ON 'E'HE BUCKLENG OF RECTANGULAR
PLATES WSTH iNTERNAL PQENT SUPFQR?

Thesis for H1. Dawn 01" M. S.
MECHtGAN STA'E‘E COLLEGE
James Andrew Gusack

T954

THESIS

This is to certify that the

thesis entitled

ON THE BUCKLING OF RECTANGULAR PLATES

WITH INTERNAL POINT SUPPORT

presented by

JAMES ANDREW GUSACK

has been accepted towards fulfillment
of the requirements for

MASTER OFLSQIENQE. degree in MECHANICS

flame

 

ajor professor

Date mm};—

ON THE BUCKLING OF RECTANGULAR PLATES
WITH INTERNAL POINT SUPPORT

BY

James Andrew ggsack \

A THESIS

Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements

for the degree of

MASTER OF SCIENCE

Department of Applied Mechanics

1954

THESR

 

in

 

w

*‘J

(‘9

l
\N to

V‘

ACKNONLEDGJENTS

The author wishes to express his sincere appreciation
to Dr. Lawrence E. Malvern for the suggestion of this
problem and for his patient guidance and assistance over
the past year.

He also wishes to thank the many members of the
faculty of Michigan State College who have contributed to
the academic background necessary for the preparation of
this paper.

Grateful acknowledgment is due to Dr. and Mrs. Leland
M. McKinley, whose inepiration and friendship have given
the writer confidence to continue his education.

The author is finally indebted to his parents, Mr. and
Mrs. Andrew Gusack, who have been, through personal sacri-

fice, reSponsible for this as well as all his achievements.

338670

CHAPTER

I.

II.

III.

IV.

V.

TABLE OF CONTENTS

INTRODUCTION . . . . . . . . . . .
Statement of the Problem . . .
Critical Buckling Load . . .
Method of Solution of the Problem
Assumptions . . . . . . . . . .
Applications of the Theory . . .

GENERAL THEORY . . . . . . . . . .
Stationary Potential Energy . .
The Extended Ritz Method . . . .

SIMPLY SUPPORTED PLATE . . . . . .

Application of the General Theory

Graphical Solution of the Characteristic

Equation 0 O O O o O O O O O O 0
Example of the Numerical Solution
CLAMPED PLATE . . . . . . . . . .

Application of the General Theory

Solution of the Characteristic Equation

SUMMARY AND CONCLUSIONS . . . . .

BIBLJOGRAPHY . . . . . . . . . . . . . .

PAGE

01' N N‘ N P P

()3

Cf:

LIST OF TABLES

TABLE PAGE
1. Critical values oflfit for the simply
supported plate . . . . . . . . . . . . . . . 17
11. Boots of the transcendental equation . . . . 30
III. Evaluated constants of the characteristic
equation . . . . . . . . . . . . . . . . . . 30
IV. Critical values of/fig,for the clamped plate . 31

LIST OF FIGURES

FIGURE
1. Schematic view of the problem . . . . . . . . 7
2. Graph of yifi)VB/g . . . . . . . . . . . . l4
3. Doubly infinite array . . . . . . . . . . . . lb

4. Schematic view of the problem . . . ... . . . 18

1. INTRODUCTION
Statement of the Problem

The problem considered in this paper is to determine
the effect of an internal point support on the critical
elastic buckling load of a rectangular thin plate under

various edge support conditions.
Critical Buckling Lead

A plate subjected to an edge load in its plans is
said to be on the verge of buckling when the plate is in a
condition of neutral equilibrium. In this condition the
edge load may produce either strain in the initial flat
equilibrium configuration or, more important, a laterally
bent equilibrium configuration. This second configuration
is called a buckling mode. The edge load necessary to
produce a condition of neutral equilibrium in a plate is
called the critical buckling load. A buckling mode cor-
reSponding to a higher critical buckling load is possible,
but in practice the plate will buckle in the first mode
corresponding to the lowest critical buckling load unless

constrained.

Method of Solution of the Problem

The extended Ritz method was chosen for the solution
because an explicit solution of the buckling differential
equation of the plate was not available with the additional
restraint of the point support. A discussion of this

method is given in the General Theory,Chapter 2.
Assumptions

The usual classical theory assumptions are made:

a. The material is homogeneous, isotropic, and
follows hooke's Law.

b. Normals to the undeformed middle plane of the
plate remain straight and normal to the deformed
middle surface.

c. The cross section thickness is constant and small
compared to the length and width of the plate.

d. The plate is loaded in plane stress before

buckling.
Applications of the Theory

Two applications of the General Theory of Chapter 2 are
made in this paper. The rectangular plate with simply
supported edges and an arbitrary point support is discussed
in Chapter 3. The rectangular plate with clamped edges and

an arbitrary point support is discussed in Chapter 4.

