riil
_DucKllNG'or
prywooDptATts rN compRrssroN,
':
s]lfaR, oR coilBtNtD eoilppfssloN AND s]rfAR
::
,':-
''"'_t\
.'
.
.*f€'|rriffrr.H<llitr,
..r,,,.--
Llood hgt*e*r{ag
Basc.rch
Far*t
Fr**t*
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lledl.rcn Sl*corntn
53105
UNI?EDSTATESDEPARTMENT
OF A,GRICULTURE
FOREST SERVICE
FOREST PRODUCTS LABORA,TO-R.Y i.
MadisonS,
Wisconsin
:'
BUCKLING OF FLAT
PLYWOOD PLATES IN COMPRESSION,
SHEAR, OR COMBINED COMPRESSION AND SHEAR!
By'
H. W. MARCH, a Head Mathernatician
Introduction
In this report are presented the results of rnathernatical analyses oi
several buckling problems for flat plates under various typ.es of loadirg.
The critical stresses given were frequently obtained by the use
Although they rnust be considered as tentaof approxirnate rnethods.
tive until confirrned by tests, they are given at this tirne in the belief
In a number of cases, the plates are taken to be
they will be useful.
Shown in such cases are
infinitely long to simplify the mathernatics.
the effects of differences in orientation of the grain of the face plies
with respect to the direction of loading and the effects of differences
in the structure of the plywood.
For finite plates, the critical stress will not be less and will usually
When
be larger than that for the corresponding infinitely long plate.
should
plate,
it
long
infinitely
the critical stress is known only for an
be possible to rnake a satisfactory estimate of the critical stress for
fffri"
is one of a series of progress reports prepared by the Forest
Rerelating to the use of wood in aircraft.
Products Laboratory
and rnay be revised as addisults here reported are prelirninary
tional data becorne available.
Original report published April 1942,
)
:8.
W. Kuenzi, Jr., Junior Engineer, attended to rnany detaile in the
preparation of this report and checked the integrations and matheleading to the final forrnulas.
matical rnanipulations
Report No. I316
,l-
Agriculture
- Madison
a finite plate of the sarne construction by considering the effect of the
length-breadth ratio in cases that have been cornpletely worked out.
The approxirnate rnethods lead to critical stregses that are too large.
In cases where a comparison with exact values was possible, the errors
were found to be less than 8 percent.
It is believed that in other cases
the errors rnay be considered to be of this order of magnitude.
In all cases, the angle between the grain of the wood in adjacent pliee
is 90o. The grain of the face plies rnakes an angle of 0", 45o, or 90b
with the edges of the plates, which are taken to be sirnply supported.
The state of stress is uniform compression, uniforrn shear, or cotrrbined uniforrn cornpression (or tension) and uniforrn shear.
In all calculations, the values of the elastic constants of Douglas-fir
were used, as given in the appendix, and the plies were assurned to be
rotary cut. But the resuLts for plywood with sliced veneer that is approximately edge-grain should not be greatly different frorn those obtained for plywood of rotary-cut veneers.
The critical stresses are
exPressed in a form containing the modulus of elasticity along the grain
E" as a factor.
rt is to be expected that differences in species, denllty,
and. moisture content are taken into account to a considerable
tent in the choice of this factor.
ex-
rn the case of plates under uniforrn shear and having the grain of the
face plies either parallel or perpendicular to the edges, the results
obtained by E. Seydela for plates of orthotropic rnaterial are used. It
is shown how to calculate the constants for plywood that are needed to
utilize the farnily of curves given in his paper and reproduced here.
Notation
The choice of axes is shown in figure l.
parallel to the Y axis as shown in figure
1 z e i t ". h .
The direction
4.
of loading is
fur F1ugtech. und Motorluftschif f.ahrt ?4, ?g-93,
Report No. l3 f 6
-z-
1933.
a
= width of plate measured
b
= half wave length of the buckled surface in the case of an
infinitely
long platd; or the length of a finite plate.
h
= thickness
p
= uniforrn cornpressive stress parallel to OY.
sive stress is consid.ered to be positive-
P"t
= value of p for which buckling
q
= uniform shearing strese.
shown in figure 4.
parallel
to OX.
of plate.
occurs.
q is positive when directed as
q^_
q for which buckling
- c r = value of €
occurs.
Q
= angle between grain of face plies and
+
= angle between wrinkles
y
= tan
= elope of wrinkles.
0
,=\
k"
and OX.
a
= coefficient
in forrnula
p""
h
= O"
""
= coefficient
in formula
9""
= k" EL
a
= coefficient
Other
eyrnbols
Report No. f3l6
z
z
hz
a
f
A compres-
z
in forrnula g = fp
are defined in the Appendix.
-3-
ox.
l.
Rectangular P]"ywood Plates Under Uniforrn Compreselon
in a Direction ParalleL to One Palr of Edges
6ase l:
9
= 0' or 0 = 90".
Unlforrn compressive stress parallel.to
oY.
