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Behaviour of Thin Walled Stiffened Plates under In-Plane Axial Loads
B.Tech Project Report
by Abhijeet Singh (Y4008) Manish Kumar (Y4210)
supervised by Dr. Ashwini Kumar
Department of Civil Engineering
Indian Institute of Technology Kanpur
April, 2008
Certificate
It is certified that the work contained in this report titled “Behaviour of Thin Walled Stiffened Plates under In-Plane Axial Loads” is the original work done by Abhijeet Singh (Y4008) & Manish Kumar (Y4210), and has been carried out under my supervision.
Prof. Ashwini Kumar Supervisor Department of Civil Engineering Indian Institute of Technology Kanpur
Abstract
Unlike other slender structural members like columns, plates have the capability to resist significant amount of compressive axial load after they start to buckle. Advantage can be taken of this special characteristic of plates while determining the allowable load-deflection criteria for the purpose of design of structures. Some established approximate methods of analysis have been used in this project to analyze orthotropic plates for post buckling behaviour. The approach can be easily extended to the analysis of structures where the properties are different in two orthogonal directions because of provision of stiffeners.
Firstly effective width of a stiffened plate has been found out considering it to be an orthotropic plate and then the result has been compared with the codal provisions. Then expression for critical load for buckling of a stiffened plate has been derived.
Post buckling strength of orthotropic has been considered and expression for effective width of the loading in post buckled state has been found out by two methods: 1) Stress Method 2) Successive approximation method. Load versus deflection curve has been drawn to see the buckling mode and post buckling behavior of the plate. Then comparison has been done between the effective width found out by the two methods. Finally a conclusion has been drawn based on results and graphs obtained.
Acknowledgement
We are immensely grateful to our guide, Dr. Ashwini Kumar for the guidance he provided to us during this project. Dr. Kumar has always been ready to help us in any difficulty we faced and explained the correct approach to the problem. He has not only helped us by giving extremely helpful comments and suggestions regarding the work but also by providing the required material to study and have a better understanding of the subject.
We would also like to thank all the faculty and staff members of the Department of Civil Engineering, IIT Kanpur who made us skilled enough to work on this project by providing a highly fruitful training at this institute.
Abhijeet Singh
Manish Kumar
Table of Contents
Certificate Acknowledgement Table of Contents Chapter 1: Introduction
1.1 IS Code Provisions ……………………………………………………………………1 1.2 Developing the differential equation………………………………………………….2 1.3 Effective thickness of a stiffened plate ……………….................................................8 1.4 Comparison………………………………………………………………………......10 1.5 Corrugated Plate …………………………………………………………………….11
Chapter 2: Post Buckling Strength: Stress Solution
2.1 Effective width concept in thin walled steel section…………………………………13 2.1.1 Post Buckling Strength………………………………………………………...13
2.2 Stress Solution……………………………………………………………………….14 2.2.1 Compatibility…………………………………………………………………..14
2.3 Post buckling behavior of axially compressed plates………………………………..16 2.3.1 Boundary conditions…………………………………………………………...16
2.4 Using Galerkin method for finding f ………………………………………………..18 2.5 Finding Effective width: Stress Method…………………………………………….20 Chapter 3: Post Buckling Strength: Method of Successive Approximations
3.1 Introduction……………………………………………………………………….…21 3.2 Problem background…………………………………………………………………21 3.3 Compressive load problem…………………………………………………………..26
3.3.1 Boundary conditions…………………………………………………….……..26 3.3.2 Homogeneous solution…………………………………………………………31 3.3.3 Particular solution……………………………………………………………...33
3.4 Calculation of effective width……………………………………………………….36 3.5 Non-dimensionalization of expressions………………………………….…………..37 3.6 Graphical Analysis…………………………………………………………………...40
3.6.1 Load deflection curve…………………………………………………………40 3.6.2 Comparison of results of effective width by two methods……………………42
3.7 Conclusion…………………………………………………………………………...44
References…………………………………….……………………………………………….45
1
Chapter 1
Introduction
1.1 IS Code Provisions
IS: 801-1975 suggests an expression to calculate the effective thickness of a stiffened plate. If the intermediate stiffeners are spaced so closely that the flat width ratio between stiffeners does not exceed (w/t)lim , all the stiffeners may considered effective. Only for the purposes of calculating the flat width ratio of entire multiple-stiffened element, such element shall be considered as replaced by an element without intermediate stiffeners whose w is the whole width between webs or from wed to edge stiffener, and whose equivalent thickness heff is determined as follows:
heff = 312
wI s
(1.1)
Where Is = moment of inertia of the full area of the multiple stiffened element, including the intermediate stiffeners, about its own centroidal axis
(w/t)lim is given by
“Maximum allowable overall flat width ratio w/h disregarding intermediate stiffeners and taking h as the actual thickness of the element shall be as follows:
For stiffened compression element with both longitudinal edges connected to other stiffened elements= 500”
Fig 1.1: Buckled shape
A closer examination of the behavior of the element in compression reveals that the given formula by code needs to be reviewed. Introduction of Intermediate stiffeners transfer the plate into an orthotropic plate having much higher flexural rigidities. Plate is being analyzed considering different flexural rigidities in two directions.
2
1.2 Developing the differential equation
The equilibrium equation governing the buckling of a thin plate is given by
2
2
xM x
∂∂
- 2yx
M xy
∂∂
∂ 2
+ 2
2
yM y
∂
∂+ 2
2
xwN x ∂
∂ + 2
2
ywN y ∂
∂ + yx
wN xy ∂∂∂ 2
2 = 0 (1.2)
Fig: 1.2: Rectangular plate under axial load
We have the following relationships
][1 yyx
yx
xxx
Eενε
ννσ +
−= (1.3a)
][1 xxy
yx
yyy
Eενε
ννσ +
−=
(1.3b)
xyxyxy G γσ = (1.3c)
The strains xε , yε and xyγ can be expressed in terms of displacements as
3
xwzu∂∂
−= ywzv∂∂
−= (1.4a)
xu
x ∂∂
=ε yv
y ∂∂
=ε yv
xu
xy ∂∂
+∂∂
=γ (1.4b)
Substituting the values of u and v in strain equations and then strain values in stress train
relationships we get 2
2
xwzx ∂
∂−=ε 2
2
ywzy ∂
∂−=ε
yxwzxy ∂∂
∂−=
2
2γ (1.5a)
Therefore
][
1 2
2
2
2
yw
xwzE
yyx
xxx ∂
∂+
∂∂
−−= ν
ννσ (1.6a)
][1 2
2
2
2
xw
ywz
Ex
yx
yyy ∂
∂+
∂∂
−= ν
ννσ (1.6b)
yxwzGxyxy ∂∂
∂−=
2
2τ (1.6c)
Consider rectangular plate axially compressed in one direction and simply supported along all the edges.
