73015-Estimation of Local Buckling

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73015ESTIMATION OF THE LOCAL BUCKLING STRESS UNDER BIAXIALCOMPRESSION OF AN ISOTROPIC SKIN WITH FIBRE REINFORCED INTEGRALUNFLANGED STIFFENERS

1. INTRODUCTION

This Item illustrates by reference to a specific example the use of Item No. 73007 for estimating the localbuckling * stress under biaxial loading of an isotropic skin stiffened by orthotropic unflanged stringers.

2. NOTATION

The notation used is the same as that of Item Nos. Struct 02.01.28, 73007 and 94003. The suffixes listedbelow are added to facilitate description of the calculations.

It should be noted that the sign convention of Item Nos. Struct 02.01.28 and 73007 is compressive stressespositive whereas the sign convention of Item No. 94003 is tensile stresses positive.

3. EXAMPLE

The panel shown above is subjected to a transverse compressive stress of 20 N/mm 2 and to a longitudinalloading such that the stringers remain unbent in their own plane. It is required to find the longitudinal loadat which local buckling will occur. The material of the plate and stiffeners is aluminium alloy which has aYoung's modulus of 73 000 N/mm 2 , a shear modulus of 28 100 N/mm 2 and a Poisson's ratio of 0.3. Thestiffeners are reinforced by layers (5 each side) of 0.04 mm thick carbon fibre reinforced plastic. Theproperties of an individual layer of reinforcement under direct in-plane stresses are: Young's modulus of

* Local buckling is described and discussed in Item No. 72012 "Information on the use of Data Items on the buckling of plates andcompression panels manufactured from isotropic materials".

refers to panel and to x-direction

refers to skin and to x- and y-directions respectively

refers to stiffener and to x-direction

px

skx sky,

s tx

Issued November 1975with Amendment A


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73015207 000 N/mm 2 along the fibres and 7600 N/mm 2 across the fibres, shear modulus parallel and normal tothe fibres of 5000 N/mm 2 and Poisson's ratio under load along the fibres of 0.3. The fibres are arranged,starting with the innermost layer and going outwards, at angles of 0, 15, 0, +15, 0 to the longitudinalaxis of the stiffeners.

4. DESCRIPTION OF THE CALCULATION PROCEDURE

4.1 General

The basic method of determining the local buckling stress of panels with unflanged integral stiffeners isdescribed in Section 4.1 of Item No. Struct 02.01.28 and is shown to be dependent upon the relationship

(4.1)

between the unbalanced couple at the junctions of component flats and the angular rotations at the junctions. The stiffness A1 denotes a value of for A and B2 a value of for B. The values of and

depend upon the buckle half wavelength, the longitudinal and transverse stresses and the platestiffnesses.

Under the action of the longitudinal compressive stress local buckling occurs when the value of inEquation (4.1) is zero. The calculation procedure consists of a search for the wavelength that yields theminimum stress at . This minimum stress is then the local buckling stress. Notes on the initialchoice of the stress and wavelength appear in Item No. Struct 02.01.28 and in Section 5.6 of this Item.

4.2 Particular Calculations for the Example

The reinforced stiffener may be represented as a single orthotropic plate. The Example of Item No. 94003calculates the in-plane and flexural stiffness matrices A and D respectively.

Owing to the different in-plane stiffnesses of the skin and the stringer and the transverse load in the skinthe stresses acting on these members will not, in general, be equal for a given panel strain. For this reasonit is simpler in this particular example to investigate a range of panel loads in terms of the stresses actingin the component flats.

5. THE CALCULATION

5.1 Data Calculated in Item No. 94003

From the example of Item No. 94003 for the stiffener

N/mm, mm/N

and N mm.

T / A1 2 B 2+=

T 1 2 12

T /

T / 0=

A159.2 26.92 0

26.92 83.48 0

0 0 32

103= A 1

0.00664 0.00214 0

0.00214 0.0127 0

0 0 0.0312

103=

D

35.4 3.03 0.351

3.03 7.87 0.0340.351 0.034 3.76

103

=


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73015The terms D 13 , D 23 , D 31 and D 32 of the matrix D defined in Item No. 73007 are zero. In the above Dmatrix these terms are sufficiently small in relation to the other terms of D for Item No. 73007 to beapplicable.

