fracture mechanics analysis of an attachment lug
TRANSCRIPT
Fftac. OrpiCk PmU4 CaP)
AFFDL-TR-75-51ADR o z i m
FRACTURE MECHANICS ANALYSISOF AN ATTACHMENT LUG
AEROELASTIC AND STRUCTURES RESEARCH LABORATORYDEPARTMENT OF AERONAUTICS AND ASTRONAUTICSMASSACHUSETTS INSTITUTE OF TECHNOLOGYCAMBRIDGE, MASSACHUSETTS 02139
JANUARY 1976
TECHNICAL REPORT AFFDL-TR-75-51
Approved for public release; distribution unlimited
AIR FORCE FLIGHT DYNAMICS LABORATORYAir Force Systems CommandWright-Patterson Air Force Base, Ohio 45433
NOTICE
When Government drawings, specifications, or other data are
used for any purpose other than in connection with a definitiely
related Government procurement operation, the United States Govern-
ment thereby incurs no responsibility nor any obligation whatsoever;and the fact that the Government may have formulated, furnished, or
in any way supplied the said drawings, specifications, or other data,
is not to be regarded by implication or otherwise as in any manner
licensing the holder or any other person or corporation, or convey-
ing any rights or permission to manufacture, use, or sell any patented
invention that may in any way be related thereto.
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and is releasable to the National Technical Information Service
(NTIS). At NTIS, it will be available to the general public,
including foreign nations.
"This technical report has been reviewed and is approved forpublication."
Jý 4EL. RUDDP oject Engineer
FOR THE COMMANDER
ROBERT M. BADER, Chief Gerald G. Leigh, Lt Col, USAFStructural Integrity Branch Chief, Structures DivisionStructures Division
Copies of this report should not be returned unless return isrequired by security considerations, contractual obligations, or
notice on a specific document.AIR FORCE - 13-2-76 - 250
UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)
TDOCUMENTATION PAGE READ INSTRUCTIONSREPORT DBEFORE COMPLETING FORM
I. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER
AFFDL-TR-75-51 N N/A4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED
INTERIMFRACTURE MECHANICS ANALYSIS OF AN May 1974-October 1974ATTACHMENT LUG 6. PERFORMING ORG. REPORT NUMBER
ASRL TR 177-17. AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(s)
Oscar Orringer F33615-74-C-30639. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK
AREA & WORK UNIT NUMBERS
Aeroelastic and Structures Research Lab.
Department of Aeronautics & AstronauticsMIT, Cambridge, Massachusetts 02139 13670315
I I. CONTROLLING OFF'ICE NAME ARD ADDRESS 12. REPORT DATE
Air Force Flight Dynamics Laboratory January 1976Air Force Systems Command 13. NUMBER OF PAGES
Wright-Patterson AFB, Ohio 45433 8314. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15. SECURITY CLASS. (of this report)
Same Unclassified15a. DECLASSI FICATION/DOWNGRADING
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16. DISTRIBUTION STATEMENT (of this Report)
Approved for public release; distribution unlimited.
17. DISTRIBUTION STATEMENT (of the abstract entered In Block 20, if different from Report)
Same
18. SUPPLEMENTARY NOTES
None
19. KEY WORDS (Continue on reverse side if necessary and identify by block number)
Fracture Mechanics Hybrid ElementsStress Intensity Factors Aircraft Structural IntegrityFinite Element Analysis Stress Analysis
Computer Structural Analysis
20. ABSTRACT (Continue on reverse side if necessary and identify by block number)
This report documents a finite-element analysis procedure forcomputation of Mode I and Mode II stress intensity factors asso-ciated with a sharp crack in an attachment lug detail. The proce-dure is a complete FORTRAN-IV program which generates and para-metrically analyzes the lug, based on designer-oriented input data.The formulation of a special crack-containing element is reviewedand its performance is summarized. A detailed description of the
FORM
DD JAN 73 1473 EDITION OF 1 NOV 65 IS OBSOLETE UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)
UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE(When Data Entered)
lug analysis procedure covers the physical problem, modeling, pro-gram flow and options, input/output conventions, execution timesand limitations which must be observed. Results from exampleanalyses of some attachment lugs are presented.
UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE(When Data Entered)
FOREWORD
The developments documented in this report were carried out
at the Aeroelastic and Structures Research Laboratory (ASRL),
Department of Aeronautics and Astronautics, Massachusetts Institute
of Technology, Cambridge, Massachusetts 02139, under Contract No.
F33615-74-C-3063 (Project 1367, Task 136703) from the U.S. Air
Force Flight Dynamics Laboratory. Mr. James L. Rudd (AFFDL/FBE)
served as technical monitor. The contractor's report number is
ASRL TR 177-1.
iii
TABLE OF CONTENTS
Section Page
1. INTRODUCTION .................. ................... 1
2. BASIC ELEMENTS AND METHODS .......... ..... .... 3
2.1 Element QUAD4 ................ ................ 3
2.2 Element PCRK59 ........... ... ............... 5
3. ATTACHMENT LUG PROGRAM ........ .............. 13
3.1 Lug Structure Model ...... ............. 13
3.2 Input Conventions ........ .............. 14
3.3 Required Subprograms and Other Features . . 17
3.4 Model Generation and Program Flow ...... .. 18
3.5 Output Conventions and Error Messages . . .. 20
3.6 Visual Interpretation of Output ......... .. 21
3.7 Program Status ....... ................. . .. 22
4. RESULTS OF EXAMPLE ANALYSES ..................... 23
4.1 Analysis of Wing Root Attachment Lug . . .. 23
4.2 Analysis of Engine Pylon Truss Lug ..... 24
4.3 Example Application ...... ............. 28
5. CONCLUSIONS ... ...................... .... 30
REFERENCES .............. .................... 32
FIGURES ............. ............... .... 34
APPENDIX A . ............... ................... 67
iv
LIST OF FIGURES
Figure Page
1 Combination of Assumed-Displacement Elementswith Hybrid Crack-Containing Element .... ...... . .. 34
2 Conventions for ASRL QUAD4 Assumed-DisplacementElement ................... ....................... 34
3 ASRL PCRK59 Hybrid Crack-Containing Element . .... 35
4 Application of PCRK59 Element to SymmetricAnalyses .................. ....................... 36
5 Attachment Lug Detail ........... ................ 37
6 Crack Parameters ... ................ ......... 37
7 Program LUG Input Conventions... ... . . . ......... 38
8 Finite Element Mesh for NT=3 ........ ............. 39
9 Hypothetical Wing Root Attachment Lug ..... ........ 40
10 Input Data for Analysis of Lug shown in Figure 9 . . . 41
11 Program LUG Flow Chart ... .............. ...... 42
12 Printout of Input Data .......... ............... 45
13 Numbering Conventions Illustrated for SampleMesh shown in Figure 8 .. ................ 46
14 Overlay for Stress Intensity Analysis ... ........ 48
15 Force and Displacement Tables from Stress Analysis . . 49
16 Part of Stress Table .......... ................. 50
17 Stress Intensity Factor Table from Analysis withNT=4 .................... ......................... 51
18 Stress Contours for Wing Root Lug ..... .......... 52
19 Butterfly Plot for Wing Root Lug Stress Intensities . 54
20 Interpretation of Butterfly Plots ..... .......... 55
21 Scale Mesh Plan for Engine Pylon Truss Lug ........ .. 56
v
LIST OF FIGURES (continued)
Figure Page
22 Stress Contours for Engine Pylon Truss Lug(Tension Bearing, Cosine Pressure) ...... ........ 57
23 Butterfly Plot for Engine Pylon Truss Lug(Tension Bearing, Cosine Pressure) ...... ...... .... 58
24 Scale Mesh Plan for 7-Inch Engine Pylon Truss Lug 59
25 Effect of Shank Length on Stress Intensity(Tension Bearing, Cosine Pressure) ........ .......... 60
26 Comparison of Cosine and Uniform PressureDistributions (Tension Bearing) ..... ........... 61
27 Stress Contours for 7-Inch Engine Pylon Truss Lug(Tension Bearing, Uniform Pressure) ..... ......... 62
28 Stress Contours for 7-Inch Engine Pylon Truss Lug(Positive Shear Bearing, Cosine Pressure) ... ...... 63
29 Comparison of Finite Element Results with EngineeringBeam Theory (Positive Shear Bearing, Cosine Pressure). 64
30 Butterfly Plots for 7-Inch Engine Pylon Truss Lug(Positive Shear Bearing, Cosine Pressure) .. ...... 65
31 Butterfly Plots for 7-Inch Engine Pylon Truss Lug(Positive Shear Bearing, Uniform Pressure) ........ .. 65
32 Butterfly Plots for 7-Inch Engine Pylon Truss Lug(Compression Bearing, Cosine Pressure) ... ........ 66
33 Butterfly Plots for 7-Inch Engine Pylon Truss Lug(Compression Bearing, Uniform Pressure) .. ....... 66
vi
Section 1
INTRODUCTION
The application of linear elastic fracture mechanics analysis
to structural details in new aircraft designs has received growing
emphasis from the Air Force and from aircraft manufacturers during
the past few years. Not only have fracture mechanics data become
more readily available in recent years [1,2], but also, there has
been a trend toward treatment of the problem of fracture in its
own right, distinct from fatigue. Within the past year, this
trend has culminated in the establishment of structural design
criteria by the Air Force which require an aircraft designer to
take specific design-analysis actions to protect structures against
fracture [3]. Required design calculations now include, generally,
comparison of load-induced stress intensity factors to material
fracture toughness and assessment of crack growth rates, based on
assumptions concerning the size and location of possible cracks in
the structure. Both types of calculation require prior estimation
of the load-induced stress intensity factor. Hence, there has also
been considerable emphasis on adding to the body of available fracture
mechanics solutions.
Because so few geometrical configurations are amenable to a
purely analytical solution of the equations of elastic fracture
mechanics, the finding of new solutions depends upon development
of numerical analysis techniques. Extensive contributions have
been made by Bowie and his colleagues [4,5,6] using the complex
variable formulation of elasticity in combination with conformal
mapping, analytic continuation and boundary collocation methods.