For this case of plate having thickness “h” and given dimensions, we have
zdzM
h
hxxx ∫
+
−
=2/
2/
σ
⎥⎦
⎤⎢⎣
⎡∂∂
+∂∂
−= 2
2
2
23
)1(12 yw
xwzh
Ey
yx
x ννν
(1.7)
or
4
xM = - xD [ 2
2
xw
∂∂ + yν 2
2
yw
∂∂
] (1.8a)
yM = - yD [ 2
2
yw
∂∂
+ xν 2
2
xw
∂∂ ] (1.8b)
Similarly
zdzM xyhhxy σ2/
2/+−∫=
xywhGxy ∂
∂−=
23
61 (1.8c)
or
xywDM xyxy ∂
∂−=
2
(1.8d)
where )1(12
3
yx
xx
hED
νν−= (1.9a)
)1(12
3
yx
yy
hED
νν−= (1.9b)
3
6h
GD xy
xy = (1.9c)
are called flexural rigidities.
Now substituting the values of moments in governing differential equations
2
2
xM x
∂∂
- 2yx
M xy
∂∂
∂ 2
+ 2
2
yM y
∂
∂+ 2
2
xwN x ∂
∂ + 2
2
ywN y ∂
∂ + yx
wN xy ∂∂∂ 2
2 = 0
2
2
xM x
∂∂
= - xD2
2
x∂∂ [ 2
2
xw
∂∂ + yν 2
2
yw
∂∂
]
2
2
xM x
∂∂
= - xD [ 4
4
xw
∂∂ + yν 22
4
yxw∂∂
∂] (1.10a)
Similarly
5
2
2
yM y
∂
∂= - yD [ 4
4
yw
∂∂
+ xν 22
4
yxw
∂∂
] (1.10b)
And
yxM xy
∂∂
∂ 2
= - xyD 22
4
yxw
∂∂
(1.10c)
Substituting for all the expressions
- xD [ 4
4
xw
∂∂ + yν 22
4
yxw∂∂
∂] -2 xyD 22
4
yxw
∂∂ - yD [ 4
4
yw
∂∂
+ xν 22
4
yxw
∂∂
] + 2
2
xwN x ∂
∂ + 2
2
ywN y ∂
∂ +
yxwN xy ∂∂
∂ 2
2 = 0
or
2
2
xwN x ∂
∂ + 2
2
ywN y ∂
∂+
yxwN xy ∂∂
∂ 2
2 = xD [ 4
4
xw
∂∂ + yν 22
4
yxw∂∂
∂] + 2 xyD 22
4
yxw
∂∂
+ yD [ 4
4
yw
∂∂
+
xν 22
4
yxw
∂∂
] (1.11)
In this case we are considering rectangular plate axially compressed in one direction and simply supported along all the edges.
Boundary Conditions:
w = 0 for x = 0, a (1.12a)
w = 0 for y = 0, b (1.12b)
xM = 0, yM = 0 at edges
Hence
2
2
xw
∂∂ + yν 2
2
yw
∂∂
= 0 for x= 0, a (1.13a)
2
2
yw
∂∂
+ xν 2
2
xw
∂∂ = 0 for y= 0, b (1.13b)
Also since curvature of plate will be zero at the boundaries,
6
2
2
yw
∂∂
= 0 for x = 0, a (1.14a)
2
2
xw
∂∂
= 0 for y = 0, b (1.14b)
Substituting from (1.14a) in (1.13a) & from (1.14b) in (1.13b)
2
2
xw
∂∂ = 0 for x = 0, a (1.15a)
2
2
yw
∂∂
= 0 for y= 0, b (1.15b)
Hence from all the conditions derived, it can be said that all the four edges of the plate will be undeflected as well as curvature along an axis perpendicular to edge will be zero at the edge.
As yN and xyN are zero, the final equation can also be written as
02 2
2
22
4
4
4
22
4
22
4
4
4
=∂∂
+⎥⎦
⎤⎢⎣
⎡∂∂
∂+
∂∂
+∂∂
∂+⎥
⎦
⎤⎢⎣
⎡∂∂
∂+
∂∂
xwN
yxw
ywD
yxwD
yxw
xwD xxyxyyx νν (1.16)
Let the solution be
∑∑∞
=
∞
=
=1n 1
mnA(b
ynSina
xmSinx,y) wm
ππ for m = 1,2,3…..∞ ; n = 1,2,3…….∞ (1.17)
Substituting in our governing differential equation
( )( ) 0
21 122242224222444
422244
=⎥⎥⎦
⎤
⎢⎢⎣
⎡
−++
++∑∑∞
=
∞
=n m xyxy
yxmn b
ynSina
xmSinmnmDnmnD
nmmDA ππ
πμπλπλνπλ
πλνπ (1.18)
where xNa 22 =μ and aspect ratio of plate ba
=λ
As Amn = 0 gives trivial solution, Amn ≠ 0 gives
( ) ( ) 02 22242224222444422244 =−++++ πμπλπλνπλπλνπ mnmDnmnDnmmD xyxyyx (1.19)
or
7
( ) 2222
222
44
2
2222
2
2
2 naD
nm
na
Dnma
DN xyxyyxx λπλνλπλνπ
+⎟⎟⎠
⎞⎜⎜⎝
⎛+++= (1.20)
As n increases, Nx increases. Hence for the lowest value of Nx, n must assume one as the numerical value. This implies that the plate buckles with one half sine wave along the y-direction. The number of half sine waves along x-direction that correspond to minimum value of Nx can be obtained by taking derivative of Nx w.r.t. m, with n = 1.
0=∂∂
mN x => ( ) 022 3
4
=⎟⎟⎠
⎞⎜⎜⎝
⎛−+
mDmD yx
λ
4
x
y
DD
bam = (1.21)
Substituting the value of m,
( ) ( )[ ]xyxyyxyxcrx DDDDD
bN 222
2
+++= ννπ (1.22)
For non-integer values of λ , the buckling load is higher than that for integer values. For such cases, equation (1.20) can be written as
ψπ2
2
bDN xx = (1.23)
where ψ =buckling load coefficient
After putting the value of n=1, ψ can be written as
( )
( )⎥⎦⎤
⎢⎣
⎡++−
+++=x
xx cc
ccc
mcm
νν
νλλ
ψ21
142
2
2
2
2
2
(1.24)
where x
y
x
y
EE
cνν
== = ratio of properties in two orthogonal directions.
Hence the buckling load coefficient ψ can be plotted for integer values of m and the minimum value for each aspect ratio can be taken as the governing buckling load coefficient.
Here min value of ψ is plotted against aspect ratioλ for 25.0=xν and 2.0=c .
8
Fig1.3: variation of buckling load coefficientψ with apect ratio λ for 25.0=xν and 2.0=c
1.3 Effective thickness of a stiffened plate
Now consider a stiffened plate with given dimensions as shown in the figure.
formula for effective thickness of this stiffened plate heff is derived.
Fig 1.4: Stiffened Plate Let I1 be the moment of inertia of plate and I2 be the moment of inertia of stiffener on two sides with portion of plate between them.
