METHODS AND EQUATIONS
PLATE AND SHELL BUCKLING
Figure 1.1 - HyperSizer-generated panel buckling mode shape of a curved panel under pure shear load. This buckling
deformation plot is only available if the SS8 Rayleigh-Ritz panel buckling solution is executed.
April 25, 2018
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Table of Contents
1
SUMMARY ..........................................................................................................................................................4
2
SYMBOLS............................................................................................................................................................5
3
TERMINOLOGY ..................................................................................................................................................6
3.1
BOUNDARY CONDITIONS ............................................................................................................................6
3.2
LAMINATE CLASSIFICATION .......................................................................................................................6
3.2.1
ORTHOTROPIC .....................................................................................................................................6
3.2.2
UNSYMMETRIC, CROSS-PLY ...............................................................................................................7
3.2.3
ANISOTROPIC ......................................................................................................................................7
3.3
4
FLAT PLATE – ISOTROPIC ................................................................................................................................8
4.1
5
6
SSSS, SHEAR ..............................................................................................................................................8
FLAT PLATE - ORTHOTROPIC ..........................................................................................................................9
5.1
GOVERNING EQUATION ..............................................................................................................................9
5.2
SSSS, BIAXIAL ...........................................................................................................................................9
5.3
SSSF, UNIAXIAL .........................................................................................................................................9
5.4
SSSS, SHEAR ..............................................................................................................................................9
FLAT PLATE - UNSYMMETRIC, CROSS-PLY ..................................................................................................11
6.1
7
LOADING .....................................................................................................................................................7
SSSS, BIAXIAL .........................................................................................................................................11
CYLINDER - UNSYMMETRIC, CROSS-PLY ......................................................................................................13
7.1
SP-8007 ....................................................................................................................................................13
7.1.1
7.2
8
SS, BIAXIAL ..............................................................................................................................................14
CURVED PLATE – ISOTROPIC .........................................................................................................................16
8.1
9
AXIAL COMPRESSION .......................................................................................................................13
SSSS, SHEAR ............................................................................................................................................16
CURVED PLATE – ORTHOTROPIC ..................................................................................................................18
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9.1
10
SSSS, SHEAR ............................................................................................................................................18
CURVED PLATE – UNSYMMETRIC, CROSS-PLY ............................................................................................19
10.1
