Bruhn - Analysis And Design Of Flight Vehicles Structures Episode 2 Part 5 ppt

Since the thin skin has little bending stiffness, it cannot support the lateral pressure as a beam "plate", more correctly and hence must deflect to develop some tensile membrane stresse

A168 8

In examining the figure to determine what sort of canceling stress system must be supplied,

we see that the tangential hoop stresses border-

ing the cutout cannot be canceled by a self-

equilibrating set since they have a radial com-

ponent However, the radial component of these

stresses will actually be supplied by the door

or window pressing outward against its frame

Hence, it {s only the component of the hoop

stresses along a chord which need to be canceled

(Fig Al6-14b)*

The immediate problem becomes one of de- signing a structure to effectively support a set

of uniformly distributed self-equilibrating

stresses acting in the plane of the chord con-

necting the upper and lower edges of the opening

Fig A16.15

All that appears necessary to support the stress system is to provide horizontal headers

at the top and bottom of the cutout, which, 45

beams, will carry the loads across to the sides

of the frames where the loads cancel (Fig Alé6-

15b) For cutouts of usual sizes in pressurized

fuselages, the stress system to be supported in

this manner is quite large and it proves un-

economical to design a single horizontal frame

member of sufficient bending stiffness to resist

them Instead, the shell wall itself is em-

Ployed to help carry these loads across The

skin 1s used to form a beam of considerable

depth, the skin being the web of this beam, with

the horizontal frame member and one or more

longitudinals forming the beam flanges (Fig Alé-

15e)

Because of the heavy shear flows and direct stresses developed, the skin is usually doubled

in this region Additional stringers may also

be added to relieve the stresses The rings

pordering the cutout (and forming part of the

frame) are extended some distance above and be-

low the cutout proper (unless they coincide with

a reguiar ring location, in which case they

carry all the way around)

*Clearly one of the design requirements will be to make the

frame sufficiently stiff in bending against radial forces so

that the door or window can bear up evenly against the

frame

MEMBRANE STRESSES IN PRESSURE VESSELS

Fig Al6-16 shows the typical cutout structural arrangement While analytical approaches have been tried, it is probably safe to say that the true elastic stress diS~

tribution in such a configuration cannot be computed The necessity for avoiding nigh intensity stress concentrations (with their attendant fatigue likelihood) makes empirical information most useful in such cases Ôn the other hand, a simple rational analysis, based on principles outlined above, will very likely suffice for a static strength check and for most design purposes {Also see refersnce

8, pp 16-23)

The above discussion has concentrated attention on the problems of carrying the hoop

stresses around a cutout The longitudinal `

pressure stresses, while being smaller them~

selves, are intensified by bending stresses from the tail loads Hence, the longitudinal stresses across the cutout may make this con- dition (or the combination) most severe

Fig Al6.16 Structural arrangement

around a cutout Most or all of the

Shaded skin area would probably be

doubled

LARGE DEFLECTIONS OF PLANE PANELS; "QUILTING" The use of flat skin panels in a pressur- ized fuselage cannot always be avoided Since the thin skin has little bending stiffness, it cannot support the lateral pressure as a beam ("plate", more correctly) and hence must deflect

to develop some tensile membrane stresses which will then carry the loading The resultant bulges of the rectangular skin panels between their bordering stiffeners give a "quilted" ap- pearance to the surface

Even in the case of curved skin panels quilting will occur: if the internal stiffening framework (transverse rings and frames and longitudinal stringers) is relatively rigid and

is everywhere tightly fastened to the skin, then each skin panel is restrained along its

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES four sides (borders) against the radial expan-

