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W-A102 "S5 NAVAL POSTGRADUATE SCHOOL MONdTEREY CA F/f 11/6AN ELASTIC-MASTIC FINITE ELEMENT ANALYSIS OF NOTCHED ALUMINi7 -ETC(U)R SI M KAIR
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NAVAL POSTORADUATE SCHOOLMonterey, California
THESISAN ELASTIC-PLASTIC FINITE ELEMENT ANALYSIS
OF NOTCHED ALUMINUM PANELS
by
Michael John Kaiser
March 1981
Thesis Advisor: G. H. Lindsey
t, Approved for public release; distribution unlimitedC-1
j/
UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE 010140 0... Ente.. _________0________
REPORTf DOCUMENTATION PAGE OREAD C01PSTING 1FlNS1. REPORT NUMUEN 12 OVT ACCESSION N0'OR r R IN' AAO uS
/AD ..~A 4 0~ a ~ I'EI CALO NkMEER
4.TI TLE (and Subhie) i&Z"u.JCOE1
(L An 21asti-cP7iiic F.inite Element AnalysisMse's7eiof Notched Aluminum Panels. March- -196/i
6. PERFORMING Ono. RIEPORT NDER
S. CONTRACT OR GRANT MUMMOERI()
( ,~Michael John i Kaiser
*. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMN"T. PROJECT. TASK
ARA & WORK UNIT NUM0B1RS
Naval Postgraduate SchoolMonterey, California 93940
I I. CONTROLLING OFFICE NAME ANC ADDRESS _ _ _ _ _ _ _ _ _ _ _
Naval Postgraduate School reI 8* 1UBRO ~lMonterey, California 93940 155 pages14. MONITORING AGEINCY NAMIE 6 AOO6g4E 1(iraf 1imms Ca.,maitnd 01116) IS. SECURITY CLASS. (of this tpr)
Unclassified
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Approved for public release; distribution unlimited
17. OISTRIOUTION STATEMENT (of five o~slrast witeredt Woo "I.. It diffe"0 hme RepseV)
IS. SUPPLEMENTARY NOTES
It. It EY WORDS (C~imes an rere of"i It Rnmefm dad idIdeft 6v No"0 num.r)
Finite element analysis, stress, stress concentration factor,notched panels, residual stress, plastic zone, ADINAextrapolationelastic-plastic stress.
f.ATSTRACT (Can~mun ON e'mwO Oid, it 01061411F Md 04110111611 6F 5611 mal..)
Finite element, elastic and plastic analyses of various alumi-num panels, containing holes and notches, were conducted for com-parison with photoelastic experimental results. A FORTRAN IVprogram, ADINA (Automatic Dynamic Incremental Nonlinear Analysis),was used for both linear and nonlinear analyses. Mesh refinementswere used for each panel and the monotonically convergent resultswere extrapolated using Richardson's method. Stresses were
S/ 14732-6)EO4OF INOV11 IS B D . T ' \ ; UNCLASSIFIED A e(M PD ; -w "OIO2.O~4. 603 .- _8SCURNTY CLASSIFICATION OF THIS PG U. e.Uad
UNCLASS IFIED.,tu' laegwaYpef of Y.4 .
Item 20 (contd).
ocally smoothed from the Gauss integration points to the nodalpoints. Eight noded, isoparametric elements were used throughout.Modification to an ADINA preprocessor program, also coded inFORTRAN IV, was made for use with a VERSATEC plotter.
Comparisons were made to the elastic, analytic series solutionby Howland for a circular hole in a finite strip. The finite ele-ment results varied by less than one percent from Howland's solu-tion. Handbook values for the elastic stress concentration factorsof the geometries investigated differ from finite element resultsby less than one percent in all cases. The hotoelastic works ofFrocht were also compared where applicable. Stresses in the plasticrange obtained from slip-line theory for a r gid-perfectly-plasticmaterial show excellent correlation to a fin lte element analysis ofsuch a material. Comparisons to elastic an plastic experimentaldata were made for the panels analyzed and how good correlationto finite element results.
AcCOc~ onyr_
Avai a - .
1473~
DO1Ior% 1732 UNCLASSIFIED-5/N OF f u e e V'Ies% Se* RMO'ee
Approved for public release; distribution unlimited
An Elastic-Plastic Finite Element Analysisof Notched Aluminum Panels
by
Michael John KaiserLieutenant Commander, United States NavyB.S., St. Cloud State University, 1969
Submitted in partial fulfillment of therequirements for the degree of
MASTER OF SCIENCE IN AERONAUTICAL ENGINEERING
from the
NAVAL POSTGRADUATE SCHOOLMarch 1981
Author: ____
Approved by: • -// /Thesis Advisor
Chairman, Devrmen of Aeronautics
Dean of Science and Engineering
3
ABSTRACT
Finite element, elastic and plastic analyses of various
aluminum panels, containing holes and notches, were conducted
for comparison with photoelastic experimental results. A
FORTRAN IV program, ADINA (Automatic Dynamic Incremental
Nonlinear Analysis), was used for both linear and nonlinear
analyses. Mesh refinements were used for each panel and the
monotonically convergent results were extrapolated using
Richardson's method. Stresses were locally smoothed from
the Gauss integration points to the nodal points. Eight
noded, isoparametric elements were used throughout. Modi-
fication to an ADINA preprocessor program, also coded in
FORTRAN IV, was made for use with a VERSATEC plotter.
Comparisons were made to the elastic, analytic series
solution by Howland for a circular hole in a finite strip.
The finite element results varied by less than one percent
from Howland's solution. Handbook values for the elastic
stress concentration factors of the geometries investigated
differ from finite element results by less than one percent
in all cases. The photoelastic works of Frocht were also
compared where applicable. Stresses in the plastic range
obtained from slip-line theory for a rigid-perfectly-plastic
material show excellent correlation to a finite element
analysis of such a material. Comparisons to elastic and
plastic experimental data were made for the panels
analyzed and show good correlation to finite element results.
4
TABLE OF CONTENTS
I. INTRODUCTION -------------------------------------- 13
II. MATERIAL PROPERTY TESTING OF 7075-T6 ALUMINUM-----15
A. TESTS FOR AXIAL LOADING ----------------------- 15
B. CHARACTERISTICS OF 7075-T6 ALUMINUM PANELS ---- 16
1. Young's Modulus --------------------------- 16
2. Poisson's Ratio --------------------------- 16
3. Yield Stress and Strain Hardening Modulus-16
4. Ramberg-Osgood Coefficients --------------- 17
III. MODIFICATION TO GRAPHICAL PREPROCESSOR ------------ 18
A. PSAPI MODIFICATIONS --------------------------- 18
B. USE OF PSAPI ---------------------------------- 19
IV. FINITE ELEMENT ANALYSIS --------------------------- 21
A. DESCRIPTION OF MESHES USED -------------------- 21
1. Length to Width Ratio and BoundaryConditions -------------------------------- 21
2. Element Meshes ---------------------------- 22
B. COMPUTATIONAL PROCEDURES ---------------------- 22
1. Using ADINA ------------------------------- 22
2. Richardson Extrapolation ------------------ 24
3. Optimal Stress Locations and LocalSmoothing --------------------------------- 25
4. Computational Times ----------------------- 26
V. RESULTS OF ANALYSIS ------------------------------- 27
A. CIRCULAR HOLES IN LINEAR MATERIAL ------------- 27
5 4'
B. OPPOSITE U NOTCHES IN LINEAR MATERIAL ---------28
1. Shallow Notch Panel------------------------ 28
2. Deep Notch Panel--------------------------- 29
C. OPPOSITE U NOTCHES IN NONLINEAR MATERIAL ------30
1. Shallow Notch Panel------------------------ 31
2. Deep Notch Panel -------------------------- 32
3. Rigid-Perfectly-Plastic Panel -------------33
VI. CONCLUSIONS AND RECOMMENDATIONS------------------- 35
APPENDIX A: PSAP1 JCL----------------------------------- 94
APPENDIX B: Local Least Squares Smoothing---------------95
APPENDIX C: ADINA JCL----------------------------------- 97
APPENDIX D: PSAPi Listing------------------------------- 98
LIST OF REFERENCES--------------------------------------- 152
INITIAL DISTRIBUTION LIST-------------------------------- 155
4
LIST OF TABLES
I. MTS AND REIHLE 5 GAUGE TEST RESULTS ------------ 77
II. MTS SPECIMEN A TEST RESULTS --------------------78
III. MTS SPECIMEN B TEST RESULTS --------------------79
IV. MTS SPECIMEN C TEST RESULTS --------------------80
V. REIHLE SPECIMEN TEST RESULTS ------------------- 81
VI. X=0.2 HOWLAND DATA -----------------------------82
VII. X=0.25 HOWLAND DATA ---------------------------- 82