SSSS, BIAXIAL .........................................................................................................................................19
11
SS8 RAYLEIGH-RITZ.......................................................................................................................................20
12
APPENDIX I ......................................................................................................................................................25
12.1
13
GOVERNING EQUATIONS – ANISOTROPIC PLATE BUCKLING ...................................................................25
REFERENCES....................................................................................................................................................27
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1 Summary
This document provides the general equations and assumptions for the buckling plate and shell buckling routines
called within HyperSizer. These routines are routinely called to solve panel buckling, local buckling, and pressure
deflection problems.
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2 Symbols
a
Length of panel
b
Width of panel
L
Length of panel
r
Radius of curvature
m
Number of buckling half waves in the x direction
n
Number of buckling half waves in the y direction – plates and curved plates
n
Number of buckling full waves in the y direction – cylindrical shells
t
Laminate thickness
 xy , yx
Laminate Possion’s ratios
Aij
Membrane stiffness of the plate
Bij
Membrane-bending coupling stiffness of the plate
Dij
Flexural stiffness of the plate
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3 Terminology
3.1 Boundary Conditions
The convention used to describe the plate boundary conditions is shown in Figure 3.1 and is the same found in
Leissa (1985). According to the convention the plate in the figure as SFSC boundary conditions (simple-freesimple-clamped).
Free = F
y
Simply
Supported = S
x
Clamped = C
Figure 3.1 - Schematic of plate boundary condition convention. According to the convention, the above plate has
“SFSC” boundary conditions.
3.2 Laminate Classification
Laminates are classified by the form of the ABD matrix. Unfortunately no standard terminology exists and the terms
‘orthotropic’ and ‘anisotropic’ can have varied meanings.
3.2.1 Orthotropic
In this context orthotropic means that there are no normal-shear coupling terms (A13, A23, D13 or D23) and no Bij
terms. The orthotropic equation is exact for a single orthotropic ply and for a symmetric and cross-ply laminate.
The equation is also approximate for symmetric and balanced (angle-ply) laminates (D13 and D23 terms approach
zero).
 A11
A =  A12
 0
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A12
A22
0
0 
0 
A33 
0 0 0 
B = 0 0 0 
0 0 0 
 D11
D =  D12
 0
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D12
D22
0
0 
0 
D33 
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3.2.2 Unsymmetric, Cross-Ply
In this context unsymmetric, cross-ply refers to a laminate with the form shown below (A16 = A26 = B16 = B26 = D16
= D26 = 0). All other ABD terms may be non-zero.
 A11
A =  A12
 0
A12
A22
0
0 
0 
A66 
 B11
B =  B12
 0
B12
B22
0
0 
0 
B66 
 D11
D =  D12
 0
D12
D22
0
0 
0 
D66 
3.2.3 Anisotropic
This is the most general case in which the ABD matrix is fully populated.
 A11
A =  A12
 A16
A12
A22
A26
A16 
A26 
A66 
 B11
B =  B12
 B16
B12
B22
B26
B16 
B26 
B66 
 D11
D =  D12
 D16
D12
D22
D26
D16 
D26 
D66 
3.3 Loading
•
•
•
•
Uniaxial – Nx load
Biaxial – Nx and Ny load
Shear – Nxy
All – Nx, Ny, Nxy
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4 Flat Plate – Isotropic
4.1 SSSS, Shear
Design curves found in NACA TN 1222 Figure 1 are used for this case. The figure is reproduced below.
The buckling cofficient is taken from the line representing the smallest buckling cofficient (solid). If the aspect ratio
(a/b) is less than one, the length and width values are switched.
N xy ,cr = ks
 2D
b2
(4.1)
Figure 4.1 - Critical shear stress cofficient for flat, isotropic plate. Reference NACA TN 1222 Fig. 1.
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5 Flat Plate - Orthotropic
5.1 Governing Equation
Equation (5.1) is the governing equation for buckling of an orthotropic rectangular plate.
D11
2w
4w
4w
2w
2w
2w
+
2
D
+
2
D
+
D
=
N
+
2
N
+
N
(
)
12
66
22
x
xy
y
x 4
x 2y 2
y 4
x 2
xy
y 2
(5.1)
5.2 SSSS, Biaxial
Equation (5.1) can be solved for SSSS case under biaxial loads by assuming a double sine series for the out-ofplane displacement distribution. Doing so results in the commonly used equation which is readily available in
Leissa (1985) and MIL-HDBK-17-3F:
N x ,crit
2

