A PARAMETRIC STUDY OF DYNAMIC RESPONSE OF

RECTANGULAR WINDOW GLASS PLATES

by

SRIDHAR KAMINENI, B.E. (Hons.)

A THESIS

IN

CIVIL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

CIVIL ENGINEERING

Approved

Accepted

May. 1989

' 11^

T3

ACKNOViLEDGEMENTS

The author would like to thank Dr. C. V. G. Vallabhan,

Professor and Chairperson of his advisory committee, for

selecting the topic for this thesis. The author deeply

appreciates his patience, direction, and criticism

throughout the preparation of this thesis. Utmost gratitude

is extended to Dr. Y. C. Das, Visiting Professor, who looked

after the progress of the research in the absence of

Dr. C. V. G. Vallabhan.

Finally, the author is greatly indebted to his family

and friends for their encouragement and support during his

entire graduate program at Texas Tech University.

11

TABLE OF CONTENTS

Page

ACKNOV^LEDGEMENTS i i

LIST OF TABLES v

LIST OF FIGURES vi

LIST OF SYMBOLS vii

CHAPTER

1. INTRODUCTION 1

1.1 Review of Previous Related V^ork 2

1.2 Objective and Scope 5

2. NONLINEAR DYNAMIC PLATE EQUATIONS 7

2.1 Introduction 7

2.2 Dynamic Nonlinear Plate Equations 7

2.3 Boundary Conditions 12

3. NONDIMENSIONAL PARAMETRIC ANALYSIS 15

3.1 Nondimensional Parameters 15

4. NUMERICAL ANALYSIS USING THE FINITE DIFFERENCE METHOD 18

4.1 Finite Difference Model 18

4.2 Equations of the Dynamic Plate Model 20

4.3 Computational Procedure 21

5. DYNAMIC RESPONSE USING NONDIMENSIONAL PARAMETERS 24

5.1 Input Scheme for Developing Nondimensional Parameters 24

5.2 Results 25

111

6. CONCLUSIONS AND RECOMMENDATIONS 33

6.1 Conclusions 33

6.2 Recommendations 34

LIST OF REFERENCES 35

IV

LIST OF TABLES

5.1 Nondimensional Pj - V max/' ^ ^^^ Plates with Different Aspect Ratios (v = 0.22) 2b

5.2 Nondimensional P^ - Tj (Period) for Plates with Different Aspect Ratios ( v = 0.22) 27

5.3 Nondimensional P- -a (max). Maximum Principal Stress for Plates with Different Aspect Ratios (v = 0.22).. 28

LIST OF FIGURES

2.1 Displacement at x, y, z in a Plate 9

2.2 Boundary Conditions for a Simply Supported Rectangular Plate (only quarter plate shown) 13

5.1 Nondimensional Maximum Deflection Vs. Nondimensional Pressure 29

5.2 Nondimensional Period Vs. Nondimensional Pressure ... 30

5.3 Nondimensional Maximum Principal Tensile Stress Vs. Nondimensional Pressure 31

VI

LIST OF SYMBOLS

[A]

a

[B]

b

D

d

E

{F}

fl(V*.F)

f2(V*)

h

[M]

^x' ^ y ^xy

"n

{P}

t

u

V

a square matrix containing the biharmonic operator of the out-of-plane deflections

half length of the plate in x - direction

a square matrix containing the biharmonic operator of the stress function

half length of the plate in y - direction

flexural rigidity of the plate

nondimensional time step

Young's modulus of elasticity of the plate material

Airy stress function vector

vector representing L(W,0)

vector representing L(W,W)

thickness of the plate

diagonal mass matrix representing discrete lumped masses at each node

moment resultants per unit length

nondimensional pressure

discrete load vector

lateral pressure on the plate which is a function of time

nondimensional period

time

nondimensional time

displacement component in the x - direction

displacement component in the y - direction

Vll

w

vi

V

w

{vi}

{V*}

^max

x,y,z

X

Y

V*

^ .

i>

$

( r r

<^l

^x

4

^

' <

<

m ' 'xy

4

- 6

'n

displacement component in the z - direction

acceleration component in the x - direction

acceleration component in the y - direction

acceleration component in the z - direction

out-of-plane displacement vector

discrete nodal acceleration vector

nondimensional maximum deflection

coordinates of a point within the plate

X / a

y / a

mass of the plate per unit volume

Poisson's ratio

biharmonic operator

interpolation parameters

Airy stress function

(p/ Eh 2

membrane stresses

bending stresses

nondimensional membrane stresses

nondimensional bending stresses

nondimensional stress

• • • Vlll

CHAPTER 1

INTRODUCTION

For glazing of modern buildings, rectangular window

glass units are being used extensively. When these units

are subjected to high wind loads, the maximum lateral

displacement of the plate can exceed several times the

thickness of the plate. As the value of the maximum lateral

displacement exceeds half of the plate thickness, it is

categorized as large, and the classical linear plate theory

described by Kirchoff is no longer applicable. In such

cases, in-plane or membrane stresses are produced in

addition to conventional bending stresses, and these

membrane stresses stiffen the plate, making it behave

nonlinearly. This additional stiffening is the result of

the in-plane forces generated by the large displacements

counteracting with the bending moments produced by the

applied lateral pressure. The nonlinear behavior is further

complicated by the in-plane constraints at the boundaries of

the plate.

