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AREAS OF CONTACT AND PRESSURE DISTRIBUTION
IN BOLTED JOINTS
H.H. Gould
B.B. Mikic
Final Technical Report
Prepared for
George C. Marshall Space Flight Center
Marshall Space Flight CenterAlabama 35812
under
Contract NAS 8-24867
June 1970
DSR Project 71821-68
Engineering Projects Laboratory
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
https://ntrs.nasa.gov/search.jsp?R=19700032666 2018-05-19T08:44:34+00:00Z
-2 -
ABSTRACT
Whentwo plates are bolted (or riveted) together these will be incontact in the immediate vicinity of the bolt heads and separatedbeyond it. The pressure distribution and size of the contact zone isof considerable interest in the study of heat transfer across boltedjoints.
The pressure distributions in the contact zones and the radii atwhich flat and smooth axisymmetric, linear elastic plates will separatewere computedfor several thicknesses as a function of the configurationof the bolt load by the finite element method. The radii of separationwere also measuredby two experimental methods. Onemethod employedautoradiographic techniques. The other method measured the polishedarea around the bolt hole of the plates caused by sliding under loadin the contact zone. The sliding was produced by rotating one plateof a mated pair relative to the other plate with the bolt force acting.
The computational and experimental results are in agreement andthese yield smaller zones of contact than indicated by the literature.It is shownthat the discrepancy is due to an assumption madein theprevious analyses.
In addition to the above results this report contains the finiteelement and heat transfer computer programs used in this study.Instructions for the use of these programs are also included.
-3-
ACKNOWLEDGMENTS
This report was supported by the NASAMarshall SpaceFlight Center
under contract NAS8-24867 and sponsored by the Division of Sponsored
Research at M.I.T.
-4-
TABLEOF CONTENTS
Title Page
Abstract
Acknowledgments
Table of Contents
List of Tables and Figures
Nomenclature
Chapter I: Introduction
Chapter II: Analysis
A. Problem Statement
B. Method of Analysis
Chapter III: Experimental Method
Chapter IV: Results
A. Pressure Distribution and Radii of Separation fromSingle Plate and TwoPlate Finite Element Models
Bo Radii of Separation from Experiment and Their Pre-dicted Values from the TwoPlate Finite ElementComputation
Chapter V:
Chapter VI:
References
Appendices
A.
Application
Conclusions
Finite Element Analysis of Axisymmetric Solids
Page
i
2
3
4
6
8
i0
15
15
18
23
28
28
29
32
35
37
39
-5 -
Bo
C.
D,
Finite Element Program for the Analysis of
Isotropic Elastic Axisymmetric Plates
Finite Element Program for the Analysis of
Isotropic Elastic Axisymmetric Plates --
Thermal Strains Included
Steady State Heat Transfer Program for Bolted
Joint
44
68
94
-6 -
LIST OFTABLESANDFIGURES
Table
1
2
Separation Radius Comparison- Single and TwoPlate Models
Test and Analytical Results for Radii of Separationof Bolted Plates
i00
I01
Figure
i
2
3
4
5
6
7
8
9
i0
ii
12
13
14
Bolted Joint
Roetscher's Rule of Thumb for Pressure Distribution in
a Bolted Joint
Furnlund's Sequence of Superposition
Finite Element Idealization of Two Plates in Contact
Finite Element Models
Examples of Unacceptable Solutions
Plate Specimen, Bolt and Nuts, Fixture and Tools
Footprints on the Mating Surface of 1/16 - 1/16,
i/8 - 1/8, 3/16 - 3/16, 1/4 - 1/4, and i/8 - 1/4 Pairs
Footprint of Nut on Plate
X-Ray Photographs of Contamination Transferred from
Radioactive Plate to Mated Plate. 1/16, 1/4, 3/16,
1/4 Inch Pairs
Free Body Diagram for Two Plates in Contact
Single Plate Analysis-Midplane
Single Plate Analysis-Midplane
Single Plate Analysis-Midplanez
Stress Distributionz
Stress Distributionz
Stress Distribution
102
102
103
107
108
109
ii0
iii
116
117
119
120
121
122
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
- 7 -
Interface Pressure Distribution in a Bolted Joint
Interface Pressure Distribution in a Bolted Joint
Interface Pressure Distribution in a Bolted Joint
Finite Element Analysis Results for 1/4 Inch Plate Pair
Pressure in Joint, Triangular Loading
Variations of Loading and Boundary Conditions
Pressure in Joint, Uniform Displacement Under Nut
Deflection of Plate Under Nut
Finite Element Analysis Results for
Finite Element Analysis Results for
3/16 Inch Plate Pair
1/8 Inch Plate Pair
GapDeformation for Free and Fixed Edges -- FiniteElement Analysis, 1/8 inch Plate Pair.
Finite Element Analysis Result for 1/16
Finite Element Analysis Results for 1/8Mated With 1/4 Inch Plate
ComparisonBetweenTested and MeasuredSeparation Radii
Location of Nodes -- Steady State Heat Transfer Analysis
Inch Plate Pair
Inch Plate
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
-8-
NOMENCLATURE
A, B, C
D
E
G
hc, hf
H
k, kl, k2
P, P
r
Ro
u_w
x
Xc
Y
!Y
z
radii
thickness
modulus of elasticity
shear modulus
heat transfer coefficients
hardness
thermal conductivities
pressure
coordinate
radius of separation
displacement in r and z
coordinate
length of contact
coordinate
slope
coordinate
directions
5
g
er,e t , erz
O, _i' _2
_r' Ot Z
deflection
dilation
strains
standard deviations
stresses
T
e
Subscripts
r
t
z
-9 -
Lame's constants
Poisson's ratio
shear stress
an gl e
radial direction
tangential direction
z-direction
- i0 -
Chapter I
INTRODUCTION
When two plates are bolted (or riveted) together, these will be
in contact in the immediate vicinity of the bolt heads and separated
beyond it. The pressure distribution in the contact area and the
separation of the plates is of considerable interest in the study of
heat transfer across joints. Cooper, Mikic and Yovanovich [i] show
that with assumed Gaussian distribution of surface heights, the micro-
scopic contact conductance is related to the interface pressure,
surface characteristics and the hardness of the softer material in
O. 985tan 6 t)
h = 1.45 k(_) (i.i)c O
where
2klk 2= (1.2)
k - kl+k 2
and k 1
contact;
and k2 represent the thermal conductivities of two bodies in
o is the combined standard deviation for the two surfaces
which can be expressed as
1/22 2
O = (oI + 02) (1.3)
- Ii -
where oI and _2 are the individual standard deviation of height for
the respective surfaces; tan _ is the meanof the absolute value of
slope for the combinedprofile and it is related, for normal distribu-
tion of slope, to the individual meanof absolute values of slopes as
1/2tan e = (tan 01 + tan e ) (1.4)
whereL
IY'.I dx; i = 1,2 (1.5)l_tan e. = lim L
i L÷oo oJ q _ u
and y' is the slope of the respective surface profiles; P represents
the local interface pressure; and H is the hardness of the softer
material.
Relation (i.i), as written above, is applicable for contact in a
vacuum. One can modify the expression by simply adding to it
hf conductivity of interstitial fluid- average distance between the surfaces (1.6)
in order to account approximately for the presence of the interstitial
fluid.
All parameters in relation (i.i), except for the pressure, are
functions of the material and geometry and can be easily obtained. The
determination of the pressure distribution and the extent of the contact
area between two plates present both mathematical and experimental
- 12 -
difficulties. From the mathematical point of view, the difficulty stems
from the fact that the theory of elasticity will yield a three dimensional
(axisymmetric) problem with mixedboundary conditions. Experimentally,
the discrimination between contact and gaps of the order of millionths
of an inch is required.
Roetscher [2] proposed in 1927, a rule of thumb that the pressure
distribution of two bolted plates, Fig. l, is limited to the two frustums
of the cones with a half cone angle of 45 degrees as shownin Fig. 2 and
that at any level within the cone the pressure is constant. Also, for
symmetric plates, according to Roetscher, separation will occur at the
circle which is defined by the contact plane and the 45 degree truncated
cone emanating from the outer radius of the bolt head.
Since 1961 Fernlund [3], Greenwood[4] and Lardner [5] amongothers
reported solutions based upon the theory of elasticity. Although their
solutions also yield separation radii at approximately 45 degrees as in
Roetscher's rule, their solutions yield a muchmore reasonable pressure
distribution as comparedto Roetscher's constant pressure at each level
of the frustrum. These investigators have madeuse of the Hankel
transform method demonstrated by Sneddon[6] in his solution for the
elastic stresses produced in a thick plate of infinite radius by the
application of pressure to its free surfaces. The basic assumption in
their approach is that two bolted plates can be represented by a single
plate of the samethickness as the combined thickness of the two plates
under the sameexternal loading. It then follows that the z-stress
distribution at the parting plane can be approximated by the z-stress
distribution in the sameplane of the single plate. It also follows
that separation will occur at the smallest radius in that plane for which
- 13 -
the z-stress is tensile. In the case of two plates of equal thickness
the O stress at the midplane of the equivalent single plate is thez
stress of interest.
Fe_nlund [3], for example, used the method of superposition in the
sequence shownin Figs. 3(a) to 3(c) to obtain annular loading. Then
by superposition of shear and radial stresses at radius A, Figs. 3(d)
and 3 (e), opposite in sign of those due to the annular loading at the
free surfaces, Fernlund obtained the solution for a single plate with a
hole under annular loading (Fig. 3(f)).
Experimental work in this area included Bradley's [7] measurements
of the stress field by three dimensional photoelasticity techniques, and
the use of introducing pressurized oil at various radii in the contact
zone and measuring the pressure at which oil leaks out from the joint
[3,8]. Both of these experimental methods have uncertainties as indi-
cated by the authors.
Because of the cumbersomnessof the Hankel transform solution and
experimental difficulties, the body of work in this area has been very
limited and definite verification of analytical results by experiment is
not cited in the literature.
The research described in the succeeding chapters was undertaken
with the following primary objectives:
a) To provide a method of solution for the case of two bolted
plates without the simplifying assumption of the single
plate substitution.
b) To devise a test to validate the two plate analysis.
c) To test the validity of the single plate substitution.
- 14 -
A finite element computer program has been assembled for the
analytical solution of two-plate problems. Experiments have been
performed to verify the analytical results. Since in heat transfer
calculations the extent of the radius of contact is of primary
importance, and since by restricting the experimental effort to the
verification of only this parameter, (rather than the verification of
the entire pressure distribution,) manyexperimental uncertainties
should be eliminated, the experiments were designed only for the deter-
mination of the contact area.
Agreementbetween analysis and experiment was obtained and the
results showthat the single plate substitution is not justified and
the 45 degree rule is not valid for the flat and smooth surfaces
studied.
- 15 -
Chapter II
ANALYSIS
A. Problem Statement
The objective of the anlaysis was to solve the linear elasticity
problem of two plates in contact defined mathematically by the following
equations for each plate:
The equations of equilibrium
_T(rO) - Ut + r -- = 0Dr r _z
(r T) + _ (r 0_r _-%- Oz ) =
(2.1)
where T = T = T and T = Ttr = T = T = 0.rz zr rt zt tz
The stress - strain relations, using standard notation for stress and
strain,
Or = lg + 2 _gr
ot = le + 2 Ne t
Oz = le + 2 Us z
(2.2)
T = 2 UErz
- 16 -
where % and _ are Lame's constants and
_= G
2 G_
i - 2_
(2.3)
if G is the modulus of elasticity in shear and _ is Poisson% ratio;
and e the volume expansion is defined by
_u u _w= _r + --r + --_z (2.4)
where u is the displacement in the radial direction and
displacement in the axial direction.
The strain - displacement relations
w is the
_u
gr _r
_w
ez _z
i ._u _w= + )rz _r
(2.5)
The above equations can be combined to yield the equilibrium
equations in terms of displacements
V2u u + 1 _e = 02 i - 2_) _r
r
V2w + i _e = 0i - 2_ _z
(2.6)
- 17 -
The applicable boundary conditions are (see Fig. ii)
O(1)(A,z) = 0(2) (A,z) = 0r r
T(1)(A,z) = T(2) (A,z) = 0
O(1)(C,z) = 0 (2) (C,z) = 0r r
_(1)(c,z ) = _(2) (c,z) = o
T(1)(r,DI ) = T (2) (r,-D 2) = 0,
T(1)(r,0) = T (2) (r,0), A < r < R
o(1)(r,Dl ) = o (r, -D 2) = 0, B < r < CZ Z -- --
T(1)(r,0) = T(2)(r,0) = 0, R < r< C
o_l,th(r,0) = o_2_t_(r,0) A < r < RZ Z _ -- -- O
O (1)(r,0) = 0 (2)(r,0) = 0, R < r < RZ Z O-- --
O (1)(r,Dl) = 0 (2)(r, -D 2) = P(r), A < r < BZ Z -- --
w (1)(r,0) = w (2)(r,0), A<r <R-- -- O
(2.7)
BF
27 _ Pr dr = 27
JA
R
/oA
pr dr
- 18 -
Inspection of the above equations shows that the above constitutes
a mixed boundary value problem and the most appropriate technique for
solution is the finite element method.
