INTRODUCTION TO ANALYSIS AND DESIGN OF LIGHT
STRUCTURES.PART 2
NOTES
BY
AMOS KOLA
2022
i
, TABLE OF CONTENTS
1. STATICAL INDETERMINACY ...................................................................................................... 1
1.1 Introduction ................................................................................................................................. 1
1.2 Stress Analysis of Stratically Determinate Plane Structures ..................................................... 1
1.2.1 Plane Trusses .............................................................................................................. 2
Problem Sheet 1.1 .................................................................................................................................. 13
1.3 Definition ................................................................................................................................... 15
1.4 Degree of Indeterminacy........................................................................................................... 19
1.5 Kinematic Indeterminacy ......................................................................................................... 22
1.6 Analysis of Statically Indeterminate Structures ....................................................................... 24
1.7 Force-Displacement Relations .................................................................................................. 24
1.8 Description of Methods ............................................................................................................. 25
1.9 Statically Indeterminate Frames ............................................................................................... 28
1.9.1 Moment-Distribution Method ................................................................................ 28
Problem Sheet 1.2 .................................................................................................................................. 36
2 COLUMNS ...................................................................................................................................... 37
2.1 Buckling of Slender Columns .................................................................................................... 37
2.2 Slenderness Ratio ....................................................................................................................... 38
2.3 Axially Loaded Timber Columns. ............................................................................................. 43
2.4 Axially Loaded Steel Columns................................................................................................... 47
2.5 Axially Loaded Concrete Column ............................................................................................. 47
2.6 Eccentrically Loaded Timber and Steel Columns .................................................................... 49
2.7 Plain and Centrally Reinforced Concrete Walls ...................................................................... 49
Problem Sheet 2 ..................................................................................................................................... 52
3 THICK AND COMPOUND CYLINDERS AND SHELLS ............................................................ 54
3.1 Derivation of the Hoop and Radial Stress Equations for a Thick Walled Circular Cylinder. 54
3.2 Lame’ Line .................................................................................................................................. 57
3.3 Compound Cylinders................................................................................................................. 60
3.3.1 Introduction ............................................................................................................. 60
3.4 Thick Spherical Shells ............................................................................................................... 66
Problem Sheet 3 ..................................................................................................................................... 69
4 COMPOSITE BEAMS .................................................................................................................... 70
4.1 Introduction ............................................................................................................................... 70
4.1 Elastic Behaviour of Composite Beams .................................................................................... 71
4.1.1 No Interaction Case ................................................................................................. 71
4.2 Two-Material Composite Beams -Axial Load .......................................................................... 73
4.3 Two-Material Composite Beams -Moment Load ..................................................................... 74
4.4 Neutral Axis (NA) Location ...................................................................................................... 75
ii
, 4.6 Alternative Method - Equivalent Area ..................................................................................... 78
Problem sheet 4 ..................................................................................................................................... 86
5 DEFORMATION AND BENDING BEYOND ELASTIC LIMITS ................................................ 88
5.1 Plastic Bending of Mild-Steel Beams ........................................................................................ 88
5.1.1 Elastic-Plastic Bending of a Rectangular Mild-Steel Beam.................................... 91
5.1.2 Fully plastic moment of an I-section; shape factor ................................................ 94
5.1.3 More general case of plastic bending ...................................................................... 95
5.1.4 Comparison of elastic and plastic section moduli................................................... 98
5.1.5 Regions of plasticity in a simply-supported beam ............................................ 99
Problem Sheet 5 ................................................................................................................................... 104
6 SPACE STRUCTURES ................................................................................................................. 105
6.1 Determinancy and Stability of Space Trusses .............................................................................. 105
6.2 Analysis of Space Trusses .............................................................................................................. 107
Problem sheet 6 ................................................................................................................................... 112
7 FRACTURE .................................................................................................................................. 115
7.1 Fracture Modes ........................................................................................................................ 115
7.2 Fracturing Laws ....................................................................................................................... 115
7.3 Mechanism of fracture ............................................................................................................ 116
7.3.1 Griffith Cracks........................................................................................................ 116
7.3.2 Stress Intensity Factor............................................................................................ 119
7.3.3 Modes of Crack Growth......................................................................................... 122
7.3.4 Stress at Crack Tip of a Ductile Material .............................................................. 124
7.3.5 Characteristics of K1, K11, K111 ............................................................................... 125
7.3.6 The Griffith Criterion ...................................................................................... 127
7.3.7 Methods of Determining K10 and G10 .................................................................... 128
7.3.8 Crack opening Displacement (COD) Criterio....................................................... 134
7.3.9 J-Contour Integral and R-Curves .......................................................................... 136
7.3.10 Impact Testing for Toughness ............................................................................... 141
Problem Sheet 7 ................................................................................................................................... 145
8 NON STATIC LOADS.................................................................................................................. 147
8.1 Impact Loads ............................................................................................................................ 147
8.1.1 High-Velocity Impact ............................................................................................ 147
8.1.2 Low-Velocity Impact ............................................................................................. 148
8.2 Fatigue ...................................................................................................................................... 149
8.2.1 Description of Failure ............................................................................................ 149
8.2.2 S-N Curve ............................................................................................................... 150
8.2.3 Endurance Limit .................................................................................................... 151
8.2.4 Non Reversible Cycles of Loading......................................................................... 153
iii
, 8.3 Creep and Relaxation .............................................................................................................. 156
8.3.1 Creep 156
8.3.2 Relaxation ............................................................................................................... 159
8.4 Influence Lines ........................................................................................................................ 160
8.4.1 Influence Lines for Trusses .................................................................................... 162
Problem Sheet 8 ................................................................................................................................... 166
