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Basic structural theory
Statics
Things don’t 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 loadsw = 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 connections
Misleading pin connections
Column – Vertical Load
Axial load – Compression & Tension
Column – Lateral Load
Non-axial (lateral) load – Buckling in compression
Beam – Vertical Load
Non-axial load – Deflection
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
CN
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)
Dmax = Pl 3/48EI
Elastic modulus of materials
Structural Steel = 200 GPa (29,023,300 lb/in2)
Titanium = 110 GPa (15,962,850 lb/in2)
Aluminum = 70 GPa (10,158,177 lb/in2)
Concrete = 21 GPa (3,047,453 lb/in2)
Douglas Fir = 13 GPa (1,886,518 lb/in2)
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/ft3
Titanium = 282 lb/ft3
Aluminum = 169 lb/ft3
Concrete = 150 lb/ft3
Douglas Fir = 32 lb/ft3
1 lb/in2 = 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. it’s centroid = width x height3 /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
Racking
Frame – Lateral load
Racking
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