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 Al6~15a)
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 Ơn 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
Trang 2ANALYSIS 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
A16-12"
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
skin
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 (4) (5) Designs by Krafft Ehricke of Convair
for Space Travel
Outer Space Vehicles will Present Many New Problems to the Aeronautical Structures Engineer
* American Petroleum Institute - American Society of
Mechanical Engineers
Trang 4CHAPTER 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 couples shown by right hand vector rule.) x ye Fig Al7.1
AS in the case of a beam, the curvature in the x,2 plane, a*w/ax?, is proportional to the moment My applied The constant of proportion~ 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 is at w 12 age” Bee My ~ Mk) These two equations are usually rearranged to
give the moments in terms of the curvature
They are written - a?w m0 (8+ = atw ew) LL £ wy 20(2 y -ằ 1) - (2) where D = st°/i2 (1 - 4")
Trang 5
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
(a)
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/Ơydx, as is proven in the calcuius,
BENDING OF PLATES
In a similar manner, a moment summation about the x axis ylelds
5 2 OM, Oo My
oy ay ax
(Equations (4) and (5) correspond to V = aM/dx in beam theory)
One final equation is obtained by summing forces in the z direction on the element:
âQy 2 Q
Q* ox tay crc crc ccs cc
Equations (4), (5) and (6) provide three additional equations in the three additional quantities Qy, Qy and q The plate problem is thus completely defined
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 At BEAM CLASS: Tat PLATE THECRY THEORY Coordinates xy x Geometry Deflections w Ỹ Em 2 Distortions|So> » S53 » Seay tt
Structural Bending 7 Et? z1
Characteristic) Stiffness RBa- ”
7 Couples Ms My» Mey H Loadings Shears Rr ey v Lataral q q or w ate aw “Hooke?s Moment- My #0 (a +h Oot Law" Distortion ah Relation my 22 (28+ 2 SNe st oy =p(3 3, 3Í „ạị $ = Ta: Mụey ZD (1 ~ ”) my 3My acy gee - ox ay an Equilibrium Moments ve ix ay 2 Me 2y ox = Or, HY av Forces at ax * oy 4q*xy
Finally, one very important equation is obtained by eliminating all internal forces (Mx, My, Mxy, Qx, Gy) between the above six
equations The result (which the student should obtain by himself as an exercise) is a
Trang 6ANALYSIS 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) m,nđn=1, 9, 3; ~ -— a(x) r a —_" b Po z_
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 mn? n@nđ nộn m*n* Aon =e + 2 Am Wg bì * Amn Dt > =am/D * or mđx nn1ÿy *the common (actor sin —T— sin ~— has been divided out AlT.3 San aaa An 71 # n „t( 4 D n* az + 3) n
Hence the required deflection surface (and the solution to the problem ") has the equation
1
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 sx (= %) = Het I ar - 8% t? and hence a = 8 don (Bs * ie) 1 Semax 77a a(my, my) Cố 2) vtat(h+ 3)
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
Trang 7
A1T 4 BENDING OF STRESSES
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 TABLE AlT.1
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
must be solved by numerical ‘s Clamped 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 0906 | 5172 | 0658 | 654 | 0273 | 500 | 0251 | 4680 «1017 1 $688 | 0799 | 690 | 0279 | 302 ; 0267 | 4872 1106 | 6102 | 0987 | 714 | 0284 | 504 | 0277 | 4974 } .1436 | 7134!,128 | 750 - - - -
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 “t#l*#lrlrlrlr|r Slelelel=|=|>|»s|s ot Rectangular Plates Under Various Loadings @ J P Den Hartog, "Advanced Strength of Materials", pp 132-134
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 2 as L Vartou dine:
’ =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
aw › of its thickness) We will re-examine this
q = — statement here to show that this is not so
Dae much a definition as it is an accuracy limita~
Le “—< tion imposed by one of the assumptions made in
Fig A17.4 the plate analysis
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
Trang 8
ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES AlT.5
However, the beam theory assumptions do not
place 4 very severe restriction on the magnitude gS 4
of deflections permitted Deflections of sev~ at
eral times the plate thickness would be per- Las >
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)
was derived -
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
ca)
£ (a) Perce 2 2) oh
Fig A17.5
Fig 417-Sb shows the desired one-dimen- sional problem which now resembles a loaded cable The differential equation of equilib- rium ts obtained by summing vertical forces on the element of Fig Al7-Se (draw with all quantities; loads, deflections, slopes and curvatures shown positive) One obtains * aw aw = Sti -& +®qdx=0 x+ dx x or daw _ fa = - gt ctr ttt ttt te (14)