Tada, et al. [71 have recently collected and classified a compre-
hensive body of solutions based on the-semi-analytical methods
(complex variables, boundary collocation, Fourier transforms, etc.)
among which appear many new solutions by Tada. However, the semi-
analytical methods have not as yet proved capable of application
to the irregular geometrical configurations which are found so often
--1--
in real airframe structural details. Stress analysis of irregular
structure has been the province of the finite-element methods for
the past decade. Within the last six years, numerous contributors
have extended the finite-element technique to fracture mechanics
analysis.
Work on finite-element fracture mechanics analysis at MIT has
followed the path of assumed-stress hybrid elements, first proposed
by Pian [8] for ordinary continuum elements. The hybrid method was
subsequently extended by Tong, Pian, Luk, and Lasry [9,10,11,12] to
formulation of rectangular elements which incorporate an elastic
crack-tip singularity, but which also have an assumed linear or
quadratic variation of displacements along the element edges and
are thus compatible with ordinary elements (Figure 1). Numerical
experiments have demonstrated that the hybrid crack-containing ele-
ments are capable of producing estimates of KI and KII with less
than 1 percent error, using 20 to 50 total degrees of freedom in
the analysis, for simple geometrical configurations. Hence, it
becomes possible to create economically practical analysis procedures
for structural details by refining the mesh or ordinary elements to
pick up the stress gradients caused by nonuniform loading and com-
plicated boundary geometry, leaving to the special hybrid element
the task of picking up the local gradients caused by the crack-tip
singularity.
This report summarizes recent developments at the MIT Aero-
elastic and Structures Research Laboratory (ASRL) in which the crack-
containing hybrid element has been applied for the first time to some
typical structural details, found in current production aircraft,
with geometries too complicated for economical solution by other
techniques. The "PCRK59" crack element used in these analyses is
a generalized version of the original Lasry element [12] which was
formulated and programmed by Tong and subsequently modified for
greater utility by the ASRL computing staff.
-2-
Section 2
BASIC ELEMENTS AND METHODS
2.1 Element QUAD4
The ASRL QUAD4 four-node quadrilateral element (Figure 2) is
used as the basic building block in the analysis procedure. QUAD4
is the well-known bilinear isoparametric assumed-displacement ele-
ment which has been used for continuum stress analysis for many
years [13]. The ASRL version has been programmed as an independent
subroutine which includes the options of individual rotation trans-
formations at each node and calculation of a "B" matrix for stress
analysis.
The nodal coordinates XI, YI' X2 '' ''Y4' element thickness T
and the elastic constants matrix:
rCU C12i C13 ) [ 1CI IEy
. C12, Cz2 C-23 6, Yy - I
c C23 c35j (r Xy , (1)
comprise the required basic input information. For isotropic
materials
L -o Vr (1+ )(_IV) 0-- p (2)0 0 2 ;___L
(plane (planeS+Vess)
srIn
Subroutine QUAD4 allows for general plane orthotropic behavior, a
capability which is included in the attachment lug program. Plane
stress is assumed in the analysis. The lug program does not use
the rotation transformation option.
The element stiffness matrix k is calculated by numerical
area integration, using 3x3-point Gaussian quadrature [14] for
-3-
E LEM ENT
k ff DTT -D (XJy (3)
AREA
where D contains the interior strain-nodal displacement relations:
f•, E yX EYYýE]= D (X) Y) i$ 'ý "'---; " (4)
The stiffnesses are returned in Lower Triangle Vector (LTV) form:
k = L [k kl k2 k31 k3z *** e] (5)
For the purpose of stress analysis, Eqs. 1 and 4 may be combined
to give:
QUAD4 also returns the matrix B(Xc ,Yc), formed at the fifth
Gaussian station, for later calculation of stresses at the element"centroid", defined in terms of the nodal coordinates:
4+ S•= Yz (7)
The behavior of QUAD4 has been studied extensively on other
projects and is well understood. Uniform or nearly uniform stress
fields can be picked up to within the roundoff accuracy of the digi-
tal computer being used for the analysis. The inability of the
bilinear assumed displacement fields to follow the quadratic deflec-
tion of the neutral axis of a cantilever beam loaded by an end moment
has been well documented elsewhere [15] and constitutes a limitation
on the QUAD4. In practical terms, this requires that the element
aspect ratio (Figure 2) be held close to unity for models of struc-
tures which are expected to have quadratic or higher-order displace-
ment behavior. In some cases, even an aspect ratio of unity is not
sufficient to insure convergence of the solution. For example, the
-4-
QUAD4 element was used recently to model a thick-walled cylinder
subjected to centrifugal loading from its own mass, due to rotation
[16]. The analytical solution of this axisymmetric problem includes3an r term in the radial displacement field which the bilinear ele-
ment is unable to pick up; errors of 25% were found for a cylinder
with a 2:1 ratio of outside-to-inside radius, using four unit-aspect-
ratio QUAD4 elements through the wall thickness.
The misbehavior of the bilinear element in the presence of
higher-order gradients requires the use of many elements to model
complicated geometries. Also, "calibration" of the finite-element
model is a good idea, where possible, by comparing the numerical
results with independent solutions. Calibrations for this project
have included comparisons with the classical elasticity solution
for stresses and displacements near a circular hole in a semi-
infinite strip under tension [17] and with finite-element analyses
using higher-order assumed-displacement elements.
2.2 Element PCRK59
Formulation of the assumed-stress hybrid finite-element method
begins with the Principle of Minimum Complementary Energy:
7T f=- __c J (8)
whereE indicates summation over the element set.
n
S u = part of the element boundary over which displace-ments are prescribed.
V = element volume.
u = vector of prescribed displacements on Su
a = stress vector.
S = compliance constants = C_1 (C = Sa).
T = vector of surface tractions = Na, where N is amatrix of surface normal directlon cosines.
-5-
If R is used directly, only the stress field is assumed, subjectcto admissibility criteria requiring that the assumed stresses
satisfy:
(i) Interior equilibrium ýa i/ýx. + F. = 0, in V,1J J 1
where F. are prescribed body forces.1
(ii) Mechanical boundary conditions Na = T on
that part of the element boundary over which
the surface tractions T are prescribed.
(iii) Equilibrium of surface tractions Na across the
interelement boundaries S, which are distinct
from S and Su a
Formal application of the variational calculus to Eq. 8 leads to
two sets of Euler equations:
(iv) Interior compatibility, So = c in V, where
6ij = 1/2 (Dui/axj + DUj/Dxi).
(v) Displacement boundary conditions, u = u on Su
If assumed stress functions are substituted and Eq. 8 is integrated
before H is varied, there results a linear equation system in whichc
the generalized coordinates to be solved for are forces. This
approach leads to a Matrix Force Method analysis which brings with
it the programming problem of systematic identification and elimina-
tion of redundant quantities.
The assumed-stress hybrid approach avoids the complications of
force redundancy by modifying H so that the primary unknowns in ac
finite-element application become displacements once again. The
Principle of Minimum Complementary Energy is modified by addition
of Lagrange multiplier terms [8,181 which change admissibility
criteria. Specifically, conditions (ii) and (iii) above are relaxed
and confition (v) is enforced. Under the new principle, stress func-
tions satisfying only the interior equilibrium conditions (i) may be
assumed, and displacement functions which satisfy interelement com-
patibility and conditions (v) must now be assumed as well. The
modified energy principle which replaces Eq. 8 is:
-6-
7T• i •j f .1 - T (9)
where "@V" represents the entire element boundary, S + Su + S "
To convert H1 into a finite-element formulation, the stress vector
a is assumed within each element and the displacement vector u is
assumed on the boundary, @V, of each element:
a = P(xyz) u = L(x,y,z)q (10)
where P and L are matrices of interpolation functions. Vector 8
contains generalized stress coordinates, while q is a vector of
nodal displacements. Matrices P and L are assumed independently,
with L defined only along the element boundary 3V. Substitution
of Eq. 10 into Eq. 9 then leads to:
_•T TH A
where r= (N (P•rL ofS H= f pTSp'jrVVV
f TConsisten'd nodal -Force vec~or (2
and where Na = NPý has been substituted for the surface traction
vector T in the expression for G.
Direct assembly and solution of the equation system represented
by 111 is possible, but results in a mixed matrix method, with both
force and displacement unknowns. A more versatile formulation is
obtained by recognizing that, since the stresses are assumed inde-
pendently within each element, Pf for one element is not coupled
with any other elements. Therefore, the unknowns f may be formally
eliminated by applying the variational calculus to Il:
61(ý for only one element varied) - 6 T{ fli/ l } =
6j (Gq - Hý) = 0
-7-
which leads to-1.
H Cr (13)
Substitution of Eq. 13 back into Eq. 11 then yields*:
V Z T(14)
The alternate expression for H1 given by Eq. 14 represents a
pure Matrix Displacement Method. The quantity GT H G can be recog-
nized as an element stiffness matrix. The Z notation may now ben
identified as a conventional Matrix Displacement Method structure
model assembly procedure. Hence, structure models can be created
by assembling conventional and hybrid elements, provided only that
compatibility across the interelement boundaries is maintained by
proper choices for the assemed displacement fields. The versatility
of the hybrid method lies in its ability to provide special-purpose
elements, for restricted regions, which may be coupled into a model
containing conventional elements in the remaining regions which are
free of singularities or other unusual behavior.
The original hybrid crack elements [9,10] were derived from
Hi by assuming a stress field containing r-I/2 terms, where r
measures radial distance from the crack tip and, by assuming dis-
placement fields which vary linearly from one node to the next,
along the element boundary. However, subsequent analysis of error
sources [19] has indicated that the area integration required for
computation of H (see Eqs. 12) gives poor results for the r
terms since some of the Gaussian stations are close to the crack
tip. This situation may be remedied by increasing the number of
Gaussian stations, but the computation of k then becomes too costly.
A better approach, used for the second-generation crack elements
[11,12] has been followed in the present work. The energy principle
F11 may be further modified by introducing two displacement fields:
u assumed on WV and u assumed in V. There now arises another com-
patibility condition, u = u on WV, which is relaxed by the Lagrange
Note that H and H- are symmetric matrices.
-8--
multiplier method. At the same time, the condition that u must
satisfy the interior equilibrium equations, as well as the strain-
displacement relations, is enforced. As a result, the area inte-
gration is converted to a boundary integral and H1 is modified to
the form:
7/= f T77i S -. 5 (TTu, TU) J 5 - f s (15)
The same boundary displacement field u can be used for both H1 and
Hi However, the interior assumed fields in H2 must be a completeelasticity solution: stresses a and displacements u which satisfy
all of the equations of elasticity, with T = Na a derived quantity.
