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Calculation in the unsymmetrical bending problem of thin plates by spline boundary layer method
Computers
d Srru~r~res Vol. Britain.
34. No. 4. pp. 663668.
1990
004s7949190 $3.00 + 0.00 Pergamon Press plc
Printedin Great
CALCULATION IN THE UNSYMMETRICAL BENDING PROBLEM OF THIN PLATES BY SPLINE BOUNDARY LAYER METHOD DE-GAN Gu Department of Engineering Mechanics, Wuhan University of Technology, Wuhan, The People’s Republic of China (Received 21 February 1989) Abstract-The weighted-residual and perturbation methods are usually used to solve the unsymmetrical bending problem of thin plates. Here a mixed method of the weighted-residual and singular perturbation is considered. It is known that there are many ways from which a combination of the two methods can be. chosen to obtain numerical solutions which are uniformly valid throughout the region, but they are very different in theory. In this article the main purpose is to give a description of a mixed method for calculation in the unsymmetrical bending problem of plates, so only the singular perturbation problem for the differential equation involving a small parameter in higher derivatives will be discussed. Called the spline boundary layer method for short, boundary layer solutions are constituted with the spline function in order to overcome the difficulties of obtaining the boundary layer solution. An example is
given. Not only is the method simple and convenient for computations with a computer, but also the expression for the outer solution is offered simultaneously, and if necessary the expression of the boundary layer solution can also be given without using any complicated matching principle. Therefore the applications of the singular perturbation method in engineering calculations can be expanded.
THE BASIC METHOD
To illustrate the main characteristic of the spline boundary layer method, we consider the two point boundary-value problem on the region [a, 61: &[a]
+ 4_,[U] =f(x)
(0
&[u] = rpi (i = 0, 1,. . . , k - I),
(1) (2)
where Lk and Lk_, are order k and k - r linear differential operators, respectively, and BE is the boundary differential operator. The problems of degeneration and perturbation of all orders can be given by (3)
L-,[%I =.m) B,,,[u,,]=cp,
(i=O,l...,
k-l)
(4)
(n b 1)
(3
and
Lk-rb4 = -L&-II 48,i[%l= O-
(6)
The order of eqns (3) and (5) of both degeneration and perturbation are lower by r than that of goveming eqn (l), and the r boundary conditions must have been lost at least. If we assume that the boundary layer exists in the neighbourhood of x = a and x = b, all the boundary conditions will be discarded when we constitute the outer solution by means of degener663
ation and perturbation solutions. Certainly, we may retain partial boundary conditions when the boundary layer only exists in the neighbourhood of x = a (or x = b), and the problem will be simpler. Now we consider the more general problem in which two boundary layers, the inner and the outer, are to be constituted. Assuming that the outer solution is s-l 24= c &i&(X)+ O(E’),
(7)
i=O
where uj(x) contains k - r arbitrary constants, the number of integral constants is s(k - r). Only in the region distant from boundaries x = a and x = b does the expression (7) satisfy eqn (1) approximately. The method of spline collocation point can be taken in the neighbourhood of x = a and x = b to obtain boundary layer solutions, but there is no need for the outer region in the method of spline collocation point as the outer solution has already been obtained. Accuracy of the outer solution can be obtained if we choose a small number of points in the outer region for spline interpolation. It is unsuitable to constitute a uniform solution in the whole region with the combination of an equidistant knot 6-spline function as is done in many papers [l]. Since the requirement of spline knots is different in different parts of the interval [a, b], we may choose the expression of unequidistant knots to produce a uniform solution. For this purpose, we let the discrete points be, in ascending order as A, a=x,
664
D. Gu According to eqns (lo)-( 13) the successive derivatives of Nm,j (x) can easily be determined by the recurrence relation. The solution of eqn (1) can be expressed approximately by
the extension of A is A,. x-,-C x-nl+I <...
We introduce extra knots outside the interval [a, b], so that the uniform solution throughout the region can be expressed as a linear combination of spline functions. Assuming that the 6-spline degree m (usually m is an odd number larger than three) is {N,.j(x), j = -m,
-m
+
u=
WI (‘3,
where
[Nl=[N,,-,,-,(x)N,,-,(x).