( ) ( )[ ]xyxyyxyxcrx DDDDDb
N 222
2
+++= ννπ (1.25)
)1(12
3
ν+=
EhDxy = )1(
1
ν+wEI (1.26a)
0
0.5
1
1.5
2
2.5
3
3.5
4
0 2 4 6 8
ψ min vs λ
ψ min
9
where 1I = 12
3wh
)1(12 2
3
ν−=
EhDx = )1( 2
1
ν−wEI
(1.26b)
)1(12 2
3
ν−=
EhDy + w
EI 2
where 12
3
2tHI =
yD = )1( 2
1
ν−wEI
+ w
EI 2 (1.26c)
)1(2 2
1
ν−=+
wEIDD yx +
wEI 2
= ( )⎥⎦
⎤⎢⎣
⎡−+
−2
1
22
1 12)1(
νν I
Iw
EI (1.27)
yx DD = ( )2
1
22
1 11)1(
νν
−+− I
Iw
EI
Substituting in the expression for ( )crxN
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−++⎟⎟
⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−
= )1(2)1(2)1(12)1(12
2
1
22
1
22
3
2
2
ννννν
πII
IIEh
bN crx
Comparing with the expression for isotropic plate,
⎥⎥⎦
⎤
⎢⎢⎣
⎡+−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−+=⎟⎟
⎠
⎞⎜⎜⎝
⎛1)1(
2)1(1
21 2
1
22
1
2
3
νννII
II
hheff (1.28)
10
1.4 Comparison
Closely spaced stiffened plate has been considered here and then treating that as an orthotropic plate code gives the following formula.
heff = 312
wI s
Where Is = moment of inertia of the full area of the multiple stiffened element, including the intermediate stiffeners, about its own centroidal axis.
For our case Is = I2 + I1
The above equation can be written as (heff)3= 12Is/w
Also we have I1 = wh3/12
So (heff/h)3 = 1+ I2/ I1
Also as derived earlier ⎥⎥⎦
⎤
⎢⎢⎣
⎡+−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−+=⎟⎟
⎠
⎞⎜⎜⎝
⎛1)1(
2)1(1
21 2
1
22
1
2
3
νννII
II
hheff
This formula is compared with the one derived earlier for stiffened plate. Typical values have been taken to calculate ratio. Although two formulae can not be compared directly as there are restrictions on code formula under which it can be applied.
The curve of heff/h vs H/h for code formula is as shown on the next page.
11
Fig1.5: heff/h vs. H/h for code formula
As we can see two values are close when H/h (I2/ I1) is low but diverges as we increase the ratio H/h(I2/ I1)
1.5 Corrugated Plate
A stiffened orthotropic plate can be modeled as a corrugated plate. Approximate flexural rigidities have been considered in different directions and then final formula for effective width of corrugated plate has been calculated.
Fig 1.6: Corrugated Plate
0
0.5
1
1.5
2
2.5
3
0 5 10 15
exact
code
12
Modeling as a corrugated plate and taking flexural rigidity different in X and Y direction
We have
Ds
bDy 2≈
DsDx /12bhI3≈
DDxy )1(sb ν−≈
DsbDyx νν ≈
DsbDD xyyx ≈+ν
Now substituting these values in the equation
( ) ( )[ ]xyxyyxyxcrx DDDDDb
N 222
2
+++= ννπ
And simplifying, following result is derived
31
21
3
32 ⎥
⎥
⎦
⎤
⎢⎢
⎣
⎡⎟⎠
⎞⎜⎝
⎛+≈sh
Is
bh
h seq
13
Chapter 2
Post Buckling Strength: Stress Solution
2.1 Effective width concept in thin walled steel section
The theoretical critical load for plate is not necessarily is a satisfactory basis for design, since the ultimate strength can be much greater than the critical buckling load. A plate loaded in uniaxial compression will undergo stress redistribution as well as develop transverse tensile membrane stresses after buckling that provide post buckling support. Thus additional load often be applied without structural damage.
2.1.1 Postbuckling Strength
Postbuckling strength in plates is mainly due to the redistribution of axial compressive stresses. Local buckling causes a loss of stiffness and a redistribution of stresses. Uniform edge compression in the longitudinal direction results in nonuniform stress distribution after buckling and buckled plate derives almost all of its stiffness from the longitudinal edge supports.
The fact that much of the load carried by the region of the plate in the close vicinity
of the edges suggests the use of “effective width concept”. The maximum strength of plates can be estimated by the use of effective width concept.
This concept makes a simplifying assumption that maximum edge stress acts uniformly over two “strips” of the plate and the central region remains unstressed. Thus only a fraction of the width is considered to be effective in resisting the applied compression.
Fig 2.1: Postbuckling Stress Distribution
14
2.2 Stress Solution
The equilibrium equation in the z-direction is obtained as
( ) 0222
2
2
2
2
4
4
22
4
4
4
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂+
∂∂
+∂∂
−∂∂
+∂∂
∂+++
∂∂
yxwN
ywN
xwN
ywD
yxwDDD
xwD xyyxyxyxyyxx νν
(2.1)
2.2.1 Compatibility
The compatibility relations are obtained as
20
0 21
⎟⎠⎞
⎜⎝⎛∂∂
+∂∂
=xw
xu
xε (2.2a)
2
00 2
1⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
=yw
yv
yε (2.2b)
and yw
xw
yv
xu
xy ∂∂
∂∂
+∂∂
+∂∂
= 000γ (2.2c)
Where 0xε , 0yε and 0xyγ are middle surface strains in x and y directions and in shear respectively. 0u and
0v are displacements at the middle surface and w is the displacement in transverse direction.
