Revised Notes 2009

Revised Notes 2009 - Structural Mechanics and Dynamics 2nd...

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1 Structural Mechanics and Dynamics 2 nd Year Course Notes Professor M H Aliabadi 2009
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Contents 3 CONTENTS 1 Introduction. ....................................................................................... 5 1.1 Influence of the Aircraft Shape on Aircraft Structural Analysis . .................. 6 1.2 Components of a Stressed Skin Structure. ..................................................... 8 1.3 Loadings on the Structure. ........................................................................... 11 1.4 Structural Idealisations. ................................................................................ 11 1.5 The Layout of the Lecture Course . .............................................................. 12 2 Beam and Tube Theory. .................................................................. 15 2.1 Loads on the Airframe . ................................................................................ 15 2.2 Beam Equilibrium equations. ....................................................................... 16 2.3 Classical Beam Theory . ............................................................................... 17 2.3.1 Classical Beam Theory Assumptions. .................................................. 18 2.3.2 Modifications to Classical Beam Theory for Thin Walled Beams and Tubes 18 2.4 Sign Conventions. ........................................................................................ 19 2.5 Resolving Shear forces and Bending Moments. .......................................... 20 2.6 Loadings and Stress Resultants. ................................................................... 22 2.7 Direct Stress Distribution on the Cross-Section . ......................................... 22 2.8 Alternative Method for Finding the Direct Stress Due to Bending . ............ 23 2.8.1 Direction of the Principal Axes. .......................................................... 24 2.9 The Neutral Axis. ......................................................................................... 25 2.10 Finding Second Moments of Area for Thin Walled Sections. ..................... 26 2.10.1 Simple Example – Second Moments of Area of a Single Inclined Face 26 2.10.2 Example – Second Moments of Area of a Section with Straight Sides 29 2.10.3 Principal Axes of a Z-Section . ............................................................. 30 3 Shear Stress and Shear Strains in Thin Walled Tube Bending Theory . ..................................................................................................... 31 3.1 Shear Stress Distribution Assumption . ........................................................ 31 3.2 The Shear Flow q. ........................................................................................ 33 3.3 Equilibrium of Skin Stresses. ....................................................................... 34 3.3.1 Example – The Shear Flow in an Open Channel Section Subject to a Vertical Shear Force Sy. ...................................................................................... 39 3.3.2 Example – The Shear Flow in an Open Channel Section Subject to a Horizontal Shear Force Sx. .................................................................................. 41 3.4 Comments upon the Shear Flow Distribution in an Open Section. ............. 44 3.5 Sign of the Shear Flow. ................................................................................ 44
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Contents 4 3.6 Shear Flow Distributions for Open Sections with more then Two Free Edges 44 3.7 Resultants of the Shear Flow . ...................................................................... 45 3.8 Shear Centre for an Open Section. ............................................................... 47 3.8.1 Shear Centre of a Channel Section. ..................................................... 48 3.9 Torsion of a Single Cell Closed Tube – The Bredt-Batho Theory . ............. 49 3.10 Shear Flow Distribution in a Single Closed Cell. ........................................ 50 3.10.1 Example – Shear Flow Distribution in a Rectangular Wing Box. ....... 52 4 Deflections of Beams and Tubes Including Shear Strains. .......... 55 4.1 Twisting of Single Cell Closed Tubes . ........................................................ 55 4.1.1 Bredt-Batho Rate of Twist. ................................................................... 56 4.2 Shear Centre of a Single Cell Tube. ............................................................. 56 4.3 Warping of the Cross-Section. ..................................................................... 57 4.4 Bending Deflection of Beams Including Shear Strains. .............................. 58 5 Effect of Stiffeners on the Shear Flow Distribution . .................... 61 5.1 Smeared Stiffener Idealisation. .................................................................... 61 5.2 Lumped Stiffener Idealisation. ..................................................................... 62 6 Simple Idealisation of the Beam/Tube Model . .............................. 65 6.1 Boom Area Idealisation . .............................................................................. 66 6.1.1 Example – Shear flow and Shear Centre of an Idealised Section Subject to a Vertical Shear Force. ...................................................................... 67 6.2 A Comment upon the Idealised Section Properties . .................................... 69 7 Structural Failure and Assessment . ............................................... 71 7.1 Stress Distribution. ....................................................................................... 72 7.2 Stress Invariants. .......................................................................................... 75 7.2.1 First Stress Invariant . .......................................................................... 75 7.2.2 Second Stress Invariant. ....................................................................... 76 7.2.3 Third Stress Invariant . ......................................................................... 76 7.3 Mohrs Circle . ............................................................................................... 76 7.4 Strain Transformations. ................................................................................ 79 7.5 Strain Gauge Rosettes. ................................................................................. 79 8 Buckling Analysis. ............................................................................ 83 8.1 Simple Buckling Example . .......................................................................... 83 8.2 The Buckling of Beams and the Pin-Ended Strut. ....................................... 85 8.3 Strut with One End Built in and One End Pinned . ...................................... 88 8.4 The Imperfect Pin-Ended Strut. ................................................................... 90 9 Combination of Failure Mechanisms – Failure Envelopes.
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This note was uploaded on 03/24/2010 for the course AE A.213 taught by Professor S.pinhoandm.aliabadi during the Winter '09 term at Imperial College.

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Revised Notes 2009 - Structural Mechanics and Dynamics 2nd...

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