Bruhn - Analysis And Design Of Flight Vehicles Structures Episode 3 part 8 potx

C8.16 in Chapter ca, Rp = ratio of In using Equation C9.5, the value of is negative as inward pressure is opposite the inward acting outward pressure, Rp to C9.7 Shear Buckling Stress

Pig C9.4 Shear Buckling Coefficient for Long

Simply Supported Curved Plates

Zp

Ly

Fig C9.5 Shear Buckling Coefficient for Wide

Simply Supperted Curved Plates

BUCKLIN ULTIMATE STREN G STRENGTH 0 C9.8

GTH OF STIFF

where Ro = fo/Foon

applied internal pressure external inward pressure that would buckle the cylinder of

which the curved panel is a

section Found by use of Flg C8.16 in Chapter ca,

Rp = ratio of

In using Equation C9.5, the value of

is negative as inward pressure is opposite

the inward acting outward pressure, Rp to

C9.7 Shear Buckling Stress of Curved Sheet Panels with Internal Pressure,

As in the case of monocoque cylinders,

internal pressure increases the shear buckling

Stress of the curved sheet panel Brown and

Hopkins (Ref 5) solved the classical

equilibrium equations to determine the effect

of radially outward pressure upon the shear

buckling stress of curved panels and obtained

fair agreement with test data by Rafel and

Sandlin (Ref 3) The test data also

correlates with the interaction curve used

for the effect or internal pressure upon

cylinders in torsion (see Chapter C8) The

interaction equation 1s;

Rg* + Rp = bo + eee ee LL C9.6

where Rg = Ÿs/P So

~ applied internal pressure

Rp = ratio of external inward pressure that

would buckle the cylinder of

which the curved sheet panel

1s a section Found by use

Fig C9.6 illustrates a circular fuselage

Section with longitudinal stringers represented

by the small circles The area of each

stringer is 15 sq in The skin thickness is +04 inches

All material is aliminun alloy with Ey = 10,700,000 The fuselage frame

Spacing (a) 1s 15.75 inches The fuselage

Section is subjected to the fo

F CURVED SHEET PANELS

ENED CURVED SHEET STRUCTURES,

Fig C9.6

The problem is to determine whether skin

Panels (A) and (B) will duckle under the given combined loading on the fuselage section,

Solution

To find the bending stresses, the moment

of inertia of the cross Section about axis y-y

is necessary, which axis is also the neutral axis since all material is effective The moment of inertia will equal 4 times that due to material in one quadrant,

Ty due to stringers is,

To find the shear buckling stress, we use

Fig C9.2 Z is the same as calculated above

or 32.9 Thus from Fig C9.2 we read for

a/p = 3, that Kg = 20

204

Por =~ Te TT — 3%) Gags) ~ 12800 pst

The bending stress at midpoint of Panel

{A) will be calculated:

fp 2 M/ly = (600,000 x19.7)/1725 = 6850 pst

Thus stress ratio Re = fo/Fo,, = 6850/7850

= 874

Shear stress on Panel (A) due to torsion is,

fg = T/2at, where A is inclosed area of

fuselage cell

fs = 210,000/2 x m x 20° x 04 = 2090 psi

The panel is also subjected to shear

stress due to transverse shear of Vz, = 5i75 lbs

The shear flow equation is,

a= 7) BZA = > Trap ZA = - 5 IZA

The shear flow will be zero on Z axis

The shear flow at top edge of Panel (A) will

be due to effect of one-half the area of

stringer (1)

Gina 7 3 X 0075 x 20 = 4.5 1b./1n

Area of skin between stringers (1) and (2)

18 5.25 X 04 = 21 =A,

Distance from centroid of Fanel (A) from

neutral axis 1s Zr sin a/a a= 15° which

This shear stress has the same sense or

direction as the torsional shear stress so we

add the two to obtain the true shear or:

„B74 + ,217 = 918 This is less than 1.0 so

panel will not buckle

2

1.8 5 Taya + fara? + ax ele 7 1 = «08

Consider Skin Panel (8)

