124026509 aircraft structures

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CONTENTS

A Definitions 8

A1 Important Properties of Engineering Materials 8

A2 Simple Stresses and Strains 9

A3 General Definitions 10

B Conversion Tables 26

B1 Introduction 26

B2 List of Units and Abbreviations 26

B3 Conversion factors Grouped by Category 28

B4 Miscellaneous Data 33

C Section Shear Center 34

C1 Location of Shear Center Open Section 35

C2 Shear Center Curved Web 36

C3 Shear Center Beam with Constant Shear Flow Between Booms 38

C4 Shear Center Single Cell Closed Section 39

C5 Shear Center Single Cell Closed Section with Booms 40

C6 Shear Center for Standard Section (Tables) 41

D Warping Constant 46

D1 Warping Constant Typical Sections (Tables) 48

E Mechanical Properties 53

E1 DATA Basis 53

E2 Stress Strain Data 54

E3 Definitions of Terms 54

E4 Temperature Effects 57

E5 Fatigue Properties 57

E6 Generalised Stress - Strain Equation 59

E6.1 Problems 61

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1 Prismatic Bar In Tension 63

1.1 Limit Stress Analysis 63

1.2 Ultimate Stress Analysis 63

1.3 Problem 64

2. Prismatic Bar In Compression 65

2.1 Compression Stress Analysis 65

2.1.2 Problem 65

2.2 Euler Buckling 67

2.3 For Other Buckling Cases 67

2.3.1 Buckling For Eccentric Loading 67

2.3.2 Beam-Column 68

2.3.2.1 Notations And Conventions 68

2.3.2.2 Terms Used 69

2.3.2.3 Calculation Of Bending Moment 69

2.3.2.4 Allowable Stress 74

2.3.3 Local Buckling (Thin Walled Structures) 75

2.3.3.1 Flanges 75

2.3.3.2 Thin Webs 75

2.3.3.3 Plates And Shells 76

2.3.3.4 Analysis Of Thin Walled Structures 80

2.4 Problems 80

2.4.1 Problem For Euler Buckling 80

2.4.2 Problem For Beam Column 83

2.4.3 Problem For Local Buckling 86

2.4.3.1 Problem For Combined Bending And Shear 87

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3 Lug Analysis 88

3.1 Failure In Tension 88

3.2 Failure By Shear Tear Out 89

3.3 Failure By Bearing 90

3.4 Lug Strength Analysis Under Transverse Loading 90

3.5 Lug Strength Analysis Under Oblique Loading 91

3.6 Lug Analysis (Avro Method) 92

3.7 Problems 93

4. Member In Bending 97

4.1 Introduction To Bending 98

4.2 Stress Analysis Of Beam 100

4.3 Bending Stresses 101

4.3.1 Limit Stress Analysis 101

4.3.2 Ultimate Stress Analysis 101

4.3.3 Rectangular Moment Of Inertia and Product of Inertia 102

4.3.4 Parallel Axis Theorem 103

4.4 Elastic Bending Of Homogeneous Beams 114

4.5 Elastic Bending Of Non Homogeneous Beams 128

4.6 Inelastic Bending of Homogeneous Beams 133

4.7 Elastic Bending Of Curved Beams 146

4.8 Miscellaneous Problems 148

4.8.1 Bending Stress In Symmetrical Section 151

4.8.2 Bending Stress In Unsymmetrical Section 153

4.8.3 Beams Of Uniform Strength 159

4.8.4 Beams Of Composite Section 160

4.8.5 Bending Of Unsymmetrical Section About Principal Axis 163

4.8.6 Bending Of Unsymmetrical Section About Arbitrary Axis 164

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5 Member In Torsion 172

5.1 Torsion of Circular Bars 173

5.1.1 Elastic and Homogeneous 176

5.1.2 Elastic and Non-homogeneous 178

5.1.3 Inelastic and Homogeneous 180

5.1.4 Inelastic and Non-homogeneous 182

5.1.5 Residual Stress Distribution 184

5.1.6 Power Transmission 185

5.2 Torsion of Non-Circular Bars 189

5.3 Elastic Membrane Analogy 190

5.4 Torsion of Thin-Wall Open Sections 194

5.5 Torsion of Solid Non-Circular Shapes 198

5.6 Torsion of Thin-Wall Closed Sections 202

5.7 Torsion-Shear Flow Relations in Multiple-Cell Closed Sections 208

5.8 Shear Stress Distribution and Angle of Twist for Two-Cell Thin-Wall Closed Section 209

5.9 Shear Stress Distribution and for Multiple-Cell Thin-Wall Closed Section 214

5.10 Torsion of Stiffened Thin-Wall Closed Sections 215

5.11 Effect of End Restraint 217

5.12 Miscellaneous problems 222

5.12.1 Problem For Torsion On Circular Section 222

5.12.2 Problem For Torsion In Non Circular Section 225

5.12.3 Problem For Torsion In Open Section Composed Of Thin Plates 226

5.12.4 Problem For Torsion On Thin Walled Closed Section 228

5.12.5 Problem For Torsion In Thin Walled Unsymmetrical Section 233

6 Transverse Shear Loading Of Beams With Solid Or Open Sections 236

6.1 Introduction 237

6.2 Shear Center 238

6.3 Flexural Shear Stress And Shear Flow 239

6.4 Shear Flow Analysis For Symmetric Beams 261

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6.5 Shear Flow Analysis For Unsymmetric Beams 286

6.6 Analysis Of Beams With Constant Shear-Flow Webs (I.E., Skin-Stringer Type Sections) 298

7. Transverse Shear loading Of Beams With Closed C.S 310

7.1 Single-Cell Unstiffened Box Beam: Symmetrical About One Axis 311

7.2 Statically Determinate Box Beams With Constant-Shear-Flow Webs 325

7.3 Single-Cell Multiple-Flange Box Beams. Symmetric And Unsymmetric Cross Sections 331

7.4 Multiple-Cell Multiple-Flange Box Beams. Symmetric And Unsymmetric Cross Sections 338

7.5 The Determination Of The Flexural Shear Flow Distribution By

Considering The Changes In Flange Loads (The Delta P Method) 346

7.6 Shear Flow In Tapered Sheet Panels 357

8. Combined Transverse Shear, Bending, And Torsion Loading 360

9 Internal Pressure 360

9.1 Membrane Equations Of Equilibrium 360

9.2 Special Problems In Pressurized Cabin Stress Analysis 368

10 Analysis Of Joints 373

10.1 Analysis Of Riveted Joints 373

10.1.1 Shear Failure Of Rivets 375

10.1.2 Tensile Failure Of Plate Along Rivet Line 375

10.1.2.1 Single Row And Rivets Having Uniform Diameter And Equal Pitch375

10.1.2.2 More Than One Row And Rivets With Equal Spacing 376

10.1.2.3 Rivet Line With Varying Number Of Rivets 377

10.1.2.4 Rivets Of Varying Diameter, Sheet Thickness And Pitch Distance 378

10.1.3 Failure By Double Shear Of Plate 378

10.1.4 Rivets Subjected To Tensile Loads 379

10.1.5 Rivets Subjected To Eccentric Loads 380

10.1.6 Inter Rivet Buckling 381

10.1.7 Calculation Of Fir Using ESDU Data Sheets 382

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10.1.8 Problems 382

10.2 Analysis Of Bolted Joints 397

10.2.1 Failure By Bolt Shear 397

10.2.2 Failure By Bolt Bending 398

10.3 Analysis Of Welded Joints 399

10.3.1 Shear Failure In Weld 399

10.3.2 Tensile Failure Of Weld 400

11 Combined Stress Theories For Yield And Ultimate Failure 401

11.1 Determination Of Yield Strength Of Structural Member 401

11.1.1 Maximum Shearing Stress Theory 401

11.1.2 Maximum Energy Of Distortion Theory 402

11.1.3 Maximum Strain Energy Theory 402

11.1.4 Octahedral Shear Stress Theory 403

11.2 Determination Of Ultimate Strength Of A Structural Member 404

11.2.1 Maximum Principal Stress Theory 404

11.3 Problem 404

12 Cutouts In Plane Panels 409

12.1 Framing Members Around The Cutouts 409

12.2 Framing Cutouts With Doublers And Bends 412

12.3 Ring Or Donut Doublers For Round Holes 414

12.4 Webs With Lightning Holes Having Flanges 416

12.5 Structures With Non Circular And Non Rectangular Cutouts 417

12.6 Cutouts In Fuselage 418

12.6.1 Load Distribution Due To Fuselage Skin Shear 419

12.6.2 Load Distribution Due To Fuselage Bending 421

12.6.3 Load Distribution Due To Fuselage Cabin Pressurization 422

12.6.3.1 Panels Above And Below The Cutout 423

12.6.3.2 Panels At Sides Of The Cutout 424

12.6.3.3 Corner Panels Of The Cutout 424

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12.7 Problem 425

13 Tension Clips 430

13.1 Single Angle 433

13.2 Double Angle 433

13.3 Problem 435

14 Pre - Tensioned Bolts 438

14.1 Optimum Preload 439

14.2 Total Bolt Load 440

14.2.1 Mating Surfaces Not In Initial Contact 440

14.2.2 Mating Surfaces In Initial Contact 440

14.3 Preload For A Given Wrench Torque 441

14.4 Bolt Strength Requirements 441

14.5 Ultimate Stress Analysis 442

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A. DEFINITIONS:

