Lecture 25: Equilibrium of a Rigid Body ENGI 1313 Mechanics I Lecture 25: Equilibrium of a Rigid Body

Lecture Objective to illustrate application of 2D equations of equilibrium for a rigid body to examine concepts for analyzing equilibrium of a rigid body in 3D

Example 25-01 Determine the force P needed to pull the 50-kg roller over the smooth step. Take θ = 60°.  =

Example 25-01 (cont.) What XY-coordinate System be Established? y x  =

Example 25-01 (cont.) Establish FBD y x  NB  = NA w = mg = (50 kg)(9.807 m/s2) = 490 N

Example 25-01 (cont.) Determine Force Angles Roller self-weight y y x  x  70  = 20  = NB  = 20 NA w = 490 N

Example 25-01 (cont.) Determine Force Angles Normal reaction force at A y x y x  90  = NB NA w = 490 N NA

Example 25-01 (cont.) Determine Force Angles Normal reaction force at B y x y x  r = 0.6 m yB = (0.6 m – 0.1 m) = 0.5 m NB  = NB NA w = 490 N

Example 25-01 (cont.) Draw FBD y y x x w = 490 N P  = 20  = 60 NB   = NB NA w = 490 N NA= 0 N

Example 25-01 (cont.) What Equilibrium Equation should be Used to Find P? MB = 0 w = 490 N  = 20 P x y  = 60 xB = 0.3317 m yB = 0.5 m NB  NA

Comprehension Quiz 25-01 If a support prevents rotation of a body about an axis, then the support exerts a ________ on the body about that axis. A) Couple moment B) Force C) Both A and B D) None of the above. Answer: A

3-D Equilibrium Basic Equations Moment equations can also be determined about any point on the rigid body. Typically the point selected is where the most unknown forces are applied. This procedure helps to simplify the solution.

Application to 3D Structures (cont.) Engineering Design Basic analysis Check more rigorous methods

Application to 3D Structures (cont.) Axial Forces Design of Experimental Test Frame Lateral Loads Couple Forces For Bending

3-D Structural Connections Ball and Socket Three orthogonal forces

3-D Structural Connections (cont.) Single Journal Bearing Two forces and two couple moments Frictionless Circular shaft Orthogonal to longitudinal bearing axis

3-D Structural Connections (cont.) Journal Bearing (cont.) Two or more (properly aligned) journal bearings will generate only support reaction forces

3-D Structural Connections (cont.) Single Hinge Three orthogonal forces Two couple moments orthogonal to hinge axis

3-D Structural Connections (cont.) Hinge Design Two or more (properly aligned) hinges will generate only support reaction forces

Rigid Body Constraints What is the Common Characteristic? Statically determinate system

Redundant Constraints Statically Indeterminate System Support reactions > equilibrium equations

Improper Constraints Rigid Body Instability 2-D problem Concurrent reaction forces Intersects an out-of-plane axis

Improper Constraints (cont.) Rigid Body Instability 3-D problem Support reactions intersect a common axis

Improper Constraints (cont.) Rigid Body Instability Parallel reaction forces

References Hibbeler (2007) http://wps.prenhall.com/esm_hibbeler_engmech_1