buckling behavior of a rectangular plate (web)

under axial compression

Luis Mesquita
October 2024

Contents
1 Introduction 2

2 Problem Setup: Elastic Critical Buckling of a Simply Sup-

ported Plate Subjected to Axial Compression for Any Buckling
Mode 3
2.1 Displacement Function Representation . . . . . . . . . . . . . . . 3
2.2 Energy Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Strain Energy Due to Bending . . . . . . . . . . . . . . . . . . . 3
2.3.1 Second Derivatives of w . . . . . . . . . . . . . . . . . . . 3
2.3.2 Strain Energy Simplification . . . . . . . . . . . . . . . . . 4
2.4 Work Done by Compressive Forces . . . . . . . . . . . . . . . . . 4
2.4.1 Work Simplification . . . . . . . . . . . . . . . . . . . . . 4
2.5 Total Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . 4
2.6 Critical Buckling Condition . . . . . . . . . . . . . . . . . . . . . 4
2.7 Expression for Critical Buckling Stress . . . . . . . . . . . . . . . 4

3 Problem Setup: Elastic Critical Buckling of a Clamped Rect-

angular Plate Subjected to Axial Compression 5
3.1 Displacement Function Representation . . . . . . . . . . . . . . . 5
3.2 Energy Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.3 Strain Energy Due to Bending . . . . . . . . . . . . . . . . . . . 5
3.3.1 Second Derivatives of w . . . . . . . . . . . . . . . . . . . 6
3.3.2 Simplified Strain Energy Expression . . . . . . . . . . . . 6
3.4 Work Done by Compressive Forces . . . . . . . . . . . . . . . . . 6
3.5 Total Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . 6
3.6 Critical Buckling Condition . . . . . . . . . . . . . . . . . . . . . 6
3.7 Expression for Critical Buckling Stress . . . . . . . . . . . . . . . 7

1
4 Problem Setup: Elastic Critical Buckling of a Plate with Mixed
Boundary Conditions 7
4.1 Displacement Function Representation . . . . . . . . . . . . . . . 7
4.2 Energy Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.3 Strain Energy Due to Bending . . . . . . . . . . . . . . . . . . . 7
4.3.1 Second Derivatives of w . . . . . . . . . . . . . . . . . . . 8
4.3.2 Simplified Strain Energy Expression . . . . . . . . . . . . 8
4.4 Work Done by Compressive Forces . . . . . . . . . . . . . . . . . 8
4.5 Total Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . 8
4.6 Critical Buckling Condition . . . . . . . . . . . . . . . . . . . . . 8
4.7 Expression for Critical Buckling Stress . . . . . . . . . . . . . . . 9

5 Problem Setup: Buckling of a Rectangular Plate with Rota-

tionally Restrained Edges 9
5.1 Boundary Conditions for Rotational Restraint . . . . . . . . . . . 9
5.2 Displacement Function Representation . . . . . . . . . . . . . . . 10
5.3 Determining the Coefficients Am and Bm . . . . . . . . . . . . . 10
5.4 Boundary Condition at y = 0 . . . . . . . . . . . . . . . . . . . . 10
5.5 Boundary Condition at y = b . . . . . . . . . . . . . . . . . . . . 10
5.6 Critical Buckling Condition . . . . . . . . . . . . . . . . . . . . . 10
5.7 Expression for Critical Buckling Stress . . . . . . . . . . . . . . . 11

6 Conclusion 11

7 Calculation of Rotational Stiffness K0 and Kb 11

7.1 Rotational Rigidity Based on Flange Torsional Stiffness . . . . . 11
7.2 Including the Effect of the Root Radius . . . . . . . . . . . . . . 12

1 Introduction
This document presents a comprehensive analysis of the buckling behavior of
a rectangular plate (web) under axial compression, considering various edge
conditions. The plate represents the web of a steel H-section, where the rota-
tional restraints at the web edges are provided by the flanges. Different cases
are explored, including simply supported, clamped, and partially rotationally
restrained edges. The rotational stiffness values K0 and Kb are calculated based
on the torsional rigidity of the flanges, including the effect of the root radius.
Elastic Critical Buckling of a Simply Supported Plate Subjected to Axial
Compression for Any Buckling Mode

