Bruhn - Analysis And Design Of Flight Vehicles Structures Episode 2 Part 6 pps

2 Rectangular plate with two opposite edges simply Supported, the third edge free and the fourth edge built-in or simply supported.. A18.14 Cambined Bending and Tension or Compression

TỶ ==:

The expressions for bending and twisting

moments are not so quickly convergent To

improve the solution another series solution

can be developed as follows:

b) Levy alternate Single series solution:

The method will be developed for uniform

load q, = const Levy suggested a solution in

the form:

20

W= 2 Ym (y) sin ⁄" - (16)

m=)

where Ym is a function of y only Each term

of the series satisfies the boundary conditions

a*w

W320, 35a 3 0 at xa It remains to

determine Yq so as to satisfy the remaining ^

ow =0aty=b, two boundary.conditions w = 0, =— ay*

A further simplification can bs made if

we take the solution in the form

where

(x* = 2ax* + a3x)

* 24D

is the deflection of a very long strip with

the long side in the x-direction loaded by 4

uniform load q, supported at the short sides

x = 0, X =a, and free at the two long sides,

Since (17b) gatisfies the differential

equation and the boundary conditions at x = 0,

xX =a, the problem is solved if we find the

Solution of:

with w, in the form of (16) and satisfying

together with w, of eq (17b) the boundary

conditions w =o, ahs Oatyst 2 (see

where for symmetry m=1,3,5

This equation can be satisfied for all values

sinh =" sin =

where m2 1,3,5 Substituting this

expression into the boundary conditions:

where dy = mnb/2a

From these equations we find:

n°D mel,3,s m° 2 cos hay

The summation of the first series of terms

corresponds to the solution of the middle of

This series converges very rapidly Taking a

square plate, a/b = 1, we find from (21a):

We observe that only the first term of the

series in (2le)} need to be taken into con-

can represent w in double Pourler Series:

Vị -42/ Vodxay = ab

9

Let us mow examine the deflection of the plate

of Fig 6 with a concentrated load P at the

point with co-ordinates x =€ , y=n The

increment of strain energy due to the incre-~

ment of the deflection by:

OV, can Gert pa)” 6 Cun ~ - - ~ (240)

The increment of the work of the load P is:

OW = P6Can, sin SỐ sin SL

From 6V, - OW = 0 we obtain:

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES

which is 3-1/2% less than the correct value

(2) Rectangular plate with two opposite

edges simply Supported, the third edge free

and the fourth edge built-in or simply

supported

y Fig 8 8

Assume that the edges x = 0 and x = 4 are

simply supported, the edge y = b free and the

edge y = 0 built-in (Fig 8) In such a case

the boundary conditions are:

we write the deflection in the form:

WW + Wa

where w, is the deflection of a simply supported strip of length, a, which for the system of axes of Fig 8 can be written (see Levy’s method in previous section):

oo

Wa > a Ym (vy) sin TT” -~ ~ - (260)

m=1,3,5 where w, being a solution of

ow, = aw, 3*wW axe * ays * ® axtay? 7 o

is found as in the previous section:

Yn (Am cos has + By == sink +

Cy sinn +p, BY cosh MY) - ~(264)

It is obvious that the two first boundary

conditions are identically satisfied by w =

W, + W, The coefficients Am, Bg, Cm, Dy must be determined sc as to satisfy the last

four boundary conditions Using the conditions

occurs at the middle of the free adgs

A18.14 Cambined Bending and Tension or Compression of Thin Plates,

In developing the differential equations

of equilibrium in previous pages, it was assumed that the plate is bent by transverse

loads normal to the plate and the deflections

were so small that the stretching of the middle Plane can be neglected If we consider now the

case where only edge loads are active coplanar

A18, 1ậ THEORY OF THE INSTABILITY OF COLUMNS AND THIN SHEETS

with the middle surface (Fig 9} we nave a

plane stress problem If we assume that the

Stresses are uniformly distributed over the

thickness and denote by Ny, Nyy Nyy, Nyy the |

resultant force of these streSses ber Unit

length of linear element in the x and

y-directions (Fig 9) it 1s obvious that:

Ny = hoy , Ny * hay

Nxy = Nyx = hơxy = hoy,

Fig 9

The equations of equilibrium tn the absence

of body forces can be written now in terms of

these generalized stresses Ny, Ny, Nyy by

substituting from eq (27) to the equations

of equilibriun

aNx Nyy BNy êNxy

s+

ax * ay S9; ay ax

On the other hand tf the plate is loaded by

transverse loads the stresses give rise to

pure bending and twisting moments only The

equations of equilibrium for the latter have

been given in before (see eqs 6, 7, 8) If

both transverse loads and coplanar edge loads

are acting simultaneously, then for small

vertical deflections the state of stress is

the superposition of the stresses due to

Ny, Ny, N and My, My, Myy For large

were đễrlactien oF the plate, however,

there is interaction of the coplanar stresses

and the deflections These Stresses give rise

to additional bending moments due to the non-

zero lever arm of the edge loads from the

deflected middle surface, as in the case of

beams When the edge loads are compressive

this additional moments might cause instability

and failure of the plate due to excessive

vertical deflections

In this chapter the problem of instability

of plates will be examined

When the edge loads are compressive and give rise to additional bending moments sq (8)

Fig 10a

Consider an element of the middle surface

dx, dy (Fig, 104) The conditions of forse equilibrium in the x, y-directions are given

by eq (28) Consider now the projection of

the stresses Ny, Ny, Nyy in the z-direction:

(a) Projection of Ny: From Fig loa it follows that the resultant projection is:

hy ay

and neglecting terms of higher order:

aw aNy aw,

Oe BF + ax ex) TTT TTT (4)

(b) Projection of Ny: By similar argument

we find that this projection 1s equal to:

aw SS aNy aw ee ee ee Le

Fig 10b {c) Projection of Nxy and Nyy: From Fig lob

we find:

ew ON ow 37x

~ Nxy Ft (xy + TS ax) Sy Sxay 4x)

and neglecting terms of higher order:

ay Gxay * “gx: Gp) duty - - - fe)

Simtlarly we find the projection of

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES

ô3w ,„ôNxy ôw,

(xy Sxoy cây ox? axdy

Thus in eq (6a) the terms given by (a), (b},

(¢) and (d) should be added (divided of course

by dxdy):

+ ye ge Ny Se py SH oy

ay 9 * Nx xt * By ays XY axay

aN, Ay, BW, Ny, Way aH gL

Se Sy! ae Sy * Sea TO te)

But due to the equation of equilibrium (28) the

two terms inSide the parentheses in (e) are

zero Thus:

Se = _

Ny or Ta + #Ñxy quay Say?

Eq (29) replaces eq (6a) when edge loads are

present fqs (6b) and (6c) are, however, still

valid since they exnress moment equilibrium

of the element dxay in wnicn tne contriouvion

of Ny, Ny, Nyy is zero Thus eliminating Q,,

Qy between (6b), (6c) and (29) we find:

Viw = ae * 3y? S5 ax ay

T (q+ Ny Set ly Sự + 2Nxy Ea - - - -(30)

Eq (30) replaces eq (128) when edge loads

are present and the deflections are large so

that instability might occur

The distribution of the coplanar stresses

Nx; Ny, Nyy can be found from eqs (28) by

solving the plane stress problem In the

following the above theory will be applied to

rectangular plates

A18.15 Strain Energy of Plates Due to Edge

Compression and Bending

The energy expression for pure bending,

eq (11), must be complemented to include the

contribution of the edge coplanar loads

Assume that first the edge loads are applied

Obviously the strains due to the stresses

Nx, Ny, Nyy are:

ix -YNy), ey > Te (Ny -UNy),

Á18 19

Vin sass (Nxex + Nyey +Nxy/Ƒ xy) axdy =

shelf (Md + Ny ~ 2nE NY ROY 8UNNy +2(1/)0Nvy`) TOIT '¬ = (2)