2
  m 2
n
n  a  
− 2  D11   + 2 ( D12 + 2 D66 )   + D22     
b
 b   m  
  a 
=
2
2
 N  a   n 
1 +  y    
 Nx   b   m 
(5.2)
Equation (5.2) must be minimized by selecting the combination of integer half-waves (m and n) that result in the
lowest critical load.
5.3 SSSF, Uniaxial
For uniaxially loaded plates with one long edge free, the following closed-form expression is used which can be
found in MIL-HDBK-17-3E as Eq. 5.7.1.4. This expression is used to determine the local buckling load of flange
objects.
N x ,crit =
12 D66  2 D11
+ 2
b2
a
(5.3)
5.4 SSSS, Shear
The following commonly used equations are taken from NASA TN D-8257 found in Table III.
(5.4)
Multiple equations are used to solve for the critical shear load. The use of each equation depends on the value of
the previously defined parameters.
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If   1 and   1:
If   1 and   1:
N xy ,cr
4
= 2
b
4
 aa


 + bb  
5.05



D11 D 8.125 +
+ 10



(5.5)
N xy ,cr
4
= 2
a
4
 aa


 + bb  
5.05



D11 D 8.125 +
+ 10



(5.6)
3
22
3
22
(5.7)
(5.8)
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6 Flat Plate - Unsymmetric, Cross-Ply
6.1 SSSS, Biaxial
A thorough discussion of this case can be found in Chapter 6.1 of Leissa (1985). For an unsymmetrically laminated
plate, the meaning of a “simply supported” edge is not clear. Assuming that, as in classical plate theory, the edge
must have zero transverse displacement and bending moment, there remain yet four possible combinations of
“simple” (i.e., not elastically restrained) boundary conditions, depending upon the in-plane constraints.
S1: w = M n = un = ut = 0
S 2 : w = M n = N n = ut = 0
(6.1)
S 3 : w = M n = un = N nt = 0
S 4 : w = M n = N n = N nt = 0
Equation (6.1) shows all four possible interpretations of a simply-supported edge for an unsymmetric laminate.
Subscripts n and t refer to load/displacement directions that are normal and tangent to the edge respectively. All
four options specify zero out-of-plane displacement and zero edge moment as in the classical theory.
A closed-form solution is available for option S2 which specifies zero edge force and zero tangential displacements.
Buckling displacements normal to the edge are permissible. The S2 boundary condition is used in this analysis.
Assuming sinusoidal displacement functions, the governing equation takes the form:
C
C12
 11
C21 C22

C31 C32
  A
 
C23
 B = 0

C33 + ( N x 2 + N y  2 )  C 
C13
(6.2)
Where,
C11 = A11 2 + A66  2
C22 = A22  2 + A66 2
C33 = D11 4 + 2 ( D12 + D66 )  2  2 + D22  4
C12 = C21 = ( A12 + A66 ) 
C13 = C31 = B11 3 + ( B12 + 2 B66 )  2
C23 = C32 = ( B12 + 2 B66 )  2  + B22  3
=
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m
,
a
=
n
b
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For a non-trivial solution, the determinant of the coefficient matrix of Equation (6.2) must be set equal to zero,
yielding the solution for buckling stresses (Nx and Ny are positive in tension):
− N x 2 − N y  2 = C33 +
2C12C13C23 − C11C232 − C22C132
C11C22 − C122
(6.3)
It is seen that for a symmetrical laminate (Bij = 0), the right-hand-side of Equation (6.3) reduces to C33, and the
equation is the same the solution for a biaxially loaded, orthotropic plate having simple edge conditions on all four
sides.
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7 Cylinder - Unsymmetric, Cross-Ply
7.1 SP-8007
The equations used are listed in section 4.3 of NASA SP-8007 – Orthotropic Cylinders. Although the cylinders are
referred to as orthotropic, Bij terms are accounted for. The SP-8007 term ‘orthotropic’ is equivalent the
‘unsymmetric, cross-ply’ in this report.
The nomenclature for the stiffness terms in SP-8007 are converted to the HyperSizer ABD convention as:
Ex = A11 ;
Dx = D11
Cx = − B11
E y = A22 ;
Dy = D22
C y = − B22
Exy = A12 ;
Dxy = 2 ( D12 + 2 D33 )
Cxy = − B12
Gxy = A33
K xy = − B33
7.1.1 Axial Compression
The buckling load is given by the following equations (Eqs. 37-43 in the original report).
A11
A12
A13
A21
A22
A23
2
 L  A31 A32 A33
Nx = 

A11 A12
 m 
A21 A22
(7.1)
Where,
 m 
n
A11 = Ex 
 + Gxy  
 L 
r
2
2
n
 m 
A22 = E y   + Gxy 

r
 L 
2
(7.2)
2
(7.3)
 m 
 m   n 
n
A33 = Dx 
 + Dxy 
   + Dy  
 L 
 L  r
r
2
E y 2C y  n  2C xy  m  2
+ 2 +
  +


r
r r
r  L 
4
2
A12 = A21 = ( Exy + Gxy )
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4
(7.4)
m n
L r
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(7.5)
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 m  n E y n
n
A23 = A32 = ( Cxy + 2 K xy ) 
+ Cy  
 +
 L  r r r
r
2
3
Exy m
m
 m 
A31 = A13 =
+ Cx 
 + ( Cxy + 2 K xy )
r L
L
 L 
3
(7.6)
n
 
