Statics

1. Total degrees in a triangle:2. Three angles of the triangle below:3. Three sides of the triangle below:4. Pythagorean Theorem:

x2 + y2 = r2

180

A

B

C

x, y, and r

y

x

r

HYPOTENUSE

A, B, and C

Trigonometric functions are ratios of the lengths of the segments that make up angles.

Q

y

x

r

sin Q = =opp. y hyp. r

cos Q = =adj. x hyp. r

tan Q = =opp. y adj. x

sin A = opposite

hypotenuse

cos A = adjacent

hypotenuse

tan A = opposite adjacentsin A = 1

2

cos A =

tan A =

√3 2

12

3A

B

C

1 √3

For <A below, calculate Sine, Cosine, and Tangent:

ac

A

B

Cb

Law of Cosines:c2 = a2 + b2 – 2ab cos C

Law of Sines:sin A sin B sin C a b c

= =

Statics

Using 2 index cards and a piece of tape:Create the tallest structure you can.

Scoring:1 pt for each cm higher than 51.5 pts for each 5 cm2 of material (cards and tape)

saved

Structures must not break, twist, deform, collapse

1. Static structure

2. Static structure with some moving parts

3. Movable structure

4. Moving structure

What is a structure?• A body that can resist applied forces without

changing shape or size (apart from elastic deformations)

What’s its purpose?• Transmit forces from one place to another• Provide shelter• Art

Types of Structures

• Mass

• Framed

• Shells

Types of Load

• Concentrated

• Distributed

What is Mechanics?

• “Branch of science concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment.”

Statics

What is Statics?Branch of Mechanics that deals with objects/materials that are stationary or in uniform motion. Forces are balanced.

Examples:1. A book lying on a table (statics)2. Water being held behind a dam (hydrostatics)

Chicago

Kentucky & Indiana Bridge

Dynamics

Dynamics is the branch of Mechanics that deals with objects/materials that are accelerating due to an imbalance of forces.

Examples:1. A rollercoaster executing a loop (dynamics)2. Flow of water from a hose (hydrodynamics)

1. Scalar – a variable whose value is expressed only as a magnitude or quantityHeight, pressure, speed, density, etc.

2. Vector – a variable whose value is expressed both as a magnitude and directionDisplacement, force, velocity, momentum, etc.

3. Tensor – a variable whose values are collections of vectors, such as stress on a material, the curvature of space-time (General Theory of Relativity), gyroscopic motion, etc.

Properties of Vectors1. Magnitude

Length implies magnitude of vector2. Direction

Arrow implies direction of vector3. Act along the line of their direction4. No fixed origin

Can be located anywhere in space

Magnitude, Direction

Vectors - Description

45o40 lb

s

F = 40 lbs 45o

F = 40 lbs @ 45o

magnitude direction

Hat signifies vector quantity

Bold type and an underline F also identify vectors

Free-body diagrams

• #1 on Statics problem sheet

1. We can multiply any vector by a whole number.2. Original direction is maintained, new magnitude.

Vectors – Scalar Multiplication

2

½

1. We can add two or more vectors together. 2. 2 methods:

1. Graphical Addition/subtraction – redraw vectors head-to-tail, then draw the resultant vector. (head-to-tail order does not matter)

Vectors – Addition

Vectors – Rectangular Components

y

x

F

Fx

Fy

1. It is often useful to break a vector into horizontal and vertical components (rectangular components).

2. Consider the Force vector below. 3. Plot this vector on x-y axis.4. Project the vector onto x and y axes.

Vectors – Rectangular Components

y

x

F

Fx

Fy

This means:

vector F = vector Fx + vector Fy

Remember the addition of vectors:

Vectors – Rectangular Components

y

x

F

Fx

Fy

Fx = Fx i

Vector Fx = Magnitude Fx times vector i

Vector Fy = Magnitude Fy times vector j

Fy = Fy j

F = Fx i + Fy j

i denotes vector in x direction

j denotes vector in y direction

Unit vector

Vectors – Rectangular Components

y

x

F

Fx

Fy

Each grid space represents 1 lb force.

What is Fx?

Fx = (4 lbs)i

What is Fy?

Fy = (3 lbs)j

What is F?

F = (4 lbs)i + (3 lbs)j

Vectors – Rectangular Components

If vector

V = a i + b j + c k

then the magnitude of vector V

|V| =

Vectors – Rectangular Components

F

Fx

Fy

cos Q = Fx / F

Fx = F cos Qi

sin Q = Fy / F

Fy = F sin Qj

What is the relationship between Q, sin Q, and cos Q?