0]

2. GENERAL THEORY
Stationary Potential Energy

The total potential energy of a mechanical system is
said to be stationary for a given equilibrium configuration
of the system if the first order change in the total
potential energy is zero for any arbitrary small diaplace-
ments from the given configuration. The Theorem of
Stationary Potential Energylstates that at an equilibrium
configuration of a system, the total potential energy is
stationary.

Let U = V + UW (1)
where \l is the strain energy of bending of the plate in
the buckled configuration and the symbol UW is the change
of potential energy of the external loads when the plate
buckles into the buckled configuration. The Theorem of
Stationary Potential Energy requires that U be stationary

for any buckled equilibrium configuration of the plate.

The Extended Ritz Method

2
The extended Ritz method is used to solve the plate

buckling problem. In this method the lateral deflection

 

l. Friedrich Bleich, Buckling Strength gf_Metal
Structures, McGraw-Hill, New York: 1952. pp. 70,71.
2. Ibid. pp. 77-81.

of the plate is expressed as a sum of suitably chosen

coordinate functions.

For the rectangular plate

osxso. , 0S5$b,

we choose 0°, .9
w(x’sg)=~‘;‘ nzst AM“ ¢m(”)°'e'“(g) (2)

where the functions ¢M(XI) are the complete set of
eigenfunctions of‘a beam with no internal support subject
to and conditions at 19-0 , 1,: Q which are the same as
the and conditions of the rectangular plate under consider-
ation. The functions «9“(3) are the complete set of
eigenfunctions of a beam with no internal support subject
to and conditions at S30 a 3“ b which are the same as
the and conditions of the rectangular plate under consider-
ation. It is known that any arbitrary deflection configur—
ation of a rectangular plate can be represented by an
infinite double series of eigenfunction products of the
form chosen.3 The coefficients Amn are to be determined so
that the constraint condition of the internal point support
is satisfied, and the total potential energy of the system

is stationary. The constraint condition for an internal

point support at an arbitrary point with coordinates (£31.)

is _
w(€,l)=0 (is)

— 3. R. Courant and D. Hilbert, Methoden Der Mathematischen
Physik, Vol. 1, Berlin: Springer, 1931. p. 47.

 

The problem of making the expression for the total
potential energy of the system stationary and simultaneously
satisfying the subsidiary conditions of constraint can be

4
solved by the Lagrange multiplier method. In this method,

the expression

U=V+Uw-Aw(§,’l) m
is introduced. The parameters AInn and the values )L that

make U stationary also make U [Eq. (1» stationary and

satisfy the subsidiary constraint condition. The necessary

- 5
conditions for lJ to be stationary are
- au - o
E’IKInvt '

For the applications to be made, V and Uw [in FRI-(43
are given by certain double integrals over the plate. When
the series for W EEq. (2313 substituted in these integrals,
and the integrals are evaluated, V and on are obtained
as quadratic expressions in the coefficients Amn'

In practice, if N coefficients are to be determined,
the N Equations (6) together with Equation (3) form a system
of Nrtl linear homogeneous algebraic equations for the n
coefficients and the multiplier 1L.. The solution of this
system is obtained only up to an undetermined constant
multiplier. Hence the shape of the deflected equilibrium

configuration is determined, but not its amplitude.

 

-' 4. F. Bleich, Op. Cit. pp. 77-81.
5. I. S. Sokolnikoff and E. S. Sokolnikoff, Higher
Mathematics for Engineers and Physicists, McGraw-nill, New
York: 1941. pp. 165-167.

In the following Chapter the buckling problem of a
simply supported rectangular plate with lateral point support

is solved using the method discussed in this Chapter.

\‘1

3. SIMPLY SUPPORTED PLATE

Application of the General Theory

A rectangular plate with simply supported edges on

four sides and an arbitrary point support with coordinates

(§,'L) is considered.

-111 .

 

a—s—A

 

f
”L
.L ‘—

3fll_ ,

F

T

 

 

tHF’

3 Figurel

In Figure l

a:

b a

length of the plate
length of the plate
uniform compressive
acting in the plane

- uniform compressive

acting in the plane

in the x-direction,

in the y-direction5

force per unit length

of the plate on the edges

force per unit length

of the plate on the edges

y = 0, y a b‘where “is a dimensionless constant.