where s is the membrane stress in psi
Eq (14) 1S the differential equation of @ parabola Its solution is
4x
2st
(a-x)-+ - (15)
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) Thus: a o-| ds-a * a a =} ydw? + dx” - a a ——m— < aw _ = Jr +(¥) dx -a °
Since the slope dw/dx is small compared with unity, we use the binomial theorem to write
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
Trang 9A1T.6 Hence a w (1 +3 (8) ]e-s Baw 2 zl j fw “(@ a 2
Substituting through the use of eq
tegrating we find (15) and in- a3 5= 245 € Now by elementary considerations ga oF Equating these last and solving we find s+ ce ($2)"Jo. - (as)
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., Wax = " „ m— vl a a sỊ we » Lo ¥ ~ ° 1 ‘ ‘ ' ' ' ' ' ' + m 0 SMAX
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 a/b 1.0 1.5 2.0 2.5 3.0 4.0 5.0 ny 3J18 | ,228 | 16 125 | 10 068 | ,053 nạ „3586 | 3T 336 | 304 | 272 | 23 „208
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
Trang 10ANALYSIS 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"
-i st? 1 BE, *
Tan Os awa? "MAK + + + - (20)
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
Tê <3 (Ha a), 1 fy)”
Ett ° 9G) nề
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 250 exact eprer, / 200 „ 150 ayy Fe" 100 15 30 yinesa PLA? -—— 9 oO 065 w/t 1.0 1.9 2.0 Fig Al7.6 Deflections at the midpoint of a simply
supported square panel by two large-deflection
theories
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) » exc — approx _ 20 = sa +” nt Ee? ta 100 3 150 200 250 at Bt
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) ereware Coetficient, gà ws T a 30 7 100 | 1242 | 150 | tm | 200 | 250 a wh | 440| 0| 830| 1.13| t.28 li wat fae | ea] ate opb?/m71 3.89 [5.80 | 6.10 | 10.90 [13.80 xi Dediection afo | or Stress Coettieiand 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 amo wrt a} 146 | 1.34 | ise] too} Laz] tee | toe} Ros) azo 18.0 [a6 jae {23.60 a.aa [1.10 | 10.90 } 12.29 sgb2/BỞ| 44T |1.16 [10,30 | 12,80 | 14.00 | 18,40 +eS/E2|t.29 [2.40 | 414 | 3.61| %.94} 4.10 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 =
Trang 11
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
Deflections, NACA TR 740, 1942
ad) Levy, 8 Bending of Rectangular Plates With Large Deflections, NACA TR 737, 1942, © Chi-Teh Wang, Nonlinear Large Deflection
Boundary-value Problems or Rectangular Plates, NACA TN 1425, 1948
Trang 12CHAPTER 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:- P q-a) Qt {oS E—z —* Fig A181 On the lower portion of the column My On the upper portion MỸ = 208) (2) ~ ee ee ee (b) Due to the deflection u(z), the axial load P contributes to the bending moment by
the amount:-
Thus the total bending moment at section 2 will be:-
- M= pu + 282 on upper portion - =~ - - (4)
M= Pu + ạ0-4)0-z) on lower portion (e) From mechanics of simple bending, we have the deflection equation, d7u M wtcny TTT TTT (2) Thus the deflection u(Z) of the colum is, atu Qaz BIQge 27 Pur > (9 =221- a) ay (1-2) (1-2) B1 qua “~ Pu- 9 1 , (leaSa2til) - (2) If we tntroduce the notation, Bae -+ - (3) The general solution of eq (2) is: u= 0, cos Kz + Cy sin K2 - 2 > (Of21- a) a= Cs cos ka +0, sin ke - q Grae) (leasz22l) - (4b)
where C,, Cg, Cs, and C, are constants of integration to be determined from boundary conditions
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,
Trang 13
Al8.2 THEORY OF THE INSTABILITY OF COLUMNS AND THIN SHEETS
-Ưị 9 sin Ka = Q sin K(1-a) (g) the body would cause only infinitesimal changes
Ca* Be Sin kT? Ce PK tan KL & in the displacements and the body recovers if
Substituting (f) and (g) tnto eq (2), we obtain: Q sin Ka “7 ĐK sin KỊ S11 ĐZ - ẤT 2 , (OSz28l-a) -2 8 (5a) = @sin K(1-a) 91-4) (1-2) US "PR sin xt $!" K(l-z) - NT, (leaSzgs1)+ -. -2 (5b)
If load Q is applied at the middle of the column the maximum deflection is:~ K1 Grane 2 PK gl - ÁP tmax “ It 1s obvious that fo: es KI
tan 3 —— oe, Thus the maximum deflection of Column becemes infinite for KI = 7 and trom
eq (3)
Equation (7) is an important equation derived first by Euler It gives the critical compressive load which causes infinite deflect- ton in a column and it specifies the ultimate strength of a column tn compression
It is obvious from eq (7) that Euler’s
critical load is independent of the magnitude
of the transverse load Q It seems, therefore, that even in the absence of the transverse load Q, the maximm deflection becomes infinite under the action of only a compressive load as eSiven by eq (7)
A18.3 Elastic Stability of a Column
The above conclusion as to the critical load was based on purely mathematical reasoning We have found a critical value of a compressive load which causes infinite deflection
Far more important, however, 1s an in- vestigation of the stability of a column which Should be based on physical arguments The question arises as to what happens before the load P reaches its critical value as given by eq (7) and also how the column behaves if this critical value is exceeded