1,. . . , n - l},
(14)
. Nm,n-,(x)1
{Cl = [C-,c-q+, c-q+* . . . C”,,l’.
where
G,(t -x)
G,,(x) = x:,
= (t -x)“,
There are II + 1 spline knots. Substituting eqn (14) into eqn (1) and boundary condition (2) yields
.
Hence, L[N[
j+m+l Gm(xj-x,.
. .,Xj+m+l -x)=
1 wiGm(xi-x), i=,
+
Lr[Nll{C) =f(x,,)
(15)
(8)
[B~,i[N]]{C)=~i (i=0,1,2...,k-l),
(16)
where where q=
n
-.
/=j
xi-x,
(9)
/+i
The successive derivatives of N,,,j (x) should be calculated, denoted by Nb,j, NG,, . . . , N$,j (1 < m), and can be written as 1 Nk, = m PNtt-1.j xj ( Xjtm-
1 Nm-2.j
( Xj+m-l-Xj
1 Nm-z,j+l
xj+ni -xi+1 Nh-l,j+I =(m-1) -
’ ( xjtm-xjtl 1
(11)
>
Nm-*,j+i
(12)
Nm-2.jt2 ; xnt*tl-xjtZ
>
consequently, Nk,j = m ( Xj+m -
..
>
- 1) -
= L,Wm, -Q-,(xNL-~[N~,-&)I. . . . LMn.n-,(xM,=xp
(10)
Nm-i,j+i
Xj+m+l-Xj+I
N:_i,j=(m
LVV
.
[Be,i[Wl= [Be,i[Nm, -~q-~(x)Pct[Nrn. -~q(x)I.. .
1
-
Lb’1 = M% -~-~(x)lL[Nm,-&)I. . . UN~,.-I(XL~~
1 Nit- 1.j
where xP is some determined point. It is noted that letting all x satisfy the equation is impossible, and in eqn (16) xg is a boundary point (B = 0 or n). Assuming that the width of the inner boundary layer in the neighbourhood of x = a is x, - x0, that of the outer boundary layer in the neighbourhood of x = b is x, - x,; the spline knot in the inner boundary layer is x = x,(0 < c( < c), that in the outer boundary layer is x = x,(d < /I < n), and the spline knot in the outer region is x = x, (c < e c d). A system of linear equations can be formed from eqns (7) and (14)-( 16), and the number of algebraic ecluations is 2q + 1 + n. We may write
-xj
{C> = if,1
1 *
N,Li,j+r xj+mtl-xjtl
[NJ =
(13)
(17)
(18 . * . . . . . . Nm,-,-I(x,) Nm,-zq-,(x ..Nm,.-I(x I. >
where
Nn,-2q-I(XO)
Ntn.
X,-,-,(x,)
Nw-2,(x,)
-2g(xo)
. . .
K,.-,(xo)
. . .
Kn,,-,(x,)
Spline boundary layer method for bending of thin plates
Hence, c + 1 algebraic equations are involved in eqn (17). It is noted that L,[N,] and L,_,[N,] denote ‘substituting the x value after derivation’, and the following expressions have the same meaning. ’ AxX
{f,} = W%)f(x,).
(19)
(20)
wk[~~ll+ Lr[N,HCI = u31*
665
polation conditions are identical; and (2) the integral constants in the outer solution cannot be determined by the boundary conditions. In this case it is necessary to establish supplemental equations from the matching principle. For example, if there are two integral constants which cannot be determined by the boundary conditions, we may establish four equations by the methods of both spline collocation point and interpolation at x = x, and x = x,, respect-
(21 . . . * . . . Nm, -+,(A) Nm, -&J Nnv-I W 1
which contains n - d + 1 algebraic equations, where
Nm,-,+ I (xc4 Nn,-2q-,h+,
IN,1= :
K, -x&d
)
Nm,n-~(d
Nm,-2&+,)
{.&@I = Lmflf(Xdfl)~ . ~f(X,Yl
(22)
Nm-I&,+,)
ively, so that the additional
equations are:
s-l
P,.,[N,.ll{C}= {cp*;>(i = 0, 1,. . . , k - 1). (23)
P’cI{CI = c ~‘~i(~)I.v=.~< i=o r-l [Nc,{C)= 1 &bL.w
Equation (23) is obtained from boundary conditions; it should contain k algebraic equations, where N,,-,-,(x0) IN’]= [ N,,,,+i(x,)
1
N,,-&,) N,,,,_,(x,)
. . . N,,n-,(x,) . . . N,,,-I (x,) ;
(24)
(2% (30)
,=O
where
WI=
[N,,-,,-,(x)N,,-,,(x)...
