plate buckling notes

8/13/2019 Plate Buckling Notes

1/12

Plate_Buckling_Notes.doc p1 Copyright J.W. Butterworth August 2005

INTRODUCTION TO PLATES AND PLATE BUCKLING

[Reading Bulson, P.S. The Stability of Flat Plates, Elsevier, New York, 1969; Timoshenko and Woinowski-

Krieger, Theory of Plates and Shells, 2ndEd., McGraw-Hill, NY, 1959; ]

Plates are a type of structural element commonly used to span areas and support

vertical loads e.g. floor or roof slabs. They are bounded by parallel plane

surfaces and are usually of a uniform thickness that is small compared with the

plan dimensions. They also constitute major components of I-beams, plate girders

and box girders, and it is because of this role that we are studying their behaviour

in this course. Plate behaviour is a relatively advanced topic in structural

mechanics and design, so the treatment here is necessarily abbreviated in many

places.

INTERNAL ACTIONS UNDER TRANSVERSE LOADING

The figure to the right shows a rectangular plate, simply

supported on all edges (i.e. knife-edge supports resisting

up and down movement but allowing rotation perpendicularto the edge).

Loading may consist of point loads, W, line loads and

distributed loads, q(x,y), all acting perpendicular to the

plate surface.

Considering the deflection of the two shaded (beam-like)

strips it can be seen that the element defined by their

intersection will bend to different radii of curvature in

the xz and yz planes, and the four corners of the element

will have different deflections.

y

x

z

My

Mx y

x

SxSy

y

x

MyxMxy

w

BENDING MOMENT TWISTING MOMENT TRANSVERSE SHEAR

ACTIONS IN LATERALLY LOADED PLATE

The resulting internal actions will consist of:

Bending moments Mxand My

Similar to bending moments in a beam. They are measured as moments per unit length of plate,

kN-m/m.

Twisting moments Mxyand Myx

These result from the fact that adjacent imaginary strips deflect and therefore rotate by

different amounts hence tending to cause relative rotation between the side faces of the strips

with the twisting moments resisting this tendency. Also measured in kN-m/m.

Through-thickness shear force, Sxand Sy.

Similar to shear force in a beam, but generally small in magnitude.

LATERALLY LOADED PLATE

Simply

-supp

orted

edge

s

Wq(x,y)

y

x

z


2/12


3/12


GOVERNING EQUATION FOR LATERALLY LOADED PLATE

BEAM

For comparison we first consider a beam under transverse load

q(z)

Differentiating (1):3

3

dz

vdEI

dz

dM= ,

Subst. SdzdM = ,

3

3

dzvdEIS =

Differentiating again,4

4

dz

vdEI

dz

dS= ,

Subst. )z(qdZ

dS= , )z(q

dz

vdEI

4

4

= (4)

The governing equation for a beam under transverse loading.

PLATE

With suitable assumptions, a similar governing equation can be deduced for the bending of a plate undertransverse load. One of the simpler derivations follows from the Kirchhoffassumptions:

1. Deflections are small (less than the plate thickness)

2. The middle plane of the plate does not stretch during bending and remains a neutral surface

(similar to the neutral axis of a beam).

3. Normals to the middle plane remain straight, normal and inextensional (so that transverse normal

and shearing strains may be neglected). The equivalent of the plane sections remain plane

assumption in beam bending.

4. Transverse normal stresses are small compared with other normal stresses and may be

neglected.In addition to (2) and (3), the twisting moments are related to plate deformation by

yx

w)1(DM

2

xy

= (5)

For equilibrium can show that 0Sy

M

x

My

yxy=

+

(6)

and 0Sx

M

y

Mx

xyx =

+

(7)

Differentiating and combining (6) and (7) leads to

y

S

x

S

y

M

yx

M2

x

M yx2

y2

xy2

2x

2

+

=

+

+

(8)

and for loading q(x,y),y

S

x

Sq

yx

+

= , so that (8) becomes

qy

M

yx

M2

x

M2

y2

xy2

2x

2

=

+

+

(9)

Finally, substituting from (2), (3) and (5):

D/qy

w

yx

w2

x

w4

4

22

4

4

4

=

+

+

(10)

q(z)

z

v


4/12


(10) is the celebrated biharmonic equation the governing equation for elastic plate bending analysis. It is

the plate equivalent of the beam equation (4). The first and third terms represent bending of longitudinal

and transverse strips, whilst the middle term accounts for twisting action. It can also be written using

the bi-harmonic operator,

D/qw4 = (11)

The notation is built on repeated application of the Laplacian operator 2 (Nabla squared):

)y,x(qyyx

2x

)y,x(qyxyx

)y,x(q)(q

4

4

22

4

4

4

2

2

2

2

2

2

2

2224

+

+

=

+

+

==

Unfortunately the plate equation is much more difficult to solve than the corresponding beam equation.