B. Method of Analysis
A finite element computer program was assembled for the analytical
solution of bolted plates. Descriptions of the finite element method
are given in references [9,10], but for completeness, an outline of the
mathematical formulation for this case is presented in Appendix A. A
listing of the cumputer program and instructions for its use maybe
found in Appendix B. Appendix C contains user's instructions and a
listing of the finite element program modified to include thermal strains.
As in the previous work axial symmetryand isotropic linear elastic
material behavior were assumed. However, the computer programs accom-
modate plates with different material properties in a bolted pair.
The basic concept of the finite element method is that a body may
be considered to be an assemblageof individual elements. The body then
consists of a finite numberof such elements interconnected at a finite
number of nodal points or nodal circles. The finite character of the
structural connectivity makes it possible to obtain a solution by means
of simultaneous algebraic equations. Whenthe problem, as is the case
here, is expressed in a cylindrical coordinate system and in the
presence of axial symmetry in geometry and loa_ tangential displacements
do not exist, and the three-dimensional annular ring finite element is
then reduced to the characteristics of a two-dimensional finite element.
- 19 -
The analysis consists of (a) structural idealization, (b) evaluation
of the element properties, and (c) structural analysis of the assem-
blage of the elements. Items (b) and (c) are covered in the appendices
and in the references quoted. The structural idealization and the
criteria for acceptable solutions will be described in this chapter.
Fig. 4(a) shows two circular plates in contact under arbitrarY
axisymmetric loading. The plates are subdivided into a number of
annular ring elements which are defined by the corner nodal circles
(or node points when represented in a plane) as shownin Figs. 4(b) and
4(c). Unlike the cases described in Chapter I, which have been solved
by the Hankel transform method, all plates solved by the finite element
method have finite radii. The cross sections of each annular ring
element is either a general quadrilateral or triangle. To improve
accuracy smaller elements are used in zones where rapid variations in
stress are anticipated than in zones of constant stress; thus the
different size elements shownin Fig. 4(b). (However, the total number
of elements allowable are subject to computer capacity.)
Figure 4(b) shows the two plates in contact for the radial distance
X and separated beyond it. It is to be noted that the nodal pointsC
on the parting line and within the length of contact X are common toc
elements in both plates. The other elements adjacent to the parting
line on each plate are separated from their corresponding elements in
the mating plate and these elements have no common nodal points.
Physically, it is equivalent to the welding together of the two plates
in the contact zone. Mathematically, we are imposing the condition that
- 20 -
in the contact zone the displacements in the z and r directions be
identical for both plates. In the case of bolted plates of equal
thickness, i.e. in the presence of symmetryabout the parting plane,
these conditions apply exactly. Furthermore, because of this symmetry,
one needs to analyze only one plate, as shownin Fig. 5(b), with the
imposedboundary conditions on the contact zone of zero displacement in
the z-direction and freedom to displace in the r-direction. It can
also be observed that the solution of two plates with symmetry about the
parting plane is equivalent to the solution of one of these plates under
the sameloading conditions, but resting on a frictionless infinitely
rigid plane. Also, under the above conditions the shear stress in the
contact zone is identically zero.
In the case of bolted plates of unequal thickness the model includes
both plates as shown in Fig. 5(c). This model is an approximation
becaus_ in general, two plates of unequal thickness do not have the same
displacement in the r-direction on the contact surface. The solution
yield_ therefore, a shearing stress distribution in the contact zone.
The solution, however, should be exactly compatible with the physical
model if the frictional forces in the joint prevent sliding.
The critical aspect of the approach used herein is the determination
of the largest nodal circle on the parting plane which is commonto an
element on each plate. This nodal circle defines the contact zone and
the radius, R , at which separation occurs.o
The output of the finite element computer program includes the
displacement of each node in the r and z directions and the average
- 21 -
O , _ , a t and T stresses for each element.z _ rz
The computation is iterative and the objective is to achieve the
lowest possible compressive @ stress in the outermost elementsz
bordering the contact zone. Unacceptable solutions are shown in Fig. 6(a)
and 6(b). If R for a given external load distribution is too small,o
then the solution will show that the two plates intersect (Fig. 6(a)).
On the other hand, if R is assumed too large, the solution will showo
that the outer portion of the contact zone sustains a tensile _ stressz
(Fig. 6(b)). Neither of these two situations is physically feasible.
In general, the procedure employed was to commence the iterations with
a value for R which would yield a tensile o stress in the outero z
elements adjacent to the contact zone and then move R inward. Theo
iteration ended as soon as no tensile _ stress was present at thez
contact zone. For example, for the case shown in Fig. 5(b), if the oz
stress for the element in the last row and to the left of the last roller
is tensile, then the following iteration will proceed without the last
roller. Thus, the resolution is one nodal interval. Finer resolution
can be obtained by reducing the interval between nodal circles by
introducing more elements or shifting the grid locally. The same
criteria apply to tile model shown in Fig. 5(c).
In the finite element analysis of the Fernlund (3) model, i.e.
single plate with external loads at the faces z = + D no iteration is
required and the rollers shown in Fig. 5(c) would extend to the outer
radius of the plate. (Although Fernlund's computations are based on
infinite plates, computations show that there is no distinction between
infinite plates and plates of radius greater than five times of the outer
- 22 -
radius, B, of the load. See Fig. 5(a).
Convergence was tested by subdividing elements further, with nodal
points in the coarser grid remaining nodal points in the finer grid.
Changing the mesh from 180 elements to 360 elements have shown no
improvement in accuracy. Meshes from 180 to 300 elements were used
in this analysis. Typical spacings between nodal points were 0.015
inch radially and 0.03 inch in the z-direction.
- 23 -
Chapter III
EXPERIMENTALMETHOD
The objective of the experiment was to determine the extent of
contact between two plates whenbolted together. Sixteen type 304
stainless steel plates, 4 inches in diameter, were machined to
nominal thicknesses of 1/16, i/8, 3/16 and 1/4 inch, 4 plates
for each thickness. After rough machining these plates were stress
relieved at 1875°F and ground flat to 0.0002 inch. One side of each
plate was then lapped flat to better than one fringe of sodium light
(ii micro-inches) in the case of the 1/8, 3/16 and 1/4 inch plates,
and to better than two fringes in the case of the 1/16 inch plates.
Disregarding scratches, the finish of the lapped surfaces was 5 micro-
inches rms. Each plate had a central hole, 0.257 inch in diameter,
for a 1/4 - 20 bol_ and two notches and two holes on the periphery
(see Fig. 7). Two techniques were employed in determining the area of
contact when two of these plates were bolted together. The first
technique entailed the following procedure (see Fig. 7):
(a) The plates were cleaned with alcohol and lens tissue.
(b) One plate was placed on the base of the fixture shownin
Fig.7, lapped surface up and the two holes on the periphery
of the plates engagedwith two pins on the fixture. Spacers
between the fixture base and plate prevented the pins from
extending beyond the top surface of the plate.
- 24 -
(c) A second plate was placed on top of the first plate, lapped
surfaces mating. The notches on the two plates were lined
up with each other and with notches in the base of the
fixture. Thus, rotation of the plates was prevented.
(d) A standard 1/4 - 20 hex-nut with its annular bearing surface
(0.42 inch 0.D.) lapped flat was engagedon a high strength
1/4 - 20 bolt. The nut was located about two threads away
from the head of the bolt and served in lieu of the bolt head.
The lapped surface of the nut faced away from the bolt head
and since the nut was not sent homeagainst the bolt head, the
looseness of fit between nut and bolt offered a degree of self
alignment.
(e) The bolt and nut assembly described in (d) above was then
inserted through the 1/4 inch central holes of the two plates
and a second 1/4 - 20 lapped nut was engagedon the bolt.
Thus the two plates were captured by the two 1/4 -20 nuts
with the lapped surfaces of the nuts bearing against the plates.
(f) With the torque wrench shownon the right in Fig. 7, the nuts
were torqued downto 70 pound-inches of torque to yield a
ii00 pound force in the bolt [ii].
(g) The position of the keys was changed to engagewith only the
lower plate and the fixture and a special spanner wrench, as
shownin Fig. 7, was engagedwith the top plate. The spanner
wrench was restrained to move in the horizontal plane and it
was set into motion by the screw pressing against the wrench
handle.
- 25 -
(h) With the aid of the spanner wrench the upper plate was
rotated relative to the lower plate several times approxi-
mately + 5 degrees.
Thus, the above procedure allowed for the rubbing of one plate
relative to its mate while under a bolt force of approximately ii00 ibs.
The remaining steps were the disassembly and the measurementof the
extent of the contact zone which was defined by the shine due to the
rubbing in the contact zone. It is to be noted that the boundaries of
the contact zone as measuredby the naked eye and by searching for
marks of "polished" or "damaged"surface under a 10.5 power magnifi-
cation are essentially the same.
The above test was performed on
0.07 in. plate mated to ai. One
2. One
3. One
4. One
5. One
5 pairs of specimen. Thesewere
0.65 in. plate
0.126 in. plate mated to a 0.126 in. plate
0.191 in. plate mated to a 0.192 in. plate
0.253 in. plate mated to a 0.256 in. plate
0.124 in. plate mated to a 0.257 in. plate
The identical tests were repeated for
i. One 0.124 in. plate mated to a 0.126 in. plate; and
2. One 0.191 in. plate mated to a 0.192 in. plate,
but in lieu of the 1/4 - 20 nuts in direct contact with the plates
special washers, 1.000 in. O.D., 0.257 in. I.D. and 0.620 in. high,
were interposed between the bolt head and nut.
The diameters of the contact zones were measuredwith a machinist
ruler with i00 divisions to the inch and with a Jones and Lamston
Vertac 14 Optical Comparator.
- 26 -
then approximately a
radioactive isotope.
2 millicuries.
The second technique used the sameparts and fixture, but it
involved autoradiographic measurements.
Four plates, 1/4, 3/16, 1/8 and 1/16 inch thick were sent to
Tracerlab, Inc., Waltham, Mass., for electrolytic plating with radio-
active silver Ag ii0 M (half life of 8 months). Eachplate was
maskedexcept for an area on the lapped face one inch in radius. The
plates then received a plating of copper about 5 microinches thick and
5 microinch plating of silver containing the
The resultant activity on each plate was about
These plates were then mated to plates of equal thickness (not
plated) and assembled in a shielded hood as indicated in steps (a) to
(h) above except that in the case of the pair of 1/4 inch plates care
was taken not to rotate the plates during and after assembly and in the//
remaining cases the rotation specified in step (h) was done only once
in one direction.
The plates were then disassembled and the radioactive contamination
on the plates which were in contact with the radioactive plates measured.
The transferred activity was:
1/4 in. plate approximately
3/16 in. plate approximately
1/8 in. plate approximately
1/16 in. plate approximately
0.05 microcuries
3. microcuries
0.i microcuries
0.4 microcuries
It was also observed in handling that the adhesion of the silver on
the 3/16 in. plate was poor.
-27 -
Kodak type R single coated industrial x-ray film was then placed
on the contaminated plates under darkroom conditions. The sensitive side
of the film was pressed against the radioactive sides of the plates
with a uniform load of about five pounds and left for exposure for three
days. After three days, the film was removedand developed. The
results are shownin Fig. i0.
- 28 -
Chapter IV
RESULTS
A. Pressure Distribution and Radii of Separation from Single Plate
and TwoPlate Finite Element Models.
Using the finite element procedure described in Chapter II, the
midplane stress distribution of single circular plates of thickness 2D,
outer radii of 1.54 in., inner radii of 0.i in., Poisson ratio of 0.3,
and loaded by a constant pressure between radii A and B, Fig. 3(f),
was computed. Computations were performed for D values of 0.i, 0.1333
and 0.2 in. For each value of D the radius B, which defines the
region of the symmetric external load, assumed the values of 0.31, 0.22,
0.16 and 0.13 in. The O stress distribution at the midplane, fromz
the inner radius to the radius at which the above stress is no longer
compressive, is shown in Figs. 12, 13 and 14 as a function of radius.
The identical cases were then recomputed, using again the finite
element method, in accordance with the two plate model shown in Figs. 4(b)
and 5(b). These results are given in Figs. 15, 16 and 17.
Inspection of the above figures show that the two plate model yields
a somewhat different stress distribution in the contact zone than the
stress distribution approximated from the single plate model, and more
significantly, from the heat transfer point of view, the two plate
model yields a lower value for the radius of separation, R whichO'
- 29 -
results in a reduction in area for heat transfer. Table i gives a
comparison of the values for R obtained from the two models.o
It may be observed that the single plate result of Fernlund (Ref. 3,
pp. 56, 124) is in fair agreement with the finite element results
obtained for the single plate model.