r
2
(7.7)
The buckling load is found by selecting the m and n wave combination that minimizes the right-hand side of
Equation (7.1). Prebuckling deformations are not taken into account. Cylinder edges are simply supported.
7.2 SS, Biaxial
The following formula is found in Jones (2006) it is based on the Donnell-type shell buckling equations and is
similar to the SP-8007 method. The advantage is that biaxial loads are handled without interaction equations. The
nomenclature in Jones will be used here. The cylinder edges are simply supported and the ABD is fully populated
except the normal-shear coupling terms (13 and 23).
S2 simply supported boundary conditions are assumed,
 Nx =  v =  w =  M x = 0
The following buckling displacements are assumed where uo, vo, and wo denote the indeterminate amplitudes of the
mode shape. Note that for cylindrical shell buckling, n denotes the number of whole buckling waves across the
circumference of the cylinder. This is in contrast to plates where n is the number of half waves. This distinction is
necessary because the symmetry of a full cylinder requires that the number of half waves be even.
 m x 
 ny 
 cos  
 L 
 r 
 m x   ny 
 v = vo sin 
 sin  
 L   r 
 u = uo cos 
(7.8)
 m x 
 ny 
 cos  
 L 
 r 
 w = wo sin 
The following buckling criterion results,
2T12T13T23 − T11T232 − T22T132
 m 
n
Nx 
 + N y   = T33 +
T11T22 − T122
 L 
r
2
2
(7.9)
The overbar on the load terms denote that compressive load is taken to be positive. The coefficients are defined as,
 m 
n
T11 = A11 
 + A33  
 L 
r
2
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(7.10)
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𝑇12 = (𝐴12 + 𝐴33 ) (
𝐴12
𝑇13 = (
𝑟
𝑚𝜋
)(
𝐿
𝑚𝜋
𝐿
𝑛
)( )
𝑟
(7.11)
𝑚𝜋 3
) + 𝐵11 (
𝐿
𝑚𝜋
) + (𝐵12 + 2𝐵33 ) (
 m 
n
T22 = A33 
 + A22  
 L 
r
2
𝐿
𝑛 2
)( )
𝑟
2
(7.13)
 m   n 
 1  n 
n
T23 = ( B12 + 2 B33 ) 
   + A22    + B22  
 L  r
 r  r 
r
2
3
 m 
 m   n 
n
T33 = D11 
 + ( 2 D12 + 4 D33 ) 
   + D22  
 L 
 L  r
r
4
2
2
 2 B   m 
 2  n 
1
+  12  
 + B22     + A22  
 r  r 
r
 r  L 
2
(7.12)
2
(7.14)
4
2
(7.15)
Note that this equation is practically the same as Equation (6.3) with b replaced by r.
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8 Curved Plate – Isotropic
8.1 SSSS, Shear
Design curves found in NACA TN 2661 Figure 30 are used for this case. These figures are reproduced below.
The panel dimensions in the NACA report are listed as d and h to match the shear beam terminology. The NACA
notation maps to the conventional notation as follows.
d =a
(8.1)
h=b
The the formulas for the critical shear stress cofficient ks and the curvature parameter Z are dependent on whether
the panel is long (a/b > 1) or wide (a/b < 1).
If a / b > 1:
b2
Z=
1 − xy yx
Rt
N xy ,cr = ks
(8.2)
 2 Eb2t
(8.3)
12 R 2 Z 2
Equation (8.3) can be simplified into a more compact form by substituting Equation (8.2) for Z.
N xy ,cr
 2D
= ks 2
b
(8.4)
If a / b < 1:
Z=
a2
1 − xy yx
Rt
N xy ,cr
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(8.5)
 2D
= ks 2
a
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(8.6)
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Figure 8.1 - Critical shear stress coefficients for long panels. Reference NACA TN 2661 Fig. 30 (a).
Figure 8.2 - Critical shear stress coefficients for wide panels. Reference NACA TN 2661 Fig. 30 (b).
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9 Curved Plate – Orthotropic
9.1 SSSS, Shear
A curvature correction factor is used in conjuction with the orthotropic flat plate shear solution to get the critical
load for an orthotropic curved plate. The correction factor is derived using the NACA shear solutions for flat and
curved isotropic panels.
K curve =
N xy ,iso ,curved
(9.1)
N xy ,iso , flat
Nxy,iso,curved is the curved plate critical load from Equations (8.4) and (8.6), and Nxy,iso,flat is the flat plate critical shear
load from Equation (4.1). The isotropic bending stiffness D used in the isotropic shear buckling equations is the
minimum of the laminate D11 and D22.
Next the curvature correction Kcurve is applied to the orthotropic flat plate critical load Nxy,ortho,flat. The flat plate
