中厚板生产车间工艺设计外文翻译

．

Shang

[18]

studied

the

rectangular

plates

with

bidirectional

linear

stiffness

with

two

opposite

edges

simply

supported

and

the

other

two

edges

arbitrarily supported under the distributed loads

．

Yang

[19]

investigated the structural

analysis of the plates with

unidirectionally varying flexural

rigidity by the Galerkin

line

method

．

Liu

et

a1.

[20]

analyzed

the

free

vibration

of

a

Functionally

graded

isotropic

rectangular

plate

with

in

—

plane

material

in

homogeneity

using

the

Levy-type so1ution

．

Uymaz et a1

．

[21]

Considered the functionally graded plates with

properties

Varying

in

an

in

—

plane

direction

based

on

a

five

< br>—

degree

—

of- freedom

shear deformable plate theory with different boundary conditions

．

In this paper

，

the Levy-type solution[22-23]is presented for the bending of a thin

rectangular plate with in

—

plane variable stiffness under the distributed load

．

2 Basic equations

Consider a thin rectangular plate of length A and with B with in-plane Variable

stiffness

，

as

shown in Fig.1. Introduce a Cartesian coordinate system 0

—

XYZ such

that

0

≤

X

≤

A

，

0

≤

Y

≤

B

We assume. That the flexural rigidity of the plate D=D(X,Y)is a function of X

and Y. The governing equation of the plate with in

—

plane Variable stiffness can be

obtained as

Where W is

the transverse displacement

，

V is

Poisso

n

′

s ratio

，

and Q is

the

normal pressure on The plate

．

第

2

页

It

is

assumed

that

the

flexural

rigidity

of

the

plate

varies

only

along

the

Y-direction according to the following power form:

Where Y and P are two material parameters describing the in homogeneity of D,

Do is the flexural rigidity at

，

Y=0, and D

b

is the flexural rigidity at Y=b. In

this ‘case

,

Eq. (1) can be reduced to

3 Solution

The

rectangular

plate

is

assumed

to

be

simply

supported

along

two

opposite

edges parallel to the Y-direction. To solve the governing equation with the prescribed

boundary conditions, a generalized Levy-type approach is employed as

where Y

m

(y) is an unknown function to be determined.

Substituting Eq. (4) into Eq. (3) yields the following differential equation:

第

3

页

The solution of the above equation consists of two parts, i.e,the general solution

Y

m

o

of the homogeneous differential equation and the particular solution Y

m

o

of the

no homogeneous differential equation

．

Thus

，

the solution of Eq.(5) can be expressed

as .

3.1 General solution of governing equation (5)

To

find

the

solution

of

Eq.

(5),

we

first

consider

the

following

fourth-order

homogeneous differential equation with variable coefficients:

The

above

equation

can

be

transformed

into

a

Whittaker

equation,

and

the

solution can be expressed as

with

for p

≠

1 and p

≠

1/v

for p=1

，

and

第

4

页

for p=1/v

．

W

k,g

(t) is the Whittaker function

，

is the exponential integral function,

I(t) and Kg(t) are the gth- order modified Bessel

functions

of

the

first

and

second

kinds,

respectively,

and

c1,

c2,

c3,

and

c4

are

the

constants to be determined.

3.2 Particular solution of governing equation (5)

From the solution of the homogeneous differential equation (7), the solution of

the nonhomogeneous differential equation (5) can be expressed as

Where c

l

(t), c

2

(t), c

3

(t), and c

4

(t) satisfy the following equations:

from which we obtain

第

5

页

where

Then. we have

where C

1

, C

2

,C

3

, and C

4

are the constants to be determined.

By substituting Eq. (15) into Eq. (12), the particular solution of Eq. (5) can be

written as

where

From Eqs.(8) and (16)

，

the solution of Eq.(5) can be expressed as

The coefficients dl, d2, d3, and d4 are determined from the boundary conditions.

In view of Eqs.(4) and (17)

，

the transverse displacements of the rectangular plate

can be expressed as

3.3 Boundary conditions

It is assumed that the two opposite edges parallel to the y-direction are simply

supported,and

the

other

two

edges

have

arbitrary

boundary

conditions

such

as

free,

第

6

页

simply supported, or clamped conditions.

The boundary conditions for the remaining two edges(Y=0 and Y=b) are given

as follows:

The simply supported boundary conditions (S) are

The clamped boundary conditions (C) are

The free boundary condition 8(F) are

We consider the simply supported conditions in Eq. (19) as an example

．

Thus,

we have

From

Eq.

(22),

dl,

d2,

d3,

and

d4

can

be

determined,

and

the

results

are

given

in

Appendix A.

4 Results and discussion

In this section, we make numerical studies on the static behavior of a rectangular

plate

with

in- plane

variable

stiffness

under

a

uniform

pressure

(a=b=1

m,v=0.3,

Do=100 Nm,Db=500 kNm, and q=100 kN/m2). The variation of the bending stiffness

is

described

by

Eq.

(2).

For

different

parameters

p

in

Eq.

(2),

we

calculate

the

deflection w and the bending Moments M

X

and M

Y

along the y-direction (at x=a/2) of

a

four-edge

simply

supported

rectangular

plate

and

a

two- opposite-edge

simply

supported plate with the other two edges clamped, respectively. The results are shown

in Figs.2-7, and the following observations can be made:

(i) The present results for a homogeneous plate (p=0) are exactly coincident with

those

obtained

from

the

classical

plate

theory,

which

verifies

the

correctness

of

the

present solution.

(ii) If the flexural rigidities Do and Db satisfy Do

第

7

页

with the stiffness parameter p >0 is smaller than that of the plate with P <0 (see Figs.

2 and 5). This is because the average bending rigidity of the plate with p >0 is larger

than that of the plate with p <0. It can also be seen that the maximum deflection of the

plate is no longer at the center of the plate when p

≠

0.

(iii)

It

can

be

observed

from

Figs.2

and

5

that

the

deflections

of

the

homogeneous plates with the stiffness Do and Db are the upper and lower limits of

the deflections of a plate with variable stiffness from Do to Db.

(iv) For different stiffness parameters p, the variations of the bending moments

M

X

and M

Y

with the coordinate y (at x=a/2) shown in Figs.3, 4, 6, and 7 are similar.

However,

the

maximum

magnitudes

of

the

bending

moments

M

and

My

decrease

with the increase of P(when p >0), and they increase with the increase of the absolute

value of p (when p <0). The maximum bending moments of the plate with different

stiffness parameters p do not occur at the same point.

第

8

页

5 Conclusions

An exact solution of the bending problem is obtained for a thin rectangular plate

subjected to the distributed loads by assuming that the flexural rigidity is of the power

form, and

Poisson’s ratio

is a constant. The validity of the present results is shown by

comparing the present results with those of the classical solution. The influence of the

variable parameters on the deflection and bending moments is studied by numerical

examples. The obtained

solution

can be

used to

assess the validity

and

accuracy of

various approximate theoretical and numerical models of plates with in-plane variable

stiffness.

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