 

The series expression EJq.(ZD for the deflection

of the plate is then

wtx, 9): Z Z: A..." sin 11: nwa (6)

mg. ”8| b

 

where the functions sm I'm are the complete set of eigen-
¢t ' 6

functions of a simply supported beam of length'a, and the

functions sin 31"! are the complete set of eigenfunctions

b
of a simply supported beam of length b. The coefficients

Amn are the set of parameters to be determined.
7

The strain energy of bending of a rectangular plate is
\,,.L:D‘j!‘£u(:(:i ‘MI +1ylVV)z'
' 2 0 7‘2... ‘3‘-

}‘u Y’s)
-zu-WE 3-2.. 5—3 -(3:——;3)‘J} 43:43 (7)

 

where

.D = E i? <8)
ETC-:79")
is the flexural rigidity of the plate. In Equation (8)
E : modulus of elasticity,
h.= thickness of the plate, and
‘D = Poisson's ratio.
Ne assume that the limited bending that occurs when the plate

enters the buckled equilibrium configuration takes place

 

6. F. B. Hildebrand,Advanced Calculus for Engineers,
Prentice-Hall, New York: 1949. p. 215.

7. S. Timoshenko, Theory g£_Elastic Stability,
McGraw-Hill, New York: 1936. pp. 305-307.

with negligible stretch or compression of the middle plane
of the plate.8 If we then take the datum configuration of

zero potential energy to be the flat configuration of the

plate just before buckling occurs, the quantity \I is the

total strain energy in the buckled configuration.

The appropriate derivatives of W @qdéfl are taken and

..- M... ~—

substituted in V EEqJ'in. If we observe the orthogonality
9

of the eigenfunctions, namely

 

n.
Sein' m‘wx.'.sin “MT" = I
0 " " 0 Cer- m #7".
a. SL‘QOW’ nn|=='11,
‘SCHMI UB:!:2?,*¢01.1~\ffl’1E., Z. ’
d. g -

\l is found to be
2.b g, 21
‘1? t
V=—- * 4: Z A"m [$421) In] (9)
msl na\ b ‘
We also assume that the edge loads do not change during
the buckling of the plate. Then the work done by the external

10
compressive forces Nx and 0(prduring buckling is

c: b
2 l.
-'- 1" (1”)3
LNxSSEz$)+.‘%3 AfiAS.
c: o
The change in the potential energy of the external loads

during buckling is then

‘0 = —- N, S S [(2.41) ix‘%‘§)‘] “43310)

 

8. S. Timoshenko, Op. Cit. ,p. 325.
9. Friedrich Bleich, Op. Cit.,pp. 65-69.
10. S.Timoshenko, Op.Cit.,pp.308-314.

 

10

The appropriate derivatives of WEEq.(6) are taken and

substituted in UwfiSq.(lOB. The expression 1‘0er when inte-

grated is

Uwz- 8“- -,‘N Z Z Amn [mt-t- K(%)‘h‘].(ll)

M:\ “H

 

The extended hitz method as discussed in Chapter 2 is

now applied. The expressions round for V and Uware sub-

stituted in DE Lq. (43. Then

0 'Ii'b'No Z 5:. Am" .[m‘+(%)‘v\j‘

NV! ha!

-‘{-"-Nx 5: '2': A.“ E‘Hefl

"A Z Z A...“ s.~m“§s;~2_-1n')
“3 I

mat

(12)

' 11".:
where N. = F‘ load per unit width ), the Euler critical

load for a column with flexurlall rigidity equal to'D .

The necessary conditions for U to be stationary are

‘20
)Amn n (g ‘1) e

The derivatives of U @quzfl with respect to the parameters

Amn produce the equations

- 2
2.9.. : .N A [Mt ‘9’ — ‘hJ
1A.“ 4 a ° h" (b)

_ ‘b I 1.1. 2'
14:1.NgAmn [In +a(($b) In]

-J\.smm1r!.s'm51rn Q.
a. 5
11. See Lagrange's multiplier method in Chapter 2.