Nm,.-,(xk=.~~
we note that Bc.i [N,] denotes ‘substituting x = x0 and x = x, after derivation’. Since there is no mathematical criterion to determine
bP,,l= b&&&,)1~.
(25) the width of the boundary layer, x, and x, are all that
According to the interpolation conditions, d - c - 1 algebraic equations can be established as ]Nel(G) = 1~1,
(26)
where
can be determined freely. The outer solution could be used for interpolation if it had a sufficiently high accuracy, otherwise expanding the domain of the spline collocation point and increasing its points could be adopted. A higher accuracy in calculation could be reached if the supplemental equations were chosen correctly.
[Ne] =
{%} = [
:z: EiUi(Xc+I) :i
EiUi(Xc+2)
. . .
s-l
. . . 1 E$(x~_,) i-0
1 ‘.
(28)
As the number of equations included in eqn (26) is n + 2q + 1, n + 2q + 1 coefficients need to be determined. There are two factors which usually result in the number of equations being less than that of the unknown coefficients, i.e. (1) some of the expressions obtained from boundary and inter-
Incorporating systems (17), (20), (23) and (26) and the corresponding supplemental equations [such as eqns (29) and (30)] into one expression, we have
[Gl{Cl = V’J;
(31)
this should be a system of linear equations including n + 2q + 1 + m algebraic equations, where m is the number of unknown integral constants and [G] is a so-called spline matrix. In general, eqn (31) is nonhomogeneous, [G] is a nonsingular matrix, and so
666
D. Gu
the column matrix (C‘f constituted by unknown constants can be determined uniquely. THE UNSYMMETRICAL A PRESTRESSED
BENDING PROBLEM ANNULAR PLATE
OF
As an exampIe, we consider solving the problem of unsy~et~cal bending of prestressed annular plates by the linear combination of an unequidistant knot of a quintic 6-spline function. Assuming that the inner and outer radii of an annular plate are r,, and r,, respectively, the outer boundary is clamped, and the inner boundary is fixed on a rigid body which rotates about an arbitrary diameter by a small angle. The annular plate is subjected to uniformly radial and tangential pull forces in the plate midsurface, and we may write N, = N, = N (const~t). This problem was investigated first by Algheimer and Davis [2] in 1968. Using the method of matched asymptotic expansions they obtained the first approximation. The boundary layer solution was also constituted by Kiang [3] in 1980 with the method of two-variable expansion. Now we consider obtaining the boundary layer solution with a spline function to obtain numerical solutions. Assume that the nondimensional equation of the plate is &2V4W - v2w = 0,
(32)
where
D = E/?/12(1 -v*), h is the plate thickness, E and v are the plate Young’s moduli and Poisson’s ratio respectively. The boundary conditions of the plate are w =bucos@
r = b,
aw =ucose,
(33)
a;
r = 1,
dnv
W=:-=O c%
where b = r,,/r,, and r and 0 are the polar coordinates of any point lying in the plate midsurface. We assume w = u(r)cos 6.