The material of the glass plate can be assumed to be

linearly elastic and the above nonlinear behavior is

attributed to nonlinear geometry according to the theories

in solid mechanics. The equation governing the tranverse

motion of the plate is coupled with the equation governi.ig

the in-plane stretching, resulting in a pair of nonlinear

equations. To study the dynamic behavior of these window

glass units, the dynamic nonlinear von Karman plate

equations are to be used.

1»1 Review of Previous Related Work

Most of what is known about nonlinear dynamic analysis

of thin window glass plates is theoretical. The dynamic

behavior being nonlinear, even the theoretical computations

are relatively complex. Due to the nonlinearity in the

governing differential equations of the plate the

conventional modal decomposition is not possible. Various

glass plate investigators have used different finite element

models for solving the response of thin rectangular glass

plates subjected to uniform lateral pressure. They have

used a step-by-step integration procedure with or without

iteration for each load increment. These solutions consume

a considerable amount of computer time even for a single

dynamic analysis. Before presenting the equations for

dynamic behavior of window glass plates, a brief review is

given for corresponding equations for static analysis.

These equations are subsequently modified for the dynamic

analysis.

Kaiser (1) solved a simply supported, square plate

subjected to a uniform lateral pressure using the finite

difference method. He transformed each of the fourth-order

partial differential equations into two second-order partial

differential equations and solved those equations using the

finite difference method. Researchers at Texas Tech

University have developed several methods for solving the

nonlinear plate equations. Al-Tayyib (2) developed a very

sophisticated, higher order finite element approach to solve

the nonlinear plate problem. Season (3) and later Vallabhan

and Ku (4) used a Galerkin type solution for the solution of

the von Karman equations of a rectangular plate with simply

supported boundary conditions. The results obtained from

the Galerkin model come closest to the exact solution for

the given von Karman equations. It is observed that this

method is computationally expensive and requires a large

amount of memory space on the computer. Anians (5)

determined central and edge displacements (translations and

rotations) experimentally under different support and load

conditions. Vallabhan (6) established an efficient iterative

procedure to solve the von Karman nonlinear plate equations

using the finite difference method for a rectangular plate.

Later, Vallabhan and Wang (7) showed that the von Karman

nonlinear plate equations are indeed very useful to

represent the static response of thin rectangular plates

subjected to uniform pressure. They showed that their model

is very simple for input data preparation and

computationally efficient, as well. In their analysis they

used a unique iterative procedure with two acceleration

parameters used to interpolate quantities in successive

iterations. While keeping one of the acceleration

parameters as a constant, the other parameter is varied

nonlinearly with respect to the extent of the nonlinearity

of the plate. The simplicity and computational efficiency

of the model have made it possible to analyze a wide variety

of rectangular glass plates at low computational costs.

Using this method, Chou (8) and Vallabhan and Chou (9) have

developed nondimensional parameters for calculating the

displacement and maximum principal stress in a rectangular

plate subjected to uniform lateral loads. Vallabhan and

Minor (10) have indicated excellent correlation between

experimental and theoretical results obtained from the use

of von Karman's nonlinear theory of plates for static

loading conditions. These experiments conducted at Texas

Tech University created confidence in the use of the von

Karman theory for the nonlinear analysis of window glass

plates.

Chu and Hermann (11) have used the perturbation

technique to study the free vibrations of rectangular plates

with hinged immovable edges. Vendhan and Das (12) using

Rayleigh-Ritz and Galerkin methods studied free vibrations

of the plate and showed that the in-plane inertia forces are

negligible compared to transverse inertia forces.

Neglecting the in-plane inertia forces, Vallabhan and Selvam

(13) developed a numerical finite difference model to solve

the dynamic nonlinear von Karman plate equations using the

finite difference method for space and the Newmark beta

method for the time discretization, combined with an

iterative technique to obtain a convergent solution. Pal

(14) measured the nonlinear dynamic response of rectangular

glass plates kept in an enclosed chamber and subjected to

impulsive type loadings. In his experiments, the

interaction that occurred between the response of the glass

plate with that of the trapped air was very complex to

model. In the nonlinear dynamic analysis of thin window

glass plates, the frequency of vibration increases with the

amplitude of vibration. Several methods have been used to

study the free vibrations of the plate and the dependence of

the periods of vibration on amplitudes. The nonlinear

dynamic response of thin plates has received very little

attention in the past and, hence, it is the opinion of the

author that in the future special attention is necessary for

both theoretical analysis and experimental validations.

