|Table of Contents|
Front Cover 1
Front Cover 2
IFCMULAS IfCI [FLAT PLAITS COf
PLYWUOOU UNIDU UNIFOCM Cu
Information IPc i cd and IPcaffirmcd
INFORiM.ATION REVIEWED AND
REAFFIRMED JUNE 1959
DATE OF ORIGINAL REPORT
FEBRUARY 13, 1942
ro E5 -
S ---- ~ No. 1300
UNITED STATES DEPARTMENT OF AGRICULTURE
FOREST PRODUCTS LABORATORY
Madison 5, Wisconsin
In Cooperation with the University of Wisconsin
A basic study of plywood that is under way at the Porcjt
Products Lboh:-.tory has included as one p.ase the mat iematical
analysis of the deflection of flat plates under uniformly dis-
tributed or 'cnccritrated loads. This theoretical analysis has
progressed sufficiently to permit the publication of the fornmlas
presented in this mimeograph. Some of these formulas have been
checked against test results, and the others are believed to
afford reasonably accurate results.
Other phases of the study of plywood relate to basic
strength in compression, tension, bending, and shear; resistance
to combined stress; criterion for buckling in flat and curved
plates and shells and behavior after -uc- ling; and methods of re-
inforcing. It is planned tl-at as rapidly as significant results
become available, they will be presented in this series of reports.
Fore t Products Lcboratory
Digitized by the Internet Archive
STWMEY 0;W' 7OR.ITrLAS FOR FLAT :'L=T, OF FPLYWTOOD
LDER UNIFORII OR C CUf?'..ThD LOADS
I. 1,. MARCH
Special Consu.tling !,thcr.atician, 2
Forest Products Laborrt.-ry, Forest Service-
U. S. Department of Agriculture
The material herewith presented comprises a summary of the
principal results of a more extensive report soon to be issued by the
Forest Products Laboratory. Reference srr&c5, be made to the extensive
report for the derivation and discussion of the results contained in
Rcct.-ngular plywood plates will be considered in which the direc-
tions of the grain of the wood in adjacent plies are mutually perpendicular,
and per-)endicular or parallel to the edges of the plate. The plies are
assumed to be either flat grain or edge grain. The choice of axes is shown
in figure 1.
The effect of the glue other than that of securing adherence of
adja-,ent plies is assumed to be negligible. Consequently, the formulas
and methods of this summary are not intended to apply directly to partially
or completely impregnated plywood or compregnated wood, although it is to
be expected that many of the results of the extensive report apply to such
a = width of plate
b = length of plate
h = thickness of pic .e
ro = def.lection at center of plate
p = load per unit area
P = concentrated load (section 5)
P = pa4/ELh4 (sections 6 and 7)
W = deflection at center of an infinitely long plate
c width a under a specified type of loi.d
B = b where E1 and E2 are defined in
2 /section 1___ _
Ihis mimeograph is one of a series of progress reports issued by the Fo. zt
Prof.ucts Laboratory to aid the !ation's defense effort. Results hcre re-
porLed are preliminary and may be revised as additional dat- become available.
2Mainaincd at I adison, :s, in c. re:.-ation with tt-. University of .h'isconsin.
Agricult re -I-bdison
Report liNo 1300
1. Stiffness in bending of strips of plywood
Consider a strip of plywood with its edges either parallel or
perpendicular to the grain of the face plies and denote by x the direc-
tion parallel to the length of the strip. The stiffness of the strip is
determined by a nmodculus E, defined by the equation
1 (E i
where the summation is extended over all of the plies numbered, for example,
as in figure 2: (Ex)i is the Young's modulus of the ith ply measured in a
direction parallel to the length of the strip; Ii is the moment of inertia,
with res.:ect to the neutral axis, of the area of the cross section of the ith
ply made by a plane perpendicular to the length of the strip; and I is the
moment of inertia of the entire cross section of the strip with respect to
its central line, that is, I = h3/12 for a strip of unit width. An approxi-
mate formula in which the error is very slight is obtained for ElI by taking
the sum of the products (Ex)iIi formed for only those plies in which the
grain is parallel to the length of the strip. Exception is to be made of a
three-ply strip having the grain of the face plies perpendicular to the
length of the strip.
In the case of a rectangular plate with sides a and b, the modulus
E uld determine the stiffness of a strip cut from the plate with its
edges parallel to the side a as in figure 1. The modulus E2 similarly
determines the stiffness of strips parallel to the side b.
As in the case of E' the calculation of E2 can be based on the
parallel plies only, except in the case of a thr-ee-ply strip having the
grai-. of the face plies perpendicular to the length of the strip.
2, Young's modulus of a strip of plywood in tension or compression
As in section 1, consider a strip of plywood whose edges are
pararlal to the X-axis or to the side a of the rectangular plate of figure 1.
