Buckling of Thin Laminated Plates

7/30/2019 Buckling of Thin Laminated Plates

1/85

Uncontrolled When Printed

Buckl ing of Thin Laminated Plates

DOCUMENT NUMBER: RELEASE/REVISION: RELEASE/REVISION DATE:

SDM-25350 A July 1, 2008

CONTENT OWNER:

Certification Methods & Allowables (43-33-N730)

All future revisions to this document shall be approved by the content owner prior to release.

WARNING - This document contains technical data whose export is restricted by the ArmsExport Control Act (Title 22, U.S. C., and Sec 2751, et seq.) or the Export Administration Act

of 1979, as amended, Title 50, U.S.C., App. 2401 et seq. Violations of these export laws are

subject to severe criminal penalties. Disseminate in accordance with provisions of DoDDirective 5230.25.

THE INFORMATION HEREIN CONTAINS EXPORT CONTROLED DATA

UNDER EXPORT CONTROL CLASSIFICATION NUMBER EAR99.

The information contained herein is PROPRIETARY to the Spirit AeroSystems, Inc.and shall not be reproduced or disclosed in whole or in part or used for any purpose except when the user possesses direct, written

authorization from The Spirit AeroSystems, Inc...

BUCKLING OF THIN LAMINATED PLATES

Rev A SDM-25350 Page 1of 85

CAGE Code 4ATM5 PROPRIETARY ECCN EAR99 Technical


2/85


Document Approval Information


Rev A SDM-25350 Page 2 of 85



3/85





REVISION NOTICE

This revision notice provides a brief description of the changes made within thisstandard. This standard should be reviewed in its entirety for the total extent ofthe changes. Technical changes are noted here and in the margin of thestandard with a revision bar. Editorial changes may not be noted.

Revision Description:

This revision includes technical and editorial changes to correct errors andimprove clarity.

Document organization was revised;

Section 3.1 was updated;

Solution for non-orthotropic plates was added in Section 2.4.4;

Solutions for plates under shear loading were removed for methodology revision.


4/85





Table of Contents

Table of Contents.......................................................................................................................4

List of Figures............................................................................................................................6

List of Tables .............................................................................................................................7

1. Introduction..........................................................................................................................9

1.1 Buckling Nature .....................................................................................................9

1.2 Assumptions and Restrictions for Laminates.........................................................9

1.3 Analytical Approach.............................................................................................10

1.4 Plate Loading........................................................................................................11

1.5 Boundary Conditions............................................................................................11

1.6 Manual Organization............................................................................................11

2. Theoretical Basics..............................................................................................................13

2.1 Energy Method .....................................................................................................13

2.2 Boundary Conditions............................................................................................15

2.2.1 Simply Supported Edges .............................................................................15

2.2.2 Clamped Edges............................................................................................15

2.2.3 Free Edges ...................................................................................................16

2.3 Applied Loads ......................................................................................................17

2.3.1 Uniformly Distributed Loads ......................................................................172.3.2 Linearly Distributed Normal Loads ............................................................17

2.4 General Approach.................................................................................................20

2.4.1 Simply Supported Plates .............................................................................20

2.4.2 Plates with Mixed Clamped and Simply Supported Edges .........................22

2.4.3 Plates with One Free Edge ..........................................................................24

2.4.4 Simply Supported Plates with Bending-Twisting Coupling .......................27

2.4.5 Simply Supported Plate Subjected to Linearly Distributed Normal Load ..28

3. Buckling Solutions for Orthotropic Plates.........................................................................30

3.1 Assumptions and definitions ................................................................................303.2 Uniaxial Compression ..........................................................................................35

3.2.1 Analytical Solution for Simply Supported Plate .........................................35

3.2.2. Analytical Solution for Plate with Clamped Edges.....................................36

3.2.3. Analytical Solutions for Plates with Mixed Simply Supported, Clamped

and Free Edges ............................................................................................38

3.3 Biaxial Loading ....................................................................................................49

3.3.1 Analytical Solutions for Simply Supported Plate........................................49


5/85





3.3.2 Analytical Solutions for Plate with Clamped Edges................................... 53

3.3.3 Analytical Solutions for Plates with Mixed Simply Supported and

Clamped Edges ........................................................................................... 57

3.4 Linearly Distributed Normal Loading.................................................................. 63

4. Buckling Analysis of Plates with Bending-Twisting Coupling ........................................ 65

5. Computational Procedure .................................................................................................. 68

6. Example Problems............................................................................................................. 74

6.1 Example 1 Analysis of Simply Supported Plate under Uniaxial Uniformly

Distributed Compression ..................................................................................... 74

6.2 Example 2 Analysis of Plate with Two Loaded Simply Supported Edges and

Non-Loaded Clamped Edges under Uniaxial Uniformly Distributed

Compression ........................................................................................................ 79

6.3 Example 3 Analysis of Plate with Two Opposite Simply Supported and TwoClamped Edges under Biaxial Compression........................................................81

References ...............................................................................................................................85


6/85


List of Figures

Figure 2.1-1. Forces applied to edges of rectangular plate................................................13

Figure 2.2-1. Boundary conditions for plates with simply supported edges. ....................15

Figure 2.2-2. Boundary conditions for plates with clamped edges....................................16

Figure 2.2-3. Boundary conditions for free edges. ............................................................17

Figure 2.3-1. Plate subjected to uniformly distributed loads and .......................18xN yN

Figure 2.3-2. Plate Subjected to linearly distributed normal load. ....................................18





7/85


List of Tables

Table 2.3-1. Shapes of Normal Load Linear Distribution.................................................19

Table 2.4-1. Deflection Modes for Plates with Simply Supported Edges.........................21

Table 2.4-2. Deflection Modes for Plates with Simply Supported Edge and Clamped

Opposite Edge. ..................................................................................................................24

Table 2.4-3. Deflection Modes for Plates with Clamped Edges. ...................................... 25

Table 2.4-4. Approximate expressions for parameters ,,, 431 and 5 . ................26

Table 3.1-1. Plate Boundary Conditions and Loading Configurations ............................. 32

Table 3.2-1. Simply Supported Plate under Uniaxial Compression..................................35

Table 3.2-2. Plate with Clamped Edges under Uniaxial Compression ............................. 36

Table 3.2-3. Plate with Two Loaded Simply Supported Edges and Non-Loaded

Clamped and Simply Supported Edges under Uniformly Distributed Compression........38Table 3.2-4. Plate with Loaded Simply Supported and Clamped Edges and Two Non-

Loaded Simply Supported Edges under Uniformly Distributed Compression .................39

Table 3.2-5. Plate with Two Loaded Simply Supported Edges and Non-Loaded Simply

Supported and Free Edges under Uniformly Distributed Compression............................40


Clamped Edges under Uniformly Distributed Compression.............................................41

Table 3.2-7. Plate with Two Loaded Clamped Edges and Non-Loaded Simply

Supported Edges under Uniformly Distributed Compression...........................................42

Table 3.2-8. Plate with Loaded Simply Supported and Clamped Edges and Non-Loaded

Simply Supported and Clamped Edges under Uniformly Distributed Compression ........43

Table 3.2-9. Plate with Loaded Simply Supported and Clamped Edges and Non-Loaded

Simply Supported and Free Edges under Uniformly Distributed Compression ...............44


Clamped and Free Edges under Uniformly Distributed Compression..............................45


Supported and Clamped Edges under Uniformly Distributed Compression.....................46

