STRESSED-SKIN PANEL

DEFLECTIONS AND STRESSES

USDA FOREST SERVICE

RESEARCH PAPER

FPL 251

1975

U. S. DEPARTMENT OF AGRICULTURE

FOREST SERVICE

FOREST PRODUCTS LABORATORY

MADISON, WISCONSIN

ABSTRACT

This paper presents a mathematical analysis based on "shear flow" and "shear lag" theories to determine deflections and stresses for stressed-skin panels wherein skins are rigidlybonded to stringers. Experimental examination of several panel constructions showed that the analysis can provide a rational basis for the design of stressed-skin panels. Designs based on the analysis will result in more efficient utilization of materials.

CONTENTS

Page

Introduction . . . . . . . . . . . . . 1

Theoretical Analysis . . . . . . . . . 1

Experimental Examination . . . . . . . . 19

Test Results . . . . . . . . . . . . . 26

Conclusions. . . . . . . . . . . . . . 40

Suggested Design Procedure . . . . . . 41

STRESSED-SKIN PANEL DEFLECTIONS AND STRESSES

By ED WARD W. KUENZI, Engineer

and

JOHN J. ZAHN, Engineer

Forest Products Laboratory,1 Forest Service U. S. Department of Agriculture

INTRODUCTION

Design procedures for s t ressed- skin panels (wherein thin facings are r igidly bonded to s t r ingers thus producing l ightweight , s t i f f panels) have been clouded by " rule- of- thumb"procedures regarding shear s t resses in s t r ingers . Also, the possible effects of " shear lag" which may cause the skin normal s t resses to diminish with increasing dis tance from the s t r inger needed clar i-f icat ion and evaluat ion. This research s tudy sought eff ic ient ut i l izat ion of mater ia ls by developing more rat ional design procedures for s t ressed- skin panels especial ly with regard to " shear f low" and " shear lag" effects . In-cluded also in analyt ical work are complete der ivat ions for panel def lect ions, effects of shear ing deformations, and panel s t resses . Experimental research was carr ied out to substant ia te the analyt ical resul ts and underlying assumptions.

THEORETICAL ANALYSIS

This analysis fol lows Reissner 2 except that the three components ( top skin, bot tom skin, and s t r ingers) are made diss imilar and unequal . In addi t ion, the shear deformation of the s t r ingers is included s ince i t can be a s ignif icant par t of the total def lect ion for wood s t ressed- skin panels .

Figure 1A shows the cross sect ion of some s t ressed- skin panels . Figure 1B shows the smallest repeat ing uni t of which general s t ressed- skin panel cross sect ions are composed. This repeat ing uni t is shown crosshatched on the cross sect ions shown in f igure 1A.

1 Maintained at Madison, Wis. , in cooperat ion with the Universi ty of Wisconsin. 2Reissner , E. Analysis of Shear Lag in Box Beams by the Principle of Minimum

Potent ia l Energy. Applied Math. Quarter ly 4(3) : 268- 278. Oct . 1946.

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Figure 1. Stressed- skin panel cross sect ions A and notat ion, B.

M 142 555

Figure 2. I l lustrat ion of how plane t ransverse cross sect ion rotates and warps during bending in the presence of shear .

M 142 563

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Coordinate axes x, y, and z form a r ight- hand set with the x axis a long the span and the z axis ver t ical . The corresponding displacements are u, v, and w . The method of analysis is to assume a reasonable form for the displace-ments u and v and der ive an equat ion for the displacement w from the pr inciple of minimum potent ia l energy.

Assumed Displacements

Previous analysis 3 has es tabl ished that the displacement v can be ignored with l i t t le effect on the resul ts for ver t ical def lect ion. i t i s assumed that

For s implici ty , then,

(1)

and

(2)

In other words, this is a beam analysis with special a t tent ion to shear de-formations. In e lementary bending theory the cross sect ions are assumed to remain plane (u = 0) . The term " shear lag" implies a warping of the cross sect ions associated with the exis tence of shear s t ra ins in the skins . The resul t is a reduct ion in normal s t ress on the skin cross sect ion from that predicted by elementary theory and an increase in ver t ical def lect ion. In this analysis the warping of the skin cross sect ions is assumed to be a parabola in the y direct ion (f ig . 2) . Let subscr ipt i denote skin 1 ( top) , skin 2 (bot tom), or s t r inger 3 . Assume that the u displacement has the form

(3)

(4)

where z1 equals c 1 ' z 2 equals -c2 , c1 and c 2 are dis tances f rom neutral axis

to centroids of skins; U1 , U2 , and U3 are funct ions of x; and the αi are constants . The z coordinate is measured from the neutral axis . The locat ion of the neutral axis is assumed to be unaffected by cross- sect ion warping.

