This document summarizes research on the elastic buckling of plates with circular and rectangular holes subjected to axial compression and bending moments. Key findings include: - Plates with smaller circular holes (d/b=0.1) exhibited buckling coefficients close to an unperforated plate. Larger holes (d/b=0.3, 0.5) had higher coefficients when the hole center was at a nodal point versus a maximum point. - Rectangular plates generally had higher coefficients than square plates. Certain load configurations (Ψ=0.5,0,-0.5,-1) produced less stability issues than others (Ψ=1). - Rectangular holes oriented

1. Elastic stability of plates with circular and
rectangular holes subjected to axial
compression and bending moment
Emanuele Maiorana,Carlo Pellegrino ,Claudio Modena
Presented by : Anwar Al-Assaf

2. • Subject
Linear buckling analysis of square and
rectangular plates with circular and
rectangular holes in various positions
subjected to axial compression and
bending moment.

3. • Purpose
To give some practical indications of the
best position of the circular hole and the
best position and orientation of the
rectangular holes in steel plates.
This is studied when axial compression
and bending moment acting together.

4. Type of load
Ψ=1
Ψ=0 , .5,-.5
Shape of plate
rectangular
square
Ψ=-1
Shape of hole
Dimension of hole
Simply supported
circular
rectangular
d/b
d
0.25d
1.5d
RL
RS
Distance of hole varying along x
Eccentricity of hole
y/b
displacement
x/b
nodal
maximum

5. Assumption for rectangular holes
• Holes with major dimension parallel to the
vertical plate axis (RS holes)
• Holes with major dimension parallel to
horizontal plate axis (RL holes).

6. Work space
•The effect of bending moment on the stability of
the plate is studied and some differences with
respect to the uniform compression load case
are shown.
•Some design suggestions on the best
Orientation of rectangular holes for stability
purposes are given.
•The influence of dimension and position of
Perforations on linear buckling behavior and ,in
particular, on buckling coefficient of the plate is
observed.

7. Some practical design formulations for the calculation of the buckling
coefficient have been proposed taking into account
(a) dimensions and shape (square and rectangular) of the plate.
(b) dimensions and shape (circular and rectangular) of the hole,
(c) position of the hole (centre in the ‘‘maximum’’ and in the
‘‘nodal’’ point) ,
(d) orientation (RS and RL) of the rectangular hole ,and
(e) load configuration (uniform compression, combinations of axial
compression and bending and pure bending).

8. problem
• The problem of stability of plates is a
typical problem of steel structures. A wide
literature review on elastic buckling of
perforated plates has been reported.

9. concepts
• In this work the problem of elastic stability
of square and Rectangular perforated
panels with simply supported edges
Subjected to axial compression and
bending moment is studied.

10. Buckling load of simply supported plates subjected to
uniform compression is analytically determined by solving
(1)

11. (2)
(3)
Solving for Nx
(4)
The lower value of Nx occurs when n=1

12. (5)
The basic instability occurs with the single half wave
in y direction for the value
(6)
(7)
Where k is the buckling coefficient

13. Nodal and maximum points
• The ‘‘nodal point’’ is assumed as the
position where the sinusoidal shape does
not show any out-of-plane displacement
while the ‘‘maximum point’’ is assumed as
the position where the sinusoidal shape
shows the maximum out-of-plane
displacement.