The distributions for a and u are obtained from a complex variable
solution of the equations of elasticity near a crack tip or equiva-
lently, by solving the biharmonic equation for an Airy stress func-
tion. Computation of the element stiffness matrix is the same as
for HI1 except that H is now computed by a boundary integral:
H Y [Nj P)1A + ANP~d (16)
where A is a matrix of shape functions corresponding to the interior
assumed displacement field, u = Aq.
The principle H 2 also possesses the advantage of convenience
for treatment of arbitrary shapes, since only boundary integration
is required. Figure 3 illustrates the PCRK59 element based on H2.
Input information required by this element is similar to the infor-
mation required by QUAD4:
(i) Geometry: global coordinates of the crack tip
Xt' Yt; global coordinates of each node XI,YI,X 2,... Y9 .
(ii) Material properties: the shear modulus G = E/2(I+v)
and a second constant n = (3-v)/(l+v) for plane stress
(3-4v for plane strain)
PCRK59 is programmed only for isotropic material and does not
incorporate the rotation transformations available in QUAD4. In
fact, rotation transformations cannot be applied Once the PCRK59
-9-
stiffness matrix has been formed. This limitation is caused by
the appearance of the crack tip coordinates in the numerical inte-
gration scheme, but the restriction does not affect many practical
fracture mechanics problems. The numerical integration is by
five-point Gaussian quadrature [14] between each pair of nodes,
except that the crack surfaces are skipped. Omission of the crack
surfaces is justified because they constitute SG, over which T = 0,
and because the derived tractions T satisfy this stress-free con-
dition at least in an average sense.
The PCRK59 element has two other important features. First,
unit thickness is assumed. Second, a symmetric "half-element"
option is available, under which nodes 1,5 and the crack tip are
assumed to lie on a line parallel to the global X-axis, while the
element and applied loading are assumed to be symmetric about this
line. Under these conditions, a half-model of a structure may be
analyzed to obtain Mode I stress intensity solutions only; e.g.,
for the coupon in uniform tension with edge cracks, shown in Figure
4. The "half-element" consists of nodes 1,2,.. .5, only, with node
1 requiring a roller restraint to maintain the assumed symmetry.
Another input parameter determines which option is executed:
KEY = 1 for "half-element"
2 for full element
The "half-element" option is used mainly for illustrative examples
and performance testing. The full element option has been used
exclusively in the present work.
Element PCRK59 computes and returns a stiffness matrix in LTV
form (see Eq. 5) for either the 10 degree-of-freedom "half-element"
or the 18 degree-of-freedom full element. In addition, a special
B matrix for calculation of stress intensities is returned. Eq. 13,
used in the derivation of the stiffness matrix, can also be used to
compute the generalized stress coordinates ý after the element nodal
displacements q have been obtained. For the PCRK59,
-(17)
-10-
where kl,k 2 , are the Mode I, Mode II stress intensity factors,
defined by:
t&) + (fefl'Qs inX- 4 (e) + --f )(12r (18)9-y= etc.
The functions fl,f 2 , are from the classical crack tip solution,
and the other generalized coordinates 82,83, .... 18 represent
far-field behavior. Thus, B is formed by extracting the first and
tenth rows of H- G, so that the stress intensities may be calculated
from:
(ais &1 3 ~r (19)
for the full element. Only the first row of H G is extracted if
the "half-element" option is in effect:
tkj X 10) tZ (20)
NASA/ASTM standard stress intensity factors may be computed after
Eq. 19 or Eq. 20 by:
KI 1(3r -• • (21)
If a structure model with thickness T # 1 is to be analyzed, this
may be done simply by scaling the PCRK59 stiffness to:
k' = Tk (22)
Performance of the PCRK59 element has been tested extensively
by comparison with classical and boundary collocation solutions
[19]. Solutions for KI accurate to better than 1 percent have been
obtained with a rectangular crack element' surrounded by only a few
QUAD4 elements. Other tests have shown that solution accuracy
within 3 percent is maintained when the crack element shape is dis-
torted by relocating some nodes as much as 0.3 x (length of crack
within element) away from the positions they occupy for a rectangle.
Also, the 3 percent accuracy limit can be maintained with the crack
tip located anywhere from 20 to 70 percent across a line between
-11-
nodes 5 and 1, with the element shape kept rectangular. The dis-
tortions of element shape and crack tip location required for the
structure models analyzed in the present work are well within these
limits.
The PCRK59 element possesses one unavoidable quirk which arises
from its linearity. If the element is placed in a region with com-
pressive stress normal to the crack, a negative value of KI is
obtained. In a real structure, the crack would close and cease to
be a problem in this situation. Therefore, negative KI values
should be interpreted as signaling the absence of Mode I stress
intensity. On the other hand, the solution for KII will be posi-
tive (negative) according to whether the crack is being subjected
to positive (negative) shear stress, as defined by the standardconventions of elasticity. In this case, the correct interpreta-
tion is to take the absolute value of KII.
In summary, the PCRK59 element permits efficient computation
of stress intensity factors by well established procedures of the
Matrix Displacement Method. The unusual features of the element
are internal to its subroutine. The element subroutine requires
familiar input information and returns k and B matrices like a con-
ventional element. The structure model is assembled and a global
displacement solution is computed by standard techniques. Computa-
tion of either the centroid stresses in the conventional elements
or the stress intensity factors in the crack element is then merely
a matter of extracting the element displacements q from the global
solution and performing a straightforward matrix multiplication.
-12-
Section 3
ATTACHMENT LUG PROGRAM
Program LUG has been developed for analysis of stresses or
stress intensity factors in an attachment lug typical of many
structural details found in current aircraft. This section describes
the lug structure model and explains how the program is used. Results
obtained from some example analyses are presented in Section 4.
3.1 Lug Structure Model
Figure 5 illustrates the structure which Program LUG models.
The detail consists of a straight shank, built in at the foot and
a rounded ear whose outer edge is concentric with a bearing pinhole.
Provision is made to treat the lug as a two-material system composed
of an isotropic bushing ring surrounding the bearing pinhole, and
the lug proper, which may be treated as either isotropic or plane
orthotropic. A perfect mechanical bond between the bushing and
lug is assumed. A monolithic single-material lug is obtained if
identical isotropic material properties are specified for the
bushing and the lug proper.
Bearing loads are assumed to be applied to the structure at
the bearing pinhole surface. Tension, compression, positive shear
or negative shear may be applied. These loads are defined in
Figure 5. Each load component is represented as a radial bearing
pressure over one-half the circumference of the bearing pinhole,
with the pressure distribution centered on and symmetric about the
line of action of the load. Options for a cosine pressure distri-
bution or a uniform pressure distribution are available.
The attachment lug is assumed to be under plane stress, with
two analysis options allowed. Under option 1, a model of an
uncracked lug is assembled, using only QUAD4 elements, and a con-
ventional stress analysis is executed. Under option 2, a small
radial crack is assumed to emanate from the bearing pinhole surface,
-13-
with the crack tip located in the bushing. The length of the crack
is specified by the program user (Figure 6). Program LUG auto-
matically executes a sequence of solutions in which the crack'loca-
tion is varied step-wise around the entire bearing hole circumference.
3.2 Input Conventions
The input data conventions for Program LUG are summarized in
Figure 7. Formats for all numerical data have been standardized
to 15 fields for integers and El0.0 fields for floating point num-
bers. Integer data and floating point data supplied in E format
should be right-justified in the field. However, floating point
data may also be given in F format, if desired, without changing
the program code. F format data need not be right-justified. Also,
the implied decimal point location for floating point data may beoverridden. A maximum of 3 decimal figures may be input under E
format and up to 7 decimal figures may be input under F format.
A series of independent cases may be analyzed in one run.The first input data card specifies the total number of cases which
follow. The remainder of the input deck consists of six cards per
case which give the program a complete description of the case.
The conventions for these cards are as follows:
Card 2 - may contain any alphanumeric information whichidentifies the case. This information is printedas a heading title.
Card 3 - specifies the options selected by the user forfour control parameters:
IANL = 1 (Conventional stress analysis with-out crack).
2 (Stress intensity analysis).
LOAD = 1 (Cosine pressure distribution).
2 (Uniform pressure distribution).
MODE = 1 (Lug treated as isotropic).
2 (Lug treated as orthotropic).
NT = Total number of QUAD4 elements wantedper 450 arc around the bearing pinhole.A minimum value of 3 is recommended.
-14-
Card 4 - specifies the lug dimensions and crack size.
DI = Inside diameter of bearing pinhole.
DB = Outside diameter of bushing.
W = Lug width.
H = Total (root to tip) length of lug.
T = Lug thickness (lug and bushing assumedto have equal thickness).
CSIZE = Length of crack.
Card 5 - specifies the material properties of the bushing,which is always assumed to be isotropic:
E = Young's modulus.
V = Poisson's ratio.
Card 6 - specifies the lug material properties. IfMODE = 1 on card 3, the convention is:
E = Young's modulus.
V = Poisson's ratio.
If MODE = 2 on card 3, the convention is:
EL = Longitudinal modulus.
ELT = Cross-coupling modulus.
ET = Transverse modulus.GLT = Shear modulus.
0 = Angle between lug XY axes and materialLT axes (degree measure, positive CCWfrom X to L).
Card 7 - specifies the bearing force value:
TENSN = Tension or compression bearing force.
SHEAR = Positive or negative shear bearing force.
The lug dimensions and crack size were defined in Figures 5
and 6. Any value of thickness may be specified. Program LUG
rescales the model internally to unit thickness. Figure 8 illus-
trates a finite element mesh which might result when NT = 3 ele-
ments per 450 arc is specified on card 3. The positive convention
-15-
for the relationship between the lug XY axes and material LT axes
is also shown. The quantities EL, ELT' ET, GLT are the conventional
plane-orthotropic moduli for; e.g., a fiber composite laminate. The
stress-strain relations take the form:
GL " E L E LT 0 C L
T j= ELT ET 0 G T (23)
O 0 0 GL CL
in the LT axis system. For 0 3 00 the stress-strain relations in
the XY axis system take a more complicated form:
aX Cll C12 C13 CXX
GM=y• C12 C22 C2 3 Cyy = Cs (24)
CC £
!XY) jC13 C23 C3 3J XY
where, in general, C1 3 , C2 3 3 0. The matrix C in Eq. 24 is com-
puted from EL, ELT,...,O by ASRL subroutine CTFORM.