Strains can be related to forces as
( )yxxx
x NNhE
νε −=1
0 (2.3a)
( )xyyy
y NNhE
νε −=1
0 (2.3b)
xy
xyxy G
Nh1
0 =γ (2.3c)
Differentiating (2.2a) twice w.r.t. y, (2.2b) twice w.r.t. x and (2.2c) successively w.r.t. x and y and combining the results
2
2
2
2220
2
20
2
20
2
yw
xw
yxw
yxxyxyyx
∂∂
∂∂
−⎥⎦
⎤⎢⎣
⎡∂∂
∂=
∂∂
∂−
∂
∂+
∂∂ γεε
(2.4)
15
This is deformation compatibility equation. To reduce the number of equations that must be solved, a stress function is introduced. Let the in-plane forces be defined in terms of a function ( )yxF , as
2
2
yFhN x ∂
∂= (2.5a)
2
2
xFhN y ∂
∂= (2.5b)
yx
FhN xy ∂∂∂
−=2
(2.5c)
Using equations (2.3a) to (2.3c)
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂∂
= 2
2
2
2
01
xF
yF
E xx
x νε (2.6a)
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂∂
= 2
2
2
2
01
yF
xF
E yy
y νε (2.6b)
yx
FGxy
xy ∂∂∂
−=2
01γ (2.6c)
substituting from (2.6a), (2.6b) and (2.6c) into (2.4) and from (2.5a), (2.5b) and (2.5c) into(2.1) gives
2
2
2
222
22
4
22
4
4
4
22
4
4
4 11yw
xw
yxw
yxF
Gh
yxF
xF
EyxF
yF
E xyy
yx
x ∂∂
∂∂
−⎥⎦
⎤⎢⎣
⎡∂∂
∂=
∂∂∂
+⎥⎦
⎤⎢⎣
⎡∂∂
∂−
∂∂
+⎥⎦
⎤⎢⎣
⎡∂∂
∂−
∂∂ νν
or 2
2
2
222
4
4
22
4
4
4 11yw
xw
yxw
yF
EyxF
EEGh
xF
E xy
y
x
x
xyy ∂∂
∂∂
−⎥⎦
⎤⎢⎣
⎡∂∂
∂=
∂∂
+∂∂
∂
⎥⎥⎦
⎤
⎢⎢⎣
⎡−−+
∂∂ νν
(2.7a)
and
( ) 02222
2
2
2
2
2
2
2
2
4
4
22
4
4
4
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂∂
∂−
∂∂
∂∂
+∂∂
∂∂
−∂∂
+∂∂
∂+++
∂∂
yxw
yxFh
yw
xFh
xw
yFh
ywD
yxwDDD
xwD yxyxyyxx νν
(2.7b)
16
2.3 Post buckling behavior of axially compressed plates
Fig2.2: Simply supported plate compressed in x-direction
2.3.1 Boundary conditions
Transverse boundary conditions corresponding to simply supported edges are
02
2
=∂∂
=xww at ax ,0=
02
2
=∂∂
=yww at by ,0=
Thus
Average value of applied compressive stress ∫−=a
xxa dyNah 0
1σ (2.8)
Assuming lateral deflection function as b
yna
xmfw ππ sinsin= (2.9)
and substituting in the equation (2.7a)
2
2
2
222
4
4
22
4
4
4 11yw
xw
yxw
yF
EyxF
EEGh
xF
E xy
y
x
x
xyy ∂∂
∂∂
−⎥⎦
⎤⎢⎣
⎡∂∂
∂=
∂∂
+∂∂
∂
⎥⎥⎦
⎤
⎢⎢⎣
⎡−−+
∂∂ νν
17
which reduces to
⎥⎦⎤
⎢⎣⎡ +=
∂∂
+∂∂
∂
⎥⎥⎦
⎤
⎢⎢⎣
⎡−−+
∂∂
byn
axm
banmf
yF
EyxF
EEGh
xF
E xy
y
x
x
xyy
πππνν 2cos2cos2
1122
4222
4
4
22
4
4
4
(2.10)
Solution of the equation consists of a homogeneous part and a particular part i.e.
ph FFF +=
For homogeneous part of solution, RHS has to be set equal to zero. But this will be equivalent to letting0=w . Therefore the homogeneous solution corresponds to the in-plane stress distribution in the plate
just prior to buckling. But at that instant xN will be uniform and yN and xyN will be zero. Therefore a
homogeneous solution of the equation will be
2AyFh = (2.11)
as hN xax σ−=
we get
2
2yF xa
hσ
−= (2.12)
Now corresponding to the RHS, the particular solution can be written as
b
ynCa
xmBFpππ 2cos2cos += (2.13)
Substituting back in the differential equation and comparing the coefficients of the terms a
xmπ2cos and
bynπ2cos ,values of B and C are obtained as
22
222
32 bmnafE
B y=
and 22
222
32 anmbfE
C x= (2.14)
using which complete solution of F is obtained as
2
2cos32
2cos32
2
22
222
22
222 yb
ynanmbfE
axm
bmnafE
F xaxy σππ−+= (2.15)
18
2.4 Using Galerkin method for finding f
b
yna
xmfw ππ sinsin=
The Galerkin equation is
( ) ( ) 0,0 0
=∫ ∫ dxdyyxgfQa a
(2.16)
Where on arranging the terms Q(f) is obtained as
( )( )
byn
axm
amhf
bnyCos
amhfE
amxCos
bnhfE
banmDDD
bnD
amD
f
fQ
xaxy
xyxyyxyx
ππ
πσππππ
ννπ
sinsin2
82
8
2
2
22
4
443
4
443
22
22
4
4
4
44
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
+⎟⎟⎠
⎞⎜⎜⎝
⎛+
+⎟⎟⎠
⎞⎜⎜⎝
⎛++++
=
(2.17)
Substituting the value of ( )fQ in Galerkin equation, taking ( )
byn
axmfyxg ππ sinsin, = and integrating
with proper limits the following is obtained
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡++
⎥⎥⎦
⎤
⎢⎢⎣
⎡++++= 4
4
4
4222
22
22
4
4
4
4222
162
bnE
amEaf
banmDDD
bnD
amD
ham yx
xyxyyxyx
xaπννπσ
which can also be expressed as
⎥⎥⎦
⎤
⎢⎢⎣
⎡++= 4
4
4
4222
16 bnE
amEaf yx
crxaπσσ
(2.18)
where
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡++++= 22
22
4
4
4
4222
2banmDDD
bnD
amD
ham
xyxyyxyx
cr ννπσ (2.19)
19
Now as
2
2
yF
xx ∂∂
−=σ
substituting the value of F , xxσ is obtained as
xax
xx bnyCos
afmE
σππσ +=
28 2
222
(2.20)
Also we have
⎥⎥⎦
⎤
⎢⎢⎣
⎡+
−=
4
4
4
4
2
2
22 )(28
bnE
amE
ma
f
yx
crxx σσπ
so substituting the value in the equation (2.20), xxσ is obtained as
bnyCos
EE
bn
ma
x
y
crxxxaxx
πσσσσ 2
1
)(2
4
4
4
4
⎥⎦
⎤⎢⎣
⎡+
−+=
Now let Ma
m=
π
and N
bn
=π
Then the equation becomes
bnyCos
EE
MN
x
y
crxxxaxx
πσσσσ 2
1
)(2
4
4
⎥⎦
⎤⎢⎣
⎡+
−+= (2.21)
20
2.5 Finding Effective width: Stress Method
∫=b
xxe dyb0
max σσ (2.22)
maxσ =
⎥⎦
⎤⎢⎣
⎡+
−+=
=
x
y
crxxxayxx
EE
bn
ma
4
4
4
40
1
)(2 σσσσ
now,
∫∫ +=b
xa
b
xx dybnyCosbdy
00
2(........) πσσ
so by xa
b
xx σσ =∂∫0
hence
( )ησσ
σ
σ
+−
+=
1)(2 crxa
xa
xae
bb
(2.23)
where x
y
EE
bn
ma
4
4
4
4
=η (2.24)
Now we have crxx
xa
PP
=σσ
so
( )η+
−+
=
1
)1(2cr
cr
cre
PP
PP
PP
bb
(2.25)
21
Chapter 3
Post Buckling Strength: Method of Successive Approximations
3.1 Introduction
In the method of successive approximations, the set of Von Karman large deflection equations for the plates are converted from three nonlinear partial differential equations into infinite number of linear partial differential equations by expanding the displacement terms into a power series form of a parameter. The first few equations of the infinite set of equations are obtained as small deflection equations which give the solution for pre-buckling stage. Further solution of more equations gives approximate solution for post-buckling range.
