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Eigenfunction analysis for bending of clamped rectangular, orthotropic plates
ht.
J. Solids
Structures,
1973. Vol. 9, pp. 481 to 493. Pergamon
Press.
Printed
m Great
Britain
EIGENFUNCTION ANALYSIS FOR BENDING OF CLAMPED RECTANGULAR, ORTHOTROPIC PLATES B. S. RAMACHANDRA RAO and G. L. NARASIMHAM Indian Institute of Technology, Bombay, India
and S. GOPALACHARYULU Loughborough University of Technology, England
Abstract-Eigenfunctions have been developed for the clamped, rectangular orthotropic plate undergoing small deflections. The familiar Fadle-Papkovitch [l] functions for the isotropic case are shown to form a special case of the “orthotropic eigenfunctions”. An analysis for orthotropic plates using these functions yields extremely accurate solutions.
1. INTRODUCTION ONE of the useful methods for problems of bending of clamped, rectangular isotropic plates is the method ofeigenfunctions. The literature on this is extensive; see [l-9]. Tentative solutions of the biharmonic equation are postulated and a transcendental eigenvalue equation is obtained by satisfying homogeneous boundary conditions at the clamped edges. Solution of the eigenvalue equation gives rise to expansions in terms of eigenfunctions, a linear combination of which is generally sufficient to satisfy the boundary conditions on the remaining edges. The purpose of this work is to develop eigenfunctions for the clamped, rectangular, orthotropic plate undergoing “small” deflections, and to present an analysis of the problem using these functions. These functions have not been developed before; the FadlePapkovitch [l, 21 functions, well-known for the isotropic case, form a special case of the functions developed here. The clamped orthotropic plate problem can, of course, be treated by other more familiar techniques discussed in Timoshenko [lo] and Lechnitzky [ll]. The present analysis with the eigenfunctions yields an extremely accurate solution and is well suited for machine computation.
1.1. Statement of the problem We consider a clamped, rectangular, orthotropic plate of size a x b and thickness, h. The material has three planes of symmetry with respect to its elastic properties. Following Timoshenko [lo] we choose these planes as the co-ordinate planes (Fig. 1) and assume the 481
B. S.
482
RAMACHANDRA RAO, G. L. NARASIMHAMand S. GOPALACHARYULU Y
I
i
b
[--“““““~“‘~~
I_,.,,,,r ,,,,,,,,,,,, i x
O-f
I-
FIG.
1. The clamped, rectangular, orthotropic plate.
stress-strain relations for the case of plane stress in x-y plane as : ox = E&v + E”E, gy = E&,+ E”E, TXY =
%Y.
The governing differential equation for the deflection, w, of the plate under transverse load Q, is
D,$+ZH &+Dy$ =Q(x, Y)
(2)
where D, = E;h3/12;
D, = E;h3/12;
D, = E”h3/12;
D,, = Gh3/12
H = D,+2D,,
The bending moments are given by
(3)
The boundary conditions are w=“=() ay
ony = &b/2
w=$J=Clonx=
&a/2.
(4.1) (4.2)
2. EIGENFUNCTIONS Consider the homogeneous
equation
Dx$+2H
a4w w
+D”‘“=O. y ay4
(5)
Eigenfunction anafysis for bending of clamped rectangular, orthotropjc plates
483
Assume tentatively a solution in the form
and let i?W - = 0 at y = *b/2. ay
w”=
F is then determined by
(6) under the conditions F = dF/dy = 0 at y = t_ b/2. The constants C, and C, in equation (6) are given by C, = H/D, = (Dr + 20,,,),‘4 = (E” + 2G~/E~ C, = &WY
= &%‘E;)
-
On the basis F = ewy,it is seen that w = $$ kpfiq) where P = JKCZ - C,)izl
The even eigenfunction can now be written as (9.1) where p is given by the transcendental
equation
sint2w)+ sin&W = 4
P
o
(10.1) .
The odd eigenfunction is
(9.2) and the transcendental
equation is (10.2)
The eigenfunctions for the isotropic case may be obtained as a special case from the above expressions by putting El = ES, = E/(1 -/+, E” = &‘(i --p2) and G = E/2(1 +&, so that C1 = C, = 1, from which if follows p = 0, q = 1. In the limit of p -+ 0 and q = 1, the expressions (9.1), (9.2), (10.1) and (10.2) reduce to the familiar Fadle-Papkovitch functions [ 1,2].
B. S. RAMACHANDRA
484
RAO, G. L. NARASIMHAMand S. GOPALACHARYULU
Indeed, the condition p = 0, i.e. C, = C, is adequate to reduce the problem to an isotropic case. For, with the new variable Y, = yJC,, equations (5) and (6) assume the familiar forms of the isotropic problem (page 367, Timoshenko [lo]). In the above analysis it is implied that C, > C,, which makes p a real quantity. At least for plywood materials, assuming the elastic constants given in Timoshenko [lo], we have found p to be a real quantity. Even if p is imaginary, as it may happen for other materials, the above relations are valid. The transcendental equations (10.1) and (10.2) have infinite number of complex roots expressible as ( +cr,fiBx), K = 1,2,. . . . The asymptotic form for the Kth root of equation (10.1) (symmetric case) is found to be PK = 2(pz l+ 42) {[P hT(Ph) For the antisymmetric
+ (4K -
case, equation
1)v/4+ i[(4K - lbP/2 - 4 wPldl
1
(11.1)
(10.2), it is given by
PK = 2(pzl+ q2) {[P lW(P/Y) + (4K +
1)dA + i[(4K
+ lbP/:! -4 log(Plq~l)
(11.2)
Starting with the above expressions, the roots can be easily determined by Newton’s iteration method in complex form, which is quite rapidly convergent. For the isotropic case, such a technique has been given by Buchwald [3]. In a similar manner, we develop another set of eigenfunctions G,(x) which satisfy the homogeneous conditions G, = dG,/dx = 0 at x = &a/2. Let
On substituting
this in equation
(5) we obtain
where C; = H/D, = (Dl +2D,,)/D,
= (E”+2G)/E;.
