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Equilibrium
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Lecture Objectives
• In this lecture, we will learn about drawing free-
body-diagrams (FBD)
• FBDs are the most important step in both static and
dynamic analysis
• Using FBDs, we will investigate the condition in
which the resultant of a system of forces acting on a
body is zero (i.e., static equilibrium condition)
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Representing Forces as Vectors
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Planar Supports & Connections
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Cable in tension
Pin supportWeld connection
Rocker support
Freely sliding surface Pin support
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Representing Forces as Vectors
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3-D Supports & Connections
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Thrust bearing
Journal bearing
Ball-&-socket
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Engineering Mechanics Analysis Methodology
• In performing a static or dynamic analysis of a
system, we must
1. Define the mechanical system we are analyzing
• Clearly identify all known and unknown quantities of the system
• The system should include all unknown quantities we are seeking
2. Isolate the system from its surroundings using a free-
body-diagram (FBD)
• Draw an external boundary around the system being analyzed and
remove all other bodies that are not a part of the system
The FBD is the most important single step in the solution
of mechanics problems
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Engineering Mechanics Analysis Methodology
3. Identify all external forces (contact & body) acting on the
system by marking them on the FBD
• Make sure to add a force to the FBD for every contacting or
attracting body that was removed
• Mark all information (magnitude, line of action, sense) readily
available on these external forces
• If the sense of the force vector is not known, make an arbitrary
assignment
4. Define & mark a coordinate system suitable for the
problem
• Make a clever choice for the moment centers to simplify
calculations (e.g., choose moment centers with as many unknown
forces passing through as possible)
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Engineering Mechanics Analysis Methodology
5. Also indicate pertinent dimensions
6. But, avoid cluttering the FBD with unnecessary and
unrelevant information
7. Identify and state the appropiate force and moment
equations governing the problem
8. Match the number of unknown quantities to the number
of independent equations
9. Carry out the solution and check your results
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Examples of FBDs
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Exercise: Completing FBDs
MCH2008 Eng. Mechanics
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Exercise: Constructing FBDs
MCH2008 Eng. Mechanics
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Equilibrium
• A body is said to be in complete equilibrium when
the resultant forces (R ) and couples (MO, any point
O) acting on the body is zero (necessary & sufficient
conditions)
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, , ,
0
0, 0, 0
l
l
x l y l z l
l l l
M M
M M M
, , ,
0
0, 0, 0
k
k
x k y k z k
k k k
R F
F F F
Equilibrium
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Two-Force & Three-Force Members
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For a two-force member, the forces
acting on the member are only along
the line joining the two ends of the
member
Two-Force & Three-Force Members
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Link AB is a two-force member
Link ABC is a three-force member
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• Alternative 1:
– Force balance in one direction and moment balances
about two points A and B
Alternative Formulations for Equilibrium
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0, 0, 0 x A B
F M M
M A = 0 suggests that R
must pass through A
M B = 0 suggests equilibrium
if AB is not to x axis
• Alternative 2:
– Moment balances about three points A, B, and C
Alternative Formulations for Equilibrium
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0, 0, 0 A B C
M M M
When equilibrium is expressed using
a set of dependent equations (more
equations than necessary), you mayend up with a trivial force or moment
balance equation of the form 0 = 0
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Equilibrium
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Example: Equilibrium in 2-D
• Determine the forcesC and T acting on the bridge-
truss
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Example: Equilibrium in 2-D
• Ignoring the weights of the pulleys, determine the
tension T in the cable
• Find the the total force on the bearing of the pulley
C
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Example: Equilibrium in 2-D
• Determine the tension P in the cable for lifting the
100 kg beam’s point B 3 m above A
• Determine the reaction at support A
• What is the angle made with the horizontal
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Example: Equilibrium in 2-D
• Determine the magnitude T of the force in the cable
supporting the I-beam with a mass of 95 kg/m
• What is the reaction force at support A
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Example: Equilibrium in 3-D
• Determine the forces exerted at the ball supports at
points A and B on the 200 kg beam
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Example: Equilibrium in 3-D
• Determine the mass m that can be supported by the
200 N force applied at the handle
• Compute the radial force exerted on the shaft by
each bearing
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Example: Equilibrium in 3-D
• Ignoring the weight of the frame shown below,
determine the tension in the cable CD
• Determine the reaction forces at loose-fitted ring B
•
Calculate the reaction forces at the ball-and-socket joint at point A
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Constraints
• A constraint is a restriction of motion
– Example: A roller is free to move horizontally
(no horizontal constraint)
A pin cannot move vertical or horizontal
(horizontally constrained)
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• The force and moment balance equations are
necessary and sufficient for equilibrium but may not
be adequate to determine all the unknown forces
Statical Determinacy
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• A mechanical system that has more constraints than
necessary to maintain an equilibrium is termed
statically indeterminate
Statical Determinacy
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Statical Determinacy
• Constraints that can be removed without disturbing
equilibrium are termed redundant constraints
• The number of redundant constraints is termed
degree of statical determinacy
• We will mostly deal with statically determinant
problems
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Degree of Statical Unknown External Number of IndependentEquilibrium EquationsDeterminacy Forces
Unknown External Number of IndependentEquilibrium EquationsForces
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Adequacy of Constraints
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Adequacy of Constraints
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Next Lecture
• Lecture topics
– Structures
• Reading assignment: Ch.5 in textbook
• Questions?
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