A1. Import Properties of Engineering Material:

Its necessary to know from engineering point of view the following import

properties of solids.

a. Elasticity: It is property by virtue of which the body regains its original shape and

size after the removal of the external force acting on the body.

b. Isotropic Material: It is the material, which is equally elastic in all directions.

c. Ductility: It is the property of the material by virtue of which it can be drawn out

into wires of smaller dimension e.g., copper, Aluminium.

d. Plasticity: It is the property of the material wherein the strain does not disappear

even after the load or stress is not acting on the material. The material becomes

plastic in nature and behaves like a viscous liquid under the influence of large

forces.

e. Malleability: It is the property possessed by the material enabling the material to

extend uniformly in a direction without rupture. As such the material can be hot

rolled, forged or drop stamped. A malleable material is highly plastic.

f. Brittleness: A material is said to be brittle when it cannot be drawn out into thon

wires. This due to lack of ductility and the material breaks into pieces under

loading.

g. Toughness: It is property of the material due to which it is capable of absorbing

energy without fracture.

h. Hardness: It is the property of the material by virtue of which the is capable of

resisting abrasion or indentation.

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A2. Simple Stresses and Strains.

Allowable Stress (Working Stress):

If a member is so designed that the maximum stress as calculated for the expected

condition of service is less than some certain value, the member will have a proper margin of

security against damage or failure. This certain value is the allowable stress of the kind and for

the material and condition of service in question. The allowable stress is less than the damaging

stress because of uncertainty as to the conditions of service, non uniformity of material, and

inaccuracy of stress analysis. The margin between the allowable stress and the damaging stress

may be reduced in proportion to the certainty with which the condition of service are known, the

intrinsic reliability of material, the accuracy with which the stress produced by the loading can

be calculated and the degree to which failure is unaffected by danger or loss. (Compare

Damaging stress, Factor of Safety; Factor of utilization; Margin of safety)

Apparent Elastic Limit:

The stress at which the rate of change of strain with respect to stress is 50 percent

greater than at zero stress. It is more definitely determine from the stress strain diagram than is

the proportional limit, and is useful for comparing material s of the same general class(Compare

Elastic limit, Proportional limit, Yield point; Yield strength).

Apparent Stress:

The stress corresponding to a given unit strain on the assumption of uniaxial

elastic stress is called apparent stress. It is calculated by multiplying the unit strain by the

modulus of elasticity and may differ from the true stress because the effect of transverse stresses

is not taken into account.

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A3. General Definitions:

Bending Moment:

The bending moment at any section of the beam is the moment of all forces that

act on the beam to the left of that section, taken about the horizontal axis of the section. The

bending moment is positive when clockwise and negative when counter clock wise; a positive

bending moment therefore bends it so that it is concave downward. The moment equation is an

expression for the bending moment at any section in terms of x, the distance to that section

measured from a chosen origin, usually taken at the left end of the beam.

Boundary Conditions:

As used in strength of materials, the term usually refers to the condition of stress,

displacement, or slope at the ends or edges of a member, where these conditions are apparent

from the circumstances of the problem. Thus for a beam with fixed ends the zero slope at each

end is a boundary condition are apparent from the circumstances of the problem. Thus for beam

with fixed ends, the zero slope at each is a boundary condition: for a pierced circular plate with

freely supported edges, the zero radial stress at each edge is a boundary condition.

Brittle Fracture:

The tensile failure with negligible plastic deformation of an ordinarily ductile

metal is called brittle fracture.

Bulk Modulus of Elasticity:

The ratio of a tensile or compressive stress, triaxial and equal in all direction(e.g.,

hydrostatic pressure), to the relative change it produces in volume.

Central Axis (Centroidal Axis):

A central axis of an area is one that passes through the centroid; it is understood

to lie in the plane of the area unless contrary is stated. When taken normal to the plane of the

area, it is called the central polar axis.

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Centroid of an Area:

That point in the plane of the area about any axis through which the moment of

the area is zero; it coincides with the center of gravity of the area materialized as an infinitely

thin homogenous and uniform plate.

Corrosion Fatigue:

Fatigue aggravated by corrosion , as in parts repeatedly stressed while exposed to

salt water.

Creep:

Continuous increase in deformation under constant or decreasing stress. The term

is ordinarily used with reference to the behaviour of metals under tension at elevated

temperatures. The similar yielding of a material under compressive stress is usually called plastic

flow, or flow. Creep at atmospheric temperature due to sustained elastic stress is sometimes

called drift, or elastic drift. Another manifestation creep, the diminution in stress when

deformation is maintained constant, is called relaxation.

Damaging Stress:

The least unit stress of a given kind and for a given material and condition of

service that will render a member unfit for service before the end of its normal life. It may do

this by producing excessive set, by causing creep to occur at an excessive rate, or by causing

fatigue cracking, excessive strain hardening, or rupture..

Damping Capacity:

The amount of work dissipated into heat per unit of strain energy present at

maximum strain for a complete cycle.

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Deformation (Strain):

Change in the form or dimensions of a body produced by stress. Elongation is

often used for tensile strain, compression or shortening for compressive strain, and detrusion for

shear strain. Elastic deformation is such deformation as disappears on removal of stress;

permanent deformation is such deformation as disappears on remains on removal of stress.

Eccentricity:

A load or component of a load normal to a given cross section of a member is

eccentric with respect to that section if it does not act through the centroid The perpendicular

distance from the line of action of the load to either principal central axis is the eccentricity with

respect to that axis.

Elastic:

Capable of sustaining stress without permanent deformation; the term is also used

to denote conformity to the law of stress strain proportionality. An elastic stress or strain within

the elastic limit.

Elastic Axis:

The elastic axis of a beam is the line, lengthwise of the beam, along which

transverse loads must be applied in order to produce bending only, with no torsion of the beam at

any section. Strictly speaking, no such line exists except for a few conditions of loading. Usually

the elastic axis is assumed to be the line that passes through the elastic center of every section.

The term is most often used with reference to an airplane wing of either the shell or multiple-

spar type (Compare Torsional center; Flexural center; Elastic center).

Elastic Center:

The elastic center of given section of a beam is that point in the plane of the

section lying midway between the flexural center and center of twist of that section. The three

points may be identical and are usually assumed to be so. (Compare Flexural center; Torsional

center; Elastic axis).

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Elastic Curve:

The curve assumed by the axis of a normally straight beam or column when bent

by loads that do not stress it beyond the proportional limit.

Elastic Limit:

The least stress that will cause permanent set (Compare Proportional limit;

Apparent elastic limit; Yield point; Yield strength).

Elastic Ratio:

The ratio of the elastic limit to the ultimate strength

Ellipsoid of Strain:

An ellipsoid that represent the state of strain at any given point in a body; it has

the form assumed under stress by a sphere centered at that point.

Ellipsoid of Stress:.

An ellipsoid that represents the state of stress at a given point in a body; its semi

axes are vectors representing the principal stresses at that point, and any radius vector represents

the resultant stress on a particular plane through the point. For a condition of plane stress (one

principal stress zero) the ellipsoid becomes the ellipse becomes the ellipse of stress.

Endurance Limit(Fatigue strength):

The maximum stress that can be reversed an indefinitely large number of times

without producing fracture of a material.

Endurance Ratio:

Ratio of the endurance limit to the ultimate static tensile strength is called

endurance ratio.

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Endurance Strength:

The highest stress that a material can withstand with repeated application or

reversal without rupture for a given number of cycles is the endurance strength of that material

for that number of cycles. Unless otherwise specified, reversed stressing is usually implied

(Compare Endurance Limit)

Energy of Rupture:

The work done per unit volume in producing fracture is the energy of rupture. It

is not practical to establish a definite energy of rupture value for a given material because the

result obtained depends upon the form and proportion of the test specimen and manner of

loading. As determined by similar tests specimen, the energy of rupture affords a criterion for

comparing the toughness of different materials.

Equivalent Bending Moment:

A bending moment that, acting alone, would produce in a circular shaft a normal

(tensile or compressive) stress of the same magnitude as a maximum normal stress produced by

a given bending moment and a given twisting moment acting simultaneously.

Equivalent Twisting Moment:

A twisting moment that, acting alone would produce in a circular shaft a shear

stress of the same magnitude as the shear produced by a given twisting and a given bending

moment acting simultaneously.

