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Basic structural theory

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Statics Things dont continue to move if forces are resisted – Static Equilibrium What resists the force? Equal and opposite Reaction Things deflect if forces are resisted Elastic and Plastic Deformation

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Basic loads (forces) Vertical (y only) Lateral (x only) Rotational (moment) Concentrated loads Distributed loads w = P/ l force-couple

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Basic components Linear – Column, Beam Planar – Wall, Floor

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Basic connections Simple (constrain y in direction of gravity, rotate freely)

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Basic connections Roller (constrain y, rotate freely)

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Basic connections Pin (constrain x & y, rotate freely)

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Basic connections Pin (constrain x & y, rotate freely)

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Basic connections Cable (Pin with tension only)

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Basic connections Cable (Pin with tension only)

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Basic connections Fixed/Rigid (constrain x, y, rotation)

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Basic connections Fixed/Rigid (constrain x, y, rotation)

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Basic connections Fixed/Rigid (constrain x, y, rotation)

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Basic connections Fixed/Rigid (constrain x, y, rotation)

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Basic connections Misleading pin connections

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Column – Vertical Load Axial load – Compression & Tension

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Column – Lateral Load Non-axial (lateral) load – Buckling in compression

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Beam – Vertical Load Non-axial load – Deflection

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Basic loads (forces) Reactions are the same for Concentrated loads and Distributed loads Beam stresses are different w = P/ l

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Greater deflection Greater max. moment w = P/ l

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C N T Beam – Stresses Compression, Tension, Neutral axis

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Beam – Concentrated Vertical Load Resist bending with Moment connection Greater deflection Greater max. moment

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Beam – Distributed Vertical Load Resist bending with Moment connection Greater deflection Greater max. moment

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Factors influencing deflection: P = load l = length between supports E = elastic modulus of material (elasticity) I = Moment of inertia (depth/weight of beam) D max = P l 3 /48EI

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Elastic modulus of materials Structural Steel = 200 GPa (29,023,300 lb/in 2 ) Titanium = 110 GPa (15,962,850 lb/in 2 ) Aluminum = 70 GPa (10,158,177 lb/in 2 ) Concrete = 21 GPa (3,047,453 lb/in 2 ) Douglas Fir = 13 GPa (1,886,518 lb/in 2 ) Why are titanium and aluminum used in aircraft?

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Yield Strength of materials Structural Steel=350-450 MPa Titanium (Alloy)=900-1400 MPa Aluminum=100-350 MPa Concrete=70 MPa (compressive) Douglas Fir= N/A Density of materials Structural Steel = 489 lb/ft 3 Titanium = 282 lb/ft 3 Aluminum = 169 lb/ft 3 Concrete = 150 lb/ft 3 Douglas Fir = 32 lb/ft 3 1 lb/in 2 = 6891 Pa

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Moment of Inertia of beam Dependent on cross-sectional geometry Not dependent on material properties Icc = Moment of inertia of a rectangle about the neutral axis – i.e. its centroid = width x height 3 /12 Ixx = Moment of inertia of a rectangle about an axis parallel to the neutral axis = Icc + width x height x (distance between axes) 2 Centroid = S (Area x distance to bending axis)/(Total area)

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Triangulated frame (Truss) – increase depth of beam Triangulated – all members axially loaded (truss) – no moments

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Triangulated frame (Truss) – increase depth of beam Triangulated – all members axially loaded (truss) – no moments

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Rigid Frame – Vertical load Reduce deflection: Rigid connection Columns resist force and deflect

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Rigid Frame – Vertical load Thrust develops at base of columns and must be resisted (beam / foundation / grade beam)

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Cantilever Moment connection

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Cantilever Moment connection tension compression moment (force-couple)

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Cantilevered Beam – Vertical load Greater deflection Greater max. moment

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Simple Frame – Vertical load Reduce deflection at mid- span: Cantilever Lesser deflection Lesser max. moment

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Cantilever Deflection - Resist bending with counterweight

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Frame – Lateral load Racking

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Frame – Lateral load Racking

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Frame – Lateral load Triangulated – all members axially loaded (truss) – no moment connections

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Frame – Lateral load Triangulated – all members axially loaded (truss) – no moment connections

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Frame – Lateral load Rigid (moment-resisting) frame

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Frame – Lateral load Rigid (moment-resisting) frame

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Frame – Lateral load Shear-resisting (force in plane)

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Frame – Lateral load Pre-engineered shear panel

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Frame – Lateral load Pre-engineered shear panel

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Frame – Lateral load Shear-resisting (force in plane) Non-structural partitions

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Frame – Lateral load Shear-resisting (force in plane) Masonry must be grouted and steel- reinforced

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Funicular structures Tension (Cable) Compression (Arch)

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Funicular structures Tension (Cable) Compression (Arch)

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Funicular structures Tension (Cable) Compression (Arch)

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Non-Funicular structures

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Materials - Wood Tension & compression, no rigid connection

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Materials - Wood Unpredictable failure mode (non-uniform material – organic)

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Materials - Reinforced Concrete Wide range of possible forms

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Materials - Reinforced Concrete Compression and some tension (steel), rigid connection through rebar

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Materials - Reinforced Concrete Catastrophic failure mode

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Materials - Reinforced Concrete Catastrophic failure mode

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Materials - Reinforced Concrete Lab testing

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Materials - Steel Tension & compression

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Materials - Steel Rigid connection through welding

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Materials - Steel Plastic failure mode

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