STRESS ANALYSIS OF THE ADHESIVE
SCARF JOINT BETWEEN DISSIMILAR
ADHERENDS
Thesis for the Degree of Ph. D.
MECHlGAN STATE UNIVERSITY
PETHINAEDU SURULINARAYANASAM!
1968
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thesis entitled
STRESS ANALYSIS OF THE ADHESIVE SCARF JOINT BETWEEN
DISSIMILAR ADHERENDS
presented by
Pethinaidu Surulinarayanasami
has been accepted towards fulfillment
of the requirements for
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ABSTRACT
STRESS ANALYSIS OF THE ADHESIVE SCARF
JOINT BETWEEN DISSIMILAR ADHERENDS
by Pethinaidu Surulinarayanasami
This study determines the adhesive stress distribu-
tion in scarf joints between elastically-dissimilar adherends
(joined members). Results are presented for five scarf
angles; four levels of adherend dissimilarity; three levels
of adhesive flexibility in the range apprOpriate for the
bonding of metals and plastics; and for both tensile and
bending loading of the joint. The adherends are treated
using plane stress, and the adhesive is capable of resist-
ing shear, and normal stress perpendicular to its plane,
with strains assumed to be uniform through its (small)
thickness. Only linearly elastic behavior is considered.
The Rayleigh-Ritz method is employed to obtain the
extensive stress tables presented. Systems of 177 linear
equations are solved. This corresponds to the representa—
tion of each of the four components of displacement (two
elastic bodies) by the sum of all homogeneous polynomials
in x and y through the eighth degree. The convergence of
the solutions is examined, and the adhesive stress
Pethinaidu Surulinarayanasami
distributions are discussed exhaustively. Also tabulated
are those values of the adhesive combined stresses which
are critical for elastic design by some of the common failure
criteria. The use of these results in design is outlined.
STRESS ANALYSIS OF THE ADHESIVE SCARF
JOINT BETWEEN DISSIMILAR ADHERENDS
BY
Pethinaidu Surulinarayanasami
A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Civil Engineering
1968
ACKNOWLEDGMENTS
The author eXpresses his sincere appreciation to
Dr. James L. Lubkin for suggesting the present research
tapic, and for his dedicated efforts in advising the author
throughout the preparation of this thesis. The author
wishes to thank Dr. Robert K. Wen, Dr. William A. Bradley,
and Dr. Lawrence E. Malvern for serving on his doctoral
guidance committee. Thanks are also extended to the
Division of Engineering Research, for supporting the early
phases of this research.
ii
TABLE OF CONTENTS
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . .
LIST OF
LIST OF
LIST OF
CHAPTER
I
II
III
TABLES O O O O O O O O O O O O O O O O
FIGURES O O O O O O O O O O O O O O O O
APPENDICES O O O O O O O O O I O O O 0
INTRODUCTION 0 C O O O O O O O O O O O
1.1 General . . . . . . . . . . . . .
1.2 Literature Review and Background of
the Present Hypotheses . . . . .
1.2.1 Lap Joints . . . . . . . . .
1.2.2 Scarf Joints . . . . . . . .
1.2.3 Butt Joints . . . . . . . . .
1.3 The Present Investigation . . . . .
1.4 Notation . . . . . . . . . . . . .
METHOD OF ANALYSIS . . . . . . . . . .
2.1 Formulation of the Problem . . . .
2.2 The Rayleigh- -Ritz Method . . . . .
2.2.1 Derivation of Equations . .
2.3 The Sherman-Lauricella Integral
Equation Approach . . . . . . .
2.3.1 General . . . . . . . . . . .
2. 3. 2 Notation for Integral Equation
Me thOd O O O O I O O O
2. 3. 3 The Problem Analyzed by Integral
Equations . . . . . . .
2.4 Numerical Data Assumed in the
Calculations . . . . . . . . . .
2.5 Computer Programs . . . . . . . . .
CRITERIA FOR ACCEPTABILITY OF RESULTS;
PARTIAL DISCUSSION OF RESULTS . . .
3.1 General . . . . . . . . . . . . .
3.1.1 Stress Boundary Condition
Check . . . . . . . . . .
iii
Page
ii
vii
ix
H F‘
16
18
19
20
25
25
29
34
46
46
48
50
62
65
66
66
67
CHAPTER Page
3.1.2 Comparison of Adhesive Stresses
Calculated Several Ways From
the Results . . . . . . . . 75
3.1.3 Double- Precision Check of
Roundoff Errors . . . . . . . 79
3.1.4 Convergence of the Approximating
Sequence . . . . . . . . . . 81
3.1.5 Overall Equilibrium Check . . . . 98
3.2 Confirmation by the Integral Equation
MethOd O O O O O O O O O O O O O O O 104
IV ENGINEERING SIGNIFICANCE OF THE RESULTS . . 107
4.1 Adhesive Normal and Shear Stress
Distributions . . . . . . . . . . 108
4.1.1 Case of Tensile Loading . . . . . 110
4.1.2 Case of Bending Load . . . . . . 124
4.1.3 Identical Adherends in Bending . 124
4.1.4 Bending Load (General Case) . . . 135
4.2 Adhesive Combined Stresses . . . . . . 145
4.3 Construction of Design Curves for
Scarf Joints with Linearly-
Elastic Adhesives . . . . . . . . . 149
V CONCLUSIONS AND SUGGESTIONS FOR FURTHER
RESEARCH O O C O O O O O O O O O O O C O 154
5.1 Conclusions . . . . . . . . . . . . . . 154
5.2 Future Research . . . . . . . . . . . . 155
BIBLIOGRAPHY O O O O O O O O O O O O I O O O O O C 157
APPENDICES O O O O O O O O O O O O O O O O O O O O 160
iv
TABLE
10
11
F1
F2
F3
LIST OF TABLES
Largest boundary condition error
(bending) o o o o o o o o o o o o o o 0
Adhesive stresses calculated three ways .
Adhesive stresses for 6th-, 7th- and 8th-
order polynomials, tensile loading
a=3OO,B=20,Y=4........
TenSion. O. = 10°, 8 = 20' Y = 4 o o o o 0
Adhesive stresses for 6th-, 7th- and 8th-
order polynomials. Tension.~a = 5°,
8 = 20' Y = 4 o o o o o o o o o o o o
Bending.-a = 30°, 8 = 20, y = 4 . . . . .
Adhesive stresses for 6th-, 7th- and 8th-
order polynomials. Bending.-a = 10°,
8 = 20' Y = 4 o o o o o o o o o o o 0
Bending. a = 5°, 8 = 20°, Y = 4 . . . . .
Disequilibrium of forces and moments . . .
Sample overall equilibrium checks . . . .
Maximum values of the combined stresses
N1, N2, T1 and Toy O O O O O O O O O 0
Adhesive normal and shear stresses . . . .
Largest values of adhesive normal stress
(Nmax). Locations may be estimated
from tables of stress distributions . .
Average difference in adhesive normal and
shear stresses between polynomial
solutions of orders 8 and 7, or of
orders 8 and 6 (latter designated by *)
Page
72
77
83
84
9O
9O
93
93
100
103
147
182
197
198
TABLE Page
F4 Largest differERce between 8th- and 7th-
order, or 8 - and 6t h—order polynomial
solutions for adhesive stresses . . . . . 199
F5 Root- Mean— —Square values for percentage
differences between 8 th-order and lower—
order polynomial solutions. . . . . . . . 201
vi
FIGURE
la
1b
1c
1d
10
ll
12
13
14
LIST OF FIGURES
Lap joint . . . . . . . .
Scarf joint . . . . . . .
Butt joint . . . . . . .
Double lap joint . . . .
Adhesive-air boundary in a lap joint
Mylonas' test . . . . . .
William's model . . . . .
Geometry of Mylonas, McLaren and MacInnes
Scarf joint geometry . .
Geometry and loading . .
Boundary stress errors, tensile loading
a = 10°, 8 = 20, y = 4
Boundary stress errors, bending load
a = 10°, 8 = 20, y = 4
Shear stress (T) by Ritz and integral
equation methods . . .
Normal stress (N) by Ritz and integral
equation methods . . .
Shear stress concentration factor (T )
in tensile loading . .
(T in tensile loading
C)max
Shear stress concentration factor (TC)
in tensile loading . .
vii
Page
11
ll
11
11
26
26
68
69
105
105
112
112
115
FIGURE
la
1b
1c
1d
10
11
12
l3
14
LIST OF FIGURES
Lap joint . . . . . . . . . . . . . . . .
Scarf joint . . . . . . . . . . . . . . .
Butt joint . . . . . . . . . . . . . . .
Double lap joint . . . . . . . . . . . .
Adhesive-air boundary in a lap joint . .
Mylonas' test . . . . . . . . . . . . . .
William's model . . . . . . . . . . . . .
Geometry of Mylonas, McLaren and MacInnes
Scarf joint geometry . . . . . . . . . .
Geometry and loading . . . . . . . . . .
Boundary stress errors, tensile loading
a=10°,8=20,'¥=4........
Boundary stress errors, bending load
(1:100, 8:20, Y: 4 o o o o o o o o
Shear stress (T) by Ritz and integral
equation methods . . . . . . . . . . .
Normal stress (N) by Ritz and integral
equation methods . . . . . . . . . . .
Shear stress concentration factor (TC)
in tensile loading . . . . . . . . . .
in tensile loading . . . . . . .
)
(TC)max
Shear stress concentration factor (T
in tensile loading . . . . . . . ?
vii
Page
11
ll
11
11
26
26
68
69
105
105
112
112
115
FIGURE
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Shear stress concentration factor (T )
in tensile
Normal stress
in tensile
(NC max
Normal stress
in tensile
Normal stress
in tensile
Normal stress
loading . . .
concentration
loading . . .
) in tensile loading
concentration
loading . . .
concentration
loading . . .
concentration
C
factor (NC)
factor (NC)
factor (NC)
factor (NC)
in tensile loading . . . . . . . . . .
Shear stress (T) in tensile loading . . .
Shear stress (T) in bending load . . . .
Normal stress (N) in bending load . . . .
Normal stress (N) in bending load . . . .
Shear stress (T) in bending load . . . .
Maximum shear stress (T ) in bending
load 0 O O O O O O Om?x O O O O O O O 0
Maximum shear stress (T ) in bending
max
load 0 O O O O O O I O O O O O O O I 0
Normal stress concentration factor (NC )
in bending load . . . . . . . . . . .
Normal stress concentration factor (NC )
in bending load . . . . . . . . . . .
Elastic "design curves" in dimensionless
form, for various failure criteria
(failure in N , Tl' T ). Tensile
or compressivé logd . . oy . . . . . .
Elastic "design curves" in dimensionless
form, for two failure criteria
(N , TO ). Bending loads in two
senses . . . . . . . . . . . . . . . .
viii
Page
115
117
117
121
121
123
123
126
129
129
136
136
138
140
143
151
153
APPENDIX
A
LIST OF APPENDICES
FORMULATION OF THE EXPRESSION FOR TOTAL
POTENTIAL ENERGY . . . . . . . . . . . .
GENERATION OF RITZ EQUATIONS . . . . . . .
SELF-EQUILIBRATED POLYNOMIAL STRESSES . . .
DISPLACEMENT DETERMINATION FOR THE INTEGRAL
EQUATION APPROACH . . . . . . . . . . .
RIGID BODY DISPLACEMENT CONSTANTS . . . . .
TABLES OF STRESS DISTRIBUTIONS AND
AUXILIARY TABLES . . . . . . . . . . . .
COMPUTER PROGMM O O O O O O O I O O O O 0
ix
Page
160
162
167
170
176
179
202
CHAPTER I
INTRODUCTION
1.1. General
The analysis of adhesive joints has growing impor-
tance because of the increasing application of adhesive in
industrial and aerospace technology. The applications are
usually inconspicuous, but vital to the performance of the
bonded objects, whether these be automobile brake linings,
the wings of a jet liner, or the ultra-high-strength sheet
metal rings of a solid-fuel rocket case. It is essential
to pay most attention to the adhesive stresses, because the
adhesive is usually the weakest material involved in the
joint. With proper design, however, the joint need not
necessarily fail in the adhesive.
The adhesive, of course, constitutes a fastening
medium between the two adherends (joined members). Some
types of adhesive can reasonably be idealized as linearly
elastic, and this assumption is used here. This hypothesis
is entirely unrealistic12
for many other adhesives, but
even for those with complex rheological behavior, the elas-
tic analysis may offer the designer useful guidance.
Adhesives are employed in many different joint con-
figurations,to bond such materials as metals, plastics, wood
and glass to themselves or to each other. The assumptions
of the present study make the results applicable primarily
to cases where the adherends are considerably more rigid
than the adhesive layer, e.g., the bonding of metals and
plastics. Some modification of the present approach is re-
quired to accommodate glue joints for wood. This might be
worthy of study, since the "finger joint" of wood technology
is quite similar in geometry to the one considered in this
thesis.
The commonest joint configuration is the "lap joint"
(Fig. la, next page), which is also the easiest one to manu-
facture. Most previous studies deal with this (see section
1.2, "Literature Review and Background of the Present Hypo-
theses"). Also of great technical importance is the "scarf
joint," with which the present thesis is concerned. The
scarf joint connects two bars or sheets on an inclined plane
(see Fig. 1b). The scarf angle is usually in the range of
10° to 30°. This joint has the advantage over the lap joint
of avoiding the bending action due to the offset of the two
members, when the loading is tensile, and the disadvantage
of being harder to make. The scarf joint presents a much
larger surface area of adhesive for bonding than the con-
ventional "butt joint" (Fig. 1c). The latter is the special
case when the "scarf angle" is 90°. A perfect butt joint
E t Ea’ Ga
¢
F 1 VH1”: //
t2 E2 7*}
Fig. 1a Lap joint.
F‘<4———’ El r-—£> F
Fig. 1b Scarf joint.
G ,Ea
Fug—L E1 E2 ._> F
Fig. 1c Butt joint.
H/z
H/Z
11117711171
Fig. 1d Double lap joint.
offers a purely tensile loading of the adhesive, but mis-
alignment (which is hard to avoid) can give rise to substan-
tial bending action and corresponding stress concentrations.
The scarf joint configuration, with the usual 10°-30° angle,
loads the adhesive largely in shear at a moderate stress
level.
This thesis follows up a previous study of the scarf
joint between identical adherends by investigating the case
of dissimilar materials. Prior work, discussed later (Sec.
1.2), indicates that the overall features of the adhesive
stress distribution may be obtained by treating the thin
adhesive layer as a complex elastic foundation. The latter
is assumed to be capable of transmitting shear stress, and
normal stress perpendicular to the plane of the adhesive
layer. Certain local "edge effects" (stress concentrations)
are neglected, as explained subsequently.
The complexity of the present problem, formulated
as a boundary-value problem of elasticity theory, appears
to preclude an exact solution. Accordingly, the Rayleigh—
Ritz method has been used, as the most appropriate for
finding answers expeditiously. The method of finite dif-
ferences is difficult to formulate for this problem, and
was abandoned after considerable investigation. An alter—
nate method, described in Muskhelishvili,4 involves the
use of the Sherman-Lauricella integral equation in the
complex plane. As utilized here, this approach seems to
require even more computer memory than the considerable
amount available in Michigan State University's CDC 3600
computer installation. Thus the method has not been en-
tirely successful, but the computations performed seem to
support the results of the Rayleigh-Ritz method.
1.2. Literature Review and Background of
the Present Hypotheses
This review is fairly brief because a number of
recent surveys are available.8’9’lo’12
Moreover, very little
of the work is immediately relevant to the present problem,
since it appears that only one previous analysis of scarf
jointshas been conducted. There is some literature12'16’
24’26’30 on experimental work for such joints, mostly from
the aerospace field and from wood technology. However most
of these involve tests to destruction and it is likely that
correlation with the present elastic analysis would only be
of the roughest sort.
The studies reviewed, however, serve to validate
some of the present hypotheses. They also indicate some of
the factors which are neglected in the present type of
analysis.
The literature of the "peel test" is not covered
here; see References (8,9,10). This test (to destruction)
is used for quality control in joint manufacture. It in-
volves large deformations of the adhesiVe and at least one
adherend. The latter is usually peeled off a drum to which
it has been bonded. Since many of the peel test analyses
involve the assumption of an elastic adhesive, they are
probably not even relevant to the actual peel test, much
less to the present study. For convenience, the various
past investigations which deal with the stress distribu-
tion in adhesive joints have been grouped as follows:
a) lap joints, b) scarf joints, and c) butt joints.
1.2.1 Lap Joints
It should be noted that all stress calculations
have neglected the adhesive layer thickness in consider—
ing the geometry of the problem. This reflects practice;
adhesive layers are usually very thin compared to the
thickness of the adherends. This sometimes fails to be
accurate in the case of the very thin sheets encountered
in aircraft construction, but most analyses are readily
modified to accommodate the necessary changes. Most theo-
retical work has been done on the stress distribution in
adhesive lap joints. One of the earliest investigators,
Volkersen,ll developed a one-dimensional, elastic-adhesive
theory by treating the adhesive joint as an approximate
substitute for (limiting case of) the multirow riveted lap
joint. He indicates in his analysis, after neglecting
the bending of adherends, that the largest adhesive
shear stress occurs toward the ends of the overlap region.
His analysis includes the case of dissimilar adherends,
and actually applies most accurately to symmetrical double
lap joints (Fig. 1d, p. 3), since these involve comparatively
little bending. N. A. de Bruyne's12 analysis is essentially
that of Volkersen. Hartman26 supported some of the theo-
retical investigations in his tests of double lap joints.
N. A. de Bruyne12 also argues persuasively for the advantages
of the bevelled (tapered) lap joint in reducing adhesive
shear stress concentration. Sazhin19 studied the tapered
lap joint analytically, and found that it leads to a hyper-
geometric differential equation. The problem is actually
solved using a variational method, but with very few terms
and without a discussion of convergence. His reported good
agreement with experiments is somewhat suspect, because the
experiments are not described and sound suspiciously like
ultimate strength tests (in the translation of the original
article). It would have to be coincidental that an elastic
analysis predicts the behavior of a test to destruction in-
volving the amount of inelastic behavior normally found in
adhesive joint failure tests. The tapered lap joint has
been investigated experimentally by Hartman,30 who observed
in tests that a tapered lap joint does have a moderately
larger ultimate load capacity than the uniform one.
Although the neglect of adherend bending and adhe-
sive normal stress reduces these analytical studies to just
one step above "dimensional analysis," they still offer
designers considerable guidance as to what to expect. Their
results also explain why adhesive lap joints invariably
start failing at the ends,even though, as elastic studies,_
they cannot be expected to be of much help in studying
ultimate strength tests. Nevertheless, these early studies
suggested ways of plotting strength test data so as to
minimize the total amount of experimental work required to
establish system properties (de Bruyne's'ﬁoint factor").
Goland and Reissner13 published a considerably more
rigorous analysis. This included the effects of adherend
bending, inside and outside the joint, and appears to be
the first study to show that large "tearing stresses"
arise, concentrating at the ends of the joints. The latter
are direct stresses normal to the plane of the adhesive
layer. They considered two limiting cases, one of which is
relevant to a joint in which the adhesive is much more rigid
than the adherends. This applies for certain joints in wood,
paper, cardboard, and low modulus plastics. The other limit-
ing case is the one where the adhesive is much more flexible
than the adherends, as in metal-to-metal joints. Under this
hypothesis, and based upon a consideration of the strain
energies of the problem, they argue that it is sufficient
to consider only the adhesive shear stress, and the adhesive
direct stress normal to the adhesive 1ayer.. Most subsequent
work (including this thesis) has neglected the longitudinal
component of adhesive direct stress, essentially because of
the low modulus of the adhesive compared to the adherends.
Their second stress analysis problem was formulated
(with minor inconsistencies) as one of cylindrical bending
of thin plates (i.e., practically speaking, using elementary
beam theory). The first problem, not discussed here because
it applies primarily to joints in wood, used plane strain
theory. Exceptiknrthe Goland—Reissner stiff-adhesive (plane
strain) case, these investigations all model the adhesive as
a uniform elastic foundation capable of transmitting shear
(or shear and normal stress) from one adherend to the other.
Plantemal4 modified the Volkersen theory by consid-
ering the effect of bending deformation on the adhesive
shear stress, arriving at a refined shear stress concentra-
tion factor. However,the neglect of normal stress appears
undesirable, in View of the results of Goland and Reissner.
Cornell15 studied the brazed—tab fatigue specimen
as a lap joint. His work is closely related to that of
Goland and Reissner, although the geometry of the problem
is somewhat different. This study constitutes both an ex-
tension and a validation of their analysis. His assumption
that the two adherends act like beams and that the elastic
cement layer behaves like an infinite number of infinitesi-
mal shear and tension springs is simply a restatement of
the Goland-Reissner hypothesis. His "cement" was actually
a thin layer of braze compound, which perhaps cannot be
considered to be flexible enough, relative to metal ad-
herends, to qualify as "much more flexible" than the latter.
10
His experimental work, however, indicates good agreement
with the analysis. This, in turn, simply indicates that
Goland and Reissner's energy argument for deciding when
the "elastic foundation" model for the adhesive layer will
break down is somewhat conservative. It can probably suc-
cessfully be used for stiffer adhesives than they indicate.
Cornell compared his analytical results for adhesive
stresses to photoelastic and brittle lacquer experiments.
His study was also one of the first to consider a
very significant factor which is entirely neglected in the
type of approach used by Goland and Reissner (and in the
present thesis). This is the "free-edge effect" at the
ends of the joint. The "free—edge effect" is the stress
disturbance caused by the complex boundary conditions at
the ends of the joint, where adherends and adhesive are
adjacent to a stress-free boundary, usually air. Figure 2
(next page), shows such a free boundary in terms of Cor-
nell's geometry. A proper consideration of this problem
is enormously complex from the point of view of elasticity
theory. This is further complicated by our ignorance of
the precise shape of the adhesive-air boundary in practical
situations, because the actual boundary shape depends upon
the details of the production process, the actual adhesive
used, etc. Moreover, the free-edge effect is significant
at the point where shear and normal stresses in the adhe-
sive usually take on their largest values, according
11
’(Tab) El Adherends
Ea'Ga (:r
E
2 (Base Bar) 1
\\\\; \Ly \\ \
Fig. 2 Adhesive-air boundary in a lap joint.
3 PF
Fig. 3 Mylonas' test.
Elastic Adhesive
"Rigid" Adherend
Fig. 4 William's model.
F El
- /J////////J//11/// // U1 /Lz / //// // / ///////jj
Fig. 5 Geometry of Mylonas; McLaren and MacInnes.
12
to those approaches which ignore these local effects. As
Cornell puts it, the analysis neglecting the free-edge
boundary problem has a "built-in fillet," r > 0 in Fig. 2.
But whether r = n, 1.7n, 0.2m or any other value depends
very much upon manufacturing details. The value of r can
of course be investigated statistically for any particular
problem important enough to warrant the research costs.
We know, at least, that if the adhesive joint is a good
one, the fillet radius will be in the sense indicated.
This is because a proper bond requires low surface tension:
the adhesive must "wet" both of the adherends well.
In Cornell's work, his "adhesive" (brazes and
solders) in practice had a radius r about equal to n in
Fig. 2, and he bases his discussion on this observation.
Though he found fairly good agreement between supporting
experiments and the analysis, the main discrepancy was
precisely at the free edge. He points out that the stress
concentration factor becomes infinite if the tab and the
base bar form a right angle, i.e., if the radius of curva-
ture (r in Fig. 2) is zero. This is just another way of
saying that stress singularities must be expected at a 90°
boundary if shear stress is present on one side and an
adjacent edge is stress-free.
To better understand the local effects at free
edges, Mylonasl6 conducted photoelastic experiments on
transparent plastic layers bonded to "rigid" (steel) plates,
l3
simulating the ends of adhesive lap joints. In effect, his
work validates the neglect of free-edge effects, as far as
the interior of the joint is concerned. This is what we
would also expect from St.-Venant's Principle, since the
thickness of the adhesive is normally small compared to the
length of overlap. At the joint ends, however, Mylonas
finds that the stresses do vary across the thickness of the
layer and depend strongly on the shape of the free boundary
of the adhesive. He studied models having concave edges in
the shape of a circular arc (Fig. 3, p. 11), with ratios of
radius r to adhesive thickness n ranging from 0.5 (semicir-
cular edge) to w (straight edge). For the load sense shown,
he found that when r/n < 1.25 (e < 50°), the maximum stress
develops on the adhesive boundary but away from the adhesive-
adherend interface ("cohesive failure" expected in the ad-
hesive). For a larger radius, the highest stress level is
much larger and develops at the corner, A ("adhesive fail-
ure" expected). Generally speaking, his experiments show
that the local effects at the free edges are critically
dependent upon the shape of these edges.
Mylonas' study correlates well with an investiga-
tion, unrelated to adhesive joints, due to Williams.17 He
analyzed thin plates in extension, using generalized plane
stress, in order to estimate the strength of the stress
singularities which can be expected at the vertex of a
semi-infinite triangle (wedge) under various edge boundary
14
conditions. A typical metal-to-metal joint involves a
"nearly-rigid" adherend bonded to a comparatively low—
modulus adhesive, so that Fig. 4 (p. 11) approximates the
local situation at the adherend-adhesive-air corner in
Mylonas' experiments. This corresponds to Williams'
boundary-condition case of one edge free and the other
edge fixed. He formulated an eigenvalue problem for the
rate-of-decay parameter A, of stress with distance r from
point A. All stresses behave as rA-l, and all displace-
ments as rx. The calculations give A as a function of 6,
the wedge angle. According to his results, no stress sin-
gularities arise for angles 8 less than 63°, but singulari-
ties do arise (for general loading) when 6 > 63°. This
trend is quite similar to what Mylonas found in his experi-
ments. The latter differ in that the steel plates only
approximate the ideal "rigid" boundary conditions of
Williams.
Misztal20 studied lap joints in flat sheets loaded
by shear flow perpendicular to the plane of the drawing in
Fig. la (p. 3). He assumed that the shear stress is uni-
form across the adhesive and adherend thickness, and they
deform only in shear, obtaining an adhesive shear stress
distribution similar to that of the Volkersen problem. He
also examined double lap joints of this type.
McLaren and MacInnes18 performed photoelastic ex-
periments on lap joints. They found Goland and Reissner's
15
analytical results to be generally correct, observing an in-
crease in shear and "tearing" stresses towards the ends of
the joint. The shear stress, of course, obeys the actual
boundary conditions and drOps to zero at the ends. They also
studied the effect of adhesive-adherend contact angle at the
end of the joint, in tests similar to Mylonas'. In this
work the adherends are considerably more flexible than in
Mylonas' tests (although still stiffer than the adhesive).
As "8," the contact angle, is reduced to about 40°-50° or
less, the largest tensile stress moves out of the "leading
corner" (adherend—adhesive-air point, A in Fig. 3 (p. 11),
on the loaded side) to a point C on the adhesive-to-air
boundary. This supports both Mylonas and Williams quite
well, considering the somewhat different range of the elas-
tic constants. It is also noted by Mylonas, and by McLaren
and MacInnes, that the largest magnitude of stress occurs
at B (Fig. 5, p. 11), for the load sense of Figs. 3 and 5.
This is compressive in nature, but of course becomes ten-
sile if load F reverses. This is quite consistent with
Williams' analytical results, since the stress level ex-
ponent (A-l) in r)".1 increases with wedge angle. If AB in
Fig. 5 is straight and 6 is small enough to avoid stress
singularities in the adhesive wedge cornering at A, then
the obtuse adhesive wedge corner at B is surely large
enough for singular stress at B.
16
Lubkin and Reissner21 produced an extensive analysis
of lap joints between thin, circular, cylindrical shells in
axial tension. The investigation was carried out using the
linear theory of axisymmetric bending and stretching of thin,
isotropic shells. It was again assumed that the adhesive
layer is elastic and considerably more flexible than the ad-
herends. The work is basically along the lines of the Goland
and Reissner approach, with due allowance for the new geome-
try. The paper contains considerable discussion of the
following: (1) effect of amount of overlap on adhesive
stress concentration values, (2) position of maximum adhe-
sive normal and shear stress, and (3) effect of flexibility
of adhesive layer on stress concentration values. An in-
structive comparison is made between tubular and flat-plate
lap joint theories.
Sherrerz2 has investigated the stress distribution
in lap joints when the adherends are dissimilar, as an ex—
tension of Goland and Reissner's analysis. He obtained a
series solution for the stresses in the joint, but had dif-
ficulties because of slow convergence. Sazhin19 also
studied the lap joint, apparently unaware that he was
duplicating the work performed by Goland and Reissner 20
years earlier-—at least he does not acknowledge priority.
1.2.2 Scarf Joints
Lubkin7 considered an adhesive scarf joint between
elastically-identical adherends, loaded in tension. With
17
the assumption that the thickness of the adhesive is negli-
gibly small compared to the depth of the adherends, it is
found that the adhesive stress distribution is uniform from
end to end. Moreover, the uniformity of adhesive shear and
normal stress (both of the latter are considered here) is
independent of the scarf angle and of the thickness of ad-
hesive and adherend. The stresses themselves, of course,
depend on the scarf angle. This simple state of stress can
therefore be calculated directly from equilibrium considera-
tions. Its simplicity arises from the symmetry of the
identical adherends,and of the purely tensile load. (If
the loading is pure bending, for example, this symmetry is
lost and it can be shown that it is impossible for the ad-
hesive stress to vary linearly along the joint.) He pre-
sented results useful to the designer within the elastic
range. These can probably be applied in the "wood" range
of elastic constants also, because both of the adhesive
normal stresses have been takenjxux>account. Due to the
fortunate uniformity of the adhesive stresses along the
adhesive layer, it is possible to speculate that they remain
sensibly uniform when the adhesive no longer behaves elasti-
cally. The paper therefore had some success in correlating
actual failure tests, using what was originally intended to
cover only the elastic range. It is not to be expected
that the present thesis can be used in this manner, since
the stress distributions found here are generally not uni-
form along the joint.
18
The present investigation is an extension of Lubkin's
work, covering the more complex case of dissimilar adherends.
COOper12 measured the adherend strains in a scarf
joint with an extensometer. He estimated a stress concen—
tration factor of 1.45 in a joint with a scarf angle of 6°
(identical adherends), but the definition of this factor is
not clear enough to permit a comparison with Ref. 7. Miiller24
and Hartman26
performed purely experimental work on scarf
joints tested to destruction. Hartman's work was correlated
with theory in Ref. 7.
1.2.3 Butt Joints
The "butt" joint is the name given to the special
case of the 90° scarf joint (Fig. 1c, p. 3). de Bruyne27
formulated a relation based on viscous flow theory to indi-
cate that,for very thin adhesive layers, the joint strength
is inversely proportional to its thickness. This is found
to have good agreement with experimental results, although
perhaps not for the theoretical reasons adduced. Shield,28
using limit analysis, investigated bounds on the joint
strength of a butt joint. Norris29 assumed that the adhe-
sive in the bond is isotropic, and that the strains in the
adhesive, parallel to the plane of the bond, are equal to
those in the adherends. He develOped a method for the de-
termination of the elastic properties of adhesives as
they actually exist in bonds. He substituted these properties
19
in the formula for determination of the stress at which in-
stability becomes general throughout the bond, and this
stress is compared with the results of tests.
