flange length decreases. Also the stresses are small in that area. The shape of the flange has probably no greater influence on the stresses in the adhesive. The critical area, from the stress point of view, is located in the middle of the console. The objective function, TSAI in the adhesive, decreased by 51% from 1.85 to 0.94, a reasonable figure. The stresses in the console decreased from 1.35 to 1.0. Hence a much more efficient structure was obtained. Weight as objective function Objective function: Other model data:
rJ),~
)
minimize the total weight. the same as for the previous model with Tsai-factor as objective function. The initial geometry, with Tsaifactor, is shown in Fig. 15(a).
Optimization results
~_~
/
\
Levels Min 0.00 A 0.12 B 0.23 C 0.35 D 0.46 E 0.58 F 0.69 G 0.81 H 0.93 I 1.04 J 1.16 K 1.27
Fig. 1 5 Console model: (a) initial geometry with Tsai-factor; (b) final geometry after 1 0 iterations with Tsai-factor. Tsai-factor as objective function.
The final geometry, with Tsai-factor, is shown in Fig. 16. The total weight decreased 34% in 15 iterations. The Tsai-factor decreased rapidly from 1.85 to 1.0 and became permitted after eight iterations. The oscillations that appear on the bottom flange in Fig. 16 may be due to too many design variables on this flange, as in the previous case. Reducing the number of design variables on the short flange was found to solve this problem. Discussion and conclusions The results presented in the preceding sections illustrate that it is theoretically possible to apply shape optimization techniques to structural bonded joints. However, the question arises whether it is possible to actually use the type of 'optimal joints' that have been presented in this paper. From a practical point of view it was found very difficult to obtain the correct profiling of the adherends by means of machining relatively thin metal adherends and no results from manufactured optimal joints were obtained. Nevertheless, one possible application area for this technique could be extruded aluminium profiles for bonded applications due to, at least, two reasons: it is easy to produce 'tailor made' aluminium sections and aluminium bonding technology is well known.
INT.J.ADHESION AND ADHESIVES OCTOBER 1991
211
The following conclusions can be drawn from this study. 1) Shape optimization of bonded joints often gives joints with much lower stress levels and weights than that of the initial geometry, and therefore much stronger and more efficient joints. 2) This technique gives a good hint on what kind of design that should be considered for further studies. 3) It may be difficult to handle material and geometrical singularities in the optimization process. In the case of the bonded console, no such problems occurred. This because the singular points in that model were located in 'unloaded' areas. 4) It is best if a 'possible initial optimal geometry' is given as input, otherwise the objective function may find a local (not global) minimum value. 5) It is important to select the number of variables defining the geometry in a proper way. This can be very difficult to do, as the optimal result depends on this choice. 6) A more powerful computer than the VAXstation 2000 is needed for the analysis. It took about 40 hours for a optimization involving 15 iterations with 10 design variables and 175 elements. 7) It would be very interesting to test joints manufactured according to the design obtained. This is an area suggested for further studies. Acknowledgements
The authors hereby acknowledge The Swedish Board for Technical Development (STU) and The Nordic Fund for Technology and Industrial Development (NI) for financial support, and Dr Dan Holm at ALFGAM Optimering AB for advice on the optimization program and for supplying computer time for the analysis.
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INT.J.ADHESlON AND ADHESIVES OCTOBER 1991
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3 4 5
6
7 8 9
10
Hart-Smith, L.J. 'Adhesive-bonded single-lap joints' NASA CR 112236 (NASA Langley Research Center, January 1973) Adams, R.D. 'Mechanics of adhesive lap joints' Proc European Mechanics Colloquium 227 "Mechanical Behaviour of Adhesive Joints; Saint-Etienne, France, 31 August-2 September 1987 (Editions Pluralis, Paris. 1987) pp 27-50 Ojalvo, I.O. 'Optimization of bonded joints" A/AA Journal 23 No 10 (October 1985) pp 1578-1582 Cherry, B.W. and Harrison, N.C. 'The optimum profile for a lap joint' JAdhesion 2 (April 1970) pp 125-128 Adams, R.D., Chambers, S.H., Del Strothe¢, P.J.A. and Peppistt, N.A. 'Rubber model for adhesive lap joints' J Strain Anal 8 No 1 (1973) pp 52-57 Sage, G.N. 'Aspects of bonded joint design in carbon-fibre reinforcedplastic' in "Adhesion 3" edited by K.W. Allen (Applied Science Publishers, London, 1979) pp 123-141 Adams, R.D, and Wake, W.C. "Structural Joints in Engineering' (Elsevier Applied Science Publishers, London, 1984) Lees, W.A. "Adhesives in Engineering Design" (The Design Council, London and Springer-Verlag, Berlin, 1984) "OASIS-ALADDIN User's Manual, VAXoVMS Version" (Alfgam Optimering AB, Stockholm, Sweden, 1989) Eaping, B.J.D. 'Structural optimization using numerical techniques' Report 85-8 (Department of Aeronautical Structures and Materials, The Royal Institute of Technology, Stockholm 1985)
Authors
Hans. L. Groth, to whom correspondence should be directed, was formerly a Senior Research Engineer at the Department of Aeronautical Structures and Materials, The Royal Institute of Technology, S-100 44 Stockholm, Sweden. He is now with Avesta AB, Research and Development, Mechanical Metallurgy, S774 80 Avesta, Sweden. Peter Nordlund is a Research Student in the Department of Aeronautical Structures and Materials at The Royal Institute of Technology.
