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Uncovering the finite difference model equivalent to Hencky bar-netmodel for axisymmetric bending of circular and annular plates
H. Zhang , C.M. Wang , N. Challamel , Y.P. Zhang
PII: S0307-904X(18)30204-XDOI: 10.1016/j.apm.2018.04.019Reference: APM 12258
To appear in: Applied Mathematical Modelling
Received date: 16 December 2017Revised date: 6 April 2018Accepted date: 24 April 2018
Please cite this article as: H. Zhang , C.M. Wang , N. Challamel , Y.P. Zhang , Uncovering the finitedifference model equivalent to Hencky bar-net model for axisymmetric bending of circular and annularplates, Applied Mathematical Modelling (2018), doi: 10.1016/j.apm.2018.04.019
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Highlights:
• Energy preserving finite difference model (FDM) is discovered for axisymmetric
bending of circular/annular plates.
• Exact bending solutions for circular plate using Hencky bar-net model (HBM) or
equivalent FDM are obtained for the first time.
• Bending solutions for annular plates which may be tediously obtained from
continuum theory are efficiently obtained via HBM.
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Uncovering the finite difference model equivalent to Hencky bar-net
model for axisymmetric bending of circular and annular plates
H. Zhanga, C.M. Wang
b,*, N. Challamel
c and Y.P. Zhang
d
a,b School of Civil Engineering, The University of Queensland, St Lucia, Queensland 4072,
Australia (*Corresponding author’s e-mail: [email protected])
c Université de Bretagne Sud, IRDL – UBS – Lorient, Institut de Recherche Dupuy de Lôme,
Centre de Recherche, Rue de Saint Maudé – BP 92116, 56321 Lorient cedex – FRANCE
(email: [email protected])
d School of Mechanical and Mining Engineering, The University of Queensland, St Lucia,
Queensland 4072, Australia (email: [email protected])
Abstract
As there are a few finite difference models in the literature for axisymmetric bending of
plates, only one of these models is equivalent to the Hencky bar-net model (HBM) that
comprises a finite number of rigid circular arcs and straight radial segments joined by
frictionless hinges with elastic rotational springs. This paper is concerned with uncovering
the one finite difference model (FDM) that is equivalent to the HBM. Based on the energy
formulation, the governing equation for HBM is derived and it will be used to identify the
FDM that has the same discrete set of equations. By using this equivalency between the HBM
and the identified FDM, the expressions of edge spring stiffnesses of HBM are derived for
various boundary conditions. As illustrative examples, the HBM is used to solve the bending
problems of circular plates under uniformly and linearly increasing distributed loads. The
analytical solutions of HBM avoids the singularity problem faced in FDM at the plate center.
This paper also presents some benchmark bending solutions for annular plates with and
without an internal ring support for different boundary restraints by using the HBM.
Keywords: finite difference; Hencky bar-net; distributed load; axisymmetric bending; circular
plate; annular plate.
1. Introduction
Circular and annular plates are very important structural elements in aerospace, ocean,
mechanical and civil engineering applications [1]. There have been many studies conducted
on bending, buckling and vibration of circular and annular plates using the finite element
method [2, 3], differential quadrature method [4], Ritz method [5, 6], perturbation method
[7], the differential transformation method [8]. However, the finite difference method is
rarely adopted for analyses of circular and annular plates. Researchers such as Atkatsh et al.
[9] and Melersk [10] incorporated energy approach into the finite difference method for
circular plate analyses, which may be regarded as a finite element method. Chakravorty and
Ghosh [11] developed a finite difference method for analyzing a circular plate on semi-
infinite elastic foundation but they divided the entire plate into a series of elements and
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solved the problem in the region of elements instead of focusing on the nodes. Turvey and
Der Avanessian [12] treated the elastic large deflection problem of circular plates using
graded finite-difference method. Dey and Rao [13] obtained the transient response of circular
plates and membranes by discretizing the equations using the finite difference method.
Karamooz-Ravari and Shahidi [14] employed the finite difference method to solve the
axisymmetric buckling problem of circular and annular nanoplates. The most relevant
research work to this study is the paper by Bazaj and Kissel [15] who solved the bi-harmonic
equation of circular plates and sectorial plates using finite difference approximations.
However, the central deflection of the circular plate which is a singularity in FDM was not
directly determined and it had to be approximated by the deflection of the neighbouring point
which is half the nodal spacing from the plate center.
It is widely known that the governing ordinary differential equation for axisymmetric
bending of circular and annular plates could be discretized using the first order central finite
difference method in polar coordinates. However, there are various discretized versions
because one can fully differentiate the differential governing equation and then apply the
finite difference approximation on each derivative [14, 15] or one may choose to approximate
the derivatives by half nodal spacing before fully differentiating the governing equation [16].
