Maximum deflection of a blast loaded circular flat plate
by George Thomas Nolan
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE in Mechanical Engineering
Montana State University
© Copyright by George Thomas Nolan (1969)
Abstract:
The purpose of this paper is to present an analytical method for calculating the maximum deflection of
a simply supported circular flat plate which is subjected to a blast loading. The method is based on the
Modified Galerkin Method, and uses a non-linear stress-strain relationship together with non-linear
geometry. Equations of equilibrium are reduced to an ordinary differential equation which, when
solved, yields an approximation of the maximum plate deflection.
Calculated maximum deflections are compared to experimental test results of permanently deformed
plates.
The calculated deflections are approximately 72% of the magnitude of the test results. In presenting this thesis in partial fulfillment of the require?
ments for .an .advanced degree at Montana State University, I agree that
the Library shall make it freely available for inspection.
I further
agree that permission for extensive copying of this thesis for scholarly
purposes .may be granted by my major, professor, .or, in his absence, by
the Director of Libraries.
It is understood that any copying or publica­
tion of this thesis for financial gain shall not be allowed without my
written permission.
Signature
Date
julV 24, 1969
M A X I M U M D E F L E C T I O N OE A BLAST LOADED
• C I R C U L A R E L A T PLATE
"by
GEORGE THOMAS N O L A N , 'JR.
A thesis submitted to the Graduate F a c u l t y in partial
fulfillment of the r e q u i rements for the degree
of
MASTER
OF
SCIENCE
in
M e c h anical Engineering
Approved:
Head,
Major Department
C h a i r m a n , E x a m i n i n g Committee
MONTANA S T A T E U N I V ERSITY
Bozeman, Montana
A u g u s t , 1969
iii
ACKNOWLEDGEMENT
The author is indebted, to the N a t ional Science
Foundation wh i c h provided f inancial aid for this project.
Grateful a c k nowledgement is also made of the
assistance of Dr. Dennis 0. B l a c k k e t t e r , director of this
research project, who provided invaluable guidance during
the preparation of this thesis.
I
TABLE OF CONTENTS
Pag e
I
INTRO D U C T I O N A N D S T A T E M E N T OF PRO B L E M 0o#oeo*000<>oo0
2
THEORETICAL ANALYSIS
2.1
2.2
^
2.4
2.5
'3
.......
4
E q u i l i b r i u m Equations ................. .
Stress - S t r a i n R e l a t i o n s h i p .......... .
.
.
I^oaU 3-n
-Fn.nc*tjroTi
....poo...
Governing, D i f f e r e n t ial,Equations ..............
Modified Galerkin Method ........................ .
PLATE D E F L ECTION
3.1
5.2
5.5
3.4
5.5
3.6
17
Integration of Governing Differential
Equations
Governing D i f f e rential E q u ation 0 - 0 0 9 0
S o l ution of the D i f f e r e n t i a l Equation
Results and Discussion
Conclusions
Fur t h e r Studies
L I T ERATURE CONSULTED
4
7
1^0
11
11
0
0
0
..
0
0
0
0
9
0
0
0
0
0
0
0
0
0
*
6
0 0 0 0
0
9
0
0
0
0
0
0
0
0
0
0
0
0
0
0
*
0
0
*
0 * 0 0 0 0 0 6
0
0
0
0
0
6
*
0
*
0
6
0
0
0
0
*
*
*
0060
0
* 0 0
0 0
0
0 0 0
0 0
0
0 0 0
6 0
0
* 0 0 0 0
0 6 0 * 0 0 0
0 6 0 6 0 * 6 6
17
18
20
21
22
23
52
L IST OF TABLES
Page
T A B L E I.
De f l e c t i o n of S i m p l y Supp o r t e d Circular
1 A l u m i n u m (6061-T6; Plate Subj e c t e d to
Blast Loading e e e » « i » < ) # e » o o # 0 o e e e e o » e # # e e » # e
21
LIST O L -FIGURES
Fig.
I.
Page
'2(5
C y l i n d r i c a l coordinate system ................
Fig. 2.
Free b o d y ' d i a g r a m of plate element
Fig.
D i s p l acement of an element in the r a d i a l a n d ’"
transverse directions
^
. 26
3.
.........
25
Fig. 4.
Displacement of a line in the r a d i a l
’ " clr 2-*ec*tioTi ................................
Fig.
A c tual and n o n - linear stress-strain c u r v e s ..
27
Fig. 6.
Pressure-time curve of blast
28
Fig. ?.
Displacement mode r e l a t i o n s h i p
5.
Fig. 8.
Fig. 9.
......
Calculated vi r tu a l w o r k in second '
' ' displacement mode .................. .
Modified v i r t u a l w o r k -curve
(
I
29
‘
...... .......... .
3©
31
iX
- vii ABSTRACT
The purpose of this- paper is to present an analytical
method for calculating the m aximum deflection of a simply
supported circular flat plate wh i c h is subjected to a blast ■
loading.
The method is based on the M o d ified Galerkin Method,
and uses a non - l i n e a r s t r ess-strain r e l a t ionship together w i t h
non - l i n e a r geometry.
