CHAP 5 Equilibrium of a Rigid body

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CHAP 5 Equilibrium of a Rigid body
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• 5.1 Conditions for Rigid body equilibrium

Consider a rigid body which is at rest or moving
with x y z reference at constant velocity
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Free body diagram of ith particle of the body
External force (??) gravitational,
electrical, magnetic or contact force
j
Internal force (??)
i
Force equilibrium equation for particle i
(Newton first law)
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Force equilibrium equation for the whole body
(Newtons 3rd law ????????)
Moment of the forces action on the ith particle
about pt. O
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• Moment equilibrium equation for the body

Equations of equilibrium for a rigid body are
???
????
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5.2 Equilibrium in Two Dimensions

• 1. Free-body Diagram
• (1) F.B.D
• A sketch of the outlined shape of the body
  represents it as being isolated or free from
  its surrounding , i.e ., a free body.

(2) Support Reactions A .Type of support
see Table 5-1
B . General rules for support reaction
If a support prevents the translation of a body
in a given direction, then a force is developed
on the body in that direction . Likewise, if
rotation is prevented, a couple moment is exerted
on the body.
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(a) roller or cylinder support
Examples
(b) pin support
(c) Fixed support
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(3) External and Internal forces

• A. Internal force
• Not represented on the F.B.D. became their net
  effect on the body is zero.
• B. External force
• Must be shown on the F.B.D.
• (a) Applied loadings
• (b) Reaction forces ????
• (c) Body weights ??

(4) Weight and the center of gravity
The force resultant from the gravitational field
is referred as the weight of the body, and
the location of its point of application is the
center of gravity G.
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(No Transcript)
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2. Equations of Equilibrium for 2D rigid body
(1) Conditions of equilibrium
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(2) Alternative equilibrium equation
(A)
When the moment points A and B do not lie on a
line that is perpendicular to the axis a.
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(B)
Points A, B and C do not lie on the same line
a
C
A
B
a
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(3) Example
3 unknown Ax, Bx, By
Equations of equilibrium
3 equations for 3 unknowns
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5.3 Two-and Three-Force Members
1. Two-Force member
A member subject to no couple moments and forces
applied at only two points on the member.
FA
A
A
B
B
FB
Equations of Equilibrium
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2. Three-Force member
A member subject to only three forces, which are
either concurrent or parallel if the member is in
equilibrium.
(1)Concurrent
(2)parallel
(3???O?)
(3???????)
F2
F1
o
F3
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5.4 Equilibrium in Three Dimensional Rigid Body

• 1. Free Body Diagrams
• (1)F.B.D
• Same as 2D equilibrium problems
• (2)Support Reactions
• A. Types of supportsee Table 5-2
• B. General rules for reaction
• Same as two-dimensional case

• Examples
• (a) Ball and Socket joint
• No translation along any direction
• Rotate freely about any axis

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z

• (b) single journal bearing
• Rotate freely about its longitudinal axis
• Translate along its longitudinal direction

Fz
Mz
y
Mx
Fx
(c) single pin Only allow to rotate about a
specific axis.
two unkown forces and couple moments
x
z
Mz
Fz
My
y
Fy
Fx
Three unkown forces and two couple moments
x
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2. Equations of Equilibrium

• A. Vector equations of equilibrium
• B. Scalar equations of equilibrium

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5.6 Constraints for a rigid body

• 1. Redundant constraints
• (1) Redundant constraints
• Redundant supports are more than
  necessarily to hold a body in equilibrium.

Ex
Equation of motion3 5 unknown reactions gt3
equation of motion there are two support
reactions which are redundant supports and more
than necessarily.
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• (2) Statically indeterminate ???
• There are more unknown loadings on the body
  than equations of equilibrium available for the
  solution.

F.B.D of above example
unknown loadings AX,AY,MA,BY,CY5 Equations of
equilibrium SFX0,SFY0,SMA 03 5gt3 Statically
indeterminate structure
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• (3)Solutions for statically indeterminate
  structure
• Additional equations are needed ,which are
  obtained from the deformation condition at the
  points of redundant support based on the
  mechanics of deformation, such as mechanics of
  materials.

Equations of Equilibrium for above example are
SFX0 AX0 SFY0
500-AY-BY-CY0 SMA0 MA-2-DYBY-DCCY0 Nee
d two more equations to solve the five unknown
forces.
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2.Improper Constraints

• (1) Reaction force equations of equilibrium
• If this kind of improper constraint occurs
  then system is instable
• A. The lines of action of the reactive
  forces intersect points on a common axis
  (concurrent).

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body will rotate about Z-axis or point O
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B. The reactive forces are all parallel
F.B.D
Body will translate along x direction.
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• (2) Reaction forces lt equations of equilibrium
• If the body is partially constrained then
  it is in instable condition

F.B.D
Not in equilibrium