The bearing load conventions were indicated in Figure 5. The
value of TENSN or SHEAR supplied on card 7 refers to total bearing
force; the corresponding pressure distributions are computed inter-
nally. A positive (negative) value TENSN has the effect of apply-
ing a tension (compression) bearing load to the structure. A posi-
tive (negative) value for SHEAR similarly applies a positive (nega-
tive) shear bearing load.
Figure 9 illustrates a portion of the actual finite element
mesh generated for a hypothetical large all-aluminum wing root
attachment lug. Since the "bushing" diameter does not have any
physical significance in this single-material case, it is used to
control the mesh so that the tip of a 0.5-inch long crack lies at
the middle of the PCRK59 element. The crack is shown with a finite
opening for clarity. However, nodes 5 and 6 of the crack element
(Figure 3) actually overlap to provide the correct model of a sharp
crack. The PCRK59 element has replaced a group of four adjacent
-16-
QUAD4 elements in the mesh. When analysis option 2 is in effect,
a series of structure models are generated and analyzed one after
the other, with the PCRK59 shifted circumferentially by one pair
of QUAD4's after each analysis. Thus, for the case shown in Figure
9 (NT = 3), 24 stress intensity solutions are obtained with the
crack located successively at e = 00, 15', 30o,...,3450. Figure 10
summarizes the input data deck required to run a stress analysis
(case 1) and a stress intensity analysis (case 2) for the hypothetical
lug detail.
3.3 Required Subprograms and Other Features
Program LUG requires the following FORTRAN-IV subroutines to
form an executable load module:
(i) ASRL FEABL-2 subroutines ASMLTV, BCON, FACT, ORK,SETUP, SIMULQ, and XTRACT [20,21].
(ii) ASRL element and utility library subroutinesQUAD4, PCRK59, and CTFORM.
(iii) IBM Scientific Subroutine Package routines MFSDand SINV which are required by the PCRK59 elementsubroutine.
The entire source deck is supplied in IBM 029-punch format.
The following features of Program LUG may cause machine-
dependence problems on non-IBM hardware:
(i) The 20A4 format for input of case title informa-tion may be incompatible with some systems. Thismay be remedied by changing FORMAT statement 502to 80Al and redimensioning vector TITLE to 80.
(ii) FORTRAN unit numbers 5 and 6 are assumed for thecard reader and line printer respectively. Pro-gram LUG may be converted to other hardware stan-dards simply by reprogramming the two lines ofcode:
KR = 5
KW = 6
which appear shortly after the FORMAT statementsnear the beginning of the program.
-17-
(iii) Program LUG requires a sequential-access scratchdataset, designated as FORTRAN file 20, when anal-ysis option 1 (uncracked structure stresses) isin effect. The file must consist of (30 single-precision words per record) x (records = maximumnumber of QUAD4 elements expected). A total of600 records should be adequate for most analyses.A job control instruction, specific to the instal-lation where the program is being executed, isrequired to create this file on a system disk.However, Program LUG may be executed withoutcreating this file if only stress intensity solu-tions are sought.
(iv) IBM/SSP subroutines MFSD and SINV m'ay not be com-patible with other systems. If this problem arises,reprogramming or substitution will be required.
3.4 Model Generation and Program Flow
Program LUG automatically generates the geometrical information,
element interconnections, etc., which are required to compute and
assemble the element stiffnesses, restrain the structure properly,
apply the bearing load and execute a stress or stress intensity
analysis. The program flow is summarized in Figure 11. Parenthe-
sized numbers in the figure refer to FORTRAN statement numbers in
the program listing (Appendix A).
After the input data has been read for a case and some auxiliary
values have been calculated, the case title and input data are printed
for checking. A sample output from this section of the program is
shown in Figure 12. The number of QUAD4 elements required radially
in the bushing and lug and the number required axially in the lug
shank are then computed by rounding off to the nearest whole number
which gives an average element aspect ratio closest to unity for
each region. The total number of elements, total degrees of freedom
and some additional parameters are then calculated, and the vectors
which will contain the K-solutions are erased.
The major section of the code, a loop over the crack locations,
then follows. The location loop is executed 8*NT times for a stress
intensity analysis, but only once for a conventional uncracked struc-
ture stress analysis. Previous results are erased and the inter-
connections for an uncracked structure are generated. Figure 13
-18-
illustrates the node and element numbering conventions, using the
example mesh from Figure 8. The numbering patterns are as follows:
(i) Nodes are numbered, globally, radially outwardfrom the bearing pinhole on each ray. The raysare taken in counterclockwise order, beginningat e = 0'. Vertical lines of nodes in the lugshank are numbered afterward, from the top downand from right to left. The last line of nodesis restrained.
(ii) Degrees of freedom are numbered 2n-1 (displace-ment parallel to X) and 2n (displacement parallelto Y) at each node n.
(iii) Elements in the bushing are numbered radially out-ward and counterclockwise, partially following thenode numbering pattern.
(iv) Elements in the lug ear are numbered radially out-ward and counterclockwise after the bushing elements.
(v) Elements in the lug shank are numbered last, fromthe top down and from right to left.
If a stress intensity analysis is being executed, the location
of the PCRK59 element is now computed from the crack location loop
index and connections for this element are generated. As shown in
Figure 14, the PCRK59 element overlays four QUAD4 elements. The
central node of this group of elements is transferred to the bear-
ing pinhole to accommodate the PCRK59. The element numbers of the
four overlaid QUAD4 elements are also flagged.
The global XY coordinates for each node in the model are now
computed, assuming an uncracked structure. If a stress intensity
analysis is being executed, the transferred node coordinates are
adjusted and global coordinates are computed for the crack tip.A
The area corresponding to the global force vector QG in the FEABL-2
storage system is used as temporary storage for the-node coordinate
data.
After auxiliary storage for element-level data has been pre-
pared, a loop over all QUAD4 elements is executed. The node coor-
dinates for each element are extracted from the global data, other
required input is provided from auxiliary storage and k and B are
-19-
computed for the element. Also, centroid coordinates Xc, Yc are
computed for the element; B, Xc, Yc are stored in FORTRAN file 20
(stress analysis option only) and k is assembled into the global
stiffness matrix. If a stress intensity analysis is being executed,
these procedures are skipped for the four flagged elements, while
k and B are computed and k is assembled for the PCRK59.
After assembly, Q G is erased and replaced by prescribed nodal
forces which are statically equivalent to the specified bearing
load and the assumed (cosine or uniform) pressure distribution.
For stress intensity analysis, the two nodes at the crack opening
each receive one-half the nodal force which would have been applied
to a single node at that location in an uncracked structure.
The final section of the code executes a solution of the global
equation system and either a stress or a stress intensity analysis.
In the latter case, the stress intensity factors are saved and a
complete table is printed after the crack location loop has been
completed.
3.5 Output Conventions and Error Messages
If a stress analysis has been executed, nodal forces, nodal
displacements and element stresses are printed. The table of
forces and displacements appears immediately below the problem
input data and merely lists the force or displacement value for
each degree of freedom ("ROW" in the table heading). The stress
table contains one line of information for each element:
Element No., Xc, Yc' axx' ayy, axy, arr, a0 0 , arO
The stress values are computed for the element's centroid location
Xc? Yc" Figures 15 and 16 present samples of these output tables.
If a stress intensity analysis has been executed, only a table
of K-solutions is printed. Each line of the table corresponds to
one crack location, containing:
Angle to crack opening, Ki, KII
A sample is shown in Figure 17.
-20-
If abnormal conditions occur during execution, certain error
messages may be printed by Program LUG. The messages and actions
required are as follows:
(i) Insufficient core memory available for storageof the problem data causes the message:
THE LENGTH OF THE "DATA" VECTOR FOR THIS CASEIS xxxxx WHICH EXCEEDS yyyyy = THE MAXIMUMALLOWED IN THE DIMENSION STATEMENT.
The entire run will be terminated if this con-dition occurs. The dimensions of vectors REand IN (line 2 of the program code) are yyyyy.Redimension these vectors to 1.15 (xxxxx).
(ii) Ill-conditioning of the structure model causes
the message:
INDEFINITE MATRIX; THIS CASE CANCELLED.
Execution continues with the next case. Themost likely cause is misplacement of the cracktip, relative to the bushing O.D. Recheck theinput data to make sure that the crack tip doesnot extend beyond the bushing, even if a single-material lug is being analyzed. Material pro-perty errors are another probable source. Ill-conditioning may result if the bushing is toostiff, compared to the lug, or vice versa.Errors may also results from incorrect specifi-cation of orthotropic material properties.
3.6 Visual Interpretation of Output
Level contour plots are recommended as the best means of
visually interpreting the output from a stress analysis case. For
this purpose, a scale plan of the lug outline should be prepared
and the element centroid positions marked on the plan. The stress
values may then be transferred and a contour plot prepared. Plots
of arr, aee, are in the region around the bearing pinhole and of
a xx, ayy, axy in the shank region are recommended. The nodal dis-
placement solution table may be used to provide a plot of the
deformed structure, if desired. The output from a stress intensity
analysis is best treated by means of polar plots for KI and KII.
These plots are discussed in detail with examples in Section 4.
-21-
3.7 Program Status
At the date of this report, the following program options have
been exercised successfully:
(i) Stress and stress intensity analysis.
(ii) Bearing load: tension, compression, positiveshear, and negative shear.
(iii) Cosine and uniform pressure distribution.
(iv) Isotropic, single-material lug.
(v) NT = 3, 4, and 6.
The following options have not been exercised to date:
(vi) Isotropic, two-material lug.
(vii) Isotropic bushing with orthotropic lug.
(viii) NT > 6.
-22-
Section 4
RESULTS OF EXAMPLE ANALYSES
Two example analyses were run to demonstrate the program.
The first was limited to stress and stress intensity analysis of
the hypothetical wing root attachment lug shown in Figure 9. Second,
a detail similar to the aft engine support pylon truss -lug in the
C-5A was subjected to a more extensive analysis. Experience with
the program to date, on IBM S-370/165 and S-370/168 computers, indi-
cates that approximately 1.8 to 3.6 CPU seconds per Kit K II solution
pair are required, depending upon the amount of detail in the model.