Shape optimization of bonded joints was performed by use of numerical shape optimization techniques. The aim of this study was to obtain joints that are as strong and light as possible under static loading conditions by changing the profiling of the adherends. Joint types included are the single-lap, the double-lap and the double-strap; a console bonded to a rigid wall was also examined. Shape optimization was found to give a substantial decrease in the stress levels in the adhesive layer and in many cases a much lighter joint was obtained.
Key words: bonded joints; shape optimization; single-lap joint; double-lap joint; doublestrap joint; joint design
Notation E R/-/O S
Young's modulus of elasticity = $1/$2 (RHO = 1 for isotropic materials) yield stress (if only S is given:
SIc, SIT $2c, S2T
SIT = SIC = S2T = $2C = S12 = S )
Sl $2 Sl2
yield stress in x-direction (if only Sl is given: S I x = S i c = $1) yield stress in y-direction (if only $2 is given: S2T = $2C = S2) yield stress in shear
v Crx
O'y r
yield stresses in compression and tension in x-direction yield stresses in compression and tension in y-direction density Poisson's ratio normal stress in the x-direction normal stress in the y-direction shear stress
The stress distribution in adhesive joints has been studied, among many, by Hart-Smith,I A d a m s2, Ojalvo 3 and Cherry and Harrison.4 Theories and ideas have been presented on how one could decrease and smooth out the stress concentrations that appear at the joint ends in lap joints, and how to obtain joints with evenly distributed shear stresses over the total overlap length. Ojalvo 3 and Cherry and Harrison 4 have made proposals on how to shape the adherends in order to obtain a constant shear stress in the adhesive layer. An alternative way, not investigated in the present paper*, of obtaining a constant shear stress and removing stress peaks in the adhesive is to vary the adhesive layer thickness. This has been demonstrated by Adams et al. 5 and Sage.6 Further 'ideas' for decreasing the shear stress peaks may be found in the books by Adams and Wake 7 and Lees 8. Structural shape optimization, using numerical
(finite element) methods, is a recently developed technique in the area of design and analysis that has been applied during the last few years to real structural problems. However, the authors could find no reference in the literature to the application of this technique to adhesive joints. Thus the present paper uses the structural shape optimization program OASIS-ALADDIN 9 to examine theoretically different ways to optimize the geometry of some bonded joints and a bonded structure, in order to achieve joints that are as light and strong as possible. This has been done for static loading conditions only. The joint types treated are the single-lap joint, the double-lap joint and the double-strap joint. One example of a console bonded to a rigid wall was further examined.
*This method is used in the present paper, however, to reduce the stress peaks locally in a double-strap joint model.
