Bruhn - Analysis And Design Of Flight Vehicles Structures Episode 3 part 5 docx

CHAPTER C5 BUCKLING STRENGTH OF FLAT SHEET IN COMPRESSION, SHEAR, BENDING AND UNDER COMBINED STRESS SYSTEMS C5.1 Introduction.. Chapter A1l8, Part 2, introduced the student to the theo

STRENGTH & DESIGN OF ROUND, STREAMLINE, OVAL AND SQUARE TUBING C4 28 IN TENSION, COMPRESSION, BENDING,

no restrictions on type of truss or

arrangement of members, however, the goal

is the lightest truss Omit consideration

of weight of any gusset plates at truss

Same as problem (6), but instead of a

cantilever truss use a simply supported truss with supports at points (A) and (B)

Fig 5 shows a front beam and front lift

strut in an externally braced monoplane

The wing beam and lift strut are in the Same vertical plane The ultimate design

loads on the beam for the critical

conditions are w = 50 ib./in and w =

ibs per inch

down

(a)

-30

Minus means load is acting

Design a streamline tube to act as

the lift strut Material {ts 2024-T5

aluminum alloy

Same as (a) but made from alloy steel Fry = 75000 Compare the weights of the two designs

(9) Pig, 6 illustrates the strut and wire

bracing for attaching float to fuselage of

a seaplane Determine the necessary sizes for the streamline struts AC and BD for the following load conditions

Condition 1 ¥V = -32000 lbs.,

H = - 8000 lbs

Condition 2, V = ~ 8000 lbs.,

H = ~28000 lbs

Material 2024-T3 aluminum alloy

Tube size 2 - 065 round L = 44 in.,

C's 1.5 Material alloy steel Fry =

95000, welded at ends Design ultimate loads equal 22000 1b compression and

28000 lbs tension Find margin of safety

(11) Same as Problem 10 but heat treated to

Fey = 150000 after welding

compression L=30 in Use C=l

Design the lightest round tube from the following materials and compare their weights

{a) Aluminum alloy 2024-T3

(b) Alloy steel Fty = 180,000 (c) Magnesium alloy Fey = 10,000 Same design load as in problem (10) but design the lightest streamline tube from 2024-T3 aluminum alloy material

A round tube is to carry an ultimate pure bending moment of 14000 in lbs Select the lightest tube size from the following materials and compare their weights

(a) Alloy steel Fry = 240,000, (b) 2024-73

aluminum alloy, (c) Magnesium alloy

Fey = 30000, (4) Titanium 6AL¬4V

alloy

A round tube 20 inches long is to carry

an ultimate torsional moment of 15000 in

Ib Select the lightest tube size from the following materials and compare their weights

(4) Alioy steel Fey = 180,000, (b) Aluminum alloy 2024-T3, (c) Mag- nesium alloy Fry = 36000

Determine the lightest 2024-TS aluminun alloy round tube 10 inches long to carry

a combined bending and torsional design load of 4500 and 3000 in.lbs respectively Same as Problem 16, but change material

to alloy steel Fry = 95000

A 1-1/2 - 065 2024-T3 round tube 50 inches

long is used as a beam-column The distributed load on beam is 12 lb per inch and the axial load is 700 lbs What

is the M.S under these loads

If the tude in problem (19) was also subjected to a torsional moment of 1400 in.1b., what would be the M.S

CHAPTER C5

BUCKLING STRENGTH OF FLAT SHEET IN COMPRESSION, SHEAR,

BENDING AND UNDER COMBINED STRESS SYSTEMS

C5.1 Introduction

Chapter A1l8, Part 2, introduced the

student to the theoretical approach to the

problem of determining the buckling equation

for flat sheet in compression with various

edge or boundary conditions A similar

theoretical approach has been made for other

load systems, such as shear and bending, thus

the buckling equations for flat sheet have

been available for many years This chapter

will summarize these equations and provide

design charts for practical use in designing

sheet and plate structures Most of the

material in this chapter is taken from (Ref 1),

NACA Technical Note 3781-Part I, "Buckling of

Plat Plates” by Gerard and Becker This report

is a comprehensive study and summary of

practically all important theoretical and

experimental work published before 1957 The

report is especially useful to structural design

engineers

C§.2 Equation for Elastic Buckling Strength of

Flat Sheet in Compression

From Chapter Al8, the equation for the

elastic instability of flat sheet in compression!