(13)

 

ll

In these equations the multiplier A may be zero or differ-
ent from zero. If 5K.is zero the Equations (13) require that
for each possible combination of the two indices m,n, either

Amn is zero or

[ne‘+($)‘ at...) No —z"[m +o<(—b1n)n]= Q (14)

If the load ratio d and the side ratio 9b have been chosen
in advance, the critical value ”I is then determined by
Equation (14) for one choice of m,n. The choice made is that
which gives the lowest critical value of ”x and still set-
isfies the constraint condition w(§,‘1)=o. All of the
other coefficients Amn except the one corresponding to this
choice of m,n must be zero in order that Equations (13) can
all be satisfied. Thus only one term of the series $or vv
Eq. (6a survives when A is zero. Physically this means
that the internal point support is located on a nodal point
of one of the buckling modes of a simply supported plate.
In fact an expression similar to Equation (14) is given by
S. Timoshenkolzto find the critical buckling load for a
simply supported plate.

When Jl is not zero, Equations (13) require that for

each choice of the indices m,n,

4-1
Am=W1”—:.- [Sin‘b W\__Tg $310111]

E“ *c t)‘ "‘J “/4- Em” * ($76

12. S. Timoshenko, 0p.Cit., p. 518.

(15)

 

12

Na:

where/q, = —— , a dimensionless load parameter. The other
1V0

necessary condition is that w(§,"‘[)=o. Substitution
of AmnEQ- (157] in the expression for NEEq. (63 at the
point(€,fl)results in the characteristic equation,

4 A a. °° ’° 3 . g,

g” °E° Z Z Sin in"; . 5'” 'ELR
1" . mu as " b - C(16)
“fl“fi)‘ at] ‘7‘ [’3‘ § 40-3333

Since the characteristic equation was obtained for the con-

 

dition that 1 is not zero, and the coefficients

1|" ”a b
then

V94) = if s‘~‘m:—‘- “‘13—"
an “al E,‘+(£é)‘n‘]‘-js["‘* 4(37-3

must equal zero for each critical buckling value of/;; . If

(1'7)

the load ratio aflthe point support location, and the side
ratio i have been chosen in advance, the values of/Q that
satisfyyy):o are the critical buckling dimensionless load

parameters of that simply supported rectangular plate with

a point support at (£31).

13

Graphical Solution

of the Characteristic Equation

A graphical method can be used to solve V94): 0
After the load ratio ci , the side ratio fig , and the co-
ordinates of the point support(§’11)are selected, a graph
is constructed with values ofl/fip on the abscissa axis and
the values of the sum}@§g)on the ordinate axis ( See Fig.2).
This graph has a vertical asymptote at each value of;/4;
corresponding to a buckling mode of a simply supported
plate with no point support. Since there is no root of the
Equation ”AL-.0 before the first asymptote ( The terms
are all positive.), the lowest critical value of/‘b for the
point supported plate where V94): 0 , will lie between the

first two asymptotes. The accuracy of the determination cf

the root ( where W} crosses the/e - axis ) can be im-

proved by numerical interpolation, using a suitable number
of terms of the sumW).

Example of the Numerical Solution

Let us consider as an example the problem of a simply
supported square plate with equal loads in the two directions
(e(’| and $2| ) and a point support at ($8 % ,1: i).

The terms of We) can be represented in the doubly infinite

array ( See Fig. 3 ))

l4

 

 

9’99) |
+3 GRAPH a
of I
WW VS /~ I |
+1 , for the case ‘ .
.. - . ed / l
a- . 2.
+.l 5 3 3’ ‘L 2. I
I / |
. l . I
u /r I
_J I l
-1 //: / //|
/lst asymptote | ~4d aSymptotel
A = 2.0 I ~ A, 5.0 I
-.3
I / I
| I
I Z .3 4- 5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m/m/m....m-
367%.
/

I. Figure 3 ..

 

wherey‘mn is the term corresponding to the summation
indices m,n in the sumEqu'ID. In practice a finite number
of terms of the infinite array are summed to evaluate W) ,
starting at the upper left hand corner and moving diagonally
through the array as shown by the arrows.

V For the particular square plate of this example the

first six non-vanishing terms of the sum are

.15 .75 .15 .
-r -————-' -P’ -——-—- 4P
4-Z_A 25‘971. zoo-1°,"
.15 , .15 .15
—— + -——-— * ————-—' O
H.9- 03/; ass-rye (.76- 25,4.
The first asymptotes of the graph’Y/y0are found by equating

 

the denominators of the doubly infinite array ( Fig. 5 )

to zero. The first two asymptotes are/(a 2 and/(1:5(See
Fig. 2 ). The term'V'll and the term that will make the

largest contribution to the sum W) for a value of/Q.

between the asymptotes are now summed and equated to zero.

16

 

.5 o
w r. ,::/, ,;_,/,.-. .

