Statics Concepts

Force Classifications

1. Contact (push/pull)

2. Body (gravitation, magnetic)

Force Application

1. Concentrated

2. Distributed

Force Location

1. External - change motion or develop reactions

2. Internal - cause deformations

Force Characteristics

Magnitude + Direction + Point of Application + Unit

Transmissibility

Either force has the same external effect

Force Systems

1. Concurrent - LOAs intersect at a point —> can replace with on force (Fr = F1 + F2 + F3)

2. Non-concurrent: LOAs don’t intersect

Mo in scalar and vector notation

1. Scalar: Mo = Fd

2. Vector: Mo = r x F

Varignon’s Theorem

The forces along a same LOA create the same moment

Characteristics of a Moment of a Couple

1. Equal magnitude

2. Opposite direction

3. Parallel LOAs

4. Non-collinear

Newton’s First Law

An object remains at rest if there is no unbalanced force acting on it

Equations of Equilibrium (Concurrent)

ΣFx = 0

ΣFy = 0

Equations of Equilibrium (Non-concurrent)

ΣFx = 0

ΣFy = 0

ΣM_{Any Point} = 0

Required Items of a Free Body Diagram

1. Body detached from its surroundings

2. Coordinate axis system

3. Reactions & applied forces

4. Dimensions and member labels

Support Conditions

1. Roller (positive-y)

2. Pin (positive-y and positive-x)

3. Fixed (positive-x, positive-y, and positive-z)

4. Cable (tension)

5. Frictionless Slot (positive-y)

Fr (in terms of a resulting force in centroid problems)

Equivalent concentrated force/area under the load curve

Centroid

Geometric center of a line, area, or volume

Conditions of a centroid

1. Same as center of mass/gravity for homogenous material

2. Not necessarily on the shape

3. Always located on the axis/axes of symmetry

Fluid Pressure

p = ρgh *valid only for incompressible fluids

Two-Force Members

Member with forces applied only at its ends

Characteristics of Two-Force Members

1. Neglect self-weight

2. No applied moments or couples

3. Frictionless pins at the ends

4. Positive = tension; Negative = compression

5. The forces are

   1. Equal

   2. Opposite

   3. Collinear (same LOA)

Truss

structure composed of two-force members joined at their ends by frictionless pins (basically interlocking triangles)

Where and Why Trusses?

Where: bridges, roofs, cranes

Why: efficient, long-span, beautiful

Assumptions of trusses

1. Members connected at the ends only

2. Loads are applied only to truss joints

3. Self-weight neglected or placed at joints

Truss cutting procedure

1. ID members of interest

2. Solve external forces, if necessary

3. ID zero-force members

4. Cut truss - joint/section method

5. Solve for internal forces

Truss cutting rules

1. Cut members of interest

2. Don’t cut through joints

3. Cut through entire truss

4. # unknowns <= # equations of equilibrium

5. Assume tension

Zero-Force Member

A member that carries no load

Zero-force members Rules of Thumb (2 Member Joints)

Both members are zero-force members if:

1. Two non-collinear members form a joint

2. No external loads or reactions at joint

3. Members do not have to form a right angle

Zero-force members Rules of Thumb (3 Member Joints)

The non-collinear member is the zero-force member if:

1. Two collinear and one non-collinear member form a joint

2. No external loads or reactions at joint

3. Members do not have to form a right angle

Frame

A structure with at least one multi-force member

Multi-force Member

A member with either

• 3 or more forces acting on it, not just at its ends

• 2 or more forces and one or more couples acting on it

Frame & Machine procedure

1. Solve for external reactions, if necessary

2. Pull the pins

3. ID two-force members

4. ID multi-force members

5. Apply Newton’s 3rd Law

6. Include MDP+U & on which member with results

Beams

Members with loads applied perpendicular to the longitudinal axis

Statically Determinate Beam Classifications

• Simply supported

• Cantilever

• Combination

Statically Indeterminate Beam Classifications (# unk > # eq)