1.2 Objective and Scope

The objective of this study is to develop a

mathematical model for the nonlinear dynamic analysis of

simply supported rectangular window glass plates subjected

to uniform lateral pressure using nondimensional parameters.

The dynamic von Karman equations are developed using

D'Alembert's principle. Only the effect of the inertia

forces developed due to lateral motion is considered; in

other words, the inertia forces developed due to in-plane

acclerations are considered negligible. The nondimensional

parametric analysis forms a significant tool in presenting

analytical results intelligently. But, as the problem is

nonlinear, the parametric analysis can be performed only for

a certain type of loading. In particular, here, the loading

is taken as a suddenly applied uniform pressure on the

surface of the plate, even though any kind of loading can be

applied in the mathematical model. The boundary conditions

are taken as those of a simply supported plate, with no in-

plane forces on the edges and are discussed in chapter 2

along with the main governing equations. These equations

are nondimensionalized in chapter 3. The nondimensional

response of the plate is obtained by the procedure developed

in reference (13). A brief introduction to the numerical

analysis used in this technique is presented in chapter 4.

For the above loading condition, rectangular glass plates

having aspect ratios 1, 1.5, 2, and 3 are solved, and the

maximum amplitudes along with the corresponding periods of

vibration and the maximum tensile stresses are computed

using the nondimensional parameters. These are shown in

tabular form and also in graphical form in chapter 5.

Summary and conclusions developed from this study are

presented in chapter 6.

CHAPTER 2

NONLINEAR DYNAMIC PLATE EQUATIONS

2.1 Introduction

When the maximum lateral deflections of a thin plate

are no longer small in comparison with the thickness of the

plate but are nevertheless small compared with the remaining

plate dimensions, the classical linear plate theory

described by Kirchoff is no longer applicable. These large

displacements induce in-plane or membrane forces within the

plate. The developed in-plane tensile forces stiffen the

plate adding considerable resistance to further lateral

displacement, and this behavior is not predicted by the

small deflection bending theory. For such situations, the

large deflection theory of plates proposed by von Karman is

used. Researchers have demonstrated on glass plates that

the von Karman theory of thin plates is applicable to this

nonlinear behavior of thin glass plates and the pertinent

references are given in chapter 1. These equations are

modified to represent the dynamic behavior of the plate by

introducing inertia forces developed in the plate. In the

following sections the dynamic nonlinear von Karman plate

equations are developed.

8

2»2 Dynamic Nonlinear Plate Equations

The assumptions made for the development of the

dynamic plate equations used in this study are the same as

those made by von Karman for his static plate equations.

The assumptions for the von Karman plate theory are

presented in detail by Szilard (15), Timoshenko (16), and

Vallabhan and Wang (7). To these von Karman equations, the

inertia forces due to lateral motion are introduced using

D'Alembert's principle to achieve the dynamic response

conditions. The D'Alembert's principle states that a mass

develops an inertia force proportional but opposite to its

acceleration.

There are three displacement components u, v, and w at

every generic point P(x,y,z) as shown in Figure 2.1. These

displacement components introduce accelerations u, v, and w

and corresponding inertial effects in the x, y, and z

directions, respectively.

At a general point x, y, z in the plate as shown in

Figure 2.1, the lateral displacement w in the z-direction is

assumed to be equal to the lateral displacement w of the

mid-plane. The corresponding inertia forces per unit area

of the plate can be represented by /jhw, where p is the mass

of the plate per unit volume.

The following considerations are made regarding the

inertia forces due to the in-plane accelerations in the

plate. When bending phenomena are predominant, the u, v

X - axis

Figure 2 . 1 . Displacement at x, y, z in a Plate

10

displacements at the top and bottom of a particular point in

the mid-plane will be in the opposite directions, and so

will be the corresponding accelerations, resulting in

relatively negligible in-plane displacements and

accelerations at the mid-plane. However, when the plate

begins to undergo very large displacements, the in-plane

displacements and corresponding accelerations vary from top

to bottom of the plate. Consequently, a rotational inertia

force also exists in the plate. These in-plane and

rotational inertia forces make the analysis of the plate

extremely complicated, and the analysis is beyond the scope

of this study. Also, Vendhan and Das (12), after

considerable study on nonlinear dynamic plate behavior, have

found that the in-plane inertia forces can be ignored. In

this study, it is assumed that the in-plane and rotational

accelerations are small and negligible when compared to the

accelerations in the lateral directions. Based on the above

assumptions, the nonlinear dynamic plate equations in the

domain of the plate are derived as

DV^w = p + hL(w,</>) - phw (2.1)

and

vV = -EL(w,w) (2.2) 2

where

D = flexural rigidity of the plate

= Eh^/12(l-i/^)

11

w = lateral displacement of the middle surface of the plate

p = lateral pressure on the plate which is a function of time

h = thickness of the plate

(t> = Airy stress function

E = Young's modulus of elasticity of the plate

^ = Poisson's ratio of the plate 4

V = biharmonic operator

P = mass of the plate per unit volume

L(w,0) = w,xx' ,yy " 2w^xy^,xy + "^,Yy^,xx •

The "comma" in front of the subscripts represents

differentiation with respect to the variables following

them. The membrane action in the plate induced by large

displacements is represented by the Airy stress function

(16). From inspection, it can be seen that the functions

L(w, ) and L(w,w) are nonlinear. The above equations

represent conditions of equilibrium for bending action and

compatibility for the membrane action, respectively. The

above nonlinear equations have to be solved by considering

appropriate boundary conditions.