The critical buckling,stress
given by the forrnula3
zf
r
Tr
P"" = tr^ l"t
L^
-z
fi
no ^o
for a plate buckling
in n half wavee is
z a z1.z
lh
n
+ zA* Ez';
u
bu
The constants that occur in this formula
For a plate of given wi.dth and thickness,
(r)
la
)a-
are defined in the appendix.
p""
will
be least when
lEr\ 4
b =natE-l
-t
\
(z)
/
In this caee
P
'When
(3)
cr
n = l,
t h e l e a s t c r i t i c a L s t r e s e (3) will be found for a plate for
which-
1s"" for example s. Tirnoshenko, Theory of Erastic stability,
(Mccraw-Hill
Book co. I936) p. 3Bl.
The forrnula there given for
a plate of homogeneous orthotropic materiar can be used for plywood which is nonhornogeneous frorn the standpoint of its elastic
propertles,
if the constants occurring in it are given the val.ues
that are found ira the differential
equation for the bending of a plywood plate. see u. s. r'orest products Laboratory Report r3rz,
r'tr'Iat Plates of Flywood
Under uniform and Concentrated Loads. rr
Report No. l316
-4-
I
ln,\4
b=al:l
\"t/
(4)
The buckled forrn of the plate will
half waves when
change frorn n half wavea to n * I
.r+
r
b=aIt"*t!t
(5)
It will be convenient for
to write (t) in the form
D
'cr
=k
c
E
comparison with other resulte
hz
L^2
( 6)
where
n"=fr
1+w*'+.#,?J
z [-s .z
LL
n
E
(?)
z il
a
.f L fot
a
3-ply plates having the thickneee of the flTe pties equal to one-hElf
= 0" and
= 90".
that of the center ply and havlng
Gurveg for the
9
9
In tables 2 and 3 are given the valuee of k" for certain
factor
k" as a function
plates li-fig,rt.s
of the ratio L
a """
5 and 5.
When a plate is very long,
ratioe
shown for theee two types of
it buckles into a large
number
The half wave length approaches asymptotically the value
of waves.
I
/r^\r
" l=a
I
\"0 I
as the length of the plate is lncreased.
This rnay be called the ideal
half wave length.
It corresponds to the rninimurn critical load.
When
the length of the plate is greater than three times the ideal half wave
Report No. l3l6
-)-
load differs but little frorn the rninirnurn critical
length, the critical
load.
In the curves of figures 5 and 6 the greatest difference between
the critical stress for a given plate and the minirnum critical stregs
is about 3 percent when the length of the plate is greater than three
tirnes the ideal half wave trength.
Case 2:
Infinite
strip.
(long narrow plate. ) I = nS"
equation governing deflection becornes
In thie case, the differential
more cornplicated than the corresponding equation for the cases in
which the direction of the grain of the face plies is parallel or PerPendicular to the edges. Its solution requires different and apparently
an approxirnate solution
lnore cornpLicated analysis.
Accordingly,
was obtained by the energy method.3
In order to obtain solne inforthe plate u'as
mation quickly by avoiding long nurnerical calculations,
assurned to be very long in the direction of Loading. When the results
are compared with those obtained in case I for long plates having the
to the direction
grain of the face pli.es either parallel or perpendicular
of the
of loading, an estirnate can be forrned of the effect of inclination
plates
of
grain to the direction of loading that will be useful in the case
of finite length.,
Let the deflection
w=H
sin
of the buckled plate be represented
lTX
a
srn
ft'-
by the equatio#
(8)
Yx)
?
:See for example S. Tirnoshenko,
6
:The
Theory
of Elastic
Stability,
p.
325.
forrn of the buckled surface assurned in equation (8) was suggested by equation (k) p. 360, Theory of Elastic Stability, by S.
Tirnoshenko.
That equation was rnade the basis for the application of the energy rnethod to determine the critical
shearing
stress of an infinite strip of isotropic rnaterial.
Report No.
f 3f 6
-6-
where
= the half wave length of the buckled surface, that is, one-half
the distance within which the phenomenon repeats itself.
y = tanE,
where 0 is the inclination
See figure 7.
of the wrinkles
to the X-axis.
The strain energy of bending due to the buckling over a half wave length
was equated to the work done by
L, a length which is to be determined,
the load in producing the shortening resulting frorn the buckling of this
portion of the plate.
The resulting'expression
can be solved for the
critical stress p as a functlon of b and y. These two quantities, b and 1.
were deterrnined by requiring that p should be a rninirnum ae a function
of these variables, that is, that
ff,=o and f$=o
(e)
The integral of the expression for the strain energy in bending? t"k"r,
over one half wave length of the buckled surface is first written down,
the axes of reference being parallel and perpendicular
to the grain of
the face plies.
In this integral a transformation
is then made which rotates the axes to the position OX and OY of figure 7.
Using the abbreviation
z
D
^cr =A
l2).
- ;b'
z _
t"
it is found that
z
+R(1*L)
4
(,? * 6yz+ tjft
z
z
hz
J
7
-For
(ro)
the integrand, see equation (3. 2ll of.appendix 3 of Forest Products
Labonatory Report No. L3LZ.
