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Chapter 7 Geometrically Nonlinear Analysis of Plate Bending Problems 7.1 Introduction In the previous six chapters, we have discussed the application of T-elements to various linear problems. This approach can also be used to solve both geometri-cally [8-10,12] and materially [1,18] nonlinear problems. The former are treated in this chapter and the latter are addressed in Chapter 8. The first application of T-elements to nonlinear plate bending problems was by Qin [8], who presented a family of hybrid-Trefftz elements based on a modified variational principle and the incremental form of field equations. In his report, exact solutions for the Lamé-Navier equations are used for the in-plane intra-element displacement field and an incremental form of the governing equations is adopted. With the aid of the incremental form of these equations, all nonlinear terms may be taken as pesudo-loads. Moreover, some modifications have been made on nonlinear boundary equations to simplify the corresponding derivation. As a result, the in-plane and out-of-plane equations are uncoupled, and then the derivation for the HT FE formulation becomes very simple. In this chapter, a family of hybrid-Trefftz elements is presented for nonlinear analysis of plate bending problems which include post-buckling problems of thin plates and large deflection of thick plates with or without elastic foundations. 7.2 Basic equations for nonlinear thin plate bending Consider a thin isotropic plate of uniform thickness t, occupying a two-dimensional arbitrarily shaped region Ω bounded by its boundary Γ (Fig. 7.1). The plate is subjected to an external radial uniform in-plane compressive load
)boundary at thelength unit (per 0 Γp . We use a Cartesian coordinate system in which the axes 1x and 2x lie in the middle plane of the plate. The field equations governing the post-buckling behavior can be written as follows [11]:
,
,0
,2
04
,
ijij
jij
wNwpwD
N
=∇+∇
= (7.1)
where ijN represents the membrane force component which can be expressed in terms of displacements in the form:
nij
lijij NNN += , (7.2a)
Fig. 7.1: Geometry and loading condition.
},12{
)1(2 ,,, ijkkijjilij uuuEtN δ
μ−μ
++μ+
= (7.2b)
}1
{)1(2 ,,,, ijkkji
nij wwwwEtN δ
μ−μ
+μ+
= . (7.2c)
Substituting the constitutive equation (7.2) into eqn (7.1) yields the following differential equations in terms of three displacement functions
:},,{ 21 wuu
,
,
,
34
22312
12211
PwL
PuLuL
PuLuL
=
=+
=+
(7.3)
where
22,111,1 ) () () ( dL += , 12,22 ) () ( dL = ,
11,122,3 ) () () ( dL += , ), )( () ( 2244 ∇λ+∇= DL
Dp /02 =λ , 2/)1(1 μ−=d , 2/)1(1 μ+=d , (7.4)
in which 21 and uu represent in-plane displacement components, and
21 , PP
3 and P are three components of the so-called distributed pseudo-load defined by [11]
,)( 12,2,222,111,1,1 wwdwdwwP −+−= (7.5a)
ΩΓ
0p
1x
2x
o
,)( 12,1,222,11,12,2 wwdwwdwP −+−= (7.5b)
,))(1(
)})(2/())(2/{(
12,2,1,1,22,1
22,11,22,2,222,11,
21,1,13
wwwuuJ
wwwuwwwuJP
++μ−+
+μ++μ++= (7.5c)
with
21 μ−=
EtJ . (7.6)
The boundary conditions are
niin unuu == , (on nuΓ ), (7.7)
siis usuu == , (on suΓ ), (7.8)
ww = , (on wΓ ), (7.9)
niin wnww ,,, == , (on nwΓ ), (7.10)
njiijn NnnNN == , (on nNΓ ), (7.11)
nsjiijns NsnNN == , (on nsNΓ ), (7.12)
RRRR nl =+= , (on RΓ ), (7.13)
njiijn MnnMM == , (on nMΓ ), (7.14)
where ii sn and are components of the outward normal and the tangent to the boundary Γ ( =Γ )
nnnssnn MwRwNuNu Γ∪Γ=Γ∪Γ=Γ∪Γ=Γ∪Γ , repeated indi-ces i, j and k imply the summation convention of Einstein, and these indices take values in the range of {1,2}. The equivalent shear forces nl RR and are [8]
jisijijijl snMnMR ,, += , snsnn
n wNwNR ,, += . (7.15)
Finally, the relationship between moments and deflection is given by
])1[( ,, kkijijij wwDM μδ+μ−= . (7.16)
Noting that eqns (7.3), (7.7)-(7.14) are not, in general, suitable for HT FE analysis, an incremental form of these equations must therefore be adopted to linearize these nonlinear terms. Denoting the incremental variable by the superimposed dot and omitting those infinitesimal terms resulting from the product of incremental variables, one obtains
,
,
,
34
22312
12211
PwL
PuLuL
PuLuL
=
=+
=+
(7.17)
where 321 and , PPP are given by [13]
,)()( 12,2,212,2,222,111,1,22,111,1,1 wwdwwdwdwwwdwwP −−+−+−= (7.18a)
,)()( 12,1,212,1,222,11,12,22,11,12,2 wwdwwdwwdwwwdwP −−+−+−= (7.18b)
}.)()){(1(
)})(2/())((
))(2/())({(
12,2,1,1,22,112,2,1,2,1,1,22,1
22,11,22,2,222,11,2,2,2,2
22,11,21,1,122,11,1,1,1,13
wwwuuwwwwwuuJ
wwwuwwwwu
wwwuwwwwuJP
++++++μ−+
+μ+++μ++
μ+++μ++=
(7.18c)
The related boundary conditions become
niin unuu == , (on nuΓ ), (7.19)
siis usuu == , (on suΓ ), (7.20)
ww = , (on wΓ ), (7.21)
niin wnww ,,, == , (on nwΓ ), (7.22)
*nji
nijnji
