basic structural theory

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

Cable (Pin with tension only)

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

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


Triangulated – all members axially loaded (truss) – no moment

connections


Rigid (moment-resisting) frame


Shear-resisting (force in plane)


Pre-engineered shear panel



Non-structural partitions



Masonry must be grouted and steel-reinforced

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


Compression and some tension (steel), rigid connection through rebar


Catastrophic failure mode


Lab testing

Materials - Steel

Tension & compression

Materials - Steel

Rigid connection through welding

Materials - Steel

Plastic failure mode

basic structural theory

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