Equations of e q uilibrium are reduced to
an ordinary d i f f e rential equation which, whe n solved,
yields
an approximation of the m aximum plate d e f l e c t i o n .
Calculated m a x i m u m deflections are compared to e x peri­
mental test results o f p e r m a n e n t l y deformed plates.
The calculated deflections are a p p r o x i m a t e l y 72% of the
m a gnitude of the test r e s u l t s .
CHAPTER I
INT R O D U C T I O N A N D S T A T E M E N T OE P R O B L E M
This pa p h r s t u d i e s .the r e s ponse of s i m p Iy-supported
circular plates w h i c h are subjected to' a blast pressure
applied tb one side of the plate.
Large plate deflections
may be obtained if the applied blast pressure is of adequate
magnitude and if it is imposed on the plate for a sufficient
period of time.
D u r i n g the deflection of a plate,
any element of the
plate is subjected to both b e n d i n g moments,
forces,
either of w h i c h m a y be dominant,
the deflection is small or large.
large deflections,
and membrane
d e p e n d i n g on w h e t h e r
Also, when u n d e rgoing
the plate material may be stressed beyond
its yield point into the n o n - linear stress - s t r a i n region.
The complete analysis of the blast
loaded plate deflection
problem should consider deformations through the regions of
b o t h elastic and no n - l i n e a r strains while also recognizing
b e n ding and stretching effects.
Wan g
(l)* d eveloped a method of calculating the per­
manent defor m a t i o n of a simply supported circular p l a t e .
subjected to a blast loading based on a plastic rigid theory.
The method considered b e n d i n g moments dominant,
and did not
include membrane forces.
Hof f m a n
(2 ) used W a n g ’s plastic rigid t h eory in his
thesis to calculate the permanent deflection of simply
supported circular' a l u minum plates subjected to air blasts.
^Numbers w i thin parenthesis, re f e r to references.
- 2As part of his thesis, Hof f m a n also did experimental wor k
and fired explosive charges on several test plates, then
compared the actual deformation w i t h the t h e o r e t i c a l r e s u l t s ,
He found that W a n g ’s theory g r e a t l y overestimates the plate
deflection,
e s p e cially for small plate thicknesses.
May (3), in his thesis pres e n t e d to Mon t a n a State
University,
developed an analytical method i n c o r p o r a t i n g a
non - l i n e a r stress-strain rel a t i o n s h i p to predict deflections
of circular plates subjected to blast loads.
The analysis
considered ben d i n g e f f e c t s , but did not consider stretching
of the middle plane of the p l a t e .
estimated the plate deflection,
M a y ’s analysis also over­
and he suggested that both
b e n ding and s t r e tching effects be considered.
Jones
(4-) extended Wang's w o r k and c o n s i d e r e d ■both
bending moments and membrane forces in his analysis of a
simply supported circular plate..
small deflections,
The analysis considered
and used a p l a s t i c rigid theory.
The purpose of this paper is to present the d evelop­
ment of an a n a l ytical method for calculating the deflection
of a simply supported circular flat plate s ubjected to a
blast type loading.
The method recognizes, large deflections
and uses e q u i l i b r i u m equations t o g e t h e r w i t h a non-linear
stress-str a i n r e l a t i o n s h i p while considering b o t h bending '
and stretching e f f e c t s .
The M o d i f i e d Galerkin Meth o d was
use d to reduce the governing partial dif f e r e n t i a l equations
of equilib r i u m to a n o n - linear o r d i n a r y .differential equation
in time.
A solution to the o r d i n a r y dif f e r e n t i a l equation
expressed the de f l e c t i o n as a f u n c t i o n of time.
The time
derivative .of. the deflection s o l ution was .used to express’,
the plate v e l o c i t y as a function of time.
.By solving the
d e f l ection and. v e l o c i t y equations simultaneously,
mum plate deflection could be calculated.
the m a x i ­
This method determines only the m a x i m u m deflection
of a blast loaded circular flat plate,
A m e t h o d of p r e ­
d i c t i n g the p e rmanent deform a t i o n resulting from a given
m a x i m u m de f l ection has not been developed.
CHAPTER' II
T H E O R E T I C A L ANALYSIS
2 . I E quili b r i u m Equations
A cylindrical coordinate system (r, 0,
z) wit h its
origin at the center of the un d e f o r m e d middle surface of
the plate was chosen w i t h the positive z axis directed
downward (Fig.
I).
D'Alembert's p rinciple was used to express the
dynamic pro b l e m as a static problem.
The principle states
that a m e c h anical system in m o tion is always in a state of
dynamic e q uilibrium under all the forces,
i n cluding the '
inertia forces, which, act on the system ( 5) .
An element of unit thickness from the deformed
plate in an equilibrium position is shown in Fig.' 2 w ith
all the forces acting on it including an inertia force
and an applied p r e s s u r e ,
are force p e r uni t length,
respectively.
The forces and moments shown ■
and moment per unit length,
Subscripts r and t signify the radial and
tangential directions.
■'
,
Loa d i n g is assumed to be. symmetric about the center
of the p l a t e ; t h e r e f o r e , shear forces are not developed
on the t a n g ential f a c e .
