F-1998-44/D-1048 June 20, 1998

TO: All Holders of NASA/TM-1998-206542

FROM: F/Technical Publications Office

SUBJECT: Errata Sheet for NASA TM-1998-206542

The report titled Mechanical- and Thermal-Buckling Behavior of Rectangular

Plates With Different Central Cutouts by William L. Ko has an error in figure 43 on

page 36.

Please make the following change to this document. Delete the original pages 35 and

36. Then insert the attached pages 35 and 36 to replace it. Staple this errata sheet

inside the front cover of the document.

Thank you for your cooperation.

Camilla F. McArthur

Administrative Operations Specialist

National Aeronautics and

Space Administration

Dryden Flight Research Center

P.O. Box 273

Edwards, California 93523-0273

Reply to

Attn of:
35

Figure 41. Buckling temperatures as functions of hole size; square holes.

(a) 4S fixed.

(b) 4C fixed.

Figure 42. Buckled shapes of uniformly heated square plates with square holes; l/w = 1.

4C

4C

4S

/w = 1.0

c

w

4S

0 .5 .6 .7 .8 .1 .2 .3

970947

.4

c/w

50

100

200

150

∆T

∆Tcr,

°F

Symmetric buckling

Antisymmetric buckling

c/w = 0.2

∆T

970948a c/w = 0.6 970948b

∆T

∆T

c/w = 0.2 970949a

∆T

970949b c/w = 0.6
36

Change 1, 06/20/98

Figure 43. Buckling temperatures as functions of hole size; square holes.

(a) 4S fixed.

(b) 4C fixed.

Figure 44. Buckled shapes of uniformly heated rectangular plates with square holes; l/w = 1.5.

4C

∆T

4C

4S

/w = 1.5

c

w

4S

0 .5 .6 .7 .8 .1 .2 .3

970950

.4

c/w

5

10

20

25

35

40

45

50

30

15

∆Tcr,

°F

Symmetric buckling

Antisymmetric buckling

∆T

c/w = 0.2 970951a

∆T

970951b c/w = 0.6

∆T

c/w = 0.2 970952a

∆T

970952b c/w = 0.6
NASA/TM-1998-206542

Mechanical- and Thermal-Buckling 

Behavior of Rectangular Plates

With Different Central Cutouts

William L. Ko

Dryden Flight Research Center

Edwards, California

March 1998
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NASA/TM-1998-206542

Mechanical- and Thermal-Buckling 

Behavior of Rectangular Plates

With Different Central Cutouts

William L. Ko

Dryden Flight Research Center

Edwards, California

March 1998

National Aeronautics and

Space Administration

Dryden Flight Research Center

Edwards, California 93523-0273
NOTICE

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Price Code: A16 Price Code: A16
iii

CONTENTS

Page

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

DESCRIPTION OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Mechanical Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Thermal Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

FINITE-ELEMENT ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

MATERIAL PROPERTY ITERATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Solution Accuracy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Plates With Circular Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Mechanical Buckling for Fixed-Boundary Cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Mechanical Buckling for Free-Boundary Cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Thermal Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Plates With Square Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Mechanical Buckling for Fixed-Boundary Cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Mechanical Buckling for Free-Boundary Cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Thermal Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Comparison of Buckling Strengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

DISCUSSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
ABSTRACT

Mechanical- and thermal-buckling analyses were performed on rectangular plates with central

cutouts. The cutouts were either circular holes or square holes. The finite-element structural analysis

method was used to study the effects of plate-support conditions, plate aspect ratio, hole geometry, and

hole size on the mechanical- and thermal-buckling strengths of the perforated plates. By increasing the

hole size, thermal-buckling strengths of the plates could be enhanced. The compressive-buckling

strengths of the plates could also be increased considerably only under certain boundary conditions and

aspect ratios. The plate-buckling mode can be symmetrical or antisymmetrical, depending on the plate

boundary conditions, aspect ratio, and the hole size. For the same cutout areas (i.e., same plate weight

density), the buckling strengths of the same-sized plates with square holes generally surpass those of the

plates with circular holes over the range of hole sizes. The results and illustrations provide vital information for the efficient design of aerospace structural panels.

NOMENCLATURE

c side of square hole, in.

d diameter of circular hole, in.

E Young’s modulus, lb/in2

E33 triangular combined membrane and bending element

E43 quadrilateral combined membrane and bending element

G shear modulus, lb/in2

JLOC joint location (or grid point or node)

l length of rectangular plates, in.

compressive force intensity in y-direction, lb/in.

SPAR structural performance and resizing finite-element computer program

T temperature, °F

 assumed temperature for material, °F

room temperature, = 70 °F

t thickness of plates, in.

w width of plates, in.

x, y rectangular Cartesian coordinates

α coefficient of thermal expansion, in/in-°F

∆T temperature increase, °F

ν Poisson’s ratio

Subscripts

( )cr critical value at buckling

( )n n

th iteration 

Ny

Ta

Tr Tr

( ) n = 123 , , ...
2

INTRODUCTION

In aerospace structures, cutouts are commonly used as access ports for mechanical and electrical

systems, or simply to reduce weight. Structural panels with cutouts often experience compressive loads

that are induced either mechanically or thermally, and can result in panel buckling. Thus, the buckling

behavior of those structural panels with cutouts must be fully understood in the structural design. 

For an unperforated rectangular plate of finite extent (i.e., with finite length and finite width) under

uniform compression, the closed-form buckling solutions are easily obtained because the prebuckling

stress field is uniform everywhere in the plate. When a finite rectangular plate is perforated with a central

cutout (e.g., a circular or square hole), however, the buckling analysis becomes extremely cumbersome

because the cutout introduces a load-free boundary that causes the stress field in the perforated plate to be

nonuniform. Hence, the closed-form buckling solutions are practically unobtainable, and various approximate methods had to be developed to analyze such perforated plates.

The buckling of flat square plates with central circular holes under in-plane edge compression has

been studied both theoretically and experimentally by various authors (refs. 1–12). The methods of theoretical analysis used by most of the past investigators (refs. 1–3, 5) were the Rayleigh-Ritz minimum

energy method and the Timoshenko method (ref. 13). However, except for Schlack (ref. 3) and Kawai

and Ohtsubo (ref. 5), the theoretical analysis methods used do not allow the boundary and loading conditions to be precisely defined for larger hole sizes because the stress distributions of the infinite perforated

plate are used as the prebuckling stress solution for the finite perforated plate. Thus, most of the earlier

buckling solutions are limited to small hole sizes, and are not fit for studying the effects of different plate

boundary conditions on the buckling strengths of the finite plates with arbitrarily sized holes using those

approximate solutions.

