CALIFORNIA STATE UNIVERSITI, NOR1HRiffiE
PARAJ.1ETRIC STIJDY OF FINITE ELEMENT DISPLACEMENT
TYPICAL SOLUTIONS 1D THE THESIS "STRESSES AROUND
SHARP-CORNERED RECTANGULAR CENTRAL HOLE OR CRACK
IN RECTANGULAR FLAT ISOTROPIC SUPPORTED PLATE IN
BENDING UNDER UNIFORM NORJv1AL PRESSURE'' BY AN
I OR1HOGONAL :tvlETIIOD I
WHICH HAS BEEN PROVEN NOT m
CONVERGE 1D A PRACTICAL MA'I1-IEI>IATI CAL GENERAL
SOLUTION
A graduate project submitted in partial satisfaction
of the requirements for the degree of
:tvmster of Science
in
Engineering
by
Henry Joseph
De Barro
August 1984
(l
The Graduate Project of Henry Joseph De Barro is approved:
California State University, Northridge
ii
PREFACE
This work was started in January 1983 to prepare a thesis
titled "Stresses Around Sharp-cornered Rectangular Holes and Cracks in
Supported Flat Isotropic Rectangular Plates in Bending Under Various
Distributions of Normal Pressure". The general mathematical solution was
to be obtained under this thesis by an 'Orthogonal' method of deriving
'Free-edge' conditions at any point in the plate.
These conditions would result from the linear superposition
of two Fourier expressions, (131) and (133) of Reference 3, ru1d of a
similar new one, yet to be developed, for a couple concentrated at any
point in the plate. This method of 'cutting' the hole in the plate would
have improved upon the classical 'Polar' solution which does not readily
define sharp corners and straight lines (see References 1 ru1d 2) .
The initial development of the simpler case of cylindrical
bending was concluded and submitted by the 15 April 1983. Its completion
was stymied by the need for a solution of a matrix of four algebraic
equations, each one half page long, for four unknowns. This method was
considered impractical, if at all feasible, especially in bi-axial
bending.
iii
On the 15 May 1983, an alternate approach was suggested. It
It was intended to derive parametric coefficients from a spectrum of
finite element (FE) models of various sizes of plates and holes, using
the SAPIV, Reference 9, FE displacement program which is solely on-line
at the CSUN computing center. This phase of the work was continued
through February 1984 and from the 15 May through the end of June 1984.
The typical results, subsequently requested in January 1984, of the
first four programs out of this FE work are presented and discussed in
this project.
iv
ACKNOl~.uEDGEMENTS
~~
special thanks are due to all my professors and in
particular to:
Professor T. f\1. (Tom) Lee, my Graduate Connni ttee first chairman and my
principal advisor on my thesis;
Professor Stephen A. Gadomski, for his accessibility and help with
debugging SAPIV outputs, seemingly any day and time;
Professor Edward M. Dombourian, my new Graduate Connnittee Chairman and
my principal advisor on this project;
and Professor Arnold Roe, my Graduate Committee Coordinator.
!'-~
thanks are also due to the consultants of the Computing
Center and to the young assistant-consultants who seemed always on-hand
in the Engineering Building computing room, whose help with the CSUN
computing system is greatly appreciated.
Last, but not least, Mrs.
Charlotte Oyer, the University Thesis Advisor, and Mrs. Frances Foy of
the Office of Graduate Studies, share my thanks for their ever-helpful
advice in their respective fields.
v
CONTENTS
Section
Page
Preface
iii
Acknowledgements
v
Abstract
xii
Introduction
1
Plan Of Spectrum Of Finite Element Models
2
Plate Discretization
3
Input/Output Execution
6
Parametric Study Of The Results
16
Discussion Of The Calculated Results
34
Conclusions
36
References
37
Appendices
A. Input List And Output Results For 14 x 14 Plate
39
B.
"
"
"
"
"
" 14
X
20
"
62
c.
"
"
"
"
"
" 14
X
17
"
87
D.
"
"
"
"
"
"
14
X
12
"
108
E.
"
"
"
"
"
" 14
X
12
"
125
Modeled· With BRICK Elements
Table
1. Non-dimensional Parameters And Coefficients
vi
21
Figures
Page
1. Analysed Quadrant And Focal Point Identification
viii
2. Typical Quadrant Primarily MOdeled With BRICK Elements
5
3. Plots Of Deflection And Reaction Coefficients Along
7
X-axis For A Typical BRICK Element Mbdel
4. Plots Of Deflection And Reaction Coefficients Along
Y-axis For A Typical BRICK Element
8
~bdel
5. Undeflected Rod Geometry
11
6. Deflected Rod Geometry
11
7. Plots Of Deflection And Bending Mbment Coefficients
22
8. Plots Of Shear And Reaction Coefficients
23
9. 14 x 14 Plate Quadrant MOdeled With Shell Elements
24
10. Isometric View Of Deflected 14 x 14 Plate Quadrant
25
11. Isometric View Of Deflected 14 x 14 Plate Hole X-wise Edge
26
12. Isometric View Of Deflected 14 x 14 Plate Hole Y-wise Edge
27
13. 14 x 20 Plate Quadrant Modeled With SHELL Elements
28
14. Isometric View Of Deflected 14 x 20 Plate Quadrant
29
15. Isometric View Of Deflected 14 x 20 Plate Hole X-wise Edge
30
16. Isometric View Of Deflected 14 x 20 Plate Hole Y-wise Edge
31
17. 14 x 17 Plate Quadrant Modeled With SHELL Elements
32
18. 14 x 12 Plate Quadrant
33
~bdeled
vii
With SHELL Elements
Nomenclature
Figure 1.
Flat Isotropic
Plate supported all
r--.. ,------
-~
1
·~
I
Around With Central Hole.
