Eng G 130

Engineering Mechanics - statics

Section E by Yuxiang Chen

Lecture #5

1

Previously

Newtons laws of motion

2

Particle Equilibrium

The formal definition of equilibrium:

A particle is in equilibrium if

(i) it was originally at rest and remains at

rest, or

(ii) was originally moving in a straight line

with constant velocity and remains moving in

a straight line with constant velocity.

In this course, we are mainly concerned with

static equilibrium, which is the at rest condition

Newtons 2nd Law

Newtons 2nd Law tells us that:

Equilibrium, in vector notation:

The at rest condition requires zero acceleration, so

the net force F on the particle must also be zero.

(i.e. there is no unbalanced force on the particle)

aF m

0 aF

m

Example of Force Equilibrium

Bills wrist?

What force is transmitted

through Bills leg?

Jim

Bill

WBill

WJim

Wbill + WJim

Diagrams as Solutions

One of the most important steps in engineering design or solving

engineering problems is drawing a diagram.

In engineering mechanics, we make

extensive use of FREE BODY DIAGRAMS.

The diagram is an important component of

the solution, and cannot be left until the end.

Free Body Diagrams

Free Body Diagrams

What are they?

A FBD is a sketch of a body, or part of a

body, that has been removed from its

surroundings,

AND,

with the effects (i.e. forces) of the

surroundings shown explicitly on the

sketch

I.e. FBD = a diagram of Forces

Drawing a FBD of a Particle

To understand how to construct a FBD,

lets consider the simple case of a point in space, treating it as a particle:

Steps in Drawing a FBD

1. Isolate the particle (or body) and make a

sketch of it.

Imagine cutting

through the

ropes attached

to the ring.

Steps in Drawing a FBD

2. Draw all the forces which act on the

particle (or body):

- Own weight

- Reactions at supports

- External forces (Loads):

- Objects contacting the body

- Springs, cables, rods, links, etc

Steps in Drawing a FBD

3. Label all the forces:

Known

Unknown

label

magnitude

direction

name it! assign letter(s) to it.

assume a direction

F1

F3F2

= 45o

y

x

Neat, Scaled Diagrams

All FBD should take the form of diagrams and

NOT casual sketches:

- Use a straight edge and a sharp pencil

- Draw approximately to scale

- Use neat labels (print instead of writing)

Your calculations and diagrams will tell a

story make sure it is clear!

Sign Conventions for FBD

Tension (T)

When a load on a body (or part of a body)

attempts to make the body (or part) LONGER,

we say it is under tension.

Compression (C)

When a load on a body (or part of a body)

attempts to make the body (or part) SHORTER,

we say it is under compression.

Sign Conventions for FBD

Unknown Forces

Which way to draw an unknown force?

1. Assume a direction (T or C)

2. Solve the problem

3. If the answer is +ve, the assumed direction in Step 1 was correct!

4. If the answer is -ve, the assumed direction in Step 1 was wrong!

?? ??

Sign Conventions for FBD

Unknown Forces (contd)

A simple approach is always to assume tension.

Then, a positive answer is tension and a

negative answer is compression.

But

Not everyone does this. So,

if your friend showed you

these answers, would you

interpret the forces as

tension or compression?

F = 10 kN (T)

F = -35 kN (C)

Small Deformation Assumption

We just defined tension and compression by

the member becoming LONGER or SHORTER.

However, this change of length is typically very

small compared to the original length.

Thus, we can usually assume the overall

geometry does not change as the member

forces increase.

(There are a few important exceptions that we will soon see)

Support Conditions

We have seen that a FBD can be visualized by

cutting the object of interest out of the structure.

Where external supports are cut, we must

understand the restraint they provide to the

structure.

Each degree of freedom of restraint gives rise

to a support force or reaction.

= == =

Support Idealizations

1. Pin Support

Pins allow rotation, but prevent translation.

2. RollerRollers allow rotation, and allow translation.

Do they prevent uplift?

Support Idealizations

3. Fixed Support

Fixed supports prevent rotation, and prevent translation.

Key: To figure out the forces to include in your FBD, think

about the movement the support prevents!

Ask yourself: If we push on the structure, will it push back? If

so, theres a support reaction involved.F

R?

F

R?

YES NO

Reaction Forces

Arrows indicate forces that the supportapplies onto the attached member.

The arrows replace the support pictures in the FBD

Reaction Forces: Schematic Example

SNOW

FBD??

Cut Line

Forces are effects of snow and supports on the beam!

Special Member Types

1. SpringsAssume spring response is

linear-elastic

where:

Note: Spring elongation can be large, so change in

member geometry must be considered.

xF k

F = spring force

k = spring constant

x = compressed or

stretched distance

Special Member Types

2. PulleysAssume:

- pulleys have no friction

- pulleys have zero mass

Note: May need to consider total cable length over

pulley, but cable length is assumed constant

Special Member Types

3. Cables and Ropes

Assume:

- cables have zero mass

- cables do not stretch

under load

You cant push on a rope. Rope always in tension

Equilibrium in

Co-Planar Force Systems

Requirement for Equilibrium

For equilibrium of a particle (or body), the necessary

and sufficient condition is:

This is a general expression which implies that the

sum of forces in ALL directions must equal zero!

0F

We will initially look at coplanar systems, and

expand to other cases later.

Coplanar Force System

In a coplanar system, all of the forces act in the same plane:

(ex. the x, y plane)

0F

yx FFF

ji

yx FF

Coplanar Equilibrium

Since the x and y vector components are

perpendicular, each component must equal zero to

satisfy the above condition.

0 xF

0 ji

yx FF

Hence, the above vector equilibrium condition is

equivalent to the following two scalar conditions:

0 yF

Solution Process

1. Draw a FBD

0 xF2. Apply the equilibrium equations

0 yF

3. Solve the equations

- Gaussian elimination

Courtesy of Pearson Education

FBD Example

The sphere has a mass of 6kg and is supported as shown. Draw

a free-body diagram of the sphere, the rope CE and the knot at

C.

Solution

FBD of Sphere

Two forces: weight and rope

tension CE.

W = 6kg (9.81m/s2) = 58.9N

Rope CE

Two forces: sphere and knot at C

By Newtons 3rd Law:

FCE is equal but opposite to FEC

FCE and FEC pull the cord in tension

For equilibrium, FCE = FEC

Solution

FBD at Knot

3 forces acting: rope CBA,

rope CE and spring CD

Note: weight of the sphere

does not act directly on the

knot.

Instead, it is transmitted

through rope CE

Example 1: Coplanar Particle Equilibrium

Find the magnitude and direction of F so

that the particle is in equilibrium.

x

y

q

30o

250 N

F

150 N

Example 2: Coplanar Springs

A 30 kg block is supported by 2 springs having the

stiffness shown. Determine the unstretched length of

each spring after the block is removed.