C; = J(D,/D,) = J(E;IE:). On the basis G = em’“, it is found that w’ = Y(*p’*iq’) where
Jw; q’ = Jw
P’ =
- c;)Pl + c;)Pl.
Now if
c, > ~,,J(E~/E;) > (E~+~G)/E;
Eigenfunction
analysis
for bending
of clamped
rectangular,
orthotropic
plates
485
i.e.
J(E:E;)
> r+2G,
it follows C; > C;. Consequently the transcendental equations for p’ are the same as before for p with p’, q’ replacing p, q respectively.? The function set G,(x) can be obtained by inserting p’, q’, p’, a and x in the places of p, q, p, b and y respectively, in the expressions (9.1) and (9.2). The even eigenfunction in the set GK(x) is G =
cos(2q’p’x/a)
cosh(2p’p’x/a)
cos(q’p’)
cosh(p’p’)
sin(2q’p’xla)
sinh(2p’p’xla)
sin(q’p’)
sinh(p’p’)
-
1
(12.1)
where p’ is given by sin(2q’p’) + sinh(2p’p’) = o,
The odd eigenfunction G =
(13.1)
PI
4’ is
sin(2q’p’xla)
cosh(2p’p’xla)
sin(q’p’)
cosh(p’p’)
and the transcendental
equation
-
cos(2q’p’xla) sinh(2p’p’xla) sinh(p’p’) cos(q’p’)
1
(12.2)
is sin(2q’p’) 4’
sinh(2p’p’) = o
P’
-
(13.2)
.
The infinite number of complex roots of the equations (13.1) and (13.2) are also expressible as + c& f i&, K = 1,2, . . . The asymptotic forms for the kth root of p’ is obtained by putting p’, q’ in the places of p and q in equations (11.1) and (11.2).
3. SOLUTION Having orthotropic of Gaydon
FOR ANY GIVEN
LOADING
the two sets of eigenfunctions F,(y) and GK(x), a solution for the clamped plate for any given loading Q(x, y) can be developed on lines similar to that [6]. Assume that a particular integral wO of equation (2) can be found and write w =
where O(x,y) is a polynomial reasons discussed later. Then
solution
w,+n+w, of the homogeneous
equation
Dx$+*H+$+D,$ =0. ax ay t The authors
are indebted
to Professor
F. A. Gaydon
for this proof.
(5) introduced
for
(14)
486
B. S. RAMACHANDRA RAO, G. L. NARASIMHAM and
S. GOPALACHARYULU
The conditions (4.1) and (4.2) give at x = +u/2
W 1= -(w()+R) ' aw 1- -f$w,+R) ax aty=
+b/2
>.
(15)
Wl = -(w,+Q) aw, -= ay
-$(wo+Q)
/
Now consider another function w,(x, y) which satisfies
saw, aaW, i)hw, D~~+2HaX2ayZ+Dydy4 = 0
(16)
and the following conditions, at x = +a/2
a%, = -ggw,+i2) ay2 a%, ---z axay -&(WO+n) and at y = + b/2
(17)
azw2 -$(wo+Q) g= a2W2 ---z -&(Wo+n, alcay It is not difficult to see that the relation between wi and w2 is w1 = w,+k,x+k,y+k,
(18)
where k,, k, and k, are constants to ,be determined in any particular problem. If w1 and w2 are even in x, then kl = 0. If they are even in y, k2 = 0 and if odd in either coordinate k, = 0. They are determined from known conditions, say, at corners. The function w2 can now be developed into a series of eigenfunctions F,(y), GK(x). Assuming that the loading is symmetric about both the axes, we write w2 = z A, cosh(2p,x/b)F,(y) m
+ F B, cosh(2p;y/a)G,(x) m
(19)
where A,, B, are complex constants, pK = aK+ ibK, pk E cc; + iPt and A-,, B-, , p-K etc., are the respective complex conjugates of A,, B,, pK etc. Note that the sets of roots - c&+ i/t& do not yield any new solutions. Applying the conditions (17) it will -u,+&, be seen that the problem is reduced to the determination of the complex constants A,, B,
Eigenfunction analysis for bending of clamped rectangular, orthotropic plates
487
satisfying the following equations : (20.1)
f 1
(20.2)
B,$ =$wO+R) =gl(x)
y=+b/2
(20.3)
m
B&Z = - &(“o+n) (20.4) = h(X). Iu The purpose of the R will now be explained. The eigenfunctions F and G satisfy the following conditions : a/2 b/2 b’2 dF a’2 dG F.ydy = -dy = G.x.dx = (21) I -b/2 s -b/2 dy i -a/2 s _a,2 ZdX = O ;lf
and the function set is complete with respect to these self-equilibrating conditions, though this is not proved here. Consequently for the expansions in equations (20.1) to (20.4) to be valid, we should impose that b/2 s -b/2 Furthermore
fiv dy = j-yL2h dy = j-;12gix
dx = J::,gl
dx = 0.
(22.1)
remembering the properties of the eigenfunctions viz., dF, ---=Oaty= dy
ib/2,d$=Oatx=
*a/2,
and in view of expansions (20.2) and (20.4), we require that f21,=
fb/2
=
g21x=*.,2
=
0.
(22.2)
Fortunately all these requirements on the functionsf, ,f2, g, and g, can be met by a proper choice of Q generated in polynomials forming solutions of (5).
