﻿ Summary of formulas for flat plates of plywood under uniform or concentrated loads

# Summary of formulas for flat plates of plywood under uniform or concentrated loads

## Material Information

Title:
Summary of formulas for flat plates of plywood under uniform or concentrated loads
Physical Description:
Book
Creator:
March, Herman W ( Herman William ), b. 1878
Forest Products Laboratory (U.S.)
University of Wisconsin
Publisher:
U.S. Dept. of Agriculture, Forest Service, Forest Products Laboratory ( Madison, Wis )
Publication Date:

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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 29509035
oclc - 757539080
System ID:
AA00020681:00001

Front Cover
Front Cover
Foreword
Foreword
Main body
Page 1
Page 2
Page 3
Page 4
Page 5
Page 6
Page 7
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Full Text

SUMMAIAY OC IFCIMULAS II= FIlAT PILATEIS

OUf PLYWCOII UNiEl UNIFUIM Ci

PJ!

11 *.

October 1941

2

4,
4 r441

Nj

~'

TuliS r1OIT IS ONE Or A SEr1IES ISSUE)
TO AlL Tll NATION'S WAIR IPIOGIRAM

.No. 1300
L- t LL i;.,,'i,

UNITED STATES DEPARTMENT OF AGRICULTURE
FOREST SERVICE
FOREST PRODUCTS LABORATORY
In Cooperation with the University of Wisconsin

oftEpST E
f *SJ

Foreword

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

3y
W. IARCH
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
this summary.

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.

Notation
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
F,"' 1,/4
B = ) where EI Pnti E3, are Jefined in
section 1

1. Stiffness in bending of strips
of olywood

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-
come available.
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,

B21

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
figure 1.

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

-2-

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
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
this case.

It is assumed throughout that the corners of the plate are held
down.

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

1300

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
Small deflections

In this case the central deflection of the corresrondir:--
infinite strip is given by
4 4
-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

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.

130o

at the center. Edges simply supported.
Small deflections

The central deflection of the corresponding infinite strip is
given by
1/4
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
the formula

w f 7 (M)
o

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
supported

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
strip.

In addition to the conditions stat.-' in tht heifin., the c-es
are assumed to be restrained from moving in,, ar..

UTsin7 the notation

340 (9)

1300 -

the following approximate relation connects the load and the central
deflection W.

P H )+ K 3 (10)

'where

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 plate.

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
a akv)

-6-

1300

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).
cl amLp e d

The edges are further assumed to be restrained from moving
inward.

Using the notation

Spa

Lh

the following approximate equation holds

P H +J K 3()

where

H = 32.3 EL (17)
K 32-a- (17)
E
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
formula

g= 2.51 a (a (19)

and the maximum bending stress in a face -ply from the aprroximate
formula

S 1.o1 a E (^ ( (1Y0)

-7-
1300

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.

l3c,,

a
0 X

FIG. I
SIDE AND AXES DESIGNATIONS
FOR A RECTANGULAR PLATE

4

FIG. z
SECTION OF A,4 PLYWOOD STRIP
SHOWING NUIMBERING OF PLIES

0.9

0.8

0.7 i

% 0.6

0.5
LEGEND:
EXACT METHOD
ALL PLIES SAME 7-THICKtVESS
0.4- 0-3X
0-3Y
tA-5X
A-5Y
0.3

0.2
O.Z --- ----------------

o./ !

0.1
0 I 2 3 4 5 6
5/a, 4 FU/NCT/ON OF THE DIMENSIONS OF THE PLA7

FIG. 3
FAqCTOR f [FORMULA (2) ] ,4S 4 FUNCTION OF B/a, WHERE
2 UL 1 ,*. F

7 8
'E

B S=b (E,/E,)

t U.b

LQ<
0.5-
LEGEND:
EXACT METHOD
,4ALL PLIES SUY7IE THICKNESS
0.4 0-3 X
9-3Y

A-5y
0.3

0.2 -

0.I

0
0 I Z 3 4 5
6/a, A FUNCTION OF THE DIMENSIONS OF TIHE PLATE

FIG. 4
FACTOR f [FORMULA (5) ]JS FUNCTION OF 5/a,
WHERE B =b (E,/E2). UNIFORM LOAD. EDGES CLAMPED.
Z & 397o3 7

t I -T

a
.4->

0.41 i .---

Io --

LEGEND:
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

FIG. 5
FACTOR f [FORMULA (6) ] AS FUNCTION OF 8/a, WHERE
B=b(E,/E). CONCENTRATED LOAD. EDGES SIMPLY SUPPORTED.
Z ': ". 17o4 F

0.8

0.7

S0.6

K ^
f ?

Al I- I

LL

0 2 4 6 8 10
VALUE OF

FIG. 6
THE COEFFIC/ENVT a IN THE FORMULA (/4)
,S A FUNCTION OF '7

50------

4 30

S/0

0

0 2 4 6 8 I0
VALUE OF y7

FIG. 7
THE COEFFICIENT a IN THE FORMiULA (20)
AS A FUNCTION OF 77
z 5)7o5 F

UNIVERSITY OF FLORIDA

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