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SUMMAIAY OC IFCIMULAS II= FIlAT PILATEIS
OUf PLYWCOII UNiEl UNIFUIM Ci
TuliS r1OIT IS ONE Or A SEr1IES ISSUE)
TO AlL Tll NATION'S WAIR IPIOGIRAM
L- t LL i;.,,'i,
UNITED STATES DEPARTMENT OF AGRICULTURE
FOREST PRODUCTS LABORATORY
In Cooperation with the University of Wisconsin
A basic study of plywood that is under way at the Forest
Products Laboratory has included as one phase the mathematical anal-
ysis of the deflection of flat plates under uniformly distributed
or concentrated loads. This theoretical analysis has progressed
sufficiently to permit the publication of the formulas presented
in this mirieofrar'h. 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 buckling; and methods of re-
inforcing. It is planned that as rapidly as significant results
become available, they will be presented in this series of reports.
Forest Products Laboratory
ST-'I.!'.Y OF FORMJLAS FCT. FLAT PLATES OF PLYWOOD
tDER TYIFORT OR CONC::rRA:ED LOADSI
Special Consulting Mathematician,
Forest Products Laboratory, Forest Service,
U. S. Department of Agriculture
Tr.e material herewith presented comprises a summary of the
principal results of a more extensive report soon to be issue, by the
Forest Products Laboratdry. Reference should be made to the extensive
report for the derivation and discussion of the results contained in
Rectan-ular olywood plates will be considered in which the
directions of the grain of the wood in adjacent plies are mutually
perpendicular, and perpendicular or parallel to the edg:e of the plate.
The plies are ac-u-ed to be either flat -rain or eodge rain. The
choice of axes in Thov.wn in figure 1.
The effect of the glue other than that of securin, adherence
of adjacent plies is assumed to be negligible. Consequrntl>, the
formulas and methods of this summary are not intended to -nn ly directly
to partially or completely impregnated plywood or coimpregnatel wooL,
although it is t- be expected that many of the results of the extn-
sive report apply to such material.
a = width of plate
b = length of plate
h = thickness of plate
wo = deflection at center of plate
p = load per unit area
P = concentrated load (section 5)
P = pa/!F,- (sections 6 and 7)
W = deflection at center of an infinitely long plate
of width a under a specified tnpe of load
B = ) where EI Pnti E3, are Jefined in
1. Stiffness in bending of strips
Consider a strip of plywood with its edges either parallel or
perpendicular to the gr?-irn of the face plies and denote by x thr iir,/c-
tion parallel to the lenrIth of the strip. The stiffness of the strip is
'This mimeo-r-r.,':-. is one of a series of progress reports issued by t.e
Forest Products Laboratory to aid the Nation's defense effort. Results
here reported are preliminary and may be revised as additional data be-
Mimeo. No. 1300
determined by a modulus E, defined by the equation
Bj I = S^W i
where the summation is extended over all of the lies numbere-1, for ex-Jle,
as in figure 2; (F9)i is the Young's modulus of the ith liy mesured in a
direction parallel to the ler.n-th of the strio; Ii is the momnnt of inertia,
with respect to the neutral axis, of the area of thb cross section of the ith
rI m'de by a lane perprendicular to the length of the string; and I is the
moment of inertia of the entire cross section of the strip wvith respect to
Its central line, that is, I = h9/12 for a strip of unit width. An orproxi-
mate formula in which the error is very sliht is obtained for E I : t-kin!
h s'um of the products (E ).I. formed for only those plies in 4hich the
;rain is parallel to the length of the strip. Exception is to be made of a
txree-ply strip having the grain of the face plies perpendicular to the
lergth of the strip.
In the case of a rectangular plate with sides a and b, the
modulus E1 would 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 defined, namely,
determines the stiffness of strips parallel to the side b.
As in the case of E1 the c-alculation of E2 can be based on the
parallel plies only, except in the case of a three-ply strip having the
grain of the face plies perpendicular to the length of the strip.
2. Young's modulus of a strip of
plyn'ood in tension or compression
As in section 1, consider a strip of lywood 'hose ,-d-es are
parallel to the X-axis or to the side a of the rectangular plate of
The mean modulus V in tension or compression may be defined
by the equation
-1 = -(Ex)ihi
E~h = _
where (2,)i has the same -naning as in section 1, hi is the thickness of
the ith ply and h is the thickness of the strip.