3 Reissner , E. Least Work Solut ions of Shear Lag Problems . J . Aero. Sci . 8 : 284- 291. 1941.

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S t r a i n s

The motivat ion for the assumed displacements is more apparent when one ex-amines the shear s t ra in dis t r ibut ion . The shear s t ra in in the s t r ingers is

(5)

Note this is independent of z. The shear s t ra in in the skins is

(6)

Note this is l inear in y. This s t ra in dis t r ibut ion is s t rongly suggested by elementary shear f low analysis in which the shear var ies only very s l ight ly in the s t r ingers and var ies l inear ly throughout the width of the skins .

The extensional s t ra ins a t the midplane of the skins are

Cont inui ty a t Gluel ines

At the junct ion of s t r inger and skin we require cont inui ty of displacement and of shear f low . For convenience, the junct ion wil l be taken to be at z = z i(centroid of skin) and the comer wil l be ideal ized to a point with no cross- sect ional area. Then at y = b, z = c 1

(8)

-4-

(9)

and a t y = b , z = -c 2

(10)

(11)

where b is half of s t r inger spacing, ti are thicknesses ( f ig . lA), the Gi are shear moduli of e las t ic i ty in the x-y plane of the skins or x-z plane of the s t r inger , and γ denotes shear s t ra in .

From equat ions (8) through (11) , using equat ions (3) through (6) , one easi ly obtains the fol lowing relat ions:

(12)

(13)

(14)

Thus we can drop the subscr ipt on U 1

hereaf ter and seek two unknown funct ions of x, namely w and U.

Potent ia l Energy

The potent ia l energy of the load is

(16)

where M is the bending moment on the cross sect ion as shown in f igure 3 and L is the total length. Figure 3 shows the s ign convent ion for M and V.

- 5-

Figure 3 . - - S i g n convent ion f o r i n t e r n a l stress r e s u l t a n t s on c r o s s s e c t i o n .

M 142 566

The s t ra in energy of the s t r inger is

(17)

where E3 equals modulus of e las t ic i ty in x- direct ion of s t r inger and

(18)

(19)

The s t ra in energy of the skins is

(20)

where E i equals modulus of e las t ic i ty in x- direct ion of skin i. Subst i tut ing equat ions (6) and (7) and integrat ing over y, equat ion (20) becomes

(21)

- 6-

where pr imes denote different ia t ion with respect to x and

(22)

The total potent ia l energy is

(23)

Let

(24)

and use equat ions (14) and (15) . Then the total potent ia l energy can be wri t ten

(25)

where

(26)

(27)

(28)

(29)

(30)

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Equil ibrum Condit ion

According to the minimum potent ia l energy theorem, the condi t ion of equi l i-brium requires that the f i rs t var ia t ion of the total potent ia l energy vanish:

(31)

Using equat ion (25) and integrat ing by par ts , equat ion (31) can be wri t ten

(32)

f rom which one obtains two different ia l equat ions

and the boundary condi t ions

and the cont inui ty condi t ions

(33)

(34)

(35)

(36)

Since the curvature Q is cont inuous there , equat ion

(37)

a t a concentrated load. These resul ts reduce to agreement with Reissner 2

a t a concentrated load. (36) implies

when the skins are s imilar and isotropic and G3 is inf ini te .

-8-

Different ia l Equat ion for U

Assume A is greater than B2. Then el iminat ing Q between equat ions (33) and (34) yields

where

(38)

(39)

(40)

(41)

Once U has been found, equat ion (33) gives Q, that is w", from which w can be obtained by double integrat ion. Thus, wri t ing B in terms of n and ρ f rom equat ion (40) , equat ion (33) becomes

(42)

f rom which the def lect ion w can be obtained in the form of the elementary bending solut ion plus a correct ion for shear lag.

The s t ra ins in the skins can be obtained from U and w as

(43)

(44)

which are a lso in the form of the elementary solut ion plus a correct ion for shear lag. These s t ra ins are a t the midplane of the skins . More accurate expressions for maximum strains can be obtained by replacing c in the ele-i mentary par t of the solut ions with the dis tance to the outer

surface. Then equat ion (43) becomes

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(45)

using equat ion (42) . Similar ly equat ion (44) becomes

(46)

The shear s t ra ins can be obtained from U as

(47)

(48)

(49)

Applicat ion of this theory is next i l lustrated by two examples .

