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 . . .