4.1 Analysis of Wing Root Attachment Lug
Figure 18 summarizes the stress distribution in the hypotheti-
cal wing lug. Stress contours for a rr' 0 ee, and a rO are shown. A
survey of the numerical data confirmed that a rr, a ee were symmet 'ric
about the lug centerline. Hence, only half-plots are shown for
those contours. The survey also indicated that a r8 behaved anti-
symmetrically, as shown in the second part of Figure 18.
Figure 19 presents polar "butterfly" plots for K I and K II as
functions of angle to the crack opening. The crack was 0.5 inches
long and oriented radially. Again, the data behave symmetrically
about the lug centerline (crack at 0' and 1801 locations). The
interpretation of these polar plots is explained in Figure 20.
If the origin of the plot is identified with the center of the
bearing pinhole, a radius vector through the assumed crack location
may be constructed. The length of the vector between the origin
and the K-plot then gives the corresponding stress intensity value.
If the crack size is small compared with the lug dimensions,
there follows an intuitive hypothesis that the stress intensities
ought to behave in the same manner as the uncracked structurestresses. This hypothesis can be confirmed, for the present case,
by comparison of Figures 18 and 19. For a radi~lly-oriented crack,
-23-
KI should be influenced primarily by a 0 0 , while KII should beinfluenced primarily by arO* Maxima of a and KI are observed
to occur near 0 = 900, 2700. Maxima of ar0 occur approximately
at 0 = 450, 1350, 2250, and 315', while KII maxima occur approxi-
mately at 600, 1200, 2400, and 3000. The apparent discrepancy
between KII and Yrr can be explained by recognizing that the crack
tip actually lies near the 1.5 and 2.0 ksi stress contours. Local
maxima for ar0 in those regions are less sharply defined.
4.2 Analysis of Engine Pylon Truss Lug
Figure 21 is a scale plot of the structure model used to
analyze a detail similar to the C-5A engine pylon aft truss lug.
The actual lug has two tongues to place the bearing pin in double
shear. It can be reasonably assumed that the load transferred into
the engine pylon at this point is borne equally by both tongues.
Hence, the lug program has been used to analyze one tongue. The
model is 0.19 inch thick, with:
DI = 1.75 inches DB = 2.35 inches
H = 10.5 inches W = 3.5 inches
A 0.15-inch long crack was assumed, with radial orientation.
Material properties for high-strength steel alloys were used:
E = 30 x 106 psi v = 0.295
The "bushing" O.D. was chosen merely to locate the crack tip at
the middle of the bushing region. Models were run with 24 elements
(NT = 3, "coarse mesh") and 32 elements (NT = 4, "fine mesh") around
the bearing pinhole. The fine mesh model is shown in Figure 21.
All runs were made with a 1,000-pound bearing load, as a standard
for plotting the results.
Figure 22 summarizes the stress distribution near the hole,
as obtained from a fine mesh model of the uncracked structure, with
a cosine bearing pressure distribution. The symmetries discussed
in Subsection 4.1 were observed again. Three additional checks
were made to assure that the model accurately reflects the stress
-24-
gradients caused by the lug geometry. First, in a typical section
through the shank, the tension stress a was found to be uniformxx
and statically equivalent to the bearing load, to better than one
percent accuracy, for both models. Second, a section was taken
through the bearing pinhole center (6 = 900, 2700) and a66 was
plotted. The radial variation of a66 was found to agree generally
with Timoshenko's classical solution for an eye bolt under tension
bearing [17]. Also, numerical evaluation of Tfa 6 dr gave the bear-
ing load, with 1.7 percent error for the coarse mesh and 1.4 percent
error for the fine mesh. Finally, the solution for a was comparedrrwith the bearing pressure distribution. The peak value of the bearing
pressure is given by p, = P/TDIT, where P is the bearing load. For
the present case, p. = 3.83 ksi and acts at 6 = 0' The two elements
with centroid locations nearest to r = D1 /2, 6 = 00 were found to
have arr = 3.5 ksi, and the radial stress could be extrapolated to
a value close to p, at the peak point. Based on these results and
the measured performance of the PCRK59 element (Subsection 2.2),
the fine mesh model was accepted as giving a converged solution for
KI, having a cumulative error of 5 percent.
Butterfly plots for KI and K are shown in Figure 23. Again,
the stress intensities behave symmetrically to better than 1 percent
accuracy for both models, and KI follows a 6 , while K follows arO"
The data for KI, shown in the upper half of the figure, demonstrate
that convergence has been obtained. The data for KII, in the lower
half of the figure, indicate that additional refinement of the mesh
might be required to demonstrate Mode II stress intensity conver-
gence. However, since the K values are generally smaller than
Ki, and since they tend to decrease as the solution converges, no
further refinements were made. The coarse model contained 408
degrees of freedom and took 48 CPU seconds to compute a complete
set of 24 pairs of KI and KII solutions. The fine model contained
608 degrees of freedom and took 102 CPU seconds to compute 32
solutions.
The length of the lug detail was reduced from 10.5 inches to
7.0 inches for the remaining analyses, to eliminate superfluous
elements in the shank and thus reduce computation costs. Fine
-25-
mesh models (and a few "very fine" models) were analyzed in the
remaining series. The shortened lug fine mesh model is illustrated
by Figure 24.
The stress intensity analysis for cosine tension bearing was
repeated to assess the influence of the change in shank length.
Figure 25 compares the KI and KII butterfly plots from Figure 23
with corresponding plots for the shortened lug. A slight increase
in stress intensities with decrease in shank length can be observed.
Figure 26 compares butterfly plots for the 7-inch lug under cosine
and uniform bearing. Three significant differences can be observed:
(i) The increased ability of a uniform bearingpressure to spread the part outward changesthe hoop stress from compression to tensionat e = 00. Compare the uncracked structurestress contours for cosine bearing (Figure22) with the contours for uniform bearing,shown in Figure 27.
(ii) The maximum KI value changes from 5 ksi /ih.at 8 = 850 (cosine bearing) to about 4.7 ksi /ih.at 6 = 1070 (uniform bearing).
(iii) Mode TI stress intensities are lower for
uniform bearing.
The third series of runs analyzed the case of positive shear
bearing. Figure 28 illustrates the stress contours obtained for
shear bearing with a cosine distribution. The behavior of arrnear the bearing pressure peak (now at e = 900) is similar to the
tension bearing case (compare with Figure 22). The Cartesian stress
components in the shank region were surveyed to provide additional
equilibrium checks. Figure 29 compares the finite element stress
distributions for aXX' 0Xy through a typical shank section and for
0XX axially, with engineering beam theory calculations:
G M(X)Y a 3V(X) [1 _2Y)2 (25)XX I -XY 2A
-26-
where
M(X) = Section bending moment at X
V(X) = Section shear at X
I = Section moment of inertia = TW 3/12
A = Section area = TW
It is evident from the figure that the finite element results are
within 1 or 2 percent of engineering beam theory. The only excep-
tion is the axial behavior of aXX which exhibits some stress
concentration effects:
(i) Due to the cantilever restraints, as the leftend of the shank is approached.
(ii) Due to the influence of the hole, as the shank/ear interface is approached.
Based on these results, the fine mesh model was judged to be capable
of giving stress intensities for shear bearing which are comparable
to the tension bearing results (5 percent error).
Figures 30 and 31 present KI and KII butterfly plots for cosine
and uniform pressure distributions, respectively. Data for a "very
fine" model (NT = 6, 48 elements around the hole) as well as for
the fine mesh model, are shown in Figure 31. The refined model
was run to improve the fairing of the curves, after plots of the
fine mesh model were seen to have large gaps between KI data points.
The refined model data indicate that the fine mesh has not quite
converged the KI solutions. Two interesting features are illustrated
by these plots. First, the stress intensity maxima and minima no
longer coincide with the stress distribution. Apparently, even a
small crack is sufficient to change the stress distribution signifi-
cantly when the bearing load is shear. Second, the significant
difference between cosine and uniform pressure now occurs at the KI
maxima, which are about 10 percent larger for uniform pressure. This
arises from the fact that the KI maxima are located near and nearly
opposite to the bearing load line of action.
-27-
A fourth series of analyses treated the case of compression
bearing, using only the fine mesh model. Both the stresses and
stress intensities were found to behave symmetrically, similar to
the case of tension bearing. Equilibrium checks and stress con-
tour plots have been omitted, in view of the results already pre-
sented. Figures 32 and 33 show KI and KII butterfly plots for com-
pression bearing with cosine and uniform distribution, respectively.
The most interesting feature is the extreme sensitivity to load
distribution when the crack is at or opposite to the load center.
The Mode I stress intensity for uniform bearing increases by factors
of 2 at the first location and 4 at the second. This extreme sensi-
tivity results from the high hoop stresses which are present in
these regions.
4.3 Example Application
To provide an example of how the butterfly plots may be applied
to structural integrity verification analysis, the following data
have been abstracted from load calculations for the original C-5A
engine pylon truss design [22]:
Load Condition Tension (Compression) Shear3
"Maximum Compression" (MC) -221 x 10 lb. -340 lb.3
"Maximum Tension" (MT) 148 x 10 lb. 220 lb.
The values in the above table represent total load transferred
through the attachment lug, and must be divided by 2 to obtain
the loads per tongue. Since shear bearing can obviously be ignored
for the above conditions, there results:3
Condition MC: 110.5 x 10 lb. Compression Bearing
Condition MT: 74 x 103 lb. Tension Bearing
Assuming that a cosine pressure distribution is representative,
the following calculations can be made for 0.15-inch cracks assumed
to be located at 00, 450, 90°, and 1800:
-28-
Crack Condition MC Condition MTLocation (K values in ksii/Thn.) (K values in ksi/i-n.)
K0 K 0.4 x 110.5 = 44.2 KI K II
K 0II6 x 74 = 192.4
K4 0.3 x 110.5 = 33.2 K 2.6 x 74 = 192.4450
K =0 KII= 1.25 x 74 = 92.5
9 0 KI II = 0 KI 5 x 74 = 370
K II 0.2 x 74 = 14.8
= •01800 KI K 1.63 x 110.5 = 180.1 KI K II
KII =0
"Unit" K values are read from Figure 26 for Condition MT and from
Figure 32 for Condition MC. The actual values are then computed
by using the actual load to scale the unit values.