lijn NnnNNnnNN =−== , (on
nNΓ ), (7.23)
*nsji
nijnsji
lijns NsnNNsnNN =−== , (on
nsNΓ ), (7.24)
*RRRRR nl =−== , (on RΓ ), (7.25)
njiijn MnnMM == , (on nMΓ ). (7.26)
It should be pointed out that lij
nij NN << , ln RR << in practical problems. So
we may move these non-linear terms to the right-hand side of the above boundary equations. In this way, the in-plane and out-of-plane boundary equations are uncoupled. As a result, the ensuing derivation becomes quite simple, but an iterative approach is required to evaluate the nonlinear terms . and , , *** RNNP nsni
7.3 Assumed fields and Trefftz functions 7.3.1 Assumed fields In this application the internal fields have two parts. One is the in-plane field uin (={u1,u2}T) and the other is the out-of -plane field uout (=w). They are identified by subscripts “in” and “out” respectively, and are assumed as follows:
=inu inininininuu
cNucNN
u +=⎭⎬⎫
⎩⎨⎧
+=⎭⎬⎫
⎩⎨⎧
2
1
2
1 , (7.27)
outout ww cNu 3+== , (7.28)
where cin and cout are two undetermined coefficient vectors and 3 and , , NNu inin w are known functions which satisfy
,2
1
32
21
⎭⎬⎫
⎩⎨⎧
=⎥⎦
⎤⎢⎣
⎡PP
LLLL
inu 02
1
32
21 =⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡NN
LLLL
, (in eΩ ), (7.29)
,34 PwL = 034 =NL , (in eΩ ), (7.30)
and where 3 and NNin are formed by a suitably truncated T-complete system of the governing equation (7.17), which is given at the end of this section. All that is left is to determine the parameters c so as to enforce on u (={ )} , , 21
Twuu inter-element conformity ( )on e ffe Γ∩Γ= uu and the related boundary conditions, where ‘e’ and ‘f ’ stand for any two neighbouring elements. This can be completed by linking the Trefftz type solutions (7.27) and (7.28) through an interface displacement frame surrounding the element, which is approximated in terms of the same DOF, d, as used in the conventional elements
dNu ~~ = , (7.31)
where
Tin }~ ,~{~
outuuu = , (7.32)
Tin uu }~ ,~{~
21=u = indNN
⎥⎥⎦
⎤
⎢⎢⎣
⎡
2
1~~
inindN~= , (7.33)
outT
nout ww dNNu
⎥⎥⎦
⎤
⎢⎢⎣
⎡==
4
3, ~
~}~ ,~{~
outoutdN~= , (7.34)
Toutin } ,{ ddd = , (7.35)
and where outin dd and stand for nodal parameter vectors of the in-plane and out-
of-plane displacements, and iN~ (i=1-4) are the conventional FE interpolation functions. For example, along the side A-B of a particular element (see Fig. 7.2), a simple interpolation of the displacements may be given in the form
ABinAB
ABABin )(~
2
1)( d
NN
u ⎥⎦
⎤⎢⎣
⎡= , (7.36)
ABoutAB
ABABout )(~
4
3)( d
NN
u ⎥⎦
⎤⎢⎣
⎡= , (7.37)
where
⎥⎥⎦
⎤
⎢⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
2
2
1
1
2
1 ~0
0
~~0
0
~
NN
NN
AB
AB
NN
, (7.38)
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−=⎥
⎦
⎤⎢⎣
⎡
22
61
21
625
12
41
11
423
4
3~2
~~2
~
0
~2~2
~~2
~
0
~221
NnlNn
NnlNnN
NnlNn
NnlNnN ABABABAB
AB
AB
NN
, (7.39)
TBBAAABin uuuu } {)( 1121=d , (7.40)
TByBxBAyAxAABout wwwwww })( )( )( )( {)( ,,,,=d , (7.41)
in which iN~ are defined in eqn (2.27) for i=1,2, and in eqn (4.27) for i=3-6.
Fig. 7.2: A typical HT element in post-buckling analysis of thin plates. The generalized boundary forces and displacements can be easily derived from eqns (7.19)-(7.26), and these expressions are as follows:
inins
nin u
ussnn
uu
cQvv 12
1
21
21 +=⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡=
⎭⎬⎫
⎩⎨⎧
= , (7.42)
•
••
ABΔ
A• •
B
1+=ξ0=ξ1−=ξΔ
• 2,1,21 , , , , wwwuu
1
1n
2n n
innns
nin
NNN
snsnnn
snn
snn
NN
cQtt 21
12
22
11
1221
21
22
22
11
21 2
+=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎦
⎤⎢⎣
⎡+
=⎭⎬⎫
⎩⎨⎧
= , (7.43)
outoutn
out ww
cQvv 3,
+=⎭⎬⎫
⎩⎨⎧
= , (7.44)
outoutn
out
MMM
nna
na
na
MR
cQtt 4
12
22
11
21
322
221
1
2+=
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎦
⎤⎢⎣
⎡−−−
=⎭⎬⎫
⎩⎨⎧
−= , (7.45)
ins
nin u
ussnn
uu
cQv 52
1
21
21~~
~~
~ =⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡=
⎭⎬⎫
⎩⎨⎧
= , (7.46)
where
⎥⎦
⎤⎢⎣
⎡=
2
1
i
ii Q
QQ , (i=1-5), (7.47)
Tsnin uu } {=v , T
nout ww } { ,=v , Tnsnin NN } {=t , T
nout MR }{ −=t , (7.48)
,// 11111 ssnxna ∂∂+∂∂= ,// 22222 ssnxna ∂∂+∂∂= (7.49a)
ssnsnxnxna ∂∂++∂∂+∂∂= /)(// 122121123 . (7.49b)
7.3.2 Trefftz functions The T-complete functions corresponding to the first two lines of eqn (7.17) have been given in expressions (3.38)-(3.41). What follows is to derive the Trefftz functions related to the third line of eqn (7.17). In doing so, consider the homogeneous equation
0)( 2224 =λ+∇∇= ggL . (7.50)
To find the solution of eqn (7.50), we set
Ag =∇2 , (7.51)
Bg =λ+∇ )( 22 . (7.52)
Substituting eqns (7.51) and (7.52) into (7.50), one obtains
0)( 22 =λ+∇ A , (7.53)
02 =∇ B . (7.54)
The Trefftz functions corresponding to eqns (7.53) and (7.54) are, respectively
,cos)( θλ= mrJA mj θλ=+ mrJA mj sin)(1 , (7.55)
and
,cos θ= mrB mj θ=+ mrB m
j sin1 , (7.56)
where 2/122
21 )( xxr += and )/arctan( 12 xx=θ .