A s u mmation of forces in the z .direction was
expressed as follows:
.N ^ r d O ■sinC-yS) - (N^*
d r ) ( r t d r ) d O s i n (-^9 +. d/9)
JQ
-Qx TdQ cos(~^3) + (Qr+
^/rd r d Q w + PrdrdQ
Equat i o n
- Q
.
'
dr) (r + dr)d© c o s (-^ + d/9)
■
■
(I)
(l) was then simplified b y n e g l e c t i n g higher
than second order terms and r e - w r i t t e n in the following
■
form:
JN
r
^ f2 ~
o*0
+^ ~
sin(-y5) -
r sin(-y9) + Qr
cos(-y3)
(2 )
r cos(-y5) ■y’wr + P r ^ drd9
Where, V 9 is the mass of the plate per unit area.
w is the a c c e leration of the plate in the z
direction„
is the angle that the radial centerline of the
element makes w i t h the radial a x i s .
Wr is the norm a l force per unit length on. the
r a d i a l face of the element.
Qr is the shear force p e r unit length on the
r a d i a l face.
P
is the applied blast p r e s s u r e .
To simplify the summation,
equal to angle
angle
~y3J ■
-yS+ AJJ was considered
/
, .
A summation of forces in the r a d i a l direction
which,
for simplicity,
n e g l e c t e d the h o r i z o n t a l compon­
ents of the pressure and inertia, forces gave:
aw -Nr rd© cos (-yd) t (Nr +
'
' - dr) (r + dr)d© cos(-y^-fd^)
©Q '
dr) (r + dr)d© sin(-y5+ d/d)
-Qr rd© sin(-yf) + (Qr +
- 2N_|_
dr = O ■
Equat i o n
($)
($) was also simplified b y n e g l e c t i n g higher
than second order terms and was re-written as:
n
©N.
!Pr = [Nr cos (-^) + r
+^ —
Where
cos(^d) + Qr sin (-yd)
sin (-yd)- W t ^drdG
is the n o rmal force p e r unit' length acting on
(4 )
the tangential face.
A summation of moments about the r i g h t-hand r a d i a l
face of the element was expressed as follows:
M^rd© -N rd©dr si n (-^/S)-f. N ^ r d Q d w cos-(-^)
t Q r rd©dr cos
(-y9) + Q rdGdw sin (-^1)
-(Mr + ™ — dr) (r + dr)d© + E N ^ d r ^ + 2M.b ~
-PrdrdG
Equa t i o n
= 0
rj^wrdrd© 4^-
(5)"
(5) was then simplified by d i s c a r d i n g the h i g h e r
order terms,
Q
dr
and r e a r r a n g e d to express the shear force
in terms of the b e n d i n g moments as:
SM
Qr -
,
c o s ( - ^ ) + ~ Mr cos<-^)
,
- -g
(6 )
cos(-^)
Where Mr and ' M^ are the moments per unit length acting
on the r a d i a l and ta n g e n t i a l faces, respectively.
could then be eliminated from equations
Qr
(2) and (4) b y
expressing it in terms of the moments.
The r a d i a l and tan g e n t i a l forces p e r unit length
were then expressed in terms of the r a d i a l and t a n g e n ­
tial stresses as follows:
yidz'
Nt -
Kdz=
..f'
(7)
(8)
i
Where h is. the thickness of the plate.
The r a d i a l 'and t a n g ential ben d i n g moments per unit
length were d efined by summing moments about the center-
line of the plate in the f o l l o w i n g manner;
I
I/
(9)
7^z*dz*
2
2
(10)
Where z* is the distance fro m a radial center-line to the
element where the normal or t a n g ential stress is acting.
z* is at all times norm a l to the plate c e n t e r - l i n e „
The e q uilibrium equations
(2) and (4) are the
governing par t i a l dif f e r e n t i a l equations of the system.
2.2 Stres s - S t r a i n Relationship
The r a dial strain was expressed in terms, of the
1
displacement.
In Pig-. 3) line A - B of length dr is oriented in an
r-z c o o r d i n a t e ■system.
'Line A - B is d isplaced in the
plane of the coordinate syst e m to position A " -B'
the final position has attained the length d s .
and in
The d i s ­
placement components of p o i n t .A with res p e c t to the r-z
coordinate system are u and w, r e s p e c t i v e l y . . The elonga"hi n n
nf
"H n a
A- B
i n
BVia
t*
rl t t -a a +H n n
i a —— S r>
a r ir l
Bh a
line A 1- B 5 is calculated by the p y thagorean theorem as*.
ds -
(dr -f -gg dr )2 B
d r )2
By pe r f o r m i n g the internal mult i p l i c a t i o n and then ex­
p r e s s i n g ds w i th.a b i n o m i a l e xpansion c o nsidering terms
no hi gher than second order,
ds may be expressed ah: '4
T h e .strain in the r a dial direction,.
£ , could then,
be defined in terms of the displacements u and w as:
" 2^ ) ^
dr
Because of the symmetry of deformed shapes that
have resulted from blast induced deflections,
the tan­
gential strain was 'expressed as a function of the r a d i a l
displacement only.
In Fig.
4,
line C-D of length rdQ is
oriented in an r -6 coordinate system.
Line C-D is d i s ­
placed in the radial direction an amount u to the
positi o n ' C - D t .
