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Equilibrium and Equation of Equilibrium:2D

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  • 1. WELCOME TO MY PRESENTATION
  • 2. AHSANULLAH UNIVERSITY OF SCIENCE AND TECHNOLOGY PRE-STRESSED CONCRETE SESSIONAL
  • 3. PRESENTED BY MUNSHI MD. RASEL ID: 10.01.03.075 SECTION B 4th YEAR AND 2nd SEMESTER DEPARTMENT OF CIVIL ENGINEERING
  • 4. PRESENTATION TOPIC EQUILIBRIUM AND EQUATION OF EQUILIBRIUM:2D
  • 5. Equilibrium A body is said to be in equilibrium if it is at rest or moving with uniform velocity.
  • 6. Newton’s First Law of Motion: If the resultant force on a particle is zero, the particle will remain at rest or will continue at constant speed in a straight line.
  • 7. Factors that affect equilibrium   Area of the base: the bigger the area of the base, the more the stable the object is Weight: the heavier the object is, the more stable it is
  • 8. Types of Equilibrium  Static Equilibrium  Dynamic Equilibrium
  • 9. Static Equilibrium If some forces are acting on a body horizontally or vertically, and the body remains it states of rest is called Static Equilibrium. Example: A book lying on a table.
  • 10. Dynamic Equilibrium If some forces are acting on a body horizontally or vertically, and the body remains it states of motion is called Dynamic Equilibrium. Example: A train is moving with uniform velocity.
  • 11. Equations of equilibrium Consider an object moving along the xaxis. If no net force is applied to the object along the x-axis, it will continue to move along the x-axis at a constant velocity with no acceleration. We can extend this to the y- and z- axes.
  • 12. In static systems, where motion does not occur, the sum of the forces in all directions must always equal zero (otherwise, it's a dynamics problem). This concept can be represented mathematically with the following equations:
  • 13. ∑Fx=0 ∑Fy=0 ∑Fz=0
  • 14.  The concept also applies to rotational motion.  If the resultant moment about an axis is zero, the object will have no rotational acceleration about the axis. Again, we can extend this to moments about the y-axis and the z-axis. This is represented mathematically with the following.
  • 15. ∑Mx=0 ∑My=0 ∑Mz=0
  • 16. ∑F=0 : The algebraic sum of all the horizontal or vertical forces acting on a body which is in equilibrium must equal zero.
  • 17. ∑M=0: The algebraic sum of the moments of all the forces acting on a body which is in equilibrium, about any point in the plane of those forces, must equal zero.
  • 18. There are six equations expressing the equilibrium of a rigid body in 3 dimensions. ∑Fx=0 ∑Fy=0 ∑Fz=0 ∑Mx=0 ∑My=0 ∑Mz=0
  • 19. In two dimensions one direction of force and two directions of moments can be ignored. When forces exist only in the x and y directions, there cannot be a moment in any direction except z. The equations of concern when forces only exist in the x and y directions are shown below:
  • 20. ∑Fx=0 ∑Fy=0 ∑Mz=0
  • 21. How to apply equations of equilibrium?  First draw a free body diagram of the structure or its member.  If a member is selected, it must be isolated from its supports and surroundings.  All the forces and couple moments must be shown acting on the member.  Then apply the equations of equilibrium.
  • 22. y FBD at A FD A Area to be cut or isolated FB A 30˚ x
  • 23. Example  Find the reactions at support of the following beam:
  • 24. Applying eqn of equilibrium: ∑Fx=0; Ax=0 ∑Fy=0; Ay+By-10-20*4=0 Ay+By=90kn………(1)
  • 25. Considering Z axis passing through A and taking moment of all the forces about Zaxis (taking clockwise –ve and anticlockwise +ve) ∑Mz=0; By*10-10*8-20*4*2=0 By=24kn Putting this value in eqn (1) we get, Ay=66kn.
  • 26. 2D Equilibrium - Applications Since the forces involved in supporting the spool lie in a plane, this is essentially a 2D equilibrium problem. How would you find the forces in cables AB and AC?
  • 27. 2D Equilibrium -- Applications 2D Equilibrium Applications For a given force exerted on the boat’s towing pendant, what are the forces in the bridle cables? What size of cable must you use? This is again a 2D problem since the forces in cables AB, BC, and BD all lie in the same plane.
  • 28. Summary • In order for an object to be in equilibrium, there must be no net force on it along any coordinate, and there must be no net torque around any axis. • 2D equations of equilibrium: ∑Fx=0 ∑Fy=0 ∑Mz=0
  • 29. Thanks for your kind attention Any Questions?