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05. Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics β Statics 5.01 Equilibrium of a Rigid Body
Chapter Objectives
β’ To develop the equations of equilibrium for a rigid body
β’ To introduce the concept of the free-body diagram for a rigid
body
β’ To show how to solve rigid-body equilibrium problems using
the equations of equilibrium
Engineering Mechanics β Statics 5.02 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§1. Conditions for Rigid-Body Equilibrium
- The force and couple system acting on a body can be reduced
to an equivalent resultant force and resultant couple moment
at an arbitrary point π
- A rigid body is in equilibrium βΊ πΉ π = βπΉ π = 0, ππ π= βππ = 0
β’ The sum of the forces acting on the body is equal to zero
β’ The sum of the moments of all forces in the system about
point π, added to all the couple moments, is equal to zero
Engineering Mechanics β Statics 5.03 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
βΊ
Β§1. Conditions for Rigid-Body Equilibrium
- The two equations of equilibrium for a rigid body
πΉ π = βπΉ = 0
ππ π= βππ = 0
where π is an arbitrary point
- Equilibrium in two dimensions
πΉπ₯ = 0
πΉπ¦ = 0
ππ§ = 0
- Note
πΉ ππππππ‘πππ, π πππ πππππππ‘π’ππ
known value β
unknown value ?
Engineering Mechanics β Statics 5.04 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2. Free-Body Diagram
- A Free-Body Diagram: a sketch of the isolated or free body
which shows all the pertinent weight forces, the externally
applied loads, and the reaction from its supports and
connections acting upon it by the removed elements
- General rules for support reactions
β’ If a support prevents the translation of a body in a given
direction, then a force is developed on the body in that
direction
β’ If rotation is prevented, a couple moment is exerted on the
body
Engineering Mechanics β Statics 5.05 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2. Free-Body Diagram
- Support Reactions
β’ Roller: prevents the beam from translating in the vertical
direction, the roller will only exert a force on the beam in this
direction
Engineering Mechanics β Statics 5.06 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/13/2013
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Β§2. Free-Body Diagram
- Support Reactions
β’ Roller: prevents the beam from translating in the vertical
direction, the roller will only exert a force on the beam in this
direction
πΉ β₯ ππππ
?
Engineering Mechanics β Statics 5.07 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2. Free-Body Diagram
β’ Pin: prevents translation of the beam in any direction
Engineering Mechanics β Statics 5.08 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2. Free-Body Diagram
β’ Pin: prevents translation of the beam in any direction
πΉ ??= πΉ π₯
β₯ ππ₯?
+ πΉ π¦ β₯ ππ¦?
Engineering Mechanics β Statics 5.09 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2. Free-Body Diagram
β’ Fixed Support: prevents both translation and rotation of the
beam
πΉ ??= πΉ π₯
β₯ ππ₯?
+ πΉ π¦ β₯ ππ¦?
π β?
Engineering Mechanics β Statics 5.10 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2. Free-Body Diagram
- Supports for Rigid Bodies Subjected to 2D Force Systems
β’ Cable: the reaction is a tension force which acts away from
the member in the direction of the cable
πΉ β?
β’ Weightless link: the reaction is a force which acts along the
axis of the link
πΉ β?
Engineering Mechanics β Statics 5.11 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2. Free-Body Diagram
β’ Roller: the reaction is a force which acts perpendicular to the
surface at the point of contact
πΉ β?
β’ Roller or pin in confined smooth slot: the reaction is a force
which acts perpendicular to the slot
πΉ β?
Engineering Mechanics β Statics 5.12 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
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Β§2. Free-Body Diagram
β’ Rocker: the reaction is a force which acts perpendicular to
the surface at the point of contact
πΉ β?
β’ Smooth contacting surface: the reaction is a force which acts
perpendicular to the surface at the point of contact
πΉ β?
Engineering Mechanics β Statics 5.13 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2. Free-Body Diagram
β’ Member pin connected to collar on smooth rod: the reaction
is a force which acts perpendicular to the rod
πΉ β?
β’ Smooth pin or hinge: the reactions are two components of
force, or the magnitude and direction of the resultant force
πΉ ??
Engineering Mechanics β Statics 5.14 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2. Free-Body Diagram
β’ Member fixed connected to collar on smooth rod: the
reactions are the couple moment and the force which acts
perpendicular to the rod
πΉ β? , π
β?