sion normally associated with the shell membrane

stresses The result is a sort of three-dimen-

sional-case of the behavior depicted in Fig

From a structural viewpoint, the unfortun-

ate aspects of quilting lie in the high concen-

tration of stresses occurring near the panel

edges and in the tensile loadings on the rivets

which join shell to stiffeners The aerodynam-

ic characteristics of a quilted surface are

highly undesirable in a high performance air-

plane; hence again quilting is to be avoided

Computations of stresses in quilted panels,

inasmuch as they involve large, nonlinear de-~

flections, are difficult An additional (and

quite necessary) complication is that of having

to introduce the stiffness properties of the

bordering members The reader is referred to

Chapter A.17 for a further discussion of the

problem A simplified approach, indicative of

trends, is given there along with further ref-

erences to the literature

Al6.5 Shells of Revolution Under Unsymmetrical Loadings

Problems in which the shell of revolution

experiences unsymmetrical loadings are not un-

common in aircraft structural analysis The

nose of a fuselage, the external fuel tank and

the protruding radome are shelis of revolution

which may be leaded unsymmetrically by external

aerodynamic pressures Again, the same external

fuel tank shell receives an unsymmetric internal

hydrostatic pressure load from the weight of

fuel directed normal to the shell axis

Because of the unsymmetry of the problem,

membrane shear stresses are now present and 30

the analyst must solve not two, but three equa-

tions in three unknowns (Ng, Np and Ng) More-

over, these become differential rather than

algebraic equations

Because the derivation of the differential

equations of equilibrium is rather lengthy, and

because their general solution cannot be written

(rather, only specific solutions for certain

cases may be be found), no details are repro-

duced here The reader is referred to pp 373-

379 of reference 2 for the derivation of the

equations and for an example problem

© One design which reauces quilting in the curved skin,

fastens rings and irames to the inner surface of "hat" section

stringers only Thus the ring is not directly fastened to the

skin which is therefore not continuously restrained around

each ring circumference The result is a modified floating

A18,9 REFERENCES

API-ASME* Unified Pressure Vessel Code

isSl Edition, et seq

Timoshenko, S "Theory of Plates and

Shells"

McGraw-Hill, N.¥., 1940 (3) Watts, G and Lang, H., Stresses in a

Trans ASME, vol 74, 1952, pp 315-324

» Stresses in a Pressure Vessel With a Flat Head Closure, Trang ASME, vol 74, 1952, pp 1083-1090

» Stresses in a

Pressure Vessel With a Hemispherical Head,

Trans ASME, vol 75, 1953, pp 83-&89

Roark, R J "Formulas for Stress and

Strain", McGraw-Hill, N ¥ S€d Edition,

1954

Howland W and Beed, C Tests of Pres-

surized Cabin Structures, Journ Aero

Sci vol 8, Nov 1940

for Space Travel

Outer Space Vehicles will Present Many New Problems to

the Aeronautical Structures Engineer

* American Petroleum Institute - American Society of

Mechanical Engineers

CHAPTER A-17 BENDING OF PLATES ALFRED F, SCHMITT

Al7.I Introduction

It was seen in the last chapter that thin

curved shells can resist lateral loadings by

means of tensile-compressive membrane stresses

As will be seen later, thin flat sheets, by de-

flecting enough to provide both the necessary

curvature and stretch, may also develop mem=

brane stresses to support lateral loads In the

analysis of these situations no bending strength

is presumed in the sheet (membrane theory)

In contrast to the membrane, the plate is

a two-dimensional counterpart of the Deam, in

which transverse loads are resisted by flexural

and shear stresses, with no direct stresses in

its middle plane (neutral surface)

The skin may also be classified as either a

plate or a membrane depending upon the magnitude

of transverse deflections under loads Trans-

verse deflections of plates are small in compar-

ison with the plates’ thicknesses - on the order

of a tenth of the thickness On the other hand,

the transverse deflections of a membrane will be

on the order of ten times its thickness.*

Unfortunately for the engineers’ attempt at an orderly

cataloging of problems, most aircraft skins fall between the

above two extremes and hence behave as plates having some

membrane stresses

Plate bending investigations have for a

longtime been important in aircraft structural

analyses in their relation to sheet buckling

problems Recently they have assumed new im—

portance with the introduction of thick skinned

construction and still more recently with the

use of very thin low aspect ratio wings and

control surfaces which behave much like large

plates, or even are plates in some cases

It is the purpose of this chapter to pre-

sent briefly the classic plate formulas and

some applications Appropriate references are

cited in lieu of an exhaustive treatise, which

could hardly be presented in one chapter (or

even one volume) as witness the voluminous

literature on the subject

AlT.2 Plate Bending Equations **

Technical literature in this field abounds

with many excellent and elegant derivations of

the plate bending equations (references 1 and 2,

for instance) Rather than labor the subject

* As will be seen later, the presence or absence of mem~

brane stresses is not wholly dependent upon the magnitude

of deflections, but is also determined by the form of de~

flection surface assumed by the sheet (in turn dependent upon

the shape of boundary and loading),

**the assumptions implicit in the following analysis are

spelled out in detail in Art Al7,5, beiow AlT.1

with another such, we write the equations down

by a direct appeal to past experience and intuition

Fig Al7-1 shows the differential element

of a thin, initially flat plate, acted upon by bending moments (per unit length) My and My about axes parallel to the y and x directions respect- ively Sets of twisting couples My (2 - Myx) also act on the element