4. RE-ORTHOGONALISATION
OF EIGENFUNCTIONS
Unfortunately the functions F,(y), GK(x) do not seem to possess suitable orthogonality relations and indirect methods are to be employed for determining the constants in the expansions (20.1) to (20.4). Although the usual methods such as least-squares etc., can be used, they do not possess special merits in the present problem. The techniques having certain advantages are (a) the method of biorthogonal functions developed by Johnson and Little [7], and Little [8], (b) the method of reorthogonalization developed by Gaydon and Shepherd [5], and Gaydon [&I. The latter method is adopted here. This consists of expanding each of the eigenfunctions into the orthogonal set of normalized clamped beam functions Y, defined by : ~-(21,,b)4Ym
= 0
B. S. RAMACHANDRA RAO. G. L. NARASIMHAM and S. GOPALACHARYULU
488
under
the conditions Y,=S=Oaty= dy
In the symmetric
case, the functions
*/J/2.
are
1 where 1, is given by tan 1, + tanh 1, = 0. The 2,s are well-known FK = by virtue of the orthogonality
il
[5]. Writing (23)
%nYm
m=l
relation b/2
s we
_b,,
_
LK
=
4nn
find that b/2 URm =
i -b/2
F, . Y, dJ
The expression for aKmis given in Appendix. Also, it follows that F_,
= Conjugate
of F, =
f m=l
where CI-~~ = Conjugate Furthermore, because
of a,,. of the following
sv b/2
d2
s -b/2 the functionsf,,
fi in equations
2
d!,
relations
[5]
y-‘&y
-L-!! dy = (212,/b)” 6,, dy’
-b,2
b’2 dY
orthogonality
U-K,Y,
d3Y
dy = -(2&,/b)”
2
6,”
dy3
(20.1) and (20.2) can be expanded
.fl(Y) = i:
1
c-3
in the following
manner (23.1)
rL*
f,(y) = c 4,x% 1
where,
(23.2)
Eigenfunction analysis for bending of clamped rectangular, orthotropic plates
489
If we substitute the above expressions in equations (20.1) and (20.2), we obtain
K=-a,
(m = 1,2,. fbPKaKm
=
(24)
.)
dm
m
which are the infinite system of equations required for the determination of A,. Similarly by expanding the set G,(x) into the orthogonal set X,(x) defined by c3
- (21*,*)4X,
x,
= 0,
+a/2
“=Oatx= = “d”,
we obtain the following system of equations for B, ;
B,b,,
=
1
c:,
K=-a-
(m =
a!
h’;bKm
1,2,. . .)
.
(25)
= 4, i
The expression for bK, is obtained by putting a, p’, p’, q’ in the places of b, p, p, q respectively in the expression for aKm(Appendix). The expressions for CL and d& are: d2X
“dx g1 dx2 d3Xndx
g2=
5. NUMERICAL
’
EXAMPLE
As a numerical example, we consider the problem of a clamped plate under uniform normal pressure. A particular solution is w
0
=
L &(b2,&y2)2. 24 Dy
Since this automatically satisfies the conditions at y = k b/2, the function set GK(x) is not required. It is also seen that the function R is not required, as the functionsf, = - ~2wo/~y2 ; f2 = - 82wo/Jx8y satisfy the requirements (22.1) and (22.2). Equations (20.1) and (20.2) give 1 K
cosh(PKa/b) AKaKm
cash pK
= C,
sinh(PKa/b) = c AKaKmPK cash PK K
where
o
490
B. S. RAMACHANDRA
RAO. G. L. NARASIMHAMand S. GOPALACHARYULU
These are the required infinite system of equations, which may be solved by the usual truncation procedure, using a computer. Since complex algebra can be handled on modern high speed computers, the separation of real and imaginary parts can be carried out on the computer itself. For numerical work, we consider the plywood materials. The elastic constants are listed in Timoshenko [lo] and Lechnitzky [ll]. We reproduce these values in Table 1, and also present the values of p and 4, defined by equation (8). Tables 2 to 6 give the eigenvalues, for symmetric case, for these materials. For a square plate (a = b), the numerical work has been carried out, by retaining only the first ten A, and taking m up to 10. This means, each of the eigenfunctions is expanded into the first ten clamped beam functions resulting in a square matrix of size 20 x 20, for the ten unknown real parts, ak, and the ten unknown imaginary parts, b,, of the complex constants A,. The diagonal terms of the matrix of the coefficients of ukand b, are found to be the largest, which justifies the truncation of the infinite system of equations. The coefficients y0 to y4 occurring in the following expressions for the deflection and moments are presented in Table 7. WI,= 0 = yoQoa4/D, y=o
M,l,=o = y1Qoa2, y=o
M,;/,=y=o foi2
=
My/,=, = YZQO~’ y=o
M,lx=o
Y3Qoa2,
y=
=
~4Qoa'.
