FLEXURAL ANALYSIS OF SIMPLY SUPPORTED RECTANGULAR
MINDLIN PLATES UNDER BISINUSOIDAL TRANSVERSE LOAD
Clifford Ugochukwu Nwoji
1, Benjamin Okwudili Mama
1, Hyginus Nwankwo Onah
1and
Charles Chinwuba Ike
21
Department of Civil Engineering University of Nigeria, Nsukka, Enugu State, Nigeria
2Department of Civil Engineering Enugu State University of Science & Technology, Enugu State, Nigeria
E-Mail: [email protected]
ABSTRACT
This study considered the solution of the system of coupled partial differential equations of equilibrium of rectangular Mindlin plates subjected to bisinusoidal transverse load over the entire plate domain. The method of undetermined parameters was used. Solutions were assumed in terms of unknown parameters for the unknown displacements in the governing equation that apriori satisfied all the simply supported boundary conditions on the plate edges. The unknown parameters were obtained by solving a system of algebraic equations. The in plane displacement, transverse displacement, inplane stresses and transverse stresses were obtained from the determined parameters using the equations of Mindlin plate theory. Comparison of the results of this study with the literature results shows agreement with the previous work of Reddy and Pagano. It was also found that for thin plates, the results agree remarkably well with classical plate theory.
Keywords: mindlin plate theory, classical plate theory, method of undetermined parameters, bisinusoidal transverse load. 1. INTRODUCTION
Plate problems are encountered frequently in engineering applications like bridge decks, floor slabs, retaining walls, raft foundations, aircraft and spacecraft panels. In general, plates are classified into three major categories - thin plates, moderately thick plates and thick plates - depending upon the ratio of thickness to the governing in plane dimension of the plate. They can be further categorized based on the nature of the deformation as small-deformation plates or large deformation plates; and the material properties as anisotropic or isotropic, homogeneous or non-homogeneous [1, 2]. The plate problem in general, is a three dimensional elasticity problem that is solved by solving the differential equations of elasticity theory [3]. Solutions are impossible even in most simple problems of plate geometry and loading. Hence, simplifications are made, yielding the various theories or models of plates. Kirchhoff developed the first complete theory of plate flexure. Based on Bernoulli’s hypothesis, for thin beams, Kirchhoff derived the same partial differential equation of equilibrium for plate bending as obtained by Navier, however using a different energy method [4, 5, 6].
Kirchhoff plate theory also called the classical plate theory (CPT) or the Kirchhoff - Love plate theory is a linear infinitesimal theory. The assumptions of the CPT are [7]:
a) points on the plate lying initially on a normal to the plate middle surface remain on the normal to the middle surface even after bending deformation.
b) the normal stresses in the transverse direction can be
neglected i.e.
zz0,
xz yz0
c) the middle surface of the plate is free of deformation, and remains neutral during bending.
Flexure and stretching which are the two dominant behaviours of plates are uncoupled in the CPT, but can be analysed together by linear addition [8]. This is a merit of the CPT. Another merit of the CPT is that the resulting governing equation which is a fourth order partial differential equation is linear and has only one unknown; the transverse displacement w(x, y). Despite the merits of the CPT, it has the following limitations:
a) CPT is applicable to thin plates with small deflections where the shear stress can be disregarded. For moderately thick plates, shear stress contribution must be accounted for and the CPT becomes inadequate. Also, for large deflections, inplane stresses need to be incorporated yielding non-linear equations
b) Solutions for transverse shear stresses are inconsistent with the theory of elasticity results
c) CPT presents complications in the application to finite element analysis and other computer applications [4].
1.1 Research aim and objectives
The research aim is the analysis of simply supported rectangular Mindlin plates under bisinusoidal transverse loads. The objectives are:
b) to obtain maximum values of inplane displacements
and stresses
xx,
xz,
xy for various ratios of platethickness to least inplane dimension for square Mindlin plates.