Report No. l3f6
-7-
where z arrd y satisfy
the simultaneous
equations
(v3 + v)
, 4 = | + v 4 + Rvz tut
G
(rI )
(rz)
and
Et * Ez+ ZA
( r3 )
fi=
R=(r(Er*Ez)-A
(14)
M=EI-Ez
(15)
The syrnbols Er,
EZ, and A are defined in the appendix'
to be taken fo-r strTll parallel
grain of the face plies"
and perpendicular,
E, and Erare
respectively,
to the
Equation (f 0) wifl be written in the form
P""=*""tF
,z
n
(16)
where
zf-4
+rfrl
n"=# +- L L lor"'+6tz
Z
+R(rr{r-M(3v-+|
zzJ
Report lilo. f 316
-8-
(r?)
The value s of z and 'y to be used in (l?) are the solutions
ana (f Z).
taneous equati-ons (t!
of the simul-
a r e g i v e n f o r c e r t a i n types of plates in tables 5 and 5
c
and at points on the horizontal axis in f i g u r e s L 3 t o 2 6 .
The values of k
2.
Rectangular
Plywood
Plates
Under Uniforrn
Shear
Casel:9=0"or0=90o.
The results for this case can be obtaingd from formulas and a farnily of
These curves are reProduced
curves contained. in a paper by Seydet.9
in figure 8. A certain arnount of approxirnation was involved in drawing
these curves, but it appears that they can be condidered sufficiently accurate for practical purposes.
In order to rnake use of the farnily of curves of figure 8 to determine the
critical stress for a given plate, it is necessary to know the values of
two quantities,
o and p^, which are defined in terms of the constants Err
a
E^, and A and the dimensions a and b by the following equations:
z-
A
(18)
l(El E
zlz
,r'
F"= *(
E
(re)
I)*
8
:Edgar
24,
fur Flugtechnik und Motorluftschiffahtt
Seydel, Zeitschrift
t
h
e
b
y
p
u
b
l
i
s
h
e
d
w
a
s
78-83, 1933. A translation of this paper
National Advisory Cornmittee for Aeronautics as Technical Memorandum No. 705.
Report No. l316
-9-
It is to be observed that, in accordance with the definitions given in the
are nreasured Paralappendix, the rnean rnoduli in bending, E, and E'
Le1to the sides a and L, respectively.
The quantities
o and F" being known,
gI
of figur"
the curves
The critical
the quantitY c" can be nead frorn
shearing
q- c r can then be calculated
stress
frorn the forrnula
I
(20)
"rlz
Equation (20) c a n be written
hz
=k
"EL
where,
in the forrn
a
(zt)
z
after introducing
for convenience
the ratios
",
q
(z2l
k=
EI
To illustrate the procedure, the crltical shearing stress for a 3'ply
panel of Douglas-fir
plywood will be calculated.
A11 the plies are taken
to be of the sarre thickness"
The grain of the face plies is assurned to
be parallel to the side b, which is twice as long as the side a. See figure l.
Report No. l3l6
- 10-
= 0 . 0 9 2 ?E
IL
A = 0.1451 t
E = 0 . 9 6 5 1E
ZL
L
0. l45l
o=
= 0.485
( o . 9 6 s lx o . o g z T l z
F"=d)tfr##l
c
a
I
4=o.8eB
= 15.6
r 5 .6
s=T
orrrl' o.rrr,] + = O . 8 7 4
[to.
Hence
= 0.874 E
crLz
A difftculty
uZ
=-
(23'l
a
will be met in this type of plywood if the eide b in figure
I
l s l e s s t t a n 1' o$.l l - 1 n
09z7'
^ = t.797a.
To illustrate the procedure in this case, let b = l. 5 a. It will be found
that p_
' a is greater than I and c bannot be forfrd frorn figure 8. It ie
a
onlyilceosary
to lnterch*rrg.-t-h" eides and the corresponding conatantg
E, and E- as in figure 9.
IL
Report No. l316
-ll-
Then
rt
D
2 ,
=TA
t r ' = 0 . 9 6 5 1E l
E z = o . 0 9 2 7E l
A = 0. t45l EL
o = 0.485
I
o
o . o 9 nr 4
" ' ,tojmr
P" -= 6T
= 0 . 5 5 5 6*
= o. 8ls
b'
From
figure
c
a
8,
= L4.7
Then
= -t 4 . 7
s3r
+ = 2.659
rrurl' o.oez?]
[to.
and
cr
= 2.659E
h
L
2
^t2
But
at=b=1.5a
Report No. 1315
-12-
in terme
Hence,
of the side a of the given plate
-tJ-=
= 2.659
cr
'
z
"- .zs^"
l . 1 8 2E
L
hz
-z
(z+l
a
long plat. F" =
The corresponding value c" is the
intercept on the axis p" = 0To-r the proper value of a in figurff.
For an infinitely
Thus, for an infinit.flfong
plate of 3-ply,
plies of the same thickness and 0 = 90",
Douglae-fir plywood wlth atl
a = 0.485
c
a
k=
s
= 10.7
i
[to.orrrl'o.e65l
13*
= 0.600
Then
= 0.600 E
h
z
(z5l
L 2a
tr9= o
o= 0. 485
c
a
= 10.7
_ 10.7
3I
=
[to.rrurl' o.osz|I
L.935
Then
z
= 1.935 Eh
L ,rZ
Report No. 1315
(26)
-l3-
Gase 2z
Infinite
strip
(long narrow
plate).
0 = 45"
h thie case, the rnathernatical analysis leading to the resultg of case I
of thie section does not apply.