I
I
The Analysed Quadrant Is
Shown In Solid Lines
PC
b
With Focal Points.
r:-+- ___ ~---,
HY HC
PHY
:_!!ole . _,_.;:.:HX,;;,.;.__....J _P!..,_
v
I
l__l
I
I
I
I
L___ i___ j
I
I
I
I
l
I
I
I
:I
-'-------4------J
I
~
a
= Width of plate = 14 inches
A
= Cross-sectional area
b
= Length of plate
D
= Eh3 I 12(1 - v2)
e
= Elongation
E
= Young's ·modulus
E
= Strain
h
= Plate thickness
in
= Inch
K
= Coefficient
L
= Length
M
= Moment
viii
l-u j
a= 14
I
,..j
X
N
= Normal
n
= Rod
P
= Uniformly distributed transverse pressure
PSID = Lbf. per square inch of differential pressure
Q
= Shear
R
= Reaction
T
= Traction
U
= Local axis
u
= Width of hole
V
= Poisson's ratio
v
= Length of hole
w
= Local axis, deflection
X
= Global axis (horizontal)
y
= Global axis (vertical)
z
= Global axis (perpendicular above paper, not shown)
Symbols
6
= Finite increment
a
=
oo
= Infinity
<
= Smaller than
>
= Greater than
Partial derivative
ix
Coefficients
~
= Deflection coefficient parallel to Z at point HY
~c
=
II
II
II
II
II
II
"
HC
~
=
"
"
II
"
II
II
II
HX
\roN
= Moment
"
in XZ plane
\roic =
II
"
II
\M-Ic =
"
"
" YZ
"
\1YHX
=
"
II
II
II
KQXHC
=
Shear
II
II
xz
"
"
II
II
YZ
"
"
parallel to X
KQYHC =
KrXHY
KrXHc
= Traction
=
~c =
Kmx =
II
"
"
"
II
"
ii
ii
II
II
"
II
y
"
"
"
II
II
II
II
II
z
~y
= Reaction
~c
=
II
II
"
II
II
~X
=
II
II
"
II
II
~ =
II
"
II
II
II
~HX
II
II
.fl
II
II
=
Acronyms
CSUN
= California State University in Northridge
FE
= Finite Element
X
Subscripts
py
=
Plate outer edge intersecting the Y-axis
hole X-edge
PHY =
II
II
II
PC
=
II
II
"
at its corner
PHX
=
II
II
"
intersecting the hole Y-edge
II
II
II
"
II
X-axis
II
II
II
Y-axis
II
"
" X-axis
PX =
HY
= Hole
HC =
II
HX =
II
II
II
corner
~1X
= Moment in the XZ plane (per unit length)
MY
=
"
II
QX
=
Shear
II
II
"
QY =
" YZ
II
II
II
II
" xz
"
il
II
li
" YZ
"
II
"
"
TX
= Traction parallel to the X-axis (per unit length)
1Y
=
R
= Reaction
"
w = Deflection
"
"
II
Y-axis
"
II
II
II
"
"
Z-axis
"
"
II
"
"
II
X
= X-axis direction
y
= Y-axis
II
z
= Z-axis
II
u
=
U-axis
"
w
=
W-ax is
"
xi
"
ABSTRACT
PARA.t-1ETRIC STIJDY OF FINITE ELEMENT DISPLACEMENT
TYPICAL SOLUTIONS TO THE THESIS "STRESSES AROUND
SHARP -mRNERED RECTANGULAR CENTRAL HOLES OR CRACKS
IN RECTANGULAR FLAT ISOTROPIC SUPPORTED PLATES IN
BENDING UNDER A UNIFORM NORMAL PRESSURE" BY AN
'ORTHOGONAL METIIOD' WHI rn HAS BEEN PROVEN NOT TO
ffiNVERGE TO A PRACTICAL MATHPMATICAL GENERAL
SOLUTION
by
Henry Joseph De Barro
Master of Science in Engineering
The non-convergence to a Practical Mathematical General
Solution was followed by a request for a spectrum of finite element
solutions by the Displacement Method as coded in the SAPIV program of
analysis, solely on-line at CSUN.
The SAPIV program can yield satisfactory solutions, but has
an unusually high potential for a new user's time wastage by churning
out masses of totally erroneous output, instead of terminating the
execution and diagnosing the cause, as do professional programs. Three
Xii
comparative output studies had to be made to isolate causes and effects
and to eliminate them. A set of rules have been written, derived from
these studies, to ensure against a variety of SAPIV pitfalls. The CSUN
computing hardware, too, takes its own massive toll of the user's time.
It is still incompatible with the CDC NOS 2.2 system introduced in
January 1984.
The objectives sought under this project have been achieved.
All the calculated results plot into notably scatter-free smooth curves.
~bre
plate aspect ratios are needed to establish upper and lower trends.
Any linear finite element solution; however, is inherently inadequate to
deterrni11e concentration levels. Additional methods beyond the scope of
this project are suggested for this pu1pose.
xiii
INTROIXJGriON
Two alternate mathematical methods have been explored, so
far, 1'li thin the context of this work, namely, the 1 Orthogonal 1 method
-~
and the method of 'Continuous Plates'. The first is the subject matter
of my initial thesis that is yet to be completed or concluded. The
second method is developed in page 229 of Reference 3. Both methods
require the formulation of new expressions and lead to cumbersome
matrices of very large non-numerical algebraic equations. A general
mathematical solution by either method seems too unwieldy, if not
impossible. A numerical solution of such a matrix of expressions is
possible, however, for a given set of plate and hole parameters.
A spectrum of sets of parameters was, then, requested and
the numerical solution of each to be obtained for both plates with
virtual and real holes. A finite element displacement method by the
SAPIV program of analysis, Reference 9, on-line at CSUN computer center
was to be used. This work was required to check the expressions derived
eventually in the thesis. The typical parametric results of the first
four FE programs for plates with real holes and their parametric study
have been subsequently requested for the purpose of this project. They
are presented and discussed herein.
1
p •
PLAN OF lliE SPECTRUM OF FINITE ELEMENT IDDELS
It was anticipated at the outset that plotting the results
might be necessary to derive intermediate and extrapolated values. A
minimum of four (4) points are necessary to plot a non-reversing gradual
non-circular curve. The plate variable parameters, see Figure 1, are:
Plate width, a
Plate lenght/width ratio, b/a
Hole/plate ratio of widths, u/a
Hole length/width ratio, v/u
The resulting total of 2 x 44 = 512 solutions was prohibitive. The
plate width and the hole/plate ratio of widths, consequently, had to be
fixed at a
= 14 and u/a = 0.5 respectively. The four values selected for
each of the remaining variables were:
b/a = 10/7, 17/14, 1.0 and 6/7
v/u = 1.0, 9/14, 2/7 and 0/7 (a through-crack).
