deflection and stresses in a uniformly loaded, simply supported

•

1111111,

DEFLECTION AND STRESSES IN A

UNIFORMLY LOADED, SIMPLY SUPPORTED,41.1

RECTANGULAR SANDWICIl PLATE

Original report dated December 1955

Information Reviewed and Reaffirmed

September 1962

LOAN COPYNo. 1847

PLEASE RETURN TO:

Wood Engineering ResearchForest Products LaboratoryMadison, Wisconsin 53705

This Report is One of a SeriesIssued in Cooperation with theANC-23 PANEL ON SANDWICH CONSTRUCTIONof the Departments of theAIR FORCE, NAVY, AND COMMERCE

FOREST PRODUCTS LABORATORY UNITED STATES DEPARTMENT OF AGRICULTURE

FOREST SERVICEMADISON •5, WISCONSIN

In Cooperaton with the University of Wisconsin

DEFLECTION AND STRESSES IN A UNIFORMLY LOADED,

SIMPLY SUPPORTED, RECTANGULAR SANDWICH PLATE!

By

MILTON E. RAVILLE, Engineer

Forest Products Laboratory, .2 Forest ServiceU. S. Department of Agriculture

Summary

A theoretical solution is presented for the deflection and stresses ina uniformly loaded, simply supported, rectangular sandwich plate.The solution is applicable to sandwich plates having an orthotropiccore of arbitrary thickness and isotropic facings. The facings maybe of equal or unequal thickness. Numerical results and curves areincluded.

Introduction

The purpose of this report is to obtain formulas from which the de-flection and stresses in a uniformly loaded, simply supported, rec-tangular sandwich plate may be computed. The sandwich plate is as-sumed to consist of isotropic facings separated by and bonded to an

!This report is one of a series (ANC-23, Item A-7) prepared and dis-tributed by the Forest Products Laboratory under U. S. NavyBureau of Aeronautics Order No. NAer 01684 and U. S. Air ForceNo. AF-18(600)-102, Amendment A6(55-286). Results here re-ported are preliminary and may be revised as additional data be-come available. Original report dated December 1955.

—Maintained at Madison, Wis. , in cooperation with the University ofWisconsin.

Report No. 1847

orthotropic core. The core is considered to have such a small load-carrying capacity in the plane of the plate as compared to that of thefacings that the normal stresses in the core in the plane of the plateand the shear stresses in the core on planes perpendicular to the fac-ings and in directions parallel to the facings may be neglected. Theanalysis of the facings is based on the usual small deflection theoryof laterally loaded plates.

Notation

x, y, z rectangular coordinates (fig. 1)

a width of sandwich plate

b length of sandwich plate

a

c thickness of core

t 1thickness of upper facing

thickness of lower facingt2

E modulus of elasticity of facings

v Poisson's ratio of facings

modulus of elasticity of core in z directionEc

modulus of rigidity of core in xz planeGxz

modulus of rigidity of core in yz planeyz

q intensity of uniform external lateral loading

crznormal stress in core in z direction

.rxz ,xz yz shear stresses in core

Report No. 1847 -2-

E normal strain in core in z direction

shear strains in core

displacements of core in x, y, and z directions

normal forces and shear force per unit length ofupper facing

normal forces and shear force per unit length oflower facing

bending moments and twisting moment per unitlength of upper facing

bending moments and twisting moment per unitlength of lower facing

Yrs

u , v , wC C C

N , N ,x y N xy

N' N' N'x y xy

M M , Mx y xy

Mx, MY, y

Q,,Qy transverse shear forces per unit length of upperfacing

10' y transverse shear forces per unit length of lower facing

u, v, w displacements of upper facing in x, and z direc-tions, respectively

, v', w' displacements of lower facing in x, E, and z direc-tions, respectively

E x, E y , • y xy normal strains and shear strains in upper facing

normal strains and shear strains in lower facing

integersm, n

Arnn' Bmn Crnn 1'mn' I mn' Kmn Lmn constants

Report No. 1847 -3-

k

Sx

Sy

16qa 4 (1-v2)ir 6 E/

(1+2)

ti t 2 )( t1 + tz)

c+

w z Ect 1 t 2

G a 2 (1-v 2) (t1 + t

2)xz

z Ect 1 t 2

G a 2 (1-v 2 ) (t1 + t

2)yz

Theoretical Analysis

The dimensions of the sandwich plate and the coordinate system usedin the analysis are illustrated in figure 1. The method of analysisconsists of determining expressions for the core displacements thatsatisfy the core equilibrium equations and the boundary conditions.The arbitrary constants that appear in these expressions for the coredisplacements are then evaluated from consideration of the equili-brium of the facings in conjunction with the requirement that the dis-placements of the core and facings be equal at their mutual interfaces.

Equilibrium of the Core

A differential element of the core is shown in figure 2. In accordancewith the assumptions outlined in the Introduction, a. , cr and r

in the core are assumed to be zero. From the Summation of forces inthe x, y, and z directions, respectively, the following three equili-brium equations of the core are obtained:

2

Report No. 1847 -4-

(4)

(5)

(6)

and

„,.