The optimization problem can mathematically be formulated for continuous design variables as to:
Analysis
0143-7496/91/040204-09 © 1991 Butterworth-Heinemann Ltd 204
INT.J.ADHESION AND ADHESIVES VOL. 11 NO. 4 OCTOBER 1991
Problem, P:
min w(x)
Subjected to:
gi(x) < 0 xj < x j < ~ j
(1) i = 1,m j = 1,n
The problem, P, is a non-linear optimization problem since in general both the objective function, w, and the constraints, gi, are implicit, non-linear functions of the design variables, xj, and x_j and ~-j represent the lower and upper limits of the design variables, respectively. P may in general have multiple minima. The design variables, x j, are point coordinates describing the shape of the adherends or the console. The optimization problem is solved numerically with one stress analysis module, using finite element analysis and one optimization module in an iterative way l°. A VAXstation 2000 was used for the computations. The constraints used were stresses and linear geometric constraints; the objective functions used were stresses and weight. The aim of the study was to optimize the geometry for some bonded joints, this also including stress and/or weight optimization. In order to do this it was necessary to investigate suitable constraints and objective functions for the joint optimization process and further to study the influence of the selection of design variables. The finite element analyses carded out were twodimensional and linear with respect to both the geometry and the materials. Eight-noded, plane strain, isoparametric membrane elements with second-order Gauss integration were used. Singularities, such as crack tips and sharp corners, appearing in the finite element models were, if possible, excluded from the optimization process. This was done by assigning the elements close to the singular stress concentrations a high yield stress, which means that they will not be critical in the optimization process. The models consisted of a base contour made of straight lines and Ferguson splines with a number of reference points. The coordinates of the reference points on and at the end points of the lines were either master or slave design variables. They were allowed to move within given limits during the optimization process. The slave variables were linearly dependent on one, or a number of, master design variables. One problem that may appear when using splines is illustrated in Fig. l(a). As can be seen, the spline crosses the lower line, despite the fact that the limits of the design variables are arranged so this should not be the case. This may give degenerated finite elements with poor properties. Sometimes the optimization process was interrupted due to this problem. The problem can be solved by limiting the freedom of the variables with so-called explicit linear constraints. This can be formulated, for instance, so that no master variable obtained a value larger than that for the variable to the left as shown in Fig. l(b). As another example, one may, as illustrated in Fig. l(c), give the next variable a limited movement for both the lower and the upper limit. The Tsai-Hill yield criterion is a general form of the von Mises yield hypothesis and this yield criterion has been used in the present study. Yielding occurs when the Tsai-factor, here denoted TSAI, becomes larger than, or equal to 1. Only stresses below the yield limit are permitted. For two dimensions, the Tsai-factor is written as:
.. ~te~.v.~...abJr.~
....
a
(~.U..p.pe..r.limit for P2 F2
~
A
Uoper limit for P4
~f~....up.~r.Ijmit for PS
P3 P4
b
P5 Upper limit for P2 / e. . . . L. . . . .
=P5"~
r ..........
.............
~
.... ~ r ~ ~
/
I
c
........
.~ . . . . . . . . . . .
Lower limit for P2
Fig. 1 Master variables describing the spline that gives the adherend shape. (a) The spline reaches below what was thought as being the limiting line. (b) Linear explicit constraints limiting the design variable movement in the y-direction. (c) Linear explicit constraints limiting both the lower and upper limit for the design variables
TS.
=
"
"
" ,2,
If one needs a single stress component as an objective function, this may be obtained by giving the yield stresses SIT, Sic . . . . different values. Then the Tsaifactor will become dependent on one, or several, stress components that may be of interest. The different stress components may have different influences, or different 'weights'. When TSAI (r) is written as objective function, this means that the shear stress, r, dominates the Tsai-factor. As an example: S i r = Sic = SET = $2c = 10 000 MPa, Si2 = 20 MPa and RHO = 10 000 gives the Tsai-factor dependent (in practice) only on the shear stress. The adherends were either of steel with E = 210 000 MPa, v = 0.3, S = 300 MPa and ( = 7800 kg m -3, or aluminium with E = 70 000 MPa, v = 0.3 and ( = 2800 kg m -3. The adhesives used were: an epoxy with E = 3300 MPa and v -- 0.3; an epoxy with E = 3300 MPa and v = 0.35; an epoxy with E = 2000 MPa and v = 0.30; and a polyurethane with E = 690 MPa and v = 0.3. The density of all the adhesives was 1000 kg m -3. All materials were assumed to be isotropic. If stresses were used as constraints or an objective function, they were evaluated in the Gauss points of the finite elements. For this reason stresses may appear in figures showing a Tsai-factor higher than one for an -~ optimal design. Finally, the optimization was made for the case of static loading only, and no creep or dynamic effects were accounted for.