Table 3.2-12. Plate with Loaded Clamped and Simply Supported Edges and Two Non-

Loaded Clamped Edges under Uniformly Distributed Compression................................47


Supported and Free Edges under Uniformly Distributed Compression............................48

Table 3.3-1. Simply Supported Plate under Biaxial Compression....................................49Table 3.3-2. Simply Supported Plate under Biaxial Load - Compression and Tension....51

Table 3.3-3. Plate with Clamped Edges under Biaxial Compression................................53

Table 3.3-4. Plate with Clamped Edges under Biaxial Load - Compression and Tension55

Table 3.3-5. Plate with Two Opposite Simply Supported and Two Clamped Edges

under Biaxial Compression ............................................................................................... 57





8/85






under Biaxial Load - Compression and Tension................................................................59


under Biaxial Load Tension and Compression...............................................................61Table 3.4-1. Simply Supported Plate under Uniaxial Linearly Distributed Load .............63

Table 4-1. Simply Supported Plate with Bending-Twisting Coupling under Uniaxial

Compression ......................................................................................................................66


9/85





1. Introduction

This manual is dedicated to buckling analysis of thin laminated plates subjected to either

uniaxial compression or biaxial loading. Buckling of plates can cause redistribution of loads inthe structure or even its complete failure.

1.1 Buckling Nature

The critical buckling load for a thin flat plate is a function of plate dimensions, boundary

conditions and laminate properties. If the magnitude of the applied in-plane loads is low, the

median surface of the plate will remain flat and in a state of equilibrium. If such a plate is

subjected to a transverse load, it will induce transverse plate deflection. After removal of thetransverse load the plate will return to its initial condition. Such a form of equilibrium is called

stable.

With increase of applied in-plane loads magnitude, the plate equilibrium becomes unstable.Under this condition very small transverse loads or plate imperfections can cause largetransverse deflections of the plate. This transition of the plate stable equilibrium to unstable is

called buckling. The load under which this phenomenon occurs is called the critical buckling

load.

A plate can have several buckling modes depending on the plate aspect ratio, each having adifferent buckling load. However, only the lowest value of the buckling load, called critical

buckling load, is a matter of practical interest.

1.2 Assumptions and Restrictions for Laminates

Only buckling of flat thin laminated rectangular plates is considered in this manual. Solutionspresented in this document are valid only for symmetric balanced laminates; i.e. laminates with

plies on both sides of the plate symmetric about plate midplane having identical properties andorientation. The main assumptions and restrictions for the laminate buckling analysis are [6]:

The laminate is presumed to consist of perfectly bonded layers (lamina).

Each layer (lamina) of the laminate is quasi-homogeneous and orthotropic.

Interlaminar bonds are assumed to be infinitesimally thin and non-deformable by shear.

The laminate acts as a single layer of material.

The length and width of the laminate is much larger than its thickness.

The laminate is loaded in its plane only (i.e. no through-thickness loads).

The laminate and its layers are in a plane stress state (except the edge area).

All displacements are small in comparison with the thickness of the laminate.

Displacements are continuous through the laminate.


10/85


In-plane displacements (u and v displacements in thex- andy-directions) vary linearly

through the thickness of the laminate: i.e. they are linear functions of thez-coordinate.

A line straight and normal to the middle surface of the undeformed laminate remainsstraight and normal to the middle surface after deformation. This is equivalent to the

assumption that the transverse shear strains are equal to zero.

Strain displacement and stress strain relations are linear.

Normal distances from the middle surface dont change. It is equivalent to theassumption that the transverse normal stress is equal to zero.

1.3 Analytical Approach

The energy method is used in this manual for buckling analysis of rectangular composite platessubjected to either distributed uniaxial compression or in-plane biaxial loads. In-plane biaxial

loads can be either biaxial compression or compression in one direction and tension inperpendicular direction.

The analytical procedure is based on a Ritz approximation of the plate out-of-plane

displacements to obtain the plate stiffness matrix eigenvalues. Critical buckling loads are

calculated as linear functions of these eigenvalues. This analytical approach can be applied tosolve buckling problems for the most combinations of plate boundary conditions and loading

and can be analyzed in general case by commercial software.

However, for the overwhelming majority of the mentioned combinations, which are of practical

interest, the precise values of the critical buckling loads can be directly obtained, or theacceptable engineering estimationscan be made without extensive numerical algorithms:

For the orthotropic simply supported plate not loaded by shear the analytical expressionof critical buckling load contains only one member of Ritz deflection approximation.

Elements and of D-matrix and in-plane behavior of laminate depend on the

distance between (+) and (-) layers. If laminate is constructed with adjacent to each

other (+) and (-) layers the distance between their mid planes is very small and suchlaminate is close to orthotropic.

16D 26D

So, solutions for orthotropic plates give good approximations of deflections and buckling loadsfor non-orthotropic plates that have balanced symmetrical lay-ups and minimal distance

between (+) and (-) layers.

For laminate with ratio of more accurate analysis methods must be used.

When , special analytical approach [8] is developed for simply supported plate

under uniaxial compression to consider bending-twisting coupling effect.

1.0/ 1116 DD0, 2616 DD

Also the case of linearly distributed in-plane loads normal to the plate edge is considered for

simply supported orthotropic plate using the energy method.





11/85





1.4 Plate Loading

The following types of in-plane loading applied at the plate midplane are considered in this

manual: Uniaxial load normal to the plate edge and uniformly distributed along the edge. Uniaxial load normal to the plate edge and linearly distributed along the edge. Combination of normal uniaxial loads in two directions (biaxial loading). In this case

one of loads can be tensile, but another one must be compressive.

1.5 Boundary Conditions

The following plate boundary conditions are considered in this manual:

Simply Supported. Clamped, or Built-In. Free.

Analysis cases for orthotropic plates are combined into following sections according to the

loading:

Uniaxial compression. Biaxial loading. Linearly distributed loads.

The first two sections are divided into subsections according the boundary conditions:

Plates with all four edges simply supported. Plates with all four edges clamped. Mixed boundary conditions.

The following designation for boundary conditions is used in this manual: S for a simply

supported edge, C for a clamped edge, and F for a free edge. For example, SCSF

describes a plate with a simply supported first edge, a clamped second edge, a simply supportedthird edge, and a free fourth edge, i.e. a plate with two opposite simply supported edges, with

clamped and free other edges. In this document, the edges are labeled in counter-clockwise

direction starting from the coordinate system origin.

1.6 Manual Organization

The manual includes the following main parts: Brief description of the analytical approach to the solution of buckling problems for flat

composite plates.

Available solutions of buckling problems for flat composite plates presented in tabularform.

Step-by-step computational procedure for the definition of critical buckling loads forflat composite plates.


12/85





Example problems:o Plate with all four edges simply supported and subjected to in-plane uniaxial

uniformly distributed compression.

o Plate with two opposite edges simply supported and two edges clamped subjected toin-plane uniaxial compression uniformly distributed along simply supported edges.

o Plate with two opposite edges simply supported and two edges clamped subjected toin-plane biaxial uniformly distributed compression.


13/85


2. Theoretical Basics

2.1 Energy Method

The energy method is widely used for plate buckling analysis ([1] [5]). For an illustration of

this method the rectangular simply supported plate with dimensions aand b (Figure 2.1-1) and

symmetric layout ([ ] ) is considered.[ ]0=B

The plate in-plane loading is shown in Figure 2.1-1. The loads applied to the edges include

uniformly distributed compression loads in two directions and . The internal in-plane

distributed forces are proportional to the edge loads:

xN yN

xyyx qqq ,,

xyxyyyxx NqNqNq === (2.1.1)

Figure 2.1-1. Forces applied to edges of rectangular plate.

