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On the design and analysis of simply supported and clamped thick rectangular plates
Comparers & Srruclures Vol. 38, No. S/b. pp. 589-596. 1991 Printed in Great Britain.
0045.7949/91 $3.00 + 0.00 0 1991 Pergamon Press plc
ON THE DESIGN AND ANALYSIS OF SIMPLY SUPPORTED AND CLAMPED THICK RECTANGULAR PLATES S. F. NG and N. BENCHARIF Department of Civil Engineering, University of Ottawa, Ottawa, Ontario, Canada KIN 6N5 (Received 5 Februury 1990)
Abstract-The finite difference method, although well known as an efficient numerical method, was applied in the past, in the case of plate problems, only for the solution of thin plates. In the present study, the suitability of the method for problems involving thick plates is studied. The finite difference method as applied here is a modified finite difference approach to the ordinary finite difference method generally used for the solution of thin plate problems. Thin plates are treated as a particular case of the corresponding thick plates. The method is first applied to investigate the behaviour of clamped, square isotropic homogeneous thick plates. After the validity of the method is established, it is then extended to the solution of similar problems for simply supported square plates. Once the solution for a thick plate with a particular plate aspect ratio and boundary condition is obtained using a limited number of mesh sizes, a more refined solution to investigate the amracy and convergence of the problem is then extended by providing more detailed functions satisfying the mesh sizes generated automatically by a computer program. Whenever possible results of the present method are compared with existing solutions in the technical literature obtained by much more laborious numerical techniques, and close agreements are found. The submatrices involved in the formation of the finite difference equations from the governing differential equations are generated directly by the computer program. Simplicity in formulation and quick convergence are the obvious advantages of the method in comparison with other numerical methods requiring extensive computer facilities.
NOTATION
x, .v,2 u, v, w
NV Q.vQ,
A = (1 + v)(l _ 2v)
rectangular co-ordinate system displacements of the nodes in x, y, and z directions respectively longitudinal dimension of the plate transverse dimension of the plate thickness of the plate axial forces parallel to the x-axis per unit length of a section of plate perpendicular to y-axis axial forces parallel to the y-axis per unit length of a section of plate perpendicular to x-axis shearing forces parallel to the z-axis per unit length of a section of plate perpendicular to x- and y-axis, respectively inplane shearing intensities per unit length bending moment intensities per unit length twisting moment intensity per unit length transverse external force per unit normal stress components in x, y, and z directions, respectively shearing stress components in rectangular co-ordinates modulus of elasticity in shear Lame coefficient rectangular sizes
finite difference mesh
589
I= *h/2
Poisson’s ratio lower and upper face of a plate, respectively Laplacian operator co-ordinates
in rectangular
Laplacian ordinates
in space co-
operator
INTRODUCTION Plates are classified by their geometrical forms or by
their physical characteristics. Thick plates are a type of plates where the thickness is considerable and the approximate theories of thin plates are not applicable. In thick plate theory, the plates are treated as a three-dimensional problem of elasticity. The stress analysis becomes, consequently, much more involved and, until now, the problem has been solved only for a few particular cases. The difficulty of these colutions is a direct result of the complexities of the partial differential equations governing the behaviour of thick plates. Furthermore, the boundary conditions for thick plates have to be satisfied in three directions, hence three functions determining the three displacements are required. Finite difference methods are often used for solving structures governed by complex differential equations. In applying the finite difference method to plate problems, a mesh size must first be chosen. The
590
S. F. NG and N. BENCHAIUF
lation [ 11, the differential equations governing deflection of thick plates can be obtained as
the
Fig. I. Forces acting on thick plate element.
mesh size covers the entire domain of the plate with n points yielding n simultaneous equations. Convergence is easily investigated by increasing the mesh size covering the plate domain. In the present study, the finite difference method is applied to the three-dimensional problem of thick plates where the governing differential equations of equilibrium of such plates are expressed in terms of the displacements U, v, and w following the co-ordinates axes x, y, and z, care being taken to modify the finite difference expressions to satisfy the appropriate boundary conditions and the symmetry of a particular problem. GOVERNING DIFFERENTIAL EQUATIONS
The formulation of the governing differential equations of an isotropic rectangular thick plate under vertical load q(x, y) is presented in the three principal directions x, y, and z. The equilibrium conditions are satisfied by equating the internal and external forces in each principal direction. Figure 1 shows all the forces on the plate element, and Fig. 2 shows the deformation of the plate in its general form. After considerable mathematical manipu-
I
- \
The general form of the stress components yields three equations of equilibrium along the x, y, and z directions, respectively. !%+%+!!+ ay
a7,
soy
z+-+z=o ay
(4)
a’syz
!%+9++ ay
(5)
(6)
Thus, the three equations of equilibrium, expressed in terms of displacements, are
Fig. 2. Deformations of thick plate.