This equation is now solved for the first approximate value

01%., , which 18/4,: 4.14. The first approximate value of
‘/Al is now inserted in the first six contributory terms

9‘) and the sum taken. The sum of m 4.14 ) is
+.015. Since this sum is positive, a value of/l‘ less than
A: 4.14 is chosen. We choose/4: 3.85 arbitrarily and
sum the first six terms again with this value. The sum of
m 3.85 ) 13-.049. Since this sum is negative, the crit-
ical value car/4 lies within the interval 3.85. (,4. (4.14.
Further numerical interpolation will give the value of/Q
with greater accuracy. The dritical value of/AQ for this
example is approximately/1;: 4.05 ( See Fig. 2 ).

Critical values of/lv. for other values of load ratios
4, side ratios #2 , and point support locations (€31)

for the simply supported plate are tabulated in Table I.

'The lowest critical value of/a, occurs for A not zero
except for the case that the point support is at the middle
of the plate (5: i ,1: %_ ). The values tabulated were
therefore all obtained by solving Equation (16), except the
values for the simply supported plate with a point support in the
middle of the plate. In these cases the lowest critical
value of/g, occurs forA‘O, and the denominator of the

Equation (16) was equated to zero to find the lowest value.

TABLE I

CRITICAL VALUES OF A
FOR THE SIMPLY SUPPORTED PLATE

 

 

 

 

 

 

 

 

 

 

5 ”L 1.; x ,9.
{3 1;: l o 6.25
5: -§; 2: 0 16.00
g % "'i 0 4.00
g ‘32"- l l , 5.00
9i ‘52": . 2 1 8.00
1,: 3;: i i 2.00
g? 5% 1 0 5.75
’§’ ii 1 1 4.05

 

 

 

 

 

l7

18

4. CLAMPED PLATE
Application of the General Theory

A rectangular plate with clamped edges on four sides

and an arbitrary point support with coordinates (5,11) is

 

 

 

 

considered.
at
. 1v») .,
_. 3,...
b—e- L..—
-__q~“__'gf'_d'4 "75
H1
5 FIGURE4
In Figure 4

a a length of the plate in the x - direction,

b 3 length of the plate in the y -- direction,

Nx «13 uniform compressive force per unit length acting
in the plane of the plate on the edges 1.: 0,1'3‘,

o(.Nx a’uniform compressive force per unit length acting

in the plane of the plate on the edges 5:9, 3:5
where 0‘18 a dimensionless constant.

The series expression Eq.(2) for the deflection

of this plate is

19

W(x,3)=z Z [Ann '91-“. 3n 4' BM" FM.G'I

"3| “3‘

+ Cm“ 'Pm° G" ‘.’ Em“ Fm'sn] (18)
3

where the functions

fm=(I-Cosam11'£) (19)
a.

Fm:(umg “‘5” “In £)_(l -c°‘ amt)

km -Jin “In I ’C-O‘ um

and
(20)
are the complete set of eigenfunctions of a clamped beam of

13
length a. The functions

3“: (1-Cossn1rti) (21)
b

and ear-(“"3 .. sin 11,, %)_(I - cos a. i) (22)
an - Sin a” I- COS “n

are the complete set of eigenfunctions of a clamped beam of

1ength b. The numbers a... and an respectively in the

functions Fan, and (3" are the positive rootsl4of the

transcendental equation

«-9.
tang-L.

The coefficients Amn: an, Cmn: and Emn are the multiple
set of parameters to be determined.
The same expression for the strain energy of bending

[Eq.(7)] is used as was used in Chapter 3. We observe the

13. F.B. Hildebrand, Op.Cit. p. 217.
14. See Table II.

 

15
orthogonality properties of the eigenfunctions, namely

4
S'Pm"pm|d%
o :0 $or mine,

I: 4' as
m m‘ #0 germ-3m.

4:“. like!»

9 ”D 0"),

A fourth orthogonality property,

. “;¢ £:u
. . ‘1
S) m m, 7"

=0 $9? m¢m\

#0 #or m=m,

is obtained by integration by parts as follows:
a» u I l‘ a t |
S'gm'¥m‘Jx =(‘$m°“m) 0-S'cm'.¥m.dx.
o , o

The integrated part vanishes because of the and conditions
at «so, QL‘P-a . Since the remaining integral is the same
as the second orthogonality property listed above, the
fourth orthogonalitylproperty is proved. The same ortho-
gonality properties of the eigenfunctions are true if ‘Pm
and‘n. are replaced by Pm and Pen. . If, however, only
one of the two functions is replaced ( *m by pm or .9”.
by anbut not both ), then the integrals vanish whether or
not finsvw\,. Similar orthogonality properties are true for

the eigenfunctions 3“ and CD“ .