14. Circular holes

15. Hole position Varying along X in square
plates a/b=1 (square plate)
a
x
Observations:
b
As d/b reduces k inceases
x
y

16. Square plate
Ψ=-1
Ψ=0
Ψ=1
Ψ=-0.5
Ψ=0.5

17. Hole position Varying along X in rectangular plates
a/b=2 (rectangular plate)
a
x
Observations:
b
As d/b reduces k increases
x
y

18. Rectangular plate a/b=2
Ψ=-1
Ψ=0
Ψ=-0.5
Ψ=0.5
Observations
Ψ=-1,k >>for d/b
=0.1 and 0.3
until x/b=0.6
Ψ=1
For other values of Ψ ,k increases for all hole dimension until
x/b=1 also k is not influenced by the position ratio x/b for hole
with d/b=0.1

19. Rectangular plate a/b=2
Hole position Varying along X in
rectangular plates a/b=3
a
x
Observations:
b
As d/b reduces k increases
x
y

20. Rectangular plate a/b=3
Ψ=-1
K
Ψ=0
Ψ=1
K
Ψ=-0.5
K
K
Ψ=0.5
K
Moving the hole towards the center of the plate in the x direction ,the
rectangular plate shows similar values of k for different diameters
greater than that of the square plate.

21. Nodal points for plates without holes

22. Holes in nodal points
Ψ=-1
Ψ=0
Ψ=1
Ψ=-.05
Ψ=0.5

23. Holes having the center in the maximum points

24. Ψ=-1
Ψ=0.5
Ψ=1
K is influenced
more than that
in nodal case
Ψ=-0.5
Ψ=0

25. Eccentricity on plates with aspect
ratio a/b =3
a
y
b

26. Ψ=-1
Ψ=0
Ψ=1
Ψ=-0.5
Ψ=0.5

27. K vs y/b with the center in a maximum point
Ψ=-1
Ψ=0
The center of
hole is in a
maximum
point
Ψ=-0.5
Ψ=0.5

28. Rectangular holes

29. RL holes
RS holes

30. Dimension of holes
d
0.25d
1.5d

31. RS holes having the center in nodal point
Ψ=-1
Ψ=-0.5
Min k
Ψ=0
Ψ=1
Min k
Ψ=0.5
iKt
cn h
r
sae h na
se t at nu t ap
o frep e l
f o
r
ta
de

32. .RS hole having the center in maximum point
Ψ=-1
Ψ=0
Ψ=1
Ψ=-0.5
Ψ=0.5
K almost
constant with
varying d/b for
a/b =2 or =3

33. RL holes having the center in nodal point
K decreases when d/b
increases for Ψ=-1 &-0.5
Ψ=-1
Ψ=0
Ψ=-0.5
Ψ=0.5
K remains
almost the
same as in
unperforated
plate for
Ψ=1
Ψ=1 ,0.5
and 0

34. conclusions
• In this work linear buckling behavior of plates with
perforations subjected to axial compression and bending
moment was studied according to the following
parameters:
–
–
–
–
Shape of plate
Dimension of hole
Shape of hole
Load configuration

35. Results for plates with circular holes
• For d/b =0.1 buckling coefficient does not
change with that of unperforated plates.
• For d/b =0.3 and 0.5 ,k for perforations with the
center in nodal points is greater than that of
center in maximum point.
• With holes in the center of the plate maximum k
were obtained with d/b=0.1 when Ψ=-0.5 and -1
and with d/b=0.5 when Ψ=0,0.5 and 1
• Holes near panel edges have to be avoided for
all load conditions since the values of k are
always smaller.

36. circular holes…
• K for rectangular plates with a/b=2 or 3 is
generally greater than for square plates.
• Loads with Ψ=0.5,0,-0.5,-1 give minor
stability problems than for Ψ=1.

37. Rectangular holes results
• RS holes allow to obtain higher values for
k than RL holes except for d/b =0.1 for
which the results were as in unperforated
plates.
• RS and RL holes having their center in
nodal points show higher k of those having
their center in maximum points.
• Both RL and RS holes near the edges of
the plate should be avoided.

38. Limitations of the study &future
work
•
•
•
•
Only one boundary condition was studied
Only two types of holes were studied
One hole in the plate was studied at a time
Orientation of rectangular holes were RS
and RL only.

39. Related work
On Buckling and Ultimate Strength of
Perforated Plate Panels under Axial
Compression: Experimental and
Numerical Investigations with Design
Formulations057.pdf