B. Radii of Separation from Experiment and Their Predicted Values from
the Two Plate Finite Element Computation.
As described in Chapter III, stainless steel circular plate
specimen (Fig. 7) were bolted together, rotated relative to each other
with the bolt force acting, and after disassembly the contact area of
the joint was determined by measuring the footprints (the shiny, polished
areas) on each plate due to the plates rubbing against each other.
Photographs of these footprints are shown in Fig. 8. Fig. 9 also shows
a typical footprint of the annular bearing surface of the 1/4 - 20 nut
against a plate. All plates tested were of 304 stainless steel,
4 inch O.D., .257 I.D., and the nominal thicknesses of the plates were
1/16, 1/8, 3/16 and 1/4 inch. In addition to the plates fastened
with standard nuts which gave a loading circle of radius B (Fig. 5) of
0.211 inch, plates fastened by the special nuts described in Chpater III
for which B was 0.5 inch were also tested.
Figure i0 shows the results of the autoradiographic tests described
in Chapter III. For all plate pairs tested, i.e. 1/16, 1/8, 3/16 and
1/4 inch nominal, the value of B was 0.211 inch.
The pressure distributions and radii of separation for all the
- 30 -
above test cases were computedindependently by the two plate model
finite element analysis. Table 2 gives the test and analytical results
for all test cases. The test results are an average of all measurements
(minimumof six readings). A description of the analyses follows.
Figure 18 shows the results of a two plate and a single plate model
analysis for the 0.253 inch bolted test specimen.For Figure 19
the external pressure distribution between radii A and B is triangu-
lar. (The total force, however, is equal to the force exerted in the
case of uniform pressure.) In one case, the peak external pressure is
at A, Fig. 20(a), and in the other case at B, Fig. 20(b). Results
of another computation which assumeda uniform displacement of 50
microinches under each nut is shownin Fig. 21. It is interesting to
note that the point of separation obtained by using the two plate model
for all variations of loading given above occurs in the range of r/A
values of 2.73 to 2.93 while the two plate model yields separation at
a value for r/A of 3.5. The computeddeflections under the nuts are
given in Fig. 22.
The finite element analysis results for the 0.191 in. plate pair
specimen are given in Fig. 23. Figures 24 and 25 show the computed
pressure distribution and deflection patterns in the joint, respectively,
for the 1/8 in. plate pair. In order to investigate the possible
influence misalignments of the spanner wrench, i.e. vertical forces or
restraints exerted at edge of plate, mayhave on the results of the
experiment, the extreme case of fixing the outer edges of the plate as
shownin Fig. 20(c) was considered. As Fig. 24 shows, within the
- 31 -
resolution of the finite element gridsize, the effect is negligible.
This model, Fig. 20(c), and result also indicate that the influence of
additional fasteners 2 inches awaywould not have an influence on the
contact zone for the geometry considered. (However, if the distance
between bolts is considerably reduced, then the contact area should
increase.) The computedresults for the 1/16 inch plste pair is
given in Fig. 26.
Figure 27 gives the finite element analysis results for the asym-
metric case of a 1/8 in. plate bolted to a 1/4 in. plate. The model
shownin Fig. 5(c) was used and as discussed in Chapter II, this
model is strictly valid only if the friction in the joint prevents
sliding between the plates. Nevertheless, the percent discrepancy
between the computedvalue and tested value (see Table 2) falls within
the range of the symmetric cases analyzed and tested.
In summary, the results obtained from the two plate finite element
model and from experiment are in good agreement (Fig. 28).
- 32 -
Chapter V
APPLICATION
An application of the above results for the evaluation of the
thermal contact conductance, hc, and the determination of the heat
transferred in a specific, but typical, lap joint section is illus-
trated in this chapter.
An aluminum lap joint in a vacuumenvironment, the relevant
section and boundary conditions as shownin Fig. 29, was analyzed by
meansof a nodal analysis. The plate thickness was 0.i in. and the
hole diameter, 2A, was 0.2 in. The bearing surface of the bolt, 2B,
was 0.26 in. in diameter. Becauseof the high conductivity and small
thickness of the plates, no z dependence(see Fig. 29) was assumed
for the temperature in the main body of the plate. However, heat flow
in the z-direction in the nodes above and below the contact zone is
considered. Qualitatively, the heat flow in the joint proceeds in the
x-y plane from tile left end (Fig. 29) toward the 0.2 in. diameter
hole. In the vicinity of the hole, a macroscopic constriction for heat
flow is encountered because the flow is being channeled toward the
small contact zone. The flow of heat then encounters the microscopic
constrictions at the contacting asperities (which determine h ) in thec
contact zone; spreads out in the x-y directions in the second plate;
and continues to the right edge of the lap joint.
- 33 -
The material properties assumedwere (refer to equation i.I):
H = 150,000 psi
k = i00 Btu/hr-°F-ft
O = 5.9 x 10-6 ft.
tan e = 0.i
(kI = k2 = i00)
(O1 = 02 = 50 x 10-6in.)
Assuming further, a uniform load of 46,500 psi on the loading surface
(#10 screw; 1000 lb. bolt force) and referring to Fig. 15, curve
BA 1.3, the following interface stresses, O , contact heat transferz
coefficient, h , and conductance, (area)'(h), were obtained as ac c
function of inner and outer radii. (These radii define increments of
area, the sum of which define one quarter of the contact zone.):
route r rinne r Oz h Area x hC c
inch inch psi Btu/hr -° F-ft 2 Bt u/hr -° F-ft 2
.13 .i 27,900 446,000 16.6
.16 .13 14,000 223,000 10.6
.175 .16 3,950 63,100 1.7
The conductance between nodal points were then computed and with the aid
of the steady state heat transfer program listed in Appendix D, the
nodal temperatures for the conditions given in Fig. 29 were computed.
The heat transferred from the edge maintained at 20°F to the edge at
0°F (Fig. 29) for this case was 2.88 Btu/hour. The same computation
was repeated for the case of a bearing surface between the plate
- 34 -
andthe bolt (2B) of 0.44 in. in diameter, but the bolt force was
left unchanged. The heat transferred from the 20°F edge to the
0°F edge in this case was 3.15 Btu/hour. In the absence of the
joint the heat transfer along an equivalent 7 inch length of solid
aluminumwould have been 3.58 Btu/hour. This data shows that the
thermal resistance of the contact zone (not entire 7 inch lap joint)
was decreased from 1.52 to 0.92 °F-hr/Btu by the increase of the
effective bolt head diameter from .26 to .44 in. It should be observed
that the change in thermal resistance of the joint is primarily due to
the increase in contact area and the resulting decrease in macroscopic
constriction resistance at the hole. Also, the heat flux in this
example is mainly controlled by the 7 inch length and 0.i inch
thickness rather than the joint resistance. This emphasizes the import-
ance of a balanced thermal design.
For large heat fluxes where thermal strains mayhave an influence
on the radii of separation, the finite element program given in
Appendix C maybe used. Also, in a non-vacuumenvironment the effect of
the interstitial fluid is added in two ways. Firstly, equation (1.6)
is applied to account for the presence of interstitial fluid in the
contact zone, and secondly, conduction across the gaps between the plates
and convection from the plates is considered. (Radiation heat transfer,
if applicable, should also be included.)
- 35 -
Chapter VI
CONCLUSIONS
The finite element technique used in this work for the analysis
of the pressure distribution and deformation of smoothand flat bolted
plates under conditions of axial symmetrypredicts contact areas in
joints considerably lower than reported p_eviously in the literature.
These results were verified experimentally. The discrepancy between
the previously reported results and the results reported here is due
to the simplifying assumption madeby earlier _esearchers that a joint
can be modeled as a single plate.
The computer programs listed in the appendices will also accom-
modate joints madeup of plates of dissimilar materials and the presence
of thermal gradients.
Of the eleven tests performed, only one (case 3, autoradiographic)
yielded inconsistent results. (This data point could probably be
ignored because of the poor adhesion of the plating material which
manifested itself by the high radioactive contamination count during
test.)
The finite element analysis performed for the test specimen show
that the gap between the 1/4 inch bolted steel specimen is 98.6
microinches at the outer radius of the plate of 2 inches, and 1/32
of an inch away from the radius of separation (0.35 in.), the gap is
- 36 -
only 3 microinches for the test load. This data indicates the
difficulties previous workers have encountered in their experiments.
(This also explains the oval shape of several of the footprints.)
Furthermore, this data shows that the effects of surface roughness and
the lack of flatness could have a significant effect on the size of
contour area.
An application of the above work to a heat transfer problem is
illustrated in Chapter V.
- 37 -
REFERENCES
1
me
Be
e
.
e
e
e
e
i0.
ii.
12.
Cooper, M.G., Mikic, B.B., and Yovanovich, M.M., "Thermal Contact
Conductance," International Journal of Heat and Mass Transfer,
Vol. 12, No. 3 (March 1969), 279-300.
Roetscher, F., Die Maschinenelemente,Erster Band. Berlin: Julius
Springer, 1927.
Fernlund, I., "A Method to Calculate the Pressure Between Bolted
or Riveted Plates," Transactions of Chalmers, University of Tech-
nolo__y. Gothenburg, Sweden, No. 245, 1961.
Greenwood, J.A., "The Elastic Stresses Produced in the Mid-Plane
of a Slab by Pressure Applied Symmetrically at its Surface,"
Proc. Camb. Phil. Soc. Cambridge, England, Vol. 60, 1964, 159-169.
Lardner, T.J., "Stresses in a Thick Place with Axially Symmetric
Loading," J. of Applied Mechanics, Trans. ASME, Series E, Vol. 32,
June 1965 (458-459).
Sneddon, I.N., "The Elastic Stresses Produced in a Thick Plate by
the Application of Pressure to its Free Surfaces," Proc. Camb. Phil.
Soc. Cambridge, England, Vol. 42, 1946 (260-271).
Bradley, T.L., "Stress Analysis for Thermal Contact Resistance
Across Bolted Joints," M.S. Thesis, Massachusetts Institute of
Technology, Mechanical Engineering Department, Cambridge, Mass.
August 1968.
Louisiana State University, Division of Engineering Research, The
Thermal Conductance of Bolted Joints. NASA Grant No. 19-001-035,
C.A. Whitehusrt, Dir. Baton Rouge: LSU Div. of Eng. Res., May 1968.
Zienkiewicz, O.C., The Finite Element Method in Structural and
Continuum Mechanics. London: McGraw-Hill Publishing Co., Ltd., 1967.
Przemieniecki, J.S., Theory of Matrix Structural Analysis. New York:
McGraw-Hill Book Co., 1968.
Cobb, B.J., "Preloading of Bolts," Product Engineering, August 19,
1963 (62-66).
Wilson, E.L., "Structural Analysis of Axisymmetric Solids," AIAA
Journal, Vol. 3, No. 12, (Dec. 1965), 2269-2274.
- 38 -
13. Jones, R.M. and Crose, J.G., SAAS II Finite Element Stress Analysis
of Axisymmetric Solids. United States Air Force Report No. SAMSO-
TR-68-455 (Sept. 1968).
14. Christian, J.T. and others, FEAST-I and FEAST-3 Programs. Cambridge:
Computer Program Library, M.I.T. Department of Civil Engineering,
Soil Mechanics Division, 1969.
15. Clough, R.W., "The Finite Element Method in Structural Mechanics,"
Ch. 7, Stress Analysis, Zienkiewicz, O.C. and Holister, G.S.,
editors. London: John Wiley and Sons, Ltd., 1965.
- 39 -
APPENDIXA
FINITE ELEMENTANALYSISOFAXISYMMETRICSOLIDS
The finite element method and the equations which govern the
stresses and displacements in axisymmetric solids is given in the
literature [9,10,12,13,15] and the procedure will be briefly summarized
in this appendix.
The procedure for the standard stiffness analysis method is as
follows [15]:
(a) The internal displacements, v, are expressed as
{v(r,z)} = [M(r,z)] {_} (A.I)
where M is a displacement function and _ are the generalized coordi-
nates representing the amplitudes of the displacement functions.
(b) The nodal displacements vi are expressed in terms of the
generalized coordinates
{vi} = [A] {_} (A.2)
where
into
A is obtained by substituting the coordinates of the nodal points
M.
(c) The generalized coordinates are expressed in terms of the nodal
displacements
{_} = [A] -I {vi} (A.3)
- 40 -
(d) The element strains, g, are evaluated
{g} = [B(r,z)] {_} (A.4)
where B is obtained from the appropriate differentiation of M.