The study of post-buckling behaviour of a simply supported orthotropic plate subjected to longitudinal compression is presented here.
3.2 Problem background
For a plate with no lateral loads, Von Karman equations for large deflection can be written as
0=∂
∂+
∂∂
yN
xN xyx (3.1a)
0=∂
∂+
∂
∂
yN
xN yxy (3.1b)
( ) 0222
2
2
2
2
4
4
22
4
4
4
=∂∂
∂−
∂∂
−∂∂
−∂∂
+∂∂
∂+++
∂∂
yxwN
ywN
xwN
ywD
yxwDDD
xwD xyyxyxyxyyxx νν (3.1c)
The strain force relations are the following
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
y
yy
x
xx E
NEN
hνε 1
(3.2a)
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
x
xx
y
yy E
NEN
hνε 1 (3.2b)
22
xy
xyxy G
Nh1
=γ (3.2c)
From equation (2a), (2b) & (2c), forces can be expressed in terms of strains as
yx
xx
hEN
νν−=
1)( yyx ενε + (3.3a)
yx
yy
hEN
νν−=
1)( xxy ενε + (3.3b)
xyxyxy hGN γ= (3.3c)
Strain-displacement equations are
2
21
⎟⎠⎞
⎜⎝⎛∂∂
+∂∂
=xw
xu
xε (3.4a)
2
21
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
=yw
yv
yε (3.4b)
yw
xw
xv
yu
xy ∂∂
∂∂
+∂∂
+∂∂
=γ (3.4c)
substituting these relations we get the force-displacement relations as
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎟⎠⎞
⎜⎝⎛∂∂
+∂∂
+∂∂
−=
22
21
1 yw
xw
yv
xuhE
N yyyx
xx νν
νν (3.5a)
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎠⎞
⎜⎝⎛∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
−=
22
21
1 xw
yw
xu
yvhE
N xxyx
yy νν
νν (3.5b)
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+∂∂
+∂∂
=yw
xw
xv
yuhGN xyxy (3.5c)
23
Now assuming that u , v & w may be expanded in a power series in terms of an arbitrary parameterα , they can be expressed as
∑∞
=
=2,0n
nnuu α (3.6a)
∑∞
=
=2,0n
nnvv α (3.6b)
∑∞
=
=3,1n
nnww α (3.6c)
Here u and v are assumed to start from a term having zero power of α whereas w is assumed to start with non-zero power ofα . The reason is that just prior to buckling, the in-plane deflections u and v are expected to have finite value whereas the transverse deflection w is expected to be zero until the plate buckles.
The plate can buckle in either direction but ( )yxw , should be independent of that except for a sign. Hence for + ve to –ve values ofα , w should just only change its sign and therefore only odd powers of α are assumed in the power series expansion of w .
Opposite to this, the in-plane deflections u and v should be independent of the direction of buckling (and hence of the sign of α ). So only even powers of α are assumed in the power series expansion of u and v .
Substituting these in equations (5a) to (5c)
nmnmy
nm
yx
x
m n
nny
n
n yx
xx y
wy
wx
wx
whEyv
xuhE
N +∞
=
∞
=
∞
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+∂∂
∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
−+⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
−= ∑ ∑∑ αν
νναν
νν 121
1 3,1 3,12,0
nmnmx
nm
yx
y
m n
nnx
n
n yx
yy x
wx
wy
wy
whEx
uyvhE
N +∞
=
∞
=
∞
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+∂∂
∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
−+⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
−= ∑ ∑∑ αν
νναν
νν 121
1 3,1 3,12,0
∑ ∑∑∞
=
∞
=
+∞
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
=3,1 3,12,0 m n
nmnmxy
n
nnnxyxy y
wx
whG
xv
yu
hGN αα
24
Expressing these equations as sum of two terms as
∑ ∑∑∞
=
∞
=
+∞
=
+=3,1 3,1
),(
2,0
)(
m n
nmnmx
n
nnxx NNN αα (3.7a)
∑ ∑∑∞
=
∞
=
+∞
=
+=3,1 3,1
),(
2,0
)(
m n
nmnmy
n
nnyy NNN αα (3.7b)
∑ ∑∑∞
=
∞
=
+∞
=
+=3,1 3,1
),(
2,0
)(
m n
nmnmxy
n
nnxyxy NNN αα (3.7c)
where
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
−=
yv
xuhE
N ny
n
yx
xnx ν
νν1)( (3.8a)
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+∂∂
∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
−=
yw
yw
xw
xwhE
N nmy
nm
yx
xnmx ν
νν121),( (3.8b)
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
−=
xu
yvhE
N nx
n
yx
yny ν
νν1)( (3.8c)
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+∂∂
∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
−=
xw
xw
yw
ywhE
N nmx
nm
yx
ynmy ν
νν121),( (3.8d)
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
=xv
yu
hGN nnxy
nxy
)( (3.8e)
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
=y
wx
whGN nm
xynm
xy),( (3.8f)
Now as stated earlier in this chapter, for a plate with no lateral loads, Von Karman large deflection equations can be written as
0=∂
∂+
∂∂
yN
xN xyx
0=∂
∂+
∂
∂
yN
xN yxy
25
( ) 0222
2
2
2
2
4
4
22
4
4
4
=∂∂
∂−
∂∂
−∂∂
−∂∂
+∂∂
∂+++
∂∂
yxwN
ywN
xwN
ywD
yxwDDD
xwD xyyxyxyxyyxx νν
Substituting the power series expansions of all the terms in these equations, three equations are obtained each of which equates a power series of α to zero. For the equation to hold true for all values ofα , coefficients of all the exponents of α in each of the three equations should be equal to zero.
Equating the coefficients of 00 =α ; the following relations are obtained
0
0
)0()0(
)0()0(
=∂
∂+
∂
∂
=∂
∂+
∂∂
yN
xN
yN
xN
yxy
xyx
} (3.9a)
and equating the coefficients of 01 =α ;
( ) 022 12
)0(2
12
)0(2
12
)0(4
14
221
4
41
4
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂+
∂∂
+∂∂
−∂∂
+∂∂
∂+++
∂∂
yxwN
ywN
xwN
ywD
yxwDDD
xwD xyyxyxyxyyxx νν
(3.9b) Similarly equating the co-efficient of 02 =α
0
0
)1,1()2()1,1()2(
)1,1()2()1,1()2(
=∂
∂+
∂
∂+
∂
∂+
∂
∂
=∂
∂+
∂
∂+
∂∂
+∂
∂
xN
xN
yN
yN
yN
yN
xN
xN
xyxyyy
xyxyxx
} (3.9c)
and equating the co-efficient of 03 =α the next equation is obtained as
( )
[ ] [ ] [ ]yx
wNNywNN
xwNN
yxw
Nyw
Nxw
Nyw
Dyx
wDDD
xw
D
xyxyyyxx
xyyxyxyxyyxx
∂∂∂
++∂∂
++∂∂
+
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂+
∂∂
+∂∂
−∂∂
+∂∂
∂+++
∂∂
12
)1,1()2(2
12
)1,1()2(2
12
)1,1()2(
32
)0(2
32
)0(2
32
)0(4
34
223
4
43
4
2νν
(3.9d)
Similarly equating higher coefficients of α to zero, more equations can be obtained.