2
2 Problem Setup: Elastic Critical Buckling of
a Simply Supported Plate Subjected to Axial
Compression for Any Buckling Mode
Consider a thin rectangular plate of length a, width b, and thickness t, simply
supported along all four edges. The plate is subjected to uniform compressive
stress σx along the length direction (axial compression). The boundary con-
ditions are that the deflection w and bending moments are zero along all four
edges.

2.1 Displacement Function Representation

We define the deflection w(x, y) as a double Fourier series:
∞ X
X ∞  mπx   nπy 
w(x, y) = Wmn sin sin
m=1 n=1
a b

where:
• Wmn are the amplitudes of the buckling mode shapes corresponding to
different values of m and n.
• m and n are integers representing the number of half-waves in the x- and
y-directions, respectively.

2.2 Energy Expressions

2.3 Strain Energy Due to Bending
The strain energy U is given by:
Z a Z b " 2 2  2 2  2 2 #
1 ∂ w ∂ w ∂ w
U= D 2
+2 + dx dy
2 0 0 ∂x ∂x∂y ∂y 2

Substitute w(x, y) into the expression for U , and calculate the second deriva-
tives.

2.3.1 Second Derivatives of w

∞ X ∞ 
∂2w X mπ 2  mπx   nπy 
= − W mn sin sin
∂x2 m=1 n=1
a a b
∞ X ∞ 
∂2w X nπ 2  mπx   nπy 
2
= − Wmn sin sin
∂y m=1 n=1
b a b
∂2w
=0 (f orsimplysupportedboundaryconditions)
∂x∂y

3
2.3.2 Strain Energy Simplification
Substitute the second derivatives into the strain energy expression:
∞ X ∞  4 4   4 4 
1 X m π n π 2
U= Dab 4
+ 4
Wmn
4 m=1 n=1
a b

2.4 Work Done by Compressive Forces

The work W done by the compressive stress is:
Z a Z b  2
1 ∂w
W = − σx dx dy
2 0 0 ∂x

where
∞ X ∞
∂w X mπ  mπx   nπy 
= Wmn cos sin
∂x m=1 n=1
a a b

2.4.1 Work Simplification

After integrating, we obtain:
∞ X ∞  2 2
1 X m π 2
W = − σx ab 2
Wmn
4 m=1 n=1
a

2.5 Total Potential Energy

The total potential energy Π is given by:
∞ X ∞  4 4   4 4  ∞ X ∞  2 2
1 X m π n π 2 1 X m π 2
Π = U +W = Dab 4
+ 4
Wmn − σx ab 2
Wmn
4 m=1 n=1
a b 4 m=1 n=1
a

2.6 Critical Buckling Condition

The critical buckling load occurs when the total potential energy is stationary,
i.e., δΠ = 0. This leads to the equation:
 4 4
n4 π 4 m2 π 2

m π
D 4
+ 4
= σx 2
a b a

2.7 Expression for Critical Buckling Stress

Solving for the critical buckling stress gives:

π 2 D m2 n2
 
σcr = 2 + 2
a a2 b

4
 Et3
Substitute D = 12(1−ν 2 ) :

 2  2
π2 E n2

t m
σcr = +
12(1 − ν 2 ) a a2 b2

This equation gives the critical buckling stress for any buckling mode defined
by integers m and n, representing the number of half-waves in the x- and y-
directions, respectively. article amsmath amsfonts amssymb
Elastic Critical Buckling of a Clamped Rectangular Plate Subjected to Axial
Compression

3 Problem Setup: Elastic Critical Buckling of a

Clamped Rectangular Plate Subjected to Ax-
ial Compression
Consider a rectangular plate of length a, width b, and thickness t, where all four
edges are fixed (clamped). The plate is subjected to uniform axial compressive
stress σx . In clamped boundary conditions, both the deflection w and its slope
∂w
∂n (normal to the boundary) are zero along the edges.