Using the Rayleigh-Ritz method, Schlack (ref. 3) analyzed the buckling behavior of a simplysupported square plate with a circular hole, subjected to uniform edge displacements with three arbitrary

displacement functions, and calculated the buckling displacements. The buckling loads were then calculated using the stress-strain relationships. Ritchie and Rhodes (ref. 7) studied the buckling behavior of

both square and rectangular simply-supported perforated plates. Their theoretical analysis employed an

approximate approach using a combination of Rayleigh-Ritz and finite-element methods. These methods

are reasonably accurate for small holes, but lose accuracy when dealing with larger holes. The results of

the analysis show that the buckling behavior of perforated rectangular plates is quite different from that of

perforated square plates, and that the buckling mode is dependent on the hole size. Kawai and Ohtsubo

(ref. 5) also studied the perforated square plates using the Rayleigh-Ritz procedure with the prebuckling

stress distribution determined by the finite-element method. To reduce the labor of numerical calculations, the double integrations in the energy procedure for each finite element were transformed into line

integrals around the element boundary using the well-known Gauss theorem.

To minimize the mathematical complexities, Nemeth (refs. 8–11) analyzed perforated square

orthotropic plates by converting the classical two-dimensional buckling analysis into an equivalent onedimensional analysis by approximating the plate displacements with kinematically admissible series. In

the analysis, the two unloaded edges were assumed simply supported, and the loaded edges were either

simply supported or clamped. This approximate buckling analysis predicted the buckling loads to within

10 percent of those calculated using the finite-element method. Nemeth’s analytical and experimental

results (ref. 10) indicated that increasing the hole size in a given plate does not always reduce the
3

buckling load. Using the finite-element method, Lee et al. (ref. 12) examined the buckling behavior of a

square plate with a central circular hole. The studies, however, were limited to small hole sizes.

As mentioned earlier, most past theoretical analyses of perforated plates considered mainly the square

plates under simply-supported boundary conditions to minimize the mathematical complexities. Likewise, most past experimental studies on perforated plates were limited to square plates under simplysupported boundary conditions because of the need to compare the test data with the existing theoretical

results. The past investigations, in general, lack comprehensive information on the buckling behavior of

rectangular plates of arbitrary aspect ratios, containing holes of arbitrary sizes, and supported under any

specified boundary conditions. 

With the availability of powerful tools such as the well-developed, finite-element structural analysis

computer programs, it is now possible to calculate the prebuckling stress fields and the buckling eigenvalue solutions quite accurately for the finite rectangular plates of any aspect ratios, containing cutouts of

any geometry and any hole sizes, under any specified boundary and loading conditions.

This report investigates the mechanical- and thermal-buckling analyses of rectangular plates containing arbitrarily-sized central circular holes or square holes. A finite-element method was used to study the

effects of plate aspect ratio, hole geometry, hole size, and plate boundary conditions on the mechanicaland thermal-buckling strengths of perforated plates.

DESCRIPTION OF THE PROBLEM

 The geometry of the perforated rectangular plates and different boundary conditions used in the

finite-element analysis are described as follows.

Geometry

 Figure 1 shows the geometry of two types of perforated rectangular plates with length l, width w,

and thickness t. The central cutout is either a circular hole with diameter d (fig. 1(a)), or a central square

hole with side c (fig. 1(b)). Table 1 lists the dimensions of various perforated rectangular plates analyzed.

Notice that all the plates have the same width, w = 20 in., and the same thickness, t = 0.1 in. 

In table 1, the range 0 ~ 0.7 covers 0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6,

0.65, 0.7. For a given plate aspect ratio, the 15 hole-size cases will provide 15 data points for plotting

each buckling curve in the ( )cr vs. d/w (or c/w) plots. If the buckling curve has any sharp bends in

certain regions, however, additional data points for d/w (or c/w) values not listed in table 1 were generated to define those sharp bend regions more accurately.

Table 1. Dimensions of perforated plates.

w, in. t, in. l/w d/w c/w

20 0.1 1.0 0 ~ 0.7 0 ~ 0.7

20 0.1 1.5 0 ~ 0.7 0 ~ 0.7

20 0.1 2.0 0 ~ 0.7 0 ~ 0.7

Ny
4

Boundary Conditions

The various boundary conditions considered in the mechanical- and thermal-buckling analyses are

described as follows.

Mechanical Buckling

For mechanical buckling (uniaxial compressive buckling), the four edges of the perforated plates are

either simply supported or clamped. The lower edge of the plate is kept stationary and the upper edge is

allowed to move freely in the loading direction (y-direction). The two unloaded edges are either constrained from the transverse in-plane motions (figs. 2(a) and 3(a)) (called fixed case), or unconstrained

from the transverse in-plane motions (figs. 2(b) and 3(b)) (called free case). The four cases of boundary

conditions considered in the analysis are as follows:

1. 4S fixed—four edges simply supported; the two side edges can slide freely along the lubricated

fixed guides (fig. 4(a)).

2. 4S free—four edges simply supported; the two side edges can slide freely along the lubricated

guides, which can have free in-plane transverse motions (fig. 4(b)).

3. 4C fixed—four edges clamped; the two side edges can slide freely along the lubricated fixed

clamping guides (fig. 5(a)).

4. 4C free—four edges clamped; the two side edges can slide freely along the lubricated clamping

guides, which can have free in-plane transverse motions (fig. 5(b)).

Thermal Buckling

For thermal buckling, the plates were subjected to uniform temperature loading. Two types of plate

boundary conditions were considered:

1. 4S fixed—four edges simply supported by fixed-edge supports (fig. 6(a)). 

2. 4C fixed—four edges clamped by fixed-edge supports (fig. 6(b)).

FINITE-ELEMENT ANALYSIS

In the finite-element analysis, the structural performance and resizing (SPAR) finite-element computer program (ref. 14) was used. Because of symmetry, only one quarter of the perforated plates were

modeled. The plates with circular holes were modeled with both triangular combined membrane and

bending elements (E33 elements) and quadrilateral combined membrane and bending elements (E43 elements). For the plates with square holes, only the square-shaped E43 elements were used. Two typical

quarter-panel finite-element models generated for a typical circular cutout case (l/w = 1.5; d/w = 0.2) and

a typical square cutout case (l/w = 1.5; c/w = 0.2) are shown, respectively, in figures 7 and 8.

For the circular hole cases, models with aspect ratio l/w = 2 were generated by adding 100 additional

square E43 elements to the square models (l/w = 1); and models with aspect ratio l/w = 1.5 were generated by properly modifying the aspect ratio of those additional E43 elements in the models with l/w = 2.
5

Models with circular holes ranging from small to moderate sizes (called the basic models) were generated

by modifying the hole sizes without changing the number of elements in the annular domains around the

holes (fig. 7). By using this simple method of generating multiple models, the tangential-to-radial aspect

ratios of the elements in the annular domains will gradually increase with the increasing size of the holes.