(I+G);+GV’U=O
(7)
(i+G)~+GV%=O
(8)
(I.+G);+GVlw=O.
(9)
Design and analysis of thick rectangular plates
a4
The system of equations above can be expressed in explicit form in terms of the derivatives
GS
= [U(Z - ZK) - 2U(Z)
!2+($ (i+2Gg$+Gay2 +(A +G)-
(l+G)-
a% axay
a% axay
+ U(Z + ZK)] $
= [V(Z) - V(Z + 1)
@+G)&,
(IO)
+(i+G)~=O
- Y(Z + ZJ) + V(Z + ZJ + 1)
+fi(, +2G)$ ax*
x(A+G) &S
+G$+(l +G)-$=o
(11)
= [W(Z) - W(Z + 1) - W(Z + ZK)
(1 +G)g (n+G)g+(i+~)&+~$
(2 +G) + W(Z + ZK + l)] 11 x 1
+G$+(I
+2G)$=o.
(12) = [U(Z + 1) - U(Z + ZJ)
(1 +G)g With the co-ordinate system located at the midsurface of the plate, the boundary conditions will be satisfied at x = *a/2, y = &b/2, z = &h/2.
+ qz+z.I+
x+5 2
J-i;:
u = 0,
v = 0,
w=o
u =o,
v =
w
=
a,
=
0,
I)]9 x 2
Clamped plate
z=+-:-2
591
simply supported
0
6, = 0.
7xy =
0,
w
=
0
by =
7xy =
0,
w
=
0
0,
h 7,,
=
0,
7yz =
0,
&;
7xr =
0,
FINITE DIFFERENCE REPRESENTATION
G2 Due to the symmetry of the rectangular plate, only a quarter of the plate need be considered. In the application of finite difference, the thick plate is divided into rectangular mesh sizes connected at their nodes, resulting in a system of simultaneous equations defining the deflection of the plates. Here, due to space limitations, only the formulations of two typical nodes are shown, namely, a general point and the central point inside the thick plate. Detailed formulations of other nodes can be found elsewhere [I].
= [U(Z - 1) - 2U(Z) + U(Z + 1)] $ 1
(1 + 2G) $
= [ V(Z - ZJ) - 2 l’(Z)
+ V(z+zJ-)]~ Y G$=(Y(Z-ZK)-2V(Z)
+ V(Z + ZK)] ; i
General point inside the plate (A + G) 2 (A +2G)$=[U(Z-
= [W(Z) - W(Z + ZJ) - W(Z + JK)
l)-2U(Z) + W(Z + ZJ + JK)]
+ U(Z + I)]?
(A+G)
n,~,
x G$
= [U(Z - ZJ) - 2U(Z) + cJ(Z + ZJ)] $
(A+G)g=[U(Z)-U(lfI)-U(ZfZK)
+ U(Z + ZK + l)] v Y
x 1
S. F. NC and N.
592
(2 + G)
$ =[V(Z) -
G$=[2V(Z-I)-2y(I)];
?‘(I + ZJ) - v(z + JK)
-I- V(Z + ZJ) + JK)]
(A+@ 11
Y
C$=[W(Z-
BENCHARIF
I[ (A + G) $
= [2V(Z - ZJ) - 2 V(Z)] y
2
l)-2w(Z)
G$=[V(Z-ZK)-2V(Z)
+ W(I + l)] ;
+ V(Z + ZK)] ;
X
I
G$=[W(Z-Z.Z)-2H’(Z)
(A + G) -j$
= [W(Z) - W(Z - ZJ) - U(Z + JK)
+ W(Z + Z.Z)];
+ U(Z-ZJ+JK)]T (1+2G)$=[W(Z-ZK)-2W(Z)
+
Y 2 (1 + G) -$
W(ZfZK)]q2
- U(Z + ZK)
Central point inside the plate (A.+2G)$=[211(1-
= [U(Z) - U(Z - 1)
+ U(Z+ZK-
I)-2u(Z)]y
l)]F XL
I (A + G) &
G$=[2U(Z-ZJ)-2U(Z)];
= [V(Z) - V(Z - ZJ)
> G$
- ?‘(I + JK)
= [U(Z - ZK) - 2U(Z)
(2 +G) + V(Z - ZJ + JK)] 11 y *
+ U(Z + ZK)] ; ‘
G+G)&=[V(i) -
G$=[ZW(Z-I)-2W(Z)]; x
V(Z - 1) G$=[2W(Z-ZJ)-2W(Z);
- V(Z - ZJ) +
Y(Z-ZJ-Z)]W x ?