The strain in the ta n g e n t i a l direction
-
was then defined in terms of the radial displacement u as:
(r d- u )dQ - rdQ
rd©
ct
(12)
r
The following n o n - l i n e a r stress-strain relation
ship was used to approximate the actual stress-strain
relation of the plate material:
9
- r tanh
&.£
:(13)
B y selecting the proper constants a and b, the linear
slope and overall position of a certain stress-strain
curve could be a p p r o x i m a t e d .
Fig.
aluminum,
5 shows a stress- s t r a i n curve for 6061-T6
the mater i a l used in the test plates.
Fig. 5
also shows the n o n - linear stress-strain curve as com­
pared to the a c tual curve for a one-dim e n s i o n a l stressstrain relationship.
Data for the actual stress-strain curve (6 ) is a
composite o f var i o u s rates of loading f rom
s e c . through 2700 i n . / i n . - s e c .
.07 in./in.-
This i ndicates that the
test material is not strain-rate sensitive at r e l a tively
low loading rates.
Test data, however, was not available
for higher rates of strain.
The foll o w i n g relationships were a ssumed to express
the stresses in terms of the strains for the two-dimension­
al stress system:
_ 9 “- ,
for the r a d i a l direction,
Tr = F tanh f l T y f S r + VZtjJ
(14)
"
and for the tangential direction,
V t = F tanl> 6 5 v 2 [ e t + v e j }
Where a value of 'U=
«3 was a ssumed for P oisson's ratio.
By i n c o r p o r a t i n g equations
and
os)
(15 ) into equations
(7 ) and
(ll),
(12),
(13),
(14)
(8 ) ,'the r a dial and
tangential forces per unit length could be redefined in
terms of the displacements as:
Also,
equations
(9) and (10),
the radial and t a n g ential
moments per u nit length could be.redefined in 'terms of
the displacements as:
-
2
Thus the gove r n i n g equations
(2) and (4)
can be expressed
in terms of the displacements, u and w»
The analysis up to this- point may be summarized
. .
as follows:
■
INI'
—
The equil i b r i u m equations
10
—
(2) and
(4-) were developed to
form the gove r n i n g partial differential equations of
the system and are in terms of the forces acting on the
plate e l e m e n t .
All inter n a l plate forces and moments w ere then expressed
in terms of r a d i a l and t a n g ential s t r e s s e s ,
Relati o n s h i p s were deve l o p e d to define-the strains in
terms of radial and transverse displacements u and w.
A non- linear stress-strain r e l a t ionship was then used
together with strain-dis p l a c e m e n t relationships .to ex­
press the internal plate forces in terms of the d i s ­
placements u and w.
By subs t i t u t i n g equations
(6 ) and (16)
through (19),
into the g overning pa r t i a l d i f f e rential equations
and (4),
(2 ),
all terms except the pressure and inertia
terms could be expressed in terms of the displacements
u and w.
2 «3 Bonding .Function
The applied blast pressure
loading P was r e p r e ­
sented b y the foll o w i n g d e c aying exponential function:
P =
Pig,
(20)
6 shows the p r essure-time rel a t i o n s h i p of an actual
bomb blast compared to the exponential function.
As part of his research, H o f f m a n
(2)
e x peri­
ment a l l y d e t e rmined the p eak pressure p and the impulse
I, for each of a series of blast loadings on plates of
various t h i c k n e s s e s .
The impulse I is defined as the
positive area u n d e r the pressure-time curve of an actual
bomb b l a s t .
Por this a n a l y s i s ,■ Hoffman's -impulse data
was rede f i n e d in terms of a decaying exponential f unc­
tion as:
I
H o f f m a n 1s pressure and impulse data could then be used
to define the constant K in equation (21)
as:
K = $
'
(22)
2 . 4 , Governing D i f f e rential Equations
The -elemental e quilibrium equations
(2) and (4)
form the gove r n i n g d i f f erential equations of the system.
A simultaneous solution of these equations w o u l d describe
the plate d e f l e c t i o n .
and
(4) become,
In a b r e v i a t e d form,
equations (2)
for the ho r i z o n t a l or radi a l direction:
2 :Fp = o
- .(23 )
and for the v e r t i c a l or transverse direction:
S s z = 0
Equat i o n
.
(24)
(24.), if rewritten to separate pressure and
inertia terms b e c o m e s :
- W y ^ r z= - Pr
Ay
(25)
1
wh e r e the F 0 summation signifies that the p r e s s u r e ,and
inertia terms have been s eparated from the total F
summation.
The gove r n i n g partial d i f f e rential equations as
expressed above are not r e a d i l y solvable b y conventional
techniques.
",
2,5 Modifie d Galerkin Method
The M o d i f i e d Galerlcin Method.was u s e d to reduce
the governing par t i a l d i f f e r e n t i a l equations to an
ordin a r y n o n - l i n e a r dif f e r e n t i a l equation in time which
w h e n solved w o u l d permit the p l a t e ■d i s p l acement t o 'be
determined.
-12The Modified Galerkin Method is one version of
the meth o d of w e i g h t e d r e s i d u a l s .
