5.2 Free-Body DiagramsFBD is the best method to represent all theknown and unknown forces in a systemFBD is a sketch of the outlined shape of thebody, which represents it being isolated fromits surroundingsNecessary to show all the forces and couplemoments that the surroundings exert on thebody so that these effects can be accountedfor when equations of equilibrium are applied
5.2 Free-Body DiagramsSupport Reactions If the 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 Consider the three ways a horizontal member, beam is supported at the end - roller, cylinder - pin - fixed support
5.2 Free-Body DiagramsSupport ReactionsRoller or cylinder Prevent the beam from translating in the vertical direction Roller can only exerts a force on the beam in the vertical direction
5.2 Free-Body DiagramsSupport ReactionsPin The pin passes through a hold in the beam and two leaves that are fixed to the ground Prevents translation of the beam in any direction Φ The pin exerts a force F on the beam in this direction
5.2 Free-Body DiagramsSupport ReactionsFixed Support This support prevents both translation and rotation of the beam A couple and moment must be developed on the beam at its point of connection Force is usually represented in x and y components
5.2 Free-Body Diagrams Cable exerts a force on the bracket Type 1 connections Rocker support for this bridge girder allows horizontal movements so that the bridge is free to expand and contract due to temperature Type 5 connections
5.2 Free-Body Diagrams Concrete Girder rest on the ledge that is assumed to act as a smooth contacting surface Type 6 connections Utility building is pin supported at the top of the column Type 8 connections
5.2 Free-Body Diagrams Floor beams of this building are welded together and thus form fixed connections Type 10 connections
5.2 Free-Body DiagramsExternal and Internal Forces A rigid body is a composition of particles, both external and internal forces may act on it 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 and must be shown on FBD FBD for a system of connected bodies may be used for analysis
5.2 Free-Body DiagramsWeight and Center of Gravity When a body is subjected to gravity, each particle has a specified weight For entire body, consider gravitational forces as a system of parallel forces acting on all particles within the boundary The 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 DiagramsWeight and Center of Gravity Center of gravity occurs at the geometric center or centroid for uniform body of homogenous material For non-homogenous bodies and usual shapes, the center of gravity will be given
5.2 Free-Body DiagramsIdealized Models Needed to perform a correct force analysis of any object Careful selection of supports, material, behavior and dimensions for trusty results Complex cases may require developing several different models for analysis
5.2 Free-Body DiagramsIdealized Models Consider a steel beam used to support the roof joists of a building For force analysis, reasonable to assume rigid body since small deflections occur when beam is loaded Bolted connection at A will allow for slight rotation when load is applied => use Pin
5.2 Free-Body DiagramsSupport at B offers no resistance to horizontal movement => use Roller Building code requirements used to specify the roof loading (calculations of the joist forces) Large roof loading forces account for extreme loading cases and for dynamic or vibration effects Weight is neglected when it is small compared to the load the beam supports
5.2 Free-Body DiagramsIdealized Models Consider lift boom, supported by pin at A and hydraulic cylinder at BC (treat as weightless link) Assume rigid material with density known For design loading P, idealized model is used for force analysis Average dimensions used to specify the location of the loads and supports
5.2 Free-Body DiagramsProcedure for Drawing a FBD1. Draw Outlined Shape Imagine body to be isolated or cut free from its constraints Draw outline shape2. Show All Forces and Couple Moments Identify all external forces and couple moments that act on the body
5.2 Free-Body DiagramsProcedure for Drawing a FBD Usually due to - applied loadings - reactions occurring at the supports or at points of contact with other body - weight of the body To account for all the effects, trace over the boundary, noting each force and couple moment acting on it3. Identify Each Loading and Give Dimensions Indicate dimensions for calculation of forces
5.2 Free-Body DiagramsProcedure for Drawing a FBD Known forces and couple moments should be properly labeled with their magnitudes and directions Letters used to represent the magnitudes and direction angles of unknown forces and couple moments Establish x, y and coordinate system to identify unknowns
5.2 Free-Body DiagramsExample 5.1Draw the free-body diagram of the uniformbeam. The beam has a mass of 100kg.
5.2 Free-Body DiagramsSolution 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
5.2 Free-Body DiagramsExample 5.2Draw the free-body diagram ofthe foot lever. The operatorapplies a vertical force to thepedal so that the spring isstretched 40mm and the forcein the short link at B is 100N.
5.2 Free-Body DiagramsSolution Lever loosely bolted to frame at A Rod at B pinned at its ends and acts as a short link For idealized model of the lever,
5.2 Free-Body DiagramsSolution Free-Body Diagram Pin support at A exerts components Ax and Ay on the lever, each force with a known line of action but unknown magnitude
5.2 Free-Body DiagramsSolution Link at B exerts a force 100N acting in the direction of the link Spring exerts a horizontal force on the lever Fs = ks = 5N/mm(40mm) = 200N Operator’s shoe exert vertical force F on the pedal Compute the moments using the dimensions on the FBD Compute the sense by the equilibrium equations
5.2 Free-Body DiagramsExample 5.3Two smooth pipes, eachhaving a mass of 300kg, aresupported by the forks of thetractor. Draw the free-bodydiagrams for each pipe andboth pipes together.
5.2 Free-Body DiagramsSolution For idealized models, Free-Body Diagram of pipe A
5.2 Free-Body DiagramsSolution For weight of pipe A, W = 300(9.81) = 2943N Assume all contacting surfaces are smooth, reactive forces T, F, R act in a direction normal to tangent at their surfaces of contact Free-Body Diagram at pipe B
5.2 Free-Body DiagramsSolution*Note: R represent the force of A on B, is equaland opposite to R representing the force of B on A Contact force R is considered an internal force, not shown on FBD Free-Body Diagram of both pipes
5.2 Free-Body DiagramsExample 5.4Draw the free-body diagramof the unloaded platform thatis suspended off the edge ofthe oil rig. The platform has amass of 200kg.
5.2 Free-Body DiagramsSolution Idealized model considered in 2D because by observation, loading and the dimensions are all symmetrical about a vertical plane passing through the center Connection at A assumed to be a pin and the cable supports the platform at B
5.2 Free-Body DiagramsSolution Direction of the cable and average dimensions of the platform are listed and center of gravity has been determined Free-Body Diagram
5.2 Free-Body DiagramsSolution Platform’s weight = 200(9.81) = 1962N Force components Ax and Ay along with the cable force T represent the reactions that both pins and cables exert on the platform Half of the cables magnitudes is developed at A and half developed at B
5.2 Free-Body DiagramsExample 5.5The free-body diagram of each object isdrawn. Carefully study each solution andidentify what each loading represents.