Potential fracture sites may be assessed by comparing KI with
KIC for a proposed lug material. Since high strength steel alloys
have fracture toughness generally below 100 ksi u/n., the above
data indicate that a 0.15-inch crack is longer than critical size
if the crack is located at 450, 900, or 1800. If a criterion that
0.15-inch cracks be less than critical is to be met, the designer
might do this by increasing the lug thickness. Since the numerical
data result from a linear analysis, the design can be scaled. For
example, a revised thickness
Tv = 370 x T = 370 x 0.19 = 1.41 in. (26)KIC 50
can be calcuiated, assuming that protection against a 0.15-inch
crack at 90, in a material with KIC = 50 ksi .uT, is required.
At other points; e.g., 0 = 450 (Condition MT), KI and KII are
comparable, and interaction formulas such as:KI2 (II) 2
(K- ) 2 + ( 1 2 < 1 (27)IC KIIC
may be used to assess structural integrity.
-29-
Section 5
CONCLUSIONS
A finite-element analysis program for computation of Mode I
and Mode II stress intensity factors in attachment lug details has
been presented. Since the program is based on an assumed-stress
hybrid crack element, relatively crude structure models can be used.
Performance tests of the crack element and the lug program have
indicated that KI and KII solutions can be obtained to + 5 percent
accuracy, for 1.8 to 3.6 CPU seconds Der solution pair on current-
generation large computers.
A series of demonstration examples, involving a lug detail
similar to the C-5A engine pylon aft truss attachment lug, served
to illustrate a number of important features of the K solutions.
With the crack size held at 0.15 inch and the crack orientation
kept radial, parametric analyses were conducted for KI and KII with
the lug subjected to tension, shear and compression bearing forces.
In each case, data were obtained for both a cosine and a uniform
pressure distribution, to represent possible extremes of load trans-
fer across the bearing surface. The parametric capability of the
program was used to compute for each case a number of KI and KII
values corresponding to location of the crack at various positions
around the bearing pinhole. Polar plots of KI and KII versus angle
to the crack location were presented to provide a concise picture
of the parametric behavior.
The following specific conclusions can be drawn from the
results of the analysis. First, uniform bearing pressure has more
tendency than cosine pressure to spread the lug apart, and this is
reflected by increased KI values. This effect interacts with the
relation between the crack location and the line of action of the
bearing load. The most significant sensitivity to pressure dis-
tribution occurs when a KI maxima coincides with or is close to the
line of action of the load, or when a maximum lies opposite to the
-30-
load line. Second, the most critical locations for a given crack
often lie in unexpected places or correspond to unexpected load
conditions. For example, cracks at +901 to the lug axis appear to
be most critical in tension bearing. However, cracks at +450 may
actually be the most critical if the lug material happens to have a
low Mode II fracture toughness. A significant Mode I stress intensity
value for a crack at 1800, under compression bearing, is another
unexpected result. Finally, the maxima and minima of KI and K
sometimes tend to follow local maxima and minima of the stress
distribution in an equivalent uncracked structure, if the crack is
small compared to the structure detail dimensions. However, the
coincidence of maxima and minima occurs only for some load conditions,
while significant discrepancies occur under other load conditions.
One is, therefore, led to conclude that a stress analysis of an
uncracked structure does not always provide a good map of where to
expect the most critical stress intensities, even for small cracks.
-31-
REFERENCES
1. Campbell, J. E., et al., "Damage Tolerant Design Handbook",Metals and Ceramics Information Center, Battelle ColumbusLaboratories, 505 King Avenue, Columbus, Ohio, 1972.
2. Matthews, W. T., "Plane Strain Fracture Toughness (KData Handbook for Metals", Army Materials and Mechani sResearch Center, Watertown, Massachusetts 02172, AMMRCMS 73-6, December 1973.
3. Anon., "Airplane Damage Tolerance Design Requirements",MIL-A-83444, U.S. Air Force Aeronautical Systems Division,Wright-Patterson AFB, Ohio 45433, June 1974.
4. Bowie, 0. L., J. Appl. Mech., Vol. 31, pp. 208-212 (1964).
5. Bowie, 0. L. and Neal, D. M., Int. J. Fracture Mech., Vol. 6,pp. 199-206 (1970).
6. Bowie, 0. L., Freese, C. E. and Neal, D. M., "Solution ofPlane Problems of Elasticity Utilizing Partitioning Concepts",ASME Paper No. 73-APM-C, 1973.
7. Tada, H., Paris, P. C., and Irwin, G. R., The Stress Analysisof Cracks Handbook, DEL Research Corporation, Hellertown,Pennsylvania, 1973.
8. Pian, T. H. H., AIAA J., Vol. 2, pp. 1333-1336 (1964).
9. Tong, P. and Pian, T. H. H., Int. J. Solids Struc., Vol. 9,pp. 313-321 (1973).
10. Luk, C. H., "Assumed Stress Hybrid Finite Element Methodologyfor Fracture Mechanics and Elastic-Plastic Analysis", Aero-elastic and Structures Research Laboratory, MIT, ASRL TR 170-1,December 1972.
11. Tong, P., Pian, T. H. H. and Lasry, S. J., Int. J. Num. Meth.Eng., Vol. 7, pp. 297-308 (1973).
12. Lasry, S. J., "Derivation of Crack Element Stiffness Matrixby the Complex Variable Approach", Aeroelastic and StructuresResearch Laboratory, MIT, ASRL TR 170-2, AFOSR-TR-73-1602,February 1973.
13. Zienkiewicz, 0. C. and Cheung, Y. K., The Finite Element Methodin Structural and Continuum Mechanics, McGraw-Hill, 1967.
-32-
14. Abramowitz, M. and Stegun, I. A. (ed.), Handbook of MathematicalFunctions, National Bureau of Standards, U.S. Department ofCommerce, Ninth Ed., November 1970.
15. Desai, C. S. and Abel, J. F., Introduction to the Finite Ele-ment Method, Van Nostrand Reinhold Company, 1970.
16. Orringer, 0., "FEABL (Finite Element Analysis Basic Library)Documentation of Sample Program for Export Version", Aero-elastic and Structures Research Laboritory, MIT, ASRL TM162-3-4, February 1974.
17. Timoshenko, S. and Goodier, J. N., Theory of Elasticity,McGraw-Hill, 1961.
18. Washizu, K., Variational Methods in Elasticity and Plasticity,Pergamon Press, 1968.
19. Lin, K. Y., Tong, P. and Orringer, 0., "Effect of Shape andSize on Hybrid Crack-Containing Finite Elements", submittedfor presentation at ASME Pressure Vessels and Piping Con-ference, San Francisco, California, June 1975.
20. Orringer, 0. and French, S. E., "FEABL (Finite ElementAnalysis Basic Library) User's Guide", Aeroelastic andStructures Research Laboratory, MIT, ASRL TR 162-3, AFOSR-TR-72-2228, August 1972.
21. Orringer, 0. and French, S. F., "FEABL (Finite Element AnalysisBasic Library) Version 2/Release 2", Aeroelastic and StructuresResearch Laboratory, MIT, ASRL TM 162-3-3, February 1974.
22. Neilson, J. A., Shockley, E. J. and Dallas, C. H., "C-5A PylonEvaluation and Redesign Recommendations", Lockheed-GeorgiaAircraft Company, Marietta, Georgia, Report No. SRD 72-21-775,p. 29.
-33-
A53SUMED S-rtiEss
- ASSUMEDDiSFLAcEMErNT
A~sUlmrD DlsLACCMEt*T
FIG. 1 COMBINATION OF ASSUMED-DISPLACEMENT ELEMENTSWITH HYBRID CRACK-CONTAINING ELEMEN!T
6
(~Y*) '7l
z Z A-sPECr RATIO =aL/1
y 44
FIG. 2 CONVENTIONS FOR ASRL QUAD4 ASSUMEDýDISPLACEMENT ELEMENT
34
I zJ H.
00
41-
CLL
2: H
0 tin,
Its Y.
35U
tftVg '< z
U 0
wL lAS
w u
-I In '
H H
r- iw
U) I 0IE-z
'E-1
LIIO
I-
z
zz 0
Z• E-1
H
36 H
UJ a
0 o
H
36
BUILT-ZN EDGE
COSIN4E% %
UNIFORM
EARNI-TENSION BEARIN
2- POSTT2VE SHEARDISTnWTIONS j COMPRESSION VEARING
4- NEGATIVE SHEAR
FIG. 5 ATTACHMENT LUG DETAIL
CRACKSIZE
CRAkCK
FIG. 6 CRACK PARAMETERS
37
-- ~ ~ ~ ~ ~ ~ -. -T -- ~I~-- ~ ~ >
iJ Ij 'u Tb_ I I
I -TI
Z'IJJ
U-1-
-I li 7 - -A,41i v
u~ --- ----0 ~'K ~ '-~---r n-- rV
zz Lr zz --- r-LI.--,
_2i 4 I1 L'iý 1- {T zi j L
fl 1~ i Z7zff 41* F
38
ixI
E-,4
z
044
z
W
•,t[ H
0 --t
39
wH
a V)V) 0 0
ix I
0- 0
H
E-t
0
4P
40o
40
w
I--
.1 --- --- En
Q141
TCASE iNCASES
READ INpur DATA
HOUSEKEEPJING
PRINT INPUT D~ATA
COMPUTE: # ele+wS raid~alyIn bvs")hI, IUS ; It 0606t in
etc.- er~ase ~z, K1
S~or,%e areas (2.01)
LOOP ovER CRACK LOCATION b(200)
0#04T If Er se peiucase;
a4+0l~ PCK- - 'IraConflectiom, ns; fa 3 n
COMPUE:- It ovedlAy slobalXY node coor~inactes forun cracke4 s+ruacture;
FIG. 11 PROGRAM LUG FLOW4 CHART
42
1ANLW
Move interior Poje tocrack openlnSý $+oreleca+1061 'Angle ; compiset+Ip c@oEJs X, 1p,YrI,,
ra-
Erase element 'force
Vec~r- ~ (24)
III A N141 = total *'elem dS
TANL N~i OFT- I uIAIOL-Z
U2
Extract XY c~orJt fromcompute Xe,)Yc, for~
I uQAb4; allemble k+* X&
FIG. 11 (CONTINUED)
43
Comapte k B. fvr
P-CRK~r, ~asSemqble k I
CowfoA+c vnodsi Aorces
cit~ivalcewt to specifie4t
0 beauhil IaAJ opfloyts;
z (20!50
C ~Solve: -I
Cev~!:roe ERROR¶I k1imfrIiu(eu VY)
~ ~,!(~O2Cr) Si- kNoK
I oAu4L= f 22 ,I
____________ i+FIG.u 11(OCLDD
ceptri a44
in
EII
ontp
0
I--
0z
HH
H 0+
+0
C) ID C, oiM n p.44
C) .- = 04H) 0 0)
0 1 0 E-(N H 0 z 01 11
41 0 0 0 0 v ~H r4 $4 Mn K E-
0 H H H )coctH 0 i 4 0 0 4
A 04 In H rfl pa px- 0 .