This m e t h o d uses
assum e d solutions to the gove r n i n g equations which
satis f y the displacement b o u n d a r y conditions,
ne c e s s a r i l y the force b o u n d a r y c o n d i t i o n s .
but not
Assumed
solutions to the governing differential equations
prob a b l y wil l not satisfy the equations at every loca­
tion in the plate;
result.
t h e r e f o r e , errors,
or residuals w i l l '
Since the terms of the governing equations
r epre s e n t a summation of f o r c e s , the r esiduals will be
the errors in t h o s e ■f o r c e s .
If t h e .force residuals are-
then given v i r t u a l displacements considered to be
w e i g h t i n g functions,
in the assumed displacement m o d e s ,
the force times displacement products m a y be considered
to have the units of w o r k .
The Modified Galerkin M e thod
may be i n terpreted as r e q u i r i n g that the v i r t u a l work
done by all forces during a v i r t u a l d i s p lacement in each,
of the assumed modes from an assumed displacement shape
be i d e n t i c a l l y ■z e r o »
The modif i e d method is different
from G a l e r k i n 1s conventional m e t h o d in that the assumed
displacement mode shapes,
or rather the ass u m e d solu­
tions to the d i f f e rential equations do not have to
satis f y the force bound a r y c o n d i t i o n s .
O n l y the. d i s ­
place m e n t b o u n d a r y conditions must be satisfied.
I f .the
assumed solution does not satisfy .the force boundary
conditions,
boundary,
then the v irtual w o r k done by forces at the
or plate edge, d u r i n g the v i r t u a l displace­
ment must be added to the v i r t u a l work d o n e .b y the .
internal f o r c e s ,
'
Test results
.
(2) have indicated that a simply
supported circular plate w h i c h has been p e r m a n e n t l y d e ­
f orme d by a blast
loading acquires a d i s h e d shape
symmetric about the plate center.
The test results also
"13™
showed that for a n y large transverse deflection,
there is
an acc o m p a n y i n g radial deflection directed inward towards
the center of the plate.
This implies that the g o verning
partial d i f f e rential equations
(24) and (25) are each
functions of b o t h the radial and the transverse d i s p l a c e ­
ments u and w.
Equations were selected to act as a set of assumed
solutions to the set of g o v e r n i n g partial differential
equations.
The assumed solutions for the transverse and
r a dial d i s placements are:
w so qg cos
u - P1
Where
ocr
(26)
(2 ?)
l"Zf Z*j sin ex r
, and where R is the radius of the simple
support circle.
These assumed solutions a p proximate the d e f o r m e d plate
shape and also sat i s f y the d i s p l a c e m e n t .b o u n d a r y condi­
tions.
For the transverse deflection w, equation .(26)'
permits max i m u m de f l e c t i o n to be obtained at the plate 1 .
center
(r = 0 ), and satisfies the displacement boundary
conditions by a l l o w i n g no de f l e c t i o n to- occur at the
radius of the simple support
(r = R ) .
It also describes
deflections as be i n g symmetric about the r a d i a l center
of the plate.
Fro m equation
(26),
the vir t u a l displace- '
ment or w e i g h t i n g function for the vertical mode',
was defined as:
.Q
a
s- cos CKr
OM1O ,
'
■
(28)
Internal b e n d i n g moments were d e c l a r e d ident i c a l l y
zero in the development of an expression f o r the shear
force Qr .
However,
b e n d i n g moments might.be calculated
at the plate edge if the ass u m e d solution did not s atisfy
the edge b o u n d a r y condition,V To account for this error,
—14—
the v i r t u a l w ork was 1 also calculated at the plate edge
for a vir t u a l rotation.
The v i r t u a l d i s p lacement in the
ro t a t i o n a l mode was defined as:
&
-Lm =
The angle
d (<jr) =
<x t
-<xsin
(29 )
that the radial center line of the
deformed plate makes w i t h the radial axis, as shown in
Fig.
2,
could then be described from the assumed solution
as:
^w)
-I
C ” o>r) ™ tan*"
*-J? = tan
' ’
(q^ ^ sin <=< r)
(30 )
The sine and cosine of a n g l e J3 were then d e f i n e d in terms
of the assumed solution (26).as:
_ 3*,
.
sin
(^)
~ \ dr^ t
Sg =<sin ( x r
- \|l + qg^c x ^ s i n c<r
2r
cos
(-y?) =
(31)
I
tOr^-f O w ^
The assumed solution
K
11 t
c^2 S in^cx r ■
(32)
(26) was also u sed to define
the acc e l e r a t i o n for the i n e rtial force t e r m in equation
(25) as:
.
w
S2
.
.
q.2 cos or r
(33)
St2
In the assumed solution to the radi a l d i s place­
ment , equation
(27 )? the sine function per m i t s no radial
displacement to occur at the plate center,
and describes
a m a x i m u m radial displacement at the support radius.
value of
equal to minus one-fourth was selected to
account for the additional r a d i a l displacement,
bending,
during
of any plate element located above or below
the r a d i a l center-line
of the plate.
The magnitude of
Zf was selected to represent an average plate d e flec­
tion of about two inches. 1 It should be .-noted that the
A
-15v a r yin g . o f
by several orders of magnitude had no
noticeable effect on the end result of the deflection
calculation.
