ME13A: CHAPTER FOUR

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ME13A CHAPTER FOUR

• EQUILIBRIUM OF RIGID BODIES

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4.1 INTRODUCTION

• Equilibrium of a rigid body requires both a
  balance of forces, to prevent the body from
  translating with accelerated motion, and a
  balance of moments, to prevent the body from
  rotating.
• For equilibrium ?F 0 and
• ?Mo ?(r x F) 0
• In three dimensions ?Fx 0 ?Fy 0
  ?Fz 0
• ?Mx 0 ?My 0 ?Mz 0

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4.2 SOLUTION OF REAL PROBLEMS IN TWO DIMENSIONS

• Step 1 Examine the space diagram and select
  the significant body - the free body diagram.
  Detach this body from the space diagram and
  sketch its contours
• Step 2 Identify and indicate all external
  forces acting on the body. The external forces
  consist of the externally applied forces, the
  weight and reactions of the body at the point of
  contact with other external bodies.
• Step 3 All the important dimensions must be
  included in the free body diagram.

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Types of Reactions Contd.
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Support Reactions

• If a support prevents the translation of a body
  in a given direction, then a force is developed
  on the body in that direction.
• Likewise, if rotation is prevented, a couple
  moment is exerted on the body.

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4.4 STATICALLY DETERMINATE AND INDETERMINATE
REACTIONS. TYPES OF CONSTRAINTS IN TRUSSES
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Example Find the Reactions in the Truss Below
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Difference Between Improper and Partial
Constraints

• In improper constrained case, one equation of
  equilibrium has been discarded but in the partial
  case, there are only two unknowns so the two
  equations were used to determine them.

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Conclusion on Supports
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Example

• A truck mounted crane is used to lift a 3 kN
  compressor. The weights of the boom AB and of
  the truck are as shown, and the angle the boom
  forms with the horizontal is ? 40o. Determine
  the reaction at each of the two (a) rear wheels
  (b) front wheels D

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Solution
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4.5 TWO AND THREE-FORCE MEMBERS

• The solution of some equilibrium problems are
  simplified if one is able to recognise members
  that are subjected to only two or three forces.
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• 4.5.1 Two-Force Members When a member is
  subject to no couple moments and forces are
  applied at only two points on a member, the
  member is called a two-force member.

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Two-Force Members

• Consider a two-force member below in (a). The
  forces in the two points are first resolved to
  get their resultants, FA and FB. For
  translational or force equilibrium

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Two-Force Members Contd.

• (? F 0) to be maintained, FA must be of equal
  magnitude and opposite direction to FB.
• For rotational or moment equilibrium
• (? Mo 0) to be satisfied, FA must be collinear
  with FB.
• See other two force members in (c) below.

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Three Force Members
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Example

• One end of rod AB rests in the corner A and the
  other is attached to cord BD. If the rod
  supports a 200-N load at its midpoint, C, find
  the reaction at A and the tension in the cord.

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4.6 EQUILIBRIUM IN THREE DIMENSIONS

• 4.6.1 Reactions at Supports See Fig. 4.10 of
  text book for possible supports and connections
  for a three dimensional structure.
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• 4.6.2 Equilbrium of a Rigid Body in Three
  Dimensions
• Six scalar equations are useable (see section
  4.1)

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Equilbrium of a Rigid Body in Three Dimensions
Contd.

• Problems will better be solved if the vector form
  of the equilbrium equations is used i.e.
• ?F 0 ?Mo ? (r x F) 0
• Express the forces, F and position vectors, r
  in terms of scalar components and unit vectors.
  Compute all vector products either by direct
  calculation or by means of determinants.

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Solution Concluded

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Example

• A 35-kg rectangular plate shown is supported by
  three wires. Determine the tension in each wire.

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