As the load increases, it reaches the value under which the plate buckles. For a buckled plate

the load parameteris denoted cr and is obtained by the energy method.

The plate strain energy is

[

] dydxyx

w

y

wD

yx

w

x

wD

y

w

x

wD

yx

wD

y

wD

x

wDU

a b

+

+

+

+

+

=

02

2

02

26

02

2

02

162

02

2

02

12

0 0

202

66

2

2

02

22

2

2

02

11

222

42

1

(2.1.2)

where - bending stiffness matrix components.261612662211 ,,,,, DDDDDD





14/85


The potential energy of the external in-plane loads is:

dydxy

w

x

w

Ny

w

Nx

w

N

a b

xyyx

+

+

= 0 0

002

02

0

22

(2.1.3)

where the deflection of the plate mid-plane.0

w

The deflection of the plate mid-plane is presented in the form of double series

( ) ( )yYxXww nmM

m

N

n

mn= =

=1 1

0(2.1.4)

where - unknown amplitude,mn

w

( )xXm and - displacement functions in directions parallel to the plate edgesand satisfying the boundary conditions,

( )yYn

M,N - number of series terms chosen to obtain the reasonable analysis accuracy.

For different boundary conditions functions ( )xXm and ( )yYn usually adopt the shape of afreely vibrating beam.

Constants are defined using the principal of stationary potential energy:mnw

0)(

=

+=

mnmn w

U

w

P(2.1.5)

Substituting (2.1.4) into the expressions for0w U (2.1.2.) and (2.1.3) and differentiating

(2.1.5) results in the system of algebraic equations for determination of the eigenvalues :

=

==

= = Nnj

MmiwbG mnijmn

M

m

N

n

ijmn,,3,2,1,

,,3,2,1,0)(

1 1 K

K

(2.1.6)

or

( )

=

==NM

l

lklklkl NMkwbwG1

,,3,2,10 K (2.1.7)

where ( )

=

=+=

Nn

MmnNmk

,,3,2,1

,,3,2,11

K

K

( )

=

=+=

Nj

MijNil

,,3,2,1

,,3,2,11

K

K





15/85


In general case the values offor the buckled plate are the eigenvalues of (2.1.6), and can becalculated by commercial software.

2.2 Boundary Conditions

2.2.1 Simply Supported Edges

The following boundary conditions must be satisfied for simply supported edges (Figure 2.2-1)

Deflection in direction normal to the plate (z-direction, Figure 2.2-1) must be equal to zeroalong the simply supported edges:

00 =w

Moments about the plate edges must be equal to zero:

For 0=x and ax = 002

02

122

02

11 =

+

=ywD

xwDMx

For and0=y by = 002

02

222

02

12 =

+

=

y

wD

x

wDMy

Figure 2.2-1. Boundary conditions for plates with simply supported edges.

2.2.2 Clamped Edges

The following boundary conditions must be satisfied for clamped edges (Figure 2.2-2):

Deflection in direction normal to the plate (z-direction, Figure 2.2-2) must be equal to zeroalong the clamped edges:

00 =w

Angle of rotation of the clamped edge must be equal to zero:





16/85


For 0=x and ax = 00

=

x

w

For and0=y by = 00

=

y

w

x

y

z

(0,0)

(0,b)

(a,0)

(a,b)

C

C C

C

w0=0

00 = yw

w0=0

00 = yw

w0=0

00 = xw

w0=0

00 = xw

Figure 2.2-2. Boundary conditions for plates with clamped edges.

2.2.3 Free Edges

The following boundary conditions must be satisfied for free edges (Figure 2.2-3):

Moment about the plate free edge must be equal to zero:

For 0=x and ax = 002

02

122

02

11 =

+

=

y

wD

x

wDMx

For and0=y by = 002

02

222

02

12 =

+

=

y

wD

x

wDMy

Shear force on the plate free edge must be equal to zero:

For 0=x and ax =

( ) 020 203

66123

03

11 =++= yxwDD

xwDQxz

For and0=y by =

( ) 0203

03

222

03

6612 =

+

+=

y

wD

yx

wDDQyz





17/85


x

y

z

(0,0)

(0,b)

(a,0)

(a,b)

SorC

FMx=0

Qxz=0

SorC

SorC

y

z

(0,0)

(0,b)

(a,0)

(a,b)

SorC

FMy=0

Qyz=0

SorC

SorC

x

Figure 2.2-3. Boundary conditions for free edges.

2.3 Applied Loads

2.3.1 Uniformly Distributed Loads

The following in-plane plate uniformly distributed loading is considered in this manual

(Figure 2.3-1):

Normal load (compression or tension),xN

Normal load (compression or tension),yN

Combinations of the above loads (if one of loads is tension another one must becompression).

Under biaxial loading, if one of normal loads or is tension, the sign before the

buckling load formula component containing this tensile load should be changed to minus.

xN yN

2.3.2 Linearly Distributed Normal Loads

Linearly distributed in-plane load is a combination of uniformly distributed compression (or

tension) and in-plane bending (Figure 2.3-2).





18/85


Ny

b

y

x

Ny

Nx

a

Nx

Figure 2.3-1. Plate subjected to uniformly distributed loads and .xN yN

a

b

x

y

Nx Nx

Uniformly

Distributed

Compression

In-plane

Bending

Combined

Load

Load Components

Figure 2.3-2. Plate Subjected to linearly distributed normal load.





19/85


The load linearly distributed along the edge can be expressed as [3]:

=

b

ykNNx 10 (2.3.1)

Different cases of the linear load distribution are shown in Table 2.3-1.

Table 2.3-1. Shapes of Normal Load Linear Distribution.

Parameter kShape of Load Distribution

along the EdgeLoad Case

0=

k

Uniformly Distributed

Compression




10


20/85


2.4 General Approach

2.4.1 Simply Supported Plates

For the simply supported plate, the deflection should be zero along the edges:

=

=

=

=

=

axandby

axandy

byandax

byandx

atw

0

00

0

00

00 (2.4.1)

The boundary conditions (2.2.1) are satisfied by the following functions of deflection:

( ) a

xm

xXm

sin=

( ) b

yn

yYn

sin=

( )mXm sin=




or ( )nYn sin=

b

y

a

x== , - relative coordinates.

So, the deflection (2.1.4) for the simply supported plate is:

b

yn

a

xmww

M

m

N

n

mn

sinsin

1 1

0 = =

= (2.4.2)

or

( ) ( nmwwM

m

N

n

mn sinsin1 1

0 = =

= )

Indexes mand n indicate the modes of displacement, i.e. number of half waves inx- andy-directions (Table 2.4-1).

The general form of matrix elements and (2.1.7) is given in [klG klb 2] and eigenvalues can

be determined from the condition for existence of the nontrivial solution of system:

NMlkbG klkkl == ,...,3,2,1,0)(det

For orthotropic laminates, the bending twisting components of matrix [ ]D are:

02616 ==DD


21/85


Table 2.4-1. Deflection Modes for Plates with Simply Supported Edges

a

xm

Xm

sin=

Harmonic Distribution

of Displacement mX

Boundary Conditions:SSSimply Supported Edges SS




Mode 1, 1=m

Mode 2, 2=m

Mode 3, 3=m

For this type of laminate, the components of the matrix are:klG

( )

+

++

=

4

22

22

6612

4

114 22

4

1

b

nD

b

n

a

mDD

a

mDabGkl (2.4.3)

When the plate is subjected only to normal loads and , i.e.yN 0=xyNxN , the components of

the matrix are:kl

b

+

=

22

2

4

1

b

nN

a

mNabb yxkl (2.4.4)

With these limitations only one member of the series (2.4.2) for is sufficient to obtain the

exact solution and the set of eigenvalues can now be calculated directly.