Factory of Safety:

The ratio of the load that would cause failure of a member or structure to the load

that is imposed upon it in service. The term usually has this meaning; it may also be used to

represent the ratio of breaking to service value of speed, deflection, temperature variation, or

other stress producing factor against possible increase in which the factor of safety is provide as

a safeguard (Compare Allowable stress; Margin of safety).

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Factor of Strain:

In the presence of stress raisers, localized peak strains are developed. The factor

of strain concentration is the ratio of the localized maximum strain at a given cross section to the

nominal average strain on that cross section. The nominal average strain behaviour of the

material. In a situation where all stresses and strains are elastic, the factors of stress

concentration and strain concentration are equal (Compare Factor of stress concentration).

Factor of Stress Concentration:

Irregularities of form such as holes, screw threads notches, and sharp shoulders,

when present in a beam, shaft, or other member subject to loading may produce high localized

stresses. This phenomenon is called stress concentration, and the form irregularities that causes it

are called stress raisers. The ratio of the true maximum stress to the stress calculated by the

ordinary formulas of mechanics (flexural formula, torsion formula etc.), using the net section but

ignoring the changed distribution of stresses is the factor of stress concentration for the

particulars type of stress raiser.

Factor of Stress Concentration in Fatigue:

At a specified number of loading cycles, the fatigue strength of a given geometry

depends upon the stress concentration factor and upon material properties. The factor of stress

concentration in fatigue is the ratio of the fatigue strength without a stress concentration. It may

vary with the specified number of cycles as well as with material

Factor of Utilization:

The ratio of the allowable stress to the ultimate strength. For cases in which stress

is proportional to load, the factor of utilization is the reciprocal to the factor of safety.

Fatigue:

Tendency of material to fracture under many repetitions of a stress considerably

less than the ultimate static strength.

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Fibre Stress:

A term used for convenience to denote the longitudinal tensile or compressive

stress in a beam or other member subject to bending. It is sometimes used to denote this stress at

the point or points most remote from the neutral axis, but the term stress in extreme fibre is

preferable for this purpose. Also, for convenience, the longitudinal elements of filaments of

which a beam may be imagined as composed are called fibres.

Fixed (Clamped, built-in, encastre):

A conditioned of support at the ends of a beam or column or at the edges of a

plate or shell that prevents rotation and transverse displacements of the edge of the neutral

surface but permits longitudinal displacements.

Flexural Center (Shear Center):

With reference to a beam, the flexural center of any section is that point in the

plane of the section through which a transverse load, applied at that section, must act if bending

deflection only is to be produced with no twist of the section (Compare Torsional center; Elastic

center; Elastic axis).

Form Factor:

The terms pertains to a beam section of a given shape and means the ratio of the

modulus of rupture of a beam having a section adopted as standard. The standard section is

usually taken as rectangular or square; for wood it is a 2 by 2 in square, with edges horizontal

and vertical. The term is also used to mean, for a given maximum fiber stress within the elastic

limit, the ratio of the actual resisting moment of a wide-flanged beam to the resisting moment the

beam would develop if the fiber stress were uniformly distributed across the entire width of the

flanges. So used, the term expresses the strength reducing effect of shear lag.

Fretting Fatigue:

Fatigue aggravated by surface rubbing, as in shaft with press fitted collars.

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Guided:

A condition of support at the ends of a beam or column or at the edge of a plate or

shell that prevents rotation of the edge of the neutral surface but permits longitudinal and

transverse displacement (Compare Fixed; Guided; Supported).

Held:

A condition of support at the ends of a beam or columns or at the edge of a plate

or shell that prevents longitudinal and transverse displacement of the edge of the neutral surface

but permits rotation in the plane of bending (Compared Fixed; Guided Supported).

Influence Line:

Usually pertaining to a particular section of a beam, influence line is a curve

drawn; so that its ordinate at any point represent the value of the reaction; vertical shear, bending

moment, or deflection produced at the particular section by a unit load applied at the point where

the ordinate is measured. An influence line may be used to show the effect of load position on

any quantity dependent thereon, such as the stress in a given truss member, the deflection of a

truss, or the twisting moment in a shaft.

Isoclinic:

A line (in a stressed body) at all points on which the corresponding principal

stresses have the same directions.

Isotropic:

Having the same properties in all properties in all direction. In discussions

pertaining to strengths of materials, isotropic usually means having the same strength and elastic

properties (modulus of elasticity, modulus of rigidity, and poissons ratio) in all directions.

Korn (Kornal):

The korn is that area in the plane of the section through which the line of action of

a force must pass if that force is to produce, at all points in the given section, the same kind of

normal stress, i.e., tension throughout or compression throughout

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Margin of safety:

As used in aeronautical design, margin of safety is the percentage by which the

ultimate strength of a member exceeds the design load. The design load is the applied load, or

maximum probable load, multiplied by a specified factor of safety. [The use of the term margin

of safety and design load in this sense is practically restricted to aeronautical engineering]

Mechanical Hysteresis:

The dissipation of energy as heat during a stress cycle, which is revealed

graphically by failure of the descending and ascending branches of the stress-strain diagram to

coincide.

Modulus of Elasticity (Youngs Modulus):

The rate of change of unit tensile or compressive stress with respect to unit tensile

or compressive strain for the condition of uniaxial stress within the proportional limit. For most,

but not all, materials, the modulus of elasticity is the same for tension and compression. For non-

isotropic material such as wood, it is necessary distinguish between the moduli of elasticity in

different directions.

Modulus of Resilience:

The strain energy per unit volume absorbed up to the elastic limit under the

condition of uniform uniaxial stress.

Modulus of Rigidity (Modulus of Elasticity in Shear):

The rates of change of unit shear stress with respect to unit shear strain for the

condition of pure shear within the proportional limit. For non-isotropic materials such as wood,

it is necessary to distinguish between the moduli of rigidity in different directions.

Modulus of Rupture in Bending (Computed ultimate Twisting Strength):

The fictitious tensile or compressive stress in the extreme fiber of a beam

computed by the flexure equationI

Mc= , where M is the bending moment that causes rupture.

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Modulus of Rupture in Torsion (Computed Ultimate twisting moment):

The fictitious shear stress at the surface of circular shaft computed by the tension

formula J

Tr= , where T is the Twisting moment that causes rupture.

Moment of Area (First Moment of an Area, Statical Moment of an Area):

With respect to an axis, the sum of the products obtained by multiplying each

element of the area dA by its distance from the axis y; it is therefore the quantity dAy . An axis in the plane of the area is implied.

Moment of Inertia of an area (Second Moment of an Area):

The moment of inertia of an area with respect to axis is the sum of the products

obtained by multiplying each element of the area dA by the square of its distance from the axis

y; it is therefore the quantity 2dAy . An axis in the plane of the area is implied, if the axis is normal to that plane, the term polar moment of inertia.

Neutral Axis:

The line of zero fiber stress in any given section of a member subject to bending;

it contains the neutral axis of every section.

Notched Sensitivity Ratio:

Used to compare stress concentration factor kt and fatigue-strength reduction

factor kf, the notch sensitivity ratio is commonly defined as the ratio ( ) ( )1/1 tf kk .It varies from 0, for some ductile materials, to 1, for some hard brittle materials.

Neutral Surface:

The longitudinal surface of zero fibre stress in a member subject to bending; it

contains the neutral axis of every section.

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Plasticity:

The property of sustaining appreciable (visible to the eye) permanent deformation

without rupture. The term is also used to denote the property of yielding or flowing under steady

load.

Poissons Ratio:

The ratio of lateral unit strain to longitudinal unit strain under the condition of

uniform and uniaxial longitudinal unit within the proportional limit.

Principal Axes:

The principal axes of an area for a given point in its plane are the two mutually

perpendicular axes, passing through the point and lying in the plane of the area, for one of which

the moment of inertia is greater and for the other less than for any other coplanar axis passing

through that point. If the point in question is the centroid of the area, these axes are called

principal central axes.

Principal Planes; Principal Stresses:

Through any point in stressed body there pass three mutually perpendicular

planes, the stress on each of which is purely normal, tension, or compression; these are the

principal planes for that point. The stresses on these planes are the principal stresses; one of them

is the maximum stress at that point, and one of them is the minimum stress at that point. When

one of the principal stresses is zero, the condition is one of uniaxial stress.

Product of Inertia of an Area:

With respect to a pair of rectangular axes in its plane, the sum of the products

obtained b multiplying each element of the area dA by its coordinates with respect to those axes

x and y; it is therefore the quantity dAxy .

Proof Stress:

Pertaining to acceptance tests of metals, a specified tensile stress that must be

sustained without deformation in excess of a specified amount.

