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

• Subject

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

• 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.

Ψ=-1Ψ=0 , .5,-.5Ψ=1

Type of load Shape of plate

square rectangular

Shape of hole

rectangularcircular

Dimension of hole

d

1.5d

0.25dd/b

Distance of hole varying along x

x/b

Eccentricity of hole

y/b

RLRS

displacement

maximumnodal

Simply supported

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).

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.

(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).

Some practical design formulations for the calculation of the buckling coefficient have been proposed taking into account

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.

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.

Buckling load of simply supported plates subjected to uniform compression is analytically determined by solving

(1)

Solving for Nx

(2)

(3)

(4)

The lower value of Nx occurs when n=1

(5)

(6)

(7)

Where k is the buckling coefficient

The basic instability occurs with the single half wave in y direction for the value

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.

Circular holes

x

Hole position Varying along X in square plates a/b=1 (square plate)

b

a

y

x

Observations:

As d/b reduces k inceases

Ψ=-1

Ψ=0

Ψ=-0.5

Ψ=0.5

Ψ=1

Square plate

x

Hole position Varying along X in rectangular plates a/b=2 (rectangular plate)

b

a

y

x

Observations:

As d/b reduces k increases

Ψ=-1 Ψ=-0.5

Ψ=0Ψ=0.5

Ψ=1

Observations

Ψ=-1,k >>for d/b =0.1 and 0.3 until x/b=0.6

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

Rectangular plate a/b=2

b

a

y

x

x

Hole position Varying along X in rectangular plates a/b=3

Observations:

As d/b reduces k increases

Rectangular plate a/b=2

Ψ=-1

Ψ=0

Ψ=-0.5

Ψ=0.5

Ψ=1

K

K

K

K

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.

Rectangular plate a/b=3

Nodal points for plates without holes

Ψ=-1

Ψ=0

Ψ=-.05

Ψ=1

Ψ=0.5

Holes in nodal points

Holes having the center in the maximum points

Ψ=-1Ψ=-0.5

Ψ=0.5Ψ=0

Ψ=1

K is influenced more than that in nodal case

Eccentricity on plates with aspect ratio a/b =3

yb

a

Ψ=-1

Ψ=0

Ψ=1

Ψ=-0.5

Ψ=0.5

The center of hole is in a maximum point

K vs y/b with the center in a maximum point

Ψ=-1

Ψ=0

Ψ=-0.5

Ψ=0.5

Rectangular holes

RL holes

RS holes

Dimension of holes

1.5d

d0.25d

RS holes having the center in nodal point

Ψ=-1

Ψ=0

Ψ=-0.5

Ψ=0.5

Ψ=1

Min k Min k

K incre

ases

than that o

f

unperfora

te

d plate

RS hole having the center in maximum point.

Ψ=0.5

Ψ=-0.5

Ψ=0

Ψ=-1

Ψ=1

K almost constant with varying d/b for a/b =2 or =3

RL holes having the center in nodal point

Ψ=1

Ψ=0.5

Ψ=-0.5

Ψ=0

Ψ=-1

K decreases when d/b increases for Ψ=-1 &-0.5

K remains almost the same as in

unperforated plate for

Ψ=1 ,0.5 and 0

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

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.

…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.

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.

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.

Related work

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