Substituting
(35)
eqn (35) into eqn (32) yields
>( -
d2u 1 du p+;Ti;--f
u >
Spline boundary layer method for bending of thin plates
According to eqns (33) and (34), it follows that u(b) = ba,
u(l)=;
du(r)
-
dr
=a
(37)
r-b
_ = 0. r-l
(38)
Taking two terms, the outer solution of the equation is
A0
u=y+BOr+s
Al
(
-+B,r r
> ,
(39)
where A,, B,,, A, and B,, are four integral constants. Since there are two boundary layers, the inner and the outer, the integral constants can be determined merely by matching conditions. As we use expression (39) for interpolation, the integral constants to be determined by algebraic method will dwindle to two, i.e. A,,+eA, and B,,+&B,. Letting E = 0.05, b = 0.4, a = 0.1; the extension of the discrete points in ascending order as a is: r_5 = 0, r-,=0.1, rc3 = 0.2, r-r= 0.3, r_, = 0.35, r. =0.4, r, = 0.45, rr= 0.5, r,= 0.55, r,= 0.6, rs =0.7, r, = 0.8, r, = 0.85, r8 = 0.9, r, = 0.95, r,. = 1.0, rl, = 1.05, r,r = 1.1, r,3 = 1.2, r,4 = 1.3, rr5 = 1.4. For the convenience of calculation, we may take the spline collocation point on a knot, and choose three points r,, r, and rs for spline interpolation. Then the spline matrix [G] can be constituted as follows:
667
D. Gu
668
{F} and {C} in eqn (27) are {F)=[00000000000000.040.100]’ {C] = E-2 c- I c 0c I c 2 c 3 c 4 c 5 c 6 c 7 c 8 c 9 c
10
c
,I
c,*c,3c,,]r,
result is respectively. The c_2 = -0.0545, c_, = 0.03194, co= 04315, C, = 0.04058, c, = 0.03507, C, = 0.02819, c, = 0.02144, c5 = 0.01575, c, = 0.01077, C, = 0.0063449, C, = 0.003033, C, = 0.0007592, Cl0 = - 0.0005866, c,, = 0.000517, Cl2 = 0.01237, C,, = 0.02438, Cl4 = -0.02695, where C,, and Cl4 are two integral constants in the outer solution, C,j = A, + &A,, C,4 = B, + EB,. Using the method of matched asymptotic expansions, we may obtain A, + &A, = 0.02508, B. + EB, = -0.02699 by the Van Dyke matching principle [4], they are close to C,3 and C,, as above. The values of function u can be calculated by the following expression:
A composite expansion obtained by Alzheimer and Davis is
--
2&U &bZexp
--
[
2&b2u
+1-bZexp
-[
r-b 8
l-r e
1 1
+ 0(E2).
(41)
Substituting b = 0.4, CI= 0.1, E = 0.05 into eqn (41) yields a result which can be compared with the numerical result obtained by the presented method as shown in Table 1. CONCLUSION
As the error of the composite solution is 0(e2) = 0(0.0025), the accuracy of the solution is very different in different parts of the region. In the neighbourhood of the outer boundary, the values of function u are very small and upper bound of the corresponding relative error becomes very large. In fact, when we add the constraint of connecting the outer and inner solution ‘smoothly’ to constitute the composite expression, the main purpose is only to obtain a solution which has a good form, and usually
its accuracy is not good; in general, lower by one order than that of the outer solution. However, the spline solution is given by the spline collocation point with the spline knots possessing proper density in a narrow boundary layer region and by interpolation fitting the outer solution, so its accuracy shouldn’t be lower than that of the composite solution. The results of the above calculation show that the rough estimation is correct. When we use the mixed method to solve problems of thin plate small deflection unsymmetrical bending, of large deflection bending and of thin shell bending and stability in which the boundary effect exists, it is discovered that using the mixed method presented above to consider the problems of the boundary layer is more efficient than using the any other weightedresidual method. It is predictable that the accuracy of the spline solution will be much higher than that of the composite solution, if we do well. The use of the outer solution, and also the introduction of the spline function as the boundary layer solution, permits one to take a simpler method to treat the singular perturbation problems. Our next goal consists in the further application of the spline boundary layer method to establish with more accuracy the spline solution in bending problems of various composite material laminated plates and shells with reinforced fibre. The mathematical elements of the mixed method are based on the theories of spline function and singular perturbation, which have been developed considerably in the last two decades and can be seen in many references, e.g. [5-71. The topics in mathematics including the existence, uniqueness, and stability of the spline solution and its analysis of error will be discussed elsewhere.
REFERENCES
1. Zhuo-qiu Li, Influence of effect to buckling and vibration of unsymmetric angle-ply rectangular plates. J. Wuhan Univ. Technol. 4, 419-424 (1987). 2. W. E. Algheimer and R. T. Davis, Unsymmetrical bending of prestressed annular plates. Proc. J. Engng Mech. Div., ASCE 4, 905-917 (1968). 3. Fu-ru Kiang, Some applications of perturbation method in thin plate bending problems. Appl. Math. Mech. 1, 37-53 (1980). 4. A. H. Nayfeh, Perturbation Methods. John Wiley, New York (1973). 5. T. N. E. Greville (Ed.), Theory and Applications of Spline Functions. Academic Press, New York (1969). 6. M. H. Schultz, Spline Analysis. Prentice-Hall, Englewood Cliffs, NJ (1973). 7. R. E. CSMalley, Introduction to Singular Perturbations. Academic Press, New York (1974).
d Srru~r~res Vol. Britain.