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Proportional Limit:

The greatest stress that a material can sustain without deviating from the law of

stress-strain proportionality. (Compare Elastic limit; apparent elastic limit; Yield point; Yield

strength)

Radius of Gyration:

The radius of gyration of area with respect to a given axis is the square root of the

quantity obtained by dividing the moment of inertia of the area with respect to that axis by the

area.

Reduction of Area:

The difference between the cross sectional area of a tension specimen at the

section of rupture before loading and after rupture.

Rupture Factor:

Used in reference to brittle materials, i.e. materials in which failure occurs

through tensile rupture rather than through excessive deformation. For a member of given form,

size, and material, loaded and supported in a given manner, the rupture factor is the ratio of the

fictitious maximum tensile stress at failure, as calculated by the appropriate formula for elastic

stress, to the ultimate tensile strength of the material, as determined by a conventional tension

test.

Section Modulus (Section Factor):

Pertaining to the cross section of a beam, the section modulus with respect to

either principal central axis is the moment of inertia with respect to that axis divided by the

distance from that axis to the most remote point of the section. The section modulus largely

determines the ability of a beam to carry load with no plastic deformation

Anywhere in the cross section (Compare plastic section modulus)

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Set (Permanent set, Permanent deformation, Plastic strain, Plastic deformation):

Strain remaining after removal of stress.

Shakedown Load (Stabilizing Load):

The maximum load that can be applied to a beam or rigid frame and on removal

leave such residual moments that subsequent application of the same or a smaller load will cause

only elastic stresses.

Shear Lag:

Because of shear strain, the longitudinal tensile or compressive bending stress in

wide beam flanges diminishes with the distance from the web or webs, and this stress diminution

is called shear lag.

Singularity Functions:

A class of function that, when used with some caution, permit expressing in one

equation what would normally be expressed in several separate equations, with boundary

conditions being matched at the ends of the intervals over which the several separate expressions

are valid.

Ultimate Strength:

The ultimate strength of a material in tension, compression, or shear, respectively,

is the maximum tensile, compressive, or shear stress that the material can sustain calculated on

the basis of the ultimate load and the original or unstrained dimensions. It is implied that the

condition of stress represents uniaxial tension, uniaxial compression, or pure shear, as the case

may be.

Unit Stress:

The amount of stress per unit of area. The unit stress (tensile, compressive, or

shear) at any point on a plane is the limit, as A approaches 0, AP / where P is the total tension, compression, or shear on an area A that lies in the plane and includes the point.

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23

Vertical Stress:

Refers to a beam, assumed for convenience to be horizontal and loaded and

supported by forces that all lies in a vertical plane. The vertical shear at any section of the beam

is the vertical component of all the forces that act on the beam to the left of the section. The

shear equation is an expression for the vertical shear at any section in terms of x, the distance to

that section measured from a chosen origin, usually taken at the left end of the beam.

Yield Point:

The lowest stress at which strain increase in stress. For some purposes it is

important to distinguish between the upper yield point, which is the stress at which is the stress-

strain diagram first becomes horizontal, and lower yield point, which is the somewhat lower and

almost constant stress under which the metal continues to deform. Only a few materials exhibit a

true yield point; for other materials the term is sometimes used synonymously with yield

strength (Compare yield strength; Elastic limit; Apparent elastic; proportional limit)

Yield Strength:

The stress at which a material exhibits a specified permanent deformation or set.

The set is usually determined by measuring the departure of the actual stress-strain diagram

from an extension of the initial straight portion. The specified value is often taken as a unit strain

of 0.002.

Slenderness Ratio:

The ratio of length of a uniform column to the least radius of gyration of the cross

section.

Slip Lines:

Lines that appear on the polished surface of crystal or crystalline body that has

been stressed beyond the elastic limit. They represent the intersection of the surface by planes on

which shear stress has produced plastic slip or gliding.

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24

Strain:

Any forced change in the dimensions of a body. A stretch is tensile strain; a

shortening is a compressive strain; and angular distortion is a shear strain. The word strain is

commonly used to connote unit strain.

Strain energy:

Other wise called as elastic energy, potential energy of deformation. Mechanical

energy stored up in the stressed material. Stress with in the elastic limit is implied; therefore, the

strain energy is equal to the work done by the external forces in producing the stress and is

recoverable.

Strain rosette:

At any point on the surface of a stressed body, strains measured on each of the

intersecting gauge line make possible the calculation of the principal stresses. Such gauge lines

and the corresponding strains are called strain rosettes.

Stress:

Internal force exerted by either of two adjustant parts of a body upon the other

across an imagined plane of separation. When the forces are parallel to the plane, stress is called

shear stress; when the forces are normal to the plane, the stress is called normal stress; when the

normal stress is directed towards the part on which it acts, it is called compressive stress; and

when it is directed away from the part on which it acts, it is called tensile stress. Shear,

compressive, and tensile stresses, respectively, resists the tendency of the parts to mutually slide,

approach, or separate under the action of applied forces.

Stress solid:

The solid figure formed by the surfaces bounding vectors drawn at all points of

the cross section of a member and representing the unit normal stress at each such point. The

solid stress gives a picture of the stress distribution on a section.

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25

Stress trajectory (isostatic):

A line in a stressed body tangent to the direction of one of the principal stresses at

every point through which it passes id called stress trajectory.

Torsional center:

Otherwise called as center of twist or center of torsion or center of shear. If a

twisting couple is applied at a given section of a straight member, that section rotates about some

point in its plane. This point which does not move when the member twists, is the torsional

center of that section.

True stress:

For an axially loaded bar, the load divided by corresponding actual cross section

area. It differs from the stress as ordinarily defined because of the change in area due to loading.

Twisting moment (torque):

At any section of the member, the moment of all forces that act on the member to

the left of that section, taken about a polar axis through the flexure center of that section. For the

sections that are symmetrical about each principal central axis, the flexural center coincides with

the centroid.

Ultimate elongation:

The percentage of permanent deformation remaining after tensile rupture

measured over an arbitrary length including the section of rupture.

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26

B Conversion Tables:

B1. Introduction:

The conversion factors in this section include the System International (S.I) Units

as these are now widely accepted as the standard units of measurement.

There are six basics units in the S.I. system. These are follows

The main advantage of the S.I. System is the ease with which mechanical units expressed in

mass length and time can be related to electrical units.

B2. List of Units & Abbreviations:

27

27

Decimal Prefixes:.

The use of double prefixes should be avoided when single prefixes are available.

Basic Units in Stress Analysis

28

28

B3. Conversion Factors Grouped by Category:

The following pages give a list of various conversion factors used grouped by

category

Length:

Area:

Volume:

Force:

29

29

Mass:

Density:

Velocity:

30

30

Force Length Combination:

Moment of Inertia:

Mass Moment of Area:

31

31

Energy:

Acceleration:

Electrical:

Angular:

32

32

Power:

Stress or Pressure:

Temperature:

33

33

Time:

Viscosity:

B4. Miscellaneous Data:

34

34

C. Section Shear Centre:

The shear centre can be defined as that point in the plane of a cross-section

through which an applied transverse load must pass for bending to occur unaccompanied by

twisting. To obtain the shear centre a unit transverse load is applied to the section at that point.

The shear stress distribution, as thus shear flow distribution, thus shear flow distribution is

obtained for each element of the section. Equating the torque given by the shear distribution to

the torque given by the transverse load will produce position of shear centre.

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35

C1. Location of Shear Centre-Open sections:

Section Constants

2xyyyxx

xxx III

IZ =

2xyyyxx

yyy III

IZ =

2xyyyxx

xyxy III

IZ =

Shear flow in an element

( )[ ] ++= oyyixxixyxiyitf qdsZVyZVxZVyVxtq Where 0q is value of fq at s = 0

xi = horizontal distance from the element centroid to the section centroid

yi = vertical distance from the element centroid to the section centroid

Shear force in an element

dsqF if = Torque/Moment given by element

iii RFT = Centroid chosen as datum point, but it could be any convenient point.

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36

Horizontal Location of shear Centre:

To Locate the horizontal position of the shear centre, we apply a vertical unit

force through the unknown position of the shear centre x0,y0 ie Vy = 1, and Vx = 0.

( ) 0qdstZyZxq tyixyii += Shear forces and total torque iT are calculated as defined previously.