Subtracting eqn (7.51) from eqn (7.52) and using eqns (7.55) and (7.56), the Trefftz functions of eqn (7.50) can be given by
θ= mrfg mj cos)( , θ=+ mrfg mj sin)(1 , (7.57)
where ).()( rJrrf mm
m λ−= As a consequence, the T-complete system of eqn (7.50) may be taken as
Τ }{}sin)( ,cos)( ),({ 0 imm Tmrfmrfrf =θθ= . (7.58)
7.4 Particular solutions The particular solutions inu and w in eqns (7.27) and (7.28) are obtained by means of a source function approach. The source functions corresponding to eqn (7.17) can be found in [13]:
])1(ln)3([41)( ,,
*jPQiPQPQijPQij rrr
Eru μ++δμ−−
πμ+
= , (7.59)
)](ln2[4
1)( 02*
PQPQPQ rYrD
rw λπ−λπ
= , (7.60)
where )(*PQij ru represents the ith component of in-plane displacement at the field
point P of an infinite plate when a unit point force (j=1,2) is applied at the source point Q, while )(*
PQrw stands for the deflection at point P due to a unit transverse force applied at point Q. Using these source functions, the particular solutions inu and w can be expressed as
Ω⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
= ∫Ω duu
Pj
jjin
*2
*1u , (7.61)
Ω= ∫Ω dwPw *
3 . (7.62)
7.5 Modified variational principle The HT FE formulation for post-buckling analysis of thin plates can be obtained by means of a modified variational principle [7]. The related functional used for deriving the HT FE formulation is, in this case, in the form
,~)(
)({
94
2
*
*)()(
dsdsuNN
dsuNN
inTinsnsns
nnne
ineinm
ee
e
vt∫∫
∫∑
ΓΓ
Γ
−−+
−+Π=Π
(7.63)
,~)(
)({
98
6
*
,
)()(
dsdswRR
dswMM
outTout
nnne
outeoutm
ee
e
vt∫∫
∫∑
ΓΓ
Γ
−−+
−−Π=Π
(7.64)
where
dsuNdsuNdU snsnninineeee∫∫∫ ΓΓΩ
−−Ω=Π31
)( , (7.65)
dswRdswMdUeee
nnoutoute ∫∫∫ ΓΓΩ−+Ω=Π
75 ,
)( , (7.66)
**
21
621
ijijklklin NNEt
NNEt
U μ++
μ−= , (7.67)
)])(1(2)[()1(2
12211
212
222112 MMMMM
DUout −μ+++
μ−= , (7.68)
ijkkijij NNN δ−=31* , (7.69)
and where eqns (7.17) are assumed to be satisfied, a priori. The boundary Γe of a particular element consists of the following parts:
, 987965
943921
eeeeee
eeeeeee
Γ+Γ+Γ=Γ+Γ+Γ=
Γ+Γ+Γ=Γ+Γ+Γ=Γ (7.70)
where
,1 nuee Γ∩Γ=Γ ,2 nNee Γ∩Γ=Γ ,3 suee Γ∩Γ=Γ (7.71a)
,4 nsNee Γ∩Γ=Γ ,5 nwee Γ∩Γ=Γ ,6 nMee Γ∩Γ=Γ (7.71b)
,7 wee Γ∩Γ=Γ Ree Γ∩Γ=Γ 8 , (7.71c)
and 9eΓ represents the inter-element boundary of the element. Consequently, we discuss some properties and their proof of these two functionals. They are:
(a) The modified complementary principles
⇒=Πδ 0)(inm Conditions (7.19), (7.20), (7.23), (7.24) and
fin
ein vv = , 0=+ f
inein tt , ( )on fe Γ∩Γ . (7.72)
⇒=Πδ 0)(outm Conditions (7.21), (7.22), (7.25), (7.26) and
fout
eout vv = , 0=+ f
outeout tt , ( )on fe Γ∩Γ . (7.73)
(b) Theorems on the existence of extremum
(b1) If the expression
dsdsuNdsuNdU ine
Tinsnsnnin
ensNnN
vt ~ 9
2 δδ−δδ−δδ−Ωδ ∑∫∫∫∫ ΓΓΓΩ (7.74)
is uniformly positive (or negative) at the neighbourhood of )} {( 20100Tuu=u ,
where 0u is such a value that )( 0)( uinmΠ is the stationary value of functional ,)(inmΠ we have
)()( 0)()( uu inmininm Π≥Π , [or )()( 0)()( uu inmininm Π≤Π ]. (7.75)
(b2) If the expression
dsdswRdswMdU oute
Toutnnout
eRnM
vt ~ 9
,
2 δδ−δδ−δδ+Ωδ ∑∫∫∫∫ ΓΓΓΩ (7.76)
is uniformly positive (or negative) at the neighbourhood of 0w , where 0w is such a value that )( 0)( woutmΠ is the stationary value of functional ,)(outmΠ we have
)()( 0)()( ww outmoutm Π≥Π , [or )()( 0)()( ww outmoutm Π≤Π ]. (7.77)
PROOF. From the first, we derive the stationary conditions for the functional .)(inmΓ To this end, performing the variation of )(inmΓ and noting that eqns
(7.17)1,2 hold, a priori, by the previous assumption, one obtains
,]~)~[(
)()(
)()(
e
*
*
)17.7( )(
9
2,1
∑∫
∫∫
∫∫
Γ
ΓΓ
ΓΓ
δ−δ−+
δ−−δ−−
δ−+δ−Πδ
e
nsNnN
sunu
ds
dsuNNdsuNN
dsNuudsNuu
inTinin
Tinin
snsnsnnn
nsssnnninm
vttvv
(7.78)
where the constrained equality2,1)17.7(
represents eqns (7.17)1,2 being satisfied, a priori. Therefore, the Euler equations for expression (7.78) are eqns (7.19), (7.20), (7.23), (7.24) and
fin
ein vv = , 0=+ f
inein tt , ( )on fe Γ∩Γ . (7.79)
The principle (7.72) has thus been proved. The theorem on the existence of extremum may be proved by means of the so-called second variational principle [2,15]. In doing so, taking the variation of
)(inmΠδ again and using the constrained conditions (7.17)1,2, one obtains
=Πδ )(2
inm ∫∫∫ ΓΓΩδδ−δδ−Ωδ
nsNnN
dsuNdsuNdU snsnnin
2
dsine
Tin
e
vt ~9
δδ−∑∫Γ =expression (7.74). (7.80)
Hence, the theorem has been proved from the sufficient condition on the existence of a local extreme of a functional [15]. In the same way, we can easily prove the above properties for )(outmΠ , and we omit those details here.