VIII. X=0.2 FEA RESULTS - NODAL OUTPUT --------------- 83
IX. X=0.2 FEA RESULTS - GAUSS OUTPUT --------------- 83
X. 'k=0.25 FEA RESULTS - NODAL OUTPUT -------------- 84
XI. X=0.25 FEA RESULTS - GAUSS OUTPUT -------------- 84
XII. SHALLOW NOTCH FEA LINEAR RESULTS - NODAL ------- 85
XIII. SHALLOW NOTCH FEA LINEAR RESULTS - GAUSS ------- 85
XIV. DEEP NOTCH FEA LINEAR RESULTS - NODAL ---------- 86
XV. DEEP NOTCH FEA LINEAR RESULTS - GAUSS ---------- 86
XVI. SHALLOW NOTCH FEA NONLINEAR 60,000 LB LOAD ----- 87
XVII. SHALLOW NOTCH FEA NONLINEAR 65,000 LB LOAD ----- 87
XVIII. SHALLOW NOTCH FEA NONLINEAR 70,000 LB LOAD ----- 88
XIX. DEEP NOTCH FEA NONLINEAR 30,000 LB LOAD -------- 88
XX. DEEP NOTCH FEA NONLINEAR 35,000 LB LOAD -------- 89
XXI. DEEP NOTCH FEA NONLINEAR 40,000 LB LOAD -------- 89
XXII. RIGID-PERFECTLY-PLASTIC RESULTS ---------------- 90
XXIII. EXPERIMENTAL DATA X=0.25 HOLE LINEAR LOADING--- 90
XXIV. EXPERIMENTAL DATA SHALLOW NOTCH LINEAR LOADING-91
7
XXV. EXPERIMENTAL DATA SHALLOW NOTCH 60,000 LB LOAD- 91
XXVI. EXPERIMENTAL DATA SHALLOW NOTCH 65,000 LB LOAD- 92
XXVII. EXPERIMENTAL DATA SHALLOW NOTCH 70,000 LB LOAD- 92
XXVIII. EXPERIMENTAL DATA DEEP NOTCH LINEAR LOADING---- 93
XXIX. EXPERIMENTAL DATA DEEP NOTCH 30,000 LB LOAD---- 93
8iJ........... .......... III ..II....... I............. -. '
LIST OF FIGURES
1. 2 GAGE SPECIMEN ------------------------------------- 37
2. 5 GAGE SPECIMEN ------------------------------------- 38
3. 1 GAGE SPECIMEN ------------------------------------- 39
4. 7075-T6 ALUMINUM STRESS-STRAIN CURVE ---------------- 40
5. CALCOMP AND VERSATEC PLOTTER AXES .-------------------41
6. NODAL LOADING DIAGRAM ------------------------------- 42
7. COURSE MESH FOR CIRCULAR HOLES ---------------------- 43
8. FINE MESH FOR CIRCULAR HOLES ------------------------ 44
9. COURSE MESH FOR SHALLOW NOTCH ----------------------- 45
10. FINE MESH FOR SHALLOW NOTCH ------------------------- 46
11. COURSE MESH FOR DEEP NOTCH -------------------------- 47
12. FINE MESH FOR DEEP NOTCH ---------------------------- 48
13. EXAMPLE OF COMPLETE PANEL MESHES -------------------- 49
14. COMPUTATIONAL FLOW CHART ---------------------------- 50
15. CIRCULAR HOLE X=0.2 LINEAR RESULTS ------------------- 51
16. CIRCULAR HOLE X=0.25 LINEAR RESULTS ----------------- 52
17. SHALLOW NOTCH LINEAR RESULTS ------------------------53
18. DEEP NOTCH LINEAR RESULTS ---------------------------- 54
19. SHALLOW NOTCH 60,000 LB LOAD ELASTIC-PLASTICRESULTS ---------------------------------------------55
20. SHALLOW NOTCH 65,000 LB LOAD ELASTIC-PLASTICRESULTS ---------------------------------------------56
21. SHALLOW NOTCH 70,000 LB LOAD ELASTIC-PLASTICRESULTS .---------------------------------------------57
22. SHALLOW NOTCH 60,000 LB LOAD PLASTIC ZONE ----------- 58
9
23. SHALLOW NOTCH 65,000 LB LOAD PLASTIC ZONE ------------ 59
24. SHALLOW NOTCH 70,000 LB LOAD PLASTIC ZONE ------------ 60
25. SHALLOW NOTCH RESIDUAL a0 FROM 60,000 LB LOAD -------- 61
26. SHALLOW NOTCH RESIDUAL ur FROM 60,000 LB LOAD -------- 62
27. SHALLOW NOTCH RESIDUAL oa FROM 65,000 LB LOAD -------- 63
28. SHALLOW NOTCH RESIDUAL a FROM 65,000 LB LOAD -------- 64r
29. SHALLOW NOTCH RESIDUAL ar FROM 70,000 LB LOAD -------- 65
30. SHALLOW NOTCH RESIDUAL a r FROM 70,000 LB LOAD ----- 66
31. DEEP NOTCH PLASTIC LOADING RESULTS ------------------- 67
32. DEEP NOTCH e RESIDUALS ------------------------------ 68
33. DEEP NOTCH ar RESIDUALS ------------------------------ 69
34. DEEP NOTCH 30,000 LB LOAD PLASTIC ZONE --------------- 70
35. DEEP NOTCH 35,000 LB LOAD PLASTIC ZONE --------------- 71
36. DEEP NOTCH 40,000 LB LOAD PLASTIC ZONE --------------- 72
37. RIGID-PERFECTLY-PLASTIC RESULTS ---------------------- 73
38. RIGID-PERFECTLY-PLASTIC INITIAL PLASTIC ZONE --------- 74
39. RIGID-PERFECTLY-PLASTIC INTERMEDIATE PLASTIC ZONE---- 7 5
40. RIGID-PERFECTLY-PLASTIC FINAL PLASTIC ZONE ----------- 76
10
SYMBOLS AND ABBREVIATIONS
ADINA Automatic Dynamic Incremental Nonlinear Analysis
b Half width of strip
CPU Central processor unit
E Young's Modulus of Elasticity
Et Strain hardening tangent modulus
FEA Finite element analysis
JCL Job control language
K Stress concentration factor referenced to reducedcross-section a/a
MVS Multiple virtual storage
n Ramberg-Osgood exponent
O(hm ) Order of the discretization error
r Radius of hole or notch
VM Virtual machine
Ramberg-Osgood coefficient
£ General representation for strain
XNon-dimensional size parameter X=r/b
v Poisson's Ratio of transverse strain
a General representation for stress
U n Principle stress in 6 direction (hoop stress)
rPrinciple stress in radial directionr
a Nominal stress in reduced cross-sectionn
aFar-field stress
a Yield stress by 0.2% offset methody
11
ACKNOWLEDGEMENT
I would like to thank all the people who assisted me
in my education at the Naval Postgraduate School.
In particular, I would like to thank Professor G. H.
Lindsey for his guidance in completing this work. I would
also like to thank Professor G. Cantin for introducing me
to the finite element method; Mr. Bob Besel and his staff
for their support; and the staff of the W. R. Church
Computer Center of the Naval Postgraduate School for their
assistance.
I thank my wife and family for their sacrifices while
I have been in pursuit of my career.
12
I. INTRODUCTION
Development of on-board fatigue monitoring systems
for Naval aircraft have made it possible to record exten-
sive structural loading data in flight. The strain gages
used in such a system must be located away from stress
concentration areas to prevent their fatigue; however,
these areas are of the greatest interest in analyzing
and predicting fatigue life of the structure. Understand-
ing the relationship between nominal, far-field stresses
and local stresses in critical areas thus becomes vitally
important. Recent experimental investigations into the
effect of uniform, far-field loads on stress concentration
areas have been made at the Naval Postgraduate School
(NPS) using photoelastic techniques (Refs. 1, 2 and 3].
These experiments involved loading 7075-T6 aluminum into
both the elastic and plastic regions, as well as measure-
ments of residual stresses resulting from plastic yielding.
Finite element analyses (FEA) of the aluminum panels
used in the experiments of Stenstrom [Ref. 1] were con-
ducted. The panels used in the experiments of Engle
[Ref. 2) and Stuart [Ref. 3] have similar geometry. The
finite element programs available at NPS were surveyed
and ADINA [Ref. 4] was chosen for its proven ability to
produce the nonlinear analyses required for plastic
13
yielding of aluminum. To provide increased accuracy,
each panel was modeled using two meshes. The results
obtained for the coarse and fine meshes were extrapolated
to a final result using the Richardson extrapolation
technique [Refs. 5 and 61.
Along with ADINA, a preprocessor program, PSAPI LRef. 7],
was used to verify mesh connectivity prior to analysis by
ADINA. PSAP1 provides a graphical output of the finite
element mesh and was coded for the CALCOMP plotter installed
at NPS prior to 1978. For this thesis PSAPI was adapted
for use with the VERSATEC plotter now installed at NPS.
The stress-strain material properties of the 7075-T6
aluminum actually used to make the panels had to be estab-
lished to provide an accurate material model for use with
ADINA. Material testing was conducted to establish the
Young's Modulus (E), Poisson's Ratio (M), yield stress (ay),y
strain hardening modulus (E ) and the Ramberg-Osgood co-tefficients and n.
Comparisons were made to other works, in addition to
the experiments conducted at NPS. The initial analysis
involved a comparison of FEA to the results of Howland
[Ref. 8], for a circular hole in a finite strip, to vali-
date the methods used. A comparison of FEA to plane
stress, slip-line theory, for rigid-perfectly-plastic
material was also included as a validation for the plastic
analyses.