Therefore, the models with large circular holes were generated from the basic models by carving out

some elements in the hole boundary regions of the basic models. Thus, creating larger holes would hardly

disturb the aspect ratios of the elements lying in the remaining annular domains. When the circular holes

diminish, 20 additional E33 elements were used to fill each hole cavity so that the models could represent

unperforated solid plates. 

For the square-hole cases, all the models were generated from the solid plate models (basic models;

c/w = 0), which have the highest number of joint locations (JLOCs) and E43 elements. The models with

different hole sizes were then generated from the basic models by removing the proper number of the

square E43 elements around the square-hole boundaries. Also, to increase the model aspect ratios, more

square E43 elements were added to the square models. 

Table 2 lists the sizes of the basic finite-element models from which numerous perturbed models

were generated.

The material properties used in the finite-element analysis are those of monolithic Ti-6Al-4V

titanium alloy (ref. 15), listed in table 3.

The data in table 3 are plotted in figure 9 to show the nonlinearity of the temperature-dependent

material properties.

Table 2. Sizes of the basic finite-element models.

Circular hole (0 < d/w < 0.6) Square hole (c/w = 0)*

l/w JLOC E33 E43 JLOC E43

1.0 951 32 271 441 400

1.5 1061 32 371 651 600

2.0 1061 32 371 861 800

*Number of JLOC and E43 is maximum when c /w = 0. 

Table 3. Material properties of Ti-6Al-4V titanium alloy (ref. 15).

70 °F 200 °F 300 °F 400 °F 500 °F 600 °F 700 °F 800 °F 900 °F 1000 °F

E, lb/in2

× 106 16.0 15.28 14.80 14.40 14.02 13.63 13.15 12.64 11.84 10.56

G, lb/ in2

× 106 6.20 5.83 5.65 5.50 5.37 5.20 5.02 4.82 4.52 4.03

ν 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31

α, in/in-°F × 10–6 4.85 5.00 5.10 5.19 5.27 5.36 5.44 5.52 5.59 5.62
6

MATERIAL PROPERTY ITERATIONS

In the thermal-buckling analysis, calculations of buckling temperatures require material property

iterations. Figure 10 shows a graphical iteration process for finding the buckling temperatures for a

typical case of a square panel (l/w = 1) with circular hole size d/w = 0.7 under a 4S free boundary condition. The buckling temperature is plotted against the assumed material temperature . The 45-deg

line represents the solution line for the buckling temperature . 

For example, if the assumed material temperature agrees with the calculated buckling temperature

 + ( = 70 °F), then the data point of will fall right on the 45-deg solution line. In the first

iteration, the material properties at room temperature, = , were used to calculate the first buckling temperature . The second iteration then uses the material properties at any other temperature,

say = 200 °F, to update the input material properties for the calculations of the second buckling

temperature . In the third iteration, the two buckling data points and were

connected with a straight line to locate the intersection point with the 45-deg solution line. Then, this

intersection-point temperature was used to update the material properties for the calculations of the third

buckling temperature . This iteration process is to continue until the n

th calculated buckling

temperature ( )n data point falls right on the 45-deg solution line. 

From the geometry of figure 10, may be expressed as a function of and as

(1)

For the present material, the data point (less than 200 °F) would fall practically on the

45-deg solution line, giving an acceptable solution for . Namely, the value of calculated

from the third material iteration would practically agree with that obtained from equation (1) because the

material property curves (especially E curve, fig. 9) are almost linear in the range 70 < T < 200 °F. 

RESULTS

The following sections present the results of the finite-element mechanical- and thermal-buckling

analysis of rectangular plates with circular and square holes.

Solution Accuracy

For checking the finite-element solution accuracy, the finite-element buckling solutions for simplysupported solid plates (no holes) of different aspect ratios under uniaxial compression were compared

with the corresponding classical buckling solutions (ref. 13). Table 4 shows the results. The classical

cases actually correspond to the 4S free cases, which are under uniaxial loadings. The 4S fixed cases are

slightly under biaxial loadings and, therefore, were not used for solution-comparison purposes.

∆Tcr

∆Tcr Ta

∆Tcr

Ta

∆Tcr Tr Tr ∆Tcr

Ta ( )1 Tr

∆Tcr ( )1

Ta ( )2

∆Tcr ( )2 ∆Tcr ( )1 ∆Tcr ( )2

∆Tcr ( )3

∆Tcr

∆Tcr ( )3 ∆Tcr ( )1 ∆Tcr ( )2

∆Tcr ( )3

∆Tcr ( )1

1

∆Tcr ( )2 ∆Tcr ( )1 –

Ta ( )2 Ta ( )1 – – --------------------------------------------

= -----------------------------------------------------

∆Tcr ( )3

∆Tcr ∆Tcr ( )3
7

The very close correlation of the finite-element and the classical buckling solutions shown in table 4

indicates the adequacy of the finite-element modeling, and provides great confidence in the accuracy of

the finite-element buckling solutions for the perforated plates presented in this report.

Plates With Circular Holes

For plates with circular holes, three cases are considered: mechanical buckling fixed, mechanical

buckling free, and thermal buckling.

Mechanical Buckling for Fixed-Boundary Cases

Figures 11–16, respectively, show the compressive buckling loads plotted as functions of hole

size d/w (figs. 11, 13, and 15), and the associated buckling mode shapes (figs. 12, 14, and 16) for the

plates with circular holes under 4S fixed and 4C fixed boundary conditions. Only figure 11 shows typical

finite-element buckling solution data points.

For the 4S fixed cases, the antisymmetrical buckling curves always lie considerably above the symmetrical buckling curves for all plate aspect ratios (figs. 11, 13, and 15), and therefore, the buckling mode

shapes are always symmetrical (figs. 12(a), 14(a), and 16(a)). The actual buckling loads (symmetrical) for all plate aspect ratios monotonically decrease slightly from their respective solid plate values as the hole sizes increase (figs. 11, 13, and 15). 

For the 4C fixed cases, the symmetrical and antisymmetrical buckling curves entangle each other (or

mutually intersect), causing the actual buckling curves (for the lowest values) to be composite

curves consisting of symmetrical and antisymmetrical buckling curves segments. Thus, the buckling

mode shapes could be either symmetrical or antisymmetrical (figs. 12(b), 14(b), and 16(b)) depending on

the hole sizes and the plate aspect ratios. For larger holes, the secondary or local buckling modes start to

appear at the hole boundaries (figs. 12(b), 14(b), and 16(b)). 

For a square plate (fig. 11) of the 4C fixed case, the lowest buckling load decreases from that

of the solid plate (d/w = 0) as the hole starts to initiate and grow in size. The value of reaches its

minimum value in the vicinity of d/w = 0.2, and then continues to increase and changing buckling mode

at larger hole sizes. Beyond d/w = 0.4, the buckling load becomes greater than that of the solid

square plates (d/w = 0).