x (Twisting

ality is 1/EI, the reciprocal of the bending

stiffness For a unit width of beam I = t*/l2

In the case of a plate, due to the Poisson effect, the moment My also produces a (negative) curvature in the x, Z plane Thus, altogether,

with both moments acting, one has

a?w 12

ax? 7 Bee Mk ~My)

where uy is Poitsson’s ratio (about 3 for alumin- um) Likewise, the curvature in the y, z plane

give the moments in terms of the curvature

They are written

AlT.2

(and visa versa)***, It is proportional to the

twisting couple Myy A careful analysis (see

references 1 and 2) gives the relation as

) 3® w

H xây

Myy =D (1 -

Equations (1), (2) and (3) relate the applied

bending and twisting couples to the distortion

of the plate in much the same way as does

M = El d*y/dx® for a beam

While a few highly instructive problems may

be solved with these equations (see reference 1,

pp 45-49 and reference 2, pp 111-113), they

are of little technical importance Hence we

move on to consider bending due to lateral loads

Pig Al7-2 shows the same plate element as

in Fig Al7-1, but with the addition of internal

shear forces Q, and Qy {corresponding to the "Vv"

of beam theory) and a distributed transverse

pressure load q (psi) With the presence of

these shears, the bending and twisting moments

now vary along the plate as indicated in Fig

Al7-2a (For clarity, the several systems of

forces on the plate element were separated into

the two figures of Fig Al7-2 They do, of

course, all act simultaneously on the singie

e@lement)

ayraay

Fig Al7.2 The differentials are increments which

should be written more precisely as, for instance,

aa, = (aa /ay)ay

The next relations are obtained dy summing

moments in turn about the x and y axes For ex-

ample, we visualize the two loading sets of Fig

Al7-2 acting simultaneously om the single ele-

ment, and sum moments about the y axis

My dy + (Myy + d My) dx + (Qe +d Qy) dx dy =

(ty +d My) dy + Myy ax Dividing by dx dy and discarding the term of

higher order gives

“e* If w, the deflection function, is a continuous function of

Xand y (aa it must be, of course, in any technically im-

portant plate problem) then at each point d4w/dxdy =

3®w/Ôydx, as is proven in the calcuius,

To summarize, we tabulate below the quan- tities and equations obtained above For com- parison, the corresponding items from the engineering theory of beams are also listed

Structural Bending 7 Et? z1

Characteristic) Stiffness RBa- ”

7 Couples Ms My» Mey H

equations The result (which the student should obtain by himself as an exercise) is a

relation between the lateral loading q and the deflections w*:

* the corresponding equation for a simpie beam is

Q/EI = dty/ax+

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES The plate bending problem is thus reduced

to an integration of eq (7} For a given

lateral loading q (x, y), 4 deflection function

w (x, y) is sought which satisfies both eq (7)

and the specified boundary conditions Once

found, w (x, y) can de entered into eqs (1) to

(5) to determine the internal forces and stress—

es

AIT.3 An Dlustrative Plate Bending Analysis

Assume a lateral loading applied to a rec-

tangular plate having all edges simply supported

(hinged) The coordinates are chosen as in Fig

Al7-3 With foreknowledge of the general use-

fulness of the result, we assume 2 sinusoidal

loading of the form

4 = Qn sin SE* sin RAY - (8)

Fig A17.3 Sinusoidal loading on a

rectangular plate Sections through

the loading shown for m=3, n=2

To find the resulting deflected shape of

the plate we try a solution of the form

Woe Any sin ans sin aay Yo o. + - (9)

where Ag, is the unknown deflection amplitude

This trial deflection function 1s kmown to sat-

isfy the boundary conditions on the plate since

at x = 0, a and at y= 0, D We Have

w=0O (zero deflection at the supported

edges }

aawl twig (zero moment at the hinged

ax? "ay edges: see eqs 1 and 2)

t remains only to find the value of Ag, which

will satisfy eq (7) Substituting (&) and (9)

into (7) one obtains

we om mm” DH ait 5? ny? sin 22% sin DEY - (10) a b

The maximum deflection is seen to occur

where the trigonometric functions have values

of unity and q is also a maximum

If eq (10) is substituted into eqs (1), (2) and (3) one obtains

In a similar manner the transverse shears may be found from eqs (4) and (5)