Case 1. Uniformly Distr ibuted Load and Simple Support

Figure 4A shows the notat ion, Let the total load be W and the total span be 2a. The dis t r ibuted load is

(50)

The bending moment is

(51)

Therefore equat ion (38) becomes

(52)

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Figure 4 .-- Notat ion for panel loadings, A, case 1, uniformly dis t r ibuted load. B, case 2, loads at quarter- span points .

M 142 560

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By symmetry

(53)

The solut ion of equat ion (52) subject to equat ions (53) and (54) is

Str inger and skin s t ra ins may now be computed by subst i tut ing

and equat ion (51) into equat ions (45) through (49) . To obtain the center def lect ion, subst i tute equat ions (51) and (55) into equat ion (42) and get

The solut ion of equat ion (57) subject to the condi t ions

and

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and s ince are zero at the supports , equat ions (45) and (51) imply

(54)

(55)

(56)

(57)

(58)

(59)

is

(60)

The center def lect ion is

(61)

Case 2. Quarter- Point Load and Simple Support

Figure 4B shows the notat ion. Here the total load is 2P and the total length is 4a, The bending moment is

(62)

hence equat ion (38) becomes

(63)

whose solut ion, subject to the condi t ions

(64)

(65)

(66)

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is

(67)

Str inger and skin s t ra ins may now be computed by subst i tut ing

(68)

and equat ion (62) into equat ions (45) through (49) .

Deflect ions can be obtained by solving equat ion (42) subject to the con d i t i on s

(69)

(70)

(71)

The resul t ing center def lect ion is

(72)

Remark :

Since equat ion (38) possesses a nonzero homogeneous solut ion, U does not necessar i ly vanish when M' does . This means that cross- sect ional warping, which is necessi ta ted by ver t ical shear wherever M' i s not zero, can " spi l l over" into regions where the bending moment M is constant and there

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is no net ver t ical shear force. in this s tudy as can be seen by examining the case of quarter - point loading in tables 1 through 4. Note that the skin s t ra ins a t x =

12 inches (where bending moment i s constant) are smaller between s t r ingers

This effect was experimental ly ver i f ied

than they are a t s t r ingers .

However this theory contains one incongruous resul t . By equat ion (49) the s t r inger shear s t ra in is nonzero wherever the cross- sect ional warping ampli-tude U is nonzero. Yet surely the s t r inger s t ra in must vanish wherever there- is no net ver t ical shear force. The above- mentioned “spi l l- over” thus puts s t r inger shear s t ra in where there should not be any. This apparent contradict ion is the resul t of only approximating equi l ibr ium by minimizing the potent ia l energy rather than imposing exact equi l ibr ium equat ions . To do a bet ter job i t would be necessary to introduce more degrees of f reedom in the assumed displacements than was done here . However , this theory does an excel lent job of model ing the maximum values of s t ra ins and def lect ions in s t ressed- skin panels and is ent i re ly adequate for design purposes .

L i m i t i n g Cases--Reduction t o Elementary Theory

For s implici ty , only the case where al l E values are equal is considered here . There is an elementary theory of shear def lect ions of beams which uses Cast igl iano’s theorem. The shear s t ress energy is obtained by integrat ing the elementary shear s t ress dis t r ibut ion given by

(73)

where τ equals shear s t ress ,

V equals total shear force on cross sect ion, and

A'z equals f i rs t moment about neutral axis of area bounded by free surfaces and longi tudinal cut t ing plane.

In the skins

(74)

so that

(75)

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and in the s t r inger A'y is near ly constant so that , approximately,

The shear s t ress energy is therefore

where

and by Cast igl iano 's theorem the shear def lect ion is

where Q is a dummy force at the point where def lect ion is desired, and δ sis the shear def lect ion.

Applying this theory to cases 1 and 2 above yields

where δb equals e lementary bending def lect ion at center and

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(76)

(77)

(78)

(79)

(80)

(81)

I t wil l now be shown that the Reissner- type shear lag analysis of this report reduces to the elementary resul t above under appropriate l imit ing assumptions.

