LOW SPEED IMPACT OF PLATES OF COMPOSITE MATERIALS. Pei Chi Chou ...... L23=A44*TERMB. L33={A55*XNX)*TERMA2+( A44+YNY)*TERMB2.
REPORT NO. NADC-78259-60
Go CERTIFICATION OF COMPOSITE AIRCRAFT STRUCTURES UNDER IMPACT, FATIGUE AND ENVIRONMENTAL CONDITIONS PART I
0
LOW SPEED IMPACT OF PLATES OF COMPOSITE MATERIALS
Pei Chi Chou, William J. Flis and Harry Miller DREXEL UNIVERSITY Philadelphia, Pennsylvania 19104
DDC )
JNUARY 1978 S.MAR
291979
c_-) LU
FINAL CONTRACT REPORT
JulY 1976 - 31 Decmer 11977
A-J-1
CONTRACT NO. N62269-76-C-0378
LA.
i•
Q1-r
IAPPROVED FM PUDLC
I'ZJLSt WTSTRYE
W WLZNXMD
Prepared for
NAVAL AiR DEVELOPMENT CENTER Waminster, Pennsylvania 18974 "79
03 28 008
Uý C "
NO0T ICE S REPORT NUMBERING SYSTEM -The numbering of tedinical project reports issm by tho Naval Air Development Center is van*ne for Wjcfifc idetification purposs Each number con=~t of the Center acronym, fth calendar ywesn wic te ~number a sinK msewenumbeot er ow re wihinth seccificcand ar year,e the official 2*dki correspondence cAd of the Command Offic or the FunctiaWi Directorate re~sibrls~ for the repor. For eiunpls Repor No. NADC-78015-20 iricates the fiteed Center reprt for the yea V378, Wnpmepre by ýia systew Diucntort. The namer od es ame as fodowr cam1
OflHVl OR DiRECIORATE
a) 01 02 10
Commander, f"v Air Ou ~paren Center TatNicW Diwwm Nadit Air Wv*m an~p t Cant Cm"e MDIfetut Comuna Roomt 20Y& iKtrerte &M akmwcs Tekdmcig Directaes Senor 40 Comrnuication & "atgatr TWncdny Oirerzcrts 50SoftWw Cempm Dmectomta WAircaft & Crew Systm Tueioaog Ehracew 70 f~anv* AssesmMn R"== OD 6wing &wort GrW4 PR~IDIiT E WWNT - Ne dwiscm or kum ~n aw ros w1*gamnarc pdulto herm do rwf amst~n in mnw awb h Gvm o aw o y yor i te k o ri* to su~~dpa ts
APM f.
ATE. J7
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'7
/W
vK? .t9
UNCLASSIFIED REPORT NO.
P7
~NAC 78259-6
4ERTIFICATION OFPOHPOSITh IRCRAFT •TRUCTURES UNDER IMPACT, JATIGUE AND ENVIRONMNTAL CONDITIONS. Y PART I
;/
'1
•::
OW JPEED IMPACT OF PLATES OF
OOSI
MATERIALS
/• .
-.-
\
K
HHarry Miller *
..
,
William J./Flis /.'... ~~~~
Philadelphia, Pennsylvania".
*...
Prepared for NAVAL AIR DEVELOPMENT CENTER Warminster, Pennsylvania 18974
UNCLASSIFIED
N"
!03
28
fI• [
_._________"______,_
!1NCTASSj-T-ýT) SECURITY CLASStFICAT',;N OF TriS PAGE t)hen Da.,
REPORT PEFOREMCRM,
EntoredI
[OCU",.ENTATION PAGE U.
-REPORT NUMAD R
CO PLET.NG
GOVT ACCESSION NO.
3.
RECIPIENT'S CATALOG NUMBLA
NADC-78259-60 Certification of Composite Aircraft Structures Under Impact, Fatigue and Environmental Conditions. Part I-Low Speed Impact Qf.Ilates of Composite Materials
S. TYPE OF REPORT I PEIOD COVERE;
1. AUT$OR(s)
6. CONTRACT OR GRANT NUMBER(s)
14. TITLE (and Subtitle)
* Final Contract Report
July 1 1976 - Dec. 31 1977 4. PERFORMING ORG. REPORT NUMBER
Pei Cb3 Chou
William
..
Flis
N62269-76-C-0378
Harry ":iller 5. PERFORMING ORGANIZATION NAME AND ADDRESS
0.. PROGRAM ELEMENT. PROJECT. TAS*(
AREA 4 WORK UNIT NUMBERS
Drexel University Philadelphia, PA 19104 It. CONTROLLING OFVICE NAME AND ADDRESS
I2. REPORT DATE
Aircraft and Crew Systems Technology Directorate
January 1978
t.Naval Air Development Center
13. NUMBER OF PAGES
Warminster, PA 18974 f1:
14. MONITORING AGENCY NAME
219 ADORESS(If dilleront fro
IS. SECURITY CLASS. (of this Popottf
Controlling Offce)
UNCLASSIFIED , IS*. DE LASSIFICATION, DOWNGRACING SCI4EDULE 16. -ISTRIOUTION
STATEMENT (at this Report)
Approved for Public Release; distribution unlimited
17.
IN
DIST R19UTION ST ATEmENT (of the, abotroct entered In Block 20. it d~iffeent, from Z..) Ir.
1. .
SUPPLEMENTARY NOTES
It
KEY WORDS (Continue on toveu. side if noees
y and id
y by black numbo..
.
..
Composites Structural Response !'
Impact. Plte
Experiments
20.AGTRCT(Cntnu owhich reere give ld I ncosa dn Ienif b boc sreport asere of design urves the' peak strain in several types of prset simple structures (beams atI4 plates) due o low velocity impact. These curves have been developed
based on both analytical solutions of impact and numerous impact experiments. It is sahown that all impact problems may be divided into two regimes based on the, ratio of the mass of the structure to the mass of the impactor. For large impactors multiple impacts occur and the structural response depends chiefly on the kinetic energy of the impactor. For small impactors only a single impact occurs. Its force and duration determined chiefly by contact effects and the
StCURITY CLA3iIFICATION OF TIlS I AGE't11hen Jo•ae-S
c-94
20.
impactor momentum governs the response. are required for each regime.
fIj VP
Thuis different formi of design curves
NADC,- 78259 -60 TABLE OF CONTENTS Page List of Figures List of Tables
[
I.
INTRODUCTION
II.
DESIGN CURVE FOR IMPACTS BY LARGE IMPACTORS
7
A. Generalized One-Degree-of-Freedom Model
7
F
1. 2. 3.
Simply Supported Beam Simply Supported Plate Clamped Plate
10 11 12
4. 5.
Comprehensive Impact Strain Curve Critical Impact Velocity Curve
13 15
B. Timoshenko Solution of Transverse Plate Impact
V
I
1.
Description of Solution Method
18
2.
Dimensionless Form of Timoshenko Solutions
20
a. b.
20 22
3. C.
•:3. I'
17
Plate Impact Beam Impact
Parametric Study for Large Impactors
24
Impact Experiments
27
1. 2.
28 32 39
Specimens Strain-Measurement Impact Experiments Impact-to-Failure Experiments
D. Design Curves for Shear Failure Due to Impact 1.
Simply Supported Beam a. b.
2.
40 46
Relative Importance of Shear and Bending Critical Impact Velocity Curve
47 49
Simply Supported Plate
50
III. IMPACTS BY SMALL IMPACTORS
56
A. Simply Supported Beam 1 1. 2.
56
Approximate Analysis Timoshenko Solution
56 60
B. Simply Supported Plate 1. 2. IV.
63
Approximate Analysis Comparison vith Experimental Results
63 69
DESIGN PROCEDURE
70
A. Use of Design Curves in This Report
70
B. Construction of Nev Design Cutves by Experiments
80
C. Construction of New Design Curves by Analytical Tools
62
S...
.m
.
......
l
NADC-78259.60
V.
SUMMARY
84
VI.
REFERENCES
85
VII. APPENDICES
APPENDIX A STATIC SOLUTION OF A SIMPLY SUPPORTED ORTHOTROPIC PLATE
87
APPENDIX B STATIC DEFLECTION OF A CLAMPED RECTANGULAR ORTHOTROPIC PLATE DUE TO A CENTRAL POINT LOAD
91
APPENDIX C COMPUTER PROGRAll FOR CALCULATING TIMOSHENKO SOLUTION OF PLATE IMPACT
100
APPENDIX D SIMLY SUPPORTED BEAM CENTRA.Y LOADED BY HALF-SINE PULSE
105
APPENDIX B DETILED RESULTS OF IMPACT EXPERIMENTS
110
APPENDIX F
215
THE EFFECT OF THE APPROXIMATION U0N THE PREDICTED IMPACT RESPONSE OF AN ORTTOTROPIC PLATE
JI.
Ai 'A'a
NADOC-7-8259-60
(4
List of Figures
#2
Figure
Page
1. 2. 3. 4.