Table 4. Comparison of finite-element and classical buckling solutions for simply

supported rectangular solid plates under uniaxial compression.

, lb/in.

l/w Buckling mode

Circular hole

model (d = 0)

Square hole

model (c = 0)

Timoshenko

(ref. 13)

1.0 Symmetrical 145.5983 145.5858 145.5854

 1.5 Antisymmetrical 157.9806 157.9707 157.9703

 2.0 Antisymmetrical 145.5891 145.5866 145.5854

Ny ( )cr

Ny ( )cr

Ny ( )cr

Ny ( )cr

Ny ( )cr

Ny ( )cr

Ny ( )cr
8

For rectangular plates (figs. 13 and 15) of the 4C fixed cases, the composite buckling curves are wavy

and bent upward as the hole size increases. The lowest buckling loads increase initially as the

holes start to grow in size, then changed mode and reached their respective minimum values near the hole

size d/w = 0.2. Beyond hole size d/w = 0.3, the values exceed their respective values of the solid

rectangular plates (d/w = 0). The degrees of increases in the values for the rectangular plates

(figs. 13 and 15), however, are not as pronounced as the square plate case (fig. 11).

Mechanical Buckling for Free-Boundary Cases 

Figures 17–22, respectively, show the changes of the compressive buckling loads with hole

size d/w (figs. 17, 19, and 21) and the associated buckling mode shapes (figs. 18, 20, and 22) of the plates

with circular holes under 4S free and 4C free boundary conditions. The symmetrical and antisymmetrical

buckling curves intersect at certain hole sizes (figs. 17, 19, and 21). For the 4C free cases, the buckling

mode shapes of the plates with larger holes (figs. 18, 20, and 22) show complex buckling modes consisting of global and pronounced local buckling at the hole boundaries.

For a square plate (fig.17, which represents the most extensively studied geometry), the buckling

loads for both 4S free and 4C free cases decrease monotonically with the hole size and never

increase with the hole size like the 4C fixed case (fig. 11). 

For rectangular plates (figs. 19 and 21), the buckling loads for both the 4S free and 4C free

cases increase at larger hole sizes. This effect is more conspicuous for the 4C free cases (especially for

aspect ratio l/w = 2, shown in fig. 21) than for the 4S free cases. 

Thermal Buckling 

Figures 23–28 show the buckling temperatures , plotted as functions of the hole size d/w

(figs. 23, 25, and 27), and the associated buckling mode shapes (figs. 24, 26, and 28) for the plates with

circular holes. The lowest buckling temperatures decrease slightly, as the hole size grows initially,

and then increase at larger hole sizes. This effect is more pronounced for the 4C cases than for the 4S

cases. The increase of at larger hole sizes is the most pronounced for the square plates under 4C

boundary condition (fig. 23), and the value of at hole size d/w = 0.7 reaches as high as five times

the value of of the unperforated square plate (d/w = 0).

Plates With Square Holes

Analyses of plates with square holes were also made for the three case types: mechanical buckling

fixed, mechanical buckling free, and thermal buckling.

Mechanical Buckling for Fixed-Boundary Cases

Figures 29–34, respectively, show the variations of the compressive buckling loads with the

increase of the hole size c/w (figs. 29, 31, and 33) and the associated buckling mode shapes (figs. 30, 32,

and 34) for the plates with square holes under 4S fixed and 4C fixed boundary conditions. The overall

Ny ( )cr

Ny ( )cr

Ny ( )cr

Ny ( )cr

Ny ( )cr

Ny ( )cr

∆Tcr

∆Tcr

∆Tcr

∆Tcr

∆Tcr

Ny ( )cr
9

buckling behavior of plates with square holes is very similar to that of the plates with circular holes. In

comparison with the circular hole cases, however, the square hole cases have slightly higher buckling

loads at larger hole sizes, especially for the aspect ratio l/w = 1.5 (compare 4C fixed cases in figs. 13 and

31). At large hole size (figs. 30, 32, and 34) for 4C cases, the buckling mode shapes exhibit both global

and local buckling modes.

Mechanical Buckling for Free-Boundary Cases 

Figures 35–40 show the compressive buckling loads plotted against the hole size c/w

(figs. 35, 37, and 39) and the associate buckling mode shapes (figs. 36, 38, and 40), for the plates with

square holes under 4S free and 4C free boundary conditions. Again, the buckling behavior of the square

hole cases is very similar to that of the corresponding circular hole cases. Similar to the fixed cases, the

local buckling modes show up clearly at larger hole sizes under the 4C boundary condition.

Thermal Buckling 

Figures 41–46, respectively, show the buckling temperatures as functions of hole size c/w

(figs. 41, 43, and 45) and the associated thermal-buckling modes (figs. 42, 44, and 46) for the plates with

square holes. Again, the 4C boundary condition causes secondary local buckling modes to show up at

large hole size for the square plate (fig. 42). For the rectangular plates (figs. 44 and 46), the local buckling

modes are almost invisible. 

Comparison of Buckling Strengths

The buckling strengths of the plates with circular holes will be compared with those of the plates with

square holes under the same weight density conditions. Therefore, for a given aspect ratio of the perforated plates (no change in width w), the area of the square hole was set equal to that of the circular hole by

adjusting the side c of the square hole according to the relationship

(2)

Thus, before the comparison of buckling strengths could be made, the abscissa c/w of figures 29, 31,

33, 35, 37, 39, 41, 43, and 45 must first be converted to the equivalent d/w using equation (2). Namely,

the data points for the square hole cases must be shifted slightly toward the right because the equivalent

d/w is slightly greater than c/w in view of equation (2).

Figures 47–49 compare the compressive buckling strengths of the two types of perforated plates

under 4S fixed and 4C fixed conditions. At large hole sizes, the square hole cases exhibit higher buckling

strengths than the respective circular hole case.

For the 4S free and 4C free boundary conditions, the square plates with square holes (fig. 50) have

higher compressive buckling strengths than those with circular holes. For the perforated rectangular

plates (figs. 51 and 52), the square hole cases give slightly higher buckling strengths at moderate hole

sizes than the circular hole cases. The reverse is true, however, for hole sizes greater than d/w = 0.5.

Ny ( )cr

∆Tcr

c

d

-- π

2 = -------
10

Figures 53–55 compare the thermal-buckling strengths of the two types of perforated plates. Again,

the plates with square holes exhibit higher thermal-buckling strengths than those with circular holes for

all the hole sizes and boundary conditions.