0w

( )

22

4

22

22

6612

4

112

22

+

+

++

==

bnN

amN

b

nD

b

n

a

mDD

a

mD

yx

mnk (2.4.5)

k crThe lowest eigenvalue , denoted , gives the lowest, or critical, buckling load

ycrcryxcr

crx NqNq == (2.4.6)


22/85


In the case for which one of the normal loads, or , is compression and the other is

tension, the tensile load should be used in formula (2.4.5) with a minus sign.

yNxN

The solutions for orthotropic plate under the uniformly distributed uniaxial load, biaxialcompression load, biaxial with compression in one direction and tension in another are shownin Table 3.2-1, Table 3.3-1, and Table 3.3-2 accordingly (Configuration numbers 1, 14, 15 in

Table 3.1-1).

2.4.2 Plates with Mixed Clamped and Simply Supported Edges

In this section, plates with mixed clamped and simply supported edges are considered. Each

edge can be either clamped or simply supported. The plate lay-up is symmetric and balanced.

The plate is subjected only to in-plane normal loads and .yNxN

0===xyyyxx

qNqNq (2.4.7)

The eigenvalues solution for this type of boundary condition is:

2

5

2

4

4

43

222

5

2

46612

4

111 )2(2

bN

aN

bD

baDD

aD

yx

mn

+

+++

= (2.4.8)




Parameters ,,, 431 and 5 in the above expression (2.4.8) are:

4

1

0

2

2

2

23

1

=

dY

c

n

y

4

1

0

2

2

2

21

1

=

dX

c

m

x

(2.4.9)

=

21

0

24

1 m

x

X

c

=

21

025

1 n

y

Y

c

where

=

1

0

2mx Xc =

1

0

2ny Yc (2.4.10)

- displacement functions depending on the type of edge supports.nm YX ,

b

y

a

x== , - relative coordinates.


23/85


For a plate with one simply supported edge and the opposite edge clamped the displacement

function inx-direction is:

( ) ( ) ( ) ( )

mmmmmmmX sinhsincoshcos += (2.4.11)

mThe exact values for can be obtained from the equation

0tanhtan = mm

+

4

1mmApproximate value:

m are defined from the conditionValues




0=mX for 1=

and will be

mm

mmm

coscosh

sinhsin

=

Deflection modes for this type of boundary conditions are shown in Table 2.4-2.

A similar solution can be obtained for the displacement function in they-direction.nY

When both opposite edges of the plate are clamped the displacement function inx-direction is:

( ) ( ) ( ) ( ) mmmmmmmX coshcossinhsin ++= (2.4.12)

mExact values for can be obtained from the equation:

( ) ( ) 1coshcos =mm

+

2

1mmApproximate value:

m are defined from the condition:The values

0=mX for 1=

and will be

mm

mmm

sinsinh

coshcos

=

Table 2.4-3.Deflection modes for this type of boundary conditions are shown in


24/85





C S

Approximate expressions for parameters ,,, 431 and 5 are given in Table 2.4-4 and

used for obtaining of the critical buckling loads for the plates with all clamped or with a mix of

clamped and simply supported edges (Table 3.1-1, configuration numbers 2-4, 6-8, 11, 12, 16-

20).

2.4.3 Plates with One Free Edge

Like in the previous section the lay-up of the plate is orthotropic and symmetrical. The plate is

subjected only to in-plane normal loads . The one edge parallel toy is free. Other edges can

be simply supported or clamped.xN

If three edges are simply supported (SFSS designation for boundary conditions) the criticalbuckling load can be obtained using one member approximation of the deflection of the plate

by Ritz method. Following L.P. Kollar and G.S. Springer [2], the buckled shape in relative

coordinates is:

( )mXm sin= =nY

and

( ) mww mn sin0 = (2.4.13)

Table 2.4-2. Deflection Modes for Plates with Simply Supported Edge and Clamped OppositeEdge.

( ) (( ) ( )

)

mm

mmmmmX

sinhsin

coshcos

+=Harmonic Distribution

of Displacement mX

Boundary Conditions:

One Edge is Simply Supported and

the Opposite Edgeis Clamped - CS

Mode 1, 1=m

Mode 2, 2=m

Mode 3, 3=m


25/85


Table 2.4-3. Deflection Modes for Plates with Clamped Edges.

( ) (

( ) ( )

)

mm

mmmmmX

coshcos

sinhsin

+

+=Harmonic Distributionof Displacement mX

Boundary Conditions:

C CClamped Edges CC

Mode 1, 1=m

Mode 2, 2=m

Mode 3, 3=m

Substituting (2.4.13) into (2.1.5) results in the approximate closed-form expression for criticalbuckling load:

662112

2 12D

bD

aN

crx +=

(2.4.14)

The solutions, which satisfy boundary conditions on the edges SFSS, SFCS, SFSC, CFCSaccording to [2], are given in Table 3.2-5, Table 3.2-9, Table 3.2-10, Table 3.2-13. In Table3.1-1 configuration numbers of the solutions examined in this Section are 5, 9, 10, 13

accordingly.





26/85





Table 2.4-4. Approximate expressions for parameters ,,, 431 and 5 .

Parameters 1 4 mEdge Supports

m 22m 1,2,3,

+

4

1m ( )111 1,2,3,

4.730 ( )211 1

+

2

1m ( )211 2,3,4,

Parameters

Edge Supports 3 5 n

n 22n 1,2,3,

+

4

1n ( )133 1,2,3,

4.730 ( )233 1

+

2

1n ( )233 2,3,4,

x

y

SS

xCS

xCC

x

y

S

S

y

xS

C

y

xC

C


27/85


2.4.4 Simply Supported Plates with Bending-Twisting Coupling

0, 2616 DDNon-orthotropic plate with bending-twisting coupling ( ) is considered in this

section. The plate is subjected only to in-plane normal loads . The solution by singlemember approximation series by Ritz cant be obtained, but some estimation of coupling effectcan be made [

xN

8].

Let the deflection (2.1.4) for this plate is defined by function:

( ) 1...,3,2,1,sincos0 ==

= nmkyx

a

m

b

yww mn

(2.4.15)

( ) ( ) ( ) ( )[ ]kykykyxb

yww mmmmmn

cossincossincos0 =or

a

mm

= and value kis much smaller than 1, i.e. 1


28/85


2

222 6

= mm bkwhere

)2(2 66122

2

2224

4

112

0 DDb

Db

DL

m

m +++=

262

2

162

01 3 Db

DL m

+= ,

( ) ( ) )2(6266 66122222222

112

1 DDDb

DkL mmm +++++=

The critical buckling load

( ) 1,,3,2,1min === nmNN mnxcrx K

The above approach takes into account the influence of bending-twisting coupling on buckling

load of simply supported plate under uniaxial compression with certain degree of accuracy.

More accurate estimations can be obtained using multiple members of approximation or

other numerical methods.

0w

02616 ==DDThis solution is given also in Table 3.4-1. Note, if then (2.4.17) givesclassical solution for orthotropic plate.

