6161103 5.2 free body diagrams

5.2 Free-Body Diagrams5.2 Free-Body Diagrams

� FBD is the best method to represent all the known and unknown forces in a system

� FBD is a sketch of the outlined shape of the body, which represents it being isolated from body, which represents it being isolated from its surroundings

� Necessary to show all the forces and couple moments that the surroundings exert on the body so that these effects can be accounted for when equations of equilibrium are applied


Support 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 � 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


Support ReactionsRoller or cylinder

� Prevent the beam from translating in the vertical translating in the vertical direction

� Roller can only exerts a force on the beam in the vertical direction


Support ReactionsPin� The pin passes through a hold in the beam

and two leaves that are fixed to the groundPrevents translation of the beam in any � Prevents translation of the beam in any direction Φ

� The pin exerts a force F on the beam in this direction


Support Reactions

Fixed Support

� This support prevents both translation and rotation of the beamtranslation 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


� 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



� Concrete Girder rest on the ledge that is assumed to act as a smooth contacting surfacesurface


� Utility building is pin supported at the top of the column



� Floor beams of this building are welded together and thus form fixed connections

� Type 10 connections� Type 10 connections


External 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 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


Weight and Center of Gravity� When a body is subjected to gravity, each

particle has a specified weight

� For entire body, consider gravitational forces as � 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


Weight and Center of Gravity

� Center of gravity occurs at the geometric center or centroid for uniform body of homogenous materialhomogenous material

� For non-homogenous bodies and usual shapes, the center of gravity will be given


Idealized 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


Idealized 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 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


Support 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 � 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


Idealized Models� Consider lift boom, supported by pin

at A and hydraulic cylinder at BC (treat as weightless link)(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


Procedure for Drawing a FBD

1. Draw Outlined Shape

� Imagine body to be isolated or cut free from its constraintsconstraints

� Draw outline shape

2. Show All Forces and Couple Moments

� Identify all external forces and couple moments that act on the body


Procedure for Drawing a FBD� Usually due to

- applied loadings- reactions occurring at the supports or at points of contact with other bodypoints 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 it

3. Identify Each Loading and Give Dimensions� Indicate dimensions for calculation of forces


Procedure for Drawing a FBD

� Known forces and couple moments should be properly labeled with their magnitudes and directionsand 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


Example 5.1

Draw the free-body diagram of the uniform

beam. The beam has a mass of 100kg.beam. The beam has a mass of 100kg.


Solution

Free-Body Diagram


Solution

� Support at A is a fixed wall

� Three forces acting on the beam at A denoted as Ax, A , A , drawn in an arbitrary directionAy, 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


Example 5.2

Draw the free-body diagram of

the foot lever. The operator the foot lever. The operator

applies a vertical force to the

pedal so that the spring is

stretched 40mm and the force

in the short link at B is 100N.


Solution

� Lever loosely bolted to frame at A

� Rod at B pinned at its ends and acts as a short linkshort link

� For idealized model of the lever,


Solution

� 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


Solution

� Link at B exerts a force 100N acting in the direction of the link

� Spring exerts a horizontal force on the lever� 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


Example 5.3

Two smooth pipes, each

having a mass of 300kg, are having a mass of 300kg, are

supported by the forks of the

tractor. Draw the free-body

diagrams for each pipe and

both pipes together.


Solution

� For idealized models,


of pipe A


Solution

� 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 forces T, F, R act in a direction normal to tangent at their surfaces of contact

� Free-Body Diagram at pipe B


Solution*Note: R represent the force of A on B, is equal and opposite to R representing the force of B on A� Contact force R is considered an internal force, not

shown on FBDContact force R is considered an internal force, not shown on FBD

� Free-Body Diagram of both pipes


Example 5.4

Draw the free-body diagram

of the unloaded platform that of the unloaded platform that

is suspended off the edge of

the oil rig. The platform has a

mass of 200kg.


Solution

� Idealized model considered in 2D because by observation, loading and the dimensions are 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


Solution

� Direction of the cable and average dimensions of the platform are listed and center of gravity has been determinedhas been determined



Solution

� Platform’s weight = 200(9.81) = 1962N

� Force components Ax and Ay along with the cable force T represent the 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


Example 5.5

The free-body diagram of each object is

drawn. Carefully study each solution and drawn. Carefully study each solution and

identify what each loading represents.


Solution

6161103 5.2 free body diagrams

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