18,

nk, E

oy = ——— @* - (05.1)

12 (1-Uạ) Ô

Where kg = buckling coefficient which depends

on edge boundary conditions and

sheet aspect ratio (a/b)

E = modulus of elasticity

Vg = elastic Poisson’s ratio

bd = short dimension of plate or loaded

edge

t = sheet thickness

CS5.3 Buckling Coefficient ke

Fig C5.1 shows the change in buckled

shape as the boundary conditions are changed

on the unloaded edges from free to restrained

In Fig (a) the sides are free, thus sheet

acts as a column In Fig (bd) one side is

restrained and the other side free, and such a

restrained sheet is referred to as a flange

In Fig (c) both sides are restrained and this

restrained element ts referred to as a plate

supports are added along unlaaded edges Note changes in

buckle configurations

Fig C5.2 gives curves for finding the puckling coefficient kK, for various boundary

or edge conditions and a/b ratio of the sheet

The letter C on edge means clamped or

fixed against rotation The letter F means a

free edge and SS means simply supported or hinged Fig C5.3 shows curves for k, for

various degrees of restraint (e) along the

sides of the sheet panel, where ¢ is the ratio

of rotational rigidity of the plate edge stiffener to the rotational rigidity of the plate

Fig C5.4 shows curves for k, fora

flange that has one edge free and the other

with various degrees of edge restraint Fig

CS.5 illustrates where the compressive stress varies linearly over the length of the sheet,

a typical case being the sheet panels on the upper side of a cantilever wing under normal flight condition

Fig CS.6 gives the ky factor for a long

sheet panel with two extremes of edge stiff-

ener, namely a zee stiffener which is a torsitonally weak stiffener and a hat section

je

Fig CS.4 (Ref 1) Compressive-buckling-stress coefficient

of flanges as a function of a/b for various amounts of edge

Fig C5.3 (Ret 1) Compressive-buckling-stress coefficient

of plates as a function of a/b for varicus amounts of edge rotational restraint

with linearly varying axiai load ke p17 E 2

which 1s a closed section and, therefore, a Substituting in Eq C5.1,

relatively torsionally strong stiffener File

c5.6a gives the compression buckling Jan = mn? x 4.0% 10,700,000 (204? = 2480 pst

coefficients ky for isosceles triangular cr 12 (1 - 3*) 5 x psi

range, then E and YV are not the same as for

elastic buckling, thus a plasticity correction factor is required and equation C5.1 is

written,

NV ke BO

STIEFEWER 7 H T 12(1 - 1⁄2?)

Fig C5 8 (Ref 1} Compressive-buckling coefficient for long

rectangular stiffened panels as a function of b/t and stiffener

Fig, C5 6a (Ref 1) Uniform Compression

Illustrative Problem Find the compressive

buckling stress for a sheet panel with (a) = 10

and b = 3 inches, thickness t = 04 and all

edges are simply supported Material is

Thus

using curve (c) for a/b = 2, we read ky = 4.0

The values of k, and Uy are always the

elastic values since the coefficient 7 contains

all changes in those terms resulting from

inelastic behavior

A tremendous amount of theoretical and “ experimental work has been done relative to

the value of the so-called plasticity cor-

rection factor Possibly the first values

used by design engineers were n = E,/E or <

N= Esec/E- Whatever the expression for ] it

must involve a measure of the stiffness of the

material in the inelastic stress range and

since the stress-strain relation In the plastic range is non-linear, a restrt must be made to

the stress-strain curve to obtain a plasticity correction factor This complication is

greatly simplified by using the Ramberg and Osgood equations for the stress-strain curve which involves 3 simple parameters (The

reader should refer to Chapter Bl for information on the Ramberg-Osgood equations.)