2.4.5 Simply Supported Plate Subjected to Linearly Distributed Normal Load

The plate lay-up is orthotropic, symmetric and balanced. The plate is subjected only to in-plane

normal loads (Figure 2.3-2), defined by expression (2.3.1)

=

b

ykNNx 10

0k where




The plate linearly distributed normal loading is shown in Table 2.3-1 for different values k.

The energy method is used for definition of the plate buckling load. According to Lekhnitskiis

approximation ([4]) the functions of deflection in (2.1.4) are defined as:0wnm YX ,

=

a

xmXm sin


29/85


+

=

b

ynw

b

ynwYn

2sinsin 21

so

+

=

b

ynw

b

ynw

a

xmw

2sinsinsin 21

0

(2.4.18)

Substituting (2.4.18) and (2.3.1) into the expressions for0w xN U (2.1.2) and (2.1.3) and

minimizing the total potential energy (2.1.5) with respect to the equation to determine

the critical value ([

1, ww 2cr

xN 0 3],[4]) was obtained:

( )2

2

2222

2

0

9

1642

1

=

kk

DbN

m

x

( )( ) ( ) ( )

++

2

221

2

21

2

219

161622

kaaaakaak (2.4.19)

222

12 168 rc

r

ca ++=222

11 2 rc

r

ca ++= where

21

=

b

a

mr K,3,2,1=m

22

66122

2

D

DDc

+=

22

111

D

Dc =

The critical buckling load

( ) K,3,2,1min 00 == mNN mxcrx (2.4.20)

Table 3.4-1 and has the configuration number 21 in Table 3.1-1.This solution is given in





30/85


3. Buckling Solutions for Orthotropic Plates

3.1 Assumptions and definitions

All buckling solutions presented in this section are given for rectangular plates (Figure 2.1-1)

with dimensions a and b in directionsx andy respectively. The following boundary conditionsare considered (see Section 2.2): simply supported, clamped or free edges in differentcombinations.

The main assumptions are:

Plate dimensions: ba The laminate is balanced, i.e. the following components of [ ]ABD matrix are equal to

zero: 026162616 ==== DDAA

The laminate is symmetric, i.e. matrix

[ ]B is a zero matrix and there is no coupling

between in-plane loads and out-of-plane deformations: [ ] [ ]0=B

The plate is loaded in-plane by uniformly or linearly distributed compression (in some cases

tension) forces and .yNxN

Some practical combinations of boundary conditions and loading, for which the solutions are

presented in this section, are given in Table 3.1-1.

The following notation for boundary conditions is used here:

S Simply Supported Edge, C Clamped Edge,

F Free Edge.

The plate boundary conditions are described by a combination of the above symbols reflecting

the boundary conditions on the plate four edges, starting from the edge 0=x and continuingcounterclockwise around the plate. For example, SFSC denotes a plate with a Simply

Supported Edge at 0=x by = 0=x, Free Edge at , Simply Supported Edge at , and Clamped

Edge at .0=y

Formulas for buckling loads in tables of Section 3 contain indexes and , which

are the numbers of half waves along edges a and b. The critical buckling load depends on the

plate aspect ratio

yx NN , m n

bas = . The minimum number of half waves n is supposed to occuralong the short edge b, i.e. .1=n

The number of half waves along edge a can be evaluated [7] as:

( ) ( )1111

22

2

2

+


31/85


[ ]Dwhere are elements of stiffness matrix2211 ,DD .

For example:

2=s 1/ 1122 =DD

( ) ( 1224122 +< )


32/85


Table 3.1-1. Plate Boundary Conditions and Loading Configurations

Config. SolutionLocationIn-Plane Loading Boundary Conditions

No.




1 Compression xN SSSS Table 3.2-1S

S

S

S

2 Compression xN CCCC Table 3.2-2C

C

C

C

3 Compression xN SSSC Table 3.2-3S

S

C

S

4 Compression xN SSCS Table 3.2-4S

C

S

S

5 Compression xN SFSS Table 3.2-5F

S

S

S

6 Compression xN SCSC Table 3.2-6C

S

C

S

7 Compression xN CSCS Table 3.2-7S

C

S

C

8 Compression xN SSCC Table 3.2-8S

C

C

S


33/85


Table 3.1-1 (continued)

Config. Solution

LocationIn-Plane Loading Boundary ConditionsNo.




9 Compression xN SFCS Table 3.2-9

10 Compression xN SFSC Table 3.2-10

11 Compression xN CSCC Table 3.2-11

12 Compression xN CCSC Table 3.2-12

13 Compression xN CFCS Table 3.2-13

14Compression xN

Compression yNTable 3.3-1

15Compression xN

Tension yN

SSSS

Table 3.3-2

16Compression xNCompression yN

Table 3.3-3

17Compression xN

Tension yN

CCCC

Table 3.3-4

a

C

S

S

F

S

C

S

S

C

C

C

CS

C

C

F

C

S

C

S

S

S

S

CC

C

C


34/85



Config. Solution

LocationIn-Plane Loading Boundary ConditionsNo.




18Compression xN

Compression yNTable 3.3-5

19Compression xN

Tension yNTable 3.3-6

C

S

C

SSCSC

Tension Nx20 Table 3.3-7

Compression yN

21

Linearly

Distributed

xN SSSS Table 3.4-1

S

S

S

S


35/85


3.2 Uniaxial Compression

3.2.1 Analytical Solution for Simply Supported Plate

Table 3.2-1. Simply Supported Plate under Uniaxial Compression

x




Plate Shape Rectangular

Laminate Type Symmetric, Balanced

Loading Uniaxial Uniformly Distributed Compression Load xN

Boundary

Conditions:,0 ax = 00 == xMw :,0 by = 00 == yMw

Solution

+

++

=

=

24

22

2

6612

2

112 )2(2

m

a

b

nD

b

nDD

a

mD

Nmn

x

,...3,2,1, =nm

Critical Load ( ) K,3,2,1,min == nmNN mnxcrx

Reference[1], p. 304

[2], p. 123

Nx bS S

S

S

Nx

a

y


36/85


3.2.2. Analytical Solution for Plate with Clamped Edges

Table 3.2-2. Plate with Clamped Edges under Uniaxial Compression

x






Loading Uniaxial Uniformly Distributed Load xN

Boundary

Conditions

:,0 ax = 00 ' == xww

:,0 by = 00 ' == yww

Parameters

=

+

==

K,4,3,22

1

1730.4

1 mm

m

( )2114 =

=

+

==

K,4,3,22

1

1730.4

3 nn

n

( )2335 =

Solution

+++

=

4

3222

5

2

46612

4

111

4

2)2(2

bD

baDD

aDaN

mnx

Nx bC C

C

C

Nx

a

y


37/85



( ) K,3,2,1,min == nmNN mnxcrxCritical Load

Reference [2], p. 119-121





38/85


3.2.3. Analytical Solutions for Plates with Mixed Simply Supported, Clamped and Free Edges

Table 3.2-3. Plate with Two Loaded Simply Supported Edges and Non-Loaded Clamped and

Simply Supported Edges under Uniformly Distributed Compression

x







Boundary

Conditions

:,0 ax = 00 == xMw

:0=

y 00

' ==yww

:by = 00 == yMw

Parameters

m=1 ( )2

4 m=

+=

4

13 n ( )1335 = ,...3,2,1, =nm

Solution

+++

=

4

3222

5

2

46612

4

111

4

2

)2(2

b

D

ba

DD

a

Da

Nmn

x


Reference [2], p. 119-121, 123

Nx bS S

S

C

Nx

a

y


39/85


Table 3.2-4. Plate with Loaded Simply Supported and Clamped Edges and Two Non-Loaded