+b/2
TABLE 1. ELASTIC CONSTANTS FOR PLYWOOD IN BENDING (x-axis is parallel to the face grain [9]) Unit lo6 psi Material
Maple, 5-ply Afara, 3-ply Gabbon (Okoume), 3-ply Birch, 3- and 5-ply Birch with bakelite membranes
E:
E;
E”
G
P
4
187 1.96 1.28 2.00 1.70
0.60 0.165 0.11 0.167 0.85
0.073 0.043 0.014 0.077 0.061
0.159 0.110 0.085 0.17 0.10
0.746 0.964 0.932 0.693 0.742
1.100 1.588 1.591 1.725 0.926
TABLE 2. EVENEIGENVALUES FOR MAPLE~-PLY
1.38189480 3.34143423 5.29769735 7.25395011 9.21020294 11.16645576 13.12270859 15.07896142 17.0352 1424 18.99146707
1.12401050 2.44260144 3.76932012 5.09601519 6.42271029 7.74940539 9.07610049 1040279559 11.72949069 13.05618579
Eigenfunction analysis for bending of clamped rectahgular, orthotropic plates
491
TABLE 3. EVENEIGENVALUESFOR AFARA ~-PLY
1.00986320 2.46011174 3.90570665 5.35131412 6.79692157 8.24252902 9.68813647 11.13374392 12.57935137 14.02495882
0.77871471 1.650545 11 2.52813040 340569048 4.28325067 5.16081086 6.03837105 6.91593124 7.79349143 8.67105162
TABLE 4. EVEN EIGENVALUESFOR GAB~~N (OKOUME) 3-PLY
1.02382621 2.49943556 3.96953534 5.43965707 6.90977871 8.37990035 9.85002198 11.32014362 12.79026526 14.26038690
0.77664605 1.63218568 2.49340025 3.35459019 4.21578024 5.07697029 5.938 16034 6.79935039 7.66054043 8.52173048
TABLE 5. EVEN EIGENVALLJES FOR BIRCH 3- AND ~-PLY
1.06875796 2.65281775 4.22091405 5.78903801 7.35716445 8.92529087 10.49341729 12.06154371 13.62967013 15.19779655
0.70045830 1.33025887 1.96003851 2.59001708 3.21999486 3.84997260 4.47995035 5.10992809 5.73990584 6.36988358
The numerical work has been carried out on CDC-3600. Further increase in the number of terms has shown no appreciable effect on the results. As a check on the accuracy of the solution, the deflection and the slopes at the centre of the clamped edges at x = a/2 are evaluated and found to be negligible. The ratio of the deflection at the edge to the central deflection is found to be less than O-7x 10e5 in all the cases. Numerical results are also derived by applying a Galerkin method for a few cases; the results of the two methods agree with the limits of accuracy of the Galerkin method.
B. S. RAMACHANDRA RAO. G. L. NARASIMHAM and
492
S. G~PALACHARYULU
TABLE 6. EVEN EIGENVALUESFOR BIRCH WITH BAKELITE MEMBRANES
1.49197169 3.55724328 5.62333449 7.68940848 9.75548245 11.82155643 13.88763040 1595370437 18.01977835 20.08585232
1.32261523 2.97003726 4.62556816 6.28110539 7.93664198 9.59217858 11.24771518 12.90325177 14.55878837 16.21432497
TABLE 7. DEFLECTIONAND BENDING MOMENTSFOR A CLAMPED SQUAREPLATE UNDER UNIFORM U)AD
Material Maple, 5-ply Afara, 3-ply Gaboon (Okoume) 3-PlY Birch 3- and 5-ply Birch with bakelite membranes
)10
1’1
Yz
73
Y4
0.00023176 000001872
0.0369 1146 0.04431386
0.01595847 OX)0877378
- 0.07638084 - 0.0862 1326
- 0.03274248 -0.01638792
0~00001976 0.00001799
0.04307108 0.04636339
0.00889494 0.01409167
-0.08569530 -0.08558040
-0.01665159 -0.01645233
0.00052536
0.03321223
0.01864018
-0.07014675
-0.041%2136
6. CONCLUSIONS Eigenfunctions have been developed for the problem of clamped, rectangular, orthotropic plate undergoing small deflections. An analysis using these functions to obtain extremely accurate solutions is presented. The reasons for the accuracy in the solutions are that all the loads are made self-equilibrating and that it is the second derivatives that are matched at the boundary. These factors cannot be easily incorporated in the familiar methods such as multiple Fourier and energy methods. It is hoped that the present results provide a basis for comparison in the development of approximate techniques such as finite elements. Acknowledgemenrs-The authors wish to thank Professor F. A. Gaydon of Bristol University suggestions in this work. One of the authors (S.G.) wishes to express his gratitude to Professor Loughborough University for his encouragement and facilities provided to the author. Thanks the Referees for their comments on an earlier version of the paper.
[I] [2] [3] [4] [5j [6j
J. FADLE, Ostenn. Ig. Archiu. 2, 125 (1941). P. F. PAPKOVITCH, Dokl. Akad. Nauk SSR. 27.337 (1940). L. S. D. MORLEY, Q. JI Mech. Appl. Math. 16, 109-l 14 (1963). V. T. BUCHWALD, Proc. R. Sot., Lond. 277,383-400 (1964). F. A. GAYWN and W. M. SHEPHERD,Proc. R. Sot., Lond. 281, 184206 F. A. GAYDON, Q. JI Mech. Appl. Math. 14,65-78 (1966).
(1964).
for his helpful D. J. Johns of are also due to
Eigenfunction
analysis
for bending
of clamped
rectangular.
orthotropic
plates
[7j M. W. JOHNSON and R. W. LITTLE,Q. appl. Math. 22,335-344 (1964). [8] R. W. LITTLE,J. upp/. Mech. 36 (2), 320-323 (1969). [9] R. C. T. SMITH. Ausi. J. Sri. Res. 5. 227-231(1952). [IO] S. TIMOSHENKO and S. WOINOWSKY~KREIGER. Theory of Plates and Shells, ch. 11. McGraw-Hill [I l] S. G. LECHNITZKY. Anisofropicplutes, 2nd edition, Moscow (1957).
493
(1959).
APPENDIX b/2
a Km
=
FK
Y,
dY
J-b/Z
-zl =
{A, tan/2, . q(cos 2qpK+ cash 2ppK)/sin(2qpK) - pK(p2+ 4’))
'2
~~2+(~+‘,/~K)2~~~2+(~-~~~K)2}x(q2+(~+~~~K)2~{q2+(p-~,/~K)2).