Tlc mean modulus E. in tension or compression may be defined by
the -quLation -
Eah = (Ex) ihi
where (Ex)i has the same meaning as in section 1, hi is the thickness of
the ith ply and h is the thicknesG of the strip. -
Report No. 1500
In like mra:'-ner for a s *-ip parallel t- ";'he side b
Ebh = (Ey)h
The moduli Ea and E az-< rt-deIdf in ci3se3 "'..r,
addition to Lending stressc-_d ar." c.-. 'e .ej T e mod'ili can be calcuA-tI:
with little error by considc: 22'- oy those plies vh.ch are parallel to the
length of the strip,
3. Rectangular pLate. Un:,.formy o.stributed .5.
Edges simply support d,. c.a=.3 deflectio-ns
The method presented b-lowi may be taken to apply if the loads are
such "hat the deflections do rnot exceed the thickness of the plate.
Appreciable direct stresses develop at deflections of the order of ag-
nituC'e of the thickness of the plate and the deflection will be less tlan
that found by the method presented. For a plate whose length exceedS its
breacdth by a moderate amount the method of section 6 can be applied in
It is assumed throughout that the corners of the plate are held
To find the deflection wo at the center of a given plate of width
a and length b, calculate first the central deflection W of a similarly
loaded very long plate (infinite strip) of width a and of the same con-
w X 0.99 Za (1)
W 32 = o.l547 IL (1)
El E1h3 Elh3
Except for the factor 0.99, which expresses the ..-...te effect (in wood
practically negligible), this is the formula for the central deflection
of a beam of unit width under a uniformly distributed load.
Then the deflection at the center of -.he given plate can be
found approximately from the formula
wo = f W(
where f is a factor to be tV-?. from the curve of fiche 3 corresponding
to the argument
3 b (3)
a a \E *
Repo 'I No, 1500
The points shown in figure 5 were determined by an exact method,
using the elastic constants of spruce, for various types of plywood. The
curve is merely a smooth average curve determined by these points. A
consideration of the extended analysis discloses the fact that the essen-
tial factors determining the central deflection of a plate under the con-
ditions of loading and support of this section are the two moduli E1 and
E2 that enter into the determination of W and B. Variations in other
elastic constants will account for variations of the order of magnitude
shown by the points in figure 3.3 Hence it appears that this curve may
be used for plywood of the type described at the beginning of this sum-
mary, independently of the species of wood used. The constants E1 and E2
must be known. They can be determined by calculation or by static bending
tests of strips of matched material.
The maximum shown in figure 5 in the vicinity of B/a = 2, which
at first sight appears to be impossible, is found in the exact analysis.
It is associated with a wave form of slight amplitude that is assumed
by the deflected surface of a plywood plate.
A presentation of the results of an approximate analysis in
essentially the form (2) was made by C. B. Norris.- Because of the
approximations involved, the deflections calculated from his results are
too small, a fact which he recognized would be the case.
4. Rectangular plate. Unifor.rjly distributed load.
Edges clamped. Small deflections.
In this case the central deflection of the corresponding infinite
strip is given by
w= 0.99 = 0.0309 (4)
32 Elhh Elh3
The deflection of a finite plate can be found from the formula
wo f W (5)
in which f is to be taken from the curve of figure 4 where it is shown as
a function of B/a, B being related to b by (5). The points shown near
the curve in figure 4 are the exact values of f for various types of ply-
wood. The elastic constants of spruce were used, but the curve may be
usd for wood of other species as pointed out in section 3.
3It was convenient to calculate the points shown in some of the figures for
pl^t-s in which the plies are all of the same thickness but the formulas
and curves of this report are not restricted to such plates.
-Hardwood Record, May 1957.
Berort No. 13:0
The actual deflections will usually be considerably larger than
those calculated by (5) because perfect clamping of the edges is rarely
realized in practice. If the edges are restrained from moving inward,
direct stresses will develop at moderate deflections. In this case the
methods of section 7 are available for a plate whose length is moderately
greater than its breadth.
5. Rectangular Tlate. Load concentrated at the center.
Edges simply supported. Small deflections.
The central deflection of the corresponding infinite strip is
S1.051 x 6 x 0.99 jE)(6) Pa
0.8 Tr 2 E(6)
0.252 E) 1 a p (7)
The central deflection of a finite )plate can be found from the
wo = f W (8)
where f is to be taken from the curve of figure 5. In formula (6) the
number 0.8 is an approximate figure for a constant whose values may rangr-e
from 0.76 to 0.86, for various types of ordinary plywood. A means of
calculating this constant is to be found in the extended report.
6. Infinite str..p. (Long, narrow, rectangular plate).
Uniformly distributed load. Edges simply supported.
The formulas of this and the next section are applicable when the
maximum deflection is small in comparison with the width of the strip,
although possibly equal to several times the thickness of the strip.