DISCUSSION

The buckling behavior of plates with central holes as presented is quite peculiar because, under

certain boundary conditions (especially cases with clamped edges) and plate aspect ratios, the mechanical- and thermal-buckling strengths of the perforated plates, contrary to expectation, increase rather than

decrease as the hole sizes grow larger. The conventional wisdom is that, as the hole sizes increase, the

plates lose more materials and become weaker. Therefore, the buckling strengths were expected to

decrease as the hole sizes increase. This was not the case. Such peculiar buckling phenomenon of the

perforated plates may be explained as follows.

When the hole size becomes considerably large relative to the plate width, most of the compressive

load is carried by the narrow side strips of material along the plate boundaries. As is well known, a

stronger plate boundary condition (e.g., clamped rather than simply-supported boundaries) increases

the buckling strength, while the higher stress concentration decreases the buckling strength. Thus, which

effects become dominant will determine the increase or decrease of the buckling strengths of the

perforated plates. 

For the square-hole cases, the load-carrying narrow side strips along the plate boundaries are

practically under uniform compressive stress fields. For the circular-hole cases, the narrow compressed

side strips are under stress concentration, which reduces the buckling strengths. This fact may explain

why, for most of the cases studied (except figs. 51 and 52), the buckling strengths of the plates with square

holes increase more at larger hole sizes than the plates with circular holes having the same weight density.

The unusual buckling characteristics of the perforated plates offer vital applications in aerospace

structural panel design. Namely, by opening holes of proper sizes in aerostructural panels for weight

saving, their buckling strengths can be boosted simultaneously. Thus, with a single stone, one can shoot

down two birds.

CONCLUDING REMARKS

Finite-element mechanical- and thermal-buckling analyses were performed on plates containing

centrally located circular and square holes. The effects of plate aspect ratio, hole geometry, hole size, and

plate support conditions on the mechanical- and thermal-buckling strengths and buckling mode shapes

were studied in great detail. The key findings of the analysis are as follows:

• The buckling mode shapes of the perforated plates can be symmetrical or antisymmetrical

depending on the hole sizes, plate aspect ratios, and plate boundary conditions.

• Increasing the hole size does not necessarily reduce the mechanical- and thermal-buckling

strengths of the perforated plates. For certain plate aspect ratios and plate support conditions,

mechanical- and thermal-buckling strengths increase with the increasing hole sizes.
11

• For most cases under the same weight density conditions, the mechanical- and thermal-buckling

strengths of the plates with square holes are slightly higher than those of the corresponding plates

with circular holes at increasing hole sizes.

• The clamped boundary conditions more effectively enhance the mechanical- and thermalbuckling strengths of the perforated plates at larger hole sizes than the simply-supported boundary

conditions.

Dryden Flight Research Center

National Aeronautics and Space Administration

Edwards, California, May 27, 1997
12

REFERENCES

1. Levy, Samuel, Ruth M. Woolley, and Wilhelmina D. Kroll, “Instability of Simply Supported Square

Plate With Reinforced Circular Hole in Edge Compression,” Journal of Research, National Bureau

of Standards, vol. 39, research paper no. RP1849, Dec. 1947, pp. 571–577. 

2 Kumai, Toyoji, “Elastic Stability of the Square Plate With a Central Circular Hole Under Edge

Thrust,” Proc. Japan Nat. Cong. Appl. Mech., 1951, pp. 81–88. 

3. Schlack, A. L., Jr., “Elastic Stability of Pierced Square Plates,” Experimental Mechanics, June 1964,

pp. 167–172.

4. Schlack, Alois L., Jr., “Experimental Critical Loads for Perforated Square Plates,” Experimental

Mechanics, Feb. 1968, pp. 69–74.

5. Kawai, T. and H. Ohtsubo, “A Method of Solution for the Complicated Buckling Problems of Elastic

Plates With Combined Use of Rayleigh-Ritz’s Procedure in the Finite Element Method,” AFFDLTR-68-150, 1968.

6. Yu, Wei-Wen and Charles S. Davis, “Cold-Formed Steel Members With Perforated Elements,”

J. Structural Division, ASCE, vol. 99, no. ST10, Oct. 1973, pp. 2061–2077.

7. Ritchie, D. and J. Rhodes, “Buckling and Post-Buckling Behaviour of Plates With Holes,” Aeronautical Quarterly, vol. 26, Nov. 1975, pp. 281–296.

8. Nemeth, Michael Paul, “Buckling Behavior of Orthotropic Composite Plates With Centrally Located

Cutouts,” Ph. D. Dissertation, Virginia Polytechnic Institute and State University, May 1983.

9. Nemeth, Michael P., A Buckling Analysis for Rectangular Orthotropic Plates With Centrally Located Cutouts, NASA TM-86263, Dec. 1984.

10. Nemeth, Michael P., Manuel Stein, and Eric R. Johnson, An Approximate Buckling Analysis for

Rectangular Orthotropic Plates With Centrally Located Cutouts, NASA TP-2528, Feb. 1986.

11. Nemeth, M. P., “Buckling Behavior of Compression-Loaded Symmetrically Laminated Angle-Ply

Plates With Holes,” AIAA Journal, vol. 26, no. 3, Mar. 1988, pp. 330–336.

12. Lee, Y. J., H. J. Lin, and C. C. Lin, “A Study on the Buckling Behavior of an Orthotropic Square

Plate With a Central Circular Hole,” Composite Structures, vol. 13, no. 3, 1989, pp.173–188.

13. Timoshenko, Stephen P. and James M. Gere, Theory of Elastic Stability, 2nd ed., McGraw-Hill Book

Company, New York, 1961.

14. Whetstone, W. D., SPAR Structural Analysis System Reference Manual: System Level 13A, vol. 1,

Program Execution, NASA CR-158970-1, Dec. 1978.

15. U.S. Government, Military Handbook 5B (MIL-HDBK-5B), Aug. 31, 1973.
13

(a) Circular cutout.

(b) Square cutout.

Figure 1. Rectangular plates with central cutouts.

w 970887

t

d

x

y

w 970888

t

c

x

y
14

(a) Fixed case—two vertical edges with no in-plane transverse motions.

(b) Free case—two vertical edges with free in-plane transverse motions.

Figure 2. Fixed and free boundary conditions for two vertical edges; circular cutout case (simply

supported or clamped).

Fixed

Fixed

970889

Ny

Ny

x

y

Free

Free

970890

Ny

Ny

x

y
15

(a) Fixed case—two vertical edges with no in-plane transverse motions.

(b) Free case—two vertical edges with free in-plane transverse motions.

Figure 3. Fixed and free boundary conditions for two vertical edges; square cutout case (simply supported

or clamped).

Fixed

Fixed

970891

x

y

Ny

Ny

Free

Free

970892

x

y

Ny

Ny
16

(a) 4S fixed.

(b) 4S free.

Figure 4. Two types of boundary conditions for simply-supported vertical edges.

(a) 4C fixed.

(b) 4C free.