7.6 Element matrix The element stiffness matrix may be established by setting 0)( =Πδ inme and
0)( =Πδ outme . To simplify the following calculation, we first transform all domain integrals in eqns (7.63) and (7.64) except loading terms into boundary integrals. In fact, by reason of the solution properties of the intra-element trial functions, the functionals )(inmeΠ and )(outmeΠ can be converted to
,~
21)(
)(21
94
231
*
*
)(
∫∫∫
∫∫∫∫
ΓΓΓ
ΓΓΓΩ
−+−−
−−−−Ω=Π
eee
eeee
dsdsdsuNN
dsuNNdsuNdsuNduP
ininininsnsns
nnnsnsnniiinme
vtvt (7.81)
.~
21)(
)( 21
98
675
*
,
,
3)(
∫∫∫
∫∫∫∫
ΓΓΓ
ΓΓΓΩ
−+−−
−+−+Ω=Π
eee
eeee
dsdsdswRR
dswMMdswRdswMdwP
outoutoutout
nnnnnoutme
vtvt (7.82)
The substitution of expressions (7.27), (7.28), (7.31) and (7.42)-(7.46) into eqns (7.81) and (7.82) leads to:
21)( 21 rdrcdSccHc T
inTininin
Tininin
Tininme +++−=Π
+ terms without inc or ind , (7.83)
43)( 21 rdrcdSccHc T
outToutoutout
Toutoutout
Toutoutme +++−=Π
+ terms without outc or outd , (7.84)
in which
dsdsdseee
TTTin ∫∫∫ ΓΓΓ
++−=42
1222
1121
12 22 QQQQQQH , (7.85)
dse
Tin ∫Γ−=
9 52 QQS , (7.86)
,])([
])([
)(21) (
21
22
*12
21
*11
22
212
122
111
23
1
∫
∫∫
∫∫∫
Γ
ΓΓ
ΓΓΩ
−−+
−−+−
−++Ω+=
e
ee
eee
dsuNN
dsuNNdsu
dsudsdPP
sT
nsnsT
nT
nnT
sT
nT
inT
inTTT
QQQ
QvQtQNNr
(7.87)
dsQ inT
e
tr ∫Γ−=9
52 , (7.88)
dsdsdseee
TTTout ∫∫∫ ΓΓΓ
++−=86
3141
3242
34 22 QQQQQQH , (7.89)
dse
outT
out ∫Γ−=9
4~NQS , (7.90)
dsoutTout
e
tNr ∫Γ−=9
4~ , (7.91)
.])([
])([
)(21
21
8
67
5
41
*31
,4232
41
,424
3
333
∫
∫∫
∫∫∫
Γ
ΓΓ
ΓΓΩ
−−+
−−+−
−++Ω=
e
ee
eee
dswRR
dswMMdsw
dswdsdP
TT
nT
nnTT
nT
outT
outTT
QQQ
QvQtQNr
(7.92)
It should be pointed out again that all terms not involving c and d are of no significance for an approximate solution and are therefore not listed explicitly. The vanishing variation of the two functionals (7.83) and (7.84) with respect to c and d, at the element level, straightforwardly leads to the standard force-displacement relationship
ininin PdK = , (7.93)
outoutout PdK = , (7.94)
where
ininTinin SHSK 1−= , outout
Toutout SHSK 1−= , (7.95)
211 rrHSP += −
inTinin , 43
1 rrHSP += −out
Toutout , (7.96)
and where Kin and Kout can be calculated in the usual way, while Pin and Pout contain the unknown variables wuu , , 21 . An iterative procedure is thus required. The procedure is described below. 7.7 Iterative scheme Before describing the scheme, let us study some properties of Pin. It is obvious from eqns (7.87) and (7.96)1 that Pin depends only upon w . So only an initial value 0w is required. As long as the value of w in Ω is known, we can calculate the pseudo-load Pin, and then all unknown variables in eqn (7.93) are in-plane displacements { }., 21 uu We may solve eqn (7.93) for them. As a consequence, Pout can be evaluated from the current values of { }. ,, 21 wuu An iterative scheme may be established according to the above analysis. Specifically, suppose that
kkk wuu and , 21 stand for the kth approximations, which can be obtained from the preceding cycle of iteration. The (k+1)th solution may be evaluated as follows:
(a) Assume initial values 00 and ww in Ω. If the current loading step is not the first one, but the (k+1)th step, 00 and ww are taken as kk ww and , where
kk ww and stand for the incremental and the total deflection at the kth loading step, respectively.
(b) Enter the iterative cycle for i=1,2, . Calculate Pin in eqn (7.93) by means of eqn (7.96)1, solve eqn (7.93) for the nodal displacement vector )(i
ind , and then
determine the values of )(1
iu and )(2iu in Ω ;
(c) Calculate Pout using the current values of { } , , 21 wuu , then solve eqn (7.94) for )(i
outd and determine the value of )(iw in Ω ;
(d) If
ε≤−
=ε −−
−−
)1()1(
)1()1()()(
iTi
iTiiTii
dddddd , (7.97)
where ε is a convergence tolerance, proceed to the next loading step and calculate
,)(11
11
ikk uuu +=+ )(1
11
ik uu =+ , (7.98)
,)(22
12
ikk uuu +=+ )(2
12
ik uu =+ , (7.99)
,)(1 ikk www +=+ )(1 ik ww =+ , (7.100)
otherwise, set
,)(0 iww = )(0 ik www += , (7.101)
and go back to Step (b). In fact, the iterative approach belongs to the modified Newton method. Its order for the rate of convergence is about 2. However, it should be noted that the required stiffness matrices obtained from eqn (7.95) do not change through each step of computation. Hence, once the matrices have been formed, they can be stored in the core and used in each cycle of iteration. Obviously, this can save a large amount of computing time. 7.8 Extension to post-buckling thin plates on elastic foundation As treated in Section 4.9, in this case the left-hand side of the third line of eqn (7.17) and the boundary equation (7.26) must be augmented by the terms wK and
wGPα , respectively:
34 PwKwL =+ , (7.102)
nPjiijn MwGnnMM =α−= , (7.103)
where ,α K, GP are defined in Section 4.9.
The T-complete solution for eqn (7.102) can be obtained by considering the corresponding homogeneous equation
0))(()()( 22
12224
4 =+∇+∇=+∇η+∇=+ gbbgSgKL , (7.104)
where
22 λ=η , DkS w /= , (for Winkler type foundation), (7.105)
,/22 DGP−λ=η DkS P /= , (for Pasternak type foundation), (7.106)
,4421 Sb −η−η= Sb 442
2 −η+η= . (7.107)
To find the solution of eqn (7.104), set
,)( 12 Agb =+∇ Bgb =+∇ )( 2
2 . (7.108)
Substituting eqn (7.108) into (7.104), one obtains
,0)( 12 =+∇ Bb 0)( 2
2 =+∇ Ab . (7.109)
The Trefftz functions for eqn (7.109) are given by
,sin)( 22 θ= mbrJA mm ,cos)( 212 θ=+ mbrJA mm ( ,2,1=m ), (7.110)
,sin)( 12 θ= mbrJB mm ,cos)( 112 θ=+ mbrJB mm ( ,2,1=m ). (7.111)
Hence, the Trefftz functions of eqn (7.104) can be expressed as
,sin)(2 θ= mrfg mj θ=+ mrfg mj cos)(12 , (7.112)
where ).()()( 21 brJbrJrf mmm −= As a consequence, the T-complete system of eqn (7.104) may be taken as
