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Large deflection of circular plates with affine imperfections
LARGE DEFLECTION OF CIRCULAR PLATES WITH AFFINE IMPERFECTIONS G. J. Tmvm Dcpamm of Enginheg, Universityof Lumster. Railr& Lac8stc-rLA1 4YR, Rnghnd (Rmived 11 October1977;in nwisedfotm 16 February 197%;rdvdfor
pdcation 16 March 1978)
circular plates are commonly utilised as primary load catrymg elements in stm&res. usually,
these plates are of constant thickness, but sometimes it is convenient to use plates, which taper in thickness. The design of these plates is generally based on small dekction theory. Whilst this approach is adequate in most circumstances, it is neverthdess conservative. A more rational approach would be to utilise elastic m delkction theory and to account, as well, for initial imperfections, which may be present fortuitously or by design. Unfortunately, the initial imperfectkms, which may be preset& may take a variety of forms and this serves to complicate any study of theii influence on the plate b&&our. However, if the restriction is imposed that the initial imperfection should always remain a&e to the additional plate deflection, then a parameter study to assess the Muence of the initial imperfection on the plate response in the large de&&ion regime may be more readily accomplished. In the present context, the type of atlInity selected for the initial imperfection is linear (see eqn 3 of the next section). This choice implies that.it is of the same form as the additional deflection resulting from the applied loading. Thus as only uniform lateral pressures are considered herein, then the implied initial imperfection is a sin& half-wave. The linear relationship between the initial and additional detktions implies the further consequence that the initial imperfection is load dependent, a fact which is commonly assumed to be atypical of real imperfections. Nevertheless, this.feature is thought not to be too serious[l], and, moreover, the results of the present computations may be expected to reveal tlje same general trends as those based on real imperfections. 559
G. J. TURVEY
Nylander[ l] has shown that provided certain boundary condition similarity conditions are satisfied, then the perfect or flat plate large deflection results may be used to construct the corresponding results for plates with tine imperfections. Amongst the many results presented by Nylander are those for a uniformly loaded, constant thickness, simply supported, circular plate with a&e imperfections. However, no results were presented for clamped circular plates and, moreover, Nylander has not shown that the same form of construction can be applied to plates, which taper in thickness. Since the usefulness of this technique relies heavily on the availability of the corresponding flat plate solution, the author has not seen fit to examine whether this technique may be extended to tapered plates, even though some perfect tapered plate results for the large deftection regime are available[2,3]. Instead, the author believes that a more reliable approach is to solve the large deflection equations of tapered plates with a5ne imperfections directly by numerical integration. The particular integration procedure selected is known as Dynamic Relaxation (DR)[4]. PLATE GEOMETRY The current study seeks to examine the influence of two particulargeometric meters on the plate response in the presence of a uniform lateral load. These two parameters are: the thickness taper ratio, a, and the imperfection or deflection a#lnity parameter, k In practice, only linear thickness tapers are likely to arise and heace the plate thickness at an arbitraryradius, r, may be expressed as,
/I, = II& - 2aiu-1)
(1)
where r = 0-4~. Thus for the present purposes, it is considered sufUcient to restrict the computations to three distinct a values, namely: -1, 0 and +{. The total plate de&&on may be expressed as, w= wo+w1
(2)
where w. and w1 are the initial aud additional deflections respectively. If the initial deflection is to remain a&e to the additional deflection, then the followiug relationship is implied, w,, = kw,
(3)
and hence by varying the value of k a whole rouge of initial imperfection ma&u&s may be exam&I. The partlcular k vducs chosen for comprttptioltplpurposes are: 0,&l and 2.
ANALYSIS
(i) Goorrning QQwtions The DR method is suited to the numerical in@uation of low order diflerential equatious. Because of this, it is convenieut to retain the separate identities of the compatiMity, coustitutive and e&Mum equa&s throughout the analysis. These equations assume the followitttj forms for aziayatmetric deformations: (a) CmputibUify quahns. Utilising eqns (2) and (3) the compatibility equations may be expressed as, @,EU’+f(l+2k)Wi2 e, = r-‘u
k,=-wi* kl = - r-‘wi. (b) Constitntive equations.