1.3. The Present Investigation
The purpose of this study is to obtain detailed in-
formation on the stress pattern in the adhesive of scarf
joints between elastically-dissimilar materials, for a
variety of parameters and loading conditions. The problem
is treated as one of plane stress, with the assumption (re-
flecting most practical cases) that the adhesive layer is
negligibly thin when compared to the adherends. A con-
siderable number of different approaches have been attempted;
only the relatively successful ones are reported.
As formulated here, the problem consists of finding
a set of unknown internal boundary conditions for two dif—
ferent plane elastic bodies of trapezoidal shape. It ap-
pears that the method of finite differences is not well
suited, partly because of the present complexity of shapes
and boundary conditions. Of itself, this is not so bad.
The major problem is that the model adopted for the adhesive
almost demands that the solution be carried out in terms of
displacements, which implies four Navier equations in two
adherends. Moreover, the nature of the Navier equations
is such that interlocking nets of node-points are required
in each elastic body. The primary method selected, there-
fore, is the Rayleigh-—Ritz method, a direct approach to
20
variationally-formulated problems. There is no reason to
expect this to yield results inferior to the method of fi-
nite differences, and perhaps some reason to expect it to
be better.
In addition, the problem has been studied using an
elegant approach based upon the Sherman-Lauricella integral
equation in the complex plane.4 This method is so radi-
cally different in concept from the Ritz method that good
agreement would constitute an independent check on the re-
sults of the Ritz method. Unfortunately, just when agree-
ment appears to be getting good, the integral equation
formulation exceeds the capacity of the computer. It has,
therefore, not been pursued extensively.
The details of the method of analysis, including
the mathematical formulation of the problem and the various
parameters arising in the investigation, are given in Chap-
ter II. Chapter III is devoted to a discussion of the
checks used to validate the results.
1.4. Notation
The symbols used in this thesis are defined in the
text while they first appear. For convenience, they are
also listed here in alphabetical order, with English let-
ters preceding Greek letters. There are many symbols which
are common to both the Rayleigh-Ritz method and the integral
equation approach, but these occasionally represent slightly
21
different quantities. A separate notation section is there-
fore given for the integral equation method in section 2.3.2.
The present section gives the list of all symbols common to
both methods, and conveying the same meaning. Other symbols
used in Chapters III and IV are also included here.
Subscripts l and 2 almost always represent quantities
defined for adherends l and 2, respectively. Figure 7, p. 26
shows many of the geometric quantities.
Am’n, B ,
I I
C , D = Coefficients of displacement functions
(ul,vl),(u2,v2) of lst and 2nd adherends,
respectively.
D = Coefficients of dimensionless displacement
functions (U1,V1),(U2,V2) of let and 2nd ad—
herend, respectively.
c = h(2 + cot a)(see Fig. 7).
C = 2 + cot a, (dimensionless value of c for
h = l).
E = Young's modulus.
E1,E2 = Young's moduli.
Ea = Young's modulus of adhesive.
F = Resultant axial tensile force per unit width
of adherend (Figs. 1-7).
Z
Nl’NZ
22
Shear modulus of adhesive.
Adherend half—thickness.
See Fig. 7.
Bending moment per unit width of adherend
(Figs. 6,7).
Highest order of homogeneous polynomials.
Coordinate directions (Fig. 7).
Adhesive normal stress (dimensionless).
Adhesive normal stress concentration factor.
N evaluated at xj'Yj°
Adhesive principal stresses.
s/h cot a = fraction of joint length along
adhesive interface, measured from midpoint
(origin).
Adhesive shear stress (dimensionless).
Adhesive shear stress concentration factor.
T evaluated at Xj’Yj'
Adhesive principal shear stress.
Adhesive octahedral shearing stress.
Strain energy of whole system, adhesive,
lst adherend, 2nd adherend.
23
Dimensionless strain energy of whole system,
adhesive, lst adherend, 2nd adherend.
Displacement components in x- and y- direction.
Displacement components of adherends in x-,
y- directions.
Dimensionless displacement components in
X-, Y- directions.
Total potential energy of the system.
Coordinates.
x/h, y/h (nondimensional coordinates).
Coordinates of particular points along ad-
hesive interface.
Surface tractions in X-, Y- directions.
Poisson's ratio.
Poisson's ratios.
Scarf angle.
nEl/Ea h(l - vi) = relative stiffness of
adhesive and adherend l.
E2(1 - VI)/El(l - v3) = relative stiffness
of adherends.
Adhesive film thickness.
Total potential energy.
ns
com
24
I
(l - vi)0 /Elh2 = dimensionless total
potential energy.
Adhesive normal and shear strains.
External tensile stress loading adherends
(=F/2h), or outer-fiber value of bending
stress loading adherends ("MOh/I").
Usual components of stress.
ox(ic,y)/El = dimensionless end stresses
loading scarf joint system.
Adhesive normal stress.
Adhesive shear stress.
Miscellaneous
(Eq. 4.1.1).
Any one of the adhesive combined stresses
T .
N1' N2' T1' 0y
l/Ncom = OXO/Za°
.Any allowable (design) value for the adhe-
sive combined stresses.
CHAPTER II
METHOD OF ANALYSIS
This chapter further describes the physical problem
and the methods used to investigate it.
2.1. Formulation of the Problem
The scarf joint considered here is shown in Fig. 6
(p. 26). It consists of two elastic adherends, joined to-
gether by a thin film of adhesive along the inclined face.
The depth of the adherends away from the joint is uniform
and equal to 2h, while that of the adhesive is n, also uni-
form and assumed to be very small compared to 2h. The
values of Young's modulus and Poisson's ratio for adherends
l and 2 are El’ v1, and E2, v2. The adhesive is assumed to
be elastic, with Young's modulus Ea and shear modulus Ga.
The scarf angle is a, and the joint is subjected to either
tensile force F or bending moment M0, both per unit width.
These are typical loadings for this type of joint.
The actual geometry selected for the boundary is
shown in Fig. 7 (p. 26). The practical reason for using
an adhesive scarf joint is to increase the size of the ad-
hesive area, so that the adhesive--a weak material--can
25
26
y s
E v
El’vl ///// 2’ 2
A n 1 4K .
Mo (:> (2) M0
F 2 F
0 x
2h h
n
Fig. 6 Scarf joint geometry.
F = Resultant of o L
0
YIY S 1
”Coy ) 00 A D
1 E H
n
Adherend l Adherend 2
F
x,X ;:>4>
(r
M F
0 =1 Mo
G
M0 = Resultant «a c,C C'C ——-—£>
of -00Y NF
Ll h cot a
Tns'T “v~ A/ zero . . .
T T P051t1ve Adhe31ve
zero _ﬂ_ //’ ns' Stress Defin1tions
Gn’N Fig. 7 Geometry and loading.
27
sustain lower stresses (stress concentration, of course,
can defeat this objective). For this reason a is usually
less than 25-30°. The lower limit, perhaps in the range
5—10°, is occasioned by the difficulty of manufacturing
straight, finely-tapered edges, especially in thin adherends.
The geometry chosen for the mathematical study of
the problem is inherently a compromise. Remote from the
joint, there is uniform tension parallel to x. Near the
joint, the stress pattern is greatly disturbed, except in
the case of identical adherend materials. It is therefore
judged important to allow a certain distance Ll for this
disturbance to reduce to the remote uniform bending or
tensile field, which is the ultimate end-boundary condi-
tion.‘ A trapezoidal shape for each adherend therefore
appears to be essential, if the problem is not to be ideal-
ized out of existence. The latter would be the case if.
only triangles adjacent to the adhesive interface were con-
sidered (L1 = 0). Conceivably, a parallelogram shape could
also be used.
As explained in the introduction, the present type
of study attempts only to describe the overall behavior of
the joint, and is admittedly imprecise at the free ends of
the adhesive. The adhesive-adherends-air boundary must
have a clearly-specified geometry before this complex local
problem can be attempted. Reasonable assumptions here de-
pend very much upon the actual materials used and precise
28
manufacturing details. Some of the questions which must
then be asked are as follows: Was surplus adhesive wiped
off before curing? Machined off after curing? Left alone?
Did bonding pressure squeeze any adhesive out? Was there
enough adhesive initially? How much adhesive shrinkage was
there? At best, these can only be answered statistically,
in individual applications whose importance warrants the
expense. Nevertheless, the present type of study offers
the designer a comparative framework into which he can fit
empirically-determined constants (stress at yield, or some
such index) for his own particular case. It also offers
information about general trends, and the effects of varia-
tion of physical parameters.
It is assumed that the adhesive is very thin, and
quite flexible compared to the adherends in the metal-to-
metal joints at which this study is aimed. The adhesive
strains and stresses are therefore taken to be uniform
across its thickness, and the direct adhesive stress in
the joint axial direction (5 — direction in Fig. 7, p. 26)
is ignored. To justify this neglect, note that the adhe-
sive is assumed to have a much smaller Young's modulus than
that of the adherends. The model used for the adhesive is
thus that employed by most previous investigations, and
validated by the experiments of Cornell, Mylonas and others.
Essentially, it is an elastic foundation, capable of trans-
mitting shear as well as transverse normal stress. The
29
chief discrepancy in treating the adhesive is the neglect of
the local free-edge effects at the ends of the boundary,
which is a proper subject of separate study. The idealized
problem thus consists of two two-dimensional elastic trape-
zoids with different elastic moduli, interacting with each
other through a complex elastic foundation, the adhesive
layer. The exterior boundary conditions (Fig. 7, p. 26)
are that the tOp and bottom surfaces of the system are free
of stress (oy = Txy = 0). In addition, the ends are loaded
by uniform tension (00) or pure bending with linear stress
distribution (-oO-y). The quantities sought are the unknown
adhesive shear and normal stress distributions, which con-
stitute a set of unknown interior boundary conditions.
This, plus the fact that we are dealing with two different
elastic solids of trapezoidal shape, accounts for the pecu-
liar difficulty of obtaining good solutions to this problem.
2.2. The Rayleigh-Ritz Method
Boundary-value problems in the linear theory of elas-
ticity may be solved using the Theorem of Minimum Potential
Energy in conjunction with the variational calculus. This
theorem states that: "Of all continuous (compatible) dis-
placement fields satisfying the given boundary conditions,
the actual, equilibrium state of displacement is such as to
minimize the total potential energy of the system." Thus,
the input to this theorem must be a compatible displacement
30
field, satisfying (as forced conditions) internal continuity
and any prescribed displacements on the boundary. Then the
statement of the theorem itself furnishes the conditions of
equilibrium for the problem to which it is applied. For a
linear problem, it is known that the single stationary value
of total potential energy is a minimum, so that rendering it
stationary is equivalent to minimizing it.
In using this theorem, the total potential energy
is normally expressed in terms of displacements, which must
be differentiated to find the stresses when the latter are
sought. The loss of significance associated with differen-
tiation makes it desirable to work with other minimum prin-
ciples, in most cases where approximate stress solutions
are contemplated. Here, however, the stresses sought are
those in the adhesive, which will soon be expressed directly
as differences in the displacement of the adherends at the
adhesive-adherend interfaces. Since no differentiation is
required in the present case, the use of the Minimum Poten-
tial Energy Theorem appears to be quite appropriate.
The total potential energy 9' of a region R in
plane stress can be expressed as
Q = U + W 2.2.1
I I
where US is the strain energy and W is the potential energy
of the external forces:
31
2
+ __
812.
3x
' E
U =
5 2(1 - v2) JLLR<-
2
11.2321 (22 + 3V) dx dy 2.2.2
+ By 5;
__ __ I
W = j’, (uX + vY) ds
C .
I
Here C represents that portion of the boundary where ex-
ternal forces are specified and displacements are not, and.
s. is an arc variable on the boundary C. of region R. The
notation is otherwise a standard one: u, v are displace-
ments, and Y, Y are the x- and y- traction components.
In terms of the calculus of variations, the equili-
brium content of the Theorem of Minimum Potential Energy
may be expressed as the vanishing of the first variation,
60' = 0. The Rayleigh-Ritz method is commonly used for
finding approximate solutions in such variational problems.
It consists of the following steps. First, select a set
of functions fi(x,y) which satisfy the necessary continuity
conditions and the essential or forced boundary conditions.
This set fi must be "complete" in the mathematical sense.
From the fi’ an "approximating sequence" on is constructed.
Form:
¢1 = fo + c1f1
¢2 = fO + lel + sz2 2.2.4
11
¢n = f0 + .2 cnfn
1-1
32
where f0 is used to satisfy all forced boundary conditions
and the rest of the terms of on can therefore satisfy homo-
geneous conditions. The ci are undetermined parameters.
Next, these functions are inserted into the functional to
be rendered stationary (here, the total potential energy,
9'), and any necessary integration is carried out. Finally,
the functional is minimized with respect to the parameters
ci. Thus the best approximation possible within the family
of ¢n is obtained from the minimizing conditions
—'_= 0 i: 1,2,oooo,n 20205
In a linear problem with a quadratic functional, such as
the present one, the above procedure generates a symmetric
system of n simultaneous linear equations, if n parameters
(ci) are used. An approximate solution for the given prob—
lem is arrived at by substituting the values of the param-
eters thus determined into the assumed function ¢n° The
procedure is essentially the same if (as in the present
case) the functional being minimized depends upon several
functions (four displacement components for two bodies).
The critical question is always one of convergence.
It is necessary to check that the desired quantities ap-
proach a limit as n is increased, and to verify as well as
possible that this limit is theoretically the true solution
for the problem in question. It is also important to check
33
that roundoff errors do not accumulate, in calculations
such as these, which may involve large numbers of equations.
Ideally, the approximating sequence in the Ritz
method consists of functions orthogonal over the region of
interest, to simplify the inevitable integrations. The pre-
sent problem involves two trapezoidal regions, and suitable
orthogonal functions are not easily found. They can be
constructed, but it seems more practical to trade simplicity
of computer programming and more equations against the sub—
stantial difficulties of constructing a set of orthogonal
functions. Accordingly, homogeneous xy-polynomials are
used below.
In assuming a purely polynomial solution, it is
recognized that no account is taken of the possibility of
stress singularities at the four adherend "wedge corners".
adjacent to the adhesive layer. Singular stresses do not
necessarily imply singular displacements, of course. If
the proper displacement variation corresponding to stress
singularities can be introduced as part of the assumed
Ritz function, relatively few equations must be solved.
Unfortunately, the eigenfunction method used by Williams17
does not seem to extend readily to the present case, which
is considerably more complex. Where he dealt with a single
wedge, the present problem involves two adjacent wedges
coupled by an elastic foundation of a complex type. It
appears that the elastic foundation model is too distant
34
an idealization of an elastic solid to permit a treatment
along his lines. It has not been possible thus far to de-
duce the correct stress singularities (if any are present,
which the results show to be likely). As a result, very
large numbers of polynomial terms have had to be introduced,
in order to obtain good approximations for the locally-high
displacement gradients which accompany stress singularities.
Other variational methods have been considered,
principally the method of Kantorovich.3 This does not seem
well adapted to a trapezoidal region, because it appears to
require the solution of a large system of simultaneous,
ordinary differential equations with variable coefficients.
2.2.1 Derivation of Equations
Only the principal features are given here; addi-
tional details appear:h1Appendices A and B. Primes are
used at first to denote dimensional quantities, and are
later removed during the changeover to non-dimensional
variables.
The x- and y- displacements of adherend l are
v
designated u 1 (respectively), and for adherend 2, as
1’
u2, v2 (Fig. 7). When resolved in the n- and s- direc-
tions along.adherend-adhesive interfaces, the displacement
components are referred to as u With the
nl’ usl' un2' usZ’
assumption that the two adhesive strains considered are
uniform across its thickness
35
y = 32 2.2.6a
(u -u )
e = “1 “2 2.2.6b
a 7]
(Huastrainenergy of the adhesive, U , is obtained by con-
sa
sidering an infinitesimal length ds along the inclined ad-
hesive face. The corresponding volume is = (n°ds) l,where
n is the thickness of the adhesive. Allowing for adhesive
transverse normal strain and shear strain, and integrating
along the joint from end to end:
s 2 2
. f 0 Eaea GaYa
sa
Substituting equations 2.2.6a, 2.2.6b for Ya' 8a into 2.2.7,
we obtain
U'—_E_ISOE(u -u)2+G(u - )2d
sa _ 2n -s a n1 n2 a 52 usl S
0 2.2.8
But
un1 - un2 = (vl - v2) cos a - (ul - uz) s1n a 2.2.9a
usZ - uSl = (u2 — ul) cos a + (v2 - V1) s1n a 2.2.9b
Substitution of equations 2.2.9a-b into 2.2.8 results in
36
s
U. = —£- J[ O G (v — V') sin 2a + (u - u )2 cos 2a
2n a 2 1 2 1
. 2 2
+ 2(v2 - v1)(u2 - ul) cos a s1n a] + Ea[(v1 - v2) cos a
2 O 2 O
+ (u1 — uz) s1n a - 2(vl - v2)(ul - u2) s1n a cos d] ds
0 o s 0 2.2.10
Rearrang1ng and subst1tut1ng ds = dy/s1n a, we get
+h
' _ l 2 . 2 2
+ (v - v )2(E cos 2a + G sin 2a) + 2(u - u )(v - v )-
1 2 a a l 2 l 2
cos a sin a(G - E )] dy 2.2.11
a a
The remaining strain energy terms consist of two expressions
of the form 2.2.2, with subscripts appropriate for adherends
1 and 2.
On the exterior boundary, only tractions are speci-
fied: the top and bottom surfaces of the adherends are
stress-free, and the outer ends of the trapezoids are loaded
by either pure tension or pure bending. Thus there are no
forced conditions on displacement, other than the normal
requirement that the rigid displacement of the system be
properly specified. Therefore, using polynomials, the
four unknown displacements are taken in the (dimensional)
form
M M-m . n
m ,
u = A x y 2.2.12a
mgo ngo m,n
37
1E1 M-z'm .
v = B m n
1 m=0 n=0 m’nx y 2.2.12b
M M-m . m n
u2 = E Z Cm,nx y 2.2.12c
m—O n=0
% M-m ' m n
v = D x y 2.2.12a
2 m=0 n=0 m'“
Note that these double sums actually represent the sum of
all polynomials homogeneous in x and y, from a constant to
the highest order M. If the double sum went to M on both
upper limits, a great many additional terms would be in-
cluded. However, it is likely that these would contribute
little to accuracy, and difficulties with Ritz matrix con-
dition could well be anticipated. These displacement
functions are next substituted into the total potential
energy per unit width of joint in the z- direction. See
Appendix A for details; the main item omitted in the deri—
vation to this point is the potential energy of the external
loading.
' E1 3“1 2 av1 1 1
9=‘ 2[[(ﬁ‘ +(sy— +2V1ﬁ—ey—
2
(1 - v ) Bu 3v E Bu
+ 2 l (a l +~——£) ]dx dy + 2 .[j'(——£
. y 3x 2(1 2 3
2 C)
38
1 ”1 2 2 2
+ mlh [“11 ‘ u2’ (Ea 51“ 0‘ + Ga °°° °"
2 2 . 2
+ (v1 v2) (Ea cos a + Ga S1n a)
+ 2(ul - u2)(vl - v2) sin a cos a (Ga - Eaildy
+h +h
+ f Ox(-C:Y)ul(-C:Y)dy 'f 0X(CIY)u2 (CIY)dY
-h -h
c = 2h + h cot a 2.2.13
The double integrals, one for each adherend, are strain
energy expressions of the form 2.2.2; the next integral
represents the adhesive strain energy 2.2.11; and the last
two terms are the potential energy of the only nonvanish-
ing external tractions, cx(ic,y), at the end boundaries of
the joint.
The energy expressions are now converted to a non-
dimensional form. Let
111 V1
'11:?" Vi=h_
x - h , Y h
2.2.14
Bui = BUi . Sui = BUi
ex ex ’ 3y aY
3V = 3V1 . 8Vl = 8V
3x 3X ’ ay ay
I
00(Y) OX(:CIY) . w = (l - vi) -Q—§
1 E h
39
The following dimensionless quantities are defined for com-
pact presentation of the long expressions which result.
They can all be calculated directly from the primary and
secondary parameters governing the physical problem.
32(1 - vi)
C = 2 + cot.a ; Y = 2
El(1 - v2)
(E s1n a + G cos 2a)
H = (l _ v2) a a h
l 1 El n-sin a
2.2.15
(E - G )
_ 2 h a
H2 — (1 v1) E E1 cos a
(E cos 2a + G sin 2a)
H = (1 - V2) a a r}
h 1 El n°s1n a
Substituting these quantities into the various energy ex—
pressions, we define a dimensionless total potential energy:
2
w = 1f 301) + avl + 2V eul evl + (1 - v1) BUl
’2‘ ® ex BY 1 ex BY 2 a
‘ evl)2 Y. auz 2 ev2 2 8U2 av2
+ ‘h‘ex dx C” + 2' Te + W + 2\)2 """ax """3Y
(1 - v2) 3U2 3V2 2 1 +1 2
+ —
2 BY + 52— dx dY + i’j:l H1(Ul U2)
40
+ Hh(vl - v2) - 2H2(Ul — U2)(Vl — v2) dY
2 +1
+ (1 - v1)j:l 00(Y)U1(-C.Y)dY
2 +1
- (l - v1).j:l 00(Y)U2(C,Y)dY 2.2.16
The displacement functions of 2.2.12 a-d, in suitable
dimensionless form, are now taken as
M M-m n
U1 = Z 2 Am nxmy 2.2.l7a
m=0 n=0 '
M M-m n
v1 = 2 2 B me 2.2.l7b
m=0 n=0 m'n
% M-m m n
U = c x Y 2.2.17c
2 m=0 n=0 m,n
M M-m
v2 = Z 2 D xmyn 2.2.l7d
m=0 n=0 m,n
After substitution of these dimensionless displacements
and their derivatives, and the evaluation of all integrals,
the expression for total potential energy reduces to a sys-
tem with 2M(M + 1) - 3 degrees of freedom. These consist
of the 'generalized co-ordinates' (unknown parameters)
Am,n’ Bm,n' Cm,n and Dm,n' The term -3 appears because
plane rigid motion is suppressed by setting certain con-
stants to zero.
The values of Am,n’ Bm,n' Cm,n and Dm,n are deter-
mined at this stage by using the principle of minimum
41
potential energy. The Ritz equations now take the form
2.2.18
These four relations yield four sets of linear equations.
On expansion, this produces as many equations as there are
undetermined coefficients. The detailed derivation of
these equations, which follow, is given in Appendix B:
3w
0:—
aAm,n
MM-k
km k+m-l .
= Z A - _ ¢ - (-C) f(n+j+l))
kgo j=0 k,j[k + m 1 ( 0
(1 - v‘) .
, 1 nj 2 _ _ k+m+l . _
+ 2 k+m+l(¢OCOta (C) f(n+j 1))
Vijm l — V1 kn a
+ H1°1 + Bk,j E‘I‘ﬁ'+ 2* k + m °o °°t
- (-C)k+mf(n + j)
' H2¢1]‘ Ck,jH1¢1 + Dk,jH2¢1
2 +1 m n
+ (l - v1) jf 00(Y)(—C) Y dY 2.2.19
-1
42
3w
aBm,n
bid MIR kn l-Vl mj H
A . v + ¢ .cot a
k=0 j=0 k,j[( l k + m 2 k + m 0
_ k+m . _ ‘ nj 2
( C) f(j + n4 Hzol] + Bk,j[k + m + 1 (oo cot a
(1-V)
k+m+l . 1 km
”('C’ f(n+3'l)+ 2 k+m-l(¢0
k+m-l .
- (-C) f(n + J + 1)) + Hh¢l] + Ck,jH2¢l
- Dk,th¢l 2.2.20
8w
acm,n
M M-k
-A .H +B .H
k=0 jéo k,3 1¢1 k,3 2¢1
km k+m-l .
+ Ck,j[yk + m _ l (C f(n + j + 1) - ¢0)
(l-V) .
+ Y 2 2 k +n% + l Ck+m+lf(n + j - l) - ¢0 cot 2 a)
k+m . m'
+ Hl¢l] + Dk,j[(c f(n+j) - ¢0 cot d)b'v2 E—ﬁjﬁ
+ 1-v2 kn -Hd> —(1—v2)j+10(Y)CmYndY
Y 2 k + m 2 1 1 _1 0
2.2.21
43
0 _ 8w
_ 35...
m,n
M M-k I .- kn
= A .H ¢ - B .H + C . v
1 - v .
2 m3 k+m . _ _
+ 2 k.+'HJ (C f(n + j) 60 cot a) H261]
nj k+m+l . _ _ 2
+ Dk,j[Yk + m + 1 (C f(n + j 1) 60 cot a)
l - v
+ - .
+ y 2 2 k +k$ _ 1 (Ck m lf(n + j - l) - ¢0)
+ Hh¢l] 2 2.2.22
The new symbols are defined below:
f(R) = [l - (-1)R] = % : R odd
R
= 0 R even
¢0 = (cot d)k+m-lf(k + m + n + j)
$1 = (cot a)k+mf(k + m.+ n + j + 1)
It can be verified that these equations comprise a symmet-
ric system, in accordance with the general theory of the
Ritz procedure for quadratic functionals.
In equations 2.2.19 and 2.2.21 the integrals rep—
resent the loading conditions which are to be considered.
In case of purely tensile loading at the ends of the ad-
herends we take
44
00(Y) = 00 = constant 2.2.23
and in pure bending:
with 00 also a constant. In the tensile loading case the
integral of equation 2.2.19 becomes
(1 - vi)00(-C)mf(n + 1) 2.2.25
Similarly, the integral of equation 2.2.21 reduces to
- (1 - vi)oO(C)mf(n + 1) 2.2.26
When there is pure bending, the integral of equation 2.2.19
becomes
- (1 - vi)oO-(-C)mf(n + 2) 2.2.27
and 2.2.21 results in
(1 - vi)oO(C)mf(n + 2) 2.2.28
In the computations o is always taken as unity, which means
0
that the resulting Ritz coefficients must be multiplied by
a factor oxO/El--see last line of equations 2.2.14-—to re-
store true (dimensional) stresses, displacements, etc. The
stress OX0 is the actual uniform tensile stress loading the
adherends, or the largest value of the bending stress load-
ing the adherends.
Of the undetermined displacement parameters Am n’
I
D . . _ . _
Bm,n' Cm,n’ and m,n,' three represent r1g1d body d1splace
ment choices which must be fixed to avoid a singular Ritz
matrix. These arbitrary choices are
3V1(0,0)
= 0
ex 2.2.29
‘Ul(0,0) = 0 ; Vi(0,0) = 0 ;
45
These in turn require that A0 0 = 0, = 0 and B = 0.
I
B0,0 1,0
The corresponding rows and columns of the Ritz
matrix are deleted and the remaining parameters are evaluated
by solving the surviving system of simultaneous equations,
2.2.19-2.2.22. After obtaining the displacement coefficients,
the (dimensional) adhesive normal (on) and shear (Ins)
stresses are calculated using the strains of equations 2.2.6:
(uni - un2)
2.2.30
where (as before) usl' 1182 are the displacements of the two
adherends along the adhesive film at their respective inter-
faces and unl' un2 are the normal displacements at these
interfaces. These in turn come from the Ritz coefficients
via equations 2.2.9, 2.2.14 and 2.2.17. For user convenience,
the actual quantities tabulated later are stresses N and T,
corresponding to on and Ins for unit applied tensile load—
ing, or a bending moment producing an outer-fiber bending
stress of unity. Thus the dimensional form of 2.2.30 be-
come S
In?”
ﬂvi - v2) cos a - (ul - u2) sin a] 2.2.3la
:3
:5
0
_ _2 _ - _
Tns — n [sz v1) s1n a + (u2 ul) cos a] 2.2.3lb
46
and the dimensionless version is
'E h
a. _ _ _ .
N 2ﬁIﬁ [(V1 V2) cos a (U1 U2) s1n a] 2.2.32a
Gah
T EEK (V2 - V1) s1n a + (U2 - U1) cos a 2.2.32b
The factor (Bah/Elm) is a primary dimensionless parameter
of the tabulated results (discussed later), and the Ui’ Vi
are found from the Ritz coefficients using 2.2.17. Expli-
citly, the adhesive normal and shear stresses at points
(X.,Y.) along the inclined adhesive face are calculated
3 J
from the relations
Eah M M-m n
N = E_ﬁ (Bm n--Dm n) cosa.— (Am n- Cm n) sina]XTY.
J 1 m=0 n=0 ’ ’ ’ ’ J 2
2.2.33a
Ga Eah If MEIR [
T. = —— ——— (D - B ) sin a + (C
j Ea Eln m=0 n=0 m,n m,n mn
— A ) cos a XTYS 2.2.33b
m,n J J
On the adhesive line, of course, Xj = Yj tan a: only one
variable is independent.
2.3. The Sherman-Lauricella Integral
Equation Approach
2.3.1 General
Integral equations are used quite effectively to
formulate many engineering problems. This method of attacking
47
the fundamental boundary-value problems of plane elasticity
appears in several forms in Muskhelishvili4. The version
used here was initially devised by G. Lauricella5 and ex-
tended by D. I. Sherman.4 Its power appears in the lack of
restrictions on the unknown "weight" function, for which
only modest continuity requirements are specified. A
similar, real-variable, vector-integral-equation formula-
tion has been discussed by Massonet.6 A difficulty of the
present situation is that the integral equation in question
is complex, so that it represents two real, coupled,
Fredholm-type equations. Furthermore, this set must be
solved simultaneously in each of two regions having dif-
ferent elastic properties. Fortunately, the solution for
one region can be made to depend upon the solution for the
other. However, once solved, a lengthy numerical integra—
tion for displacements must be carried out to complete the
solution. In the present case, the results for both regions
must be maintained in computer storage at the same time.
The problem thus becomes one of computer capacity, and it
has been found necessary to relegate this elegant approach
to the role of an independent check on the Ritz procedure
used for most of the calculations.
Generally speaking, it is out of the question to
use analytical methods to solve a linear integral equation.
It is usually possible to obtain good answers by solving
a large number of simultaneous linear algebraic equations.
48
The procedure is to write the equation at a set of nodal
points, suitably spaced along the boundary, carrying out
the integrations numerically.
2.3.2 Notation for Integral Equation
Method
The symbols used in the integral equation method
development are defined in the text where they first ap-
pear. For convenience, those symbols used exclusively for
the integral equation method are listed here in alphabeti—
cal order, with English letters preceding Greek letters.
The symbols having the same meaning in both the present
method and the Ritz method are listed in section 1.4. As
before, subscripts 1 and 2 normally distinguish quantities
defined for adherends l and 2. Bars over symbols have the
usual "complex conjugate" significance in this section, and
in associated appendices.
am, bm = Coefficients of dimensionless, self equili-
brated normal and shear stresses on adhesive
interface.