An elastic system is called stable under given loads when infinitesimal loads added to
the added loads are removed When the dis- placements are continuously increased with little or no further increment of loads, the system is unstable If the body will remain in the displaced position after the removal of the disturbance, the body 1s said to be in neutral equilibrium Having these definitions, we will not investigate the benavior of the column before and after the critical load is Teached,
Plg A18.2
Assume, aS shown ín F1g A18.2, that a simply supported column loaded by an axial load P is bent by a small disturbance 1? the deflection is u, the bending moment due to P igs Pu, From basic mechanics, we know that, EI xz? -M , whence BI _
at “Pu -w~ see ee LL (8a)
The exact expression for the curvatuve of the neutral axis is:-
z? $ ; Where 3 1s the are length of the deformed axis, and 9 the angle between the
tangent to the curve and the z axis Thus,
do =
El ggtPuto -+ (8b) Differentiating (8b) with respect to s and since 2 = sin @, we obtain:
4
BỊ ĐC +P sin 9= 0 wetter eee (8e)
Multiplying (8c) by d@ and noting that;~ 1°98 ag „ d9 „ ,d6, Gp? ce = as ¢ as ; and integrating d9 „ ,d@ BIf a (qs) + P sin ode = c, or El /d9,2 - 3 Ge) - Pcs gee wee ee ee (8d } , - = d6
Trang 14ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES A18.3 =/2 Po Pl? _ 4K”
Now let k= Vero -+ (8a) Pop EL OT TTT TT (8n)
Then eq (8d) becomes, L de
PS a = -k Y cos O-cos a, or
2 ds Let us now write the bending moment M =
P6 at the middle point in non-dimentional form:
ds = - —— - - - (8#)
VỀ 8K (eos @9—-cos d)
The total length of the column in the deflected shape is given by:- 49 1 1 1= đ8 =~/ ` ————D——— - ~ - (8h) $ "8 VãkVeos9~=cosq ol= (2 ———=———— (0L) 2kv sin^® 2 ~sin?9/2 4 Denoting sin 2 by p and introducing a new variable g :- sin 3 =psing Equation (81) then becomes, T 2 pe ae 2K Le eae 1-p* sin® @ oe (83) T1 where K = Le 28 is called the ¥ l+p? sin? g
complete elliptic integral of the first Kind, and it can be found in tables If a and therefore p is very small, “hen p? sin? @ can be neglected in equation (€j) and then TL =2 7? wgekan J Hb 1= z4 aspen P aa whence Pe MEL LL LLL ee (8k) 12
The deflection at the midpoint of colum is:- 90, du =ds sin 9, and from Eq (8f) u(z= ÿ#**z~(° =H—— s49 ~— ~ (B1) /sin* > - sin? 3 or in terms of Ø:- 3 x an (êm)
From equation (83) and equation (8k), we obtain
Since p = sin Ỹ is a function of a so ts the
elliptic integral K and the ratios a and Ÿ calculated from equations (8) and (80)
P 6
s— is a function of — calculated by
Por 1
means of tables giving elliptic integrals Thus m can be plotted against 6/1 as shown in Fig A18.3 Thus SA a4 Fig, A18.3
Let us mow examine the stability of various equilibrium configurations Assume
that a load P* is acting on the column and
the column has a certain maximum deflection 6 where P* does not correspond to 6
The non-dimensional maximum bending moment is:-
+ P` ,ơ a
wees ——— (8g)
The m* versus 6/1 curves are straight lines
The column is in equilibrium if m =m‘, or in other words, if the m'(6/1) curve intersects
the m(6/1) curve
We see from Fig 418.2 that these curves intersect at the origin only if P*< Pop The column, therefore, nas only one possible equilibrium form, for example, that for which 6/1 = 0, which is the straight form When
Trang 15Al8.4
one when ơ/1 = 0 and the other (point A) for which 6/1 #0 The column thus has two possible equilibrium forms, one straight and
one bent
Let us now assume that at 6 = 0, the colum is displaced by a small disturbance
and acquires a deflection 6 For P* we see from Fig 3 that m= m*
not sufficient to maintain the column in
equilibrium in the bent form and it will spring
back to its straight form Thus for P` « Pẹp,
the straight form is stable
If P* > Poy, then m+ =m Thus m* will
bend the column still further This means
that if P* » Por, the straight form of equilibrium is unstable The column will
continue to bend until m* becomes equal to m
(point A in Fig 3) If the column ts dis- placed further from A, the deflection becomes
larger than 6, and m =m” at the new position
The column will spring back to point A Point A is therefore stable
At P = Pop, the m* versus 6/1 line is tangent to the m curve at the origin There-
fore, for an infinitesimal disturbance, the column will remain in equilibrium at the displaced position since for such small
disturbances m* remains equal to m The column is therefore in neutral equilibrium
Ai8.4 The Failure of Columas by Compression
in discussing the stability of a colum in the previous section, it was shown that below the critical Euler load (Eq 7), the Straight form is stable, above the Pyy the bent form is stable and at Poy the equilibrium
is neutral By plotting the curve P/Por versus