The internal forces of the plate can be classified into

two categories: one due to membrane forces and the other due

to bending moments.

The resulting membrane stresses at any point in the

plate are:

<x ' ,yy

12

"y ^,XX

" xy ~ ~ , xy • (2.3)

The bending moments per unit length are:

Mx = -D(w^xx + w,yy)

My = -D(w,yy + z/w,xx)

Mxy = D(l-i/)w^xy (2.4)

and the resulting bending stresses are:

rP 2

b 0 y =± 6My/h''

xy =± 6Mxy/h^ . (2.5)

The superscripts 'm' and 'b' represent membrane and bending

stresses.

2.3 Boundary Conditions

For a plate, the solution of Equations (2.1) and (2.2)

requires that two boundary conditions be satisfied at each

edge. These may be a prescribed deflection and slope, or

prescribed force and moment, or some combinations.

In this study, simply supported boundary conditions are

used for the analysis of the rectangular glass plates. The

two sets of boundary conditions to be satisfied are flexural

boundary conditions and membrane restraints on the boundary.

Let 2a and 2b be the width and length of the plate and

the reference axes pass through the center of the plate and

be parallel to the edges of the plate as shown in Figure

2.2. The boundary conditions for the simply supported plate

13

Y-axis

, I

w « 0

W, a yy

TXX

^xy

>0

- 0

- 0

Comer

^•xx-0

T'yy

^ . X - axis

Figure 2.2. Boundary Conditions for a Simply Supported Rectangular Plate (only quarter plate shown)

14

are also shown in the figure. Since, the applied pressure

is assumed to be uniform, considering the symmetry along

X- and y-axes, only a quarter of the plate is used in the

analysis. The dimensions a and b are one half the shorter

and longer dimensions of the plate, respectively.

The flexural boundary conditions are:

§ X = a, w = 0, M^x = 0' i.e., w^xx = 0

§ y = b, w = 0, M^y = 0, i.e., w^yy = 0

§ x = 0 , w ^ x ^ O ' and

@ y = 0, w^y = 0 . (2.6)

The second set of boundary conditions involve membrane

restraints on the edges. It is assumed that the in-plane

restraints of the edges are negligible; i.e., the surface

tractions on the edges are assumed to be zero. Thus the

membrane boundary conditions are:

@ X = a, o^ ^ <i>^yy = 0, "Scy = -< ,xy = 0

§ y = b, a^ = (?!>,xx = 0' xy = -< ,xy = 0

@ x = 0, <^,x"0' and

@ y = 0, ^,y = 0 . (2.7)

Equations (2.6) and (2.7) give the complete set of boundary

conditions to be satisfied for the plate at all times during

the dynamic response.

CHAPTER 3

NONDIMENSIONAL PARAMETRIC ANALYSIS

In the exuberance of mathematical modelling and digital

computation in recent years, the theory and applications of

nondimensional parameters are of great help to the analyst.

The analysis using nondimensional parameters yields the same

results as dimensional analysis. In particular, it

generates coefficients in the most general form where the

coefficients can be used to solve any dynamic plate problem

of same geometry, loading, and boundary conditions. It can,

in fact, be viewed as a short-cut that retains the essence

of dimensional analysis without requiring the explicit

solution of the set of governing equations as accomplished

by Vallabhan and Selvam (13). In this chapter the nonlinear

dynamic plate equations are rewritten in a nondimensional

form. This is accomplished by making the involved variables

of the problem dimensionless.

3.1 Nondimensional Parameters

The following parameters are introduced in the von

Karman's equations in order to make the equations

dimensionless:

X = x/a

Y = y/a

W = w/h

15

16

$ = <?i/Eh^

C = t(h/a^)(yE77) (3.1)

where

W is nondimensional displacement

$ is related to the Airy stress function

C is the nondimensional time.

After the substitution of the above parameters in the von

Karman equations, the modified nondimensional equations are:

v S = Pn + L(W,$) - W (3.2)

vH = -1L(W,W) (3.3) 2

where

Pn = (p/E)(a/h)'* . (3.4)

L(W,$) and L(W,W) are differential operators as defined

earlier but differentiated with respect to X and Y. W ^, 'CC

implies that W is differentiated twice with respect to

nondimensional time C. All parameters are calculated

internally in the program except P^ and C/ which are the

nondimensional lateral pressure and time. The value of P^

has to be calculated by modifying the original lateral

pressure, p, using the plate half width, a, plate thickness,

h, and the elastic modulus of the plate material, E, as

given in Equation (3.4). Similarly, the nondimensional time

is obtained by modifying the time, t, using the mass density

p , thickness h, plate half width a, and the modulus of

elasticity of the plate material E, as given in Equation

(3.1). The value of the Poisson's ratio has to be specified

17

explicitly for each material, and cannot be eliminated from

Equation (3.2).