2. THEORETICAL FRAMEWORK
2.1 Governing equations of the Mindlin plate theory
The governing partial differential equations of equilibrium of rectangular Mindlin plates under transverse distributed loads q(x, y) are the following system of three coupled partial differential equations in terms of three unknown variables w(x, y),
x( , )
x y
and
y( , ).
x y
[9]
2 x y
Q
q
w
x
y
D
(1)2
2 2
2 2
1 1
0
2 2
y Q
x x
x
D w
x y D x
x y
(2)
2 2 2
2 2
1 1
0
2 2
y y x Q
y
D w
x y D y
y x
(3)
where w(x, y, z = 0) = w(x, y) is the transverse deflection of the plate,
x and
y are the rotations about the x andy axes of lines normal to the middle plane before deformation,
is the Poisson’s ratio of the plate material,D is the plate flexural rigidity, DQ is the shearing stiffness of the plate , x, and y are the in-plane Cartesian coordinate variables, z is the transverse Cartesian coordinate variable
and
2 is the Laplacian in two dimensions, given by2 2
2
2 2
x
y
(4)The plate flexural rigidity, D is given by:
3 2
12(1
)
Eh
D
(5)where E is the Young’s modulus of elasticity, h
is the plate thickness
The shearing stiffness of the plate is given by:
DQ = kGh (6)
where k is the shear correction factor, and G is the shear modulus.
The internal bending forces are given by:
y x
xx
M
D
x
y
(7)
y x
yy
M
D
y
x
(8)3. RESULTS
3.1 Solution to the governing equations for Bisinusoidal transverse loads
A rectangular Mindlin plate lying on the domain
0
x
a
,
0
y
b
,
h2
z
h2 as shown in Figure-1 is considered in this study.Figure-1. Rectangular Mindlin plate.
The plate is subjected to the action of bisinusoidal transverse load q(x, y) given generally by the Fourier double Sine series:
1 1
( , )
mnsin
sin
m nm x
n y
q x y
q
a
b
(9)where qmn are the Fourier coefficients of the load distribution.
By the Fourier series theory,
0 0
4
( , )sin
sin
a b
mn
m x
n y
q
q x y
dxdy
ab
a
b
(10)Figure-2. Distribution of bisinusoidal load on the plate domain.
The problem is then to solve the system of Equations (1-3) for the case of distributed transverse load given by Equation (9) for simply supported edges x = 0, x
= a, and y = 0, y = b. The boundary conditions are:
(
0, )
0
xx
M
x
y
(11)(
0, )
0
y
x
y
(12)(
0, )
0
w x
y
(13)(
, )
0
xx
M
x
a y
(14)(
, )
0
y
x
a y
(15)(
, )
0
w x
a y
(16)( ,
0)
0
yy
M
x y
(17)( ,
0)
0
x
x y
(18)( ,
0)
0
w x y
(19)( ,
)
0
yy
M
x y
b
(20)( ,
)
0
x
x y
b
(21)( ,
)
0
w x y
b
(22)The deflection functions that satisfy all the boundary conditions Equations (11-22) at the edges x = 0,
x = a, y = 0, y = b are given in terms of unknown generalized displacement parameters, and the shape functions, as:
1 1
( , )
mnsin
sin
m n
m x
n y
w x y
w
a
b
(23)1 1
( , )
cos
sin
x mn
m n
m x
n y
x y
A
a
b
(24)1 1
( , )
sin
cos
y mn
m n
m x
n y
x y
B
a
b
(25)where wmn, Amn, and Bmn are the unknown generalized displacement parameters that are sought in the method of undetermined parameters and m, and n are integers.