C2, D2 = Rigid-body translation constants, adherend 2.
f(t) fl(t) + if2(t).
f1(t),f2(t) Known real functions which depend upon the
prescribed external loading.
I = Total number of intervals on the boundary
of each adherend.
49
l' 2' 3’
I4 = Number of intervals on the various boundaries
of each adherend.
K = I1 = number of intervals on inclined adhe-
sive face.
p(t), q(t) = Real and imaginary part of w(t).
Uil)' Vil)’
Uél), Vél) = Nondimensional displacement components due
to unit applied tensile load parallel to X,
in X- and Y-directions,for adherends l and
2 [part (1) contribution].
UiZ)’ V12)'
Uéz), Véz) = Same as item above, but part (2) contribu-
tion associated with self-equilibrated ad-
hesive stress system acting on adhesive
interface.
Xn, Yn = Given tractions on adherend l boundary, in
the X- and Y- directions.
s,t = Values of the complex variable X + iY on
adherend boundary.
S = Distance along the adhesive interface.
2 = X + iY = complex variable.
ul, “2 = Shear moduli of adherends.
On'cn,j = Dimensionless adhesive normal stress, and
same when evaluated at points Xj' Yj.
50
Ins, Tns,j = Dimensionless adhesive shear stress, and
same when evaluated at Xj’ Yj.
6(2), w(z) = Analytic functions entering governing in-
tegral equation.
¢(t), w(t) = Boundary value of functions 6(2), 0(2).
x1 = (3 - v1)/(1 + v1) for plane stress.
w(t) = p(t)-+iq(t) = "density function" in defini-
tion of ¢(z).
w2 = Rigid-body rotation constant of adherend 2.
2.3.3 The Problem Analyzed by
Integral Equations
The adhesive scarf joint problem solved here is the
tensile loading case described in Section 2.2. All ex-
pressions are in the non-dimensional form ultimately used
there. The original boundary-value problem is decomposed
into two parts, (1) and (2). The first part consists solely
of the elementary solution for uniform tension parallel to
X in each member, due to a unit applied tensile stress, and
the uniform shear and normal stress on the adhesive boundary
required to equilibrate the applied stress. Part (2) is
then the wholly self-equilibrated residual problem for the
"difference" tractions on the adhesive-adherend interfaces,
now the only loaded boundary in each adherend. The adhesive
normal and shear stresses are still unknown at this stage,
and are taken to be polynomials in S, the distance along
51
the interface, with coefficients to be determined ultimately
by the matching of two different expressions for the adhe-
sive stresses at a finite set of points along the adhesive-
adherend interface.
In terms of dimensionless quantities, the self-
equilibrated adhesive normal and shear stresses [part (2)
tractions] along the adhesive inclined face are assumed in
the form
K
0(2) = Z a Sm 2.3.1a
n m
m=l
K
1:2) — 2 bmSm 2.3.lb
m=l
with undetermined coefficients am, bm(m = 1,2,...K). Here,
S = Y csc a is the dimensionless distance along the inclined
adhesive-adherend boundary, measured from the origin of co-
ordinates. The dimensions of (am, bm) are "self-adjusting"
as used here, and need not be specified. The integer K is
chosen to be odd, as explained later. It would be simple
to introduce the appropriate wedge-corner singularities as
functions of S at this stage, if these could be determined.
Since this part of the solution is self-equilibrated
for force and moment, we constrain the unknown coefficients
accordingly. Applying the three static equilibrium condi-
tions for adherend l, the equations EFX = 0, 2FY = 0 and
ZMxy = 0 are used to el1m1nate the coeff1C1ents aK-l’ aK
52
and b These are expressed in terms of the remaining
K-l'
total of (2K-3) unknown coefficients (Appendix C), so that
O(2) _ (K‘%)/2 a SZm-l _ (K + 2) (csc a)2m K-lSK]
n _ m-l 2 3 Zm-l (2m + I)
_ I I
(K-3)/2 2m K(csc a)2m-K+l K-l
+ Z a s - —— s 2.3.2a
m=l,2,3 2m 2m + l
(2) _ (K+l)/2 2m-l (K-3)/2 2m
Tns - Z me-lS + Z b2m S
m=l,2,3 m=l,2,3
K 2m-K+1 K-l
_ ﬂm+ (CSC a) S 2.3.21)
The boundary conditions of the first fundamental problem
(all-traction case), in terms of unknown analytical func-
tions of a complex variable, ¢(z) and w(z), is of the fol-
lowing form4
¢(t) + t ¢'(t) + w(t) = f(t) 2.3.3
where
¢(t), w(t) = boundary values of functions
¢(z), 0(2)
f(t) = fl(t) + if2(t)
= i‘[(xn + iYn)ds 2.3.4
X ,Y = given tractions on the boundary in
the X- and Y— directions
53
The functions fl(t) and f2(t) are known real func-
tions, which depend in a simple way upon the prescribed
external loading. Let w(t) be an unknown density function
("weight function") for points on the boundary. It is
assumed that w(t) has a derivative w'(t), which satisfies
a Halder's condition. The latter guarantees the continuity
of the functions ¢(z), ¢'(z) and w(z) up to the boundary.
The boundary condition 2.3.3 thus constrains the choice of
w(t); this constraint is the governing Sherman-Lauricella
integral equation of the problem.- Following Sherman,4 let
_ 1 w(s)
49(2) - mfmds 2.3.5a
W2) =fg_(s m-_J.-_..IM d3 2,3,5};
5 - z 2H1 s - 2
From equation 2.3.5a
A.
I
' l w(s)
o (z) = ——+jr——————— ds 2.3.6
n1 (S _ 2)
After using the Plemelj formulae for the boundary values
1
of Cauchy integrals, and an integration by parts in 2.3.6
I
for o (2), equation 2.3.3 becomes
w(t) + y-Tlr-{fmsm 10232:; — grifﬁn g. I §= f(t)
2.3.7
This may be converted to two real equations by letting
s - t = rele 2.3.8
ls-tl
H
II
54
Here 8 is the angle between the vector 5 — t and the x-axis,
measured in the positive (ccw) direction. By equation
2.3.8:
5 - t .
s - E
i ' E = cos 28 + i sin 28 2.3.9b
s - t
Equation 2.3.5a becomes
w(t) + % j2w(s) - e2165(s) d8 = fl(t) + if2(t) 2.3.10
Further, writing
w(t) = p(t) + iq(t) 2.3.11
and separating real and imaginary parts, equation 2.3.10 may
be represented in the form of two real, coupled integral
equations:
p(t) + %:[Ip(s)(l - cos 26) - q(s) sin 28]d6 fl(t)
2.3.12
q(t) - %;[[p(s) sin 28 - q(s)(l + cos 26)]d6 f2(t)
2.3.13
Equations 2.3.12 and 2.3.13 are quite simple and readily
permit numerical solution. They were derived under the
assumption of a continuously-turning tangent for the boundary
contour. Muskheliskvili remarks that corners can be included
if the contour integrations are interpreted as Stieltjes in-
tegrals.
55
The present contours have at least two critical cor—
ners per trapezoidal adherend, those along the inclined face.
In failing to interpret the integration in the Stieltjes
sense, the trapezoids are implicitly supplied with corner
radii which are, roughly speaking, comparable to the inter-
val size chosen for numerical integration. Omitting-the
Stieltjes interpretation (as in the numerical work here)
is probably equivalent to ignoring the singularities of the
problem, which is essentially what is done in choosing Ritz
trial functions which are exclusively XY—polynomials.
To perform the numerical integration, the bounda-
ries CD, DA, AB and BC of adherend 1 of the scarf joint
(Fig. 7, p. 26) are divided into 11’ I2, I3 and I4 inter-
vals, respectively. Care is taken to make I1 = K, the
number of points at which adhesive stress expression match-
ing will later take place. Before further discussion of
the numerical approach to these equations, the functions
fl(t) and f2(t) are evaluated. From equation 2.3.4
fl(t) = - Ynds ; f2(t) = [knds 2.3.14
x = _ <2) . _ (2)
where ‘n on Sln a Tns cos a 2.3.15a
Y = 0(2) cos a - T(2) sin a 2.3.15b
n n ns
Substituting equations 2.3.2a and 2.3.2b for 032) and Téi)
into 2.3.15a and 2.3.15b we obtain
56
(K-1)/2 2m K+1J
_ _ S _ (K.+ 2) 2m-K-l S
f1(t) - COS a m_122 3 a2m_l [Zn— 751T].— (CSC a) m
— I I
+ (K-§)/2 a2m[§:m;+Il - 2m 5 I (csc a)2m-K+l SE]
m=1,2,3 _
(K+l)/2 S2m (K-3)/2 [:52m+l
+ sin a b §—— + b .—7——I
m=lI2I3 2m_1 m m=l,2,3 2m M
K 2m-K+l SK
_ ____1_2m + (csc a) 7 2.3.16
(K-1V2 52m (K + 2) 2m-k—l sK+l
f (t) = - sin a a - (csc a)
2 m-l 2 3 2m-l 2m TEII_I— —K:I
" r I
+ (K-3V2 a [Szm+i _ K (csc a)2m _ K + 1 £5]
m=l,2,3 2m 2m + I 2m‘+ I K
(K+nyz S2m (K-mvz [Szm+1
- cos a -—- + b §——*‘I
m=l,2,3 2m-1 2“ m=l,2,3 2m m +
K 2 K+l SK
_. ..__._..._2m + l (csc a) m. R— , 2.3.16
Since the superscript (2) stresses were subjected to the re-
quirements of overall equilibrium, fl and f2 must be con-
tinuous and it is possible to verify this directly from the
foregoing expressions. The numerical integration is per-
formed on the assumption that p and q vary negligibly in
the intervals into which the whole boundary is divided. The
p and q terms are extracted and the remaining integrals can
be evaluated analytically. Thus, carrying out the integra-
tions described, the two equations 2.3.12 and 2.3.13 are
57
rewritten in the following form [use pj E p(tj), etc.]:
I sin 26 . - sin 26 .
p. + i Z P 9 - - 9 - - +Kl* -kl)
j n _ k +kj -kj 2
k—l
(cos 26 . - cos 26 .)
+kJ -.12 _
+ qk 2 ]— fij 2.3.18
I (cos 26 . - cos 26 .)
q- + i X p +k3 ‘k3 + q a . - e .
3 ﬂ k=1 k 2 k +kj -kj
sin 26 . - sin 26 . '
+ +k3 2 ’k3 ] = fzj 2.3.19
where I = Il + 12 + I3 + I4 = total number of intervals
along the boundary,
i
eikj = arg (sk — t )
S: _ + Hk
k ‘ Sk ‘ 7T
Hk = length of the kth interval
flj' f23' = values of the function fl(t),
f2(t) at the jth node (center of
jth interval), t = t..
3
Upon completion of the numerical integration of
equations 2.3.18 and 2.3.19, we have a numerical matrix
relating the (pj,qj)(j = l,2,3,...,I) to the still unknown
adhesive stress coefficients (ak, bm)(k = l,2,3,...,K - 2;
m = 1,2,3,..., K - 3, K - 2, K).
58
Having established this relation, the expression
for the non-dimensional components of displacement Uiz)
(2)
l
and V must now be developed.4 In terms of the functions
¢ and 6 defined previously:
(2)
l + iVi2)) = xl¢(z) — z¢'(z) - w(z) 2.3.20
2u1(U
where
Shear modulus of adherend 1.
X1 = (3 - vl)/(l + v1) for plane stress.
C
II
1 Poisson's ratio for adherend 1.
This equation can be reduced to a contour integration around
the boundary of the adherend (Appendix D):
X
2“1(Ui2) + Win) = RH [pm _ p(t)] + i[q(s) - q(t)] gig—E
+ 3%Ij'[p(s) - p(t)] + i[q(s) - q(t)] 595 E
+ %—f[p(s) + iq(s)] 9919-43—5 +txl[p(t) + iq1
r
+% [p(t) + iq(t)] - fl(t) - if2(t) 2.3.21
where
Uiz), Viz) = displacement components of adherend l.
a a(SIt) is the angle between the vector 3 - t and
the outward normal at s (unrelated to the scarf
angle a used elsewhere).
r = Is - tl 2.3.22
59
The adjustment of the rigid—body displacement constants is
made later.
Equation 2.3.21 is integrated numerically using the
trapezoidal formula, with special treatment required at
various points, such as corners (Appendix D). The integra-
tion is carried out only for the displacements Uij)' Vii),
i.e., the X, Y- displacement components of adherend l at
h
the jt boundary node point (midpoint of jth interval), of
points (Xj, Yj) on the inclined adherend face. This es—
tablishes expressions for Uii) and Vii) in terms of quanti—
ties (pk,qk) at all the node points on the boundary, which
in turn still depend upon the unknowns (am, bm). Both
relations take the form of known numerical matrices. Note
that the only elastic constant affecting the right side of
2.3.21 is x1. Thus if both adherends have the same Poisson's
ratio, the right side of 2.3.21 serves for both. This assump-
tion is made in the calculations.
Next, using the numerical coefficient matrix which
relates (pj, qj) to the (ak, bk), the final displacements
of the adherend 1 can be expressed as a single, known numeri-
cal matrix multiplying a still unknown column vector of the
(ak, bk). Careful consideration of the geometry of the
second adherend with respect to the first adherend permits
us to use the results for the first adherend to write the
(2)
2
corresponding U and Véz) displacement expressions at an
equal number (I1 = K) of points on the inclined face.
60
The principal adjustment required, if we assume the same
Poisson's ratio on both sides, is for the different shear,
modulus. This affects only the left side of 2.3.21. Some
sign changes and "mirror" reflections of coordinates in
the origin are also required.
The displacement expressions corresponding to the
uniform unit tensile fields in each member [part (1) solu-
tions] are now superposed on those due to the self-equili-
brated distributions (superscript 2), to obtain the final
displacement expressions for the two adherends.
Rigid-body displacement constants must now be
established. This is done by arbitrarily suppressing all
translation and rotation at the origin, in the first
adherend. The first adherend's final displacement com-
ponents at the desired points (xj, Yj) along the adhesive
inclined face are thus taken as (Appendix E):
_ 0‘2.) 1 _ (2)
_ V(2) V1 _ (2)
where the terms involving Xj and -V1Yj represent the total
contribution of the uniform tensile field here. For the
second adherend, the solution itself must determine the
rigid-displacement constants (c2, D wz below)--see Appen-
2’
dix E:
U<2> 1 _
U2j= (Xj ,jY ) + g Xj+ C2 wZYj 2.3.243.
61
\)
V”) _ .3. x
V2j= J(Xj Yj ) E2 Yj+ D2 + wz j 2.3.24b
where Uéi), V23) are the displacements of adherends 2 at
node point j.
Because of the presence of the three rigid-body con-
stants C2, D2 and wz, the number of unknowns has increased
All
from the (2K - 3) quantities (a bk) to 2K (K = I
k’ l)“
unknowns are now evaluated by equating two different ex-
pressions for the adhesive normal and shear stress. The
total stresses are the sum of the part (1) and part (2)
contributions. Recalling that part (1) consists of a uni-
form unit tensile field, and using 2.3.1,
K
0 . = 0(12 + 0(2i = sin2 a + Z a S? 2.3.25a
n,j n,j n,j m=l m j
K
T . = 1(1) + T(2) = sin 6 cos a + 2 b S? 2.3.25b
ns,j Tnsrj TnSIj m=l m J
The sinusoidal terms are the adhesive stresses required to
equilibrate a unit tension parallel to X, and subscript j
indicates that the stresses are calculated at S = Sj' For
the part (2) summations, the equations actually used are
those of 2.3.2, which show clearly that only 2K - 3 unknowns
appear, not the 2K values (am, bm) implied above. The ver—
sion presented above is more compact and clearer, and con-
ceptually equivalent as long as it is understood that aK—l'
aK and bK-l are linearly related to the rest of the (am,
bm). The unknowns C2, D2, “2 do not appear explicitly in
2.3.25.
62
The second expression for the stresses is developed
from equations 2.2.32, pn 46 rewritten to conform to the de-
mands of the present approach. These equations give the
adhesive stresses in terms of the relative displacements of
the adherends at the adhesive line. The subscript j implies
evaluation at S = Sj' or Xj = Y. tan a:
E h
_ a _ . — —
On,j - _ﬁ_ [Ul,j U2j] Sln a [Vlj sz] cos a
.3.26a
Ga Eah
Tns,j = E;._ﬁ—- [U2j - Ulj] cos a + [sz - Vlj] Sin a
2.3.26b
(The denominator factor of E has been removed from 2.2.32
1
because the U's and V's as defined in this section contain
the adherend moduli already.) Equations 2.3.24 show that
2' D2 “’2
as well as the 2K - 3 unknowns (am, bm) implied in 2.3.23
2.3.26 contain the unknown constants C explicitly,
- and 2.3.24. The equating of the stress expressions of
2.3.25 and 2.3.26 is therefore sufficient to determine the
2K unknowns, and hence the adhesive stresses.
2.4. Numerical Data Assumed in the
Calculations
There are a large number of dimensionless parameters
to investigate, so that it becomes necessary to divide them
into primary and secondary parameters. The latter are taken
as constant throughout the calculations. The primary
63
parameters are the scarf angle (a); the relative stiffness
of the adherends (y); and the relative stiffness of adhe-
sive and adherends (B). Treated as secondary parameters are
the Poisson's ratios of the adhesive and of the adherends.
Even so, a great many cases must be studied in order to es-
tablish the overall behavior of the dissimilar-adherend
scarf joint.
Generally speaking, the parameters are chosen so
as to simplify interpolation in the results, either on a
linear scale or as equally-spaced values on a logarithmic
scale. First of all, reflecting the practical range, the
scarf angles are chosen as 5°, 10°, 20°, 30° and 40°. The
first and last values are probably outside the usual range,
but are explored for completeness and to facilitate
interpolation.
32(1 - vi)
The primary dimensionless parameter y =
El(l - vi)
is a measure of the relative stiffness of the adherends.
The values selected are l, 2, 4 and 8; values much larger
than 8 are probably quite close to the case of one "rigid"
adherend. Adherend 2 is thus always the stiffer of the
two, except that when Y = l, we have a scarf joint with
identical adherends. The successive factors of 2 permit
interpolation with respect to this parameter at uniform
intervals on a logarithmic scale. We have taken the ad—
herends' Poisson's ratios to be 01 = 02 = 0.3 throughout
64
the major part of the work. This value is intermediate
between typical steel and aluminum values.
The next primary dimensionless parameter is the
"elastothickness" parameter 8 = nEl/Eah(l - vi), which is
a measure of relative adherend and adhesive stiffness. As
used here, large 8 implies a relatively flexible adhesive.
Assuming metal adherends and typical adhesive thickness
and moduli for the metal-bonding range, the values for B
are set so as to permit interpolation on a logarithmic
scale: 8 = 4, 20 and 100. The dimensionless ratio
(Ea/Ga) of the adhesive's Young's modulus to the shear
modulus has been fixed at 8/3 in the present computations.
Referring to Fig. 7, the value of L1 is assumed to be 2h= 2
so that the uniform portion of each adherend has the same
length as its depth. In auxiliary calculations, this
length appeared to give the best overall check of the in-
put traction boundary conditions with the polynomial Ritz
functions used. It is possible, however, that another
choice might be better by some other criterion. This check
is described later.
When using the Ritz method, the Ritz matrix is in-
dependent of the type of loading used, so that it is ex-
pedient to introduce all types of loading considered at
the same time. Results are presented here for both pure
tension and pure bending.
65
2.5. Computer Programs
The principal numerical results for this thesis were
obtained using the Ritz method of analysis, by means of a
computer program developed for the Control Data Corporation
3600 computer at Michigan State University. The program was
written in the 3600 Fortran source language. Various auxi-
liary programs were used to perform checks on the results,
described later.
A separate program was developed for the integral
equation approach. Both programs allow for full variation
of the dimensionless parameters of the problem, including
the effects of geometry, material properties and external
loading. Upon providing the few input data cards required,
the programs calculate and print the adhesive shear and
normal stresses at uniformly spaced points along the ad-
hesive's inclined face. In the case of the Ritz program,
the Ritz parameters are also punched on cards so that any
desired additional calculations can be performed later.
Stress distributions on the external boundaries of each
adherend are calculated from the approximate Ritz displace-
ment solution, in order to check against the known input
boundary conditions.
The format of the input parameters to be supplied
by the user of the program (and copies of the programs)
appear in Appendix F.
CHAPTER III
CRITERIA FOR ACCEPTABILITY OF RESULTS;
PARTIAL DISCUSSION OF RESULTS
3.1. General
In using the Rayleigh-Ritz procedure to arrive at
a solution for this problem, the validity of the results
always depends upon the convergence of the solution to the
correct limit. The few available results of the integral
equation method assume only a supporting role here: so
that Ritz solutions are the main ones to be checked for
acceptability. In this connection, the first and foremost
problem is to assess the convergence of the solution.
Since the accuracy acceptable for engineering pur-
poses varies with the demands of the particular problem,
it has been considered sufficient to state the indices of
accuracy used, and how the results behave in each case. It
is left to the reader to decide if this is sufficient for
his purposes.
It is worthwhile to note that in every case, the
analytical solution obtained by the Ritz method for the
case of identical adherends in tension verifies to high
accuracy the exact solution of Ref. 7. The identical-adherend
66
67
bending solution reported here is new; it is readily shown
that the method of Ref. 7 cannot be extended to include the
case of the bending of identical adherends.
3.1.1 Stress Boundary Condition Check
For each trapezoidal adherend's external boundary,
the input traction conditions are known. The top and bot-
tom surfaces of the system are free of stress, and the
ends are loaded by uniform tension, or pure bending with
linear stress distribution. Therefore, the ability of the
approximate solution to reproduce these is a powerful pri—
mary check. This check has been performed for all Ritz
solution cases investigated.
In this regard, a more or less intermediate case
(a = 10°, 8 = 20, y = 4) is surveyed next for both tensile
load (Fig. 8) and bending load (Fig. 9). These figures
show the boundary traction error on the exterior boundaries,
in the form [(calculated stress, from solution) - (true
stress, from boundary conditions)]. The reference level
is unity, which is either the value of the uniform tensile
load or the maximum value of the applied bending stress.
All of the calculated results show these general patterns
of boundary-traction error distribution. From this it is
possible to select C, and E or F as the critical points,
respectively, in adherends l and 2. All other errors are
either smaller, or much smaller. Point C, in particular,
68
No .1.
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a .0...
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H. H .0. A
x. I//
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wo 366% we wxp /
w \ .2... m. .. ....
Q \ ‘/ ql‘ a J!
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w .m U m
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69
.e u > .om u m .oee u a
UmoH mcﬂocmn .muonuw mmmnum mnmccsom m .mﬂm
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70
invariably shows the largest error in the case of tensile
loading of the joint. For bending load, however, the larg—
est error occurs at point E for the smaller scarf angles
and at point F for the larger angles (invariably, the values
at E and F are comparable in level). The peaking of the
boundary stress errors at the points in question is quite
clearly linked to the neglect of singularities in the Ritz
trial functions.
The value of the boundary stress errors may be taken
as one measure of the merit of the results. In the parti-
cular problem of Fig. 8, the point -C error values for
tensile loading are -0.029 for shear and -0.005 for normal
stress. For Fig. 9--bending load-—the corresponding point
-E quantities are -0.063 for shear and —0.009 for normal
stress.
It is not immediately possible to carry out a com-
parison of the type just discussed on the adhesive-adherend
interface, since we have no "true adhesive stresses" to
serve as a reference level. The purpose of the thesis is
to find these unknown stresses. Something equivalent has
been devised, however, and is discussed in the next section.
In interpreting the present "index of merit" of the
calculations, the following should be borne in mind. The
desired results in this problem are the adhesive stresses
on the inclined boundary. These are calculated using equa-
tions 2.2.32 and 2.2.33, i.e., directly from member
71
displacements. The latter are the direct output of the Ritz
method, in the form of the coefficients A B , etc. On
m,n’ m,n
the other hand, the boundary-stress distributions are much
more sensitive to error than the displacements, because these
stresses are found using derivatives of displacements. In
other words, we anticipate that the Ritz method will produce
displacements which are an order of magnitude more accurate
than the corresponding stress calculations. This makes the
boundary stress an overly-sensitive index, sometimes alarm-
ingly so. The convergence of the adhesive stresses them-
selves nevertheless appears to be quite good in most cases,
as will be seen later.
Speaking generally, then, if the user is satisfied
that the boundary stress error is small enough, he can surely
be satisfied that the corresponding adhesive stresses are
considerably better determined. And a boundary condition
error of 20% (0.2 on an applied load scale of unity) may
still mean that the corresponding adhesive stresses have
been determined to within a few percent. With this as a
background we examine Table 1.
Table 1 (next page) gives the largest errors in the
boundary stresses for those bending load cases having scarf
anglescx= 20°, 30° and 40°. These stresses have been cal—
culated at 21 equally-spaced points on each of the three
external boundaries of each adherend. To emphasize the
highly local character of large peak errors, when these
72
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73
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74
occur, the numbers in parentheses give the value of the same
stress error at the point next to the peak. When the peak
is large, the gradient is always very steep. Errors in
boundary stresses are entered only when they exceed 0.10
(where 1.0 is the reference level for applied external
stress).
In all tensile load cases these errors are smaller
than 0.06, and for all bending cases with a = 5° or 10°,
they are less than 0.10. Hence they are not represented
in Table l at all.
The largest errors in the boundary stresses are
usually observed to be in the shear stresses on side GF
of adherend 2 (Fig. 9). For scarf angle a = 20°, the larg-
est error is 0.368 when 8 = 100, y = 8. The largest error
for a = 30° is found to be 0.526 when 8 = 20, y = 8. When
a = 40°, the largest value is 0.573 fore = 4 and y = 8.
These values are observed at the lower tip of the second ad-
herend, point F in Fig. 9. Note from Table 1 that the 0.368
error value for a = 20°, 8 = 100, y = 8 falls to 0.102 in
5% of the distance along GF from G--see value in parentheses.
Likewise, the 0.526 local peak for a = 30°, 8 = 20, Y = 8
drops to 0.151 in the same distance, and the a = 40°,
8 = 4, Y = 8 value of 0.573 falls to 0.164. It is to be
anticipated that these local stress errors are associated
with much smaller errors in the displacements of the region
in question. This, of course, will be revealed by the study
75
of the convergence of the adhesive stresses themselves. It
seems possible to show by direct computation that the stress
error functions, separately, contribute displacement errors
which are quite small compared to the primary Ritz displace-
ments used to calculate the adhesive stresses. In effect,
this would establish the loose "order of magnitude" by
which the displacements are more accurately determined than
the stresses. However, this hardly seems to be worthwhile,
since the study of adhesive stress convergence effectively
does the same thing directly for the desired end product.
3.1.2 Comparison of Adhesive Stresses
Calculated Several Ways From the
Results
The adhesive shear and normal stresses reported
here are calculated from equations 2.2.32, i.e., by using
displacement differences. Displacement derivatives have
been used only to investigate the boundary stresses. It
was pointed out in the previous section that the error in
calculating adhesive stresses using derivatives is expected
to be much greater than when using differences in displace—
ments, since the Ritz method (as employed here) produces
accurate displacements, but less accurate strains and
stresses. We now suppose that the adhesive stress distri—
butionsfound using Eqs. 2.2.32 are "exact," and compare
them to the same stresses calculated from the displacement
derivatives of adherends 1 and 2. We would expect that the
76
difference between the adhesive stress distributions, as
calculated the "exact" and the two "approximate" ways,
would be comparable to the boundary stress error. This
seems to be the case.
Table 2 (next page) shows the adhesive normal and
shear distribution (for a = 10°, 8 = 20, y = 4), calculated
three ways: using the displacement derivatives of adherend
l; of adherend 2;and from the differences of the adherend
displacements (eq. 2.2.32). The quantity S is the fraction
of the joint half-length, measured along the adhesive-
adherend interface from the origin (at the center). The
value S = -1.0 corresponds to points (C,F) in Figs. 8-9,
and S = 1.0 to (D,E). Table 2 gives results for tensile
and for bending load. It can be deduced from Table 2, by
subtraction, that the characteristic differences between
the more accurate and two less accurate methods are of the
same order of magnitude as the errors in the satisfaction
of the stress boundary conditions. The largest differences
occur at the ends of the adhesive joint, corresponding to
corners of the adherend trapezoids. These are also the
regions of discrepancy on the external boundaries. Such
discrepancies are inherent in any approach which fails to
account for the stress singularities associated with the
acute and obtuse wedge corners of the adherend trapezoids.
77
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79
3.1.3 Double-Precision Check of
Roundoff Errors
In order to check the roundoff error involved in
solving the present very large system of simultaneous equa-
tions, a few computer runs have been carried out using double
precision arithmetic on systems of 141 equations. This cor-
responds to a sum of homogeneous polynomials, for each of
the four displacements, up to and including all 7th-order
terms. In all cases checked (a = 10° and 30°, 8 = 4, Y = 2),
it was found that the numerically larger values of adhesive
normal and shear stress are affected by roundoff error only
in the sixth and seventh significant figures.
Consider, for example, the tension-loaded case of
30°: 8:4! Y
a 2. The largest shear stress value at
S = -1.0 (point C or F of Fig. 7) changes from 0.4723434 (sin-
gle precision) to 0.4723424 (double precision). The normal
stresses are somewhat smaller than the shear stresses for
the angle a = 30°, but the roundoff contribution must still
be comparable to the values found for the shear stress level.
This is because the same imperfectly-determined Ritz coeffic-
ients are involved in all computations. At S = -1.0, the
normal stress changes from 0.2687167 (single precision) to
0.2687145 (double precision). Here also it is found that
the difference is in the sixth or seventh significant figures.
The final results presented here are for 8th-order
polynomials (177 equations). It is estimated that roundoff
80
error might affect the numerically larger values in the
fourth to the sixth significant figure,depending upon the
parameters a, B and Y. A precise estimate of roundoff
error for 8th
-order polynomials is not easily obtained,
because the required memory capacity for the double pre-
cision version of the program then exceeds the available
high—speed memory of the Control Data Corporation 3600
Computer. However, it does not really appear to be neces-
sary to make this check, since in most cases the 7th- and
8th-order (or 6th- and 8th-order) polynomial solutions
agree to sufficient significant figures for the latter to
be regarded as satisfactory.
Another measure of roundoff is obtainable from a
study of a few cases where an exact solution is available,
or where considerations of symmetry demand an odd function.
Any case involving tensile loading of identical adherends
1) should show adhesive stresses uniform along the
(a
joint, and independent of the parameter 8 (this is discussed
further, later). From such cases, and from those bending
problems where a = 1, it is possible to estimate that
roundoff error accumulations for 8th-order polynomial Ritz
functions (177 equations) consistently affect a few units
in the fifth decimal place, ranging occasionally up to 1
unit in the fourth place. This roundoff contribution is
independent of the absolute size of the particular stress
tabulated. However, it represents a small error in the
81
significant (i.e., numerically larger) values of any of the
tabulated stresses. It is quite clear that imperfect con-
vergence is a far more important source of error than round—
off accumulation. Roundoff error may make it impossible to
go to 9th-order polynomials, however, unless double preci-
sion arithmetic is used, or some effort is made to "purify"
the inverse of the Ritz matrix.