=
(Received 4 April 1972; revised 30 June 1972)
A6cTpaKT--Pa3pa60TaHbr
CO6CTBeHHbIe (~IYHKUI~U~JIR 3aulCMJICHHOfi. I?JflMOyrOJIbHO~ OpTOTpOnHOii npO~H60B.YKa3aHO,'lTOO6blHbIC $YHKUNM @anJIe-&HKOBH%i (1) Ann cnygas H~OT~O~IMM o6pa3ymT co6oii CneUManbHblP cnyvaii Bonee 06~wx “opToTponHblx CO6CTBCHHbIX 1$yHKI&i2i"..&anru OpTOTpOI?HblX IIJlaCTHHOK,Ha OCHOBC 3TMX +yHKUWh,AaeT Ype3Blt'diHO ~naCTHHKH,nO~Bep~KeHHO8~e~CTBe~ManblX
T4aTeJIbHbIepU.UeHHFt.
J. Solids
Structures,
1973. Vol. 9, pp. 481 to 493. Pergamon
Press.
Printed
m Great
Britain
EIGENFUNCTION ANALYSIS FOR BENDING OF CLAMPED RECTANGULAR, ORTHOTROPIC PLATES B. S. RAMACHANDRA RAO and G. L. NARASIMHAM Indian Institute of Technology, Bombay, India
and S. GOPALACHARYULU Loughborough University of Technology, England
Abstract-Eigenfunctions have been developed for the clamped, rectangular orthotropic plate undergoing small deflections. The familiar Fadle-Papkovitch [l] functions for the isotropic case are shown to form a special case of the “orthotropic eigenfunctions”. An analysis for orthotropic plates using these functions yields extremely accurate solutions.
1. INTRODUCTION ONE of the useful methods for problems of bending of clamped, rectangular isotropic plates is the method ofeigenfunctions. The literature on this is extensive; see [l-9]. Tentative solutions of the biharmonic equation are postulated and a transcendental eigenvalue equation is obtained by satisfying homogeneous boundary conditions at the clamped edges. Solution of the eigenvalue equation gives rise to expansions in terms of eigenfunctions, a linear combination of which is generally sufficient to satisfy the boundary conditions on the remaining edges. The purpose of this work is to develop eigenfunctions for the clamped, rectangular, orthotropic plate undergoing “small” deflections, and to present an analysis of the problem using these functions. These functions have not been developed before; the FadlePapkovitch [l, 21 functions, well-known for the isotropic case, form a special case of the functions developed here. The clamped orthotropic plate problem can, of course, be treated by other more familiar techniques discussed in Timoshenko [lo] and Lechnitzky [ll]. The present analysis with the eigenfunctions yields an extremely accurate solution and is well suited for machine computation.
1.1. Statement of the problem We consider a clamped, rectangular, orthotropic plate of size a x b and thickness, h. The material has three planes of symmetry with respect to its elastic properties. Following Timoshenko [lo] we choose these planes as the co-ordinate planes (Fig. 1) and assume the 481
B. S.
482
RAMACHANDRA RAO, G. L. NARASIMHAMand S. GOPALACHARYULU Y
I
i
b
[--“““““~“‘~~
I_,.,,,,r ,,,,,,,,,,,, i x
O-f
I-
FIG.
1. The clamped, rectangular, orthotropic plate.
stress-strain relations for the case of plane stress in x-y plane as : ox = E&v + E”E, gy = E&,+ E”E, TXY =
%Y.
The governing differential equation for the deflection, w, of the plate under transverse load Q, is
D,$+ZH &+Dy$ =Q(x, Y)
(2)
where D, = E;h3/12;
D, = E;h3/12;
D, = E”h3/12;
D,, = Gh3/12
H = D,+2D,,
The bending moments are given by
(3)
The boundary conditions are w=“=() ay
ony = &b/2
w=$J=Clonx=
&a/2.
(4.1) (4.2)
2. EIGENFUNCTIONS Consider the homogeneous
equation
Dx$+2H
a4w w
+D”‘“=O. y ay4
(5)
Eigenfunction anafysis for bending of clamped rectangular, orthotropjc plates
483
Assume tentatively a solution in the form
and let i?W - = 0 at y = *b/2. ay
w”=
F is then determined by
(6) under the conditions F = dF/dy = 0 at y = t_ b/2. The constants C, and C, in equation (6) are given by C, = H/D, = (Dr + 20,,,),‘4 = (E” + 2G~/E~ C, = &WY
= &%‘E;)
-
On the basis F = ewy,it is seen that w = $$ kpfiq) where P = JKCZ - C,)izl
The even eigenfunction can now be written as (9.1) where p is given by the transcendental
equation
sint2w)+ sin&W = 4
P
o
(10.1) .
The odd eigenfunction is
(9.2) and the transcendental
equation is (10.2)
The eigenfunctions for the isotropic case may be obtained as a special case from the above expressions by putting El = ES, = E/(1 -/+, E” = &‘(i --p2) and G = E/2(1 +&, so that C1 = C, = 1, from which if follows p = 0, q = 1. In the limit of p -+ 0 and q = 1, the expressions (9.1), (9.2), (10.1) and (10.2) reduce to the familiar Fadle-Papkovitch functions [ 1,2].
B. S. RAMACHANDRA
484
RAO, G. L. NARASIMHAMand S. GOPALACHARYULU
Indeed, the condition p = 0, i.e. C, = C, is adequate to reduce the problem to an isotropic case. For, with the new variable Y, = yJC,, equations (5) and (6) assume the familiar forms of the isotropic problem (page 367, Timoshenko [lo]). In the above analysis it is implied that C, > C,, which makes p a real quantity. At least for plywood materials, assuming the elastic constants given in Timoshenko [lo], we have found p to be a real quantity. Even if p is imaginary, as it may happen for other materials, the above relations are valid. The transcendental equations (10.1) and (10.2) have infinite number of complex roots expressible as ( +cr,fiBx), K = 1,2,. . . . The asymptotic form for the Kth root of equation (10.1) (symmetric case) is found to be PK = 2(pz l+ 42) {[P hT(Ph) For the antisymmetric
+ (4K -
case, equation
1)v/4+ i[(4K - lbP/2 - 4 wPldl
1
(11.1)
(10.2), it is given by
PK = 2(pzl+ q2) {[P lW(P/Y) + (4K +
1)dA + i[(4K
+ lbP/:! -4 log(Plq~l)
(11.2)
Starting with the above expressions, the roots can be easily determined by Newton’s iteration method in complex form, which is quite rapidly convergent. For the isotropic case, such a technique has been given by Buchwald [3]. In a similar manner, we develop another set of eigenfunctions G,(x) which satisfy the homogeneous conditions G, = dG,/dx = 0 at x = &a/2. Let
On substituting
this in equation
(5) we obtain
where C; = H/D, = (Dl +2D,,)/D,
= (E”+2G)/E;.