(1 + G)
& =[W(Z) - W(Z - 1)
(n+G) - 2 W(Z) + W(Z + ZK)] 12 1
- W(Z + ZK) COMPARISON
+
W(Z+ZK-Z)]q) I
=[U(Z) (i+G)-&
OF RESULTS
General
:
U(Z - 1)
- U(Z - ZJ) + U(Z-ZJ-
AND DISCUSSION
I)]9 x .L
The formulation outlined above has been applied to both clamped and simply supported thick plates. A complete study of the effects of various parameters has been carried out and extensive results obtained [l]. In general, both clamped and simply supported thick plates have been analysed for the same plate aspect ratios and the same type of mesh sizes, however the number of nodes used has been larger for
593
Design and analysis of thick rectangular plates
Table 1. Variation of the maximum centre deflection of clamped and simply supported rectangular thick plates with number of nodes used in the finite difference analysis? 0
hla
1.0
1.2
1.5
1.8
2.0
20
Nodes used for clamped plates 45 80
Nodes used for simply supported plates 40 75 120 175
125
2.1118 0.7155
47.0804 5.8572 2.1218 0.7187
47.1543 5.8662 2.1249 0.6790
47.1861 5.8701 2.1263 0.7207
10.4510 1.2950 0.4671 0.1572
31.4175 3.9050 1.4130 0.4775
31.5698 3.9235 1.4194 0.4795
31.6191 3.9295 1.4215 0.4611
31.6409 3.9321 1.3412 0.4803
5.6410 0.6976 0.2509 0.0838
5.4300 0.6713 0.2410 0.0748
17.5889 2.1822 0.7878 0.2650
17.6658 2.1914 0.7909 0.2488
17.6922 2.1945 0.7540 0.266 1
17.7042 2.1960 0.7925 0.2662
3.2490 0.4008 0.1438 0.0479
3.0168 0.3719 0.1331 0.0442
2.9060 0.3580 0.1286 0.0425
10.2358 1.2671 0.4561 0.1526
10.2707 1.2712 0.4575 0.1529
10.2840 1.2727 0.4579 0.1529
10.2903 1.2484 0.4582 0.1529
2.2050 0.2714 0.097 1 0.0323
2.0458 0.2516 0.0900 0.0295
1.9700 0.2427 0.0865 0.0285
7.3074 0.9030 0.3244 0.1081
7.3256 0.905 1 0.3163 0.1080
7.3332 0.8701 0.3253 0.1081
7.3340 0.9064 0.3254 0.1080
0.05 0.10 0.14 0.20
20.9200
2.6010 0.9412 0.3182
17.8300 2.2150 0.8010 0.2703
16.5541 2.0555 0.7428 0.2505
15.9200 1.9760 0.7139 0.2417
46.8489 5.8290
0.05 0.10 0.14 0.20
13.7400 1.7060 0.6165 0.2079
11.7000 1.4520 0.5240 0.1763
10.8649 1.3471 0.4860 0.1637
0.05 0.10 0.14 0.20
7.1590 0.8869 0.3197 0.1072
6.0720 0.7514 0.2705 0.0904
0.05 0.10 0.14 0.20
3.8540 0.4762 0.1711 0.0571
0.05 0.10 0.14 0.20
2.6280 0.3241 0.1162 0.0386
r Values given as w/q, for v = 0.30.
the analysis of the simply supported plates in order to satisfy the boundary conditions at the edges. Convergence and comparison of results
In general, the convergence characteristics for simply supported and clamped thick plates are slightly different. In order to improve the convergence and the accuracy of the method, four mesh sizes were
investigated for each case. Typical convergence results for different plate aspect ratios, a/b, are shown in Table 1. Extensive deflection results for centre deflections of various Poisson ratios. For the clamped plate, the convergence of the results for different mesh sizes is shown in Figs 3-5 and, in the case of simply supported plates, the convergence of the results due to different mesh sizes is shown in Figs 7-9. 25.0
I
pTq7q 0
rPOISSON’S
r
RATIO 0.q
0
(1 1 POISSON’S
7.0.0 -
RATIO O.Ztj
P
0 = 20 Nodes 0 = 45 Nodes A = 50 Nodes
RATlO AXIS (va)
Fig. 3. Comparison of the deflection of clamped thick plates for four mesh sizes.
Fig. 4. Comparison of the deflection of clamped thick plates for four mesh sizes.
S. F. No and N.
BENCIWFUF
,jRAflOo/b
/-
1POISSON'S RATIO O.iq
,-
D= 0= A= +=
20 Nodes 45 Nodes 80 Nodes t25 Nodes
a = Q= b= + I:
40 Nodes 75 kodrs 120 Nod*s i75 Nodes -I
“.I30
“.O>
Q.20
RATIO t%
(h/b)
0.m
0.25
0.10
Fig. 5. Comparison of the deflection of clamped thick plates for four mesh sizes.