Based on previous studies on FDM for beams [17-20], columns [21, 22], circular arches
[23, 24] and rectangular plates [25-27], there is an equivalent physical structural model which
has been named the Hencky bar-chain (for beams and circular arches) or Hencky bar-net (for
rectangular plates). The Hencky bar-chain or Henchy bar-net comprises rigid bar segments or
rigid arcs (for circular arches) connected to each other by frictionless hinges with elastic
rotational springs that allows for the flexibility of the structures. Such Hencky-type models
have also been applied to model pantographic structures that have in-plane large deformation
[28] and to serve as a base for building a second gradient continuum model for pantographic
structures [29]. In this paper, we wish to identify the FDM from a few candidates that will be
equivalent to such a Hencky bar-net model. The energy approach will be used to determine
the discrete governing equation for bending of circular or annular plate based on the HBM.
This resulting discrete equation will be used to uncover the FDM that is equivalent to the
physical HBM.
2. Problem Definition and Layout of Paper
Consider a thin circular plate of radius R, thickness h, Young’s modulus E, Poisson’s ratio υ
and flexural rigidity D = Eh3/[12(1− υ
2)] subjected to an axisymmetric transverse distributed
load q(r) where r is the radial coordinate as shown in Figure 1. The elastic rotational springs
of stiffness Ker are uniformly distributed at the edge. Special case of the elastically restrained
edge is simply supported edge where Ker = 0 and clamped edge where Ker = ∞. The problem
at hand is to determine the FDM which has a physical HBM for the axisymmetric bending of
circular plate.
First, we present various FDMs in polar coordinates for circular plates in Section 3. In
Section 4, we develop the HBM for circular plates and derive the discrete governing equation
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for HBM from the energy approach. By comparing the discrete governing equations, one can
identify the FDM that has the same discrete equation as HBM. In Section 5, we develop the
boundary conditions for the HBM based on the identified FDM. Section 6 demonstrates the
use of HBM for determining the exact bending solutions for circular plates under uniformly
distributed and linearly increasing load with elastically restrained edge, simply supported
edge and clamped edge. The application of HBM for axisymmetric bending of annular plates
is treated in Section 7. Section 8 presents the concluding remarks of this study.
Figure 1 Circular plate under axisymmetric distributed load q
3. FDM for Axisymmetric Bending of Circular Plates
According to the Kirchhoff (or classical thin) plate theory, the bending moment in radial
direction Mr, bending moment in circumferential direction Mθ and shear force Vr for
axisymmetric bending of circular plate are, respectively, given by
2
2r
d w dwM D
dr r dr
(1a)
2
2
1d w dwM D
dr r dr
(1b)
2
2
1r
d d w dwV D
dr dr r dr
(1c)
where w is the transverse deflection. Note that there is no twisting moment in axisymmetric
bending of plates.
The equilibrium equations for a plate element based on shear force balance and bending
moment balance are given by [30]
rd rV
rqdr
(2a)
r
r
d rMrV M
dr (2b)
By combining Eq. (2a) and Eq. (2b), one obtains
2
2
rd rM dMrq
dr dr
(3)
The substitution of Eqs. (1a) and (1b) into Eq. (3) furnishes the classical governing
equation for axisymmetric bending of circular plates
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2 2
2 2
1d d w d dwD r rq
dr dr dr r dr
(4)
In the FDM, Eq. (4) is approximated by the first order central finite difference method.
However, there are different ways to discretize Eq. (4) and two possible methods are
presented herein. The first method (which is mainly adopted by researchers [14, 15]) is to
fully differentiate the equation first and then apply the finite difference method on each
derivative while the second method involves discretizing the derivatives in the parentheses
first and then further discretizing the derivative outside the parentheses. The finite difference
equations obtained from the two methods are given by
2 1 1 2 2 1 1 2
4 3
1 1 1 1
2 2 3
4 6 4 2 22
2
21 1
2
j j j j j j j j j
j
j j j j j
j
j j
w w w w w w w w wD
a r a
w w w w wq
r a r a
for first method (5a)
2 1 1 2 2 1 1 2
4 3
2 2
2 2
1 1
4 6 4 2 22
2
1 1
4 4
j j j j j j j j j
j
j j j j
j
j j j j
w w w w w w w w wD
a r a
w w w wq
r r a r r a
for second method (5b)
where jr aj and a the nodal spacing in the radial direction. Note that FDM and HBM are
only equivalent provided that the nodal spacing a of FDM is equal to the segmental length a
of the HBM. More explanations about the first and second finite difference methods are given
in Appendix A. Note that other ways of finite difference approximations using half nodal
spacing [16] are unlikely candidates for matching the physical HBM.