14S
II. MATERIAL PROPERTY TESTING OF 7075-T6 ALUMINUM
The elastic and plastic material properties of the
aluminum panels were established by tensile tests of uni-
axial specimens made from the same mill run. The speci-
mens were manufactured and tested according to current
ASTM standards [Ref. 9]. MICRO-MEASUREMENTS, EA-13-125AD-
120, precision strain gages with a temperature compensated
bridge circuit were used on all specimens. Transverse
gage sensitivity errors were corrected according to the
manufacturer's recommendations [Ref. 10]. Critical cross-
section measurements were made with a micrometer.
A. TESTS FOR AXIAL LOADING
Initial tests of the two-gaged specimen, Fig. 1, in
the MTS testing machine indicated a significant bending
moment was being produced by the 30,000 lb GRIFF grips.
To investigate this problem further, tests were conducted
on both the MTS and RIEHLE test machines with a five-
gaged specimen shown in Fig. 2. The results of these
axial loading tests, shown in Table I, verified that the
GRIFF grips on the MTS test machine do not give axial
loading. An inspection of the gripped region on the speci-
men indicated that the jaws of the grip were not applying
a uniformly distributed force and thereby induced a bending
15
moment by off-axis loading as shown in Fig. 2. The grips
on the RIEHLE test machine gripped evenly and a uniform
strain distribution resulted as seen in Table I.
B. CHARACTERISTICS OF 7075-T6 ALUMINUM PANELS
The following characteristic properties were determined
from the four specimens tested.
1. Young's Modulus (E)
Tests were conducted using the specimen shown in
Fig. 3 on the MTS test machine with 10,000 lb INSTRON grips,
which gripped the specimen evenly. The results of testing
three specimens are shown in Tables II to IV. Linear
regression in the elastic range of all the test results6?
determined a Young's Modulus of 10.12 x 106 psi, with a
correlation coefficient of 0.9996.
2. Poisson's Ratio (v)
Tests were conducted using the specimen shown in
Fig. 1 on the RIEHLE test machine with 10,000 lb RIEHLE
grips. The results are tabulated in Table V. Linear
regression of these results in the elastic region deter-
mined Poisson's Ratio to be 0.3256 with a correlation
coefficient of 0.99996.
3. Yield Stress and Strain Hardening Modulus
These values, required for ADINA's bi-linear
material model, were determined graphically using the
data from Tables III and IV. Plastic region data in
16
Table II is not reliable because of excessive creep en-
countered during that test.
0.2% offset yield stress, a = 76,000 psiy
strain hardening modulus, Et = 566,000 psi
The graphical fit of these values to the test data can be
seen in Fig. 4.
4. Ramberg-Osgood Coefficients
The Ramberg-Osgood equation for elastic-plastic
stress-strain characterization is given by:
a + (())n
E E
where:
= strain
a = stress
E = Young's modulus.
The 3 and n coefficients were determined graphically from
the data of Table IV, by the method given by Rivello [Ref. 11].
The data in Table IV gave the following values which are
the best fit to the combined test data
S = 1.479 x 1043
n = 21.58
The graphical fit of these values, in Eq. (1), with the
test data is also shown in Fig. 4.
iIi 17
III. MODIFICATION TO GRAPHICAL PREPROCESSOR
The use of a graphical preprocessor program, such as
PSAPI, is vital in detecting mesh errors that may other-
wise go unnoticed. Establishing the proper node locations
and element geometry prior to analysis for a complex code
such as ADINA is of utmost importance.
A. PSAP1 MODIFICATIONS
The program PSAPI was originally coded in FORTRAN IV
for use on the NPS IBM 360/370 installation with the
CALCOMP Model 765 drum plotter. The CALCOMP system o,
installed at NPS used the +Y axis as the unlimited plotting
axis, see Fig. 5. The entire plotting logic in PSAPI uses
this orientation of axes to allow multiple plots in a con-
tinuous strip. With the VERSATEC Model 8222A electrostatic
plotter now installed at NPS, the +X axis becomes the un-
limited plotting axis, shown in Fig. 5. To avoid an ex-
tensive recoding of PSAP1 for use with the VERSATEC plotter,
a simple coordinate transformation of the plot was made in
a limited number of short subroutines. First, all installa-
tion dependent plotting calls used in PSAP1 were identified.
These involved seven plotter functions for which new sub-
routines were coded.
18
.....................
Function New Subroutine
Initialize Plotter CALCMP
Move Plotter Pen CALPLT
Letter on Plot NOTATE
Number on Plot CALNUM
Determine Current Pen Location CALWH
Draw a Line CALINE
Stop Plotter PSTOP
The subroutines listed above merely rotates the plot
to coincide with the VERSATEC axis orientation and retain
all features originally in PSAP1. Since all plotter
hardware code is now isolated in these seven subroutines,
future adaptations to other plotting systems is simplified.
To provide documentation of this update to PSAP1, a complete
listing of the new program is provided in Appendix D.
B. USE OF PSAPI
Previous use of PSAPI on the IBM 360 system necessitated
use of a load module since PSAPI took over one minute of
CPU time to compile. With the new IBM 3033 system
compilation requires eight seconds; however, use of a load
module or disk stored source code is still recommended
since PSAP1 contains roughly 2,500 lines of code. Appendix
A contains sample JCL to use PSAPI on the IBM 3033 MVS
system. With minimal effort PSAP1 could also be set up
19
for use on the IBM VM/370 system. The user's manual for
PSAPI is in Ref. 7. In addition to a mesh plot, PSAPI
provides a listing of the node coordinates, element
connectivity and several key input values used in execu-
tion of ADINA. This information provides a useful check
of the input data.
20
IV. FINITE ELEMENT ANALYSIS (FEA)
A. DESCRIPTION OF MESHES USED
1. Length to Width Ratios and Boundary Conditions
Initial finite element models of specimens had
length to width ratios near one, as used by Garske [Ref. 12]
for his FEA, but they did not provide the desired uniform
distribution at the loading boundary. Specimen length to
width ratios of 3-5 were used by Armen, Pifko and Levine
[Ref. 13] in their FEA and by Stenstrom (Ref. 1] in his
photoelastic experiments. The criteria established to
determine uniform boundary stress distribution was uni-
formity in nodal displacements along the loaded edge as
discussed by Segerlind [Ref. 14]. In the models used for
FEA in this thesis, nodal displacements were uniform to
within 0.1%, and the resulting stress distribution was
uniform axially to within 0.1% at the panel ends. In all
cases two-dimensional, eight noded, isoparametric elements
were used. These higher order elements cannot be loaded
in an "intuitive" manner as discussed by Zienkiewicz
[Ref. 15, p. 223]. Figure 6 shows the nodal loading
required to obtain a uniform surface load.
21
iJ
2. Element Meshes
Two meshes were developed for each panel analyzed.
A reasonable effort was made to keep element corner angles
as close to 900 as possible to reduce the effect of element
distortion discussed by Hopkins and Gifford [Ref. 161.
All meshes modeled a quarter of the actual panel by using
the two axes of symmetry as is common practice in FEA.
The step from course to fine element meshes was made so
that each element in the course mesh was subdivided into
four smaller elements of the same type. Such a mesh sub-
division can be expected to give monotonic convergence of
results, Cook [Ref. 173, and allow extrapolation to results
of an infinitely fine mesh. Figures 7 through 13 illus-
trate the element meshes used in this analysis as plotted
by PSAPl.
B. COMPUTATIONAL PROCEDURES
1. Using ADINA
Once the mesh has been developed, input data is
prepared in accordance with the ADINA user's manual [Ref. 4].
This same set of data is then used as input for PSAPI to
check for errors and provide a graphical display of the
element mesh. After preprocessing by PSAPI, the data is
entered into ADINA for analysis. In the case of linear
analysis, two types of stress output may be specified,
nodal point or Gauss integration point. Nodal point output
22
A1.
i' I II .... Illr " -
;l _u ii li~ I'
i " r " I
can be computed for up to eight node point stresses for
each element. Since 2x2 Gauss integration was used, four
Gauss point stresses were computed for each element. The
2x2 Gauss integration is recognized as the most efficient
integration order for this type of analysis [Ref. 15, p. 284].
The linear analysis used an isotropic linear elastic material
model (MODEL "1" in Ref. 4) which required input of E and
v material properties. The nonlinear analysis allows only
Gauss point stress outputs and uses a bilinear elastic-
plastic material model, with von Mises yield condition and
isotropic strain hardening (MODEL "8" in Ref. 4).
For static analyses ADINA uses a time function
method to apply loads in steps. Linear analysis loading
was accomplished in a single step to a nominal value of
3,000 lbs load. Nonlinear analysis loads were applied in
ten steps to a maximum value, matching the experimental
loads, and then unloaded to zero in ten steps to obtain
residual stresses. The stress output from ADINA is a
listing of nodal or Gauss point stresses for each element.
Since the only area of interest in this analysis was the
distribution of stresses along the reduced cross-section,
no large post-processing program was developed or used.
All final computations using ADINA output data were accom-
plished on a HEWLETT-PACKARD 9830A calculator, using short
programs coded in BASIC. If more extensive stress
23
distribution information were desired, some form of auto-
mated post-processing would be necessary to reduce the
computational workload. At a minimum, nodal stress outputs
by ADINA must be averaged to obtain unique values of stress
at nodes shared by more than one element.
2. Richardson Extrapolation
The use of course and fine meshes allows extrapola-
tion to an infinitely fine mesh as discussed earlier.
