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http://jvc.sagepub.com/content/17/14/2131The online version of this article can be found at:
DOI: 10.1177/1077546310395973
2011 17: 2131 originally published online 21 March 2011Journal of Vibration and ControlAnirban Mitra, Prasanta Sahoo and Kashinath Saha
Free vibration analysis of initially deflected stiffened plates for various boundary conditions
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Article
Free vibration analysis of initiallydeflected stiffened plates forvarious boundary conditions
Anirban Mitra, Prasanta Sahoo and Kashinath Saha
Abstract
Free vibration analysis of initially deflected stiffened plates subjected to uniformly distributed loading with differentflexural boundary conditions involving simply supported and clamped ends and zero displacement in-plane boundaryconditions has been presented. A domain decomposition technique depending on the number, orientation and locationof the stiffeners is employed to ensure sufficient number of computation points around the stiffeners. Geometric non-
linearity arising out of large deflection is accounted for by consideration of non-linear strain-displacement relations.Mathematical formulation is based on a variational form of energy principle, and a solution technique, where staticanalysis serves as the basis for the subsequent dynamic study, is followed. The results are validated with the publishedresults of other researchers. The dynamic behavior has been presented in the form of backbone curves in a dimen-sionless frequency-amplitude plane. Vibration mode shapes along with contour plots are provided in a few cases.
Keywords
Backbone curves, geometric non-linearity, stiffened plate, variational methods
Received: 20 August 2010; accepted: 15 November 2010
1. Introduction
Stiffened plates are essentially thin plate structures rein-
forced with slender beams or stiffeners. They provide
enhanced stiffness and stability characteristics along
with the added advantage of light weight. It is no
surprise, therefore, that they are extensively used in
many branches of modern civil, mechanical, and struc-
tural engineering. Especially in fields like marine
construction, aerospace structures, where the reduction
of weight is of primary importance, stiffened plates
have wide application. Hence, investigation of the
dynamic behavior of stiffened plates has always been
an area of immense interest to researchers and research
work on this topic has a long history. Comprehensive
review works (Bedair, 1998; Mukherjee and
Mukhopadhyay, 1986) provide an excellent idea
about different techniques and methodologies
employed over the years to determine the natural
frequencies and mode shapes of such structural
elements.
Kirk (1970) carried out an analysis of simply
supported rectangular plates stiffened with a single
integral stiffener placed along one of its centerlines
and determined the natural frequencies for a wide
range of parameters using an approximate Ritz
method. He also presented an exact solution for the
system. Szechenyi (1971) used empirical relations and
simplifying assumptions to derive simple approximate
formulae for the natural frequencies of certain stiffened
panel structures. Aksu and Ali (1976) used variational
principles along with the finite difference technique to
examine dynamic characteristic of uni-axial rectangular
stiffened plates. They minimized the total energy of the
system with respect to discretized displacement compo-
nents and obtained natural frequencies and mode
shapes as the solutions of a linear algebraic eigenvalue
problem. In a later work Aksu (1982) extended the
previous free vibration analysis including the effect of
Department of Mechanical Engineering, Jadavpur University, Kolkata,
India
Corresponding author:
Prasanta Sahoo, Department of Mechanical Engineering, Jadavpur
University, Kolkata 700032, India
Email: [email protected]
Journal of Vibration and Control
17(14) 21312157
! The Author(s) 2011
Reprints and permissions:
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DOI: 10.1177/1077546310395973
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amplitude of deflection yields the backbone curve of the
system and the results for dynamic behavior are
furnished in terms of the backbone curves in the dimen-
sionless amplitude-frequency plane. The first six vibra-
tion modes of a stiffened plate with various
combinations of clamped and simply supported bound-
ary conditions have been documented. The vibrationmode shapes are also provided corresponding to the
minimum and maximum amplitudes of vibration.