Similarly, the shear force of FDM can have two forms:
2 1 1 2 1 1 1 1
3 2 2
2 2 21 1
2 2
j j j j j j j j jr
j
j j
w w w w w w w w wV D
a r a r a
for first method
(6a)
2 1 1 2 2 2
3 2 2
1 1
2 2 1 1
2 4 4
j j j j j j j jr
j
j j
w w w w w w w wV D
a r a r a
for second method (6b)
In the next section, we shall identify which one of the two finite difference
approximations matches the physical HBM.
4. HBM for Axisymmetric Bending of Circular Plates
Next, we shall establish a HBM for the axisymmetric bending analysis of circular plate as
shown in Figure 1. Consider a bar-net system which comprises a finite number of rigid
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circular arcs and straight radial segments joined by frictionless hinges with elastic rotational
springs as shown in Figure 2. The stiffness of internal rotational springs in radial direction is
Cr = D/a and in circumferential direction is Cθ = D/s where a = R/n is the length of straight
segments, n is the number of straight segments and s the length of circular arcs which varies
with respect to the joint location. The rotational springs of stiffnesses Cer and Ceθ are
presented for modelling different boundary conditions. Note that the joint number in radial
direction j starts from 0 at the center point to n at the edge.
Referring to Figure 2, the rotational angle of the straight element between joint j−1 and
joint j (i.e. ψj-1) and the angle between joint j and j+1 (i.e. ψj) are given by
1
1
j j
j
w w
a
(7a)
1j j
j
w w
a
(7b)
The curvature in the radial direction r
j and the curvature in the circumferential direction
j
can be expressed as
1 1 1
2
2j j j j jr
j
w w w
a a
(8a)
1 1
2
j j
j
j
w w
r a
(8b)
Therefore, the lumped radial bending moment r
jM and lumped circumferential bending
moment jM at joint j of HBM can be obtained as
1 1 1 1
2
2
2
j j j j jr r
j r j j
j
w w w w wM C a C s D
a r a
(9a)
1 1 1 1
2
2 1
2
j j j j jr
j r j j
j
w w w w wM C a C s D
a r a
(9b)
The Marcus moment (or moment sum) jM at joint j of HBM, defined as the sum of radial
bending moment and circumferential moment, is given by [31]
1 1 1 1
2
2 1
1 2
r
j j j j j j j
j
j
M M w w w w wD
a r a
M (10)
Figure 2 HBM for axisymmetric circular plate
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The governing equation for HBM can be obtained from energy approach. The strain
energy U due to internal deformed rotational springs can be expressed as the summation of
products of bending moments and segmental curvatures, i.e.
1
1
1 12
2 2
nr r
j j j j j
j
U M M r a
(11)
while the work done W by the transverse distributed load acting on the internal joints is given
by
1
1
2n
j j j
j
W q w r a
(12)
Therefore, the total energy function Π is given by
1
1
1 12
2 2
nr r
j j j j j j j
j
U W a M M q w r
(13)
By substituting Eqs. (8) and (9) into Eq. (13), the total energy function can be written as
221 1
1 1 1 1 1 1 1 1
2 21 1
2
2 22 1
2 2 2
n nj j j j j j j j j j j
j j j
j jj j
a
Dr w w w w w w w w w wq w r
a r a a r a
(14)
The variation of the total energy function δΠ is given by
11 1 1 1 1 1 1 1
2 21
11 1 1 1 1 1 1 1
2 21
2 2 1
2 2 2
2 2
2 2
nj j j j j j j j j j
j
j j
nj j j j j j j j j j
j j j
j
w w w w w w w w w wD r
a a a r a a
w w w w w w w w w wq r w
a a a a
(15)
By minimizing the total energy with respect to wj (i.e., δΠ/δwj=0), one obtains the
following discrete governing equation
2 1 1 1 1 2 2 2
1 14 4 4 2 2
1 1
2 1 1 2 2 1 1 2
3 3 3 3 3
2 2 2 1 12
4 4
2 22
2 2 2 2 2
j j j j j j j j j j j j j
j j j
j j
j j j j j j j j j j j j
j j
w w w w w w w w w w w w wD r r r
a a a r a r a
w w w w w w w w w w w wq r
a a a a a
(16)
which can be simplified to
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2 1 1 2 2 1 1 2
4 3
2 2
2 2
1 1
4 6 4 2 2
1 1
4 4
j j j j j j j j j
j
j j j j
j j
j j
w w w w w w w w wD r
a a
w w w wr q
r a r a
(17)
The governing equation (17) for HBM is exactly the same as the governing equation for
FDM (see Eq. (5b) that is discretized by using the second method). Therefore, we can
conclude that the FDM discretized as in the second method has a clear physical structural
model behind it.