During bending due to transverse loads or/and due to buckling we assume that the edge loads and consequently Ny, Ny, Nyy, remain constant Its variation 1s thus zero and we

do not consider it in the following Let us

apply now the transverse load that produces

bending (We can also consider bending due to other transverse disturbance, which is the case of buckling) If u, v, are the displace- ments of the middle surface due to the coplanar loads (which are assumed constant across the thickness) and w the bending deflection of the plate it can be shown that the strains are:

surface the energy is:

Jf (Nxex + Nyey + Nxy Syy) dxdy = - - - (c)

Introducing (b) into (c) and adding the strain

energy due to bending, eq (11), we find the

total change of strain energy due to bending which ts:

2(1-v) [= a ee} dxdy - - (4) Here u, v are the additional coplanar displace-

ments after bending has started It can be

shown by integrating by parts that the first

integral is the work done during bending by the edge loads For instance taking a rectangular

plate this integral becomes:

A18 20

Obviously the first two integrals represent

the work done by the edge loads while the

second integral is zero due to the equilibrium

equations (28) Thus the work of the edge

plate is negligible (This is the so-called

inextensional theory of plates) In this case

by zeroing the strains in eq (b) and substi-

tuting in (f) we find:

aw aw

Nụ ==È // [Xx Bat ery Bt + amy & Blanay

In the strain energy expression, eq (a) the first two terms cancel each other and the

strain energy 1s due only to bending:

„1 320, o2Wy a

Vị =2D0// lt-šÐ -

2a-) (oe oe cả} đxếy - - (32h) xẻy

In the absence of transverse loads the work

of external forces is simply due to the edge

leads:

Wom Wy

Expressions (32b) and (c) will be used in

Solving the buckling problem by means of the

principle of virtual work

Al8.16 Buckling of Rectangular Plates with Various Edge

Loads and Support Conditions

General discussion

In calculating critical values of edge

loads for which the flat form of equilibrium

becomes unstable and the plate begins to

buckle, the same methods and corresponding

Treasonings as for compressed bars will be

employed

The critical values can be obtained by assuming that the plate has a slight initial

curvature or a small transverse load These

values of the edge loads for which the lateral

deflection w becomes infinite are the critical

values (see Part 1 for similar treatment in

columns)

Another way of investigating such

instability is to assume that the plate buckles

due.to a certain external disturbance and then

to calculate the edge loads for which such a

THEORY OF THE INSTABILITY OF COLUMNS AND THIN SHEETS

buckled configuration (deflections different from zero) is possible It was found in the

case of column that this latter solution

(Euler’s solution) approaches asymptotically

the first at the limit where the deflections

become extremely large but for even small

deflections the edge load acquires a value

very near the Zuler’s critical value The

latter technique is mathematically more

convenient and it gives for plates also a very

good estimate of their compressive strength

In the following we shall use this latter

approach by assuming a plate with edge loads

and no transverse load zq (30) becomes in

this case:

atw atw 1w lạy SỔN aẦw aw y

ax 2 Otay? ay? — 0x y1! Ny ay? * Bhy Bay?

By solving eq (33) we will find that the

assumed buckling made is possible (w # 0) for

certain definite values of the edge loads, the

smallest of which determines the critical icad

The energy method can also be used in investi-~

gating buckling problems In this method we assume that the plate is initially under the plane stress conditions due to the edge loads

and the stress distribution is assumed as known

We then consider the buckled state as a possible

configuration of equilibrium The change of the work i8 given by eq (22a) We interpret here was @ Virtual displacement though we do not use the variation symbol 6 Thus the increment of work 6W is given by (32a) and the increment of strain energy 5Vi is given by eq (32b) If Gw<5V; for every possible shape of buckling the flat equilibrium is stable If 6W=éVy for a certain shape of buckling then the flat con- figuration is unstable and the plate will buckle under any load above the critical load If