The number of solutions was thus reduced to
2 x 42 = 32 SAPIV programs.
Fracture .1\fechanics, however, being topical for the
durability of mechanical structures, further crack ratios were added,
as follows:
v/u = 0/3, 0/1.5 and 0/0.2,
i. e. 2 x 3 x 4 = 24 solutions. These raised the number of models to a
2
3
total of 32 + 24 = 56 SAPIV programs for plates with virtual and real
holes. They are numbered from HJDTHOl through HJDTHS6 respectively.
The following constant parameters were selected to minimize
the effects of numerical tolerances and truncations on the results:
h = 0.1 inch, E = 30 x 10 6 PSI, V = 0.3 and P = 10.0, subsequently
reduced to 1.0 PSID.
PLATE DISCRETIZATION
The mathematical model of one plate quadrant is sufficient
in bi-axial symmetry. The mesh is intended to be as fine as possible
around the hole corner or the crack lip for a better approximation at
this focal point. The discretization gradually becomes as coarse as
possible to minimize the input time and the cost of the solution. Also,
the number of nodes is limited by the SAPIV capability. The ratio of
contiguous element areas is limited to about four to avoid numerical
instability which may lead to ill-conditioning.
The SAPIV SHELL or 'Plate' element was initially an
obvious choice for the first four pairs of programs. However, this
element does not output out-of-plane shears which are main components
of our problem.
~fully
elements of this first mesh concept were unduly
elongated and irregular. Also, the input of such a mesh was timeconsuming. All four models with real holes, HJDTH02, 04,06 and 08, are
shown in Figures 9, 13, 17 and 18 respectively.
A new time-saving mesh of all rectangular BRICK elements
was, therefore, conceived for all subsequent programs. The BRICK
element was expected to better model a real plate. A reproduction of
HJDTH09 in Figure 2 typifies this mesh.
4
tn
INPUT /OUTPUT EXEUJTION
Each pair of solutions, i. e. for plate with virtual hole
(plain) and with real hole, initially took over eight hours of fast
~ontinuous
work from mesh preparation to the final outputs. This time-
cycle was achieved after the initial familiarization with the CSUN
computing system and hardware. Subsequently, the time was reduced to
over two hours by the use of modified duplicate programs. When other
associated considerations are taken into account, this program of FE
work has absorbed several hundred hours of continuous work. By the
6 December 1983, the Cyber 750 was used for 131 000 SRU and 438 336 CP
seconds or 122 hours of computer time alone.
All fifty-six FE solutions had been obtained by mid-October
1983. The SAPIV program, however, does not provide options for the
automatic output of the nodal force balance and of the reactions at the
supported nodes. It became necessary to expand the input of each model
with stiff boundary elements to obtain the reactions and with
fictituous flexible beamlets to obtain some of the loads at selected
nodes. The output of these dummy beamlets is solely valid when they are
placed along free edges, where Poisson's ratio is inoperative.
Further, the results from the first pair of models with the
BRICK elements, HJDTH09 and 10, when reduced and plotted, Figures 3
and 4, gave alternating reactions. The latter sharply contrasted with
the scatter-free regularity of the deflection curves parallel to both
axes. In consultation on SAPIV, it was suggested that to overcome this
6
7
Figure 3. Non-dimensional Coefficients Of Transverse Deflections
And Reactions Plotted Parallel To The X-axis Obtained From Mbdels
JUDTI-I09 And JUDTIUO Primarily Made Of SAPIV BRICK Elements.
Q
8
··----
···---.
u:: =-AI-r ii;rr.zz:F"a~~-(29 ·wr::P.;.-r-~ w.;.,...;,<.:&i.4ri
_______;_;.;.---=-
'--_:: -':::j-;7 ~ '' -' ·:::' - ='''AND ,...;r~tc::"Our.,i>ar "<'-t PM7£ !.<..7•=-rz--tfJ:Oi..£:
::::: --- - .
-::-:E"= ':.:.:.:=;-
..
~--
::~-~
:
---
..
. --- :;.;:_·-·:.·:,;.c.-·
---
----:~---
-:::.. ~::=: .. ::::c:_·-.
::_~------,- ·-;::: U
~==~ -----~-~~;..E,-=--:
-~
:. :_ ::· :::u:-·:.. :_
··:.-=.:;..:·:
-=~
:-- .·-:=t==
:: ::.·: •o uO
~~a-,:-··
---···:.::... . . .
-.:.~
. ..
--·-·-:-:--
-----------
.. -· __ ...... -----
ftj-c-L~~~--;4=~~~~=-c_c-·
'====!··
,o;:.:~~
---;t;r;'j¥-"E~;r==:
;:
*''m>5
~A
~&=
=
..
=
-- ----- --
- --··.
--
.
-·: ;: .
--------~
Figure 4. Non-dimensional Coefficients Of Transverse Deflections
And Reactions Plotted Parallel To The Y-axis Obtained From Mbdels
IUD1H09 And lUDTI-IlO Primarily Made Of SAPIV BRICK Elements.
'
9
irregularity of the reactions, the user's manual main option for the
boundary element orientation should be replaced by the alternate device
of 'artificial' nodes. This device logically failed to affect the
reactions. Several iterations to isolate and to eliminate the cause of
this reaction distribution had been tried before the last one showed
that the kinematic incompatibility inherent to any 8-noded solid
element low order shape function or stiffness matrix was the cause. In
this last iteration, the boundary elements were replaced by short stiff
'posts' or pinned columns, thus modeling a real structure throughout.