Comparison of large and small impactor cases One-degree--of freedom model, momentum conserved One-degree-of-freedom model, energy conserved Functions of aspect-orthotropy parameter n~ for constructing impact design curves 5. Comprehensive impact strain curve for simply supported and clamped rectangular plates 6. Generalized strain curves 7. Exploded view of plate clamping device 8. Typical c vs. H curve for clamped orthotropic plate 9. Typical c~ vs. H curve for simply supported orthotropic plate 10, Strain curve for clamped plates impacted by large masses
3 8 8 14
11. 12. 13. 14.
38 41 42 43
Strain curve for simply supported plates impacted by large masses Typical critical velocity curve for clamped plates Critical velocity curve for clamped and simply supported plates Typical zeflectoscope trace of plate specimen
15. 16.
Typical ref lectoscope trace of plate specimen Typical ref lectoscope trace of plate specimen 17. Shear stress curve for simply supported beam 10. Critical impact velocity for shear and bending failure 19. Edge shear stress curve for simply supported rectangular orthotropic plate. 20. function of aspect-orthotropy ratio qjfor shear stress curve of simply supported plate 21. Generalized strain 0~ vs. usess ratio H for Impacts of simply supported beam by small impactor# 22. Plot of g(8i,r) showing weak dependence on 8
lb23. I.25. 24.
16 26 33 35 36 31
44 45 48 51 54 55 62 67
Strain curve for Impact of simply supported rectangular plate
68
Selected design curve for impacts of simply supported beamsn by large impactors
71
Selected design curve for Impacts of simply supported beane
72
bysalimatr
WNAC-7825960
'I
26.
Selected design curve for impacts of simply supported plates Impacted by large impactors
76
27.
Selected design curve for clamped plate impacted by large impactors
77
28.
Selected design curve for impact of simply supported rectangular plate by a small impactor
78
NADC-78259 60 List of Tables
Table
1A1
Page
I.
Parametric Study: Plate Impact Cases Calculated by the T1oshenko Solution
25
II.
GraphiteiBpoxy Plate Specimens
29
III.
Shear and Flexural Stiffnesses of Graphite/Epoxy Plate Specimenu
30
IV.
Design Curve Parameters for Graphite/Epoxy Plate Specimens
31
V.
Computer-Calculated Values of f(B)
59
VI.
Calculations Using Timoshenko Solution for Large Nasa Ratio N
61
NADC-76259-60
Foreword
This is part I of the final technical report for Contract go. N6226976-C-0378, which is sponsored by the Naval Air Development Center, Warmuister, Pa. The work was performed during the period of July 1, 1976 through December 30, 1977. Mr. Lee W. Gause was the contract monitor. The contracted study is under the title "Certification of Composite Aircraft Structures under Impact, Fatigue and Environmental Conditions"; parts I and II of the s'udy are under the supervision of Dr. P.C. Chou, while part III is under Dr. A.S.D. Wang, both of Drexel University, This report concerns part I of the contract, low speed impact of plates of composite materials. It is a self-contained report, including definitions of all nomenclature used, and its own Introduction and conclusions. Ilia authors wo4ld like to thank Dr. Edward J. McQuillen, Dr. James L. Huang and Hr. Lee W. Gause for the frequent technical discussions. They would also like to thank Mr. Frank Patota and Mr. George Chou who helped conducting part of the experiments.
SNADC-78259-60
I,
k
INTRODUCTION Aerospace structures frequently undergo impacts by blunt objects, in-
cluding dropped tools, hail, and runway stones impinging on exposed aircraft components,
and foreign objects entering jet engines.
component subject to such an impact is
often of the structural type, rather
t -an due to local penetration or indentation. (e.g.,
The failure of a
A number of numerical techniques
fl~ite-eltment method, lumped-parameter models) are available for calcu-
lating structural response to impact, but these are typically complicated, time-consuming, and usable only on a problem-by-problem basis.
Designers need
a quick, convenient, widely applicable method for this purpose. In a previous report [1],
a method was developed for constructing a de-
sign curve which predicts the response of a givev type of structure to impact loading.
This curve gives the maximum strain in the structure, which may have
various dimensions and material ptoperties, due to impacts involving a certain range of impact masses and velocities.
An example of the bending response of
a simply supported beam under central impact was prevented in detail.
Both
experimental results and numerical calculations involving s&veral solutions of beam Impact were used in establishing the design curve. studied in [1I
were liaited to large impactor mes,
The Impact cases
where the impactor has
uses roughly equal to or greater than that of the beia. In the present report, plates, and iacts
the desigw-cutve approach is extended to aniaotropic
by smeall iapactor4.
Shear failures are also studied.
elemntary wmdel of impaet on beano presented in all structures.
).I
This modal is
(I) is
The
generalized to eabrace
then applied to the cases of claped and
NADC-78259 -60
simply supported anisotropic rectangular plates and to predicting impact
failure due to shear effects in both beams and plates.
Design curves are
also developed for predicting the response of beams and plates to impact by small impactors.
In each instance, both analytical methods and impact
experiments are employed in generating the design curves.
In dealing with low-velocity impact problems, it is convenient to divide them into two domains, based on the ratio M of the mass of the structure to The motion of the impactor and the
the mass of the impactor (see Fig. 1).
response of the structure for a large impactor differ greatly from those for a small impactor.
The two domains will be treated separately; the assumptions
and final design curves are also different for the two domains. In impacts where the mass ratio M is
small (large impactor),
it
has
been observed that multiple collisions occur and that the general motion of the impactor follows the path of the structure at the impact point (Fig. la). The final rebound of the impactor takes place after the structure has reached its maximum deflection.
Further, the entire event is generally more pro-
longed than the fundamental period of structural vibration.
On the other hand, when M is large (small impactor), only a single,
S!-
.udden collision occurs, after which the impactor rebounds and the structure continues to vibrate freely, reaching its maximum deflection at a later time (Fig. lb).
The duration of actual contact is
than the fund.ment&l period of vibration. 4,
characteristically much shorter
The sudden rebound of the impactor
is due chiedly to the elastic resistance of the structure and the iqmator to local indentation,
i.e., contact effects between the two bodies.
2
NADC-i,6259 4,.,
~44 0u
Q 4%
41
"4
41
E4.
0
0
NADC -78259 .60
In [1], six analytical models of impact were examined, including two classical one-degree-of-freedom models; a two-degrees-of-freedom model which accounts for contact effects; a solution by Clebsech [2], which assumes plastic impact and treats the beam with attached impactor as a free-vibration problem;
a mudified Clebsch solution due to McQuillen et al. [3]; and Timoshenko's solution 14], which couples the impactor displacement with the beam deflection using Hertz's law for contact deformations.
The dimensionless parameters
which determine the peak impact response according to each solution method were derived. It was shown in
[1] that, for those models which neglect contact effects,
the maximum strain due to impact, when properly normalized to i, is a function of the mass ratio K only and is
independent of any other parameters.
These
solution methods, however, are generally not suited for treating problems involving small impactors.
Conversely,
the two models that include contact
effects indicate a dependence of C on M and another parameter, and are useful over the entire range of M.
For large impactors, it was demonstrated that the
dependence on the other parameter is weak, so that a single c vs. M curve approximately represents all impact situations,
in addition, when data from
"assorted impact experiments were plotted in coordinates of c and M, a single curve could be drawn through all data points, with about + 303 verlation, for
...
large impa.tr.
kNADC-78259-6 it.this report, several approaches are taken in extending the designturve concept to impacts of other structures by large impactors (Section II).
rVirst,
the one-degree-of-freedom model described in (l] is generalized to encompass all structures and is
then applied to anisotropic laminated rectang-
ular plates with clamped and simply supported boundary conditions. order to identify additional significant parameters,
Then, in
the structural impact
cquatiGni of Timoshenko are cast in a dimensionless form
the influence of
each of these parameters is appraised by a series of calculations using
Timoshenko's solution in which the values of the parameters are systematically %aried. No
--
we discuss two series of impact experiments performed on lam-
inated graphite/epoxy nlates; in one series, the bending strains in the plates we.e wviurzJ, and in the o'41er,
the plates were repeatedly impacted at grad-
ually increasixg velocities until fsilure occurred. periments is
The purpose of these ex-
tvofold* co verif:' thcA unalytically constructed design curves, and
to dvior.3t•oe now deig~a curves may also be constructed experimentally.
Finally,
"design curves for predicting she!i: failure due to impact are developed, and the relative importance of shear und beLAng effects in
"i•
impact failure is did-,
cussaed. For cate"
in which the mass of the impiactor is
mass of the structure (largo H), the on tij
.N°endence
s&all with respect to the
of the generalized strain c
was ratio M is not axclusives an. ther dimensionless parameter whi,*1
Involves the Seometric and materia4 properties governing the contact effv,•,ts between the two Widies can also be shown to be aibnificaut,
This coantact
paramter, denoted A, and the mass ratio H together determine tae duration of contact between the struzture and impactor (relative co the fundamental period of vibratict), Wdch Is cypical&y quite ihort. is
Clearly, the duration of contact
Ljortan. In daterululug the responso of the structure.