With respect to the above defined parameters, the

nondimensional membrane stresses are:

^X ~ ^,YY

< Y =^,XX

^XY = •^,XY ^^'^^

and the nondimensional maximum bending stresses are:

^X =-2(1-1/2) (V ,XX + ^W^yy)

O^ = ±__J—-- (W YY + VW XY) y . 2(1 - u^) '^^ ' ^ T^Y = + ' V YY • ^3.6) ^^ " 2(1 + 1/) ' ^

The nondimensional stresses can be derived as

an = (a/E)(a/h)2 (3.7)

where

Cj is the nondimensional stress, and

a is the actual stress.

The numerical finite difference model and the

computational procedure used for solving the simply

supported rectangular plate, subjected to the suddenly

applied uniform lateral pressure, are described in the next

chapter.

CHAPTER 4

NUMERICAL ANALYSIS USING THE FINITE

DIFFERENCE METHOD

For nonlinear analysis of rectangular glass plates

subjected to uniform static pressure, Vallabhan (6)

developed a numerical finite difference model for solving

the von Karman equations. Later, Vallabhan and Selvam (13)

extended this to solve dynamic pressures. Their numerical

model consists of two parts: one part involves the space

discretization of the domain of the plate and the other part

involves the time discretization. The space discretization

is achieved by the finite difference method and the time

discretization by the Newmark-Beta method (13). The same

model is used here for the nondimensional analysis of the

rectangular plates by using a systematic input scheme.

4.1 Finite Difference Model

Using the central difference equations, the two

nonlinear dynamic plate equations (2.1) and (2.2) are

replaced by two sets of algebraic equations of the type

[A] (W) = {P} + {fi(W,F)} - [M] {W} (4.1)

and

[B] {F} = {f2(V^)} . (4.2)

In these equations matrices [A] and [B] are square matrices

representing the corresponding biharmonic operators. {W},

18

19

{F}, and {P} are vectors representing discrete nodal

displacements. Airy stress function, and loads at each nodal

point, respectively. The vectors f and £2 are appropriate

vectors representing L(W,$) and L(W,W) in Equations (3.2)

and (3.3), respectively. The matrix [M] represents discrete

lumped masses at each node and is a diagonal matrix. {W} is

the discrete nodal acceleration vector.

The matrices [A] and [B] are symmetric and are

developed as banded matrices by incorporating the

appropriate boundary conditions. As mentioned earlier in

this study, simply supported conditions with zero edge in-

plane forces are incorporated. Only the half banded portion

of the matrix is used on the computer. The details of the

matrices [A] and [B] and the vectors f and £2 are available

in Vallabhan (6). Since the functions L(W,$) and L(W,W) in

the von Karman equations are nonlinear, the solution can be

achieved only by an iteration technique.

For the dynamic analysis of the plate, the Newmark-Beta

method was used. In this method, the velocity and

displacement at time C+dC are computed from the values of

displacement, velocity, and acceleration at times ( and C+dC,

where dC is a small increment of time, called time step.

The equations, respectively, are

and

W^+dC = W^ - W^dC + [(--/3)W + /3W^+d^](d(;)2. (4.4)

20

Here 7 and /? are two interpolation parameters. In this

study, the values of 7 and /3 were kept as 1/2 and 1/4,

respectively. Thus, in this model, the acceleration between

the time intervals becomes a constant and equal to the

average of the values of accelerations at the beginning and

end of the time step dC. With these values the method is

numerically stable for all discrete time steps. The details

of this method, its system properties, and convergence

characteristics are described in Vallabhan et al. (17).

4.2 Equations of the Dynamic Plate Model

In Equations (4.1) and (4.2) the unknown vectors at any

time C are the displacement vector W., the acceleration

vector W , and the Airy stress function vector F . The

values of these vectors coupled nonlinearly. A recursive

set of equations suitable for iteration was developed, where

the acceleration and the Airy stress function vectors are

calculated at time C+ciC* Considering Equations (4.1) and

(4.2) at time C+dC/ substituting for W^+d^ using Equations

(4.3) and (4.4), and rearranging, we get the following set

of recursive equations:

and

[B] {F +c } = {f2(V*(;fdC > ^^-^^

where

{W^} = [A] (W^ + W.dC + [(^-/3)W.(dC)2} (4.7)

21

and

[M] = [^(dC)^A + M] . (4.8)

The vector { } is a constant vector at time C containing

displacement, velocity, and accelerations at time C only and

premultiplied by the [A] matrix. Hence, this vector is

computed once in the beginning of the iterative procedure

made at time C+<iC'

For a complete dynamic analysis of the plate, initial

conditions of the plate have to be established. In this

study, the plate is assumed to be flat, stress-free, and

stationary at time ( = 0. Thus, the dynamic nonlinear von

Karman plate equations are solved by using the finite

difference method in space and the Newmark-Beta method in

time. Also, in this study, the lateral load is assumed to

be uniform and suddenly applied.