(1)

6Tyz = 0Z

and

6cr 31'

z xz ♦ yz'...-••••••••• = 0

6z 6x 6y

On the basis of Hooke's law, the following stress-strain equations areapplicable:

crz = E ec z

T G y

xz xz xz

and

T = G y

yz yz yz

Also, the strains and displacements are related as follows:

6we z

6u 6w6wcY =

xz 6x

(2)

(3)

6vc 6 wc= — +Yz 6z 6y

Report No. 1847 - 5 -

Equations (4) through (9) enable the equilibrium equations of the core,equations (1), (2), and (3), to be expressed as follows:

w a2 u• = 0 (10)a xaz az2

62 wcv• = 0 (11)a yaz

and

az w az _ 62 u a w 32 vE c G c) G c) . 0 (12)c xz

az' ax2 33caz Yz 6y2 a y az

The expressions for the core displacements are assumed to be of thefollowing form:

uc = (z) cos mirx sin 213- rY1 a(13)

2 (z) sin rn" cosa (14)

and

mint nor3 (z) sin sina

(15)

It is noted that the above expressions satisfy th= e0b,boundary conditionsthat we = 0 at all boundaries and that (M

(u.c)m = 0, and (vc )x=0 = 0. The three functions of z in equations (13),x=a

(14), and (15) are determined, as follows, from the requirement that

Report No. 1847 -6-

these equations satisfy equilibrium equations (10), (11), and (12). Ifequations (13), (14), and (15) are substituted into equations (10), (11),and (12) and it is specified that the resulting equations be valid for allvalues of x and y, the following equations are obtained:

f(z) + 3 (z) = 0

(z) = 0

(16)

(17)

1 a

11 nir f2 b(z) + — 3and

2 2E f3 Gc 3 xz

MIT f iz%11f3 (z) + a 1

2 2 '-Yz

[11 f(z) f2 (z) = 0 (18)3b

where the primes denote derivatives with respect to z. From equations(16) and (17)

f (z) = - 1-111r f3 (z) + Amn1 a 3and

(z) = - mr f (z) + Bmn2 b 3

where Amn and Bmn are constants of integration. The substitution ofthe above values of f 1 (z) and f 2 (z) into equation (18) yields the follow-ing differential equation:

Gyz nnGxz—EcAmn mn + Ec Bmn mn

Integration of the above equation yields:

(19)

(20)

f3 (z)

Report No. 1847 -7-

Gf3 (z) =

2-

ff ( -E

Gxcz Amn ÷c -E Bmn z2 + Cmn + Fmn c (21)

The functions f 1 (z) and f2 (z) can now be determined by substituting the

above value of f3 (z) into equations (19) and (20) and performing the in-

dicated integrations. The results are:

w 2 m2 Gxz 3f (z)=A z- —z -B Tr 2 mn z3

1 mn —6 a2 E

c mn 6 ab Ec

m- C - mc2 a mn a n z Hmn c

and1. 2 [ n2 ri Z Gyz

f (z) = - Amnmn Gxz z 3 + Bmn z2 ab E

c 6 b2 Ec

n nc- 7 I-3- Cum z2 - -F Fmn z + L mn c

The functions of z that appear in equations (13), (14), and (15) havingbeen determined, it is possible by redefining the arbitrary constantsto express the core displacements as follows:

2marrc % A 3u = >2" 4 z%tnn 3 + Bmn-2• +Cmn--)•

m=1 n=1 c c

+ Fmn -- + Hrnn In"

] cos sin mil' (24)ca

Report No. 1847 -8-

V =C nitct • 4 - z 3z2_ A „6„, C

b 3 rnn 3 mn 2 inn c

+ Km • +n Lmn sin niing cos Di-a (25)a

wc

= [4 Amn 7• + Bum

+ Cmn sin nla" sin 111—T1

(26)

The above expressions for the core displacements satisfy the equili-brium equations of the core if

G F + G np Km -8a Axz mn yz n c mn

(27)

where p = t.

Thus it is seen that there are actually only six arbitrary constantspresent in the expressions for the core displacements.

Since, from equations (4) through (9),

6v1cz = EC 6z

alt 6VVT G f C+ cixz X z

6z 6x

Report No. 1847 -9-

GxzT =

XZ=TX

sin Eacosmn a

(29)

and

and

6ArT = G ( C 4. --2)yz yz 31.

the core stresses may be expressed as follows: •

Eccr =z c

'RX(8 AArun + Brnn)

Msin sina (28)

03

T > Kmn sin M1TXcos ril–a

Yz c m= f n=1a

In the analysis which follows it is shown that the displacements andstresses in the facings may be expressed in terms of these same arbi-trary constants and that these constants can be evaluated from consid-eration of the eqttilibrium of the facings.

Equilibrium of the Facings

A differential element of the upper facing of the plate is shown in fig-ure 3; the forces in the plane of the facing are shown in figure 3(a),and the remainder of the forces and the moments are shown in figure3(b). The summation of forces in the x, y, and z directions yields thefollowing three equations, respectively:

aNx

M•x

= - T6x yy

xz

(30)

(31)

Report No. 1847 -10-

(34)

(35)

3N dN

ay axyz

xy _ T

and

aQ3QY = q cr) z = -

)3c 6y

Also, the summation of moments about the x and y axes, respectively,

yields

2,/s/L dM I 1 %xy Q = - T /y yz 2

and, since MYx

= M ,xy

t61vIx 6Mx Qx = - Txz (T)

If equations (34) and (35) are solved for Q and Qx and these values are

substituted into equation (33), the result is:

(32)

(33)

2d Mx

2

2 a

Mxy 6 m

Y = _ q (o`z)z

ax ay 3y2

2

t , dTxz- + -/2 ax 6y

The equilibrium equations of the upper facing are thus reduced to equa-tions (31), (32), and (36).

The equilibrium equations that apply to the lower facing are obtained ina similar manner on the basis of figure 4. The summation of forces inthe x, y, and z directions yields


(36)

6N' 61‘1'x =

6,1t 6y X Z

6N' 6N'

_/ =y

and

aca;c U1'+ = c

6x 6y z 2

The summation of moments around the x and / axes yields

am.' am' t2_ _ 1 = T6y 6x

y yz 2

and, since Me - ,yx y

WVI;c 6111,4

6x 6y QX1 = TXZ ("124(41)

The substitution of the values of Q ' and obtainedd from equations

(40) and (41) into equation (39) results in:

62 MX

6xz

M;cv 622 = a.) c

6x6y 6y2 z = 2-

t 2 )Txz aT yz

?)r(42)

(37)

(38)

(39)

(40)


The forces and moments per Unit length of the facings are related tothe displacements of their respective middle surfaces by the followingequations:

Et_ 2( au +vv

1 av au%N vy 2 dy ax1-v

Et1 I au , av%N =xY 2(1+0 ay ax

auiNx

Et2

Yo-7 v o )-

Et ,N1 = 2 , ( w'r + v aui)

Y 2 ay

1-v •

Et2 ( +

Y 2(14-0 ay 6x

Et

Et 3 2 2 Et 3 2 21 ( a .w +v W ) 2 ( 6 w + v

12(1-v2) ax2

ay2

12(1-10 )2

6y2

3Eti 3 2 Et 2 2M - w + v 6 w) M

I - (1-M- + v )

12(1-v2) ay2 ax 12(1-v2) ay2ax2

Et 1 3a

2w )Myv

12(1+v) axay

Et 3 2

Mr = 2 w 1 )

xy 12(1+v) axay(43)

When the foregoing expressions for the forces and moments in the fac-ings are substituted into the equilibrium equations of the facings, theequilibrium equations of the upper facing become


_ T

xz(44)

Et l a2u 1-v 6

2 u 2v]

) (1 -2v ox2 6

y2 ) ax6y.

Et 1 [622 1 1-v % 62v / 1+v%

Zu6y1-v 6Y i)x

= - TYZ

(45)

and

34Et 1 t 6T 6T1 xz yV w) = q (Tz ) z = - .G. + 2 ( 6x + a

z )

12(1-v 2 ) 2

and the equilibrium equations of the lower facing become

Et2 62u , 1 , Zu" (-1+v ) 313cy

2v,= T ,Lz1-v 2

ax z 6y

Et 2 aZvi.)2 1+v Z 13.

= T yz1.-10 y Z

6x2,l/Z ax.y

andEt

32 4 t 6Txz 6T yz7 (V w ) = ((rz )z + -2- (Th;c12(1-v-) 2

Matching Displacements at the BoundariesBetween the Core and the Facings

The equilibrium equations of the facings, equations (44) through (49),may be expressed in terms of the core displacements by equating theinterface displacements of the facings to the corresponding'interfacedisplacements of the core and then expressing the middle surface die-placements of the facings in terms of the interface displacements. Inso doing, it is assumed that w and w' are constant through the facingthicknesses and that u, u', v, and ;T-vary linearly through the thick-__nesses. Thus,

(46)

(47)

(48)

( 4 9 )


o I

in +..-..... La

0to

iltOa

UINII

N..-..U,

I

0 INII

N

tot)

..4Ual

44

ou N

egu ltg

N 4.444 I

et IA'4,N IN

440110

hN

II

t

II

Iu IN

ION u IN

ill0 u IN U IN

II IIII 0

UN

....0 .-.6 ....'

N

Uof

7214

N

0.•••n0 to

,ta):11

II II II

4.%3

Lel NIn

u..O 0" td 60

44 44/4 4.,O rY00 0

014/ w

14

.g

ed , w

'5

0,-

0 UIN IN 'S 'C:l%

I I.4414 .....

II II 42 XIN N

IXO og 0

tt tI6 eguietr1 ,,r4m4.n

0 lo

•nn••M ...n••ID All

44o-o IN

4.)1 —

o-1 ,, I.0...--

+ +XO to

0UIN UIN U IN V 0

• ZI

1 II

0 td

II IIII 4 0

m 04

N N .....N

oS) 03)

......i j ...-.6

u i.,1 .5O > lit 0 ot

II II II0 41)

O fit .1


gi. ;n ill',...

in tr) Ln ...0in 4 1tio......• 0 cd -O 4.1 . 0k 1.5 5 mn•••n '4

Tr 43

v.... .4 5 1

tr, to 1O N. 44 1O 0 u,.,4 4.:,14 E 0O a) V,

4-'

NN a) vi 2

u)I- I- -0 0

II II 11 (15 11a) +; A

..... I.) IN • ..i 5N u I N II

et,t10 to

t

---,

NN u 3 LI ) ri :, i

cu — .---II

..-.-...I)

. Ncl,to 8

al ...1+ i 1

eoI- k

04N ,U

N /0›, k.

il ,45, 44al

I

+ rt) I

..•

et) W

li•

At M

N+1,g AI tr, Acn 7.i

q .0 4)N

VINNON

I rq NI I N4 tf) 5CO 0 zr 0 0

I I I P..a) A

N Nifip Ai

, V r°

V

t.)

to+

co Ina

dNN

(..)11

b(1) M II

—i..,to t../.2,

%re 1:1 S13 Q

♦ SINSIN N•••••". N

N wil r:10 004 b .1.1 —I

..... ...... 0 4.b

li +L) 1 if

+ ta.) .004

1 N

en to0 U 1 N

N

5. „t

0 I r.,4 II

0 INI,.., g

•ri

0 I N

II 0....... 11 ... 5

.

1 No I

.... N IO

MI

at cli 0

IS I N ..---. N .0440 N..... %or' 1') • r1 • Pi Ch

Ttl + + 14co 4-, 0

rn..--,

AN N U I r,3

/0

c°) 2 0tU

14 0

0. 14 "rd•ri..I

Nen to ...-. K II 0

> I

N

en 0 N 13./ k 0:1)

N.-4 gil .24

I 1%1 N

I1 N>

N 1N1 ":

N.—.

I 14

4%1 8444

rd0- 0ii4 •-n N

1


i ofTI il) 0 IN 01%1cd 0

• 0 .g 114 w...I el 1,-• co

.to • 1-11 in tri1 k pci No. N o.

k • V "1:11.4 0 N NII

kg 0 14O S

44..+

O 1:11N

N

t' a t .. s.o ......•..I 94 iv ......

in 0 1.4 +k 0,0 0 .w3 11 m rd N I IN

44 to (1.1 aa 5....-•”I r..)

•,.• MIk g i----1

g

tv >..pc, tu .,9,

$4 4.I + 5 W .0 0 al 1 4i N14.4 "I:1 Or n••••0 I 6'4O 4) $4 ....... , I-1

IN U

..--..N3 03 rd (I)

X 1 6°1 N4.1

to oi 0 1 g to I (.15 71, it 0 + , t.) + ..-ik a) 1 5 1 {0:3 - I iNi

.—o

re 4-7°PI IN ...r.