$
NADC-78259 6O The most sophisticated solution method applicable to problems involving small impactors, the Timoshenko solution, involves a nonlinear integral
equation which can only be solved numerically on a problem-by-problem basis, and is thus not convenient for developing a design curve.
However, by making
the simplifying assumption that the structure does not appreciably deflect
during the short period of actual contact, the problem uncouples into two parts - the elastic Hertzian impact of a sphere on a semi-ihfinite body, and the vibration response of the structure to a dynamic load (i.e., the contact force).
This leads to an approximate relationship (for each type of structure)
between the parameters Z, M, and X, which may be further simplified to a simple equation in the mass ratio H and a new generalized strain for small impactors e* (Section III).
By comparison with experimental data and with
calculations involving the complete Timoshenko solution, the form of this relationship is shown to be useful as a design guide. In Section IV, procedures for using the design curves presented in this report are described,
and a few examples are giveta.
Also included are methods
for constructing new design curves for structures not treated in this report, based on either analytical tools or impact experiments.
, .:
_
.•.6
#NADC: II.
782594
DESIGN CURVE FOR IMPACTS BY LARGE IMPACTORS
For impact cases where the mass of the structure is small compared to the mass of the impactor,
the structure may be assumed to behave as a simple
spring, or as a one-degree-of-freedom mass-spring system.
Details of this
approach as applied to simply supported beams were presented in Ref.[l]; in this section, this impact model is briefly reviewed, because it will be useful
lis
in the development of parameters in plate impact problems.
These parameters
are employed in constructing a design curve for predicting the response of
plates to impacts by large masses.
Finally, by examining data from a large
number of experiments and by studying impact solutions using more sophisticated analytical techniques,
Generalized One-Denree-of-Freedom
A.
In this model, ! I
the validity of the design curve is established and the Model
the impactor and the structure are considered attached to-
gether after contact and move together as a single mass m, and the motion of the structure is governed by an equivalent spring.
The initial velocity of
the combined mass, v 0 , may be determined by two methods: the conservation of momentum (Fig. 2), (Fig. 3).
k
wheat
One is based on
the other on the conservation of energy
In the firset.case.
m2 is
the impactor mass, and eam
is an "equivalent"
ases of the structure.
The equivalent mase may be obtained by matching the total kinetic energy of the structure with the kinetic enegy of the unknown equivalent mass traveling at the .velocity of tha impt
polit in the structure, assmumnn
&hape is the static ifefletlon Cwove.
that the defacti=•
oe o
Yoc., soply supported'beamp,
". •
.
.
OX44D.25949a
V2
w
K,
(a) betore itapact
(b) immodfately alter imptact
21. One-degree-af-f readow mdel of
7jp-r.r
v
qat
etmopuvd
S
(a) before Impact VSture
'3.
(b)at wax±izaum deflect ion
Ona-degr*ec-of-freedo
akodel of
6qw~et# sumu O*Rvwvw*
#INADC76259.60 By equasting the momentum of the impactor before impact, m2 v,
e - 17/35.
with the momentum of the combined mass m, we have
v 0 - v/(1 + eM).
(2) (3)
M =rn/rn.
where
Note that according to Eq. (2), the energy before and after impact is not conserved, i.e., mv 22 In the energy conserved case, we assume e
v
-
0, so that
(4)
v
As a result, the kinetic energy in the impactor is conserved, and will be entirely converted to strain energy in the structure.
For small values of
M, the difference between these two approaches is small, The spring constant K 1is the force per unit deflection, with respect to a force acting at the impact point in the direction of the impactor evelocity, or
P-K w After acquiring the initial velocity vO, to perform a free vibration.
S=
the mass-spring systea is assumed
The maximum deflection is then VoV
"In modeling the structure by a mass-ep-ring system, it is inplied that only "the first mode of vibration is retained and that its mode shape is the ae
i
4•'
as the static deflection distribution under a concentrated force.
•ext, 1
we shall consider the most critical strain
occurs at a knIow
$
vI ,un
:i~i'•.9
be found.
point 2.
c2
In the structure which
Assume that a proportional relation between c 2 and
S
NADC.78259 60 2
(( d l
(6)
where d 1 2 can be determined from the static deflection distribution, or from experimental static measurements.
Combining Eqs. (5) and (6), we obtain the
maximum strain as VO •.
or, combining Eqs. (1),
(7)
(2), and (7),
mV 2 2 I.
-- d
(8)
-
d1 2 F(1+:) -Y,
Simply Supported Beam The next step of defining a generalized strain c is best illustrated by
considering a specific type of structure.
For a simply supported beam of
l'Ugth L and depth h, impacted at middle span, we have
d
(9)
L216h
and3 X K
I
48I/L
(10)
•
Therefore,
V
V4 M(l+eM)
Cb
fS
where cb
M
., the velocity of longitudinal waves in a bar.
Defining the generalited strain as
~~ma h(12)
v
we obtain finally, -i17.
.m .I .
V4H10
....
(13)
which is the equation of the design curve for simly supported beamsimpacted
by aemvassw,
e given in (I.
to
NADC-78259 60
The corresponding expressions for d 1 2 , K1 and e for simply supported and clasped orthotropic plates will be given in 2.
the following sections.
Simply Supported Plate The impact design curve for simply supported, rectangular,
orthotropic plates may be generated by applying the generalized one-degreeof-freadou model.
For transverse impacts at the center of the plate, we first
the corresponding static problem involving a centrally applied con-
S€consider
centrat*ed load.
In this case, the maximi
deflection occurs at the center of
the plate, and the maximn
strain on the surface opposite the applied load.
shonu in Appendix A
that subject to a mild approximation, the spring
It
is
constant of a plate so loaded may be expressed as
w~4 (14)
4 D
.
-
S•there (15) u-l,3.5
n-l,3.5 Co
(17)
*
Ca 4 + 2
2
(1
4 2
n 2 n+
n
The strain-deflection constant d 1 2 releting the straln due to bending in the x-direction to the uaxina deflection vi
y
d12.
,2a
1
L•
(18)
NADC-78259-60 where
2 m1l,3,5 By combining Eqs. (14) and (18) with Eq.
no1,3,5 (8),
and assuming that e
(19) -
0
(conservation 'of energy), we obtain
C
f iW
2
h
h)
D
gr
fl(n)j
If we define the generalized strain
ex mCQAX
2 m 1/2.21...,..20)
cl2" ( f (01
(20)
as
k
(21)
maex Vhere ep- iV7-its the speed of flexural waves in the plate in the 11 x-direction, and k is the radius of gyration of the plate, then, !k
(22)
where
81(n)
This function is plotted in Fig.. 4. C
(23)
f(n)lIM f
,
Similarly, we have 24)
1'Z
yy
are functions not only of the structure-to-impactor mas ratio Mbut also of the 3.
spect-orthotropy ratio n.
Clam&ed Plate We my develop the design curve for clamped-edge. rectangular,
orthotropie plates In a similar manner.
The spring constant of such a plate
with respet to a central point load my be expressed approzxiately (see 12
I
NADC-76259 -60 Appendix B) as b DI1
K1 "
3 a f 3 Wn)
(25)
The strain-deflection constant d1 2 relating the strain in the "-direction with the maximum deflection is
a2 f 3n)
d12
h
_
(26)
where f 3 (n) and f 4 (W) are given in Appendix B. By combining Eqs.
(25) and (26) with Eq.
(8),
and assuming that e
-
0
(conservation of energy), and defining the generalized strain according to Eq. we obtain 2 (n)
S8
(27)
where h2 (n) -
This function is
f (q)/lF
also plotted in Fig, 4.
(28)
Similarly, we have
g2
Equations (27)
and (29)
rectangular plate.
(29)
represent the impact design curves for a clsmped
Again, the generalized straine.
C and c are functions Xn y
of both the mass ratio M and the aspect-orthotropy ratio n. 4.
Cqpprehensive Impact Strain Curvy
ExamLnation of the design-curve equations developed above, Eq&. (22)o (24), (27), and (29),
hows that a different
vo. K cur
mugt be prePaed
not only for each boundary condition but also for each vale of the aspect orthotropy parameter q.. :*,'-'
prehenswv
These curves can be combined into a
s design curve if we introduce a nea CmsnlxwL
13
Single, coo80eral'sed 14s
(21)
NAOC.78259 60
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.-
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0
-.. **-..
t
0
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'.-.
C,
bO 54 a0
I
'C,
I
N .
£
U _______
Q 54
I
--
J
us,..-
I
14
II
-
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o
I
..
t.
q Ig .
4.
I
0
--
.4
/
I ii
"0
j
tJ
*0
C'
34
00
.
e
NADCJ-78259-60 strain, c, defined as V
,
i
1,2
(30)
where gl(n) applies to simply supported plates and is g 2 (n) applies to clamped plates and is
given by Eq.
given by Eq.