34. No. 4. pp. 663668.
1990
004s7949190 $3.00 + 0.00 Pergamon Press plc
Printedin Great
CALCULATION IN THE UNSYMMETRICAL BENDING PROBLEM OF THIN PLATES BY SPLINE BOUNDARY LAYER METHOD DE-GAN Gu Department of Engineering Mechanics, Wuhan University of Technology, Wuhan, The People’s Republic of China (Received 21 February 1989) Abstract-The weighted-residual and perturbation methods are usually used to solve the unsymmetrical bending problem of thin plates. Here a mixed method of the weighted-residual and singular perturbation is considered. It is known that there are many ways from which a combination of the two methods can be. chosen to obtain numerical solutions which are uniformly valid throughout the region, but they are very different in theory. In this article the main purpose is to give a description of a mixed method for calculation in the unsymmetrical bending problem of plates, so only the singular perturbation problem for the differential equation involving a small parameter in higher derivatives will be discussed. Called the spline boundary layer method for short, boundary layer solutions are constituted with the spline function in order to overcome the difficulties of obtaining the boundary layer solution. An example is
given. Not only is the method simple and convenient for computations with a computer, but also the expression for the outer solution is offered simultaneously, and if necessary the expression of the boundary layer solution can also be given without using any complicated matching principle. Therefore the applications of the singular perturbation method in engineering calculations can be expanded.
THE BASIC METHOD
To illustrate the main characteristic of the spline boundary layer method, we consider the two point boundary-value problem on the region [a, 61: &[a]
+ 4_,[U] =f(x)
(0
&[u] = rpi (i = 0, 1,. . . , k - I),
(1) (2)
where Lk and Lk_, are order k and k - r linear differential operators, respectively, and BE is the boundary differential operator. The problems of degeneration and perturbation of all orders can be given by (3)
L-,[%I =.m) B,,,[u,,]=cp,
(i=O,l...,
k-l)
(4)
(n b 1)
(3
and
Lk-rb4 = -L&-II 48,i[%l= O-
(6)
The order of eqns (3) and (5) of both degeneration and perturbation are lower by r than that of goveming eqn (l), and the r boundary conditions must have been lost at least. If we assume that the boundary layer exists in the neighbourhood of x = a and x = b, all the boundary conditions will be discarded when we constitute the outer solution by means of degener663
ation and perturbation solutions. Certainly, we may retain partial boundary conditions when the boundary layer only exists in the neighbourhood of x = a (or x = b), and the problem will be simpler. Now we consider the more general problem in which two boundary layers, the inner and the outer, are to be constituted. Assuming that the outer solution is s-l 24= c &i&(X)+ O(E’),
(7)
i=O
where uj(x) contains k - r arbitrary constants, the number of integral constants is s(k - r). Only in the region distant from boundaries x = a and x = b does the expression (7) satisfy eqn (1) approximately. The method of spline collocation point can be taken in the neighbourhood of x = a and x = b to obtain boundary layer solutions, but there is no need for the outer region in the method of spline collocation point as the outer solution has already been obtained. Accuracy of the outer solution can be obtained if we choose a small number of points in the outer region for spline interpolation. It is unsuitable to constitute a uniform solution in the whole region with the combination of an equidistant knot 6-spline function as is done in many papers [l]. Since the requirement of spline knots is different in different parts of the interval [a, b], we may choose the expression of unequidistant knots to produce a uniform solution. For this purpose, we let the discrete points be, in ascending order as A, a=x,
664
D. Gu According to eqns (lo)-( 13) the successive derivatives of Nm,j (x) can easily be determined by the recurrence relation. The solution of eqn (1) can be expressed approximately by
the extension of A is A,. x-,-C x-nl+I <...
We introduce extra knots outside the interval [a, b], so that the uniform solution throughout the region can be expressed as a linear combination of spline functions. Assuming that the 6-spline degree m (usually m is an odd number larger than three) is {N,.j(x), j = -m,
-m
+
u=
WI (‘3,
where
[Nl=[N,,-,,-,(x)N,,-,(x).
1,. . . , n - l},
(14)
. Nm,n-,(x)1
{Cl = [C-,c-q+, c-q+* . . . C”,,l’.
where
G,(t -x)
G,,(x) = x:,
= (t -x)“,
There are II + 1 spline knots. Substituting eqn (14) into eqn (1) and boundary condition (2) yields
.