2 ;a 0 0 0 c
r) 1~410 0 C U Pr4
(N 9-40 2 I I P
0 P, 04 +D H Z en(
0 E-4H W 0 n eU) 0 c0 4e r4 0 0
en H V) W n o n 11 mE-c4 H-~ H Mn en MC34 11e n N 0 0 mt
H1 0 0 04 0H('2 z 01 N.' w 4 1P: ,4c pq .03 E- n
cn C) inE4 0 m 0 0 p
o ;& 00~4 C) 04 C n
un HE-o w
0 0 0~( H .45
H0
'.4~~. P. . z
46-
V4~
'-Vo
r4H
-r4
47
E-4
H
I4
H
H
48z
CA se It I case I It O8CE04RER 1974 -
ANALYSIS OPTION- It LOAD OPTION. 2- _____________________________________
1% 0NODE 1 It-ISOTROPIC. 2. TýOPTIITOPIC LUGITOTAL OF 32 FLE"N841 AP~JUND PIN HOLEHOLEI8.0.- 0.875E010 BUS..ING O.0.. 0.235E0*0l. LUG WIDTH- 0.3500OI.0 LUG LENGTH. 0.?7000.08 TH4ICKNESS-. 0. 190E.GO00
SUSMING MATERI ALS 6-0.300f-08,40-J O.Z%96D__________________________________________
1p ISOTROPIC LUG, 8- 0.370E06- 11U. 0.29S0.00
- APPLIED TENSION- 0.I008*04, LB. &ND SIROAR. 0.0 _ Lt. _
VALUE 0.982E-02 -0.IZ4E-03OO._ 0.0 _._ O-. .0.0-0.0 _
VALUE 0.94)E-02 0.002 0.0 0. 00 0.0 0.0 0.00 00
01O" 21 22. 23 24 25 26 2? 26 29 30- VALUE 0.90110D2 .0.176E-02 -00.....O0.... 0.0 - --000.0 - .00.0 - .0.0 . 0.0
-ROW 3 32__ D3 31,..i...... A. .. L I tVALUE 0.616E#02 0.545(-02 0.0 0.0 0.0 0.0 0.0 9.0 0.0 0.0
Nov 4 42 43 44 41 ý4 47s6o43VALUE 0.694t.02 0.6941*02 0_.0 . ..... .......... .O 0..00.... .. . .O 0.0 - .__
VA.LU1 0.45.0 0.6168.0 0.0 0.0 0.0 0.0 -0.0 0.0 9.0 0.0
ROW 61 62 63 64 AS *6 * 8. - 70VALUE _9..3T6E-02_. 0.4070.02 -0.0.-.... -0.0 00-. ........ 0.......... - 0.0.-... .--. O...... -. .
low' -- 7j --... 72 - I. t1. 0. .. . .VALUE 0.9921E602 0.963E-02 4.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
VLUE 0.07EE .580 .580 .680 0.43-0 .6080 n.10 0.*01-0 0.511101 0447G
It AU .&T-3 02)-0 .507 0.22)8-04 0.T17o-k 0.2 10 4 6 .5 80 3 0.78o .0 73 _0.51E-
VALUE 0.- _. _. ... _OO .0...07Z -0
P ~ ROW - o 18. 4. 31.. 4S4.. 455__ 46 .. 67 468 44 .q 10...VALUE 0.9803 0.AI04 .60478 0.6020 0.4803 0.6050 0.430A 0.6804 0.40-0 0.70-0
*. ROW I 2 13 3s 6 to56544VALUE 0.5704*E-0) 0.6350.-04 0.441-0) 0.5607-04 0.S428-03 0.6135-04 0.3970-03 _ 0.1 -04 .516F-03 0.11ME-06
R.. OW - AL_ ' *2 ' _ Al __ 14- IS� ' 6 __ 7 Is t9 to
.Nqe19 _kOEff 3150f 32 73747 7 8......79PVALUE-- .0.3660-03 - .0.41470-4......0.1A303 O33-4* . 4-0,0.230E-04 0)3-03......01 -04 F)..027E0 ... CIA-E4.......
I8OW. 21 SI.... 67 25 434 __ 84 0,99-27 63.. 54.. ST .S*E. .68 94.. 4VALUE 0.S3E6-03 0.8968-04 0.53140-03 0.1412E-04 0.9820E-01 -. 4-U 0.9 -0 -. 16ON 07703 -0.68C
ROW 91 32 333k 3.63" 9,9 49 400!VALUE _ O 31-0. .00-3.jAE0 0.36~ *- 06_0 - . 8-1 060___ .ý8- _.TS0
, VALUE __0.3118-0) -0.6228-04 .?10 .0E0 .5E0 .010, 043-,1 066-4O5s-1 07%-4
FIG. 15 FORC AND D SPLCMN TALE FRO STRS ANALYSIS
VALE .4SE03 0.1S!74 .41E01 0SOE- 4 9 .2E0 _ .SS:O P3D-) OV fO _ .bfO ..1EU
I * 0 cc m 0r 0t 0n 0' It 0 0 n 0 0 - 0- 10 10 N n 0 10 0~ 0u m 0' 0- 0 01 04 lr ,MN -f-* C -++ 4 + 4 +++Y+N , 4+4 -4 rj,4 -4+4+0 n0 tc .4t+%Ul 4
LU *
z orý I * LU iu iu iu LULLU UJWWUL u iUI ULU WWWWWIUJ LU wU. jWu iuiu LUW WW LU0n 4- - a'ý, 0o1-0o a '0L -L)t-a
LU 0r I * o 0 * 0 * 0 a * 9 * * 0 0 * * * * 0 0 . 4 V
I * 0000000000OC)C0000000000000000000C.( ..JI .1 11 . ......... t
LU CU *iu iu iu iu iUIL iu U ju jL L jL iL jwLJ
z f* 00o0CO)000OOOOoO0OoO00 00000000000
mw * LU wLIu w w uL U)WWiLUWU-LU L i LU WLJ u LUWWWWW.ui U U LJWWj U JWuo- or* z lr )mm01 ,.4' ru z 4
U.X. lM0N-4CV ' ,NOTLO- 4- Z Mr a, 0 k 0 LLC)ON9 'lff 4-44
of VI* oo o o o o o ~ o ~ o O U c O OIc * 9 9 9 9 9 9 9 9 9 9 9 *9
Zf V)0 0 0 0 0 0 0 00 0 C 0 0 0 0 0*jIr4r I T- n()I r' f t I 7I ne*U c ý00C - 0C )0C lC
- )-* -0000 1 0 0 0 m0 a 0 0 m -00
x4 )(. E-4N~mm -
.. . .I * . .0. . .0..0. . .Q.0.. .0.00. .0.. .04uju t UU J.I4 11 11 1 iw Uj i L jU L1L L iu UI LL i l t u WL It 0
LU~ ~ 0 0 * 00 0 00900 0 00 0 000 0 0 00 0 0"~ ~-1* + ++++ 4+4 44~4 444+ +44+ 4P4
0-0 * 00000000000000000.0)0000000000000I * . .4 . . . .4.. .. .. .. 4.. . + . . . .
;eI * LUWWJ WW LU u iu L iUjLUWWWWL U)u LujLU LIUW U wJUWW L UWW W 0ju iu j
I* 0000000000000000000000000000000 P>C)
) *D 0 00 0C)0 00 00 0 00 000)0 000C0 00C00 0 00 00 0
I) ( * L iu L w:u uWW WW WUw LU UwWW j WWj j iwiUJW w W LU WUJiWWW L
4 4 L 0 0 90 0 0 9 0 9 0 9 9 0
V) ~-j C .L 00 0 0 0 0 0 0 0 0 0 0 0 0 0
- 0 ~z+4 ++4+ ++4 +++4 ++4+ 4*450+4
PH
E-1
zE-1H
zS~H
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V)
l~i •"I .I .- -. ." ., ,. .I ,. .I .I .I .- . . -. I . . . . . . . . . . . . . . . -7
M,. r . M4 0• N-' 0 0• M M: m 0) ; m r- CN 0N 0 CO r-. r.D V) 0.,) m ) Ln ) m .M 00 %- o%-O* O*0 C k r¢ ) N rl0 C- m C-> nc Ow W 0 00 C) 4 m 00a N a) k N
!~ • 2
M M1 M) M o y o M o ý 'Il M ) M M P-) N N 04 ry (Y (Y ¢) oy M CYu ) t) (Y) o (E H
gt) */,t+ ++ ÷+*+ + + + +4 +*+ +++ I + ++4+*++ + + +1+ + +
rf) .'4,, . .• . to..o . ,. .- .i . . . -- o . . . .I . . . . . . . . .-
o N *Oc(Y)NHQ w m m 0
) ý c -;-I C; . 09 C 9 9D. ý n9 C,: C9) C, 9 9to51
c.~ IP4 W*
* -
E-4 M Ln to '0 W~ C,) MN Lm %0 CO m 1:.) N' M LO r-~ '. O ma I C) 0 r- N i)--
Z * II
E-
0 *
M~ g N ,Uor )v 1 . O10r-XM0 MZ or NP5- * N H NN( r
c,~v~j ++++++4*4+++*+++++++51+
E-0
z
0
P4
0
> 6
0 U)
040
a 2 H
52
00
53
90
£1 1+12 1 8 6 4 .A• ,,.. * •it lo s 6 0 LZ 14 16
/\
II270
DATA FAIRING
KX .