(28).
(23) and
Thus, the
equation for the design curve in c vs. H coordinates becomes I
-
-.1/2
cM-(31)
All impact data, regardless of the boundary condition (simply supported or clamped) or the value of the parameter q may be presented on a single curve. curve is shown in Fig. 5,
This
in which are included data from numerous impact
experimts which are discussed in detail in a later section. 5.
Critical Imact Velocity.Curve The design curves in term of cx or c vs H give the =aimua strain in the
plate for a given impact situation.
This strain .•ay be compared to the
ultimate failure strain of the plate (assuming a nsximum-straii failure criterion) to determine whether the plate fails as a result of t given impact. SHowever, by introducing a new parameter,
the dimensionlesa
i-act
velocity,
we can construct an alternate form of the design curve which tan be directly used to deteraine whether a given impact causes failure of the plate. the curve
ay be developed either by analysis or by expervamt.
Agoin,
To do this,
a dimensLonlesa impact velocity, v, to defined a
V4 h-
(32)
p
w(* a rv Is the 1sact wlocIty, b Is theaplate thickn2ess, c
to the speed of
fle=&ra waves In the plates and k to the radius of gyration of the plate.
K.I'ADC -78259 -60
,
n
LU !
0-0G0
0 -. wL*
~ £ u
4 CI
LU -
Q-
LU (D
0
w C-w x: -- VI
I U..
LU
xkJ
Wi
I-U)
t~a.
P~ 0W 0
I
to'-) 0
0
a.. w m
m
0
w w04
L
~41 L41
oc 0000> >
64 03
00
100
LUU
uiu
z1
..... .
Cd
NADC.78259 40 Then, by Eq's (21),
(22) aua (32)
the critical dimensionless velocity •
(i.e., the lowest value of v ct which failure occurs) is related to the failure attain Cf by
(33)
f~ In v vs. M coordinates, the curve v
-
v (M) divides the piane into a safe
region (btlow the curve) in which failures due to impact do not occur (according
a
to the aasumed maximum-strain failure criterion) and an amsafe region (above the vuxre) in which failures do occur.
k
That is,
if
the point (v,M) corresponding
to a given impact situation falls above the curve, then failure occurs; if the point falls below, no failure occurs. sign curve equation. A typical v
Thus, Eq.
(33) is
useful as a
Le-
vs. M curve is shown in kig. 12.
Further, if we define a new dimensionless impact vi4locity v such that
,
;
,i(n) then the equation of the alternate design curve, in terms of v vs
This single curve, v
v,(M),
M, is
may be used to predict impact-induced failure
for all place structures, regardless of the boundary conditions or the value of the aspect-orthotropy ratio n (provided these structures have the saea value of failure strain Cf). nominal value (say,
Note that the value of c
can be taken as the
the static ultimate strain) of the given material or can
be determined by performing a single iWpact-to-fallure test.
A typical
vs. M curve is shown in Fig. 13. B.
Tioshenko Solution of Treneverre Plate
gmet,
Tim. hanko's approach for solving transverse impast problem on simple
beamn by coupling Hertz's law of contact with the Suler beam equation has beas 17
NADC 76259 60 extended to the case of a simply supported rectangular plate by Bringen (5] and more recently to the case of a simply supported anisotropic laminated plate by Sun and Chattopadhyay (6].
In this section we review the salient
points of the latter solution and show how it may be used to develop an impact design curve for anisotropic plates.
In addition, a normalized form of the
equations corresponding to a special case of this solution is derived, so that the dimensionless parameters governing the peak strain response may be identified.
Finally, a series of numerical computations is performed, in
which each of the derived parameters is systematically varied in order to demonstrate the sianificance of each parameter in describing the response to a given impact. L.
Description of Solution Method The anisotropic plate equations of Whitney and Pagano [7) are used
to predict the plate motiou.
Neglecting effects of rotatory inertia, the
deflection of a symmetric crons-aly laminatel plate due to a centrally applied force F(t) is 2.f-2 V(X.y.t)-•
¥(-) sinWVM(t-r) dV F'"1(•)
sin where vi' a. and b are the seas w
,USto 1,,.5..,-O
*in
(35)
and planar dimensions of the plate, and
are natural frequencies dependant on a and n and the properties of the
£-I
's law of contact is
"wead
to hold
I -2312
18
(36)
NADC-78259 60
where k2 is a constant and a is
the indentation of the impactor relative
to the plate surface, or
Q
(ai b
for central impact, where w2 is
0
UM3/
the impactor displacement.
Newton's law applied to the impactor
vt -
-
t)
m2
is
,f Of dt dt
38)
These four equations (35-38) may be combined into a single nonlinear integral contact force F between the plate and the impactor,
equation in terms of te
O
--vt-
1 mu
,3,5
t
t
1
p12/3
•:
n=1,3,5
dt dt
P
m
F(T) sin wmn(t-T)dT O
(39)
which is solved numerically by applying the small-increment method suggested by Timoehenko,
in which the contact force F is assumed to be constant or
linearly varying during any time increment Ar. to calculate the force P
Expanding the above integrals
during the ith time interval, we obtain
i--( l l~~i'i
'
•5:i'
whereL2
2,11
-
3
coe[w m(t!-j+i)&r],
If the coutact force is
(40) 3
"
upo
-
a
l,3,S,...,e.
approximated as a piecewlse linear continuous
function of tun, with an average value of IF during the i
I
time step$ then
NADC.*78259 -60 I
SDi-j+l Fj
2[(-l)F1 + (i-2)(F 2 -F1 )
-
+ (i-3) (F 3 -F 2 +F1 )+. (F1
1 -F 1
_2 +F1 _3
"+(i-I) (Fj-F00)+F_
F1 )] +
1 -F 1
For computing the solution of Eq.
2
"+
+
.+F_2'... -(F -T-F1 )
(40),
Sun and Chattopadhyay have
suggested a recursion method, but we have found that such a time-saving approximation is unnecessary.
A listing of the computer program for solving
this equation is presented in Appendix C. 2.
Dimensionless Form of Timoshenko Solutions a.
Plate Impact
By expressing the Timoshenko solution in a particular normalized (dimensionless) form, the parameters governing the impact response of plates may be identified.
Specifically, it
is
shown that the generalized strain
depends on three dimensionless parameters: the mass ratio M, the aspectorthotropy ratio n, and also another paraumeter which involves the geometric and material properties governing the contact effects between the plate and the impactor. For a simply supported rectangular plate impacted at the center, let us consider the governing equation (39)
20
and the strain equation,
x
NADC-78259.60 I,(.)
S2hv X
a2
sin * wv(t-.T)dT
n
(41)
=
If we assume that the plate is specially orthotropic in bending (i.e., D16 D26
0) and further that the flezural rigidities are related by 26i
D
)Fl
+ D
(42)
%
(see Appendix F) then the natural frequencies are
wm-
,
Am(n)
where
11•2
D
. ,2 + 2 14'm
A
I
(43)
(,b) •Dll
n-
(44)
Note that this expression for the natural frequencies Is a special case of that employed in the Sun and Chattopadhyay solution, In hicbh the approximation of Eq. (42) is not used. In order to uormalize Eqs. (39) and (41), the followlng dimmasonless variablas are definad, t
i *-F
S"
tu
a3 w /D vb
(45)
Recalling the deftnition of generalitLd strain,
'•:-* X then Eq&.
%'Za
€_k
hv
P
(21)
(39) and (41) may be writtn as 1/3 4.2/
-
1)X
10 L r A 6
o
t
it
.•"•oa W i•.. ) 8•m
(i)[dt
(46)
,NADC -78259 60 22
(t
-
ff
-4( )Z-,,t(47d J1+f) 3i
-(")sin
A(
'0
where M-
/(:)
and A - (
3b)
(48)
.-
a3
vk2
and ; is a dummy variable for dimensionless time
t. It can be seen from
these normalized equations that the generalized strain ex depends only on the mass ratio H, aspect-orthotropy ratio n, and the parameter Ap; that is
(48) may also be expressed as
The parameter IP defined in Eq.
p
p
(49)
x(MHnrI p)
x
v )225
",6 A
Thus, the parameter XP can be described as the product of several dimensionleos quantities.
.
{ e~~i fQ-•
xp
..
(plate thickness\ ( .plate
b.
e
L j
, impact Velociety, .
r
x [function of aspect-orthotropy ratio]
lea-th
the equatious for the Timoabenko solution of beau impact
presented and discussed.
above procedure.
These equations may also be normlized by the
If we Aefine
t,
,,
.
.
Beam Impact
"InRef. (11. wvet
.__!ave speed
force\ 2 elasticforce,/ Aconsant)x (constant) x ~. (contact
•aL~ml'
,L
•..,.
/E.,
NADC-78259.60 then the equations governing the response of a beam, corresponding to
,4
Eqs.