Hence, L[N[
j+m+l Gm(xj-x,.
. .,Xj+m+l -x)=
1 wiGm(xi-x), i=,
+
Lr[Nll{C) =f(x,,)
(15)
(8)
[B~,i[N]]{C)=~i (i=0,1,2...,k-l),
(16)
where where q=
n
-.
/=j
xi-x,
(9)
/+i
The successive derivatives of N,,,j (x) should be calculated, denoted by Nb,j, NG,, . . . , N$,j (1 < m), and can be written as 1 Nk, = m PNtt-1.j xj ( Xjtm-
1 Nm-2.j
( Xj+m-l-Xj
1 Nm-z,j+l
xj+ni -xi+1 Nh-l,j+I =(m-1) -
’ ( xjtm-xjtl 1
(11)
>
Nm-*,j+i
(12)
Nm-2.jt2 ; xnt*tl-xjtZ
>
consequently, Nk,j = m ( Xj+m -
..
>
- 1) -
= L,Wm, -Q-,(xNL-~[N~,-&)I. . . . LMn.n-,(xM,=xp
(10)
Nm-i,j+i
Xj+m+l-Xj+I
N:_i,j=(m
LVV
.
[Be,i[Wl= [Be,i[Nm, -~q-~(x)Pct[Nrn. -~q(x)I.. .
1
-
Lb’1 = M% -~-~(x)lL[Nm,-&)I. . . UN~,.-I(XL~~
1 Nit- 1.j
where xP is some determined point. It is noted that letting all x satisfy the equation is impossible, and in eqn (16) xg is a boundary point (B = 0 or n). Assuming that the width of the inner boundary layer in the neighbourhood of x = a is x, - x0, that of the outer boundary layer in the neighbourhood of x = b is x, - x,; the spline knot in the inner boundary layer is x = x,(0 < c( < c), that in the outer boundary layer is x = x,(d < /I < n), and the spline knot in the outer region is x = x, (c < e c d). A system of linear equations can be formed from eqns (7) and (14)-( 16), and the number of algebraic ecluations is 2q + 1 + n. We may write
-xj
{C> = if,1
1 *
N,Li,j+r xj+mtl-xjtl
[NJ =
(13)
(17)
(18 . * . . . . . . Nm,-,-I(x,) Nm,-zq-,(x ..Nm,.-I(x I. >
where
Nn,-2q-I(XO)
Ntn.
X,-,-,(x,)
Nw-2,(x,)
-2g(xo)
. . .
K,.-,(xo)
. . .
Kn,,-,(x,)
Spline boundary layer method for bending of thin plates
Hence, c + 1 algebraic equations are involved in eqn (17). It is noted that L,[N,] and L,_,[N,] denote ‘substituting the x value after derivation’, and the following expressions have the same meaning. ’ AxX
{f,} = W%)f(x,).
(19)
(20)
wk[~~ll+ Lr[N,HCI = u31*
665
polation conditions are identical; and (2) the integral constants in the outer solution cannot be determined by the boundary conditions. In this case it is necessary to establish supplemental equations from the matching principle. For example, if there are two integral constants which cannot be determined by the boundary conditions, we may establish four equations by the methods of both spline collocation point and interpolation at x = x, and x = x,, respect-
(21 . . . * . . . Nm, -+,(A) Nm, -&J Nnv-I W 1
which contains n - d + 1 algebraic equations, where
Nm,-,+ I (xc4 Nn,-2q-,h+,
IN,1= :
K, -x&d
)
Nm,n-~(d
Nm,-2&+,)
{.&@I = Lmflf(Xdfl)~ . ~f(X,Yl
(22)
Nm-I&,+,)
ively, so that the additional
equations are:
s-l
P,.,[N,.ll{C}= {cp*;>(i = 0, 1,. . . , k - 1). (23)
P’cI{CI = c ~‘~i(~)I.v=.~< i=o r-l [Nc,{C)= 1 &bL.w
Equation (23) is obtained from boundary conditions; it should contain k algebraic equations, where N,,-,-,(x0) IN’]= [ N,,,,+i(x,)
1
N,,-&,) N,,,,_,(x,)
. . . N,,n-,(x,) . . . N,,,-I (x,) ;
(24)
(2% (30)
,=O
where
WI=
[N,,-,,-,(x)N,,-,,(x)...