4.3 Computational Procedure

To achieve maximum efficiency in the computations, the

following procedures are taken initially.

1) Establish a suitable time step d to yield a

satisfactory solution.

2) Develop matrices [A], [B] and [M]. As these

matrices are symmetric and positive definite,

they are formed in half band width to save

computer storage. Matrices [B] and [M] are

decomposed using Cholesky type decomposition

22

to minimize the number of computations for

simultaneously solving Equations (4.5) and

(4.6).

3) At time C = 0 , W>.=0, W. = 0, W. = 0, and

F. = 0.

4) Start the step-by-step integration procedure,

C = n d^, for n = 1,N, where N is the desired

total number of time steps.

5) At the beginning of time C+dC* first calculate

W. and assume ^(+dC " ^C ^"^ C+<3C " ^'' ^^

first estimates for use in computing the

vector f^.

6) Start the iteration at time C+dC/ for i = 1,

NITER, where NITER = maximum number of

iterations specified.

For the (i + 1) th iteration,

7) Calculate w|. " V using Equation (4.5).

8) Calculate wl "ti using Equation (4.4). The C+d(

superscript i corresponds to the i-th

iteration.

9) Calculate F +d'* using Equation (4.6). 10) Check convergence using the following as an

acceptable tolerance:

E ||ur(»+i)_pT/(0 I

number of components in the vector < e

23

Repeat steps 7 through 10 until convergence is

achieved.

11) Repeat step 4 until the desired number of time

steps is completed.

The most important criteria to obtain satisfactory

response data is to select the optimum time step d< . Even

though some general guidelines have been established in the

literature in the case of linear analysis, some experience

is desirable to achieve good results. The time step

selected should be at least 1/20th of the period of the

first oscillation. The dynamic response of the plate using

the nondimensional parameters and the numerical finite

difference model is given in chapter 5.

CHAPTER 5

DYNAMIC RESPONSE USING NONDIMENSIONAL

PARAMETERS

In this chapter, the nonlinear dynamic responses of a

simply supported rectangular plate subjected to a suddenly

applied uniform pressure are presented using the

nondimensional parameters. In Vallabhan's finite difference

model for simply supported plates subjected to uniform

lateral pressure, the input data consisted of the actual

plate dimensions, material properties, and loads. To solve

the nondimensional nonlinear dynamic plate Equations (3.2)

and (3.3), a systematic input scheme is developed, without

causing any modification of the original finite difference

program.

5.1 Input Scheme for Developing Nondimensional Parameters

In the nondimensional plate equations, nondimensional

load, PQ, is a function of the actual lateral pressure p,

plate half width a, thickness of plate h, and modulus of

elasticity E, of the plate material as given by Equation

(3.4). The maximum value of X representing half of the

plate dimension, a, in the x-direction is set equal to one,

and the corresponding maximum value of Y is set equal to the

aspect ratio, n, of the plate. The input data are prepared

by modifying the lateral pressure, dimensions of the plate,

24

25

mass density of the plate, and modulus of elasticity of the

plate material in the following manner:

Pn = (p/E)(a/h)4

X = 1

Y = n

h = 1

E = 1

A> = 1 .

The quantities h, p, and E represent the thickness,

mass density, and modulus of elasticity of the plate in the

dimensionless version only. In order to select the optimum

time step, d^, an understanding of the plate behavior is

required. As a criterion, it should be at least l/20th of

the period of the first oscillation.

5.2 Results

For plates having aspect ratios equal to 1, 1.5, 2, and

3, nondimensional curves for maximum deflection, W ^ ^ ,

corresponding periods of vibration, Tj , and maximum

tensile stresses, o'j (max), as related to the nondimensional

load Pj , are given in a tabular form as Tables 5.1, 5.2, and

5.3, respectively. They are also given in graphical form as

Figures 5.1, 5.2, and 5.3, respectively. In these

computations, the Poisson's ratio is taken equal to 0.22,

representing glass material. For practical purposes the

influence on the nondimensional quantities due to variation

26

Table 5.1. Nondimensional P^ - VJitiax/*^ ^^^ Plates with Different Aspect Ratios ( 1/ = 0.22)

