Lecture #6Normal stressesin thin-walled beam

FOREWORD

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The calculation of stresses in the wing is generally a statically indeterminate problem, involving three basic mechanical equations: equilibrium, compatibility, constitutive.

BASIC EQUATIONS OF SOLID MECHANICS

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Equilibriumequations

Constitutiveequations

Compatibilityequations

This is not only the sum offorces or moments, but appliesfor elementary volume as well

Physical law, expressesthe relation between

stress and strain

Solid body shouldremain continuous

while being deformed

WAYS TO FIND NORMAL STRESSES IN THE WING

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There are two major possible ways:

1. General energy-based methods for statically indeterminate problems (effective only for some specific problems). E.g.: Papkovich theorem (will study later).

2. Beam bending theory, accordingly modified to take account of the physical law.

WAYS TO SOLVE A SOLID MECHANICS PROBLEM

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Displacements are set as unknowns

Strains are derived

Stresses are derived

Compatibility equations

Constitutive equations

Equilibrium equations are

solved

Equilibrium equations

Stresses are set as

unknowns

Strains are derived

Constitutive equations

Equilibrium equations

Compatibility equations are

solved

Compatibility equations

BASIC EQUATIONS IN BEAM THEORY

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Equilibriumequations

Constitutiveequations

Compatibilityequations

Internal force factors are equal to appropriate integrals of moments

Taken into account using the reduction coefficients (a.k.a.

effectiveness factors)

Hypotesis of planar cross sections

LIMITATIONS OF BEAM THEORY

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Beam theory gives unsatisfactory results in following cases:1. Low aspect ratio of the wing. 2. Zone of sweptback angle change. 3. Cut-outs or any other irregularities.

NORMAL STRESSES IN THIN-WALLED BEAMS

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The distribution of normal stresses obeys the hypothesis of planar cross sections:

,w x y a b x c y

For the case of uniform linear material, it comes to be (for a right coordinate triad):

, yz xz

y x

MN Mx y x y

A I I

CROSS SECTION DISCRETIZATION

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(CE – conservative estimation, OE – optimistic estimation)

The discretization of real cross section is usually used to possess the calculations of moments of inertia and other geometrical properties:

- small but complex elements like stringers are substituted by point areas positioned at stringer center of gravity (CE) or at the skin surface (OE);

- skins are substituted by center lines (CE);

- complex center line is substituted by polygonal curve;

- portions of skins are substituted by point areas (CE).

CROSS SECTION DISCRETIZATION

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The problem is to find the moment of inertia.Dimensions:a = 60 mm; h = 22 mm; 1 = 4 mm; 2 = 6 mm;H = 120 mm.

CROSS SECTION DISCRETIZATION – variant #1

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One option is to substitute the real cross section by center lines with appropriate thick-nesses – usually used in FEA.

Ix, cm4 Iy, cm4

Exact 975 2478

Discrete 1000 2494

Error,% +2.5 +0.6

CROSS SECTION DISCRETIZATION – variant #2

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Another option is to use concentrated areas instead of stiffeners and webs – very good discretization.

Ix, cm4 Iy, cm4

Exact 975 2478

Discrete 972 2477

Error,% -0.3 -0.0

CROSS SECTION DISCRETIZATION – variant #3

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Finally, all thin members could be substituted by point areas – ideal for manual calculations.

Ix, cm4 Iy, cm4

Exact 975 2478

Discrete 972 2419

Error,% -0.3 -2.4

CROSS SECTION DISCRETIZATION – variant #4

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If point areas are positioned at the skin surface, the moment of inertia is overestimatedwhich is not recommended.

Ix, cm4 Iy, cm4

Exact 975 2478

Discrete 1092 2419

Error,% +12 -2.4

WAYS TO TAKE THE ACCOUNT OF PHYSICAL LAW

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There are many possible ways, some of them are:

1. Method of sequential loading.

2. Method of variable secant modulus.

3. Method of reduction coefficients (a.k.a. effectiveness factors).

The last one is widely used since it is convenient for cross sections with multiple materials, and to study a post-buckling behavior.

METHOD OF REDUCTION COEFFICIENTS

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The reduction coefficient is the ratio between the real stress in the member and the fictitious stress obeying the Hook law :

,

ii

i f

, ,i f f i fE

The reduction coefficient is introduced to maintain the condition that fictitious strain is equal to real one:

,i i f

METHOD OF REDUCTION COEFFICIENTS

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The method of reduction coefficient is iterative one. At the first iteration, we set the initial reduction coefficient as the ratio between Young moduli:

Then we calculate fictitious geometrical properties:

,

ii

i f

EE

,i f i iA A

METHOD OF REDUCTION COEFFICIENTS

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Next we calculate the fictitious stress and find the real stress, forming a new iteration of reduction coefficient:

The process stops when values of reduction coefficients are converged for all points in the section.

,

ii

i f

METHOD OF REDUCTION COEFFICIENTS

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Calculation of red.coef.

Fictitious geom. properties

Fictitious and real stresses

Initial value is the ratio between Young moduli

Area and axial moments of inertia are calculated using fictitious areas of members

New reduction coefficients

Resultant real stresses

The process is stopped upon convergence of red.coefficients

Real stresses are derived from material physical law or

compression diagram for panel

WHERE TO FIND MORE INFORMATION?

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Megson. An Introduction to Aircraft Structural Analysis. 2010Chapter 15 – beam bending theory;Chapter 19 - discretization of the cross section.

Method of effectiveness factors (reduction coefficients) is explained inBruhn. Airplane Design HandbookChapter A19.11

… Internet is boundless …

TOPIC OF THE NEXT LECTURE

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Shear stresses in thin-walled beam

All materials of our course are availableat department website k102.khai.edu

1. Go to the page “Библиотека”2. Press “Structural Mechanics (lecturer Vakulenko S.V.)”