5. Edge Spring Stiffnesses of HBM
Based on the identified FDM, we can calibrate the edge spring stiffnesses Cer and Ceθ of
HBM. The analogy between HBM and FDM is based the transverse deflection, bending
moment and characteristic length equivalence given by
H Fw w , H FM M and H Fa a a (18)
where the superscripts H and F stand for quantities belonging to HBM and FDM,
respectively.
For HBM, the bending moments are affected by the rotational springs in both r and θ
directions but the springs in r and θ directions have different influences on Mr and M
θ.
Therefore, we have the bending moments written as
1 1r r n n n nn err n e r n err e r
w w w wM C a C s C C s
a Ra
(19a)
1 1r n n n nn er n e n er e
w w w wM C a C s C C s
a Ra
(19b)
where Cerr represents the portion of rotational spring stiffness Cer in the r direction on Mr
while Cerθ represents the portion of Cer on Mθ; Ceθr represents the portion of rotational spring
stiffness Ceθ in the θ direction on Mr while Ceθθ represents the portion of Ceθ on M
θ.
While for the FDM, the boundary condition for a circular plate with rotational springs of
stiffness Ker is given by
1 1 1 1 1 1
2
2
2 2
r n n n n n n nn er
w w w w w w wM D K
a Ra a
(20)
By eliminating the deflection of fictitious node wn+1 in Eq. (20), the edge radial and
circumferential bending moments for FDM can be expressed as
12
21
r n nrn
r
er er
w wCM
C D a
K RK
(21a)
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2
1 1 1 1 1
2
12
22 21
r r
n n n n n er n nn
r
er er
DC C
w w w w w RK w wDM
C Da aR a
K RK
(21b)
By comparing Eqs. (19a), (19b) and Eqs. (21a), (21b), respectively, the edge spring
stiffnesses of HBM for a circular plate with rotational springs having stiffness Ker located at
the edge are given by
2
21
rerr
r
er er
CC
C D
K RK
, 0e rC (22a)
2
1
rer
r
er er
CC
C D
K RK
, 212
1
re
r er
er er
C DC
C D RK
K RK
(22b)
For some special cases of boundary conditions, the above formulae reduce to the following
forms:
For clamped edge (Ker = ∞), the expressions reduce to Cerr = 2Cr, Ceθr = 0 while Cerθ =
Cr and Ceθθ = 0.
For simply supported edge (Ker = 0), the formulae reduce to Cerr = 0, Ceθr = 0 while
Cerθ = 0 and 21
2
r
er
CC
C R
D
. This indicates that for simply supported edge, there is
no rotational spring in r direction and the spring in θ direction has no effect on the
radial bending moment Mr. This is so because the radial bending moment M
r has to be
zero at the simply supported edge but the circumferential bending moment Mθ is non-
zero.
6. Exact Bending Solutions of Circular Plates based on HBM
Consider a circular plate under a distributed load q. In the following, a second order
difference governing equation for HBM will be derived in view to obtain the analytical
solutions.
By observing the relationship between Marcus moment and bending moments for HBM
given by Eq. (10), r
jM and jM in Eqs. (9a) and (9b) could be expressed as
1 11
2
j jr
j j
j
w wDM
r a
M (23a)
1 1
2
21
j j j
j j
w w wM D
a
M (23b)
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It is interesting to note that Eq. (17) can be expressed with the bending moments given by
1 1 1 1 1 1
2
2
2
r r r
j j j j j j j j
j j
r M r M r M M Mr q
a a
(24)
By substituting Eqs. (23a) and (23b) into Eq. (24), we obtain
1 1 1 1 1 1
2
2
2
j j j j j j j j
j j
r r rr q
a a
M M M M M (25)
Note that the equilibrium equation (25) in terms of Marcus moment jM and external load qj
for HBM applies to both circular plates and annular plates.
6.1 Circular Plate under Uniformly Distributed Load
Consider a circular plate under a uniformly distributed load q = q0 and the edge is elastically
restrained with rotational spring having stiffness Ker. From Eq. (25), jM is readily to be
known as
2
0 0
1
4j jq r M M (26)
where 0M represents the Marcus moment at the center of circular plate (i.e. at j = 0).
