Introduction to Aircraft Structure - 4

7/29/2019 Introduction to Aircraft Structure - 4

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Introduction to Aircraft Structure

References: Strength Of Materials (Shanley)

Structural Principles and Data (R.Ae.S. Handboook)

-T.G.A.Simha


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Structural Analysis

Objective

To ensure Airframe has adequate Strength

To ensure Airframe has adequate durability

Adequate Strength

No Yielding at limit Load

No Failure at ultimate Load

Adequate Durability

Achieve Design Service Life


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Determination of Strength

Strength

Based on Material Properties

Based on Structural Geometry

Strength expressed as an Allowable Stress

Analysis to determine the applied stress

Adequacy expressed as

Reserve Factor = (Allowable Stress)/ (Applied Stress)

Margin of Safety = Reserve Factor 1.0


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Fundamental Principles

Equilibrium

Compatibility

Saint Venants Principle

Conservation of Energy


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Stresses

In 3 D there are 3 direct and 3 Shear stresses.

Complimentary Shear Stresses

Plane Stress State

xy = yx

yz =zy

zx = xz

xyy

xy

x

z

yz

xy

xz

x

yx


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Stress Transformation

Mohr Circle

Stress Transformations


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Stress Strain Relations


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Strain Transformation


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Basic Structural Elements

Classification of Force Transmission


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Axially Loaded Structures

Examples :

Tubular Fuselage structure of Light Airplanes

Undercarriage side braces

Control Rods

Stress = P = LoadA Area of cross - section

Trusses

Simple Structure

Light Weight and Good Stiffness

Must be loaded at joints (predominantly)


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Analysis of Structures

Criteria for stability and determinacy

2D Truss m = 2j 3

3D Truss m = 3j 6

Where

j ---- No. of Joints

m---- No. of members


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Method of Analysis

Method of Joint

Equilibrium equation at each joint

Solution of joints in succession

Determine load in each member

Method of Section

Consider a section through the structure

Section with 3 members -2D truss

Section with 6 members -3D truss

Obtain Loads in members using equilibrium equations.


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Deflection of trusses

Methods for determination of deflection

Influence coefficient method

Unit load method

Principle of Virtual work

Castiglianos Theorem

= U/PWhere, --- Deflection

U----Strain EnergyP ----Applied Force

Unit Load method assumes application of a unit load at the point where

is required.Calculate the change in internal energy

= Pi Ui Li

Ai Ei

i = m

i = 1


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Bending of Beams

Beams are loaded transversely

Reacted at the support

Support condition

Simple support

Deflection prevented

Rotation allowed

Fixed support (Clamped)

Deflection prevented

Rotation prevented


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Definitions :-

Bending Moment of a Section

Sum of moment of all forces acting on one side of the section (including

reactions)

Shear Force at a section

Sum of all forces acting on one side of the section. (including reactions)


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Examples of SF and BM diagrams.

Examples of pure Bending Cantilever Beam with concentrated Loads


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Shear and Bending-moment Curves for Cantilever Beams


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Shear and Bending-moment Curves for Cantilever Beams contd...


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Shear and BM diagrams for simple (pinned-end) beams


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Shear and BM diagrams for simple (pinned-end) beams contd


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Stress due to Bending

Basic Equation: M/ I = / Y = E/ R

And = (MY) / I

Strains in a Bent Beam


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Bending of Unsymmetrical Section


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Inelastic Bending

Modulus of Rupture Form Factor


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Bending of Curved Beams

Correction Factor for Maximum Stress in curved

Beams (rectangular c/s)

Effect of Initial Curvature on Strain Distribution


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Shear Stress Distribution

Concept of Shear Flow : q = .t

Shear Flow in beam cross section

q = (V /I) y dAVariation of Shear Stress for Various C/S

0

y


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Concept of Shear Centre


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Deflection of Beams

Beam Differential Equation

d2y = - M

dx2 E I

Therefore y = -M dxdx + Ax + BI

Conjugate Beam method

Unit Load Method

Maxwells Reciprocal Theorem ij = ji


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Torsion of Circular Shafts

Basic Equation:

T/J = /r = G/L

where, J = Polar Moment of Inertia

r and L

Max. Shear force occurs at the outer surface

Applicable for Hollow Shafts also

Nature of Basic Assumptions for Torsion of Solid Round Bars


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Torsion of Non-Circular Shafts Rectangular Section

Shear Stress at corner = 0

Max. Shear stress, = T / (bt2)

The twist, = (TL) / (bt3G) = (TL) / (GJ)

where, J, Torsion Constant = bt3

, constants depend on b/t

b/t 1.00 1.50 1.75 2.00 2.50 3.00 4 6 8 10

0.208 0.231 0.239 0.246 0.258 0.267 0.282 0.209 0.307 0.313 0.333

0.141 0.196 0.214 0.229 0.249 0.263 0.281 0.290 0.307 0.313 0.333


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Torsion of Thin Walled Closed Sections

The shear flow q = T/ (2A) (Bredt Batho Equation)

Where, A is the enclosed area of cross-section

q = Shear Flow, Shear Stress = q / t

J = 4A2 / ds/t

Thin Walled Torsion box


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Thin Walled Open Tubes

Section with constant Thickness continuous

= T/ (befft2)beff= bi

For sections such as I, T

Jeff= JiShear Stress, i = T * L * Ji * 1

Jeff G biti2


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Comparison of Closed and open tube

Closed Tube Open Tube

= T / (2R2t) = 3T / (2Rt2)

= TL / (2R

3

tG)

= 3TL / (2Rt3

G)


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Torsion Bending

General Torsion Equation

T = GJ d/ dz - E d3/ dz3 - Torsion Bending Constant

T = -E d2/ dz2 w*

T = E d3/dz3 . (1/ t) . w* t ds0

s

W* = p t ds (Warping Function)S

0


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Columns

Concept of Buckling - stability

Pb/ = 4Pa/L = K


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Spring Constant k= 4 P/L

This stiffness is provided by Bending stiffness.