iv
STRUCTURES.PART 2
NOTES
BY
AMOS KOLA
2022
i
, TABLE OF CONTENTS
1. STATICAL INDETERMINACY ...................................................................................................... 1
1.1 Introduction ................................................................................................................................. 1
1.2 Stress Analysis of Stratically Determinate Plane Structures ..................................................... 1
1.2.1 Plane Trusses .............................................................................................................. 2
Problem Sheet 1.1 .................................................................................................................................. 13
1.3 Definition ................................................................................................................................... 15
1.4 Degree of Indeterminacy........................................................................................................... 19
1.5 Kinematic Indeterminacy ......................................................................................................... 22
1.6 Analysis of Statically Indeterminate Structures ....................................................................... 24
1.7 Force-Displacement Relations .................................................................................................. 24
1.8 Description of Methods ............................................................................................................. 25
1.9 Statically Indeterminate Frames ............................................................................................... 28
1.9.1 Moment-Distribution Method ................................................................................ 28
Problem Sheet 1.2 .................................................................................................................................. 36
2 COLUMNS ...................................................................................................................................... 37
2.1 Buckling of Slender Columns .................................................................................................... 37
2.2 Slenderness Ratio ....................................................................................................................... 38
2.3 Axially Loaded Timber Columns. ............................................................................................. 43
2.4 Axially Loaded Steel Columns................................................................................................... 47
2.5 Axially Loaded Concrete Column ............................................................................................. 47
2.6 Eccentrically Loaded Timber and Steel Columns .................................................................... 49
2.7 Plain and Centrally Reinforced Concrete Walls ...................................................................... 49
Problem Sheet 2 ..................................................................................................................................... 52
3 THICK AND COMPOUND CYLINDERS AND SHELLS ............................................................ 54
3.1 Derivation of the Hoop and Radial Stress Equations for a Thick Walled Circular Cylinder. 54
3.2 Lame’ Line .................................................................................................................................. 57
3.3 Compound Cylinders................................................................................................................. 60
3.3.1 Introduction ............................................................................................................. 60
3.4 Thick Spherical Shells ............................................................................................................... 66
Problem Sheet 3 ..................................................................................................................................... 69
4 COMPOSITE BEAMS .................................................................................................................... 70
4.1 Introduction ............................................................................................................................... 70
4.1 Elastic Behaviour of Composite Beams .................................................................................... 71
4.1.1 No Interaction Case ................................................................................................. 71
4.2 Two-Material Composite Beams -Axial Load .......................................................................... 73
4.3 Two-Material Composite Beams -Moment Load ..................................................................... 74
4.4 Neutral Axis (NA) Location ...................................................................................................... 75
ii
, 4.6 Alternative Method - Equivalent Area ..................................................................................... 78
Problem sheet 4 ..................................................................................................................................... 86
5 DEFORMATION AND BENDING BEYOND ELASTIC LIMITS ................................................ 88
5.1 Plastic Bending of Mild-Steel Beams ........................................................................................ 88
5.1.1 Elastic-Plastic Bending of a Rectangular Mild-Steel Beam.................................... 91
5.1.2 Fully plastic moment of an I-section; shape factor ................................................ 94
5.1.3 More general case of plastic bending ...................................................................... 95
5.1.4 Comparison of elastic and plastic section moduli................................................... 98
5.1.5 Regions of plasticity in a simply-supported beam ............................................ 99
Problem Sheet 5 ................................................................................................................................... 104
6 SPACE STRUCTURES ................................................................................................................. 105
6.1 Determinancy and Stability of Space Trusses .............................................................................. 105
6.2 Analysis of Space Trusses .............................................................................................................. 107
Problem sheet 6 ................................................................................................................................... 112
7 FRACTURE .................................................................................................................................. 115
7.1 Fracture Modes ........................................................................................................................ 115
7.2 Fracturing Laws ....................................................................................................................... 115
7.3 Mechanism of fracture ............................................................................................................ 116
7.3.1 Griffith Cracks........................................................................................................ 116
7.3.2 Stress Intensity Factor............................................................................................ 119
7.3.3 Modes of Crack Growth......................................................................................... 122
7.3.4 Stress at Crack Tip of a Ductile Material .............................................................. 124
7.3.5 Characteristics of K1, K11, K111 ............................................................................... 125
7.3.6 The Griffith Criterion ...................................................................................... 127
7.3.7 Methods of Determining K10 and G10 .................................................................... 128
7.3.8 Crack opening Displacement (COD) Criterio....................................................... 134
7.3.9 J-Contour Integral and R-Curves .......................................................................... 136
7.3.10 Impact Testing for Toughness ............................................................................... 141
Problem Sheet 7 ................................................................................................................................... 145
8 NON STATIC LOADS.................................................................................................................. 147
8.1 Impact Loads ............................................................................................................................ 147
8.1.1 High-Velocity Impact ............................................................................................ 147
8.1.2 Low-Velocity Impact ............................................................................................. 148
8.2 Fatigue ...................................................................................................................................... 149
8.2.1 Description of Failure ............................................................................................ 149
8.2.2 S-N Curve ............................................................................................................... 150
8.2.3 Endurance Limit .................................................................................................... 151
8.2.4 Non Reversible Cycles of Loading......................................................................... 153
iii
, 8.3 Creep and Relaxation .............................................................................................................. 156
8.3.1 Creep 156
8.3.2 Relaxation ............................................................................................................... 159
8.4 Influence Lines ........................................................................................................................ 160
8.4.1 Influence Lines for Trusses .................................................................................... 162
Problem Sheet 8 ................................................................................................................................... 166
iv