In addition to the conditions stated in the heading, the edges are
assumed to be restrained from moving inward.
Using the notation
p I2 (9)
the following approximate relation connects the load and the central
P = H ( W h (10)
Report No. 1300
H = 6.46 El/EL (11)
K = -.8 E (12)
where 1 denotes Young's mcdulus in the direction of the grain of the
wood., T the plywood is made of wood of mcre tanr. one species, equation
(10) can be multiplied through by EL. There results a relation in which
only the aod;AlJ E1 and Ea enter.,
t..;.< mean direct stress is given by the formula
h )2 W 2
g = 2.60 E ( ~ ( ) (13)
This is the mean direct stress averaged over the thickness of the plate.
The direct stress in any ply can be calculated by observing that the
str=-;s in any ply is proportional to the E of that ply in a direction
parallel to the X-axis. This follows from the fact that the strain
associated with the direct stress is constant across the thickness of
Th' maximum bErlur-- stress in a face ; ly -an be calculated by
the approximate formula
s = 1.01 a E, a (14)
x a E
where a is to be taken from the curve of figure 6. In this figure the
argument n is connected with the deflU-tion hw the formula
i =2.778 f--/2 (15)
The bending stress at any point in any other ply can be calculated
by ,-ioting tbut the associa-cd strain varies linearly with the distance
from the "Iutral plane ar. that the corresponding stress is the Tprod1ct of
this strain ;rd Ex at the point under -'cr Ideration. The formulas 'ust
given can be used for a plate whose length exceeds its breadth by a
modercte amcunt and for small as well as large de'ILe. tions. An inspection
of figure 5 indicates that these formulas can be used with small error if
B b '
a b E2 is greater than 1.75. It is to be expected that the
Report -1. 15CO
stresses clIculated in this way will be -atisfactory approximations to
the stresses in the central oron of E..ch. a plate.
7. Infinite strip. ( '
Uniformly distributed load. Lde.5 i, ed.
The edges are further assued to be restrained from moving
Using the notation
the following approximate equation holds
P = (h) +K () (16)
H = 32.3 E (17)
K 23.2 E (18)
Equation (16) may be multiplied thrc-_gh by EL and thus be made applicable
if the plywood is made of wood c: two or more different sLecies. LSee
discussion following (10) .
The mean direct stress is obtained from the approximate formula
g : 2.51 E ( 2 ) 2 (19)
and the -naximum beading stress in a face ply from the approximate
S 1.01 a E ( a ) ( ) (20)
x a h
Report No. 1300
where a is to be taken from the curve of figure 7. In this curve, the
argument 'n is connected with the deflection by the formula
1 2.732 (-y w
An inspection of figure 4 justifies the conclusion that the
formulas just written can be used to calculate approximately the central
deflection and the stresses in the central portion of a plate for which
( E1 /4
is greater than 1.5.
RcT-ort No. 1300
FOR A RECTANGULAR PLATE
SECTION OF A PLYWOOD STRIP
SHOWING NUMBERING OF PLIES
Z l 397oi F
S 0.6 -
F,4CTOR f[FORMULA (2) ] AS A FUNCTION OF B/5a, WHERE B=b(E,1/E%.
UNIFORM LOAD. EDGES SIMPLY SUPPORTED.
Z U 39762 F
/ 2 3 4 5 6 7
5/a, 4 FUNCTION OF THE DIMENSIONS OF THE PL,4TE
- 0.5- -
ALL PLIES SftYME THICKNESS
O I -- -----------
0 / 2 3 4 5
8/a, 4 FUNCTION OF THE DIMENSIONS OF THE PLATE
FACTOR f [FORMULA (5) ] 4S 4 FUNCTION OF B/a,
WHERE B = b (E,/E2). UNIFORM LOAD. EDGES CLAMPED.
Z h 39753 F
0 }ALL PLIES SAME THICKNESS.
0 0-FACE PLIES ONE-H4ALF AS THICK
/ AS REMAINING PLIES.
O I I -----------
0 I Z 3 4 5
b/a, f FUNCTION OF THE DIMENSIONS OF THE PLATE
FACTOR f/[FORMULA (8) ] AS A FUNCTION OF 8/a, WHERE
B=b(E,/E2). CONCENTRATED LOAD. EDGES SIMPLY SUPPORTED.
Z ) 47o4 F
n -------f*-A-- --U ----- 0 -----
0 2 4 6 8 10
VALUE OF ;7
THE COEFFICIENT d IN THE FORMULA (14)
4S A FUNCTION OF y
4 0 --------
0 -4 6 8 /0
THE COEFFICIENT a IN THE FORMULA (20)
AS A FUNCTION OF 7
Z 1 39765 F
UIJIVERSITY OF FlI OROIDA
3 1262 08866 5913