6/1 as shown in Fig Al8.4, we observe the following behavior (a) Below P/Por = 1, there is only one equilibrium position, 6/1 = 0 P/Per Fig Al8.4
THEORY OF THE INSTABILITY OF COLUMNS AND THIN SHEETS
(b) At P/Pey = 1, or at point (A), a bifurcation
of equilibrium occurs and the colum starts
to acquire two possible neighbor positions of equilibrium, the straight and the bent {c) Above P/Poy = 1, the column has two possible
equilibrium positions 6/1 = 0 and 4/1 # 0 Thus as far as initiation of instability is concerned, the Buler load as given by &q 7 can be considered as the critical load The question arises whether this load has a practical use for design purposes A logical design criterion is obviously the maximum load which a column can Sustain We observe from Fig Al8.4 that the load P increases for increasing displacement 6 This behavior is due to the development of large deflections due to bending However, over a considerable range of deflections 5, the P + 6 curve is practically horizontal (for instance, between points A and B the ratio 6/1 vartes from zero to 0.4) For such large deflections for which the column load does not change
practically, it is obvious that the colum ceases to function properly Therefore, from this point of view, the Euler load can be con- sidered that which characterizes the maximum strength of the column
The rising part of ths curve BD holds ag long as the material behaves elastically At some point 3, however, inside the almost flat portion of the curve the imner fibers of the column acquire maximum stress equal to the yield stress If we carry out an elastic-— Plastic analysis of the subsequent behavior, we observe that the curve drops almost
immediately Again this maximum load Pg is very near the Euler load For design purposes, therefore, the Euler load, which is a buckling load, is a very good approximation to the ultimate load which the column can sustain
Another argument will confirm the above conclusion In discussing the ouckling of
columns in the previous paragraphs, we have assumed that the column is initially straight,
Pp centrally loaded and made of homogeneous material Actual
columns, however, are imperfect
due to initial crookedness (for instance, due to unavoidable tolerances in their manufacture), due to slight load eccentricities and due to lack of complete 1 homogenity Therefore, a certain
amount of bending is always
ope present even for small loads
Let us now examine the
behavior of such initially tm- perfect columns by assuming a certain initial deflection ug of the column axis (see Fig xt
Fig
A18 8 Al8.5) For small deflections,
Trang 16ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES the change cf curvature due to subsequent
vending {after loading) {s:-
1, „du @u
9 ( “re
In the differential equation of deflection one can prove that 1/R 1s the change of
curvature which for an initial straight column coineides with the curvature itself Thus in the present case, where the bending moment is Pu, the equation of deflection becomes:~ đu dz* 7 azz TS” TT (9) Let us express ug in Fournter series:~ = TUZ uạ^šỔ sin—~ 9 = B Oy T1 -+ (10) a
Substituting (10) in (9), we find the solution which satisfies the boundary conditions (u = 0 for z 20, z22#1) is:- oo ust On sin “ -e (11a) n=] 8 where ơn = Mo -—-—¬ =——- (110) 1~P/Pn aia =~ ARE fa The deflection of the column at the center 181~ Omax = 2 - 63 + 5s nh nan (12)
If we plot the deflection versus the load we obtain the curve (Fig Al8.6), which P/Por Fig A18.6 Al8, 5 approaches the horizontal line P/Pgy = 1 asymptotically This curve, however, is valid for small deflections for which the approxi- mation:-
8 & = is valid
By a treatment similar to that in the previous paragraph, we will find that for
large deflections the load deflection curve
raises after the point I (curve FIH) Due to the onset of plasticity, the actual curve drops at the point I* (curve FII*H*) The
failing load at I* can be either greater or
smaller than Por, but it is usually very near to it
In the above discussions we have shown that for all practical purposes the Euler buckling load can be considered as the ulti- mate load which a real or practical column can sustain Besides its closeness to the actual ultimate load, the critical load can be easily calculated from equation (7) with- out the necessity of carrying out a lengthy calculation which will include the initial
imperfections and plastic effects
It should be noted, however, that the buckling load given by equation (7) is valid when the uniform stress due to 4 compressive load (go = P/A, where A is cross-sectional
area) is below yield stress Ifo is above
the yield stress, the theory of plasticity predicts another value for the buckling lead Referring now to equations (11) we find:-
Py = 1 #Per (Per from equation 7) ễ 6) = ——2 E +~ n*Por Sie S28 4 , 8 4578 ate 3 Thus 6, >= 6,>= 6, and:-
In a buckling test we measure 5 = Spay