(39) and (41) for the plate, may be written as
1/3 -2/3
¼
'.
ttPd ~
Jldtd
._t-:
(51)
•) sin i2(t-_;)dr 1?::
12
4
00
.f
and
' 7
I
--
'
F(•) sin i (t-T)d;
2(7 (52)
where
¼ .
which way be
)() 4
(
2
f,
1 2 W
h2(3
(13
txpressed as
on force deflect contact force,
Xb=(constant x wrave
pe.ed
x (radius
VNip&ct velocity It can be seen from Eqs. (51)
2
of jratton
depth of beau
and (52)
thOt in this case the generalized
strain c depends only on the mass ratio A and the contact parameter
7+
',
or
-
in Ref.
(1)
it
was shown that for large impact masses (M 7).
S(Opn) is plotted over n for two widely different values of
8 ,7 and 500;
f
is practically
IIL
7il
17
I
-
-1 -
____
j\QV.TM";
*
__
L
-T
_____
*~..
-
%4
4
S--..
.
-
_0 T
~-77.
14--
~--
4.1 1L :~V4
ro
____
NADC -78259.60)
f
Approximate solution
N._____
......
.
. X ..Dexe .
7t
--
's
eperient
IV
I
l
L
LLzJi__~ft~
~4444j.j4z FigueCuve 2. Sra~ or upat ~ SiplySupot4ý
Retgua T=6.
laeb
aSal
aa
NADC-76259-0 Cotaparison with F:perimental Results
2.
In Fig. 23, the design curve for impact of plates by small masses is presented along with the results from experiments performed at Drexel.
Two
graphite-epoxy plates which were also used in the large Impactor experiments, F1 and F2, were impacted by aluminum projectiles.
These experiments were
performed using an air Sun with the projectile velocity recorded by a photodiode system.
Data from theae experiments are tabulated in Appendix B. As
can be seen in Fig. 23, the agre.ment between the theoretical design curve and the design curve based on experimental results is of the same order as in the large impactor case;
that is, the experimental curve is about 40% below the
theoretical curve and has approximately the same slope. Also tabulated in Appendix E are experimental data found in SchvioSer
A
These experiments were paerformed on a square Duxaluminum (AlCWSgl)
-:[11].-
'
plate. 550 x 550 x 4.97 m~, centrally impacted by stme-I sphere:% of ZO- aud 30- = *
radius.
These data, vhan plotted tl E* vs. N coo dAtes, do not coi"a:a
well vith either the Drexel experiments or the'theory.
One
discrepancy !ies in the site of the plate tested by Schuieger.
planeton of this Iu tho Drexel
ixperisents. tha ratio of the plate span to thickness (1./h) has a value -, 18 whareeas in the izxparimants performed by Schiegar, thio ratio has a value of 11. Thus, actordlng to Zener [121,
the responsa of the plate used by Sc, wioger 1s
governed by veubrane and wave propagation effects. not treated by me curteza on the 4eslz:- cu•v
A6,
S69
Since these effects are
theory, the dat3 firom Schwieger are uot incleded
NADC-7s259. IV.
DESIGN PROCEDUWE In this section, 'procedures are recoutended for usiug the design curves
presented in this report and for constructing new design curves for structures not treated here based either on analytical calculations or on experimental data.
It is
only the A.
emphasized that these design curves are limited to predicting
aximum str.tural
respoase to low-velocity impact.
Use of Desian Curves in This Report The design curves in this report predict the peak banding istrain in simply
supported and clamped plates impacted by large impactors, and in simply supported beams and plates impacted by small impactors.
Also, in
beams impacted by large masoes were treated in detail.
(11, simply supported Methods for the use of
each of these design curves will be discussed separately here.
•.•.
1.
SipySportedj.m
"Desiga urves for simply aupport&A bvviea by small impactors are repeated here for tively).
impacted by large impactors and
onyve
nce (Fige, 24 a4d 25, repec-
In each case, we have selected a devigu Purve fit
experimental raulte as the one we
ill
by eye to the
use.
The first step in using thetse curve. ts t* vo)ute the mass ratio
H
-
W2
iq#"tor mass
(101)
If H is leas than 2, the ispact case fals into the domain of Large impactors, it H a.
a~U ia$Cir
LSWJE..agltoara From Fi$. 24, read the value of the generalized strain c corre-
spodlng to the value of H. i°ct
Then, using this value c, compute the
strain,
10
aximis
NADC-76259 .60
4&A
0
WO
ot
NADC-7q259'60
0.5
0.1 H
0.001.1
is.l
100.
KMISS lIO,
Figure 25.
OSLD.- DESIGN
m
Selected design curve for Impacts of simply supported beams by small impactors.
72
300o.
NADC-78259,60
C -hv
(102)
kc
max
h - beam depth
where
k - radius of gyration of beam cross-section
V w impact velocity speed of flexural waves
c - AT-
in the beam The use of this curve will be illustrated by an example. Example #1 For a simply supported steel (density, 7.9 g/cm3) beam of rectangular
cross-section with dimensions L x h x b - 500x25420 mm impacted at 10 M/s by a mass of 4.0 kg, we have pL bh
M
(7900)(.50)(0.025)(0.020) Q
4.0
m
2
= 0.494 From Fig. 24,
the value of • corresponding to M - 0.494 is
by Eq.
we have
(102),
• •
,
CV
,•,
(0.025)(10)
-(1.20)
210oc1 o9
SXo0.025
"7900
Ai
£
-
1.20.
Then,
9 112 J
- 0.0081
"b. Small Impactors P•rom Fig. 25, read the value of the generalized strain ponding to the value of N.
Then compute the maxiwim of Ipact strain,
C : ="
(103) •
_where
8
sad
2 -Hertzian
-
* corres-
2
1/5Tk
(0.09897)
contact stiffness.
73
NADC-78259-60
Example #2 Consider the beam of Example #1 impacted by a steel sphere of radius
20 am at 20 l/s.
Then, by Eq. (101), we have (7900) (0.50) (0.025) (0.020) 4 (0.020)3 (7900)
"1M X2
- 7.46 From Fig. 25, the value of e* corresponding to M - 7.46 is
c* - 0.060.
Hertzian contact stiffness ia
4
2
1-
V2]-
210x3w 10
2L.-
= 2.176101 0 Ncm3/2
o .15
S•o~so2 .
so that
0099)0*0220(2.176xi010
0
210,,1o 9 1-7900
:i•-(00997
0.025
L
21 1/5
0265j
*= 28.1 sTherefore, the waxiu bending etrain is C
*
(0.060)
ii9-' fL
7-900 .
•2104 o0
(0.060)427
.(.05(0
MAXA
'I-
74 _:
7' 9
(0.025)
The
NADC-78259.60
2.
Simply Suoported and Clamped Plates The procedures for using the design curves for rectangular plates is
similar to those outlined above for the beam.
For convenience, we have re-
peated here the design curves for simply supported and clamped plates impacted by large impactors (Figs. 26 and 27)s and for simply supported plates impacted by small impactors (Fig. 28).
The first step in using any of these curves is
to compute the mass ratio M according to Eq.
(101) and the aspect-orthotropy
ratio,
(a 2
7wn(=
)
where a and b are the plate dimensions in the x and y directions.
2
(104) For the
plates treated here, impact cases in which M < 4 fall into the domain of large impactors, while cases where H > 4 are in the domain of small impactors. The use of the design curves in each of these domains will be discussed separately here. a.
Large Impactors The procedures for the clamped and simply supported plates are the
sae but different curves are employed.
For simply supported plates$ read
10~)from the lover curve of Fig. 4 and 7 corresponding to the value of X from Fig. 26; for clamped plates, read g2 (n) from the upper curve of Fig. 4 and " from Fig. 27.
Then, the maximum strain may be computed from the foiaula EC 'ma
where
v)
g
v
impact velocity
h - plate thickness p - plat* density D U - plate flexural rigidity
..
•
75
(105)
NAOC -7,8259 .6O
___ _N
-
-
__
to
-____________
z
14
0
0
.9.4
co.
C14
Lhe
76~
NADC-78259Q60
u 4n
'rid C4
77
NADCq259 60
0.2:-
0.1 Selected
Design Curve
0.05
0.02
0.01
0.005
4:0
L
1o.o
20.o
50.0
100.0
Mas Ratio, it Figure 28.
Selected Design curve for Impact of Silply Supported Rectangular
Plate Uy a Small Impactor.
78
. .. . . .
.
,
Z•.
T,• .......
NADC-78259 60 This procedure will be illustrated by an e-asle.
Mxale #3 For a rectangular simply supported plate with the following properties
D
-
D22-
h- 7
680. N-z
2490.
p
-m-
-
1.7 glcu
a - 170 m
E - 8.96 Gpa
b - 350m
v-
0.3
impacted by a projectile with the following properties R-
25
m
n2 - 1.42 kg
v-
0.33
v-
3.0
/s
E - 2.1.L02 Gpa we first compute the values of the mass ratio and aspect-orthotropy ratio
M- 0.5 tF
n-0.451
From Fig. 4, we find 81(0.451) - 0.26, and from Fig. 26, Z- 0.91 corresponds to M - 0.5.