3.1.4 Convergence of the Approximating
Sequence
Convergence is studied primarily in the quantities
wanted as an end result, the adhesive stresses. From the
discussion of the preceding sections, this also amounts to
an examination of the convergence of the adherend displace-
ments. To do this, a number of cases are studied in which
the adhesive normal and shear stresses are obtained by
successively assuming displacement functions consisting of
the sum of all homogeneous polynomials through the 6th, 7th
and 8th degree. Considering that four displacements are
involved and subtracting the three rigid-body constants,
this amounts to solving 109, 141 and 177 simultaneous equa-
tions, respectively. Typical cases examined includef3= 20
(an intermediate level of flexibility), Y = 4 (a 4:1, or
substantial level of adherend dissimilarity), for a = 5°,
10°, and 303 in both tensile and bending load. The tensile
loading level here is a unit stress parallel to X, and the
82
largest bending stress is also unity (Fig. 7). For tensile
load, and in most cases for bending load, it is found that
the adhesive stresses appear to approach a definite limit.
h—, 7th- and 8th-degree polynomials, the
In using 6t
larger magnitudes of stress are often so close to each other
that it is usually not practicable to represent them in
terms of tables.. Smaller values of stress, of course, are
not determined as well as large ones, but they are of less
significance precisely because they are small. Tables 3—8
will be used as the framework of this portion of the con-
vergence discussion. Most of the final tabulated results
have been treated along the lines of the three samples to
be discussed exhaustively in the rest of this section, but
in most cases only 6th- order and 8th-order polynomial re-
sults have been compared (not the full 6-7-8 sequences as
in what follows). In a few cases, only 8th-order results
are available. Thus the user usually has one or more in-
dices from which to judge for himself whether he considers
the adjacent Ritz solutions to be close enough for the re-
sults to be meaningful in his application.
Consider, as a successful example, the tensile load
case for the angle a = 30° (upper end of the practical angle
range), in Table 3 (next page). The largest shear stress
level is of order 0.45, and all orders of polynomial con-
sidered give the same stresses to three or more significant
figures, usually to better than 0.01%. There appear to be
83
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85
"random" differences in the fourth and fifth significant
figures. At some points in the table the stress level goes
down slightly with increasing polynomial order; at others,
up slightly; and in still others changes very little. For
example, at s = -l.0, the value goes from 0.45137 (for 6th-
order) to 0.45133 (7th-order) to 0.45131 (8th-order). At
S = 0.9, the trend is reversed, and the respective (6,7,8)
figures are 0.41696, 0.41700, 0.41702. At S = -0.6, there
are no changes at all in any of the figures given. These
changes can be shown to be systematic rather than random.
h-order
The difference between the 7th- order and the 6t
solutions can be plotted to reveal a 7th-order polynomial.
of very small maximum amplitude, affecting only the last
significant figure of the five given in Table 3. In the
same way, the difference between the 8th-order and the 7th-
order solutions is an 8th
-order polynomial of even smaller
maximum amplitude, affecting the last significant figure
to a somewhat smaller extent. All differences between adja-
cent polynomial solutions have this character, so that it
is only worthwhile to examine the larger differences.
These differences are invariably expressed as a percentage
of the highest-order solution, in the discussion which fol-
lows. Effectively, it seems fair to say that very good
convergence has been achieved for the shear stress in this
particular example.
86
Another measure of convergence has also been examined.
It is difficult to inspect the small-amplitude "difference
polynomials" just described and decide how much the calcu-
lated shear or normal stress function has shifted as a whole
with increase in number of terms. To assess this shift,
the average shear or normal stress along the length of the
joint may be calculated by integration, and the values for
adjacent Ritz solutions compared. The most convenient way
to do this is to integrate the difference polynomial along
the length of the joint. This is a small quantity when
convergence is adequate, and its general order of magnitude
is about as informative as an accurate absolute value. This
index, therefore, has been computed directly from the tabu-
lated output data, using numerical integration and Simpson's
rule, rather than by analytical integration. A sample of
the "difference polynomials"is given later.
Thus, referring to Table 3, the average difference
in shear stress between the 7th-order and 6th-order results
7
is -2(10- ), and between the 8th-order and 7th-order is
2(10'6
). For reference, the typical shear stress level is
about 0.43. The present index also seems to support the
assertion that practical convergence has been obtained for
the case under consideration. The fact that the (8-7) dif-
ference is larger than the (7-6) difference can probably
be ascribed to the rounding of figures to five, for the
purposes of tabulation. Some small contribution may also
87
be present due to accumulated roundoff in the Ritz matrix
inversion, which is believed to have a small but measurable
effect in the case of 8th-order polynomials.
For this relatively large value of a, the adhesive
normal stress is smaller than, but still comparable to,
the shear stress. Generally speaking, in all calculations
it appears that the normal stress converges more slowly
than the shear stress, possibly as a result of different
physical mechanisms involved. Joint end-values are parti-
cularly uncertain, but these are not always the points of
greatest adhesive normal stress. This general behavior is
seen to a slight extent in Table 3, where three significant
figures are absolutely stable, but the fourth may vary as
much as 3.5 units as we sweep through the polynomial orders
6-7-8. Nevertheless, it still seems fair to say that good
adhesive normal-stress convergence has been achieved in
this particular example (other examples are not nearly as
favorable). The (7-6) normal stress difference averages
to -l(10-6) along the joint, while the (8-7) figure is
5(10-7), both to be compared to a typical normal stress
level of about 0.25. This index also implies that the
overall solutions are in good agreement, despite local
variations. Thus, even 109 equations deliver average
stresses which agree well with the average values for 141
and 177 equations. The index in question is perhaps a good
measure of overall equilibrium, but says very little about
the accuracy with which distributiomshave been determined.
88
The tensile load case in Table 4 (a = 10°, 8 = 20,
Y = 4), page 83, shows the same general behavior. Here the
angle a = 10° is near the lower end of the practical range,
and the shear stress is much smaller than forIx = 30° (Table
3). The shear stress agreement between 7th- and 8th-order
solutions is very good. This is particularly easy to see
in Table 4, because the "difference polynomials" have been
included in this table as a sample, under such headings as
105[7-6]. The normal stress is now 10 times smaller than
for the 30° case, and the slower rate of convergence always
observed for this stress component means that a good deal
more variation is found as the order of the polynomials is
increased. Over most of the joint, this still amounts to
th th
changes of less than 1% between 7 - and 8 -order results.
At or near the ends, as much as 5% change can be detected
in going from the 7th-order to the 8th
-order solution (e.g.,
at the end s = 1.0). It must be remembered that the normal
stress at this point is not the largest value along the ad-
hesive joint, and also that it is always less than one-
quarter the size of the shear stress at the same point. A
modest uncertainty in its determination does not preclude
the general statement that satisfactory convergence seems
to have been achieved. This is particularly true if we
think in terms of the combined stress picture for the ad-
hesive, which will be dominated by the rather well-determined
and much larger shear stress. The (7-6) average adhesive
89
shear stress difference for this case is -2(10-6
), and the
(8-7) value is the same, both referred to average stresses
of about 0.17. The respective data for the normal stress
are -2(10-6) and 1(10"6
), implying better overall "adjacency"
than the local variations at the joint ends might at first
appear to indicate.
The tensile load example of Table 5 (a = 5°, Y = 4,
B = 20) page 90, has been included here aS‘a particularly
poor example: the Ritz method is approaching the limits of
its effectiveness for simple polynomial inputs. (The 5°
angle is also probably smaller than normally attempted in
most technological applications, since such a scarf joint
is hard to make. It has been included primarily to see how
far we can push the Ritz method, particularly in connection
with bending--discussed later). The shear stress now falls
below 0.1 over most of the joint, which means that the
changes with polynomial order usually observed in the fourth
and fifth decimal place affect the third significant figure
rather than the fourth, as before. Nevertheless, the 7th-
and 8th-order shear stress results agree to better than
0.5% at all points, and better than 0.35% at all points
where the shear stress is large. This is considered to be
adequate convergence for most purposes. The corresponding
average-difference indices are -l(lO-6) for (7-6) and
-5(10-7) for (8-7), on a reference scale of about 0.09
average shear stress.
polynomials.
90
Table 5.--Adhesive stresses for 6t
h th
_'7
- and 8th-order
50’ B = 20’ Y = 4 Tensile Loading
Shear Stress Normal Stress
6(109) 7(141) 8(177) 6(109) 7(141) 8(177)
0.13756 0.13785 0.13785 0.00121 0.00659 0.01201
0.12497 0.12484 0.12493 0.01327 0.01210 0.01019,
0.11441 0.11395 0.11410 0.01185 0.00923 0.00805
0.10625 0.10569 0.10573 0.00609 0.00478 0.00544
0.10043 0.10002 0.09988 0.00112 0.00192 0.00328
0.09667 0.09654 0.09624 -0.00083 0.00152 0.00242
0.09450 0.09466 0.09433 0.00050 0.00316 0.00319
0.09337 0.09375 0.09352 0.00417 0.00590 0.00525
0.09273 0.09320 0.09317 0.00872 0.00875 0.00794
0.09208 0.09249 0.09268 0.01270 0.01093 0.01042
0 0.09098 0.09126 0.09160 0.01499 0.01208 0.01203
0.1 0.08913 0.08925 0.08961 0.01507 0.01217 0.01247
0.2 0.08635 0.08635 0.08661 0.01312 0.01145 0.01183
0.3 0.08261 0.08256 0.08262 0.00990 0.01033 0.01054
0.4 0.07799 0.07796 0.07782 0.00665 0.00917 0.00915
0.5 0.07267 0.07269 0.07242 0.00464 0.00820 0.00807
0.6 0.06690 0.06694 0.06670 0.00481 0.00740 0.00737
0.7 0.06092 0.06090 0.06082 0.00705 0.00658 0.00668
0.8 0.05493 0.05477 0.05491 0.00948 0.00553 0.00553
0.9 0.04897 0.04880 0.04899 0.00757 0.00437 0.00412
1.0 0.04288 0.04329 0.04308 -0.00697 0.00411 0.00488
6,--Bending a = 30°, 8 = 20, Y = 4
0.05807 0.05899 0.05948 0.29183 0.28525 0.28131
0.05451 0.05538 0.05587 0.27289 0.27181 0.27144
0.04987 0.05046 0.05074 0.24611 0.24730 0.24808
0.04442 0.04466 0.04471 0.21350 0.21518 0.21598
0.03838 0.03830 0.03819 0.17770 0.17821 0.17865
0.03194 0.03163 0.03143 0.13798 0.13853 0.13863
0.02524 0.02477 0.02457 0.09780 0.09782 0.09775
0.01840 0.01790 0.01772 0.05759 0.05736 0.05729
0.01150 0.01106 0.01093 0.01834 0.01814 0.01820
0.00463 0.00432 0.00423 —0.01911 -0.01906 -0.01883
0 -0.00217 -0.00237 -0.00235 -0.05400 -0.05392 -0.05327
0.1 -0.00883 -0.00878 -0.00876 -0.08551 -0.08487 -0.08449
0.2 -0.01530 -0.01508 -0.01501 -0.11302 -0.11232 -0.11203
0.3 -0.02153 -0.02118 —0.02107 -0.l3579 -0.l3530 -0.13522
0.4 -0.02745 -0.02705 -0.02689 -0.15301 -0.15304 -0.15326
0.5 -0.03296 -0.03260 -0.03245 -0.l6383 -0.l646l —0.16514
0.6 —0.03797 -0.03775 -0.03767 -0.16726 -0.l6879 -0.16956
0.7 -0.04253 -0.04236 -0.04241 —0.16214 -0.16408 -0.16490
0.8 -0.04589 -0.04624 -0.04646 -0.l47l6 -0.l4860 -0.l4910
0.9 -0.04844 -0.04912 -0.04945 -0.12075 -0.12001 -0.ll957
1.0 -0.04973 -0.05065 -0.05083 -0.08110 -0.07544 -0.07309
91
The adhesive normal stress is very small at this
angle, which is fortunate, for the Ritz process can hardly
be said to have converged. Only the leading figure can be
considered significant at many points, and at most of the
others there is uncertainty in the second significant fig-
ure. If it were of technical importance to determine the
adhesive normal stress very accurately, the present methods
would have to be abandoned in favor of other approaches.
The normal stress average differences here are 6(10-6)(7-6)
and 7(10-6)(8-7), where the reference level is about 0.012.
In general, bending load seems to place a far
greater strain on the capabilities of polynomial displace-
ment functions than tensile load, and convergence is not
always clearly established. The bending load case of Table
6 (a = 30°, 8 = 20, Y = 4), page 90, shows variations af-
fecting the second significant figure of the adhesive shear
stress as the order of polynomials is increased, where the
corresponding tensile load case is affected in the fourth
figure. Since both shear and normal stress change sign for
loading by moments, it seems useful to study convergence
for the numerically larger positive and negative values of
h- and
stress only. The largest difference between the 7t
8th-order results occurs between S = -0.9 and S = -1.0,
and amounts to less than 0.9%; all other differences rep-
resent smaller percentages than this for stress values
above the magnitude 0.024 (where 0.059 is the largest
92
absolute value). The normal stress pattern for this case
offers some novelty because larger normal stresses than
shear stresses are encountered for the first time. Compar-
ing polynomial orders 7 and 8 for the larger stress range
(above 0.119 where 0.28 is the maximum), it is found that
the differences are below 0.5% everywhere except at S = -l.0,
the joint end. (The difference at S = 1.0 is 3.2%, but this
is a point of rather small normal stress.) A difference of
1.4% is observed at the end S = -1.0, where the normal
stress level takes on its largest value, 0.281. This
probably can be characterized as adequate convergence, de-
pending upon the needs of the user. The average-difference
6
indices for shear stress are 1.4(10-5)(7-6) and 1.5(10- )
(8-7), where the largest stress value is about 0.06. The
5)
respective results for normal stress difference are -3(10-
(7-6) and 1(10-5)(8-7), referred to a largest datum level
of 0.28.
The bending load case of Table 7 (a = 10°, 8 = 20,
y = 4), page 93, shows slightly better convergence of the
adhesive shear stress values than the preceding case. For
values larger than 0.04 (where 0.1 is maximum), the 8th-order
result never differs from the 7th-order result by more than
0.3%. The normal stress calculation convergence is inferior.
If attention is confined to values above 0.02 (where 0.04 is
the maximum), then the largest difference is less than about
2%. If the lower limit considered is 0.016, however,
93
Table 7.--Adhesive stresses for 6th-, 7th- and 8th-order
polynomials.
Bending a = 10°, 8 = 20, Y = 4
Shear Stress Normal Stress
6(109) 7(141) 8(177) 6(109) 7(141) 8(177)
0.10148 0.10110 0.10098 0.00707 0.01379 0.01632
0.09122 0.09106 0.09104 0.02661 0.02599 0.02548
0.07998 0.08015 0.08030 0.03812 0.03587 0.03513
0.06780 0.06822 0.06843 0.04177 0.04030 0.04009
0.05486 0.05538 0.05554 0.03863 0.03847 0.03866
0.04150 0.04194 0.04196 0.03039 0.03115 0.03142
0.02809 0.02833 0.02821 0.01899 0.02002 0.02015
0.01502 0.01500 0.01479 0.00639 0.00715 0.00712
0.00268 0.00242 0.00219 -0.00565 -0.00542 -0.00549
-0.00860 -0.00903 -0.00920 -0.01574 -0.01599 -0.01595
0 -0.01857 -0.01905 -0.01911 -0.02293 -0.02342 -0.02320
0.1 -0.02708 -0.02745 -0.02739 -0.02677 -0.02715 -0.02682
0.2 -0.03391 —0.03418 -0.03402 -0.02734 -0.02737 -0.02710
0.3 -0.03921 -0.03928 -0.03909 -0.02519 -0.02478 -0.02478
0.4 -0.04305 -0.04292 -0.04278 -0.02123 -0.02054 -0.02094
0.5 —0.04560 -0.04533 —0.04530 -0.01661 -0.01600 -0.01669
0.6 -0.04711 -0.04681 -0.04689 -0.01247 -0.01240 -0.01298
0.7 -0.04784 -0.04766 -0.04770 -0.00974 -0.01048 -0.01043
0.8 -0.04809 -0.04811 —0.04821 -0.00882 -0.01009 -0.00917
0.9 -0.04808 -0.04830 -0.04828 -0.00921 -0.00968 -0.00884
1.0 -0.04797 -0.04819 -0.04810 -0.00914 -0.00576 —0.00873
Table 8.--Bending a = 5°, 8 = 20, Y = 4
0.09976 0.09976 0.09974 0.01232 0.01159 0.00826
0.08395 0.08392 0.08376 0.00704 0.00717 0.00831
0.06888 0.06894 0.06876 0.00652 0.00697 0.00761
0.05448 0.05462 0.05460 0.00739 0.00769 0.00732
0.04079 0.04095 0.04115 0.00781 0.00774 0.00707
0.02780 0.02800 0.02839 0.00704 0.00661 0.00622
0.01591 0.01593 0.01629 0.00504 0.00447 0.00448
0.00499 0.00491 0.00512 0.00223 0.00179 0.00201
—0.2 -0.00474 -0.00489 -0.00491 -0.00079 -0.00087 -0.00069
-0.01317 —0.01335 -0.01358 -0.00341 -0.00309 -0.00305
0 -0.02022 -0.02038 -0.02073 -0.00521 -0.00457 -0.00462
0.1 -0.02586 —0.02596 -0.02628 —0.00597 -0.00527 -0.00523
0.2 -0.03011 -0.03013 -0.03031 -0.00575 —0.00529 -0.00508
0.3 -0.03302 -0.03297 -0.03297 -0.00485 -0.00489 -0.00459
0.4 -0.03472 -0.03463 —0.03447 -0.00378 -0.00434 -0.00419
0.5 -0.03535 -0.03526 -0.03505 —0.00303 -0.00390 -0.00414
0.6 -0.03511 -0.03505 ~0.03492 -0.00303 -0.00371 -0.00420
0.7 -0.03420 -0.03420 -0.03421 -0.00378 -0.00372 -0.00418
0.8 -0.03283 -0.03288 -0.03298 -0.00466 -0.00369 -0.00332
0.9 -0.03120 -0.03124 -0.03130 -0.00403 -0.00320 -0.00224
1.0 -0.02948 -0.02936 -0.02938 -0.00113 -0.00169 -0.00426
94
differences of about 4% can be observed, and the values for
end S = -1.0 differ by 15.5%. (Again, the stress level here
is only 40% of the maximum.) The shear stress difference
indices are 2(10-6)(7-6) and 3(10_6)(8-7), where 0.1 is the
maximum, while for normal stress the respective figures are
3(10-6) and -4.5(10-6), referred to a maximum of about 0.04.
The bending load case (a = 5°, 8 = 20, Y = 4) of
Table 8, page 93, is an example where the convergence is
quite good in the shear stress. The largest error is about
l/4%, if we consider stresses larger than about half the
maximum value. The 6th-order polynomial results, surpris-
ingly, seem to agree with the 7th-order results somewhat
h with the 7th. It is tempting to guess
better than the 8t
that roundoff has affected the 8th-order results. However,
the Ritz matrix inverse is exactly the same for both the
bending and tensile loading. In the latter case the 7th-
order and 8th-order results are quite close. Hence, ques-
tions of convergence rather than of roundoff accumulation
seem to be involved here. Overall, the convergence of the
shear stress seems reasonably good. The average shear
stress difference indices are 3(10-7) for (7-6) and 7(10-7
)
for (8-7), referred to largest a shear stress of about 0.10.
The normal stress is very small for this case
(Table 8, p. 93). This is just as well, because the Ritz
method produces only one significant figure; at a few points,
even this is uncertain. The normal stress difference
95
averages are 3(10_5) for (7—6) and -7(10-6) for (8-7), on
a scale of perhaps 0.008. Since the reference stress is
small, these are relatively larger average differences than
generally encountered before, so the index in question is
not completely insensitive to convergence. To summarize
the results, it appears that the Ritz procedure with x-y
polynomials produces results ranging from very good to
"acceptable," for all cases of tensile loading. The adhe-
sive shear stress is always more reliably determined than
the normal stress, but where the latter is not well deter-
mined, it is usually small enough to be of no great signif-
icance for the overall stress pattern. In tensile loading,
convergence seems to improve as the scarf angle increases,
while in bending, the trends seems to be opposite.
On the other hand, not all of the bending results
are reliable. It becomes necessary to put down some "figure
of merit" to characterize these cases for the user. These
have been chosen as the largest difference between the re-
sults for 7th-order and the 8th-order polynomial solutions,
expressed as a percentage of the latter, with attention
confined (usually) to the larger levels of stress. Where
7th-order data have not been computed (to limit the total
computer time involved), the 6th-order results are compared
to the 8th
-order ones. This is a considerably rougher ver-
sion of the index.
The second index of merit is the average difference
between "adjacent" Ritz solutions. This consists of the
96
average along the joint of the difference between the adhe-
sive stress calculated from the 8th-order and the 6th-order
polynomial solutions (7th-order substituted where available).
This index is provided for both the adhesive shear stress and
the normal stress. All of these indices are in App. F tables,
since there is no room to place them on the sheets tabulating
the raw results (Appendix F). As a supplementary warning to
the user, an asterisk is placed on each table in Appendix F
where the local error in the satisfaction of the stress
boundary conditions exceeds 0.1 on a scale of "largest applied
stress" = 1.0. As mentioned in section 3.1.1, this is a con—
siderably less reliable indicator than the "percentages" of
App. P. It may be of some interest to note how the boundary
stress error behaves as the order of the Ritz polynomials
is increased. The discussion here is confined to the vicinity
of the peak errors in the stress component showing the largest
error.
Consider, for example, the tension-loaded case cor-
responding to Table 4 (a = 10°, 8 = 20, Y = 4). The largest
shear stress boundary errors at C (Fig. 8) are —0.0256,
—0.0247 and -0.0246 for 6th-, 7th- and 8th-order polynomials
respectively. (The reference level is a largest applied
stress of unity.) This clearly indicates that the boundary
stress errors do not decrease very rapidly as the polyno—
mial order increases. What is interesting is that the
gradient down from the peak gets larger with polynomial
97
order, i.e., the error effect becomes more localized. The
Ritz solution is attempting to conform to the local stress
singularity ignored in the analysis, as well as a polynomial
can while still satisfying the remaining boundary conditions
of a stress-free edge. Possibly, if the order of the poly-
nomials could increase greatly and a solution could still
be obtained, the error "spike" would grow to very large
values, but becomes very narrow.
The shear stress boundary condition errors for the
bending-load case at E (Fig. 9) are -0.0676 for the 6th-
order, -0.0654 for the 7th-order and -0.0629 for the 8th-
order polynomial. This indicates that the level of error
for the bending load case likewise does not decrease much
in magnitude as the degree of the polynomials used is in-
creased. As in the case of tensile loading, the significant
trend observed is a steeper gradient in the shear stress
boundary condition error with increasing polynomial order.
» At the same points, the normal stress CY is much
smaller and thus closer to the desired zero boundary values.
Its general behavior is similar. In the case of tensile
loading, the point-C values are -0.00343 (6th-order),
-0.00353 (7th-order), -0.00337 (8th-order). For the bend-
ing load case, at E the values are -0.00999, -0.00946 and
th
-0.009l4 (6th-, 7 - and 8th-order polynomials respectively).
98
3.1.5 Overall Equilibrium Check
As another check of the results, a few sample cases
have been examined for various aspects of overall equili-
brium. In one approach, this involves finding the resultant
forces produced on each of the four trapezoidal boundaries
of adherend l, and writing the three equations of static
equilibrium. In each case, the stresses used are the actual
ones produced by differentiating the polynomial solution,
not the exact input boundary conditions. On the inclined
adhesive interface, the stresses chosen are also the less
exact ones produced by differentiation, for consistency.
All integrations reported here are carried out analytically,
using the 8th-order polynomial solutions.
Since these calculations have the error level of
the stresses, which is substantially larger than that of
the displacements, no great perfection of the results is
anticipated. The integration process for finding resul-
tants may improve the situation somewhat, however. The
resultant forces should theoretically be zero, of course,
so that a reference level has to be devised to help evalu-
ate the numbers representing resultant force and moment.
For tensile loading of the adherends, the logical reference
level is the input force derived from the unit stress act—
ing on a member height of 2h = 2.0. The representative
length required for examining overall moment equilibrium
is somewhat more of a puzzle. As one possibility, it could
99
be thought of as the adherend height 2h = 2, producing a
moment of 2(2) = 4 units. However, since error tractions
act on long sides of the adherends, as well as the short
sides, a better choice (perhaps) might be the average length
of the adherend in the X—direction, (2 + cot 0), making the
reference level for moment angle-dependent. The latter
length ranges to about 10 for the smallest a considered (5°),
producing a reference level for moment of 2(10) = 20; it is
2(3.73) = 7.5 for a = 30°.
For bending load, a suitable force and moment refer-
ence level are about as hard to choose. For moment, the
applied moment could be used. The input bending stress for
this case varies from +1 to -l as y ranges from -h to +h
(nondimensional Y goes from -1 to +1). The resultant force
from Y = 0 to 1 can be chosen as a reference level for
force (= l/2 unit). The corresponding applied moment is
then l/2(4/3) = 2/3. However, the argument about error
tractions acting on the long sides is just as valid here
as before, in which case the reference levels for moment
become angle-dependent values perhaps 0/2V2.0 = 1/4 as
large as those estimated for the tensile—loaded system.
With these uncertainties in mind, we examine a few sample
cases which have been tabulated below. The calculations
are carried out using the separate stress distributions for
adherends l and 2, in bending and tensile load. Also
tabulated are the resultants for the overall equilibrium
100
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101
of the two joined members, with the interior adhesive inter-
faces making no contribution at all.
Considering the tensile load case, with datum force
2.0, the force resultants tabulated seem generally good. No
error exceeds l/2% significantly, and most values are a good
deal smaller. Moment equilibrium discrepancies are compar-
ably small, in adherends l and 2 taken separately, if the
angle—dependent reference level criterion is adopted. For
example, the 10° (adherend 1) figure of 0.0537 is small on
a scale of 20, and not even too bad on the more conserva-
tive scale of 4 units. The "overall" columns seem to pro-
vide the same sort of error level also, but perhaps here
the moment reference level at 10° can be enlarged to 40,
since the largest dimension of the system doubled.
Bending-load force resultants are not quite as good,
for a reference level of 0.5, but this is to be expected
from the rather large boundary stress errors in Table l,
and the generally less satisfactory results observed for
the bending solution. Even so, no error is as large as
10% of the nominal datum level in either adherend considered
alone. This 10% level occurs for a = 30°, 8 = 20, Y = 4.
The Table 1 data for this case shows a largest stress error
of 0.428 (on a scale of unity, or 42.8%) in the shear
stress which directly affects the present error quantity.
Thus there appears to be some gain in the process of inte-
grating to resultants. The "overall" columns do not change
the pattern appreciably.
102
Moment equilibrium in bending is again hard to in-
terpret because of the uncertainty regarding the reference
level. If the angle-dependent length is chosen, the refer-
ence level for moment becomes(1/2K10) = 5 for individual
adherends with a = 10°, and perhaps twice this for the over-
all case. On this Optimistic basis, even the worst cases
appear to be fairly satisfactory.
Another sort of overall check is possible, one
where significance should be very much greater. The tabu-
lated adhesive stresses of this thesis are presumably much
more accurate than the stresses calculated by differentiat-
ing displacements (discussed in Section 3.1.1). Therefore,
it is reasonable for us to test how well they hold the
25322 input boundary tractions in equilibrium. The trac-
tions are the pure tensile stress at the end of each ad-
herend, or the linearly-distributed stresses at the same
ends (bending load case). To be explicit, these should be
supplemented by the vanishing of stress at the top and
bottom of each adherend, and of the shear stress on the
extreme ends. Tabulated in Table 10, for the cases treated
in the preceding table, are the resultant forces and moment
produced when the "more exact" adhesive stress resultants
are equilibrated against the input forces and moments.
103
Table 10.--Sample overall equilibrium checks
Either Adherend
a B Y X—Force Y-Force Moment
Tension*
10 20 4 0.0000055 0.0000054 0.000029
20 20 4 0.000025 -0.0000080 —0.0000005
30 20 4 -0.0000048 0.0000047 -0.0000004
Bending**
10 20 4 0.0000016 —0.0000025 0.00022
20 20 4 —0.0000025 0.0000017 0.00246
30 20 4 0.0000031 0.0000011 0.00631
4.0, or
ll
*Force reference = 2.0; Moment reference
larger, angle-dependent values.
**Force reference 2 0.5; Moment reference = 2/3
(applied) or angle-dependent from = l.85(30°) to 5(10°).
For results of this caliber, it is probably not im-
portant which of the many possible reference levels dis-
cussed before are used, with possible exception of the
moment imbalance for bending load. Even using the most
conservative reference level of 2/3, the applied moment,
the worst error is now less than 1% (a = 30°, 8 = 20,
Y = 4, bending). This, incidentally, is for a case show-
ing very large boundary—stress errors in Table 1. Whatever
else may be said about the precision with which the adhesive
stress curves are determined by the Ritz process, the
104
overall resultants of these stresses seem to equilibrate
the input forces in satisfactory fashion.
3.2. Confirmation by the Integral
Equation Method
The approach outlined in Section 2.3 is quite com-
plex and radically different from the Ritz method. Any
reasonable degree of agreement between solutions obtained
these two ways constitutes a further useful check on the
present results. As noted before, the largest problem
which can be run in the high-speed memory of the CDC 3600
computer involves just enough boundary points to begin to
support the Ritz method. The character of the solution,
however, is such that it does not seem worthwhile to ex-
plore larger-sized problems, by attempting to utilize the
slow-speed computer memory. It appears that the integral
equation approach is more sensitive to the neglect of
wedge-corner singularities than the Ritz method.
This is seen in Figs. 10 and 11, comparing the ad-
hesive shear and normal stresses obtained by the Ritz and
the integral equation methods. Over most of the length of
the adhesive joint, the agreement is quite good. Near the
ends, the integral equation method shows large sudden de—
partures from the general trend of the curves (and from
the Ritz results). These are believed to be associated
with the neglect of stress singularities, or to treating
the sharp direction changes at the corners as continuously
0035‘
105
a 20°, 8 = 20, Y = 4
Integral Equation
Ritz
0.30t
: i E 4
-l.0 -0.5 0 S E 0.5 1.0
Fig. 10 Shear stress (T) by Ritz and integral equation
methods.
