Journal of Vibration and Control 2011 Mitra 2131 57

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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

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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

2134 Journal of Vibration and Control 17(14)



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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

2136 Journal of Vibration and Control 17(14)



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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

Journal of Vibration and Control 2011 Mitra 2131 57

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