Fig. 7. Comparison of the deflection of simply supported thick plates for four mesh sizes.
Figures 6 and 10 show the influence of the plate aspect ratio a/b on plate displacements for both clamped and simply supported thick plates. Also, in order to show that the modified finite difference approach is a good numerical scheme for the design and analysis of thick rectangular plates, a few examples of rectangular thick plates were analysed and results were compared with those obtained by other investigators. Figures 11 and 12 show the
excellent agreement between the results obtained by present method and those obtained by other researchers. CONCLUSIONS
A finite difference method for computing the displacements of thick rectangular plates has been presented. Equilibrium equations in terms of displacements have been derived, The numerical
r 1pinmE-l L’OISSON’S RATIO 0.3 1
i \ \ \ \ D =
0= 1) = += x=
RATtO q,‘b=l.O RATIO qb1.2 RATE0 o/%=1.5 RATIO 4~1.8 RATIO a/b&O
Fig. 6. Influence of the plate aspect ratio on the deflection of clamped thick plates.
\
l
0 = 40 Nodes
75 Nodes h = l20 Nodes + = i75 Nodes 0 =
\
Fig. 8. Comparison of the deflection of simply supported thick plates for four mesh sizes.
Design and analysis of thick rectangular plates
595
IRATlOa/brl [POISSON’S
L7micqy
RATIO 0.V.j
IPOISSON’S
RATIO 0.1
\ \
q = 40 Nodes Q = 75 Nodes d = 120 Nodes + = 175 Nodes
O.Ot
0.30
l
0.0 10
0.01
RATIO AXIS (h/a)
0.10
0.11
0.20
0.21
0.30
RATIO AXIS (h/a)
Fig. 9. Comparison of the deflection of simply supported thick plates for four mesh sizes.
Fig. 11. Comparison of the deflection of clamped thick plates with other investigators.
examples indicate that the method presented herein yields excellent results. Even with a crude mesh size of 4 x 4, the method yields results of acceptable accuracy for both the clamped and simply supported thick plates. In general, the agreement between the finite difference results and those obtained by other methods is well shown in Figs 11 and 12. As a result of this investigation the following conclusions may be drawn.
(1) The modified finite difference method presented in this paper yields good convergence characteristics. (2) For square thick plates, deflection results are in excellent agreement with those obtained by other investigators [2-51. (3) General results for the deflection of thick rectangular plates are meagre, and as a result data obtained in this investigation for thick
Iptnicqy
tTlmiq 1 POISSON’S
RATIO 0.3
)
1 POISSON’S
RATIO 0.31
aAn a/a=10 0 = RATIO a/b=12
0.00
0.01
0.10
0.11
0 =
0 = Reistner
0 = RATIO a/b=lJ + = RATIO a/b=13 x = RATIO a/b=2.0
0 A + X
0.20
0.2,
0.30
RATIO AXIS (h/a)
Fig. 10. Influence of the plate aspect ratio on the deflection of simply supported thick plates.
Y.“V
0.0,
0.10
0.1,
0.20
= = = =
1 Reistne Srinlvos Q MIF[5] Present Study
0.35
,
0
RATIO AXIS (h/a)
Fig. 12. Comparison of the deflection of simply supported thick plates with other investigators.
596
S. F. NC and N. BENCHARIF
rectangular plates cannot as yet be verified. It is believed, however, that those results for square thick plates compare well with those of other investigators. (4) The method is simple to apply and accurate results are obtained even with a relatively crude mesh size. (5) With the detailed computer program which can automatically generate the system of simultaneous equations [l] and the facility of flexibility of input, the designer can easily input various loadings, plate aspects and boundary conditions to obtain accurate results for the bending of all types of simply supported and clamped rectangular thick plates.
REFERENCES 1. N. Bencharif, Application of finite difference method to the deflection of clamped and simply supported thick rectangular plates. M.A.Sc. thesis, Ottawa University, Ontario, (1988). 2. S. Srinivas, A. K. Rao and C. V. Joga Rao, Flexure of simply supported thick homogeneous and laminated rectangular plates. Z. angew. Math. Mech. 49,449-458 (1969).
3. S. Srinivas and A. K. Rao, Flexure of thick rectangular plates. J. appl. Mech. 40, 298-299 (1973). 4. S. Desmukh’ Rajjabau and R. Archer, Numerical solutions of moderately thick plates. J. Engng Mech. Div. AXE. lOON, 903-917 (1974). 5. K. T. Sandara Raja Iyengar, K. Chandrashekhara and V. K. Sebastian, On the analysis of thick rectangular plates. Ing. Arch. 43, 317-330 (1974).