Exactly ae in section I, the energy
rnethod can be used to obtain an approximate solution for an infinitely
long plate.
The forrn of the deflected surface was taken to be given by
(8) of section l.
The same reasoning that was used. in that section (except that the strain energy of bending is equated to the work done by the
Ioad in producing the shearing deformation associated with the bendingZ)
leads to the following expression for the critical shearing etress:
(z7l
=ft
where
z
1r
(uz* 6yz
*+,
z4|\
+ R(l -4,
(28)
In thls equation z and r1 satisfy the simultaneous
equations:
,4 = t+ y4+ nvz - U(v3 * v)
(ze)
z zc (r - v4) - rra(v - v3)
(30)
z=
6cyz - R
The quantities G., }v!, and R are defined by equations (13), (I4), and (15)
of section l.
It is to be observed that here, as in case Z of section I,
E, and E" are associated with strips parallel and perpendicular,
respec'- L L
tively, to the grain of the face plies.
q
3See for exarnple
S. Tirnoshenko,
Report No. 1316
Theory
-14-
of Elastic
Stability.
p.
357.
It will be found that equations (29) ana (30) yiefd two values of Y, one
poqitive and the other negative.
The positive value corresponds to the
situation of figure 10, while the negative value corfesponds to that of
figure ll.
Evidently g.* should be expected to be rnuch greater, numerin the second i-a"..
ically,
This is found to be true in tabtre 6 and in figures 14, 17, 20, 23, and 26.
As a check on the probable accuracy of the energy method that was used
in obtaining forrnuta (27), this method was used to find the critical
shearing stress for infinitely long plywood plates in which the grain of
the face plies is either parallel or perpendicula'r to the edges of the
plate.
Using the energy rnethod together with equation (8) for the buckled surface, the following formula was found for the critical shearing stress:
o' c r s =
Lk
E
h
z
(31)
za
where
2+
+z A n2
(3zl
EL
and z and y satisfy
4
the simultaneous
*rr zavz+ r,
equations
4
(33)
Et
2
ur-nrv4
(34)
z=
lnrvz-A
Report N o . 1 3 1 5
- I5-
In these formulas
E, and Erare
to be taken as corresponding
to strips
parallel to the X an?-y a*6
respectively.
A cornparison of the values
of k- as calculated by f ormula (32) and as calculated from Seydelts
s
.rri7"" is shown in table 4for a nurnber of types of plates.
It appears that the energy rnethod leads to results for the cases 0 = 0"
Irr the case of-the
and 0 = 90" that are from 6 to 8 percent too high.
isotiopic infinite st1ip it also leade to an approximate value of the critiThe fact that a
cal shearing stressru
that is about 7 percent too high.
useful approximation is attained in these cases by the energy rnethod
lends support to the belief that the same is true for the infinite plywood
strip with 0 = 45" .
An estirnate of the variation of critical shearing stress with length that
rnay be expected to be useful for the case 0 = 45" can be rnade by studying the curve of figure 12. In this figure the ratio of the value of the
factor k of forrnula (21) for a plate of length b to the value of k for an
s
forrTTion of
infinitet'
; plotted
long plate of the
constructi"r
""rrr"
"" "
the ratio of the length b of the finite plate to the half wave length b! of
the buckled surface of The infinitely
long plate.
Points are plott.tfo"
several types of plywood with 9 = 0" or 0 = 90".
They are Been to lie
almost exactly on a comrnon curve.
The values of k" were calculated
with the aid. of the curves of figure 8, which was ta6n frorn Seydelr e
paper, to which reference has been rnade.
The fact that maxirna and
rninirna sirnilar to those in figures 5 and 6 do not appear in figure LZ is
a consequence of the approxirnations involved in Seydelrs curves in figo
ure 8, which are intended to lead to curves for k, as a function of
which pass through the rninirna of the exact curv-es, which would bEslmilar to those of figures 5 and 6, except that the ordinates of the
itt"t""".".
rninirnurn points decrlase to a limiting vitue as the ratio
$
t0-.-€ee
for example S. Tirnoshenko,
Report No. 1316
Theory of Elagtic Stability,
-f6-
p. 36I.
3.
Cornbined
Uniforrn
Plates
Under
and Uniform
Shear
Long Plywood
Cornpression
0 = 0" or 0 = 90o.
Case l:
Let
of tnfinitely
Buckling
(35)
q=fP
where f ie either poeitive or negative.
The sense of q positive
compr&eion.
figure 4.
A positive value of g denotes a
is indicated by the arrowe in
The energy rnethod was used and the forrn of the buckled eurface was
taken to be that given by (8). The strain energy of bending of the plate
in buckling to this form was equated to the work done.by Pand q in produclng the deforrnation
associated with the bending, that is, the shortening of the plate and thq shearing strain.
This equation was solved for g
after making use of equation (35). Then, p"" i" the minimurn value of g
as a function
of r = 9"rd
It was found TE-t
y.
z
P""=n"""5
( 36)
where
k
z
=Jc l2I
t
t
L
t-rEl]r
{
+ zeyz + e,
.b
and z - : and y satisfy
@2*6yzl+zA+4fr,
z-
.,tl
the sirnultaneous
(3?)
equations:
7_4
E ^ + Z A \f + E , .y^
,4=
LL
Report No. l316
(38)
El
-t7-
Z
A y + n , V 3 + E l y4 f. - r ' zf.