Single-lap joint A model of a single-lap joint, shown in Fig. 2, was first made, since this may be regarded as a 'reference case'
INT.J.ADHESION AND ADHESIVES OCTOBER 1991
205
Objective function: Constraints: Design variables:
10.0 1.61
[
Load
I
77.0
Geometry:
1.6 ~
141.3
Fig. 2 The single-lap joint, geometry and boundary conditions. All dimensions in mm
'A
~Master
Boundary conditions: Load:
vadables 1
Number of elements: Material data: Slave variables Fig. 3
Design variables in the single-lap joint model
and a start for all types of stress analysis of adhesive joints. The objective function was to minimize the Tsai-factor in the adhesive layer. The design variables were the y-coordinates for the master variables on the spline marked A in Fig. 3. An initial profiling was made by assigning the master variables initial coordinates, corresponding to a linearly decreasing adherend thickness as shown in Fig. 4(a). This pre-profiling also made the optimization execute faster. Every slave variable was connected directly to the antisymmetrically located master variable (cf., Fig. 3). The master variables had explicit linear constraints that implied that every master variable in the upper part of the model must have a smaller y-coordinate than the preceding point to the left. The explicit linear constraints must be present to ensure that the finite element mesh did not fail during the optimization process. This was found to be a proper assumption after some initial investigations. Otherwise problems may occur, partly because of the geometrical non-linearity of the joint that is not accounted for in the analysis.
minimize the maximum Tsai-factor, TSAI, in the adhesive layer. explicit geometric constraints. eight master variables on the adherend that may vary in the y-direction. see Fig. 2. The joint width is 25.4 ram. Pre-shaped adherends according to Fig. 4(a). according to Fig. 2. the loading is a distributed load along the adherend edge in the positive xdirection (see Fig, 2) with a magnitude of 322,58 N. This correspondes to a nominal shear stress in the adhesive layer of 1 MPa. 48. adherends -- aluminium. adhesive -- epoxy, E = 2000 MPa, v = 0.3. $1 - - S 2 = 2 M P a . Si2 = 1.01 MPa, RHO = 10000.
Optimization results The optimal profiling obtained is shown in Fig. 4(b). The maximum Tsai-factor in the joint decreased by 25%, from 4.61 to 3.4 after 15 iterations. This gives an increase in the load-carrying capacity of 33%. It should be noted that the geometry of the singlelap joint is geometrically non-linear due to the eccentricity of the load path. This is not accounted for in the stress analysis. Hence this may not be the ideal example of joint optimization, using a linear elastic analysis.
Double-lap joint The double-lap joint was simpler to optimize. One reason for this is that the joint has not the same geometric non-linearity as the single-lap joint. The model is shown in Fig. 5.
• ---I
I
I a
•
[
1 ~y,.~,>,%,,, ~,J,,,,J,),,,,,; .....................
Fig. 4
206
Geometry of the single-lap joint: (a) initially; (b) after 15 iterations
INT.J.ADHESION AND ADHESIVES OCTOBER 1991
Fig. 5
The double-lap joint and the equivalent model
for a joint without profiling 7. The final adherend geometry is weakest in the ends of the adhesive layer. This reduces the shear stresses there and increases the stresses in the middle part of the joint. The objective function converged after six iterations. The maximum shear stress in the adhesive layer decreased by 23%. This gives a corresponding increase in the loadcarrying capacity of 30%. Note that stresses in Fig. 6 for the optimal shape may be larger than 1.0 in the nodes. In the Gauss-points they are always < 1.0.
S h e a r stress as objective f u n c t i o n
The double-lap joint model appeared as the model for the single-lap joint, but without slave variables and with other boundary conditions, as shown in Fig. 5. The explicit linear constraints present within the single-lap joint model were removed as the rotation of the double-lap joint is reduced. Also, only one adherend in the model is pre-shaped (the other side is locked with boundary conditons due to symmetry). Objective function:
minimize the maximum shear stress, TSAI(r), in the adhesive layer. none. see Figs 5 and 6(a). The joint width is 25.4 ram. as for the single-lap joint in Fig. 2. In addition the symmetry line is locked in the y-direction. 322.58 N (Fig. 2), this corresponding to a nominal shear stress in the adhesive layer of 1 MPa. 90. adherends - - aluminium. adhesive - - epoxy, E = 2000 MPa, v = 0.3. Sl = $2 = 10 000 MPa. S12 = 1.01 MPa, RHO = 10000.