Τ }{}cos)(,sin)(),({ 0 imm Tmrfmrfrf =θθ= . (7.113)
The variational functional used to derive the HT FE formulation is the same as those of eqns (7.63) and (7.64), except that the energy density function
outU
should be replaced by *outU :
** VUU outout += , (7.114)
where
⎪⎩
⎪⎨
⎧
+=
).foundation rnak type(for Paste),(21
),foundation peWinkler ty(for ,21
,,2
2
*
iiPP
w
wwGwk
wkV (7.115)
7.9 Geometrically nonlinear analysis of thick plates 7.9.1 Basic equations Consider a Mindlin-Reissner plate of uniform thickness t, occupying a two-dimensional arbitrarily shaped region Ω bounded by its boundary Γ. The nonlinear behavior of the plate for moderately large deflection is governed by the following equations [6]:
(a) Equilibrium equations in Ω:
12211 PuLuL =+ ,
22312 PuLuL =+ ,
qPLLwL +=ψ+ψ+ 323513433 ,
024514443 =ψ+ψ+ LLwL ,
025515453 =ψ+ψ+ LLwL . (7.116)
(b) Constitutive relationship in Ω:
nij
lijij NNN += ,
},12{
)1(2 ,,, ijkkijjilij uuuEtN δ
μ−μ
++μ+
=
}1
{)1(2 ,,,, ijkkji
nij wwwwEtN δ
μ−μ
+μ+
= ,
}12){1(
21
,,, kkijijjiij DM ψδμ−μ
+ψ+ψμ−= ,
)( , iii wCQ ψ−= . (7.117)
(c) Natural boundary conditions
njiijn NnnNN == , (on nNΓ ),
nsjiijns NsnNN == , (on nsNΓ ),
njiijn MnnMM == , (on nMΓ ),
nsjiijns MsnMM == , (on nsMΓ ),
RwNwNnQR snsnnii =++= ,, , (on RΓ ). (7.118)
(d) Essential boundary conditions
niin unuu == , (on nuΓ ),
siis usuu == , (on suΓ ),
niin n ψ=ψ=ψ , (on nψΓ ),
siis s ψ=ψ=ψ , (on sψΓ ),
ww = , (on wΓ ), (7.119)
where L1, L2, L3 are defined in eqn (7.4), P1, P3, P3 in eqn (7.5), q represents the transverse distributed load, and
,233 ∇= CL ,) () () ( 1,4334 CLL −=−= ,) ( 2,5335 CLL −=−=
,144 CDLL −= 25445 DLLL == , CDLL −= 355 . (7.120)
For the reason stated earlier, the nonlinear equations (7.116)-(7.119) are not suitable for HT FE analysis, and an incremental form of these equations is required to linearize the above nonlinear terms. The linearized process of eqns (7.116)-(7.119) leads to
12211 PuLuL =+ , (7.121)
22312 PuLuL =+ , (7.122)
qPLLwL +=ψ+ψ+ 323513433 , (7.123)
024514443 =ψ+ψ+ LLwL , (7.124)
025515453 =ψ+ψ+ LLwL , (7.125)
together with
niin unuu == , ( on nuΓ ), siis usuu == , (on
suΓ ), (7.126)
ww = , (on wΓ ), (7.127)
niin n ψ=ψ=ψ , (on nψΓ ), siis s ψ=ψ=ψ , (on
sψΓ ), (7.128)
*nji
nijnji
lijn NnnNNnnNN =−== , (on
nNΓ ), (7.129)
*nsji
nijnsji
lijns NsnNNsnNN =−== , (on
nsNΓ ), (7.130)
*RRRnQR nii =−== , (on RΓ ), (7.131)
njiijn MnnMM == , (on nMΓ ), (7.132)
nsjiijns MsnMM == , (on nsMΓ ), (7.133)
where snsnnn wNwNR ,, += .
7.9.2 Assumed fields As noted before, the HT FE model is based on assuming two sets of distinct displacements, the internal field u and the frame field u~ (see Fig. 7.3). The internal field u fulfills the governing differential equations (7.121)-(7.125) identically and is assumed over each element as
⎭⎬⎫
⎩⎨⎧
=out
in
uu
u = Ncucc
NN
uu
+=⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡+
⎭⎬⎫
⎩⎨⎧
out
in
out
in
out
in
00
, (7.134)
where
,},{ 21T
in uu=u ,},,{ 21T
out w ψψ=u ,},{ 21T
in uu=u Tout w },,{ 21 ψψ=u , (7.135)
and where , , outin uu Nin, Nout are known functions which satisfy
,2
1
⎭⎬⎫
⎩⎨⎧
=PP
ininuL ,02
1=
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
=N
NLNL ininin (on eΩ ), (7.136)
,00
3
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧ +=
qP
outoutuL 0
5
4
3
=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧=
NNN
LNL outoutout , (on eΩ ), (7.137)
with
,32
21⎥⎦
⎤⎢⎣
⎡=
LLLL
inL ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
555453
454443
353433
LLLLLLLLL
outL . (7.138)
The interpolation functions Nin and Nout are formed by suitably truncated complete systems (3.38)-(3.41), (5.32) and (5.33). Furthermore, in order to enforce on u the conformity, fe uu = on fe Γ∩Γ (where ‘e’ and ‘f ’ stand for any two neighboring elements), an auxiliary conforming frame field of the form
dNu ~~ = , (7.139)
is defined at the element boundary, eΓ , in terms of parameter d, where
⎭⎬⎫
⎩⎨⎧
=out
in
uu
u ~~
~ , ⎭⎬⎫
⎩⎨⎧
=out
in
dd
d , (7.140)
inin uu
dNNu⎥⎥⎦
⎤
⎢⎢⎣
⎡=
⎭⎬⎫
⎩⎨⎧
=1
1
2
1~~
~~
~ , outout
wd
NNN
u⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
ψψ=
4
4
3
2
1 ~~~
~~~
~ , (7.141)
and where iN~ (i=1-5) are the usual interpolation functions.
Fig. 7.3: Typical thick plate p-element and its assumed fields.
In the development of the present p-method elements, the following assumptions are adopted. First of all, the element may be of a general quadrilateral shape or a triangular shape with 5 DOF ( ) , , , , 212 ψψwuu at each corner node and with 1 DOF ( w ) at each mid-side node (Fig. 7.3). In this case, the in-plane displacements and rotations are linear along each side of the element
•
• •
••
•
Δ
ΔΔ Δ
Δ 11n
2n n
CA B
A C B
1−=ξ 0=ξ 1+=ξ
), , , ,() , , , ,(
111
2121
pbawwuu
Δψψ•
Ncuu +=
dNu ~~ =
boundary, and the boundary deflection is quadratic. Secondly, to achieve higher order variations, an optional number of extra hierarchic modes is introduced along with the hierarchic DOF, ),( ii ba for ( )~,~
21 uu , ( ),, iii rqp for ( )~,~,~21 ψψw ,
which are conveniently associated with the mid-side node C (Fig. 7.3). Thus, along the side A-C-B of a particular element, the frame functions are finally defined by
∑ −δ++=i
iii
BA aMuNuNu 112111
~~~ , (7.142)
∑ −δ++=i
iii
BA bMuNuNu 122212
~~~ , (7.143)
∑ ξδ+++=i
iii
CBA pMwNwNwNw 543~~~~ , (7.144)
∑ −δ+ψ+ψ=ψi
iii
BA qMNN 112111
~~~ , (7.145)
∑ −δ+ψ+ψ=ψi
iii
BA rMNN 122212
~~~ , (7.146)
where 21~ and ~ NN are given in eqn (2.27), δ is defined in eqn (3.123), ξ is
shown in Fig. 7.3, and
)1(21~
3 −ξξ=N , )1(21~
4 +ξξ=N , 25 1~
ξ−=N , )1( 21 ξ−ξ= −iiM . (7.147)