With such results as these the plates’

stresses may be determined as desired For example, the maximum direct bending stresses are seen to occur where the shear stresses (due

to Myy) are zero, Thus

The reader having a familiarity with Fourier series

methods will recognize immediately that the above analysis

provides the key to the solution of the problem of any general

loading q (x, y} on the same plate Such an application is

made by determining the proper combination of sinusoidal pressure terms (each of the form of eq 3) such that their sum will closely represent the desired loading The sum of the

corresponding deflection functions (each of the form of eq 10)

gives the desired solution Details of this type of analysis are

to be found in reference 1 on pp 113-176 and 199-256,

In common with all problems which are formulated in terms of a partial differential

** the uniquenegs of solutions to the differential equation of the form of eq (7) is a classical proof appearing in num- erous advanced texts on mathematics and mathematical physics Since the equation is known to have a unique

solution, then any solution found for it is the one ana oniy correct solution

equation, the solution of the plate

lem depends strongly upon the Sound:

(both the shape of the boundary an¢ Stress and Deflection Coefficients for a Uniformly

support provided there} The above example may Loaded Rectangular Plate Having Various Edge

be said to nave been deceptively easy because of Conditions i

both the simple shape of the boundary and the | Long Sides | Short Sides |

type of support Plate problems rein the | All Sides Pinned, Pinned, | All Sides

plate planform is not a simple geometric figure ¡_ Pinned Short Sides | Long Sides | Clamped

type of support, 2 full discussion of boundary b/a | 4 B Gq 8 qatpioa I 8

conditions for plates is to be found in refer-

ence 1, 2D 69-95 .0443 | 2874 | 0209 | 420 | 0209 | 420 | 0138 | 3078

0616 | 3756 | 0340 | 522 | 0243 | 462 | 0188 | 3834 0770 | ,4518 | 0502 | 600 | 0262 | 486 | 0226 | 4356

AlT.4 Compilations of Results for Piate Bending Problems

Fortunately for the practicing engineer, it

is not necessary to perform analytic computations

as discussed above for the sreat majority of

practical plate problems Problems of the type

tllustrated above, plus the myriad variations „1400 | 7410 - te : - - -

possible, became very fashionable exercises „14168 | 74T16 - is - : -

amongst mathematicians following the discovery oo [1422 | 7500 | 1422] 750 | 0284 | 498 | 0284 | 498

by LaGrange of eq (7) in the year 1811 The

results of many researchers’ lebors have Deen

compiled in various forms for handy reference @ S Timoshenko, "Theory of Plates and Shells",

A common and important case is that of 4 pp 113-176, 199-256

uniformly loaded rectangular plate (Fig Al7-4)

The major engineering results are the values of

the maximum deflections and the maximum stresses

developed These may be put in the form (a is ® 8 J Roark, "Formulas for Stress and

the length of the short side): Strain", pp 202~207

’ =a 4 $ ~ oe ee te ee eee eee (12) Circular Plates Under Various Loadings

(same three references, in order)

Sux = B ae da (13) PP 55-64, 257-287

PP 129-132

where the coefficients a and B are given in ® pp, 194-201, 209-211

Table Al7.1 for the four most common edge

conditions A17.5 Deflection Limitations in Plate Analyses

In the introductory remarks of this chap- ter it was stated that a plate may be distin- guished from 4 membrane by the small order of its deflections (on the order of a few tenths

q = — statement here to show that this is not so

Le “—< tion imposed by one of the assumptions made in

There are several familiar assumptions from beam theory which, of course, carry over here, inasmuch as the plate analysis resembles Similar presentations may be made for many the beam analysis rather closely These “bean

dozens of other cases With the ready availa- theory assumptions” are:

bility of comprehensive catalogings of these

problems in references devoted to the purpose, 1 - elastic stresses only are presumed,

there appears to be little virtue in duplication 1i + small slopes (so that 3*w/3x* and

here Hence the following list of selected 37w/ay” are-good approximations to

references 1s presented Additional references the curvatures),

are to be found in turn within these works We

note that, because of the linearity of the plate

bending problem, superposition of solutions is

possible to extend even further the usefulness

of these extensive listings

tii - at least one transverse dimension (length or width) be large compared to the thickness so that shear deflections may be neglected