Richardson extrapolation [Ref. 5] was used in this analysis
where:
0c (hF)m - GF(hc)mjextrap h m -hm (2)
where
Gextrap extrapolated solution
a solution obtained with h=hc c
F - solution obtained with h=hF
h - linear dimension of course elementc
hF linear dimension of fine element
m = 2 (for this analysis)
The exponent m is determined by the order of the discre-
tization error O(h m). Since h represents the length of an
2element the element area is represented h2 . In a two
mdimensional problem such as this O(h ) is of the order ofh2
, the area of an element. In the mesh refinement scheme
24
S2 h _ 1 Equation (2) can be rewrittene c h 2c
(hF 2 h c 2ch)- IF (R- )
c c (3)aextrap hF 2 h 2
(s-) -
c c
thus 1_ F -4 Oc
extrap 3 (4)4
Equation (4) then becomes the relation to obtain extrapolated
stresses from coarse and fine mesh results in a two dimen-
sional analysis. Better extrapolations can be obtained by
using three or more refined meshes, buti the computational
effort increases significantly.
3. Optimal Stress Locations and Local Smoothing
It is generally accepted that the most accurate
sampling points for stresses are the Gauss integration
points within the element [Ref. 15, p. 281, and Ref 18].
In this analysis, the nodal points are of the greatest
interest; thus a technique of local smoothing must be
applied to the integration point stresses to obtain nodal
stresses as reported by Hinton and Campbell [Ref. 19].
The formulation of this local smoothing technique for ADINA
elements is developed in Appendix B. The nodal values
obtained must then be averaged if shared by two or more
elements.
25
ai
4. Computational Times
Because of the extremely large size of ADINA (about
17,000 lines of code in the NPS version) and the out of core
solver, it does not adapt well to time sharing systems.
Using the IBM 360 system at NPS, ADINA required 31 user
defined overlays to create a manageable load module in
about 30 minutes of CPU time. With the new IBM 3033 MVS
system at NPS, ADINA is compiled without overlays in about
one minute. When using a load module, the program execu-
tion took less than 2 minutes CPU time on the IBM 3033.
In addition to ADINA, the preprocessor (PSAPl)
and post-processing techniques involve considerable time
and effort. Figure 14 is a flow chart of the computational
procedure used in this analysis. An example of the JCL
to use ADINA in load module form on the IBM 3033 MVS
system with use of the mass storage facilities is shown
in Appendix C.
2
26
A
V. RESULTS OF ANALYSIS
A. CIRCULAR HOLES IN LINEAR MATERIAL
The FEA results for a circular hole in a finite width
strip were used to validate the elastic computational pro-
cedure discussed earlier. The results of Howland [Ref. 8]
were compared to both the Gauss point smoothed results and
the nodal output results in Fig. 15. The stress concentra-
tion factor a/cu is referenced to the far-field stress.
The smoothed results give the best match to the results
of Howland at the edge of the hole, and the only signifi-
cant variation between the two FEA methods occurs within
the first 0.25 inches from the edge. In order to obtain
the 0.25 inch stress value for the coarse mesh, in the
Gauss point smoothed result, a midside node value had to
be obtained by the averaging method discussed in Appendix
B. The linear distribution of smoothed stresses along the
sides of the element, [Ref. 19], appears to produce a
less accurate result in this area of extreme stress
gradient, when compared to ADINA's nodal output result.
This tendency was noted in all cases; however, the peak
stress values from smoothed results consistently gave
better correlation with other investigators [Ref. 20].
A circular hole with X=0.25 was also analyzed and
compared to the experimental results of Stenstrom [Ref. 1]
27
along with an interpolation of Howland's results. The a
experimental data correlates well with the FEA results;
however, the a r experimental data shows significant varia-
tion between 0.125 and 0.375 inches from the edge of the
hole, as seen in Fig. 16.
B. OPPOSITE U NOTCHES IN LINEAR MATERIAL
The results for linear analysis are presented in non-
dimensional stress concentration form; however, the
normalizing stress changes. For U notches KT is the
stress concentration factor referenced to the theoretical,
nominal stress (an) in the reduced cross-section where
n = Load/Area of Reduced Cross-Section.n
1. Shallow Notch Panel
The FEA results plotted with the experimental
data of Stenstrom are shown in Fig. 17. Once again both
FEA results are shown and the variation for the two methods
occur within 0.25 inches from the notch edge; however,
there was less variation than was seen in the circular
hole analyses.
For this panel the experimental data appear to be
uniformly below the FEA results for a0 . The ar data shows
significant variation at the 0.625 inch point but follows
the proper trend within 0.5 inches from the notch edge.
According to data collected by Peterson (Ref. 20], this
notch geometry should yield a maximum KT = 2.74. The
28
Gauss point smoothed results matched this value exactly.
Stuart [Ref. 31 reported a KT = 2.69 for this same notch
geometry, with a standard deviation of 0.187 in the 14
samples he measured by photoelastic methods. Early photo-
elastic work by Frocht [Ref. 21] determined KT = 2.7 for
this notch geometry. The FEA results appear to be in good
agreement with other investigators, for this notch geometry.
2. Deep Notch Panel
Results of the FEA for a deep notch were plotted
with Stenstrom's experimental data in Fig. 18. The results
of the two FEA methods again diverge within 0.5 inches from
the notch edge. FEA stress values at the 0.25 inch point
have spread farther apart in this case since the stress
gradient is very severe at that point. The experimental
data correlates well for both a and ar; however, the maxi-
mum experimental a9 is considerably lower with a KT = 3.83.
Results reported by Stuart for this notch geometry was a
K = 4.05 with a standard deviation of 0.219 for 14 speci-T
mens measured by photoelastic methods. Frocht [Ref. 22]
reported a photoelastic KT = 3.9 for this notch geometry,
but concluded that the result was 5-10% low, giving a
corrected range of KT from 4.1 to 4.3. The Gauss smoothed
FEA result gave a KT = 4.24 which compares well to an
empirical relation given by Peterson [Ref. 20] for
r/d < 0.25.
29
_ ___I___ I1
T(1Lt) (0.78 + 2.243V.t/r) [0.993 + 0.18 Lt 1.06(~ + 1.71 (D ]5
where
t = notch depth (3.9375)
r = notch radius (0.625)
d = minimum width (15.625)
D = maximum width (23.5)
Inserting the values above into Eq. (5) gives a KT = 4.26,
which is 1/2% above the FEA result. Other FEA results
reported by Armen, Pifko and Levin [Ref. 13] and Griffis
[Ref. 23], using linear strain triangle (LST) elements,
produced KT values within 5% of those produced by use of
Eq. (5). The rectangular elements used in this analysis
are known to give better results than LST elements as
noted by Clough [Ref. 24]. It is clear that the FEA
results obtained for this notch are in good agreement
with other works.
C. OPPOSITE U NOTCHES IN NONLINEAR MATERIAL
The analysis for loading into the plastic region of the
7075-T6 aluminum was made using the bilinear material
model discussed earlier. The loads used were selected to
match those used in the experiments of Stenstrom; thus
allowing direct comparison. The strains obtained in those
experiments were used to solve for stresses by use of the
Prandtl-Reuss plastic flow equations.
30
1. Shallow Notch Panel
The results of FEA for the three load cases, 60,000,
65,000 and 70,000 lbs are presented along with the experi-
mental results in Figs. 19 through 21. The aa results
compare well although no trend for peak ae stress away from
the notch edge is shown in the experimental data. In all
cases the FEA determined the peak a stress to occur near
the yield boundary, and the gradient of the a stress to
fall off dramatically in the plastic zone. This character-
istic behavior of the a 6 stress was reported by other
investigators [Refs. 13 and 23] using FEA on 2024-T3 aluminum.
Plane elastic-plastic stress distributions reported by
Frocht [Ref. 25] show similar trends. The experimental
data also shows a marked change in the gradient of a0 stress
within the plastic region. The growth of this plastic
region is approximated using the FEA results for this notch
in Figs. 22 through 24.
Experimental data for the a r stress distribution
matches the FEA results closely except at the notch edge
where the measured a does not go to zero as it should.r
The characteristic peak value of a r near the plastic
boundary as seen in the FEA results is also shown by the
test data.
The FEA residual stress computed upon unloading
from the three load cases are shown in Figs. 25 through 30.
The characteristic distributions of the a residual stress
31
agrees with those reported by others [Refs. 25, 26 and 27].
The experimental residual stress distributions reported by
Stenstrom show similar trends but significant variations
when compared to the FEA results.
2. Deep Notch Panel
Three load cases were computed to plastic loading
levels; however, only limited experimental results are
available for this notch as seen in Fig. 31. The 30,000 lb
load is just at the onset of yield in the notch root area.
Limited residual experimental data [Ref. 3] was available
for comparison in Figs. 32 and 33 which are plots of the
residual ae and a stress distributions as a result of the
three loading cases. The 30,000 lb load has caused yielding
in a small region at the root of the notch as seen in Fig. 34.
Figures 35 and 36 illustrate that the plastic zone does not
grow to the extent it did in the shallow notch. The stress
gradients are very severe in the deep notch and as a result
the 0.25 inch sampling points used in the FEA may be useful
in only showing gross trends close to the notch edge. The
trends appear to be much the same as in the shallow notch,
with the peak a and ar stresses occurring near the yield
boundary. The yield boundaries shown in Fig. 31 are
approximations based on qualitative analysis of the finite
element results. The a experimental data for the 30,000
lb load case correlates well with the peak stress, again
32
A . . . .. . ... ..IU F n '
appearing low as it did in the linear analysis. The residual
a stress distribution in Fig. 32 follow much the same trends
as seen in the shallow notch but with much higher gradients
within the first 0.5 inches from the notch. The FEA results
for the residual ar stresses shown in Fig. 33 indicate a
limitation in the element size used since the a r value of
the notch edge does not return to zero as it should. Because
of this problem the data may be questionable for showing
proper trends in the first 0.5 inch from the notch edge.