4:4(1) k (13 0 N

NitLA44 rn d g rti W

ta 4N3en•r4 0 i

F4 .Wa./ ... M N

'd b.0 ,-' U M U ,.. ka) re)in as C 1.3

4-444 q11

I I

it,go 0o

....I4

...

" I °

44.46

X IMII

cd ..4 4.1 NI. .1 .0)t 1. .4.

ty jO a)tI.) 04 g 4 0."! -1:1

--i I vIl "4 1144.0 V g t.r H I

♦

>. 0 ..-tco

u 0gcd ...4

0 48 cril G3 x.. re] IN

tel

jl

g 1 giI

• Pi CI

4(14 r:14+.1 QJO 14

+ K +i

a 11 .', LI Ali i 1) a. L-TetIa tr) 0

fd ..-, G., 6 + 5 +g Z ;.,al to 11. 44 kr)

0.4 1%0

XI - I 0 t1) 0 IN + INT .!....1 ...,

E (r) 81^ 0II 1:14.1 II g j

..... ......> 'N

44 + ♦...'aih, 1 0 0.-4 t_i_ j 1.0 m 1.n..1

.-4 cd 01 1:4 glA II 0

5511 ay

i.4 id r7- Kr 4

x ..-- .2 col-1...4.1 CD

0. I :^ g ,,r4

INi

....0 #0 41 N

0 14g ,U 44

P'g 4 II '--.. 4-0N

..+ +

N41 0J

C71tn 0(1/ A

E

..• 641: $4 bto

0 t113 11

0al 0En I-'

Ut El Id

c.) 1.8 : -4 '.8 5 til


Z: IIN S' gay IN ....P.i

N

CI. N0.,N

Ng0 N

1 0 +....••-• el

i I N N

1r 4.1

id +cr 41 N

#.1:I N .0 IN

0 cr, o 1__1I___.;:-1p.4

Nli wg

W

+)

.......44...

r.---14--

0 .4.,N311nni

1 u ...N Irtiu+

..4 0 ..4 I N

..... 44

1 .1 N + . N--i1

nt4...i I N ....... .1-a

N • W0

Ncs1

I U

e

m

II

1%1 NN 0 k 0 a...-... ..........i N 0 .......1 + 0 0>

i Utn1 ,_,44 ..—., +

II...... n f—s t4 .N 3

a.....

rnI FLI ......

v 4.1 ›.• M. + .

471 irlcdel0 I=

W ....P.-4114 +

Wu II+ +...4 I nti

+ g0N 0 0.-4

1 0 01 0 IN a INgt.1 + W 1:11 44

•••n +E a

^

0a.

4-. 44 VI

+ g .,_,N I g.I a+U P.4 1.0

g %moo

...I Nfnnn1 0

't IN —1 1 so.t N..I

I 14 1 AP 1%MO

I a 1

ill mai 4.b g

A re'o,------. .-74 • •.14I

a N

4 .4

.7 i

a. gl gr 411

a.+

N + +N 0 No

N+

O. N +N

O 0 Ns

+

NEg IW 0 Ia Ili 10

0


2 12 Ec a4 (1-v2)+ n2 p 2) (Amn + Bmn + Cain)) (4 A

ir 4 E t 23 c

mn

6 Gxz a 3 (1-y

2)

+ 2 Brnn) + m Fmn3E t2 2 c

6 G a3 (1-v2 )yz nP Kmn = ° (62)

Tr3 E t c

A literal solution for the constants Amn , B mn , Cmn , Finn, Kmn , Hmn,and L

mn obtained on the basis of equation 27)and equations (57) through

(62) is very lengthy and contains too many parameters to be of practicalvalue for design purposes. These equations can be simplified enough torender a practical solution possible if certain additional assumptionsare made. The amount of error introduced by making further simplify-ing assumptions can be determined in any particular case by obtaininga numerical solution based on the foregoing general system of equations.

To obtain, the aforementioned simplification. it is assumed that the flex-ural stiffnesses of the individual facings are negligible and that the mod-taus of elasticity of the core in the z direction (E) is infinite. This ad-

ditional assumption in regard to the core results in a core analysis thatis identical with that obtained on the basis of the so-calledmethod commonly used in sandwich analysis. The neglect of the flex-ural stiffnesses of the facings is known to be justifiable for most prac-tical sandwich constructions. As a result of these assumptions, thesystem of equations, equations (27) and (57) through (62), reduces tothe following:

8aG m + np K = — Exz mn yz mn irc c A mn (271)


t.,/ I I Ni4 @

II„Th +

1 u44.....,

•Ila

cd

N›,

eSji

N# N

0 0 >.'

+11

m'II 0

4.1

Z°I E

• N N..—..

Om%

titn

MA IN

+.-4

-,—, q1/4/

.1.1

+

I

j

1,n..

.....,

1 N

gj

.

E,--.

•••••nsi

rilF

n-iiNI...... N

N INQ. ,--1

I

+N

0 + a

+414

Ne Tr

Cd

IMNN

N-4

1 J

4.

^ 1 u

••n••

IN

Na.

N


..... --.... .....a ;LI A-3.0 JJ ...0

Epa

Ur4

Vli03

d

.....0 E

+

+ rf

a0 I r'.1 ..a

0

Wqr4 iII

Ei44 X•

N N 0rci0. ...... a. 0..4 I I

NN i iN 0

I

NA

).

I U 0

..+•n ,4- ."

-.... + ...... saNi X

IN Nrd 41

4.863 N NI W Ct. '`=4

••=r, 0(0 N 1:1NN

t.)0 IN SI.)t 0 I NI

I

.74

E

•

in

L............–.-1 04 t 0 oig (D

Ncl.