However, the beam theory assumptions do not

missible if these were the only restricting

assumptions

In deriving the plate bending equations it

was assumed that no stressas acted in the middle

(neutral) plane of the plate (no membrane

stresses) Thus, in summing forces to derive

eq (6), no membrane stresses were present to

help support the lateral load Now in the sol-

utions to the great majority of all plate bend-

ing problems (solved as in Art Al17.3), the de-

flection surface solution found ts 4 non—de-

velopable surface, i.e., a surface which cannot

be formed from a flat sheet without some strech-

ing of the sheets’ middle surface* But, if

appreciable middle surface strains must occur,

then large middle surface stresses will result,

invalidating the assumption upon which eq (6)

Thus, practically all loaded plates deform

into surfaces which induce some middle surface

stresses It is the necessity for holding down

the magnitude of these very powerful middle

surface stretching forces that results in the

more severe rule-of-thumb restriction that plate

Dending formulae apply accurately only to prob-

lems in which deflections are a few tenths of

the plates’? thickness

ALT.6 Membrane Action in Very Thin Plates

There is still another source of middle

surface strains in plates: this is the re-

straint against in-plane movements offered by

the edge supports while not important in prob-

lems wherein deflections are limited in accord-

ance with the restriction of the last article,

such restraint doas assume great importance in

the case of large deflections of very thin

plates which support a major share of the load

by membrane action It is, in fact, useful to

consider the limiting case of the flat membrane

which cannot support any of the lateral load oy

bending stresses and hence has to deflect and

stretch to develop both the necessary curvatures

and membrane stresses

The two-dimensional membrane problem is a

nonlinear one whose soluticn has proven to be

very difficult Rather than attempt to treat

the complete problem, we can study a simplified

version whose solution retains the desired

general features The one-dimensional analysis

of a narrow (unit width) strip will be treated

This strip is cut from an originally flat mem-

brane whose extent in the y-direction is very

great (Fig Al7-Sa)

* The cone and cylinder are examples of developable sur-

faces, the sphere is a nondevelopable one It is a familiar

experience that the skin of an orange cannot be developed in-

to a flat sheet without tearing

where s is the membrane stress in psi

Eq (14) 1S the differential equation of

@ parabola Its solution is

4x

2st

The (as yet) unknown stress in eq (15)

can be found by computing the change in length

of the strip as it deflects This “stretch” is given by the difference between the curved arc length and the criginal straight length (a)

aw\?\*/* - 1 faw\?

("=i

*here "ds" is the differential arc length of the calculus

and has no kinship with the s which denotes the membrane

stress throughout the remainder of the analysis

Substituting through the use of eq

If eq (16) is substituted into eq (15)

one gets for the maximum deflection (x = 3 )

a/s

wax * -560 4 #‡) — (17)

Equations (16) and (17) display the essen-

tial nonlinearity of the problem, the stress and

the deflection both varying as fractional ex-

ponents of the lateral pressure q

Solutions of the complete two-dimensional

nonlinear membrane problem have been carried

out*, the results being expressed in forms iden-

tical with those obtained above for the one~

dimensional problem, viz.,

Here "a" ts the length of the long side of the

rectangular membrane and ni and ne are given in

Table Al7.2 as functions of the panel aspect

ratio a/b

The maximum membrane stress (SMAx) oceurs

“The work of Henky and Foppl is summarized in reference 3,

pp 258-290 and in reference 4 The partial differential

equation solved is given in reference 1 on p 344 (eq 202)

and the approximate method of solution usually employed is

sketched out on pp 345, 346 of this same reference The

reader who would compare presentations amongst these ref-

erences should note the differences in the definitions of the

plate dimensioning symbols ‘a" and "b"'