(4)
Large detection of circular
plntcswitha5ncimperfections
561
N, = A#, + A+, N, = A,e,+A,e, M,= D,k,+D,k,
(5)
M, = D,k,+ D,k,.
Equations (5) apply to polar orthotropic materials, but all the plates considered herein are assumed to be fabricated from an isotropic material (e.g. steel, for which v = 0.3) and hence, A,=A,=Eh(l-vs)-‘,
Ar=vA,
and
(6) D, = Dt = EI?(l - v3-‘112, D1 = vD,
(c) Equilibrium equations. Only the transverse equilibrium equation is affected by the afhne imperfection, so that these equations are N;+r-‘(N,-
N,)=O
M;‘+ r-*(2M;-M;)+(l+
k){N,(wi*+ r-‘wi)+ N&i}+ q ~0.
(7)
(ii) Boundary conditions The study is restricted to circular plates with either a simply supported or a clamped boundary and in each case full in-plane restraint is assumed. Thus the appropriate boundary condition equations are: (1) Simply supporrcd (I = la).
w,=o M,=DA,+D,k,=O
(W
u =o. (2) Clamped (I==fa).
w,=o Wi3:0
(W
u so.
(3) DR procedure. As the DR method is fully documented ekewhere[4], a description of its application to qns (4)-(8) is omitted. Neverthekss, it is worth pointing out that the present application does make use of rationally demrmined fictitious densitks[s] tog&her with a unit time increment. Thus only the two dampii factors require to be determined by &itrary means and their e&x&ion and adjustment to achieve rapid solution convergence is a relatively simple matter. This approach has been found to be superior to the more usual approach, in which both the fktitious densities and the damp& factors are deWmined by trial and error. RESULTS
A preliminary set of computations was undertaken in order to: (a) verify the computer program, and (b) to determine a suit&k mesh size for the main computations. Table 1, which compares the DR results with those of Nylander for a uniformly loaded, constant thichmss, simply supported, circular plate, ckarly demonstr&es objective (a). Rased on these and other results, it was conch&d that objective (b) would be met by usin8 a 105 interval inWkci~ mesh for the main computations of the parameter study. In order that the computer results may enjoy their widest possibk applkation, they are presented in Figs. 1-5 in nomiimensional graphical form, which allows intupoktion with reasonable accuracy.
562
G. J. TURVEY
Table 1. Comparisonbetween Nylander’sand the DR prm results at the ctntre of a uniformlyloaded, constant thickness, simply supported,circularplate with an alEneimperfection(v = 0.25) ~r.WUre i
l
Aifi8ltt
IYl&Or*e
k
R..alte*
Zl
zm
DR Promam =I
r,t
Romalts qt
20
0.36
1.40
2.90
1.3684
2.9331
20
1.oo
1.00
2.75
1.0020
2.7640
20
4.44
0.45
2.25
0.4828
2.1456
The Iiylm~der Remlte uod are-therefore
tire
boon
evalurrtrd
by rcsling
to two decimal
only g1v.n
. Fig. I(a).
k-05
Fig. I(b).
placer.
gnphe
563
Jsgedefkztionofcircuiar phteswitba5aeimperfections
k-l -
ZOO
0
400
600
600
1000
ii
lc).
Fii.
k0
t 3.0
#.
/.C.-.
=, 20 / .’ ,’ /H
1.0
./.. .C ~/y. ./
.C.C
.I./
_ _--
_-
__--
i/---
001
0
1
200
I
400
B
I
I
I
60
600
am
Fig,l(d). Fig.1. &Ldj&al
centml defkctionvs lataalpmnuri for a simplyv (a)k-O,(b) k =0.5,(c)k-l,(d) k = 2.
