Transcript Document 7235237

Procedure for drawing a free-body
diagram - 2-D force systems
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Imagine the body to be isolated or cut “free” from its
constraints and connections, draw its outlined shape
Identify all the external forces and couple moments that act
on the body
– Applied loads
– Reactions occurring at the supports or at points of contact with
other bodies
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Most common: roller, pin, fixed
Review additional supports, Table 5-1, pg 210
– Weight of the body
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Applied at the center of gravity, G
When a body is uniform or made of homogeneous material, G will be
located at the body’s geometric center or centroid (if not, it will be
specified)
Procedure for drawing a free-body
diagram - 2-D force systems (continued)
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Indicate the dimensions of the body necessary for computing
the moments of the forces
Forces and couple moments that are known should be labeled
with their proper magnitudes and directions
Use letters to represent magnitudes and direction angles of
forces and couple moments that are unknown
– Need to assume a sense for the unknown forces and couple
moments
– Correctness of assumed sense will become apparent after solving
equations of equilibrium for the unknown magnitudes (negative
values – vector’s sense is opposite to that assumed in the freebody diagram)
2-D equilibrium
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Equilibrium requires both a balance of forces, to prevent the
body from translating with accelerated motion, and a balance
of moments, to prevent the body from rotating
Equations of equilibrium
– ∑F=0
– ∑ M(any point) = 0
– In the x-y plane
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∑ Fx = 0, algebraic sum of the x components
∑ Fy = 0, algebraic sum of the y components
∑ M(any point) = 0, algebraic sum of the couple moments and the
moments of the force components about an axis perpendicular to
the x-y plane and passing through any arbitrary point, which may lie
either on or off the body
2-D equilibrium – procedure for analysis
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Draw the free-body diagram
Apply equations of equilibrium
– Establish the x,y,z axes
– Apply equations of equilibrium: ∑ Fx = 0, ∑ Fy = 0, ∑ Mo = 0
– To avoid having to solve simultaneous equations apply the
moment equation, ∑ MO = 0, about a point “O” that lies at the
intersection of the lines of action of two unknown forces – the
moments of these unknowns are zero about “O” and then able to
obtain a direct solution for the third unknown SHOW
Two- and three-force members
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Two-force members
– Forces are applied at only two points and the member is not
subjected to any couple moments
– In order to maintain force (translational) equilibrium, ∑ F = 0, F’
must be of equal magnitude and opposite sense to F
– In order to maintain moment (rotational) equilibrium, ∑ MP = 0,
F’ must be collinear (i.e. have the same line of action) with F
– SHOW
– Examples include a cable or bar attached at two points
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Three-force members
– If a member is subjected to three forces, then it is necessary
that the forces be coplaner and either concurrent or parallel if
the member is to be in equilibrium
– SHOW
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EXAMPLES (pgs 234 - 243)
Free-body diagram - 3-D force systems
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Support reactions, Table 5-2, pg 246
Hinge and bearing supports
– Single hinge and single bearing supports produce both force and
couple moment reactions
– When hinges and bearings are used in conjunction with other
hinges and bearings, respectively, force reactions at these
supports may alone be adequate for supporting the object (i.e.
neglect couple moments on the free-body diagram)
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The supports must be properly aligned when connected to the object
The object has to maintain its rigidity when loaded
3-D equilibrium
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Vector equations
– ∑F=0
– ∑ M(any point) = 0
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Scalar equations:
– ∑ F = ∑ Fxi + ∑ Fyj + ∑ FZk = 0
– ∑ M(any point) = ∑ Mxi + ∑ Myj + ∑ MZk = 0
– Since i, j, k components are independent from one another
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∑ Fx = 0, ∑ Fy = 0, ∑ FZ = 0
∑ Mx = 0, ∑ My = 0, ∑ MZ = 0 (forces that are parallel to an axis or
pass through it create no moment about the axis)
The set of axes for the force summation and moment summation do
not have to coincide
Statically indeterminate objects
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Some bodies may have more supports than are necessary for
equilibrium - redundant supports
Some bodies may not have enough supports or the supports
may be arranged in a particular manner that could cause the
body to collapse - improper constraints
Redundant supports
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Object has more supports than is needed to maintain it in equilibrium
SHOW
Difference between the number of reactions and the number of
independent equilibrium equations is called the degree of
redundancy
Improper constraints
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Object with improper constraining by the supports will not remain in
equilibrium under the action of the loads exerted on it
– The supports can exert only reactions that intersect at a common axis
and this axis is perpendicular to the plane of the loads SHOW
– The supports can exert only concurrent reactions SHOW
– The supports can exert only parallel reactions SHOW
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EXAMPLES (pg 262 - 267)