 

15. Friedrich Bleich, 0p.Cit, pp.55-59.

' —.‘ fi’er‘J-uz— 1‘. -‘A 'w.
w. .A i . _
v I.

2.1

With these properties in mind,‘wc first evaluate the

contribution of the curved bracket

4 b 2,
SS 47%,) 4“:

in Equation—(7). This expression can be expanded to
a z 2. t- 2. 1- 1.
SS (3:: + 2. 2‘21 3.22;) deal
0 o 3%.. 3* 33 ‘33 O

The only terms that will not vanish due to orthogonality

 

9: L'Jfl. ‘ ‘33“ tax-“u'o'crmc

when the approPriate derivatives of w ESquBfl are substi-

tuted in the expressions above are found to be

 

G;- ._,

b_ co oo 43 I: .‘

as enagsspmtmisi +

B:»(Fv:)" G: + Ci... (4.3.7- G:+E:~n(‘:)z3:}d“l
J

’

Z.

 

b a a. u ” ‘5 u “
§(3 w,%',)de43:{: SS inn$m$h3“1“*

‘3 11‘ Inst rm

0(a’59

E3::aFin'i: (Ancsz -* <::;n‘¢u.$£: <3,,c5:‘?

5‘

'E:k"|FinflFi: fSh‘S’e ‘d‘fld3)

a. b .‘_ z a. .° 1‘ " g it 0| 1
§S(-.:-:.)Ma=§.§ SS (Am Man) +

E3:hvv Fi::6§3;sz+'C::un'Fs:(£3;:)}4pEah"'F::e§:§}dkdl

22

Similar expressions for the Square bracket of Equation (7)
are also found. The apprOpriate derivatives of the eigen-
functions 4‘“, F,“ ,3“ , and G“ are calculated from
Equations ( 19,20,21, and 22 ), and substituted in Equation
(7). We find that the square bracket vanishes and the curved

bracket when integrated produces

“7

v: -'-N
t, a“ hunt!

L”"“" :(Aam ‘;:;l*'2(EE§L‘(nK‘-rt*1fiijr12n{‘:v\) +-

3.3.")

EM“(%;F—1 "m + ‘(gb‘zn'z Km +8(%)4'"‘4Pz. :M)} (33)
3

 

 

 

 

where
Nozw‘D
a. ’ h
a.
33nslfiuun z,
Hm=u.'.. (aim 4- )_ 5‘“ “m

z Z {41" A: ”(3m +£92.32 mg n ‘+3(%)n 4}!-

 

 

(“In - Sun. an)‘ (a... ~$au um)(l-COS a ”7.)...

Z-

“In shall-M...)
< .. 4, -

_, 7—71—47 ..‘_:‘_'_ ‘

" h

23

u. u 3 I.“
"I: ‘ (3-35 4-25;" HM+ZUmCosam+ f-%__’.n)*
0:; (u... - Sin um)‘

(— at, + Z. a... Site “on - Sin‘um)
(um - 53" “m)( i - c°‘ “m) a
l

(-1! Urn-LS"; um+* Strata-m) )and
(I - Cos 14,“):-

Km: a,” ("22, uM-Z. Sinumia-fishzum)

(“m- Sin km)‘

 

(L C03 um '- Z. + ain‘um)
(um - sin um)(1- cos u...)
C “I?” -$ Sin 2. am)
(| — Cos 04...)" ’
Note that Q“: H“ , 3'“: P“ ,and L”: K“ . Different

symbols have been used to avoid confusion when one of the

 

quantities is evaluated for an index m, and the other one
is evaluated for a different index n. These quantities
have been evaluated for m,n 8 1,2, and 3,and are tabulated
in Table III.

The same expression UW for the change in the potential
energy of the external loads during buckling is used as in
Chapter 3 [134410]. The appropriate derivatives of wutqdlea

are taken and substituted in U“. The expression for ”we

24

when integrated is

 

U": -éT—;:.Nfiz mu “zf‘éAm”(m‘ +d(gtn‘)+
vw‘
1:.:;n, (.ZL\~JL :5. +-«-((% kuf) +

5:“ (i—zi'“ Km +- 46:)!- Zn,” 3;”ng )(‘4

where Km ,3... , L... , and PM are the quantities defined
following Equation (23).