Vertical Location Of Shear Centre:

To locate the vertical position of the shear centre, we apply a horizontal unit

through the unknown position of the shear centre x0,y0 ie Vy = 0, and Vx = 1

( ) 00 qdstZxZyq txixyii +=

C2. Shear Centre-curved Web:

Horizontal Location:

=

0

1 tdsI

yqxx

cosRy = Rdds = = RdRItq xx cos sin2R

Itqxx

=

Moment of force of element about 0

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37

Rqds= RqRd =

dI

tRx

sin4

Total Moment

= 04

sin dI

tRxx

[ ] 04 cosxxItR

Vertical component of force of element

sinqds Total Vertical Force

= dI tRxx 23

sin

xxItR

2

3 =

Taking Moment about 0

xxxx I

tReItR 43 2

2=

Horizontal Location,

Re 4=

Alternatively for a solid section:

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38

with q = web shear flow

A = Enclosed area

h = web depth

Vertical Equilibrium

Vyqh = Moment Equilibrium

eVyAq =2 . hAe /2=

C3. Shear centre-Beams with constant shear flow between boom members:

xyyyxx III ,, evaluated using boom areas only(skin shear carrying only) and xyyx ZZZ ,, using

above values of xyyyxx III ,,

Shear flow in each element

( )[ ] AZyVZxVZyVxVq yyxxxyxyi +=

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39

Horizontal location:

Firstly set Vy = 1 and Vx = 0

Giving ( ) ( )[ ] ( )1+= iqAZyZxq yxyi Where x and y are locations to element area A

Vertical location:

Firstly set Vy = 0 and Vx = 1

Giving ( ) ( )[ ] ( )1+= iqAZxZyq xxyi Where x and y are locations to element area A

C4. Shear Centre Single Cell Closed Section:

For Horizontal Location:

Cut element at 0 and assume 0=oq Evaluate shear flow distribution around section.

Take moment about 0to give the out of balance moment Mo.

Balance Mo with balancing shear flow qb.

Where A

Mqb 2

0= With resultant shear flow distribution; take moments about convenient point to give moment MR.

y

R

vMe =

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40

For Vertical Location:

Cut element at a and assume 0=aq Evaluate shear flow distribution around section.

Take moment about 0to give the out of balance moment Mo.

Balance Mo with balancing shear flow qb.

Where A

Mqb 2

0= With resultant shear flow distribution; take moments about convenient point to give moment MR.

y

R

vMe =

C5. Shear centre single cell closed section with booms:

Initially the member is dealt with as a beam i.e. a cut is made at appropriate skin

element thus reducing it from a single cell to an open beam section. Having established the open

beam element shear floes, moments are taken about a convenient point and out of balance

moment Mo established.

Mo is balanced by shear flow A

Mq ob 2

= With resultant shear flow distribution; take moments about a convenient point to give MR.

Shear centre horizontal location and vertical location:

y

R

VMe = ,

x

R

VMe =

MR calculated independently for unit Vy and Vx.

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41

C6. Shear Centre for Standard sections:

42

42

43

43

44

44

45

45

46

46

D. Warping Constant:

When a torque, is applied to a non-circular beam, the beam will twist with each

section of the beam rotating about torsional centre. The sections of the beam will not remain

plane and will wrap out of plane. Depending on how the beam is constrained the effects of the

applied torque, T will vary.

Consider a torque, T, applied to the section to the right.

The diagrams below illustrate the effects of the torque with different constraints.

The warping constant is calculated by adding 1 and .2 The constant 1 relates to the strain in the middle plane of the walls arising from variations of warping of the cross-section along

length due to twist. The constant 2 relates to the strain across the thickness, t, of the walls. Therefore the warping constant is defined by

( )21 += Consider the following section

We can now show equations for the warping constants based on the generalised section shown

above.

The primary warping constant 1 is obtained from the formula

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47

( ) 20 0

21

1 = s AWdAAtdsW Where

dsrWs

t= 0 The secondary warping constant 2 is obtained from the formula

( )= A n dAtsr0 22 121 in6 where A = Total cross-section area

s = distance from any free end along middle of wall.

rt = the perpendicular from the shear centre to the tangent of the point defined by s.

rn = distance from the point of rotation to the normal of the tangent line of the

middle of the wall.

A = Total cross-section area

rt = distance from the point of rotation to the tangent line of the centre line of

the section wall at s

s = distance from any free end along the middle of the wall.

Constraint Warping about a Point

It is sometimes necessary to know the warping constant ( )01 may be obtained by means of the formula

[ ] xyyyxx IyxIyIx 002020101 2++= The numerical method of calculating for open section is shown,

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48

D1 Warping Constant Tables for Typical Sections:

49

49

50

50

51

51

52

52

53

53

E. Mechanical properties:

E1. DATA Basis:

A-values:

When applied loads are eventually distributed through a single member within an assembly, the

failure of which would results in the loss of the structural integrity of the component involved,

the guaranteed minimum design mechanical properties (A - values) must be met.

At least 99% of the populations of values are expected to equal or exceed the A-value

mechanical property allowable, with a confidence of 95%

Examples of such items are:

Single members such as drag struts, push-pull rods, etc.

B-values:

Redundant structures, in which the failure of individual elements would result in applied loads

being safely distributed to other load-carrying members, may be designed on the basis of 90%

probability.

B-values are above which at least 90% of the population of values is expected to with a

confidence of 95%.

Examples of such items are:

Sheet stiffener combinations.

Multi rivet or multiple bolt connections.

Thick skin type wing construction.

S-values:

The S-value is the minimum value specified by the governing industry specification of federal or

military standards for the material. For certain products heat treated by the user the S- values

may reflect a specified quality control requirement. Statistical assurance associated with this

value is not known.

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54

E2. Stress-Strain Data:

The significant properties of material for structural design purposes are

determined experimentally. Tension and Compression coupon tests provide the simplest

fundamental information on material mechanical properties. Experimental data is recorded on

stress strain curves that record the measures units strain as a function of the applied force per

unit area. Data of importance for structural analysis is derived from the stress strain as shown

above.

E3. Definition of terms:

Modulus of Elasticity:

Slope of initial straight-line part of the stress-strain curve, E (force per unit area)

Secant Modulus:

Slope of secant line from the origin to any point (stress) on the stress-strain curve

a function of the stress level applied, Es

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55

Tangent Modulus:

Slope of the line tangent to stress-strain curve at any point (stress) on the stress-

strain curve, Et. The tangent modulus is the same as the modulus of elasticity in the range.

Proportional Limit:

Stress at which the stress-strain curve deviates from linearity, indication of plastic

or permanent set is called proportionality limit. Sometimes defined arbitrarily as stress at 0.01%

offset, Fpl (psi

Yield Strength or Stress:

At 0.2% offset, Fcy, Ftu (psi)

Ultimate Strength or Stress:

Maximum force per unit area, Ftu (psi)

Poissons Ratio:

Ratio of the lateral strain measured normal to the loading direction, to the normal

strain measured in the direction of the load. Poisson ratio varies from the elastic value of around

0.3 to 0.5 in the plastic range.

Secant stress:

Stress at intercept of any given secant modulus line with the stress strain curve

is called secant stress. F0.7, F0.85 are secant lines of 0.7 and 0.85 % of youngs modulus.

Elongation:

Permanent strain at fracture (tension) in the direction of loading e(%). Sometimes

this property is considered a measure of ductility/brittleness.

Fracture Strain:

Maximum strain at fracture of the material, e (in/in)

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56

Elastic Strain:

Strains equal to or less than the strain a t the proportional limit stress, e or the strain to the projected youngs modulus line of any stress level,

Plastic Strain:

Increment of strain beyond the elastic strain at stresses above the proportional

limit.

Shear Properties:

The results of torsion tests on round tubes or round solid section s are sometimes

plotted as torsion stress-strain diagrams. The modulus of elasticity in shear as determined from

such a diagram is a basic shear property. Other properties, such as proportional limit and

ultimate shearing stress, cannot be treated as basic properties because of the form factor.

Bearing Properties:

Bearing strength are of value in the design of joints and lugs. Only yield and

ultimate values are obtained from bearing tests. The bearing stress is obtained by dividing the

load on a pin, which bears against the edge of the hole, by the bearing area, where the area is the

product of the pin diameter and sheet thickness.

The bearing test requires the use of special cleaning procedures as specified in ASTM E 238. In

the various Room Temperature Property Tables. When the indicated values are based on tests

with clean pins, the values are footnoted as dry pin values. Designers should consider the use

of a reduction factor in applying these values to structural analysis.

In the definition of bearing values, t is sheet thickness, D is the hole diameter and e is the edge

distance measured from the hole centre to the edge of the material in the direction of applied

stress.

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57

E4. Temperature Effects:

Temperature below room temperature generally causes an increase in all strength

properties of metals. However ductility usually decreases below room temperature.

Temperatures above room temperature have the opposite effect which is dependent on many

factors such as time of exposure and the characteristic of the material.

E5. Fatigue Properties:

Practically all materials will break under numerous repetitions of a stress is not as

great the stress required to produce immediate rupture. This phenomena is known as fatigue.

Fatigue crack initiation is normally associated with the endurance limit of a material below

which fatigue does not occur is usually presented in the form of S-N curves, where the cycles

stress amplitude is plotted versus the number of cycles to failure. Another major factor affecting

the fatigue behaviour is the stress concentration caused by detail design of the structure.