26
3.3 Compressive load problem
The plate under compressive loading along x- direction has the length a , width b and thickness h. The origin lies at one corner of the plate as shown in figure. The edges are all simply supported with compressive load of xN per unit width.
Fig: 3.1- Loading condition on the plate
3.3.1 Boundary Conditions
Zero transverse deflection at the edges
( ) ( ) ( ) ( ) 0,0,,,0 ==== bxwxwyawyw
Zero moment at edges
0),(
2
2
)0,(2
2
),(2
2
),0(2
2
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
bxxyay xw
xw
xw
xw
Constant displacement along an edge
0),()0,(),(),0(
=⎟⎠⎞
⎜⎝⎛∂∂
=⎟⎠⎞
⎜⎝⎛∂∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
bxxyay xv
xv
yu
yu
27
Zero shear stress at edges
0),(),0(),()0,(
=⎟⎠⎞
⎜⎝⎛∂∂
=⎟⎠⎞
⎜⎝⎛∂∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
yaybxx xv
xv
yu
yu
Loaded edges
( ) PdyNb
x −=∫0
at ax ,0=
Unloaded edges
( ) PdxNa
y −=∫0
at by ,0=
The load is equal to or greater than the buckling load. Substituting the relations from equation (8a) & (8c) into the condition for loaded edge, the following is obtained
n
n
nPP α∑∞
=
=2,0
)( (3.10)
where
For loaded edges
( )dyNPb
x∫−=0
)0()0( at ax ,0= (3.11a)
( )dyNNPb
xx∫ +−=0
)1,1()2()2( at ax ,0= (3.11b)
( )dyNNPb
xx∫ +−=0
)3,1()4()4( 2 at ax ,0= (3.11c)
(as )1,3()3,1(xx NN = )
28
Similarly for unloaded edges
( ) 00
)0( =∫ dxNa
y at by ,0= (3.12a)
( ) 00
)1,1()2( =+∫ dxNNa
yy at by ,0= (3.12b)
( ) 02
0
)3,1()4( =+∫ dxNNa
yy at by ,0= (3.12c)
Now using equations (1a) and (1b)
0)0()0(
=∂
∂+
∂∂
yN
xN xyx
and 0)0()0(
=∂
∂+
∂
∂
yN
xN yxy
Substituting the relations from equations (3.8a), (3.8c) and (3.8e)
01
0000 =⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
∂∂
+⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
−∂∂
xv
yu
hGyy
vx
uhEx xyy
yx
x ννν
and 01
0000 =⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
−∂∂
+⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
∂∂
xu
yvhE
yxv
yu
hGx x
yx
yxy ν
νν
rewriting these
011
02
20
2
20
2
=∂∂
∂⎟⎟⎠
⎞⎜⎜⎝
⎛+
−+
∂∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
− yxv
GE
yu
GxuE
xyyx
yxxy
yx
x
ννν
νν (3.13a)
0
110
2
20
2
20
2
=∂∂
∂⎟⎟⎠
⎞⎜⎜⎝
⎛+
−+
∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
−+
∂∂
yxu
GE
yvE
xv
G xyyx
xy
yx
yxy νν
ννν
(3.13b)
29
Solutions of these equations satisfying the boundary conditions will be of the form
⎟⎠⎞
⎜⎝⎛ −=
200axUu
⎟⎠⎞
⎜⎝⎛ −=
200byVv
Using the loading boundary conditions the values of 0U and 0V are obtained as
hbEPU
x
)0(
0 −=
and hbE
PV
x
x)0(
0ν
=
so that
⎟⎠⎞
⎜⎝⎛ −−=
2
)0(
0ax
hbEPu
x
(3.14a)
⎟⎠⎞
⎜⎝⎛ −=
2
)0(
0by
hbEP
vx
xν (3.14b)
and therefore
0)0()0(
)0()0(
==
−=
xyy
x
NNb
PN (3.15)
Now 1w can be obtained from the equation (3.9b) which has the solution of the form
b
yna
xmWw ππ sinsin11 = (3.16)
NyMxW sinsin1=
which satisfies the boundary conditions. Putting this solution in the equation (3.9b) and substituting the results of the equation (3.15)
30
( )
0sinsinsinsin
sinsin2sinsin
12
22)0(
14
44
122
422
14
44
=⎟⎟⎠
⎞⎜⎜⎝
⎛−
++++
byn
axmW
am
bP
byn
axmW
bnD
byn
axmW
banmDDD
byn
axmW
amD
y
xyxyyxx
ππππππ
πππννπππ
which gives the value of )0(P as
( )
2
22
4
44
22
422
4
44
)0(
2
am
bnD
banmDDD
amDb
Pyxyxyyxx
π
ππννπ⎥⎦
⎤⎢⎣
⎡++++
= (3.17)
Now using equation (3.9c)
0
)1(2)1(
1122
21
2122
=⎥⎦
⎤⎢⎣
⎡∂∂
∂∂
∂∂
+⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
∂∂
+⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎟⎠⎞
⎜⎝⎛∂∂
−∂∂
+⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
−∂∂
yw
xwhG
yyv
xuhG
y
yw
xwhE
xyv
xuhE
x
xyyxy
yyx
xy
yx
x
ν
ννν
ννν
or
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
⎟⎠⎞
⎜⎝⎛∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
⎟⎠⎞
⎜⎝⎛∂∂
−−=
∂∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−+
∂∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
−
21
211
21
12
12
12
122
22
2
22
2
)1()1()1(
xw
xw
yxw
yw
G
yxw
yw
xw
xwE
yxv
GE
yu
GxuhE
xy
yyx
xxy
yx
yxxy
yx
x ννννν
ννν
Substituting the value of 1w and simplifying the equation becomes
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟
⎟⎠
⎞⎜⎜⎝
⎛
−
+−⎟
⎟⎠
⎞⎜⎜⎝
⎛
−
−
=∂∂
∂⎟⎟⎠
⎞⎜⎜⎝
⎛+
−+
∂∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
−
NyCosNGENM
MENMM
MxSinw
yxv
GE
yu
GxuhE
xyyx
xy
yx
xy
xyyx
yxxy
yx
x
2)1(2
)()1(2
)(2
2
)1()1(
222222
1
22
22
2
22
2
ννν
ννν
ννν
νν
(3.18a)
31
Similarly after simplifying, the second equation of (3.9c) becomes
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟
⎟⎠
⎞⎜⎜⎝
⎛
−+
−⎟⎟⎠
⎞⎜⎜⎝
⎛
−−
=∂∂
∂⎟⎟⎠
⎞⎜⎜⎝
⎛+
−+
∂∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛
−+
∂∂
MxMGEMN
NEMNN
NySinw
yxu
GE
yu
GyvhE
xv
G
xyyx
yxx
yx
yx
xyyx
xyxy
yx
xxy
2cos)1(2
)()1(2
)(2
2
)1()1(
222222
1
22
22
2
22
2
22
2
νννν
ννν
ννν
νν
(3.18b)
Now the total solution of the equation is given by
hp uuu 222 +=
hp vvv 222 +=
Total solution = Particular solution + homogeneous solution
3.3.2 Homogeneous solution
As the homogeneous part of the equation (3.18a) and (3.18b) in 2u and 2v is same as that in ou and
ov in equation (3.13a) and (3.13b), the homogeneous solution will of the same form.