5.2 Distribution of Applied Loads Between Skin and Stiffeners

For a given panel load intensity, N px, the stiffener and skin loads must sum to N px and their strains must beequal. The stiffener strain is obtained from the relationship (see Equation (4.6) and Section 6.3.1of Item No. 94003). Since the stiffener is loaded only in the x-direction by N stx the only relevant term of A-1 is the leading term of 0.00664 10 3 mm/N so that estx = 0.00664 10

3 N stx . The skin strain isgiven by

.

Equating the skin and stiffener strains and putting yields the skin and stiffener loadsfor a given N px . Hence, equating the strains gives

Then, summing the load contributions gives

from which N/mm

giving N/mm 2.

Similarly, N/mm 2.

5.3 Calculation of Expressions for and for the Skin

The skin is isotropic so that, using Section 1.2 of Item No. 73007,

N/mm 2

and, noting that compressive stresses are positive when using Item No.73007,

.

The stress parameter is given by

.

e A 1 N=

e skx N skx( N sky ) / Et =

N stx h N skx b N px b=+

N s tx N skx 0.3 20( ) 1.5( ) / 0.00664 103

73000 1.5( )=

N skx 9+( ) / 0.727 N/mm .=

N sk x 9+( ) 16 / 0.727 N sk x 40+ N px 40= N sk x N px 4.95( ) / 1.55= f sk x N px 4.95( ) / 2.32= f s tx 0.887 N px 7.98+( ) / 1.4=

I 2 730003 1 0.3

2( )

----------------------------- 1.540------- 2 371= =

I 1 2 + 20 / 371( ) / b / ( )2

1 0.108 / b / ( )2= =

I

I f sk x

371 I -------------- 1

4--- b---

2 I 1 I -----

+=


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730155.4 Calculation of Expressions for and for the Stiffener

The stiffener may be treated as an orthotropic plate (see Section 5.1 ) so that, using Section 1.3 of Item No.73007,

N/mm 2

and

D o = (3.03 + 2 3.76) 103 = 10.6 10 3 N mm

giving, since no transverse stress exists in the stiffener, and . Thestress parameter is given by

.

Also, .

5.5 Stiffness Scale Factors for and

For the skin

N mm/mm.

For the stiffener

N mm/mm.

5.6 Tables 5.1 and 6.1 and the Selection of the Applied Load N px and Wavelength

Table 5.1 gives and those terms of that depend upon wavelength only. The results of Table5.1 are then used in Table 6.1 , into which are introduced the applied loads. Table 6.1 then uses the figuresof Item No. 73007 to obtain and and subsequently .

Some guidance on the choice of an initial value of N px may be obtained from use of Item No. 70003 whichgives, for the unreinforced panel,

N/mm

at a of about 40 mm. (In cases where an orthotropic component flat is attached at both sides ItemNo. 80023 may be used to give equivalent figures.) Thus, the buckling load of 603 N/mm provides areasonable starting value. The value of = 40 mm cannot be relied upon. It is usually necessary to extendthe range of investigation of to cover a small range of sub-multiples and multiples of the stiffener pitch;it is also advisable to check values approaching the panel length. Table 6.1 commences with a of 40 mmand changes it in steps of 10 mm.

o 4 2

7.87 103( ) / 16 2 1.4( ) 867= =

o 10.6 / 7.87 1.35= = o1 2

1.16=o

o f st x

1.35 867---------------------------14--- b---

2 1.35 35.4/7.871.35

----------------------- + f

1170------------ 0.495

b--- 2= =

D 12 / D o 3.03 / 10.6 0.286= =

1 2

E 't 3 / 3b 73000[ / 1( 0.3 2 )]1.5 3 / 3( 40 ) 2260==

4 D 22 b 4 7.87 103

16 1970= =

1 2 b / ( )

1( A1 ) 22( 2 B2) T /

N px 2.85 73000 1.5 / 40( )2 40 1.5 16 1.4+[ ] 1 / 40( ) 603= =


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73015

6. RESULTS AND CONCLUSIONS

Figure 2 shows that the minimum panel buckling load is 570 N/mm at a half-wavelength of about 50 mm.Actual buckle wavelengths may be influenced by the finite length of the panel and by initial irregularities.Notes on the effect of the initial irregularities are provided in Item No. 72012.

