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
8/13/2019 Plate Buckling Notes
2/12
8/13/2019 Plate Buckling Notes
3/12
Plate_Buckling_Notes.doc p3 Copyright J.W. Butterworth August 2005
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
8/13/2019 Plate Buckling Notes
4/12
Plate_Buckling_Notes.doc p4 Copyright J.W. Butterworth August 2005
(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
8/13/2019 Plate Buckling Notes
5/12
Plate_Buckling_Notes.doc p5 Copyright J.W. Butterworth August 2005
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.
8/13/2019 Plate Buckling Notes
6/12
Plate_Buckling_Notes.doc p6 Copyright J.W. Butterworth August 2005
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
8/13/2019 Plate Buckling Notes
7/12
Plate_Buckling_Notes.doc p7 Copyright J.W. Butterworth August 2005
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 /
8/13/2019 Plate Buckling Notes
8/12
Plate_Buckling_Notes.doc p8 Copyright J.W. Butterworth August 2005
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)
8/13/2019 Plate Buckling Notes
9/12
Plate_Buckling_Notes.doc p9 Copyright J.W. Butterworth August 2005
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
8/13/2019 Plate Buckling Notes
10/12
Plate_Buckling_Notes.doc p10 Copyright J.W. Butterworth August 2005
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.
8/13/2019 Plate Buckling Notes
11/12
Plate_Buckling_Notes.doc p11 Copyright J.W. Butterworth August 2005
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
8/13/2019 Plate Buckling Notes
12/12
Plate_Buckling_Notes.doc p12 Copyright J.W. Butterworth August 2005
Flange outstand:
)fromTable,14(48.5250
280
25
5.129
250t
bey
y1e