3. Rigid-Perfectly Plastic Panel
A stress distribution for the theoretical material
used in slip-line theory was desired. By using ADINA's
bilinear material model such a material could be approxi-
mated reasonably weli. For this analysis a Young's Modulus
(E) of 10 26 psi was used to model perfect rigidity. The
strain hardening modulus (Et ) was set to zero to model
perfect plasticity. Poisson's ratio (%;) was 0.4999999,
as close to 0.5 as the computer would allow. The E = 1026
also represents a computer limit in approaching an infinitely
large E. Figure 37 illustrates the results obtained and
compares them to a slip-line solution. The results are
normalized to the arbitrary 73,000 psi yield stress used
in the analysis. The a0 values obtained agree exactly
with slip-line theory. Conversely the a r results do not
reflect the same values as slip-line theory, but do show
33
a similar trend. The growth of the plastic zone obtained
is shown in Figs. 38 through 40.
34
i'I
VI. CONCLUSIONS AND RECOMMENDATIONS
The results obtained from the FEA have proven useful
in determining the validity of experimental data gathered
by photoelastic techniques. The FEA results varied by
less than 1% when compared to published analytical results
and handbook values. Experimental data from Stenstrom's
photoelastic work correlated well with the FEA results.
The primary exception was the residual stress experimental
values, which varied significantly from the FEA results.
The variation may be in transforming the residual strains
measured photoelastically into stresses for comparison
with the FEA results, since ADINA only provides a stress
output. Limitations of ADINA's bilinear material model,
initially considered severe, do not appear to have hampered
this investigation. A possible exception is the residual
analyses where the transition region from elastic to
plastic strains becomes especially important. The Gauss
point smoothed results gave the best correlations at the
edge singularities in all cases; however, due to the
limitations noted at 0.25 inches from the edge, the
results at that specific point may not be as accurate for
this method. The nodal output results gave consistently
higher stress values at the edge singularities. Because
35
hri
of the severe stress gradients near the deep notch analyzed
the use of a finer element mesh near the notch would probably
produce better results.
The effort involved in developing two meshes such as
4those used in this thesis is considerable. An automatic
mesh generation capability would reduce the workload and
allow experimentation with several types of element meshes.
ADINA proved to be a useful and powerful program, as
expected, but something simpler and less awkward to use may
be all that is required for two dimensional analysis. Such
a system is already in use at NPS but does not offer non-
linear capabilities. If use of ADINA is to be continued in
this type of investigation, a post-processing program should
be adapted. There are programs available to post-process
ADINA data at NPS [Refs. 28 and 29] but they would require
modifications to work with two dimensional analyses and
the VERSATEC plotter.
Standardized material property testing would ease the
inevitable task of obtaining basic material properties for
use in analysis or experiments. Some form of automatic
data collection with use of the MTS testing machine would
allow testing of a larger sample population and provide
statistically more accurate information.
36
FIGURE 1
2 GAGE SPECIMEN
AXIAL STRAINGAGE
TRANSVERSESTRAIN GAGE
11.72"
thickness: 0.0800"
j 1.
37
STRAIN GAGESj
REGION OF GRIP3MARKS BY 30,000 LB. .LJGRIFF GRIPS
thiCkness: 0.080010
1.5"
FIGURE 2
5 GAE SPECIMEN
38
-- 0.5"
AXIAL STRAIN
4.0" GAGE
8.25"
0.75"i R
Thickness :0.0795"
FIGURE 3
1 GAGE SPECIMEN
39
,i77
F I URE H
707S-TE RLUII!NUM STRE5S-5TRRIN CURVE
IB E
Et=E566,000psi!t
02a76,000 psi
y/79
,___ ______
E- 10.12 x10 6 psiuL = 1.479 x 10
wJ n=z 21.58cr.I-Ln
13- SPEC I MEN 3
2 - SPE IMEN C
SOL D L INE - RRSERS-0550D____ _ ________CURVE
III
0.2 % offset
2 4 a 12 I 2
STRR IN, , I03 in/in
40
;-j
Q- Paper Width GIN
0
0,0
CALCOMP PLOTTER+
0'0 - Paper Width +y
CL
VERSATEC PLOTTER
FIGURE S
CALCOMP AND VERSATEC PLOTTER AXES
41
P~uniform load over4element surface
L
2P 2 p3 3
3
L
FIGU E.6
NODAL LOADING DIAGPAM
42
28 ELEMENTS (ISOPARAMETRIC)
111 NOE192 DEGREES OF FREEDOM
DIMENSIONS
W L RADIUS
0.2 5 25 1
0.25 4.0625 20 1
RADIUSw
SYMMETRYAX3S
FIGURE 7
COURSE MESH FOR CIRCULAR HOLES
43
A SUBDIVISION OF COURSE MESH
112 ELEMENTS (ISOPARAMETRIC) I
389 NODES i
720 DEGREES OF FREEDOM
*I
.2
j SYMMETRYAXIS
FIGURE 8
FINE MESH 'OR CIRCULAR HOLES
44
iS -... . i -. i
FIGURE 9 COURSE MESH FOR SHALLOW NOTCH_ ~L. 7s,-5-
i I
60 ELEMENTS
(ISOPARAMETRIC)
219 NODES
404 DEGREES OF
FREEDOM
45"0
1. 8125" Radius
18125"
AXES8.875" 2.875"
45
FIGURE 10 FINE MESH FOR SHALLOW. NOTCH
iliKSLBDIVISION OFCOURSE MESH,
240 ELE NTS
(ISOPARAMETRIC)
797 NODES -1-I---
1528 DEGREES OFFREEDOM
SYMMETRYAXES
46
11.75"
. .... ........
FIGURE 11
I COURSE MESH FORDEEP NOTCH
60 ELEMENTS(ISOPARAMETRIC)
219 NODES
45"404 DEGREES OF FREEDOM
4 I
RADIUS OF NOT, 0.625"
.625"
7.8125" 3.9375"
47
I
bib-
I I INE ESHFIGURE 12
_____FINE___MESH FOR DEEP NOTCH
[SUBDIVISION OF CURSE MESH
________________240 ELEMENTS (ISOPARAMETRIC)
I 797 NODES
i 1528 DEGREES OF FREEDOM
SYMMETRY AXES
48
1528~~ DEGEESOFFREDO
I I -
FIGURE 1.3
EXAM~PLE OF COMPLETE PANEL MESHES
£ 49
FIGURE 14 COMPUTATIONAL FLOW CHART
DEFINE MODEL GE M ETRY
Course < E 6SH FineESH?
DEFINE 7DEFINENODE NODEPOINTS OR POINTS
OUTPUT MESHLIST PSA PLOT
No PROPER NoMESH
Yes
ADINA7Non-linear ANALYSIS Linear
TYPEGauss LIT
OR TION
NodaL 2-Li
GAUSS PT. STRESSES NODAL STRESSES
LOCALSMOOTH IN G
COURSE MESH Course ODAL Fine FINE MESHSTRESSES VERAGIN STRESSES
FINAL STRESS
RICHARDSON EXTRAPOLATION VALUES
50
F I SURE IT
CIRCULAR HOLE X=0.2 LINEAR RESULTS
3.5 2 - SOnED FRDHGRUSS PO3INTS
3.0 _IRULAR SOLID LINE 15 HRLND.3S1LUT I ON
le
C 3I-'
LaJ
'I .
.0 0. 1.0 1.5 2. 2.N
DI5T.NCE FROM HOLE i nches
51
-"'-- 2.4,
FIGURE IE
CIRCULRR H13LE X=0.2S LINERR RESULTS
0-NODAL MITPUT
3.5 2 N NTEFROM5HM5 PEI INTS
R + H13WLFIND SOLUITIOCN
3.9+ -STENSTRUM DRTR
b8 2
2.0- +
+Lai +
v + +
I
1. r + 2 U U _
1+
DISTRNCE FROM1 HOLE, inches
521
-lz
1
FISURE 17
SHRLLDW NOTCH LINERR RESULTS
4. 0
3 -NDRL OUTPUT
3.5 ~~FM THTO E
Gi llU55POINTS+ -
3.0
b2_ +b
0
I +
Ln Z
t.,J3 1.1i3
-= .I 4
____ + +1-Ui ' I g i ¢
0,.+
110r +cr ,A
0.0 0.5 1.3 IS~ 2.3 .
DISTRNCE FROM NOTCH, inches
53
F ISURE 13
DEEP NOTCH LINERR RESULTS
4.SI
2
+ 13 - NDL MUTPUTI ~ ~+ FER -,=,= a2- SMOTHED FRON
_ _ _ _ RUS' POINTS
"l + - STENTRM DRTR
€3.0
+
" LnI-
w" 2Z,., I.E [
C: + 21.0 2
in 2
il~ ___+ 2__ 222
0..
{ I
3.1 1~ l.1fl 2.0 .