The reactions obtained from the internal axial loads in the posts
remained unchanged. Both the input list and the output results for this
iteration of HJDTHlO are reproduced in Appendix E. Evidently, the BRICK
elements were improperly loading the columns due to their kinematic
incompatibility. No 8-noded solid element should be used without a
system of other fictituous elements which would complement its
kinematic deficiency. The SAPIV user's manual does not warn its new
user against this pitfall. Its next version, SAP S, hints at the latter
stating: 'For 8-node elements without incompatible modes use element
type 8 (a higher order BRICK element)'. This version includes, also,
other improvements on SAPIV and the very helpful plotting option, but
is not on-line at CSUN. Consequently, this writer decided by midJanuary 1984 to shelve the bulk of his BRICK-based FE work, i. e.
HJDTH09 through flnJ.QHS6.
The afore-mentioned reasons curtailed this project to a
parametric study of the results of the four initial models which are
based on SAPIV SHELL elements and identified by HJDTH02, 04, 06
&08.
10
These four models had been expanded, as previously described, with
boundary elements to obtain the reactions and fictituous beamlets along
the free edges to obtain the nodal loads, especially at the hole focal
corner. The plate elements in the first two models were, also, doubled
with MEMBRANE elements. This was to investigate the SAPIV option which
occurs in the membrane elements only: 'Non-zero numerical punch will
suppress the introduction of incompatible displacement modes'. The
linear SAPIV solution was not expected to output in-plane stresses in
the membranes of this flat plate when no external in-plane loads were
applied. Why, then, were the incompatible displacement modes not
automatically suppressed, as in the case of the SHELL elements ?
It was hoped that having linearly solved the deflections
on the basis of a balance of all forces about the undeflected shape,
the program would, then, calculate the in-plane loads in the membranes
from the out-of-plane deflections. This calculation would be part of
the post-processed output. It would err conservatively and procede as
follows:
1) Consider the simplest case of a rod element, n, of stiffness
EA (or any edge) in the X-Z global plane, shown in Figure 5, which
extends from (X1 , z1 ) to (Xz, z2) and is subjected to a global
deflection, 6Z 2 , all other deflections being null.
The rod, n, true length is
L ;_,· [(Xz - Xl)2
+ (Zz - Zl)2J0.5
The displacement components along and across the rod are
respectively;
11
(&.1
z
t
\w~~~//
!
A/ ---r
/
.(
~
1
\
\
'
/
~z2
CX 2 ~j
___i_
w
z2-zl
X
~
Figure 5.
Figure 6.
2) In the rod local plane U-W, consider both similar triangles,
shown in Figure 6, formed by the rod of length, 1, and the normal
deflection component, 6N, and write
to obtain
ll1N p (t;N)2 I 21
and
t;1N = (t>Z 2) 2 (X2 - X1 )2 I 2 [(X2 - X1 ) 2 + (Z2 - z1 )2] 1 ·
3) Then, the total elongation of the rod,
its strain,
s = e I 1 = (611
and
+ ~)
I 1
5
12
its internal force,
F
= £EA
= EA.
[l'.Z2
z2 - zl
2
2
(X2 - Xl) + (Z2 - Zl)
+
(l'.Z2)2
2
and the rod element stiffness matrix coefficient for a normal
displacement, l'.Z 2 , at node 2,
k(n) = EA [
44
(X
2
z2 - zl
- X )2 + (Z
1
2
-
z1 )2
(X2 - Xl)
+
2
l'.Z2 --------~--------~]
(Z _ z )2]2
2 [
- X )2
cx 2
1
+
2
1
though the second term cannot be included for a linear solution of this
two-dimensional case;
4) Since both the rod and the flat plate of our problem lie in
the global X-Y plane, the value of (Z 2 - z1) vanishes and the first
term of the above force expression, solely calculated in a linear
solution, disappears but still leaving an in-plane small force,
'
which the SAPIV and similar FE programs could next calculate in
preference to the null value that is normally output in this case from
the basic linear solution.
13
The output seemed at first sight rewarding because all inplane and out-of-plane likely loads were given. However, closer scrutiny
disclosed that all the results including the reactions were erroneous.
Here again, several tentative runs were made to isolate and to eliminate
--~
the causes of the erroneous output. It emerged during consultation that
the SAPIV program had been under scrutiny for two years at CSUN and
three days had been spent to identify one such cause. In another case,
two nodes, 106 and 116, in the HJDTH08 grid, see its input list in
Appendix D, had been inadvertently mislocated one inch Y-wise, a
common occurrence. Both nodes were away from the edges and caused the
generation of compressed and stretched elements above and below their
locations respectively, but no negative areas about which SAPIV issues
a warning. Every section of the output was affected including the
reactions. The plotting option of SAP 5, if this had been on-line at
CSUN, would have exhibited the moderately distorted grid after the
first run. Now, either ill-conditioning had resulted from the proximity
of large and small elemental areas or the SAPIV element shape functions
deteriorate with the irregularities of the elements. This is an
important consideration in the professional use of SAPIV, because
complex models frequently must depart from the ideal.
The results of faultless runs from the first two models
gave either null membrane stresses or very small numbers throughout.
The source of these very small numbers is obscure because they are much
smaller than would be anticipated if post processing by the aforedeveloped theory had been executed. Surprisingly, they were output when
the boundary element orientation was made with 'artificial' nodes,
14
their sole effect and a most obscure relationship. For these reasons,
no
~ffiRANE
elements were input in the next two models. The Input list
and the Output results of each of the four models, HJDTH02, 04,06 and
08 are reproduced in Appendices A, B, C and D respectively.
Two comparative studies of several outputs had to be made
to relate causes and their effects on the results. These studies led
to the derivation of the following guidelines which may ensure first
successful static solutions from SAPIV:
1) The grid should be as uniform as possible or gradually
change from fine to coarse at a ratio of adjacent areas of up to four;
2) The grid input should be checked without reliance on the
SAPIV geometry check;
3) The external loading multipliers should not be coded for
elements that are incompatible with such loadings;
4) The element property set identifier should be, preferably,
input in the element data, regardless of the user's manual statement of
automatic selection of a unique set of properties;
5) The BEAM element compatible nodal load set identifier is
most critical and should not be referenced in the BEAM element data, if
such a set were non-existent;
6) The external loading should be commensurate with the
assumption of small deflections, basic to a linear solution;
7) The 8-noded BRICK element should not be used without a
system of artificial elements which complement its kinematic deficiency.
