# Basic structural theory. Statics Things dont continue to move if forces are resisted – Static Equilibrium What resists the force? Equal and opposite Reaction.

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Basic structural theory

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

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

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

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

Basic connections Roller (constrain y, rotate freely)

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

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

Basic connections Cable (Pin with tension only)

Basic connections Cable (Pin with tension only)

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

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

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

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

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

Greater deflection Greater max. moment w = P/ l

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

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

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

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

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?

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

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)

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

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

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

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

Cantilever Moment connection

Cantilever Moment connection tension compression moment (force-couple)

Cantilevered Beam – Vertical load Greater deflection Greater max. moment

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

Cantilever Deflection - Resist bending with counterweight

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

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

Frame – Lateral load Rigid (moment-resisting) frame

Frame – Lateral load Rigid (moment-resisting) frame

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

Frame – Lateral load Pre-engineered shear panel

Frame – Lateral load Pre-engineered shear panel

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

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

Funicular structures Tension (Cable) Compression (Arch)

Funicular structures Tension (Cable) Compression (Arch)

Funicular structures Tension (Cable) Compression (Arch)

Non-Funicular structures

Materials - Wood Tension & compression, no rigid connection

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

Materials - Reinforced Concrete Wide range of possible forms

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

Materials - Reinforced Concrete Catastrophic failure mode

Materials - Reinforced Concrete Catastrophic failure mode

Materials - Reinforced Concrete Lab testing

Materials - Steel Tension & compression

Materials - Steel Rigid connection through welding

Materials - Steel Plastic failure mode

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