Nondimensional Maximum Deflection

Pn

0 . 0

0 . 4

0 . 6

0 . 8

1.0

1.5

2 . 0

3 . 0

4 . 0

5 .0

6 . 0

7 . 0

8 .0

9 .0

1 0 . 0

n = 1

0 . 0

0 . 5 8 0

0 . 8 3 4

1 .070

1 .281

1 .701

2 . 0 5 9

2 . 5 7 2

3 .017

3 .347

3 .665

3 .942

4 . 2 2 1

4 .520

4 .782

n = 1.5

0 .0

1 .042

1.446

1.787

2 . 0 8 9

2 . 7 3 4

3 .272

4 .116

4 . 7 6 3

5 .277

5 . 9 6 1

6 .422

6 .853

7 .238

7 . 5 6 1

n = 2

0 . 0

1 .403

1 .961

2 .450

2 .867

3 .760

4 . 4 9 4

5 .687

6 .655

7 . 5 0 5

8 .253

8 .924

9 . 5 2 5

1 0 . 0 8 6

1 0 . 5 9 2

n = 3

0 . 0

1 .878

2 . 7 0 3

3 . 4 3 6

4 . 1 2 1

5 . 5 9 4

6 . 7 9 3

8 . 7 3 3

1 0 . 2 7 3

1 1 . 5 6 9

1 2 . 7 3 3 •

1 3 . 8 1 7

1 4 . 8 4 7

1 5 . 8 4 0

1 6 . 7 6 3

27

Table 5.2. Nondimensional P^ - T^ (Period) for Plates with Different Aspect Ratios ( u- 0.22)

Nondimensional Period

Pn

0 . 4

0 . 6

0 . 8

1.0

1.5

2 . 0

3 . 0

4 . 0

5 .0

6 . 0

7 . 0

8 .0

9 . 0

1 0 . 0

n = 1

4 . 2 8 4

4 . 2 8 4

4 . 0 8 0

3 .876

3 .672

3 .672

3 .264

3 .060

2 .856

2 .856

2 . 6 5 2

2 . 6 5 2

2 .652

2 . 4 4 8

n = 1.5

5 .712

5 .508

5 .304

5 .100

4 . 6 9 2

4 . 4 8 8

4 . 0 8 0

3 .672

3 .468

3 .200

3 .100

3 .100

3 .000

2 .900

n = 2

6 . 9 3 6

6 . 7 3 2

6 . 5 2 8

6 .120

5 .712

5 .304

4 . 9 5 0

4 . 1 2 5

3 . 8 2 5

3 .600

3 .525

3 . 3 7 5

3 .300

3 . 2 2 5

n = 3

7 . 7 5 2

7 . 3 4 4

7 . 1 4 0

6 . 9 3 6

6 . 3 2 4

5 . 6 2 5

5 .300

5 . 2 5 0

4 . 9 5 0

4 . 7 5 0

4 . 6 0 0

4 . 5 0 0

4 . 3 5 0

4 . 2 5 0

28

Table 5.3. Nondimensional P^ - crj (max). Maximum Principal Stress for Plates with Different Aspect Ratios ( z/ = 0.22)

Nondimensional Maximum Principal Stress

Pn

0.0

0.4

0.6

0.8

1.0

1.5

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

n = 1

0.0

0.976

1.423

1.840

2.205

2.912

3.426

4.484

5.685

7.000

8.363

9.577

10.688

11.783

13.155

n = 1.5

0.0

1.492

2.032

2.452

2.851

3.617

4.294

6.181

7.900

9.678

11.917

13.720

15.564

17.498

18.996

n = 2

0.0

1.927

2.643

3.218

3.708

4.667

5.420

7.749

10.727

13.311

15.765

18.033

20.208

22.214

24.589

n = 3

0.0

2.469

3.542

4.489

5.400

7.576

8.890

11.410

12.932

16.108

18.962

22.063

26.970

30.402

33.665

29

c .2

I Q £ s X CO

c .2 c E •o c o

Nondimensional Pressure

Figure 5.1. Nondimensional Maximum Deflection Vs. Nondimensional Pressure

30

9 u >0.22

1 -

1 2 3 4 5 6 7

Nondimensional Pressure

igure 5.2. Nondimensional Period Vs. Nondimensional Figu

Pressure

31

GL

'o

Q.

E E «> X w

c 2 og <D

E c o

Nondimensional Pressure

Figure 5.3. Nondimensional Maximum Principal Tensile Stress Vs. Nondimensional Pressure

32

in the value of Poisson's ratio can be ignored. In the

subsequent chapter, the conclusions and recommendations

derived from this study are given.

CHAPTER 6

CONCLUSIONS AND RECOMMENDATIONS

The main objective of the research was to develop a

mathematical model for the nonlinear dynamic analysis of

simply supported rectangular window glass plates subjected

to uniform lateral pressure, using nondimensional

parameters. This objective was accomplished successfully.

Using this model, rectangular glass plates having aspect

ratios 1, 1.5, 2, and 3 are solved and the maximum

amplitude, maximum principal tensile stress, and the period

of oscillation with respect to the suddenly applied uniform

lateral pressure are computed. The conclusions derived

from these results are presented in the following section,

followed by the recommendations.