From eqn (7.134), the conjugate vectors of the in-plane generalized boundary displacements and tractions have been given in eqns (7.42) and (7.43), while the out-of-plane boundary forces and displacements at any point of the element boundary eΓ are readily defined, in this case, as
outout
s
nout
w
ssnn
wcQvv 3
2
1
21
21
00
001+=
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
ψψ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
ψψ= , (7.148)
outout
ns
n
n
out
MMQ
cQtAGt 4+==⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
= , (7.149)
out
s
nout
w
ssnn
wdQv 6
2
1
21
21~~~
00
001
~~~
~ =⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
ψψ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
ψψ= , (7.150)
where
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧=
3
2
1
i
i
i
i
QQQ
Q , (i=3,4,6), (7.151)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+=
1221
21
22
22
11
21
21
2000
00
00
snsnnn
snn
snn
nnA , (7.152)
TMMMQQ } , , , ,{ 12221121=G . (7.153)
7.9.3 Particular solutions The in-plane particular solution inu can be calculated through use of eqns (7.59) and (7.61), while the source functions used for calculating the particular solutions of deflection and rotations outu are now as follows [6]:
⎥⎥⎦
⎤
⎢⎢⎣
⎡−λ
λ−λ
μ−λπ−= )1)(ln(
4)ln(
12
21)(
22
2*
PQPQ
PQPQ rr
rD
rw , (7.154)
]2/1)[ln(4
)( 1,*1 −λ
π−=ψ PQ
PQPQPQ r
Drr
r , (7.155)
]2/1)[ln(4
)( 2,*2 −λ
π−=ψ PQ
PQPQPQ r
Drr
r , (7.156)
where 22 /)1(10 tμ−=λ . Hence, the particular solution outu is given by
Ω⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
ψψ+=
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
ψψ= ∫Ω d
wqP
w
out*2
*1
*
3
2
1 )(e
u . (7.157)
7.9.4 Element matrix The functionals used for deriving the HT FE formulation of nonlinear thick plates can be constructed as [7]
,~21)(
)(21
114
231
*
*
)(
∫∫∫
∫∫∫∫
ΓΓΓ
ΓΓΓΩ
−+−−
−−−−Ω=Π
eee
eeee
dsdsdsuNN
dsuNNdsuNdsuNduP
ininininsnsns
nnnsnsnniiinme
vtvt (7.158)
,~21
)()()(
)(21
11
1086
975
*
3)(
∫∫
∫∫∫
∫∫∫∫
ΓΓ
ΓΓΓ
ΓΓΓΩ
−+
ψ−−ψ−−−−
ψ−ψ−−Ω+=Π
ee
eee
eeee
dsds
dsMMdsMMdswRR
dsMdsMdswRdwqP
outoutoutout
snsnsnnn
snsnnoutme
vtvt
(7.159)
where
, 1110911871165
11431121
eeeeeeeee
eeeeeee
Γ+Γ+Γ=Γ+Γ+Γ=Γ+Γ+Γ=
Γ+Γ+Γ=Γ+Γ+Γ=Γ (7.160)
with
,1 nuee Γ∩Γ=Γ ,2 nNee Γ∩Γ=Γ ,3 suee Γ∩Γ=Γ
,4 nsNee Γ∩Γ=Γ ,5 wee Γ∩Γ=Γ Ree Γ∩Γ=Γ 6 ,
,7 nee ψΓ∩Γ=Γ ,8 nMee Γ∩Γ=Γ ,9 see ψΓ∩Γ=Γ nsMee Γ∩Γ=Γ 10 , (7.161)
and 11eΓ representing the inter-element boundary of the element. The vanishing variation of these two functionals leads to the same force-displacement relationships as in eqns (7.93) and (7.94), except that the auxiliary matrices Hout, Sout, 3r and 4r are now replaced by
dsdsdsdseeee
TTTTout ∫∫∫∫ ΓΓΓΓ
+++−=1086
3343
3242
3141
34 222 QQQQQQQQH , (7.162)
dse
Tin ∫Γ−=
11 64 QQS , (7.163)
,])([])([
])([
)(21)(
21
106
897
5
4233
41
*31
4232
43
42
414
3
333
∫∫
∫∫∫
∫∫∫
ΓΓ
ΓΓΓ
ΓΓΩ
ψ+−−+−−
ψ+−−ψ−ψ−
−++Ω+=
ee
eee
eee
dsMMdswRR
dsMMdsds
dswdsdqP
sT
nsnsTTT
nT
nnT
sT
nT
Tout
Tout
TT
QQQQ
QQQQ
QvQtQNr
(7.164)
dsoutT
e
tQr ∫Γ−=11
64 . (7.165)
It should be mentioned that the above formulations hold true also for the case
Fig. 7.4: Illustration for β and φ .
of sandwich plates if the following replacements are made [9]:
)( thGC c +⇒ , )1(2
)(2
2
μ−+
⇒tthED ,
)()1(42
thEtGc
+μ+
⇒λ .
where cG and h are shear modulus and thickness respectively of the core in a sandwich plate, and t is the face-sheet thickness. 7.9.5 Extension to thick plates on elastic foundation In the case of thick plates on an elastic foundation, the formulation presented in this section holds true provided that following modifications have been made. (a) The interpolation function outN should be formed from a suitably truncated
complete system of eqns (5.32) and (5.85), rather than eqns (5.32) and (5.33). (b) The source function ) , ,( *
2*1
* ψψw , used in calculating the particular solution
outu , is now replaced by [7]
)/1)(()/1)(()( 21012202* CDCCrYBCCDCCrKACrw PQPQPQ ++−= , (7.166)
)cos()]()([)( 212111*1 φ−β+−=ψ CrKCACrYCBr PQPQPQ , (7.167)
)sin()]()([)( 212111*2 φ−β+−=ψ CrKCACrYCBr PQPQPQ . (7.168)
where φβ and are defined in Fig. 7.4, 21 and CC are defined in Section 5.6, and
,)(2
1
21 CCDA
+π=
)(41
21 CCDB
+−= . (7.169)
Q φ
ns
β
P
PQr
1x
2x
o
7.10 Numerical examples To test the applicability and accuracy of the Trefftz finite element formulation described in this chapter, some numerical examples are presented in this section. In all calculations, one quarter of the solution domain is analyzed. L is the side-length for a square plate and a is the radius for a circular plate. The convergence tolerance is taken to be ε = 0.0001. Example 1: Post-buckling analysis for a simply supported circular plate.
Consider a circular plate subjected to a uniform compressive load 0p at the boundary Γ . The geometric and material parameters are
,3.0=μ ,cm/kg102 26×=E 48/ =ta .
To study the convergence properties of the proposed formulation, three meshes, as shown in Fig. 7.5, are used in the analysis. The numerical results describing the relationship between the maximum deflection twm / occurring at the centre and the compressive load coefficient crpp 00 /=β (β >1) are shown in Table 7.1, where crp0 stands for the linear buckling load of a plate (p0cr=
2/198.4 aD in this example). Comparison is made with other methods [13]. As is shown in Table 7.1, the results obtained by the present approach seem to be better than those of BEM, even with 20 elements. It should be pointed out that the BEM results were obtained using 8 constant boundary elements and 16 internal cells, and the values in [17] were obtained from Fig. 68 in p. 169. Table 7.1. Maximum deflection twm / for Example 1.