In cases 1 and 2 the correct ion for shear lag has the form

where

and

Unless the span a is extremely short and G is extremely low, the quant i ty Ψ (ka) wil l be legs than 0.05 and can be neglected with error less than 5 -percent . Using the def ini t ions of k and n (eqs. (40) and (41)) and neglect ing Ψ (ka) equat ion (82) becomes

where B and C are given by equat ions (28) and (29) . Assuming that

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(82)

(83)

(84)

(85)

(86)

i t fol lows that β = 1 and B and C become

(87)

(88)

where

(89)

The contr ibut ion of the s t r inger to shear def lect ion can be obtained by le t t ing G 1 approach inf ini ty . Then

(90)

(91)

and

(92)

which essent ia l ly agrees with the f i rs t term of equat ion (80) except for a factor which is within the approximation error of these approximate theories . That they are different is not surpr is ing, for the theories differ in small respects . For example, one matches shear f low at the junct ion of s t r inger and skin and the other does not .

To obtain the contr ibut ion of the skins to the shear def lect ion, le t G3 approach inf ini ty . Then from equat ions (87) and (88)

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(94)

and

(95)

which agrees exact ly with the second term of equat ion (83) . Thus t h e t h e o r y presented here reduces to e lementary theory under appropriate l imit ing assump-tions. I t should be noted that without these l imit ing assumptions the theory is considerably more accurate than elementary theory. The o n l y r e a s o n for making these reduct ions is to check the form of the theory and to increase confidence in i t .

EXPERIMENTAL EXAMINATION

An experimental examinat ion of s t ressed- skin panels was conducted to deter-mine whether the theoret ical analysis could be ut i l ized to predict panel def lect ions and s t resses or s t ra ins .

Panel Mater ia ls and Fabricat ion

Six panels were fabricated with plywood skins and plywood s t r ingers . Detai ls of construct ion are shown in the sketches of f igure 5. Panels were 96 inches in length, thus avoiding spl ic ing of thin, tension skins for longer panels . The panel depth, however , was determined by bending def lect ion cr i ter ia for panels 14h inches long; namely that a panel should def lect no more than l /360 of the span under a uniformly dis t r ibuted load of 40 pounds per square foot . The effects of shear ing deformations and shear ing s t ress were thus somewhat accentuated by tes t ing panels with rather thin s t r ingers on a short 93- inch span.

All plywood used to construct the panels were grademarked as sanded, A- C, exter ior , Group 1 of U.S. Product Standard PS 1- 66.

The thin 1/4- inch skins had three veneers of Douglas- fir 4 of equal thickness .

The 1/2- inch plywood s t r ingers were f ive- ply with face veneers about 0 .08 inch thick and crossband and core veneers each about 0 .11 inch thick. Douglas- fir veneers were used for panels 1 , 2 , 3 , and 4. Str ingers for panels 5 and 6 had one face ply and the core ply of Douglas- fir ; the other face ply was of western hemlock; and the crossbands were of ponderosa pine. 4 Veneer species were ident i f ied by Dr. B. Francis Kukachka of the Forest

Products Laboratory.

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Figu

re

5 .-

-C

onst

ruct

ion

deta

ils

of

stre

ssed

-sk

in p

anel

s te

sted

.

M

142

562

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Note that a double s t r inger of two nai l- glued 1/2- inch plywood s t r ips was used in panels 5 and 6. Panel 6 was cut f rom panel 5 af ter panel 5 had been tes ted to determine s t i f fness and s t ra ins without loading to fa i lure .

The thick, 5/8- inch, f ive- ply skin of panel 1 had 0.08- inch face pl ies and a 0 .16- inch core ply of Douglas- fir , one crossband of 0 .15- inch white f i r , and the other crossband of 0 .15- inch ponderosa pine . The thick skin of panel 2 was seven- ply, 5/8- inch, with face pl ies of 0 .06- inch Douglas- fir , outer crossbands and core of 0 .10- inch white f i r , and inner pl ies of 0 .10-inch Douglas- fir . The thick, 3/4- inch skin of panels 3 , 4 , 5 , and 6 had face pl ies about 0 .08 inch thick and core and crossbands about 0 .19 inch thick . Al l veneers were of Douglas- fir in the thick skins of panels 3 , 5 , and 6. Panel 4 had face and core veneers of Douglas- fir and crossbands of ponderosa pine in the thick skin.

The thick skin of panels 1 , 2 , and 3 were spl iced at midlength with inner spl ice plates of plywood the same thickness as the thick skin. The spl ice plates were 16 inches long in the dimension paral le l to the panel length. These nai l- glued spl ices were made pr ior to assembling the s t ressed- skin panels .