15
In conclusion, the SAPIV program gives satisfactory static
linear solutions within its limitations, but has an unusually high
potential for a new user's time wastage by churning out masses of
erroneous output, instead of terminating the execution and diagnosing
the cause as do professional FE programs. It is obsolesced by its next
though compatible version, SAP 5, which would be an asset at CSUN.
However, two advanced professional versions, COSMOS 6 and 7, which are
incompatible with both earlier versions, are available, References 12
and 13.
PARAMETRIC S1UDY OF TilE RESULTS
The end product of this project is a parametric study of the
results at eight focal points, identified by PY, PHX, PC, PHY, PX, HX,
HC and HY in Figure 1, in each of four flat plates with a central
constant square sharp-cornered hole. The size of the latter is fixed at
7 x 7 and the width and thickness of the former at 14 and 0.1
respectively, but their lengths are 14, 20, 17 and 12. As previously
stated, only the quadrant of each plate shown in Figure 1 was
discretized for a FE element analysis.
The FE models of the four plate
quadrants are reproduced in Figures 9, 13, 17 and 18 and identified by
HJDTH02, 04, 06 and 08. Their results are reproduced in Appendices A,
B, C and D respectively.
The results are reduced for the purpose of this study to
non-dimensional coefficients for a common basis. Their derivations are
based on the classical Navier's Fourier expression for the deflection
of a plain plate, see Equation 131 in Reference 3, and its derivatives
for bi-axial moments and shears, but the derivation of the reaction
coefficients is based on lengths, as follows:
00
wz
00
L:
= 16P [ 12(1-vl) ] L:
m=l n=l
7T6
Eh3
. mTIX s1n
. ..1!!!:L
s1n-b
a
2
2
n
m
)2
nm( - 2 +
a
b2
mTIX
.
00
00
4
s1n -a- sin
Pa
16
L:
= -D- [ ( ) L:
m=l n=l
na
7T
nm. [ m2 + (
b
Pa 4
= --D--
. Kwz
16
....!!2!Y_
b
) 2 ]2
]
17
sin ~ sin ~
a
b
=+
2
16
= Pa [ 1T
00
00
L:
L:
4 m=l n=l
[....!!!.... +
n
v (2-) 2 ]
--1L
m
b
·..nr-
. ln
mrrx- s1n
S
a
]
( Solely correct for V = 0.3 )
Similarly,
va 2w
I'-1-y=-D(
= Pa
...
Qx
2
+
ax 2
a2w
aY
-::::2 )
( Solely correct for V = 0.3 )
. KMY
I'>\r
ISw=2
Pa
= - D
C
a~v
ax 3
+
r
~
= + _ill_
3
m=l
n=l
rr a
a\r
axaY 2
2
c...!!L...
2
na
)
+ .JL_)
b2
Cos ~ s1"n ~
a
b
18
= Pa .
Kqx
Similarly,
=
Pa . KQY
Logically, since the traction and the shear have the same dimensions,
we write
TX
= Pa
Ty =
Pa .
:. KrY
=
and for the distributed reactions
RZN =
~a ( Coordinaten-l - coordinaten+l ) . KRZN
19
•••
KRZN =
2RzN
Pa ( Coordinaten-l - coordinaten+l )
Similarly, for the concentrated corner forces, we write
2
Pa
b
Rzc =
z-
(a) . KRZC
noting that this mathematically concentrated corner load must in
reality be distributed over a discrete area or length on each edge of
the plate from its corner.
The non-dimensional coefficients for the deflections,
bending moments, shears and reactions are calculated by the above
equations from the respective computer output results. These are
reproduced in Appendices A, B, C and D. The results and their
corresponding coefficients for each aspect ratio are also listed for
convenience in Table 1. Included in the latter are the corresponding
node and element numbers, the hole aspect ratio and the hole to plate
ratio of their widths. Note that the nodal shears and the moments are
taken from the discrete beamlets placed along the hole edges. These
beamlets have performed well, again, while the plates have provided
only the general stiffness for the deflection solution. The non- - - -Climensional coefficients are. next parameTrically protTed-InFigures T------and 8 against an ordinate of plate aspect ratio,
b
a
The FE models of the first two plates with a central hole,
HJDTH02 and 04, and isometric views of their deflected shapes and of
20
both deflected edges of the hole have been plotted and are, also,
reproduced in Figures 9 through 16. These plots were produced from a
development of SAP 5 which includes a plotting sub-routine, while it
was made available at CSUN.
Table l.
. ._-.. .o"
--8~
<S--,v
. o'?
<:)jl>"'-'
Parrureter
Program
Function
Coe1fic't
,___,_
b I ______
a
t--·
i
u I a
r-----c----- t--·-·
vI u
KwlLX
. o~"' K\\1-IC
..._.,
!-;;-·--
r;;:..ec,
..
--
f<r.1XHY
..._,
~'Q$'~
WD/Pa4
r--·
~c
~M-lC
100
100
~
I PaZ
My /jPal
~e~~
Qx
c.,'<.."'
IKQYHC
100 Qy I Pa
113
KTXffi·
Tx I
106
1---'----
KnllC
t-:--
100
Pa
113
100 Ty I Pa
- o¢'
KRPX
KRPIIY
~R~/(CN- -cN+tlPa
KRPc
12R:c/Pa2 (1 /a)
1------
225
120
~Rz~/(CNKRPHX
r:=-::
__ !--""
KRPY
113
218
Kn1fX
c,'<..-,;
'¥-'3'
113
113
jiPa
-o~"' Krnrc
'\'{'3'
---
113
100
--
!-uD1H02
_ _ _ .--..: H.J~TIHO_.:,_ _ _ _ b _____ l~IDTH~~---- - - - c ; - - __!:fJD1110 8
Output
.