Nm,.-,(xk=.~~
we note that Bc.i [N,] denotes ‘substituting x = x0 and x = x, after derivation’. Since there is no mathematical criterion to determine
bP,,l= b&&&,)1~.
(25) the width of the boundary layer, x, and x, are all that
According to the interpolation conditions, d - c - 1 algebraic equations can be established as ]Nel(G) = 1~1,
(26)
where
can be determined freely. The outer solution could be used for interpolation if it had a sufficiently high accuracy, otherwise expanding the domain of the spline collocation point and increasing its points could be adopted. A higher accuracy in calculation could be reached if the supplemental equations were chosen correctly.
[Ne] =
{%} = [
:z: EiUi(Xc+I) :i
EiUi(Xc+2)
. . .
s-l
. . . 1 E$(x~_,) i-0
1 ‘.
(28)
As the number of equations included in eqn (26) is n + 2q + 1, n + 2q + 1 coefficients need to be determined. There are two factors which usually result in the number of equations being less than that of the unknown coefficients, i.e. (1) some of the expressions obtained from boundary and inter-
Incorporating systems (17), (20), (23) and (26) and the corresponding supplemental equations [such as eqns (29) and (30)] into one expression, we have
[Gl{Cl = V’J;
(31)
this should be a system of linear equations including n + 2q + 1 + m algebraic equations, where m is the number of unknown integral constants and [G] is a so-called spline matrix. In general, eqn (31) is nonhomogeneous, [G] is a nonsingular matrix, and so
666
D. Gu
the column matrix (C‘f constituted by unknown constants can be determined uniquely. THE UNSYMMETRICAL A PRESTRESSED
BENDING PROBLEM ANNULAR PLATE
OF
As an exampIe, we consider solving the problem of unsy~et~cal bending of prestressed annular plates by the linear combination of an unequidistant knot of a quintic 6-spline function. Assuming that the inner and outer radii of an annular plate are r,, and r,, respectively, the outer boundary is clamped, and the inner boundary is fixed on a rigid body which rotates about an arbitrary diameter by a small angle. The annular plate is subjected to uniformly radial and tangential pull forces in the plate midsurface, and we may write N, = N, = N (const~t). This problem was investigated first by Algheimer and Davis [2] in 1968. Using the method of matched asymptotic expansions they obtained the first approximation. The boundary layer solution was also constituted by Kiang [3] in 1980 with the method of two-variable expansion. Now we consider obtaining the boundary layer solution with a spline function to obtain numerical solutions. Assume that the nondimensional equation of the plate is &2V4W - v2w = 0,
(32)
where
D = E/?/12(1 -v*), h is the plate thickness, E and v are the plate Young’s moduli and Poisson’s ratio respectively. The boundary conditions of the plate are w =bucos@
r = b,
aw =ucose,
(33)
a;
r = 1,
dnv
W=:-=O c%
where b = r,,/r,, and r and 0 are the polar coordinates of any point lying in the plate midsurface. We assume w = u(r)cos 6.