By substituting Eq. (10) into Eq. (26), we obtain the following second order governing
difference equation for HBM or FDM:
1 1 1 1 2
0 02
2 1 1
2 4
j j j j j
j
j
w w w w wD q r
a r a
M (27)
The general solution of Eq. (27) solved from Mathematica 11.1 [32] is known to be
2 2 201 2 2 4
024 2
ˆ ˆln 4 1 2 1 1
4 128
ˆˆ 10,
256 8 2
j
qw C C j j j
n n
qC j
n n
M
M (28a)
with 0.5772 and
1'
1 20,
12
2
j
j
j
(28b)
where 4
0ˆ /q q R D ,
2
0 0ˆ /R DM M , γ is the Euler’s constant and ψ(m, z) is the
PolyGamma function of order m which is defined as the (m+1)th derivative of the logarithm
of the Gamma function Γ(z), that is ψ(m, z) = dm+1
lnΓ(z)/dzm+1
. Specifically, ψ(0, z) =
dlnΓ(z)/dz = 1/Γ(z)∙dΓ(z)/dz as indicated in Eq. (28b) where Γ’(z) denotes dΓ(z)/dz.
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For axisymmetric bending of circular plates, the slope at the center point is zero where
w1−w-1 = 0. As a result, one obtains 02 4 2
ˆˆ
256 8
qC
n n
M. Thus, the general solution for
deflection in Eq. (28a) becomes
2 2 20 01 4 2 2 4
ˆ ˆˆ ˆln 4 1 2 1 1
256 8 4 128j
q qw C j j j
n n n n
M M (29)
Based on Eqs. (6b) and (29), the shear force of HBM or FDM is given by
0
2
j
j
q rV (30)
For a circular plate with rotational elastically restrained edge, the boundary conditions
are given by
0nw , 1 1 1 1 1 1
2
2
2 2
n n n n n n ner
w w w w w w wD K
a aR a
(31)
The substitution of general solution (29) into Eq. (31) yields the deflection of circular HBM
with rotational elastically restrained edge as
4 2 2
0
2 2 2
1 251
64 1 1
rrj
r r
Kq R Kj jw
D n K n K n
(32)
where /r erK K R D .
The Marcus moment at the center of the plate with elastic rotationally restrained edge
according to the HBM is
2
00 2
13
8 1 4
rr
r
q R KK
K n
M (33)
The Marcus moment at any joint of HBM based on Eqs. (26), (33) and bending moments
based on Eqs. (23a), (23b), (32) are given by
2 2
0
2 2
3 1 2
8 1 4 1
r rj
r r
q R K K j
K n K n
M (34a)
2 22
0
2 2
2 1 3 3
16 2 1
r rr
j
r
n K K jq RM
n K n
(34b)
2 2 22
0
2 2
2 1 3 1 1 3
16 2 1
r r
j
r
n K K jq RM
n K n
(34c)
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The maximum deflection wmax of HBM with elastic rotationally restrained edge is located
at the plate center, i.e. j = 0,
4
00 2
1 25
64 1 1
rrmax
r r
Kq R Kw w
D K n K
(35)
The maximum radial and circumferential bending moments are also at center point where j =
0,
22
00 2
2 1 3
16 2 1
r rr
r
n K Kq RM
n K
(36a)
2 22
00 2
2 1 3 1
16 2 1
r r
r
n K Kq RM
n K
(36b)
Figure 3 shows the variation of the deflection at center point w0D/(q0R4) of the HBM with
respect to the number of radial segments. The Poisson’s ratio is taken as υ = 0.25 and edge
rotational spring has a stiffness Kr = 10. The deflection converges to the continuum result
from the top. This is because the HBM (or FDM) makes the plate more flexible.
Figure 3 w0D/(q0R4) of HBM with υ = 0.25 and Kr = 10 for different n
Figure 4 shows the variation of w0 with respect to the rotational spring stiffness Kr. The
Poisson ratio is taken as υ = 0.25 and n = 100. It can be seen that the deflection decreases as
Kr increases. This is because the increasing of Kr will lead to a stiffer plate.
Figure 4 Variation of w0D/(q0R4) with respect to Kr for HBM with n = 100 and υ = 0.25
For a circular plate with clamped edge (Kr = ∞), the deflection given by Eq. (32) reduces
to
4 2 2
0
2 2 2
21 1
64j
q R j jw
D n n n
(37)
The Marcus moment at the center point of HBM is given by
2
00 2
11
8 4
q R
n
M (38)
The Marcus moment and bending moments at any joint of HBM are expressed as
2 2
0
2 2
1 21
8 4j
q R j
n n
M (39a)
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2 2
0
2 2
11 3
16 2
r
j
q R jM
n n
(39b)
2 2
0
2 21 1 3
16 2j
q R jM
n n
(39c)
The maximum deflection wmax is at center point where j = 0,
4
00 2
21
64
q Rw
D n
(40)
When n approaches infinity, we obtain the maximum deflection of continuum circular plate
under a uniformly distributed load [33] which is equal to q0R4/(64D). The deflection, radial
and circumferential bending moments given by Eqs. (37), (39b) and (39c), respectively also
converge to the continuum results obtained by Szilard [33] when n approaches infinity.