0045.7949/91 $3.00 + 0.00 0 1991 Pergamon Press plc
ON THE DESIGN AND ANALYSIS OF SIMPLY SUPPORTED AND CLAMPED THICK RECTANGULAR PLATES S. F. NG and N. BENCHARIF Department of Civil Engineering, University of Ottawa, Ottawa, Ontario, Canada KIN 6N5 (Received 5 Februury 1990)
Abstract-The finite difference method, although well known as an efficient numerical method, was applied in the past, in the case of plate problems, only for the solution of thin plates. In the present study, the suitability of the method for problems involving thick plates is studied. The finite difference method as applied here is a modified finite difference approach to the ordinary finite difference method generally used for the solution of thin plate problems. Thin plates are treated as a particular case of the corresponding thick plates. The method is first applied to investigate the behaviour of clamped, square isotropic homogeneous thick plates. After the validity of the method is established, it is then extended to the solution of similar problems for simply supported square plates. Once the solution for a thick plate with a particular plate aspect ratio and boundary condition is obtained using a limited number of mesh sizes, a more refined solution to investigate the amracy and convergence of the problem is then extended by providing more detailed functions satisfying the mesh sizes generated automatically by a computer program. Whenever possible results of the present method are compared with existing solutions in the technical literature obtained by much more laborious numerical techniques, and close agreements are found. The submatrices involved in the formation of the finite difference equations from the governing differential equations are generated directly by the computer program. Simplicity in formulation and quick convergence are the obvious advantages of the method in comparison with other numerical methods requiring extensive computer facilities.
NOTATION
x, .v,2 u, v, w
NV Q.vQ,
A = (1 + v)(l _ 2v)
rectangular co-ordinate system displacements of the nodes in x, y, and z directions respectively longitudinal dimension of the plate transverse dimension of the plate thickness of the plate axial forces parallel to the x-axis per unit length of a section of plate perpendicular to y-axis axial forces parallel to the y-axis per unit length of a section of plate perpendicular to x-axis shearing forces parallel to the z-axis per unit length of a section of plate perpendicular to x- and y-axis, respectively inplane shearing intensities per unit length bending moment intensities per unit length twisting moment intensity per unit length transverse external force per unit normal stress components in x, y, and z directions, respectively shearing stress components in rectangular co-ordinates modulus of elasticity in shear Lame coefficient rectangular sizes
finite difference mesh
589
I= *h/2
Poisson’s ratio lower and upper face of a plate, respectively Laplacian operator co-ordinates
in rectangular
Laplacian ordinates
in space co-
operator
INTRODUCTION Plates are classified by their geometrical forms or by
their physical characteristics. Thick plates are a type of plates where the thickness is considerable and the approximate theories of thin plates are not applicable. In thick plate theory, the plates are treated as a three-dimensional problem of elasticity. The stress analysis becomes, consequently, much more involved and, until now, the problem has been solved only for a few particular cases. The difficulty of these colutions is a direct result of the complexities of the partial differential equations governing the behaviour of thick plates. Furthermore, the boundary conditions for thick plates have to be satisfied in three directions, hence three functions determining the three displacements are required. Finite difference methods are often used for solving structures governed by complex differential equations. In applying the finite difference method to plate problems, a mesh size must first be chosen. The
590
S. F. NG and N. BENCHAIUF
lation [ 11, the differential equations governing deflection of thick plates can be obtained as
the
Fig. I. Forces acting on thick plate element.
mesh size covers the entire domain of the plate with n points yielding n simultaneous equations. Convergence is easily investigated by increasing the mesh size covering the plate domain. In the present study, the finite difference method is applied to the three-dimensional problem of thick plates where the governing differential equations of equilibrium of such plates are expressed in terms of the displacements U, v, and w following the co-ordinates axes x, y, and z, care being taken to modify the finite difference expressions to satisfy the appropriate boundary conditions and the symmetry of a particular problem. GOVERNING DIFFERENTIAL EQUATIONS
The formulation of the governing differential equations of an isotropic rectangular thick plate under vertical load q(x, y) is presented in the three principal directions x, y, and z. The equilibrium conditions are satisfied by equating the internal and external forces in each principal direction. Figure 1 shows all the forces on the plate element, and Fig. 2 shows the deformation of the plate in its general form. After considerable mathematical manipu-
I
- \
The general form of the stress components yields three equations of equilibrium along the x, y, and z directions, respectively. !%+%+!!+ ay
a7,
soy
z+-+z=o ay
(4)
a’syz
!%+9++ ay
(5)
(6)
Thus, the three equations of equilibrium, expressed in terms of displacements, are
Fig. 2. Deformations of thick plate.