Figure 5. Two types of boundary conditions for clamped vertical edges.

x

Fixed support

Lubricated

970893

x

Movable support

Lubricated

970894

x

Fixed support

Lubricated

970895

x

Movable support

Lubricated

970896
17

(a) 4S fixed.

(b) 4C fixed.

Figure 6. Two types of boundary conditions used for thermal buckling.

970897

x

y

∆T

970898
18

Figure 7. Quarter-panel, finite-element model for rectangular plate with circular hole; l/w = 1.5, d/w = 0.2.

Figure 8. Quarter-panel, finite-element model for rectangular plate with square hole; l/w = 1.5, c/w = 0.2.

970899

x

y

970900

x

y
19

Figure 9. Temperature-dependent material properties of Ti-6A1-4V titanium alloy.

Figure 10. Graphical representation of buckling temperature iteration process.

17 x 106

16

15

14

13

12

11

10

7 x 10–6

α 6

E

E,

lb/in2

α,

in/in–°F

5

4

3

2

1

0

0 100 200 300 400 500

T, °F

600 700 800 900 1000

960431

First iteration

Second iteration

Third iteration solution

Solution line

970901

100

(∆Tcr)1

(∆Tcr)1

∆Tcr,

°F

(∆Tcr)3 =

1 –

(∆Tcr)3

(∆Tcr)2

(∆Tcr)2 – (∆Tcr)1

(Ta)2 – (Ta)1

(Ta)2

(Ta)1 = Tr

 = 70

Ta, °F

100

200

0 300

1

1

200
20

Figure 11. Compressive buckling loads as functions of hole size; circular holes; fixed edges.

(a) 4S fixed.

(b) 4C fixed.

Figure 12. Buckled shapes of square plates with circular holes under compression; l/w = 1; fixed edges.

Symmetric buckling

Antisymmetric buckling

4C

4C

4S

Ny

Ny

/w = 1.0

d

w

4S

0 .5 .6 .7 .8 .1 .2 .3

970902

.4

d/w

50

100

150

200

300

350

450

400

250 (Ny)cr,

lb/in

Ny

Ny

d/w = 0.2 970903a

Ny

Ny

d/w = 0.6 970903b

d/w = 0.2 970904a

Ny

Ny

d/w = 0.6 970904b

Ny

Ny
21

Figure 13. Compressive buckling loads as functions of hole size; circular holes; fixed edges.

(a) 4S fixed.

(b) 4C fixed.

Figure 14. Buckled shapes of rectangular plates with circular holes under compression; l/w = 1.5; fixed

edges.

Symmetric buckling

Antisymmetric buckling

4C

4C

4S

Ny

Ny

/w = 1.5

d

w

4S

0 .5 .6 .7 .8 .1 .2 .3

970905

.4

d/w

50

100

150

200

300

350

400

250

(Ny)cr,

lb/in

d/w = 0.2

970906a

Ny

Ny

d/w = 0.6 970906b

Ny

Ny

d/w = 0.2 970907a

Ny

Ny

d/w = 0.6 970907b

Ny

Ny
22

Figure 15. Compressive buckling loads as functions of hole size; circular holes; fixed edges.

(a) 4S fixed.

(b) 4C fixed.

Figure 16. Buckled shapes of rectangular plates with circular holes under compression; l/w = 2; fixed

edges.

Symmetric buckling

Antisymmetric buckling

4C

4C

4S

Ny

Ny

/w = 2.0

d

w

4S

0 .5 .6 .7 .8 .1 .2 .3

970908

.4

d/w

50

100

150

200

300

350

400

250

(Ny)cr,

lb/in

d/w = 0.2 970909a

Ny

Ny

d/w = 0.6 970909b

Ny

Ny

d/w = 0.2 970910a

Ny

Ny

d/w = 0.6 970910b

Ny

Ny
23

Figure 17. Compressive buckling loads as functions of hole size; circular holes; free edges.

(a) 4S free.

(b) 4C free.

Figure 18. Buckled shapes of square plates with circular holes under compression; l/w = 1; free edges.

4C

4C

4S

Ny

Ny

/w = 1.0

4S

0 .5 .6 .7 .8 .1 .2 .3

970911

.4

d/w

100

150

200

300

350

450

400

250

(Ny)cr,

lb/in

d

w

Symmetric buckling

Antisymmetric buckling

d/w = 0.2 970912a

Ny

Ny

d/w = 0.6 970912b

Ny

Ny

d/w = 0.2 970913a

Ny

Ny

d/w = 0.6 970913b

Ny

Ny
24

Figure 19. Compressive buckling loads as functions of hole size; circular holes; free edges.

(a) 4S free.

(b) 4C free.

Figure 20. Buckled shapes of rectangular plates with circular holes under compression; l/w = 1.5; free

edges.

Symmetric buckling

Antisymmetric buckling

4C

4C

4S

Ny

Ny

4S

0 .5 .6 .7 .8 .1 .2 .3

970914

.4

d/w

100

150

200

300

350

400

450

500

250

(Ny)cr,

lb/in /w = 1.5

d

w

d/w = 0.2 970915a

Ny

Ny

d/w = 0.6 970915b

Ny

Ny

d/w = 0.2 970916a

Ny

Ny

d/w = 0.6 970916b

Ny

Ny
25

Figure 21. Compressive buckling loads as functions of hole size; circular holes; free edges.

(a) 4S free.

(b) 4C free.

Figure 22. Buckled shapes of rectangular plates with circular holes under compression; l/w = 2; free edges.

Symmetric buckling

Antisymmetric buckling

4C

4C

4S

Ny

Ny

/w = 2.0

4S

0 .5 .6 .7 .8 .1 .2 .3

970917

.4

d/w

100

150

200

300

350

400

450

500

550

250

(Ny)cr,

lb/in

d

w

d/w = 0.2 970918a

Ny

Ny

d/w = 0.6 970918b

Ny

Ny

d/w = 0.2 970919a

Ny

Ny

d/w = 0.6 970919b

Ny

Ny
26

Figure 23. Buckling temperatures as functions of hole size; circular holes.

(a) 4S fixed.

(b) 4C fixed.

Figure 24. Buckled shapes of uniformly heated square plates with circular holes; l/w = 1.

Symmetric buckling

Antisymmetric buckling

4C

4C

4S

/w = 1.0

d

∆T

w

4S

0 .5 .6 .7 .8 .1 .2 .3

970920

.4

d/w

25

50

100

125

175

150

75

∆Tcr,

°F

∆T

d/w = 0.2 970921a

∆T

d/w = 0.6 970921b

d/w = 0.2 970922a

∆T

∆T

d/w = 0.6 970922b
27

Figure 25. Buckling temperatures as functions of hole size; circular holes.

(a) 4S fixed.

(b) 4C fixed.