The maximum radial and circumferential bending moments are also at center point where
j = 0,
2
00 2
11
16 2
r q RM
n
(41a)
2
00 2
116 2
q RM
n
(41b)
Figure 5 shows the deformed shape of the clamped HBM with n = 5 and 30. Note that wj
in z-axis is normalized by w0. It can be seen that when n is larger (i.e., n = 30), the deformed
shape of HBM is smoother and when n is very large, the deformed shape of HBM will be the
same as that of its continuum plate counterpart.
Figure 5 Deflection of clamped HBM with (a) n = 5 and (b) n = 30
For a circular plate with simply supported edge (Kr = 0), the deflection given by Eq. (32)
reduces to
4 2 2
0
2 2 2
5 1 21
64 1 1j
q R j jw
D n n n
(42)
The Marcus moment at the center point of HBM is given by
2
00 2
13
8 1 4
q R
n
M (43)
The Marcus moment and bending moments at any joint of HBM are expressed as
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2 2
0
2 2
3 1 2
8 1 4 1j
q R j
n n
M (44a)
2 2
0
23 1
16
r
j
q R jM
n
(44b)
2 2
0
2 2
13 1 3
16 2j
q R jM
n n
(44c)
Note that when n approaches infinity, the deflection, radial and circumferential bending
moments of HBM given by Eqs. (42), (44b) and (44c) respectively converge to the solutions
for the continuum circular plate [34].
The maximum deflection wmax for simply supported case is also at center point where j =
0,
4
00 2
5 1 2
64 1 1
q Rw
D n
(45)
It is found that the deflection of circular HBM with simply supported edge is dependent on
the Poisson’s ratio υ, which is not observed in the clamped case.
The maximum radial and circumferential bending moments are also located at the center
point where j = 0, i.e.
2
00 3
16
r q RM (46a)
2
00 2
13
16 2
q RM
n
(46b)
6.2 Circular Plate under Linearly Varying Distributed Load
Consider a circular plate under linearly varying distributed load q = q0r/R as shown in Figure
6. The Marcus moment at joint j of HBM may be assumed as a polynomial function given by
3 2
0 0 0
0
j j j
j
q r q r q r
R R R M M (47)
where ξ, ζ, λ are constants.
Figure 6 Circular plate under linearly varying distributed load
By substituting Eq. (47) into Eq. (25), one could obtain the values of ξ, ζ, λ, i.e.
1
9 , 0 ,
2
9
a (48)
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The governing difference equation for HBM or FDM can thus be obtained by substituting
Eq. (10) into Eq. (47) and considering the values of ξ, ζ, λ. The resulting equation is
3 2
1 1 1 1 0 0
02
2 1
2 9 9
j j j j j j j
j
w w w w w q r q r aD
a r a R R
M (49)
The general solution of Eq. (49) solved from Mathematica 11.1 can be finally simplified
to the following form by considering w1−w-1 = 0:
2 2 201 2 4
ˆ ˆln 4 2 2 1 4
8 225j
qw C j j j j
n n
M (50)
Based on Eqs. (6b) and (50), the shear force of HBM or FDM is given by
2
0
3
j
j
q rV
R (51)
For a circular plate with rotational elastically restrained edge, the substitution of general
solution (50) into Eq. (31) gives the deflection of HBM as
24
2 2 2 20
5
12 1 4 3 6 8 1 5 4
450 1j r r r
r
n nq Rw j j j n K K j K
n D K
(52)
The Marcus moment at the center point of HBM is given by
2
0
0 2
2 4 11
45 1
r
r
q R K
K n
M (53)
The Marcus moment at any joint of HBM based on Eqs. (47), (48), (53) and bending
moments based on Eqs. (23a), (23b), (52) are given by
22
0
2 3
5 12 4 11
45 1 2
rj
r
j jq R K
K n n
M (54a)
22
0
2 3
14 11 1 4
45 1
r rj
r
j jq R KM
K n n
(54b)
22
0
2 3
14 11 1 1 4
45 1
rj
r
j jq R KM
K n n
(54c)
The maximum deflection wmax is at the center point where j = 0,
4
00 2 2
8 111 3 6
450 1
r
r
r
Kq Rw K
K D n n
(55)
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When n approaches infinity, we obtain the maximum deflection of continuum circular plate
4
0
0
6
150 1
r
r
K q Rw
K D
subjected to a linearly varying distributed load q = q0r/R.