BENDING OF PLATES

at the middle of the long side of the panel

We note that the limiting case, a/b = 0, cor- responds to the one-dimensional case analyzed earlier - Unfortunately, an extrapolation of these two-dimensional results to that limit does not show agreement with the one-dimen- sional result Presumably the discrepancy may

be traced to the excessive influence of inac~

curacies im the assumed deflection shape of the membrane as used in the approximate two-dimen- sional solutions

Experimental results reported in reference

4 show good agreement with the theory for square panels in the elastic range

TABLE A1T.2 Membrane Stress and Deflection Coefficients

A1T.T Large Deflsctions in Plates**

In the previous articles of this chapter the results of analyses were outlined for the two extreme cases of sheet panels under lateral loads At one extreme, sheets whose bending stiffness ts great relative to the loads applied (and which therefore deflect only slightly} may

be analyzed satisfactorily dy the plate bending solutions, At the other extreme, very thin Sheets, under lateral loads great enough to cause large deflections, may be treated as mem~ branes whose bending stiffness is ignored

As it happens, the most efficient plating designs generally fall between these two ex- tremes On the one hand, if the designer is

to take advantage of the presence of the tn~

terior stiffening structure (rings, bulkheads, stringers, etc.), which is’ usually present for other reasons anyway, then it 1s not necessary

to make the skin so heavy as to behave like a

"pure" plate On the other hand, if the skin

ts made so thin as to necessitate supporting all pressure loads by stretching and developing membrane stresses, then permanent deformation results, producing “quilting” or "washboarding" The exact analysis of the two-dimensional plate which undergoes large deflections and thereby supports the lateral loading partly by its bending resistance and partly by membrane action is very involved A one-dimensional

** The discussion to follow will be concerned primarily with problems dealing with the support of a uniform pressure load on a flat skin panei It may, therefore, help the

reader to tix his ideas if he visualizes the discussion as applied to the probiems of analysis of a single rectangular

skin panel taken between the stringers and bulkheads of a

seaplane huil bottom Equaily useful is the picture of the

very nearly flat panel between rings and stringers in the slightly curved side of a large pressurized fuselage

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES analysis, parallel to that of Art Al7.6, is to

be found in reference 1, pp 4+10 A more

elaborate two-dimensional analysis is shown on

pp 347-350 of this same reference

An approximate solution of the large de-

flection plate problem can be obtained by adding

together the flat plate and membrane solutions

in the following way:

Solve eq (12), the plate bending relation,

for q; call it q',

qt = max B ĐỀ

a at Now solve eq (18), the membrane relation, for

a; call it q",

qr = “ax Et n,? a The sum of these two pressures gives the total

lateral pressure, called simply, q

q=q'+q"

Eq (20), we see,.is based upon summing the in-

dividual stiffnesses of the two extreme be-

havior mechanisms by which.a flat sheet can

support a lateral load No interaction between

stress systems is assumed and, since the system

is nonlinear, the result can be an approximation

only

£q (20) is best rewritten as

Fig Al7-6 shows eq (21) plotted for a

square plate using values of a and nm, as taken

from Tables Al7.1 and Al7.2 Also plotted are

the results of an exact analysis (reference 5)

AS may be seen, eq (21) is somewhat conserva-

tive inasmuch as it gives a deflection which is

too large for a given pressure

Fig Al7.6 Deflections at the midpoint of a simply

supported square panel by two large-deflection

ALT.7

The approximate large-deflection method outlined above has serious shortcomings insofar

as the prediction of stresses is concerned

For simply supported edges the maximum combined stresses are known to occur at the panel mid- point Fig Al7-7 shows plots of these Stresses for a square panel as predicted by the approxi- mate method (substituting q’ and q” into eqs

(13) and (14) respectively and cross plotting with the aid of Flg Al7-6*) Also shown are

the maximum stresses computed by the exact

large-deflection theory (reference 5)

Fig Al7.7 Large-deflection theories’ mid-

panei stresses; simply supported square panel

Because of the obvious desirability of using the results of the more exact theory, some

of these are presented in Table Al?7.3 The treatment of additional cases (other types of edge support) may be found in reference 6, pp

221, 222,

TARLE AIT.3 Large Deflection Rectangular Plate Comfflcieote (Gnitorm Pressure Load (4), Siemoiy $mggortad Edges)

18.49 [11.00 | 18, 20 | Độ, 30 yom} ozo | 1.60 | 3.ao | 4.00} 80g} sun | 7.00 | 7.98 | 8.60 | 10.20