m= 1, 2, 3, …
n= 1, 2, 3, …
Applying the method of undetermined parameters, Equation (23-25) and (9) are substituted into Equations (1-3) to obtain conditions for the unknown generalized parameters to be solutions to the problem. Thus,
2
1 1 1 1
sin sin cos sin
mn mn
m n m n
m x n y m x n y
w A
a b x a b
1 1
sin
cos
mnm n
m x
n y
B
y
a
b
1 1
1
sin
sin
mn Q m n
m x
n y
q
D
a
b
(26)2 2
2 2
1 1 1 1
1
cos sin cos sin
2
mn mn
m n m n
m x n y m x n y
A A
a b a b
x y
2
1 1 1 1
1
sin cos cos sin 2
Q
mn mn
m n m n
D
m x n y m x n y
B A
x y a b D a b
1 1
sin
sin
0
mn m n
m x
n y
w
x
a
b
(27)2 2
2 2
1 1 1 1
1
sin cos sin cos
2
mn mn
m n m n
m x n y m x n y
B B
a b a b
y x
2
1 1 1 1
1
cos sin sin cos 2
Q
mn mn
m n m n
D
m x n y m x n y
A B
x y a b D a b
sin
sin
0
mn
m x
n y
w
Simplification of Equations (26-28) yield:
2 2
1 1
sin sin
mn mn mn
m n
m n m n m x n y
w A B
a b a b a b
1 11
sin
sin
mn Q m nm x
n y
q
D
a
b
(29)2 2 1 1
1
2
Q mn m nD
m
n
A
a
b
D
1cos sin 0
2
Q
mn mn
D
m n m m x n y
B w
a b a D a b
(30) 2 2 1 1
1
2
Q mn m nD
n
m
B
b
a
D
1sin cos 0 2
Q
mn mn
D
m n n m x n y
A w
a b b D a b
(31)
From Equations (29-31), we obtain the system of three algebraic equations:
2 2
mn
mn mn mn
Q
q
m n m n
w A B
a b a b D
(32)
2 2 1 1
0
2 2
Q Q
mn mn mn
D D
m m n m n
w A B
a D a b D a b
(33) 2 2 1 1 0 2 2 Q Q
mn mn mn
D D
n m n n m
w A B
b D a b b a D
(34)
Solving Equations (32) – (34), we obtain:
2 2
2
2 2
1
Qmn mn
D
m
n
D
a
b
w
q
m
n
D
a
b
(35) 2 2 2 mn mnm
q
a
A
m
n
D
a
b
(36) 2 2 2 mn mnn
q
b
B
m
n
D
a
b
(37)The transverse deflections are then obtained as follows:
2 2
2
2 2
1
( , ) sin sin
Q
mn m n
D m n
D a b m x n y
w x y q
a b m n D a b
(38)2 2 2
2 2
sin sin sin sin ( , )
mn mn
m n m n
Q
m x n y m x n y
q q
a b a b
w x y
m n
m n D
D a b
a b
(39)( , )
K( , )
Q( , )
w x y
w
x y
w
x y
(40)where 2 2 2
sin
sin
( , )
mn K m nm x
n y
q
a
b
w
x y
m
n
D
a
b
(41)2 2
sin
sin
( , )
mn Q m n Qm x
n y
q
a
b
w
x y
m
n
D
a
b
(42)wK(x, y) is the Kirchhoff thin plate solution for deflection of plates under bisinusoidal load and wQ(x, y) is the additional deflection due to the effect of shear deformation.