C; = J(D,/D,) = J(E;IE:). On the basis G = em’“, it is found that w’ = Y(*p’*iq’) where
Jw; q’ = Jw
P’ =
- c;)Pl + c;)Pl.
Now if
c, > ~,,J(E~/E;) > (E~+~G)/E;
Eigenfunction
analysis
for bending
of clamped
rectangular,
orthotropic
plates
485
i.e.
J(E:E;)
> r+2G,
it follows C; > C;. Consequently the transcendental equations for p’ are the same as before for p with p’, q’ replacing p, q respectively.? The function set G,(x) can be obtained by inserting p’, q’, p’, a and x in the places of p, q, p, b and y respectively, in the expressions (9.1) and (9.2). The even eigenfunction in the set GK(x) is G =
cos(2q’p’x/a)
cosh(2p’p’x/a)
cos(q’p’)
cosh(p’p’)
sin(2q’p’xla)
sinh(2p’p’xla)
sin(q’p’)
sinh(p’p’)
-
1
(12.1)
where p’ is given by sin(2q’p’) + sinh(2p’p’) = o,
The odd eigenfunction G =
(13.1)
PI
4’ is
sin(2q’p’xla)
cosh(2p’p’xla)
sin(q’p’)
cosh(p’p’)
and the transcendental
equation
-
cos(2q’p’xla) sinh(2p’p’xla) sinh(p’p’) cos(q’p’)
1
(12.2)
is sin(2q’p’) 4’
sinh(2p’p’) = o
P’
-
(13.2)
.
The infinite number of complex roots of the equations (13.1) and (13.2) are also expressible as + c& f i&, K = 1,2, . . . The asymptotic forms for the kth root of p’ is obtained by putting p’, q’ in the places of p and q in equations (11.1) and (11.2).
3. SOLUTION Having orthotropic of Gaydon
FOR ANY GIVEN
LOADING
the two sets of eigenfunctions F,(y) and GK(x), a solution for the clamped plate for any given loading Q(x, y) can be developed on lines similar to that [6]. Assume that a particular integral wO of equation (2) can be found and write w =
where O(x,y) is a polynomial reasons discussed later. Then
solution
w,+n+w, of the homogeneous
equation
Dx$+*H+$+D,$ =0. ax ay t The authors
are indebted
to Professor
F. A. Gaydon
for this proof.
(5) introduced
for
(14)
486
B. S. RAMACHANDRA RAO, G. L. NARASIMHAM and
S. GOPALACHARYULU
The conditions (4.1) and (4.2) give at x = +u/2
W 1= -(w()+R) ' aw 1- -f$w,+R) ax aty=
+b/2
>.
(15)
Wl = -(w,+Q) aw, -= ay
-$(wo+Q)
/
Now consider another function w,(x, y) which satisfies
saw, aaW, i)hw, D~~+2HaX2ayZ+Dydy4 = 0
(16)
and the following conditions, at x = +a/2
a%, = -ggw,+i2) ay2 a%, ---z axay -&(WO+n) and at y = + b/2
(17)
azw2 -$(wo+Q) g= a2W2 ---z -&(Wo+n, alcay It is not difficult to see that the relation between wi and w2 is w1 = w,+k,x+k,y+k,
(18)
where k,, k, and k, are constants to ,be determined in any particular problem. If w1 and w2 are even in x, then kl = 0. If they are even in y, k2 = 0 and if odd in either coordinate k, = 0. They are determined from known conditions, say, at corners. The function w2 can now be developed into a series of eigenfunctions F,(y), GK(x). Assuming that the loading is symmetric about both the axes, we write w2 = z A, cosh(2p,x/b)F,(y) m
+ F B, cosh(2p;y/a)G,(x) m
(19)
where A,, B, are complex constants, pK = aK+ ibK, pk E cc; + iPt and A-,, B-, , p-K etc., are the respective complex conjugates of A,, B,, pK etc. Note that the sets of roots - c&+ i/t& do not yield any new solutions. Applying the conditions (17) it will -u,+&, be seen that the problem is reduced to the determination of the complex constants A,, B,
Eigenfunction analysis for bending of clamped rectangular, orthotropic plates
487
satisfying the following equations : (20.1)
f 1
(20.2)
B,$ =$wO+R) =gl(x)
y=+b/2
(20.3)
m
B&Z = - &(“o+n) (20.4) = h(X). Iu The purpose of the R will now be explained. The eigenfunctions F and G satisfy the following conditions : a/2 b/2 b’2 dF a’2 dG F.ydy = -dy = G.x.dx = (21) I -b/2 s -b/2 dy i -a/2 s _a,2 ZdX = O ;lf
and the function set is complete with respect to these self-equilibrating conditions, though this is not proved here. Consequently for the expansions in equations (20.1) to (20.4) to be valid, we should impose that b/2 s -b/2 Furthermore
fiv dy = j-yL2h dy = j-;12gix
dx = J::,gl
dx = 0.
(22.1)
remembering the properties of the eigenfunctions viz., dF, ---=Oaty= dy
ib/2,d$=Oatx=
*a/2,
and in view of expansions (20.2) and (20.4), we require that f21,=
fb/2
=
g21x=*.,2
=
0.