β’ Fixed support: the reactions are the couple moment and the
two force components, or the couple moment and the
magnitude and direction of the resultant force
πΉ ??
, π β?
Engineering Mechanics β Statics 5.15 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2. Free-Body Diagram
- Some typical examples of actual supports
Engineering Mechanics β Statics 5.16 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2. Free-Body Diagram
- Internal Forces
Engineering Mechanics β Statics 5.17 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2. Free-Body Diagram
- Weight and the Center of Gravity
π = ππ
π: mass, ππ
π : gravity acceleration, π/π 2
Engineering Mechanics β Statics 5.18 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
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Β§2. Free-Body Diagram
- Idealized Models
Engineering Mechanics β Statics 5.19 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2. Free-Body Diagram
- Example 5.1 Draw the free-body diagram of the uniform
beam. The beam has a mass of 100ππ
Solution
Engineering Mechanics β Statics 5.20 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2. Free-Body Diagram
- 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 36ππ. and the force in the
short link at π΅ is 90π
Solution
Spring force
πΉπ = 36 Γ 3.5 = 126π
Engineering Mechanics β Statics 5.21 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2. Free-Body Diagram
- Example 5.3 Two smooth pipes, each having a mass of
300ππ, are supported by the forked tines of the tractor. Draw
the free-body diagrams for each pipe and both pipes together
Solution
Engineering Mechanics β Statics 5.22 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§2. Free-Body Diagram
- 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 200ππ
Solution
Engineering Mechanics β Statics 5.23 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Problems
- Prob.5.1 Draw the free-body diagram of the 50ππ paper roll
which has a center of mass at πΊ and rests on the smooth
blade of the paper hauler. Explain the significance of each
force acting on the diagram
Engineering Mechanics β Statics 5.24 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
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Problem
- Prob.5.2 Draw the free-body diagram of member π΄π΅, which
is supported by a roller at π΄ and a pin at π΅. Explain the
significance of each force on the diagram
Engineering Mechanics β Statics 5.25 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Problem
- Prob.5.3 Draw the free-body diagram of the dumpster π· of
the truck, which has a weight of 5000π and a center of gravity
at πΊ . It is supported by a pin at π΄ and a pin-connected
hydraulic cylinder π΅πΆ (short link). Explain the significance of
each force on the diagram
Engineering Mechanics β Statics 5.26 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Problem
- Prob.5.4 Draw the free-body diagram of the beam which
supports the 80ππ load and is supported by the pin at π΄ and a
cable which wraps around the pulley at π· . Explain the
significance of each force on the diagram
Engineering Mechanics β Statics 5.27 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Problem
- Prob.5.5 Draw the free-body diagram of the truss that is
supported by the cable π΄π΅ and pin πΆ. Explain the significance
of each force acting on the diagram
Engineering Mechanics β Statics 5.28 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Problem
- Prob.5.6 Draw the free-body diagram of the bar, which has a
negligible thickness and smooth points of contact at π΄, π΅, and
πΆ. Explain the significance of each force on the diagram
Engineering Mechanics β Statics 5.29 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3. Equations of Equilibrium
- The conditions for equilibrium in two dimensions
βπΉπ₯ = 0
βΉ βπΉπ¦ = 0
βππ = 0
- Alternative sets of equilibrium equations
β’ The first alternative set
βπΉπ₯ = 0
βππ΄ = 0
βππ΅ = 0
β’ The second alternative set
βππ΄ = 0
βππ΅ = 0