From equation
(27),
the r a d i a l mode,
f-
the virtual displacement in
was def i n e d as:
(54)
(1 - Zt Z ) sin oc r
Sq-
The radi a l and t a n g ential force and moment
expressions,
equations
(.16)-' t hrough (19), had p r e v iously
been d efined in terms of the displacements u and w.
assumed solutions for u and w, equations
The
(26) and (27 ),
were then used to express the displacement components of
the force and moment equations as:
Ir ”
(55)
-Sg^sinefr
r
(36)
S r ' ” cI1 c^ C1"2^ 2 ) cosexr
(57)
Mir)
™ i
Mlr)^= 5
Z^ 2 Z* 2 )cosc<r
(58)
It was the n obvious that all force and moment
expressions were functions of b o t h the r a d i a l and t a n g e n ­
tial displacements q^ and qg.
The g o verning differential
equations were coupled by the two displacement modes.
The v i r t u a l w ork for the radial or first d i s p l a c e ­
was defined in terms of equations (25)
m e n t mode,
and
(54) as:
swI - A o f h
V
v o l T e d g F h - ex t e r h Inter. edge
e d g J V, ^(39)
This expression also considered the force b o u n d a r y condi­
tions not s atisfied by the assumed solution in conform­
ance w i t h the M o d ified Galerkin Method. .
“ 16-
ext:er^al5 at the,
The external radial forces,
plate edge were known to he identi c a l l y zero.
However,
any error in the assumed solution would cause a non-zero
internal r a d i a l f o r c e ,
the plate e d g e »
i n t e r n a 1 ’•
In a s imilar manner,
second d i s p l acement mode,
tions
(25 ),
<£wp =
-f
the v irtual w o r k for the
S W p , was expressed using equ a ­
(28) and (29 ) a s ;
Wy^ r J p d v o l +
yVol
Jpdvol + / ^ ( F
y vnlz ~
>z- internal Iedge^2 ''d e d S e t edge
-M
'b'
e calculated at
ySdge 3- external
external
internal e<jge-“ 'd e a Ee = V a r e I r ^2 •da r e s
<4 0 )
This expression again considers any error in the assumed
solution b y evaluating the v i r t u a l w o r k done at the edge
of the plate b y the edge moments and f o r c e s .
External
moments and forces are i d e n t i c a l l y z e r o 5 however,
inter­
nal moments and forces 1could he calculated at the '-plate
edge because of the inaccuracy of the assumed solution.
CHAPTER III '
PLATE DE F L E C T I O N
3.1 Integration of Governing; Differential Equations
The gove r n i n g equations of the syst e m are each
functions of both
and
.
Since the pur p o s e of this
analysis was to determine only the transverse deflection ■
of the plate pp,
it was nece s s a r y that.the second mode
v i r tua l work expression,
function of qg a l o n e »
equation
(4-0), be r educed to a
This was accomplished by numeri­
cally i n tegrating the first mode virtual w o r k expression,
equation
(39),
through, a range of a r bitrarily selected
radial and transverse d i s p l a c e m e n t s .
Because of the
complexity of the e q u a t i o n s , nume r i c a l i n tegration was
p e rfor m e d w i t h the aid of a dig i t a l computer.
The
relationship b e t w e e n the two assumed displacement modes
was obtained by n o ting wh i c h arbitr a r i l y a s s igned nu m e r ­
ical values of q-^ and qg wo u l d cause the calculated
v i r tua l wor k
to be i d e n t i c a l l y z e r o .
Fig.
7 shows
the line of zero virtual w o r k calculated t h r o u g h the
assumed radial and transverse displacement ranges for
plate thickness of 1/8,
curve of Fig.
1/4U and 3/8 i n c h e s .
F r o m the
7, q,^ could be expreshed as a polynomial
function of qg of the form
= - . 0 0 0 4 6 4 ^ 2 5 + , -.OOGO^^g^ -.OZlSqg^+,
(41)
This r e l a t ionship was valid for the three plate thick­
nesses considered by this analysis.
With the radial displacement
then defined as a
functi o n of the transverse displacement p g , .the second
mode vir t u a l w o r k expression,
equation (40),
expressed as a function of.q- only.
could be
— 18—
The virtual wor k in the second d i s p l acement mode,,
was then calculated,
u s i n g the digital computer and
n u me r i c a l t e c h n i q u e s , over a range of transvere d i s ­
placements for several different plate thicknesses.
The
results are i l lustrated in Fig. 8 .
The v i r t u a l w o r k as a function of the assumed
disp l acement was considered comparable to a n o n - linear
force v s .. deflection r e l a t i o n s h i p of a spring,
as wo u l d
be part of a vibrati o n s - t y p e problem.