DISTRNCE FROM NOTCH, inches
54
, q- ,-
FIGURE 19
SHALLOJ N13TCH 699931. LGRI) ELR5T I C-PLR5T I C RESULTS
so i I'~- MTER IRL M13DEL
9+ +1I
79 __ _ _ _ +
Oe +
b 0 3
LiJ
CI - FER RESULTS
3 + - STENSTROM DRTR
+~ + 0 03 C30Is + _ _3_
+ +
DI5TRNCE FROM NOTCH, inches
55
FIGURE 20
SIrRLL I N1TCI I 2 I1 LOC ELJI5TI ¢--PUILSTI C IESULTS
11I IE I FI MEI3
+ 1: 13 YE I LD 13MI]LNDORi
+0+
+ 13
so o +4-3
+
63 03
310
29'r + + +r1 13
0. 1. 1T2. 2.
Lii i
D+STRN-E FR1M NOTCH, inches
56
33 _ _A
F I URE 2I ISHALLOW NOTCH 72MU LB LORD ELAIST I C-PLFIST I C RESULTS
D 0 ZIATER IAL MODEL+ + + YE ILD BOUNDAIRY
93 +
0* 1+
700
51 0
bu 13 - FER RESULTS
+ - STEWSROM DRTh
20 + 31
3. BS 1.1 IS.E2.A
D ISTRINE FROM NOTCH, inches
57
_ _ _ _ _ _ ___ _ P lastic~Zone
FIGURE 22
SHALLOW NOTCH 60,000 LB LOAD PLASTIC ZONE
58
i
,,.11._.Plast ic
Zone
FIGURE 23
SHALLOW" NOTCH 65,000 LB LOAD PLASTIC ZONE
59
r..- Plastic
!.-.--_-Zone
FIGURE 24
SHALLOW NOTCH 70,000 LB LOAD PLASTIC ZONE
60
FIGURE 2S
SHRLLO3N NO3TCH REIDURL a. FROM MaM LN LORD
2+
w I + +
an +
LJ2-31
-48
- STEN5TRUN DRTii
0.E IS 1. I 2.a3.
DISTRNCE FROM NOTCH, inches
61.
F:SURE 2S
9tPLQI NOTCH RESIDURL (rFROM Mom11 L13 LORD
31
-E +
+
-4TRC FROM NOCRcesL
1 62
+4
FI5URE 27
qmLam NOTCI RE, I O. 0 FRUN M U LU35
-a
LuL0 1 0
13 D D
w +
Ln +
-j +
ul +W 3- FR RESULTS
Cx +
-41 + - STENSTRIN DRTR
-6 .. 1 u.5 1.0 1.5 . .
DISTRNCE FROM NOTCH, inches
63
En
FIGIURE 28
SHRLLO NOTCH RESIDURL (Yr FROMI E1 LB LOD
3 0
K~ +
LflL -1
Ln +
- -
_ _ ++ _ _
1 +-"3
D - FER RESULTS
+ - STENSTRU DRTR
-7 09.9 3.5 1.0 I
DISTRN'CE FROM NOTCH, inches
64
-l
11
13 E3__ C3
3 13
20
- 0
-3+
U'i +
3 - FMREMT
MLII+ + -5TENTAONlDmT
0.2 IS Is~ 2.13 2.
DISTRNCE FROM NOTCH, inches
65
FIGURE 30
HIRLLfW NnTCH RESIDLURL 0r FR1I3 701000 LB L3RD
5r
3 13
- I
Lni+
Lf +a IL.,J -I +Ln +
-3_ Li _
-4 [3 -FER RE ULT5
+ - STENSTR13M DRTR
-7
-7 11I I I
0.1 I.9 1.3 I.5 2.fl
DISTRNCE FROM NOTCH, inches
66
F I SURE 31
DEEP NOTCH PLR5TIC LORDINS RESULTS
MMRTE I RL HELE YEILD SOMNDRRIES
so 123,IUS~lq FER 0-lBus iq
5u ~~+ 1..I,3LLR
- -
+ - STEW6RON ORTH
"/H[ r -31b 0 LZ 3
+ 03
tn" +n 2
+02
Lrl [] QL13 2 U
+ 03 2
+ 4l+
0.0 21 IS [. --
DISTRNCE ROM NOTCH, inches
67
O'r +WT -='
F1IRE 32
DEEP NOTCH 0e RE5IDURLS
200
.12-I
= FER LMRS
W -30U 3 - 32420 LS
+ I2 -M "Jul Los___~ -_ 41 000eD LS
-40,0001lb EquivalentResult from Stuart
*1DISTRNCE FROM NOTCH, inches
68
EC
FIGLURE 33
DEEP NOTCH 0-r RESIDURLS
3I i
U,0I I
w a 2 a- a
II
0:
Ll [
-FE L13BiS
*R U * mmLU,9 Las
Q - ,4& 00 LO
-3
"LI I i
LUn *. I .E . .
DISTRNCE FR13M NOTCH, inches
69
I i
I
Plastic Zone
FIGURE 34
DEEP NOTCH 30,000 LB LOAD PLASTIC ZONE
70
FIGURE 35
DEEP NOTCH 35,000 LB LOAD PLASTIC ZONE
71
71/
I
------ ----- Plastic Zone
FIGURE 36
DEEP NOTCH 40,000 LB LOAD PLASTIC ZONE
72
C3 a
in -
14 4
aa a a i
CL.
-a
a. T
a -
73
SPlastic Zone
FIGURE 38
RIGID-PERFECTLY-PLASTIC INITIAL PLASTIC ZONE
74
Loom,,
- "-"Plastic Zone
FIGURE 39
RIGID-PERFECTLY-PLASTIC INTERMEDIATE PLASTIC ZONE
75
r~
-- I Plastic Zone
FIGURE 40
RIGID-PERFECTLY-PLASTIC FINAL PLASTIC ZONE
76
TABLE' I. YITS AND REIHLE 5 GAGE TEST RESULTS
All Strains are 10-6 in/in
MTS Test Machine
LOAD STRAINSibs 2 C3 E4 C5
1000 -3 421 814 1235 1638
2000 338 1003 1619 2279 2903
3000 960 1714 2431 3186 3908
REIHLE Test Machine
LOAD STRAINS
lbs £1 £2 £3 £4 £5
1000 800 763 755 741 726
2000 1564 1540 1546 1543 1535
3000 2370 2334 2363 2377 2388
77I
" /7A _______-_
I4TAIBLE II. MTS SPECIMEN A TEST RESULTS
Cross-section 0.03975 in 2
Load, lbs. Strain, F,1-6 in/in
5 0.31 1 1.11l
j C~
1255 30I.'42
150 6I5
tin, 4-2 5
'9w92
225 2
2508
78~
905
'98 Cl 4., 5
TABLE III. MTS SPECIMEN B TEST RESULTS
Cross-section = 0.03975 in2
Load, lbs. Strain,e , 10-6 in/in5 I: :
1 75 0 0"":2
2 C, CI n C ?
1UU- -
ILLC'
l cn
1 C'C I i-'
I 5l C.T I 15Z- ' D
..' . I.J" :4 :-7
2. 0 - -"
• -,U..,L110 155
i 7- t ' = 12 ::312,5
"79
TABLE IV. MTS SPECIMEN C TEST RESULTS
Cross-section =0.03975 in 2
Load, lbs. Strain,E-,10Q in/in
V i=
-7DI
I2 CJ Cl
71~C~
:310 Ii so~
TABLE V. REIHLE SPECIM4EN TEST RESULTS
Cross-section =0.12 in 2
LOAD STRAN,Ej STRAIN,C 2
lbs. l0 6 jin/in 10in/in
5 &0
I~ .
41
0I I
L1 4
Clc.
31-
TABLE VI. X - 0.2 HOWLANL DATA
DISTANCE
FROM HOLE / ain.
U.L1 14
0.5 1
1.
5I . 7,
TL4. 5 tL 0DA
DISTANC8FROM HOLE a / o
in.
0. 42 .5
1 , 41 .[
2.0 47 I Li 7
. .... 0. 35 ;
82
TABLE VIII. X = 0.2 FEA RESULTS - NODAL OUTPUT
a = 5000 psiDISTANCE
FROM HOLE, in. a8 , psi r
0j Zl 5 -7-,
C1.75 ~ :7
5 0 1 -. 4
1 5 ....
TABLE IX. =0.2 FEA RESULTS -GAUSS OUTPUT
-'a - 5000 psi
DISTANCEce#s sFRM HOLE, in. a: ,-i. ar
0. -0. 1559. 4
c.25 942. Wi.
0.5 9~:.5 -5 -7 .
2. ;' A ="- c:,- 4:
T . =0. ARESL - U I
1.50 5000:~ 1s
FRO 0 5 49 9. 4 ,
83
77 A
TABLE X. X = 0.25 FEA RESULTS - NODAL OUTPUT
DISTANCE a = 4500 psi
FROM HOLE, in. ae #psi or ,psi
cl .UC1 14745. 54, 4 ., C
O. :'5 '9 4. '-4" :-4!
6 4 . 7 5i:.0.774
1.25 5471 . 9.2
1.... 5269 *. i.0
1.75 5 091.7 4S9, 1
2.00 4940.2 5.
TABLE XI. X 0.25 FEA RESULTS - GAUSS OUTPUT
am = 4500 psiDISTANCE
FROM HOLE, in. ae ,psi ar ,psi
. .'_ cl c 14 4 4 2 2 -':. .2
0 50 1 17 1..