critical load is computed using Equations (5.4)-(5.8).
N xy ,cr = K curve N xy ,ortho , flat
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(9.2)
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10 Curved Plate – Unsymmetric, Cross-Ply
10.1 SSSS, Biaxial
The full cylindrical shell solution from Section 7.2 is approximated into plate form by dictating that the
circumferential waves terminate at the width of the plate.
The half wavelength of a mode across the curved edge is defined as,
h =
b
nh
(10.1)
where b is the length of the curved edge and nh is the number of half waves across the curved edge. The half
wavelength is converted into the number of cylinder full waves by dividing through by the circumference of the
equivalent cylinder.
n* =
2 r
2h
(10.2)
The cylinder full waves n* is not required to be an integer; hence this solution is approximate. The approximate
number of cylinder full waves is then substituted into Equations (7.9) through (7.15) to find the critical load.
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11 SS8 Rayleigh-Ritz
HyperSizer contains a robust Rayleigh-Ritz solution for anisotropic plate and shell buckling using the method
developed by Wilkins (1973). The Rayleigh-Ritz method is a direct energy method used to get approximate
solutions to a number of structural analysis problems such as static, buckling, and vibration analysis. It is important
to clarify that energy principles themselves are exact, and the assumed displacement functions are what makes the
solution approximate.
The following assumptions are made in this formulation:
•
•
•
•
•
•
•
•
Flat plate, cylindrical shell, or full cylinder geometry
Any combination of fixed, simple, and free boundary conditions
Fully anisotropic material properties
Vlasov shell theory (strain-displacement relations)
The shell is thin compared to the thickness
Displacements are small compared to the thickness
Transverse shear effects are negligible
No imperfections assumed
The Rayleigh-Ritz method works by assuming a set of displacement functions that satisfy the geometric boundary
conditions. Energy principles are using to compute the total potential energy directly. With the total potential energy
known, the principle of stationary potential energy is used to solve for equilibrium and the principle of minimum
potential energy is used to find the buckling load.
The basic energy principle involved is the theorem of stationary potential energy. In the present case it may be
written as:
V + U + Q − T = constant
(11.1)
where
V = strain energy
U = potential energy of membrane loads
Q = potential energy of lateral loads
T = kinetic energy
For a static deflection problem (used in HyperSizer flat pressure panel analysis), the kinetic energy is zero and
Equation (11.1) takes the form:
V + U + Q = constant
(11.2)
For the buckling problem, Equation (11.1) becomes:
V + U = constant
(11.3)
where λ is the buckling eigenvalue.
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For the frequency problem, including membrane loads, Equation (11.1) is now:
V + U − T = constant
(11.4)
As noted above, each of the problems of concern is governed by Equation (11.1), where the variations can be
replaced with the problem of finding the minimum of Equation (11.1) by assuming the displacements (u, v, w) in
the form of a finite series:
u=
mf
nf
 a
m = mi n = ni
v=
mf
nf
 a
m = mi n = ni
w=
1mn
mf
2 mn
X 1m ( x ) Y1m ( y ) sin ( )
X 2 m ( x ) Y2 m ( y ) sin ( )
nf
 a
m = mi n = ni
3 mn
(11.5)
X 3m ( x ) Y3m ( y ) sin ( )
where
mi = ix ;
m f = i x + nx − 1
mi = iy ;
n f = iy + ny − 1
The aimn are the undetermined constants, and the functions Xim(x), and Yin(y) are chosen to satisfy the geometric
boundary conditions on u, v, w. Introducing the assumed series into Equation (11.1) reduces the problem to finding
the minimum of Equation (11.1) with respect to the undetermined constants, aimn. Thus, Equation (11.1) is now a
function of only the undetermined constants, and is equivalent to the following conditions:

(V + U + Q − T ) = 0
aimn
(11.6)
where
i = 1, 2, 3
m = mi , . . . ., m f
n = ni , . . . ., n f
such that Equation (11.6) denotes a set of 3 (nx * ny) simultaneous algebraic equations. The assumed series (11.5)
always involves additional constraints on the energy criteria beyond the physical constraints on the problem, so that
the solution obtained by the Raleigh-Ritz method is always in the direction of a stiffer structure. However, if the
assumed series is complete and satisfies the geometric boundary conditions, then the consecutive solutions obtained
by including additional terms in the assumed series must approach the correct solution.
In this method, the strain-displacement relations of the Vlasov shell theory are used. Commas denote partial
differential with respect to the x, y, or z coordinate. R is the radius of curvature of the cylindrical shell. If the
structure is a flat, the radius of curvature goes to infinity and the strain-displacement relations degenerate into the
typical flat plate equations.
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 xo = u, x
 yo = v, y + w R
 xyo = u, y + v, x
(11.7)
 x = − w, xx
 y = − w, yy − w
 xy = −2w, xy −
R2
u, y
R
+
v, x
R
Figure 11.1 - Shell geometry sign convention.
The boundary conditions considered are the classical conditions are clamped, simply supported and free. all
combinations of these three may be specified, that is, any edge of a panel may be specified as clamped, supported
or free (in the Buckling Tab). A distinct advantage of the Rayleigh-Ritz method is that only the geometric boundary
conditions (displacement and slope) need to be satisfied to insure convergence of the solution (although
convergence is improved by the satisfaction of the force boundary conditions). The Rayleigh-Ritz method does
require a set of assumed modal function, each of which satisfies the geometric boundary conditions. The functions
chosen are a series of simple beam vibration modes. These functions form a complete orthogonal set, and are all of
the same general form. the used of these functions allows the normal deflection, w, to satisfy the following
conditions (n denotes a normal to the particular edge):
1. clamped edge: w = 0; w,n = 0
2. simply supported edge: w=0; w,nn = 0
3. free edge: w,nn = 0; w,nnn = 0
In additions to these conditions, which apply to flat or curved plates and the ends of a cylinder, the normal deflection
in the circumferential direction of a cylinder is taken to be:
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 2n 
Y3n ( y ) = cos 
y
 b 
(11.8)
An assumption is made concerning the form of u and v. In the x direction, it is assumed that the mode shape function
for v is the same as that for w and that the mode shape function for u is the derivative of that for w. Mathematically,
X 1m ( x ) = X 3m, x ( x )
(11.9)
X 2 m ( x ) = X 3m ( x )
Since the roles of u and v are reversed in the y direction, it is also assumed that,
Y1n ( y ) = Y3n ( y )
(11.10)
Y2 n ( y ) = Y3n , y ( y )
These assumptions on the form of u and v allow them to always satisfy their required geometric boundary
conditions.
The beam mode shapes can be written as a sum of four terms:
4
Z m ( z ) =  Cmj  jm
(11.11)
j =1
where
1m = cosh (  m z )
 2 m = cos (  m z )
(11.12)
3m = sinh (  m z )
 4 m = sin (  m z )
and the Cmj are constants for a particular mode shape m and the appropriate boundary condition. The εm is the
corresponding natural frequency of the mth mode. The successive derivatives of Zm(z) are also of this form with
changes in the Cmj due to the repeating nature of the derivatives of the ρjm. the z-notation used here is replaced by x
or y depending on the plate direction being integrated. With this special form of the beam mode shapes all of the
various integral (required to compute the energy) may be obtained in closed form.
In approximate methods like Rayleigh-Ritz, the accuracy of the solution is depending on the number of terms used
to approximate the deflection or mode shape. In the HyperSizer implementation 25 terms are used in the x and y
direction (nx = ny = 25).
The generality of this method comes at relatively high computational cost. Run times may be on the order of ¼
second. The HyperSizer optimization algorithm has been optimized to execute this method only after faster methods
been executed with postive margins. Even so, this method can slow down sizing optimizations significantly. It is
for this reason that it is recommended that this method be used only when the design space has been narrowed.
Failure Analyses using SS8 Rayleigh-Ritz
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•
•
•
AID 011 – Panel Buckling, Curved or Flat, All BC
AID 033 – Beam Buckling, Cylindrical, Axial and Bending, Rayleigh-Ritz
Flat Panel Pressure Loading (only in selected cases, see Panel Pressure HME)
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12 Appendix I
12.1 Governing Equations – Anisotropic Plate Buckling
The following section presents the governing differential equations for plate buckling. These equations can be found
in Leissa (1985) and Turvey (1995).
A general anisotropic plate refers to any general laminate (all ABD terms populated). The equations governing the
bifurcation buckling of a general anisotropic plate may be expressed in terms of the displacements in matrix operator
form as:
 L11