Thus using the Ramber-Osgood parameters (Ref.1)

presents Figs C5.7 and CS.8 for finding the compressive buckling stress for flat sheet panels with various boundary conditions for both elastic and inelastic buckling or in-

~ The sketch shows a 3x9 Ess

inch sheet panel The sides ss 4 ga are simply supported The

material is aluminum alloy SA TTT

2024-73 The thickness is van 094" = 10,700,000, H1 :

Fig C$.7% Chart of Nondimensional Compressive Buckling Stress for Long

Hinged Flanges 7] = (Eg/E)(1 - Ve*)/(1 - VU)

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES

Ve = 0.3 Pind the buckling stress oor

Solution: We use Fig C5.8 since it covers

the boundary conditions.of our problem The

parameter for bottom scale is,

The use of Fig C5.8 involves the use of

o,,, and n the Ramberg-Osgood parameters

Referring to Table Bl.1 of Chapter Bl, we find

for 2024-T3 aluminum alloy that o,., = 39000

and the shape factor n = 11.5 —

Tei phoosy

Substituting in (A):-

4,0 1° x 10,700,000 (2084) so)" #985

12 a - 3°) 39, 000 ~

From Fig C5.8 using 98 on bottom scale

and n = 11.5 curve, we read on left hand scale

that Gor/o,,, = 84.-

NN Then doy’ = 39000 x 84 = 32800 pst ~

If we neglected any plasticity effect, then!

we would use equation C5.2 with ne =1,0, oy

n? x4 0x10,700, 000 cán"

12 - 3°)

Whereas the actual buckling stress was 32800,

or in this case the plasticity correction

factor is 328/384 = 954

The sheet thickness used in this example

of 094 is relatively large If we change the

sheet thickness to 051 inches the results

would be practically no correction within the

accuracy of reading the curves, and the buckling|

Stress Ogr would calculate to be 11200 psi,

which is below the proportional limit stress

and thus no plasticity correction

C5.6 Cladding Reduction Factors

Aluminum alloy sheet is available with a

thin covering of practically pure aluminum and

is widely used in aireraft structures Such

material is referred to as alclad or clad

aluminum alloy, The mechanical strength

properties of this clac material is consider-

ably lower than the core material Since the

clad is located at the extreme fibers of the

alclad sheet, it is located where the strains

attain their highest value when buckling takes

place Fig C5.9 shows make up of an aliclad

Sheet and Fig C5.10 shows the stress-strain

curves for cladding, core and alclad combina-

tions

C5.5

Thus a further correction must be made for

alclad sheets because of the lower strength

clad covering material Thus the buckling stress for alclad sheets can be written:

by use of equation C5.3, using values of

and Alclad Combinations Ø/Œ„are = Ì - Ý +01; 0 *ƠglØcore:

Tabie C5 1 (Ref 1) Summary of Simplifted Cladding

Reduction Factors

BUCKLING UNDER SHEAR LOADS

C5.7 Buckling of Flat Rectangular Plates

Under Shear Loads

The critical elastic shear buckling stress for flat plates with various boundary

conditions is given by the following equation:

Where (b) is always the shorter dimension of

the plate as all edges carry shear k, is the

shear buckling coefficient and is plotted as a

function of the plate aspect ratio a/b in Fig

CS.11 for simply supported edges and clamped

Test results compare favorably with the

results of equation (5.5 if 7], * Gg/G where G

18 the shear modulus and Gg the shear secant

modulus as obtained from a shear stress-strain

diagram for the material

A long rectangular plate subjected to ' pure shear produces internal compressive

stresses on planes at 45 degrees with the

plate edges and thus these compressive stresses

cause the long panel to buckle in patterns at

an angle to the plate edges as illustrated in

Fig C5.12, and the buckle patterns have a half

wave length of 1.25b

fae 1 25b —o

Fig C5 12 (Ref 7) Fig C5.13 is a chart of non-dimensional shear buckling stress for panels with various