(e) The element stresses are expressed in terms of the stress-
strain relation D
{_(r,z)} = [D] {g} = [D] [B] {_} (A.5)
(f) Assuminga virtual strain g and a generalized virtual
coordinate displacement _ the internal virtual work, Wi, in the
differential volumn, dV, is given by
dW. = {_}T {O} dV = {_}T [BIT [D] [B] {_} dV (A. 6)i
and the total internal virtual work is
W.I -- {--IT [V_o 1 [B]T [D] [B] dV ] _ (A.7)
(g) The external work,
displacement _ is
W . associated with the generalizede
W = {.}T {B} (A.8)e
where _ are generalized forces corresponding with the displacements _.
- 41 -
(h) After equating W. and W and setting thei e
to unity
f{ _ } = [ I [BIT [D] [B] ] _ =
Vol
[ _ ] {_}
where [ k ] = [B]T [D] [B] dV
Vol
m
c_ displacement
(A.9)
(A.10)
and which transforms to the nodal point surfaces
k = [A-1] [ k ] [A-1] (A.II)
(i) The stiffness matrix for the complete system is then
11
[K] = Z [k] mm=i
(A.12)
where n
becomes
equals the number of elements and the equilibrium relationship
{Q} = [K] {v.} (A.13)1
wheren
{Q} = Y. {R} (A.14)m
m=l
{R} = f [A-I]T [M]T {P}m dA (A.15)
Area
and P are the surface forces.
The above procedure applies with minor modification to problems
with thermal and body force loading.
- 42 -
The expression
{Q} = [KI {v.}i
(A. 16)
represents the realtionship between all nodal point forces and all
nodal point displacements. Mixed boundary conditions are considered
by rewriting this equation in the partitioned form
IQa K , u_ _ = aa ! Kab _ _
Kba ', Ebb!
!
where v. = u.i
The first part of the partitioned equation can be written as
(A. 17)
{Qa } = [Kaa] {Ua } + [Kab] {Ub} (A. 18)
and then expressed in the reduced form
where
{Q*} = [Kaa] {Ua } (A.19)
{Q,} = {Qa} - [Kab ] {Ub} (A.20)
The matrix equation (A. 19) is solved for the nodal point displacements
by standard techniques. Once the displacement are known the strains
are evaluated from the strain displacement relationship and the stresses
in turn are evaluated from the stress strain relations.
Both triangular and quadrilateral elements are used. The displace-
ments in the r-z plane in the element are assumed to be of the form
- 43 -
Vr = _I + _2 r + e3 z
Vz = _4 + _5 r + _6 z
(A. 21)
This linear displacement field assures continuity between elemen_ since
lines which are initially straight remain straight in their displaced
position. Six equilibrium equations are developed for each triangular
element.
A quadrilateral element is composed of four triangular elements
and ten equilibrium equations correspond to each element.
- 44 -
APPENDIX
FINITE ELEMENTPROGRAMFORTHEANALYSISOFISOTROPIC
ELASTICAXISYMMETRICPLATES(ref. 13,14)
Input Instructions:
Card
Sequence It em
i
2
Title
Total number of nodal points
Total number of elements
Total number of materials
Normalizing stress (NORM)
Number of pressure cards
Format
18A4
15
15
15
15
I5
Columns
1-72
1-5
6-10
11-15
16-20
21-25
(If NORM= 0, put in value of E in material card;
if NORM = i, put in value E/Overtical;
if NORM = -i, put in value E/Ooctahedral;
NOTE: Use NORM = 0 for this application.)
(Material property cards - one set of
(a) and (b) for each material)
(a) ist card
Material No.
Initial _Z
Initialr
15 1-5
stress FI0.0 6-15
stress FI0.0 16-25
(b) Second Card
E FI0.0
FI0.0
i-i0
11-20
- 45 -
CardSequence It em
Nodal point information (One for
each node)
Node number
CODE
r-coordinate
z-coordinate
XR
XZ
If the number in column i0 is
0 XR is the specified R-load and
XZ is the specified Z-load
XR is the specified R-displacement and
XZ is the specified Z-load
XR is the specified R-load and
XZ is the specified Z-displacement.
XR is the specified R-displacement and
XZ is the specified Z-displacement.
Fo rmat Co lumn
215,4F!0.0
1-5
6-10
11-20
21-30
31-40
41-50
Condit ion
free
fixed
Remarks
The following restrictions are placed on the size of problems which
can be handled by the program.
Item Maximum Number
Nodal Points 450
Elements 450
Materials 25
Boundary Pressure Cards 200
-45A-
All loads a_e considered to be total forces acting on a one radian
segment. Nodal point cards must be in numerical sequence. If cards
are omitted, the omitted nodal points are generated at equal intervals
along a straight line between the defined nodal points. The boundary
code (column i0), XR and XZ are set equal to zero.
If the number in columns 6-10 of the nodal point cards is other
than 0, i, 2 or 3, it is interpreted as the magnitude of an angle in
degrees. The terms in columns 31-50 of the nodal point card are then
interpreted as follows:
XR is the specified load in the s-direction
XZ is the specified displacement in the n-direction
The angle must always be input as a negative angle and may range from
-.001 to -180 degrees. Hence, +i.0 degree is the same as -179.0
degrees. The displacements of these nodal points which are printed by
u = the displacement in the s-directionr
u = the displacement in the n-directionz
Element cards must be in element number sequence. If element cards
are omitted, the program automatically generates the omitted information
by incrementing by one the preceding I, J, K and L. The material
identification code for the generated cards is set equal to the value
given on the last card. The last element card must always be supplied.
Triangular elements are also permissible; they are identified by
repeating the last nodal point number (i.e. I, J, K, K).
One card for each boundary element which is subjected to a normal
pressure is required. The boundary element must be on the left as one
the program are
- 46 -
progresses from I to J. Surface tensile force is input as a negative
pressure.
Printed output includes:
i. Reprint of input data.
2. Nodal point displacement
3. Stresses at the center of each element.
Nodal point numbersmust be entered counterclockwise around the
element when coding element data.
The maximumdifference between the nodal point numberson an
element must be less than 25. However, on a nodal diagram elements
and nodes need not be numberedsequentially.
- 47 -
0 o000o000000o00o0o00oooooo oo oo00000oO0 O0 oo 000o0ooo o00o 0ooooo o ooooo o0oo0o_ _ _ _ _ _ _H _ __ _P _F _ _ _ _F P _._ __ _ _ZZZZZZZZZ_ZZZZZZZZZZZLZZZZZZ ZZ ZZZZZZ
.+
- 48 -
000000000000000OO O00000000000 O00C_O0000000000000000000 CO O00000CO0000000 O0
ZZZ_ZZZZZ_ Z_Z_ZZ_ZZ_Z_ZZZZ_ZZZZ_ZZZ_
C
C
C
C
170
180
3OO290
200
210
325
340
IX(N,4)=I X(N-I,4)+I
IX(N,5)=IX(N-I,5)
WRITE (6,2C17) N,(IX(N,I),I=I,5)
IF (M-N) i£.0,180,140
TF (NU_4EL-N} 300,300,130
READ AND PRINT THE PRESSURE CARDS
IF(NU_APC) 290,210,290
WR I TE (6,9000 )
OQ 200 L=I,NU_PC
RFAD(5,gOOl) IF_C(L),JSC(L),PR(L)
WRI TE(6,9002) IBC(L),JRC(L),PR(L)
CONT INUE
DETERMINE PANg WIDTH
J = 0P_O 340 N:I,NUMEL
DO 340 I=i,4
0[3 325 I_=I,4
KK=IX(N, II-IX(K,L)
IF (KK.LT.O) KK=-KK
IF (KK.GT_.J) J=KK
CC}NTINUE
C[1NT INUE
_RANO:2_
SF1LVE FO
K S W=O
CAt. L STIFF
IF (KSW.NE.O) g_ TO 900
,I +2
P fiISPLACEMENTS AND STRESSES
CALf. RANSCL
_R I TE(6,2052 I
FENTO073
FENTO074
FENTO075
FENTO076
FENTO077
FENTOCT8
FENTO079
FENTO080
FENTO081
FENTO082
FENTO083
FENTO084
FENTO085
FENTO086
FENTO087
FENTO088FENTO089
FENTO090
FENTOOgl
FENTOOg2
FENTOOg3
FENTOOg6
FENTO095
FENTOOg6
FENTOOg7
FENTOOg8
FENTOOgq
FENT01CC
FENTOt01
FENTOt02
FENT0103
FENTOIOA
FENTOtC5
FEkT0106
FENT0107
FENTOIOR
I
I
C
C
C
450
c_O0
ci0
q20
_750
1000
I001
1002
1003
1004
IOD5
1006
1007
WRITE (6,2025) (N,B (2*N-1I,B (2_N), N= I, NUMN P )
CALL STRESS(SPLOT)
PROCESS ALL DFCKS EVEN IF ERROR
GO TC glO
WRITE (6,4C00)
WRITE (6,4001) NED
READ (5,1000) CHK
IF (CHK.NE.STRS) GO TO 920
GO TO 50
CONTINUE
WRITE (6,4002)
C_Lt EXIT
F_RM_T (18A4)
FORMAT (1215)
FRRMAT ( 15,2F10.0)
FORMAT(2FIOoO)
FORMAT (2F10°0)
FORMAT (3FIO.O)
FORMAT (215,4FI0.0)
FC]RMAT (615I
2000 FORMAT (IHI,ZOAZ+)
2006 FOPMAT (28HOqU ,BER OF NODAL POINTS ..... 13/
i 2RH NUMBER OF ELEMENTS 13)
2007 FORMAT (20HOMAIERIAL NUMBER .... [3/
i 25H INITIAL VERTICAl STRESS= FIO°3 ,5X,
2 26HINITIAL HCRIZONTAL STRESS: FIO.3)
2013 FORMAT (12HIHi-IDAI. POINT ,4X, 4HTYPE ,4X, IOHR-@RDINATE ,4X,
1 ]OHZ-ORCINATE ,IOX, 6HR-I_qAD ,IOX, 6HZ-LOAD I
20].4 FORM_T (II2,18,2F14.3,2E16.5)
FENTOIOS
FENT0110
FENTOI11
FENT0112
PENTOII3
FENTOI14
FENT0115
FENTOI16
PENT0117
FENT0118
FENTOII9
FENT012C
FENTOI21
FENTOI22
FENTOI29
FENTOI24
FENTOI25
FENTOI26
FENTOI27
FENTOI28
FENTO12g
FENTOI30
FENTO131
FENTO132
FENTOI33
FENTOI34
FENT0135
FENTOI36
FENTOI37
FENTOI38
FENT0139
FENTOI40
FENTOI41
FENTOI42
FENTOI43
FENTOI44
I
o
I
2015
2016
2017
20,25
2041
2051
2052
C
BOO3
C
4000
4001
4002
C
9000
9001
9002
C
FORMAT (26HONO
FORMAT (49N1EL
FORMAT (ii13,4
FORMAT (12HONO
lENT / (I12,1P2
FORMAT (76HOMLq
IITIAL VERTICAL
FORMATILHO,IOX
FORMAT(LHI)
DAL POINT CARD ERROR N: ISl
EMENT NO. I J K L MATERIAL )
16,ill2)
DAL POINT ,6X, 14HR-DISPLACEMENT ,6X, 14HZ-DISPLACEM
D20.7))
CULUS AND YIELD STRESS NORMALIZED WITH RESPECT TO IN
STRESS )
,'E',SX,'NU',/,BX,FII.I,FIO.4/}
FORMAT (1615)
FORMAT (11/I ' ABNORMAL TERMINATION')
FORMAT (//// ' END OF PROBLEM ' 20A4)
FQRM4T (////' END OF JOB')
FORMAT(2gHOPRESSURE BOUNDARY CONDITIONS/ 24H I J PRESSU
RE }
FGRMAT(215,FIO,O)
FORMAT(216,F[2°3)
ENDSURROUTINE STIFF
IMPLICIMPLIC
COMMON