The texts cited at the beginning of these notes present solutions for a range of plate shapes, boundary

conditions and loading. We turn now to problems of buckling and failure the .major objectives of this

excursion into plate behaviour.

IN-PLANE LOADING AND BUCKLING

INTERNAL ACTIONS

If forces are applied at the edges of a plate, possibly in addition

to lateral loading, the possibility of buckling arises, with

substantial changes in behaviour.

The figure shows normal and shearing forces applied to the plate

edges. Their effect will be to cause in-plane deformations and

corresponding actions (unless buckling occurs).

The internal actions shown are sometimes referred to asmembraneactions and consist of

In-plane normal forces, Nxand Ny

Similar to axial force in a column. Expressed as force per unit

length of plate, kN/m.

In-plane shearing forces, Nxyand NyxAlso expressed as force/unit length, kN/m.

BUCKLING OF A SIMPLY-SUPPORTED PLATE UNDER COMPRESSIVE EDGE

LOADING

This case corresponds roughly to that of a pin-ended

Euler column buckling under axial compressive loading.

The elastic buckling load is given by

b

DKN

2

CR

= (12)

where)1(12

EtD

2

3

= , the plate rigidity,

2

nb

a

a

nb

K

+= ,

=n number of buckle half-waves.

y

x

z

Ny

Nx

MEMBRANE ACTIONS DUETO IN-PLANE LOADING

Nxy

Nyx

PLATE BUCKLING - 1 HALF-WAVE

a

bN

N

all edgessimply supported

rigid end bars

t

IN PLANE LOADING

N

N

S

S

S

S


5/12


0

4

8

12

0 1 2 3 4

Plate aspect ratio, a/b

Buckling

coefficient,

K

n=1 n=2 n=3 n=4

The variation of K with aspect ratio, a/b, is shown above for various numbers of buckle waves

It can be seen that the minimum value of K is 4, and this occurs whenever the plate length a, is n x b, the

plate width. Thus a long plate prefers to buckle into roughly square segments as shown in the figure

below.

a = n x b

b

t

NCR

BUCKLING MODE OF A LONG PLATENCR

all edges simply supported

Similar solutions can be found for plates with different support and loading conditions (e.g. some edges

clamped or free, shear loading rather than direct compression, etc).

The lefthand figure on the next page shows buckling coefficients for various support conditions, and the

figure on the right shows buckling coefficients for a plate loaded in pure shear.


6/12


Buckling coefficients for axially loaded plates with Buckling coefficients for plate subject to

various support conditions in-plane shear loading

Buckling Load

)1(b12

EtKN

2

32

CR

=

)1(b12

EtKS

2

32

CR

=

Buckling Stress

2

2

2CR

CR b

t

)1(12

EK

bt

N

==

2

2

2CR

CR b

t

)1(12

EK

bt

S

==

POST-BUCKLING BEHAVIOUR

Commencement of buckling in a thin elastic plate does not immediately result in failure. The buckled plate

remains stable and can resist loads well above the elastic buckling limit without deflecting excessively (in

contrast to a slender column which can carry little more than its elastic critical load before lateral

deflections become excessive), as illustrated in the next figure. This is because the plate buckling

deformations are accompanied by stretching of the middle surface. The post-buckling stress distribution

and the extent to which the middle surface stretching influences post-buckling behaviour depends on the

support conditions at the edges of the plate. Two cases are illustrated in the next figure.

(a) The unloaded edges are free to move horizontally but constrained to remain straight.

(b) The unloaded edges are free to move horizontally


7/12


Distribution of post-buckling stress and load-deflection behaviour for different edge conditions

If:

cr= the (uniform) applied stress, Ncr/bt, at the critical load,

av = the average applied stress after buckling,

cr= the longitudinal strain just prior to buckling, and

av= the average longitudinal strain after buckling,

a plot of av/cragainst av/crreveals the characteristicchange in the apparent elastic modulus as the plate moves into

the post-buckling range.

The apparent post-buckling modulus of the plate, E*, (i.e. the

post-buckling stiffness) is significant, about 0.4E for plate

sides free to wave, 0.5E for straight sides free to move, and

0.75E for plate sides straight and not free to move.

Consequently it is not unusual for plates to be designed to

operate in the post-buckling range. The only disadvantages are

the modest reduction in stiffness and visible buckling

deformation.

Ultimate strength, failure

Redistribution of in-plane stresses after buckling continues with increasing applied load. Stress in the

stiffer sections of the plate, near the supported edges, continues to increase, while stress in the buckled

sections, such as the middle region shown in the next figure, fails to increase. The process continues until

yield stress is reached near the plate edges or as the result of bending stress associated with the

buckling deformation. Yield then tends to spread rapidly and the plate soon fails.