2. Analysis
Figure 1 shows a stiffened plate of length a, width b and
thickness tp along with notations of other significant
dimensions and the coordinate system for the present
analysis. As shown in the figure the plate is orthogo-
nally stiffened by eccentric stiffeners parallel to the
edges of the plate. Although the figure shows only
one stiffener along each of the coordinate directions,
the mathematical formulation is carried out for multi-
ple stiffeners having generalized location and spacing in
both directions, and the number of stiffeners in x- and
y- direction are represented by nsx and nsy, respectively.
The location of the stiffeners are given by xqstf , y
pstf
where the superscripts denote the number of the
stiffener starting from the origin (O). In the present
scenario only the rectangular cross-section of the stiff-
eners is considered and bpsx, bqsy represent width of p-th
stiffener along x-direction and q-th stiffener along
y-direction, respectively, whereas thickness of the
abovementioned stiffeners are denoted by tpsx,tqsy. The
eccentricity (epx , eqy) of the stiffeners are quantified by
the perpendicular distance between the mid-plane of the
plate and the centroid (O) of the stiffener cross-section.
The stiffeners are rigidly attached to the plate such that
no field discontinuity can occur between these two.
A compatible strain distribution at the interface
between the plate and the stiffeners is assumed. It is
also assumed that the plate and stiffener materials areisotropic, homogeneous and linearly elastic. Finally,
the effects of shear deformation and rotary inertia
have been neglected as the plate thickness is taken
sufficiently small compared to its lateral dimensions.
For the stiffeners the effect of shear deformation
while imposing large deflection and the effect of
rotary inertia during dynamic analysis have been
neglected.
The present analysis studies the effect of vibration
amplitude on the dynamic behavior of stiffened plates.
Non-linear vibration frequency is dependent on ampli-
tude of vibration and the nature of the deflected profile
as in both the cases the strain energy stored in the
system changes, thereby effecting a change in the
system stiffness and its dynamic behavior. However,
the deflection profile of a system is a function of its
boundary conditions and the nature of loading.
Different boundary conditions may arise from the
different combinations of clamped (C), simply
supported (S) and free (F) end conditions of a plate
but in the present paper only the combinations of
simply supported (S) and clamped (C) boundary
conditions are considered. But other types of boundary
conditions, including free edges (F) can also be simu-
lated through the present methodology.
Figure 1. Schematic drawing of a stiffened plate with notations for significant dimensions and coordinate system.
Mitra et al. 2133
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Large amplitude free vibration analysis of the
non-linear system is considered equivalent to its free
vibration analysis, subjected to a static load producing
same magnitude of large amplitude deflection
(Crandall, 1956). Hence, the present free vibration
analysis of stiffened plates is performed in two steps.
First, large displacement is statically imposed by apply-ing transverse loading and then the free vibration prob-
lem is solved as an eigenvalue problem. The
mathematical formulation for both the static and
dynamic problem is based on a variational form of
energy principle. Consideration of non-linear strain
displacement relations implies geometric non-linearity
in the system. Appropriate start functions are assumed
and necessary higher order constitutive functions are
generated through two dimensional a Gram-Schmidt
orthogonalization procedure.