Let ⎟⎠⎞
⎜⎝⎛ −=
222axUu
& ⎟⎠⎞
⎜⎝⎛ −=
222axVv
Now ⎥⎦
⎤⎢⎣
⎡∂∂
+∂∂
−=
yv
xuhE
N yyx
xx
22)2(
)1(ν
νν
[ ]22)1(
VUhE
yyx
x ννν
+−
=
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎟⎠⎞
⎜⎝⎛∂∂
−=
21
21)1,1(
)1(2 yw
xwhE
N yyx
xx ν
νν
32
Substituting the value of SinMxSinNyWw 11 =
[ ]21
21
)1,1( )()()1(2
SinMxCosNyNwCosMxSinNyMwhE
N yyx
xx ν
νν+
−=
similarly
⎥⎦
⎤⎢⎣
⎡∂∂
+∂∂
−=
yu
xvhE
N xyx
yy
22)2(
)1(ν
νν
[ ]22)1(
UVhE
yyx
y ννν
+−
=
[ ]21
21
)1,1( )()()1(2
CosMxSinNyMwSinMxCosNyNwhE
N xyx
yy ν
νν+
−=
Using the relation from equation (3.11b)
( ) dyNNP
ax
b
xx,00
)1,1()2()2(
=∫ +−= for loaded edges
[ ] NySinWM
hEVU
hENN
yx
xy
yx
xaxxx
221
222,0
)1,1()2(
)1(2)1( ννν
νν −++
−=+
=
Similarly from equation (3.12b)
( ) 0
,00
)1,1()2( =+−=
∫ dxNNby
a
yy
for unloaded edges
[ ] MxSinWN
hEUV
hENN
yx
yx
yx
y
byyy22
12
22,0
)1,1()2(
)1(2)1( ννν
νν −++
−=+
=
Using the integrals ∫ =b bNydySin0
2
2
& ∫ =a aMxdxSin0
2
2
33
The following is obtained
For loaded edge
[ ] bWMhE
bVUhE
Pyx
xy
yx
x 21
222
)2(
)1(4)1( ννν
νν −++
−=−
For unloaded edge
[ ] aWN
hEaUV
hE
yx
yx
yx
y 21
222 )1(4)1(
0νν
ννν −
++−
=
Solving these two equations
( )( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
−
−+−=
41
21
22)2(
2WNM
hbEPU
yx
y
x ννν
( )( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
−−
+=41
21
22)2(
2WNM
hbEP
Vyx
x
x
x
νννν
3.3.3 Particular Solution
The particular solution will be of the form
[ ]NyBCosAMxSinu p 222 +=
[ ]MxDCosCNySinv p 222 +=
Substituting in two equations and comparing the co-efficient of terms
( )M
NMWA y222
1
16ν−
−= (3.19a)
(Comparing the co-efficient of MxSin2 )
( )N
MNWC x222
1
16ν−
−= (3.19b)
34
(Comparing the co-efficient of NySin2 )
Again comparing coefficient of NyMxCosSin 22
( ) ( )( )( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡+
−−
+=
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟
⎟⎠
⎞⎜⎜⎝
⎛
−+
⎥⎥⎦
⎤
⎢⎢⎣
⎡+
−2
22212
2
11811MNG
ENMMWGE
MNDGNEM
B xyyx
x
yx
yxy
yx
yxxy
yx
x
ννννν
ννν
νν
comparing coefficient of MxNyCosSin 22
( ) ( )( )( ) ( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡+
−−+
=⎥⎥⎦
⎤
⎢⎢⎣
⎡+
−+
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟
⎟⎠
⎞⎜⎜⎝
⎛
−NMG
EMNNWGM
ENDG
EMNB xy
yx
y
yx
xxy
yx
yxy
yx
xy 2222
122
11811 ννννν
ννννν
Solving above two equations B and C are obtained as
16
21MWB = (3.19c)
16
21NWD = (3.19d)
Therefore the complete expressions for 2u and 2v can be written as
( )( )
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡+
−−+⎟
⎠⎞
⎜⎝⎛ −⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
−+−= NyCosMW
MNMWMxSinaxWNM
hbEPu y
yx
y
x
21616
2241
21
2221
21
22)2(
2
νννν
(3.20a)
and
( )( )
( )⎥⎦
⎤⎢⎣
⎡+
−−+⎟
⎠⎞
⎜⎝⎛ −⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−
+= MxCosNWN
MNWNySinaxWNMhbE
Pv x
yx
x
x
x 21616
2241
21
2221
21
22)2(
2ν
νννν
(3.20b)
where
amM π
= and b
nN π=
35
Now from equation (3.9d)
( )
[ ] [ ] [ ]yx
wNNywNN
xwNN
yxw
Nyw
Nxw
Nyw
Dyx
wDDD
xw
D
xyxyyyxx
xyyxyxyxyyxx
∂∂∂
++∂∂
++∂∂
+
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂+
∂∂
+∂∂
−∂∂
+∂∂
∂+++
∂∂
12
)1,1()2(2
12
)1,1()2(2
12
)1,1()2(
32
)0(2
32
)0(2
32
)0(4
34
223
4
43
4
2νν
(3.21)
R.H.S. of the equation (3.21) is the sum of three terms
R.H.S = Term 1 + Term 2 + Term 3
Analyzing term by term
Term 1 21
221
2122
21
)1( xw
yw
xw
yv
xuhE
yyyx
x
∂∂
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎟⎠⎞
⎜⎝⎛∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
−= νν
νν
substituting the value of 122 &, wvu in Term 1 and simplifying we get co-efficient of SinMxSinNy in Term1 as
( ) ( )[ ][ ] 12
221 2)(21
WMDCNVBAMUE
C yyx
x +++++−
−= ννν
Similarly co-efficient of SinMxSinNy in Term 2
( ) ( ) [ ][ ] 12
222 )(221
WNBAMUDCNVE
C xyx
y +++++−
−= ννν
and co-efficient of SinMxSinNy in Term 3
( )[ ]3113 8 MNWWDNBNhMNGC xy −+=
Hence co-efficient of SinMxSinNy in Term 2 in the R.H.S of equation
321 CCCC ++=
36
So the equation becomes
( ) +++= SinMxSinNyCCCSHL 321.. A term free of SinMxSinNy
The homogeneous part of equation (3.21) is same as that in (3.9b). Hence SinMxSinNy will also be a homogeneous solution to the equation (3.21). But a term of SinMxSinNy appears on the right-hand-side. Therefore no solution to the equation (3.21) will be possible that satisfies the boundary conditions unless co-efficient of SinMxSinNy on R.H.S =0
0321 =++∴ CCC
Substituting the values of coefficients and solving this gives the value of 21W as
( ) ( )⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
+−+
−
−=
)1()1()2(
822
44
2)2(2
1
yx
xyyxyx
yx
yx NMEENEME
MbhPW
νννν
νννν
(3.22)
3.4 Calculation of effective width
Axial shortening is given by
),(),0( yauyu −=Δ
),(),(),( 20 yxuyxuyxu +=
Now ⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−=−
22),(),0( 0000
aUaUyauyu
= aU 0−
hp uuu 222 +=
0),(),0( 22 =− yauyu pp
aUyauyu hh 222 ),(),0( −=−
Hence the Total shortening
),(),0( yauyu −=Δ
37
aUaU 20 −−=Δ
Substituting the values of U0 and U2
( )( ) aWNM
hbEPa
hbEP
yx
y
xx ⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
−++=Δ
41
21
22)2(2
)0(
ννν
α
( ) ( )( ) 41
21
222)2(2)0( WNMhbE
PPa yx
y
x ννναα
−
−+
+=
Δ
( )( ) 41
21
222 WNMhbE
Pa yx
y
x νννα
−
−+=
Δ
Effective width is given by ha
EPbx
e Δ=
so
( )( ) 41
21
222 WNMhbE
PhbE
P
bb
yx
y
x
xe
νννα
−
−+
= (3.23)
3.5 Non dimensionalization of expressions
The relations obtained in this chapter are non-dimensionalised in this section so that all the expressions can be evaluated using just the aspect ratio of plate and the ratio of properties in two orthogonal directions.
