Simplification of a Force Group

Abstract

In this chapter, we will learn how to simplify a force group. First, the concept of planar couple is given, and then the theorem of parallel translation of a force is introduced. Based on these theories, a complex force system can be simplified, and can also be further simplified.

Exercises

• 4.1 As shown in Fig. 4.14, there is a triangle ABC with a right angle, and the angle A = 30°. The length of BC equals a. There are four forces applied at the three points, with the same magnitude P. Please give the final result of the simplification on this force group.

  Fig. 4.14

  

  A triangle force group with concentrated forces

• 4.2 As shown in Fig. 4.15, there is a triangle ABC with a right angle, and the angle A = 30°. The length of BC equals a. There are two forces applied at the point A and B, respectively. The magnitude for these two forces is P = qa, and there is a uniformly distributed force q applied at AC side. Please give the final result of the simplification on this force group.

  Fig. 4.15

  

  A triangle force group with one distributed force and two concentrated forces

Answers

• 4.1 Select point A as the simplification center, and one has:

  $F_{x} = \frac{1}{2}P - \frac{\sqrt 3 }{2}P,\,F_{y} = \frac{\sqrt 3 }{2}P - \frac{1}{2}P,\,M_{A} = - \left( {2 + \sqrt 3 } \right)Pa$

• 4.2 Select point A as the simplification center, and one has:

  $F_{x} = \frac{3}{2}qa,\,F_{y} = - \frac{\sqrt 3 }{2}qa,\,M_{A} = - \frac{7}{2}qa.$