βππΆ = 0
Engineering Mechanics β Statics 5.30 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
βπΉ = 0
βπ = 0
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Β§3. Equations of Equilibrium
- Example 5.5 Determine the horizontal and vertical
components of reaction on the beam caused by the pin at π΅
and the rocker. Neglect the weight of the beam
Solution
Free-body Diagram
Equations of Equilibrium
+β βπΉπ₯: 600πππ 450 β π΅π₯ = 0
+ β βπΉπ¦: π΄π¦ β 600π ππ450
β100 + π΅π¦ = 0
+βΊ βππ΅: β7π΄π¦ +600π ππ450 Γ5
β600πππ 450 Γ0.2
+100Γ2 = 0
βΉπ΄π¦ = 319π, π΅π₯ = 424π, π΅π¦ = 405π
Engineering Mechanics β Statics 5.31 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3. Equations of Equilibrium
- Example 5.6 The cord supports a force of 100π and wraps
over the frictionless pulley. Determine the tension in the cord at
πΆ and the horizontal and vertical components of reaction at pin π΄
Solution
Free-body Diagram
Equations of Equilibrium
+β βπΉπ₯: βπ΄π₯ + ππ ππ300 = 0
+ β βπΉπ¦: π΄π¦βππππ 450β100=0
+βΊ βππ΄: 100Γ0.5βπΓ0.5= 0
βΉπ = 100π, π΄π₯ = 50π, π΄π¦ = 187π
Note: The tension remains
constant as the cord passes
over the pulley
Engineering Mechanics β Statics 5.32 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3. Equations of Equilibrium
- Example 5.7 The member is pin-connected at π΄ and rests
against a smooth support at π΅. Determine the horizontal and
vertical components of reaction at the pin π΄
Solution
Free-body Diagram
Equations of Equilibrium
+β βπΉπ₯: π΄π₯ βππ΅π ππ300 = 0
+ β βπΉπ¦: π΄π¦ βππ΅πππ 300 β 60 = 0
+βΊ βππ΄: ππ΅Γ0.75β60Γ1β90=0
βΉππ΅ = 200π
π΄π₯ = 100π
π΄π¦ = 233π
Engineering Mechanics β Statics 5.33 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3. Equations of Equilibrium
- Example 5.8 The box wrench is used to tighten the bolt at π΄.
If the wrench does not turn when the load is applied to the
handle, determine the torque or moment applied to the bolt
and the force of the wrench on the bolt
Solution
Free-body Diagram
Equations of Equilibrium
+β βπΉπ₯: π΄π₯ β 525
13+ 30πππ 300 = 0
+ β βπΉπ¦: π΄π¦ β 5212
13β 30π ππ600 = 0
+βΊ βππ΄: ππ΄ β 5212
13Γ 0.3 β
30π ππ600 Γ 0.7 = 0
βΉ ππ΄ = 32.6ππ, π΄π₯ = 5π, π΄π¦ = 74π
Engineering Mechanics β Statics 5.34 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3. Equations of Equilibrium
- Example 5.9 Determine the horizontal and vertical components
of reaction on the member at the pin π΄ , and the normal
reaction at the roller π΅
Solution
Free-body Diagram
Equations of Equilibrium
+β βπΉπ₯: π΄π₯ βππ΅π ππ300 = 0
+ β βπΉπ¦: π΄π¦ β 500 + ππ΅πππ 300 = 0
+βΊ βππ΄: β500 Γ 3 + ππ΅πππ 300 Γ 6
βππ΅π ππ300 Γ 2 = 0
βΉ ππ΅ = 536π
π΄π₯ = 268π
π΄π¦ = 286π
Engineering Mechanics β Statics 5.35 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3. Equations of Equilibrium
- Example 5.10 The uniform smooth rod is subjected to a force
and couple moment. If the rod is supported at π΄ by a smooth wall
and at π΅ and πΆ either at the top or bottom by rollers, determine
the reactions at these supports. Neglect the weight of the rod
Solution
Free-body Diagram
Equations of Equilibrium
+β βπΉπ₯: πΆπ¦β²π ππ300+π΅π¦β²π ππ30
0βπ΄π₯ = 0
+ β βπΉπ¦: β300+πΆπ¦β²πππ 300+π΅π¦β²πππ 30
0 =0
+βΊ βππ΄: βπ΅π¦β² Γ 2 + 4000 β πΆπ¦β² Γ 6
+300πππ 300 Γ 8 = 0
βΉ π΅π¦β² = β1000.0π, πΆπ¦β² = 1346.4π
π΄π₯ = 173π
Engineering Mechanics β Statics 5.36 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
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Β§3. Equations of Equilibrium
- Example 5.11 The uniform truck ramp has a weight of 400π
and is pinned to the body of the truck at each side and held in
the position shown by the two side cables. Determine the
tension in the cables
Solution
Free-body Diagram
Equations of Equilibrium
βππ΄ =βππ ππ100Γ0.165+400Γ0.125πππ 300 =0βΉπβ² =π/2=755.6π
Engineering Mechanics β Statics 5.37 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§3. Equations of Equilibrium
- Example 5.12 Determine the support reactions on the
member in the figure. The collar at π΄ is fixed to the member