2.1. Static analysisIt is known from the principle of minimum potential
energy that for a conservative system, of all the kine-
matically admissible displacement fields, the one corre-
sponding to the stable equilibrium minimizes the total
potential energy of the system. The above statement is
expressed mathematically as,
U V 0: 1
Here, Vis the work function or potential of the exter-
nal forces and U is the total strain energy stored in the
system, which comprises two components, namely,strain energy of the plate (Up) and total strain energy
stored in all the stiffeners (Us). The expression for Uscan be written as, Us
Pnsxp1 U
psx
Pnsyq1 U
qsy, where
Upsx, Uqsy are strain energies stored in p-th stiffener
along x-direction and q-th stiffener along y-direction,
respectively. In case of large displacement analysis with
geometric non-linearity, both bending and stretching
effects are taken into consideration. Assuming a
compatible strain distribution at the line joining
the plate and the stiffener, axial strain of a stiffener
along x-direction is derived from the expression
"p
sx "
pxj
ztp=2 z t
p=2 w,
xx y
sv,
xx, where the
plate strain at z tp/2 denoted by "pxjztp=2 is substi-
tuted. From these expression of strains the total strain
of an x-direction stiffener is obtained as,
"psx u, x 0:5 w, x2 zs w, xx ys v, xx. So, the axial
strain of a stiffener along x-direction includes
axial strain due to bending action about major
axis (zsw, xx), stretching of the neutral axis
(u, x 0:5 w, x2) and bending action about minor
axis (ysv, xx). It should be pointed out that the effect
of torsion has not been taken into consideration while
calculating the total axial strain. Substituting this total
strain in the generalized expression of strain energy,
Upsx Es=2 R
vol"psx
2dV, the final expression is
obtained as,
Upsx Esa
2 Z1
0
Ipy Apye
px
2
a4w,
2
24
Ipyz Apy b
pstf
2 a4
v, 2
Qpy2
a3u,
w,
1
a4w,
w, 2& '
Apy1
a2u, 2
1
4a4w, 4
1
a3u,
w, 2& '#
d
2
Similarly, the strain energy expression of a y-direc-
tion stiffener is given by,
Uqsy Esb
2
Z10
Iqx Aqxe
qy
2
b4w, 224
Iqxz Aqx a
qstf
2 b4
u, 2
Qqx2
b3v,
w,
1
b4w,
w, 2& '
Aqx1
b2v,
2
1
4b4w,
4
1
b3v,
w,
2
& '#d
3
Here, Es is the elastic modulus of the stiffener
material. Iqx bqsyt
q3
sy = 12, Ipy b
psxt
p3
sx= 12 and
Iqxz bq3
sy tqsy= 12, I
pyz b
p3
sxtpsx= 12 are moment of iner-
tia about the major and minor axis of the stiffener
cross-section. Qpy Aqxe
qy, Q
qx A
pye
px are the first
moment of area about the plate mid-plane and Apy, Aqx
are the cross-sectional areas of the p-th x- and q-th
y-direction stiffeners, respectively.
Strain energy of the plate (Up) also consists of two
parts: strain energy due to pure bending (Ub) and strain
energy due to stretching (Um
) of its mid-plane. The
expressions of Ub and Um are well known for rectangu-
lar plates (Saha et al., 2004) and are indicated here for
ready reference.
Ub Ept
3p
12 1 2
ab
2
Z10
Z10
1
a2w,
1
b2w, & '2"
2 1 1
a2b2w, 2
w,
w, n o#
dd
4
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Appropriate start functions for these orthogonal sets
are selected in such a way that they satisfy the flexural
and membrane boundary conditions of the plate. The
higher-order functions are generated from the selected
start functions following a two-dimensional implemen-
tation of the Gram-Schmidt orthogonalization scheme.
The details of Gram-Schmidt orthogonalization princi-ple and its two dimensional implementation have been
presented previously by Saha et al. (2004) and Das et al.
(2009). It should be noted that the generated functions
are for the total domain and they need to be broken
down in terms of the sub-domains. mni , mni ,
mni
represent these sets of functions for the sub-domains
derived by carrying out interpolation operation on the
total functions, where m 1, . . . , nsy 1 and
n 1, . . . , nsx 1.
The displacement fields associated with the plate
presented in equation (7) are two dimensional in
nature. But the stiffeners are one dimensional elements
and hence the energy functional expressions related to
them include single integrations (as shown in equation
(2) and (3)). So, the plate displacement fields in their
original form cannot be used in these expressions and
to make the functions compatible with the stiffeners,
plate displacement function is evaluated at the stiffener
location (stf or stf, depending on the orientation of the
stiffener). For example, in the case of a y-direction stiff-
ener the transverse displacement function is taken as,
w , jqstf
w qstf, .