The ma•i•um strain in the x-direction is
0)(.00o7)
~~3,0
(0.26)(0.91)
CR,,-
3
02
- 6.57 x 1074
b.
Small I-Sctgors The design curve for impact of plates by small impactors can handle
only simply supported boumdary conditions. value of generalized strain e Fig. 22.
(Note that g(Bpvq)
from Pig. 28,
To us* the design curve, read the end the value of S(Opvp)
does not depend significantly on 0. for 0p > 7).
p Then, the maxism strain my be computed frost the foruiA& "" ,5ma -
fros
*xnOP 4(1.068) g(,OSOp
79
P
p
(106)
-NADC-78259 -60 ý15
2
where ep2
- 0.09897
h
a
2/ This procedure is illustrated by an example. Exmaple #4 Consider the same impact situation as in Example #3 but change the
impactor mass to m2 - 0.035 kg. Now we have M-20, a small impactor case. From Fig. 28, we read c - 0.025; and from Fig. 22, g(D ,0.451) x p
0.35.
Computing 8p, we get
p
(ý1700),(0.007)
8-(0.098971) (0.170)
680
1+0.451
L
Q3.0) (2.QO8xl X )0
(0.035)
-
6.54
r
Therefore, the maximum strain is
Sx~ms
-
4(0.025)(1.068)(0.35)(6.54)
-6.79
B.
(0.007213.02
x 10-
Construction of New Desgn Curves by .xperiment. For beans and plates with boundary condiltiona different from those con-
f
sidered in this report, desigu curves way also be constructed based on impact experiments.
For these structures, the parameters governing the impact re-
&sponehave the same form, but the relationships (and thus the design curves)
80
NADC-78259 -60 are different.
For still
other structures (e.g., shells, rods), new param-
eters may have to be defined. 1.
Impact St.ain Curves To construct new design curves for large impactors, one may perform a
series of impact experiments and measure the maximum bending strain which occurs in the structure during each impact.
Then, compute the value of the
generalized strain according to the appropriate formula: cb k Er for beams c .
i
m.a
-
k for plates
Plot one point for each impact in e vs. M coordinates, Logarithmic scales are most convenient.
Finally, estimate a curve to fit these points.
This
design curve may then be used to predict the response of similar structures having different dimewious subject to various impact conditious, as described in Szection A above. 2.
Critical Impact Velocity Curves In simple impact-to-failure experiments when the strain is not simul-
taneously measured, all that is determined Is the lowest impact velocity (or critical velocity) at which the structure failA.
However. this is sufficient
to construct a design curve useful in predicting failures due to other impact situations. t•
For each exporimat, the dimensionless critical velocity v, defiaed for plates in Eq.
(33) and for beamu
a4 Eq, (62),
is plotted vs. the *As&
ratio M. Then these points may be connected to form the design curve.
-
"81
NADC-78259 -60 Note that for different impact conditions, different mechanisms (bending, shear, torsion, etc.) may be responsible for failure; consequently,
several
design curves may be required, especially to completely describe a series of widely varying impact experiments. Finally, note that since the contact effects depend on the impact velocity, through a non-linear relation, this approach is not useful in the small impactor domain. C.
Construction of New Design Curves by Analytical Tools Using analytical methods, design curves for some structures which are
not treated in this report may be constructed.
We can cffer a few suggestions
on approaches to follow in order to construct such curves.
1.
Design Curves Based on Generalized One-Degree-of-Freedom Model For impacts of other structures in the large-ipactor regime,
the gener-
alized one-degree-of-freedow wodel presented in Section II my be applied.
"To follo ithis appro4ch, the equivalent structural stiffness
1
and the strain-
displacement factor d 1 2 wust be found; these may be either derived from the exact solution of the corresponding static problem or couputed numerically using some approximate solution (such as the finite-differance or finiteelement methods).
I
Then the equation of the design curve may be derived
directly from Sq. (8). -"the
"aes aa in Eq. can be employed.
For beatm.
the farm of thbc-generalized strain will be
(12); for plates, the generalized strain defined in
Eq.
For other structures. a new form may have to be defined.
82
(21)
NADC-78259-60 2.
Destiu Curves Based on Impact Calculations Several dynamic solution methods are available for determinizg impact
response of structures on a probltm-by-problem basis. finite-element method (in particular, NASTRAN; see Ref.
These include the (1]),
Timoshenko-
type solution methods (application of this approach to several beau and plate
structures is outlined in (5]), and possibliy some approximate solutions based on the Timoshenko method (similar to those presented in this report for small impactors).
Design curves vay be constructed by plotting the result. of a
few calculations using one of these methods in coordinates of the appropriate generalized strain (i for laige impactors, c* for small mass ratio N o'•er the range of M of interest.
impactors) vs. the
These plotted points way then
be connected to form the design curve.
183
V
I/!:;. :;"llI
83
NADC .7,6259.60
V.
SUMMARY In this report, a method for generating a design curve which predicts
the peak response of a structure subjected to low-velocity impact has been Only a few experiments or analytical calculations are required
presented.
to construct such a curve for a given type of structure. such a curve is
*
that all impact cases,
The importance of
involving various impact velocities,
structural dimensions, and material properties, fall on the same curve within
*
a variation tolerable to the designer. To facilitate our study, we have divided all impacts into two large and small impactors.
egimes.
each of vhich exhibits different kinegmatic behavior.
Specifically, massive impactors have been observed to produce mult4plt impacts, tthe peak response of the structure depending on the initial kinetic energy of the impactor; in contrast, small impactors strike the structure only once, the maximum response being governed chiefly by the impactor's Initial moyaentum and by contact effects at the impact point.
These differences in impact-:
response are accounted for in the distinct forms of thi, design curses for each domain. Design curves have been developed here for predicting the impact response of simply supported beams and simply supported and clamped-edge anisvopic Theae curves have been constructed using both results of analytical
plates.
solutions and data from impact experiments.
Inaddition. detailed procedures
have been described for using the design curves presented in this report, as well as for generating new curves for structures not treated here.
84
': "" .T• •m• .: '7 :
:7•
.
.
.•,
'
.
..
7:'-
-'No
Shear Failure
0
,0.2
S~~Bending Fallurei•
1
•
0.1
NO Beading Failure
0.02 0.02
. .. 0.05
'... 0.1
0.2
0.5
1.0
Kass R~tio, H --
'--'-Critical
S....
V"Critical
Velocity for Shear Failure. Velocity for Bending Fzilure
Experizerits on Graphite-Epoxy b~eas..•
14Noo
9 Failure" Failure igure'S4o
Critical Impact velocities for shear and bending failure and experimental data--a bediug-dominated. C•ae.
193
WADC -78259-60
- -, -
-
-
• --
L•~
It4_
01-
-
- --
-e
'Cq
o
J~..a.s~~..
.
-
-
----,
-
-
---
--
,_194
-n
-
F *
'IN4
.1l,
w-
-
F Lw
-
.-
-
-
-195
MADC-78259 60
I>''4 x
r
- -"
i!
! *19
-
-,
-,
-
NiADC -78259 .60
T
10.
s
X I'm
7..
;m
.THIN
-.
6.. £
ww
.w
4......... 0f
4~~.
I.f
f
fi
-~I
.. ..
fliP,
'19
. .....
..
f.
4
1
WADC725 9 .60
10~-
....... 9-
- -.. --...--
..
2.L--
........
LI I
~~~~~ D~
Ii
-----
... ----
Itao Iuw I34~~ L~t.a
it Oa..
3q.~rftuita1nDatA frcm Opwioiuw 31. B49, mA B9 moveaqs Bumdar
Oondi5.ai
1...
L.5
'1
2.5t
10
198
1.
2.5
4
WADC -78259-60
.... .... .
nr
.. . . .... .......... .
.. .. .... .-.... ......
.... ......
. A~..
.
-~~~I
......-.
.....
..
~
-F
8t
an =q~
ff.
_Zr.M
m_;
9ICT.......
F-
7_
6..~:f2
model, ;nJj
-One-degree-of-freedom
3-1
Impact-to-failure Experiments
K Clamped ?late$ * Simply-.Supported Plates
..... .. . .......
K.~ 1.2.5.A~ ~~~
~
.-.....
.
..
,
-
--
~~ *
~A91
.
.
. -
. .
-.... .
, .
'1
Figure'-SW8. Critical Im~pact Velocity Curve for Clamped and Simply-Supported Plates Impacted by Large Masses. 199
. .
.
. .
4
. .
.
-
-
7 a
Al
WADC -78259 -60
7---
4--.. 04 N
.
.....
2 4i
4
t..
... . .......... ..........
One~mdgroe-otefreslam model
+
itM
I I
Itgure
Impact Velocity Carve for Claqed Plates -4.Critical
Lb
2.5 10
200
1.5~
2.5
4
&
B
NADC .78259 480)
10.. 7...