.t I 1.4W 0. I N +

I: iN

N13

I No

a cra, como+ + .4

NEN Tr ..4a 4)

..... 1 ta .. 44

o I coc.)

0til cl

NU. 1

a N41

U 4.,•011) 4944l$ RI

h N h t4 re+ r4)4

N to t IN .. 0 8

0 II .... II + H


NU

41" UN

4.i I

u

ra

N

••••I - +

V

N

E

M ‘t L7nCI .0 ..0

f---------4_----s

U)

u,1

NN

E44

:3

Na.

Na

N

N

NE

INN

I

L:LTV

—t

N5

N


0'0'

NIN"14.1N 1NC. 41 41

NE

.4 .1

NNT%

a.

N

N5

+N.....CI

CI.. 0+ t IN S

t..) 1....N

.....LI, i

+ ...,41

I ti...NI

I Tr

t IN I•er +4'4 4.ria.....

ILI

0 I 41 ril 414 N..-s 4

N N

N N0 O.No.

5 0 NI

E 0 NIN

0 0

44 + 0W +

N ,-I N+. 5

1....

U I NI;

E.64 404.11 4+

N N

N NT.. N041

4 I +

N I + it..-4

.0 .4.0.. En

K + N44

1.4co

...4 ...1

4.) .I .4 N

a. 4.n 44

4J 4.) .1.n

rd + ..... vtrs

r. N 1.4 ... .0..

. + .1

fib. i

1-41

.0

NNT%

a.

N

N5

0N

HN

H

rei ....0.-rei

II

0IIII

Report No. 1847 -23-Report No. 1847 -23-

where

16 qa 4 (1-v2)k

EI

t 1 t 2t l t 2

- H ct 1 + t 2

Sy =xz 2 (1- 2 ) (t + t2 )— G a v 1

SGa2 (1-v 2)(t 1 + t2)yz

Lateral Deflection

Since, under the assumptions used in obtaining equations (63) through(69), Amn and Bmn are equal to zero, the expression for the deflection

given by equaticTria6) becomes

w z Ec t 1 t2

w2 E c t l t 2

co

wc = w = w' =>m=1

Cmn sin max sin r114 (70)

m and n are odd

The maximum deflection occurs at the center of the plate, and, with thesubstitution of the expression for Cmn given by equation (65), it may be

expressed as


max0.nm 1 + ( 1-2v ) n2 p 21 sx+

= k 7. (-1) 2m=1 n=1 mrx (m2 + n2 p 2) 2 [1

mz + n2 p .2] s + ( 1-v ) (m22 1 Y 2

7 7 2+ S S

x Y(7 1 )

+(1-v)(m2S _2 p 2 sx)]2

m and n are odd

If the moduli of rigidity of the core are set equal to infinity, S and Sxare zero, and the above solution reduces to the classical Navier solu-tion for the deflection of a homogeneous plate provided that the momentof inertia is taken to be that of the spaced facings of the sandwich plate.

Core Stresses

The expressions for the core stresses are obtained by substituting thevalues of the constants given by equations (63), (64), (66), and (67) in-to equations (28), (29), and (30). These expressions are:

1 TTx. rnsin sin 11P. (72)inn a

m and n are odd


rtiftT-ftif

1:1 V0 0a) a)

••n

N

14 14Id ni0 I CI I

cd cd

b0

V/NE>,... V)

El l 0 i›..

r.* N............. 0?

..--.I IN

I C•1---..

N

.5

Ns

Nn.

rqN 0 ,...

+ M 4 0♦ 1 4

co

co

u4.10

NO 1: I:-°4

0

ra

to 81A170 .5 1,..0

e7

g at

5to

1J toCI

.,...-------.1.-----Th. BI

0—\ trl

.60

N N 4.1 r%3 N

V) W TA 0" t4

r1 rd ......----_____

Eel .--. (i)M

cu NI

17.71.. 0

r.1 ..--, N 4-.1

a. a. R. O. • I A- NN N

0 0

a. + Ntda° 4.71

N N0 0

tocu 4.J AJ

N –1

+ + .-.0 ÷ + ± /-r + -I-- ri

.--; t, u uu CI)

'------.--n

rn Xi IANtil

II v i4•J

N 0V II

... •,-,I- to b

bl


II

N

ciC1)

1-i

U

U0

5

t4H $

a)0 • H

I-1

00

Id

5 „.9

a)ca

0

ci

Id

r--1

.5 o-1to

0 alal 1.1

N

.0E°5 $.1 ^

rd

rdcts &L

a) .T.I

Ni

bDII

0.1H..,

COI X 4 +• H

7.-J El .--1Sd

4.1• H g

rd

43

inN t-

C/7

N

N

0 .1

RIa Cr. c_c_i14

Cd

El N

N

5

g

'6'BA

il

84)

In

R)v.i4J

--.—i

.......

A

.1.4

...i

I I 8

5

^N

r-1›-•

tr) sd

E

V\ g

rr

Na:

Na_ Q. Q.

N N . fir N N

0 0 0 0

+ +

ci5

RI


Facing Stresses

The expressions for the forces and moments per unit length of the fac-ings are given by equations (43). Under the present assumption thatthe flexural stiffnesses of the individual facings are zero, the six mo-ment expressions are zero. The forces per unit length of the facingsmay be evaluated by first expressing them in terms of the core dis-placements by means of equations (50) and then substituting in thevalues of the core displacements given by equations (24), (25), and (26),remembering that Amn and Bmn are zero. The results of these substi-tutions may be expressed as

Et 1Et2Nx = 2 (e x + ve y) 142: = (E;c + ye )

1-v 1-v 2

Et l

v 2Et ,Ny = -

(vex

+ e y 2) N = tve + e ' ) (77)

1 1-v

Et, Et7N - (y ) N' = (y1 )xY 2 (1+ V ) XY 4 XY 1n7VT XY


04)

Id,

at

Cd

01aoaO

sru

0

•9• r1

COIII 1

§ rikir--10r-71 14 I---1

_0 4

5 1

M t I ri Z..-0

g I Cd

*dP 1 rd

M

j1 i

44

K

0 j 44 g 1 r%1g I rc4d rai

gZI3♦ SIN

MbMkid 11 1

U a 0 aE C..) 0 r41.0

4r1t

U

♦ ----. N .to0

U N4.1-I- N

+U

N1..-I

4 .1

+

c.)