2.1.2. Governing system of equations. Substitutionof the complete energy expressions and approximate
displacement fields in equation (1) gives the set of
system governing equations in matrix form,
K df g f
: 8
[K], {d} and {f} are stiffness matrix, vector of
unknown coefficients and load vector respectively.
The total stiffness matrix [K] in equation (8) is given
by, K Kb Km Pnsx
p1 Ksx p
Pnsyq1 Ksy
q, Kb
and Km
being the contributions from bending and
stretching action of the plate, whereas Ksx p and
Ksy
q are stiffness matrices of the p-th stiffener along
x-direction and q-th stiffener along y-direction, respec-
tively. The details of the stiffness matrices are as
follows:
The form of Kb is given by,
Kb
kb11 kb12 k
b13
kb21 kb22 k
b23
kb31 kb32 k
b33
264
375,
where,
kb11
Ept
3p ab
12 1 2
Xnwj1
Xnwi1
Z10
Z10
"&1
a4i,
j,
1
b4i,
j,
1
a2b2i,
j,
1
a2b2i,
j, '
1 a2b2
i,
j,
:
i,
j,
2 i,
j, #
dd
kb12
kb13
kb21
kb22
kb23
kb31
kb32
kb33
0
The form of Km is given by,
Km
km11 km12 k
m13
km21 km22 k
m23
km31 km32 k
m33
2
64
3
75,
where,
km11 Eptp
2 1 2 ab
Xnwj1
Xnwi1
Z10
Z10
"1
a4
Xnwi1
dii,
2
i,
j,
1
b4
Xnwi1
dii,
2i,
j,
1
a2b2
Xnwi1
dii,
2i,
j,
1
a2b2 Xnw
i1
dii, 2
i, j,
2
a3
Xnwnuinw1
diinw,
i,
j,
2
b3
Xnwnunvinwnu1
diinwnu,
i,
j,
2
a2b
Xnwnunvinwnu1
diinwnu,
i,
j,
2
ab2
Xnwnuinw1
diinw,
i,
j,
1
ab2
( Xnwnuinw1
diinw,
i, j,
Xnwnu
inw1
diinw,
i,
j, )
1
a2b
( Xnwnunvinwnu1
diinwnu,
i,
j,
Xnwnunv
inwnu1
diinwnu,
i,
j, '#
dd
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km12
km13
0
km21
Eptp
21 2ab
Xnwnujnw1
Xnwi1
Z10
Z10
"1
a3
Xnwi1
dii,
i,
jnw,
ab2
Xnwi1
dii,
i,
jnw,
1 ab2
Xnwi1
dii,
i,
jnw, #
dd
km22
Eptp
21 2ab
Xnwnujnw1
Xnwnuinw1
Z10
Z10
"2
a2inw,
jnw,
1
b2inw,
jnw, #
dd
km23
Eptp
21 2
Xnwnujnw1
Xnwnunvinwnu1
Z10
Z10
h2 inwnu,
jnw,
1 inwnu, jnw, i
dd
km31
Eptp
21 2ab
Xnwnunvjnwnu1
Xnwi1
Z10
Z10
"1
b3
Xnwi1
dii,
i,
jnwnu,
a2b
Xnwi1
dii,
i,
jnwnu,
1
a2b
Xnwi1
dii, i,
jnwnu, #
dd
km32
Eptp
21 2
Xnwnunvjnwnu1
Xnwnuinw1
Z10
Z10
2 inw,
jnwnu,
1 inw, jnwnu, ddkm33
Eptp
21 2ab
Xnwnunvjnwnu1
Xnwnunvinwnu1
Z10
Z10
2
b2inwnu,
jnwnu,
1
a2inwnu,
jnwnu,
dd
The form of Ksx is given by , Ksx
ksx11 ksx12 k
sx13
ksx21 ksx22 k
sx23
ksx31 ksx32 k
sx33
26664
37775; where,
ksx11
Xns
x
p1
Esa2
Xnwj1
Xnwi1
Z1
0
2 Ip
y Ap
yep2
x
a4i,
j,
Qpy
8