.20‘
0 r A
H
I"
/\ a = 20, B = 20, Y = 4
I \
I, \
0.15-- ,I \
I \
\ Integral Equation
I Ritz
\
N \\ \——————’/
0.10t \
l
\
\
\
\
\-
0005 l I l J]
-1.0 -025 d 015 1.0
_>.
S
Fig. 11 Normal stress (N) by Ritz and integral equation
methods.
106
turning tangents (which may amount to the same thing as
neglecting singularity stresses). It is considered pos-
sible (but not likely) that these effects would be reduced
if larger-sized problems could be explored. The overall
agreement is adjudged sufficient for us to be able to say
that the integral equation method does support the Ritz
method, despite the substantial differences in the two
approaches.
CHAPTER IV
ENGINEERING SIGNIFICANCE OF THE RESULTS
A considerable amount of discussion of the results
has necessarily taken place in Chapter III, with primary
emphasis on evaluating their quality. Here, attention is
centered on their physical and engineering significance.
When the scarf joint consists of elastically-dissimilar
members, which is the general case here, it becomes impor-
tant to take account of the sense of the applied loading.
Reversal of the loading may cause a numerically-smaller
stress component to become the critical one for design pur-
poses. In the following treatment of stress distributions,
we will assume that the loading is in the sense pictured
in Fig. 7, p. 26. The emphasis will then be on one set of
largest stresses, and these will form the basis for the
diagrams presented, and their discussion. If the loading
were reversed, a different set of largest stresses would
become the center of interest. This second discussion has
been omitted for brevity. The corresponding material
could be drawn from the raw data of Appendix F. When atten-
tion is finally focused on principal stresses and the design
aspects of the present results, the question of sense of
loading is taken into account properly.
107
108
4.1 Adhesive Normal and Shear Stress
Distributions
The discussion here concerns the general character
of the adhesive normal stress (N) and shear stress (T) dis-
tribution calculations as a whole. Because of the large
number of cases studied, it is impractical to attempt to
plot and crossplot all of the results, and the discussion
is carried out in terms of representative sample cases.
Even here it has been necessary to reduce the presentation
of figures to a bare minimum, in order to keep the total
number within reasonable bounds.
Primary calculated results for all cases are tabu-
lated in Appendix F, and any desired crossplots may be
constructed from these. In addition, the computed Ritz
coefficients for each case are available in punched-card
form if any further processing of the raw data appears de—
sirable in the future. The stresses T and N are already
dimensionless quantities, in the sense that they are the
result of a unit applied stress. To recover actual, dimen-
sional adhesive stresses, the user must calculate for his
case the values of the parameters a, B and y. Their defini-
tions can be found in sections 1.4, 2.4 and section 4.3.
Interpolation three ways in the tables is then required.
Finally, he must multiply the tabulated values by a "load
stress." In the case of tensile loading of the joint, the
load stress is the actual tensile stress (0X0) in the
109
adherends at points remote from the adhesive joint. For
pure bending, the "load stress" is the maximum value of
the bending stress 0 at remote points in the adherends
X0
(i.e., the usual "Mc/I").
This describes one use of the raw results; advice
on interpolation appears in Appendix F. For a discussion
of the physical significance of most of the data, however,
it is most useful to convert the raw dimensionless stress
distributions into an even more meaningful dimensionless
form: stress concentration distributions. A primary re-
sult of Ref. 7 is that in the case of tensile loading of a
scarf joint the adhesive normal and shear stresses are in-
dependent of adhesive properties and thickness, and uniform
along the joint. Hence they can be calculated from equili-
brium alone for each angle 6. Thus they comprise a con-
venient stress reference level for discussion of the
influence of adherend dissimilarity and other parameters--
the principal goal of this thesis. For tensile loading,
then, we take
where
sin a T = OX0 sin 6 cos a 4.1.2
are the identical-adherend results for tensile loading.
Equations 4.1.1 produce a stress concentration factor of
l, uniform along the joint and independent of the value of
8, for the case of identical adherends.
110
We continue to use the position variable "S," the
fraction of the joint half-length, to locate points along
the adhesive joint. Since the origin of the coordinate
system is fixed at the midpoint of the inclined joint
(S = 0 here), the location of (C,F) in Fig. 7 is S = -l.0,
and (D,E) correspond to S = 1.0. For bending, not covered
by prior theory, the stresses in the adhesive are neither
uniform along the joint nor linear, even for the case of
identical adherends. Moreover, even these distributions
are now B-dependent. Thus the treatment of bending does
not benefit greatly from the method of non-dimensionaliza-
tion now under discussion, and it will be handled differently.
To convert the "stress concentration factor" type
of dimensionless stress (N T0) back to dimensional form,
OI
it is necessary to multiply by NO or T as well as the
O I
II N
load stress CXO'
In discussing the results to follow, we attempt to
follow the rule that any behavior pattern pointed out holds
for other cases of similar type, unless otherwise noted.
The difference for other cases is thus one of degree, not
of general trend.
4.1.1 Case of Tensile Loading
Some representative samples of the adhesive shear
stress distribution TC defined in the preceding section
are plotted in the main portion of Fig. 12 (the smaller
111
inset is mentioned later). This shows the stresses due
to tensile loading for scarf angle a = 10°, for B = 20 (an
intermediate level of adhesive flexibility), and for three
of the four values of the dissimilarity parameter: Y = l,
2 and 8 (essentially, ratio of adherend 2 Young's modulus
to that of adherend 1). Thus only the dissimilarity param-
eter Y changes. A curve is sometimes omitted, Y = 4 in
this case, when it is so close to another as to confuse the
diagrams. Figure 12 indicates that the shear stress is not
uniform when the adherends are dissimilar, with the depar-
ture from uniformity increasing smoothly as Y increases.
This seems reasonable on a physical basis. The omitted
curve for Y = 4 lies between the cases pictured for Y = 2
and Y = 8, but closer to Y = 8 than to Y = 2. This implies,
perhaps, that Y = 8 is probably rather near the limiting
case Y = m (adherend 2 "rigid"). Note that the largest
stress concentration factor is about 1.35, for Y = 8.
The largest shear stress for all cases of ten—
sile loading follows the pattern shown: it is always at
S = -1. This can be supported by physical reasoning, be-
cause of the simple model adopted here for adhesive
strains in terms of relative displacement of the ad—
herend-adhesive interfaces. At S = -l, adherend 2
has its smallest stiffness, because it has a sharp point.
It therefore deforms very readily. In the same region, ad-
herend 2 is at its stiffest and deforms less readily, by
112
1.5-4. 0, = 100' B
Y = 8 Y = 1,2,8
20,
100
1.0 .
2 l
l 4 Y 72
0 i H: I
-l.0 —0.5 o S 0.5 +1.0
Fig. 12 Shear stress concentration factor (TC) in tensile
loading.
_._._. 50 - = 200
(TC)max a (TC)max a
3.0+ 2.0»
a - 30° 6 = 40°
(TC)max (TC)max
1.2L 1.2r
B = 4
1.1- 1.11 B = 4
20
1.0 . . . 100 1.0 . . = 20
1 2 4 Y 8 1 2 4 Y 8
Fig. 13 (T ) in tensile loading.
C max
113
contrast. The situation is geometrically identical but
elastically different at the other end. Since the adhe-
sive acquires stress on the basis of relative displacements
of adherends, it follows that the local stiffness of tip
regions dominates end-region stress generation in the adhe-
sive. Flexible tips yield readily if the opposing member
displaces, and do not allow the local relative displacement
to become very large. The tip of adherend 2 is the stiffer
one whenever Y > 1, since Y = E2/E1 (essentially). Hence
the largest shear stresses are invariably found at this tip,
or S = -1. This reasoning should apply to bending load as
well as tensile load, and it is offered to account for the
asymmetry of the adhesive stresses when the adherends are
dissimilar. A few exceptions are discussed later, in con-
nection with bending load cases.
The same pattern is observed with a relatively more
flexible adhesive (8 = 100), except that there is much less
stress concentration and the adhesive shear stress is al-
most uniform along the joint for all values of y: TC 2 1.
This behavior is found in all prior studies of the stresses
in adhesive joints: a very flexible interlayer permits
smooth and uniform load transfer. On the other hand, a
relatively stiff adhesive layer (B = 4) results in a sub—
stantial exaggeration of the trends of Fig. 12, with the
largest shear stress concentration factor (for Y = 8) reach-
ing 1.84 at the same scarf angle. This number comes from
the tables in Appendix F, and can also be deduced from
114
the small inset diagram in Fig. 12. The latter shows how
the maximum shear stress (TC)max’ which always occurs at
S = -l.0, varies with dissimilarity Y for the fixed value
8 = 20 of Fig. 12. Also shown are the behavior of (TC)max
for the other values of B at the same scarf angle, a = 10°.
This simply restates the foregoing discussion pictorially,
for the special case of the largest shear stress. The
similar diagrams which obtain for the other scarf angles
all appear in Fig. 13.
If dissimilarity Y is held fast, adhesive shear
stress concentration drops off rapidly with increase in
adhesive flexibility, B. This is seen in Fig. 14, for
a = 10°, Y = 4, and B = 4, 20, 100. No inset diagrams of
(TC)max are used here, because the rapid drop off of the
maximum shear stress with B is readily visualized from the
inset of Fig. 12 and the four parts of Fig. 13.
Figure 15 shows what happens to the adhesive shear
stress distribution T for intermediate B(=20) and Y(=4),
C'
when the scarf angle a is varied. It is found that stress
concentration increases smoothly but suddenly when a is
reduced below 20° (30° and 40° curves conform to the pat-
tern, but are omitted for clarity). Remember, however,
that the absolute value of the shear stress T becomes very
small as a is made small. Therefore, the stress concen-
tration factor TC of about 1.59 for a = 5° is applied to
a reference stress of small magnitude, T0 = sin 6 cos a.
115
-' 10°: Y = 4r
- 4, 20, 100
100
0.5.-
0 . I L I
-1.0 -0.5 0 0.5 1.0
S
Fig. 14 Shear stress concentration factor (TC) in tensile
loading.
(‘1
I B = 20, Y = 4
a = 5°, 10°, 20°
l'05°10° 20° 30° 40° 9*
-1.0 -015 0 015 1.0
s
Shear stress concentration factor (TC) in tensile
loading.
Fig. 15
116
This means that the halving of angle a from 10° to 5° has
a much larger effect on the stress magnitudes than the
change in local end-stress concentration from 1.59 to 1.28
exhibited at S = -1.0. Within the "manufacturable" range,
we are always gaining ground if we reduce the scarf angle,
apparently. Figure 15's trend applies to all other cases
of a-variation at constant B and Y. The inset diagram of
Figure 15 shows some typical variations of (TC)max with a,
for the constant B and Y of the main part of the figure,
and the two other sets of B and Y which give the worst
shear stress concentration in tensile loading.
The most obvious feature of the adhesive normal
stresses of Fig. 16 is their wavy pattern (case of a = 10°;
8 = 20; Y = 2,4,8). This phenomenon has been observed in
the results of prior studies21 and probably arises from the
very nature of the model used for the adhesive. The latter
has been treated as an elastic foundation. It is well
known that a uniform beam on an elastic foundation (which
adheres when the beam attempts to lift) will exhibit a
damped sinusoidal displacement pattern if the foundation
modulus is large enough. The present problem is complicated
by the fact that the "beams" are tapered, but this explana-
tion appears to account for the wavy distribution of normal
stress. When the "foundation modulus" decreases (B in—
creases here), the waves become longer and the effect less
noticeable. However, when 8 = 4 the waves shorten and the
117
a = 10°, 8 = 20
1.5. Y = 1,2,4,8
NC
Y = 1
1.0 2
4
3 = 4 8
0 5“(NC)max
° 1.5 20
1.0 g : Hy-r
l 4 8
0 I r I
-1'0 -005 O S 0.5 1.0
Fig. 16 Normal stress concentration factor (NC) in
tensile loading.
(NC)max a = 5° (NC)max a = 20°
3.01 1.5L
2.0- B = 4
100
1.0 1 ; 1
2 4 8
1 Y
(NC)max a = 30°
1.11
B = 4
20
1°0- i i a
1 2 8
Y
Fig. 17 (NC)max 1n ten51le loading.
118
normal stress oscillates very rapidly. This occurs to such
a degree that high-order polynomial interpolation in the
stress distribution tables of Appendix F is sometimes re-
quired, because the 21 points at which the adhesive stresses
have been tabulated are not always sufficient to define the
normal stress curve properly for the user. Waviness is not
observed in shear stress distributions. A crude explanation
is as follows: Shear stress is governed by "axial" displace-
ment of the tapered adherend "beams" or, in another analyti-
cal formulation, by second-order differential equations for
displacement. Normal stress is governed by the bending of
these "beams," i.e., by fourth-order differential equations
in displacement. The latter can be expected to show damped
quasi-sinusoidal waves.
The dimensionless presentation of Figure 16, p. 117,
hides the fact that the normal stress is much smaller than
the shear stress when a is small, as in the 10° case plotted.
It should also be observed that increasing dissimilarity
(Y) causes an increase in the largest values of normal stress
(wave amplitude). Figure 16 also illustrates the difficulty
of stating where the peak normal stress is found. Often it
is at or near S = -1, but a secondary peak occurs in the
ranges S = 0 to 0.4 (considering all results, not just those
pictured). In many cases the differences between the peaks
are so small that imperfection of convergence, or even the
estimated roundoff error in the calculations, could affect
119
In other
a decision as to the location of Nm or (NC)
ax max“
cases, Nmax can be located fairly distinctly. For the
large value 8 = 100, the normal stress is often so nearly
uniform that the location of a peak value is of no great
significance.
For practical stress calculations, the significant
factor is the adhesive combined-stress situation. This
resolves the difficulty, because the relatively large ad-
hesive shear stress usually dominates all types of combined
stress of engineering interest, and its location is always
at S = -l. The uncertainty in N x does complicate any at—
ma
tempt to crossplot normal stress maxima. To aid the user,
Appendix F has an auxiliary table giving the values of
N .
max
The inset diagram in Figure 16 shows a plot of
(NC)max as a function of dissimilarity Y, for the values
8 = 20 and a = 10° governing the main part of the diagram.
Also shown is the curve for B = 4 (B = 100 is close to
NC 5 l, and is omitted). Remember that NC.has been nor-
malized with respect to N0 = sin 2 a, which tends to distort
the fact that 5° cases exhibit absolute stresses basically
l/4 as large as 10° cases (ignoring the stress concentration
effects exhibited in the inset). Values of (N are
C)max
plotted without regard to location S. Figure 17 furnishes
the same information as the inset diagram just mentioned,
for the other four scarf angles.
120
Figure 18 shows (for fixed a = 10° and Y = 4) the
expected result that the normal stresses become much more
nearly uniform as adhesive flexibility B is increased.
Finally, Figure 19 shows that normal stress "waves" acquire
increasing amplitude (at constant B = 20, Y = 4) as scarf
angle a is reduced below 20° (same behavior as adhesive
shear stress). The 30° and 40° angles are scarcely distin-
guishable from N = l on this scale. Note once again that
C
the absolute stress level N goes down with a as sin2 a, so
that the modest effect of increasing stress concentration
is normally overwhelmed by the gross decrease in magnitude.
For example, the 5°, 10° and 20° intercepts on an N basis
(not N as in Figure 19), are actually 0.01201, 0.03881,
C'
0.12285, respectively, from the tables in Appendix F. The
30° and 40° results are 0.25024 and 0.41293. Thus the
stress concentration type of presentation in this case re-
veals the increased amplitude of oscillation with decreas-
ing a, but distorts the picture of the magnitude of the
stresses.
Before going on to the discussion of the sample
bending load cases, one additional set of results is pre-
sented, for the case of the butt joint in tension (a = 90°).
This puts in perspective the enormous influence which the ad-
hesive's relatively large flexibility exerts in the case of
metal-to-metal bonds. Figure 20 shows the adhesive normal
stress in a butt joint for the stiffest adhesive considered
121
a = 10°,Y = 4
1.5.. B = 4
NC
100
1.0. '-' \‘
005'”-
0 If —‘r 4.
-100 -005 O S 0.5 1.0
Fig. 18 Normal stress concentration factor (NC) in tensile
loading.
1.5‘ (1.:0 8:20'Y=4
NC
20°
1.0- . ‘\
0.54
o H . .
-l.0 -0.5 S 0 0.5 1.0
Fig. 19 NOrmal stress concentration factor (NC) in tensile
loading.
122
(B = 4), and the case of identical (Y = l) and maximally-
dissimilar adherends (Y = 8). It is apparent that the two
extremes differ very little. The comparatively flexible
adhesive readily accommodates the different lateral con—
tractions of the equally—stressed adherends, and indeed,
Figure 21 shows that it acquires very little shear stress
in doing so; the shear stresses are nearly linear functions.
This diagram covers the same two cases as Figure 20. Here
it is necessary to plot dimensionless shear stresses T
(actual magnitudes for adherend tensile stress = unity),
rather than T since T0 = 0 in this case. The latter is
C’
the curve labeled Y = 1. Symmetry dictates that the adhe-
sive shear stress be an odd function of S for the butt
joint, and the normal stress an even function. It is easily
verified from first principles, by consideration of the
lateral displacements of the two axially-loaded adherends,
that this very small shear stress is of the correct order
of magnitude.
In view of the results, it is probably sufficient
to simply state here the maximum shear and normal stresses
for the butt joint cases omitted from the diagram (a = 90°,
8 = 4):
Y = 2 : leaxl = 1.0011 leax‘ = 0.0126
at S = :1
Y = 4 : leax' = 1.0027 leaxl = 0.0196
123
= 90°, 8 = 4
.1
1.004 .Y = 1,2,4,8
NC
1.0024
1.000
0.998 : ‘ i
-l.0 -0.5 0 0.5 1.0
S
Fig. 20 Normal stress concentration factor (NC) in tensile
loading.
a = 90°, 8 = 4
= 1,8
-2X10’24
-1.0 -0f5 0 0i5 1.0
S
Fig. 21 Shear stress (T) in tensile loading.
124
4.1.2 Case of Bending Load
The general conclusions from Chapter III are that
the Ritz method (as used here) finds adhesive normal
stresses with less certainty than shear stresses, and con-
verges better for tensile load than bending load. The
confidence level is therefore high for all tensile loadings,
especially in the shear stresses, which are the most sig-
nificant ones. It is less high for bending cases, particu-
larly for normal stresses. The discussion of the latter is
somewhat more tentative, because some of the trends observed
may be fictitious due to incomplete convergence.
4.1.3 Identical Adherends in Bending
Even for identical adherends (Y = 1) in pure bending,
we still have a two-parameter family of results to consider.
Adhesive flexibility (B) and scarf angle (a) are the param-
eters. The corresponding adhesive stresses must be plotted
as T- and N- type rather than as the more instructive con-
centration factors. This is because, unlike the case of
tensile loading, there is no analytical solution other than
the present one to serve as a reference level. To obtain
true, dimensional stresses from the figures to follow, or
the tables of Appendix F, multiply by the "load stress" OX0
of Section 4.1: the adhesive shear and normal stresses
tabulated and diagrammed are based on an outer-fiber bend-
ing stress ("MC/I"), remote from the joint, of unity. The
125
bending moment per unit width of adherend, in the sense of
Figures 6 and 7, is then 2/3.
Eigure 22 shows the adhesive shear stress T for a
bending case with a = 10°, Y = 1 (identical members), and
B = 4, 20, 100. The roundoff error level seems to affect
about one unit in the fourth decimal place, or less, re-
gardless of the magnitude of the stress. A typical value
at S = 0, where an odd function should be zero, is 1(10-4)
units on a reference scale of unit applied stress. (This
is not bad roundoff for "one—pass" solution of 177 equa-
tions.) Within this error level, the tables and the figures
indicate that these stresses are odd functions of the dis-
tance parameter S. The odd property can also be deduced
from a consideration of the symmetries of the identical-
adherend case. It is evident that these stresses are not
linear functions of S, in general, but this was not really
to be expected, despite the linearity of the applied bend-
ing stress.
The shear stress maximum is always at S = :1. The
expected decrease of stress level with increase of adhesive
flexibility (8 increasing) is evident. It is interesting
that the stiffest adhesive (8 = 4) represents the straightest
line. It may be speculated that as the adhesive becomes
very stiff, the identical-adherend configuration approaches
the state of a single uniform beam, with the adhesive in-
terface behaving like the imaginary line one passes to
126
0.16“
a = 10°, Y = l
B = 4,20,100
0.12.
T
0.08?
100
0.04w
0 ; I‘
—l.0 -0.5 0 0.5 l.
S
_0.04.-
Tmax B = 4
0.211
-0008‘b
20
0.1..
_o.12.. /—\
K
0 , 4 I : -f
5°10° 20° 30° 40°
-0016‘r a
Fig. 22 Shear stress (T) in bending load.
127
calculate stresses under rotation of axes. That is, the
adhesive is "not there," and the stresses can be calcu—
lated from simple beam theory. This may not be too far
from the true situation when 8 = 4, for small angles 0,
since the largest shear stress (at S = :1) is 80% of the
theoretical value sin 0 cos a which one would calculate
from beam theory. As another indication that the present
physical interpretation may be a good one for a "stiff ad-
hesive," the tangent to the B = 4 curve of Figure 22, as
estimated over the interval S = -0.1 to S = 0, would pro-
ject to S = -l.0 as an intercept of 0.144. The value of
sin 6 cos a for a = 10° is 0.171. The behavior of the
normal stress distributions also supports this interpreta-
tion; these are discussed later. However, this explanation
must not be pushed too far, because there may be some ques-
tions about the validity of the adhesive model when the
adhesive is anything but relatively "flexible."
All identical-adherend shear stresses in the bending
case follow the general pattern of Figure 22, but the level
of stress changes with angle 6. Indeed the inset diagram
in Figure 22 shows that the maximum shear stress (at S = :1)
goes up with a and then falls off, if 8 is small to moderate,
but decreases uniformly with a if B is large (100, very
flexible).
The adhesive normal stress pattern for identical
adherends (Y = 1) also shows the odd-function behavior
128
expected from symmetry. Otherwise, it is difficult to make
general statements about N, since the pattern keeps chang-
ing with the parameters. The explanations given below,
however, seem to account well enough for the calculated
behavior.
Figure 23 shows the normal stresses N for a = 10°;
Y = l; 8 = 4, 20, 100. Here the stiffest adhesive (8 = 4)
shows no oscillation, and the most flexible the largest
wave amplitudes. One would usually expect the "elastic
foundation" (adhesive) to be associated with shorter, larger-
amplitude waves as it gets stiffer, but the opposite trend
is seen here. In this case the maximum normal stress occurs
at the ends S = :1 for B = 4 only; it is found in the in-
terior for the other values of B.
It is likely that the oscillations of Figure 24 are
not solely related to the idea of a beam on an elastic
foundation, but that other mechanisms are also involved.
The following is offered as a possible interpretation of
the behavior observed. As in the case of the shear stresses,
for B = 4 (relatively stiff adhesive) the joint is not far
from being vanishingly thin and infinitely stiff (B = 0),
which we interpret as being the case of the homogeneous,
joint-free beam. In the latter situation, the "adhesive"
normal stress, calculated from elementary theory, should
be linearly distributed and have the largest value sin2 a
for the present unit "load stress." When 8 is allowed to
0 O4“ 129
a = 10°, 8 = l
- 4,20,100
0.02“
-0.021-
-0.04-
Fig. 23 Normal stress (N) in bending load.
a = 30°, Y = l
4,20,100
u:
II
"0-21
50
V
0 § 3 # ﬁJ‘ 8+
0 4 10 20 00
Fig. 24 Normal stress (N) in bending load.
130
have a finite elasticity, we must consider the mode of de—
formation of the adherends and adhesive. The adherends
are regarded as tapered beams in this argument, and since
they taper to a point, their bending stiffness drops off
rapidly as the tip is approached. In the vicinity of a
tip, the adjacent adherend beam has its maximum section
and is therefore very stiff. Adhesive normal stress is
developed by relative transverse displacement of the two
beams, one locally very stiff and the other very flexible.
When the adhesive itself is nothing but an imaginary line,
the linear stress distribution is of course transmitted
without difficulty. But when a flexible adhesive is pre-
sent, it is incapable of actually transmitting a linear
stress variation. This is because the beam tip is too
flexible to offer enough resistance to its full share of
the normal stress, precisely at the point where the latter
tends to take on its largest values. It is simply too com-
pliant, and displaces too readily. Relative to the stiff
adjacent member, the corresponding adhesive interface is
not displaced as much as a linear distribution of stress
would demand. Thus the normal stress simply falls off from
the sin2 a value. To satisfy moment equilibrium, a read-
justment of the adhesive stress distribution must take place.
It acquires larger values than the linear distribution in
the interior region of the joint, to compensate for the
dropoff at the tip. Following up on this model a little
131
further, the more flexible the adhesive, the smaller the
normal stress which the joint end is likely to experience.
Loosely speaking, a softer "spring" (the adhesive) cannot
develop a given level of stress near the tip without de-
forming the tip more, which in turn requires higher trans-
verse stress level. Instead, the actual stress develOped
falls off as adhesive flexibility 8 increases. The argu-
ment fails when the scarf angle increases to the point that
the elastic solids become compact bodies rather than slender
tapered beams. Then the load should be transferred as if
the two adherends were rigid bodies, at least when the ad—
hesive is very flexible. It follows that we should find
the adhesive normal stress almost linear along the joint,
at large scarf angles and large B. The transitional be-
havior from one model to the other should be smooth and
gradual.
This interpretation is supported by the relative
positions of the curves, and the calculated numbers. For
the stiffest adhesive in Figure 23 (B = 4), sin2 10° =
0.03022, yet the value of Nmax at S = -l is the somewhat
smaller 0.0252. To compensate for this, the convex-up
shape of the curve for B = 4 must develop. It starts out
from S = 0, where N = 0, with a slope which would project
to S = -l at a value Nmax = 0.045 (based on the interval
S = 0 to S = -O.1).
132
Considering these data, the present argument implies
that B = 4 is not too far from the "rigid" condition, as in-
dicated at the start of the discussion. For the much greater
adhesive flexibilities B = 20 and B = 100, the effect des-
cribed should be increasingly exaggerated, and the central
region of the joint must transmit an increasing proportion
of the overall bending moment. In addition, as the value of
8 increases, the intercept at S = :1 drops off. The fore-
going mechanism, of itself, may or may not be sufficient to
make the maximum normal stress actually occur toward the
interior of the joint as adhesive flexibility increases.
It does seem sufficient to account for the curve for B = 4
in Figure 23. There is another mechanism operating which
may also have an effect: the "beam-on-an-elastic-founda-
tion" idea. The oscillatory behavior of the curves for
B = 20 and B = 100 may be associated with a superposition
of these two mechanisms.
The picture for a = 5° (not shown) is entirely
consistent with the first explanation attempted above, but
in this case there is also a very clear-cut indication of
wavy behavior in the curves for B = 20 and B = 100. As
the scarf angle increases, the adherends become less and
less like tapered beams. At a = 20° there is only a small
tendency for the curve for B = 4 (not shown) to exhibit
oscillations, and little evidence of it for the other
values of B, or for any values of B at larger scarf angles.
133
Where one would expect to see wavy behavior and it does not
actually occur, this may be due to a coincidence of param-
eters, a special gradient of the normal stress, or some
aspect of symmetry. It is later found that identical-
adherend cases often do not show expected oscillations, but
that the introduction of some dissimilarity brings them out
strongly.
Besides attempting to account for the shapes of the
calculated stress distributions, there is another point to
justify this lengthydiscussion. Some sort of explanation
is demanded by a situation which has occurred very rarely
in the adhesive joint literature (if at all): the increase
of a stress component as the flexibility of the adhesive 8
is increased. The trend is normally the opposite. Indeed,
the usual smoothing effect of increasing 8 may have its in-
fluence here, as yet a third mechanism interacting with the
"flexible tip" and "elastic foundation" interpretations.
The fourth factor is the question of Ritz process conver-
gence, which may be significant because the adhesive stresses
in bending load are not as well determined as for tensile
load. The fact remains that (in these calculations) the
maximum normal stress does appear to increase with adhesive
flexibility in some cases, as shown in the inset diagram of
Figure 24.
Figure 24 is for a = 30°, Y = l, and B = 4, 100; the
case B = 20 is not distinguishable from 8 = 100 on this scale.
134
The interchange of shape of the stress distributions for
B = 4 and 100, between Figures 23 and 24, is noteworthy and
in line with the expectation that as 0 increases the large-
8 curves will begin to straighten out. The interchange
mentioned is another example of the difficulty of making
general statements about the normal stresses for bending
load. For an even larger scarf angle, a = 40° (not shown),
the pattern is similar but all curves are more nearly linear
in the interior of the joint, and the maxima are closer to
the ends S = i1. In this case the stresses for B = 100 are
almost linear from end to end, with no downturn at all.
Even the B = 20 curve has begun to straighten out.
The inset diagrams in Figure 24, incidentally, cross-
plot the values of Nmax/Sin a against a logarithmic scale of
adhesive flexibility B, for the various scarf angles. It
would theoretically be desirable to plot Nmax/sin2 a, but
then all values lie in the range 0.77 to 1.08 and the various
curves become quite confused. A plot in the present manner
separates them well. A plot of Nmax itself shows too large
a range to appear on a single diagram, since it varies essen-
tially as sin2 a.
Before leaving the case of identical adherends, the
bending of the butt joint configuration (a = 90°) should be
mentioned. For all values of B, with these compact adherend
shapes the adhesive normal stresses are perfectly linear and
are the same as those calculated from beam theory. This is
to be expected.
135
The adhesive shear stress is identically zero, as
symmetry would demand.
4.1.4 Bending Load (General Case)
Here there is a full three-parameter family again:
a, B, Y all vary. Figure 25 illustrates the variation of
the shear stress for a = 20°, 8 = 20, and Y = l, 8. The
curves for Y = 2 and 4 fall smoothly between the Y = l, 8
values and are omitted for clarity. The stress curves be-
comes increasingly asymmetric as the dissimilarity (Y) in-
creases; it is an odd curve for identical adherends (Y = 1).
The explanation of the asymmetry was taken up in section
4.1.1 and applies here also. According to the argument
there, we would normally expect the shear stress to be
largest at S = -1. This seems to hold for most of the re-
sults now under consideration. There are, however, a few
anomalous cases where the computations find Tm x slightly
a
larger at S = +1. These all have the following character:
scarf angle a is large (30°, 40° only), and adhesive flex-
ibility B is moderate (20) or large (100). In all such
cases, the adhesive normal stress is much larger than the
shear stress and the discrepancy is very small compared to
either the "load stress" of unity or the somewhat smaller
local value of normal stress. It is therefore felt that
the argument mentioned above remains valid, but that other
factors intrude to produce an opposing effect. These could
136
- 0 i t i ,
0'2 1 2 4 Y 8
Fig. 25 Shear stress (T) in bending load.
= o = ° . =
Tmax a 5 B = 4 T ax a 10 '@——£———
0,2. 0.2?////,,,H——
20 20
0.1‘ A 0.1
/ 100 100
\
0 L 2 4% 0 i . ‘
J Y ..