Blocking of 3/4- inch plywood was placed between s t r ingers a t midspan and each end of the s t ressed- skin panels .

The s t ressed- skin panels were glued with a urea- formaldehyde5 glue. Each assembled panel was al lowed to cure overnight in a cold press .

Thickness and width dimensions of the panels were measured and these data are included in tables 1 through 6.

Panel Test ing

Panels were supported on a span of 90 inches and loaded under three types of load uniformly dis t r ibuted load, l ine loads appl ied at outer quarter-span points , and concentrated point load appl ied at s t r ingers . Panels under l ine loads at quarter- span points were f i rs t tes ted inverted ( thin skin in compression) and then tes ted r ight s ide up to fa i lure . (Panel 5 was not loaded to fa i lure . )

React ion points for the panels were of 3- inch s teel pipes supported on a heavy t imber f ramework in a large- platen tes t ing machine. The pipes and port ions of the framework can be seen in f igures 6 , 7 , and 8.

Uniformly dis t r ibuted loads were appl ied to the panel through air pressure in a large plast ic bag placed between the panel and a s turdy " strongback" fastened to the movable head of a tes t ing machine. Figure 6 shows one end

5 This glue was chosen as a convenient one to produce a s t rong r igid bond in the tes t panels . A more durable glue must be used in actual panel construct ion.

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Figure 6 .- - Stressed- skin panel under uniformly dis t r ibuted load appl ied through an inf la ted plast ic bag. The bag is placed between the panel and an upper "s t rongback" at tached to movable head of a tes t ing machine.

(M 141 738-6)

Figure 7. Concentrated load appl ied through a 1- inch diameter bar under a load cel l placed at a panel s t r inger and 3 inches from react ion.

(M 141 563-10)

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of the panel with the inf la ted plast ic bag on the panel . Pressure in the bag was measured with an electronic pressure t ransducer having a pressure cal ibrat ion of 150.31 microvol ts per inch of water .

Concentrated point loads were appl ied with a 1- inch- diameters teel bar loaded by an electronic load cel l through a s teel bal l . The load cel l was fastened to the movable head of the tes t ing machine. Figure 7 shows apparatus used to apply the point load.

Line loads at quarter- span points were appl ied to the panels through 3- inch pipe pieces extending across the width of the panel . The pipes were loaded at two points through a f ramework which was loaded at i ts center with a load cel l and bal l fas tened to the upper , movable head of the tes t ing machine, The apparatus is shown in f igure 8.

Panel midspan def lect ions were measured with a dial gage micrometer fas tened to the s tem of a long- top T- bar supported at i ts ends by nai ls a t the panel react ions. The movable s tem of the dial gage micrometer was at tached to a nai l dr iven into the panel edge. Deflect ions were measured at both panel edges and the readings were averaged. Deflect ions of one s ide of the panel were within a few thousandths of an inch of def lect ion of the other s ide.

Strains a t var ious points in the panel were measured with 1- inch electr ical-resis tance s t ra in gages. Skin s t ra ins in panels 1 , 2 , 3 , and 4 were measured 12 inches from midspan. Single gages for these panels were placed midway between s t r ingers and direct ly above or below inner s t r ingers . At outer s t r ingers the s t ra in gages were placed 1 inch from the panel edge. Skin s t ra ins in panels 5 and 6 were measured with 1- inch s t ra in gage roset tes placed at 27- 1/2 inches from midspan. Widthwise, these roset te gages were placed midway between s t r ingers or 6- 1/2 inches ei ther s ide of the midpoint between s t r ingers . Strain gage roset tes were also placed on s t r ingers near one react ion. The gages were centered at s t r inger midheight and in panels 1 , 2 , 3 , and 4 roset tes were placed 35- 1/2 inches and 41- 1/2 inches from midspan. On panels 5 and 6 roset tes were placed 35- 1/2 inches from midspan.

Uniformly dis t r ibuted loads and loads at quarter- span points were appl ied to the panels in 200- pound load increments to about 1 ,400 pounds for panels 32 inches wide and 2,400 pounds for panels 48 inches wide . At each load increment , load cel l and s t ra in gage readings were recorded with a mult i-channel digi ta l data acquis i t ion system having a scanning rate of about 4 gages per second . A few readings of a l l the data were also taken during removal of load.

Concentrated point loads were appl ied at 100- pound increments to 500 pounds and s t ra in gage readings on the s t r ingers were recorded with the data acquis i t ion system.

