Chapter Objectives Chapter Outline Rigid body diagram FBD and RBD Equilibrium of RBD Chapter Outline Conditions for Rigid-Body Equilibrium Free-Body Diagrams Equations of Equilibrium Two and Three-Force Members Free Body Diagrams (3D) Equations of Equilibrium (3D) Constraints and Statical Determinacy

5.1 Conditions for Rigid-Body Equilibrium For equilibrium when considering about O For moment about any point, or in this case, point A For any point

5.2 Free Body Diagrams Support Reactions Support reaction is when there is contact and the reaction is coming from the floor or the wall The moment at the support is due to the wall Different types of supports will have different types of forces

5.2 Free Body Diagrams cable weightless link roller

5.2 Free Body Diagrams roller or pin in confined smooth slot rocker smooth connecting surface member pin connected To collar on smooth rod

member fixed connected 5.2 Free Body Diagrams smooth pin or hinge member fixed connected To collar on smooth rod fixed support

Supports for 2D system 1D Forces 2D Forces 2D forces + 1D moment

5.2 Free Body Diagrams Internal forces For a rigid body, there is both external and internal forces For FBD, internal forces within the FBD boundary is not needed For outside the boundary FBD, it is external forces

5.2 Free Body Diagrams Weight and Center of Gravity A rigid body are composed of weight of individual smaller rigid bodies, and the self-weight of the rigid body can be represented by a single weight W The location of the weight W is at center of gravity

5.2 Free Body Diagrams Procedure for Drawing a FBD 1. Sketch a body Consider if the body is free from all the connections 2. Draw all forces and moments Draw all external forces and moments acting on the body Draw self-weight if considered 3. Indicate the magnitude and the directions of forces and moments Indicate all the magnitude and directions of forces and moments if known Split all components in x, y or a convenient reference Display all known distance

Example 5.1 Draw the free-body diagram of the uniform beam. The beam has a mass of 100kg.

Solution FBD

5.3 Equations of Equilibrium For equilibrium of a rigid body in 2D, ∑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 Sets of Equilibrium Equations For coplanar equilibrium problems, ∑Fx = 0; ∑Fy = 0; ∑MO = 0 2 alternative sets of 3 independent equilibrium equations, ∑Fx = 0; ∑MA = 0; ∑MB = 0 (A, B is not perpendicular to x) ∑MA = 0; ∑MB = 0 ; ∑MC = 0 (A, B, C are not on the same line)

5.3 Equations of Equilibrium Procedure for Analysis Free-Body Diagram For unknown forces or moments, display the line of action Put known distances Equations of Equilibrium ∑MO = 0 and ∑F = 0 Indicate x, y system that simplified calculations If the result is negative, the direction of forces or moments in FBD in incorrect, hence negate the direction of the forces

Example 5.5 Determine the horizontal and vertical components of reaction for the beam loaded. Neglect the weight of the beam in the calculations. (A is a roller and B is a hinge support.)

Solution

Solution Equations of Equilibrium

5.4 Two- and Three-Force Members Two-Force Members is parts that contain on forces acting at two points, no moments The two forces are the same in magnitude but opposite directions and lie in the same line of action Not O.K. Not O.K. O.K. The two forces are in the same line of action

5.4 Two- and Three-Force Members Forces acting at 3 points The forces are either meet at a single point in space or they are parallel

Example 5.13 The lever ABC is pin-supported at A and connected to a short link BD. If the weight of the members are negligible, determine the force of the pin on the lever at A.

Solution Free Body Diagrams BD is a two-force member Lever ABC is a three-force member Equations of Equilibrium Solving,

5.5 Free-Body Diagrams (3D) Support Reactions For 3D problems Reactions forces are at the supports Some supports may have moments The directions of forces are indicated by angles α, β and γ

5.5 Free-Body Diagrams

5.5 Free-Body Diagrams

Example 5.14 Several examples of objects along with their associated free-body diagrams are shown. In all cases, the x, y and z axes are established and the unknown reaction components are indicated in the positive sense. The weight of the objects is neglected.

5.6 Equations of Equilibrium (3D) Equilibrium equations in vector notation 2 equations for rigid body ∑F = 0 ∑MO = 0 In vector Cartesian form ∑F = ∑Fxi + ∑Fyj + ∑Fzk = 0 ∑MO = ∑Mxi + ∑Myj + ∑Mzk = 0

5.7 Constraints for a Rigid Body Redundant Constraints There are more forces and moments from the supports than equilibrium equations Statically indeterminate: there are more unknown than equations

5.7 Constraints for a Rigid Body Improper Constraints There are not enough constraints, hence the system are not in equilibrium

5.7 Constraints for a Rigid Body Procedure for Analysis Free Body Diagram Sketch all bodies Draw all forces and moments Put all unknown forces Display all the known distances Equations of Equilibrium Apply equilibrium equations Apply appropriate reference system, choose the locations where many forces pass

Example 5.15 The homogenous plate has a mass of 100kg and is subjected to a force and couple moment along its edges. If it is supported in the horizontal plane by means of a roller at A, a ball and socket joint at B, and a cord at C, determine the components of reactions at the supports.

Solution Free Body Diagrams Five unknown reactions acting on the plate Each reaction assumed to act in a positive coordinate direction Equations of Equilibrium

Solution Equations of Equilibrium

Solution Solving,