3.1 Displacement Function Representation

For a plate with clamped edges, the displacement function w(x, y) is expressed
as a double Fourier series that satisfies the boundary conditions:
∞ X
X ∞ h  mπx i h  nπy i
w(x, y) = Wmn 1 − cos 1 − cos
m=1 n=1
a b

∂w
This form ensures that both the deflection w and its slope ∂n are zero at x = 0, a
and y = 0, b.

3.2 Energy Expressions

3.3 Strain Energy Due to Bending
The strain energy U for the plate under bending is given by:
Z a Z b " 2 2  2 2  2 2 #
1 ∂ w ∂ w ∂ w
U= D 2
+2 + dx dy
2 0 0 ∂x ∂x∂y ∂y 2

Et3
where D = 12(1−ν 2 ) is the flexural rigidity of the plate.

5
3.3.1 Second Derivatives of w
∞ X ∞
∂2w X  mπ 2  mπx  h  nπy i
= W mn cos 1 − cos
∂x2 m=1 n=1
a a b
∞ X ∞
∂2w X  nπ 2 h  mπx i  nπy 
= W mn 1 − cos cos
∂y 2 m=1 n=1
b a b
∞ X ∞
∂2w X  mπ   nπ   mπx   nπy 
=− Wmn sin sin
∂x∂y m=1 n=1
a b a b

3.3.2 Simplified Strain Energy Expression

Substituting the second derivatives into the expression for U , we get:
∞ X ∞  4 4
m2 π 2 n2 π 2 n4 π 4

1 X
2 m π
U = Dab Wmn + 2 +
4 m=1 n=1
a4 a2 b2 b4

3.4 Work Done by Compressive Forces

The work done by the compressive forces is:
Z aZ b 2
1 ∂w
W = − σx dx dy
2 0 0 ∂x
where:
∞ X ∞
∂w X mπ  mπx  h  nπy i
= Wmn sin 1 − cos
∂x m=1 n=1
a a b
Evaluating the integrals gives:
∞ X ∞ 2 2
1 X
2 m π
W = − σx ab Wmn
4 m=1 n=1
a2

3.5 Total Potential Energy

The total potential energy Π is:
∞ X ∞  4 4 ∞ X ∞
m2 π 2 n2 π 2 n4 π 4 2 2

1 X
2 m π 1 X
2 m π
Π = U +W = Dab Wmn + 2 + − σx ab W mn
4 m=1 n=1
a4 a2 b2 b4 4 m=1 n=1
a2

3.6 Critical Buckling Condition

Setting the derivative of the total potential energy with respect to Wmn equal
to zero gives the critical buckling condition:
 4 4
m2 π 2 n2 π 2 n4 π 4 m2 π 2

m π
D + 2 + = σx
a4 a2 b2 b4 a2

6
3.7 Expression for Critical Buckling Stress
Rearranging the equation for σx , the critical buckling stress is:
π 2 D m2 n2 n2
 
σcr = 2 +2 + 2
a a2 ab b
Et3
Substitute D = 12(1−ν 2 ) :
 2  2
π2 E n2 n2

t m
σcr = + 2 +
12(1 − ν 2 ) a a2 ab b2
This expression gives the critical buckling stress for any buckling mode with m
and n, considering clamped boundary conditions for all edges.
article amsmath amsfonts amssymb
Elastic Critical Buckling of a Plate with Mixed Boundary Conditions

4 Problem Setup: Elastic Critical Buckling of a

Plate with Mixed Boundary Conditions
Consider a rectangular plate of length a, width b, and thickness t, subjected
to uniform axial compressive stress σx along the x-direction. The boundary
conditions are:
• Simply supported along x = 0 and x = a: The deflection w = 0, and
2
bending moment ∂∂xw2 = 0.
• Clamped along y = 0 and y = b: The deflection w = 0, and slope ∂w
∂y = 0.