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Figure 8.- - Stressed- skin panel under load appl ied at quarter- span points . In the lef t foreground are s t ra in gage reading and recording equipment .

(M 141 738-8)

F igure 9.--Type of s h e a r t e s t t o de te rmine s h e a r s t i f f n e s s of t h e s t r i n g e r s .

(M 125 798)

- 24-

Panels and coupons from them were tes ted within the laboratory during the winter months. Condi t ions recorded during the tes t ing per iod showed a re la t ive humidi ty of 35 ± 10 percent and a temperature of 72° ± 6° F, to

resul t in a wood moisture content of 7 ± 1- 1/2 percent . 6

Graphs were constructed of load- deflect ion and load- strain data . Except for the f inal tes ts to fa i lure these graphs showed a l inear re la t ionship between load and def lect ion and load and s t ra in for near ly a l l data . A few non-linear data graphs were able to be represented by a s t ra ight l ine f i t t ing the general s lope of the data . The s lopes of the l ines on the graphs were used to compute the data given in tables 1 to 6 . Shear s t ra ins were computed from the s t ra in gage roset te data .

Coupon Testing

The elast ic propert ies of the skins and s t r ingers of the panels were determined by tes t ing small coupons cut f rom the panels af ter they had been fai led.

Two compression skin coupons 1 inch wide and 4 inches long and two tension skin coupons 1 inch wide and 10 inches long were cut f rom the skins in each space between the s t r ingers of the s t ressed- skin panels . The long dimen-sion was paral le l to the length of the s t ressed- skin panel . Compression coupons were loaded through a spherical- seated loading head in the movable head of the tes t ing machine. Tension coupons were gr ipped in self- al ined wedge gr ips in a tes t ing machine. Deformations of compression and tension coupons were measured by a mechanical gage that caused motion of the core in a different ia l t ransformer. The s ignal f rom the t ransformer was com-bined with the tes t ing machine load t ransducer s ignal to produce load-deformation graphs on an x- y recorder . The deformation gage had knife edges spaced 2 inches apart and was fastened to the edges of the specimen.

One s t r inger shear coupon 4 inches wide and 10 inches long was cut f rom each s t r inger . The specimen was cut so that the s t ra in gage roset tes were in-cluded in i ts length. The type of shear tes t conducted is shown in f igure 9.

Average values of the elast ic propert ies are given in the lower port ions of tables 1 through 6. These propert ies were combined with the theoret ical analysis to compute the s t i f fnesses of the repeat ing element ( f ig . 1) of each s t ressed- skin panel . Several panels did not have a t rue repeat ing element . For these panels an average s t r inger thickness was used to compute bending s t i f fnesses and def lect ions. Actual s t r inger thickness was used to compute local s t ra ins . St i f fness values are a lso given in the lower port ion of tables 1 through 6.

6 Determined from data in table 38 of Wood Handbook, Agr. Handb. No. 72. 1955.

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TEST RESULTS

Detai led data f rom tests of the s t ressed- skin panels under concentrated point load appl ied 3 inches from a react ion at one s t r inger are not included. The load- strain data were l inear up to 500 pounds. The shear s t ra ins a t st r ingers adjacent to the loaded s t r inger were about 10 percent of s t ra ins measured on the loaded s t r inger . Strains in s t r ingers two or three spaces f rom the loaded s t r inger were only about 3 percent of s t ra ins measured on the loaded s t r inger . Thus when concentrated point loads occur near a react ion very l i t t le of this load i s t ransferred to other s t r ingers .

Experimental values given in tables 1 through 6 show that data for panels inver ted ( thin skin in compression) are in general agreement with those �or panels tes ted r ight s ide up. Deflect ions were within a couple of thousandths of an inch of each other . Strains were not in as c lose agreement but the general pat tern was s imilar whether the panel was inverted or not . The skin normal s t ra ins , ε 1 and ε 2 ' were at most under 400 microinches per inch, thus represent ing s t resses less than about 700 pounds per square inch. The shear st ra ins , γ , were at most under 1 ,600 microinches per inch, thus represent ing st resses less than about 160 pounds per square inch. Thus the elast ic behavior of the panels was not great ly dependent upon their or ientat ion ( inverted or not) even though the 1/4- inch tension skin was not iceably buckled inward even without load on the panel .