 L21
 L31
L12
L22
L32
  u  0

L23   v  = 0 
( L33 − F )  w 0
L13
(12.1)
where the Lij are differential operators representing the plate stiffness; F is a differential operator representing the
in-plane stress resultants (Nx, Ny, Nxy). u and v are in-plane displacements of the midplane during buckling in the x
and y directions; and w is the transverse displacement.
Expanding the terms in Equation (12.1):
L11 = A11
2
2
2
+
2
A
+
A
16
66
x 2
xy
y 2
L22 = A22
2
2
2
+
2
A
+
A
26
66
y 2
xy
x 2
4
4
4
4
4
L33 = D11 4 + 4 D16 3 + 2 ( D12 + 2 D66 ) 2 2 + 4 D26
+ D22 4
x
x y
x y
xy 3
y
L12 = L21 = A16
L13 = L31 = − B11
2
2
2
+
A
+
A
+
A
(
)
12
66
26
x 2
xy
y 2
3
3
3
3
−
3
B
−
B
+
2
B
−
B
(
)
16
12
66
26
x3
x 2y
xy 2
y 3
3
3
3
3
L23 = L32 = − B16 3 − ( B12 + 2 B66 ) 2 − 3B26
− B22 3
x
x y
xy 2
y
F = Nx
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2
2
2
+
2
N
+
N
xy
y
x 2
xy
y 2
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It is important to note that u and v are not the in-plane displacements which occur with increasing in-plane stress
resultants. Rather these are the additional displacements which arise when the buckling load is reached and the plate
is deformed in a buckled mode shape of infinitesimal amplitude. These additional in-plane displacements arise due
to the bending-stretching coupling which is characterized by the L13(=L31) and L23(=L32) operators. The bendingstretching coupling terms primarily act to decrease the stiffness of the plate and thus decrease the critical buckling
load.
Equation (12.1) is an eighth order set of differential equations which closely resemble the form of shell buckling
equations, which are also eighth order. Since the equations are of eighth order, four boundary conditions must be
specified along each edge to define the problem physically, and to generate a proper mathematical eigenvalue
problem.
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13 References
Bruhn, E. (1973). Analysis and Design of Flight Vehicle Structures. Jacobs Publishing, Inc.
Department of Defense. (2002). MIL-HDBK-17-3F, Polymer Matrix Composites Materials Usage, Design, and
Analysis.
Jones, R. (2006). Buckling of Bars, Plates, and Shells. Blacksburg: Bull Ridge Publishing.
Kuhn, P., Peterson, J., & R, L. (1952). A Summary of Diagonal Tension, Part I - Methods of Analysis. Washington:
NACA TN 2661.
Leissa, A. (1985). Buckling of Laminated Composite Plates And Shell Panels. AFWAL-TR-85-3069 Air Force
Flight Dynamics Laboratory.
NASA. (1665). Buckling of Thin-Walled Circular Cylinders. Washington: NASA SP-8007.
Stein, M., & Neff, J. (1947). Buckling Stresses of Simply Supported Rectangular Flat Plates in Shear. Washington:
NACA TN 1222.
Stroud, W., & Agranoff, N. (1976). NASA TN D-8257 - Minimum-Mass Design of Filamentary Composite Panels
Under Combined Loads: Design Procedure Based On Simplified Buckling Equations. Washington: NASA.
Timoshenko, S. (1961). Theory of Elastic Stability. McGraw-Hill.
Turvey, G., & Marshall, I. (1995). Buckling and Postbuckling of Composite Plates. Chapman & Hall.
Whitney, J. (1987). Structural Analysis of Laminated Anisotropoic Plates. Lancaster: Technomic Publishing
Company, Inc.
Wilkins, D. (1973). Anisotropic Curved Panel Analysis. Fort Worth, TX: General Dynamics.
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