4 Y 8 l. 2 4 Y" 8
_ O
TTax a== 309, Tmax a - 40
0.2““ 0.2: B = 4
001‘” 20 0.1.?- 20
\’ '\ _—_—
.M 100__ \- 100 l
0 ,. ‘f H. 5 0 3 °
1 2 4 Y 8 l 2 4 Y 8
Fig. 26 Maximum shear stress (Tm X) in bending load.
a
137
include roundoff error accumulation, another physical phe-
nomenon, or more likely, a question of convergence. It
should be emphasized that the effect is very small, what-
ever the source. For example, the Appendix F tables show
(a = 40°, 8 = 20, Y = 8) T = 0.0372 at S = -l, and
T = -0.0396 at S = l, where N = -0.2526. In bending, it
is not believed likely that the stresses are always so well
determined as to make a discrepancy of 0.0024 significant.
The extra curve in Figure 25 is for the same scarf
angle of 20°, but 8 = 4 and Y = 8. It is introduced to
show that on many diagrams, the effect of dissimilarity
(Y > 1) is to produce a curve of this concave-up character.
It is also interesting that an increase in dissimilarity Y
may produce a decrease in the largest shear stress level
(as in Figure 25), or sometimes an increase, or even no
appreciable change. This is seen in the inset diagram of
Figure 25, for a = 20° and the three values of 8. Thus
Tmax decreases uniformly with increase of adhesive flexi-
bility B, but has varying behavior with change of Y. Figure
26 shows the same information as the inset diagram of Figure
25 for the rest of the scarf angles, and produces similar
conclusions.
Figure 27 shows how the maximum shear stress in
bending varies with scarf angle a. Each diagram holds adhesive
flexibility B constant, and allows dissimilarity Y to vary.
The interaction is quite complex.
138
T T
max max
0.3+- B = 4 0.151? B = 20
Y=1
2 O 10
0.2 4 .
8
0.1- 0.05‘
O r +5 I t 0 : a t :
5°lD° 20° 30° 40° 5°10° 20° 30° 40°
6 a
T
max
0.041
Fig. 27 Maximum shear stress (Tmax) in bending load.
139
The normal stresses in bending are shown in Figure
28, for a = 10°, 8 = 20, and Y = 1,2,4,8. The curve for
Y = l was discussed in the previous section; it represents
an odd function. For increasing adherend dissimilarity,
the curves become increasingly asymmetric. For Y > 1, all
show the largest stress Nm toward the end S = -l, but
ax
still on the interior of the joint. There is strong evidence
of oscillatory behavior, of "elastic foundation" type. All
this is to be expected, in the light of previous discussion:
for small scarf angles, wavy behavior is expected; Nmax tends
to be toward S = -1 with the present elastic asymmetry; and
Nmax does not appear at the end of the joint for small scarf
angles and moderate to large adhesive flexibility. The dia-
gram shows that increase of dissimilarity raises the peak
value of Nmax’ Going to a larger value of adhesive flexi-
bility (B = 100, not shown) leaves the general pattern of
stress much the same. For a stiffer adhesive (8 = 4), the
curves are slightly different in character toward S -l.
2
Figure 23 shows the Y = 1 case for B = 4, and the Y
curves for this value of B are somewhat similar (i.e., mono-
tomic increasing as 5 approaches -l). The latter two show
a slight upturn toward the end, with Nmax there. Only the
case of Y = 8 shows the downturn at S = -l, with a maximum
still on the interior of the joint. The inset diagram of
Figure 28 shows how N varies with dissimilarity Y. On
max
this small scale, the curves for B = 4, 100 are hard to
140
10°, 8 = 20
1,2,4,8
‘ 0 I 444
I 1 2 4
Y
0.5.
2
0 i i
71.0 -0.5 0 S 0.5 1.
—0.5-- 4
A
-1.0 — 8 /
"LST
Fig. 28 Normal stress concentration factor (NC) in
bending load.
141
distinguish and have been omitted. The trend with Y in
this inset holds for every case tabulated, but the slope of
the curve may become very small.
If a is reduced to 5° and the "tapered beams" become
more slender, the stress distribution for the stiff adhesive
(8 = 4) and identical adherends (Y = l) is roughly linear
and shows little waviness. As Y is increased to 2, 4, 8,
violent oscillations of increasing amplitude and rather
short wavelength are superimposed upon the Y = 1 case, and
the asymmetry of the stresses increases. Also at 5°, the
wavelength of the oscillations gets longer as B is in-
creased to 20 and 100 ("foundatiod‘becomes more flexible).
All this is much as one might expect, and this further in-
dicates that the identical adherend cases (Y = 1, Section
4.1.2) which show little evidence of waviness probably do
so as a result of special symmetry or a coincidence of
parameters. For a = 20°, 8 = 4 and 20, the stress distri-
butions (not shown) also have the general character of
Figure 28, although a gradual transition to new behavior
is observable. When adhesive flexibility is increased to
8 = 100, the normal stresses have evolved until they are
quite similar to Figure 29. The latter shows the normal
stresses for a = 30, B = 20, and Y = 1,2,4,8. For large
dissimilarity y, the maximum no longer occurs on the in-
terior of the joint, and thereis little evidence of elastic
foundation waviness. There is still asymmetry for Y > 1,
142
of course, and all values of Nmax occur at or near S = -1.
Some of the curves are beginning to straighten out, a trend
anticipated in Section 4.1.2 for large scarf angles a and
moderate to large B. If B is reduced to 4 (not shown), the
pattern of Figure 29 still holds. However, for B = 100
(not shown), all curves are quite close to each other for
all Y, and nearly straight, with just a small tendency to
curve toward the ends. The large scarf angle a = 40° still
shows similarity to Figure 29 when 8 = 4 (stiff adhesive).
The stresses for the case of a = 40° with B = 20 behave
much like the Y = 8 case of Figure 29, but generally
straighter and with more sudden end changes in curvature.
Finally, for a = 40° and the large flexibility B = 100, the
transition is nearly complete. All curves are nearly linear
from end to end, independent of dissimilarity Y, as befits
"compact" adherends in bending with a highly flexible ad-
hesive. These features could largely be deduced from the
general discussion of the normal stresses in the case of
identical adherends.
For the bending of butt joints between dissimilar
adherends, it is still found that the adhesive normal stress
is the linear distribution one would calculate from elemen-
tary beam theory, to four or more significant figures.
When Y # 1, small shear stresses are induced in the adhe-
sive. The distributions are self-equilibrated and even in
S, as demanded by considerations of symmetry in geometry
143
.UMOH mCHUsmn CH Hozv Houomm soHumuucmocoo mmmnum HMEHoz mm .mHm
.DoHI
INS-
m.o m o m.o.. OH.
H u o
.&.o
// H [@OH
m.e.N.H n > N o
2
ON u m .oom u s
e
Iv.m
m"
144
and loading. The largest such value found (8 = 4, Y = 8),
at S = i1, is 0.00851 for the unit "load stress" in bending.
Some of the other values of T are:
max
8 Y T
__ __ max
4 4 0.00725
4 2 0.00477
20 8 0.00178
20 4 0.00152
100 8 0.00036
145
4.2. Adhesive Combined Stresses
Since the adhesive is in a known state of combined
stress (shear and one component of normal stress), we can
at every point find the corresponding maximum principal
stresses (N1,N2), the maximum shear stress T1’ and the
octahedral shear stress Toy' All of these are quantities
which commonly enter into engineering design criteria, in
one form or another. They are found by using the standard
relations:
2
N1 = N/2 + /QN/2)2 + T 4.2.1
N2 = N/2 - /(T\I/2)2 + T2 4.2.2
T1 = f(N/z)2 + T2 4.2.3
Toy = /Q6T2 + 2N2)/9 4.2.4
The principal stress N N2 and is always the
>
1:
governing tensile stress for tensile loading of the joint.
When the loading is compressive, N2 becomes the largest
tensile stress. For bending moments in the sense of Fig,
7, the N1 stress is the critical tensile value, and if the
moments are reversed, the N stress takes on this role.
2
From a design viewpoint, there is no need to examine
entire distributions of the combined stresses. Accordingly,
the largest values of the combined stresses have been ex-
tracted from the calculated distributions by suitable in-
terpolation techniques, and only these are tabulated. In
146
Table 11 (next page), the signs given correspond to the
loading senses of Figure 7, p. 26. The "load stress" OX0,
labeled 00 in Figure 7, is always unity in Table 11.
The variation of the combined stresses is fairly
simple for tensile loading. Table 11 shows that all quan-
tities have the following behavior, with minor exception
ascribable to roundoff or other computational (rather than
physical) cause:
1. All increase monotonically with increase in dis-
similarity of the adherends, Y (at constant a, B).
2. All decrease monotonically with increase in adhe-
sive flexibility, 8(at constant a, Y).
3. All decrease monotonically with decrease in scarf
angle a (at constant B. Y).
The last item indicates that the function of a scarf joint
is being accomplished (if the loading is tensile): as the
scarf angle decreases, the combined stresses which are
likely to be critical for failure decrease, which means
that‘more load can be applied‘for smaller scarf angles.
General statements cannot easily be made for bend-
ing load; there is a complex interaction of parameters.
To keep the length of the present discussion within reason-
able bounds, the bending problem will not be treated here.
The user may deduce all the necessary information from
Table 11.
147
Table 11. --Maximum values of the combined stresses Nl T1
T0 Signs correspond to the loading sense of Fig.
7,yp. 26, with adherend 2 assumed to be the stiffer
one.
Tension Bending
a B Y N1 -N2 Tl Toy N1 -N2 T1 Toy
5 4 1 0.0907 0.0831 0.0869 0.0710 0.0853 0.0853 0.0818 0.0668
2 0.1450 0.1331 0.1390 0.1135 0.1318 0.1239 0.1278 0.1044
4 0.1971 0.1820 0.1895 0.1548 0.1745 0.1672 0.1708 0.1395
8 0.2343 0.2226 0.2284 0.1865 0.2061 0.1970 0.2015 0.1646
20 1 0.0907 0.0831 0.0869 0.0710 0.0728 0.0728 0.0670 0.0571
2 0.1215 0.1103 0.1159 0.0947 0.0921 0.0858 0.0889 0.0726
4 0.1440 0.1320 0.1380 0.1127 0.1040 0.0957 0.0998 0.0815
8 0.1591 0.1492 0.1541 0.1259 0.1085 0.0979 0.1032 0.0843
100 1 0.0907 0.0831 0.0869 0.0710 0.0469 0.0681 0.0457 0.0373
2 0.1041 0.0953 0.0997 0.0814 0.0485 0.0447 0.0466 0.0381
4 0.1127 0.1037 0.1082 0.0884 0.0468 0.0416 0.0442 0.0361
8 0.1188 0.1090 0.1139 0.0930 0.0415 0.0385 0.0400 0.0327
10 4 1 0.1868 0.1566 0.1717 0.1404 0.1557 0.1557 0.1431 0.1170
2 0.2532 0.2133 0.2332 0.1907 0.2034 0.1699 0.1867 0.1526
4 0.3035 0.2589 0.2812 0.2298 0.2341 0.1941 0.2141 0.1750
8 0.3421 0.2900 0.3161 0.2584 0.2452 0.2050 0.2251 0.1840
20 1 0.1868 0.1566 0.1717 0.1404 0.1080 0.1080 0.0990 0.0809
2 0.2171 0.1826 0.1999 0.1634 0.1142 0.0940 0.1041 0.0851
4 0.2384 0.1996 0.2190 0.1791 0.1095 0.0931 0.1013 0.0828
8 0.2538 0.2102 0.2320 0.1897 0.1048 0.0878 0.0941 0.0769
100 1 0.1868 0.1566 0.1717 0.1404 0.0464 0.0463 0.0425 0.0348
2 0.1968 0.1649 0.1808 0.1479 0.0514 0.0406 0.0380 0.0310
4 0.2025 0.1696 0.1861 0.1521 0.0536 0.0356 0.0332 0.0290
8 0.2051 0.1729 0.1890 0.1545 0.0573 0.0325 0.0335 0.0295
20 4 1 0.3852 0.2682 0.3267 0.2681 0.2443 0.2442 0.2067 0.1697
2 0.4519 0.3132 0.3826 0.3141 0.2618 0.1857 0.2238 0.1836
4 0.4979 0.3435 0.4207 0.3454 0.2710 0.1837 0.2245 0.1843
8 0.5222 0.3699 0.4460 0.3659 0.2878 0.1560 0.2174 0.1799
20 1 0.3852 0.2682 0.3267 0.2681 0.1463 0.1463 0.1013 0.0841
2 0.4076 0.2842 0.3459 0.2839 0.1537 0.1197 0.0976 0.0835
4 0.4183 0.2955 0.3569 0.2928 0.1645 0.1043 0.0978 0.0856
8 0.4218 0.3042 0.3630 0.2977 0.1875 0.0966 0.1052 0.0942
100 1 0.3852 0.2682 0.3267 0.2681 0.1028 0.1028 0.0548 0.0501
2 0.3901 0.2726 0.3314 0.2720 0.1121 0.0901 0.0585 0.0540
4 0.3923 0.2753 0.3338 0.2739 0.1317 0.0831 0.0676 0.0630
8 0.3932 0.2768 0.3350 0.2749 0.1572 0.0797 0.0796 0.0746
148
Table 11 Continued.
Tension Bending
a B Y N1 -N2 Tl T0y N1 -N2 Tl T0y
30 4 1 0.5758 0.3257 0.4507 0.3727 0.3140 0.3140 0.2253 0.1873
2 0.6256 0.3566 0.4911 0.4060 0.3391 0.2526 0.2285 0.1926
4 0.6494 0.3821 0.5756 0.4257 0.3704 0.2150 0.2320 0.1998
8 0.6554 0.4046 0.5300 0.4367 0.4317 0.1947 0.2500 0.2214
20 1 0.5757 0.3257 0.4507 0.3727 0.2256 0.2255 0.1258 0.1129
2 0.5882 0.3363 0.4623 0.3821 0.2480 0.1949 0.1333 0.1215
4 0.5935 0.3432 0.4683 0.3869 0.2934 0.1778 0.1527 0.1412
8 0.5955 0.3475 0.4715 0.3894 0.3410 0.1693 0.1742 0.1625
100 1 0.5757 0.3257 0.4507 0.3727 0.2108 0.2107 0.1066 0.1000
2 0.5783 0.3282 0.4533 0.3748 0.2457 0.1950 0.1235 0.1161
4 0.5795 0.3296 0.4546 0.3758 0.2681 0.1868 0.1344 0.1265
8 0.5801 0.3303 0.4552 0.3763 0.2804 0.1826 0.1405 0.1323
40 4 1 0.7406 0.3274 0.5340 0.4467 0.4109 0.4109 0.2490 0.2167
2 0.7683 0.3476 0.5580 0.4662 0.4523 0.3445 0.2598 0.2306
4 0.7791 0.3636 0.5713 0.4766 0.5286 0.3047 0.2874 0.2608
8 0.7820 0.3750 0.5785 0.4820 0.6146 0.2837 0.3221 0.2970
20 1 0.7406 0.3274 0.5340 0.4467 0.3506 0.3506 0.1816 0.1683
2 0.7470 0.3334 0.5402 0.4517 0.4109 0.3160 0.2090 0.1954
4 0.7499 0.3369 0.5434 0.4542 0.4595 0.2977 0.2318 0.2176
8 0.7511 0.3389 0.5450 0.4555 0.4882 0.2884 0.2455 0.2308
100 1 0.7406 0.3274 0.5340 0.4467 0.3905 0.3905 0.1956 0.1843
2 0.7419 0.3287 0.5353 0.4478 0.4133 0.3786 0.2069 0.1949
4 0.7426 0.3294 0.5360 0.4483 0.4256 0.3724 0.2129 0.2007
8 0.7429 0.3298 0.5363 0.4486 0.4319 0.3693 0.2161 0.2037
149
4.3. Construction of Designgurves for
Scarf Joints with LinearlyeElastic
Adhesives
If the adhesive is assumed to fail when a certain
combined stress attains some specified allowable value,
presumably determined empirically, it is possible to use
Table 11 in design. Let 2a be this allowable stress, whether
it is a maximum normal stress, a principal shear or an
octahedral shear. Designate the four combined-stress quan-
tities in Table 11, collectively, by Ncom‘ It is up to the
user to decide which "law of failure" he wishes to select,
and to determine the appropriate value of 2a for his chosen
law. Recall that the "load stress" 0
X0
tensile stress for tensile loading, or the maximum bending
is the actual applied
stress ("Mc/I"). Since Table 11 is constructed for OX0 = 1,
when the adhesive combined stress is equal to the "allowable"
we must have
2a = OXONcom 4.3.1
Since 2a is a known constant, the external loading 0 which
X0
the designer is allowed to introduce can be computed from
this equation. A dimensionless load quantity, convenient
for design, is
2 = oxo/Xa = l/Ncom 4.3.2
Curvesof X can readily be constructed from Table 11,
by crossplotting quantities as desired. It is always
150
necessary to interpolate three ways in a, B, Y, since we have
tabulated a three-parameter family. Some sample "design
curves" are shown in Figure 30 for tensile loading. In one
diagram, N = N
com 1’ in a second, NC
om = N2; etc. The values
actually plotted are not 2 but 2 sin 0, since these particu—
lar samples are plotted against scarf angle a. The a-
dependence of 2 is such that the curves plot over a convenient
range of ordinates when multiplied by sin a or sin 0 cos a.
The parameters chosen represent a random sampling. Note
that all cases of Y = l are independent of B, a result from
Ref. 7. For actual design use, more extensive families of
curves would be needed, and systematic interpolation schemes
are necessary. It may prove simplest to interpolate directly
in Table 11, and not work with design curves at all.
For bending load, some of the curves can be plotted
on the sin OI/NC m basis of Figure 30, and others cannot, de-
0
pending upon the parameter values. The large-'8 cases in
particular show a very large peak near a = 10-15°, shifting
toward 20° for intermediate values of B. This implies that
certain scarf angles should be favored by designers in par-
ticular flexibility ranges. The device of using a log-log
scale permits a smoother plot with all curves treated on
the same basis; further study would probably reveal an even
better device. A few samples of the log-log plot for bend-
ing load are shown in Figure 31, for two failure criteria.
151
s1n 0L/Nl sin a/N2
" 8:20 Y = l
2.0-- 3‘4
4
1.01 8
Max. Tensile Stress
w (Tensile Loading) Max. Tensile Stress
(Compressive Loading)
: ,L g i 0 3 i 3 ﬂ
5°10° 20° 30° 40° 5°10° 20° 30° 40°
6 a
Sln 0I/T0y
1.5Y
1.0‘
0.5
0.2“ Max. Shear Max. Octahedral
Shear ~
HI I I I 0 I 1% I I
5°10° 20° 30° 40° 5°10° 20° 30° 40°
0. 0.
Elastic "design curves" in dimensionless form, for
various failure criteria (failure in N1, N2, Tl'
Toy)° Tensile or compressive load.
152
The discussion of design is an appropriate place
to review the use of both Table 11 and the primary results
of Appendix F. It is necessary to align the geometry and
loading so that it coincides with Figure 7, p. 26, with
adherend 2 made the stiffer one. All tabulated quantities
assume that the applied loading has the sense of Figure 7.
The parameters
E1” E (l - V2)
_ _ 2 l
Eah(l ‘ V1) E1(l - V2)
are calculated, using the actual values. Interpolation is
then carried out as needed. The tabulated values of B
(4,20,100) and Y (1,2,4,8) are uniformly spaced on a loga-
rithmic scale, permitting 3- and 4—point Lagrangean inter-
polation formulas. The angles a = 5, 10, 20, 30, 40° permit
linear interpolation (omit 5°) or logarithmic interpolation
(omit 30°) by 4-point formulas.
The Poisson's ratios 01 = v = 0.3, together with
2
the ratio Ea/Ga = 8/3 (implying an adhesive Poisson's ratio
of 1/3), are inextricably incorporated into the Ritz matrix
and thus into all the results. To this extent the user
cannot make any adjustments. No calculations have been
performed, but the errors are not believed to be very large
if the user's Poisson's ratios differ slightly. To the ex-
tent that Poisson's ratio affects Y and B, it can be accounted
for exactly.
153
B 4
Y 8
Max. Octahedral
Shear
5° 10° a 20° 30° 40°
Max. Tensile Stress
(Moment Opposite Fig. 7)
5° {0° 20° 30°40°
a
Fig. 31 Elastic "design curves" in dimensionless form, for
two failure criteria (N2, To ). Bending loads in
two senses. y
CHAPTER V
CONCLUSIONS AND SUGGESTIONS FOR
FURTHER RESEARCH
5.1. Conclusions
The Rayleigh-Ritz method, all in all, appears to
have handled the present complex problem with quite good
results for the case of tensile loading, and fair to good
results for most of the bending load cases. It has been
possible to account for many of the phenomena observed in
the calculations by physical arguments, and the large range
of primary variables explored is probably adequate to give
the user a good idea of the overall adhesive stress distri-
bution for any case encountered in practice. The results
are valid for a linearly elastic adhesive only, but there
are enough adhesives for which this is a fair approximation
to make the present results useful. How to employ these
results in design has been outlined in sufficient detail
that the next stegsrequired are well within the grasp of
most stress analysts.
While most of the practical information a user
needs has been tabulated and analyzed here, any additional
data desired in the future may be calculated from the basic
154
155
Ritz matrix coefficients for the two types of loading con—
sidered. The latter are available in punched-card form.
The computer program in Appendix G can be used for further
studies along the present lines, and with minor changes can
cope with other types of loading. It is easily possible to
alter some of the geometric assumptions about the shape of
the adherends.
The nature of the results indicates a large stress
disturbance in the adherends, at the ends of the scarf joint.
The singularities characteristic for the adherend shapes
have been activated by a formulation of the problem which
introduces a finite shear stress on the inclined adhesive
interface, but leaves the top and bottom adherend faces
("around the corner") stress free. Some alternate formula-
tion, which avoids this difficulty in a manner consistent
with the way scarf joints are actually manufactured, would
probably greatly improve the rate of convergence of the
Ritz method.
5.2. Future Research
Several interesting possibilities can be explored
within the present framework; these were not studied here
to keep the amount of computer time within reasonable
bounds. One useful item would be to examine the effects
of varying the Poisson's ratios of adherends and adhesive.
A few additional cases might also be computed to facilitate
156
interpolation in the bending load problem for the smaller
scarf angles.
The most interesting research centers around the
stress singularities characteristic of the adherend-adhesive
corners at the joint ends. If these can be correctly de-
duced, it seems likely that the Ritz method's convergence
could be greatly accelerated, and far fewer equations would
be required to calculate any given case. This would open
the possibility of dispensing entirely with design tables
and curves. It would also make the Sherman-Lauricella in-
tegral approach a practical computing tool for this problem.
Another interesting study would circumvent the prob—
lem of stress singularities entirely, and yet still be a
practical computing tool for scarf joints in the practical
(lo-30°) range of scarf angles. This would be to treat each
adherend as a beam of variable cross section, in extension,
bending and shear. The adhesive model could be the same as
at present, and there would be little difficulty in allow—
ing for the component of adhesive normal stress not considered
here, in the simpler proposed problem. The latter could be
tested against the present results to delimit its range
of validity. An additional consideration motivating this
study is the following: it would be highly desirable to
have available a workable, simpler model of the scarf joint
for the purposes of studying adhesives with complex rheologi-
cal behavior.
10.
11.
12.
13.
BIBLIOGRAPHY
Langhaar, H. L., Energy Methods in Applied Mechanics,
John Wiley, New York, 1962.
Courant, R., and Hilbert, D., Methods of Mathematical
Physics, Interscience, New York, 1953, pp. 175-176.
Kantorovich, L. V., and Krylov, V. 1., Approximate
Methods of Higher Analysis, Interscience, 1964.
Muskhelishvili, N. I., Some Basic Problems of the
Mathematical Theory of Elasticity, P. Noordhoff,
Holland, 1963, pp. 418—422.
Lauricella, G3, "Sur l'intégration de l'equation rela:
tive a l'equilibre des plaques élastiques encastrees"
Acta Math. Vol. 32 (1909), pp. 201-256.
Flﬁgge, W., Handbook of Engineering Mechanics, McGraw-
Hill, 1962, Section 37, p. 28-29.
Lubkin, J. L., "A Theory of Adhesive Scarf Jointsf'
Journal of Applied Mechanics, June, 1957.
Eley, D. D., Editor, Adhesion, Oxford University Press,
1961.
Bikerman, J. J., The Science of Adhesive Joints, Academic
Press, 1961, Chapter 7.
Benson, N. K., "The Mechanics of Adhesive Bonding,"
Applied Mechanics Surveys, Spartan Books, Washing-
ton, D. C., 1966.
Volkersen, O., Luftfahrtforchung 15, 1938, p. 41.
de Bruyne, N. A., and Houwink, R., Adhesion and Adhesives,
Elsevier Publishing Company, New York, 1951, Chapter
4 (by Mylonas and de Bruyne).
Goland, M. and Reissner, E., "The Stresses in Cemented
Joints," Journal of Applied Mechanics, Trans. A.S.
M.E., Volume 66, 1944, p. A-l7.
157
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
158
Plantema, F. J., and Hartman, A. M., Rept. M. 1181,
Nat. Luchtvaartlaboratorium, Amsterdam, 1947.
Cornell, R. W., "Determination of Stresses in Cemented
Lap Joints," Journal of Applied Mechanics, Trans.
A.S.M.E., Vol. 75, 1953, pp. 355-364.
Mylonas, C., "Experiments with Composite Models," Pro-
ceedings of the SESA. XII, 1955, pp. 129-142.
Williams, M. L., "Stress Singularities Resulting From
Various Boundary Conditions in Angular Corners of
Plates in Extension," J. Appl. Mech., 19:4, Dec.
1952, pp. 526—528; also 20:4, Dec. 1953, p. 590.
McLaren, A. S. and MacInnes, I., "The influence on the
stress distribution in an adhesive lap joint of
bending of the adhering sheets," British Journal
of Applied Physics, 9, 1958, p. 72; AMR, 12, 1959,
Rev. 1280.
Sazhin, A. M., "Determining the Stresses in Glued
Joints Between Metal Plates," Russian Engineering
Journal, 1964, No. 11, pp. 45-49.
Misztal, F., "Stress Distribution in Glued Single and
Multi-Joints of Sheets Subjected to Shear Along the
Splice Line," Bull. DE L'ACADEMIE POLONAISE DES
SCIENCES, Cl, IV - Vol. IV, No. l, 1956, pp. 21-27.
Lubkin, J. L., and Reissner, E., "Stress Distribution
and Design Data for Adhesive Lap Joints Between
Circular Tubes," Trans. ASME, Vol. 78, 1956, pp.
1213-1221.
Sherrer, R. E., Rept. 1864, U. S. Forest Products Labora-
tory, Madison, Wisconsin, 1957; AMR, 12, 1959, Rev.
1897.
Timoshenko, S., and Goodier, J. N., Theory of Elasticity,
McGraw-Hill, New York, 1951.
Mﬁller, G., Dr.-Ing Thesis, Technische Universitat,
Berlin-Charlottenburg, 1959.
Sokolnikoff, I. 8., Mathematical Theory of Elasticity,
McGraw-Hill, New York, 1956.
Hartman, A., Rept. M. 1275, Nat. Luchtvaartlaboratorium
Amsterdam, 1948.
27.
28.
29.
30.
159
de Bruyne, N. A., Aero Research Technical Note No. 61,
February, 1948.
Shield, R. T., Quarterly of Applied Mathematics, 15,
July, 1957; AMR, 11, 1958, Rev. 128.
Norris, C. B., "Plastic Flow Throughout the Volume of
Thin Adhesive Bonds," U.S. FPL Rept. No. 2092,
Mar. 1958.
Hartman, A., Rept. M. 1475, Nat. Luchtvaartlaboratorium,
Amsterdam, 1949.
APPENDIX A
FORMULATION OF THE EXPRESSION FOR TOTAL
POTENTIAL ENERGY
For the case of plane stress, the strain energy per
unit width in each adherend takes the standard form
(i = 1,2):
= H:—
2(1 - v. i) A.
2 (avi 2 ani avi
+ _8y + 2"1 _8x —8y
2
1
l - v. Sui avi
+
where A1 is the area of the ith
adherend, the EJ.- and vi
are the usual elastic constants, and the ui and the vi are
(respectively) the x- and y- components of displacement.
The total potential energy of the scarf joint is
Q = U + U + U + W A-2
51 52 sa
I
where Usa = strain energy of the adhesive film
I
W = potential energy of the external forces
The strain energy of the adhesive is derived in Section 2.2.
160
161
+h
1 2 . 2 2.
' = _ -
Usa 20 sin a j:h [(ul u2) (Ea Sin 0 + Ga cos a)
+ ( - )(E 2 G ' 2 + 2( )(
v1 v2 a cos a + a Sln 0) ul - 112 v1
- v2) Sln a cos 0 (Ga - Ea)] dy A-3
The only nonvanishing external loading consists of the pure
tensile or bending stress ox(ic,y) on x::tc = ih(2 + cot a),
the extreme ends of the joint (Fig. 7, p. 26). The poten-
tial energy due to loading is thus
h h
W = fjlh ox(c,y)u2(c,y)dy +-j:h oX(-c,y)u1(-c,y)dy
A-4
This accounts for all terms needed for equation A-2, which
is the same as 2.2.13. Using the nondimensionalization of
2.2.14, the dimensionless total potential energy w of
2.2.16 is produced.
APPENDIX B
GENERATION OF RITZ EQUATIONS
The total potential energy expression in its dimen-
sionless form is given by equation 2.2.16, p. 39. The four
principal equations are derived from the following four
relations
8A_—_'— 0 B - 1
m,n
8w _ _
8B - 0 B 2
,n
8w
= 0 B - 3
8Cm'n
8w
8D - 0 B - 4
,n
Note that Ul contains only Am,n' V1 contains only Bm,n' U2
only Cm,n and V2 only Dm,n°
300 f [(aul) a (3'11 (”1) a (3‘11)
=0= ——+\) .—
3Am,n 3X BAm,n 3X 1 BY aAm'n 8X
+
8U 6V 8 EU
1 l 1 1
'2- (1 ‘ Vl’(—ay + 'a—x" m— ('37-'de dY
m,n
+ - - -
_1 H1(U1 U2) 5r— H2 (V1 V2) F—A ]dY
m,n m,n
162
163
2 1 BUl(-C,Y)
+ (1 - 01).]:1 00(y) BA dY B-5
m,n
Here and below, Q) and ® indicate integrations over ad-
3,n (:Zl)+ “1(xﬁ) n(§‘£
herends l and 2.