Kjjo ------
FIG. 19 BUTTERFLY PLOT FOR WING ROOT LUG STRESS INTENSITIES
54
K.,oR K/
POLAR PLOT
cRAc K •srRESSXT"EN SITY
FIG. 20 INTERPRETATION OF BUTTERFLY PLOTS
55
U)
0
'-4
- I- -T - -L-
56
z
z0
w H
zI-q
Km
E-U)
u-J
vs<M
"574S P4
z W4~
z 0
00z N
l)'
E-1 E
W.N
H
57
90\ I /
\ /
'DATA FOR K 1
IK
ABoVE AXIS
is~o I k' "'-- 3 0
DATA FOR K]:BELOW AXIS 4
22"/\
I
270A COARSE MESH0 FINE MESH
FIG. 23 BUTTERFLY PLOT FOR ENGINE PYLON TRUSS LUG(TENSION BEARING, COSINE PRESSURE)
58
ot--q
00 0
440 )
0 OnHr
594
90
I
180000I N
% %
10.5" L"U 270
FIG. 25 EFFECT OF SHANK LENGTH ON STRESS INTENSITY(TENSION BEARING, COSINE PRESSURE)
60
//135" 45"
% %
-I -
ISO 180 0
225' It%/1s
% dI
I
270-CO5INE
UNIFORM
FIG. 26 COMPARISON OF COSINE AND UNIFORM PRESSUREDISTRIBUTIONS (TENSION BEARING)
61
0 + 0
-0.5" 0
FIG. 27 STRESS CONTOURS FOR 7-INCH ENGINE PYLON TRUSSLUG (TENSION BEARING, UNIFORM PRESSURE)
62
-1 -0 + l
00
0 2.52
0
-10
2
0--
FIG. 28 STRESS CONTOURS FOR 7-INCH ENGINE PYLON TRUSSLUG (POSITIVE SHEAR BEARING, COSINE PRESSURE)
63
10
X 0 z3II I,1n
xo ,
0 2. 3
-10 F y / 0A.)
0 FINITE ELEMENT
• SOLUTION
k ENGINEERING SEANU THEORY
S|0/ -. NEUTRAL AXIS
7•6-
0
I" •I I I I I
0 i 3X' (in.)
Note: See Fig. 24 for location of x'y' axes
FIG. 29 COMPARISON OF FINITE ELEMENT RESULTS WITH ENGINEERINGBEAM THEORY (POSITIVE SHEAR BEARING, COSINE PRESSURE)
64
U~ -4
/~ OH
E-1U
141
-44
W0)
E-1 En rX
404
p..
4 H
0 En (
H4
65 1
00
In WU H-
+ H
P- E-1 AQ0
N' 0
w zI
H
CoV
IoI
/ n z-4 0
- 1*I
9 0U
E-1~J
W Z H0%~~~E 0 C) p.~ 4c0
m P, P
U *5. r14
- I % ~ 66
APPENDIX A
-x 41. x x X. N K If M If -N
E-4
z E-4 E-4 C-4 CW X 14 pqZ "4 C) w z C> 40 0 0
CY4 I V Ir4 W E-4 0 a) T- V- T- r-
V) 0 H -E-4 = P4 tn Q H PLI W N 14 GqH m H sc 0 E-4 0 W 0 H N t'o (N (N Ul) NV) I u A Pil U tf) E-4 0 E-4 P0,4 W H tr) z " 0 w (i0 z tm H E-4 n M t4 z00 0 m 00 = PLI 44 H z P =)z cn, co w H 0 C/40!4 Kc M 0 Z) 1.4 0 rx-4 "
!;M H E-4 M >4 M 040 M WC PtlV) z 099 M. O:c M Ow z wa wtn H 0 H u m < 04 z w 4
0 =) H 0 E-4 CU >4z A w P4 ýý04 M H -"4
E-4 H Pý 0 CN M H E4 OC4 v zrn 0 E-4 PQ H 03 'W 99 L) E-4 H
EA = = ag El W, 1=1 #cc W z m C4W u W E-4 P4 0 0 E-4 #,0 -14 W W *0 OCC m M H H 0 M4 z M ýdVLI 44 w H 04 m 0 &4 o -4 u
W4 0 z oc pi M 04 U 00w P-4 m 03ý p Fo po C14 H
O-C F-4 pq lki P"Aý H M a z H
m pr4pq oel m 04 z
04 oq H po E-4 po N H $4 H occP4 E4 E-4 I'll PC4 co = N 1.4 L)
z u tp4 = JX4 m = H (nM H Ir og lx4 04 E-4 Cý rZ4 u M W! 4H 0 L) m 0-4 W W4 Cr- E-4tn V) n w " H .0 C3 0 peg occ :b,4
ým V) 00 14 H 1:3 04' If) lk4 0 0
0 PO 0 mw I UpOomp p 9z 94ow gy, b-4 H rl ý1-4 W 0 C) W rxl frl Z Pq E-4 zz t- U P- - EA U IY4 1% Z 4 V = 0994 tri CD M z w 04 pi 1-4 M) 04
m M H H " M ý tf) Z U)E-4 0 H " H t-4 U M Z Z og E-4 M H N H 0Z m 0ý mpt4oz I mwww = -11 in " mPQ Ica 4 m H r-19i H m 0 u m w H 09 mZ; " w PC 0-4 Pit 3V OM 44 M = 194 a MP z cn w H %W H E-A
P p 's M 1-4 Z 0 = N E-4w = p PcC = Z E-4 0 M PL4 E-4 H w
I En E-4 E-4 OcC W W H H E-4 En 0 z M w zpi sO E-4 0 M M Qr4 W fil L) (n H
Txl F-4 M E-4 M tn " E-4 N 0 tO IZ r- % 4H E--f 134 o Z 94 H = 41 E-4 CP % L) Mx tn cý z w u m flcq w m H m m Cl p t* zH 014 W4 tT4 H U PQ E-1 W: M H W4 bz 0 FL4 H 0
04 rZ4 'N E-4 0 M H rml 0 H w = U 0 E-4 "v 04 w I ICQ w w m t-4 oc M P4 W4 P4 E-i0 ocC W P4 v 1 0 0 4 0-11 OH Wm om "I.) " x H z u H u V 0 tn 0 eel 0
V) m PO C:) tit 0 H pt P4 = 04 om rx = uH =1 W W4 oic c)i H tn W Q (n v) (n
U) PW V) rz m Pri p U Qý x 0 w 1.4 % 0 1.4 w HE-A Z P4 Z H W 00 W4 4 En ro T- N ob #cc n (n
LD H 0-4 P4 04 0 rV 0 FCC U M PQ CP- z H owZ) 1-40 0 = U 0 3M #-:4 W E-4 04 U W *4 j--4 En z C) OH 0
E4 C) &4 44 En z m pq 44 ge wo E-1 m N Z 0 E-4 Z oc4z w H N m 0 H E-4 M E-4 0 E-4 u H 114 H m 0 w H 019 z43 W &4 W4 o-4 W U 0 V) ::D ;M P1 ril Z PcC Z E-4 H C:) W W 0 W EA F4 11 04
CD Pýq 0 P1 0 p;q = :1,-4 (n 4 ý4 P- H on 11
1:4 o4 H P o-4 M 0 U) E-i (n E-4 #4 H E-1 m V) 11P4 :z C4 w 0 1-4 tm M P, 0-4 FC4 F4 04 tic PO p
4 U 0,4 u M W tz 0ý H W CD 0 W 0 0H 0 H 0 M N p;! H Z V) 014 0 ri4 P4 C%4 (n-t Ln W 04 P4 H zE-4 M F-4 4 W4 A = " JA P4 ;M R4 (Ai 0 W z 014 u H 4F4 PA u xi 04 04 H u 0 01C H Aý ý04 w H W Z H H
uuvuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu
67
4( N XK- K 1K X X K K KKK' KK~ K K K*~ K *~
E-1)
4(4
~c 0
u 444 0.HP 0
N ri 91H Hq
D1P4 0 H-
z u
r)mN00 P4
H W-4 FA
~L of)
Z 0 U P
0 e zH ri H I05 EU) M.z HH
E-4 P4 104 Z~ W4~ H u HH 1- UP4j - H w H 0 ý14
inO ul mI H u bAN
ZHO N40C4W O z Z V ) 01 C
P4 H et 0 0 " w P40 U) HP 0 1.4
zZ 04z=H H~ m~ 07 u~ ~ 0HNiH H~ WA U ~ HA H =~ =
E-1 ý m /~ OHO ON w~~ Q H U 0) 0 z Vul z 14 ac P HO0 If 1:4 0.4 9-0 EH u HV)HF P LIr)N H 04N 0 x' P4 ni 04
C4 0p 1 ) H 1 0 H. H" ~ FC9c' z z ý4 KC M H4E- U H- wV) 0 -) ZO ~ l =1 Hr ý4H V) 0 U)
:Z~) = H P Z toH 04 ~ - 4 HH UO FL
z M) 0 F1 C) zH H C) H OH " 014 P 4 P. cptc a) u C H H) EA U) .z t 0 w 0 Un
H- H O 410V . L U LH rZ Hq It b11If 114 11) H 0 m H- rA HDXC4 W
0 HC - -0 0 LA0 0 M~~ 0
CA E-4 = U W 0 tn= I II W N 0¼40
H 0 3r pt 4 0 UrPfM = 0 0O 0 0 tH) O C) V) %PIJ(~ P4z P p r4P - t=
E-1 W~f HO Z %) H4 o(N W IZI4 L r I : 4
04 M)Ul H 0 = z - H) CN,4 1 w to U0 0 OHl EZ-iZ.~ bd W-W - Pa HQ P
H ~ E-4 H)U HU HIIN PI M 1 = ý PýZ
II Ir. E-iL4 0 En u -'
I- ot N Fzl fil 0H (N
x p H =~ &I) u u 14 Pr N E-1C
UUUUUUUUUUVLUUUUUUUUUUUUUUUUUUUUuuuuu
68
* N 0000 L)0 f,-0C0 MC>r-(U) * 0 0 00c00000c 00 V-ý
0 C. c 0 C> 0000 00co000
-tr~-rY-1 0 * 0 Dca CI r- ,-P44 E- D== ==
E-4 % w H 90 pwI En (n ) 0m 0 000 C Z(N. P4 X:rZ4 W4 M t*
()HH = 0*in F 0 *HW~ P4uH E-4*
*1*
ZEZP4H N 0 *ý0f W9 *.
Pl Il 04ati
ow 04 U)0 *
= U c) w0O * Cdb m6
m0~ - HN PL 00 * vC1 rl -4H. 0 -4 u0 tN'M O
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