and can slide vertically along the vertical shaft
Solution
Free-body Diagram
Equations of Equilibrium
+β βπΉπ₯: π΄π₯ = 0
+ β βπΉπ¦: ππ΅ β 900 = 0
+βΊ βππ΄: ππ΄ β 500 β 900 Γ 1.5
+ππ΅ Γ (1πππ 450 + 3) = 0
βΉ π΄π = 0
ππ΅ = 900π
ππ΄ = β1.49πππ = 1.49ππ β»
Engineering Mechanics β Statics 5.38 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§4. Two- and Three-Force Members
- The solutions to some equilibrium problems can be simplified by
recognizing members that are subjected to only two or three forces
- Two-Force Members
β’ Forces applied at only two points on the member
β’ Force equilibrium: πΉ π΄ = βπΉ π΅
β’ Moment equilibrium: βππ΄ = 0 or βππ΅ = 0
βΉ πΉ π΄ ββ πΉ π΅
|πΉ π΄| = |πΉ π΅|
Engineering Mechanics β Statics 5.39 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§4. Two- and Three-Force Members
- Three-Force Members
β’ A member is subjected to only three forces
β’ Moment equilibrium can be satisfied only if the three forces
form a concurrent or parallel force systems
β’ If the lines of action of πΉ 1 and πΉ 2 intersect at point π, then the
line of action of πΉ 3 must also pass through point π so that the
forces satisfy: βππ = 0
β’ If the three forces are all parallel, the location of the point of
intersection π will approach infinity
Engineering Mechanics β Statics 5.40 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§4. Two- and Three-Force Members
- Example 5.13 The lever π΄π΅πΆ is pin supported at π΄ and
connected to a short link π΅π·. If the weight of the members is
negligible, determine the force of the pin on the lever at π΄
Solution
Free-body Diagram
Equations of Equilibrium
π = π‘ππβ1(0.7/0.4) = 60.30
+β βπΉπ₯: πΉπ΄πππ π βπΉπππ 450 +400 = 0
+ β βπΉπ¦: πΉπ΄π πππ βπΉπ ππ450 = 0
βΉ πΉπ΄ = 1.07ππ
πΉ = 1.32ππ
Engineering Mechanics β Statics 5.41 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Fundamental Problems
- F5.1 Determine the horizontal and vertical components of
reaction at the supports. Neglect the thickness of the beam
Engineering Mechanics β Statics 5.42 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
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Fundamental Problems
- F5.2 Determine the horizontal and vertical components of
reaction at the pin π΄ and the reaction on the beam at πΆ
Engineering Mechanics β Statics 5.43 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Fundamental Problems
- F5.3 The truss is supported by a pin at π΄ and a roller at π΅.
Determine the support reactions
Engineering Mechanics β Statics 5.44 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Fundamental Problems
- F5.4 Determine the components of reaction at the fixed
support π΄. Neglect the thickness of the beam
Engineering Mechanics β Statics 5.45 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Fundamental Problems
- F5.5 The 25ππ bar has a center of mass at πΊ . If it is
supported by a smooth peg at πΆ, a roller at π΄, and cord π΄π΅,
determine the reactions at these supports
Engineering Mechanics β Statics 5.46 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Fundamental Problems
- F5.6 Determine the reactions at the smooth contact points π΄,
π΅, and πΆ on the bar
Engineering Mechanics β Statics 5.47 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§5. Free-Body Diagrams (3D)
- Supports for Rigid Bodies Subjected to 3D Force Systems
β’ Cable: the reaction is a force which acts away from the
member in the known direction of the cable
πΉ β?
β’ Smooth surface support: the reaction is a force which acts
perpendicular to the surface at the point of contact
πΉ β?
Engineering Mechanics β Statics 5.48 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/13/2013
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Β§5. Free-Body Diagrams (3D)
β’ Roller: the reaction is a force which acts perpendicular to the
surface at the point of contact
πΉ β?
β’ Ball and socket: the reactions are three rectangular force
components
πΉ = πΉ π₯ β? + πΉ π¦
β? + πΉ π§
β?