V,
III..t
2.15 '171
7.
~
~
4)sdge~*f~dMmd'
1
I
4~~~
it
44,
.
1
3
2.5
j,, q
9
J2
2
a
B00
4;Fi.'O
6
4 --
~~
2.64
OtfrnlmpdPlt
Veoct
Impc
ritca
3
~
8 to
1.
1
.. ereoffedo
xe4m~
oe
MADC-78259.60
3.
LsUlt
of Small Impactor Strain Measuring Experiments
Plate specimns vere impacted by l"(25ua) fired from an air SuM.
diode system.
Iapactor velocities vere recorded using a photo-
Plate strain was recorded using type EA-13-062TT
and a type 565 dual beam oscilloscope.
N
radii aluminum projectiles
rosettes
MADC.78259-60
TABLE 37 DATA FROM EXPERIMENTS ON SIMPLY SUPPORTED BEAMS IMPACTED BY SHALL MASSES B aass
DiBeam Mass,
Impact
Ratio,
Dinesins and Material
0.389 kg
Contact
Velocity,
N-ml/m 2
k 2 (N/m2)
27.0
6.37 13.17
2;466x10 1 0
"
8.72
"
12.19
5.49
6.85
0.153 8
.
2
-
ca /hv
0.0795
0.0184
"
0.0814
0.0175
"
0.1229
0.0315
of
0.1489
0.0369
"
0.2388
0.0739
ft
0.2512
0.0743
3.54
6.28
*'
0.3224
0.0554
13
4l9.33
"
0.393
0.1171
10.88
"
,0.080
14.87
"0.0790
41.8
0.601 k1g
"35 1"6a 13.8 Itel11I.06 _
_
_
11.2 "
5.88
7.92
"
8.1
"0
0 ...... 7.5.,
-0...
0.1096 kg. alu,,iuu.=.
0.1033
0.01703
~ 0.1076 6
ý0.01714 1Q793
0.1800
6.16
,
1 0.2478
1.93
4.39.
0. 2662 0.475 .......
"
6.92
0.462
m
•203
.
.176.6 ..
It
---.-
0.01142 0.01092
"12.37
""8.11
'
E
.,--
13.8
steel
••"
_
Strain,
*3/2
v (m/sec)
19Z x 16.x 16 m
152 x 20 x 12. 7
Generalized
Stiffuess,
03.o 41 0.0330
o.o054 •
. 0579 . 0,1689
to.21569
NADC-78259.60
TABLE ?
Beam a Dimensions,
Mass Ratio,
and Material _l,"a
m,
Impact Velocity, 2
041.8
kg
So0.601
x 16 x 16
'305
23.8
~steel s12.53
:i
(continued)
, 11.2
Contact Generalized Strain, Stiffness
2) 3 //v (2/see) k 2 (N/m "
1.744x10 1 0
10.58
"12.71
"
9.42
"
"
12.01 5.88 i
"
0.360 kg 185 x 16 x 16='
23.0 "..
"
12.8
2 ca2/hv 0.0760 0.0761 -
5.33
-
0.01166 -
0.01146
0.1242
0.0216
0.1235
0.0208
0.1534
0.0328
0,1611
0.0317
5.39
"
0.2345
0.0570
7.71
"
0.2799
0.0656
0.,0868 0.0978
0.0203 0.0223 . . ..
0,1523
0.0421
.10.70
0.1647
0.0442
8.20
0.2563
0.081 4
10.2
0.2434
0.075,
0.3252
0.122i
"8.60
0.3203
0.1155
3190
0.458
0.2222
6.40
0.465
0,2147
1. !1131 14.42
10 2.466...
8.11
to
steel 6,34
" 3.27
0.1074 kg, steel 98x 15 x 9 s
1.89
5,73
20,
.
"
NADC-78259-60
TABLEIS DATA FROM GOLDSMITH (1960) Beam Mass, Dimensions, and Material
3.66 kg
Mass Ratio,
Impact Velocity,
Contact Generalized Stiffness, Strain 12 2 Nla iam32 Ili/v k MINIM v (Msec) 1a2 0.405
6.44
3.48'7x1.0 10 to
it1.338 762 x 25.4 x 25.4 va2.54 steel
2.12
0.408 1.271
C56.9 kg
C /0m/e)
0.262
0.0563
0.286.. 0.311
0.0500 0.0510
'0.520
0.1278
"0.564
0.1237
'
U1.777
-
4.894x10
0.584
0.1239
0.148
0.00932
57.0
2.40
23.7
2.47
0.247
0.01855
to3.00
0.237
0.03740
0.260 0.239
0.01886 0.01676
-
2900 x50x 50 m steel
-
3.40
'
2.40
'40.365
0.03173
2.40
'40.420
0.04107
'44.75
11.41 ________6.34
3.6kg tel55.0 72x5.452 5.4
tr
2.170*kg, steel
1.125
1.744x101
0.0802
0.01065
1.233xl'10
0.28
0016 0.001625
_______
266.
1.44 kg11.2 stel
45.7 45.
'40.0259
762x19x12.7 rim 0.965 kg, steel -2x12,7,x, 7mt 0.482 kg, steel
118.2
45.7
591
4.7
0.0713 1
205'
0.1578
.
0.003228 0.007144
NADC-78259-60
0.1
0.01
~ -Appro~imatia
KpTimosau
solutiona
olt
io
0.00)L LS10
t.
I'lural~51 * NWniiir, %:uItv, (.jr Impac.tc
100
of si,-ai~l
m:rijprtO. b..iats lay =
300
NADC-78259.60
'TABLE £9 Impact of Composite Plates by Small Masses
40,38
0ooo
3•ZI
3,756
A
29.7
30
F2
32.3
_____,
/8,3
"//W
-207
4, •'aw
-,03Y"
3.50 Y7
to, O__
3.75
O,,Off
S,03
_.___,
6,6,C93
MADC -78259,60 TABLE E1O Data Fr:,, Schwieger
0, Of28 O,o 3•s
170
0,25* 0,51
/9,7 21, 5
53/
&75
23,3
F
1-26'
2548
0,0S'31
/283
A75"
?7,6
6.A313
0,
20,3
ooe
•!35"
,/Z,
7"7* __7
/,o.. _
____
FS
to~o~ 0?t
V-7
0,011_
M
1417 7.-1
I •
••
•,•.
...
......
. ....
NADC-78259-60
ff.~:HL~f~ijI.~.Approximate
+PT45}.+X
solution
experiments
:'XDrexel's
.........
~V
.:'
TV
-=
___
~1~~~_.........-:2s
77.
7.
-17
1
1
.
7
1
12:
V
f
Vigsjre&520 Strain Curve for impact Of Simply Supported
Rectangular Plate by a Small Impactor.
209
NADC .78259 60 4. Typical Strain.vs. Time Oscilloscope Traces for Impact of Clamped and Simply Supported Orthotropic Plates by Large Impactors.
'Al
A)
210
NADC-78259 -60
Scale: 2000 vi M/div verticAl 0.5 msec/div horizontal
a)
Simply Supported
61-i171-O scale: 4000 V,M/div vertical
l0.5 Usec/major horizontal
b)
Clamped
Figure U53: Impact of Specimen B3, M = 0.21, Impact Velocity = 1.73 m/s
211
NAC -78259,60
Scale: 2000 )1 M/div
vertical 0.5 msec/div horizontal
a)
Simply Supported
111
,
|]_1161-7
Scale:
..
4000 P Ma/div
"vertical 0.5 iuecfmajor horizontal
b)
Figure 154:
Clamped
Impact of Specimen B3, X - 0.21, Impact Velocity - 2.45 a/see
212
-
7
+4, .. W•p.;.wt• j*, rt,,,,•
'. "',,,
.
.
-
..
....
..
MADC-78259 .60
Scale: 1000 IA '/div
vertical 0.5 msec/div
horizontal
a) Simply Supported
Scale: 1000 -P M/div vertical
0.5 msec/aijor horizontal
"b) Clam•Ped
1Jfpie E55
IftPact of Specimen B76 i-
213
2H8,
wa~act Velocity
-
1.73
NADC -78259 -60
Scale 1000 11 I:div vertical 0.5 msec/div
y
horizontal
a) Simply Supported
Scale 1000 j /i vet ~ical 0.5 asec/major horizontal
b)
Clamped
Figure B56: IMpAct of Specimen B7$ X
214
2.38o Impact Velocity
*2.49
u/see.