-Node~'iUtput
OUtput
C ff' , Node M
C ff' , Node ~
OUtput
r.oe·ff'~c , t :--lode ~
oe lC t Eler1 't u
VC~l•re
V<11ue Coeff1ct
"1t'!'l'
t ¥
v~lue
Ele!fl't #~ue
. -- ... - oe lC t l'lem' t #
- ----···
-- ·• -· ·-· ····--- 1.0
10 I_?_ ~---- t - - - - - - __17 I 14
6 I ~-- 1 - - - 0.5
0.5
0.5
0.5
- - t--- --'-----· 1.0
l.O
1.0
1.0
~18
-0.0442 -0.00316 LH!
-O.Oll93 -0.00638 U!Y
i -0. 0669 I -0.004 78 163
-0.0298
-0.00213
- - - - - 1-------- r.----------- t - - - - ----113-- - - · - - t----- - TI3- ! -72
-0.0747 -0.110535 111
-0.0318 -0.00228
-o.oi~ -0.00134
~0.0533 -0.00381
t-------"------r----106
66
-0.0.142 -0.00316 i - w - - -0.1064 -0.00761 i-05-- -0.0752 . -0.00538
-0.0253
-0.00181
106
lOS
106
66
-0.01!65 -0.0442
-0.0116
1 -0.0228
1 -0.0468 -0.0239
1 -0.1274 -0.065
218
r~MlX
1\)XHC
ParJ.met.:rs ancl Coefficient".
----.
---
KWHY
_<;!-'
:~on-Jimensional
-CN+ 1 )Pa
15
8
J.
7
0.1275
0.0651
10
-0.1275
-0.0651
113
113
16
0.0468
0.0239
218
0.1716
0.1716
113
7
0.024
10
0.024
113
106
1
113
7
113
10
218
16
29
-o. c)755
-0.2787
225
22
-2.0115
-0.2874
120
15
12.18
8
1
-
-2.t1115
-----I).
9755
0.1243
-0.2874
--0. 271!7
15
,__ii
- - r---
0.0526
0.0269
189
0.1543
-0.0987
1. I 021
111
111
10
~
10
--
-0. 705
l_
lOS
~
111
In
10
189
16
~-1803
9 i -0.16
0.0499
14
61 0.1)294
0.0199
9
0.0920
-0.0816
72
72
0.0255
l!Jj
0.210
0.1424
72
9
14
I
0.022
0.1948
9
0.0145
0.1039
163
15
25 -0. 8987
-1.4311
-0.2116
72
-1. 1687
-0. 3053
195
__zzJ -1.0234
-o.2q24
169
<2
-3.702
-0.3526
~l_l_____fl_j_~~. 900-3
-0.3315
78
15
t5.ns
0.1137
13
0.1224
13
--- ~:,32?4
-1.1181 -0.4052
13 ! 1-1.565
·r-----·-7 ! -2-7699
1
~
-2.~425
7
l. -1.289
0.0493
-0.0495
0.0273
72
72
6
-----
0.0431
66
1
6 0.0967
9 -0.097
fi
29
8
1
-0.1479
111
111
0.2~62
16
II
0.1256
-0.0755
7
15
1
6
9
.l!i
-0.2568
t------
13 ~:5qs! 0.1142
-0.3166
7
7 -2.1176
-0.242
-0.3683
1
1 -0.6599
-0.1885
,_.
N
22
Figure 7.
Deflection And Bending Moment Coefficients.
23
1
1
i
'l
1_1,1 1
IjI
I
\
I
1
~
i'
I~.
~I
I
1111
!"
I
II
. .
~,.~--····',~IAit~
liU .. .
.. .
IIIIi
I
lji
i
1ilJ.
h ',
I'I
I
~ 11Jlw,:11 :
l' IIIII i !
IIi I
f+tt++l+t=tlt+l:M~~ml=t::!++f-tt++t:J:+.WW-l-W-:t-l--W--W-W-+ID4-
1 '-++,·++++I
lit
I!
..·.. ..
I
I i Jl i : l'
l I, I I I
I
... .
l! 1IT !-4 l t~
i! ''il i j l
iiI ill'!''. 1! I .. j'-·j
11
1,
.
i
,
I
..
I
, •
r1m11trJ ~tYlffi'
.
r
1
l
i
I
11
!
l
I
-· -
-.
,., l
1'
.
--
: I.,'
1
I
I
,
l J ~ ~~~ :
1
I
.~f
r!W
.. .
•. ...-
I' Ii iI
II l
I
.
-.
'
I
.
jl
i I ; 1 II
, , , , • I I
Iii
....
!
I
,~'~ ~
. . •.
I
li I I !'II
1
....
tt
I
.
.
..
~
·
••[J.,
I
+++++_1-++'l'-R{l-I~~+++++H
I . ..
--- --- --m·~~Hd~ t
.-
. ..
I I
I'
+t!rl
ljll
I
.lt -.. .· _···· .......... t'·i
j·t . . - -
...
Figure 8.
..
Shear And Reaction Coefficients.
24
HJDTHc;! svM
rTATIC
:]~H
r::_ eot
::7Y 7 Y.I
~~-"L
H:JLt.--.~
.... )'J_
...:DL
=1-or-~i
LO\i.' C._I\';J_
6~10"1?'>
Figure 9.
[ .. J!)L"-
v~
-~
!~
25
HJCTHO~
~YM
!J!R FL Pl•7X7X.I REAL
HClf"J.~XJ.~
UDL•I.QP~.I
STATIC LOAD CASE
84/0~/1~
!AXIS• J
~LrHA•
l)[fLF.C T I Oil 5( ALF
30 DO
•
~TOR
!lUA• ·45 OQ
23.984
Figure 10.
[•JOE~
V" J
26
HJOTH02 SYH DTR ~l PL•7X7X. I REAL HOlf•J.~XJ.~ UDL•I.OPSI E•JOE6 V•.J
5TATIC LO~D CAS£
64/Qr,/16
IAXIS• J
4LP!I.~·
JO.OD 9UA• IJ~ 00
OEFLHTlO!l SCALF: F¥'TOR•
23.%4
Figure 11.