1 DEPTH
2 UZ(45
CqMMGN
COMMON
I HH(6,
2 EE(IO
COMMON
COMMSN
IT REAL_8 (A-H,O-Z)
IT INTEGER_2(I-N)
STTCP,HED(I8),SIGIR(25),SIGIZ(25),GAMMA(25),ZKNOT(25},
(25),E(!O,ZS),SIG(7),R(450),Z(450|,UR(450),
O),STOTAI_(450,G),KSN
/INTFGR/ NUMNP,NUMEL,NUMMAT,NOEPTH,NORM,MTYPE,ICODE(450)
/ARG/ RRR(5),ZZZ(5),S(IO,IO),P(IO),LM(4),DD(3,_),
IO),RR(_),ZZ(4),C(4,_),H(6,10},D(5,6),F(6,10),TP(6),XI|6),
),TX(450,5)
/B_NARG/ B(qcO),A(gOO,54),MBA_D/PRFSS/ IBC(2OO),JBC(2OO),PR(2OO),NUMPC
DIMENSION CODE(450)
FENTOI45
FENTOI46
FENTOI47
FENTOI48
FENTOI4 Q
FENTOI50
FENTOI51
FEKTOI52
FENTOI53
FEKTO154
FENTOI55
FENTOI56
FENTOI57
FENTOI58
FEkTOI59
FENTOI60
FENTOI6I
FENTOI62
FENTOI63
FENTOI64
FENTOI65
FENTOI66
FENT0167
FENT0168
FENTOI69
FENT0170
FENTOITI
FENTOI72
FE_TOITB
_ENTOI74
FFNTOI75
FENTO176
FENTOI77
FENTO178
FENTOITg
FENTO180
I
L,n
I
- 53 -
N N N N N N N Q N N N N N _i N N N N N N N N N N N N N N N N N N _ N N N
0000000000000000000000000 O000000000C
ZZ ZZ ZZ_Z ZZ ZZZZ Z ZZ ZZZZZZ ZZ _ Z_Z Z_ ZL ZZZ
_ ______.____ ___
- 54 -
0 C,_ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
_ ZZZ ZZ_Z ZZ ZZZ ZZZZZ LZZ _ ZZZ ZZ Z Z ZZ Z_ZZ_
_ _ _ _ _ _ _ _ _ _ _ _ _. _ _ _ _ _ _ _ _ _ _. _ _ _ _ _ _ _ _ _ _ _ _ _
A A
z Z Z
_0_
400
C
500
C
2C03
200z_f-
C
C
C
C
C
C
CONTINUE
RETURN
FORMAT (26HONEGATIVE AREA ELEMENT NO. [4)
FORMAT (2q_O_AND WIDTH EXCEEDS ALLOWABLE [4)
END
SlJ_ROtJT[NE OUAD(N,VQL)
IMPtlCIT RE_L_8 (A-H,O-Z)
T_PLICTT INTEG£R_2(I-N)
CO_C]N STTOP,HED(I81,SIGIR(25),SIGIZ(25),GAMMA(25},ZKNOT(25),
t OEPTNI25),E[[C°25I,S[G(7),R(650),Z(450),UR(450),
2 tIZ(450)°ST_TAL(450,4),KSW
CQHNGN /INTEGR/ NUMNP,NUNEL,NUNMAT°NDEPTH,NQRM,MTYPE, [CODE(450)
COMMON /ARG/ KRR(5),ZZZ(5),S(IO,IOI,P(IOI,LM(4},DD(3,3|,
t HH(6°IO),RR(#},ZZ(4),CI4,41,H(6, IOI,O(6,6I,F(6,10),TP(O),XI(6),
2 FE{!O}, 1X(450,5)
COMMCN /8_NARG/ B(900),A(900,54),MBAND
I=IXIN,1)
3=IX(N.2)
K=IX(N°3)
L=IX(N.4)
Tl=]
[2=2
I3=3
14=4
I5=5
I-)FTERMINE EI_AST[C CONSTANTS AND STRESS-STRAIN RELATICNSH[P
CALL _PROP IN)
FENT0289
FENTO2gO
FENTO2ql
FENT0292
FENT0293
FENT02£4
FENTO2q5
FENTO2q6
FENT02£7
FENT0298
FENTO2q9
FENT0300
FENTO30t
FENT0302
FENT0303
FENT0304
FENT0305
FENT0306
FENT0307
FENT03O8
FENT0309
FENT03IO
FENT031[
FENT0312
FENT0313
_FNT0314
FENT0315
FENT0316
FENT0317
FENTO3t8
FENTO3IgFFNT0320
FENTO32t
FENT0322
FENT0323
FE_T0326
I
_m
I
- 56 -
0 0 O0 0 0 0 O0 0 0 C) 0 0 0 0 0 O0 0 0 O0 0 O0 0 0 0 0 O0 0 0 0 0
Z_Z_ZZ ZZ ZZ _Z_Z_ZZZZZZZZZZ_ZZZZ_ZZZZ_
__ _._ _ _M __ _ _ ____ _ _ _
O0@ •
_'_cr_ Z
_ 0
0nf_ o
*+ ._,
I! ,,.,,4 II !! '?..3_-4 I---r_"
Z Z
0 0 00 0 00 o 0oJ _4 _4
u'_ I&l if,, LIA tl"_, Lk_
0 - _ _ I _-- _ e_ __ _ 0_ _
- _57 -
O O OOO OOOO OOOo OO OO O OOOO OOOOO OO OOOOOO O
_Z_Z_ZZZZ_ZZ_Z_ZZ_ZZZZ_ZZZZZ_ZZZZZZ
0
J
''3
,"I
II I1 !1
(;30 +",?,-1",--
CC3Ir-" [3 I
C,.1+
tJqJ
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x
A,0 #
0 ._
LL
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E3 C34
0 CC_-,-+ _
•4" O_
-.I" _#
tJ _
O_r4" 0-_
',4" (DtD 44- Z
a_ _I" "_ _"
_ ,--, _ _-
_ L_ C2 +_ Z
q3 t_._ £..)
II !I II II !! !! I! II II !! II
_ _ _. _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ _ _, _
tD
- 58 -
0 0 O0 0 0 0 0 0 0 0 0 0 0 0 O0 0 O0 O0 O0 0'0 0 0 0 C 0 C' 0 0 0 0
Z Z _._ Z _." Z J Z Z Z Z .__ Z _ ,L Z ::._ _ Z Z Z Z Z Z Z 2_ Z _- Z 2_/ Z Z J" Z Z _
".0 0 ,.0
,,,,,,4 ., 0 0 ,-40
II ,.-4 Q t II ¢II 0.0"30
II II II
C.30 ""3 "_ 0 --'_
0 0
cr; o- 0
,'b
hE _O ]g
Z7; m
V', .-_ D,,. Ca-.-I m r---',1 ,rb -_- z:q'J m 00
rrl ,-_ I
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0 0 ,J'l 0
Z _,__ _ Z _ &u ;...'D I --.,,,-ju-..-,+ C_.
II C ii --i",o _ II "-'-
•,-,.. b,o ,.-,, ,",,0 3::
•,, ZD - o"1 ]m,TEe _ .,,,.,-- r,J _ I'.,J
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,-.,
0_
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.Z. '.'_ C'_ '--
_. _,-_;0 ,.*'I"I
zzm
_0 m r-- 0
m
O I "
•i .... I
OO,I
3E
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Zt'9
rr!zm
;;o
z
C.) C_ m i-,,• C3 ,,",, •ol ,'1"1 ..",J II "1'1 --,. rri
m mr'rl ,'-,.
f',O
f,,j
-tl -rl "T1 "_ "1"1 -I'1 "rl -rl -rl 1! m TIm m "T1 m "rl _'1 "T1 1_. 1"1 q'l "r'l "T1 _rl 1"1 m "11 -r_ T1 -Ti T1 m "n _ -rl
m m mm rn m,m. ,mm :-nm r_ cn m m m m m m m m.,m rn m, m m i_ m,-n m rn m rn m rn rnZ Z Z Z Z Z Z Z Z Z Z Z Z 7 Z Z Z Z 2 Z Z Z 7 Z Z Z Z Z Z -.7-Z Z :7 Z Z Z
0 0 0 0 0 C.O 0 0 0 0 0 C3 CD O 0 0 0 0 0 0 0 0 O 0 0 0 0 0 _ 0 0 C_ (ID 0 0 0
,J"l ,J'1 Lm ',.D 'J'_ '.21 L,_ J'l ,....D _j"l ',...'1 '.31 '0"1 _J't ',..._ _ U'_ _ 'J'l U"I L,'I L,'-I _ '0"1 L.,'I _ '..?1 LD _J'l L."l _ 'v'l _ J"l 'J"l
- 19 -
- 62 -
0 O0 O0 CO 000000000000 O0 O00000000000000
ZZZZZ_Z_ZZZZLZZZZZZZZZZ_ZZZZZZ ZZZZ_Z
_._ __ _ _ __ _ _ __ __ __
oor r,
Z
I'-'-LL"
C(2,IZ"
- 63 -
0o000000000000000ooooo000o0o0o0ooooo
_- Z Z Z Z 7 Z Z Z Z Z Z Z z z Z z Z z _ Z 2_ z Z. L Z ;_ Z _£" Z z z _--"Z Z zLLI LIJ LLI U:' UJ U.J IU LI_ IJJ IJJ IJJ LIJ IJJ _ iJ_'Lb UJ UJ LU L_I t.L'LU LU I.IJLL, LLI _L. LIJ LU _ LIJ UJ I.I_LIJ L_J I.L!U_ Ul. U_ Ij_ IJ_ LI_ L_ IJ- U. i._.U- IJ_ IJ-LI. I_ I.I.I_L IJ- .I.IJ. 'J.4_ I.I_iJ- I_. D- LL. I.L L_ L_ U_ LL U_ L_.I.L IJ_
- 66 -
0000000000000 O000000000000000000 O0 O0
ZZZZ _ZZZ_Z_ZZZZ_Z ZZZZZZZZZZZZZZ_ZZZZ
_ W _ _ _ _ _ _ _ _ _ _ _ LU _ _ _I _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
H
- 68 -
APPENDIX C
FINITE ELEMENT PROGRAM FOR THE ANALYSIS OF ISOTROPIC
ELASTIC AXISY_ETRIC PLATES - THERMAL STRALNS INCLUDED (Ref. 13, 14)
Program Capabilities:
The following restrictions are placed on the size of problems which
can be handled by the program.
Item Maximum Number
Nodal Points 450
Elements 450
Materials 25
Boundary Pressure Cards 200
Printed output includes:
i. Reprint of Input Data
2. Nodal Point Displacements
3. Stresses at the center of each element.
I__put Data Format:
A. Identification card - (18A4)
Columns 1 to 72 of this card contain information to be printed
with results.
B. Control card - (515,FI0.0)
Columns 1 - 5 Number of nodal points
6 - i0 Number of elements
ii - 15 Number of different materials
- 69 -
C,
D.
16 - 20
21 - 25
26 - 35
Normalizing stress (see NORM, Appendix B)
Number of boundary pressure cards
Reference temperature (stress free
temperature)
Material Property informationY
The following group of cards must be supplied for each
different material:
First Card - (215, 2FI0.0)
Columns 1 - 5 Materials identification - any number
from 1 to 12.
6 - i0 Number of different temperatures for
which properties are given = 8 maximum.
ii - 20 Initial Z stress.
21 - 30 Initial R stress.
Following Cards - (4FI0.0) One card for each temperature
Columns 1 - i0 Temperature
ii - 20 Modulus of elasticity - E
21 - 30 Poisson's ratio -
31 - 40 Coefficient of thermal expansion
Nodal Point Cards - (215, 5FI0.O)
One card for each nodal point with the following information:
Columns 1 - 5 Nodal point number
i0 Number which indicates if displacements
or forces are to be specified.
ii - 20 R - ordinate
21 - 30 Z - ordinate
31 - 40 XR
41 - 50. XZ
51 - 60 Temperature
- 70 -
If the number in column i0 is
.
Condition
XR is the specified R-load and freeXZ is the specified Z - load
XR is the specified R-displacement and
XZ is the specified Z-load.
XR is the specified R-load and
XZ is the specified Z-displacement.
XR is the specified R-displacement and fixedXZ is the specified Z- displacement.
All loads are considered to be total forces acting on a one radian
segment. Nodal point cards must be in numerical sequence. If cards
are omitted, the omitted nodal points are generated at equal intervals
along a straight line between the defined nodal points. The necessary
temperatures are determined by linear interpolation. The boundary code
(column i0), XR and XZ are set equal to zero.
Skew Boundaries:
If the number in columns 5-10 of the nodal point cards is other
than 0, 1, 2 or 3, it is interpreted as the magnitude of an angle in
degrees. The terms in columns 31-50 of the nodal point card are then
interpreted as follows:
XR is the specified load in the s-direction
XZ is the specified displacement in the n-direction
The angle must always be input as a negative angle and may range from
-.001 to -180 degrees. Hence, +i.0 degree is the same as -179.0 degrees.
The displacements of these nodal points which are printed by the
program are
- 71 -
ur
uz
= the displacement in the s-direction
= the displacement in the n-direction
No Element Cards - (615)
One card for each element
Columns 1 -5
6 - i0
ii - 15
16 - 20
21 - 25
26 - 30
Element number
Nodal Point I
Nodal Point J
Nodal Point K
Nodal Point L
Material Identi-
fication
i. Order nodal points
counter-clockwise
around element.
2. Maximum difference
between nodal pointI.D. must be less
than 25.
Element cards must be in element number sequence. If element cards
are omitted, the program automatically generates the omitted information
by incrementing by one the preceding I, J, K and L. The material iden-
tification code for the generated cards is set equal to the value given
on the last card. The last element card must always be supplied.
Triangular elements are also permissible; they are identified by
repeating the last nodal point number (i.e., I, J, K, K).