0 7' 5 621'- ;L .4-l. .4t ." 5 i :': ::1.. ' 1 :
1.0l0 57179. e 5.
t 5 5.45::; ;:1.C
1 50 5 G. 1-77,'
1.75 50,:5. 49 6~~. ~ 004:3:.?5"- l ll 4:' 7 9 7:4:'. :
94
• L ,,.,
TABLE XII. SHALLOW NOTCH FEA LINEAR RESULTS - NODAL
n = 4225 psin
DISTANCEFROM NOTCH, in. a9 ,psi ar ,psi
lI. U.' 1 ::4o -4.
1. 75c ~ 5>7 65 'J. -
1..25 5134.1 137',4 9
1. 50 4:3,:3 2.4 : .
1.•75 4564.7
2.00 4.3 e, L4.4
TABLE XIII. SHALLOW NOTCH FEA LINEAR RESULTS - GAUSS
a = 4225 psi
DISTANCE
FROM NOTCH, in. a9 ,psi ar ,psi
ci 0 l.
0.50-' ' ,"1'4>|- ' ,~~71 5 Cl ::: .5. 5 I:;04 :
0.75 C209
1. 5 5tE1 22.4
.2.5 5 1 5'9':
1. 50 4':3;4 I 4.2
1 .75 4 55Z. 151. 7
0 ,14 .0
85
TABLE XIV. DEEP NOTCH FEA LINEAR RESULTS - NODAL
n = 4800 psi
DISTANCE
FROM NOTCH, in. ae ,psi ar ,psi__._________4 __._: _ r5 .
0.25 1 11'.? - 4, Z.
0. 50 :3:",4:-: 6 '? i: '- .-
Cl -5 6,
1.005Ut.. , .6. -.
1 25 .:2 .:0 4
1 . 50 51 .
1.75 4:42,4 .
TABLE XV. DEEP NOTCH FEA LINEAR RESULTS - GAUSS
n = 4800 psi
DISTANCE
FROM NOTCH, in. % psi a r psi
cl 4
0 0. ':3 ", 1 ,
1 .7, 6 :56. 2 1 1.0rt0 60n:-:.e. 7. ::I.;"J:
1 .5 54'6. .
1.75 4:3 . I . 4 -1..
2. ,0 4--2. 0
86
TABLE XVI. SHALLOW NOTCH FEA NONLINEAR 60,000 lb LOAD ,
DISTANCE NO LOAD RESIDUALS
FROM NOTCH,in. a 0 psi a r ,psi C0 ,psi ar psi
1.cc 3-:I4.7 lI'9 4. 5 -3 11. 4 4:39. 123.5 .8:1:33S:. 7 95,2"3- 5 - :
0 4:31.4 CS.99 t -7.
1.25 ='-' ... .:4 5
1. 50 49111. 1 17477.1 :: 7 115 .
15 E' OR 0.7 E.567 I. 4. j C4 G.7
00 4 397:-,:: 1567 471.-15 6'' U
TABLE XVII. SHALLOW NOTCH FEA NONLINEAR 65,000 lb LOAD
NO LOAD RESIDUALSDISTANCE
FROM NOTCH,in. ae ,psi ar ,psi ae ,psi ar ,psi
E. E90.::: :I 8 4 1D ;3 04. 6 -4 291 224.
0'. 25 3-:2'?:':9. :' 10r 229. 5 - 9.96.. -2,- -12
l. 50 8:3_ 642.7 160 54. 744.
01,, 74E22.0- 145 7 :2 •4.
1.0 '477 -. 4 2"1171 •' I2' 5 57 ! 5 4 ' 1- . . .
1. 50 5 '7 6 7 15:1:,7. ci D.' 5-I
1.75 5:32. 1:7.:- 4-. 2 17 10.
2. 00 4 -( 94::. 2 17421.4 144:': . 0
37
TABLE XVIII. SHALLOW NOTCH FEA NONLINEAR 70,0001b LOAD
NO LOAD RESIDUALSDISTANCEF OM NOTCH,in. ae ,psi 0 ,psi aq ,psi a ,psiSr r
. 1957.9 104. .
CI. C1 c:4007. 155. 14.4 - . -- ' 3
U, 7 : 017 . 4 22 2 6- 1. 2_2. . 7 .,
S1. __ , 13 .0 2: -765.2 5 1 ::: :
c 2 - 15. 731
1.50 5 :94 1. 21 , 1,5 9 -7 :- 5 6 .-
2. 00I@ 52177.0 1'4: , 144::: ' .'207.0
TABLE XIX. DEEP NOTCH FEA NONLINEAR 30,000 lb LOAD
NO LOAD RESIDUALSDISTANCE
FROM NOTCH, in. a, ,psi a r ,p si 0 e ,psi ar ,psi
0.0 cl~ 233 .4 - . -7'5.20.2 -, 2091 :3Fi. 14 02' . 5
4:3:30':I.': " 1 E~ 1 7 3370 I'D 4 :-::-'0 9 . 1 1 -: 4 0 5 2'' 4 G-:2 .
l.7 :44i2. "1 8 I 19 .; 4 -;, GK'.. ;= ''" ,4 6 7 1i -'"-
9543.7 145 7 L4- '. 11 :.',
1.5 275.0 145714.7 ~ . 15T. :7: . , . '. f' 13 - . .
1.75 2717 .3 12 195.0 5 12 6, . 32.0¢ 2:3-1 4 1 E.5.7 74 1 i .9
88
r
TABLE XX. DEEP NOTCH FEA NONLINEAR 35,000 lbLOAD
1 Ii
NO LOAD RESIDUALSDISTANCE
FROM NOTC,in. ae ,psi ,psi a ,psi a ,psi
30929.5 ~~ 59*4 :530.3 .
C. .C .4 " 61". 5
." _05657.-,,315. -63.4 I -.2i' 0
I 1 4100.6 225. 81 4 5, 195.1 7
10 456, ? 1--:-5., - 4102.5::
-1 -19 9, 1 5 1
_ 1 1582-2 1 40.. ,"
-3 14. 4 1 1:2i' D 12.0. 4"0-
_1-.:2 .4 , .
TABLE XXI. DEEP NOTCH FEA NONLINEAR 40,000 lb LOAD
NO LOAD RE.SIDUALSDISTANCE
FROM NOTCH,in. ae #psi a r ,psi a ,psi a r ,psi
0-: 4-- C1 C' ' 4::7 , 11,4 -. 2 1 4:_.1
I,~~~~~ 1:-14100 " ''. 4.
4 , 9 : 4 . -- " 2 ., -4 : :- : ,z 4 3 I .- .F
7. 2 ,164 .. 0 6 8 7' ..5 .a.:: a.2•"".
1= 0..F -46g ? 1 . ! .:2::7 ''4 :-: :'"..71 .
75 _1 12 - 16 6. 21 _ 3" *35.: 0: .. 56, _ 4.
89
TABLE XXII. RIGID - PERFECTLY - PLASTIC RESULTS
a = 73,000 psiy
DISTANCE
FROM NOTCH,in. 08 ,psi a ,psi
0I. 57 i :33
.4 4 1 02
-,. , ',-' 41n ' 1 -' ,- 4~ ,- H :,
7. 4 0 -1 .1 4 4 .:3 4H C1 .7
2. <.10 ~~::":3 .:9 5 R=: : _' ,,
TABLE XXIII. EXPERIMENTAL DATA X = 0.25 HOLE LINEAR LOADING
a = 10,749 psi
DISTANCE
FROM HOLE, in. a@ ,psi ar psi
c . . , . .
0 . 21 5 197':09.5 '.
501 16588.C3 "422, .
90
I
ji
TABLE XXIV. EXPERIMENTAL DATA SHALLOW NOTCH LINEAR LOADING
15,000 lb Load
DISTANCEFROM HOLE, in. GO ,psi 0r ,psi
0. ' , 19999 .
c.... 1727.. -
0 . 5 lC 15 579:3 .,
0-. E. 25 136-, 1.7,79.1 : -
TABLE XXV. EXPERIMENTAL DATA SHALLOW NOTCH 60,000 lb LOAD
NO LOAD RESIDUALSDISTANCE
FROM NOTCH,in. a ,psi ar ,psi %a ,psi 0r ,psi
511 . 55F-. Ii - * -.
0.125 7 I971 ' .- -: . .. -4- , . l
03 41- ' 5 01 .7:3, 1 1 2. - 1 41. 5 -1 4 1:,9 "- ID .i4 4. l
0l. 3-.,75 1771 :3 9. 3 1 1 1 121 0. 2 '
l.5 , 7i 1207.2 1l5 E '? 4 -Ucl': . -,--.
' . 625 714-36 . 4 E92 . Ei -. C; .. .1
0. 5 C3 0 4 5.: 1t5 :::3. '9
0.:75 59477.4 1 7 4:--.
I .000 57146.4 19275.4
91
TABLE XXVI. EXPERIMENTAL DATA SHALLOW NOTCH 65,000 lb LOAD
NO LOAD RESIDUALSDISTANCE
FM4M NOTCH,in a ,psi a pSi a8 ,psi a rpsi
29L1 5 ,0 .tq : 7.: -: '-: .4 7 0 : .; -:: . . .: -1 '
:- ... . ...... 7 .