Also, the rotations are:
2 2 2
cos
sin
mn x m nm
m x
n y
q
a
a
b
m
n
D
a
b
2 2 2
sin
cos
mn y m nn
m x
n y
q
b
a
b
m
n
D
a
b
(44)3.2 Bending moments
Using Equations (7) and (8), the bending moments are found to be:
2 2 2
sin
sin
mn xx m nm
m
m x
n y
q
a
a
a
b
M
D
m
n
D
a
b
2 2 2sin
sin
mnn
n
m x
n y
q
b
b
a
b
m
n
D
a
b
(45) 2 2 2 2 2 sin sin mn xx m nm n m x n y
q
a b a b
M m n a b
(46)2 2 2 2 2 sin sin mn yy m n
n m m x n y
q
b a a b
M m n a b
(47)3.3 Case of load
q x y
( , )
q
0sin
x
sin
y
a
b
For this case, m = 1, n = 1 and
2 2 2 2 0 2 2 2 1
( , ) Q sin sin
m n
D
D a b x y
w x y q
a b D a b
(48)4 2
0 0
4 2 2 2 2
sin sin sin sin
( , )
( 1) Q ( 1)
m n
y y
x x
q b q b
a b a b
w x y
D
D r r
(49)where r is the plate aspect ratio
b
r
a
(50)For square Mindlin plates, r = 1, at the center, x =
a/2, y = b/2 and the center deflection is:
4 2 0 0 4 2
1
4
2
c Qq b
q b
w
D
D
(51)4 0
4 2 2
1
4
2
c
Q
q b
D
w
D
D
b
(52)2
6(1
)
Q
D
h
D
k
(53)4 2
0
4 2 2
1
4
6(1
) 2
c
q b
h
w
D
k
b
(54)2 4
0
4 2
1
1
4
12
(1
)
c
q b
h
w
D
b
k
(55)For k = 5/6,
0.30
2 40
4 2 5
6
1
1
4
12
(1
0.3)
c
q b
h
w
D
b
(56)2 4 0 4 2
1
1
4
7
cq b
h
w
D
b
(57)For
1
4
h
b
4 0 4 21
1
1
16
4
7
cq b
w
D
(58)4 0 4 2
1
1
4
112
cq b
w
D
(59)4
0
1
1
389.636
1105.396
cq b
w
D
(60)4 4
0
0.0025665
0389.636
K
q b
q b
w
D
D
(61)4 4
4 0
0.0034712
c
q b
w
D
(63)The results of center deflection of Mindlin plate and Kirchhoff plate for values of h/b ranging from 120 to
1
3 are shown in Table-1.
Table-1. Center deflection of Mindlin plate and Kirchhoff plate under bisinusoidal load q q0sin xsin y
a b
.
h
b
4 0
q b
D
wK
4 0
q b
D
ws
4 0
q b
D
wm
% Difference
1
3
0.0025665 0.0016083 0.0041748 62.661
4
0.0025665 0.0090465 0.0034712 35.251
10
0.0025665 0.0001447 0.00271124 5.641
20
0.0025665 0.000036186 0.002602686 0.143.4 Alternative way of presentation of results
The deflection at the plate center is:
2 4
2 0
3 4 2
1
1
12(1
)
4
12
(1
)
c
q b
h
w
b
Eh
k
(64)
2
2 4 2 4
0
4 4 2 4
12(1 ) 12(1 )
4 12 (1 )
c
q h b h b
w
E h b k h
(65)
4 2 2
0
4 2
3(1
)
1
c
q h
b
h
w
h
E
b
k
(66)4 2 2
0
4 2
3(1
)
1
c
q h b
h
w
E
h
b
k
(67)For
0.3
4 2
0
4 5 2
6
2.73
1.3
cq h b
h
w
E
h
b
(68)4 2
0
4 2
2.73
1.56
cq h b
h
w
E
h
b
(69)For
1 ,
4
h
b
4 2
0
4 2
1
2.73
1.56 1
4
4
c
q h
w
E
(70)
0
1
0.028026131
0.009818815
256
c
q h
w
E
(71)4
2 0
3.7905 10
c
q h h
w
E
b
(72)4 2
0 0.037904946 3.7905 10 0
256
c
q h q h h
w
E E b
(73)
4
1.48066 10
c
qh
w
E
(74)For
1
10
h
b
4 2
0
4 2
1
2.73
1.56
1
10
10
c
q h
w
E
(75)
4 0
0.028026131
0.00158061
c
q h h
w
E
b
(76)4
2 0
2.9606742 10
c
q h h
w
E
b
(77)2
2
xh
h
u
0 2
2 2
2
2
h
q
h
a
u
D
a
b
(79)
0 0
2 2
2
4 4
2 2 2
2 2
2 1 1
1
h h
q q
h a a
u
r
D D
a b b b
(80)
0 0
2 2 2
2 4
4
4 2
(2 )
2
2
1
(1
)
h
h
q
q
h
a
a
u
r
r
D
D
b
b
(81)
4 0
4 2 2
2
2
(1
)
h
q
b
h
a
u
D
r
(82)4 2
0
4 2 2 3
12(1
)
2
2(1
)
q
hb
h
u
r
a
Eh
(83)2 3
0
2 2 3 2
6(1
)
2
(1
)
q
h
b r
u
E
r
h
(84)2 3
0