- 5, where 5 is the initial deflection at the
middle point Thus:-
6 = by, & = _ỗÄ — and
Trang 17A18.8
If in a buckling test we plot 6/P versus 6, we can obtain the critical load expert- mentally without knowing the initial deflection 6, It is simply the inverse of the slope of
this curve
A18.5 Buckling Loads of Columns with Various End
Conditions
From the conclusions reached in the previous discussion, we can consider the buckling problem as an instability problem of an initially straight column Thus we assume a certain deflected position near the straight configuration as another possible equilibrium form and seek the loads under which the non- straight form is possible Furthermore, only a small deflection analysis is necessary
The general differential equation of bending-buckling is:~
aU, ys az" dâu F1 =0
and the general solution is:-
usc, sin kz +C, cos kz +C,z+C, — ~ (15) The coefficients C,, C,, C, and C,
depend on the conditions of the end supports The various end conditions are:- đ u đệu _ free end:~ am” Oo, as 9 a
Pinends + uso , Tê 0
Fixed end:-u 20 , Sao ? dz
Thus we have 4 end conditions These give systems of four linear nomogenous equations A trivial solution of these is the zero solution For the buckling state, however, C,, Cy, Cy, Cy are not all zero The condition of non-zero solution of the above system is that the determinant of the coefficients of C,, Cz, Cy and C, ts equal to zero From this equation, we calculate the buckling load
For example, in a simply supported beam,
2
(ua 20, os 0, at both ends), the end
conditfons furnish give C,+Q 20 , 0, =0
C, sin kl + C, cos kl +C,l +, = 90 For buckling we must have:-
THEORY OF THE INSTABILITY OF COLUMNS AND THIN SHEETS 1 9 1 1 9 0 sin kl cos ki 1 1 = 0 - sin kl -+cos kl 0 90 or sin kl =0
whence Py, #S—— TT — +~ -+ (1ea)
Thus for P, equal to the right nand side of equation (16a), we have a possible form of equilibrium of the bent form The smallest value of Py occurs at n= 1, and this is the buckling load:
nr eT
Por = 1Ễ
The buckling load for other end conditions can be derived in Similar manner
INELASTIC COLUMN STRENCTH Al8,6 Inelastic Buckling introduction
Euler’s theory of buckling is valid as long as the stress in the column nowhere ex-~ ceeds the elastic limit of the column material We have seen that the analysis for perfect and
imperfect elastic columns leads to the same result, namely, equation (7)
The case of the inelastic buckling, that is, instability under axial load exceeding the elastic limit stress, presents some difficulties AS we will see, the perfect column analysis leads to 4 different expression for the critical load than for the perfect colum This is due to the fact that in the plastic stress range, the material behaves differently under loading and unloading, as illustrated in Pig 418.7 Let us now examine the two cases: Loading Stresso Deformation ¢ Fig Al8.7
Trang 18
— =
re a
ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES A18.T
the column becomes unstable we assume:- 6S, =86e, + -+-++ +-454 (19) (1) That the displacements are small so that
the relation between the radius of curvature R and the deflection u of the
elastic axis is, đâu
az?
+ R
(2) Plane sections remain plane, therefore
the change of strain due to bending at a
distance h on the plane of bending is,
(3) The stress-strain relation follows the Simple tension curve for the material (4) The plane of bending is a plane of
symmetry of the cross-section
Assume now a column with the cross-section as shown in Fig Al8.8a be compressed in the
Fig A18 8b
Fig A18 8a
plastic stress range and that the compressive stress prior to instability-beo To consider the condition of buckling, let the column be Slightly deflected transversely The stress on one side of the column will then increase due to the bending following the stress-strain curve, while on the other side the stress will decrease and will therefore follow the un- loading elastic line shown in Pig Al8.8b
Por small changes of the stress, on the first side, the variation of stress {s related to the variation of strain by:-
6S, = Ey (co) Oe
where Et (go) 1s the slope of the stress-strain curve at stress g On the second side the changes will follow the elastic relation,
that is,
The distribution of the compressive (-) stress and the tensile (+) stress due to bending is shown in Fig Al8.6a The stress becomes zero on line (a a+), which ts ata
distance e from the centroid c For
equilibrium of stresses on the cross-section we have, - £% o5,aa + £72 og,0a = 0 - - ~- (20) and for equilibrium for moments, £3" 68, (hy+e)aa + £3 6S, (Re-e)dA=Pu ~ - (21) Due to the linear distribution of stress, we have:- oa, 68, = ` hy ————— (22) 68, = TT hạ Introducing now (17b), (18), (19) and (22) in equation (20), we obtain, “ip Q + EQ, 20 where, Qa = f2 hgh, Qa = (` HạdA - - - (24)
are the moments of the cross-sectional areas to the right and left of line a a?