F. Pressure Cards - (215, IFI0.0)
One card for each boundary element which is subjected to a
normal pressure.
Columns 1 - 5 Nodal Point I
6 - i0 Nodal Point J
ii - 20 Normal Pressure
The boundary element must be on the left as one progresses from I to J.
Surface tensile force is input as a negative pressure.
Listin$:
C FINITE ELEMENT PROGRAM FOR THE ANALYSIS OF ISCTROPIC ELASTIC
C A×YSY_4METRIC PLATES RFF FEAST 1,3 SAAS 2
C
C
C
C
C
IMPLICIT
IMPLICIT
COMMON
I O[TPTH(25
2 UZI450)
3 T(450],
COMMQN /
COMMON /
I HH(6,10
2 EE(IOI,
REAL::,:8 (A-H,O-Z)
INTEGER_X2 ( I-N}
STTOP,HEDI18),SIGIRI25),SIGIZ(25),GAMMA(25},ZKNOT(25I,
), F(8,4,25),SIG(7),R(450),Z(450),UR(450),TT(3),
, STOTAL(450,4 } ,
TF,_P, @, KSW
INTFGR/ NUMNP,NUMEL,NUMMAT, NDEPTH,NORM, MTYPE, ICODE(450)
ARG/ RRR(SI ,ZZZ(5),S ( i0, i0| ,P(IO),LM(4) ,DO(B,3) ,
),RRI4|,ZZ(4!,C{4,4},HI6, IO),D(6,6),F(6,10),TP(6),XI (6},
IX(450,5)
5O
CC!MMUN tB_NARG/ B(gOOI,AIgOO,54),MBAND
COMMCNIPRESSI IBC(2OO),JBC(2OOI,PR(2OO),NUMPC
DATA STRS /'_#_'/
READ AND PRINT CONTROL :INFORMATION
READ (5.1000,END=950) HED
WRITE (6,2000) HEO
READ(5,1001I NUMNP,NUMEL,NUMMAT,NOR_,NUMPC,Q
WRITE(6,2O06) NUMNP,NUMEL,NUMMAT,NUMRC,Q
IF (NORM) 65,65,66
66 WRITE (6,2041)
READ AND PRINT MATERIAL PROPERTIES
65 CONTINUF
DO 8C M=I,NUMMAT
PEAD(5,10]2) MTYPE,NUMTC,SIGIZ(MTYPE),SIGIR(MTYPE)
WRITE(h,2OII)MTYPF,NUMTC,SIGIZ(MTYPE),SIGIR(MTYPE)
FEwTOOCt
FEWTO002
FEWTOC03FEWTO004
FEWTO005
FEWTO006
FEWTO007
FEWTO008
FEWTOOOq
FEWTOOI.O
FEWTO011
FEWTO012
FEWTO013
FEWTOOI4
FEWTO015
FEWTO016
FEWTO017
FEWTO01_
FEWTOOlg
FEWTOO2C
FEWTO021
FEWTO022
FEWTO023
FEWTO024
FEWTO025
FEWTO026
FEWTO027
FEWTO028
FEWTO029
FEWTO030FEWTOO31
FEWTO032
FEWTO033
FEWTO034
FEWTO035
FEWTO036
I
I'.)
I
P,.>0
7_II i;a
'71
.0
o
v
mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm
OOO0OOOOOOOO000000oO0OO000000OO0OOO000ooooo0o0o00 o0o0000o0oooo 0o0 O000ooO
- _L -
C
C
C
CC
C
130
140
150
170
180
BOO
290
200
210
325
B40
READ (5,1CC7} M,(IX{M,I),I=I,5)
N=N+I
IF (M.-N) !70,170,150
I×(N, i i=IX(N-1,1 }+1
IX(N,2)=IX{N-t ,2)+i
IX{N,3)=IX(N-1,3)+I
IX(N,4)=IX(N-1,4)+I
IXIN,5)=IX(N-t ,5)
WRITE (6,2C17) N,(IX(N,I),I=I,5)
IF (M-N) 180,t. 80,140
IF (NUMEL-N) 300,300,130
REAC AND PRINT THE PRESSURE CARDS
IFINUMPC) 290,210,290
WRtTE( 6,9000 )
DO 200 L=I,NUMPC
READ(5,gO01) IBC(L),JBC(L),PR(L}
WRITF(6,9002 ) IBCiLI,JBCIL),PR{L)
CONTINUE
DETER_'4INE BANg WIDTH
J=O
(30 340 N=I,NUMEL
DO 340 I=1,4
DO 325 L=l,4
KK=IX(N,II-IX(N,L)
IF (KK,LT,O) KK:-KK
IF (KK,GT,J) J=KK
CONTINUE
CDNT INUE
MBANO.:2_J*2
SOLVE FITR DISPLACEMENTS AND STRESSES
FEWTO073
FEWTO074
FEWTO075
FEWTO076
FEWTOOT7
FEWTO078
FEWTO079
_EWTO080
FEWTO081
FEWTO082
FEWTO083
FEWTO084FEWTO085
FEWTO086
FEWTOCS7
FEWTO088
FEWTOOBg
FEWTO090
FEWTOOgl
FEWTOO92
FEWTO093
FEWTOOg4
FEWTOOg5
FEWTOOq6
FEWTOCg7
FEWTOO98
FEWTCCgq
FEWTOIO0
FEWTO!OI
FEWTOIC2
FEWT01C3
FEWT0104
FEWT0105
FEWT0106
FEWT0107
FEWT0108
i
...4
i
C
C450
£CC
900cio
C920
950
CC
I000i001] 0021003100410051006
1007
101 1
i012
C
2COO
KSW=O
CALL STIFf:
IF (KSW.NEoO) GO TO 900
CALL BANSDL
WRITE (6,2052)
WRITE (6,2025) (N, B I2=N-I ) ,B (2_N) , N=I, NUMNP)
CALL STRESS(SPLOT)
PROCFSS ALL _ECKS EVEN IF ERROR
GO TO 910
WRITE (6,4000|
WRITE (6,4001) HED
READ (5,1000) CHK
IF (CHK.NE,STRS) GO TO 920
GO TCI 50
CONTINUE
WRITE (6,40021
CALf EXIT
FORMAT (18A4)
FCIRMAT(515,FIO.O)
FDRMAT ( 15,2FI0.0)
F_RMAT (2F10, O)
FORMAT (2FI0.0)
FDRMAT (%F]O.O)
F_R_AAT(21 5.5FIC,0)
FORMAT (615)
FORMAT ( 4FI O. O)
FORMAT( 2[ 5,2FIC. O)
FORMAT (IHI,ZOA4)
FEWTOIO9
FEWTOIIO
FEWTOIII
FEWT0112
FEWTOII3
FEWT0114
FEWT0115
FEWTOI16
FEWT0117
FEWTOII8
FEWTOI19
PEWTOI20
FEWT0121
FEWT0122
FEWT0123
FEWTO12k
FEWT0125
FEWT0126
FEWTOI27
FEWTOI28
FEWT0129
FEWTOI30
FEWTOI31
FEWTOI32
FEWTOI33
FEWTOI34
FEWTOI35
FEWTOI36
FEWTOI37
FEWTOI38
FEWTOI3g
FEWTOI40
FEWTOI4I
FEWTOI42
FEWTOI43
FEWTOI44
i
,.,4
i
2006 FORMAT (11,
1 30HO NUMBER OF NODAL POINTS I3 /
2 30HO NUMBER OF ELEMENTS I3 /
3 30HO NtJMIRFR _]F DIFF, MATERIALS--- 13 1
4 30HO NUMRFR OF PRESSURE CARDS .... I3 /
5 BOHO REFERENCE TEMPERATURE ....... E12.4|
2010 F{]RMAT (i5HO TEMPERATURE t5X 5HE 15X 6HNU 15X 6HALPHA 9X
1/4F20.8)2011 FORMAT (17HOMATERIAL NUMF_ER= 13, 3OH, NUMBER OF TEMPERATURE CARDS=
I 1.5,25H INITIAL VERTICAL STRESS= FIO°B,5X,
2 27H INITIAL HCRIZONTAL STRESS= F10,3)
2013 FORMAT (12HIN@DAL POINT ,4X, 4HTYPE ,4X, IOHR-ORDINATE ,4X,
1 IOHZ-ORgINATE ,IOX,6HR-LOAO ,IOX, 6HZ-LOAD,IOX,4HTEMP l
2014 FORMAT ( 112, I 8,2F14o 3, 2E16.5,F14, 3)
2015 FORMAT (26HONi]BAL POINT CARD ERROR N= I5)
2016 FORMAT (49HIELEMENT NO. I J K L MATERIAL )
2017 FORMAT (i113,416,1112)
2025 FORMAT (12HONODAL POINT ,6X, 14HR-DISPLACEMENT ,6X, 14HZ-DISPLACEM
IENT / (II2,1P2D20.7)}
2041 FORMAT (76HOMOBULUS AND YIELD STRESS NORMALIZED WITH RESPECT TO IN
IITIAL VERTICAL STRESS )
2051 FORMAT(1HO,10X,'E',8X,'NU',/,3X,F11°1,FlO°4/)
2052 FORMAT(1HI)
3003 FORMAT (1615)
4000 FORMAT (//// ' ABNORMAL TERMINATION'I
AO01 FORMAT (//// ' END OF PROBLEM ' 20A4)
40(]2 F(DRM_T (///I' END OF JCB')
9000 FORMAT(29HOPRESSURE BOUNDARY CONDITICNSI 24H I J PRESSU
1RE )
9001 FORMAT(215,FIO.O )
9002 FORMATI216,FI2.3)
FND
£UBROUT INE STIFF
FEWTOI45
FEWTOI46
FEWTOI47
FEWT0148
FEWTOI4£
FEWT0150
FEWTOI51
FEWTOI52
FEWTOI53
FEWTOI54
FEWT0155
FEWTOI56
FEWTOI57
FEWTOI58
FEWTOISg
FEWTOI60
FEWTOI61
FEWTOI62
FEWTOI63
FEWTOI64
FEWTOI65
EEWTOI66
FEWTOI67
FEWTOIeB
FEWTOI69
FEWTOI?O
FEWTOI71
FEWTOI72
FEWTOI73
FEWT0174
FEWTOI75
FEWTOI76
FEWTOIT7
FEWTOI78
FEWTOIT9
FEWTOI80
I
0"_
I
C
C
IMPLICIT REAL_8 (A-H,C-Z)
IMPLICIT INTEGER_2(I-N)
COMM{]N STTOP,HED( 181, SIGIR(25) , SIG I Z ( 251 ,GAMMA {251 , ZKNCT (25) ,
IDEPTH(25) , E( 8,4,25 },S IG(7),R(450) , Z(450),UR(450 ),TT(3) ,
2 UZ(450),STOTAL(450,4),
3 T(450),TEMP,Q,KSW
COMMON /INTEGR/ NUMNP,NUMEL,NUMMAT,N_EPTH,NORM, MTYPE,ICOOE(450)
COMMON /ARG/ RRR(5) ,ZZZ(5},S(lO,lO),P(IO),LM(4),00(3,3|,
1 HH{6,10),RR(4),ZZ(4),C(4,4),H(6,10),D(6,6),F{6,10|,TP(6),XI|6),
2 EE(10),IX(450,5)
COMMON /B_NARG/ B(900),A(900,54),MBAND
COMMON/PRESS/ IFIC (200), JBC(200 }, PR(200), NUMPC
DIMENSION CODE (450)
INITIALIZATION
NB=27
ND= 2*N B
ND2=2*NUMNP
DO 50 N=I,ND2
B(N)=O,,O
DO 50 M=I.ND
50 A(N,M)=O.O
FORM STIFFNESS MATR IX
F)O 210 N=I,NUMEL
90 CALL QUAD(N,VOL}
IF (VOL) 142,142,144
1._2 WRITE (6,2003) N
KSW=I
GO TO 210
FEWTOI81
FEWTOI82
FEWT0183
FEWTOI84
FEWTOI85
FEWTOI86
FEWT0187
FEWT0188
FEWT0189
FEWTOIQO
FEWTOlgl
FEWT0192
FEWT0193
FEWT0194
FEWTOIg5
FEWT0196
FEWT0197
FEWT0198
FEWT0199
FEWT0200
FEWT0201
FEWT0202
FEWT0203
FEWT0204
FEWT0205
FEWT0206
FEWT0207
FEWT0208
FEWTO20g
FEWT0210
FEWT0211
FEWT0212
FEWT0213
FEWT0214
FEWT0215
FEWT0216
I
,.,..I
I
- 78 -
0 C_OOO0000000oO0000000000000000000000
W_W_JUj_W_WWW_WW_WWW_WWW_I_WW_WWWWWW
r_JI
P
ix
o,J-c II,4D
C3_E3..J
43,,O
v
.-- ÷
.,_ e,j ÷ ..T r,_ .-
_. _ 0_ 0
_Z
II !! + + _ tl It + + _ _ _ 0 I! 0
_ I _ _ I _ I _ __ •00_ II CO_O _ZZ_
!III _ II II _ _ _ Z Z _
m _ mm nlm m rnm mm m mm m m m m mm m m m m m m m m ml_ m mm m m m_ _;_EEE_EEEE_EEEEEEEE_E_EEE__ E_
O OO OO OOO OOO OO Q OOOOOOQ OO O OOOOO OO OOO OO
- 6L -
:iZiii.