N N

N1

sl$

LI ..-..-..--.

N

......

N N

......

3

M11

Eol0

NNgN Ncl

0. cd a.11:1

U

LL....Li EN

o.1.._...I

E

I__I--II

Elk Pi

8v `0

II041

$4

ID III

1 024

91X-411

8k8El

II

wo>.‘

8

81"I

A

It

' N

vo

-.A

II

El

gkil

/\

It

mo

$3

tr

5

.",a,

A

II

>14

›.

it

4

.....,ri


toO TI

PO 1:1 PliC PO

.,-1 0 0 1

14 a) w 0O 14

idk ktr i rd b

0.1 ;

:II il I a I>-• w

-0 rd 1:1 PO PCI0

Pel 0 to A of t'dPO

w0 .c' 5 E l E l E l

k VO tg

cii A I al ck;

4,N1 vtd4 .5-0 i'4td 10 "

14

d 0iRI

0 1 >.+ - 113

NIA4, P 1 .0

to 5 I rd 0a*., tu1 rd

0 I .140)

+ jdP4to g

4401N . to . 0ag

0 Id.

w 1,-,-1:11

al

44 1°^ tIJ

lad

0to

k0j 8."1°-41i

01..) 114.sto .0

0 0 81"17cl

iv\rt

il -40

C.)014 co

k4

i.

N44 o).1

131 •dg

4.1

OB II

E44N

Oft.

ilki 70o 0 4—. + N

N+ u 0 Pd N .....

N 4.g N4.1

♦O ID sua 04

N II oi ID.4to 4.1

U

♦ 1 Nu

m al0

O 4.1

iiO .0

O+ ...... +

to2- N

u) ..—. 4.6N u0O 0

J..—.. ........— N

e

+O I N Er .....

kr III

0

•rirnen0.1k ("- +

NO

rdNI

4..4.b..4

r4 4.

42N3

II

.4 R. .0O az N N N

rd

Not. 0

014.a

n••4

8

n . •••n••1 .dri N 44►+

k A NO rid

/ \ 0 c^) 0

Lc'II 1 1 AIIa) II II II

>•n 44 11.14.. 1.4P.

n0

IV N 4ir, "vi>4


4.14

+ N

p41N

41

N

4.n

N

N

0

rd

o 0 02

for' Cd CO

14 1 r4 1 1I

Ts

1

ij

C Cd

El 0I 5 l

8IA 0

13 11

A

N+ N

4.4

N

•n••••

a.

NINed

),*

11 Id

0

MI 4tt)0t.)

pia01co0

)4

NNO

0.N

SIN

8 0,N

__)4 En•1I I

♦P*1

NN0.

I N4.1 N

I N♦

•••mr,

N4.1

N N0.

5 Na. +.

N0 ttl

N

N0

II 11


(78)

(79)

and

mn 2y

(m 2 + n2 p 2 ) /1--v (m2

2 k S + n2 p 2 Sx)]

1As would be expected, the absolute values of N x and N__', N and N ,x Y Y

N and N ' are equal, respectively, as shown by equations (77) andxy xythe•subsequent expressions for the strains. The maximum values of1\13c andoccur at the center of the plate, and these values may beNYobtained from the following:

Etl(Nx)2

+ ve )(Exmaxmax 1-v 'max

Et l(N ) (ve

xmax +

y 'maxmax 1-v 2

where

2 + (m2 - vn2 p 2) Sx + (- vm 2 + n2 p 2 ) S

E

t + t- - (

t2 ) (c + 1

2 2

a 2 t 1 + t2max

m+n

-1) 21

4jmn

m and n are odd

m+n2 2 t2 t

1 + t 1

= - k(t (c + > T°3 (-1) Ilmn

aYmax 22 t- + t 2 m=1 n=1

m and n are odd

yx is zero at the center of the plate and reaches a maximum at the

corners. No attempt was made to calculate values of N since suchxyvalues would be of little importance for design purposes.


(84)

Numerical Computations

Calculations were made for the maximum deflection, the maximumshear stresses in the core, and the maximum normal forces per unitlength of the facings. For purposes of calculation, equations (71),(78), (79), (75), and (76) were expressed as follows:

(w) =k= k C 1max (80)

(Nx) = ki (C2 + vC )max

(NY

) = k1 2ax

+ vC )ma

(T

XZ)

= k2 C4max

and

( TyZ )= k

2 C5max

where

16 qa4 (1-v 2 ) k -Tr 6 EI

Tr 2 k 1

t 2

tl

+ t 2 )

( a )E t

(ca (1-v

2 ) (t + t ) 2 w4

tl t2

1 2 (c + 2 )


T

16 qa2

and

k2 -

t1 + t

2n3 (c + 2 )

The coefficients C 1 through C 5 represent the corresponding double in-

finite series in equations (717 (78), (79), (75), and (76). Since theseries represented by C 1 , C2, and C 3 in equations (80), (81), and (82)

are alternating in sign when summed over either the Ws or the mt s,the values of these coefficients can be obtained with sufficient accuracyby summing a finite number of terms and using Euler' s transformationon the last few terms in cases where convergence is slow. In obtain-ing the values of C 1 , C2, and C3 given in table 1, the first 21 terms of

the double infinite series were used. The double infinite series thatappear in the expressions for the core shear stresses, represented byC4 and C5, are more difficult to sum because C 4

alternates only when

summed over then' El and C5 alternates only when summed over the ml s.

The nonalternating part of these series was summed by the methodsuggested by Gumowskil and the resulting partial sums could then besummed using Euler's transformation. In all of the numerical work,the value of Poisson's ratio of the facings was taken as 0. 3.