OW = 6V;, the equilibrium is neutral and fron

this equation we find the critical load The

critical load therefore 1s found from the equation:

Here w ts a certain assumed deflection which

satisfies the boundary conditions (virtual

deflection)

A18,17 Buckling of Simply Supported Rectangular Plates

Uniformly Compressed in One Direction

Let a plate of sides a and > (Fig 11)

simply supported around its periphery be com- pressed by load Ny uniformly distributed along the sides x = 0 and x =a From the

1 \ F— x This fraction has some intermediate value

= between the maximum and the minimum of the

fractions (a) It follows that if we wish to

make the fraction (b), which is similar to the

Fig 11 fraction of eq 35d, 4 minimum, we must take

ty 8 only one term in the numerator and the corres- obvious solution of the corresponding plane

stress problem we find that the state of

stress 1s everywhere a simple compression

equal to Ny (the load at the periphery)

The deflection surface of a simply

supported plate when bending takes place have

been found previously (see eq 14a) Its

general expression can be written in a double

series form:

co of

Wa Š 3 mg sin TC sin TC - - - (88a)

me} n=l

The increment of strain enargy found by

substituting (35a) in the right-hand side of

The increment of work done by the external

forces is found by substituting (35a) into the

left-hand side of eq (34) and Ny = const,

Here Cmn 18 arbitrary We are interested,

however, to find that values of Cyy which make

Ny minimum To that effect we use the follow-

ing mathematical reaconing:

Imagine a series of fractions:

If we add the numerators and the denominators

we obtain the fraction:

ponding term in the denominator Thus to make

the fraction of eq (35d) minimm, we must put

all the parameters C,,, Cis, Cara except one, zero This is equivalent to assuming that the buckling configuration is of simple

sinusoidal form in both Gatecttons, 1.9

Won = Can sin, sin SY ,

expression for Cyn obtained by dropping all the terms except Cy becomes:

The minimum

71%4°D m2 | n#

Ny = 42e ta)?

It is obvious that the smallest value of Nx is

obtained by taking n=1 This means that the plate buckles always in such a way that there

can be several half-waves in the direction of compression but only are half-wave in the

perpendicular direction Thus for n= 1, eq

(35e) becomes:

Wy) op 5 = m+ = op)

The value of m (in other words the number of

half-waves) which makes this critical value

the smallest possible depends on the ratio

a/b and can be found as follows:

Let us express (36a) in the form:

(N_) 2p ed

x/er b2 TTT “TT (46p)

where k 1S a numerical factor depending on bi

From (36a) and (36b) we nave:

a

«225 (a+: oe pF" ae eee eee (36c) a2

If we plot k against 2 for various values of the integer m=1, 2, 3, we obtain the curves

of Fig (12) From these curves the critical

load factor k and the corresponding number of

half-waves can readily be determined It is

only necessary to take the corresponding point

es as the axis of abcissas and to choose that curve which gives the smallest k In Fig 12 the portion of-the various curves which give the critical values of k are shown by full

lines The transition trom m to (m + 1) halr- waves occurs at the intersection of the two

(36) we find:

corresponding lines From eq

Thus the transition from one to two hal?-waves a 2 ar ¬

22/1 = Ve

from two to three for

¬ teens and so on

Tne number of half-waves increases with the

ratio a/b and for very long plates m 1s very

large

zon

This means that 4 very long plate buckles in

half-waves the lengths of which approach the

width of tha plate The buckled plate is

subdivided into squares

The critical value of the compression stress is:

= (xJor „ K"*E t®

yy = ker = 15

h 12(1-y”) b#

{t = thickness)

Á18.18 Buckling of Simply Supported Rectangular Plate

Compressed in Two Perpendicular Directions

Lat (Fig 15) Ny, Ny the uniformly distri- buted edge compressions Using the same as

before expression for the deflections (eq 35a)

and applying the energy equation (33) with

Nx, Ny = constants (which ts the solution of

the corresponding plane stress problem) we find:

Taking any integer m and n the corresponding

deflection surface of the buckled dlate is given

Dy:

stn SRY, TƯỚC

a » Sin 5

and the corresponding ox, Oy are given dy (37c)

wnich is a straight line in the diagram By, Oy

(Fig 14) By plotting such lines for various

pairs of mand n we find the region of stability

and the critical combination of ay, Oy which ts

on the periphery of the polygon rormed by the

full lines of Pte 14,

Fig 14 A18.19 Buckling of Simply Supported Rectangular Plate Under Combined Bending and Compression

Let us consider a simply supported

rectangular plate (Fig 15) Along the sides

xX = 0, X =a there are linearly distributed

edge loads given by the equation:

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES which is 4 combination of pure bending and

pure compression.- Let us take the deflection

azain in the form:

Substituting in the right-hand side of

eq (34) we find the variation of strain

energy:

abn+

while the increment of work is:

The coefficients now Cyn are so adjusted that

(Noler becomes minimum By taking the

derivative of expression (38e) with respect

to each coefficient Cyn and equating these to

We examine for each value of m the solutions of

the system (38f) Starting from m > 1 and

is of infinite number of equations (n= 1, 2, 3 ) A sufficient approximation is

obtained by taking a large but finite number

of terms and find the solution of the finite

determinant (using for example digital computers) Thus a curve of Ogr versus a/b 1s

obtained for m= 1 like that of Fig 12

Repeating the same calculation form =2,3 , atc., we find similar curves of two, three, etc

half-wave lengths The regions of the curves with minimum ordinates define the region of

stability as in Fig (12)

A18.20 Inelastic Buckling of Thin Sheets The problem of the inelastic buckling of

thin sheets has been extensively studied by

various authors The main difficulty in such

studies is in reference to the stress-strain relations of plasticity under complex states

ef stress Many controversial discusstons have appeared in literature without resolving

the theoretical difficulties For this reason

we Will not develop the theory of inelastic buckling in this chapter Some of the better references on this subject are listed below

Chapter C4 presents the plasticity

correction factors to use in calculating the inelastic buckling strength of thin sheets

A18,21 References, (1) Bleich, F: Buckling Strength of Metal Structures Book by McGraw-Hill

(2) Stowell, E.Z., A Unified Theory of Plastic

Buckling of Columns and Plates NACA Report 998, 1948

(3) Gerard and Becker: Handbook of Structurel Stability NACA T.N 3781, 1957

A18 24

a

Courtesy of The Boeing Company, Seattle, Washington

This multiple exposure photograph of a Boeing supersonic transport mode! shows the variable-sweep wing

in three configurations: forward for takeoff and landing, swept part way back for transonic flight, and swept completely back as an arrow wing for 1800-mile-an-hour supersonic cruise

SPECIFICATIONS (Basic Design) Gross Weight

Payload Range Takeotf Distance (Max gross weight) Landing Distance

Cruising Speed

Takeoff Speed

Approach Speed Wing Span (forward position)

CHAPTER Al9 INTRODUCTION TO WING STRESS ANALYSIS

A19.1 Typical Wing Structural Arrangement

For aerodynamic reasons, the wing cross-

section must nave a streamlined shape commonly

referred to as an airfoll section The aero-

dynamic forces in flight change in magnitude,

direction and location Likewise in the various

landing operations the loads change in magni-

tude, direction and location, thus the required

structure must be one that can efficiently

resist loads causing combined tension, com-

pression, bending and torsion To provide

torsional resistance, a portion of the airfoil

surface can be covered with a metal skin and

then adding one or more internal metal webs to

produce a single closed cell or a multiple cell

wing cross-section The external skin surface

Which is relatively thin for subsonic aircraft

is efficient for resisting torsional shear

stresses and tenston, but quite inefficient in

resisting compressive stresses due to bending

of wing To provide strength afflciency, scan-

wise scifgening units commonly referred to as

flange stringers are attached to the inside of

the surface skin To nold the skin surface to

airfoil shape and to provide a nedium for

trans?erring surface air pressures to the

cellular deam structure, chordwise formers and

ribs are added To transfer large concentrated

loads into the cellular beam structure, heavy

ribs, commonly referred to as bulkheads, are

used

Figs AlS.1 and Als.2 illustrate typical

structural arrangements of wing cross-sections

for subsonic aircraft The surface skin is

In general the wing structural

arrangement can be classifled into two (1) the concentrated Zlange type where material is connected directly to in- ternal weds and (2) the distributed f @ cype