-.500 7 :-: 4 1.44. -.z 132. . -
- .- '25 197C1 7
S, :- ' 'j I I7 - : ;.'4 :2: t
J. cl 1 74171:-,4. 1
TABLE XXVII. EXPERIMENTAL DATA SHALLOW NOTCH 70,000 lb LOAD
NO LOAD RESIDUALSDISTANCE
FMM NOTCH,in. ae psi a ,psi a8 ,psi a ,psi08r r1* I rci . cl I0 ::'5!9.0 -5,i> , ': 4::. 693
0i. 12549t
- E.c - 93. -4Li-.2 :4 ,. 4: 13. 4 , 4. . -:: . U
0 . 450 l . 2:3190.4
0:3.75 ...:. . C--
1.000 2-4. 1 204. C
92
TABLE XXVIII. EXPERIMENTAL DATA DEEP NOTCH LINEAR LO0ADING
15,000 lb Load
DISTANCE
FROM NOTCHi,in. aq ,psi ar ,psi
0l. 17-0 1 4 51Q7. 0' - 4ID. -
0 . 25 C"V- 210 . 0 1 tS (
20:1 7 . 10
0. 750 1 C:?1 '.:4
TABLE XXIX. EXPERIMENTAL DATA DEEP NOTCH 30,000 lb LOAD
Elastic - Plastic
DISTANEFROM NOTC,in. ae ,psi ar ,psi
0 . 0 C, . 15 6. [61 ~.
0 . 125 E 3 979 7
0 . 5 .,91 744::- 6 16 4::3 . 6-
0i 5 .61: 2 5 50 C .
93
APPENDIX A
PSAPI JCL
-STA'40APO J3B CARD -
IEXEC FRTXCLGP/IFORT.SYSPRINT DO DUMMY//FORT.SYSIN DO UNlT=3330,V'DLsSER=)ISK02tIDSN-S2939.PSAP(PSAP) ,DISP;SHRtLABEL;(tIN11)D/ D lUNIT3333 ,VOL=SEl=DISI(O2,D0 =S2939.P AP( PLOT),
I ISP=SHRfLAB EL=( t,,['4 )IDO UNIT=3333,VOL =SER=DISK02,DSN=S2939.PSAP(INIT),If SHqABL(,IfDO UNIT=3333tVOL= El=OISK02vDSN=S2939.PSAP(ELER)vIf ISP=SHR,LA3ELu( ,O.41OID UNIT= 330,V L=SER=DISK(02tDSN=S2939.PSAP(SAPF),
1DISP=SHR,LA8EL(,,,I4)fDD UN1T=3333,V L=SER=DlSK02tOSN=S2939.PSAP(ADNA)tII 1ISP=SHR, LAB EL=( ,,1,pI NIIfDO UNITa333DVOL=SER=DISKO2,0SNsS2939.PSAP(AUXL),IfDI SP-SHR, LAB EL=( 9 tI N)IDO UNITm3333,VOL=SERzOISKO2,OSN=S2939.PSAP(ADPTI,
/1DD *C *** MAIN PROGRAM**
DIMENS104 ZZZ(3203),DISPD( 5t3i830)CALL PSAPI( ZZZ,3203,DlSPOB03JSTOP
***********DELIMITER CARD(*)********/IGO.FTIOFOD1 00 UNIT=SYSOA,DISP=(,PASSJ,// SPACE=(CYL,( 2t2)) ,DSN=&ETEMPIIIGO.SYSIN DO
*~~****14SERT PSA'1 DATA HERE *****I
***********DELIMITER CARD(*)********
94
&~~ 7-:r--~-i
APPENDIX B
S LOCAL LEAST SQUARES SMOOTHING
22+ r Corner Nodes
o Midside Nodes
1 K 4 + 2x2 Gauss Points
3
Two-Dimensional Isoparametric Element from ADINA [Ref. 4]
The local smoothing expression from Hinton and Campbell
[Ref. 19] in ADINA coordinates becomes
C B B A
G2 B A C B aII
a3 A B B C X a1 I1
J4 B C A B
1 /
where A = . + - , B = - and C = 1 -1/
With a and 54 representing the smoothed corner node
stresses and 7.- 3 111 , and aIV as the unsmoothed
stresses at the Gauss integration points, this expression
can be written in an equivalent form.
53 A B C B a
a4 B A B C TIII
1 C B A B IV
a2 B C B A Ii
95
-~-
W-AL02 "S5 NAVAL POSTtRAD4JATE SCHOOL MONTEREY CA F/6 11/6AN ELASTIC PLASTIC FINITE ELEMENT ANALYSIS OF NOTCHED ALUMINUM -ETCIU)MAR 81 N J KAISER
MLASSIFIED
EEEllllllElllIEIIIIIIIIIEEII
EIIEI-IEIIEELll lIIIIII 9 ,]
The midside node stress values may be obtained by averaging
the values at the associated corner nodes, since the distri-
bution of the smoothed stresses is linear along the sides of
the element. Smoothed stress values obtained by this least
squares method should subsequently be averaged to obtain
unique values at nodal points shared by adjacent elements.
96
APPENDIX C
ADINA JCL
--------------STA'40ARD J33 CAR)----1EXEC FGRTX'LGvR7'I34=2001K
//FORT.SYSPRTNT DD20 Ai//FORT.SYSIN DD *
IMPLICIT REAL*B(A-HO-Z)RE AL ACOMMON A(120010)MA X= 120030CA~LL EXE:(MAX)STOP
****~*****DELIMITER CARD(*)********/ILKED.JSDD 2) DISP=S-RDSN=4SS.S2939.ADlNA//LKED.SYSIN DO
INCLJOE JSO)(LO04)EN TRY UAIN***********DELIMITEI CAR)(*)********
//GO.FT07FOOL DD UlIT=SYSDAvO1SP=('4EW,DELET'-),//DCB=(RECFM=VBS,3LK(STZE=400O),SPACE=(CYL,(5,1)
//GO.FT0IFOO1 DD NTSYSDApD!SP=CNEWDELETE),IfDCB=(RECF=/BS,3LK(3IZE=4tDOO),SPACE=(CYL,C5,1))
f/G0.FT)2F00. DD UjT=SYSDA,DISP=(N~EW,OELETE),IIOCB=(RECFM=VBS,BL(SIZE=4300),SPACE=(CYL,(,,1))
//GO.FT33FOOI DO U4IT=SYSDA,DlSP=(4JEW,DELETE),// DCB=(RECFM=VBS,aLKSIZE=1200),SPACE=(CYL (:;11))//GO.FTO4FOOI 9D UJJIT=SYSOADISP=HAEW,DELETE),
1/ CB=CRECFM=VBS,BLKSTZE=1000) ,SPACE0=(CYL, (5,1))/IGO.FTORF001DD UNTSYSOADSP(lEW,DELETE),
//GO.FT39F0OI 00 dNjT=SSDA,DSP=(NE'j3ELETE)l.// DCB=(RECFM=VBS,BL<STZE=400OOOSPACc=(CYL,(7,L))//GO.FTIOFOOI DO UAIT=SYSDA,OISP=(4EWd,DELETF)9// OCB=(RECFVBS,BLKSIZE=4-00O),SPACE=(CYL,(--,1))//GO.FT11PO1 OD NfT=SYSDA,0ISP=('JEW,DELETEJ,// DCB=(RECFM=VBS,3LKSIZE=1100),SPACE=(CYL,(5,1))//GO.FT12FOOI 3D J'J[T=SftSDA,DIP=(NEW,DELETE:)f// OCB=(RECFM=VBS,BLi(SIZE=1200),SPACE=(CYL,(5,1))//GO.FT13FO01 DD UNT=SYSOAvOISP=('JEWETE
I/GO.F.T56FOOL DD U'IT=3330,VCL=SER=DlSKO1,// OSN=FO099.TEMP,DIS0 =SHR,LABEL=(, ,,I4)//GO.FT57=0O1 DD UlIT=SYSDA,O!SP=(4EWqDELETE)v// DCB=(RECFM=VBS,B3L(STZE=1DDO) ,SPACE=(CYL, (5,1))IIGO.SYSIN OD
********1ISERT ADI4A DATA HERP *****
BLA4I( CARD --BLAN'K CARD --- (T40J BLANK CARDS sriP EXEC)
***********DELIMITER CARD(/) ********
97
APPEND IX D
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12. Garske, J.C., An Investigation of Methods for DeterminingNotch Root Stress from Far-Field Strain in Notched FlatPlates, Master's Thesis, Naval Postgraduate School,Monterey, California, September, 1977.
13. National Aeronatics and Space Administration (NASA)Report CR-1649, Finite Element Analysis of Structuresin the Plastic Range, by H. Armen, Jr., A. Pifko, andH.S. Levine, February, 1971.
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153
24. Clough, R.W., "The Finite Element Method in StructuralMechanics," Stress Analysis ed. by 0. Zienkiewicz andG. Holister, pp. 85-I19, Wiley, 1965.
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27. Lubahn, J.D., and Felgar, R.P., Plasticity and Creepof Metals, p. 510, Wiley, 1961.
28. Easterling, L.R., Stress Analysis of Ceramic TurbineBlades by Finite Element Method - Part I, Master'sThesis, Naval Postgraduate School, Monterey, California,March 1978.
29. Preisel, J.H., Stress Analysis of Ceramic Gas TurbineBlades by the Finite Element Method - Part II, Master'sThesis, Naval Postgraduate School, Monterey, California,March 1978.
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