Constraints: Geometry: Boundary conditions:
Load:
Number of elements: Material data:
Tsai-factor as objective f u n c t i o n
Objective function:
Constraints: Geometry:
Boundary conditions:
Load: Number of elements: Material data:
Optimization results The final geometry, shown in Fig. 6(b), seems logical if one recalls how the shear stress distribution appears
minimize the maximum Tsai-factor, TSAI, in the adhesive. Tsai-factor, TSAI, in the adherends. see Figs 5 and 7(a). The initial adherend thickness was increased from 1.6 mm to 2.15 mm (compared with Fig. 2). The joint width is 25.4 mm. as for the single-lap joint. In addition the symmetry line is locked in the y-direction. 3225 N. 148. adherends - - aluminium, S = 170 MPa. adhesive - - epoxy, E = 2000 MPa, v = 0.35. Sl = 40 MPa. Szr = 20 MPa, $2c = 60 MPa. S 12 = 40 MPa, RHO = 1. Levels Min 0.02 A 0.02 B 0.15 C 0.29 D 0.42 E 0.56 F 0.69 G 0.82 H 0.96 I 1.09 J 1.23 Max 1.50
a Levels Min 0.01 A 0.01
L_ ~-a BDN •
rl i*=
~
f'i,,,,
I
-
-------_,_~
-
C
0.23
D E
0.34 0.45
F
0.55
H
0.77
I J Max
0.88 099 1.21
b Fig. 6 levels
The double-lap joint: (a)
initial g e o m e t r y
of the adherend profiling; (b) final geometry
of the adherend
profiling a f t e r
15 iterations. Both with TSAI(c)
INT.J.ADHESION A N D A D H E S I V E S OCTOBER 1 9 9 1
207
3
Levels Min 0.01 A 0.01 B 0.07 C 0.13 D 0.19 E 0.25 F 0.31 G 0.37 H 0.43 I 0.49 J 0.55 Max 0.67
a
Levels
/ b..
a
il-/I
~
--,,
f
f
\
\
\
Min
0.00
A B C D E F G H I J Max
0.00 0.06 0.11 0.16 0.22 O.27 0.32 0.38 0.43 0.48 0.59
b Fig. 7
The double-lap joint: (a) initial geometry with Tsai-factor; (b) final geometry with Tsai4actor after 15 iterations
Optimization results First the optimization gave that the fight corner, see Fig. 7(b), in the adhesive was critical. Thus the adherend was shaped (slightly) at that end. When this was done the left corner became critical. At this end the design variables could not do much to decrease the stresses, as the stresses have to be equal to the nominal adherend thickness outside the overlap in this area. The optimization stopped as the objective function was to minimize the maximum Tsai-factor which was located in the left corner. The objective function decreased by 11% and a very minor change of the shape was obtained, as can be seen from Fig. 7(b). The corresponding increase in the load-carrying capacity is 12%.
the rest of the adhesive layer. This would make the optimization stop (compare the double-lap joint with the Tsai-factor as objective function). The initial finite element model is shown in Fig. 10(a). Shear stress as objective function
Objective function:
minimize the maximum shear stress, TSAI(r), in the adhesive layer.
//
Comparison
w i t h O j a l v o ' s results
An adherend shape similar to the one obtained by Ojalvo 3 in an optimization study was made to see if a similar optimal shape was obtained using the numerical approach. Ojalvo 3 wanted to obtain a constant shear stress in the outerpart of the adhesive layer of a double-lap joint. It should be noted that he did not try to optimize the stresses in the whole adhesive layer. The model has spew-fillet elements and the adhesive layer is made thicker to the left end, cf., Fig. 8. The two elements to the right (of., Fig. 9) in the adhesive layer have been assigned a higher yield stress, due to the stress peak there that could not be removed by the program if these elements had the same yield stress as
208
INT.J.ADHESION A N D ADHESIVES OCTOBER 1 991
aIOI
I
Jr
i
c
/J /
or
Fig. 8 Geometry of the double-lap joint used by Ovaljo 3 and the doublestrap joint, a = 4 0 . 2 mm, b = 2 0 ram, c = 0 . 7 4 ram, d = 2 0 0 mm, e = 1 0 0 ram, f = 4 1 0 mm, g = 1.2 mm, h = 0.2 ram, i = 20 mm
Fig. 9 The double-lap joint used by Ojalvo 3 and the double-strap joint, showing shaded elements having a higher yield stress
Ii
!