The extended Ritz method as discussed in Chapter 2 is
now applied. The expressions found for V EEqJZSD and Uw
@q. (24] are substituted in U «@9443

The necessary conditions1 6for £1 to be stationary are

2.9... 29.-., 2.9.... 29...,

3 Ana 2 BM“ zchfl 3 ENG
and

wCE.‘D=O.

The derivatives of \1 with reSpect to Amn are

2 : A an 2 N .‘W 4 (3 m‘i- 1(53‘m‘n14.
CL ‘b

at» '%‘”‘*°‘¢t>‘“3 mm:-

 

16. See Lagrange's multiplier method, Chapter 2.

25

The derivatives of l’ with respect to an are

10;, - 3...... _ m~3' '0!- 5 1' °
23...; : Akl}. " 34-) KTL'”

-1r'~1_. Fit) 6.90:“ o. (256)
The derivatives of ‘1 with respect to Cmn are

2.9. g h, 2’: N. [(81I“m‘- I," 4(%)"1~zm"-L. ‘9

Cm.
fit) a.) ‘V-("L "“3» *‘-‘--(8;) ""‘ufl
w‘a - £5?)- GM.‘ = - ‘2'”

The derivatives of D with respect to Em are

0’

.2 -.E.:..h:.[(!i -H _+4.(gf"'-1-n K
nun.
8(9‘.‘ 1!". ‘fm)-‘l‘/g_(-’ihK +zei($)wfl'n- p..]
(28)
-vw‘A Iflg» 3.0L)- - o -
In Equations ( 25,26,2'7, and 28 ), A may be zero or diff-

N

erent from zero. If Jl. is zero, the critical value of

is determined from one of the Equations ( 25,26,27, and 28 )
in a manner similar to that for the case of the simply
supported plate of Chapter 3. When the lowest critical value
of/‘g has been chosen in this manner» only one term of the
series for w [@4183 survives. The other coefficients of
that set and all three of the remaining sets of coefficients
must be equal to zero in order that Equations ( 25,26,27,

and 28 ) can all be satisfied for the lowest critical buck-

ling value Of/q. chosen. Physically this means that the

DRE'W‘W - And-mm

internal point support is located at a nodal point of one
of the buckling modes of a clamped plate with no internal
support.

For the case that )L is different from zero, Equations
( 25,26,27, and 28 ) require that for each choice of the

indices m,n,

A 2
Am". 1!“ b m' (-F...(€)-3.ou) ,
e «a .4 + .2 «2m 2“ «as»?

no.1“
"T N. (tam . e..m>)

E". In “’ z’(%)‘Km°Ln+(%)‘Pm'Qu-.] "'

lsrnvt=

 

 

1P;Q, E Km‘ In "’ ‘«%)‘ Pr-(LJ

 

 

 

Xe. 1“"
Cm“: b No . (‘Fm(€)‘ Gh(n))
[g ‘fl‘4'm‘. 3-“ +4(%)z,u. z mt- . L“... %(%)‘.'Qg-
, and
‘7tjyflu [:Fr‘qtfln‘-:Ivs‘*’°"%:c%:ftt";]
Jch‘flJL
Ema: 5N0. (Fm(£). 3"(n))

 

‘-‘$
[% Hawt- +(%)‘fz'n‘Km-|-8($bq"fl' 9‘ 'Pm] «—

#74: 3; K... +- d z(%)‘f‘n‘- Pm]

 

L“ -‘fiEQ . . 2".
O

27

The other necessary condition for a stationary value
of D is w(g,'o=0 . Substitution of the expressions for
Amn, an, Cmn, and Emn in this necessary condition results
in the characteristic equation

3

’23-” 2:” 41‘ (ex - GD

”0 -__-b- ( m 3» >

4.
m“ "“ 1!“ [4(: m“'+ 2.(%3‘m‘a‘+ 3%.)? )

 

.. 3/40} +42... “)3

mars-nus: 54‘ w*‘ "M. -" ‘7‘

FGX 610D
Bm-=»+F’-<%)K--‘L~* 55-"? ‘QJ

+

 

:TP‘/(', [Km-5,. «- d(%)" P”. I... J

9.2%)- elm
. . . 4
[mm + ore-w hie-m1

4‘4-

. TykLLTP‘mt-I“ + 4‘ at a)" La

 

kab- 3100
1. x ' ‘F 4' 43
[PiHm++(%)‘1r n «ma-8% 1‘ n Fm]
. 20(29)

mt... W... }.