Prom eq (21) we obtain, z„ đâu _ El Gye Pu -+ (25) where, _ ol, + Be ly E= Do ttc ccc ee (26)
E is the so-called reduced modulus, and I, and I, being the moment of inertia of the two sides
We observe that the position of the neutral axis in terms of the axial stress is given by eq (24), while the buckling eq (25) is similar to the elastic buckling eq (14) However, the value of K here is not given by eq (3), but by,
K=
ae a ‘ ' ' ' ' 1 ' + ‘ t ' ‘ ‘ Đ `
Trang 19A18.8
supported column according to eq (7), we will
have,
Since E is a function of Oop given by eq (28) and the value of Ey at the unknown der, the calculation of the critical stress requires a trial and error simultaneous solution of equations (23), (26) and (28)
A18.8 Imperfect Column Tangent=Modulus Theory The tangent-modulus theory, originally proposed by Engesser (Ref 1), 1s based on the assumption that at the critical state, no
stress reversal takes place, and the critical stress, therefore, is determined only by the tangent modulus Et This theory was abandoned early Since according to the previous dis- cussion with the classical definition of instability (perfect colum, bifurcation of equilibrium) strain reversal does take place In recent years, however, this tangent modulus theory has been proved useful
Under the assumption of no strain reversal both sides of the cross-section in Fig Al8.8a, will be characterized by the same linear stress distribution, corresponding to the tangent-—
modulus Ey Thus the buckling equation will be,
mr 8+ puso - (29) dz
and the critical stress for simply supported end conditions becomes, a 7 Đựl oy F oe A1 Since 1, + I, =I and E ~ Ey, it follows from (26) E> Ey and dp > of
The critical stress ot, therefore pre- dicted by the tangent-modulus theory is smaller than or from the reduced modulus theory Although for perfect columns, the assumption of no strain reversal is in
contradiction to the material behavior in the plastic range, most experiments have given results more closely to the results by the tangent-modulus theory
To resolve this controversy, Shanley (Ref 2), proposed the following explanation For simplicity, let a two-flange buckled column be formed by two rigid legs (see Fig Al8.9) joined in the middle by a plastic hinge Assume that this column starts to By consider- buckle as soon as o¢ is reached THEORY OF THE INSTABILITY OF COLUMNS AND THIN SHEETS P Fig Alé.9 ing the load Pt Shanley effect of a load P above the critical corresponding to tangent-modulus, proves the relation, a a where T= E/E
It must be emphasized that the buckled con- figuration is a stable one similar to that considered in the refined Euler’s theory Shanley has recognized the fact that such a stable configuration may exist after exceeding the tangent modulus load
If R= P/Py , Shanley found that the relation between the variation of stress due to bending and the compressive strain er corresponding to oy is:- se, 2 (b-R) Concave side: et == (1+) R-1 - + (33) Se, 2 (R-1) Convex side: 5 —— et Rt (+t)
Trang 20ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES là ante ' go 3S & § 8 \ " \ »a ae \ ˆ > Ob | reaper Nobles Loan) 3 hà ~ cad] Moovivs| Load a vt Fig A18 10
modulus, even though it does not actually define an unstable configuration, it repre- sents the lower limit of a spectrum of possible buckled configurations, the upper limit of which is the reduced modulus load which corresponds to infinite deflections
‘mus to summarize, sufficient experi-
mental results are available to show that the
fa1ling stress of a column in the inelastic range can be found by replacing E by the tangent modulus Ey in Eular’s equation, or,
n? fi
Ger * Tt ;Tð#ẽVÃ TT TTTxT— (L/z)® Figs Al8.11 and 12 show how experimental results check the strength as given by the Euler equation using the tangent-modulus Ey A18.9 uteaaTe stacncnd( PA) poi 9 a a F1 “ m 1m ri T1 1 mo 2 0 su eewei 64110 (4) Column teat resulta om 173-8 solid tomnd rad (ORE in, diamster) Pig Al8.il uutowars staewote() em > 2 “4 `“ iC HO MƠ 20 29 86 sizmponess nara, G4) Fala me tent ml ng 173-7 angie (146 in by 16 im, by By i) Pig Al8, 12 References: Ref 1 Engesser F., Schweezeriche Bauer Zeitung Vol 26, p 24, 1895 Shanley, F.R., Inelastic Column Theory, Jour Aeronautical Sciences,
1947, p 261 Ref 2
Trang 21A18 10 THEORY OF THE INSTABILITY OF COLUMNS AND THIN SHEETS PART 2 THEORY OF THE ELASTIC INSTABILITY OF THIN SHEETS Al18.9 Introduction
Thin sheets represent a very common and important structural element in aerospace structures since the major units of such structures are covered with thin sheet panels Since compressive stresses cannot be eliminated
in aerospace structures, it is important to know what stress intensities will cause thin sheet panels to buckle Equations for the buckling of thin sheet panels under various load systems and boundary conditions have been derived many years ago and are readily available to design engineers Part C of this book takes up the use of the many buckling equations in the practical design of thin sheet structural elements The purpose of this chapter is to introduce the student to the theory of thin plate instability or how these buckling equations so widely in use by design engineers were derived For a broad comprehensive treatment of the subject of instability of structural elements, the student should refer to some of the references as listed at the end of this chapter
A18.10 Pure Bending of Thin Plates
To derive the theory of instability of
thin plates, we must first derive the theory
of the pure bending of thin plates
