SOLID AND STRUCTURES1 (ENGG110)/EQUILIBRIUM OF A PARTICLE AT A POINT, Study notes of Engineering

Download SOLID AND STRUCTURES1 (ENGG110)/EQUILIBRIUM OF A PARTICLE AT A POINT and more Study notes Engineering in PDF only on Docsity!

• Co-planar (2-D) equilibrium of forces
• FBDs
• Concept quiz
• Support reactions in 2-D
• Quick summary

TYPICAL APPLICATION

In crane lifting operations, the

forces in the wires and chains

that hold up the load have to

be calculated to determine if

they are strong enough and in

the right position to stabilize

the load. In this application the

forces converge at a point –

the hook/pulley wheel.

How do we find the forces in

these members and at the supports?

Hook

COPLANAR FORCE SYSTEMS

This is an example of a 2-

D or coplanar force

system.

If the whole assembly is in

equilibrium , then particle

B is also in equilibrium.

To determine the

tensions in the cables for

a given weight of crate,

you need

to draw a free-body

Roller diagram and apply the

support equations of equilibrium

(E-of-E).

Conditions of equilibrium (in 2

dimensions)

Applies to

equilibrium of

a particle

y(+)

x(+)

Sum of all acting forces in x-direction must

be zero i.e. F

x

Sum of all acting forces in y-direction must

be zero i.e. F

y

Moment of force F

about point O

THE WHAT, WHY, AND HOW OF

THE FREE-BODY DIAGRAM (FBD)

Free-body diagrams is one of the most important

things for you to know how to draw and use for

statics and other subjects like Design!

O

F

What? – It’s a drawing (sketch) that shows all external

forces acting on the particle or body.

Why? – It’s key to being able to write the equations of

equilibrium—which are used to solve for the

unknowns (usually forces or angles or both).

How?

1. Imagine the particle to be isolated or cut free from its surroundings.
2. Show all the forces that act on the particle.
3. Identify each force and show all known magnitudes and directions.

Show all unknown magnitudes and/or directions as variables.

In general, for a particle in equilibrium,

F

x

= 0 and F y

These two equations of equilibrium can be used to

solve for up to two unknowns.

EQUATIONS OF 2-D EQUILIBRIUM

Write the equations of equilibrium:

+ F

y

= 0: F

BA

sin 30º – 392.4 N = 0

Solving the second equation gives directly: F BA

= 785 N

From the first equation, we then get: F BC

= 680 N

A

W =392.4 N

F

BA

F

BC

30

FBD at B

B

y

x

+ F

x

= 0: F

BA

cos 30º

F

BC

F

BA

F

BA

cos 30º

Respond at PollEv.com/grahamschley

CONCEPT QUIZ – which of these problems

can be solved using E-of-E?

F

T=

kN

A) 1 only B) 3 only D) All E) None C) 1 and 3

T=5 kN

F

x

y

60

F

1

F

2

60

5 kN

7 kN

30

W=10 kN

30 35

Note: B has 3 unknowns, F 1

, F 2

and angle. A and C have just 2

unknowns.

SUPPORT REACTIONS IN 2-D

edge

As a general rule, if a support prevents translation of a

body in a given direction, then a force is developed on the

body in the opposite direction.

Similarly, if rotation is prevented, a moment is exerted on

the body in the opposite direction.

Roller or smooth Smooth pin or hinge Fixed support

QUICK SUMMARY

• Equilibrium of a particle in 2-D is simplified to solving

the equations:

F

x

0 F

y

• The key to writing down the E-of-E is first to draw a

correct FBD of the object (particle or body) with all the

forces acting on it

• If there are more unknowns than independent E-of-E,

the problem can’t be solved by statics alone

Supporting reading/reference

Clifford – Intro to ME:

Part 1 Unit 1 Solid

Mechanics

Hibbeler – Statics &

Mechanics of

Materials

Blockley – Structural

Engineering: A Very

Short Introduction

p.5 free body diags. – p.7 Ch.4 pp.175- 194

Self-study:

• Try self-study problems (at end)
• Watch video – equilibrium at a point (on Canvas)

[Ans. F = 3.49 kN

and F = 9.45 kN]

1 2

The members of a truss (structural frame) are all

connected through a smooth pin at O. Determine the

magnitudes of F 1

and its angle if F 2

= 6 kN.

[Ans. F = 4.91 kN

and = 15.4 ]

1