(I+G);+GV’U=O
(7)
(i+G)~+GV%=O
(8)
(I.+G);+GVlw=O.
(9)
Design and analysis of thick rectangular plates
a4
The system of equations above can be expressed in explicit form in terms of the derivatives
GS
= [U(Z - ZK) - 2U(Z)
!2+($ (i+2Gg$+Gay2 +(A +G)-
(l+G)-
a% axay
a% axay
+ U(Z + ZK)] $
= [V(Z) - V(Z + 1)
@+G)&,
(IO)
+(i+G)~=O
- Y(Z + ZJ) + V(Z + ZJ + 1)
+fi(, +2G)$ ax*
x(A+G) &S
+G$+(l +G)-$=o
(11)
= [W(Z) - W(Z + 1) - W(Z + ZK)
(1 +G)g (n+G)g+(i+~)&+~$
(2 +G) + W(Z + ZK + l)] 11 x 1
+G$+(I
+2G)$=o.
(12) = [U(Z + 1) - U(Z + ZJ)
(1 +G)g With the co-ordinate system located at the midsurface of the plate, the boundary conditions will be satisfied at x = *a/2, y = &b/2, z = &h/2.
+ qz+z.I+
x+5 2
J-i;:
u = 0,
v = 0,
w=o
u =o,
v =
w
=
a,
=
0,
I)]9 x 2
Clamped plate
z=+-:-2
591
simply supported
0
6, = 0.
7xy =
0,
w
=
0
by =
7xy =
0,
w
=
0
0,
h 7,,
=
0,
7yz =
0,
&;
7xr =
0,
FINITE DIFFERENCE REPRESENTATION
G2 Due to the symmetry of the rectangular plate, only a quarter of the plate need be considered. In the application of finite difference, the thick plate is divided into rectangular mesh sizes connected at their nodes, resulting in a system of simultaneous equations defining the deflection of the plates. Here, due to space limitations, only the formulations of two typical nodes are shown, namely, a general point and the central point inside the thick plate. Detailed formulations of other nodes can be found elsewhere [I].
= [U(Z - 1) - 2U(Z) + U(Z + 1)] $ 1
(1 + 2G) $
= [ V(Z - ZJ) - 2 l’(Z)
+ V(z+zJ-)]~ Y G$=(Y(Z-ZK)-2V(Z)
+ V(Z + ZK)] ; i
General point inside the plate (A + G) 2 (A +2G)$=[U(Z-
= [W(Z) - W(Z + ZJ) - W(Z + JK)
l)-2U(Z) + W(Z + ZJ + JK)]
+ U(Z + I)]?
(A+G)
n,~,
x G$
= [U(Z - ZJ) - 2U(Z) + cJ(Z + ZJ)] $
(A+G)g=[U(Z)-U(lfI)-U(ZfZK)
+ U(Z + ZK + l)] v Y
x 1
S. F. NC and N.
592
(2 + G)
$ =[V(Z) -
G$=[2V(Z-I)-2y(I)];
?‘(I + ZJ) - v(z + JK)
-I- V(Z + ZJ) + JK)]
(A+@ 11
Y
C$=[W(Z-
BENCHARIF
I[ (A + G) $
= [2V(Z - ZJ) - 2 V(Z)] y
2
l)-2w(Z)
G$=[V(Z-ZK)-2V(Z)
+ W(I + l)] ;
+ V(Z + ZK)] ;
X
I
G$=[W(Z-Z.Z)-2H’(Z)
(A + G) -j$
= [W(Z) - W(Z - ZJ) - U(Z + JK)
+ W(Z + Z.Z)];
+ U(Z-ZJ+JK)]T (1+2G)$=[W(Z-ZK)-2W(Z)
+
Y 2 (1 + G) -$
W(ZfZK)]q2
- U(Z + ZK)
Central point inside the plate (A.+2G)$=[211(1-
= [U(Z) - U(Z - 1)
+ U(Z+ZK-
I)-2u(Z)]y
l)]F XL
I (A + G) &
G$=[2U(Z-ZJ)-2U(Z)];
= [V(Z) - V(Z - ZJ)
> G$
- ?‘(I + JK)
= [U(Z - ZK) - 2U(Z)
(2 +G) + V(Z - ZJ + JK)] 11 y *
+ U(Z + ZK)] ; ‘
G+G)&=[V(i) -
G$=[ZW(Z-I)-2W(Z)]; x
V(Z - 1) G$=[2W(Z-ZJ)-2W(Z);
- V(Z - ZJ) +
Y(Z-ZJ-Z)]W x ?