Arm 2 to midpoint of panel = 15.88 in

Shear flow q due to transverse shear load:

a> 732A = 3 ZA

Calculation of 352A at upper edge of panel:

.075 x 20 + 15 (19.3 + 17.34) 7.0

For stringers = For skin: Area = 2x 5.25 x 045 42

Vertical distance 2 to centroid of skin portion

=rsina/a a= 30% The result is Z'= 19.1

in The ZA = 19.1 x 42 = 8.03, Total IZA =

1251 psi The total shear stress fg on panel

then equals 1251 + 2090 = 3341 psi

Rg = fs/Faor = 3341/11200 = 299 Substituting in interaction equation Ry +

Rg* = 704 + 299* = 793, The result is less than 1,0, thus panel will not buckle

The student should check other panels for buckling and compare their margins of safety

BUCKLING STRENGTH OF CURVED SHEET PANELS

C9 8

General Commen:

In general the compressive stress is the

dominant factor in causing the panel buckling

Thus to increase the buckling stress of the

panels and also to give a more effective

Stringer arrangement to carry the bending

moment, the stringers should be spaced closer

in the top and bottom regions of the cross-

section and with increased spacing as the

neutral axis 1s approached,

PROBLEM 2

The fuselage section in Problem 1 is sub-

jected to an internal outward pressure of 5

psi What would be the compressive buckling

stress of a panel and also the shear buckling

stress with this internal pressure existing

Solution

From Art C9.6, the interaction equation

is,

From Problem 1, the compressive buckling

stress Poop = 7850 psi

To evaluate Rp; the external inward radial

acting pressure that would cause buckling of a

eircular cylinder having 2 radius equal to that

of the curved sheet panel must be determined

Use is made of Fig C8.16 of Chapter C8 to

determine the backing stress under such a

loading The lower scale parameter for Fig

The external radial pressure to produce

this buckling stress is,

psi increases the axial compressive buckling stress from 7850 to 13750 pst

Shear Buckling Stress Under 5 psi Internal

Radial Pressure

From Art C9.7, the interaction equation

is,

Re? + Ry = 1

From Problem 1, Pop or our panel was

11200 psi The value of Rp is determined as

C9.9 Introduction

A cylindrical structure composed of a thin skin covering and stiffened by longitudinal stringers and transverse frames or rings is a common type of structure for airplane fuselages, missiles and various types of space vehicles,

and such structures are often referred to as

the semi-monocoque type of structure The design of 4 semi-monccoque structure involves the solution of two major problems, namely, the

stress distribution in the structure under

various external loadings and the check of the structure to see if these stresses can be safely and efficiently carried by individual components