Q

Vectors – Rectangular Components

y

x

F Fx +

Fy +

When are Fx and Fy Positive/Negative?

FFx -

Fy +

FFFx -Fy -

Fx +Fy -

Vectors – Rectangular Components

III

III IV

1. Vectors can be completely represented in two ways:1. Graphically2. Sum of vectors in any three independent directions

2. Vectors can also be added/subtracted in either of those ways:1.

2. F1 = ai + bj + ck; F2 = si + tj + uk

F1 + F2 = (a + s)i + (b + t)j + (c + u)k

Vectors

Use the law of sines or the law of cosines to find R.

Vectors

F1 F2

R45o

105o

30o

Brief note about subtraction1. If F = ai + bj + ck, then – F = – ai – bj – ck

2. Also, if

F =

Then,

– F =

Vectors

Resultant Forces

Resultant forces are the overall combination of all forces acting on a body.

1) find sum of forces in x-direction

2) find sum of forces in y-direction

3) find sum of forces in z-direction

3) Write as single vector in rectangular components

R = SFxi + SFyj + SFzk

Resultant forces

• #2 and #3 Statics Problem Sheet

Resultant Forces Only!

Statics Newton’s 3 Laws of Motion:

1. A body at rest will stay at rest, a body in motion will stay in motion, unless acted upon by an external force

This is the condition for static equilibrium

In other words…the net force acting upon a body is Zero

Now on to the point…

Newton’s 3 Laws of Motion:

2. Force is proportional to mass times acceleration:F = ma

If in static equilibrium, the net force acting upon a body is Zero

What does this tell us about the acceleration of the body?It is Zero

Newton’s 3 Laws of Motion:

3. Action/Reaction

Statics Two conditions for static equilibrium:

1.

Individually.

Since Force is a vector, this implies

Two conditions for static equilibrium:1.

Resultant and Reaction forces

• #2 and #3 Statics Problem Sheet

Reaction Forces Only!

Two conditions for static equilibrium:

Why isn’t sufficient?

Two conditions for static equilibrium:

2. About any point on an object,

Moment M (or torque t) is a scalar quantity that describes the amount of “twist” at a point.

M = (magnitude of force perpendicular to moment arm) * (length of moment arm) = (magnitude of force) * (perpendicular distance from point to force)

Two conditions for static equilibrium:

MP = F * x MP = Fy * x

M = (magnitude of force perpendicular to moment arm) * (length of moment arm) = (magnitude of force) * (perpendicular distance from point to force)

PF

x

P

F

x

Moment Examples:

1. An “L” lever is pinned at the center P and holds load F at the end of its shorter leg. What force is required at Q to hold the load? What is the force on the pin at P holding the lever?

2. Tension test apparatus – added load of lever?

Let’s Return to Saving Pablo…

Trusses

Trusses: A practical and economic solution to many structural engineering challenges

Simple truss – consists of tension and compression members held together by hinge or pin joints

Rigid truss – will not collapse

Trusses Joints:Pin or Hinge (fixed)

Trusses Supports:Pin or Hinge (fixed) – 2 unknowns

Reaction in x-direction

Reaction in y-direction Rax

Ray

Trusses Supports:Roller - 1 unknown

Reaction in y-direction only

Ray

Assumptions to analyze simple truss:

1. Joints are assumed to be frictionless, so forces can only be transmitted in the direction of the members.

2. Members are assumed to be massless. 3. Loads can be applied only at joints (or nodes). 4. Members are assumed to be perfectly rigid.

2 conditions for static equilibrium:5. Sum of forces at each joint (or node) = 06. Moment about any joint (or node) = 0

Start with Entire Truss Equilibrium Equations

Method of Joints

• Problem #4 from Statics Problem Sheet

Method of Joints

Method of Joints

Method of Joints

Method of Joints

Truss Analysis Example Problems:Using the method of joints, determine the force in each member of the truss shown and identify whether each is in compression or tension.

Static determinacy and stability:

Statically Determinant: All unknown reactions and forces in members can be determined by the methods of statics – all equilibrium equations can be satisfied.

Static Stability:The truss is rigid – it will not collapse.

Conditions of static determinacy and stability of trusses:

Homework: Method of Joints