2 2 3 3
6(1
)
2
(1
)
q
h
h b r
u
E
r
h
(85)For
0.3,
r = 13 0
3 3
5.46
2
4
q h
h
b
u
E
h
(86)3 0
0.0440233
2
q h
h
b
u
E
h
(87)3
2 0
4.40233 10
2
q h
h
b
u
E
h
(88)The maximum stresses at z = 0, and
z
h2are calculated using
2
1
y x
xx
Ez
x
y
(89)2
1
y x
yy
Ez
y
x
(90)2(1
)
y x xy
Ez
y
x
(91)2(1
)
zx x x
w
E
w
G
x
x
(92)Table-2. Inplane displacement
u x
0,
y
b2,
z
h2
,
transverse displacementw x
a2,
y
b2,
z
0 ,
in-plane normal stress
a2,
b2,
h2
xx
x
y
z
and transverse shear stress
0,
b2,
0
zx
x
y
z
in isotropic square plate subjected to bisinusoidal load.
h
a
Plate theory used Plate model usedu
(q0h/E)
w
(q0h/E) xx
xy (q0)
zx (q0)1
10
Mindlin (present
study) FSDPT 0.04402 2.9340 0.1970 0.1060 0.1690 Reddy [7] HSDPT 0.044 2.9610 0.1990 0.1070 0.2380
Pagano [8]) Elasticity theory 0.0443 2.9425 0.1988 - 0.2383
Kirchhoff CPT 0.044 2.802 0.197 0.106 -
1
4
Mindlin (present
study) FSDPT 0.04402 3.6260 0.197 0.1060 0.1590 Reddy [7] HSDPT 0.0460 3.7870 0.2090 0.1120 0.2370
Pagano [8] Elasticity theory 0.0454 3.663 0.2040 - 0.2361
Kirchhoff CPT 0.0440 2.8030 0.197 0.1060 -
4. DISCUSSIONS
In this work, the problem of flexural analysis of rectangular moderately thick plates modeled using the Mindlin plate theory, under transverse bisinusoidal load distribution on the entire plate domain
(0
x
a
, 0
y
b
)
has been solved for the case of simply supported edges (x = 0, x = a, y = 0, y = b). The method of undetermined parameters was used to solve the system of three coupled partial differential equations in terms of the three unknown displacements w(x, y),( , )
x
x y
and
y( , ).
x y
Trigonometric basis (coordinate) functions that satisfy apriori all the boundary conditions of the simply supported edges were used in terms of undetermined parameters wmn, Amn and Bmn. The basis (coordinate) functions were constructed to satisfy all the boundary conditions on the simply supported plate edges, and the functions represented by Equations (23) - (25) became the trial functions of the problem for which the solutions for the unknown parameters wmn, Amn and Bmn were sought by substitution of the equations into the governing PDE of the Mindlin plate in order to find the conditions for which the unknown parameters would yield a solution. A system of three algebraic equations were then obtained as Equations (32) - (34). Solution of the system of algebraic equations yielded the unknown displacement parameters as Equations (35) - (37). The trial displacement fields or displacement functions were then completely determined as Equations (39), (43) and (44) for the general case of arbitrary distributed load. The solution was then applied to the particular case of bisinusoidal transverse load to obtain the expression for the center deflection as Equation (55) or Equation (57) for the case when the shear correction factor,k
56 and the Poisson’s ratio,
0.30.