of the structure as well as the structure acting

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES

In general, thin curved sheet panels buckle

under relatively low compressive stress and

thus {f design requirements specified no

buckling of the sheet under limit or design

loads, the sheet would have to be relatively

thick or the stringers placed very close to-

gether and the fuselage or body structure

would be unsatisfactory from a strength weight

standpoint In missile structures, internal

pressurization increases the buckling stress

greatly, thus the buckling weakness of thin

Sheet is improved, but keeping a structure

pressurized under all operating conditions nas

its difficulties

In a semi-monocoque body, the longitudinal

stringers provide efficient resistance to

compressive stresses and buckled sheet panels

can transfer shear loads by diagonal tension

field action, thus the buckling of the sheet

panels is not an important factor in limiting

the ultimate strength of the over-all

structural unit When buckling of the skin

panels takes place, a stress redistribution

over the entire structure takes place, thus it

is important to know when skin buckling begins

Furthermore, design requirements may often

specify that no skin buckling should take

place under -a certain percent of the limit or

design loads The equations and design curves

in Part 1 of this chapter can be used to

determine the buckling stress of curved sheet

panels under various stress systems

(2) Panel Instability

The internal rings or frames in a semi-

monocoque structure such as a fuselage divide

the longitudinal stringers and their attached

skin into lengths called panels If these

frames are sufficiently rigid, a semi-monocoque

structure if subjected to bending will fail on

the compression side as {llustrated in Fig

c9.7a The stringers act as columns with an

effective length equal to the panel length

which is the ring spacing Initial failure thus

occurs in a single panel and thus is referred

to as a panel instability failure In general,

this type of failure occurs in most practical

aircraft and aerospace semi-monocoque

structures because the rings are sufficiently

stiff to promote this type of failure Since

the inside of a fuselage carries various loads,

such as passengers, cargo, etc., the rings must

act as structural units to transfer such loads

to the shell skin, thus requiring rings of

considerable strength and stiffness Even

lightly loaded frames must be several inches

deep to provide conduits required in various

installations to pass through the web of the

ring cross-section, thus providing a relatively

stiff ring for supporting the stiffeners in

their column action When the skin buckles

under shear and compressive stresses, the skin

C9.9 panels transfer further shear forces by semi-

diagonal tension field action which produces

additional axial loads in stringers and also bending which must be considered in arriving

at the panel failing strength This subject

is treated in detail in Part 2 of Chapter Cll

(3) General Instability

In general instability, failure is not confined to the region between two adjacent frames or rings but may extend over a distance

of several frame spacings as illustrated in

Fig C9.7b for a stiffened cylindrical shell

in bending In panel instability, the trans- verse stiffeners provided by the frames on rings is sufficient to enforce nodes in the stringers at the frame support points as illustrated in Fig C9.7a Any additional

stiffeners in excess of this amount does not

contribute to additional buckling strength

General instability may thus occur when the stiffeners of the supporting frames is less

than this minimum value

C9.11 The Determination of the Stresses ina

Stiffened Cylindrical Structure Under

External Loads

The stresses in a stiffened cylindrical structure such as used in typical fuselage or missile design can be fairly accurately determined by the modified beam theory as pre- sented 1n Chapter A20 A more rigorous approach

is given in Chapter AS involving matrix formu~

lation but this approach requires the use of a large electronic computor to handle the required calculations For details of applying the modified beam theory, the reader should refer

to Articles A20.3 and A20.4 of Chapter A20

In the example problem solution as given in Article A20.4, the effective area to use for the curved sheet was based on the ratio of the buckling stress of the curved panel to the bending compressive stress on the panel due to bending of the entire effective cross-section

of the fuselage under the design loads In the example problem as given in Table A20.2, a conservative buckling compressive stress equal

to 3 Et/r was used for the curved panel and no consideration of the effect of shear stress on the compressive buckling stress was considered

A more accurate procedure would be to cal- culate the effective area of the curved panels taking into account the influence of combined compression and shear on the buckling strength

of the panel Thus tn Table A20.2 on page AZ0.5

of Chapter AZO, the shear stress on each curved panel should also be calculated and then the buckling streas of the panel under the combined compression and shear calculated

The buckling stress under pure compression

ANALYSIS AND DESIGN OF F and shear should be calculated using Equations

C9.1 and C9.2, The buckling stress under

combined compression and shear is given by the

Let f, be the compressive stress that will

buckle the curved sheet panel when subjected to

combined compression and shear when the ratio

of the applied compressive and shear stresses

in a constant Then,

M.S

s 2

fos fo (Qo ƯNG + đàng

These Ÿ¿ values should then replace the

values in column 5 of Table A20.2,

The author has noted that one aerospace company in their missile design uses only 90

percent of the theoretical buckling stress in

computing the effective area of the buckled

curved panels This correction assumes that

the curved sheet fails to hold the buckling

stress as the fuselage section as a whole is

further loaded and the curved sheet suffers

more buckling distortion

C9.12 Panel Instability Strength

‘ Panel instability means failure of the

stringer and its effective skin between two

adjacent frames The bending of the stiffened

shell as a whole produces a compressive load

or stress on the striuger The semi-tension

field action of the skin after buckling

produces an additional compressive load on the

stringer and also a bending moment

The compressive stress due to bending of

the stiffened shell as a whole is found by the

methods discussed in Article C9.11 The

additional stringer loads due to semi-tension

field action are determined by the theory and

procedure given in Part 2 of Chapter C11

These calculated stringer loads are then compared to the stringer strength to determine