It is of interest to note that the values for S x = S = 0 in table 1 represent

the deflection, moment, and shear coefficients for a uniformly loadedhomogeneous plate with a moment of inertia equal to that of the spacedfacings of the sandwich plate. Thus, if the proper conversion factoris used in each case, these values may be shown to agree with thosegiven by Timoshenko for the homogeneous plate problem. ±

3

—Gumowski, Igor. Summation of Slowly Converging Series. Journalof Applied Physics Vol. 24, No. 8, p. 1068. 1953.

4—Timoshenko, S. Theory of Plates and Shells. p. 133. New York.

1940.

16 cla

T •


Conclusions

A general solution for the deflections and stresses in a uniformlyloaded, simply supported, rectangular sandwich plate is contained inthis report. This solution, based on the assumptions outlined in theIntroduction, consists of expressions for the deflections and stressesin the form of double Fourier series in which the coefficients must beobtained from equations (27) and (57) through (62).

In order to reduce the amount of numerical work necessary for thepreparation of design curves, certain additional simplifying assump-tions are made. On the basis of these additional assumptions the nec-essary Fourier coefficients may be expressed as shown in equations(63) through (69). The solution for the deflections and stresses is thenrepresented by equations (70), (72 1 ), (73), (74), and (77); and the ex-pressions for the maximum deflection, the maximum shear stressesin the core, and the maximum forces per unit length in the facings aregiven by equations (71), (75), (76), (78), and (79). Numerical resultsbased on equations (71), (75), (76), (78), and (79) are given in table 1,and design curves based on these values are shown in figures 5 through16.

Report No. 1847 -35- .4-54

InoNco (x) 0 \O

lc, (7. 01C- -(-0 ( ,

• • • • •

(Nap (C) ON Ihrl ch if, 0, CV1- I. 0) (0 ((

• • • • •

\ --I- Kv r-H 0, In 0 ,-P- 1-(0 (1)

• • • • •

oCS, I0 0\t) N-00

• • • • •

\C) 0 CV (-1\ 0 In N-rXD

N-CD

(:), 0. o. 0. (3.\ 0 N.0 \ 0

o CV 0, 1:- \\ IC \

C17 ittyi R [7_,Rj 74, 10 tit -j;

ul

0\ON ON 0\ (N. . . . .

115 th ON 0\ ON ONCO. . . . . I.- It--. . . \AD tr-N trn Lc.• • • • •

U 0)

u CO CC) 0 0 01

11

ro

ro .-1

11

toX

U

U

0 0 0 0 0

HH rl rl rl\O N.D \SD \ID \I)N-0

CO st3 \ (NI0 H tIM IC\00000

0\ r-1 0 ON-1- -1- tr, N 1-1

,S3 H N-0 N- \ \ \HH0.101(11

N- O\ CV -1-CO H \CD CV O\

rr--\ N\• • • • •

H CO -1- -4-\ .0 0 CU

Hir\ rc\• • • • •

t- 0t- stD 0 \SD -4-K\ H H

• • • • •

-1 0.1 1.-1 001 HMCO (-1 NM0.1 KM -1 g•

\ID C-00CV (-1 ,() HN x-10 0 0

UU

ryU

• • • • • • • • •coal ,4_,UM

\.0 \ID U'0 H te \

• • • • •

0 0 If N \-0 -4-r-1 C- 0tv-cp Lr. 1/1H CV if \

• • • • • •

ON 0 CO0 N- H 4 UM50 \ H 0/0/. Lf1

H CV r+M

• • • • • •

ON al -A-re\ 0\ -1-

H

• • , • , • • • • •

-4- 0 -4 t- ONCU \C) Cl‘ al UM

• • • • •(-1 /-1

XroE

xN

-aU

CI\ 1-1 , \JD ON CY\co 1h

N-AO \JD In In0

(3. (-1 \H ti CV CO UM

\ 0 \ \ON

rl \0 al (Du Int- LIM Li- \

-4- NO 01 \ COON if \ H CO UM

1/4.0 \.0 Ill UM

-4' CV NCO N-In CV O\\0 LC \\ \ .0 UM In ill

• • • • • - • • - • - • . • • •

ON ON 0\ ON 0\ \ID C- r-co CO CV ON -4- 0\ CV 0 nap CO 0\

U\C3 \O 50 \ID0\0 \ CN

\C) D \OON ON ON 0\ CIN

0 0 HON CY\ ON 01 (7N

OUN H 01N- N-CO CO CO

• • • • •

IIMOD 0 C11\ D

• • • • •esi

U +tr)