(3e)
z=
Af-3Ely-snrvzf
After
p^_ has been found,
cr
-rp
glvenDyq
cr
cr
the associated
critical
ehearing
etress is
It will be convenient to write
L
q- c r = f k
Ec L
L
^^=
-k
a
Z
t
s-L
hz
z
a
(40)
where
k
=f k
sc
(4t)
The valuee of k- and k_ are listed in table 5 for eeveral types of plywood
cs
and for s"v"r"Ivalueiof
f.
Case 2z trfinite etrip.
As in case l, Iet
0 = 45".
g = fp
(35)
and let the forrn of the buckled eurface be given bV (g).
The energy rnethod leads to the forrnula
?
='k
crcLZ
E
Report No. l3f6
n
(42l-
a
- l8-
f
where
k
c
=n?
lzl
E L (1 + zyf)
t
(rzr612+r+rv41
zo
z3l
+R(r+!t-M(3v-"#ll
(43)
zzl
.b
and z = a and y eatlefy the equations
a-
(44,
,4=L+y4+nvz-U(v3*v)
G
uz -*
2+t-z
-zR +M
t-
l z G ( y + Y2 f) - 3M - zRt
Two values of f. may be expected correeponding to positive
values of q. The positlve sense of q ie shown in figure 4.
After
p
cr
( 45)
has been found,
q
cr
can be found from
or negative
(35).
Then
-2
,z
!-Lc r = f 'pc r = f I c E - \ = k
c L 2
aa
Values of k and k for several
.- c g
f are given in table 6"
ln"
E- L
a L Z
types of plywood and several
.r"rrres of k" and k" given in table 6
arefor
an infinitely
values of
long etrip.
for
itfray
As in other
be expected that they hold approxirnately
""8s,
plates whose lengths are three or more tirnes their widths except PosIlr the
sibly in a few cases where the wave lengths are extremely large.
case of shorter plates a rough estimate of the values of k. and k" can be
made by consideration
of the curve
values for the infinitely
long plate.
Report No. f3l6
of figure
- 19-
I? in corrrreE?l.ott*Iif,
th.
In many instan,^ces o f c o r n b i n e d s t r e s s e s , t h e f o r r n u l a s f o r t h e c a l c u l a z=*,
and y have two solutions and one of these solutions
tionofk,
c'a
A negative
to a negative value of p, or g, or both.
may beEonffi-l"ad
value of p indicates that the direct stress is a tension, since the convention has been adopted here that a direct corrrpressive stress is positive.
In such a case, the buckling takes place under tension cornbined with a
positive or negative shearing stress.
The direction of a positive shears
h
o
w
n
ing stress is
in figure 4.
,
tl addition to tables 5 and 6, the results of this report are shown in the
Associated,
curves of figures l3 to 26 f.or several types of plywood.
values of k and k are plotted as the rectangular coordinates of a point.
cs
Along a fiG k =J-t d.rawn frorn the origin with slope f, points nearer
SC
the origin than the point of intersection with the curve
states of cornbined stress for which buckling does not
showing the ratio of the half wave length b to the width
y of the wrinkles as functions of k", are plotted to the
As previously stated., the value"Tf
used in the calculations.
the constants
correspond to
occur.
Curves
a, and the slope
left in each figure.
of Douglas-fir
were
Conclusion
tr conclusion, attention is again called to the fact that the results of
this report have been derived by theoretical treatrnent, rnuch of it approxirnate, and that they have not yet been confirrned by Laboratory
tests.
The calculated approxirnate critical stresses will be too high,
since they were obtained by the energy rnethod.
However, these errors
rnay be expected to be at least partially offset in practical applications
by the effect of restraints at the edges.
A lirnited series of tests appears to confirrn forrnula (1) for plates having the grain of the face plies parallel or perpendicular
to the edges and
under uniforrn cornpressive stress parallel to one pair of edges, if care
is taken to avoid the effects of initial curvature.
Because of initial lack
of perfect flatness rnany, probably most, plates do not buckle sharply
under any of the types of loading considered in this report.
Instead, a
characteristic behavior, as is well known, is a gradual increase in
Report No. 1316
-20-
lateral deflection'with increasing load, then a rnore rapid increase in
the lateral deflection as the load nears the critical load.
When the
lateral deflections are of the order of rnagnitude of the thickness of the
plate -- greater than one-half the thickness -- the effect of rnernbrane
stresses becomes appreciable.
The apparent stiffness of the plate increases and prevents the occurrence of sudden buckling.
Frequently,
in this stage of loading, the load-lateral
deflection curve is observed
to be nearly a straight line inclined to the axes and not asymptotic to
a horizontal line.
For very emall initial curvatures,
the rnethod.s proposed by Southwelt.ll
and rnodified by Lundquislll
may be used. to calculate the theoretical
critical load from the observed load deflection curve.
The rnethod appears to fail when the initial curvature is so great that appreciable
rnernbrane stresses develop long before the critical load is reached.
The method when applied to plates assumes that the departures of the
rniddle surface frorn a plane are srnall in cornparison with the thicknees,
in order that the behavior of the plate can be described by a linear dif ferential equation.