whether a positive margin of safety exists

The local crippling and column strength of a

stringer plus its effective skin can be found

by the theory and analysis methods given in

Chapter C7 The bending strength of the

stringer cross-section can be found by the

theory and analysis method given in Chapter cz

The strength of the stringer in combined

compression and bending 1s found by use of the

proper interaction equation

LIGHT VEHICLE STRUCTURES C8.11 C9.13 Calculation of General Instability

“ great deal of theoretical and experi-

mental work has been done on the subject of

general instability of stiffened shells The general goal in the design of such structures

is to insure the frames have sufficient

stiffness so as to prevent the type of failure illustrated in Fig C9.7b or, in other words ,

to insure the type of failure illustrated in

Pig C9.7a which is panel instability

7 Shauley (Ref 6) has derived an expression for the required frame stiffness to prevent general instability failure of a stiffened shell

where, E = modulus of elasticity

moment of inertia of frame section

diameter of stiffened shell frame spacing

bending moment on shell

oH "

Recker (Ref 7) in a comprehensive study

of most published theoretical and experimental

material relative to the general instability of stiffened shells, summarizes the results of his studies as given in Table C9.1

Bending

For the case of bending, the constant of

4.80 in the equation given in Table C9.1 is for the condition where the frames are attached to

the skin between the stringers For frames not

attached to the skin between stringers, the

b = stringer spacing (in.)

Foor = compressive buckling stress for

curved skin panel

Fo = compressive stress at bending general instability (psi)

BUCKLING STRENGTH OF CURVED SHEET PANELS

ULTIMATE STRENGTH OF STIFFENED CURVED SHEET STRUCTURES

Table C9.1

(Ref 7)

THEORETICAL GENERAL INSTABILITY STRESSES OF ORTHOTROPIC CIRCULAR CYLINDERS

(Results are based on the assumption that spacings of longitudinal stiffeners and circumferential frames are uniform and small enough to permit

assumption that cylinder acts as orthotropic shell)

Fy = Compressive stress at bending general instability (psi)

Fy = Circumferential normal stress under external Pressure at general instability (psi}

FsT = Shearing stress at torsional general instability {psi)

b= Stringer spacing (in }

d= Frame spacing (in }

R= Cylinder radius (in }

¢ = Skin thickness (in )

Ag = Stringer area (in 7)

Ag = Frame section area (in 7)

ts = Distributed stringer area = Ag/b

tg = Distributed frame area = Ag/d

if = Bending moment of inertia of frame section (in *)

I = Distributed bending moment of inertia of frame = ff/d

Os = Stringer section radius of gyration (in )

Đr = Frame section radius of gyration (in }

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES For the frames the effective skin width

should be taken as the total frame spacing (d)

For inelastic stresses, the use of the secant

modules appears to be applicabie on the basis

of limited test data

External Radial or Hydrostatic Pressure

The effective skin width to be used in

computing the stringer and frame section

properties may be determined from the following

equation

We We

~s ft Be 2 05 Pog /Fe) / ell le c9.9

The subscript s refers to stringer and f

refers to frame The term d is the frame

spacing

Torsion

The effective skin to be used can be

determined from the following equation:—

Weg M

ey 70.5 (Fegp/Pat)*/* - 09.10

where F Ser is the shear buckling stress

for the curved skin panel Fst, the torsional

shear stress at torsional general instability

(psi)

The equations for torsion as given in

Table C9.1 would not apply to shells in which

there is a strong tension field that could

introduce appreciable secondary stresses in

the frame The reader should refer to Part 2

of Chapter Cll for a treatment of this subject

involving the effect of semi-tension field

action in the skin panels

Transverse Shear General Instability

From Ref 7, it 1s stated that a con-

servative shear general instability shearing

stress may be made by utilizing the relation

where Fg is the transverse Shear stress under

transverse shear general instability

General Instability in Combined Torsion and

Bending

From (Ref 7} the following interaction

relation may be used to compute the permissible

combinations of applied torsion and bending

Moments to a stiffened cylinder

C9, 13 where M = applied moment

Mg = moment causing bending general in- :

stability acting alone I

T = applied torsional moment

To = torsional moment causing torsional general instability acting alone

General Instability in Combined Transverse

Shear and Bending

(Ref 7) concludes there is no interaction

for this combination of transverse shear and bending loads General instability occurs only

for either type of loading acting alone and thus doth loadings may be examined separately