27
HJOTH02
5TATI~
~YM OTR FL PL•7X7X-1 REAL HOLF•J.5X3.5 UOL•J.OPSI E•JOE6 V•.J
LOAD CA5E
~~!OS!lt;
IAXI5= J
,I,LI'H~=
30.00
DEFLHTION S(ALE FACTOR•
9tTA•
00
23.964
4~
IS
OJ
Figure 12.
28
HJD1H04 SYH
~TR
STATIC lOAD CASF.
FL PL•7X!OX. I REAL
HOLF•J.~Xl-~
UOL•I.OF~I
E•JOE~
L
V• J
I
6-1/Qr,;21
I AX l S ~
~
DEFLFrTIQ~
~l P'IA·" 0. 0~
StALF. FACTOR•
'ltl A• 0. OG
Q.Oc:::EO
20
I
.J";
50
f;5
60
~::;
I0
!Lb!
2~
Figure 13.
29
HJDTH04 SYM Q•R ~l PL•7XIOX. I
~TATIC L8AD Cll5f
£>4/0J)/21
I IIX I~- 3
DlFLf(T~ON
~tAL
!lll ,.._
S(ALF
FALTCR~
~~
H~lf•J.~XJ.~
UDL•I.OF~I
OG
•1.·)7~3
Figure 14.
[•JCE~
V• 3
30
------------
HJOTH04 SYH QlR FL Pl"7X!OX.I REAL HOLF>J."XJ.S
STATIC LOAD CA5f
UOL•!.OP~l
1'>410'>121
tAXIS' J
ALP'!\> -3~ G~
DHLFrTION SfALF FACTOR"
g[TA" 13' 00
t\.710
Figure 15.
l,J0l6 V•.J
31
HJOTH:)4 ~YM 0'R Fl PL~7X!OX.I ll[AL HDLF<J,oXJ-~
';TATIC LOAD CASf.
I
611/0rl/21
!AXIS• .l
~LPIIA~
JO ~~ 9UA• 4~ 00
DEFLf[TION S(ALF FACTOR•
9.97SJ
UOL•!.OF'~I
-~
16
...................
.....
........
7J
Figure 16.
E•J0l6 V•.J
32
X
~·
ty
I
I
171:.
/7o+
1.!'~+
/.l)
ils
X
1<!9
v,
")
0~
HJDTI-!06. Quadrant modeled primarily with SAP IV Shell elenents of
a 14 x 17 x 0.1 rectangular flat isotropic plate with real 7 x 7
central square hole (HJDTHOS witl1 virtual hole)
E = 30 x 106 PSI
V = 0.3
Figure 17.
X~~' Oy
6·0
fy
z~
-t-1'·~
·-r--
/
ii:'J'
HJDTI-108
~6ly
. ..,
...
~s
J'
'
I7
66
~>J'
I .
-·
(~"'
I
I
1-t:LJ?_;;·
j-'1'
~~¥
176 ~ .f'
'
7,
o
=tti
-~ ~
I
'
Quadrant modeled
primarily with
SAPIV Shell elements
of a 14 x 12 x 0.1
rectangular flat
isotropic plate with
real 7 x 7 central
square hole
(IDD'IHO 7 with
virtual hole)
E = 30 x 106 PSI
v = 0.3
f,e3
7?+
I\ \\\'H'
. C/7 _ 9.P _ 9'9' _/oo
1.:57+
+
16~
/6J
!<')
~
ltv
/i:.r
>;
!66
_/o/
N"7
_::.t_
IM'
169-x
~
&x
Figure 18.
K
>; 7~ B-x
t.N
t.N
DISOJSSION
OF 1HE CALaJLATED RESULTS
All the calculated coefficients plot into notably scatterfree smooth curves shown in Figures 7 and 8. The range of plate aspect
ratios, however, is too small to indicate trends at both its ends. In
the symmetrical case of the square plate, where _Q_
a
=
1, the respective
curves either intersect or are symmetrical to each other at that point,
as logical.
The curves of all the deflection coefficients, shown in
Figure 7, continue their almost linear though decreasing rise up to an
aspect ratio _Q_
= 1.5 . They are expected to reverse their slight
a
curvature and to be asymptotically stabilized at 3 < - b < oo , when the
a
plate will tend to cylindrically bend and the hole influence to remaim
localized. At their lower ends, the HC and the HY coefficients are
. h and at HX to reach a m1n1mum
. .
expecte d to VanlS
Va1Ue at
ab
=
0• 5,
when the plate and the hole parallel edges coincide, i. e. the plate is
cut through into two smaller plates each
SL~ported
on three sides.
The curvatures of all the reaction plots, shown in Figure
8,
are more pronounced and their trends are clearer than those of the
deflection curves. At their upper and lower ends, they are all expected
to be stabilized at maximum or minimum values, except that PC and PY
values will vanish at the upper and lower ends respectively.
In Figure 7, the curves of the bending moment coefficients
in the YZ and the XZ planes at points HX and HY on the hole edges,
34
.I
35
,, .
i. e. YHX and XHY respectively, are similarly expected to be stabilized
asymptotically at ~
.~ = 0.5, except YHX which
> 3 and to vanish at
should reach a minimum value. In contrast, the coefficients at the hole
corner, YHC and XHC, together with the shear curves in Figure 8 well
illustrate the rapid changes that occur in both internal loads at
1 <
~
< 1. 5 . The local bi-axial concavity of the square plate
pressure surface at the hole corner rapidly increases cylindrically with
b, while it decreases orthogonally and may even turn anti-clastic if the
hole length were sufficiently increased. However, both pairs of curves
are expected to reverse themselves and to be stabilized asymptotically
at 3 < - ba <
oo
and to vanish at ~
a
= 0. 5
.
All the results so far obtained are subject to the
averaging effect inherent to any finite element solution. Therefore, a
linear FE solution by itself is inadequate to investigate loading
concentrations. In fact, special iterative FE programs for crack-growth
have been developed to assist with ·Fracture Mechanics analysis. An
approximation of such concentrations at the hole sharp corner can still
be made by extrapolation from a plot of values along the hole edges,
leaving much to the imagination. Mbre accurately, we should write at
least a 4 x 4 matrix of Chebychev-type of curve-fitting expressions,
e. g •
Y
=a
+ bX + eX
2 + dX 3 + •••
and solve for the unknown coefficients. The narrow range of only 0.05
in our models of this extrapolation may predict usefully close
approximations in the elastic range. However, this development is
outside the scope of this project.