_g
3C0
310
C
C
B70
38O
390
400
C
50O
C
2003
2004
C
C_ SAI--_-P._C_cJS ( C[I]D E (I) )
B(JJ-I)=B(Jj-I ) ÷Z X_ (COS_-DZ+ S I NA ._DR I
B(JJI=B(JJ )-ZX_(SINAm-OZ-COSA_DR)
C[)NTI NUE
CONT I NUE
DISPLACEMENT B.C.
DO 400 M=I,NUMNP
U=UR(M)
N=2mM-I
KX=ICODE(MI*I
GO TO (400,370,390,380),KX
CALL MODIFYIN,U,ND2)
G_ T_ 400
CALL MODIFYIN,U,ND2)
U=LJZIM)N=N+I
CALL MOD[FYIN,U,ND2)
CONTINUE
RETURN
FORMAT (26HONEGATIVE AREA ELEMENT NO,: I4|
FORMAT (29HORANO WIDTH EXCEEDS ALLOWABLE I_)
END
SUBROUTINE OUAE(N,VOL)
IMPLICIT REAL_8 (A-H,O-Z)
IMPLICIT INTEGER_2(I-NICOMMGN STTOP,HED(18|,SIGIR(25},SlGIZ(25),GAMMA(25|.ZKNOT(25},
1DFPTHI25|,E(8,4,25),StG(7|°R(450),Z(450|,UR(_50|,TT(3),
2 UZ(450),STOTAL(_50,4),
3 TI450I,TEMP,Q,KSW
C_MMON /INTEGR/ NUMNP,NUMEL,NUMMAT,NCEPTH,NORM,MTYPE,ICODE(450|
COMMON IARG/ RRR(SI,ZZZISi,SIIO,IOI,P(IOI,LMI_|,DD(3,3},
FEWT0289
FEWT0290
FEWTO2gl
FEWTO2g2
FEWTO2g3
FEWTO2g4
FEWT0295
FEWT0296
FEWT0297
FEWT0298
FEWTO2g9
FEWT0300
FEWT030I
FEWT0302
FEWT0303
FEWT0304
FEWT0305
FEWT0306
FEWT0307
FEWT0308
FEWT0309
FEWT0310
FEWT031I
FEWT0312
FEWT0313
FEWT0314
FEWT0315
FEWT0316
FEWT0317
FEWTO_I8
FEWT0319
FEWTO320
FEWT0321
FEWT0322
FEWT0323
FEWTO32_
I
Oo0
I
C
C
C
C
C
C
C
C
C
C
103
104
70
71
]05
£8
lIO
HH(6, I01 , RR( 4 ),ZI(4 ),C (4,4| ,H( 6, i0 ) ,D(6,6) ,F( 6, tO) , TP(6) ,Xl i61 ,
EE(IO) _IX(450,5)
COMMON /BANARG/ B(900_,A(gOO,54),MBANO
I=IX(N,I)
,1= IX( N, 2)
K=IX(N,B)
L=IX(N,4)
II=112=2
13=3I4=4
I5=5
THERMAL STRESSES
TEMP=(T(I)+T(J)+T(K)+T(L))/4-O
DO 103 M=?,8i F ( E( M, l, MTYPE)-TEMP) 103,104, i04
CONT INUE
RATI@=O.O
DEN=E( M, 1, MTYP E I-E(M-I ,I ,MTYPE)
IF{BEN) 70,71,70
RATIO= (TFMP-E( M-l, l, MTYPE | )/DEN
F)O 105 KK=I,3
FE ( KK )=E( M-I , KK+I, MTYPE )+RAT IO*( E( M,KK÷I, MTYPE )-E ( M- 1,KK+I,M TYPE ) I
TEMP=TE MP-Q
DETERMINE ELASTIC CONSTANTS AND STRESS-STRAIN RELATIONSHIP
CALL MPRC]P (N)
DE! l]O _=i,3TT(M) = IC(M,1 } +C( M,2 )+C (M,3) )_EE(3 } _,'_TEMP
FORM G[JADRILATERAL STIFFNESS M/_TRIX
FEWT0325
FEWT0326
FEWT0327FEWTO.328
FEWT0329
FEWT0330
FEWT0331
FEWT0332
FEWTOB33
FEWT0334
FEWT0335
FEWT0336
FEWT0337
FEWT0338
FEWT0339
FEWT0340
FEWT034I
FEWT0342
FEWT0343
FEWT0344
FEWT0345
FEWT0346
FEWT0347
FEWT0348
FEWTO34q
FEWT0350
FEWT0351FEWT0352
FEWT0353
FEWT0354
FEWT0355
FEWT0356
FEWT0357
FEWTOB58FEWTO35g
FEWT036C
Ioo
I
- 83 -
000000000000000 O0 O00 O0000000 O0 O00000
_I_IW___I___I____J_J__
0e
-1+
o,--'
.-r
II I!
,.,-4
C_ "r"zIZ
0
_.-_ .-_ v
!1 1! !1
_J ._ ,J
!I II II I! I! !I II II
__.__
¢,...; c.;
- 84 -
00000000000000000o00o0o0o00000000000
,_ o ,0
II 0 0"_ 0II ti II
u_ o 0
I
N
÷
÷ _ _ ....
N k- X X C x X X X X X X X X _ _ 0 _ _ III II III! IIZ z II II I II I! II II II II II It II II 0 _ ......
o
- 86 -
x
rvI--
Z0
l,.-
v :_
Z -I-
÷ _,_ ,,-4r*-4.L
'--" _ II II I
I:Y_ _'_ ")II
0.- L/_ C.b 0 -_,II .-_ _ _'
"-" C3 C" C i._-q_ UL C.._ _ 33
0 0..0 ,--
,,1-
- 87 -
o-,1r,,,
rm t'_l_ t',.,; ,-,-.
u'_ I 0 *" I 0 r_
_Z _C I _L LU"" II _ _X. _ II _ II
0 + _ _'_ _ I ,_.._d:' _ Z _ _
z _d_Z _ZZZZ
II LI._,,,__ II I..L.,,,__ ___,,,,,,,_,,,._
C_ _ C_ CX,
- 88 -
Oo o0000000000o 0o000 o0000o000000o0o00
_ _ _ _ _ _ _ __ _ _ _ _ _ _. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
- 89 -
000000000000000000000000000000000000
xxx_
IIIIIIII
@
,...4
..,,.,
÷
0,-..4
I-- 4-
LD -"_
.-, r,,-O 4-
L,_
•,.--40_._
I !IA
LL O-CZ.
P_o
_ -nv
]o.>xz ,--t 4
Q
_C4
I, I
"o-4
0
o
"nn
0i
"xJ
c_) :,.ic_o c_ i-.":L3 C) _ri
-T'I,"b .TE ..__i--,
__,C..3Jo 7X;I"rlco Z -..4_,.,,....7_ -.I --I--I Z
Z> _,.4rl-1i'Tl•-"_Z Z
r" _0C3
m Z 0
Z
i, N
NX _
N i
X if)
_ 4
_ N
X
r_
&x
-,4
_ II Ii II
ii il _ 0
I÷__1@
* Oo
o
-0 _ tt • Ii
Z ,.-,._
÷
r- ..-.
r'l-i
___ _______'_ _ ___m mm mm mm m,m m mm m m m m m m mm m m ml_ m m mm m mm mm m m m
000000000000000000000000000000000000
- 92 -
O OC) OOOOOOOOO OOOOO OOO OOOO OOOO OOOOOC °o
_ U Z _ _ __ _ X X x X
_I_ _ IIII
I_ _ _ __ __
+
Z ZZ ____ II __ I1__- _ _ _ _ X X _ X X_ _ _ _ _ __ _ X
__ _XX eXX • • e_N XX X X X_ _ _ _ I! I1 II 11 I! II !l I I_ _ O X X X X O II
Z _ _ [ _ _ X X X X X X X X X _ v 0 II _ -- _ _ 0 _:__ _ xXXXXXX xx_ __ _X
O OO O
II
oO II
fL:' _-',IZbX
OO
I
N
w
or..
+
r,.,4
I
er_
N_4
tY
4.M
I
o4w
r_
cz::
L_e
It
_jjrY,
c.3 ,.,j.
p,.
!!
oo,.,,1"
;_ :'i'l
Z
m
o 0
0 0
_ xx_xxx
il _ II li ii 11 Ii IiX X XX XX_
÷ ÷ ÷ ÷ ÷XXXXXX
m _ m I_ m m _ m m m m m
000000000000MMM M M M MM M MM M
- _6-
- 94 -
APPENDIX D
STEADY STATE HEAT TRANSFER PROGRAM FOR BOLTED JOINT
Program Capacity: 50 nodal points
Output Data:
(a) Input data
(b) Inverse of matrix
(c) Nodal temperature
(d) Given and calculated augmenting vector
and residual error
Input Data Sequence:
A. Case identification (12A4) followed by two blank cards
B. Card (Ii) with a i
C. Card (17) with dimension of matrix
D. Card (II) with a i
E. Cards (Ii, 3(213, E15.8)) with node indices started in the
first 13 field followed by conductance between these nodes.
Only input from lower node number to higher node number
required (since the conductance from node i to j equals
the conductance from j to i.) Each card has three groups
of z node numbers followed by a conductance value except
the last card. Last card could have i, 2 or 3 groups and
has a i in column i.
F. Cards (Ii, 3(16, E15.8) with number of node followed by
conductance from the node to ground node which is at
specified temperature. Each card has 3 groups of node
number followed by conductance. The Ii field is skipped
except for the last card for ground conductances which can
have i, 2 or 3 fields and the first column has a i. A
- 95 -
node can be connected to only one ground node.
G. Sameas F above, but code temperature specified for groundnode instead of the conductance value.
H. Sameas F above, but code internal power dissipation for theparticular node instead of the conductance value.
- 96 -
C) CJ C_ C: C) 0 0 C) C) _ _ "_ _ e--4 ,._ _ ,-.4 ,-4 ,,--i _ ¢,.j ¢Nl ¢'d ¢'4 r'q o,,t Od ¢xj t'N. ¢¢3 ¢¢', f¢3 ¢¢_ ¢x3 ¢_'.. r_,O O O C-. O O C_ r..D O C.:, O O O C_ O O O O O' C.P O O O O C.:. O C' O O CP O O C_ O C_ ',DC_ C, O G, O Cp O O C, C. c._ O O C, O C_ O O O G. O O C; O O O O O O O G_ O O O O O
IIIII TIIiIIIIIIIIIIIITIIIITIITIIIIII
4-I
co c,,J*-,1"
,-_C3wb-
onC3
04
t_ ,-4
_LU
K'U_
t'Mt.'h
- 97 -
A
I
Z
LL'_ +
II -") i_ II II t"d• L;J
_.-'_ _ _ _ ,0
c> 0 C'CNI c'.d
@_3
_9I
,,_ X
_4
- g
t2.)Z -"
CD _ r-4
,---t ,L u_
XL._X
,_ -Z. ,.-4
X C3 t.L
I Pd _T_.
F.- ,._ N-
CD _ C)
q3 L_'_Cq
L_
- 98 -
t.._ i,_ i,_ i.,.. i... i,..., i,... ,.jo oo oo, ec cc oo oc oo ec eo O. 0', o" _ _ _ _ _ _ _ 0 o c- o c o o 0 0
c.pO 0 C-j 0 0 (D C,. 0 0 G _ 0 0 0 0 0 0 0 0 (P C_ 0 0 Q-_ 0 r-3 <D' C" 0 0 C _ _ 0 0 0
i i i -r ]i i i i i I I I _ I I I I I I I I I I I I I i I _ I I I I I I _
÷
ut_ I
<[ Z "_ ,'-4 '_ '_" ",,_ "_
Z II II II !1 !1 !1 !1
_--_ .,_ .,-. I _ ""*
- 99 -
II _ I I I III IiIIII fill III • I I I _ III I I I I_I
I
._ _ I
ILl qD ""+ ,,-4 _ _ 0..
ti+,+ I +,_ _E ,,.4 _Z: _;- P-- t"-+I--- UJ UJ LId LU
I.-- I,.-- .¢'_ l'.,-," I,-,.,- I I
I ,-,4 _ :_
,-4 _,4 ,,I,. _ _-,,._ ;j..; _+_ _.. P,+++l_4 _'4 0 'P+4 ,..4 ,,..4 +.--4 _,..4
II II N II II II II II Z II II II II II II II 11 +! II
+'_ tl !1 _ +.u LU II LU _: II _ U+.I "_
,.u I +_. ,--+_ L _ Z !-,,0 ,-_ _ I Z ,.D IC_ ""_,_"_ ,,_- Z _'''_ L,e"_+-,,,.+_.'--' <a+- _3 Z "'_ _"_ _'_ 0 LP,-_
I_:. _'+'+ ,--, Z ,--_ _-"_ Z Z _ZO,- _Z ::_: ._Z: _Z _- +"_
Q.D
ILl
1.+..