Substituting
(35)
eqn (35) into eqn (32) yields
>( -
d2u 1 du p+;Ti;--f
u >
Spline boundary layer method for bending of thin plates
According to eqns (33) and (34), it follows that u(b) = ba,
u(l)=;
du(r)
-
dr
=a
(37)
r-b
_ = 0. r-l
(38)
Taking two terms, the outer solution of the equation is
A0
u=y+BOr+s
Al
(
-+B,r r
> ,
(39)
where A,, B,,, A, and B,, are four integral constants. Since there are two boundary layers, the inner and the outer, the integral constants can be determined merely by matching conditions. As we use expression (39) for interpolation, the integral constants to be determined by algebraic method will dwindle to two, i.e. A,,+eA, and B,,+&B,. Letting E = 0.05, b = 0.4, a = 0.1; the extension of the discrete points in ascending order as a is: r_5 = 0, r-,=0.1, rc3 = 0.2, r-r= 0.3, r_, = 0.35, r. =0.4, r, = 0.45, rr= 0.5, r,= 0.55, r,= 0.6, rs =0.7, r, = 0.8, r, = 0.85, r8 = 0.9, r, = 0.95, r,. = 1.0, rl, = 1.05, r,r = 1.1, r,3 = 1.2, r,4 = 1.3, rr5 = 1.4. For the convenience of calculation, we may take the spline collocation point on a knot, and choose three points r,, r, and rs for spline interpolation. Then the spline matrix [G] can be constituted as follows:
667
D. Gu
668
{F} and {C} in eqn (27) are {F)=[00000000000000.040.100]’ {C] = E-2 c- I c 0c I c 2 c 3 c 4 c 5 c 6 c 7 c 8 c 9 c
10
c
,I
c,*c,3c,,]r,
result is respectively. The c_2 = -0.0545, c_, = 0.03194, co= 04315, C, = 0.04058, c, = 0.03507, C, = 0.02819, c, = 0.02144, c5 = 0.01575, c, = 0.01077, C, = 0.0063449, C, = 0.003033, C, = 0.0007592, Cl0 = - 0.0005866, c,, = 0.000517, Cl2 = 0.01237, C,, = 0.02438, Cl4 = -0.02695, where C,, and Cl4 are two integral constants in the outer solution, C,j = A, + &A,, C,4 = B, + EB,. Using the method of matched asymptotic expansions, we may obtain A, + &A, = 0.02508, B. + EB, = -0.02699 by the Van Dyke matching principle [4], they are close to C,3 and C,, as above. The values of function u can be calculated by the following expression:
A composite expansion obtained by Alzheimer and Davis is
--
2&U &bZexp
--
[
2&b2u
+1-bZexp
-[
r-b 8
l-r e
1 1
+ 0(E2).
(41)
Substituting b = 0.4, CI= 0.1, E = 0.05 into eqn (41) yields a result which can be compared with the numerical result obtained by the presented method as shown in Table 1. CONCLUSION
As the error of the composite solution is 0(e2) = 0(0.0025), the accuracy of the solution is very different in different parts of the region. In the neighbourhood of the outer boundary, the values of function u are very small and upper bound of the corresponding relative error becomes very large. In fact, when we add the constraint of connecting the outer and inner solution ‘smoothly’ to constitute the composite expression, the main purpose is only to obtain a solution which has a good form, and usually
its accuracy is not good; in general, lower by one order than that of the outer solution. However, the spline solution is given by the spline collocation point with the spline knots possessing proper density in a narrow boundary layer region and by interpolation fitting the outer solution, so its accuracy shouldn’t be lower than that of the composite solution. The results of the above calculation show that the rough estimation is correct. When we use the mixed method to solve problems of thin plate small deflection unsymmetrical bending, of large deflection bending and of thin shell bending and stability in which the boundary effect exists, it is discovered that using the mixed method presented above to consider the problems of the boundary layer is more efficient than using the any other weightedresidual method. It is predictable that the accuracy of the spline solution will be much higher than that of the composite solution, if we do well. The use of the outer solution, and also the introduction of the spline function as the boundary layer solution, permits one to take a simpler method to treat the singular perturbation problems. Our next goal consists in the further application of the spline boundary layer method to establish with more accuracy the spline solution in bending problems of various composite material laminated plates and shells with reinforced fibre. The mathematical elements of the mixed method are based on the theories of spline function and singular perturbation, which have been developed considerably in the last two decades and can be seen in many references, e.g. [5-71. The topics in mathematics including the existence, uniqueness, and stability of the spline solution and its analysis of error will be discussed elsewhere.
REFERENCES
1. Zhuo-qiu Li, Influence of effect to buckling and vibration of unsymmetric angle-ply rectangular plates. J. Wuhan Univ. Technol. 4, 419-424 (1987). 2. W. E. Algheimer and R. T. Davis, Unsymmetrical bending of prestressed annular plates. Proc. J. Engng Mech. Div., ASCE 4, 905-917 (1968). 3. Fu-ru Kiang, Some applications of perturbation method in thin plate bending problems. Appl. Math. Mech. 1, 37-53 (1980). 4. A. H. Nayfeh, Perturbation Methods. John Wiley, New York (1973). 5. T. N. E. Greville (Ed.), Theory and Applications of Spline Functions. Academic Press, New York (1969). 6. M. H. Schultz, Spline Analysis. Prentice-Hall, Englewood Cliffs, NJ (1973). 7. R. E. CSMalley, Introduction to Singular Perturbations. Academic Press, New York (1974).