Statics


By Prof. Seungchul Lee
http://iai.postech.ac.kr/
Industrial AI Lab at POSTECH

Table of Contents

1. Statics and Free choice of axisĀ¶

Learning goals

  • Solve problems with objects under static equilibrium.
  • Learn how to choose the axis location when doing statics problems

Equilibrium Conditions

When we say that an object is in static equilibrium, we mean that the object does not rotate or translate. A great fraction of Mechanical and Civil engineering is (very well-developed) statics since many machines and structures have only a few parts designed to move.

Equilibrium Conditions for a Point Particle

We can see from Newton's Second Law that a point particle has to experience a zero net force to be in static equilibrium:

$\sum \vec{F}= 0$

Equilibrium Conditions for a Rigid Body

When considering rigid bodies, the body must also experience zero net torque to be in static equilibrium.

In this course, we are only considering rotation in the $x$-$y$ plane, i.e. only one axis of rotation, the $z$ axis. We can express the conditions for static equilibrium as simply:

$\sum \vec{F}= 0$ (for both $x$ and $y$ forces)

$\sum \vec \tau_z = 0$ about any axis anywhere in the plane &

If an object is experiencing no rotation or translation, all points are fixed, and we can choose to label the axis of rotation $z$ anywhere on the object. We customarily choose a location that will make calculating torques easier. Often this is a location that will result in zero moment arms for one or more forces, especially ones that are unknown in the problem.

In statics problems, choose the axis of rotation so as to make the torque calculation as simple as possible.

Equations of Equilibrium

$$ \begin{align*} \sum F_x &= 0\\ \sum F_y &= 0\\ \sum M_o &= 0 \end{align*}$$
  • Equivalent to solving the system of linear equations

Moment of a CoupleĀ¶

$$ M = Fd \quad \quad M = r \times F, \quad \text{but }\; F_{net} = 0$$



Free-Body DiagramsĀ¶





ExerciseĀ¶

  • Determine the moment of the force about point $O$

    • Magnitude

    • Direction

  • Static?


  • Describe all the static forces and moments

    • Reaction forces $F_a$ and $F_b$

    • Which one is bigger intuitively?

    • Think about extreme case


ExerciseĀ¶

The ball $D$ has a mass of 20 kg. If a force of is applied horizontally to the ring at A, determine the dimension $d$ so that the force in cable AC is zero.



1) Draw free body diagram (FBD)

2) Write down equilibrium equation

3) Determine the dimension $d$ so that the force in cable AC is zero

ExerciseĀ¶

Draw the free-body diagram of the uniform beam. The beam has a mass of 100 kg.


ExerciseĀ¶

  • Determine the horizontal and vertical components of reaction on the member at the pin A, and the normal reaction at the roller B.


2. StructureĀ¶

Tension and CompressionĀ¶

  • Sign of force vector on bar element

  • Sign of force vector on pin ?


Method of JointsĀ¶

  • Determine the force in each member of the truss and indicate whether the members are in tension or compression.


SolutionĀ¶

Method of SectionĀ¶

  • Determine the force in members GE, GC, and BC of the truss. Indicate whether the members are in tension or compression


InĀ [2]:
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