Experimental s t r inger shear s t ra in data given in tables 1 through 6 show that inner s t r ingers are s t ra ined more than outer s t r ingers of the same s ize , the difference being as much as 3 to 1 in a few cases . Panels with inner st r ingers double the thickness of outer s t r ingers had s t r inger shear s t ra ins more near ly equal .

Five of the s t ressed- skin panels were loaded to fa i lure under load appl ied at quarter- span points . Fai lures occurred suddenly in shear a t the s t r inger- skin bonds. Detai ls of the type of fa i lure are given in table 7 . For panels 1 , 2 , 3 , and 6 the load- deflect ion curve was l inear to about 95 per- cent of the maximum load. For panel 4 the load- deflect ion curve was l inear to about 60 percent of maximum load. Maximum shear s t resses a t skin- st r inger bonds ranged from 300 to 779 pounds per square inch as computed by elementary shear- flow theory and 230 to 680 pounds per square inch as com- puted by shear- lag theory. The elementary theory was about 20 percent conservat ive for the panels tes ted.

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Table 1.- -Stressed-skin panel No. 1

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Table 2.- -Stressed-skin panel No. 2

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Table 3.--Stressed-skin panel No. 3

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Table tressed-skin No. 4

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Table 5.--Stressed-skin panel No. 5

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Table 6.- - Stressed- skin panel No. 6

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Table 7.- - Fai lure of s t ressed- skin panels

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Comparis on o f The ore t i cal an d Experimental Resul ts

Data given in tables 1 through 6 include theoret ical values of def lect ions and s t ra ins determined by subst i tut ing dimensions and elast ic property values in the per t inent formulas given in the Theoret ical Analysis . The deflect ions and s t ra ins were computed for the total panel load indicated in the crossheadings. Experimental values a re given for comparison,

A comparison of theoret ical and experimental resul ts can be had by con- st ruct ing graphs showing theoret ical values as ordinates and experimental values on the abscissa . Perfect agreement between the theoret ical and experimental values would resul t in a l l points lying on a 45º l ine f rom the origin. Graphs of f igures 10 to 16 were placed in the order of importance of panel character is t ics to adequate s t ructural design.

Figure 10 shows that theoret ical and experimental values of midspan def lec- t ion agreed wel l ; hence the theoret ical analysis can be used to c losely predict the pr ime design character is t ic of panel def lect ion. The computa- t ion of def lect ion by the theoret ical analysis involving " shear lag" includes effects of shear ing deformations in the s t r ingers and skins to produce shear def lect ion of the panels . The amount of midspan shear def lec- t ion of panels under uniformly dis t r ibuted load as a percentage of total deflect ion of the panels tes ted is given as:

Panel (Pet of

Shear def lect ion panel midspan def lect ion)

28 28 49 25 34 32

The shear def lect ion was determined by subtract ing the computed bending def lect ion from the theoret ical def lect ions given in tables 1 through 6. The shear def lect ion appears re la t ively large but this is because effects of shear were emphasized in these panels by choosing short spans and thin s t r ingers .

An important difference between these resul ts and current pract ice can be seen in the values of s t r inger shear s t ra ins . I t i s current pract ice to reduce the al lowable design rol l ing- shear s t ress for plywood skins a t the gluel ine with the outer s t r inger , on the assumption that s t ress concentrat ions are produced by a panel edge- effect . However , i f a l l s t r ingers are the same s ize , e lementary shear f low theory shows that the inner s t r ingers carry twice as much shear s t ress as the outer s t r ingers . The data for panels 1 through 4 confirm that inner s t r ingers are more highly s t ressed, though not qui te by a factor of 2 . All of these panels fa i led at an inner s t r inger glue bond. A more rat ional panel design would be to have inner s t r ingers double the thickness of the outer s t r ingers . Then al l s t r ingers would be s t ressed near ly a l ike. This i s confirmed by the data for panel 6 in table 6 .

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Figure 10.--Comparison of t h e o r e t i c a l and e x p e r i m e n t a l midspan d e f l e c t i o n . δ . (M 142 565)

Figure 11.- - Comparison of theoret ical and experimental shear s t ra ins , γ 3 , of inner s t r ingers , (M 142 564)

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Figure 11 compares theoret ical with experimental values of s t r inger shear s t ra ins in the inner s t r ingers . Although there is considerable scat ter in the data points , the t rend shows good agreement between theory and experi-ment . The data scat ter could be at t r ibuted to local var ia t ion of e las t ic propert ies of the face veneers and also to inaccuracies in measuring maximum strains . The s t ra ins shown on the graph represent shear s t resses less than 200 pounds per square inch.