Figure 26. Buckled shapes of uniformly heated rectangular plates with circular holes; l/w = 1.5.

Symmetric buckling

Antisymmetric buckling

4C

4C

4S

/w = 1.5

d

w

4S

0 .5 .6 .7 .8 .1 .2 .3

970923

.4

d/w

15

20

25

30

35

10

5

∆Tcr,

°F

∆T

∆T

d/w = 0.2 970924a

∆T

d/w = 0.6 970924b

∆T

d/w = 0.2 970925a

∆T

d/w = 0.6 970925b
28

Figure 27. Buckling temperatures as functions of hole size; circular holes.

(a) 4S fixed.

(b) 4C fixed.

Figure 28. Buckled shapes of uniformly heated rectangular plates with circular holes; l/w = 2.

Symmetric buckling

Antisymmetric buckling

4C

4C

4S

/w = 2.0

d

w

4S

0 .1 .2 .3 .5 .6 .7 .8

970926

.4

d/w

10

15

20

25

30

35

5

∆Tcr,

°F

∆T

∆T

d/w = 0.2 970927a

∆T

d/w = 0.6 970927b

∆T

d/w = 0.2 970928a

∆T

d/w = 0.6 970928b
29

Figure 29. Compressive buckling loads as functions of hole size; square holes; fixed edges.

(a) 4S fixed.

(b) 4C fixed.

Figure 30. Buckled shapes of square plates with square holes under compression; l/w = 1; fixed edges.

4C

4C

4S

/w = 1.0

c

w

4S

0 .5 .6 .7 .8 .1 .2 .3

970929

.4

c/w

50

100

150

200

300

350

450

400

250 (Ny)cr,

lb/in

Ny

Ny

Symmetric buckling

Antisymmetric buckling

Ny

Ny

c/w = 0.2 970930a

Ny

Ny

c/w = 0.6 970930b

Ny

Ny

c/w = 0.2 970931a

Ny

Ny

c/w = 0.6 970931b
30

Figure 31. Compressive buckling loads as functions of hole size; square holes; fixed edges.

(a) 4S fixed.

(b) 4C fixed.

Figure 32. Buckled shapes of rectangular plates with square holes under compression; l/w = 1.5; fixed

edges.

Symmetric buckling

Antisymmetric buckling

4C

4C

4S

/w = 1.5

c

w

4S

0 .5 .6 .1 .2 .3 .7 .8

970932

.4

c/w

50

100

150

200

300

350

450

500

550

400

250

(Ny)cr,

lb/in

Ny

Ny

Ny

Ny

c/w = 0.2 970933a

Ny

Ny

c/w = 0.6 970933b

Ny

Ny

c/w = 0.2 970934a

Ny

Ny

c/w = 0.6 970934b
31

Figure 33. Compressive buckling loads as functions of hole size; square holes; fixed edges.

(a) 4S fixed.

(b) 4C fixed.

Figure 34. Buckled shapes of rectangular plates with square holes under compression; l/w = 2; fixed edges.

Symmetric buckling

Antisymmetric buckling

4C

4C

4S

4S

0 .5 .6 .7 .8 .1 .2 .3

970935

.4

c/w

50

100

150

200

300

350

450

400

250 (Ny)cr,

lb/in

c

w

/w = 2.0

Ny

Ny

Ny

Ny

c/w = 0.2 970936a

Ny

Ny

c/w = 0.6 970936b

Ny

Ny

c/w = 0.2 970937a

Ny

Ny

c/w = 0.6 970937b
32

Figure 35. Compressive buckling loads as functions of hole size; square holes; free edges.

(a) 4S free.

(b) 4C free.

Figure 36. Buckled shapes of square plates with square holes under compression; l/w = 1; free edges.

Symmetric buckling 4C Antisymmetric buckling

4C

4S

/w = 1.0

c

w

4S

0 .5 .6 .7 .8 .1 .2 .3

970938

.4

c/w

50

100

150

200

300

350

450

400

250 (Ny)cr,

lb/in

Ny

Ny

Ny

Ny

c/w = 0.2 970939a

Ny

Ny

c/w = 0.6 970939b

Ny

Ny

c/w = 0.2 970940a

Ny

Ny

c/w = 0.6 970940b
33

Figure 37. Compressive buckling loads as functions of hole size; square holes; free edges.

(a) 4S free.

(b) 4C free.

Figure 38. Buckled shapes of rectangular plates with square holes under compression; l/w = 1.5; free

edges.

4C

4C

4S

c

w

4S

0 .5 .6 .7 .8 .1 .2 .3

970941

.4

c/w

100

150

200

300

350

450

500

550

400

250

(Ny)cr,

lb/in

/w = 1.5

Ny

Ny

Symmetric buckling

Antisymmetric buckling

Ny

Ny

c/w = 0.2 970942a

Ny

Ny

c/w = 0.6 970942b

Ny

Ny

c/w = 0.2 970943a

Ny

Ny

c/w = 0.6 970943b
34

Figure 39. Compressive buckling loads as functions of hole size; square holes; free edges.

(a) 4S free.

(b) 4C free.

Figure 40. Buckled shapes of rectangular plates with square holes under compression; l/w = 2; free edges.

4C

4C

4S

/w = 2.0

c

w

4S

0 .5 .6 .7 .8 .1 .2 .3

970944

.4

c/w

100

150

200

300

350

450

500

550

600

400

250

(Ny)cr,

lb/in

Ny

Ny

Symmetric buckling

Antisymmetric buckling

Ny

Ny

c/w = 0.2 970945a

Ny

Ny

c/w = 0.6 970945b

Ny

Ny

c/w = 0.2 970946a

Ny

Ny

c/w = 0.6 970946b
35

Figure 41. Buckling temperatures as functions of hole size; square holes.

(a) 4S fixed.

(b) 4C fixed.

Figure 42. Buckled shapes of uniformly heated square plates with square holes; l/w = 1.

4C

4C

4S

/w = 1.0

c

w

4S

0 .5 .6 .7 .8 .1 .2 .3

970947

.4

c/w

50

100

200

150

∆T

∆Tcr,

°F

Symmetric buckling

Antisymmetric buckling

c/w = 0.2

∆T

970948a c/w = 0.6 970948b

∆T

∆T

c/w = 0.2 970949a

∆T

970949b c/w = 0.6
36

Figure 43. Buckling temperatures as functions of hole size; square holes.

(a) 4S fixed.

(b) 4C fixed.

Figure 44. Buckled shapes of uniformly heated rectangular plates with square holes; l/w = 1.5.