3•2 Governing Differential Equation
The system was then re a d y to be expressed as a
non-homogenous s e c o n d-order ordinary differential e q u a ­
tion h a ving one degree of freedom,
forced v i b r a t i o n s - t y p e problem.
and' r e s e m b l i n g the
The equation of motion
was of the form:
(42)
The
^ p r efix t o ,the d i s p l acement term qg is
the slope of some finite s ection of the second mode
v i r t u a l w o r k vs.
qg curve, w h i c h w ill be covered later
in this gaper,
A and 0, the prefixes to the a c c e l eration and
pressure terms respectively,
were calculated by Inte„
.
grating the elemental forces with their -virtual
displacements over the region of the plate as f o l l o w s :
S\L pressure
T[%
gdrd9, where P - pe
= 2
-kt
n (-ro< "I''--kt
o(2. -)P,e‘
Cpe-tt
(4$)
■2TT r r + I
<?¥
y ’rqgcoscxrr $ 2 drd©
2 inertia
(r + I )2,. (r + l ) s i n Q ^ ( r + l ) )
4
4<s<r
^ GdsC 1
^ Cr-M)) ~ I
8 =x 2
geX 2
= Aq2
(44)
^2
The limits of the. integrals were chosen to match
the test
C2 ) c o n d i t i o n s »
inches in diameter,
H o f f m a n ' s test plates were 26
and were s i m p l y supported on a 24-
inch diameter support ring.
However,
the plates were
shielded so that no blast pressure could be applied
outside of the support r ing of radius r, w h e r e r « 12
inches.
The i nertia of that portion of the plate w h i c h
extended one inch beyond the support circle was to be
considered,
hence,
the i n tegration limit of r plus one
inch.
■• The general, solution to the gove r n i n g d i f feren­
tial equation,
is of the f o l l o w i n g form:
q 2 = q 2 c o m plementary + q 2 p a r t icular
for which:
q 2 complem e n t a r y = D- sin
B
B
'(Se)
2 C t 4-B ops
AI Il
(45)
(46)
and;
q 2 pa r t i c u l a r
§p
,-k t '
The constants D and E were then evaluated fro m the
foll o w i n g initial c o n d i t i o n s :
I-. q 2 (o) = 0 ; at time zero the plate has no
initial displacement.
2 . q 2 (o) = 0 ; at time zero the plate starts
fro m rest.
(4?)
-20The complete displacement and v e l o c i t y solutions for
the system were then expressed as qg, and
where:
t
•q2
(48)
Cq2) t
cos
-f-
B(92)
A
sin'
t
K2
B ^q2')
(49)
S o l u t i o n of the Differential Equation
The m a x i m u m deflection of the plate wo u l d be
attained at.the same instant, in time that the v e l ocity
of. the plate had first dec a y e d to zero. ' A trial and
error simultaneous solution to the v e l o c i t y and dis­
p l ac e m e n t equations would b.e. complicated, by. the non­
linear stiffness of the. syst e m during its deflection,
As a simplification,
the s t i f f n e s s , or r a t h e r the
v i r t u a l w o r k vs. d i s p l acement curve as p l o t t e d in Pig.
8
—21—
was considered to be made up of several linear segments
each h a v i n g a different slope,
or s t i f f n e s s .
Fig. 8 was
r e plot t e d to describe the v a r y i n g stiffnesses and is
shown in Fig.
9»
A constant slope or stiffness
could then be determined b y .inspection fro m each linear
segment of the r eplotted c u r v e .
All terms of the d i s ­
placement and v e l o c i t y solutions equations
(48) and (49)
had t h e n ■b e e n ,d e f i n e d .
A series of trial and error solutions,
ering the variable stiffnesses,
consid­
were then made by
computer to determine the first non-zero time that the
plate v e l o c i t y reached zero.
The max i m u m plate d e f l e c ­
tion was then calculated for. that elapsed t i m e .
$.4 Results and Discussion
A . c o m p a r i s o n of the calculated m a x i m u m deflection
vs. permanent deformations from e x p e r imental tests (2)
is made in Table I.
TA B L E I
'DEFLECTION OF S I M P L Y SUPPORTED
C I R C U L A R A L U M I N U M (6061-T6) PLATE
S U B J E C T E D TO B L A S T L OADING
hickness ■... Pressure
(inches)
...(Psi) .....
’ .128
.128
1/4
1/4
3/8
3/8
.
185.
281. 582.
1080.
620.
2100 .
Impulse
(Psi-Sec)
.0712
.0837
.1482
.1965
.1893
.3600
Text Results
Permanent'
Deformation,
(inches)
Calculated
Max i m u m
D e f l e c t i on
(inches).
1.238
• 1.784
1.519
2.273
1.218
2.648
.89
1.12
1.06
1.57
.87
1.96
The deflection m e a s u r e d from a p e r m a n e n t l y d e ­
f ormed plate w i l l always be less than the max i m u m
deflection r e q u i r e d to obtain that p e r m a n e n t deformation.
-22The plate will re b o u n d e l astically to the final perma­
n e n t l y deformed position.
A calculated maximum d e f l ection w i l l p r o b a b l y be
less than an actual max i m u m deflection if the chosen
deflec t i o n mode shape is. not exa c t l y equal to the true
deflection s h a p e .
systems,
It has been shown that in linear
a bea m w o u l d be deformed to a p o s ition which
requires a min i m u m amount of potential' ener g y (7)•
' .
If
the m i n i m u m pote n t i a l energy concept could be extended
to app l y to the no n - l i n e a r system,
the relative magni­
tude of the calculated deflections
can be explained as
follows:
by not sele c t i n g an exact mode shape for the
calculation procedure,
the theoretical analysis requires
the plate to absorb something more than the minimum
amount of energy to deform to the shape of the selected
mode.
