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## College Of Engineering Electrical Engineering Department

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**College Of Engineering Electrical Engineering Department**Engineering Mechanics-Static Equilibrium of a Rigid Body Lecture-1 By Dr. Salah M. Swadi 2018-2019**Chapter Objectives**• Develop the equations of equilibrium for a rigid body • Concept of the free-body diagram for a rigid body • Solve rigid-body equilibrium problems using the equations of equilibrium**Chapter Outline**• Conditions for Rigid Equilibrium • Free-Body Diagrams • Equations of Equilibrium • Two and Three-Force Members • Free Body Diagrams • Equations of Equilibrium • Constraints and Statical Determinacy**5.1 Conditions for Rigid-Body Equilibrium**• The equilibrium of a body is expressed as • Consider summing moments about some other point, such as point A, we require**5.2 Free Body Diagrams**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. • If rotation is prevented, a couple moment is exerted on the body.**5.2 Free Body Diagrams**Internal Forces • External and internal forces can act on a rigid body • For FBD, internal forces act between particles which are contained within the boundary of the FBD, are not represented • Particles outside this boundary exert external forces on the system**5.2 Free Body Diagrams**Weight and Center of Gravity • Each particle has a specified weight • System can be represented by a single resultant force, known as weight W of the body • Location of the force application is known as the center of gravity**5.2 Free Body Diagrams**Procedure for Drawing a FBD 1. Draw Outlined Shape • Imagine body to be isolated or cut free from its constraints • Draw outline shape 2. Show All Forces and Couple Moments • Identify all external forces and couple moments that act on the body**5.2 Free Body Diagrams**3. Identify Each Loading and Give Dimensions • Indicate dimensions for calculation of forces • Known forces and couple moments should be properly labeled with their magnitudes and directions**Example 5.1**Draw the free-body diagram of the uniform beam. The beam has a mass of 100kg.**Solution**Free-Body Diagram**Solution**Free-Body Diagram • Support at A is a fixed wall • Three forces acting on the beam at A denoted as Ax, Ay, Az, drawn in an arbitrary direction • Unknown magnitudes of these vectors • Assume sense of these vectors • For uniform beam, Weight, W = 100(9.81) = 981N acting through beam’s center of gravity, 3m from A**5.3 Equations of Equilibrium**• For equilibrium of a rigid body in 2D, ∑Fx = 0; ∑Fy = 0; ∑MO = 0 • ∑Fx and ∑Fy represent sums of x and y components of all the forces • ∑MO represents the sum of the couple moments and moments of the force components**5.3 Equations of Equilibrium**Alternative Sets of Equilibrium Equations • For coplanar equilibrium problems, ∑Fx = 0; ∑Fy = 0; ∑MO = 0 • 2 alternative sets of 3 independent equilibrium equations, ∑Fa = 0; ∑MA = 0; ∑MB = 0**5.3 Equations of Equilibrium**Procedure for Analysis Free-Body Diagram • Force or couple moment having an unknown magnitude but known line of action can be assumed • Indicate the dimensions of the body necessary for computing the moments of forces**5.3 Equations of Equilibrium**Procedure for Analysis Equations of Equilibrium • Apply ∑MO = 0 about a point O • Unknowns moments of are zero about O and a direct solution the third unknown can be obtained • Orient the x and y axes along the lines that will provide the simplest resolution of the forces into their x and y components • Negative result scalar is opposite to that was assumed on the FBD**Example 5.5**Determine the horizontal and vertical components of reaction for the beam loaded. Neglect the weight of the beam in the calculations.**Solution**Free Body Diagrams • 600N represented by x and y components • 200N force acts on the beam at B**Solution**Equations of Equilibrium