0+,
LiJ
I----I
½+' L+UDZ_Z
Ai--i
I1 ++ !1I! uJ L:-I
:D ...2
_2:ZZ
Z_ZZ ZZZ_
- i00 -
TABLE i
Separation Radius Comparison - Single and Two Plate Models
(see Figs. 12 - 17)
A
B
.75
.5
B
A
3.1
2.2
1.6
1.3
3.1
2.2
1.6
1.3
3.1
2.2
1.6
1.3
Single
Plate
Model
4.2
3.3
2.7
2.4
4.5
3.6
3.0
2.7
5.1
4.2
3.6
3.3
R /Ao
Two
Plate
Model
3.7
2.7
2.1
1.7
3.8
2.8
2.2
2.0
4.1
3.2
2.8
2.5
Percent
Discrepancy
Between
Models
13.5
22.2
28.6
41.7
18.5
28.9
36.4
35.0
24.4
31.3
28.6
32.0
TABLE2
Test and Analytical Results for Radii of Separation of Bolted Plates (see Fig. 5)
Case
i
2
3
4
5.
6.
7.
D 2B
in. in.
.065 .422
.124 .422
.191 .422
.253 .422
Unmatch-
ed Pair
.124/ .422
.257
.124 1.0
.191 1.0
Separation Diameters, 2 R - in.o
"Rubbing Test"
Range
.42-.48
.50-.53
.58-.64
.70-.76
•54-•58
i. 06-1• i0
i. Ii-i. 17
Average
.45
.51
.62
.72
.56
1.09
1.16
Autoradiographic Test
Range
41-.46
.4 - .6
.76-. 81
.68-.73
Average
•44
.55
.78
.7
Computed
.488
.554
•620
.700
.588
i. 104
1.210
% Discrepancy Between
Computed Values and
Tested Values
Rub. Test
7.8
7.9
0
2.9
4.8
Autorad. Test
9.8
.7
25.8
0 I
o
l
Original x-ray film shows hole in plate and 0.6
area sensitized by the radioactive contamination•
test.
inch diameter zone more distinctly than remainder of
Loose radiographic contamination observed during
Assembled and disassembled radioactive and non-radioactive plates without rotating plates relative to
each other.
- 102 -
I I
I I
FIG. I. BOLTED JOINT
z
A
I
r
FIG. 2. ROETSCHER's RULE OF THUMB FOR PRESSURE
DISTRIBUTION IN A BOLTED JOINT
I
o
N
.Qv
eo
Ll-
m
\ \ \ \
\\\\
\\\\\\\\\\\\\ \\\
\\\\
\\ \ \
\\\\
\\\\
\\\\
\ \\\\\\\\ \ \ \
\\\\
\ \\\
\\\\
,@©©,\ \\\
\
\\\\\\\\
\i
\\\x_\\\x4
N
I I, ,_IZI
-
uv
I.,1_
- 107 -
I II J JJ
(a) Actual Plates
(b) Finite Element
Idealization II If:I
I! ____iXc
(c) Single Annular Ring
Element
-f."-"- _._ Nodal Circle-
Element
FIG. 4. FINITE ELEMENT IDEALIZATION OF TWO PLATES IN CONTACT
- 108 -
'IIITII
111111,V///,
TTTTII'
P
"///////,_ o"-- rr
(a) Plates of Equal Thickness Under Load
Iltiillrri ,,,,,,I IIIII III II I I
/1¢ /////////////
D
"-- r
(b) Finite Element Model for Plates of Equal Thickness
Z
I D1I
D2
y r
(c) Finite Element Model for Plates of Unequal Thickness
FIG. 5. FINITE ELEMENT MODELS
- 109 -
II
,_ z_2e _ Interference
(a) Plates Intersect, R too smallo
con.c,zone Tensile o z stress
-- Ro -.---.-_
(b) Contact Zone Sustains Tension, R too largeo
FIG. 6. EXAMPLES OF UNACCEPTABLE SOLUTIONS
-1 5-
FIG. 8(e). FOOTPRINTS ON MATED PAIR OF I/8 AND I/4 INCH PLATES.
FIG. 8. FOOTPRINTS ON THE MATING SURFACES OF 1/16 - 1/16,
I/8 - I/8, 3/16 - 3/16, I/4 - I/4, and I/8 - I/4
PAIRS. (A = .128, B = .21)
FIG. lO(c). 3/16 INCHPAlR
FIG. lO(d). I/4 INCHPAIR
FIG. 10. X-RAY PHOTOGRAPHSOF CONTAMINATION TRANSFERRED
FROM RADIOACTIVE PLATE TO MATED PLATE. 1/16, I/4,
3/16, I/4 INCH PAIRS. (A = .128 in., B = .21 in.)
- az/p
-I"1
Z
C,o rrl--..t
r-
C --I--4 rr_i--i
Z Z
I--
III
'-O
---t
0"1
Ii
II II |1 II
-'_ --' 0 0
_j0 _"
I
0
I
- 121 -
O_
N_D
1.2 ¸
1.0
.8 _=31
.6 _- = 2.2
L6
.4
.2
0 l _i
1 2 3 4_r/A
I
A =A/D =
C =
I
0.30.I in.0,751.54 in.
FIG. 13. SINGLE PLATE ANALYSIS-MIDPLANE oz STRESS
DISTRIBUTION (D = 0.133 in.)
I
¢g
I
r--.
II II II II
0
I--"
II /
II
cO
j
0
<_..S-
I--zH
0
LJA
0
0
c-
O'3
_ H
I,I
l.x.I
l.x.I
I--
_AL.L.
_+rr
,( /
- 125 -
O_
N
1.2
1.0
6
0 \\ \1 2 3 4
r/A
J=
A=AID :
C=
r0.30.I in.0.51.54 in.
FIG. 17. INTERFACE PRESSURE DISTRIBUTION IN A BOLTED JOINT
(D = 0.2 in,)
- 126 -
O-
N_D
1.2
I •
.6 _
.4
-TWO PLATE
_ S_INgLE PL_
ANALY SIS
.2
0 L I i
I. 1.5
I I
= O. 305D = .253 in.
A = .1285 in.B = .211 in.
ANALYSIS
2. 2.5 3. 3.5r/A
FIG. 18. FINITE ELEMENT ANALYSIS RESULTS FOR I/4 INCH PLATE PAIR.
- 127 -
ocDo
om--
N0
I
1.2
1.0
.8
I
TOTAL EXT- EQUIVALEN
UNIFORM P
.6
I I
ERNAL LOADT TO IO,OO0_SIRESSURE (87g LBS.)
I
D=A=B =
I0.305.253 in..1285 in..211 in.
.4
.2
-PEAK EXTERNAL PRESSURI AT A
01
_ EXTERiNAL PRESSUR E AT B
-I I I
2 3
r/A
FIG. 19. PRESSURE IN JOINT, TRIANGULAR LOADING
- 128 -
(a)
\
] Jl f I II I f I I I I I I I I II IIII] []lJ I I I J I
IIIII lllll I I I I I
ill fillllJlllIII IIIo( Qoo @ogooo oo ooo_e
@/, "I/////////////////
f_,
(b)
A¢
IZ
illlllIIIll I IllIIIII IIII I I I I
llllllllill] I I I 1ul De_oooo D_too
v
(c)
Z
I!IIIIIIIIIIII
_l/ I, "ll ill lll/ lllllll
"-- rY
FIG. 20. VARIATIONS OF LOADING AND BOUNDARY CONDITIONS.
- 129 -
ooo
o
N_D
l
1.2
1.0
.8
.6
.4
.2
I I
50 MICROIN
NUT BETWE
I=
D=A =B=
CH DISPLACE
EN RADII A
I I
2 r/A 3
I0.305.253 in..1285 in. -.211 in.
MENT UNDER
AND B
FIG. 21. PRESSURE IN JOINT, UNIFORM DISPLACEMENT UNDER NUT.
- 130 -
C)
X
h_-r"
Z
I
Z0
h_
b-.W
I00
v = 0.305- D = .253
A = .1285B = .211
in.
in.in.
TOTAL LOAD = 879#
I
DEFLECTIONUNDER NUT
DEFLECTIONSURFACE BEY
]F FREE]ND NUT,
---- PEAK EXTE _NAL PRESSUR8O
DIS FRIBUTION
-- UNIF ]RN PRESSURE
40 --/'PEAK EXTER AL PRESSL _E_' "_AT _
-- __ _,,
.0
2O
Z = m_
E AT A,
r>B.
TRIANGULAR
I
1.25 1.5 1.75r/A
i
2.0
FIG. 22. DEFLECTION OF PLATE UNDER NUT.
- az/p
I"1"1
3::"
"1'1I--.i
'-I-1
Z
'-4r-el
r-i-ii'-"m
.--I
i---<
1-17
c::r-,-t
-1-1o
..,_1
i-.-iZ
-m
0
II
_ZJ
II II II
r,o_o
(J1
/
II
°
t
b k_
I
I
I
I
• c-c- °r"-
or--
L_ "::_t" CO0 0.1 0_1
o/,-,
It
rc_
r7 U
XO
I..I.1 I,
or-,.
"7-
°o')
I I I I0 CO _ _ _ 0
L.I_II---
...J
-'r-
Z
CO
0
I---.--1
__1
Z
I---Z1.1.1
--..1
1--
Z
Ix.
,g
_A1.1_
- 133 -
t-°r--
coor=--
X
Z
0'-D
WF-"
..J
l.JZ0
LL
0
0
t_c_)
L_
200
180 -
160 -
140 -
120 -
I00 -
8O
60-
4O
20-
01
I I
= .305D = .124 in.A = .1285 in.
_--LAST TWO NODES FIi(ED AS SHOWN
IN FIG. 20(c)
2 3 4 5 6 7 8 9 I0 II
r/A
I
12
FIG. 25. GAP DEFORMATION FOR FREE AND FIXED EDGES -- FINITE
ELEMENT ANALYSIS, I/8 INCH PLATE PAIR.
I
I
@J
r,--
i
I
• °•,..-,- c.,,. _"-
• r"' 0r'-
0 C_I _.-.- _0
tl II il li
O
r""
iCO
J
d/z.0
I0
I-"'4
r,,.
L..,,..I
,,_1
.n-.-
,..0
0L.L_
1"-..,--I
COL,L.I
C_
>.,..,.--.I
I'--Z
L._,.-..t
I'--"
ZI-"4
L.z_
@.1
I,---i
L.I_.
- 135 -
r_
N_D
1.2I i I I
A=B =
1.0
.8
.4
.2
1
.._r. 124/.124 IN PAIR DA"FA FOR COMPA
I01.0 1.5 2.0 2.5 3.0
l
•305
.1285 in.
.211 in.
r/A
_ASI ON
FIG. 27. FINITE ELEMENT ANALYSIS RESULTS FOR 1/8 INCH PLATE
MATED WITH 1/4 INCH PLATE•
- 136 -
L_
U_
LLJ:E
6
5
I
R INDI
1
1 2
I I I
CATES AUTORA )IOGRAPHIC T-ST
I I I I
3 4 5
r/A COMPUTED
6
FIG. 28. COMPARISON BETWEEN TESTED AND MEASURED SEPARATION RADII.
.I-I
' ' T
[ I
• 7 I • I, I0, "15It I .....
I I
" 8 ' "III "16 1.5"I I
L I
, , .- "_L."--C ;- I
• 9 /.F/.. ;.:E"-,," '.
/ ,. ,.'_////11 \ "
12 13 43 44 14 19 46 4518 17
CI
I
22 23 3940 24 29 42 41 28 27
!
I"20°F
e
20°F
20°Fi,
iI
• I i "4It
I"2 I "5
I
.I
I
• 3 I ,6I
I
i
I
I
I--
I
I
I
I
"7"
11
21 1,26I
I
.... 1"
• 35
i' • 38II .5 '°
T
I 25
I 5 _II" "
I
• 31 [ • 34II
I
• 30 ' • 33u I
11
I
' " 37I 5"I ..... •I
I, • 36
#1 I #l ,,,
II 1 .5
2O
• 5"
• oOF
• oOF
• oOF
I
FIG. 29. LOCATION OF NODES -- STEADY STATE HEAT TRANSFER ANALYSIS