In like manner for a strip parallel to the side b
Ebh = (Ey)ihi
The moduli Ea nrid Eb are needed in cases where the deflections
of the plates under consideration are so large that direct stresses in
addition to hendinm stresses are developed. These moduli can be calculated
'"ith little error by considering only those plies which are parallel to
the length of the strip.
3. Rectangular plite. UTniformly
distributed load. Ed.es simply
supr-ort. i. S7-;i! d'1 1 1o.,
The method presented below may be t<:-n to apply if the lo',d3
are such that the deflections do not exceed the thickness of the -late.
Appreciable direct stresses develop at deflections of tne order of mar-
nitude of the thickness of the plate and the deflection will be less than
that found by the method presented. For a plate -hose len,th exceeds its
breadth 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 riven plate of
width a and length b, calculate first the central deflection 7 of a sim-
ilarly loaded very long late (infinite strip) of width a and of thr.
same construction. Now
W=5x 0.99 pa =0.1l547 ()
3 E2 Elh3
Except for the factor 0.99, whichh expresses the plat,- effect (in woo1
practically negligible), this is the formula for the central deflection
of a b, r,-, of unit with inlrr a uniformly distribut e loa.
Then the deflection at the center of the given plate can be
found approximately from the formula
o = f -7 (2)
where f is a factor to be taken from tihe curve of fi.,ure 3 corresponiin7
to the argument
B b l El l1/4
a a'. ^ / 3
The points shown in figure 3 were determined by an exact ,nthod,
using, the elastic constants of spruce, for various types of plywood. Th.
curve is merely a smooth average curve determined by those points. A
consideration of the extended analysis discloses the fact that the essen-
tial factors detr-rminin.g the central deflection of a plate under the con-
ditions of loading and support of this section are the two moduli E1 and
ET 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 3.2- Hence it appears that this curve may
be used for plyTrood of the type described at the beginning of this sum-
mary, independently of the species of wood used. Th- constants E1 and E,
must be known. Tr-'y can be determined by calculation or by static bend-
ing tests of strips of matched material.
The maximum shown in figure 3 in the vicinity of B/a = 2, 7hich
at first sight arpears 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.3 Because of the
approximations involved, the deflections calculated from his results are
too small, a fact vhich he recognized would be the case.
4. Rectangular plate. Uniformly
distributed load Edges claimed.
In this case the central deflection of the corresrondir:--
infinite strip is given by
-0.99 pa = 0.0309 pa (4)
32 Elh3 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 (3). Th- points shown
near the curve in figure 4 are the exact values of f for various types
of plywood. The elastic constants of spruce were used, but the curve
may be used for wood of other species as point'cd out in section 3.
The actual deflections will usually be considerably lar.- r 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 lerqth is moderately
greater than its breadth.
2It was convenient to calculate the points shown in some of the fi-urcs
*for plates in which the plies are all of the same thickness but the
formulas and curves of this report are not restricted to such plates.
3--ardwood Tecord, May 1937.
5. Rectangular plate. Load concentrated
at the center. Edges simply supported.
The central deflection of the corresponding infinite strip is
w = 1.051 x 6 x 0.99_E \ Pa (1)
0 .S Tr3 \^^ El-
0 .2521 _
E, \ ) 1E1h3
The central deflection of a finite plate can be found from
w f 7 (M)
where f is to be taken from the curve of fi.'ure 5. In formula (6) the
number '0.8 is an approximate figure for a constant Those values -.a" rang-e
from 0.76 to O.g6, for various types of ordinary ol.v-yood. A measns of
calculating this constant is to be found in the extended retort.
6. Infinite strip. (Long, narrow,
rectangular plate). Uniformly
distributed load. Edges simply
Th: formulas of this and the next section are applicable ,rhen
the maximum deflection is small in comparison "ith the v'idth of the
strip, although possibly equal to several ties the thi.'kness of the
In addition to the conditions stat.-' in tht heifin., the c-es
are assumed to be restrained from moving in,, ar..
UTsin7 the notation
the following approximate relation connects the load and the central
P H )+ K 3 (10)
H = 6.46 El/EL (11)
K = 20.8 Ea/EL (12)
'here Z_, denotes Young's modulus in the direction of the rain of the
wood. If the plywood is made of 77ood of more than one species, equation
(10, can be multiplied through by EL. There results a relation in -'hich
only the moduli 31 and 'Ea enter.
