6161103 5.2 free body diagrams

5.2 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
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
exerted on the body
Consider the three ways a horizontal member,
beam is supported at the end
- roller, cylinder
- pin
- fixed support


Support Reactions
Roller or cylinder
Prevent the beam from
translating in the vertical
direction
Roller can only exerts a
force on the beam in the
vertical direction


Support Reactions
Pin
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


Support Reactions
Fixed 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


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


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


Floor beams of this building
are welded together and
thus form fixed 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
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
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 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
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
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)
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
constraints
Draw outline shape

2. Show All Forces and Couple Moments
Identify all external forces and couple moments
that act on the body

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 it

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


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


Example 5.1
Draw the free-body diagram of the uniform
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,
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


Example 5.2
Draw the free-body diagram of
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 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
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
supported by the forks of the
tractor. Draw the free-body
diagrams for each pipe and
both pipes together.


Solution
For idealized models,

Free-Body Diagram
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
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 FBD
Free-Body Diagram of both pipes


Example 5.4
Draw the free-body diagram
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
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 determined
Free-Body Diagram


Solution
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


Example 5.5
The free-body diagram of each object is
drawn. Carefully study each solution and
identify what each loading represents.


Solution

6161103 5.2 free body diagrams

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6161103 5.2 free body diagrams