C9.14 Buckling of Spherical Plates Under Uniform External Pressure

Classical Theory using 0.3 for Poisson’s

ratio gives the following buckling stress for

a perfect spherical shell subjected ta 4 uniform external pressure:-

Avatlable test data on practical shells show this theoretical buckling stress to be much too high Thus to satisfy experimental results, reduced values must be used The buckling equation which is similar to that for

curved plates, under external pressure (from

Ref 2) is,

Kp tt E tye

For = 1B (1 - BA a

Fig C9.8 shows curves for determining the

buckling coefficient Kp and shows how test data falls considerably below the theoretical buckling

curve Equation C9.14 is for buckling stresses

below proportional limit stress of material

Report AS-D-S68 of the Astronautics Division

of General Dynamics Corp from a statistical

study of test data gives the following equations for the buckling stress of spherical shells under

uniform external pressure for use in preliminary design

For Mean expectsd value:-

BUCKLING STRENGTH OF CURVED SHEET PANELS

Pressure Compared with Empirical Theory

The equations are for buckling stresses

below the proportional limit stress of the

material

(1)

(2) (3)

(4)

ULTIMATE STRENGTH OF STIFFENED CURVED SHEET STRUCTURES,

PROBLEMS

The fuselage cross-section as given in Fig

C9.6 of example problem 1 is changed by increasing the skin thickness to 05 inches, The design loads are increased to the

The fuselage section as given in problem 1

above is subjected to an internal outward

pressure of 6 psi What would be the com- pressive and shear buckling stress for the skin panels under this internal pressure

REFERENCES Schildcrout &@ Stein Critical Combinations

of Shear and Direct Axial Stress for Curved

Rectangular Panels NACA T.N 1928

Gerard 2 Becker Handbook of Structural

Stability Part III Buckling of Curved Plates and Shells NACA T.N 2885,

Rafel & Sandlin fect of Normal Pressure

on the Critical Compression and Shear Stress

or Curved Sheet NACA WRL-57

Rafel Effect of Normal Pressure on the Critical Compressive Stress of Curved Sheet

NACA WRL-258

Brown & Hopkins The Initial Buckling of

a Long and Slightly Curved Panel under Combined Shear and Normal Pressure R&M

No 2766, BRITISH ARC (1949)

Shanley F.R Simplified analysis of

General Instability of Stiffened Shells in

Pure Bending Jour of Aero Sciences, Vol 16, Oct 1949

Becker H Handbook of Structural Stability

Part VI Strength of Stirren ed Curved Plates and Shells NACA T.N 3786

The analysis and design of a metal beam

composad of flange members riveted or spot-

Nelded to wed members is a frequent problem in

airclane structural design In this chapter,

the general theory zor beams with non-buckling

weds is considered In Chapter Cll, the more

common case where the beam web wrinkles and

zoes over into a semi~tenston field condition

is considered The advantages and disadvantages

of the nom-buckling and the buckling or semi-

tension field web are discussed in Chapter Cll

The general beam theory as given in this chapter

is basic to that given in Chapter Cll, thus the

student should study this chapter before Cll

gyration of the beam section as large 45 possible, and at the same time maintain a flange section which will have a high local crippling or crush- ing stress Furthermore, the flange sections for large cantilever beams which are frequently used in wing design should be of such shape as

to permit efficient tapering or reducing of the

section as the beam extends outboard This

tapering of section should also be considered

from a fabrication or machining standpoint “he most efficient flange from a strength/weight standpoint might be very costly or entirely impractical from a fabrication and assembly standpoint

Basic “lange

seccion 1s square tube

10.1 shows a few typical metal 5eam

ad wings Such