Ideal column Pcr = 2 E I (Euler load)

L2

Column Strength is affected by

End Conditions

Material Plasticity

Eccentricities


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Effect of End Conditions

Leff= L Various types of Column and End Constraint


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Effect of Material Plasticity

American Approach

Long Columns

Short Columns

Johnson Parabola

Column Yield Stress

British Approach

Replace E with an Effective Modulus Eeff

Eeff= ET

(Tangent Modulus)


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Effect of Eccentricity (e)

Secant Formula


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Beam Columns

Beam Bending moments magnified by Axial Load

Bending Moments depend on deflection of beam

Mmax = Mo / (1 - P/Pcr) (approximate)

.

Where

Mo Bending Moment of Beam without

end Load

Pcr Euler Buckling Load

P Applied compression (end load)


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Buckling of Plates

Plates subjected to compression buckle similar to columns

Deformation of the plate is characterized by wave length ,

Wave length, depends on the aspect ratio, a/b

Critical stress, = KEeff(t/b)2

K coefficient depends on a/b

Eeff= the effective modulus


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Effective Width

Plate after buckling continues to carry additional load

Stress increases at the sides over an effective width

Effective width, W = 1.71 t (E/ e), e = edge stress

This is also presented as average stress-edge stress

relation

Max. capacity is reached when e = y


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Effective area = (a/ e). bt b = (a/ e). bTangent area = (a / b) .bt b (a / b) .b

L l B kli


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Local Buckling

The individual flat elements buckle undercompression

The column has post buckle strength

The ultimate state reached is known asCrippling

Simple estimate of crippling stress

crp = yb


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Load buckling of cylinders

Buckling stress of a cylinder

b

= 0.19 E t/R

When reinforced with longitudinal stiffness

b = 0.19E t/R + K E (t/b)2 b = stiffener spacing


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Buckling of sheet stringer panels


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Initial Buckling of sheet stringerpanels


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Flexural Mode

Pcr= (2 Eeff Ieff)/L2e = Pcr /(As + (a/ e). bt )

where, Ieff= Moment of inertia of stringer and effective area of plate

Effective area of plate Aeff= (a / b) .bt

e is estimated by successive iterations

b is the stringer spacing and t is the plate thickness


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Torsional Mode of Bucking

Figure shows the torsional mode of an open section strut

T = [GJ + (/)2Eeff + (/ ) 2 k] / Ib

Torsional mode of buckling of sheet stringer panelThe torsional mode and Flextural mode interact and result in Torsional-Flextural

(Flextor) mode


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Typical Buckling of Sheet Stringer Panel


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Buckling of Plates in Shear

cr= K E (t/b)2

kli i Sh d l


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Buckling in Shear curved plates

B kli f Pl t i B di


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Buckling of Plates in Bending+ Axial Stress


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Buckling of Plates under biaxial stress and shear

x

y

Critical Stress Predicted using ESDU 81047

T i Fi ld B


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Tension Field Beams

The webs of beam can carry shear load after buckling

The shear is carried as diagonal tension

This causes additional compression in the vertical stiffeners as well as in

edge members

The failure of web or permanent deformation of web

The edge members are also subjected to bending due to diagonal tension

and act as beam columns

Vertical stiffeners act as columns


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Statically Indeterminate Structures


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Statically Indeterminate Structures

Structures such as continuous beam, Portal Frames, Fuselage frames, etc.

are classified as statically indeterminate structures

The external reactions (continuous beams) or the internal loads and stresses(fuselage frames) cannot be determined by equations of static equilibrium

Hence deformation conditions are used to derive additional equations to

solve the problem

The deformation equation can be derived using various methods such as unitload method, relaxation method, energy method, etc. (Finite Element

Method)

Examples

Ai ft St t l A l i


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Aircraft Structural Analysis

The wing and Fuselage structures are essentially beams

The c/s is subjected to

Bending about two axes

Shear about the two axes

Torsion about the longitudinal axes

The bending and shear stresses induced are obtained from simple beam

theory

Wing Box Beams


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Wing Box Beams

(Skin Stringer Panels Design)

Transport Wing (Two - cell box)

The Fuselage and Wing Loading


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The Fuselage and Wing Loading


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Wing bending moment envelope for static conditions

Wing Design Torsion envelope for Static Conditions


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Body monocoque vertical shear envelope

Body monocoque vertical Bending Moment envelope


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Body monocoque Lateral shear envelope

Body monocoque Lateral Bending Moment envelope

Body monocoque Torsion envelope

i i f di


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Determination of Bending Stresses

Section properties Ix, Iy and Ixy are computed

Allowance of skin effective area in above calculations

Allowance for taper effects

Stresses calculated using beam formula (M/I y)

Determination of Shear flows


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Determination of Shear flows

Shear flow is estimated from q = (V / I) y dA for shear

Shear flow due to torsion from q = T / (2A)

For closed sections and multiple cell sections, twist conditions are used to

determine the unknown starting shear flows

Frame Analysis


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y

Fuselage Frames are subjected to concentrated/distributed loads

The fuselage skin provides reaction to these loads

The internal loads in the frame are axial load, shear load and bendingmoment

The internal loads are computed using methods of statically indeterminate

structures

Use of charts


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Analysis for Ribs


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Analysis for Ribs

Ribs are essentially analyzed as beams

Ribs are supported by spars

The shear and bending moments are obtained as for a beam


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Thank You . . .

Introduction to Aircraft Structure - 4

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