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Equilibrium

Equilibrium. Of a Rigid Body. Objectives. To develop the equations of equilibrium for a rigid body. To introduce the concept of the free-body diagram for a rigid body. To show how to solve rigid body equilibrium problems using the equations of equilibrium. Part A.

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Equilibrium

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  1. Equilibrium Of a Rigid Body

  2. Objectives • To develop the equations of equilibrium for a rigid body. • To introduce the concept of the free-body diagram for a rigid body. • To show how to solve rigid body equilibrium problems using the equations of equilibrium.

  3. Part A

  4. Conditions forRigid-Body Equilibrium

  5. 2D Supports

  6. Support Reactions General Rule: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.

  7. Rocker

  8. Smooth Surface

  9. Pinned or Hinged

  10. Fixed

  11. Modeling

  12. Modeling

  13. Procedure for Drawing a Free-Body Diagram • Select co-ordinate axes. • Draw outlined shape isolated or cut “free” from its constraints and connections. • Show all forces and moments acting on the body. Include applied loadings and reactions. • Identify each loading and give dimensions. Label forces and moments with proper magnitudes and directions. Label unknowns.

  14. Free Body Diagrams

  15. Important Points • No equilibrium problem should be solved without first drawing the appropriate F.B.D. • If a support prevents translation in a direction, then it exerts a force on the body in that direction. • If a support prevents rotation of the body then it exerts a moment on the body.

  16. Important Points • Couple moments are free vectors and can be placed anywhere on the body. • Forces can be placed anywhere along their line of action. They are sliding vectors.

  17. Part B

  18. 2D Equilibrium Scalar Equations

  19. Procedure for Analysis • Free-Body Diagrams • Equations of Equilibrium

  20. Equations of Equilibrium • Apply the moment equilibrium equation,  MO= 0. Take the point O to be the intersection of the lines of action of two unknown forces. This allows the direct solution for the third force. • Orient the x and y axes along lines that will provide the simplest resolution of the forces into their x and y components.

  21. Direction of Forces If results are a negative scalar for the magnitude the force acts in the opposite sense that you selected on the Free-Body Diagram.

  22. Example 1 Determine the reactions at the supports.

  23. Example 2 Determine the reactions at the supports.

  24. Example 3 Determine the reactions at the supports.

  25. P 27kN 27kN 1.8m 0.6m 0.9m 0.6m Example 4 Three loads are applied to a beam as shown. Determine the reactions at A and B when P = 70kN.

  26. 300 N 100 N 200 N d 900 mm 900 mm Example 5 Three loads are applied to a beam as shown. Determine the reactions at A and B. 300 mm

  27. Example 6 Distributed load applied to AB beam. Determine the support reactions. 250 kn/m 6 m 6 m 4 m

  28. Example 7 A beam supports a distributed load as shown. Determine the equivalent concentrated load and the reactions at the supports

  29. Example 8 Determine the horizontal and vertical components of reaction at the pin A and the tension developed in cable BC used to support the steel frame.

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