The maximum radial and circumferential bending moments are also at the center point
where j = 0,
2
00 0 2
4 11 1
45 1
r r
r
q R KM M
K n
(56)
For a circular plate with clamped edge (Kr = ∞), the deflection of HBM based on Eq. (52)
can be expressed as
4
2 2 2 2 20
52 1 4 1 3 8 5
450j
q Rw j j j n n n j
n D
(57)
The Marcus moment at the center point of HBM is given by
2
00 2
2 11
45
q R
n
M (58)
The Marcus moment and bending moments at any joint of clamped HBM are
22
0
2 3
5 12 11
45 2j
j jq R
n n
M (59a)
22
0
2 3
111 1 4
45
r
j
j jq RM
n n
(59b)
22
0
2 3
111 1 1 4
45j
j jq RM
n n
(59c)
The maximum deflection wmax is at the center point where j = 0,
4
00 2 2
1 81 3
450
q Rw
D n n
(60)
When n approaches infinity, we obtain the maximum deflection of the continuum circular
plate w0 = q0R4/(150D) subjected to a linearly varying distributed load q = q0r/R [35].
The maximum radial and circumferential bending moments are also at the center point
where j = 0,
2
00 0 2
11 1
45
r q RM M
n
(61)
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For a circular plate with simply supported edge (Kr = 0), the deflection of HBM based on
Eq. (52) is given by
24
2 2 2 20
5
12 1 4 3 6 8 1 5 4
450 1j
n nq Rw j j j n j
n D
(62)
The Marcus moment at the center point of HBM is given by
2
0
0 2
2 4 11
45 1
q R
n
M (63)
The Marcus moment and bending moments at any joint of HBM are
22
0
2 3
5 12 4 11
45 1 2j
j jq R
n n
M (64a)
22
0
2 3
114 1
45
r
j
j jq RM
n n
(64b)
22
0
2 3
114 1 1 4
45j
j jq RM
n n
(64c)
The maximum deflection wmax is at the center point where j = 0,
4
00 2 2
8 111 3 6
450 1
q Rw
D n n
(65)
When n approaches infinity, we obtain the maximum deflection of continuum circular plate
4
0
0
6
150 1
q Rw
D
subjected to a linearly varying distributed load q = q0r/R [36].
The maximum radial and circumferential bending moments are at the center point where j
= 0,
2
00 0 2
14 1
45
r q RM M
n
(66)
When n approaches infinity, Eq. (66) corresponds to the result of continuum circular plate
obtained by Reddy [36].
7. Solutions for Annular Plates using HBM
Next, the axisymmetric bending problems of thin annular plates under uniformly distributed
load are solved as illustrative examples. Even though the analytical solutions for continuum
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annular plates with different boundary restraints could be derived, the solutions are tedious.
However, one can readily obtain solutions from using the HBM by solving the difference
governing equation (5b) and this method is very efficient.
7.1 Annular Plate
Consider an annular plate of outer radius R, inner radius 0.5R, Poisson’s ratio υ = 0.3
subjected to a uniformly distributed load q = q0 as shown in Figure 7. Table 1 shows the
maximum deflection wmaxD/(q0R4) for HBM (or second FDM) with different n (= R/a) and
boundary conditions where C represents clamped edge, S denotes simply supported edge, F
for free edge (e.g., CS means clamped inner edge and simply supported outer edge, SF means
simply supported inner edge and free outer edge).
As shown in Table 1, HBM (or second FDM) can well predict the deflection of annular
plates provided that the radial segmental number n is large enough.
Figure 7 Annular plate
Table 1 Maximum deflection wmaxD/(q0R4) for HBM (or second FDM) with different n and boundary conditions
Edge
conditions
wmaxD/(q0R4) for HBM (or second FDM)
n = 20 n = 100 n = 1000 Continuum resulta
FC 0.0050 0.0052 0.0053 0.0053
FS 0.0599 0.0620 0.0624 0.0624
CF 0.0081 0.0085 0.0086 0.0086
SF 0.0776 0.0816 0.0825 0.0826 a Young and Budynas [35]
7.2 Annular Plate with internal ring support
Consider an annular plate of outer radius R, inner radius 0.4R, Poisson’s ratio υ = 0.25
subjected to a uniformly distributed load q = q0 with an internal ring support of radius 0.7R as
shown in Figure 8. Table 2 shows the maximum deflection wmaxD/(q0R4) for HBM (or second
FDM) having an internal ring support for different n and boundary conditions. Figure 9
shows the deformed shape of HBM (or second FDM) having internal ring support with n =
100 for CS, FC and SF boundaries.