(1 + G)
& =[W(Z) - W(Z - 1)
(n+G) - 2 W(Z) + W(Z + ZK)] 12 1
- W(Z + ZK) COMPARISON
+
W(Z+ZK-Z)]q) I
=[U(Z) (i+G)-&
OF RESULTS
General
:
U(Z - 1)
- U(Z - ZJ) + U(Z-ZJ-
AND DISCUSSION
I)]9 x .L
The formulation outlined above has been applied to both clamped and simply supported thick plates. A complete study of the effects of various parameters has been carried out and extensive results obtained [l]. In general, both clamped and simply supported thick plates have been analysed for the same plate aspect ratios and the same type of mesh sizes, however the number of nodes used has been larger for
593
Design and analysis of thick rectangular plates
Table 1. Variation of the maximum centre deflection of clamped and simply supported rectangular thick plates with number of nodes used in the finite difference analysis? 0
hla
1.0
1.2
1.5
1.8
2.0
20
Nodes used for clamped plates 45 80
Nodes used for simply supported plates 40 75 120 175
125
2.1118 0.7155
47.0804 5.8572 2.1218 0.7187
47.1543 5.8662 2.1249 0.6790
47.1861 5.8701 2.1263 0.7207
10.4510 1.2950 0.4671 0.1572
31.4175 3.9050 1.4130 0.4775
31.5698 3.9235 1.4194 0.4795
31.6191 3.9295 1.4215 0.4611
31.6409 3.9321 1.3412 0.4803
5.6410 0.6976 0.2509 0.0838
5.4300 0.6713 0.2410 0.0748
17.5889 2.1822 0.7878 0.2650
17.6658 2.1914 0.7909 0.2488
17.6922 2.1945 0.7540 0.266 1
17.7042 2.1960 0.7925 0.2662
3.2490 0.4008 0.1438 0.0479
3.0168 0.3719 0.1331 0.0442
2.9060 0.3580 0.1286 0.0425
10.2358 1.2671 0.4561 0.1526
10.2707 1.2712 0.4575 0.1529
10.2840 1.2727 0.4579 0.1529
10.2903 1.2484 0.4582 0.1529
2.2050 0.2714 0.097 1 0.0323
2.0458 0.2516 0.0900 0.0295
1.9700 0.2427 0.0865 0.0285
7.3074 0.9030 0.3244 0.1081
7.3256 0.905 1 0.3163 0.1080
7.3332 0.8701 0.3253 0.1081
7.3340 0.9064 0.3254 0.1080
0.05 0.10 0.14 0.20
20.9200
2.6010 0.9412 0.3182
17.8300 2.2150 0.8010 0.2703
16.5541 2.0555 0.7428 0.2505
15.9200 1.9760 0.7139 0.2417
46.8489 5.8290
0.05 0.10 0.14 0.20
13.7400 1.7060 0.6165 0.2079
11.7000 1.4520 0.5240 0.1763
10.8649 1.3471 0.4860 0.1637
0.05 0.10 0.14 0.20
7.1590 0.8869 0.3197 0.1072
6.0720 0.7514 0.2705 0.0904
0.05 0.10 0.14 0.20
3.8540 0.4762 0.1711 0.0571
0.05 0.10 0.14 0.20
2.6280 0.3241 0.1162 0.0386
r Values given as w/q, for v = 0.30.
the analysis of the simply supported plates in order to satisfy the boundary conditions at the edges. Convergence and comparison of results
In general, the convergence characteristics for simply supported and clamped thick plates are slightly different. In order to improve the convergence and the accuracy of the method, four mesh sizes were
investigated for each case. Typical convergence results for different plate aspect ratios, a/b, are shown in Table 1. Extensive deflection results for centre deflections of various Poisson ratios. For the clamped plate, the convergence of the results for different mesh sizes is shown in Figs 3-5 and, in the case of simply supported plates, the convergence of the results due to different mesh sizes is shown in Figs 7-9. 25.0
I
pTq7q 0
rPOISSON’S
r
RATIO 0.q
0
(1 1 POISSON’S
7.0.0 -
RATIO O.Ztj
P
0 = 20 Nodes 0 = 45 Nodes A = 50 Nodes
RATlO AXIS (va)
Fig. 3. Comparison of the deflection of clamped thick plates for four mesh sizes.
Fig. 4. Comparison of the deflection of clamped thick plates for four mesh sizes.
S. F. No and N.
BENCIWFUF
,jRAflOo/b
/-
1POISSON'S RATIO O.iq
,-
D= 0= A= +=
20 Nodes 45 Nodes 80 Nodes t25 Nodes
a = Q= b= + I:
40 Nodes 75 kodrs 120 Nod*s i75 Nodes -I
“.I30
“.O>
Q.20
RATIO t%
(h/b)
0.m
0.25
0.10
Fig. 5. Comparison of the deflection of clamped thick plates for four mesh sizes.