Similarly, the expression for the inplane displacement u atz
h2 was obtained as Equation (85). Normal and shear stresses were calculatedusing Equations (89) - (92) and their maximum values were shown tabulated in Table-2 for the ratios of
0.1
ha
and ha
0.25.
The variation of the center deflection of the plate with the plate thickness to governing span ratio (h/a) for square plates was computed and shown in Table 1; which also displays corresponding results for the classical Kirchhoff-Love plate theory. Table1 shows that for
1
20
h
a
which falls within thecategorization of thin plate, the relative difference in the results of the Mindlin plate theory of the present work and the classical Kirchhoff plate theory is a negligible 0.14%.
For
1
10
h
a
the Mindlin plate center deflectionexceeds the center deflection predicted by the classical
thin plate theory of Kirchhoff by 5.64%. For
1 ,
3
h
a
the center deflection of the Mindlin plate is greater than that of the Kirchhoff plate by 62.66%; illustrating the significant role of shear deformation as the ratio h/a
increases and the plate becomes thicker. Table-2 also compares the results of the present study with earlier studies done by Reddy, and Pagano. The results displayed in Table-2 shows that the present work agrees with the results obtained previously by Reddy who used a Higher Order shear deformation plate theory (HSDPT) and by Pagano who used the method of the theory of elasticity.
5. CONCLUSIONS
The conclusions of this study are as follows:
satisfy all the boundary conditions at the simply supported edges.
b) the method simplifies the computation of the boundary value problem of solving a system of three coupled partial differential equations to an algebraic problem of simultaneous equations in terms of the undetermined displacement parameters.
c) the classical Kirchhoff plate theory which disregards the contributions of shear deformation under predicts the center deflection when the thickness-governing span ratio (h/a) exceeds 1/20; and for such cases is not a suitable theory for moderately thick plates.
d) the results for the transverse deflection at the center of the plate agree remarkably well with the results of the same problem solved by Reddy using the Higher order shear deformation plate theory and by Pagano using theory of elasticity.
e) for
1
,
20
h
a
the transverse deflection of the platecenter for Mindlin plate theory become significantly similar to the results obtained by Kirchhoff plate theory.
REFERENCES
[1] Ghugal Y.M. and Sayyad A.S. 2010. Free Vibration of Thick Orthotropic Plates using Trigonometric Shear Deformation Theory. Latin American Journal of Solids and Structures. 8: 229-243.
[2] Ghugal Y.M. and Pawar M.D. 2011. Buckling and Vibration of Plates by Hyperbolic Shear Deformation Theory. Journal of Aerospace Engineering and Technology. 1(1): 1-12.
[3] Chantarawichit P., Kongtong P., Sompornjaroensuk Y., Vibootjak J. 2015. Note on the Mathematical Development of Plate Theories. Advanced Studies in Theoretical Physics. 9(1): 47-55, Hikari Ltd.
[4] Kumar S.K. 2013. Analysis of Composite Plates using Element Free Galerkin Method. Master of Technology (Structural Engineering) Thesis, National Institute of Technology Bourkela.
[5] Hosseini S.B. 2016. Implementation and Analysis of Error Estimates for hp MITC Finite Elements of Reissner-Mindlin Plates. Master of Science in Technology Thesis, School of Engineering Aalto
[6] Onarte E. 2013. Structural Analysis with the Finite Element Method. Springer, Netherlands.
[7] Reddy J.N. 2004. Mechanics of Laminated and Composite Plates and Shell Theory and Analysis, 2nd Edition. CRC Press, Boca Raton F.L.
[8] Pagano N.J. 1970. Exact Solutions for Bi-directional Composite and Sandwich Plates. Journal of Composite Materials. 4: 20-34.