cimh
phbe b-0.3);
Figure 1 presents the additional deflection at the centre of a simply suppor@d plate as a function of the lateral pressure for thee values of the taper ratio and four values of the ahity parameter. It is evident that, in all cases, the additional de&&on -#lMKghUyWitb increasing taper ratio and decreases substantially as the value of the ahity parmae@ increases. Corresponding results for the clamped plate are not preaenW since they d&r only margblly from the simply supported results, i.e. from about S% at low values of the lateral pressure to rather less than 2% at higher values. Therefore, except where gteat accuracy is required, tbe results of Fu. 1 may be used for both sets of boundary conditbns. The central bending stress vs lateral pressure is plotted in Fw. 2 for a simply swporbd plate.Hmtoo,the~~nceofthebcndinestresronthevrlueoftBt~ratip~the ahity parameter is similar to that of Fu. 1, i.e. maqinal in the case of the former and rather more substantial in the case of the latter: Fw 3 compares the simply supported and clamped plate centre be&& stress for a = 20.5 and k = 1 and 2. It is evident that the stresses diverge as the taper ratio decreases, but that this effect reduces as k increases in value. SSVOL I4 NO. 1-C
G. 1. TIJRVBY
I #
19 -
00
I
0
200
1
1
400
600
1
I
boo
1000
T
6.0
I
__----
I’ _.-.
_._
s-o-
I 1.0 p
OQ-
0
I
200
1
I
400
600 7i
I
mo
1000
Large dcilection of circulnr pkes withalbe imperfections
565
In Fri. 4 the simply supportedplate care membrarkstrws vs latefal pressurecurves are plottedfor the upperand lower taperratiolimits and for each of the four k valueacoasidand herein. They demonstratethat this stress is almost indcpcat of the p&e taper ratio and, moreover,thatit~~~Uyasthevalueofthe~pu;rmettr,~~~ FiBalIY, Pi. 5 compares the plate ccntrc membranestress for both simply suppaaw and clamped~cwditIons.~compvativenapttsartshownfortbt~~of the t&perratio only, B similarbeiwioW is observed for ot&cra and k values. Thus, these cmws suggestthat the resuks presentedin Fii. 4 may be used for plates witb eitherbowdary conditionwithout all undue loss in accuracy.
G.
J.
TURVW
-k-0
hgc
defkdon
of circular pl8tcs with 8fTineimpcrfcctions
ksO-5
50
30 I-
?Tt ZC )-
11)-
0
r! 200
600
600
800
1000
s’
Fig. 4(b).
.!k! LO /
30
-
800
Fk. 4%
1000
567
G. J. TURVEY
i Fig. 4(d). Fii
Fii.
4. Central mcmbraue stress YS lateral pressure for a simply supported cifculpr plate (v = 0.3); (a) k a 0, (b) k = 0.5, (c) k = I, (d) k = 2.
5. Central
membrane
stress vs &ml
pressure for simply supported (v = 0.3).
and C&II@
circular
plates
CONCLUSIONS
conclusions to be drawn from the computed results are summa&d as follows: (1) The central de&&ion increases moderately with increasing taper ratio and reduces substantially as the alllaity parameter increases. (2) The central bending stress decreases moderately with increasing taper ratio and reduces substanti&y as the affiaity parameter imxeases. (3) The central membrane stress is reiativeiy insensitive to taper ratio and decreases only my as the a&&y parame@ increases. (4) The centraI deflection and membrane stress are reiativeiy insensitive to the boundary support conditions. The
main
Ackno~-The computing facilith
author wia&s to expressbis iadsbtcdness and whncc
with the prcpahon
of the figures.
to the Dept. of En~ring
for providin#
both
Lafgctkktioaofcirtuhrplrteswithrlharimpcrftctions
569
REFERENCES 1. H.Nyhader,I~ydcscctedtbia~~hithl~~to~ddlection.fnl.~r.forB~ond
sknctuml Eagng 11,346-374 (1951). 2. S. D. N. Murtityaad A. N. !ibabmw. Nalinarbendingofehsticphtcsofvahbkpro6k.I.&agAkAIXu., Proc. Am. Sot. ciu. hgrs. 10 (EM2). 251-265 (1974). 3. G. J. Turvey, Largeddkaion of taperedlMUIU plates by dynamicrelaxation.1. EngngMech.Do., h. Am. Sot. ciu. ERgm 1.4. EM2 (April !978). 4. A. S. Day. An iatrodudlon to dyuunic relaxation. 7he hgincer 2ml), 218-221 (19611). S. A. C. CurelI and R. E. Hobbs, Nunmicd stability of dynunic rchxahn analysis of non-lirtcaratructwcs. ht. 1. Num. Methods En.givglY6). 1407-1410(1976).