0

•-nCO CV CO 1110 \C) H

+ O 0 0 0 0 0CO \ D .4 CV H0 0 0 0 0

In 4Th \N- lf \ 0.1

CV ON 0.1 H0 \ -4' 0 CO \ 0

-4- 01H .4 CV\-0 C\ 0 CO

U 0II 0 0 0 0 0• • • • • (-1 0 0 0 0• • • • • H (-1 1--1 0 0 CV1-41--1H0

rn• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • .1 - •

II

E

U

HHHHHNN 1/4.0 \SD \D

0•

ON CV -3 UM ND-4- In LC N

N-

• • • , • • • • •

ti N- \O N Inap H tC\UMND \ ‘.0 \D

• • • • • • • • • •

0\ CI\N- CV 50 00 041-3 -3 -1- If\

• 1. • • • • • • 1. • •

-4- CY \ In 51)CV N- \13

CV NM KM NM

• • • • •

CV -1- CO 0 0 -1- 0 \ 01 ND ON -4- 0 CYN CO CO UM UM CD al In 0 \ N- tC \zIN CO 41010 KM ON

N-- ON 0 0.1 NM• • • • •

0 1-1 H (-1

N- CV t- WV:0t--- ON 0 CV K\

H I-A I-1

0 t-- -1- CN LnN.0 N-- 0\ 0 al

• •r-1 (-1

0\ \ 0 CV NMNM U\ t- CO 0

H

-4 ON KM 1-- HCV KM Inl0 CO

• • • • •

en

NI INJ

ttl U

cr. as ON CT 0\HHHHH

N-

0\ \ Os ON ONHHHHHt-- N- t--

•

t-HHHHH

• • • • • • • •

-4. -4- -4-ON CT 0 \ CN 0 \\ \ \ D D

• • • • •

• .1 • • • • • " •

UM LC\ ti-N UM If\\ 0 ND ND \ \

O

U• • • • • • , • • • •

U

0\ 0\O) O\ 0\\CD \D \D \Dcr. G. a, cr,

•

'-05) \O \\0 5.0 \as cr. G. crn

CV C CV 0.1 010 0 0 0 0cr\ ON 0\ ON

L N -t"--

N- N-• • • • •

LIM tr-N In in505050 50'-O

• • • • •CD

•• •• •• .• ••

zX

N to

11

xU 0 0 0 0 0

CO CO

a8 "8W.00 000 CD

`8SSr-1 rlH r-1 1-1 rlrl 1-1 r4 I-1

-c-11 -(-1` Z\11-CV CV CV CU CV

Val

U-X

Crr

,D'Cr

r‘lU

• • • •

5.0 \ \D \ D DN-

• • • •0

C3NON ON ON ON-4- -4- -4- -4- -1-N- t--

• •

N- t-- t•-•CO OD 00 OD COtn UM tr. n Lc.

• • • • •t- t-- N-

\ teN fC\ N.\ IC\• • • • •

• • • • •

-4- -1- 01 CV 0.1

CV CV CV CV 01• • • • •

roE

tu

.0U

CV N oCO \O -1- N0

lc\ 01 NM• • • •

0 H r-1 (-1 CV

0 CO \0 UMODN- (11.1 0 CO

H Ih CIN 01rlr1HN

1-1 \0 In 0\ OINND ON CV VD (r,

HrlH

CO H ON In K\ONCO \D I(\ -$tr••nn•0 CI\ CV Ill

H r-i

-4- OD CU H-3 \O 05 H-0.1 \0 ON H

0000 C3 CD 0000 0 0 CD C) CD 0 CD CD CD CD CD CD CD 00 LIM CD LIMO CD LC\ C) UM 0 C) LIMO In CD 0 LIMO In 0 0 UM C) 11\0

• • • • • • • • • •H •-4 CV HHOJ H 1-4 CV H H al (-1 1-1 CV

I(\ IC\ uM 0 0 0 0 0 Ill 4I-N 115 UM UM 0 0 0 0 00- 0 0 0

NN N 0) 0] UM UM In LIM LIA 0 0 0 0 0

• • • • • • • • •P-1 H rI r-I •-I

Report No. 1847

'0

q

0

ra

Er)

)-4

rd

td

4-)

0q

a)

• 4-4

rc;a)

-0

INica

.21 .4'--im u) )40 r-- iup $_, cr-- -41Pr rd

.. •-1 co 1-•4 L

'V, dq ,-1 . -1

.-I-1 H ;-c'd

rf &0 m>, . 0 M00° "-4 1-4 q0 I-I-1 4, u)

q 0 ..-1Ed tj 1- -3::1

0q rd

.5 11)1_1 0° 01-4 Ill5

4-I- rd m°

U3 0▪ $4O . gII To 'cl il ;

(I) +.1 rd .__i coco $4

A 0103 U WUQ3 0 -, n . cn

-.r 04 C.) . GIU $.14 4" CLI • r- .

.-4 4-, a)u3 '0 "cl 0 ijrd 01 Q3 0 1, 4-3 CD

q VIg g .-- -0 1:1%4 4-)CD0 r.4 N co4-, 04 I:ld 1-'W.,

=- g H I; to M al.,-,

a) .,-, at 04

co t qz4 0

4. 'IC--; 03,.., q 01I1) °•) •. 4 44 0 +.1 >.,O 'CI Pci 04 0 u ,.. 1 •

.,-,0 liq) ,4 c.r) co ti) sai 0> c) rd o ul ma) E -{L4cl ;.-1 4)c4 0 '5

- rl 4.-1W4 'V co ra,

U▪

bU

'0

>sN----.QII NO E---..,• a-- .14

C210

,.., ,-1- In j:,

P T; 0_,0

4t-1 +:u fCU ° 18 fa,

•4-4 k qcr) _0 1-,

rd" • r-I rd0 vco 4-i al 14 .--4' 1co s4 g COCD 0 44 Q.,/-1 04 u

. -'-'1::14 • r.,1U3 7:1 r'CI41 9:3 01 I3.3 0 N.-.I-Ig >, "cf0 .-4 N (4 u;4-, pi

tv, r' E ....

. .3 I'l.2 ., co a) -714.", in 41 IA

. a)V lt 0cy) -

cl -0 54 ri-1 04a) v•- .0 tfl

,,i. n u (/) en

-.--€ •

rd °

-c,

.,. NII .▪ ---.,---1 cm

t.4.70 i`.Crs• 714

/.4 41 74 1n4 COa; •--1•-7-1

e••I-I• 1:40 7,414 00 0

It

U +4{I) L-i 34tP4/.., 0 0

rom ed .-4

10 a) -cSa) 4-• rd 1-1 4

m )-4 En

a) q 4-) n .,,-)4 0., C.)• 4-' w- o p.,. , u) o re -0Li4 • 01 0.) 0 ,...

O >-, eci PIO ..-1 N Pi ,;4, r3,,-,

:d 0 d 4-,q 0 ta '-'44-I•74 • I-1 I) • .-I.) 113 ID: ;4

u . .L1 qa)" 0.1 CI -0 04

1'14 04

•...1 o tu

(J til

A "cird r°0 .; cr-1rd

c40 .5

•

deflection and stresses in a uniformly loaded, simply supported

Documents

Transcript of deflection and stresses in a uniformly loaded, simply supported