LE 2

• Free Body Diagram of Rigid Bodies


  Diagram of a "free" body with all the known and unknown external forces (represented by arrows and their corresponding point) acting on the body while maintaining the shape and dimension of the body
• External Forces


  • Applied Forces
  • Support Reactions
  • Weight
  • Reactions of other bodies previously attached to the body in consideration
• Reactions/Constraining Forces


  • Oppose a possible motion of the body and constrain a body to remain in the same position
  • Exerted at points where the body is supported by or connected to other bodies
• Reactions at Supports and Connections for a 2D Structure

  • Reactions equivalent to a Force with a known Line of Action
  • Reactions equivalent to a force of unknown direction and magnitude (sometimes expressed as Rx and Ry)
  • Reactions equivalent to a force and a couple
• Reactions equivalent to a Force with a known Line of Action

  • Prevent motion of a body in one direction only
  • Rollers, Rockers, Frictionless Surfaces
  • Short Links and Cables
  • Collars on Frictionless Rods and Frictionless Pin in slots
• Reactions equivalent to a force of unknown direction and magnitude

  • Prevent motion in all directions (movements along x- and y- axes) of a body but not rotational motion
  • Pins/Smooth Pins/Frictionless Pins in Fitted Holes
  • Hinges, Rough Surfaces
• Reactions equivalent to a force and a couple

  • Prevent all types of motion in all directions of a body (full constraint)
  • Fixed Supports
• Equilibrium of Rigid Body in 2D

  No net translational (∑Fx = 0 ; ∑Fy = 0) and rotational motion (∑M = 0)
• Two-Force Body


  • A body in equilibrium in which all forces are acting on only two points in the body with no couple moments
  • These forces can be summarized/simplified as Tension or Compressive Forces since these forces will cancel each other out to maintain the body's equilibrium
  • In the two points of application, the resultant forces should be equal in magnitude but opposite in direction
• Three-Force Body


  • A body in equilibrium in which all forces are acting on only three points in the body with no couple moments
  • The three forces are concurrent OR
  • The three forces are parallel
  • If the three forces are parallel, the location of the point of intersection of the forces will approach infinity
• Truss


  • Frameworks composed of members connected at their ends to form a rigid structure and are composed solely of two-force members
  • Ends of truss members are welded, riveted, or connected by bolts/pins resulting to a pattern of triangles
  • Plane Truss – truss in 2D | Space Truss – truss in 3D
  • Built to support loads and are usually stationary
• Assumptions of Plane Truss Analysis

  • Analysis is two-dimensional
  • All external loads are applied at the points of connection (joints)
  • Weights of the members are negligible
  • All members are two-force members subjected to tensile or compressive forces
• Truss Joints


  Points of connections of a truss formed by bolting or welding the ends of the members to a common plate
• Zero-Force Member


  • A truss member which does not experience internal force or has zero-member force
  • Case 1: If only two noncollinear members form a truss joint and no external load or support reaction is applied to the joint, the two members are both zero-force members
  • Case 2: If three members form a truss joint for which two of the members are collinear, the third member is a zero-force member provided that, there is no applied external force or support reaction on that joint
  • Special Case of Case 2: If two members form a truss joint in which one of them is collinear to an external force applied to the joint, then the other member is a zero-force member
• Method of Joints


  • Truss Analysis is done joint by joint
  • Each truss joint is treated as a particle
  • Only summations of forces can be used in each FBD (no summations of Moment)
• Method of Sections


  • Truss Analysis is done section by section
  • Each section is treated as a rigid body
  • Summations of forces and moments can be used in each FBD
  • Cutting plane passes through a member only once
  • Commonly used when we need to find the force in only a few members of the truss
• Frame


  • Composed of at least one multi-force member unlike trusses
  • Built to support loads and are normally fixed
• Assumptions of Frame Analysis

  • The joints that connect the members of a frame are not necessarily located at the ends of the members (unlike Truss)
  • The members are joined together by smooth pins
• There is a corresponding set of action-reaction forces in a point of connection where a frame member is dismembered or disconnected. These forces will not appear unless the member is dismembered from the point/s of connections.
• Machine


  • Like Frames, it is composed of at least one multi-force member
  • Built to transmit and multiply input forces
  • May or may not be stationary and will always contain moving parts
  • Have irregularly shaped members
• The procedure on Machine Analysis is the same as Frame Analysis
• Pin


  • Type of connector in structures that can resist translation but not rotation
  • If a pin connects two or more bodies, it is recommended to treat the pin as a separate body or consider it to be part of a specifically identified body
• There is a corresponding set of action-reaction forces between a pin and a member when the pin is disconnected or dismembered. If the pin is reconnected, these action-reaction forces will disappear since they will cancel each other.