28

Let WA) equal the doubly infinite sum in Equation (29).
Then the values of A that satisfy the equation

V99) = o (30)
will be the critical values of/‘ for the case that A. is

different than zero.

 

Solution of the Characteristic Equation

-»“.'.' -'-:‘-«" V‘x.
..~..-. ,

The characteristic equation of this Chapter is solved

by a numerical interpolation method similar to the method

 

r — m ._
." o . .- _
4

of solution of the characteristic equation of Chapter 8.

A graph of Vlaversus/v. for a Specific clamped plate

will have vertical asymptotes at values of/g corresponding

to the critical buckling values of/Q‘ for that clamped

plate with no internal point support. These values are found

for the case that 1 is zero. The lowest solution of the

characteristic equation will lie in the interval between the

first two asymptotes. A value or/q. within this interval

~is arbitrarily selected and is substituted in a finite

number of terms of each of the four infinite arrays. After

the sign of the sum is found, a second value of/g is selected,

lower than the first choice if the sum is positive, and higher

than the first choice if the sum is negative. When two values

of/g, have been found to give opposite signs for the sum, the

root of V“): O is found by repeated numerical interpolation.
Several critical values of/t for various load ratios d ,

side ratios % , and point support locations (g’ql) are

29

tabulated in Table IV.

The lowest critical value of jg occurs for 1 not zero
except in the case that the point support is at the middle
of the plate (9: 3;: , '1: Pi ). The values tabulated were
therefore all obtained by solving Equation (30), except for
the case of the middle support. In this case the lowest
critical value or/(‘ occurs forks O , and one of the

denominators or a term in Equations ( 25,26,27, and 28 )

was equated to zero to find the lowest value.

wfnfik - 15.5”“!‘K‘9Qm (1’.qu
‘ ‘ n

OF

TABLE II

ROOTS OF THE

TRANSCENDENTAL EQUATION

 

 

 

 

 

 

 

 

 

it».
u , 8. 9868
a," 15. 4506
«4, 21.8082
R4 28. 1324
“.3 34.4416
TABLE 111

EVALUTED CONSTANTS

 

 

 

 

 

THE CHARACTERISTIC EQUATION
m Hm Qm Km 1m Pm JIn
1 896. 11.12 .280
2 7880. 80.9 .200
3 28250. 60.8 .211

 

 

 

 

50

 

31

TABLE IV

CRITICAL VALUES OF
FOR THE CLAMPED PLATE

 

 

 

 

a.
g 7L 1; a! [L
% if 1 0 11.90
1}: '3'; 1 1 9.85

 

 

 

 

 

 

 

5. SUMMARY AND CONCLUSIONS

The problem considered in this paper was to determine
the effect of an internal point support on the critical
elastic buckling load of a rectangular thin plate under
various edge support conditions. The extended Ritz energy
method was used in the solution. The general theory of this, a

solution is discussed in Chapter 2. Two applications of

 

the theory, namely the simply supported plate in Chapter 3 g“

and the clamped plate in Chapter 4) were made. i;
The lowest critical buckling load for a rectangular

plate with a point support not in the middle of the plate

was shown to lie between the first and second critical

buckling load of the plate with no internal point support.

The lowest critical load for a plate with a point support

in the middle of the plate was shown to be the second crit-

ical buckling load of that plate with no internal point

support. Therefore the most effective location of the

point support is at the middle of the plate.
The critical values of /&g.( a dimensionless load

parameter ) can be used for plates of any material that

conforms to the assumptions made in Chapter 1, namely that

the material is homogeneous, isotropic, and follows hooke's

law. The specific material properties are only introduced

when the critical buckling load Nx is evaluated using the

dimensionless parameter/ޢ..

(s
(a

BIBLIOGRAPHY

Bleich, Friedrich, Buckling Strength g£_Metal Structures,
McGraw-hill, New York: 1952.

 

Courant, R. and Hilbert, D., Methoden Der mathematischen
Physik, Vol. 1, Springer, Berlin: 1951.

Sokolnikoff, I. S. and Sokolnikoff, E. 8., Higher Mathematics
for Engineers and thsicists, McGraw-Hill, New York: 1941.

Hildebrand, F. B., Advanced Calculus for Engineers, Prentice-
Hall, New Yerk: 1949. _

Timoshenko, 8., Theory g£_Elastic Stabilit , McGraw-Hill,
New York: 1936.

 

ls. . r
a“ 2th.... in...) astute. th/

c." !-£1Y_

 

 

 

MIC CHI GAN STATE UNI IIVERS ITY LIB

078