pdcation 16 March 1978)
circular plates are commonly utilised as primary load catrymg elements in stm&res. usually,
these plates are of constant thickness, but sometimes it is convenient to use plates, which taper in thickness. The design of these plates is generally based on small dekction theory. Whilst this approach is adequate in most circumstances, it is neverthdess conservative. A more rational approach would be to utilise elastic m delkction theory and to account, as well, for initial imperfections, which may be present fortuitously or by design. Unfortunately, the initial imperfectkms, which may be preset& may take a variety of forms and this serves to complicate any study of theii influence on the plate b&&our. However, if the restriction is imposed that the initial imperfection should always remain a&e to the additional plate deflection, then a parameter study to assess the Muence of the initial imperfection on the plate response in the large de&&ion regime may be more readily accomplished. In the present context, the type of atlInity selected for the initial imperfection is linear (see eqn 3 of the next section). This choice implies that.it is of the same form as the additional deflection resulting from the applied loading. Thus as only uniform lateral pressures are considered herein, then the implied initial imperfection is a sin& half-wave. The linear relationship between the initial and additional detktions implies the further consequence that the initial imperfection is load dependent, a fact which is commonly assumed to be atypical of real imperfections. Nevertheless, this.feature is thought not to be too serious[l], and, moreover, the results of the present computations may be expected to reveal tlje same general trends as those based on real imperfections. 559
G. J. TURVEY
Nylander[ l] has shown that provided certain boundary condition similarity conditions are satisfied, then the perfect or flat plate large deflection results may be used to construct the corresponding results for plates with tine imperfections. Amongst the many results presented by Nylander are those for a uniformly loaded, constant thickness, simply supported, circular plate with a&e imperfections. However, no results were presented for clamped circular plates and, moreover, Nylander has not shown that the same form of construction can be applied to plates, which taper in thickness. Since the usefulness of this technique relies heavily on the availability of the corresponding flat plate solution, the author has not seen fit to examine whether this technique may be extended to tapered plates, even though some perfect tapered plate results for the large deftection regime are available[2,3]. Instead, the author believes that a more reliable approach is to solve the large deflection equations of tapered plates with a5ne imperfections directly by numerical integration. The particular integration procedure selected is known as Dynamic Relaxation (DR)[4]. PLATE GEOMETRY The current study seeks to examine the influence of two particulargeometric meters on the plate response in the presence of a uniform lateral load. These two parameters are: the thickness taper ratio, a, and the imperfection or deflection a#lnity parameter, k In practice, only linear thickness tapers are likely to arise and heace the plate thickness at an arbitraryradius, r, may be expressed as,
/I, = II& - 2aiu-1)
(1)
where r = 0-4~. Thus for the present purposes, it is considered sufUcient to restrict the computations to three distinct a values, namely: -1, 0 and +{. The total plate de&&on may be expressed as, w= wo+w1
(2)
where w. and w1 are the initial aud additional deflections respectively. If the initial deflection is to remain a&e to the additional deflection, then the followiug relationship is implied, w,, = kw,
(3)
and hence by varying the value of k a whole rouge of initial imperfection ma&u&s may be exam&I. The partlcular k vducs chosen for comprttptioltplpurposes are: 0,&l and 2.
ANALYSIS
(i) Goorrning QQwtions The DR method is suited to the numerical in@uation of low order diflerential equatious. Because of this, it is convenieut to retain the separate identities of the compatiMity, coustitutive and e&Mum equa&s throughout the analysis. These equations assume the followitttj forms for aziayatmetric deformations: (a) CmputibUify quahns. Utilising eqns (2) and (3) the compatibility equations may be expressed as, @,EU’+f(l+2k)Wi2 e, = r-‘u
k,=-wi* kl = - r-‘wi. (b) Constitntive equations.
(4)
Large detection of circular
plntcswitha5ncimperfections
561
N, = A#, + A+, N, = A,e,+A,e, M,= D,k,+D,k,
(5)
M, = D,k,+ D,k,.