4C

∆T

4C

4S

/w = 1.5

c

w

4S

0 .5 .6 .7 .8 .1 .2 .3

970950

.4

c/w

5

10

20

25

35

40

45

50

30

15

Symmetric buckling

Antisymmetric buckling

∆Tcr,

lb/in

∆T

c/w = 0.2 970951a

∆T

970951b c/w = 0.6

∆T

c/w = 0.2 970952a

∆T

970952b c/w = 0.6
37

Figure 45. Buckling temperatures as functions of hole size; square holes.

(a) 4S fixed.

(b) 4C fixed.

Figure 46. Buckled shapes of uniformly heated rectangular plates with square holes; l/w = 2.

4C

∆T

4C

4S

c

w

4S

0 .5 .6 .7 .8 .1 .2 .3

970953

.4

c/w

5

10

20

25

35

30

15

∆Tcr,

°F

/w = 2.0

Symmetric buckling

Antisymmetric buckling

∆T

c/w = 0.2 970954a

∆T

970954b c/w = 0.6

∆T

c/w = 0.2 970955a

∆T

970955b c/w = 0.6
38

Figure 47. Comparison of compressive buckling strengths of square plates with different geometrical

cutouts; fixed edges.

Figure 48. Comparison of compressive buckling strengths of rectangular plates with different geometrical

cutouts; fixed edges.

S

S

S

S

S

4S

4C

c

w

A

A

0 .5 .6 .7 .8 .1 .2 .3

970956

.4

d/w

100

150

50

200

300

350

450

400

250 (Ny)cr,

lb/in

/w = 1.0

Ny

Ny

d

w

Ny

Ny

A - Antisymmetric buckling

S - Symmetric buckling

d 2

c = π

S

S

S

S

S

S

4S

A 4C A

A

A

0 .5 .6 .7 .8 .1 .2 .3

970957

.4

d/w

100

150

50

200

300

350

450

500

550

400

250

A - Antisymmetric buckling

S - Symmetric buckling

c

w

(Ny)cr,

lb/in

/w = 1.5

Ny

Ny

d

w

Ny

Ny

d 2

c = π
39

Figure 49. Comparison of compressive buckling strengths of rectangular plates with different geometrical

cutouts; fixed edges.

Figure 50. Comparison of compressive buckling strengths of square plates with different geometrical

cutouts; free edges.

S

S

S

S

S

S

4S

4C

A

A

A

A

0 .5 .6 .7 .8 .1 .2 .3

970958

.4

d/w

100

150

50

200

300

350

400

250

(Ny)cr,

lb/in

A - Antisymmetric buckling

S - Symmetric buckling

c

w

/w = 2.0

Ny

Ny

d

w

Ny

Ny

d 2

c = π

S

S

S

S

A

A

A

0 .1 .2 .3 .5 .6 .7 .8

970959

.4

d/w

100

150

200

300

350

450

400

250

(Ny)cr,

lb/in

A - Antisymmetric buckling

S - Symmetric buckling

4S

4C

c

w

/w = 1.0

Ny

Ny

d

w

Ny

Ny

d 2

c = π
40

Figure 51. Comparison of compressive buckling strengths of rectangular plates with different geometrical

cutouts; free edges.

Figure 52. Comparison of compressive buckling strengths of rectangular plates with different geometrical

cutouts; free edges.

S

S

S

S

S

S

S

4C

A

A

A

A

A

A A

0 .5 .6 .7 .8 .1 .2 .3

970960

.4

d/w

100

150

200

300

350

450

500

550

400

250

(Ny)cr,

lb/in

A - Antisymmetric buckling

S - Symmetric buckling

4S

c

w

/w = 1.5

Ny

Ny

d

w

Ny

Ny

d 2

c = π

S

S

S

S

S

4C

A

A

A

A

A

0 .5 .6 .7 .8 .1 .2 .3

970961

.4

d/w

100

150

200

300

350

450

500

550

600

400

250

(Ny)cr,

lb/in

A - Antisymmetric buckling

S - Symmetric buckling

4S

c

w

/w = 2.0

Ny

Ny

d

w

Ny

Ny

d 2

c = π
41

Figure 53. Comparison of thermal buckling strengths of square panels with different geometrical cutouts.

Figure 54. Comparison of thermal buckling strengths of rectangular panels with different geometrical

cutouts.

∆Tcr,

°F

S

S

S

S

4C

A

A

S

.1 .5 .6 .7 .8 .2 .3

970962

.4

d/w

0

25

50

100

125

175

200

150

75

A - Antisymmetric buckling

S - Symmetric buckling

4S

c

w

/w = 1.0

d

w

d 2

c = π

∆T

∆T

∆Tcr,

°F

S

S

S

S

4C

A

A

.1 .5 .6 .7 .8 .2 .3

970963

.4

d/w

0

5

10

20

25

35

40

45

50

30

15

A - Antisymmetric buckling

S - Symmetric buckling

4S

∆T

∆T

c

w

/w = 1.5

d

w

d 2

c = π
42

Figure 55. Comparison of thermal buckling strengths of rectangular panels with different geometrical

cutouts.

∆Tcr,

°F

S

S

4C

A

S A

S

.1 .5 .6 .7 .8 .2 .3

970964

.4

d/w

0

5

10

20

25

35

30

15

A - Antisymmetric buckling

S - Symmetric buckling

4S

c

w

/w = 2.0

d

w

d 2

c = π

∆T

∆T
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NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102

Mechanical- and Thermal-Buckling Behavior of Rectangular Plates With

Different Central Cutouts

WU 522 32 34 

William L. Ko

NASA Dryden Flight Research Center

P.O. Box 273

Edwards, California 93523-0273

H-2206

National Aeronautics and Space Administration

Washington, DC 20546-0001 NASA/TM-1998-206542

Mechanical- and thermal-buckling analyses were performed on rectangular plates with central cutouts. The

cutouts were either circular holes or square holes. The finite-element structural analysis method was used to

study the effects of plate-support conditions, plate aspect ratio, hole geometry, and hole size on the mechanicaland thermal-buckling strengths of the perforated plates. By increasing the hole size, thermal-buckling strengths

of the plates could be enhanced. The compressive-buckling strengths of the plates could also be increased

considerably only under certain boundary conditions and aspect ratios. The plate-buckling mode can be

symmetrical or antisymmetrical, depending on the plate boundary conditions, aspect ratio, and the hole size. For

the same cutout areas (i.e., same plate weight density), the buckling strengths of the same-sized plates with

square holes generally surpass those of the plates with circular holes over the range of hole sizes. The results

and illustrations provide vital information for the efficient design of aerospace structural panels.

Buckling mode shapes, Mechanical buckling, Plates with circular holes, Plates

with square holes, Thermal buckling A03

48

Unclassified Unclassified Unclassified Unlimited

March 1998 Technical Memorandum

Available from the NASA Center for AeroSpace Information, 800 Elkridge Landing Road, 

Linthicum Heights, MD 21090; (301)621-0390

Unclassified—Unlimited

Subject Category 39