NADC.78259.60 Appendix F The Effect of the Approximation H
-
f
on the
Predicted Impact Response of an Orthotropic Plate
To simplify the analysis of the bending and vibration of specially orthotropic plates, Timoshenko and Woinowsky-Krieger
j13] have suggested the approx-
imation
H - fDI1D22
(Fl)
where H is defined as
H E D12 + 2D66 An advantage of using this assumption is that the equation governing bending of an orthotropic plate may be transformed$ through a simple change in variables, into an equation similar to the equation governing bending of an isotropic plate. By reversing the transformation, a solution for the deflection of an Isotropic plate may be converted into the corresponding solution for an orthotropic plate. This is done in Appendix B for a centrally loaded rectangular plate with clamped boundary conditions; the approximation (F1)
is
instrumental in this instance
"because no general solution exists for the orthotropic case. Like all approximations, error into the solution.
however,
application of Eq. (11) introduces sons
In this Appendix, we investigate the influence of this
approximation on the strain predicted by the one-degree-of-freedom model of Impact of a simply supported rectangular orthotropic plate. deriving the relationship between the generalized strain c both with and without the use of Eq. (11),
215
and then cooparin
This is done by and the uses ratio M, the results.
NADC-78259,60
The static deflection of a simply supported rectangular orthotropic plate (10]
due to a central concentrated load P is
Y~)
s in W sin-2 f snmX s- nMY b b a -M4 D+2---+R 2 2 ina 4
4Pv .b~ 4'
a 2 b2
a 4 ll
b4
22
where a and b are the plate dimensions in the x and y directions.
If we define
nias in Sq. (16) an €-
2
then the maxims deflection, which occurs at the canter of the plate, may be cxpressed as
w where
w4
-aX P/1K 11 b
1
(2
(F2)
3-- -
,,..(ton)
L
u,.
odd
maa li
Similarly, the mazius x-direction bendng strain, which also occurs at the plate center, my be eupressed as 2 Ph&
,(Vn
* 2Pb D~ b 91 2h a K 1*
.!.,
-
'wtere
1 2 (ClO)a
2
i. 216
U ~
-
U2
2
a,*
odd.
At-DC-78259-60 Therefore,
the strain-deflection constant, defined in Eq. r2
d2k=2
1b
DII a K1
(6),
is
1 P2 (c•i)
(P3 (3
To apply the one-degree-of-freedom ipact model to this case, we substitute Eqs.
(F2) and (P3) into Eq. (8) with e - 0 and define the generalized strain
x
as in Eq. (21) to obtain -
G(Cn) S.(F4)
where G1(Con) Note that the approximation (Fl) corresponds to setting C this assunption is used, then the approximate relation between
1.
Thus, if
x and K,
corresponding to Eq. (22), is S 1 (0,n)
05s) Therefore,
the correction required to adjust the approximte generalized sarain
C given by Eq. (QS) to rhe exact value given by Eq.
(F4) is
This error is plotted vs. C for a fev values of n in Fig. VI.
S
the uagnitude of this error does cot exceed 17
Noes that
for the range of t values
calculated and is less than 92 for mest lay-upa cosmnty used in actual aircraft structures, which lie in the range 0.60 < c
1.6.
(Lay-tip. havig C values
outside this range [e.g., C " 0.31 for unidirectional AS/3501-61 rarely bhve practical application as plate atructures.)
Further, for the range of ; valuss
of the plate specimens used in the impact &saperwmts,this errot is always less
than 6.5Z. 2.7
"RACC-78259 60 In Table F1, strain E(•x)
the actual values of
n, i, and the error in predicted generalized
for the plate specimens are summarized.
among all of the plates is only 5.4%. motion of Eq.
The largest value of error
Therefore, we conclude that the 4pproxi-
(F1) does not contribute a significant error to the design curve
for impact of simply supported plates by large impactors. similarly swal
correction is
We my infer that a
required for orthotropic plates with other boundary
conditions. Table It Error in Predicted Generalized Strain for the Plate Impact Specimens
Plate
HFaE
'ID
1112
K
)il 0 .625
-3.61
B2
0.164
-4.2
B3
2.39
-3.7
14
0.625
-3.6
B5
0.074
-4.0
16
5.29
-5.4
!7
1.38
-3.2
B8
0.282
-4.1
B9
0.625
-3.6
0.641
-2.2
0.168
-2.6
1.430
B1
1.252
11 12
32.45
-2.3 O.64l
-2.2
1.150
-2.1
32
0.302
-2.7
13
4.41
-3.3
14 1.270
Ul
84.5
-2.1
218
I.
~VWW-78259-60
VF
-F
F- FI
-t-T
. *T
A7. .....-.
4~~
jl
4T
f
4
J.
~-
I
* ft
-
-
T
~-
f I
'J-
t*
~
..
*
t.
.
-
Vi
4
.
...
r
I
NADC-78259-60
DTSTRIBUTI[•N LIST Govenroent Activities lNo. of
v
NVAISSYSCO(, AIR-954 (2 for retention), 2 for AIR-530, I1for AIR-3.0, AIR-52032D. AIR-53029 AIR-53021, AIR-5302:5).
9
AF Mr, WA.1, O 45433. (Att.:-"Mr.P. A. P
2
aerymey).
(Attn*:
FBC/t. C. Wallace)
(Attn.
FBC/Mr.EE. E. Zink)
a.
....
•
........
. ..-
....
...............
AfL*, WAFB, OR 45433 (Attn: LAM (Technical Library)),.. . 0 (Attnm LTdI/1r. W. R. Johnston), . . . O(tto: LTI14. T. Cordel) ... 1........ (Attn: FBSC/Ur. L. Kelly) . ....... (Attu: &ACiMr.G. P. Pete'son) . . . * (Attn: Mkhtr? Fs J9 Fechek) e s (Attn. NBC/tr. ''.G. Re.tdiard,J=.). KqAPOSR, Washinrton, D.C. 20333 (Attu: Mr. J. . . *.. . . . ,..
I a.
.
*.
..
.
.
..
. .
.
.a
a . *
..
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.
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3
Centor, Cievelam, OU 44153 , Library, azu H. ?i4Rhber1g), ., ,* .a * CA 95940 ati•L ft o . 4.H.Ho anu). , 0 0 * 0 0 0 0 *
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Mrs J, P, Peterson, Mi.,
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....... *.
.
...
......
1
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Voloc) .:.:.:. .".:.:.:.
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1
*....
.
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...
2
..
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De.H
1*
. a.
,v,,,) . . .. . . a . (Attu: SW-ASUt-H@0r. J. t Stuckey) ........ ASAI, Laiiley Research Center,sRutou, VA 23365
(Attn:
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1 ,,.
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AL 35812 (Attn: S.E-A9M4-ESstr.Ir) (Attn: SSA-.,"MM/Mr. R, S
MI iAV•)HIWC
I
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RUSA (AD~4) Washirgtoi, )XC, 20546 (Attut: er. cret e ). u . . . . .a a . . . ASAp, George C. Marshall Space Flght Center, HuattvtLle,
(Attui:
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VAAUAiriwraws braw-h, ES-O120, Waahiton, D).C.
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1... 1
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,
,
Ia
1.
NADC.78259,60 Cowerment. Activities (Cont.) PLASTEC, Picatituny Arsenal, Dover, WU 07801 (Attn: Librarian, Bldg. 176, SARPA-FR-Mo-D and Mr. R. hbly). 2 Scientific & Technical Information Facility, College Park, ND (Attnt NASA Representative). *. a * * . . * .,., * ., . UMAAWThAB, Fort Eustis, VA 23603 (Attn: Hr, Re Dere•ford) . .e . . . . . q . . * *. . . . 1 +MATRES Watertown, UK (Attw" Dr. E. Lenie) . . . 1 USARS0OFC, Durham, tIC 7701. . . ... ...... . . . ... .
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1
C6tr
SBell
(Atta:
Dr.
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JUMe). ,... * a
,
.
.,
.,
,
.,
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Hr. V. ftaeau/t
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,
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1 I
.,
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.
.llerica. VA , ,, ,
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Gatat Lakes Carbon Corp.. N.Y..
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Lehigh Univernity, Bethlehem, PA (Attn.
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LTV Aerospace Corporation, Dallas, TX 75222 (Attn: Mr. 0. E. Dhonau/2-53442, C. R. Foreman). 2k• 1n Company, Baltimore, MD 21203
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Northrop Aircraft Corp.~, tiorair Divot Rawthorne,. CA 90250 (Attun Hr. R. D. "iLyes, J. V. Koyes, R. C. ise=mnn). -Dockwell International, Columbus, OH 43216 (Attu: Mr. 0. Go. Acker, K. Clayton). •. 0 ...... ROckwegl International, Los Ageles, CA 90053
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(Attn: Mr. E. Sanders, Mr. J. 1. Powell) . . . . . . . . . . . Ovens Corning Fiberglass, Granville, OR 43023
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"(Attn: Mr. D. Mettes), . . . Rohr Corporation, Riverside, CA 92503 (Attnv Dr. k. Riel and Hr. R. Elkin)
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(Attn:
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Vliversity of Oklahamt,, Norman, OK 93069 (Attn: Dr. G'. M. tiordby) . . . . . . . . . . . . Union Carbide Corporation, OH 00 44101 *S •,. o' Volk) CleveLnd, 40, .40 (Attu: D•.'.F.
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