Tests are under way to check the forrnulas presented in this report,
attention is being given to the effects of initial curvature.
-t-1T o u t h w e l l ,
LZ
-Lundquist,
R. F.,
Proc. Roy. Soc. 135A, 501-516, Lg3Z.
E. E. , N. A. C. A. Technical Note No. 658, f 938.
Report No. l3l5
-zI-
and
APPENDIX:
Elastic
Constants of Wood and Plywood
Wood
W'ood will be considered to be an orthotropic rnaterial,
that is, a material which has three rnutualLy perpendicular
planes of elastic eymrnetry.
These planes are deterrnined by the three principal directions, the longitudinal, radial, and tangential directions,
as shown in figure 3. There
are three Youngt s moduli, EL, ER, and E", associated with extensions
and tangential direcor contractions parallel to IE? toigituainiTradial,
tions, respectively.
There are three shearing moduli trLT, FLR, and
p--.
'RT
p- -is the rnod.ulus associated wiifrTsh6ing
LT
Efr-it
with respect toEaxes
In such a strain a square
oL and OT.
a
whose sid.es are parallel to Ot, ana OT,Tespectively,,becomes
rhombue.
For
exarnple,
.
There are six Poissont s ratios,
OLT,
OTL, ELR,
Thus 0, - is the ratio of the contffifi"ffi"JlTtoT
LI
parallel
to OL associated
0RL,
oRT' and or *'
t o O T to the extension
with a tension paraLlel to 9.
Under the assumption that wood has three planes of elastic syrnrnetry,
the three followlng relations hold arnong the elastic constanta:
or,t
EE'
otr,
=-'
Experirnentally
determined
given in table l.
ot,R
8Rr,
E=E;
oRr
E
hn
(1)
R
values o f t h e c o n s t a n t s f o r s o r n e e p e c i e s a r e
The values given beLow of the elastic moduli of Douglas-fir at l0 percent
moisture content were obtained frorn a lirnited nurnber of tests.
Because
of the srnall number of tests, these values are considered to be tentative.
Report No. f316
-zz-
E !L
= l, 960, 000 lb. per Bq. in.
ET = ll3' 200 lb. Per sq. in.
E r r = 1 5 5 ,8 0 0 l b . p e r s q . i n .
!L
PLT = 123,800lb' Per 3q' in'
ltLR = I10, 600 lb'
tsRT = 7, 100 lb. per sq. in.
Per sq' in.
T h e v a l u e e o f P o i s s o n r 6 r a t i o s f o r s p r u c e were used in the calculatione,
s i n c e t h o s e f o r D o u g l a s - f i r h a v e n o t b e e n determlned.
The influence of
considerable variations of these ratios on the buckling stress is sllght.
Plywood
Plates
In this appendix rectangular plates of plywood will be firet coneidered
in which the directione of the grain of the wood in adjacent plies are
rnutually perpendlcular,
and perpendicular
or parallel to the edges of
the plate.
Later it will be shown how to use the constants defined for
pLates of this type in treiting
plates ln which the grain of the wood in
the respective plies is inclined to the edges of the plate.
The dimengions and choice of axes are shown ln figure 2. The thickness of a
plate is denotea bV b.
The effect of the glue, other than that of securing adherence of adjacent
plies, was assumed to be negllgible in obtaining the forrnulas for calculating the rnean moduli i.n bending E, and
to be defined below. These
Q,
forrnuLas for E, and E, cannot be &iecteFto
apply to partially or completely irnpre!-nated p$wood. nor to compregnated, wood.
the
Further,
relevant shearing rnoduLus cannot in these cases be regarded as independent of the glue.
The quantities E, and E'
howeverr.can be determined by static bending tests and the-Thearii[rnodulus
pXy by a guitable
shear test.
To define E, consider
a strip
of unit width,
with its edges parallel
to
I
the X axis]-to be cut frorn the ptrate of figure l.
The grain of the wood
in the various pliee is assurned to be either parallel or perpendicular
to t.Le edges of the plate.
The stiffness of the strip is determlned by a
modulus E, defined by the equation
I
Report No. 13f6
-23-
E. I=E(E
I
)
x'i
(21
I.
I
where the surnrnation is extended over all of the plies nurnbered, for
exarnple, as in figure Z; .(E*). is the Youngr" .rroirrlos of the ith ply
rneagured in a direction
parallel
to the length of the strip;
I. is the
rnoment of inertia, with respect to the neutral axis, of the area of the
.th
cross section of the i"" ply made by a plane perpendicular to the length
of the strip; and I is the rnoment of the inertia of the entire cross section of the strip ilith r""pect to its central line, that is,
.3
t i o n o f t h e s t r i p w i t h r e s p e c t t o i t s c e n t r a l l i n e , t h a t i s , I = 1- for a
TZ
strip of unit width.
An approxirnate forrnula, in which the error is
very slight, is obtained for E, I by taking the surn of the products
(E*). I. forrned for only thos$lies
in which the grain is parallel
to the
I
length of the strip.
Exception is to be made of a three-ply strip having
the grain of the face p l i e s p e r p e n d i c u l a r t o t h e l e n g t h o f t h e s t r i p .