Fig. 7. Comparison of the deflection of simply supported thick plates for four mesh sizes.
Figures 6 and 10 show the influence of the plate aspect ratio a/b on plate displacements for both clamped and simply supported thick plates. Also, in order to show that the modified finite difference approach is a good numerical scheme for the design and analysis of thick rectangular plates, a few examples of rectangular thick plates were analysed and results were compared with those obtained by other investigators. Figures 11 and 12 show the
excellent agreement between the results obtained by present method and those obtained by other researchers. CONCLUSIONS
A finite difference method for computing the displacements of thick rectangular plates has been presented. Equilibrium equations in terms of displacements have been derived, The numerical
r 1pinmE-l L’OISSON’S RATIO 0.3 1
i \ \ \ \ D =
0= 1) = += x=
RATtO q,‘b=l.O RATIO qb1.2 RATE0 o/%=1.5 RATIO 4~1.8 RATIO a/b&O
Fig. 6. Influence of the plate aspect ratio on the deflection of clamped thick plates.
\
l
0 = 40 Nodes
75 Nodes h = l20 Nodes + = i75 Nodes 0 =
\
Fig. 8. Comparison of the deflection of simply supported thick plates for four mesh sizes.
Design and analysis of thick rectangular plates
595
IRATlOa/brl [POISSON’S
L7micqy
RATIO 0.V.j
IPOISSON’S
RATIO 0.1
\ \
q = 40 Nodes Q = 75 Nodes d = 120 Nodes + = 175 Nodes
O.Ot
0.30
l
0.0 10
0.01
RATIO AXIS (h/a)
0.10
0.11
0.20
0.21
0.30
RATIO AXIS (h/a)
Fig. 9. Comparison of the deflection of simply supported thick plates for four mesh sizes.
Fig. 11. Comparison of the deflection of clamped thick plates with other investigators.
examples indicate that the method presented herein yields excellent results. Even with a crude mesh size of 4 x 4, the method yields results of acceptable accuracy for both the clamped and simply supported thick plates. In general, the agreement between the finite difference results and those obtained by other methods is well shown in Figs 11 and 12. As a result of this investigation the following conclusions may be drawn.
(1) The modified finite difference method presented in this paper yields good convergence characteristics. (2) For square thick plates, deflection results are in excellent agreement with those obtained by other investigators [2-51. (3) General results for the deflection of thick rectangular plates are meagre, and as a result data obtained in this investigation for thick
Iptnicqy
tTlmiq 1 POISSON’S
RATIO 0.3
)
1 POISSON’S
RATIO 0.31
aAn a/a=10 0 = RATIO a/b=12
0.00
0.01
0.10
0.11
0 =
0 = Reistner
0 = RATIO a/b=lJ + = RATIO a/b=13 x = RATIO a/b=2.0
0 A + X
0.20
0.2,
0.30
RATIO AXIS (h/a)
Fig. 10. Influence of the plate aspect ratio on the deflection of simply supported thick plates.
Y.“V
0.0,
0.10
0.1,
0.20
= = = =
1 Reistne Srinlvos Q MIF[5] Present Study
0.35
,
0
RATIO AXIS (h/a)
Fig. 12. Comparison of the deflection of simply supported thick plates with other investigators.
596
S. F. NC and N. BENCHARIF
rectangular plates cannot as yet be verified. It is believed, however, that those results for square thick plates compare well with those of other investigators. (4) The method is simple to apply and accurate results are obtained even with a relatively crude mesh size. (5) With the detailed computer program which can automatically generate the system of simultaneous equations [l] and the facility of flexibility of input, the designer can easily input various loadings, plate aspects and boundary conditions to obtain accurate results for the bending of all types of simply supported and clamped rectangular thick plates.
REFERENCES 1. N. Bencharif, Application of finite difference method to the deflection of clamped and simply supported thick rectangular plates. M.A.Sc. thesis, Ottawa University, Ontario, (1988). 2. S. Srinivas, A. K. Rao and C. V. Joga Rao, Flexure of simply supported thick homogeneous and laminated rectangular plates. Z. angew. Math. Mech. 49,449-458 (1969).
3. S. Srinivas and A. K. Rao, Flexure of thick rectangular plates. J. appl. Mech. 40, 298-299 (1973). 4. S. Desmukh’ Rajjabau and R. Archer, Numerical solutions of moderately thick plates. J. Engng Mech. Div. AXE. lOON, 903-917 (1974). 5. K. T. Sandara Raja Iyengar, K. Chandrashekhara and V. K. Sebastian, On the analysis of thick rectangular plates. Ing. Arch. 43, 317-330 (1974).