NGhh&: - 1 sg = “bending stress” compooent of sirea

3 thự 3 “memprane stress” component of 61788684

J total atresa stg + ye

Al17.8 Considerations in the Applications of Large-De-

flection Plate and Membrane Analyses

Before concluding this chapter it is pertinent to note several serious omissions in the developments outlined above with regard to their application to flat pressure-panel analyses within a ship hull or fuselage The

* but using ng = 260 in eq (19), This value gives the

stresses at the center of a square panel whereas ng =

956 in Table Al7 2 is for stresses at the panel edge

A1T.8

large-deflection plate and the membrane analyses

were dev2loped for applications where the plate

bending analysis appeared inadequats However,

these analyses themselves presumed conditions

seidom encountered in practice

FIRST, the analyses assume unylelding sup-

ports on the boundaries of the sheet panel In

practice, the skin is stretched across an elas-

tic framework of stringers and bulkheads It

follows, therefore, that the heavy membrane

tensile forces developed during large deflec-

tions will cause the supports to deflect towards

each other thereby increasing the plate de-

flection and relieving some of the stresses

A simple one-dimenstonal analysis for a

membrane strip having elastic edge supports

(parallel to the analysis of Art Al7.6), shows

errors on the order of 25 per cent are likely if

the framework elasticity is neglected (reference

7), At this writing no two-dimensional treat-

ment of this problem is known to the writer

SECOND, it is seldom that the analyst has

to check a panel for lateral pressure loads

alone Most often, the entire "field" of panels

on the framework of stringers and bulkheads must

simultaneously transmit in-plane loadings from

the tail load bending stresses and the cabin

pressurization stresses

Inasmuch as the large-deflection plate and

membrane analyses are nonlinear, it follows that

correct stresses cannot be found by 4 straight

superposition The magnitude of the error in-

troduced by such a procedure is difficult to

estimate in the absence of an exact analysis A

one-dimensional analysis, parallel to that of

Art Al7.6, but with elastic supports and axial

load, is given in reference 7 These results,

which indicate the effect of the axial load to

be quite important, may be used as a guide in

lieu of more complete two-dimensional studies

The interested reader is referred to the orig-

inal work for details

BENDING OF PLATES

REFERENCES Timoshenko, S "Theory of Plates and Shells", McGraw-Hill, N ¥., 1940

Den Hartog, J P "Advanced Strength of Materials", McGraw-Hill, N Y., 1952

Sechler, BE and Dunn, L “Airplane Struc- tural Analysis and Design", Jonn Wiley,

N Y., 1942

Heubert, M and Sommer, A., Rectangular Sheil Plating Under Uniformly Distributed Hydro~

static Pressure, NACA TM 965

{selected large-deflection plate references) a) Moness, EB, Flat Plates Under Pressure, Journ, Aero Sci., 5, Sept 1938

b) Ramberg, W., McPherson, A and Levy, S.,

Normal Pressure Tests of Rectangular

Plates, NACA TR 748, 1942

c) Levy, S Square Plate With Clamped &

Under Normal Pressure Producing Large

f) Chi-Ten wang, Bending of Rectangular Plates With Large Deflections, NACA TN

CHAPTER A18 THEORY OF THE INSTABILITY OF COLUMNS AND THIN SHEETS

{BY DR GEORGE LIANIS)

PART 1

ELASTIC AND INELASTIC INSTABILITY OF COLUMNS

À18.1 Introduction

Part 1 of this chapter will be confined

to the theoretical treatment of the instability

of a perfect elastic column and an imperfect

elastic colum The column is the simplest of

the various types of structural elements that

are.subject to the phenomenon of instability

The theory as developed for columns forms the

pasis for the study of the instability of thin

plates, which subject is treated in Part 2

A18.2 Combined Bending and Compression of Columns

Consider a column with one end simply

supported and ths other end hinged (Pig A18.1)

under the simultaneous action of a compressive

load P and a transverse load Q Without the

load P the bending moment due to Q would be:-

Due to the deflection u(z), the axial

load P contributes to the bending moment by

For eqs (4), since u = 0 for z = 0 and

Z = 1, it follows that:

At z = (1 - a) the two portions of the deflection curve given by (4a) and (4b) respectively must have the same deflection and slope From these two conditions we determine

Ca and O,

Ala t