In the following the analysis will be confined to small deformations Let x, y be the middle plane of the plate before bending occurs and 2 be the axis normal to that plane Points of the x, y plane undergo small displace- ments, w in the z-direction, which will be referred as the deflection of the plate The Slope of the middle-surface in the x- and y-directions after bending are For small de- flections, the pag on = ow „ỒN w kk Fae ly * by curvature of the middle surface can be found ow 3y? as compared to unity, as it has been done for the curvature of beams, Thus the curvature of the deflected middle surface in planes parallel to xz and yz planes respectively are: approximately by omitting powers of ae ,
a°w
ay"
Another quantity used In the problem of plates is the so-called twist of the middle surface given by:
2._2%
Ry axe” is
Ry
The strains can now be expressed by means of curvatures and twist of the middle surface In the case of pure bending of prismatic bar @ rigorous solution was obtained by assuming that cross-sections of the bars remain plane after bending and rotate so as to remain perpendicular to th: deflected neutral axis Combination of such bending in two perpen~ dicular directions brings us to pure bending of plates Fig, 2b
Trang 22
ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES À18.11
My, My per unit length at its edges ‘These ey =e BỀN _„ 2z
moments are considered positive when they are xy 2 Sxay Ry 7” TT” TT” (3)
directed as shown in the figure, 1l.e when they produce compression in the upper surface of the plate and tension in the lower Let also (Fig 2b) be a rectangular element cut out of the plate with sides dx, dy, t The thickness t 1s considered very smail compared with the other dimensions Obviously the stress conditions at the edges of all such elements will be identical to that of Fig 2a assume now that the lateral sides of the element remain plane during bending and rotate about the axes so as to remain normal to the deflected middle surface Due to symmetry the middle surface does not undergo any extension and it is therefore the neutral surface
From the geometry of the above described form of deformation the displacements in the X, y, 2 directions can be found as follows:
A point B on the middle surface has been displaced to B* by W in the z-direction An
element of surface dzdy has rotated by an
angle equal to the slope of the deflected
middle surface in the direction so as to
remain normal to the middle surface See Fig 3
Fig 3
This angle for small displacements is obviously equal to ae «Ổ Thus the horizontal displacement Uy in the x-direction of a point at distance z trom the middle surface is:
ly 3-2 x (The sign ~ indicates
negative displacement for positive z)
In a similar manner we find the displacement in the y-direction The complete displacement system is: uy * - a uy © -23t, tạ #W(X,y) ~ - (2) The corresponding strains are: ofw Zz ax* Ry’? ý ex R-Z%
Since we treat the problem of plates as,a plane stress problem, we find by means of Hook’s law — 2= @_ wy 8x = Ty?) ‘Ry * Ry - Te Gre vs) en “) op = EZ em +4 2 z* Tv Gy hy - iy G+ EH - (4b)
These normal stresses are linearly distributed over the plate.thickness Their resultants Must be equal to My and My respectively: pv Ox Z dydz = Mydy 2 phe Gy 2 dxdz= M -h/2 vex Substituting from (4) we find: 33w 3x" ay” 2p Om My = -D Gort nh 2 12(1-U®
the plate If besides the flexural moments Wy, My, there are uniformly distributed twisting
moments My and Myx along the sides of the Plate of Fig 2a, these must be equal to the resultant of distributed ghaar forces Oxy, Og along the sides of the element of Pig 2b
My = -D (
where D = = the flexural rigidity of
From eq (3) we obtain: atw xay
#xy = G7 “Rxy T 202 quay T TT TT (4e) Myydx = 7P qua đxd2, Myzdy =
-h/8
f h/2 Ơyxz dydz, Mụy = Myx = -h/2
D(1-U) aay - (5e)
Equations (5) give the moments per unit length
for pure bending and twisting of a plate
Trang 23A18 12
A18.11 The DƯferential Equation of the Deflection Surface To develop the theory of small deflections of thin plates we make one more assumption At the boundary of the plate we assume that its edges are free to move in the plane of the plate
Pig 4
Thus the reactive forces at the edges due to the supports are normal to the plate With these assumptions we can neglect any strain in the middle plane during bending
Let us consider, FPig.4, an element dxdy of the middle plane Along its edge the moments My, My, Mxy are distributed These are the resultants of the bending and twisting stresses distributed linearly along the
thickness of the plate (see eqs 4 and 5) If the plate 1s loaded by external forces normal to the middie plane in addition to the above moments there are vertical shearing forces Qy, Qy, acting on the sides of the element of Fig 4
n/2 OyzdZ ~- (6)
n/2 „ 9x = /Ê - quár, Qy = /0⁄2 -n/2
Let q be the transverse load per unit area acting normally to the upper face of the plate Considering the force equilibrium in the z-direction of the element of Fig 4 we find: ax ay "ax Okay +R dydx + adxdy = 0 or = Bx ¿9y 3x tay 450 ~.~—~ (6a) Taking the equilibrium of the moments acting
in the x-direction we obtain: THEORY OF THE INSTABILITY OF COLUMNS AND THIN SHEETS 3Myy 3My ax 1xdy 3y dxdy + Qy đxdy = ' ° or Sợ By ax 7 ay Qy oO ~ (6b) From the moment equilibrium in the y-direction we find: Shey Me eg ay ox TC
By eliminating Qy, Qy from 6 a,b,c, we find the equilibrium relation between the moments: a My 3ˆMy ax? ay? 2 a Mxy _ TỶ axay * 7 49 (7)
To represent this equation tn terms of the deflections w of the plate, we make the