In the case of a rectangular
plate with sides a and b, the rnodulus E,
would deterrnine the stiffness of a strip cut frorn the plate with its?dges
parallel to the side a, as in figure l.
The rnodutus E, sirnilarly defined,
namely.
Ezr = > (Er), ri
deterrnines
the stiffness
As in the case of E'
(3)
of strips
parallel
the calculation
to the side b.
of E, can be based on the parallel
plies only, .*..pi-itt
the case of a threeJiy
strip having the grain of
the face plies perpendicular
to the length of the strip.
The reasorri.rgl3
regularities
l3
-March,
trated
leading to the definitions
that exist in the state of stress
H. W. , "Flat
Loads. rl
Plates of Plywood
U. S. Forest Products
Lg4Z.
Report No. l3l5
-24-
of E, and, E, neglects
"ffi"
junEions
the ir-
of plies in
Under Uniforrn or ConcenLaboratory
Report No. l3LZ,
a strip of plywood. A long senies of static bending testg at the Forest
Products Laboratory
shows that this procedure is justified.
If the plies in a plate are all of identical rnaterial and ane all either
flat-grained
or edge-grained,
the shearing rnodulus Fxy that is needed
in the theory of plates is the sarne in all plies.
If the plies are of wood
of different species, or if sorrre of the plies are rotary cut and sorne
are sliced, a mean rnodulus lr, - is needed that is defined in the sarne
LL
w a y' l za's E - a n d E ^ ,
viz.
FrzI=E(pxy).
T h e f a c t o , I - t,
xy
(4)
where
denotes Poissonrs ratio as it occurs in the
plate s, is replaced by the factor
theory of isotropic
)'.=l-0
ri
I
(5)
yx
in plywood plates having the grain of the wood parallel or perpendicular
to the axes.
The definition (5) etrictly applies to a eingle ply. But )t ie
not greatly different frorn unity if the axes of reference, x and y, a".
parallel and perpendicular to the grain of the wood and the annual rings
are either parallel or perpendicular
Withbut serious
to the XY plane.
error,
its value may be taken to be the sarne in all plies of the type
specified.
As in other reports of this series, its value will be taken to
be 0. 99, which is approxirnately correct for spruce.
This value is
taken instead 6f unity chiefly to emphasize the fact that this factor,
which is about 0.91 for rnetals, has been taken into account.
Frequently occurring in the discussion
A, which is defined as follows:
of plywood plates is a constant,
A = (Ex ay*) + zrptz
where (E' x Y x A*)
'
Report No. l3f6
(6)
is defined by an equation like
-25-
(a).
For a plate with flat-grained
o =
",
plies of the sarne species
0"" * z (L - 0", ott ) u",
(?)
when the grain of the face plies, as in figure 4, is not perpendicular
or
parallel to the edges of the plate, the constants Er, and Er, and A that
apPear in the forrnulas below are to be calculateFwith
rJ$ect
to temporary axes in a face ply, the X axis being taken to be in the direction
of the grain and the Y axis perllndicular
to it.
After these constants
have been deterrnined in,this way, the axes are rotated so that the Xaxis is perpendicular
to the edges of the plate.
In this report the face
grain is taken to rnake an angle of 45" with the edges of the plate when
it is not parallel or perpendicular
to the edges. As a consequence of
the rotation of the axes through an angle of 45", the following new constants, which are a cornbination of Er, Ez, and A., are introduced:
El * Ez+ ZA
G-
4
R=9(Er*Ez)-A
M=Er-E,
The new constants can be readily calculated
the plywood in the plate is known.
Report No. l316
-26_
as soon as the structure
of
.3-43
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Report No. 1116
Table 4.--Cornparlson of values of k.
ned. from
rYpe
z0
in formula
(rI)
s eurves 1n
tk"rk"
.a
:
:
:
----: -------:
l-ply (r:r:r)
l-pLy(r:r:r)
5-ply (r:e:r)
:
0
2,067
:
: goo
0.540
:
:
0
2
:
:
z.Lt+5 :
:
goo
5-pLy (t:t:1:l:1)
:
:
o
Report No. U15
o.6oo
2.004
go"
:
0.894
2.LI+z :
1.988
I.2Ol+
I.117
o.957
:
:
5-ply (t: z:2z2zl)
L.915
:
:
5-ply (t:z:2z2zl)
(Seytlel)
:
:
l-ply (r:z:r)
5-pLy (t:r:1:1:1)
(Enerry
nethocl)
-----------:
0
go"
r.972
:
l.Brb
:
r.665 z
L.5\9
Table
or
conprassion
For oompresslon
of long plJrrood platee under conbined
!.--Buck]lng
tension
and shear.
Edgss s1mDly supported.
= k E- h2/a2.
p
N€aatlve
= k-F-, h2/c2.
q^_
cr
sL
oqual
to
f
For
times
ttB
For
alenotes tenslon.
k
comblned
stress
the
(or
t€nslle)
oonpresslve
shear
shear
ls
stress
stress.
I
7ks 72p1y 3 Jply
: Jpry
:
: ke :
: 1:2;I
:1:1:I:1:I:1:2:2:2:l
5pry
Sply
Af
Loadlng
1:l
3 1:1:I
thear
Conpreeeloa
and shear
:
3
o
o
3:::3.