Q .
CONCLUSIONS
The following conclusions can be drawn from this parametric
study:
1) Though all the calculated coefficients have plotted into
notably scatter-free smooth curves, a few more than the four available
plate aspect ratios may have enabled a better exploration of their
upper and lower trends;
2) The finite element method of linear anal)Sis is inherently an
inadequate tool to investigate loading concentrations at the hole sharp
corner; In fact, special iterative FE programs for crack-growth have
been developed to assist with Fracture Mechanics analysis;
3) The rapid changes in out-of-plane bi-axial bending moments
and shears that occur at the hole sharp corner have been well exhibited
by the plots of their respective coefficients;
4) Further exploration of the loading concentrations at the
hole sharp corner can be done by extrapolation of the results plotted
along the hole edges, leaving much to the imagination; Alternately,
the solution of a matrix of Chebychev-type of curve-fitting expressions
would be closely accurate, especially over our narrow range of
extrapolation.
36
REFERENCES
1. Savin, G. N. Stress Concentrations Around Holes.
New York: Pergamon Press, 1961.
·~
2. De Jong, Thee. "Stresses Armmd Rectangular Holes In Orthotropic
Plates".
~
Composite Materials, Vol. 15 (July 1981),
p. 311. Technomic Publishing Co., Inc.
3. Timoshenko S. and Woinovsky-Krieger S. 1beory Of Plates And Shells.
New York: McGraw-Hill Book Co., Inc., 1959.
4. Wylie, C. R. Jr. Advanced Engineering Mathematics.
New York: McGraw-Hill Book Co., Inc., 1966.
5. Burington, R. S. and Torrance, C. C. Higher Mathematics.
New York: McGraw-Hill Book Co., Inc., 1939.
6. Frazer, R. A., Duncan, W. J. and Collar, A. R. Elementary Matrices
New York: The MacMillan Co., Inc., 1946.
7. Churchill, R. V. and Brown, J. W. Fourier Series And Boundary
Value Problems. New york: McGraw-Hill Book Co., Inc.,
1978.
8. Spiegel, M. R. Theory And Problems Of Fourier Analysis . . . .
New York: McGraw-Hill Book Co., Inc., 1974.
37
38
9. Bathe, K. J., Wilson, E. L. and Peterson, F. E. SAPIV, A Structural
Analysis Program For Static And Dynamic Response Of
Linear Systems. Berkeley, CA.: College of Engineering,
University of California, 1973.
10. Gallagher, R. H. Finite Element Analysis Fundamentals. EnglewoodCliffs, New Jersey: Prentice Hall, Inc., 1975.
11. McGuire, W. and Gallagher, R. H. Matrix Structural Analysis.
Englewood-Cliffs, N J: Prentice Hall, Inc., 1979.
12. Ramanathan, R. K. COSMOS6. Santa Mbnica, CA: Structural Research
And Analysis Corporation, 1983.
13. Lashkari, M.
COSMOS7. Santa MOnica, CA: Structural Research And
Analysis Corporation, 1983.
APPENDIX A
Input List And Output Results For Model HJDTH02
Of 14 x 14 x 0.1 Plate With 7 x 7 Square Hole.
39
II
a
113
..
..
,JI:s:J:zt:-03
I
1
I
72
71
70
69
•
0
0
•
•
•
0
- .30IOOE-01
-.21731t:-o3
-.2l074E-o3
-.16340[-01
I
I
1
55
54
I
-.17210E-03
oi7MJE-03
-.17176€-ol
oiH.,.-03
-.35515E-03
.34172£-03
-.34160£-03
.34174[-03
-.15242[-01
.311'JK-03
-.34174[-03
- o115f21:-03
• 20644«-03
1
1
- .17055[-01
-.16704[-01
-.16704[-01
"ol6340[-01
- .15242[-G1
-.10612£-01
o217JOE-o3
-.31717£-03
o2JOOJE-03
-.206371-03
I
- oi0612E-OI
I
1
-. 54507£-02
-.54!107[-02
o1110l[-03
.257--03
I
-.3112!1[-06
-.1109!1[-03
-.257al-o3
1
-.344461:-06
o34113t:-OS
i26676E-03
I
-.33!102[-01
-,3J502E-01
-.3266111E-01
-,3266111E-01
-.30118£-01
-.32454[-05
.47160£-03
-o4715JE-OJ
.4571JE-03
-.457761-03
.41743[-03
-.26103[-01
-.2610JE-01
-, 24701E-01
-,24701E-01
-.2.......-n
o.
o.
,00104«-04
-.001681-04
• 1!111331-03
.3s1.--o3
-.41737£-03
• 2:r.JIIII-03
-.1!1020[-03
-.35104£-GJ
.32!127£-0J
-.2Jl76E-oJ
- .32510[-GJ
.266661:-0l
-.2415!1[-01
I
I
56
57
58
60
61
62
63
64
67
I
1
1
I
73
0
1
1
.31457£-oJ
.27874«-GJ
-.2U60£-0l
-.31457£-0l
. -.278671-oJ
1.
I
-.2J!1861:-01
.~-OJ
.290011-o:J
-.2415!1E-G1
-.23!1861:-G1
-,JOJ42E-GJ
-.20994«-GJ
•
•
• ....
• .,
•
•
•
•
•
•
• "
•
-.211961:-01
-. 218961:-01
I
74
68
-.15040£-01
- o1 !1040£ -o 1
I
.27414[-0J
- • 27 4091: -oJ
-,JIJ25E-OJ
1
I
-.17682£-0J
.17687£-0J
-.345421:-G3
,J4SIIX-OJ
I
I
41:.
7!1
76
77
• ",.
•
•
•.
•
0
•
•
•
o.
o.
o.
o.
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o.
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.o.
o.
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o.
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