Statics - Johns Hopkins University · PDF fileStatics? Branch of Mechanics that deals with ......

Statics

• 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

“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.”

What is Mechanics?

Wikipedia.org/wiki/Isaac_Newton

Chicago

Kentucky & Indiana Bridge

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)

Statics

Using 2 index cards and a piece of tape:

Create the tallest structure you can. Scoring:

1 pt for each cm higher than 5

1.5 pts for each 5 cm2 of material (cards and tape) saved

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)

Construction is about static

equilibrium (statics)

Statics = no motion (almost).

All forces equal zero.

All torques equal zero.

• A force is a push or pull on an object.

What is a force?

• Gravity is pulling you down.

• The chair is pushing you up.

• Total forces are zero.

kg

N

lb

kglbF 8.9

2.2150

NF 668

How much force does the chair exert?

Are there forces

on you now?

Types of Load

• Concentrated

• Distributed

Let’s pull on a rope.

Sometimes the forces are

not just up and down.

How much tension is in each rope?NNF 3342668

What forces are

on each rope?

Forces are often

at an angle.

In equilibrium, net forces must be zero both

• Right & left

• Up & down

Vectors have magnitude and direction.

Trigonometry Review1. 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

A

B

C

y

x

r

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

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

sin A = opposite

hypotenuse

cos A = adjacent

hypotenuse

tan A = opposite adjacent

sin A = 1 2

cos A =

tan A =

√3 2

12

3

B

C

1 √3

1. Scalar – a variable whose value is expressed only as a magnitude or quantity

Height, pressure, speed, density, etc.

2. Vector – a variable whose value is expressed both as a magnitude and direction

Displacement, 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.

Understanding Forces

1. Magnitude

Length implies magnitude of vector

2. Direction

Arrow implies direction of vector

3. Act along the line of their direction

4. No fixed origin

Can be located anywhere in space

Properties of Vectors

Magnitude, Direction

Vectors - Description

45o

F = 40 lbs 45o

F = 40 lbs @ 45o

magnitude direction

Hat signifies vector quantity

Bold type and an underline F also identify vectors

1. We can multiply any vector by a whole number.

2. Original direction is maintained, new magnitude.

Vectors – Scalar Multiplication

2

½

We can add two or more vectors together - 2 methods:

1. Resolve into rectangular components then add

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

Vectors – Addition

y

xFx

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

xFx

Fy

This means:

vector F = vector Fx + vector Fy

Remember the addition of vectors:


y

xFx

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


y

xFx

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


If vector

V = a i + b j + c k

then the magnitude of vector V

|V| =


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


y

x

Fx +

Fy +

When are Fx and Fy Positive/Negative?

Fx -

Fy +

Fx -Fy -

Fx +Fy -


III

III IV

1. Vectors can be completely represented in two ways:

1. Graphically

2. 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 subtraction

1. 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

Find horizontal and vertical forces

• Simple triangle shows horizontal and

vertical parts.

If the angle at the top is

40o, what are the forces

(A & B)?

NNB

N

B

12.034.09.31

9.3120sin

60N

F2F1

40o

Half of the upward force comes

from each member.

NN

A

A

N

9.3194.0

30

3020cos

Look at point F1 for horizontal

member.

A

B

Practice Problem #1

The traction device is applied to a broken leg as

shown. What weight is needed if the traction force

pulling the leg straight out (right) is 165 N? (The

tension in the rope equals the weight.)

Practice Problem #2

Students want to hang a 1200 N cannon from ropes on

the football goalpost as shown. If the goalposts are 5

meters apart and the ropes are 3 meters long, would a

rope which breaks at 1000 N be good enough?

Practice Problem #3

A stop light is held by two cables as shown. If the stop

light weighs 120 N, what are the tensions in the two

cables?

Sometimes forces don't

move things, they rotate

them. How do you open a door?

Where is the best place to push on it?

Torque causes rotation.

Torque requires a force and a lever arm.

Give examples of

things you rotate?

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

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


1.


Why isn’t sufficient?


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)


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)

P

F

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?

Objects are in static

equilibrium if:

No net forces

No net torques

(moments)

Place a meter stick across

two scales. Put a 200 g

weight on it.

200 g

scale

200 g

scale

100 g 100g

150 g 50g

No net forces

Equate forces up & down.

Equate forces right and left.

No net torques

Pick any point.

Equate CW and CCW torques.

What should the

scales read?

1. Equate up and down forces.

2. Equate right and left forces.

3. Pick a pivot point.

4. Equate CW and CCW torques.

gFF 20021

x

cmFcmg 10030200 2

What should the scales

read?

1. Equate up and down forces.

2. Equate right and left forces.

3. Pick a pivot point.

4. Equate CW and CCW torques.

gggFF 50020030021

cmFcmgcmg 1006030030200 2

x

gFgF 260and240 12

Practice Problem #1

Consider a 12 m beam supported at each end by

two forces F1 and F2. A 10 N force is held by the

beam 4 m from the left side as shown. Find the

support forces F1 and F2.

Practice Problem #2

We have a board with a weight of 8 N that is

20 m long. A 5 N force is held 8 m from the left

side. Find the forces F1 and F2 that are

located at the ends of the board.

Practice Problem #3

Find F1 and F2. Find the forces on each

side of the triangle labels a, b and c.

Indicate whether these are tension or

compression forces.

Statics in bridges.

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

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:1. Sum of forces at each joint (or node) = 02. Moment about any joint (or node) = 0

Assumptions to analyze simple truss:

Consider this popsicle stick

triangle

What parts are under compression? Tension?

fishing line

Examine triangle

members.

Look at any point on bridge.

Forces = 0 and torques = 0.

Determine which members are under

tension (like a string)?

compression (like a rod)?

compression

tensioncompression compression

load

load

Trusses Joints:

Pin or Hinge (fixed)

TrussesSupports:

Pin or Hinge (fixed) – 2 unknowns

Reaction in x-direction

Reaction in y-direction

RAx

RAyRoller - 1 unknown

RDy

Reaction only in y-direction

Method of Joints Problem:

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.

Now examine truss

members.

compression

tension

compression tension

compression

Use symmetry

to examine

members. Which members are

under

tension (like a string)?

compression (like a rod)?

compression

tension

compressiontension

compression

Load

Conditions of static determinacy and stability of trusses:

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.

Use Bridge Designer

http://engineering.jhu.edu/ei/about-ei/course-materials/

Statics, Structures, and Bridge Project Section

Virtual Lab Bridge Designer

Calculate the forces on a triangle.

http://engineering.jhu.edu/ei/about-ei/course-materials/

Homework

• Use Bridge Designer for an initial design of a spaghetti bridge.

– 50 cm span (must be > 50 cm)

– <25 cm high

– Weight hung from bottom, center of bridge.

• For your design, determine the weakest and strongest members.

– Use knowledge of spaghetti properties from lab tests.

Statics - Johns Hopkins University · PDF fileStatics? Branch of Mechanics that deals with ......

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