I 111[1t111111 [ i I
I
'- L IIIIIIIII I I I
I
I
F--I
I
I
I
I
I
b Fig. 10 The double-lapjointused by Ojalvo3:(a) initialgeometry; (b}finalgeometryafter 20 iterstions ()b() () tJ (I z/i/i/////ill//////////////////////////e////////////////
Constraints:
Variables:
Geometry: Boundary conditions:
Load:
Number of elements: Material data:
(no active) explicit geometric constraints, implying that no master variable may have a larger value on the y-coordinate than the variable next to the left. eight master variables for the adherend profiling that may vary in the y-direction. see Fig. 8. The width of the joint is 20 mm. line C in Fig. 8 is locked in the y-direction, the lines A and B are locked in the xdirection. 31 800 N. The loading is a distributed load along line D (Fig. 8) in the positive xdirection. This correspondes to a nominal shear stress in the adhesive layer of 7.95 MPa. 96. adherends -- aluminium. adhesive -- polyurethane, E = 690 MPa, v = 0.3. The two elements to the right in Fig. 9: S = 10 000 MPa, RHO = I0000. The rest of the adhesive layer: Sl = Sz = 10000 MPa. S]1 = 7 MPa, RHO = 10 000.
Optimization results
The profiling changed from the smooth initial shape to a shape that had a bump as shown in Fig. 10(b). The stress levels decreased 1.9% only. This is because a near-optimum solution was used as the ~pre-profiling 3. From Fig. 10(b) it is evident that Ojalvo'gives essentially the same result as the present approach, except for a small bump that appeared on the adherend.
Fig. 11
Double-strap
joint and the equivalent
model
The two elements to the right in the adhesive layer have been assigned a higher yield stress (Fig. 9), due to the same reasons as for the Ojalvo model. Weight as objective function
A series of optimizations was made with a similar model, shown in Fig. 12(a). Four cases, denoted case 1 to 4, were examined. The only parameter that was changed was the load level. Also, the overlap length was introduced as a design variable. The shaded elements in Fig. 9 have obtained the higher yield stress. When optimizing for weight, the stresses in the joint (including the adhesive) are kept below the yield limit. Objective function: Constraints:
minimize the total weight. 1) Tsai-factor in the whole structure. 2) (no active) explicit geometric constraints, implying that no master variable may have a larger value on the y-coordinate than the variable next to the left. nine master variables for the adherend profiling that may vary in the y-direction. In addition to this, the overlap length may be
Variables:
a
~ i 1 1 1 [ 1 Ill(llll l l l I I I
,,
1 I
b
¢
flllllllll I II II I I IIHIIIIlll I ,[llllllll, I I i I I, I Ill,,lll=l,
IIIllllllll/l]llll[llllllllHIIIIII
I
]
I
IIIIIIIIII I I I I I I I 1t1111tt111111111I I
Double-strap joint
d
The basic geometry of the double-strap joint, shown in Fig. I 1, is similar to the model of the double-lap joint (cf., Fig. 5) with some additional boundary conditions introduced. The model hasspew-fillet elements and the adhesive layer is made thicker to the left end in order to the reduce the adhesive stresses locallys (cf., Fig. 8).
e
I I l [ l l l l / l l l l III]1111/111
I
,
,
LJ
F1
tlltllllllltmEtl
I I 1 1
t
I
I
tl
I
II
I
I
I I I I II
Illlll;L',l,|',',',',
', I
It
I--H
Fig. 12 Double-strap joint: (a) initial geometry for cases 1 to 4; (b) final geometry for case 1 (80%); (c) final geometry for case 2 (82,5%); (d) final geometry for case 3 (85%); (e) final geometry for case 4 (87.5%}
INT.J.ADHESION AND ADHESIVES OCTOBER 1991
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Geometry:
Boundary conditions:
Load:
N u m b e r of elements: Material data:
varied ('d' in Fig. 8). Note that for variables describing 'd' and 'e' in Fig. 8, the size of e is a "slave', moving proportional to "d'. see Figs 8 and 12(a). Actual sizes are as follows: a = 4.2 mm, b = 2 mm, c = 0.2 ram, d = 20 mm, e = 4 mm, f = 41 mm, g = 0.12 mm, h = 0.2 mm, i = 2 mm. The width of the model is 20 mm. line C is locked in the ydirection, the lines A and B are locked in the x-direction (Fig. 8). the loading is a distributed load along line D in the positive x-direction. The load is expressed in % of the tension fracture load of 84 000 N for the inner adherend, marked 'b' in Fig. 8. This correspondes to a nominal shear stress in the adhesive for the 200 m m overlap length of 21 MPa. In case 1 the load level was 80%; in case 2, 82.5%; in case 3, 85%; and in case 4, 87.5%. 175. the shaded elements in Fig. 9: S = 10 000 MPa, RHO = 10 000. adherends - - aluminium, S = 210 MPa. adhesive - - epoxy, E = 3300 MPa, v = 0.3. Si = $2 = 40 MPa. Si2 = 20 MPa, RHO = 1.