TABLE 5.1

SkinStiffener A

(mm) 371 Q P

3040506070

1.3331.0000.8000.6670.571

0.9390.8920.8310.7570.669

1.2920.9440.7290.5800.467

348331308281248

0.056 0.057 0.060 0.063 0.067

0.5330.4000.3200.2670.229

0.6180.4640.3710.3100.266

0.141 0.079 0.051 0.035 0.026

,

,

.

b / I I 1 2 b ( ) I b / o

1 2 b ( )

Q 1 / 4( ) b / ( )2 I 1 / I ( )=

P 0.495 b / ( )2= I 1 0.108 / b / ( )

2[ ]=


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TABLE 6.1

Skin B Stiffener

N px f skx*

* N/mm 2 and N/mm 2, see Section 5.2 .

Values in these columns are obtauned fromTable 5.1

Q

Values in these columns are taken from Figures 7 and 3 of Item No 73007.

f stx* P

mm N/mm N/mm 2 N/mm 2 N/rad N/mm 2

40 620600580560540

265256248239231

331 0.8010.7730.7490.7220.698

0.057 0.7440.7160.6920.6650.641

0.944 0.490.510.540.580.62

22002300240026002800

399386373360348

0.3410.3300.3190.3080.297

0.079 0.2620.2510.2400.2290.218

0

50 600580560540520 222

308 0.8310.8050.7760.7500.721

0.060 0.7710.7450.7160.6900.661

0.729 0.380.390.430.450.48

17001800190020002200 335 0.286

0.051 0.2790.2680.2570.2460.235

0

60 620600580560540

281 0.9430.9110.8830.8510.822

0.063 0.8800.8480.8200.7880.759

0.580 0.330.350.370.390.42

15001600170018001900

0.035 0.3060.2950.2840.2730.262

0

70 680660640620600

291282274

248 1.1731.1371.1051.0691.032

0.067 1.1061.0701.0381.0020.965

0.467 0.270.290.320.340.35

12001300140015001600

437424411

0.3740.3620.351

0.026 0.3480.3360.3250.3150.304

0

30 750740

730720700

321317

313308300

348 0.9220.911

0.8990.8850.862

0.056 0.8660.855

0.8430.8290.806

1.292 0.650.66

0.700.720.79

29003000

320033003600

481475

468462449

0.4110.406

0.4000.3950.384

0.141 0.2700.265

0.2590.2540.243

0

sk x N px 4.95( ) / 2.32( )= f st x 0.887 N px 7.98+( ) /1.4=

371 I f skx

371 I -------------- I I

1 2 b---

2 E 't

3 / 3b

----------------- - 2 B2 22= f stx1170------------ o o

1


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FIGURE 1

N px (N/mm)

500 520 540 560 580 600 620 640 660 680 700 720 740 760

T (N/rad)

-800

-600

-400

-200

0

200

400

600

800

1000

1200

50 60 40 70 30(mm)


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FIGURE 2

(mm)

30 40 50 60 70

N

px

(N/mm)

550

600

650

700

750


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73015THE PREPARATION OF THIS DATA ITEM

The work on this particular Item was monitored and guided by the Structures Committee which has thefollowing constitution:

The work on this Item was carried out in the Strength Analysis Group. The member of staff who undertook the technical work involved in the initial assessment of the available information and the construction andsubsequent development of the Item was

ChairmanMr I.C. Taig British Aircraft Corporation Ltd, Preston

Vice-ChairmanProf. W.S. Hemp University of Oxford

MembersMr H.L. Cox Independent

Mr R.S. Dabbs Hawker Siddeley Aviation Ltd, KingstonMr K.H. Griffin Cranfield Institute of TechnologyMr N.F. Harpur British Aircraft Corporation Ltd, FiltonDr G.Z. Harris Royal Aircraft EstablishmentMr P.J. McKenzie Hawker Siddeley Aviation Ltd, KingstonMr J.G. ten Asbroek Fokker-VFW N.V., Schiphol-OostMr F. Tyson Scottish Aviation Ltd.

Mr M.E. Grayley Group Head.

73015-Estimation of Local Buckling

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