where str rs are attached to sxin bet

internal webs

Fig AS.3 shows several structural 2 xa~

ments for wing cross-sections for superscnie aireraft Supersonic airfoil shapes are relatively thin compared to subsonic aircraft

Fig Al9.2 Common Types of W

Arranger Beam Flange

Al8.2

To withstand the high surface pressu:

obtain sufficient strength much

skins are usually necessary Modern milling

machines permit tapering of skin thicknesses,

To obtain more flange material integral flange

units are machineẻ on the thick sxin as tllus-

trated in Fig k

s and to Ker wing

In a cantilever wing, the wing bending

moments decrease rapidly spanwise frem the

maximum values at the fuselage support points

Thus thick skin construction must be rapidly

tapered to thin skin for welgnt efficiency, but

thinner skin decreases allowable compressive

stresses To promote better efficiency sand-

wich construction can be used in outer portion

of wing (Fig 1) A light wetght sandwich core

is glued to thin skin and thus the thin skin ts

capable of resisting high compressive stresses

since the core prevents sheet from buckling

A19.2 Some Factors Which Influence Wing Structural

Arrangements

(1) Light Weight: +

The structural designer always strives for

the minimum weight which 1s practical from.a

production and cost standpoint The hisher the

ultimate allowable stresses, the lighter the

structures The concentrated flange type of

wing structures as illustrated Fig (a, > and ¢}

of Fig Al9.1 permits high allowable compressive

flange stresses since the flange members are

Stabilized by both web and covering = , thus

eliminating column action, which permits design

stresses approaching the crippling stress of the

flange members Since the flange members are

few in number, the size or thickness required is

relatively large, thus giving a high crippling

stress On the other nand, this type of design

does not develop the effectiveness of the metal

covering on the compressive side, which must be

balanced against the saving in the weight of the

flange members

In the distributed type of 7 lange arrange-

ANALYSIS OF WING STRUCTURES

flange allcwatle compressive str

is not practical to space wi

12 to 18 times the Tlange stringer 7

17 there were no othsr controllin:

could easily make calculations to

which of the above would orovs general, if the torsional °0rees are small, thus requiring only 4 the concentrated

should prove tha

In general, placed to give the m the Z direction, » in general that

fla material should ? be placed ˆetuean the

and 50 per cent o ing chord from the lead

edge

be inert ta in

th

1

Tue secondary or distriouting str

f the structural box bea should be mad

light as possible and thus in general =:

forward

lighter

the rear closing web of the box

the wing structure as a whole

In the layout of the main spanwise ?lange

members tends or changes in direction should be

avoided as added weight is required in splicing

or in transverse stiffeners which are necessary

cO change the direction of the load in the flan:

members If flange members must de spliced, c2

should be taxen not to splice them in the region

of a maximum cross-section, Furthermore, in

general, the smaller the number of fittings,

nter the structure

igh wing type, the entire

continue in the way of the airplane pedy How-

ever, in the mid-wing type or semi-iow wing type, limitations may prevent extending the entire wing through the fuselage, and some of

the shear webs as well as the wing cover

must be terminated at the side of she fuselage

If a distributed flange type of cell structure were used, the axial load in the clange string-

ers would nave to oe transferred to the members

extending through the fuselage + provide for

this transfer of 2a: loads requires structural weight and thus a concentrated flange type of

box structure might prove the best type of

structure

the low wing or the

wing str acture can