For any given blast pressure ptnd impulse there
is only some finite amount of e n e r g y available to deform
the plate to the shape of the selected m o d e .
The calcu­
lated deflections are a p p r o x i m a t e l y 72% of the magnitude
of the test deflections because of the a d d i tional plate
stiffness induced by the- in a c c u r a c y of the selected mode
shape,
However,
these results are several h u n d r e d p e r ­
cent m o r e .accurate than tTiose obtained by other calcu­
lation procedures
(2 ) and
(5 ).
3.3 Conclusions
The foll o w i n g conclusions were based on the results
of this analysis:
1.
For large d e f l e c t i o n s , the p r i m a r y resistance
to deflection is offered b y m e m brane f o r c e s .
2. • The procedure u s i n g the Modified Galerkin
Method appears to have merit for analyzing
the b l a s t - l o a d e d plate deflection problem.
-233 »6 Fu rthe r Studies
Further w o r k should be undertaken to define more
accurate de f l e c t i o n mode s h a p e s .
For v e r y small trans­
verse deflections in the linear stress- s t r a i n region, no
radial deflection of the plate center-line w i l l occur.
This tends to confirm that the plate stiffness for the
small deflections is a function of ben d i n g moments alone.
Radial displacement,
however, w i l l occur w i t h i n the plate
at any finite distance away f rom the plate center-line *
*
The fa ctor Z^.Z
as used in this analysis to describe
r adial deflections of any non center-line point should,
t h e r e f o r e , be a s s o ciated wit h the transverse deflection
qg as part of an assumed solution of the form:
u = Cq 1 - q ^ Z ^ Z ) sin°< r
( 50) •
This form of a ssumed solution w o u l d create no difficulties
in the basic calculation p r o c e d u r e .
It is also suggested that a series-type assumed
solution be i n v e stigated for describing the transverse
displacements.
A solution of the f ollowing form is
proposed;'
w = qg (A coscK r + B cos 2 » < r)
mere-,
A+B„x
and
A»
(51)
B
This series for m of solution gives a b e t t e r approx i m a t i o n
to the p e r m a n e n t l y deformed plate shapes obtained by
H o f fma n
(2).
H offman's test plates were not of a pure
sinuso i d a l s h a p e , but were d i s h e d rnore d e e p l y in the
center- sections *
.
The ba s i c deflection calculation might also be
extend e d to h a n d l b -rectangular-plates and consider
-
-24different edge support Conditions.
A procedure should he developed to predict the
■permanent defor m a t i o n and/or rupture that w o u l d result
from a n y given max i m u m d e f l e c t i o n .
-25-
Fig.
I.
Cylindrical coordinate system.
z
>-r
Fig.
2.
Free bod y diagram of d a t e
/
element
—26—
A
Fig.
3.
dr
B
Displacement of an element in the radial
and transverse directions.
0'
Fig. 4 .
Displacement of a line in the r a d i a l direction.
/
Stress
ClOOO psi)
Compression
Tension
V- = A
Fig.
5.
tanh ae
Strain (in./in.)
Actu a l and no n - l i n e a r stress-strain curves.
—28-
Pressure
Bomb Blast
Time
Pig. 6.
Pressure-time
curve of b l a s t .
-29-
92 (In.)
Fig. 7.
Displacement mode relationship.
plate
plate
plate
Qo (in.)
i'ig. 8
Calculated virtual w o r k in second displacement mode.
plate
plate
10 . -
plate
Fig. 9
Modified vir t u a l w ork c u r v e .
LITERATURE CONSULTED
1.
Y/ang, A, J., "The P e r m anenet Deflection of a Plastic
Plate Under Blast Loading", Technical Report No. 7,
Grad. D i v . of Ap p l i e d Math., Brown U n i v ., Dec., 1953•
2.
Hoffman, A. J , , hThe Plastic Response Cf Circular
Plates to A i r Blasts", Master's- Thesis, D e p t . of Mectu
Engrg., U n i v . of Delaware, J u n e , 1955.
3»
May, L. E., "Deflection of a Circular Plate S ubjected
to a Blast Loading", Master's T h e s i s , D e p t . of Mech..
E n g r g ., Montana State University, December, 1968.
4.
Jones, N., "Impulsive L oading of a S i mply S u p p o r t e d .
Circular Rigid Plastic Plate", Journal of App l i e d
M e c h a n i c s . March, 1968.
5•
Anderson, R. A., Fundamentals of V i b r a t i o n s , The
Macmillan Company, New York, 196?.
6,
"Strain Rate Tests on A l u m i n u m 6 0 6 1 - T 6 " , GM Defense
Research L a b o r a t o r i e s , General Motors Corporation,
TR65-69.
7.
L a n g h a a r , H. L., E n e r g y Methods in App l i e d M e c h a n i c s ,
W i l e y , N e w York, 1962,
MOHTAtiA CT*r
r
-__
_
Iuu15095 0
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»378
N712
cop. 2
Nolan, George Thomas
Maximum deflection of s
blast loaded circular
flat plate
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