4.1 Displacement Function Representation

We express the deflection w(x, y) using a form that satisfies both boundary
conditions:
X∞ X∞  mπx  h  nπy i
w(x, y) = Wmn sin 1 − cos
m=1 n=1
a b
This form ensures that w = 0 at x = 0, a and y = 0, b, and satisfies the slope
condition at the clamped edges.

4.2 Energy Expressions

4.3 Strain Energy Due to Bending
The strain energy U for the plate under bending is given by:
Z a Z b " 2 2  2 2  2 2 #
1 ∂ w ∂ w ∂ w
U= D 2
+2 + dx dy
2 0 0 ∂x ∂x∂y ∂y 2
Et3
where D = 12(1−ν 2 ) is the flexural rigidity.

7
4.3.1 Second Derivatives of w
The second derivatives are:
∞ X ∞
∂2w X  mπ 2  mπx  h  nπy i
= − W mn sin 1 − cos
∂x2 m=1 n=1
a a b
∞ X ∞
∂2w X  nπ 2  mπx   nπy 
= W mn sin cos
∂y 2 m=1 n=1
b a b
∞ X ∞
∂2w X  mπ   nπ   mπx   nπy 
=− Wmn cos sin
∂x∂y m=1 n=1
a b a b

4.3.2 Simplified Strain Energy Expression

Substitute the second derivatives into the expression for U :
∞ X ∞  4 4
m2 π 2 n2 π 2 n4 π 4

1 X
2 m π
U = Dab Wmn + 2 +
4 m=1 n=1
a4 a2 b2 b4

4.4 Work Done by Compressive Forces

The work done by the compressive forces is:
Z aZ b 2
1 ∂w
W = − σx dx dy
2 0 0 ∂x
where:
∞ X ∞
∂w X mπ  mπx  h  nπy i
= Wmn cos 1 − cos
∂x m=1 n=1
a a b
Evaluating the integrals gives:
∞ X ∞ 2 2
1 X
2 m π
W = − σx ab Wmn
4 m=1 n=1
a2

4.5 Total Potential Energy

The total potential energy Π is:
∞ X ∞  4 4 ∞ X ∞
m2 π 2 n2 π 2 n4 π 4 2 2

1 X
2 m π 1 X
2 m π
Π = U +W = Dab Wmn + 2 + − σx ab W mn
4 m=1 n=1
a4 a2 b2 b4 4 m=1 n=1
a2

4.6 Critical Buckling Condition

Setting the derivative of the total potential energy with respect to Wmn equal
to zero gives:
 4 4
m2 π 2 n2 π 2 n4 π 4 m2 π 2

m π
D + 2 + = σx
a4 a2 b2 b4 a2

8
4.7 Expression for Critical Buckling Stress
The critical buckling stress is obtained by solving:

π 2 D m2 n2 n2
 
σcr = 2 +2 + 2
a a2 ab b
Et3
Substitute D = 12(1−ν 2 ) :

 2  2
π2 E n2 n2

t m
σcr = + 2 +
12(1 − ν 2 ) a a2 ab b2

This equation provides the critical buckling stress for any buckling mode char-
acterized by integers m and n, considering the mixed boundary conditions.
article amsmath amssymb amsfonts
Buckling of a Rectangular Plate with Rotationally Restrained Edges

5 Problem Setup: Buckling of a Rectangular

Plate with Rotationally Restrained Edges
We consider a rectangular plate of length a, width b, and thickness t, subjected
to uniform axial compressive stress σx . The boundary conditions are:
• Simply supported along x = 0 and x = a: w = 0 and Mx = 0.
• Rotationally restrained along y = 0 and y = b by rotational springs
K0 and Kb , respectively.