Metallurgical Instability:

In addition to the retention of load-carrying ability and ductility a structural

material must also retain surface and internal stability. Surface stability refers to the resistance of

the material to oxidizing of corrosive environments. Lack of internal stability is generally

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58

manifested by carbide precipitation, spheroidization, sigma-phase formation, temper

embrittlement and internal or structural transformation, depending upon the material and the

condition.

Biaxial and Multiaxial Properties:

Many structural geometries, or load applications, are such that induced stresses

are not uniaxial but are bi-or triaxial. Because of the difficulty in testing, few triaxial test data

exist. However considerable biaxial testing has been conducted. If stresses are referred to as

mutually perpendicular x, y, and z direction of the usual rectangular co-ordinates, a biaxial stress

is a condition such that there are either a positive or negative stress in the x, y directions and the

stresses in the z direction is usually zero.

Fracture Strength:

The occurrence of flaws in a structural component is an unavoidable circumstance

of material processing, fabrication, or service. These flaws may be cracks, metallurgical

inclusion or voids, weld defects, design discontinuities, or combination thereof. If severe

enough, these flaws can include structural failure at loads below those of normal design. The

strength of a component containing a flaw is dependent on the flaw size, the component

geometry, and a material property termed fracture toughness. The fracture toughness of a

material is literally a measure of resistance to fracture. It also is considered a measure of its

tolerance or lack of sensitivity to flaws. As with many other material properties, fracture

toughness is dependent on processing variables, product form geometry, temperature, loading

rate, and other environmental factors.

Fatigue-Crack Propagation Behaviour:

Between the crack initiating phenomenon of fatigue and the critical instability of

cracked structural elements as identified by fracture toughness and residual strength lies an

important fact of material behaviour known as fatigue-crack propagation.

In small size laboratory fatigue specimens, crack initiation and specimen failure may be nearly

synonymous. However in larger structural components the existence of a crack does not

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59

necessarily indicate imminent failure of the component. The structural life during the cycle crack

extension phase is the basis for damage tolerance certification of aircraft structure.

Material Failures:

Fracture of a metal can be very complex, it can occur in either ductile or brittle

state, in the same material depending on the state of stress and the environment. Fracture can

occur after elongation of the metal over a relatively large uniform length, or after a concentrated

elongation in a short length. Shear deformation will also vary dependent on metal and stress

state, because of these variations in magnitude and mode of deformation the ductility of a metal

can have a profound effect on the ability of a fabricated part to withstand applied loads.

E.6 Generalised Stress-Strain Equations:

Introduction:

This section describes a generalised method for representing the stress-strain

curves of materials which exhibits a smooth curves. The method uses a reference stress and a

characteristic index, in addition to the modulus of elasticity, and this section indicates how these

two quantities may be determined

Generalised Equations for Stress-Strain curves:

A smooth continuous stress- strain curve can be represented by the equation

m

nnn ff

mff

fE

+= 1

in which f is the stress, fn is taken to be the stress at which the tangent modulus is one half the

modulus of elasticity and the index m has been found to characterize the shape of the stress-

strain curve and is therefore referred to as the material characteristic. The equation above only

applies if f/fn is positive and this needs to be taken into account in cases where the chosen sign

convention leads to either for being of opposite sign to fn This equation can then be rearranged and the value of Tangent Modulus obtained as

( )ddfEt /=

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60

in which f is the stress, fn is taken to be the stress at which the tangent modulus is one half the

modulus of elasticity and the index m has been found to characterize the shape of the stress-

strain curve and is therefore referred to as the material characteristic. The equation above only

applies if f/fn is positive and this needs to be taken into account in cases where the chosen sign

convention leads to either for being of opposite sign to fn. This equation can then be rearranged and the value of Tangent Modulus obtained as

( )ddfEt /=

11

1

+=m

nt f

fEE

This can be rearranged to give

m

nntn ff

ff

Ef

fE

+=

The secant modulus Es, is the ratio of stress to total strain and may be determined directly from

stress strain curve equation.

1111

+=m

ns f

fm

EE

From the above set of equations the simple relationship between Es and Et follows

=

111ts E

EmE

E

The value of Poissons ratio is according to the equation

( )( ) 11

11

1

/

+

+== m

n

m

npe

epsp

ff

m

ff

mvv

vvEEvv

ev = Fully elastic value of Poissons ratio

pv = Fully plastic value of Poissons ratio

as the material yields Poissons ration will increase from its elastic value up to its fully plastic

value

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61

The plastic value of Poissons ratio pv can be assumed to be 0.5 for all aluminium alloys. The

elastic value for aluminium alloys can be assumed to be 0.3.

Determination of m and fn

The value of m, the material characteristic, can be found from standard fitting

procedure if the full stress-strain curve is known. However, it may be estimated, provided that

two points on the non-proportional region of the curve are known, from the equation.

( )( )'

'

/log/log

RR

RR

ffm =

where fR and fR are known stress values on the stress-strain curve of the actual material and R and 'R are the strain in excess of the elastic strain corresponding to these stresses. For example , if fR and fR are the 0.1% and 0.2% tensile proof stresses of the material then R and 'R are 0.001 and 0.002 respectively.

For a known value of m the reference stress, fn, may be evaluated from the equation

( )1/1

=m

R

RRn f

Emff

E.6.1. Problem:

Calculate the strain when a stress of 30000 lb.in2 is applied to sample 2024 T3 aluminium alloy

sheet using B-basis in compression.(0.010

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62

Therefore fn = 33593.97 lbin2

To calculate the strain, , we must simplify the equation

m

nnn ff

mff

fE

+= 1

To give

E

fff

mff

n

m

nn

+=

1

Which gives us a strain of 0.003.

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63

1. PRISMATIC BAR IN TENSION:

For Ductile material yield stress is to be considered during analysis. If material is

brittle or exhibits brittle fracture upon loading, ultimate stress is to be considered for analysis.

Load considered for ultimate analysis is factor of safety times limit load. t2 values are the

maximum permissible tensile stresses for the proof loading condition. ft values are given for the

ultimate loading conditions. In case of a few materials, which have high proof to ultimate

strength ratio, allowance may have to be made for the effects of stress concentration.

1.1 Limit Stress Analysis:

For limit stress analysis t2 is considered. It is then compared with limit stress or actual stress.

Limit load = P

Working stress AP=

Allowable Proof stress (Or) 0.2% Proof stress = 2t

Reserve factor =2t

If reserve factor value is greater than 1 the design is considered to be safe.

1.2 Ultimate Stress Analysis:

For ultimate stress analysis ultimate load has to be considered. It has to be compared with UTS

value of the material.

Ultimate Working stress = (factor of safety X actual proof stress)/A

Allowable Ultimate stress (Or) UTS = ft.

Reserve factor = ft / Actual Ultimate stress factor of safety is generally 1.5 according to MIL-A-8860

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64

1.3 Problem:

Material BSL 168T6511

Mpat 3702 =

Mpaft 415=

Limit load P = 1500kg

A = 10mm10mm Limit stress analysis:

Applied proof stress MpaAP 150

10015000 ===

Allowable proof stress t2 = 370Mpa

Reserve factor = 370/150 = 2.47

Ultimate stress analysis:

Actual ultimate stress = fos actual proof stress = 2251505.1 = Allowable ultimate stress = 415


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65

2. PRISMATIC BAR IN COMPRESSION:

For the stable members in pure compression an ultimate compressive stress equal

to the 0.2% proof stress (c2) is recommended. Where the c2 value is not given tensile values t2

may be used. For thin sections or flanges, consideration should be given to the possibility of

local or Torsional instability. For slender columns buckling load is to be considered. Where

magnesium alloys are used in compression, it is recommended that special test made to be

establish the compressive proof stress values, since they may be as low as 0.5 of the

corresponding proof stresses.

2.1 Compression Stress Analysis: (Block Compression)

For Stress analysis c2 is considered.

Actual Load = P

Working Stress = P /A Allowable stress = c2.

Reserve Factor = c2/

If the reserve factor value is greater than 1 design is safe.

2.1.2 Problem:


Actual load = 1500 kg

A = 10mm10mm

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66

Actual ultimate stress = 2/225100150005.1 mmN=

Allowable ultimate stress = 370


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67

2.2 Euler Buckling:

Buckling stress ACL

EIcr 2

2 = Pcr value depends upon end fixing condition C:

Hinged - Hinged C = 1

Fixed - Free C = 4

Fixed - Fixed C = (1/4)

Fixed - Hinged C = (1/2)

2.3 For other Buckling cases:

Section Modulus = Z

2.3.1 Buckling for Eccentric Loading:

In an actual structure it is not possible for a column to be perfectly straight or to

be loaded exactly at the centroid of the area. Therefore apart from normal load there will be

additional load due to moment caused by eccentricity.