Method mesh β =1.2 1.4 1.6 1.8 2.0 HT FE 20 cells 0.836 1.192 1.433 1.680 1.866 36 0.846 1.203 1.462 1.692 1.883 52 0.852 1.211 1.475 1.703 1.895 BEM [13] 0.882 1.235 1.533 1.792 2.086 [17] 0.860 1.220 1.490 1.720 1.920
Example 2: Post-buckling analysis for a clamped circular plate. This example differs from the previous one only by the boundary condition, which is now clamped. In this case 2
0 /68.14 aDp cr = . Table 7.2 compares the results using the present HT FE formulation and other methods [13]. It can be seen that the HT FE results are in good agreement with the other solutions. In the course of computations, convergence was achieved with about 10 iterations for Example 1 and 12 iterations for Example 2 at each load step (in our analysis the
loading step 0p is taken to be 0.2 crp0 ). As expected for the two examples, it was also found from the numerical results that the deflection converges gradually to the perturbation solution [17] along with refinement of the element meshes. Table 7.2 Maximum deflection twm / for Example 2.
Method mesh β =1.2 1.4 1.6 1.8 2.0
HT FE 20 cells 0.948 1.354 1.652 1.933 2.108 36 0.965 1.362 1.679 1.942 2.132 52 0.979 1.386 1.695 1.960 2.171 BEM[8] 1.015 1.431 1.765 2.052 2.327 Ref.[13] 0.990 1.400 1.710 1.980 2.210
Fig. 7.5: Three element meshes. Example 3: Post-buckling analysis for a square plate on a Winkler-type foundation. This example shows the application of T-elements to the post-buckling problem of thin plates on an elastic foundation. The plate under consideration is a square plate on a Winkler-type foundation subjected to a uniform in-plane compressive load 0p . Some parameters of the problem are assumed as
,/5 44 LDkw π= 50/1/ =Lt , 26 cm/kg102×=E .
In the analysis a 22× mesh is used in a quarter of the plate to allow for comparisons with those in [4]. The load step is now taken to be 0.02 crp0 ( crp0 is
equal to 22 /501.4 LDπ for a simply-supported plate, to 6.017 22 / LDπ for a clamped, simply-supported plate, and to 7.229 22 / LDπ for a clamped plate, where ‘clamped, simply-supported’ refers to a plate with a pair of opposite edges clamped and the other pair of opposite edges simply supported). It should be mentioned that the results in [4] were also obtained with a 22× mesh. Table 7.3
(a) 20 cells (b) 36 cells (c) 52 cells
a
shows the relationship between the maximum deflection twm / and the com-pressive load coefficient crpp 00 /=β (β >1) for the simply supported square plate. Table 7.4 presents the corresponding values for a clamped, simply-supported plate. The results of twm / for a square plate with all edges clamped are shown in Table 7.5. It can be, again, seen from these three tables that the HT FE results are in good agreement with the existing solutions [4], demonstrating that the proposed HT HE model is suitable for the post-buckling analysis of thin plates on elastic foundation. In the calculations, convergence was achieved with about 14 iterations for the simply supported plate, 11 iterations for the clamped, simply- supported plate, and 15 iterations for the clamped square plate at each loading step. Table 7.3. Maximum deflection twm / for a simply supported square plate in
Example 3.
Method mesh β = 1.02 1.04 1.06 1.08 1.10
HT FE
[4]
2× 2 0.5033 0.7194 0.8846 1.0284 1.1495 4× 4 0.5041 0.7199 0.8862 1.0304 1.1525
2× 2 0.5057 0.7205 0.8892 1.0350 1.1670 4× 4 Not available
Table 7.4. Maximum deflection twm / for the clamped, simply-supported square plate in Example 3.
Method mesh β = 1.02 1.04 1.06 1.08 1.10
HT FE [4]
2× 2 0.4413 0.6286 0.7747 0.8996 1.0185 4× 4 0.4409 0.6280 0.7739 0.8987 1.0168
2× 2 0.4401 0.6264 0.7722 0.8978 1.0109 4× 4 Not available
Table 7.5. Maximum deflection twm / for a clamped square plate in Example 3.
Method mesh β = 1.02 1.04 1.06 1.08 1.10
HT FE [4]
2× 2 0.3623 0.5145 0.6342 0.7325 0.8229 4× 4 0.3621 0.5139 0.6335 0.7313 0.8218
2× 2 0.3615 0.5128 0.6300 0.7298 0.8186 4× 4 Not available
Example 4: Large deflection problem for a clamped thin square plate. The purpose of this example is to show the applicability of the formulation presented in Section 7.9 to the case of nonlinear thin plates. The square plate is subjected to a uniform distributed load q, and the geometric and material properties of the plate are
26 cm/kg102×=E , ,316.0=μ L = 762cm, t = 7.62cm, 44 16/ EtqLQ = .
The maximum deflection twm / occurring at the centre versus the load factor Q is shown in Table 7.6. Table 7.7 shows the variation of the central stress coefficient β ( )/4 22
11 LEtc β=σ with the load factor Q. All these results are compared with the solutions obtained by the conventional FE approach in which 16 Lagrangian nine-node elements are used [5].
Table 7.6. Maximum deflection twm / for Example 4.
Method mesh Q= 17.79 38.3 63.4 95 134.9
HT FE
FEM [5]
Exact [5]
2× 2 0.2611 0.4689 0.6892 0.8987 1.1013 4× 4 0.2365 0.4695 0.6913 0.9035 1.1072 6× 6 0.2368 0.4702 0.6932 0.9062 1.1105
0.2368 0.4699 0.6915 0.9029 1.1063 0.237 0.471 0.695 0.912 1.121
Table 7.7. The coefficient β of the central stress 11σ for Example 4.
Method mesh Q= 17.79 38.3 63.4 95 134.9
HT FE
FEM [5] Exact [5]
2× 2 2.6431 5.5011 8.4243 11.314 14.109 4× 4 2.6322 5.4639 8.2972 11.205 13.945 6× 6 2.6113 5.3024 8.1529 11.152 13.723
2.6319 5.4816 8.3258 11.103 13.827 2.600 5.200 8.000 11.100 13.300
As shown in Tables 7.6 and 7.7, the numerical results converge gradually to the exact results along with refinement of the element meshes. The convergence was achieved with about 12 iterations for each loading step. Example 5: Large deflection for a clamped square sandwich plate. The plate consists of two identical facings ( ,cm/kg1074.0 26×=E ,3.0=μ
cm127=L , t = 0.381cm ) and an aluminum honeycomb core ( 41035.0 ×=cG
,cm/kg 2 h = 2.54cm), which are subjected to a uniform load q. A 4 4× element mesh is used in the analysis. Table 7.8 shows the results for central deflection versus load parameter )2/()1(3 23 EthqLQ μ−= , and compari-son is made with the results in [14] in which the same element mesh is used. The results in Table 7.8 show that the HT FE formulation in Section 7.9 can be used to solve the sandwich plate problem.
Table 7.8. Maximum deflection twm / for example 5.
