Statics (MET 2214) Prof. Simin Nasseri Equilibrium of a Rigid Body Examples

Statics (MET 2214)Prof. Simin Nasseri

Equilibrium of a Rigid BodyEquilibrium of a Rigid BodyExamplesExamples


Example ProblemExample ProblemDraw the free-body diagram of the plate.


Example ProblemExample Problem

Rx3

Ry3

Rx2

Ry2

Rx1

Ry1

Does this free-body diagram help?



Rx

Ry

You have to consider one reaction force from the gusset plate. The components of this force are Rx and Ry.


Example ProblemExample ProblemDraw the free-body diagram of the weight.

AB


Example ProblemExample ProblemDetermine the force in each supporting wire.

TATB

W=50lb

TA and TB are the forces inside the cables and equal to reaction forces at A and B.


Example Problem


Example Problem

Spanner wrench

F



Set up the following problem: Length A= 5 ft, and length B =10 ft and angle α= 30o. Determine the angle β of the smooth incline in order to maintain equilibrium.



Use line-of-action to solve the problem




Class ProblemClass Problem

Set up the problem and how would you solve for the resisting force at pin A to maintain equilibrium.



Set up the problem and how would you solve for the resisting force at pin A to maintain equilibrium.




Example - EquilibriumExample - Equilibrium

The tension force in AC is 28 kN, determine the the required values of tension in AB and AD so that the resultant force of the three forces applied at A is vertical. Determine the resultant force.


FBD - ExampleFBD - Example

Free-body diagram:

(Anything missing?)

TABTAD

TAC

RA

Add the dimensions!


Example-EquilibriumExample-Equilibrium

The vector AB is

x

y

z

16 m

48 m

12 m

d

d

d

2 2 2

16 48 12

16 48 12 52

AB i j k m

AB m

::::::::::::::



The unit vector in the AB direction is

The tension vector TAB is

AB

16 48 12 ˆ

52 52 52

ABu i j k m

AB

::::::::::::::

AB AB AB

AB AB AB

ˆ

16 48 12

52 52 52

T T u

T i T j T k

::::::::::::::



The vector AD is

x

y

z

14

48

0

d m

d m

d m

2 2 2

14 48 0

14 48 0 50

AD i j k m

AD m

::::::::::::::



The unit vector in the AD direction is

The tension vector TAD is

AD

14 48 0ˆ

50 50 50

ADu i j k m

AD

::::::::::::::

AD AD AD

AD AD AD

ˆ

14 48 0

50 50 50

T T u

T i T j T k

::::::::::::::



The vector AC is

x

y

z

16

48

24

d m

d m

d m

2 2 2

16 48 24

16 48 24 56 m

AC i j k

AC

::::::::::::::



The unit vector in the AC direction is

The tension vector TAC is

AC

16 48 24ˆ

56 56 56

ACu i j k m

AC

::::::::::::::

AC AC AC

AC AC AC

ˆ

16 48 24

56 56 56

T T u

T i T j T k

::::::::::::::



The tension vector TAC= 28 kN is

The reaction at A

AC

16 48 2428 kN 28 kN 28 kN

56 56 56

8 kN 24 kN 12 kN

T i j k

i j k

::::::::::::::

A A R R j::::::::::::::



Apply the equilibrium condition.

Combine the vectors

AB AD AC A0 0F T T T R ::::::::::::::::::::::::::::::::::::::::::::::::::::::::

AB AD

AB AD A

AB

0

16 148 kN

52 50

48 4824 kN

52 50

1212 kN

52

F

T T i

T T R j

T k


Example-EquilibriumExample-EquilibriumBreak the components of the forces:

Use (3) to solve for TAB

AB ADx

AB AD Ay

ABz

16 140 8 kN 1

52 5048 48

0 24 kN 252 50

120 12 kN 3

52

F T T

F T T R

F T

AB

5212 kN 52 kN

12T



Plug into (1), solve for magnitude of TAD

AD AB

AD

AD

14 168 kN

50 5214 16

52 kN 8 kN 24 kN50 52

85.714 kN

T T

T

T



Use the values to solve for RA

A AB AD

48 4824 kN

52 5048 48

52 kN 85.714 kN 24 kN 52 50154.286 kN

R T T

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Statics (MET 2214) Prof. Simin Nasseri Equilibrium of a Rigid Body Examples