ZEIPLPe

AP

cr

+= 2sec

This is a Transcendental in P. The solution will be taken as Pall.

Eccentricity of the load = e in mm.

Moment of inertia = I in mm4.

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68

2.3.2 Beam - Columns:

A beam is called Beam-Column when it is submitted to both compression and

bending loads. When a structural element similar to a beam is subjected simultaneously to a

normal load and bending moment, superposition theorem cannot be used to determine the

stresses. The lateral deflections appreciably change the moment arms of the compression forces

and deflections are not proportional to the loads. The verification of Beam Column will

include two steps as following

1 .The study of column.

2. The study of Beam column if axial load is lower than the critical load calculated in

the previous step.

2.3.2.1 Notations and Conventions:

1. The compressive loads are taken to be positive.

2. A positive bending moment compresses the upper fibres of the beam.

3. A distributed lateral loads leads to the positive moment.

4. A positive lateral load compresses the upper fibre of the beam.

5. The compression stresses are positive.

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69

2.3.2.2 Terms used:

Compression load =P in N.

Distributed lateral load = q in N/mm.

Applied isolated load = F in N.

Critical buckling stress = cr in N/mm2. Modulus of Elasticity in compression = Ec in N/mm2.

Moment of inertia = I in mm4.

Area of cross section = A in mm2.

Clamping factor = K.

Column effective buckling length = L in mm.

Real column length = S in mm.

2.3.2.3 Calculation of Bending Moment:

Let us consider a beam where the ends are on single supports with compression

loads and bending loads.

Beam with compression and bending loads:

The bending moment at X is ( )PyxL

MMMM AB

A

ZZZZ

+= in N-mm.

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70

Final bending moment is

+

=jxM

jx

jL

jLMM

MA

AB

Z

ZZ

Z cossinsin

cos in N-mm.

Maximum bending moment is at

=jL

jLMM

jArcxAB ZZ

sin

costan in N-mm.

Common expression for Beam Column bending moment

( )xfjjxC

jxCM Z

221 cossin +

+

= .

Where

PIEj t= and several cases can be combined as shown below.

71

71

72

72

73

73

74

74

2.3.2.4 Allowable Stress:

Two main cases must be considered when choosing allowable stress

1. Solid beams

2. Thin web beams

Solid beams:

No local buckling phenomenon occurs before general buckling, the allowable stress on

tensional side is equal to R . Thin Web Beams:

A local buckling phenomenon can occur before general buckling. This state does no give

rise to an immediate collapse but can considerably decrease the allowable breaking stress. The

allowable stress on compressive side is equal to { }Rbucklinglocalall MIN ;=

For further references refer,

1. Aircraft Structures P.J.Perry Page no 524 533.

2. Analysis and Design of flight Vehicle structures by E.F.Bruhn Supplement

page no 2

3. Refer Aircraft Structures for Engineers THG Megson Page no 162-165

75

75

2.3.3 Local buckling: (Thin Walled Structures):

If column is composed wholly or partially of thin material, local buckling may

occur at a unit load less than the required load to cause failure of column as a whole. Thin walled

sections are encountered in aircraft structures in the shape of longitudinal stiffeners. These

elements fall in to two distinct categories flanges which have a free unloaded edge and webs

which are supported by adjustant plate elements in the column cross section.

2.3.3.1 Flanges:

For long flange having one end fixed and other edge is free.

Modulus of Elasticity = E

Elastic Poissons ratio =

Thick ness of flange = t

Breadth of flange = b

Buckling stress 22 )(109.1

btEfcr =

For flange having one edge simply supported and other edge is free.


btEfc =

Also refer ESDU DATA sheet 01.01.08.

2.3.3.2 Thin webs:

Long thin web fixed along each edge

Buckling stress 21

73.5 )(2 btEfc =

Simply supported along each edge


btEfc =

76

76

2.3.3.3Plates and Shells:

1) Rectangular plate under equal uniform compression on two opposite edges:

Sides of rectangular plate = a, b

Compressive load = f

Buckling stress 22 )(1 btEKfc =

Where K depend on a/b ratio and end fixing conditions

Table for value of K for various end fixing condition:

Manner of Support Value of K

For (a/b=1)

All edges simply supported 3.29

All edges clamped 7.7

Edges b simply supported, edges a clamped 6.32

edges b and edge a simply supported and one

edge a is free 1.18

Refer Analysis and design of flight vehicle structures by E.F.Bruhn chapterC5 for K values for

various a/b ratios.

2) Rectangular plate under uniform compression fx on b and fy on a:


Compressive loads = fx, fy.

Poissons ratio =

77

77



3) Rectangular plate under linearly varying stress on edges b:


Compressive load = f0, fv.


Where K depend on a/b ratio and

Where v

= 00

Table for value of K for various :

0.5 0.75 1.00 1.25 1.50 K(a/b=1) 21.1 9.1 6.4 5.4 4.8 3.29


various a/b ratios.

4) Rectangular plate under uniform shear on all edges:


Shear stress = fs.

Poissons ratio =

78

78

Buckling stress 22 )(1 btEKf sc =



a/b 1 2 All edges simply supported K 7.75 5.43 4.40

a/b 1 2 All edges clamped K 12..7 9.5 7.38


various a/b ratios.

5) Rectangular plate under uniform shear on all edges, compression fx on b and fy on a:


Compressive loads = fx, fy.

Shear load = fs.

Poissons ratio =

Buckling stress for all edges simply supported

))(( 8122122Cf

Cf

Cf

Cf

Cf xyxysc ++=

Where 22 )(1823.0

btEC =

Buckling stress for all edges clamped

))(( 8431.234431.22

Cf

Cf

Cf

Cf

Cf xyxysc ++=

79

79

6) Rectangular plate under uniform shear on all edges and bending stresses on b:


Bending loads = fb,

Shear load = fs.

Poissons ratio =

Buckling stress for all edges simply supported

22 )(1 b

tEKfb =

Here K depends on fs/fsmax and on a/b


fs/fsmax 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

K(a/b=1) 21.1 20.4 19.6 18.5 17.7 16 14 11.9 8.2 0

80

80

2.3.3.4 Analysis of Thin Walled Structures:

Rb = (Actual Bending Stress/Allowable Bending Stress)

Rc = (Actual Compressive Stress/Allowable compressive Stress)

Rs = (Actual Shear Stress/Allowable Shear Stress)

Rl = (Actual Longitudinal Stress/Allowable Longitudinal Stress)

For combined Bending and Longitudinal Compression:

Reserve factor cb RR +75.1 For Combined Bending and Shear:

Reserve Factor 22

1

sb RR +

For Combined Shear and Longitudinal direct Stress:

Reserve Factor22 4

2

sll RRR ++

2.4 Problems:

2.4.1 Problem for Euler Buckling:

Problem 1:


MpaE 31070= t = 370

81

81

ft = 415Mpa

Actual load = 2500 kg

Ultimate load = fos2500 = 25005.1 = 3750

end condition = hinged hinged

C = 1

b = 10mm

d = 10mm

l = 100mm

33.83312

1012

43

=== bdI

NlEIPcr 57572100

33.83310702

32

2

2

===

Reserve factor = 535.13750

2.5757 ==ult

cr

PP

Problem 2:


MpaE 31070= Mpat 3702 = Mpaft 415=

l = 5000 mm

P = 1000 kg

82

82

Ultimate load = fosP = kg150010005.1 =

End condition = hinged hinged = c = 1

mmx 502

60201 =+= mmx 102

202 == mmx 502

60203 =+=

mmy 10220

1 == mmy 502100

2 == mmy 90220803 =+=

21 12006020 mmA == 22 200020100 mmA == 23 1200mmA =

mmAAA

xAxAxAX 82.31

4400140000

120020001200501200102000501200

321

332211 ==++++=++

++=

mmAAA

yAyAyAY 50

4400220000

120020001200901200502000101200

321

332211 ==++++=++

++=

233

3332

22

3222

11

311

121212hAdbhAdbhAdbI XX +++++=

= 23

23

23

18.18120012

206082.21200012

1020)18.18(120012

2060 +++++ = 1780573.22mm4

83

83

233

3332

22

3222

11

311

121212hAbdhAbdhAbdIYY +++++=

= 233

23

40120012

602012

2010040120012

6020 ++++ = 4626666.67mm4

NLl

EIPcr 492058000122.17805731070

2

32

2

2

===


5.4920 = Ultimate load = 18150N

Limit load = Nfos

121005.1

1815018150 ==

2.4.2 Problems for Beam Column:

Problem 1:

Considered a beam with both Transverse and Axial load with moment as shown

in the diagram. The data for the set up is listed below calculate the maximum bending moment

and reserve factor.

Data:

S = 100 mm, a = 30 mm, c =10 mm.

A = 100mm2, I = 833 mm4, F = 700 N, P = 11600 N.

M1 = 12000 N mm.