Simply Supported Edges under Uniformly Distributed Compression

x







Boundary

Conditions

:0=x 00 == xMw

:ax = 00 ' == xww

:,0 by = 00 == yMw

Parameters

+=

4

11 m ( )1114 =

n=3 ( )2

5 n= ,...3,2,1, =nm

Solution

+++

=

4

3

222

5

2

46612

4

111

4

2

)2(2b

Dba

DDa

Da

Nmnx



Nx bS C

S

S

Nx

a

y


40/85


Table 3.2-5. Plate with Two Loaded Simply Supported Edges and Non-Loaded Simply

Supported and Free Edges under Uniformly Distributed Compression

x





LaminateType

Symmetric, Balanced


BoundaryConditions

:,0 ax = 00 == xMw

:0=y 00 == yMw

:by = 00 == yzy QM

Critical Load662112

2 12D

bD

aNcrx +=

Reference [2], p. 125

Nx bS S

F

S

Nx

a

y


41/85


Table 3.2-6. Plate with Two Loaded Simply Supported Edges and Non-Loaded Clamped Edges

under Uniformly Distributed Compression

x







Boundary

Conditions

:,0 ax = 00 == xMw

:,0 by = 00 ' == yww

Parameters

m=1 ( )24 m= ,...3,2,1=m

=

+

==

K,4,3,22

1

1730.4

3 nn

n

( 2335 )=

Solution

+++

=

4

3222

5

2

46612

4

111

4

2

)2(2b

Dba

DDa

Da

Nmn

x

Critical Load ( ) K,3,2,1,min == nmNN mnxcrx Reference [2], p. 119-121

Nx bS S

C

C

Nx

a

y


42/85


Table 3.2-7. Plate with Two Loaded Clamped Edges and Non-Loaded Simply Supported Edges

under Uniformly Distributed Compression

x







Boundary

Conditions

:,0 ax = 00 ' == xww

:,0 by = 00 == yMw

Parameters

=

+

==K,4,3,2

2

11730.4

1 mm

m

( )2114 =

n=3 ( )2

5 n= ,...3,2,1=n

Solution

+++

=

4

3222

5

2

46612

4

111

4

2

)2(2b

Dba

DDa

Da

Nmn

x

Critical Load ( ) K,3,2,1,min == nmNNmn

xcr

x


Nx bC C

S

S

Nx

a

y


43/85


Table 3.2-8. Plate with Loaded Simply Supported and Clamped Edges and Non-Loaded Simply

Supported and Clamped Edges under Uniformly Distributed Compression

x







BoundaryConditions

:0=x 00 == xMw

:ax = 00 ' == xww

:0=y 00 ' == yww

:by = 00 == yMw

Parameters

+=

4

11 m ( )1114 =

+=

4

13 n ( )1335 = ,...3,2,1, =nm

Solution

+++

=

4

3

222

5

2

4

6612

4

1

114

2

)2(2 bDbaDDaD

a

Nmn

x



Nx bS C

S

C

Nx

a

y


44/85


Table 3.2-9. Plate with Loaded Simply Supported and Clamped Edges and Non-Loaded Simply

Supported and Free Edges under Uniformly Distributed Compression

x







BoundaryConditions

:0=x 00 == xMw

:ax = 00 ' == xww

:0=y 00 == yMw

:by = 00 == yzy QM

Critical Load( ) 662112

2 12

7.0D

bD

aN

crx +=


Nx bS C

F

S

Nx

a

y


45/85


Table 3.2-10. Plate with Two Loaded Simply Supported Edges and Non-Loaded Clamped and

Free Edges under Uniformly Distributed Compression

x







Boundary

Conditions

:,0 ax = 00 == xMw

:0=y 00 ' == yww

:by = 00 == yzy QM

Solution 6622242

2

112

2

2 12

4

5D

bD

bm

aDm

aNmx ++=

,...3,2,1=m

Critical Load ( ) K,3,2,1min == mNN mxcrx


Nx bS S

F

C

Nx

a

y


46/85


Table 3.2-11. Plate with Two Loaded Clamped Edges and Non-Loaded Simply Supported and

Clamped Edges under Uniformly Distributed Compression

x







Boundary

Conditions

:,0 ax = 00 ' == xww

:0=y 00 ' == yww

:by = 00 ==y

Mw

Parameters

=

+

==

K,4,3,22

1

1730.4

1 mm

m

( )2114 =

+=

4

13 n ( )1335 = ,...3,2,1=n

Solution

+++

=

4

3

222

5

2

4

6612

4

1

114

2

)2(2 bDbaDDaD

a

Nmn

x



Nx bC C

S

C

Nx

a

y


47/85


Table 3.2-12. Plate with Loaded Clamped and Simply Supported Edges and Two Non-Loaded

Clamped Edges under Uniformly Distributed Compression

x

Nx

a

bC S

C

C

Nx

y







Boundary

Conditions

:0=x 00 ' == xww

:ax = 00 == xMw

:,0 by = 00 ' == yww

+=

4

11 m

Parameters

( )1114 = ,...3,2,1=m

=

+

==

K,4,3,22

1

1730.4

3 nn

n

( )2335 =

+++

=

4

3

222

5

2

4

6612

4

1

114

2

)2(2b

Dba

DDa

Da

Nmn

x

Solution




48/85


Table 3.2-13. Plate with Two Loaded Clamped Edges and Non-Loaded Simply Supported and

Free Edges under Uniformly Distributed Compression

x







Boundary

Conditions

:,0 ax = 00 ' == xww

:0=y 00 == yMw

:by = 00 == yzy QM

Critical Load662112

2 124D

bD

aNcrx +=


Nx bC C

F

S

Nx

a

y


49/85


3.3 Biaxial Loading

3.3.1 Analytical Solutions for Simply Supported Plate

Table 3.3-1. Simply Supported Plate under Biaxial Compression






Loading Uniformly Distributed Compression Loads yx NN ,

BoundaryConditions

:,0 ax = 00 == xMw

:,0 by = 00 == yMw

Solution

( )

22

4

22

22

6612

4

112

22

+

+

++

=

bn

N

N

am

b

nD

b

n

a

mDD

a

mD

N

x

y

mnx

y

S

S

S

S Nx

x

Nx

Ny

b

Nya


50/85



or

( )

22

4

22

22

6612

4

112

22

+

+

++

=

b

n

a

m

N

N

b

nD

b

n

a

mDD

a

mD

N

y

x

mny Solution

,...3,2,1, =nm

( )mnxcrx NN min= orCritical Load

( ) K,3,2,1,min == nmNN mnycry






51/85


Table 3.3-2. Simply Supported Plate under Biaxial Load - Compression and Tension






LoadingUniformly Distributed Loads:

- Compression, - TensionxN yN

Boundary

Conditions

:,0 ax = 00 == xMw :,0 by = 00 == yMw

Condition ofSolution

Existence x

y

N

N

b

a

n

m> m,n = 1, 2, 3,

Solution

( )