80) = 0 ___f Ravl
33mm BY
BU av avl
1 1 a l
+ 2 (l V »)(a ax )% n\3x)1)]dx dY
fl 8V1 8V1
+ l Hh(Vl - V2) ﬁ— - H2(Ul " U2) 5T— dY
- m,n mln
B-6
a (302)
acm’n Bx
3w =~0 = er (302) a (002)+ v (av2
acmm BX BCm'n Bx 2 BY
BU
l 2 B 2
+ f (l - v2) (82— + BX’ 3cm n (BY ) dx dY
I
)
m,n 2 2 BCm'n
2 +1 BU2(C,Y)
- (l - 02) f 00(Y) —-— dY B-7
-1 '
BCmn
3V2 av
30) 2
BD =0=YII(W’ BY)+"2(:XZBDm) ”1;”
m,n ® D,mn
dX dY
1 3U2 SUZ
+11 -H1(Ul-U2) +H(Vl-V -——dY
164
[1 3V2 3V2
+ - H (V - v + H (U -U ) —————' dY
_1 h 1 m,n 2 l 2 aDm,n
B-8
Substituting the explicit double sums for U1, V and their
1
derivatives into the equation B-S, it becomes
M M-k Y cot a .
0 = [1 fc [A Ak . k m Xk+m-2 Yn+3
k=0 j= E0 -1 -c '3
. k+m-l n+j-l l _ n' A .Xk+mYn+j-2
+ v1 Bk,j 3 m x Y + 2 (1 v1) 3 k,
+ 1 (1 - v ) B k n xk‘Lm‘l yn+j'l] dx dY + J’1 H (A
2 l k,j -l l k,j
k+m n+j
- C . - H B . - . X Y d +
kpj) 2( kt] Dklj)] Y
2 l m n
+ (l - v1) j’ 00(Y)(-C) Y dY B-9
-1
In all the double integrals of equation B-9, inte-
gration is carried out first with respect to X. The result
is
M M k 1 km k+m+n+j-l k+m-1
= 2 Z Ak . k~+ m _ 1 Y (cot a)
k= 0 j= -0 '3
n+j _ k+m-l jm [ k+m+n+j-l k+m
Y ( C) l] + v1 Bk,j k + m Y (cot a)
(l - v1) .
_ n+3 l _ k+m n3 .
Y (C) ]+——2——Ak,jk+m+1
k+m+1 _ Yn+j-2 k+m+l]
(cot a)
.[Yk+m+n+j-l ('C)
165
(l - v ) .
l nk k+m+n+3-l k+m
+ —2—' Em W [Y “3°“ °"
_ Yn+j_l (-C)k+m] dY + jrl H (A - C )
'1 l klj klj
k+m j+n
- H B . - D . Y
2( k.) k.3)] X M
2 l m n
+ (l - v1) 1’ 00(Y)(-C) Y dY B-lO
-1
Equation B-lO is now integrated with respect to Y. The
final equation is in the form
M M-k
km . k+m-l
0 = Z Z A . _ [ f(k+m+n+3) (cot a)
k=0 j=0 k,j k + m I
- (-c>k”‘“‘1 f(n+j+l)]
+ v B r:m_ f(k+m+n+') (cot mkm - (-C)k+m f('+n)
l krj + m J A J
(1 - v ) .
1 DJ f . k+m+1
+ 2 ki-m-kl Ak,j[ (k+m+n+j) (cot a)
_ (_C)k+m+l f(n+j-JJ]
(l - v )
1 kn . k+m
+.____§_._ Bk,j k + m [ f(k+m+n+j) (cot a)
- (-c>k+m f(j+n) + H (A . - c .) - H (B .
1 k1: k.) _ 2 k.)
_ Dk j)](cot a)k+m f(k+m+n+j+1)
I
+
l
(l - vi) I. 00(Y)(-C)m Yn dY B-ll
—l
166
where
0
ll
2 + cot a
f(R) _ l - (-l)R = 0 : R even
‘ R Z/R : R odd
Other quantities are now defined as follows
(cot on)k+m_l f(k+m+n+j)
¢o
(cot a)k+m
¢l f(k+m+n+j+1)
After rearrangement, first set of Ritz equations, corres-
ponding to B—l, becomes
M M-k
_ km _ _ k+m-l .
0 - E .2 Ak,j[m (‘1’0 ‘ C) f‘n+3+1’)
k—O j—O
l - v .
1 n3 2 _ _ k+m+l ._
+ 2 k-Fm4-l (¢O cot a ( C) f(n+3 14
v jm l - v 4
l 1 kn
k+m .
- (-C) f(n+3) - H2¢l - Ck,jHl¢1 + Dk'jH2¢l
2 +1 n
+ (l - v1) Jf 00(Y)(-C)mY dY B—12
-1
Equation B-12 corresponds to 2.2.19. The other three basic
Ritz equations are derived similarly. Note that the fore-
going reflects the contribution of the Am,n coefficients
to the corresponding row of the final Ritz matrix. Each of
equations B-2 through B-4 makes a similar contribution, and
these appear in equations 2.2.20-2.2.22.
APPENDIX C
SELF-EQUILIBRATED POLYNOMIAL STRESSES
In equation 2.3.1, the part (2) contribution to the
adhesive normal and shear stresses is assumed to have the
form
, K
0(2) — Z a 5m c-1
n m
m=l
K m
Tns = X bmS C-2
m=l
,n undetermined coefficients am, bm. Here S is the di-
mensionless distance along the inclined adhesive boundang
or s/h in terms of Fig. 7, and is measured from the origin
of coordinates. From the diagram cited, S ranges from
-csca to csca. The integer K must be odd.
The purpose of this derivation is to eliminate
three coefficients from the set (am, bm), by enforcing the
requirement that C-l and -2 represent a wholly self-equili-
brated stress distribution. Of the three static equilibrium
conditions, the vanishing of the resultant force in the X-
direction (by integration and taking components) yields:
167
168
csca
K m+l
m£1(am Sincx+ bm cos a) mjrl = O (m even) C-3
‘CSCG
Similarly, the vanishing of the Y- resultant gives
csca
K Sm+l
m£1(-am cos a + bm Sln a) 577I = O (m even) C-4
-csca
The vanishing of the resultant moment about the origin
produces:
CSCG
M
m
B
m
H
o
(m odd) C-S
‘CSCG
After entering the limits, equations C-3 and C-4 may be
solved to obtain
m+l
K
2 am £%§§£%—— = O (m even) C-6
m=l
K (csca)m+l
mil bm -ﬁ—:fI——-= 0 (m even) C'7
After substitution of these two relations equation C-5
becomes
+1
2 (csca)m ‘_ 0 dd 8
an—T'T‘" (“‘0’ C“
169
Equations C—6, C-7 and C-8 give the desired expressions for
eliminating aK-l’ bK-l and aK, respectively:
- - K K333 (CSCG)m-K+l C.,g
aK-l am m + l
m even
K-3 m-K+l
_ (csaﬁ
bK-l ‘ K 2 bm m + 1 c-1o
m even
K-2 m-K
3K = - (K + 2) 2 am % C’ll
m odd
Expanding C-1 and C-2
K-2
1(2) — 2 b Sm + b SK-l + b SK c-12
ns m K-l K
m=l,2,3 ..
K—2
0(2) - a 8m + a SK.l + a SK C-l3
n m=l m K-l K
These become, after substitution from C-9, -10 and -11:
0(2) = (K'l%/2 a SZm-l _ (K + 2) 2m-K-lSK
n _ 2m-l m
m—l,2,3,...
+ (K—3 /2 a S2m _ K(caxnzm'k+lsK—l C-l4
_ 2m 2m + I
m—l,2’3,ooo
(K—3) 2
(2) (K+l)/2 2m-l / 2m
Tns = me-l S + g b2m S
m=l'2' 'ooo m=l,2' poo.
K(csca) 2m-K+l sK‘l]
- 2m + l 15
O
l
Equations C-14 and -15 are the same as 2.3.2.
APPENDIX D
DISPLACEMENT DETERMINATION FOR THE INTEGRAL
EQUATION APPROACH
The displacement components U1, V1 of the first
adherend are related to the analytic functions ¢(z), and
w(z) of the complex variable 2 = X + iY by the following
equation4
“'1—
Zulml + ivl) = xl¢(2) - 2¢ (2) - W(z) D-l
where
shear modulus
X1 = (3 - vl)/(l + v1) for plane stress
C
ll
1 Poisson's ratio
and the values of the functions ¢(z), 6(2) and W(z) are
defined below in terms of a "density function" w(t), pre-
sumed known at this stage. With the w(t) actually used,
(2)
l
the displacements calculated are the contributions U
and ViZ) of section 2.3.
¢(z) = 11““) ds D-2
Zni s- z
170
l7l
' _ w(S) _
¢J'—_ s - ”‘6
But
w(t) d§ _ . _ th) ds
21d 3 ft- ‘ conlugate 0f [ W —s - t]
X1 ”(HIE—E331?
- w(t)/2 D-7
x1 w(t)/2 D-8
172
]_ ' s—t—_ 1 . ' g-E
'mw(S)—-Eds— mconjfw(5)§T_t'ds
D-9
The terms of equation D-9 reduce
, upon integration by
parts, to
- —l+ con' E - E w(s)
2 3 s - t
_1 7—3- E-‘t‘ _
—-2—TT-i-J’ws (1(a) D10
since the first term vanishes. Let
s - t = re16
5 - E = re-16
5 ' E = e216 ; d i ' E =12e21ede
s - t s - t
The term in equation D-lO now becomes
%fw(s) e216d6 D-ll
But, from the basic integral equation of the problem, 2.3.14,
the expression of equation D-ll is also
1
w(t) + ?J{w(s)d6 - f(t) D-12
Upon substitution of relations D-7, D—8 and D-12 into equa-
tion D-6,
173
X _
2u1(Ul + ivl) = 1 ”(5) ' w(t) d5 + 1 ‘[w(S) - w(t) ds
Zﬁif s - t 2ni §'_‘E §¥E
+ %:[w(5)d6 - f(t) +%Xlw(t)+'%uﬂt) 0‘13
now write
w(s) = p(s) + iq(s) D-l4
and convert the integration in 6 to s by the relation
-36. _
de—FS-ds D15
Since4
33 = cos d
as r ’
d6 = COS a ds D-l6
r
where a is the angle between the vector 5 - t and the out-
ward normal at s (unrelated to the scarf angle a used
elsewhere).
After substitution of D—l4 and D-l6, equation D-13
becomes
X
2u1(Ul + iVl) = 32—13%] [9(8) - P(t)] + i[q(5) " q(t)] s-t
ds
3-3
Z—i—i—f [p(s) - p(t)] + itq~~ - q(t)]
+ %;[[p(s) + iq(s)] £2%_3 ds +%xl[P(t)4-iqﬁﬂ]
4-%[:;Kt)+ iq(t)] - fl(t)- if2(t) D-17
For numerical integration to find the displacement components,
the boundaries AB, BC, CD and DA in Fig. 7 are divided into
174
11’ 12, 13 and I4 intervals, respectively. Equation D—l7
is rewritten as a sum of integrals over each interval along
the boundary.
ﬂ?“
2111(Ulj + iV .)
E . ]cos a'k
l] k=1f pk(s) +lqk(s)_—Lr.k ds - flj
J
i I
_ ' L ' _ __ ‘
lfzj +2Xl[pj + iqj] 2n kgljf [pk(s)
ds
p- +i[q~~~~OCDO
00.0 00 0&00>0 0
O\OGJQO‘m'bLJNF‘O
ALPHA=10 DEGREES.
187
GAMMA=1
TENSION BENDING
N T N T
000000000000000000000000000000
000301 001710 000077 000423
000301 001710 000088 000406
000301 001710 000153 000386
000301 001710 000231 000360
000301 001710 000295 000327
000301 001710 000327 000287
000301 001710 000322 000239
000301 001710 000279 000185
000301 001710 000206 000126
000301 001710 000109 000064
000301 001710 -000001 000000
000301 001710 -000109 ~000064
000301 001710 “000206 -000126
000301 001710 ~000279 ~000185
000301 001710 ~000322 -000239
000301 001710 ~000327 -000287
000301 001710 ~000295 ~300327
000301 001710 ~000231 -000360
000301 001710 -000153 -000386
000301 001710 -000088 -000406
000301 001710 -000077 -000423
GAMMA=4
0000000 0000000000000000000000
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000296 001751 000082 000070
000304 001738 “000034 000028
000311 001724 “000136 “000012
000317 001711 -O00219 “000049
000321 001698 ~000274 “000082
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000321 001671 -O00299 -000134
000318 001656 ~000271 “000153
000311 001642 ~000222 ~000168
000304 001627 -000161 “000179
000295 001612 -000098 -000185
000286 001597 -000047 ~000189
000279 001582 “000083 “000192
000277 001567 -000048 ~000195
BETA=100
GAMMA=2
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N T N T
000000000000000000000000000000
000320 001801 000034 000380
000311 001791 000154 000358
000302 001780 000264 000332
000295 001770 000339 000300
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000289 001752 000347 000218
000289 001743 000287 000169
000293 001735 000197 000119
000297 001727 000091 000067
000302 4001719 “000021 000015
000307 001711 “000125 “000035
000311 001703 “000213 “000082
000314 001694 “000275 “000125
000315 001686 “000307 “000163
000314 001677 “000308 “000196
000311 001668 “000278 “000224
000306 001659 “000224 “000245
000300 001649 “000156 “000261
000293 001639 “000089 “000272
000288 001629 “000047 “000280
000286 001619 “000061 “000288
GAMMA=8
000000000000000000000000000000
000322 001883 000234 000281
000297 001864 000430 000255
000284 001845 000483 000222
000280 001827 000442 000187
000282 001810 000346 000149
000288 001793 000222 000111
000296 001777 000090 000074
000305 001760 “000034 000038
000313 001744 “000140 000005
000319 001728 “000221 “000025
000324 001711 “000275 “000051
000326 001695 “000301 “000074
000326 001678 “000298 “000093
000323 001661 “000272 “000107
000317 001644 “000228 “000117
000309 001626 “000171 “000125
000300 001609 “000111 “000129
000290 001591 “000057 “000131
000281 001573 “000020 “000131
000275 001555 “000010 “000132
000273 001538 “000038 “000132
*‘O‘3C)OC)O0(D\10Lnb'UIV*‘O
I
O
O
u.
F‘OCDC)O(DC)O(DC>O
O\DGJQ()UIb(JhJH’O
ALPHA=20 DEGREES.
188
GAMMA=1
TENSION SENDING
N T N T
0.............o.....00......00
0.1170 0.3214 0.0752 0.2032
0.1170 0.3214 0.0766 0.1937
0.1170 0.3214 0.0833 0.1819
0.1170 0.3214 0.0888 0.1673
0.1170 0.3214 0.0899 0.1496
0.1170 0.3214 0.0854 0.1292
0.1170 0.3214 0.0754 0.1063
0.1170 0.3214 0.0607 0.0814
0.1170 0.3214 0.0424 0.0551
0.1170 0.3214 0.0218 0.0278
0.1170 0.3214 “0.0000 “0.0000
0.1170 0.3214 -0.0218 -0.0278
0.1170 0.3214 -0.0424 -0.0551
0.1170 0.3214 ~0.0607 -0.0814
001170 0.3214 -000754 “001063
-001170 0.3214 -0.0854 -O.1292
0.1170 0.3214 -0.0899 -0.1496
0.1170 0.3214 -0.0888 -0.1673
0.1170 0.3214 -0.0832 -0.1819
0.1170 0.3214 -0.0766 -0.1937
0.1170 0.3214 -0.0752 -0.2032
GAMMA=4'
.0...oo.............00......00
0.1544 0.4136 0.0815 0.2207
0.1260 0.3971 0.1250 0.1983
0.1071 0.3832 0.1403 0.1727
0.0971 0.3717 0.1345 0.1451
0.0943 0.3622 0.1143 0.1164
0.0969 0.3542 0.0853 0.0873
0.1031 0.3475 0.0524 0.0589
0.1112 0.3415 0.0194 0.0316
0.1197 0.3359 -0.0110 0.0061
0.1276 0.3303 -0.0370 -0.0172
0.1339 0.3246 -0.0574 ‘0.0379
0.1381 0.3184 -0.0718 -0.0558
0.1399 0.3116 ‘0.0802 -0.0709
0.1392 0.3042 ‘0.0829 -0.0830
0.1357 0.2959 -0.0807 -0.0922
0.1297 0.2869 -0.0744 “0.0989
0.1212 0.2770 -0.0651 -O.1030
0.1109 0.2663 “0.0544 ‘0.1051
0.1000 0.2548 ~0.0445 -0.1054
0.0904 0.2429 -0.0383 -0.1043
0.0858 0.2306 -0.0402 -0.1022
BETA=4
GAMMA=2
TENSION SENDING
N T N T
.......................0......
0.1387 0.3763 0.0760 0.2205
0.1275 0.3659 0.0945 0.2033
0.1170 0.3569 0.1075 0.1836
0.1092 0.3494 0.1113 0.1613
0.1047 0.3432 0.1054 0.1366
0.1035 0.3383 0.0911 0.1101
0.1050 0.3343 0.0704 0.0826
0.1084 0.3310 0.0458 0.0548
0.1129 0.3282 0.0197 0.0273
0.1179 0.3255 ”0.0059 0.0007
0.1225 0.3228 “0.0293 ‘0.0245
0.1264 0.3199 ‘0.0494 -000478
0.1292 0.3167 “0.0652 ‘0.0690
0.1303 0.3130 ”0.0762 -0.0878
0.1298 0.3087 ‘0.0821 -0.1039
0.1272 0.3037 -0.0829 -0.1172
0.1227 0.2979 “0.0790 “0.1277
0.1164 0.2912 '0.0714 -0.1354
0.1090 0.2838 “0.0622 -0ol405
0.1021 0.2757 -0.0547 “0.1433
0.0981 0.2671 ‘000549 '001445
GAMMA=8
..................o...J%......
0.1523 0.4395 0.1228 0.2086
0.1136 0.4198 0.1692 0.1830
0.0939 0.4034 0.1714 0.1542
0.0878 0.3896 0.1470 0.1243
0.0909 0.3780 0.1087 0.0945
0.0993 0.3678 0.0656 0.0658
0.1102 0.3587 0.0238 0.0389
0.1214 0.3502 “0.0130 0.0143
0.1315 0.3419 -000428 ‘000076
0.1394 0.3336 ‘0.0648 “0.0266
0.1446 0.3249 ‘0.0790 “0.0427
0.1469 0.3158 ”0.0860 “0.0556
0.1463 0.3060 "0.0867 “0.0656
0.1427 0.2956 “0.0823 '0.0727
0.1364 0.2845 “0.0739 -0.0773
0.1276 0.2728 '0.0629 ‘0.0797
0.1168 0.2604 -0.0509 -0.0803
0.1046 0.2475 ‘000396 ’000796
0.0924 0.2342 “0.0308 '0.0778
0.0823 0.2206 “0.0266 ‘0.0754
0.0777 0.2069 '0.0293 ’0.0722
F'OCDOKDCDOCDO
. .1.. .4.. .
O\OUJQCDUIb(Jh)
O
.
O
O
01003411U151Jh1H
O
. . .1». . .
H OOOCDsJOlﬁ£>UIVF‘O
“100
“0.9
—O.8
-O.7
“006
‘005
-O.4
-O.3
‘002
'00!
HOOOOOOOOOO
OOG’QO‘UIOUNHO
ALPHA=30 DEGREES.
191
GAMMA=1
TENSION SENDING
N T N T
cocoon-coo...coo-00.00.0000...
002501 004330 001227 002165
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002500 004330 001928 001950
002500 004330 002018 001780
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002500 004330 “002018 -Ool780
0.2500 004330 -001928 -001950
0.2499 004330 ‘001667 -002080
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GAMMA=4
0000000000000.0000000000000000
002671 004980 002484 001947
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BETA=4
GAMMA=2
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N T N T
ooooooooooooooooooooooocoo-coo
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GAMMA=8
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0.2332 0.3722 “000929 “000910
002188 003631 '000645 -000909
0.2031 003536 ‘000418 -0.0891
“100
-009
-O.8
-007
“0.6
‘005
-004
*‘OCDOCDC)OCDO
0130-40\m-90JN
~‘OIDC)O<3C)OUfUF'O
ALPHA=3O DEGREES.
192
GAMMA=1
TENSION SENDING
N T N T
00.0.0.00000000000000.0000.coo
002500 0.4330 001281 000882
002500 0.4330 001750 000844
0.2500 0.4330 0.1971 0.0781
0.2500 004330 002007 000702
002500 004330 001906 000613
0.2500 0.4330 001706 000517
0.2500 0.4330 001433 000417
002500 004330 001112 000315
002500 0.4330 0.0758 000210
002500 004330 0.0383 000105
002500 004330 “000003 “000001
002500 004330 “000383 '000105
0.2500 0.4330 “000758 ‘000210
002500 004330 “001112 ’300315
0.2500 004330 '001433 '000417
0.2500 0.4330 “001706 ‘000517
002500 0.4330 “0.1906 “0.0613
002500 004330 ’002007 ”000702
002500 004330 “001971 “0.0781
0.2500 004330 “001749 -000844
002500 004330 '001281 ’000882
GAMMA=4 *
000.000.00000.0000000000000000
002502 004513 002813 000595
002491 004493 002714 000559
002485 004474 002481 000507
002484 004455 002160 000447
002487 004436 001787 000382
002491 0.4418 001386 000314
002496 0.4400 000978 000246
002501 004382 0.0573 0.0177
0.2507 004365 000182 000109
002512 004347 -000188 000042
002517 0.4329 “000533 ‘000023
002520 004311 “000845 ‘000088
002523 004294 ‘001120 “000150
002524 004276 -001352 ’000211
002523 004259 -001533 -000269
002519 004241 ’001651 “000324
002512 004224 -001696 '000377
002501 004206 ‘001649 “000424
0.2484 004189 ‘001491 “000465
002459 004170 -001196 '000495
002426 004151 “000731 “000508
BETA=20
GAMMA=2
TENSION SENDING
N T N T
ooooooooooooooooooooooo0000000
0.2520 004448 002078 000727
0.2504 0.4434 0.2282 0.0688
002495 004422 0.2274 0.0628
0.2490 004410 002119 000556
0.2489 004398 001865 000477
002490 004387 001546 000395
0.2492 0.4375 001189 000312
0.2495 0.4364 000812 000228
0.2499 004352 000430 000144
0.2503 004341 000051 0.0062
0.2507 004330 -000316 “000020
0.2510 004319 “000662 -000101
0.2513 004307 ’000982 -000180
0.2514 0.4296 “001267 '000257
002515 004285 ”001506 '000332
002514 004273 "001686 “000405
0.2510 004262 -001792 -000473
0.2504 004251 “001801 “000535
002493 004239 ’001687 '000590
0.2477 004227 '001415 -000631
002455 004214 ’000942 “000653
GAMMA=8 *
ooooooooooooooooooooooo000.000
002480 004549 003335 000505
0.2477 004525 003005 000472
0.2478 004502 002606 000428
002482 004479 002171 0.0378
0.2488 004457 001720 000323
002494 004435 001270 000266
0.2501 004414 000832 0.0208
0.2508 0.4392 000414 000150
002514 004371 000023 000092
002520 004350 ‘000338 000034
0.2524 004328 ’000665 ’000022
002528 004307 "000953 “000076
002529 004287 ‘001199 '000129
002529 004266 “001397 “000180
002526 004245 -001541 “000229
002521 004224 -001623 ‘000275
002511 004203 ‘001631 ”0.0318
0.2497 004183 “001553 -000357
002476 004162 “001371 '000390
002448 004140 ”001065 ‘000414
002409 004118 -000608 '000424
OOOOOOOOOO
\OQQO‘UIDUN“O
“OOOOOOOOOO
OOQQO‘UIDUNHO
ALPHA=3O DEGREES.
193
GAMMA=1
TENSION SEND1NG
N T N T
000000000000000000000000000000
002500 004330 002059 000234
002500 004330 002076 000223
002500 004330 001995 000204
002500 004330 001844 000181
002500 004330 001643 000155
002500 004330 001408 000129
002500 004330 001148 000103
002500 004330 000873 000077
002500 004330 000587 000051
002500 004330 000295 000025
002500 004330 “000003 000000
002500 004330 “000295 “000025
002500 004330 “000587 “000051
002500 004330 “000873 “000077
002500 004330 “001148 “000103
002500 004330 “001408 “000129
002500 004330 “001643 “000155
002500 004330 “001844 “000181
002500 004330 “001995 “000204
002500 004330 “002076 “000223
GAMMA=4 *
000000 0000000000000000000000
002500 004330 “002059 “000234
002499 004371 002673 000140
002499 004367 002434 000133
002499 004362 002167 000122
002499 004358 001882 000110
002499 004354 001587 000096
002500 004350 001286 000081
002500 004346 000984 000066
002500 004342 000684 000050
002501 004338 000388 000035
002501 004334 000099 000019
002501 004330 “000183 000003
002501 004326 “000456 “300013
002502 004322 “000717 “000029
002502 004318 “000965 “000045
002502 004314 “001196 “000062
002501 004310 “001405 “000079
002501 004306 “001587 “000096
002500 004302 “001732 “000114
002499 004298 “001827 “000130
002497 004294 “001856 “000143
002494 004290 “001794 “000150
SETA=100
GAMMA=2 *
TENSION SENDING
N T N T
000000000000000000000000000000
002501 004357 002444 000175
002500 004354 002303 000166
002500 004351 002106 000152
002499 004349 001870 000135
002499 004346 001609 000117
002499 004343 001332 000098
002500 004341 001045 000079
002500 004338 000754 000060
0025007 004335 000461 000040
002500 004333 000170 000021
002501 004330 “000118 000002
002501 004327 “000399 “000018
002501 004325 “000671 “000037
002501 004322 “000933 “000057
002501 004319 “001180 “000077
002501 004317 “001407 “000097
002501 004314 “001608 “000117
002500 004312 “001773 “000138
002500 004309 “001888 “000156
002498 004306 “001934 “000171
GAMMAze *
000000 0000000000000000000000
002497 004304 “001888 “000180
002498 004378 002799 000121
002498 004373 002505 000115
002498 004368 002200 000106
002499 004363 001888 000096
002499 004358 001574 000084
002500 004354 001261 000072
002500 004349 000951 000059
002501 004344 000647 000046
002501 004339 000350 000032
002501 004335 000061 000018
002502 004330 “000218 000004
002502 004325 “000486 “000010
002502 004321 “000741 “000024
002502 004316 “000981 -000039
002502 004311 “001203 “000054
002502 004307 “001403 “000069
002501 004302 “001575 “000085
002500 004297 “001710 “000101
002498 004293 “001795 “000116
002496 004288 “001815 “000128
002493 004283 “001744 “000134
“-0O(3C>OO
OQJQ-JO‘m-9th)ﬂ(3
*“CDOCDO
ALPHA=4O DEGREES.
194
GAMMA=1
TENSION SENDING
N T N T
00000..000.0.00..00.0.0..00..0
004132 0.4924 0.2509 0.2036
004132 004924 003089 001950
004132 0.4924 003303 001808
004132 004924 0.3250 001630
004132 004924 003007 001426
004132 004924 002635 001206
0.4132 004924 002177 000975
0.4132 004924 001667 0.0736
004132 004924 001125 000493
004132 004924 0.0566 000247
004132 004924 “000001 “000000
004132 0.4924 “0.0566 “0.0247
004132 004924 “001125 “000493
004132 004924 “001667 “0.0736
004132 004924 “0.2177 “000975
004132 004924 “002635 “001206
004132 0.4924 “003007 “001426
004132 004924 “003250 “001630
004132 004924 “003303 “001808
004132 0.4924 “0.3089 “001950
004132 004924 “002509 “002035
GAMMA=4 *
0.000.0..0.0.0.00.0.0.00.0000.
004155 005322 004823 001564
004105 0.5271 004615 001462
0.4082 0.5225 004165 001321
004078 005183 003569 001156
0.4085 005142 002898 000979
0.4100 0.5104 0.2200 300795
004117 005066 001508 000610
004135 0.5029 000841 000426
004154 004993 000211 000246
004170 004957 “000375 000071
004185 004921 “0.0912 “0.0098
004196 004885 “001396 “000262
0.4205 004850 “001821 “000418
0.4209 004814 “002180 “000566
0.4206 0.4778 “0.2461 “0.0705
004196 004742 “002650 “000835
004174 004705 “002726 “000952
0.4137 0.4668 “0.2666 “0.1056
004081 004629 “002443 “001140
003998 004587 “002028 “001199
0.3883 004541 “001388 “001224
BETA=4
GAMMA=2 *
TENSION BENDING
N T N T
.00000000000000000000000......
0.4207 0.5168 0.3589 0.1823
0.4147 0.5136 0.3869 0.1719
0.4114 0.5107 0.3791 0.1567
0.4098 0.5081 0.3473 0.1384
0.4094 0.5057 0.3006 0.1184
0.4098 0.5033 0.2452 0.0975
0.4106 0.5011 0.1854 0.0762
0.4116 0.4989 0.1243 0.0549
0.4128 0.4967 0.0637 0.0337
0.4140 ‘O.4945 0.0049 0.0129
0.4151 004923 “000515 “0.0076
0.4161 0.4902 “0.1047 “0.0276
0.4169 0.4880 “0.1540 “0.0470
0.4175 0.4858 “0.1986 “0.0658
0.4178 0.4837 “0.2371 -0.0837
0.4176 0.4814 “0.2675 -O.lOO7
0.4166 0.4791 “0.2874 “0.1165
0.4147 0.4768 “0.2934 “0.1306
0.4113 0.4742 "0.2816 “0.1425
0.4061 0.4715 “0.2469 “0.1513
0.3984 0.4684 “0.1835 -O.1558
GAMMA=8 *
OOOOOOOOCOCOCOCOOOCCCCO0......
0.4070 0.5415 0.5850 0.1349
0.4053 0.5354 0.5186 0.1255
0.4055 0.5297 0.4413 0.1130
004069 005244 003595 000988
0.4090 0.5194 0.2774 0.0834
0.4114 005146 001980 000675
0.4138 0.5099 0.1231 0.0514
0.4161 0.5053 0.0537 0.0355
0.4181 0.5007 “0.0097 0.0199
0.4199 0.4963 “0.0670 0.0047
0.4213 0.4918 “0.1179 -0.0099
0.4224 0.4874 “0.1620 “0.0237
0.4229 0.4830 -O.l993 -0.0369
0.4229 0.4786 “0.2289 “0.0492
0.4221 0.4743 “0.2498 “0.0606
0.4202 0.4699 “0.2610 “0.0712
0.4171 0.4654 “0.2608 “0.0806
0.4123 0.4609 “0.2475 “0.0888
0.4053 0.4562 “0.2193 “0.0953
0.3955 0.4513 “0.1745 “0.0997
0.3821 0.4459 “0.1112 “0.1012
”1.0
-0.9
'0.8
”0.7
“0.6
-O.5
-0.4
-0.3
-0.2
I
O O
O O
u.
HOOOOOOOOO
OVOGQO‘UIDUNHO
-100
‘0.9
‘0.8
”0.7
“0.6
”0.5
‘0.4
”0.3
-002
-001
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
ALPHA=4O DEGREES.