5.1 Boundary Conditions for Rotational Restraint

The boundary conditions at y = 0 and y = b are:

∂w ∂w
My (0) = −K0 , My (b) = Kb
∂y y=0 ∂y y=b

where the bending moment My is given by:

∂2w Et3
My = −D , D=
∂y 2 12(1 − ν 2 )

Thus, the boundary conditions become:

∂2w ∂w
D = K0
∂y 2 y=0 ∂y y=0

∂2w ∂w
D = −Kb
∂y 2 y=b ∂y y=b

9
5.2 Displacement Function Representation
To satisfy the boundary conditions, we represent the deflection w(x, y) as:
∞
X  mπx  h  mπy   mπy i
w(x, y) = Wm sin Am cosh + Bm sinh
m=1
a a a

where Wm are the mode shape amplitudes, and Am and Bm are coefficients to
be determined from the boundary conditions.

5.3 Determining the Coefficients Am and Bm

5.4 Boundary Condition at y = 0
The boundary condition gives:
 2 2
m π
D Am = K0 Bm
a2

Thus,
Dm2 π 2
Bm = Am
K0 a2

5.5 Boundary Condition at y = b

The second boundary condition gives:
 2 2         
m π mπb mπb mπb mπb
D A m cosh + B m sinh = −K b A m sinh + B m cosh
a2 a a a a
Dm2 π 2
Substituting Bm = K0 a2 Am , we obtain:

m2 π 2 Dm2 π 2 Dm2 π 2
         
mπb mπb mπb mπb
D Am cosh + A m sinh = −Kb A m sinh + A m cosh
a2 a K0 a2 a a K0 a2 a

This equation can be simplified to find the relationship between Am , K0 , and

Kb .

5.6 Critical Buckling Condition

To find the critical buckling stress σcr , we minimize the total potential energy:

Π=U +W

where U is the strain energy and W is the work done by compressive forces.

10
5.7 Expression for Critical Buckling Stress
For the first buckling mode (m = 1), the critical buckling stress becomes:
Dπ 2
" πb

πb

#
π 2 D cosh a + K0 a2 sinh a
σcr = 2
Dπ2
sinh πb πb


a a + Kb a2 cosh a

Et3
Substitute D = 12(1−ν 2 ) to get:


πb Et3 π 2 πb
 2

2
π E t cosh a + 12(1−ν 2 )K0 a2 sinh a
σcr = 2

πb Et3 π 2 πb


12(1 − ν ) a


sinh a + 12(1−ν 2 )Kb a2 cosh a

6 Conclusion
The expression for the critical buckling stress accounts for the effects of rota-
tional restraints provided by the springs K0 and Kb , and simplifies for the first
buckling mode to the form derived above.

7 Calculation of Rotational Stiffness K0 and Kb

7.1 Rotational Rigidity Based on Flange Torsional Stiff-
ness
The rotational rigidity provided by the flanges to the web can be estimated using
the torsional stiffness of the flange. For a flange with width bf and thickness tf ,
the torsional constant Jf is given by:

bf t3f
Jf = .
3
The torsional rigidity is:
bf t3f
GJf = G · ,
3
E
where G = 2(1+ν) is the shear modulus of the material.
The rotational stiffness at the web edge due to the flange can be represented
as:
GJf 2Gbf t3f
K0 = Kb = = ,
hw /2 3hw
where hw is the height of the web.
E
Substituting G = 2(1+ν) , the expression becomes:

Ebf t3f
K0 = Kb = .
3(1 + ν)hw

11
7.2 Including the Effect of the Root Radius
The root radius introduces additional stiffness to the flange, which affects the
rotational rigidity. When considering the root radius r, the effective torsional
constant Jef f is:
bf t3f 3
Jef f = + 0.5 (2rtf ) .
3
Therefore, the effective torsional rigidity is:
!
bf t3f 3
GJef f = G + 0.5 (2rtf ) .
3

The modified rotational stiffness considering the root radius becomes:

!
2GJef f E bf t3f
K0 = Kb = = + 4r3 t3f .
hw (1 + ν)hw 3

12