Equations (5) apply to polar orthotropic materials, but all the plates considered herein are assumed to be fabricated from an isotropic material (e.g. steel, for which v = 0.3) and hence, A,=A,=Eh(l-vs)-‘,
Ar=vA,
and
(6) D, = Dt = EI?(l - v3-‘112, D1 = vD,
(c) Equilibrium equations. Only the transverse equilibrium equation is affected by the afhne imperfection, so that these equations are N;+r-‘(N,-
N,)=O
M;‘+ r-*(2M;-M;)+(l+
k){N,(wi*+ r-‘wi)+ N&i}+ q ~0.
(7)
(ii) Boundary conditions The study is restricted to circular plates with either a simply supported or a clamped boundary and in each case full in-plane restraint is assumed. Thus the appropriate boundary condition equations are: (1) Simply supporrcd (I = la).
w,=o M,=DA,+D,k,=O
(W
u =o. (2) Clamped (I==fa).
w,=o Wi3:0
(W
u so.
(3) DR procedure. As the DR method is fully documented ekewhere[4], a description of its application to qns (4)-(8) is omitted. Neverthekss, it is worth pointing out that the present application does make use of rationally demrmined fictitious densitks[s] tog&her with a unit time increment. Thus only the two dampii factors require to be determined by &itrary means and their e&x&ion and adjustment to achieve rapid solution convergence is a relatively simple matter. This approach has been found to be superior to the more usual approach, in which both the fktitious densities and the damp& factors are deWmined by trial and error. RESULTS
A preliminary set of computations was undertaken in order to: (a) verify the computer program, and (b) to determine a suit&k mesh size for the main computations. Table 1, which compares the DR results with those of Nylander for a uniformly loaded, constant thichmss, simply supported, circular plate, ckarly demonstr&es objective (a). Rased on these and other results, it was conch&d that objective (b) would be met by usin8 a 105 interval inWkci~ mesh for the main computations of the parameter study. In order that the computer results may enjoy their widest possibk applkation, they are presented in Figs. 1-5 in nomiimensional graphical form, which allows intupoktion with reasonable accuracy.
562
G. J. TURVEY
Table 1. Comparisonbetween Nylander’sand the DR prm results at the ctntre of a uniformlyloaded, constant thickness, simply supported,circularplate with an alEneimperfection(v = 0.25) ~r.WUre i
l
Aifi8ltt
IYl&Or*e
k
R..alte*
Zl
zm
DR Promam =I
r,t
Romalts qt
20
0.36
1.40
2.90
1.3684
2.9331
20
1.oo
1.00
2.75
1.0020
2.7640
20
4.44
0.45
2.25
0.4828
2.1456
The Iiylm~der Remlte uod are-therefore
tire
boon
evalurrtrd
by rcsling
to two decimal
only g1v.n
. Fig. I(a).
k-05
Fig. I(b).
placer.
gnphe
563
Jsgedefkztionofcircuiar phteswitba5aeimperfections
k-l -
ZOO
0
400
600
600
1000
ii
lc).
Fii.
k0
t 3.0
#.
/.C.-.
=, 20 / .’ ,’ /H
1.0
./.. .C ~/y. ./
.C.C
.I./
_ _--
_-
__--
i/---
001
0
1
200
I
400
B
I
I
I
60
600
am
Fig,l(d). Fig.1. &Ldj&al
centml defkctionvs lataalpmnuri for a simplyv (a)k-O,(b) k =0.5,(c)k-l,(d) k = 2.