6.1 Conclusions

It was observed that the maximum displacement in the

plate always occurred near the center of the plate, and the

magnitudes increased with increase of aspect ratio and

applied load. The shortest periods of oscillation of the

plate decreased with increase of applied load, which can be

attributed to the stiffening of the plate; but their values

increased with increase of the aspect ratio of the plate.

For small pressures it was evident that the maximum

principal tensile stresses occurred at the center of the

33

34

plate; but as the nondimensional lateral pressure increased,

the location of the maximum principal tensile stresses

migrated towards the edge of the plate and then suddenly

moved to a region near the corners.

Using these nondimensional curves, the maximum

deflection, or the period of vibration, or the maximum

principal tensile stress can be predicted for any simply

supported rectangular plate having aspect ratios 1/ 1.5, 2,

or 3. For aspect ratios falling in between 1, 1.5, 2, or 3,

similar quantities can be computed fairly accurately using

appropriate interpolation techniques.

6.2 Recommendations

It is to be noted here that the type of loading used in

this study is a suddenly applied uniform lateral pressure,

which in reality, does not occur very often. It is the

opinion of the author that more research should be conducted

to study the dynamic response of window glass plates

subjected to different types of loading, and also for

different boundary conditions. The author sincerely hopes

that the mathematical model developed in this study will

be of benefit in future nondimensional analysis of thin

window glass plates.

LIST OF REFERENCES

1. Kaiser, R., "Rechnerische und Experimentelle Ermittlung der Durchbiegungen und Spannungen von Quadratischen Platten bei freier Auflagerung an den Randern, Gleichmassig Verteilter Last und Grossen Ausbiegungen," Z. F. A. M. M., Bd. 16, Heft 2, April, 1936, pp. 73-98.

2. Al-Tayyib, A. H., "Geometrically Nonlinear Analysis of Rectangular Glass Panels by Finite Element Method," Institute for Disaster Research, Texas Tech University, Lubbock, Texas, 1980.

3. Season, W. L., "A Failure Prediction Model for Window Glass," Institute for Disaster Research, Texas Tech University, Lubbock, Texas, 1980.

4. Vallabhan, C. V. G., and Ku, Y. F., "Nonlinear Analysis of Rectangular Glass Plates by Galerkin Method," Institute for Disaster Research, Texas Tech University, Lubbock, Texas, June, 1983.

5. Anians, D. C , "Experimental Study of Edge Displacements of Laterally Loaded Window Glass Plates," Master's Thesis, Texas Tech University, Lubbock, Texas, 1979.

6. Vallabhan, C. V. G., "Iterative Analysis of Nonlinear Glass Plates," Journal of Structural Engineering, ASCE, Vol. 109, No. 2, February, 1983.

7. Vallabhan, C. V. G., and Wang, B. Y-T., "Nonlinear Analysis of Rectangular Glass Plates by Finite Difference Method," Institute for Disaster Research, Texas Tech University, Lubbock, Texas, 1981.

8. David, Chou, "Nonlinear Stress Analysis of Insulating Glass Units," Ph.D. Dissertation, Texas Tech University, Lubbock, Texas, May 1986.

9 Vallabhan, C. V. G., and Chou, D., "Stresses and Displacements of Window Glass due to Wind," Proceedings, Fifth U.S. National Conference on Wind Engineering, Texas Tech University, Lubbock, Texas, November 1985.

35

36

10. Vallabhan, C. V. G., and Minor, J. E., "Experimentally Verified Theoretical Analysis of Thin Glass Plates," Proceedings, Second International Conference on Computational Methods and Experimental Measurements, Springer-Verlag, June/July, 1984.

11. Chu, H. N., and Hermann, G., "Influence of Large Amplitudes on Free Flexural Motions of Elastic Plates," Journal of Applied Mechanics, ASME, Vol. 23, pp. 532-540.

12. Vendhan, C. P., and Das, Y. C , "Application of Raleigh-Ritz and Galerkin Methods to Nonlinear Vibration of Plates," Journal of Sound Vibration, Vol. 39, pp. 147-157.

13. Vallabhan, C. V. G., and Selvam, R. P., "Nonlinear Dynamic Response of Window Glass Plates Using Finite Difference Method," Proceedings, Third Conference on Dynamic Response of Structures, University of California, L.A., March/April, 1986.

14. Pal, H. S., "Experimental Study of Glass Plate Strength at Rapid Loading Rates," Ph.D. Dissertation, Texas Tech University, Lubbock, Texas, 1987.

15. Szilard, R., Theory and Analysis of Plates - Classical and Numerical Method, Prentice-Hall, Inc., Englewood Cliffs, NJ.

16. Timoshenko, S., and Woinowsky-Krieger, S., Theory of Plates and Shells, McGraw-Hill Book Company, Inc., New York, NY.

17. Vallabhan, C. V. G., Vann, W. P., and Iyer, S. M., "Step-by-step Integration of the Dynamic Response of Large Structural Systems," Department of Civil Engineering, Texas Tech University, Lubbock, Texas, December, 1973.

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