Figure 12 shows shear s t ra in data for the outer s t r ingers . The comparison between theory and experiment is not good here . The experimental values average about 30 percent higher than theoret ical as indicated by the dashed l ine on f igure 12. I t i s not known why there is disagreement between theory and experiment but this is not a cr i t ical mat ter s ince the design must be based on possible greater s t resses in inner s t r ingers .

A comparison of thin skin normal s t ra ins a t the s t r ingers is shown in f igure 13. Although there is data scat ter the general agreement between theoret ical and experimental s t ra ins is good. The skin s t ress level represented by the largest s t ra ins in the graph was less than 800 pounds per square inch.

The thick skin normal s t ra ins a t the s t r ingers are compared in f igure 14. More scat ter of the data points is exhibi ted than for the thin skins but general agreement is shown between theoret ical and experimental values . Thick skin s t resses a t the s t ra ins shown on the f igure were less than 700 pounds per square inch.

Figure 15 shows a comparison of theoret ical with experimental skin shear s t ra ins a t s t r ingers . In spi te of large scat ter there is some agreement between theoret ical and experimental data , The s t resses corresponding to these s t ra ins were less than 90 pounds per square inch.

Skin shear s t ra ins midway between s t r ingers theoret ical ly should be zero. The measured s t ra ins , as given in tables 1 through 6, were small but not equal to zero. Since they are small s t ra ins they are of no real consequence in design. The smaller s t ra ins between s t r ingers than at s t r ingers show that " shear lag" exis ts in both thick and thin skins .

Figure 16 shows values of skin normal s t ra ins between s t r ingers . Again, there is a great deal of scat ter in the data points but theoret ical and experimental values show some agreement . Stresses represented by these s t ra ins would be under 600 pounds per square inch.

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Figure 12. Comparison of theoret ical and experimental shear s t ra ins , γ 3 , of outer s t r ingers . (M 142 561)

Figure 13 . Comparison of theoret ical and experimental thin skin normal s t ra ins , ε 2 , a t s t r ingers . (M 142 559)

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- -Figure 14 . Comparison of theoret ical and experimental thick skin normal s t ra ins , ε 1 , a t s t r ingers l

(M 142 558)

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Figure 15.- - Comparison of theoret ical and experimental skin shear s t ra ins , γ , a t s t r ingers . (M 142 557)

Figure 16 .- - Comparison of theoret ical and experimental skin normal s t ra ins , ε , between s t r ingers . (M 142 556)

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CONCLUSIONS

1. The theoret ical analysis presented can be ut i l ized as a ra t ional design procedure for s t ressed- skin panels .

2 . Elementary shear- flow theory can be used to predict s t r inger shear s t resses . From this theory i t i s seen that inner s t r ingers of s t ressed- skin panels should be twice as thick as outer s t r ingers to equal ize the shear s t resses in the s t r ingers . Experimental resul ts confirmed this qui te c losely.

3 . The theory presented here is the s implest possible theory which s t i l l incorporates the essent ia l features of shear lag and is s t i l l capable of agreement with experimental data .

U S GOVERNMENT PRINTING OFFICE 1975-650-253-25 - 40-

The usual s t ressed- skin panel involves skins with different s t i f fnesses and also s t r ingers different f rom the skins . This general i ty resul ts in too many parameters for s imple graphs. Thus i t precludes solving the design problem, namely working the analysis in reverse to determine dimensions of components for given performance cr i ter ia . The design problem can be solved by an i terat ion procedure that can be easi ly adapted to a programable calculator , The fol lowing s teps are suggested:

a . Choose skin thicknesses and elast ic propert ies to comply with def lect ion performance under concentrated load between s t r ingers or other cr i ter ia such a s customary usage, e tc . (This s tep is beyond the scope of this paper . )

b . Determine required panel bending s t i f fness (EI) f rom design loads and def lect ion l imitat ions, including an est imated 10 percent shear def lect ion. Uti l ize e lementary mechanics of mater ia ls to determine a s t r inger depth to sat isfy panel bending s t i f fness (EI) required.

c . Determine s t r inger widths based on shear s t ress calculat ions by elementary mechanics .

d . Analyze theoret ical panel performance- - deflect ions and s t resses- - by theoret ical analysis given in this paper .

e . Compare theoret ical panel performance with design def lect ions and s t resses .

f . I terate , i f necessary, af ter choosing a new set of component dimensions.

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