195
GAMMA=1
TENS1ON SENDING
N T N T
..............................
0.4131 0.4924 0.3260 0.0696
0.4131 0.4924 0.3379 0.0664
0.4131 0.4924 0.3301 0.0610
0.4131 0.4924 0.3080 0.0543
0.4131 0.4924 0.2761 3.0470
0.4131 0.4924 0.2373 0.0393
0.4131 0.4924 0.1939 0.0315
0.4131 0.4924 0.1475 0.0236
0.4131 0.4924 0.0992 0.0157
0.4131 0.4924 0.0498 0.0078
0.4131 0.4924 ”0.0005 0.0000
0.4131 0.4924 ”0.0498 ”0.0078
0.4131 0.4924 -0.0992 -0.0157
0.4131 0.4924 -0.1475 ”0.0236
0.4131 0.4924 ”0.1939 ”0.0315
0.4131 0.4924 -0.2373 ”0.0393
0.4131 0.4924 ”0.2761 ”0.0470
0.4131 0.4924 ”0.3080 ”0.0543
0.4131 0.4924 ”0.3301 ”0.0610
0.4131 0.4924 ”0.3379 ”0.0664
0.4131 0.4924 ”0.3260 ”0.0696
GAMMA=4 *
..............................
0.4129 0.5026 0.4554 0.0432
0.4127 0.5015 0.4149 0.0410
0.4126 0.5005 0.3684 0.0378
0.4127 0.4994 0.3180 0.0339
0.4128 0.4984 0.2657 0.0296
0.4129 0.4974 0.2125 0.0249
0.4131 0.4964 0.1595 0.0201
0.4133 0.4953 0.1072 0.0153
0.4135 0.4944 0.0562 0.0103
0.4136 0.4933 0.0067 0.0054
0.4137 0.4923 ”0.0410 0.0005
0.4139 0.4913 ”0.0865 ”0.00 5
0.4139 0.4904 ”0.1295 ”0.0095
0.4140 0.4894 ”0.1696 ”0.0145
0.4140 0.4884 ”0.2062 ”0.0195
0.4138 0.4874 ”0.2382 ”0.0246
0.4136 0.4864 ”0.2646 ”0.0296
0.4132 0.4854 ”0.2833 ”0.0345
0.4126 0.4845 ‘0.2921 ”0.0390
0.4117 0.4834 ”0.2874 ”0.0426
0.4104 0.4824 ”0.2647 ”0.0446
BETA=20
GAMMA=2 *
TENSION SENDING
N T N T
..............................
0.4136 0.4937 0.4039 0.0536
0.4132 0.4930 0.3850 0.0509
0.4130 0.4924 0.3540 0.0468
0.4129 0.4917 0.3149 0.0417
0.4129 0.4911 0.2704 0.0361
0.4129 0.4904 0.2226 0.0302
0.4130 0.4898 0.1731 0.0242
0.4131 0.4892 0.1229 0.0182
0.4132 0.4990 0.0727 0.0121
0.4133 0.4983 0.0231 0.0061
0.4134 0.4976 ”0.0257 0.0001
0.4135 0.4970 ”0.0729 ”0.0059
0.4136 0.4963 -0.1185 ”0.0120
0.4136 0.4956 -O.1617 -o.01ao
0.4137 0.4950 ”0.2020 ”0.0241
0.4136 0.4943 ”0.2383 -0.0301
0.4135 0.4885 ”0.2693 ”0.0361
0.4133 0.4879 ”0.2928 ”0.0418
0.4129 0.4872 ”0.3063 ”0.0471
0.4124 0.4865 ”0.3061 ”0.0513
0.4115 0.4858 ”0.2871 ”0.0537
GAMMA=8 *
..............................
0.4123 0.5045 0.4854 0.0372
0.4123 0.5032 0.4320 0.0354
0.4124 0.5020 0.3763 0.0327
0.4126 0.5007 0.3195 0.0295
0.4128 0.4995 0.2627 0.0259
0.4130 0.4983 0.2065 0.0220
0.4133 0.4971 0.1516 0.0179
0.4135 0.4959 0.0982 0.0137
0.4137 0.4947 0.0468 0.0095
0.4138 0.4935 ”0.0025 0.0051
0.4140 0.4923 ”0.0495 0.0008
0.4141 0.4911 ”0.0940 ”0.0036
0.4142 0.4900 ”0.1355 ”0.0080
0.4142 0.4888 ”0.1738 ”0.0124
0.4141 0.4876 ”0.2083 —0.0169
0.4139 0.4865 ”0.2380 ”0.0215
0.4136 0.4853 ”0.2618 ”0.0260
0.4131 0.4842 ”0.2780 ”0.0305
0.4124 0.4830 ”0.2842 ”0.0346
0.4113 0.4818 ”0.2772 ”0.0378
0.4097 0.4806 ”0.2526 ”0.0396
”1.0
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199
Table F4.--Largest difference between 8th- and 7th-order,
or 8th- and 6th-order polynomial solutions for
adhesive stresses.
Please note:
1. Differences are expressed as a percentage of the
8th-order results.
2. Omitted entries mean data not available--see
section 3.1.4 for the only information applic-
able to a = 5° cases.
3. The combination with [8-6] is designated by (*)
and is a pessimistic measure because [8-7] re-
sults are often substantially closer. [8—7] com-
bination appears in the table without (*).
4. The first value tabulated is the largest % dif-
ference for all stresses, from the largest ab-
solute value down to one-half of this maximum.
5. When there is a second error entry followed by
a value in parentheses, it means that if we ex-
tend our consideration to values a little less
than half the maximum, the error is larger, as
indicated. The number in parentheses shows how
far down one must go in the primary tables of
Appendix F to get this larger error. Interpreting
these items requires consultation of the primary
tables.
Tensile Loading
y=2 'y=4 y=8
a 8 Normal Shear Normal Shear Normal Shear
10° 4 10.4 0.24 5.7 -0.25 -7.8 0.50
-9.0
(.021)
20 -7.1* 0.08* -8.5* —O.13* -3.4 0.07
8.8
(.020)
100 -0.89* -0.006* -0.57* -0.005* 0.67* -0.005*
20° 4 2.2 -.069 1.38* 0.23* 5.5 0.10
0.21
(.207)
Table F4 (Continued)
200
Tensile Loading
a 8 Normal Shear Normal Shear Normal Shear
20° 20 0.2* 0.009* 0.43* -0.02* 0.49 -0.01
100 0.02 -0.001 0.02* 0.002* 0.01 0.001
30° 4 0.57* -0.06* 0.45 -0.05 0.54 -0.06
20 0.06* 0.003* -0.21* 0.02* -0.08* 0.004
100 -0.006* 0.000* -0.03* 0.001* -0.03* 0.002*
40° 4 0.32* 0.06* 0.14* -0.03* -0.41* 0.10*
20 -0.05* 0.006* -0.07* 0.01* —0.03 0.004
100 -.003* 0.000* -0.004* 0.000* - .005* 0.000*
Bending Load
10° 4 -2.6 0.15 -2.4 0.17 -9.7 0.16
8.3 0.22 -3.6 .35
(.016) (.082) (.189) (.102)
20 5.8* 0.77* 5.7* 0.42* 1.6 0.45
9.1
(.015)
100 -5.24* -0.60* -0.66* -0.16* 2.7* 0.50*
20° 4 7.7 0.28 -3.4* 0.37* -3.7 -0.34
20 1.4* 0.64* -3.9 0.73 2.9 0.96
100 2.5 -0.59
30° 4 -8.64* 1.1* -3.1 .071 -2.4 1.1
20 -2.4 0.81 -3.7* 2.4* 0.46 0.79
-3.1
(-.094)
100 1.3 1.5 -3.6* 5.1* 3.7* 5.4*
40° 4 -5.9* 2.6* -1.4 1.1 -2.01* 2.9*
20 -4.2* -4.4* -4.4 4.8 -1.8 1.9
100 -l.1* -5.7* -1.1* 6.3* -1.1* 6.7*
201
Table F5.--Root-Mean-Square values for percentage differ—
ences between 8 -order and lower-order poly-
nomial solutions. (Table added in proof for
convenience of user.)
The largest value of T or N in Table F1 is located and
multiplied by 0.4; stresses smaller than this "cutoff"
value are ignored. The difference between the Table F1
values and the available lower-order solution (7th- or
6th-order) are expressed as a percentage of the Table F1
values. Then the root-mean-square quantities RMSN and
RMST are formed from these larger-stress percent
differences:
RMSN = [2(normal stress % differences)2/(No. differences
considered)]1/2
RMST is formed similarly. These are probably the best
available indices of merit of the primary results in Table
F1; Table F4 is useful but excessively conservative.
Omitted cases below: no comparison solutions available.
Tension Bending Tension Bending
a B y RMSN RMST RMSN RMST a B y RMSN RMST RMSN RMST
5° 20 4 13.81 0.25 19.13 0.26
10° 4 2 2.91 0.07 2.36 0.14 30° 4 2 0.19 0.01 2.63 0.61
4 3.65 0.15 9.29 0.12 4 0.16 0.02 1.07 0.50
8 3.61 0.33 3.47 0.20 8 0.20 0.03 0.99 0.68
20 2 1.80 0.04 5.46 0.38 20 2 0.02 0.00 0.69 0.51
4 2.44 0.08 16.12 0.64 4 0.07 0.01 1.40 1.75
8 2.29 0.04 5.97 0.31 8 0.02 0.00 0.34 0.53
100 2 0.22 0.00 1.95 0.33 100 2 0.00 0.00 0.44 0.85
4 0.14 0.00 0.42 0.18 4 0.01 0.00 1.15 2.86
8 0.17 0.00 0.31 0.12 8 0.01 0.00 1.21 2.79
20° 4 2 0.06 0.05 2.64 0.23 40° 4 2 0.11 0.03 2.36 1.67
4 0.89 0.13 2.16 0.53 4 0.04 0.01 0.54 0.67
8 1.37 0.08 1.42 0.24 8 0.13 0.05 1.00 1.77
20 2 0.07 0.01 0.94 0.54 20 2 0.01 0.00 1.37 2.81
4 0.11 0.01 1.18 0.39 4 0.02 0.00 1.38 2.87
8 0.14 0.01 1.03 0.56 8 0.03 0.00 0.54 1.15
100 2 0.01 0.00 0.67 0.28 100 2 0.00 0.00 0.37 3.34
4 0.02 0.00 0.52 0.20 4 0.00 0.00 0.36 3.26
8 0.03 0.00 0.41 0.12 8 0.00 0.00 0.36 0.34
APPENDIX G
COMPUTER PROGRAM
The main computer program used in this research
is given below. It is written in Fortran for the CDC 3600
computer. The dollar sign ($) is a legal statement sepa-
rator for this computer; each time you encounter it, put
what follows on a new card in writing for most other com-
puters. Because of space limitations, the various auxiliary
programs used in this research have had to be omitted here.
These include the program for the integral equation method,
and more detailed commentary on the main program.
202
203
PROGRAM RITZ
DIMENSION X(21)0Y(4591’08(I7792)9X8(17792)9XI(21’92(45¢I)0C(4591)
190(4591)QSIGO(21)9TAUO(21)0YA(21)9YB(21)9XA(21)0XB(21)QBL(21)9
2AL(21)QSIGXA(21)0SIGXB(21)9TAVA(21)0TAVB(21)9TAHLA(21)9T12(21)0
3TAHUA(ZI)9TAHLB(21)9TAHUB(21)QSIGYUA(21)9SIGYLA(21)oSIGYUB(21)o
4SIGYLB(21)
COMMON /I/ A(16000)
C PARAMETERS OF TBE JOINT
READ 19IOUIQUZQHETAQHQSIGEQEAGQSME
READ BBOOALPHA
READ BBOgBETA
READ BBOOGAMMA
HERSBETA $ UJ=GAMMA
C GEOMETRY OF THE JOINT
CCCCC=COT(ALPHA) S CHI=20+CCCCC $ XDISI=CHI+CCCCC
631. S MJ=O
DO 895 J=lOI
895 MJ=J+MJ
NG=MJ§4
NE=MJ+I+2
ND=MJ+I
NCSMJ+2
NB=MJ*4-3
NABMJ*2-3
NHsMJ-l
NI=MJ+I-2
NO=MJ-2
NP=MJ-3
L=I+1
NR=MJ*3-3
R1=10-U1*UI
R2=lo-U2*U2
R3=Io-U1
R4=Io—U2
CSOU=CCCCC*CCCCC
Hl=(Io/BETA)*(SINF(ALPHA)**2+(CO$F(ALPHA)**2/EAG))/SINF(ALPHA)
H2=(lo/BETA)*COSF(ALPHA)*(Io‘IIo/EAG))
HH=(Io/BETA)*(SINF(ALRHA)**2+EAG*COSF(ALPHA)**2)/(SINF(ALPHA)*EAG)
M1=O
C GENERATION OF RITZ MATRIX COEFFICIENTS
DO 499 MN =IOI
EM=MN-I
Ll =I-MN+1
DO 495 NM = IoLl
EN=NM-I
7O
71
73
90
91
93
BO
81
83
“10(1
204
N1=O
DO 150 KJ = 101
EKde-l
II=KJ+MN‘I
I2=I1-I
I3=I2-l
CCC12=CCCCC**12
CCC13=CCC12/CCCCC
CII=CHI**I3
C12=CII*CHI
CIO=C12*CHI
CHI=(-CHI)**I3
CH2=-CHI*CH1
CHll=-CH2*CHI
L2 = I-KJ+I
DO 100 JK = loLZ
MM=MI+NM
NN 8 N1+JK
MM2=MM+MJ*2-3
MM3=MM+MJ*3-3
NN2=NN+MJ*2-3
NN3=NN+MJ*3~3
L3 = KJ+JK+MN+NM-3
L4=L3-l
JI=NM+JK-l
J2=Jl-l
J3=J2-I
EJ=JK-I
IF (2*(J2/2)-J2) 71970071
ASZ=O
GO TO 73
A82=Zo/J2
CONTINUE
IF (2*(J3/2)-J3) 91990991
A53=O
GO TO 93
A53=Zo/J3
CONTINUE
IF (2*(L3/2) -L3) 81980.81
AL3=O
GO TO 83
AL3=2./L3
CONTINUE
IF(2*(L4/2)-L4) 60706
AL4=O
GO TO 8
AL4=20/L4
IF (2*(Jl/2)-Jl) 29392
A$I=20/Jl
205
GO TO 4
3 A5180
4 PHII=AL3*CCC12
PHIO=AL4*CCCI3
CHIZ=AS3§CH11
C16=AS3*CIO
C15=ASZ*C12
CH21=ASZ*CH2
CH11=A$1*CH1
CI3=ASI*C11
IF (13) 19920919
19 E13=1o/I3
GO TO 21
20 EI3=O
21 CONTINUE
EI32=EI3*O.5
MS=MM
NSaNN
MM=MM-1
NN=NN-1
IF(MS-1) 259390925
25 IF (MS‘L) 2793409330
27 IF (NS-1) 299390929
29 IF (Ns-L) 319340933
31 MMlxMS+MJ-2
NNlaNS+MJ-2
GO TO 350
33 MM1=MS+MJ-Z
NN1=NS+MJ-3
GO TO 350
340 IFINS-I) 37093909373
330 MM1=MS+MJ-3
IF (N5-1) 31093909310
310 IF (Ns-L) 31593709320
315 NN1=NS+MJ-2
IF(MM1-NN1)1619161937O
320 NNl=N$+MJ-3
1F(MM1-NN1) 16191619370
350 IF(MMl-NN1) 16191619370
161 M$Y=IPOS(MM19NN1)
A(MSY)=(PHIO*CSOU-CHIZ1*EN*(EJ*19/11)+R3*EK*EM*EI32*(PHIO
l-CH11)+HH*PH11
370 IFIMM-NN) 17191719390
171 M5Y=IPOS(MM9NN)
A(MSY)=(PHIO-CH11)*EK*EM*EI3+(PHIO*CSOU"CH12)*R3*EN*EJ/(29*
111)+H1*PH11
390 CONTINUE
IF(MM2-NN2) 39193919392
391 MSY=IPOS(MM29NN21
206-
A(MSY)=UJ*EK*EM*EI3*(CI3-PHIO)+EN*EJ*UJ*(R4*095/I1)*(C16-PHIO*
1CCCCC**2)+H1*PHII
392 CONTINUE
IF(MM3-NN3) 39393939394
393 MSYBIPOS(MM39NN3)
A(MSY)=UJ*EJ*EN*(19/Il)* (CIé-PHIO*CCCCC**2)+UJ*EK*EM*(R4/2)*
1EI3*(CI3-PHIO)+HH*PﬂII
394 CONTINUE
IF(I2) 22923922
23 612:0
GO TO 24
22 E12219/I2
24 IF(MS-1) 2379539237
237 IF‘MS-L) 239945947
239 IF(NS-1) 2419539241
241 IF(NS-L) 43946948
43 MM1=MS+MJ-2
NN1=NS+MJ-2
GO TO 49
45 IF (NS‘I) 44953944
44 IF (NS-L) 52953954
52 NNI=NS+MJ—2
IF(MM-NN1) 2919291953
54 NN1=NS+MJ-3
IF(MM-NN1) 2919291953
46 MM!=MS+MJ-2
GO TO 51
47 MM1=MS+MJ-3
IF (NS-1) 55953955
55 IF(NS-L) 56951957
56 NN1=NS+MJ-2
GO TO 49
57 NN18N5+MJ-3
GO TO 49
48 MM1=MS+MJ-2
NNI =NS+MJ—3
49 IF(MM-NN1)29192919292
291 M$Y=IPOS(MM9NN1)
A(M$Y)=(PHIO*CCCCC-CH2I)*(U1*EJ*EM+(R3/2)*EK*EN)*E12-H2*PHIl
292 CONTINUE
51 IF(MM1-NN) 4019401953
401 MSY=IPOS(MM19NN)
A(MSY)=(PHIO*CCCCC-CH21)*(U1*EK*EN+(R3/2)*EJ*EM)*E12-H2*PHI1
53 CONTINUE
IF 1 MM2-NN3) 353 9 353 9 354
353 MSY=IPOS(MM29NN3)
A(MSY)=(C15-PHIO*CCCCC)*(U2*UJ*EJ*EM+UJ*(R4/2)*EK*EN)*E12-H2*PHI1
354 CONTINUE
IF I MM3-NN2) 355 9 355 9 356
207
355 MSY=IPOS(MM39NN2)
A(MSY)=(UJ*U2*EK*EN+UJ*(R4/2)*EJ*EM)*E12*(C15-PHIO*CCCCC1-H2*PHI1
356 CONTINUE
1F(MS-1) 45594909455
455 IF(MM-NN2) 49194919492
491 MSYBIPO$(MM9NN2)
A(MSY)=-H1*PHII
492 CONTINUE
IF(MM-NN3) 50195019502
501 MSY81P05(MM9NN3)
A(MSY)=H2*PH11
502 CONTINUE
IF (MS-L) 46094909465
460 MMl:MS+MJ-2
IF(MM1-NN2) 51195119512
511 MSYEIPOS(MM19NN2)
A(M$Y)=H2*PH11
512 CONTINUE
IF(MM1-NN3) 52195219522
521 MSY=IPOS(MM19NN3)
A(MSY)=-HH*PHI1
522 CONTINUE
465 MM1=MS+MJ-3
IF(MM1-NN2) 53195319532
531 MSY=IPOS(MM19NN2)
A(MSY)=H2*PHII
532 CONTINUE
IFIMMI-NN3) 54195419490
541 MSY=IPOS(MM19NN3)
A(MSY)=-HH*PH11
490 CONTINUE
100 CONTINUE
150 N1=N1+L2
495 N120
499 M1=M1+L1
C LOAD TERMS FOR TENSION AND BENDING LOADING
M1=O
DO 600 MN =191
EM=MN-1
ME =EM
L0=I+1-MN
DO 590 NM = 19LO
EN1=NM
NMP:NM+1
NM2=NM+2
EN2=NM2
ENP=NMP
MM=M1+NM
MS=MM
520
540
550
560
565
580
590
600
208
MM2=MM+MJ*2-3
MM3=MM+MJ*3-3
MM=MM-1
C1=(-CHI)**ME
C3=CHI**ME
C55=(19-(-19)**NMP)*C1/ENP
C77=(19-(-19)**NMP)*C3/ENP
C11=(19-(-19)**NM )*C1/EN1
C33=(19-(-19)**NM )*C3/ENI
B(MM291)= R1*SME*C33
B(MM292)=—R1*SME*C77
8(MM391)=0
IFIMS-l) 54095909540
IFIMS-L) 55095809560
MM1=MS+MJ-2
GO TO 565
MM1=MS+MJ-3
8(MM91)=R1*SME*C11*(-19)
8(MM92)=R1*SME*C55
B(MM191)=O
BIMM1921=0
GO TO 590
CONTINUE
B(MM91)=R1*SME*C11*(-19)
8(MM92)=R1*SME*C55
CONTINUE
M1=M1+L0
RITZ COEFFICIENT SYMMETRIC MATRIX INVERSION
CALL SYMINV(N8)
CALCULATION OF DISPLACEMENT FUNCTION COEFFICIENTS
631
621
641
D0 631 I9=I9N8
DO 631 K9=19M
X8(I99K9)=O
DO 621 18=19N8
DO 621 J8=19N8
D0 621 K8=19M
118=IPOS(189J8)
X81189K8)=X8(189K81+A(118)*B(J89K8)
DO 641 I7=19N8
DO 641 J7=19M
8(179J7)=X8(I79J7)
CALCULATION OF ADHESIVE AND BOUNDARY STRESSES
705
D0 2000 M=192
Y(191)=O
D0 790 J=19NB
IF (J-NH) 70597059710
II=J+1
Y(II91)=B(J9M)
710
715
720
725
730
735
740
745
750
790
795
209
GO TO 790
ZIII9II=O
IF (J-NI) 71597159720
II=J-N0
GO TO 730
2(L91)=O
1F (J-NA) 72597259735
II=J-NP
ZCII9I)=B(J9M1
GO TO 790
IF (J-NR) 74097409745
II=J-NA
C(II91)=B(J9M)
GO TO 790
IF (J-NB) 75097509790
II=J-NR
D(II91)=B(J9M)
CONTINUE
K=1
DELX=O
DSX=0
DX1=O
DI=O
N2=NZ+1 o
SIGO(K)=0
TAUO(K)=O
SIGXA(K)=O
SIGXB(K)=O
TAVA(K1=O
TAVB(K)=O
SIGYUA(K)=O
SIGYUB1K)=O
SIGYLA(K)=0
SIGYLB(K)=0
TAHLA(K)=O
TAHUA(K)=O
TAHUB(K)=O
TAHLB(K)=O
SIG(K1=O
TAU(K)=O
SIY(K)=O
X(K)=-19+DELX
ALIK)=-CHI+DSX
BL(K)=CHI-DSX
DELX=DELX+91
X(11)=09001
YAIK1=X(K)
YB(K)=YA(K)
XA(K)=-CHI +DX1
210
XB(K)=CHI-Dx1
XI(K)=-CCCCC+DI
IFIXIIK) 0E0. O) XI(K)=OOOOI
Dx1=Dxl+XDISl/209
DSX=DSX+(CHI-CCCCC)/209
DI=DI+CCCCC/109
MM=O
DO 825 MI=19I
LL=I+1-MI
MMN=MI-1
MT=MI-2
EM=MI-l
DO 825 N=19LL
MNN=MI+N-2
MM=MM+1
NN=1
NTzN-2
EN=N-1
MN=MI-1
NM=N-1
C ADHESIVE NORMAL AND SHEAR STRESS CALCULATIONS
XCOT=X(K)**MNN*CCCCC**MMN
SIGO(K)=SIGO(K)+((Z(MM9NN)-D(MM9NN))*COSF(ALPHA)-(Y(MM9NND-C(MM
19NN))*SINF(ALPHA))*XCOT/(HER*R1)
TAUO(K)=TAUO(K)+((2(MM9NN)-D(MM9NN))*SINF(ALPHA)+(Y(MM.NN)-C(MM
l9NN))*COSF(ALPHA))*XCOT/(EAG*HER*R1)
C BOUNDARY STRESS CALCULATIONS FOR BOTH ADHERENDS
VANM=YA(K)**NM
YBNM=YB(K)**NM
YANT=YA(K)**NT
YBNT=YB(K)**NT
XIMT=YANM*XI(K)**MT
XINT=YANT*XI(K)**MN
XAYA=YANM*((—CHI)**MT)
YAXA=YANT*((-CHI)**MN)
XBYB=YBNM*(CHI**MT)
YBXB=YBNT*(CHI**MN)
AYA=AL(K)**MT*(-H)**NM
YAA=(-H)**NT*AL(K)**MN
BYB=BL(K)**MT*H**NM
YBB=H**NT*BL(K)**MN
XYA=XA(K)**MT*H**NM
YXA=H**NT*XA(K)**MN
XYB=XB(K)**MT*(-H)**NM
YXB=(-H)**NT*XB(K)**MN
TAVA(K)=TAVA(K)+(EN*YAXA*Y(MM9NNH+EM/G*Z(MM9NN)*XAYA)/(29M(19+U ))
SIGXA(K)=SIGXA(K)+(EM*XAYA*Y(MM9NN)/G+U1*EN*YAXA*Z(MM9NN1)lRl
SIGXB(K)=SIGXB(K)+(EM*XBYB*C(MM95N)/G+U2*EN*YBXB*D(MM9NN))*UJ/R2
211
0TAV8(K)=TAV8(K)+(EN*Y8X8*C(MM9NN)+EM/G*D(MM9NN)*X8Y81*UJ/(29*(19+U
12))
SIGYLBIK1=SIGYLB(K)+(EN*YX8*D(MM9NN)+U2*EM*XYB/G*C(MM9NN)1*UJ/R2
SIGYLA(K)=SIGYLA(K)+(EN*YAA * Z(MM9NN)+U1*EM*AYA *YIMM9NN1/GI/R1
SIGYUA(K)=$IGYUA(K)+(EN*YXA * ZCMM9NN1+U1*EM*XYA *Y(MM9NN)/G)/R1
SIGYU8(K1:5IGYUB(K)+(EN*Y88*D(MM9NN1+U2*EM*BY8*C(MM9NN1/G)*UJ/R2
TAHLA(K)FTAHLA(K)+(EN*YAA*Y(MM9NN)+EM*AYA*Z(MM9NN)/G)/(29*(19+U1I)
TAHL8(K)=TAHL8(K)+(EN*YX8*C(MM9NN1+EM*XY8*D(MM9NN1/G)*UJ/(29
1*(10+U2))
TAHUAIK1=TAHUA(K)+(EN*YXA*YIMM9NN1+EM*XYA*Z(MM9NN)/G)/(29*I10+U11I
0TAHU8(K1=TAHU8(K)+(EN*Y88*C(MM9NN1+EM*8Y8*D(MM9NN1/G)*UJ/29
1*(1+U2))
825 CONTINUE
K=K+1
IF (K‘ZZ) 79598269826
826 CONTINUE
NGG=ALPHA*1809/3.l33
AGG=NGG
PRINT 9129AGG9HER9UJ9I
DO 835 II=19N897
JJ=M
8350PRINT 836 9II9JJ98(II9JJ)98(II+19JJ19B(II+29JJ).8(II+3.JJ).8(II+4.
1JJ)9B(II+59JJ)9B(II+69JJ1
PRINT 8409(X1K)9SIGO(K)9TAUO(K)9SIGXAIK)9SIGXB(K)9TAVA(K)9TAV8(K)9
1K=1921)
PRINT 8459(XAIK)9TAHUA(K)9SIGYUA(K)9K=1921)
PRINT 8509(X8(K)9TAHL8(K)9SIGYL8(K)9K=19211
PRINT 8559(ALIK19TAHLA(K)9SIGYLA(K)9K=19211
PRINT 8609(8L(K)9TAHU8(K)9SIGYUB(K)9K=1921I
1 FORMAT (I295E1498/2E1498)
836 FORMAT (2H8(9139H991292H)=97E1798)
840 FORMAT1 3X9HX916X94HSIGO916X94HTAUO916X95HSIGXA916X95HSIGX8916X94H
1TAVA916X94HTAV8//////////(3X97(E179892X)11
845 FORMAT (29X92HXA936X95HTAHUA932X96HSIGYUA9///(20X93(E1798920X11I
850 FORMAT (29X92HXB936X95HTAHL8932X96HSIGYL89///(20X93(E1798920X)1I
855 FORMAT (29X92HAL936X95HTAHLA932X96HSIGYLA9///(20X93(E1798920X11I
860 FORMAT (29X92H8L936X95HTAHU8932X96HSIGYU89///(20X93(E1798920X))1
880 FORMAT(E14981
912 FORMAT(1X96HANGLE=9F29093X95H8ETA=9F39093X96HE2/E1=9F29093X96HORDE
1R=912)
2000 CONTINUE
END
C FUNCTION FOR CONVERTING SYMM9 MATRIX T0 LINEAR FORM
FUNCTION IPOSIJ9K)
IFIJ-K) 10910911
10 IP08=(K*(K-1)1/2+J
RETURN
11 IP05=(J*(J-1))/2+K
END
21.2.
C SYMMETRIC MATRIX INVERSION SUBROUTINE
10
C
20
6
C
21
30
35
47
45
50
SUBROUTINE SYMINV (N1
DIMENSION P(177)9Q(177)91R(177)
COMMON/1/ AII6000)
N4=(N*(N+1))/2
DO 1O I=19N
IRIII=O
GRAND LOOP STARTS
DO 100 I=19N
BIGAJJ=Oo
DO 20 J=19N
IF (IR(J).NE.O) GOTO 20
M=(J*(J+1)1/2
2=ABSFIAIM11
IF (ZoLEoBIGAJJ) GOTO 20
BIGAJJ=Z
K=J
CONTINUE
IF (BIGAJJONEOOO) GOTO 21
PRINT 6
FORMAT (19H MATRIX IS SINGULAR)
RETURN
PREPARATION OF ELIMINATION STEP 1
IRIK1=1
M=(K*(K+1))/2
OIK1=19/A(M)
PIK)=10
A(M)=Oo
L=K~1
IF (LoLEoO) GOTO 35
M=(K*(K-1)1/2
DO 30 J=19L
M=M+1
PIJ1=ACMI
O(J)=A(M)*Q(K1
IF (IR(J)9NE90) GOTO 30
OIJ)=’O(J1
A1M)=Oo
IF (K+1-NoGToO) GOTO 50
L=K+1
DO 45 J=L9N
M=(J*(J-1))/2+K
PIJ1=AIM1
IF (IRIJ’OEQQO) GOTO 47
PIJ)=‘P(J1
OIJ1=-A(M)*O(K)
A(M)=Oo
DO 100 J=19N
100
213
D0 100 K=J9N
M=(K*(K-1))/2+J
A(M)=A(M)+P(J)*O(K)
END OF GRAND LOOP
RETURN
END
~~