,/
/ / / / /
Stiff wall
/ / /
560
10oI
/
/ / /
¢
/ / /
I
250
~o. Fig. 13
Geometry of console. All dimensions in mm
bonded to a stiff wall with an epoxy adhesive. Master variables were located at the ends of the flanges. The master variables to the splines describing the console flanges, shown in Fig. 14, had explicit linear constraints, implying that two adjacent variables may not vary too much in the x-direction. This was done in order to avoid an oscillating shape of the flanges. There was also one explicit linear constraint limiting the flange length to a m a x i m u m of 610 ram. Tsai-factor as objective function
Objective function:
Constraints:
Optimization results One may notice from the results shown in Figs 12(b) to 12(e) that the overlap length increases with increased loading. This seems to be logical. In these cases a b u m p appears on the adherend profiling. Large reductions in weight were made, except for case 4 where the nominal shear stress approaches the yield stress in shear of 20 MPa. The weight reductions obtained were: case 1, 55%; case 2, 51%; case 3, 43%; and case 4, an increase of 21%. Doing 15 iterations with nine design variables, one case and 175 elements for this model took about 40 hours for a optimization on the VAXstation 2000 that was used. It could be worth noting this rather long execution time.
80 000 N
Variables:
Geometry:
minimize the m a x i m u m Tsai-factor, TSAI, in the adhesive layer. 1) Tsai-factor in the whole structure. 2) explicit geometric constraints so that two adjacent variables may not vary too much in the xdirection. variable flange length and thickness of the flanges, see Fig. 14. see Fig. 13. The width of the model was 100 mm. The adhesive layer thickness was 1 m m with spew-fillet elements.
m
Console
Finally a model of a console made of steel was analysed as an application example. The geometry of the model is shown in Fig. 13. The same model was used for both weight and stress optimization. Only the objective function was changed. The console was
210
INT.J.ADHESION AND ADHESIVES OCTOBER 1991
2
b
Fig. 14 Master variables in the console model: (a} in the y-direction; (b) in the x-direction
according to Fig. 13. Console bonded to a stiff wall. the console was loaded with a distributed load, 80 000 N, in the negative y-direction according to Fig. 13. 106. adherend -- steel. adhesive -- epoxy, E = 3300 MPa, v = 0.3. S iT = 20 MPa, Sic = 60 MPa. $2 = 40 MPa, Sl2 = 40 MPa, RHO = 1.
Boundary conditions:
Load:
Number of elements: Material data:
Optimization results According to the T s a i - H i l l criterion the stress were too high, leading to fracture in both the adhesive and the
console in the first iteration. After a few iterations the stresses became permitted in the whole structure, cf., Figs 15(a) and 15(b).
7
Levels Min 0.00 A 0.12 B 0.23 C 0.35 D 0.46 E 0.58 F 0.69 G 0.81 H 0.93
~
I 1.04 J 1.16 K 1.27 L 1.39 M 1.50 N 1.62 O 1.73 Max 1.85
Levels MIn 0.00 A 0.08 B 0.16 C 0.24 D 0.31 E 0,39 F 0.47 G 0.55 H 0.63 I 0.71 J 0,79 K 0.86 L 0.94 M 1.02 N 1.10 O 1.18 Max 1.26 Fig, 1 6 Final geometry of console, after 1 5 iterations with Tsai-factor. Weight as objective function
The oscillations on the bottom flange are due to the fact that there are too many design variables within a very short distance, when the flange length decreases. Also the stresses are small in that area. The shape of the flange has probably no greater influence on the stresses in the adhesive. The critical area, from the stress point of view, is located in the middle of the console. The objective function, TSAI in the adhesive, decreased by 51% from 1.85 to 0.94, a reasonable figure. The stresses in the console decreased from 1.35 to 1.0. Hence a much more efficient structure was obtained. Weight as objective function Objective function: Other model data:
rJ),~
)
minimize the total weight. the same as for the previous model with Tsai-factor as objective function. The initial geometry, with Tsaifactor, is shown in Fig. 15(a).
Optimization results
~_~
/
\
Levels Min 0.00 A 0.12 B 0.23 C 0.35 D 0.46 E 0.58 F 0.69 G 0.81 H 0.93 I 1.04 J 1.16 K 1.27
Fig. 1 5 Console model: (a) initial geometry with Tsai-factor; (b) final geometry after 1 0 iterations with Tsai-factor. Tsai-factor as objective function.