Q 10 20 30 40
HT FE [14]
0.709 1.272 1.635 1.834 0.70 1.26 1.62 1.82
Example 6: Large deflection for a circular plate on a Winkler-type foundation. Consider a uniformly loaded circular plate resting on a Winkler-type foundation, with radius a and clamped movable edges (i.e., nsnsn NNw ==ψ=ψ= )0= . Some parameters for the problem are assumed as
,50/ =ta ,3.0=μ ,100/4 =Dakw .15/ 44 == EtqaQ
A quadrant of the plate is modelled by three meshes (Fig. 7.5) and the loading step is taken as .1=ΔQ Table 7.9 shows the deflection tw / along the radius of the plate and compares it with the boundary element solution [3]. Table 7.10 lists the corresponding results varying with M, where M is the number of hierarchical DOF. In this and the next examples, the hierarchical DOF are defined by
,1 1pM ⇒=
,,,3 111 rqpM ⇒=
,,,,,5 11111 barqpM ⇒=
,,,,,,6 211111 pbarqpM ⇒=
,,,,,,,,8 22211111 rqpbarqpM ⇒=
.,,,,,,,,,10 2222211111 barqpbarqpM ⇒=
Example 7: Large deflection for an annular plate on a Pasternak-type foundation. The annular plate is subjected to a uniform distributed load q ( )/ 44 EtqaQ = and
rests on a Pasternak-type foundation. The inner boundary of the plate is in a free edge condition while the outer boundary condition is clamped immovable. Some initial data used in the example are given by
,1/ 32 =EtaGP ,5/ 34 == EtakK P ,3/1/ =ab .3/1=μ
where a and b are the outer and inner radii of the annular plate (Fig. 7.6). In the example, a quarter of the plate is modelled by three meshes shown in Fig. 7.6. The loading step is taken as .5=ΔQ Some results obtained by the proposed method are listed in Tables 7.11 and 7.12. It can be found from the above two examples that the results obtained by the HT FE formulation agree well with the existing results. As expected for the two examples, it can also be seen from Tables 7.9 and 7.11 that the HT FE formulation yields converging values along with refinement of the element meshes. The results in Tables 7.10 and 7.12 show that the hierarchic DOF pi, qi, ri are more important than the in-plane DOF ai, bi in these two examples. In the course of computation, convergence was achieved with about 8 iterations for Example 6 and 12 iterations for Example 7 at each loading step. Table 7.9. Deflection tw / along the radius r in Example 6 (M=0).
Method mesh r/a = 0.098 0.304 0.562 0.800 0.960 HT FE 20 cells 1.090 0.948 0.581 0.169 0.009 36 1.106 0.957 0.588 0.174 0.008 52 1.110 0.962 0.591 0.180 0.008 BEM [3] 1.108 0.961 0.592 0.179 0.009
Table 7.10. Deflection tw / versus M in Example 6 (36 cells).
M 0 1 3 5 6 8 10 r/a = 0.098 1.106 1.108 1.110 1.110 1.112 1.112 1.112 0.304 0.957 0.960 0.961 0.961 0.964 0.964 0.965 0.562 0.588 0.591 0.592 0.592 0.594 0.594 0.593 0.800 0.174 0.177 0.177 0.179 0.181 0.181 0.181 0.960 0.008 0.009 0.008 0.008 0.007 0.007 0.007
Table 7.11. Maximum deflection twm / for Example 7 (M=0).
Method mesh Q = 10 15 20 25 30 HT FE 16 cells 0.491 0.725 0.920 1.082 1.227 32 0.508 0.732 0.929 1.095 1.238 48 0.513 0.738 0.935 1.105 1.243 [16] 0.510 0.740 0.930 1.100 1.240
Fig. 7.6: Three element meshes in example 7.
Table 7.12. Maximum deflection twm / versus M in Example 7 (32 cells).
M 0 1 3 5 6 8 10 Q = 10 0.508 0.512 0.513 0.515 0.518 0.518 0.519 15 0.732 0.735 0.736 0.736 0.739 0.740 0.742 20 0.929 0.933 0.935 0.936 0.940 0.942 0.943 25 1.095 1.099 1.099 1.100 1.103 1.103 1.104 30 1.238 1.242 1.244 1.244 1.248 1.250 1.251
References [1] Freitas, J.A.T. & Wang, Z.M. Hybrid-Trefftz stress elements for elastoplas-
ticity. Int. J. Numer. Meth. Eng., 43, pp. 655-683, 1998. [2] He, X.Q. & Qin, Q.H. Nonlinear analysis of Reissner’s plate by the
variational approaches and boundary element methods. Appl. Math. Modelling, 17, pp. 149-155, 1993.
[3] Katsikadelis, J.T. Large deflection analysis of plates on elastic foundation. Int. J. Solids Struc., 27, pp. 1867-1878, 1991.
[4] Naidu, N.R., Raju, K.K. & Rao, G.V. Postbuckling of a square plate on an elastic foundation under biaxial compression. Compu. Struct., 37, pp. 343-345, 1990.
[5] Pica, A., Wood, R.D. & Hinton, E. Finite element analysis of geometrically nonlinear plate behaviour using a Mindlin formulation. Compu. Struc., 11, pp. 203-215, 1980.
(a) 16 cells (b) 32 cells (c) 48 cells
a
b
[6] Qin, Q.H. Nonlinear analysis of Reissner plates on an elastic foundation by the BEM. Int. J. Solids Struc., 30, pp. 3101-3111, 1993.
[7] Qin, Q.H. Hybrid Trefftz finite element approach for plate bending on an elastic foundation. Appl. Mathe. Modelling, 18, pp. 334-339, 1994.
[8] Qin, Q.H. Postbuckling analysis of thin plates by a hybrid Trefftz finite element method. Comp. Meth. Appl. Mech. Eng., 128, pp. 123-136, 1995.
[9] Qin, Q.H. Nonlinear analysis of thick plates by HT FE approach. Compu. Struct., 61, pp. 271-281, 1996.
[10] Qin, Q.H. Postbuckling analysis of thin plates on an elastic foundation by HT FE approach. Appl. Mathe. Modelling, 21, pp. 547-556, 1997.
[11] Qin, Q.H. Nonlinear analysis of plate bending by boundary element method (Chapter 8). Plate Bending Analysis with Boundary Elements, ed. M.H. Aliabadi, Computational Mechanics Publications: Southampton and Boston, 1998.
[12] Qin, Q.H. & Diao, S. Nonlinear analysis of thick plates on an elastic foundation by HT FE with p-extension capabilities. Int. J. Solids Struct., 33, pp. 4583-4604, 1996.
[13] Qin, Q.H. & Huang Y.Y. BEM of postbuckling analysis of thin plates. Appl. Mathe. Modelling, 14, pp. 544-548, 1990.
[14] Schmit, Jr L.A. & Manforton, G.R. Finite deflection discrete element analysis of sandwich plates and cylindrical shells with laminated faces. AIAA J., 8, pp. 1454-1461, 1970.
[15] Simpson H.C. and Spector S.J. On the positive of the second variation of finite elasticity. Arch. Rational Mech. Anal., 98, 1-30, 1987.
[16] Smaill J.S. Large deflection response of annular plates on Pasternak foundations. Int. J. Solids Struc., 27, pp. 1073-1084, 1991.
[17] Thompson, J.M.T. & Hunt, G.W. A General Theory of Elastic Stability, Wiley: London, 1973.
[18] Zielinski, A.P. Trefftz method: elastic and elastoplastic problems. Comp. Meth. Appl. Mech. Eng., 69, pp. 185-204, 1988.