cimh
phbe b-0.3);
Figure 1 presents the additional deflection at the centre of a simply suppor@d plate as a function of the lateral pressure for thee values of the taper ratio and four values of the ahity parameter. It is evident that, in all cases, the additional de&&on -#lMKghUyWitb increasing taper ratio and decreases substantially as the value of the ahity parmae@ increases. Corresponding results for the clamped plate are not preaenW since they d&r only margblly from the simply supported results, i.e. from about S% at low values of the lateral pressure to rather less than 2% at higher values. Therefore, except where gteat accuracy is required, tbe results of Fu. 1 may be used for both sets of boundary conditbns. The central bending stress vs lateral pressure is plotted in Fw. 2 for a simply swporbd plate.Hmtoo,the~~nceofthebcndinestresronthevrlueoftBt~ratip~the ahity parameter is similar to that of Fu. 1, i.e. maqinal in the case of the former and rather more substantial in the case of the latter: Fw 3 compares the simply supported and clamped plate centre be&& stress for a = 20.5 and k = 1 and 2. It is evident that the stresses diverge as the taper ratio decreases, but that this effect reduces as k increases in value. SSVOL I4 NO. 1-C
G. 1. TIJRVBY
I #
19 -
00
I
0
200
1
1
400
600
1
I
boo
1000
T
6.0
I
__----
I’ _.-.
_._
s-o-
I 1.0 p
OQ-
0
I
200
1
I
400
600 7i
I
mo
1000
Large dcilection of circulnr pkes withalbe imperfections
565
In Fri. 4 the simply supportedplate care membrarkstrws vs latefal pressurecurves are plottedfor the upperand lower taperratiolimits and for each of the four k valueacoasidand herein. They demonstratethat this stress is almost indcpcat of the p&e taper ratio and, moreover,thatit~~~Uyasthevalueofthe~pu;rmettr,~~~ FiBalIY, Pi. 5 compares the plate ccntrc membranestress for both simply suppaaw and clamped~cwditIons.~compvativenapttsartshownfortbt~~of the t&perratio only, B similarbeiwioW is observed for ot&cra and k values. Thus, these cmws suggestthat the resuks presentedin Fii. 4 may be used for plates witb eitherbowdary conditionwithout all undue loss in accuracy.
G.
J.
TURVW
-k-0
hgc
defkdon
of circular pl8tcs with 8fTineimpcrfcctions
ksO-5
50
30 I-
?Tt ZC )-
11)-
0
r! 200
600
600
800
1000
s’
Fig. 4(b).
.!k! LO /
30
-
800
Fk. 4%
1000
567
G. J. TURVEY
i Fig. 4(d). Fii
Fii.
4. Central mcmbraue stress YS lateral pressure for a simply supported cifculpr plate (v = 0.3); (a) k a 0, (b) k = 0.5, (c) k = I, (d) k = 2.
5. Central
membrane
stress vs &ml
pressure for simply supported (v = 0.3).
and C&II@
circular
plates
CONCLUSIONS
conclusions to be drawn from the computed results are summa&d as follows: (1) The central de&&ion increases moderately with increasing taper ratio and reduces substantially as the alllaity parameter increases. (2) The central bending stress decreases moderately with increasing taper ratio and reduces substanti&y as the affiaity parameter imxeases. (3) The central membrane stress is reiativeiy insensitive to taper ratio and decreases only my as the a&&y parame@ increases. (4) The centraI deflection and membrane stress are reiativeiy insensitive to the boundary support conditions. The
main
Ackno~-The computing facilith
author wia&s to expressbis iadsbtcdness and whncc
with the prcpahon
of the figures.
to the Dept. of En~ring
for providin#
both
Lafgctkktioaofcirtuhrplrteswithrlharimpcrftctions
569
REFERENCES 1. H.Nyhader,I~ydcscctedtbia~~hithl~~to~ddlection.fnl.~r.forB~ond
sknctuml Eagng 11,346-374 (1951). 2. S. D. N. Murtityaad A. N. !ibabmw. Nalinarbendingofehsticphtcsofvahbkpro6k.I.&agAkAIXu., Proc. Am. Sot. ciu. hgrs. 10 (EM2). 251-265 (1974). 3. G. J. Turvey, Largeddkaion of taperedlMUIU plates by dynamicrelaxation.1. EngngMech.Do., h. Am. Sot. ciu. ERgm 1.4. EM2 (April !978). 4. A. S. Day. An iatrodudlon to dyuunic relaxation. 7he hgincer 2ml), 218-221 (19611). S. A. C. CurelI and R. E. Hobbs, Nunmicd stability of dynunic rchxahn analysis of non-lirtcaratructwcs. ht. 1. Num. Methods En.givglY6). 1407-1410(1976).