assumption that the expression (5) derived for Pure bending holds approximately also in the case of laterally loaded plates This assumption is equivalent to neglecting the effect on bending of the shearing forces and the compressive stress Sz This 1S an ex- tension of the engineering theory of vending of beams As in the case of beams it gives good approximation for bending of plates under transverse loads Introducing equation (4) into (7) we find: otw atw q "TT Tnnm— (8) otw * aye * 2 Sxeay? 7 ox*
The problem of bending of plates is thus reduced to integrating eq (8) for w The corresponding shearing forces in terms of the displacements are found from eqs 4 and 6b and e: Mey My 3 3% tw Qe Gy Ge De Get Fe) > > (6) 3M, 3M ð 323W ð*®w
dy “SỐ - TU” * =0 ay ax ay ‘ax Ca * Sử ——— (69) ay*
The above analysis is sufficient to seek solutions of specific problems The general procedure is to find approximate solution of the fourth order differential equation (8) which satisfies the given boundary displacement and force conditions
A18.12 Strain Energy in Pure Bending of Plates
Trang 24ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES Al8 13 + _ đWy + ay t Magy t Wy o* dxdy ~ + â^w a^w à2w 2 [- Maer 7 My ape * Ay Say] 7 > 7 =0) Fig 5b
The strain energy stored in a plate element is obtained by calculating the work done by the moments M,dy, and M on the element during bending Since the sides of the element remain plane during bending the work done by Mydy is obtained by taking half the product of Mydy and the relative angle of rotation of the two sides of the element Since the curvature in the x-direction ts -= » the relative angle of rotation of the sides 1 and 2 of distance dx will be - ray, Thus the work dua to Mydy is:
ofw
1
ay = - My Bye Gedy ee ee (a)
Similarly the work due to Mydx 13: -=_l 32w
Wy =~ 3 My Sandy - (b)
The twisting moment Mydy also does work against rotation of the element about the
x-axis The relative angle of rotation of the
a,
two sections 1, 2 ts obviously ox dx Thus
the work done by Myydy is:
-i 3w
Wry = £ Huy Gay OU - n mm ~ TT (e) and the work due to Myydx = Myydx is:
si ow gy, -
4Vyx = 3 ty Soy (úy (4)
(It is noted that the twist does not affect the work produced by the bending moments, neither the bending affect the work produced by the torsional moments)
Thus the total strain energy per unit
volume of the plate is:
Substituting My, My, H in terms of the dis- placement from eqs (5) we find: „an (33% 32w You gd Gar ay a*w 32w g3w °| 2(1-v) (=e oy 7 Gray? “¬—¬—~=—~ (11) This expression will be modified later when we will consider the superposition of compressive loads in the plane of the plate which are related to the problem of plate buckling
A18.13 Bending of Rectangular Plates
The general differential equation for bending plates was given in section C2.3
(eq 8) Two very useful methods of solution
have been widely used, namely, the Fourter Series Method and the Energy Method Both methods will be developed in the following
for rectangular plates and various edge-
supporting conditions The edge support conditions are classified as follows:
a) Built-in edge or Fixed: The deflection along the built-in side is zero and the tangent Plane to the deflected middle surface is
horizontal Thus tf for instance the x-axis coincides with the built-in edge these con- ditions are:
ye 79 Sy so?
bd) Simply supported edge: The deflection along the simply~supported side is zero and the bending moment parallel to this side is also zero Thus if the plate is simply supported along the x-axis we have:
“ _r3*w _ 35w -
(W) pag? (My) peo Gye? Se y=0 = 0 - - -(12b)
©) Free edge: The bending moment, twisting moment and shear force along the free side is zero Thus if the free side coincides with the straight line x = aL we have:
(Mx) a, = 0 x=a (xy) jeg =O XV'x=a (x) x = 0 However, as was proved by Kirchoff two boundary
conditions are only necessary to find a unique solution of the bending problem He has shown that the two last equations of the above conditions can ba replaced by one condition
Trang 25
Al8, 14 THEORY OF THE INSTABILITY OF COLUMNS AND THIN SHEETS
ee wy (2~-) ) =0 These boundary conditions are satisfied if we
3x® ot xea take:
Expressing the condition (Mk) pea = 0 in terms of wwe find the final form of the boundary conditions along the free edge:
ay, ¬ l# al =
đề aye esa Oo Lae * {2-v) axay*|x=a * °
In the following solutions for various edge conditions will be developed
1 Simply supported rectangular plates
Let a plate with sides a and b and axes X; Y, as shown in Fig 6, be simply supported around the whole periphery and loaded by 2
distributed load q =f (x,y)
Two methods of solutions will be developed:
a) Navier solution by means of double Fourler Series:
We can always express f (x,y) in the form of a double trigonometric (Fourier) series: oo an mx a=f(x,y) = 2 2 apn sin =~ sin == - (13a) m1 m1 bự TC " where: apn = af 4 (x,y) sins sins dxdy tet tte (130) x a b y II Fig 6 The boundary conditions are: WsO0,My=0 atx=0,x=4 WsO,My 50 aty=0, yb, or: 2 (Ql) wso (2) ST = o atx=0,xza ay (3) W=o (4) tà =0 aty=0,y=b oo oo Ws 3 8 Cạn Sin TC sín mel n=l nny Đ —~ ~ (14a) By substituting in eq (18) with q given by eq (13) we find: co So 2 $ & apy sin—= aX sin ~~ 7 nny _ many seo tcc (14d) $ 3 & am sin SS sin WY mel n=l This relation is identity if: Con = ——Syn —_— pike 3 Là xạx = = 7 3 — mn ai sin mưu sin EC am m=] n=1 & +o
In the case of a load q, uniformly distributed over the whole surface we have:
f (x,y) = aq = const 4q; ra TƯỢC nny se =a — ann = SE + 4° sin + Sin == dudy = 169 Sm 77777 (15a) where m, n are both odd integers If either