22

4

22

22

6612

4

112

22

+

++

=

b

n

N

N

a

m

b

nD

b

n

a

mDD

a

mD

N

x

y

mn

x

,...3,2,1, =nm

y

S

S

S

S Nx

x

Nx

Ny

Ny

b

a


52/85



( )K

,3,2,1,min==

nmNN

mn

x

cr

xCritical Load






53/85


3.3.2 Analytical Solutions for Plate with Clamped Edges

Table 3.3-3. Plate with Clamped Edges under Biaxial Compression







Boundary

Conditions

:,0 ax = 00 ' == xww

:,0 by = 00 ' == yww

Parameters

2

=

+

==

K,4,3,21

1730.4

1 mm

m

( )2114 =

=

+

==

K,4,3,22

1 1730.43 nn

n

( )2335 =

y

C

C

C

C Nx

x

Nx

Ny

Ny

a

b


54/85



( )

2

5

2

4

4

3222

524

6612

4

111 22

bN

N

a

bD

baDD

aD

N

x

y

mnx

+

+++=

orSolution

( )

2

5

2

4

4

3222

5

2

46612

4

111 22

baN

N

bD

baDD

aD

N

y

x

mny

+

+++

=

( )mnxcrx NN min= orCritical Load

( ) K,3,2,1,min == nmNN mnycry






55/85


Table 3.3-4. Plate with Clamped Edges under Biaxial Load - Compression and Tension








Boundary

Conditions

:,0 ax = 00 ' == xww

:,0 by = 00 ' == yww

Parameters

=

+

==

K,4,3,22

1

1730.4

1 mm

m

( )2114 =

=

+

==

K,4,3,2

2

1

1730.4

3 nn

n

( )2335 =

Condition of

SolutionExistence

2

5

4

>

b

a

N

N

x

y

y

C

C

C

C Nx

x

Nx

Ny

b

Ny

a


56/85



( )

2

5

2

4

4

3222

524

6612

4

111 22

bN

N

a

bD

baDD

aD

N

x

y

mnx

+++

=Solution







57/85


3.3.3 Analytical Solutions for Plates with Mixed Simply Supported and Clamped Edges

Table 3.3-5. Plate with Two Opposite Simply Supported and Two Clamped Edges under

Biaxial Compression







BoundaryConditions

:,0 ax = 00 == xMw

:,0 by = 00 ' == yww

Parameters

m=1 ( )2

4 m= K,3,2,1=m

=

+

==

K

,4,3,22

1

1730.4

3

nn

n

( )2335 =

y

C

S

C

S Nx

x

Nx

Ny

b

Ny

a


58/85



( )

2

5

2

4

4

3222

524

6612

4

111 22

bN

N

a

bD

baDD

aD

N

x

y

mn

x

+

+++

=

Solution or

( )

2

5

2

4

4

3222

5

2

46612

4

111 22

baN

N

bD

baDD

aD

N

y

x

mn

y

+

+++

=

( )mnccrx NN min= orCritical Load

K,3,2,1,min == nmNN mnycr

y






59/85



Biaxial Load - Compression and Tension








BoundaryConditions

:,0 ax = 00 == xMw

:,0 by = 00 ' == yww

Parameters

m=1 ( )2

4 m= K,3,2,1=m

=

+

==

K,4,3,22

1

1730.4

3 nn

n

( )2335 =


Existence

2

5

4

>

b

a

N

N

x

y

y

C

S

C

S Nx

x

Nx

Ny

b

Nya


60/85



( )

2

5

2

4

4

3222

524

6612

4

111 22

bN

N

a

bD

baDD

aD

N

x

y

mnx

+++

=Solution







61/85



Biaxial Load Tension and Compression







- Tension, - CompressionxN yN

BoundaryConditions

:,0 ax = 00 == xMw

:,0 by = 00 ' == yww

Parameters

m=1 ( )2

4 m= K,3,2,1=m

=

+

==

K,4,3,22

1

1730.4

3 nn

n

( )2335 =


Existence

2

4

5

>

b

a

N

N

y

x

y

C

S

C

S Nx

x

Nx

Ny

b

Nya


62/85



( )

2

4

2

5

4

3222

524

6612

4

111 22

aN

N

b

bD

baDD

aD

N

y

x

mny

+++

=Solution

( ) K,3,2,1,min == nmNN mnycry Critical Load






63/85


3.4 Linearly Distributed Normal Loading

Table 3.4-1. Simply Supported Plate under Uniaxial Linearly Distributed Load

N0

N

N0

N

x






Loading

Linearly Distributed Load

=

b

yNNx 10 20 (see Section 2.4.5)

BoundaryConditions

:,0 ax = 00 == xMw :,0 by = 00 == yMw

Parameters

K,3,2,11

2

=

= m

b

a

mr

22

66122

22

111

2

D

DDc

D

Dc

+==

222

12

222

11 1682 rc

r

carc

r

ca ++=++=

xx

y

S S

S

S

b

a


64/85



( )

= 2

2

2

222

2

0

9

1642

1

k

k

Db

Nmx

Solution

( )( ) ( ) ( )

++

2

221

2

21

2

219

161622

kaaaakaak

( ) K,3,2,1min 00 == mNN mxcrxCriticalLoad

[3], p. 43Reference

[4], p. 462





65/85


4. Buckling Analysis of Plates with Bending-Twisting Coupling

The solution presented in this section is given for rectangular plate (Figure 2.1-1) with

dimensions a and b in directionsx andy respectively. Plate with all edges simply supported isconsidered (see Section 2.2).

The main assumptions are:

Plate dimensions: ba The laminate is non orthotropic, i.e. the following components of [ matrix arent

equal to zero:

]D0, 2616 DD

The laminate is symmetric, i.e. matrix [ ]B is a zero matrix and there is no couplingbetween in-plane loads and out-of-plane deformations: [ ] [ ]0=B

The plate is loaded by in-plane uniformly distributed compression force . In addition to theorthotropic plate boundary conditions and loading configurations this load case will be denotedas configuration number 22.

xN

The solution is given in Table 4-1.





66/85


Table 3.4-1. Simply Supported Plate with Bending-Twisting Coupling under Uniaxial

Compression

x





Laminate Type Symmetric, Not Orthotropic

Loading Uniaxial Uniformly Distributed Compression Load xN

Boundary

Conditions

:,0 ax = 00 == xMw

:,0 by = 00 == yMw

Parameters

am

m =

2

222 6

= mm bk

)2(2 66122

2

2224

4

112

0 DDb

Db

DLm

m +++=

( ) +++= 22222

112

1 66Db

DkL mm

( ) )2(62 661222 DDm +++

262

2

162

01 3 Db

DL m += K,3,2,1=m

Nx bS S

S

S

Nx

a

y


67/85


Table 4.1 (continued)

+=

m

mx

kLLN 10

21

Solution

+

10

0110 1921

2

1

LkL

Lk

k

LL

m

m

m

( ) K,3,2,1min == mNN mxcrxCritical Load

Reference [8]





68/85


5. Computational Procedure

Initial Data

1.Plate properties:

a.Plate dimensions a, b

Plate aspect ratio bas =

b.Laminate stacking sequence:

- Number of plies n

- Orientation of plies in reference to the laminate nkk ,...,1= coordinate system

- Material IDs for plies nk ,...,1=

c.Material properties for every material type in referenceto the principal material axes

- Lamina elastic modulus in directions 1 and 2

- Lamina shear modulus

21,EE

21G

- Poissons ratio 21

1

22112E

E =

- Thicknesses of plies plyt2.Plate in-plane loading

a. Uniformly distributed compression load xN

b. Linearly distributed normal load xN

c. Biaxial uniformly distributed normal loads yx NN ,

3.Plate boundary condition code A




1A A2 3A4

Each of the code symbolsA can be (see Section 3.1):iS Simply supported edge,

C Clamped edge,F Free edge.