"
;
o
3
rr
:
O
:
Tenslonanat'
shear
:
n
.
:
"
"
.66:
.66,
.lt:
.17,
: r.5,ro,
.lti
.eai
:
.553
.89,
o :
3
:
i
:ks:
"3
tk"t
:4.O3kc:
.16:
.lg,
-e.go, -e.n,
-6.90r -6J9.
-.98, -J.JJ,
:1.!:k.,
:
:k':-I'4?''4'99t
;
o
:
3- - - - - - - - -
OO
;
90
,r",-r.oei
..,....,k",
:......r ks:
-.1i
:
.s6
.66
.2,*
I.12
.j9
r.57
-t6,95
-6.47
-S.go,
-5.9o:
-+.Zo
-4.?O
-2.9J:
'4'392
-2.1+O
-J'6r
-.71 '.
-.62
.90,
.64:
.96,
.99 t
1.24:
,58,
.29 :
.75,
.Js:
.862
,Et .
I.OO
,io
,+2'.r+2;
-I' .
.57
.57:
. 6/ - 7 'a.
.61:
.6J
.6j
.112
.rz:
.u,
.rL,
: k8 :
.55 .
.48 :
.67 ,
.60 :
.68
r.02
l+.0 3 k" 3
.16 :
.14 :
.?r z
.26 z
.t4
.60,
.89 r
l.ll
L.67
3- - -
.!:k":
: o" :
:
:
3
r . .nu I. uk o
,o",
.
,
r
.
I.!3ks:
:
:
3
:
.54t
.zl t.
:
. )+r rb
.tr;i
!
r.-
:
9o
: r
:
3
t
F
?..
^
^t. 3
,^
-6.16
-z.g+
-l.rg
; .5 ; kc :
i
;
i
1
.
4
7
,
2
.
&
,
3
iks.-J.05,
; r.o j to I -r.s1 : -r.oo , -r.e>'.
;-i:-:-__:11_
:____l:11_
;____l:13__
;___
_'_2!_',
__-_:1_
-7.29 '. -L2.27
I
3k":-r.8)3
: 1.! : k" : -.98 ,
3
:ks:-1.4?:
:3:3!:3
9o
.5L
-z.ge, -z.gz, -2.s4, -z.as
------- : -----
;
9o
.41,
r.or
.74,
3
:
:
4.0 : ko : -.27 ,
3ko3_l.OB,
:3:3:::
no. IJ16
Itro.
z v \ltzt F
-.74 '.
-1.LJ,
-4'69'
90
:
!
n:
4.o ' ko '. -.zt :
,
90
.tl,
I.093
- z t . s gt
-ro.g[,
;
90
.64:
.64,
- z, -er.rt .+g: i '-,z l . g l .
. -j '; .-r;o. - j .i o -s6, .- 1r j6. ;o 7
:..1+5|
O
n
.40,
.47 '.
t.61 '.
.71+:
,
:
;
.ee,
1.0J:
. r o,
r.59 '.
: t" :
o'
C o r n p r e o s iIo n 9 0
and ehear
:
aad
. 4' 1
| :
.45,
:
3 l.o:k.:
i r.o,t",q.gli
3
:k6:-I.8lr
C o n p r e s e l o n.
z
Teuelon
sheer
.\i '.
o
Shoar
. s 6I
.J6 '.
:
;
n
.rr',
.27 '.
:
:t
:
.rri
: r" :
:
3
"
, o.ti t.i
-I.00,
-1.65,
-z.zzi
-6.tJ
-1.>"
-I.55t
-2.222
-J.52
'.57 ,
-.853
-.92 t
-I.tg 3
-I.22 |
-I.82t
-1.65
-2.79
-.I8 :
_.lZ r
-.26 z
_1.10:
-.15 |
-I.4I,
'.5I
-Z.Ot
Table 5.--8u9kl1+g
of long plFoocl plates under conblned
tenglon and sheer.
Edgea gluply sttporbeai.
. k E- h2/a2.
p
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FIG. I
SURFACEOFA RECTANGULAR
PLYWOOD
PLATE ,HOW4N6
DtMENst2il, 0F 5/0E5 AND CHOlCEOF AXE5.
FIG. Z
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z rI 41112 F
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PRINCIPALDIRECTIONS
IN WOOOAIIO PLYWOOD
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F/6.4
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BUCKLINA UNOER U^IIFOPMCOMPPESSIVESTRESS
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LO\V6
PLATE
BUnKLTN;
UIDERA UNFIRM sHtARlNc JrRFis . styDEL's
cupvEs (rrc. a) wERE usEDttl clLCULATIN6ks.
zx4lugr
EXPLANATToN
0F FrcuREsu rc 25
lhe folrorring
figures show associated values of k" and k",
the constants in equa.tions (15) ana 4o), tor the buckling of rong
rectangular.plates
of prywood under combined,uniforn confression(or tension) and shear. I?re edges were taken to be sinpry supported.
lhe constants for rotary-cut Dougras-fir plyrood were used in the
celculations of the coordinates of the points on these curves.
Grrves showing the ratio of the harf wave rength b to the width
a and the slope Y of the wrlnkles as ftrnctions of ks are
plotted to the left in each flgure.
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