From equation (3.22)
( )( )( ) ( )[ ]2244
2)2(2
1 218
NMEENEMEM
bhPW
xyyxyxyx
yx
νννννν
+−+−
−=
Let cEE
x
y
x
y ==νν
& λ=ba
(3.24)
38
Substituting a
mM π= &
bnN π
=
The equation reduces to
( )
( )( )[ ]2224442
22
2
)2(2
1 2218
λνλνλν
π nmccnmcacm
hEPW
xx
x
x −+−
−= (3.25)
Let shortening at the start of buckling i.e. at )0(PP = be 0Δ=Δ
Then ( ) ( )yauyu ,,00 −=Δ
and ⎟⎠⎞
⎜⎝⎛ −==
200axUuu
therefore ( ) ( )yauyu ,,0 000 −=Δ
⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−=
22 00aUaU
aU 0−=
hence hbEaP
x
)0(
0 =Δ
or hbE
Pa x
)0(0 =
Δ
Now from previous analysis
( )( ) 41
21
222 WNMhbE
Pa yx
y
x νννα
−
−+=
Δ
dividing the two equations
( )( ) )0(
21
222
)0(0 41 P
hbEWNMPP x
yx
y
νννα
−
−+=
ΔΔ
39
after substituting
)2(
)0(2
PPP −
=α & ( )
( )( )[ ]2224442
22
2
)2(2
1 2218
λνλνλν
π nmccnmcacm
hEPW
xx
x
x −+−
−=
this reduces to
( )( )( )[ ]
( ))0(
)0(
2224442
2222
)0(0 22
2P
PPnmccnmc
ncmmPP
xx
x −−+−
−+=
ΔΔ
λνλνλν
Now Let
( )
( )( )[ ] Knmccnmc
ncmm
xx
x =−+−
−2224442
2222
222
λνλνλν
(3.26)
which results in
( ))0(
)0(
)0(0 P
PPKPP −
+=ΔΔ
or
KK
KPP
++
ΔΔ
+=
111
0)0(
and
( )( ) 41
21
222 WNMhbE
PhbE
P
bb
yx
y
x
xe
νννα
−
−+
=
40
which reduces to
( ) K
PPK
PP
bb
cr
cre
−+=
1 (3.27)
crPP =)0(Q
3.6 Graphical analysis
3.6.1 Load deflection curve
First Graph is drawn between the crPP
Vs 0ΔΔ
Recalling the expression ( )
)0(
)0(
)0(0 P
PPKPP −
+=ΔΔ
Where ( )
( )( )[ ]2224442
2222
222
λνλνλν
nmccnmcncmm
Kxx
x
−+−
−=
Graph is plotted for m=1 and m=2 and other values fix as shown.
m n νx c λ K η 1 1 0.25 0.2 1 0.83151 0.2 2 1 0.25 0.5 2 0.647399 0.5
Where cEE
x
y
x
y ==νν
λ=ba
and x
y
EE
bn
ma
4
4
4
4
=η
41
Fig 3.2: Load vs. Deflection curve for c=.2 and λ=1
Fig 3.3: Load vs. Deflection curve for c=.5 and λ=1
As we can see, the graph crPP
Vs 0ΔΔ
is linear, which is expected since we haven’t introduced any second
order term of the deflection although we have taken non linearity in deformation into account. Now the curves have been shown for aspect ratio λ=1 that is a square plate.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 0.5 1 1.5 2 2.5
P/P0(m=1)
P/P0(m=2)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 0.5 1 1.5 2 2.5
P/P0(m=1)
P/P0(m=2)
42
3.6.2 Comparison of results of bbe by two methods:
bbe has been derived by two methods.
3.6.2a) Successive approximation Method
( ) KPPK
PP
bb
cr
cre
−+=
1
where ( )
( )( )[ ]2224442
2222
222
λνλνλν
nmccnmcncmm
Kxx
x
−+−
−=
3.6.2b) Stress Method
( )η+
−+
=
1
)1(2cr
cr
cre
PP
PP
PP
bb
where x
y
EE
bn
ma
4
4
4
4
=η
Graph is plotted for m=1 and m=2 and other values fix as shown.
m n νx c λ K η 1 1 0.25 0.2 1 0.83151 0.21 1 0.25 0.5 1 0.647399 0.5
43
Fig 3.4: Effective width vs. deflection curve of orthotropic plate for c=0.2 and λ=1
Fig 3.5: Effective width vs. deflection curve of orthotropic plate for c=0.5 and λ=1
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5
be/b(m=1)
be/b stress(m=1)
be/b(m=2)
be/b stress(m=2)
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5
be/b(m=1)
be/b stress(m=1)
be/b(m=2)
be/b stress(m=2)
44
3.7 Conclusion
From effective width vs. deflection curves of the two methods: 1) Stress Method and 2) successive approximation method, results have been compared and it can been seen that the values by two methods agree to an extent. Little deviation can be attributed to fact that the two methods are approximate methods. Also as the value of m increases or number of half sine curves in which the plate buckle increases, the effective width decreases.
45
References
1. Stein M. , Loads and Deformations of Buckled Rectangular Plates. NASA TR R-40, 1959
2. Indian Standard Code of Practice for Use of Cold-Formed Light gauge Steel Structural Members In General Building Construction (First Revision). IS : 801-1975
3. Timoshenko S. P., Woinowsky-Krieger S., Theory of Plates and Shells. Second Edition.
4. Iyengar NGR., Elastic Stability of Structural Elements.
5. Chajes A., Principles of Structural Stability Theory.