(22.2)
Fortunately all these requirements on the functionsf, ,f2, g, and g, can be met by a proper choice of Q generated in polynomials forming solutions of (5).
4. RE-ORTHOGONALISATION
OF EIGENFUNCTIONS
Unfortunately the functions F,(y), GK(x) do not seem to possess suitable orthogonality relations and indirect methods are to be employed for determining the constants in the expansions (20.1) to (20.4). Although the usual methods such as least-squares etc., can be used, they do not possess special merits in the present problem. The techniques having certain advantages are (a) the method of biorthogonal functions developed by Johnson and Little [7], and Little [8], (b) the method of reorthogonalization developed by Gaydon and Shepherd [5], and Gaydon [&I. The latter method is adopted here. This consists of expanding each of the eigenfunctions into the orthogonal set of normalized clamped beam functions Y, defined by : ~-(21,,b)4Ym
= 0
B. S. RAMACHANDRA RAO. G. L. NARASIMHAM and S. GOPALACHARYULU
488
under
the conditions Y,=S=Oaty= dy
In the symmetric
case, the functions
*/J/2.
are
1 where 1, is given by tan 1, + tanh 1, = 0. The 2,s are well-known FK = by virtue of the orthogonality
il
[5]. Writing (23)
%nYm
m=l
relation b/2
s we
_b,,
_
LK
=
4nn
find that b/2 URm =
i -b/2
F, . Y, dJ
The expression for aKmis given in Appendix. Also, it follows that F_,
= Conjugate
of F, =
f m=l
where CI-~~ = Conjugate Furthermore, because
of a,,. of the following
sv b/2
d2
s -b/2 the functionsf,,
fi in equations
2
d!,
relations
[5]
y-‘&y
-L-!! dy = (212,/b)” 6,, dy’
-b,2
b’2 dY
orthogonality
U-K,Y,
d3Y
dy = -(2&,/b)”
2
6,”
dy3
(20.1) and (20.2) can be expanded
.fl(Y) = i:
1
c-3
in the following
manner (23.1)
rL*
f,(y) = c 4,x% 1
where,
(23.2)
Eigenfunction analysis for bending of clamped rectangular, orthotropic plates
489
If we substitute the above expressions in equations (20.1) and (20.2), we obtain
K=-a,
(m = 1,2,. fbPKaKm
=
(24)
.)
dm
m
which are the infinite system of equations required for the determination of A,. Similarly by expanding the set G,(x) into the orthogonal set X,(x) defined by c3
- (21*,*)4X,
x,
= 0,
+a/2
“=Oatx= = “d”,
we obtain the following system of equations for B, ;
B,b,,
=
1
c:,
K=-a-
(m =
a!
h’;bKm
1,2,. . .)
.
(25)
= 4, i
The expression for bK, is obtained by putting a, p’, p’, q’ in the places of b, p, p, q respectively in the expression for aKm(Appendix). The expressions for CL and d& are: d2X
“dx g1 dx2 d3Xndx
g2=
5. NUMERICAL
’
EXAMPLE
As a numerical example, we consider the problem of a clamped plate under uniform normal pressure. A particular solution is w
0
=
L &(b2,&y2)2. 24 Dy
Since this automatically satisfies the conditions at y = k b/2, the function set GK(x) is not required. It is also seen that the function R is not required, as the functionsf, = - ~2wo/~y2 ; f2 = - 82wo/Jx8y satisfy the requirements (22.1) and (22.2). Equations (20.1) and (20.2) give 1 K
cosh(PKa/b) AKaKm
cash pK
= C,
sinh(PKa/b) = c AKaKmPK cash PK K
where
o
490
B. S. RAMACHANDRA
RAO. G. L. NARASIMHAMand S. GOPALACHARYULU
These are the required infinite system of equations, which may be solved by the usual truncation procedure, using a computer. Since complex algebra can be handled on modern high speed computers, the separation of real and imaginary parts can be carried out on the computer itself. For numerical work, we consider the plywood materials. The elastic constants are listed in Timoshenko [lo] and Lechnitzky [ll]. We reproduce these values in Table 1, and also present the values of p and 4, defined by equation (8). Tables 2 to 6 give the eigenvalues, for symmetric case, for these materials. For a square plate (a = b), the numerical work has been carried out, by retaining only the first ten A, and taking m up to 10. This means, each of the eigenfunctions is expanded into the first ten clamped beam functions resulting in a square matrix of size 20 x 20, for the ten unknown real parts, ak, and the ten unknown imaginary parts, b,, of the complex constants A,. The diagonal terms of the matrix of the coefficients of ukand b, are found to be the largest, which justifies the truncation of the infinite system of equations. The coefficients y0 to y4 occurring in the following expressions for the deflection and moments are presented in Table 7. WI,= 0 = yoQoa4/D, y=o
M,l,=o = y1Qoa2, y=o
M,;/,=y=o foi2
=
My/,=, = YZQO~’ y=o
M,lx=o
Y3Qoa2,
y=
=
~4Qoa'.