This chart fs

Similar to the chart in Figs C5.7 and C5.8 in

that the values go, „ and n must be known for

the material before the chart can be used to

find the shear buckling stress

BUCKLING UNDER BENDING LOADS

C5.8 Buckling of Flat Plates Under Bending Loads

The equation for bending instability of flat plates in bending is the same as for

compression and shear except the buckling co- efficient kp is different from k, or kg When

@ plate in bending buckles, it involves relatively short wave length buckles equal to 2/3 0 for long plates with simply supported edges (see Fig C5.14) Thus the smaller buckle patterns cause the buckling coeffictent

Where Kp is the buckling coefficient and

is obtained from Fig C5.15 for various a/b

ratios and edge restraint « against rotation

In the a/b ratio the lodded edge is (b),

The plasticity reduction factor can be obtained from Fig C5.3 using simply supported

edges

BUCKLING OF FLAT SHEETS UNDER COMBINED LOADS

The practical design case involving the

use of thin sheets usually involves a combined

load system, thus the calculation of the buckling strength of flat sheets under com- bined stress systems 1s necessary The approach used involves the use of inter-action equations or curves (see Chapter Cl, Art

C1.15 for explanation of inter-action equations)

C5.9 Combined Bending and Longitudinal Compression

The interaction equation that has been widely used for combined bending and longi-

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES C5.7

Fig CS, 13 (Ref 1) Chart of Nondimensional Shear Buckling Stress for Panels With

Edge Rotational Restraint 7) = (Eg/E) (1 - Ve*/(1 - V4

ae Ver he Meas B.#vo

this equation is found in many of the

structures manuals of aerospace companies

Fig C5.15 is a plot of eq C5.8 It also shows curves for various margin of safety

values

Lt

Fig C5.15 Combined Bending

& Long Compression

Rpts Real

1.0

0 1 3 3 4 5 6 ,7 8 9 1.0 L1

5.10 Combined Bending & Shear

The interaction equation for this com-

bined leading (Ref, 1 & 2) 1s,

(cS.10)

Fig C5.16 is a plet of equation C5.9

Curves showing various M.S values are also

shown Rg is the stress ratio due to

torsional shear stress and Rgt is the stress

ratio for transverse or flexural shear stress

CS.11 Combined Shear and Longitudinal Direct Stress

Fig.-C5.17 ts a plot of equation CS.11,

If the direct stress is tension, it is included on the figure as negative compression using the compression allowable

CS.12 Combined Compression, Bending & Shear

From Ref 5, the conditions for buckling

are represented by the interaction curves of Fig C5.18 This figure tells whether the sheet will buckle or not but will not give the margin of safety

Rp:- if the value of the Re curve defined by Given the ratios Re, Rg and

the given value of Rp and Rg is greater

mumerically than the given value of Re, then the panel will buckle

1,0

Rp

9

BUCKLING STRENGTH OF FLAT SHEET IN COMPRESSION, SHEAR, BENDING AND UNDER COMBINED STRESS SYSTEMS C5 10

The margin of safety of elastically

buckled flat panels may be determined from

Pig C5.19 The dashed lines indicate a

typical application where Ry = 161, Rg = 23,

and Rp = 38 Point 1 is first determined for

the specific value of Rg and Rp The dashed

diagonal line from the origin O through point

1, intersecting the related Re/Rg curve at

point 2, yields the allowable shear and bending

stresses for the desired margin of safety

calculations (Note when R, is less than Rg

use the right half of the figure; in other

cases use the left nalf)