The final geometry, with Tsai-factor, is shown in Fig. 16. The total weight decreased 34% in 15 iterations. The Tsai-factor decreased rapidly from 1.85 to 1.0 and became permitted after eight iterations. The oscillations that appear on the bottom flange in Fig. 16 may be due to too many design variables on this flange, as in the previous case. Reducing the number of design variables on the short flange was found to solve this problem. Discussion and conclusions The results presented in the preceding sections illustrate that it is theoretically possible to apply shape optimization techniques to structural bonded joints. However, the question arises whether it is possible to actually use the type of 'optimal joints' that have been presented in this paper. From a practical point of view it was found very difficult to obtain the correct profiling of the adherends by means of machining relatively thin metal adherends and no results from manufactured optimal joints were obtained. Nevertheless, one possible application area for this technique could be extruded aluminium profiles for bonded applications due to, at least, two reasons: it is easy to produce 'tailor made' aluminium sections and aluminium bonding technology is well known.
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The following conclusions can be drawn from this study. 1) Shape optimization of bonded joints often gives joints with much lower stress levels and weights than that of the initial geometry, and therefore much stronger and more efficient joints. 2) This technique gives a good hint on what kind of design that should be considered for further studies. 3) It may be difficult to handle material and geometrical singularities in the optimization process. In the case of the bonded console, no such problems occurred. This because the singular points in that model were located in 'unloaded' areas. 4) It is best if a 'possible initial optimal geometry' is given as input, otherwise the objective function may find a local (not global) minimum value. 5) It is important to select the number of variables defining the geometry in a proper way. This can be very difficult to do, as the optimal result depends on this choice. 6) A more powerful computer than the VAXstation 2000 is needed for the analysis. It took about 40 hours for a optimization involving 15 iterations with 10 design variables and 175 elements. 7) It would be very interesting to test joints manufactured according to the design obtained. This is an area suggested for further studies. Acknowledgements
The authors hereby acknowledge The Swedish Board for Technical Development (STU) and The Nordic Fund for Technology and Industrial Development (NI) for financial support, and Dr Dan Holm at ALFGAM Optimering AB for advice on the optimization program and for supplying computer time for the analysis.
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Hart-Smith, L.J. 'Adhesive-bonded single-lap joints' NASA CR 112236 (NASA Langley Research Center, January 1973) Adams, R.D. 'Mechanics of adhesive lap joints' Proc European Mechanics Colloquium 227 "Mechanical Behaviour of Adhesive Joints; Saint-Etienne, France, 31 August-2 September 1987 (Editions Pluralis, Paris. 1987) pp 27-50 Ojalvo, I.O. 'Optimization of bonded joints" A/AA Journal 23 No 10 (October 1985) pp 1578-1582 Cherry, B.W. and Harrison, N.C. 'The optimum profile for a lap joint' JAdhesion 2 (April 1970) pp 125-128 Adams, R.D., Chambers, S.H., Del Strothe¢, P.J.A. and Peppistt, N.A. 'Rubber model for adhesive lap joints' J Strain Anal 8 No 1 (1973) pp 52-57 Sage, G.N. 'Aspects of bonded joint design in carbon-fibre reinforcedplastic' in "Adhesion 3" edited by K.W. Allen (Applied Science Publishers, London, 1979) pp 123-141 Adams, R.D, and Wake, W.C. "Structural Joints in Engineering' (Elsevier Applied Science Publishers, London, 1984) Lees, W.A. "Adhesives in Engineering Design" (The Design Council, London and Springer-Verlag, Berlin, 1984) "OASIS-ALADDIN User's Manual, VAXoVMS Version" (Alfgam Optimering AB, Stockholm, Sweden, 1989) Eaping, B.J.D. 'Structural optimization using numerical techniques' Report 85-8 (Department of Aeronautical Structures and Materials, The Royal Institute of Technology, Stockholm 1985)
Authors
Hans. L. Groth, to whom correspondence should be directed, was formerly a Senior Research Engineer at the Department of Aeronautical Structures and Materials, The Royal Institute of Technology, S-100 44 Stockholm, Sweden. He is now with Avesta AB, Research and Development, Mechanical Metallurgy, S774 80 Avesta, Sweden. Peter Nordlund is a Research Student in the Department of Aeronautical Structures and Materials at The Royal Institute of Technology.