M2 = 40000 N mm.

Material 7075 T 7651

E = 71711 N/mm2.

84

84

Ec = 73800 N/mm2.

4602.0 =c N/mm2. 4602.0 = N/mm2.

525=R N/mm2. e% = 7.

Check for beam does not buckle the action of P alone

6071001100

833738022

2

22

2

===

AKSIEc

cr N/mm2.

K = 1 for given clamping condition

To apply plasticity correction

cr =

c

t

EE=

Using Ram berg and Osgood model: cn

c

cr

c

cr

E

+=

2.0

002.0

cr

sE =

c

c

s

c

t En

En

E+= 11

411= N/mm2. Stress applied due to axial load 116==

AP N/mm2.

So, it can be deduced that column does not buckle.

Maximum bending moment expression 7281160

8337380 ===P

IEj c

For ax , Using combination of cases 1 to 4 given in above table.

81010sin

sin

sin

cos121 =+

=

jS

jbjF

jS

jSMM

C N-mm.

85

85

1200012 == MC N-mm.

jx

jxM AC cos12000sin81010 +=

For ax > , Using combination of cases 1 and 5 given in above table.

34310tan

sin

sin

cos121 =+

=

jS

jSjF

jS

jSMM

C N-mm.

32410sin12 =+= jSjFMC N-mm.

jx

jxM CB cos32410sin34310 +=

To determine the maximum bending moment

At point A

NmmM A 120000cos120000sin81010 =+= At point C

NmmM C 434408.7230cos12000

8.7230sin81010 =+=

At point B

NmmM B 400008.72100cos32410

8.72100sin34310 =+=

Between A and C

0728

cos728

12000728

sin728

81010 == xxdx

dM AC

The moment is maximum at abscissa point

mmArcx 7.1031200081010tan728 == But 103.7>XC

Therefore MAC is maximum at XC

MACmax = 43440 N-mm

Between C and B

0728

cos728

32410728

sin728

34310 == xxdx

dM CB

86

86

The moment is maximum at abscissa point

mmArcx 3.593241034310tan728 ==

mmNM CB =+= 47200728593cos32410

728593sin34310max

Therefore, the maximum moment on beam AB is

( ) mmNMMMAXM CBACAB == 47200; maxmaxmax Calculate the total stress and the reserve factor

Stress due to the axial load

2/116 mmNSP

c == Stress due to the bending moment

2/2832

mmNcI

Mf ==

Maximum compression stress

2/399 mmNfcMAX =+= Reserve factor

R.F. = 32.1399525 =

2.4.3 Problem for Local Buckling:

Problem 1:

Material BSL 168T6511 31070=E ,a = 10,b = 10,t = 1

All edges are simply supported

Compressed load f = 1000kg

87

87

Actual ultimate load kgfult 1501005.1 ==

Actual stress = 2/1500110

15000 mmN=

Allowable stress = 2

21

n

tEK = 2

2

3

101

3.01107029.3

= 2530.27N


27.2530 =

2.4.3.1 Problem for Combined Bending and Shear:

Problem 1:

Material BSL 168T6511 31070=E , a = 10, b = 10, t = 1

All edges are hinged.

Bending load = 100kg, Shear load = 1000kg

Actual bending stress = 23 /3000110125.0500 mmN

IMy =

=

Actual shear stress = 2/500110102

10100002

mmNAtT =

=

Buckling stress (bending) = NbtEK 3.4615)01.0(

91.010706

)1(

32

26 ==

Shearing buckling stress = 0769.73)01.0(91.0

10705.9)1(

32

2 ==

b

tEK s

65.03.4615

3000 ==bR , 068.069.7307500 ==BR

Reserve factor = 53.1)068.0()65.0(

1)()(

12222=

+=

+ sb RR

88

88

3. LUG ANALYSIS:

The lug refers to that portion of the fitting that involves the hole for the single

bolt that connects the male and female part of the fitting unit. The simplified assumptions

regarding failing action and the resulting equations have been widely used for quick approximate

check of the lug strength. The failure of the Lug may be due to Tension or shear tear out or due

to bearing. Loading conditions considered are

1. Axial loading

2. Transverse loading

3. Oblique loading

3.1 Failure in Tension:

The above diagram indicates how a plate can be pull apart due to tension stresses

on a section through the centre line of bolt hole. Both the male and female parts of the fitting

must transfer the load past the centreline of the hole, thus both parts must be considered in the

design of the fitting. Refer ESDU Data Sheet 91008 for further details.

Ultimate tensile strength of the material = ft. +

Breadth of the plate = 2R

Thickness of lug = t

Diameter of the hole = d

Limit load = P

Ultimate applied stress tdR

Pf tensionu )2(5.1

)( = +Allowable Tensile is obtained by detail fatigue analysis.

89

89

The above equation assumes that the tensile stress on the cross section is uniform.

This is not true as the flow of the stress around the hole causes a stress concentration. To take

care of this extra allowance has to be given in reserve factor.

Reserve factor = ft/fu (tension).

3.2 Failure by Shear Tear Out:

The above picture illustrates the manner in which failure can occur by the

shearing tear out of a plate sector in front of the bolt. Due to load P the bolt presses the plate or

lug around bolt - hole edge. Stresses are produced which tends to cause the portion (a) shown in

the figure to tear out. Refer ESDU Data Sheets 81006, 71011, 66011 for details. Usually fs for

ductile material are 0.6 UTS.

Shear out area = As.

Ultimate shear stress = fs.

Ultimate applied stressxtPf shearoutu 2

5.1)( =

It is very common practise to take the shear out area As equal to the edge distance

at the centreline of the hole times the thickness t of the plate times two since there are two shear

areas.

This is slightly conservative because the actual shear area is larger and area considered is limited

by the 40o line as shown in the picture.

Reserve factor = fs/fu (shear out)

90

90

3.3 Failure by bearing:

The pull causes the bolt to press against the wall, which in turn presses against the

plate wall. If the pressure is high enough the plate material adjustant to the hole will start to

crush and flow thus allowing bolt and bush to move which results in elongated hole as shown in

the diagram.

Ultimate bearing stress = fb.

Diameter of the bushing = d

Thickness of the plate = t

Bearing factor = Kbr. +

Ultimate applied stressdt

P5.1fu(bearing) = Reserve factor = Kbr*fb/fu (bearing)

Usually fb is equal to ft. It is good practise to require a reserve factor of 1.5 + Refer ESDU Data Sheet 81029, 83033 for further details.

3.4 Lug strength analysis under transverse loading:

Cases arise where the lug of a fitting unit is subjected to only a transverse load.

Bearing area = Abr.

Ultimate Tensile stress = ft.

Efficiency failing coefficient = Kt.

Allowable load tbrtu(tension) fAKP = Kt depends on 2R/d ratio and upon material used.

91

91

3.5 Lug strength analysis under oblique loading:

Fitting lugs often subjected to oblique loads. It will be resolved in to axial and

transverse components.

Reserve factor = 0.6251.6tr

1.6a )R(R

1+

Where

Ra = axial component of the applied load divided by ultimate allowable tensile load

Rtr = transverse component of the applied load divided by Pu (tension) of transverse loading.

Allowable stresses used in various loading conditions:

Table for value of ultimate stresses for various condition:

Condition Ultimate stresses

Tension ft.

Shear 0.6 ft.

Bearing 2 ft.

92

92

3.6 Lug analysis (AVRO Method):

The Permissible lug stresses listed down are based on the method illustrated in,

Av.P.970 chapter 404/2. Allowable stresses used in various loading conditions are listed below.

The permissible lug stress for each condition is taken as maximum of values for failing stress.

Condition Ultimate stresses

Tension 0.9 ft.

Shear 0.46 ft.

Bearing 1.85 ft.

For shear if a/d

93

93

3.7 Problems:

Problem 1:

Figure shows a single pin fitting the lug material is AISI steel heat treated to ft =

125000 Psi. The bolt AN steel ft = 125000 Psi. The bushing is steel with Ft = 125000 Psi .The

fitting is subjected to an ultimate tensile load of 15650 lb .check the fitting for the design load

diameter of bolt is

A fitting factor is considered = 1.15

Clearance gap = 1/6

Applied fitting load = lb180001565015.1 =

Check for bolt shear:

For diameter = 0.5

Ultimate double shear length of bolt is 29400

Applied load (Ultimate) = 18000 lb

Reserve factor = 29400/18000 = 1.62

Failure by bolt bending:

There is strength gap between lugs

gttb ++= 21 25.05.0 (Or) gttb ++= 42 21 218.0)64/1()8/3(25.0)3/7(5.0 =++=b

1962218.02

18000 ==M

94

94

124026509 aircraft structures

Documents

stress analysis of beam

limit stress analysis

ultimate stress analysis

compression stress analysis

bending stresses

c section shear center

combined bending

allowable stress