The following figure shows the distribution of stress at the failure load. The precise nature of the

redistributoin will depend on the edge support conditions, with stiffer supports attracting greaterproportions of the stress.

N/Ncr

1

central deflection

(a)sides remainstraight

avmax

sides freeto wave

av

(b)

slendercolumn

1

1 2

2

E 0.5E*

av cr /

av cr /


8/12


N

t

b

y

cr

y

STRESS RE-DISTRIBUTIONAT FAILURE

EFFECTIVE WIDTH CONCEPT

Theoretical calculation of plate failure loads is difficult. Consequently we introduce a simplified approach

based on the concept of effective width.Von Karman proposed that the nonlinear stress distribution across a plate at failure (diagram above and

left diagram below) be replaced by a uniform stress distributed over two reduced strips adjacent to the

supported edges, with the central buckled region ignored.

y

cr

av

actual stressdistribution

b

y

b

b /2e b /2e

contribution of centralsection ig nored

t

actual cross-section effective cross-section

He further proposed that the strips be considered together as a rectangular plate of width be, and that

failure occurs when the critical buckling stress of the equivalent plate reaches y.

From equation (12)2

2

2CR

cr b

t

)1(12

EK

bt

N

== (13)

For the equivalent width plate, at failure ycr = , so

2

e2

2

y b

t

)1(12

EK

= (14)

(13)(14) givesy

cre

b

b

= (15)


9/12


Thus be< b only when cr< y.

Considering the case of a rectangular steel

plate simply supported on all edges:

K = 4

E=200,000MPa

y=300MPa=0.3

Using (13), plot cr/yagainst b/t:

It can be seen that when 49t/b , yielding

precedes buckling and no reduction in b is

needed.

When b/t > 49 we need to reduce b such that

49t/be (the maximum value for which

yield stress can be reached without buckling).

YIELD LIMIT

The ratio b/t is known as theplate slenderness ratio, and the limiting value of 49 is known as theplateslenderness yield limit. The yield limit can be obtained directly by substituting ycr = in (13) and

rearranging to obtain

y2

2

itlimyield )1(12

EK

t

b

=

. (16)

Plate supported on both edges

Substituting K=4, E=200,000, y=300 and =0.3 gives

1.49t

b

itlimyield

=

.

Plate supported on one edge and free on the other

Other support conditions are taken into account by using the appropriate buckling coefficient, K (see p.6).

For one edge supported and the other free, K=0.5 (approximately this is for an aspect ratio a/b=5).

Other data is the same as previous case and gives

4.17t

b

itlimyield

=

.

DETERMINATION OF EFFECTIVE AREA

The effective area of a steel column, Aeis the sum of the effective area, bet, of each flat plate element

composing the cross-section.

The effective width of each flat plate element )bb(t

btb e

itlimyielde

= .

Implementation in the Steel Structures Standard, NZS3404

Plate element slenderness ratio, e:

NZS3404 uses the symbol efor the plate slenderness ratio and brings in a correction term for yieldstress other than 250MPa:

250t

b ye

= (17)

b/t

cr /y

yield before buckling -b = be

buckling before yielding -b < be

20 30 40 50 60 70

0

1

2


10/12


Plate element yield slenderness limit, ey:

Limiting values based on equation (16), and including modifications for residual stresses are tabulated in

Table 6.2.4, reproduced below.

Webs of I-beams are regarded as plates supported on two edges (i.e. at the junctions with the flanges),

whereas the flanges are regarded as plates supported on one edge (by the web) and free on the other.

The diagram below provides further explanation.

Section description: Hot-rolled

UB, UC

Heavily welded

BOX

Cold-formed

CHS

Cold-formed

RHS

Plate element widths:b1

d1

b1 b2 b1

d1

d0

b2

d1

Flange outstand b1 16 14

Flange b2supported

along both edges 35 40

Web d1supported

along both edges45 35 40

Diameter d0 82

Table of yield limit values, ey, from NZS3404.


11/12


EXAMPLES

1. Effective area of a 310UB32.

B = 149mm

T = 8.0

t = 5.5

b1= 69

d1= 282

Ag= 4080mm2y= 320MPa

Web:

)Tablefrom,45(58250

320

5.5

282

250t

dey

y1e >==

=

use mm21958

45282dd

e

ey

1e ==

=

Flange outstand:

)fromTable,16(8.9250

320

8

69

250t

bey

y1e ==

=

use mm5117.76

351120dd

e

ey

1e ==

=

b1d1

t

T

B

b1d1

t

T

B

63mm

219/2 = 110mm

Effective cross-section for axial load

219/2 = 110mm

149


12/12


Flange outstand:

)fromTable,14(48.5250

280

25

5.129

250t

bey

y1e

plate buckling notes

Documents