C§.13 Mlustrative Problems

In general a structural component com-

posed of stiffened sheet panels will not fail

when buckling of the sheet panels occurs since

the stiffening units can usually continue to

carry more loading before they fail However,

there are many design situations which require

that initial buckling of sheet panels satisfy

certain design specifications For example,

the top skin on a low wing passenger airplane

should not buckle under accelerations due to

air gusts which occur in normal every day

flying thus preventing passengers from

observing wing skin buckling in normal flying

conditions Another example would be that no

buckling of fuselage skin panels should occur

while airplane is on ground with full load

aboard in order to prevent public from

observing buckling of fuselage skin In many

airplanes, fuel tanks are built integral with

the wing or fuselage, thus to eliminate the

chances of leakage developing, it is best to

design that no buckling of sheet panels that

bound the fuel tanks occur in flying and

landing conditions In some cases aerodynamic

or rigidity requirements may dictate no

buckling of sheet panels To insure that

buckling will not occur under certain load

requirements, it 1s good practice to be

conservative in selecting or calculating the

boundary restraints of the sheet panels,

Problem 1

Pig CS.20 shows a portion of a cantilever wing composed of sheet, stiffeners and ribs

The problem is to determine whether skin panels

marked (A), (B) and (C) will buckle under the

various given load cases The sheet material

compressive axial load of 2 x 700 = 1400 lb

Since the P, loads are not acting through the

centroid of the cross-section, a bending moment

is produced about the x-x axis equal to 1400 x

3.7 = 5170 in lb = My, where 3.7 15 distance from load P, to x=x axis

Area of Zee Pr Stringer = 18

Area of Corner Member =

0.25 sq in

Fig C5 20

The sheet thicknesses, stiffener areas and

all necessary dimensions are shown on Fig

c5.20 The total cross-sectional area of beam

section including all skin and stringers is

3.73 sq in The moment of inertia about x-x

centroidal axis calculates to be 49.30 in.*

Since the beam section is symmetrical, the top panels A, B and C are subjected to the same stress under the P, load system

Compressive stress due to transferring

loads P, to centroid of beam cross-section is,

t, = 2P,/area = 1400/3.73 = 375 psi Compressive stress due to constant bending moment of 5170 in lbs is,

f, = M,2/T,, = 5170 x 4,.233/49.30 = 444 psi Total T1 3 375 + 444 = 819 pst

The skin panels are subjected to compres- sion as shown in Pig a The boundary edge conditions given by the longitudinal sti?*-rors

ANALYSIS AND DESIGN OF FLIGHT VEHICLE STRUCTURES

and the rib flanges will be con- The Re + Rg = 431 + 3097 = 826 Since

servatively assumed as simply Pris? the result is less than 1.0, no buckling

Supported (Fo, is same as dor) 4 SSF occurs

ssc

“>2 TU Ge -

(See Hq 05.1) môn pb a = rẽ 2 1= ,69

From Fig C5.2 far Case Œ, g (a) `

Po, 202 fer 4.0 1057005000 (088)* ¡on nọ; The two loads P, produce bending and

12 (1 - 0.3") 5 P - fleXural shear on the beam,

Since Foor the buckling stress is less than the applied stress fg, the panels will

moment of 500 x 16.5 = 8250 in lb on the

beam structure, which means we have added 4

pure shear stress system to the compressive

stress system of Case 1 loading

The shear stress in the top panels A, B and € 1s,

a/p = 18/5 = 3 Prom Fig C5.11, for

hinged or simply supported edges, we read

Kg = 5.8

TẾ x 6.8 x 10,700,000

ne “Ser” 2 (1 - 3? (SE) = 2760 pst 2035 ,3_

The sheet panels are now loaded in

combined compression and shear so the inter-

action equation must be used From Art C5.12

the interaction equation is Ry + Rễ =1

Rẹ =ứ, đ 3 3 é ‡ b 819/1900 = 431

Rg = s ay œ 3 3 a 854/2760 = 309

The bending moment

produces a different end compressive stress on

the three sheet panels since the bending moment

is not constant over the panel moment To simplify we will take average bending moment on

The two loads P, produce a traverse shear

load V = 200 lb The flexural shear stress must be added to the torsional shear stress ag

found in Case 2 loading

Due to symmetry of beam section and Py, loading the shear flow q at midpoint of sheet panel (B) is zero We will thus start at this

Gs = - 10.94-4.05 x 051 x 3.69% 3.69/2 = -12.34 (See Fig b for plot of shear flow)