Figure 8 Annular plate with internal ring support
Table 2 Maximum deflection wmaxD/(q0R4) for HBM (or second FDM) having internal ring support for different
n and boundary conditions
Edge
conditions
wmaxD/(q0R4) for HBM (or second FDM)
n = 30 n = 100 n = 1000 n = 2000
CC (10-5×) 2.5220 2.2975 2.2830 2.2829
SC (10-5×) 4.7911 4.5350 4.5243 4.5242
CS (10-5×) 6.0229 5.7747 5.7607 5.7605
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SS (10-5×) 5.5089 5.2488 5.2354 5.2353
FC 0.0011 0.0012 0.0012 0.0012
FS 0.0011 0.0012 0.0012 0.0012
CF 0.0020 0.0021 0.0021 0.0021
SF 0.0022 0.0023 0.0024 0.0024
Figure 9 Deformed shape and side view of HBM having internal ring support with n = 100 for (a) CS edges
(b) FC edges and (c) SF edges
8. Concluding Remarks
The FDM that is equivalent to the HBM for axisymmetric bending circular and annular plates
is identified. This FDM is obtained from discretizing the second order derivative terms twice
(see Eq. (A.1c)) and the first order derivative terms twice (see Eq. (A.1d)). Only this
particular FDM is energy preserving and should be adopted instead of the other forms of
finite difference approximations that are not supported by the energy principle. Note that the
different forms of FDM do not appear in the formulation of beams, circular arches or
rectangular plates.
For HBM, the stiffness of internal rotational spring in radial direction is Cr = D/a and the
rotational spring stiffness in circumferential direction is Cθ = D/s. Based on the equivalency
between the identified FDM and HBM, the formulae for edge spring stiffnesses of HBM are
derived. The expressions for the edge spring stiffnesses are given by Eqs (22a) and (22b).
With these general forms of the spring stiffnesses, axisymmetric bending problems of circular
plates and annular plates with any boundary conditions can be solved. In particular, the
analytical solutions for HBM modelling circular plates under uniformly and linearly
increasing distributed loads are obtained for the first time. These analytical solutions do not
contain the singularity problem faced in the FDM at the plate center. The benchmark bending
solutions for annular plates under uniformly distributed load with and without an internal ring
support are also obtained using the HBM. Such solutions which are tedious to obtain
analytically from the continuum theory, are readily obtained by the HBM; thereby
demonstrating the efficiency and accuracy of the HBM for plate analyses.
The research findings herein serve as a foundation for further development of HBM for
solving buckling and vibration problems of plates in polar coordinates. Furthermore, it could
be applied to optimize the shape of circular/annular plates and calibrate the small length scale
coefficient in nonlocal plate theory as shown for beams [20-22, 37], circular arches [23] and
rectangular plates [38, 39].
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Appendix A. Two Finite Difference Approximations for Derivatives
The approximations for the derivative terms in the governing equation are given below:
2 2 4 3
2 2 4 3
2 1 1 2 2 1 1 2
4 3
2
4 6 4 2 2j j j j j j j j j
j
d d w d w d wr r
dr dr dr dr
w w w w w w w w wr
a a
for first method (A.1a)
21 1 1 1
2 2 2 2
21 1 1
2
j j j j j
j j
w w w w wd dw d w dw
dr r dr rdr r dr r a r a
for first method ( A.1b)
Or alternatively,
2 22 1 1 1 1 2
1 12 2 2 2 2
2 1 1 2 2 1 1 2
4 3
2 2 22
4 6 4 2 2
j j j j j j j j j
j j j
j j j j j j j j j
j
w w w w w w w w wd d wr r r r
dr dr a a a
w w w w w w w w wr
a a
for second method (A.1c)
2 2
2 2
1 1
1 1 1
4 4
j j j j
j j
w w w wd dw
dr r dr r a r a
for second method (A.1d)
It can be seen that for the first term of Eq. (4), both methods furnish the same finite difference
form whereas there is a discrepancy in the second term of Eq. (4) between the two
discretization methods. However, it should be noted that the difference between the two
methods is rather small.
With the above finite difference approximations applied on Eq. (4), we have the two
distinct governing equations for FDM, i.e. governing equations given by Eqs. (5a) and by Eq.
(5b).
Fig. 1
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Fig. 2
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Fig. 3
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Fig. 4
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Fig. 5
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Fig. 6
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Fig. 7
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Fig. 8
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Fig. 9
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![Experimental Assessment of the Equivalent Strain Rate … · 3-D finite element analyses of the Charpy impact test [11-14]; • for the equivalent strain rate averaged across the](https://img.dokumen.tips/doc/110x75/5b8629317f8b9a162d8c543d/experimental-assessment-of-the-equivalent-strain-rate-3-d-finite-element-analyses.jpg)