• Continuous

• Propped Cantilever

• Fixed-fixed

Shear & Moment Diagrams

Graphs which plot internal shear and moment along the length of the member

• Plot position on x-axis and plot shear/moment on y-axis

dM/dx = V

Slope of moment curve = magnitude of shear curve at a point

dV/dx = w(x)

Slope of shear curve = magnitude of load curve at a point

M₂-M₁ = ΔM = ∫Vdx

Change in moment between two points = area under shear curve between two points

V₂-V₁ = ΔV = ∫w(x)dx

Change in shear between two points = area under load curve between two points

V&M Diagram Tips

1. Always start and end at zero

2. Concentrated loads cause jumps in shear diagram (up is +)

3. Concentrated moments cause jumps in moment diagram (c-clockwise is +); causes no change to V-diagram

4. At V=0, moment is a local minimum or maximum

5. Always work V and M diagrams from left to right

Areas

• Trapezoid = ½ (a + b) x h

• Parabolic = 2/3 (a x b)

• Triangle = ½ (b x h)

• Sub-parabolic = 1/3 (a x b)

Uses of cables

1. Suspension bridges

2. Power transmission lines

3. Tension only members

4. Strand in prestressed tension

Benefits of cables

• Flexible

• Light

• Efficient

Cross-section of a cable

Wires around a central core

Typical Configurations of Cables

• Parabolic

• Catenary

• Discrete

W (cable)

Uniformly distributed load (in weight/length)

T_o (cable)

Tension force in cable where θ = 0 (base of sag) —> horizontal component

y (cable)

Vertical distance from base of sag to desired height

Catenary Cable

• Example: Power lines

• At lower sag values, a parabolic approximation is good

• Solved with computer analysis

Discrete Cable

• Neglect self-weight

• Apply equations of equilibrium

• Solve like a truss

F (in terms of 3D Force)

Force vector

F (in terms of 3D Force)

Magnitude of the force

l, m, and n (in li+mj+nk)

Portion of direction

n (in terms of 3D Force)

unit vector = (li + mj + nk)

where l² + m² + n² = 1

Right-hand rule (3D Forces)

• Arrows indicate (+) direction

• Using the right hand, run fingers in (+) x-direction, wrapping them around to the y-direction, and thumb is z-direction

Moment (in 3D)

M_o = r x F

Cross Product

Notes about 3-Dimensional Equilibrium

• ΣF = 0 (Fx, Fy, Fz = 0) and ΣM = 0 (Mx, My, Mz = 0)

• The equilibrium conditions are independent of one another

• All conditions must be met for complete equilibrium

• Right-handed coordinate system with vector notation

• Complete free-body diagram is very important

Complete Fixity

Body is stable for translation and rotation (ΣF = 0, ΣM = 0)

Incomplete Fixity

Body is not stable in all directions for translation and rotation

Excessive Fixity

Body has redundant reactions

Notes About Statical Determinancy

• If > 6 reactions, indeterminate

• If < 6 reactions, determinate but unstable

• If = 6 reactions, determinate and stable

Types of Friction

• Dry friction (coulomb friction): occurs between two unlubricated solid surfaces

• Fluid friction: occurs between adjacent layers in a fluid or gas

• Internal friction: occurs in all solid materials subjected to cyclic loading

Impending Motion

about to do something; either slip or tip

μ_s and μ_k

coefficient of statics/kinetic friction

Belt friction

Belts/cables/ropes wrapped around sheaves, drums, or pulleys

How Belt Friction Works (Types of Cores)

1. Fixed core: provides frictional resistance to the belt passing around it

2. Rotating core: offsets the greater tension force, T₂

Examples of Pulleys

• cranes

• elevators

• exercise equipment

• sailboats

• climbing gear

• fishing reel

x:1 Mechanical Advantage

• x rope(s) “connected” to the weight

• must cut x number of ropes for weight to fall (ignore the tension-part rope)

Levers

F•d₁ = W•d₂

Inclined Plane

F = Wsinθ

• Raise weight up with little effort

• Assuming low friction (use hand cart)

Rollers

• use levers to lift the object to be moved

• place rollers (ex. pipes) under the object

• move rollers from back to front as object moves