The code symbols start from the edgex = 0 and continues counterclockwise around the plate.


69/85


Solution

>

Calculation of the components of the reduced stiffness matrix in reference to the principal

material axes (see Section 2.5 of Reference [6]) for each material

[ ]

=

66

2221

2111

00

0

0

Q

QQ

QQ

Q

The matrix components

1221

2

22 1 =

EQ

1221

1

11 1 =

EQ

1221

112

1221

221

2111

=

=

EEQ 2166 GQ =

If the same material is used for all plies this matrix will be calculated just once.

>

Calculation of the components of the transformed reduced stiffness matrix in reference to the

laminate coordinate system (see Section 2.7 of Reference [6]) for each ply

[ ]

=

666261

622221

612111

QQQ

QQQ

QQQ

Q

For an arbitrary angle of the ply orientation

6622

2122

224

114

11 42 QnmQnmQnQmQ +++=

6622

2122

224

114

22 42 QnmQnmQmQnQ +++=

6622

2144

2222

1122

21 4 QnmQnmQnmQnmQ +++=

6622

2122

223

113

61 2 QnmnmQnmnmQnmQnmQ =

( ) ( ) 6622212222311362 2 QnmnmQnmnmQnmQnmQ ++=





70/85


( ) 66222

2122

2222

1122

66 2 QnmQnmQnmQnmQ ++=

sincos == nmwhere

o0= (no transformation)

01 == nm

1111 QQ = 2222 QQ = 2121 QQ =

061 =Q 062 =Q 6666 QQ =

o90=

10 == nm

2211 QQ = 1122 QQ = 2121 QQ =

061 =Q 062 =Q 6666 QQ =

o45=

2

2== nm

( )662122112211 424

1QQQQQQ +++==

( )6621221121 424

1QQQQQ ++=

( )221162614

1QQQQ ==

( )21221166 24

1QQQQ +=

o45=

2

2

2

2== nm





71/85


( )662122112211 424

1QQQQQQ +++==

( )6621221121 4241

QQQQQ ++=

( )221162614

1QQQQ +==

( )21221166 24

1QQQQ +=

>

Calculation of the [ matrix (see Section 3.5 of Reference []D 6])

[ ] ( )=

=

=n

k

kk

k

zz

QQQ

QQQ

QQQ

DDD

DDD

DDD

D1

31

3

666261

622221

612111

666261

622221

612111

3

1

Matrix components[ ]D

( )[ ]=

=n

k

kkkjiji zzQD

1

31

3

3

1

>

Evaluation of the number of half waves m and n along edges a and b using the guidelinesgiven in Section 3.1.

>

Selection of the plate configuration from Table 3.1-1 using the plate boundary condition codeand the type of plate loading. From Table 3.1-1, using the link related to the chosen plate

configuration go to the corresponding solution table (Tables 3.2-1 3.4.1) of Sections 3 and

4).

>

Calculation of buckling loads for different buckling modes for the selected configuration of the

plate:





72/85


For plates subjected only to uniaxial uniformly distributed compression , only

calculation of buckling loads (Configurations 1-4, 6-8, 11-12) or

(Configuration 10) is required.

xN

mnxN

mxN

For plates with one edge free and the opposite edge simply supported and subjected touniaxial uniformly distributed compression (Configurations 5, 9, 13), the first

buckling mode is critical. Therefore, the critical buckling load only is calculated

for this mode.

xN

crxN

For plates subjected to biaxial uniformly distributed compression loads and ,

analysis of either set of buckling loads or (Configurations 14, 16, 18) can

be performed.

yNxN

mnyN

mnxN

For plates subjected to a combination of uniformly distributed compression and

uniformly distributed tension , calculation of buckling loads only

(Configurations 15, 17, 19) is required.

xNmn

xNyN

For plates subjected to a combination of uniformly distributed compression and

uniformly distributed tension , calculation of buckling loads only

(Configuration 20) is required.

yN

mnyNxN

For plates subjected to a linearly distributed normal load , calculation of buckling

loads only (Configuration 21) is required.

xN

mxN 0

If the plate has at least one clamped edge and the other edges are simply supported(Configurations 2-4, 6-8, 11-12, 16-20), the parameters 431 ,, and 5

calculations are required for the buckling loads analysis.

For the simply supported plates with bending-twisting coupling subjected only touniaxial uniformly distributed compression , the first buckling mode is critical. The

buckling load is calculated for this mode (Configuration 22, Section 4). The

parameters and calculations are required for the buckling loads analysis

(

xN

mxN

10 ,LL 01L

Table 4-1).

>

Calculation of critical buckling loads for corresponding load components:

( ) K,3,2,1,min == nmNN mnxcrx (Configurations 1-4, 6-8, 11-12, 15,17,19)





73/85


( ) K,3,2,1min == mNN mxcrx (Configuration 10, 22)

) K,3,2,1,min == nmNN mnycry (Configuration 20)

) K,3,2,1,min == nmNN mnycry( )mnxcrx NN min= or(Configurations 14, 16, 18)

( ) K,3,2,1min 00 == mNN mxcrx (Configuration 21)Critical buckling loads for plates with a free edge and opposite simply supported edge

(Configurations 5, 9, 13) were calculated at step 6 of the analysis.

>

Calculation of Margin of Safety:

For plates loaded byo Uniaxial uniformly distributed compression (Configurations 1-13, 22)xN

o Combination of uniformly distributed compression and uniformly distributed

tension (Configurations 15, 17, 19)

xN

yN

1.. =x

crx

N

NSM

For plates loaded by combination of uniformly distributed compression and

uniformly distributed tension (Configuration 20)y

N

xN

1.. =y

cry

N

NSM

For plates subjected to biaxial uniformly distributed compression loads and

(Configurations 14, 16, 18)

yNxN

11.. ==y

cry

x

crx

N

N

N

NSM

For plates subjected to a linearly distributed normal load (Configuration 21)xN

1..0

0 =x

crx

N

NSM





74/85


6. Example Problems

The example problems are analyzed using the procedure described in Section 5.

6.1 Example 1 Analysis of Simply Supported Plate under Uniaxial UniformlyDistributed Compression

10

x

Nx 5S SNx

y

S

S

Initial Data

1. Plate Properties:

a. Plate dimensions ina 10=

inb 5=

25

10===

b

as Plate aspect ratio

b. Laminate Stacking Sequence:- Number of plies 5

- Orientation of plies referred to the laminate coordinate system [0,45,90,-45,0]- Material IDs for plies [1,2,2,2,1]

c. Material properties for every material type referred to principal material axes:

Material IDMaterialproperties

Units1 2

psi 8,100,000 20,600,0001E

2E psi 8,100,000 1,130,000

21G psi 700,000 580,000

21 0.060 0.340

12 0.060 0.01865

plyt in 0.0085 0.0074

2. Plate Loading: Uniaxial Uniformly Distributed Compression lb / in00.40=xN

3. Plate Boundary Condition Code: SSSS





75/85


Solution

>

Calculation of the components of the reduced stiffness matrix in reference to the material principal axes:Material 1

( ) ( )psiQQ 8129265

06.01

81000002

1

22

1

11 =

== ( ) psiQ 48775606.01

810000006.02

1

21 =

= ( ) ps

Documents

Buckling of Thin Laminated Plates