The mean direct stress is given by the formula
g = 2. 0 1(]h 2
a, 1)2 (13)
This is the mean direct stress averaged over the thickness of the plate.
The direct stress in tny nly can be calculated by observing that the
stress in any ply is proportional to the 3 of that ply in a direction
parallel to the X-axis. This follows from the fact tnat the strain
associated with the direct stress is constant across the thickness of
The maximum bending stress in a face ely can be calculated by
the approximate formula
s'= 1.01 a( EB i ( U)
% a) h (l'i-)
where a is to be taken from the curve of figure 6. In this figure the
argu-i..mnt Th is connected with the deflection by the formula
= 2.77,9 E_ l/2
The bending stress at any point in any' other rlv can bo cl-
culted by noting that the associated s'trair. varies linearly V:itn the
distance from the neutral plane and that the corresronling stress is
the prolict of this strain and Ex at the roint under consideration.
The forrul-s just given can bo used for a plate whose length exceeds
its breadth by a moderate amount and for sr-.ll as well as large de-
flections. An inspection of figure 3 indicates that these formulas
11 e 1/4
can bhe used with small error if b 1 -.-roater than
1.75. It is to be expected that the stresses calculated in this way
will be satisfactory approximations to the stresses in the central T'or-
tion of such a plate.
7. Infinite strip. (Long, narrow plate).
Uniformly distributed load. Edges
cl amLp e d
The edges are further assumed to be restrained from moving
Using the notation
the following approximate equation holds
P H +J K 3()
H = 32.3 EL (17)
K 32-a- (17)
K 2 23. 2 --(
Equation (1i) may be multiplied through by EL and thus be made arnlii-
cable if the plywood is made of wood of two or more different srcies.
(See discussion following (10)].
The mean direct stress is obtained from the approximate
g= 2.51 a (a (19)
and the maximum bending stress in a face -ply from the aprroximate
S 1.o1 a E (^ ( (1Y0)
where a is to be taken from the curve of figure 7. In this curve, the
argument T is connected with the deflection by the formula
h 2732 ) 1/2 (21)
An inspection of figure 4 justifies the conclusion that the
formulas just written can be used to calculate approximately the Cantral
deflection and the stresses in the central portion of a plate for which
B b 1 1/74
a a (2) is-greater than 1.5.
SIDE AND AXES DESIGNATIONS
FOR A RECTANGULAR PLATE
SECTION OF A,4 PLYWOOD STRIP
SHOWING NUIMBERING OF PLIES
ALL PLIES SAME 7-THICKtVESS
O.Z --- ----------------
0 I 2 3 4 5 6
5/a, 4 FU/NCT/ON OF THE DIMENSIONS OF THE PLA7
FAqCTOR f [FORMULA (2) ] ,4S 4 FUNCTION OF B/a, WHERE
UNIFORM LOAD. EDGES SIMPLY SUPPORTED.
2 UL 1 ,*. F
B S=b (E,/E,)
,4ALL PLIES SUY7IE THICKNESS
0.4 0-3 X
0 I Z 3 4 5
6/a, A FUNCTION OF THE DIMENSIONS OF TIHE PLATE
FACTOR f [FORMULA (5) ]JS FUNCTION OF 5/a,
WHERE B =b (E,/E2). UNIFORM LOAD. EDGES CLAMPED.
Z & 397o3 7
t I -T
0.41 i .---
A} ALL PLIES SAME THICKNESS.
*-FACE PLIES ONE-H,4LF 4S THICK
AS REMAINING PLIES.
0" v I IJ IJ
0 I Z 3 4 5
B/a, A FUNCTION OF THE DIMENSIONS OF THE PLATE
FACTOR f [FORMULA (6) ] AS FUNCTION OF 8/a, WHERE
B=b(E,/E). CONCENTRATED LOAD. EDGES SIMPLY SUPPORTED.
Z ': ". 17o4 F
Al I- I
0 2 4 6 8 10
THE COEFFIC/ENVT a IN THE FORMULA (/4)
,S A FUNCTION OF '7
0 2 4 6 8 I0
VALUE OF y7
THE COEFFICIENT a IN THE FORMiULA (20)
AS A FUNCTION OF 77
z 5)7o5 F
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