+b/2
TABLE 1. ELASTIC CONSTANTS FOR PLYWOOD IN BENDING (x-axis is parallel to the face grain [9]) Unit lo6 psi Material
Maple, 5-ply Afara, 3-ply Gabbon (Okoume), 3-ply Birch, 3- and 5-ply Birch with bakelite membranes
E:
E;
E”
G
P
4
187 1.96 1.28 2.00 1.70
0.60 0.165 0.11 0.167 0.85
0.073 0.043 0.014 0.077 0.061
0.159 0.110 0.085 0.17 0.10
0.746 0.964 0.932 0.693 0.742
1.100 1.588 1.591 1.725 0.926
TABLE 2. EVENEIGENVALUES FOR MAPLE~-PLY
1.38189480 3.34143423 5.29769735 7.25395011 9.21020294 11.16645576 13.12270859 15.07896142 17.0352 1424 18.99146707
1.12401050 2.44260144 3.76932012 5.09601519 6.42271029 7.74940539 9.07610049 1040279559 11.72949069 13.05618579
Eigenfunction analysis for bending of clamped rectahgular, orthotropic plates
491
TABLE 3. EVENEIGENVALUESFOR AFARA ~-PLY
1.00986320 2.46011174 3.90570665 5.35131412 6.79692157 8.24252902 9.68813647 11.13374392 12.57935137 14.02495882
0.77871471 1.650545 11 2.52813040 340569048 4.28325067 5.16081086 6.03837105 6.91593124 7.79349143 8.67105162
TABLE 4. EVEN EIGENVALUESFOR GAB~~N (OKOUME) 3-PLY
1.02382621 2.49943556 3.96953534 5.43965707 6.90977871 8.37990035 9.85002198 11.32014362 12.79026526 14.26038690
0.77664605 1.63218568 2.49340025 3.35459019 4.21578024 5.07697029 5.938 16034 6.79935039 7.66054043 8.52173048
TABLE 5. EVEN EIGENVALLJES FOR BIRCH 3- AND ~-PLY
1.06875796 2.65281775 4.22091405 5.78903801 7.35716445 8.92529087 10.49341729 12.06154371 13.62967013 15.19779655
0.70045830 1.33025887 1.96003851 2.59001708 3.21999486 3.84997260 4.47995035 5.10992809 5.73990584 6.36988358
The numerical work has been carried out on CDC-3600. Further increase in the number of terms has shown no appreciable effect on the results. As a check on the accuracy of the solution, the deflection and the slopes at the centre of the clamped edges at x = a/2 are evaluated and found to be negligible. The ratio of the deflection at the edge to the central deflection is found to be less than O-7x 10e5 in all the cases. Numerical results are also derived by applying a Galerkin method for a few cases; the results of the two methods agree with the limits of accuracy of the Galerkin method.
B. S. RAMACHANDRA RAO. G. L. NARASIMHAM and
492
S. G~PALACHARYULU
TABLE 6. EVEN EIGENVALUESFOR BIRCH WITH BAKELITE MEMBRANES
1.49197169 3.55724328 5.62333449 7.68940848 9.75548245 11.82155643 13.88763040 1595370437 18.01977835 20.08585232
1.32261523 2.97003726 4.62556816 6.28110539 7.93664198 9.59217858 11.24771518 12.90325177 14.55878837 16.21432497
TABLE 7. DEFLECTIONAND BENDING MOMENTSFOR A CLAMPED SQUAREPLATE UNDER UNIFORM U)AD
Material Maple, 5-ply Afara, 3-ply Gaboon (Okoume) 3-PlY Birch 3- and 5-ply Birch with bakelite membranes
)10
1’1
Yz
73
Y4
0.00023176 000001872
0.0369 1146 0.04431386
0.01595847 OX)0877378
- 0.07638084 - 0.0862 1326
- 0.03274248 -0.01638792
0~00001976 0.00001799
0.04307108 0.04636339
0.00889494 0.01409167
-0.08569530 -0.08558040
-0.01665159 -0.01645233
0.00052536
0.03321223
0.01864018
-0.07014675
-0.041%2136
6. CONCLUSIONS Eigenfunctions have been developed for the problem of clamped, rectangular, orthotropic plate undergoing small deflections. An analysis using these functions to obtain extremely accurate solutions is presented. The reasons for the accuracy in the solutions are that all the loads are made self-equilibrating and that it is the second derivatives that are matched at the boundary. These factors cannot be easily incorporated in the familiar methods such as multiple Fourier and energy methods. It is hoped that the present results provide a basis for comparison in the development of approximate techniques such as finite elements. Acknowledgemenrs-The authors wish to thank Professor F. A. Gaydon of Bristol University suggestions in this work. One of the authors (S.G.) wishes to express his gratitude to Professor Loughborough University for his encouragement and facilities provided to the author. Thanks the Referees for their comments on an earlier version of the paper.
[I] [2] [3] [4] [5j [6j
J. FADLE, Ostenn. Ig. Archiu. 2, 125 (1941). P. F. PAPKOVITCH, Dokl. Akad. Nauk SSR. 27.337 (1940). L. S. D. MORLEY, Q. JI Mech. Appl. Math. 16, 109-l 14 (1963). V. T. BUCHWALD, Proc. R. Sot., Lond. 277,383-400 (1964). F. A. GAYWN and W. M. SHEPHERD,Proc. R. Sot., Lond. 281, 184206 F. A. GAYDON, Q. JI Mech. Appl. Math. 14,65-78 (1966).
(1964).
for his helpful D. J. Johns of are also due to
Eigenfunction
analysis
for bending
of clamped
rectangular.
orthotropic
plates
[7j M. W. JOHNSON and R. W. LITTLE,Q. appl. Math. 22,335-344 (1964). [8] R. W. LITTLE,J. upp/. Mech. 36 (2), 320-323 (1969). [9] R. C. T. SMITH. Ausi. J. Sri. Res. 5. 227-231(1952). [IO] S. TIMOSHENKO and S. WOINOWSKY~KREIGER. Theory of Plates and Shells, ch. 11. McGraw-Hill [I l] S. G. LECHNITZKY. Anisofropicplutes, 2nd edition, Moscow (1957).
493
(1959).
APPENDIX b/2
a Km
=
FK
Y,
dY
J-b/Z
-zl =
{A, tan/2, . q(cos 2qpK+ cash 2ppK)/sin(2qpK) - pK(p2+ 4’))
'2
~~2+(~+‘,/~K)2~~~2+(~-~~~K)2}x(q2+(~+~~~K)2~{q2+(p-~,/~K)2).
=
(Received 4 April 1972; revised 30 June 1972)
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