Engineering Mechanics β Statics 5.49 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§5. Free-Body Diagrams (3D)
β’ Single journal bearing: the reactions are two force and two
couple-moment components which act perpendicular to the
shaft
πΉ = πΉ π₯ β? + πΉ π§
β?
π = ππ₯ β? +ππ§
β?
β’ Single journal bearing with square shaft: the reactions are
two force and three couple-moment components
πΉ = πΉ π₯ β? + πΉ π§
β?
π = ππ₯ β? +ππ¦
β? +ππ§
β?
Engineering Mechanics β Statics 5.50 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§5. Free-Body Diagrams (3D)
β’ Single thrust bearing: the reactions are three force and two
couple-moment components
πΉ = πΉ π₯ β? + πΉ π¦
β? + πΉ π§
β?
π = ππ₯ β? +ππ§
β?
β’ Single smooth pin: the reactions are three force and two
couple-moment components
πΉ = πΉ π₯ β? + πΉ π¦
β? + πΉ π§
β?
π = ππ¦ β? +ππ§
β?
Engineering Mechanics β Statics 5.51 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§5. Free-Body Diagrams (3D)
β’ Single hinge: The reactions are three force and two couple-
moment components
πΉ = πΉ π₯ β? + πΉ π¦
β? + πΉ π§
β?
π = ππ₯ β? +ππ§
β?
β’ Fixed support: the reactions are three force and three
couple-moment components
πΉ = πΉ π₯ β? + πΉ π¦
β? + πΉ π§
β?
π = ππ¦ β? +ππ§
β?
Engineering Mechanics β Statics 5.52 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§5. Free-Body Diagrams (3D)
- Some typical examples of actual supports
β’ Free-body Diagrams
Engineering Mechanics β Statics 5.53 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
ball-and-socket joint journal bearing thrust bearing pin
Β§5. Free-Body Diagrams (3D)
- Example 5.14 Consider the two rods and plate, along with
their associated free-body diagrams. The π₯ ,π¦ ,π§ axes are
established on the diagram and the unknown reaction
components are indicated in the positive sense. The weight is
neglected
Solution
Engineering Mechanics β Statics 5.54 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Properly aligned journal bearings at π΄, π΅, πΆ
The force reactions developed by the bearings are sufficient for equilibrium since they prevent the shaft from rotating about each of the coordinate axes
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Β§5. Free-Body Diagrams (3D)
Engineering Mechanics β Statics 5.55 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Pin at π΄ and cable π΅πΆ Moment components are developed by the pin on the rod to prevent rotation about the π₯ and π§ axes
Only force reactions are developed by the bearing and hinge on the plate to prevent rotation about each coordinate axis. No moments at the hinge are developed
Properly aligned journal bearing at π΄ and hinge at πΆ. Roller at π΅
Β§6. Equations of Equilibrium (3D)
- Vector Equations of Equilibrium
βπΉ = 0
βππ = 0
- Scalar Equations of Equilibrium
βπΉ = βπΉπ₯π + βπΉπ¦π + βπΉπ§π = 0
βππ = βππ₯π + βππ¦π + βππ§π = 0
or
βπΉπ₯ = 0, βπΉπ¦ = 0, βπΉπ§ = 0
βππ₯ = 0, βππ¦ = 0, βππ§ = 0
Engineering Mechanics β Statics 5.56 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§7. Constrains and Statical Determinacy
- To ensure the equilibrium of a rigid body, it is not only
necessary to satisfy the equations of equilibrium, but the body
must also be properly held or constrained by its supports
- Redundant constraints: when a body has redundant supports,
that is, more supports than are necessary to hold it in
equilibrium, it becomes statically indeterminate
- Statically indeterminate: there will be more unknown loadings
on the body than equations of equilibrium available for their
solutions
Engineering Mechanics β Statics 5.57 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§7. Constrains and Statical Determinacy
- Example: the beam is shown together with its free-body
diagram
The beam is statically indeterminate because of additional (or
redundant) supports reactions
There are five unknown ππ΄, π΄π₯, π΄π¦, π΅π¦, πΆπ¦ for which only three
equilibrium equations can be written
βπΉπ₯ = 0, βπΉπ¦ = 0, βππ = 0
Engineering Mechanics β Statics 5.58 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§7. Constrains and Statical Determinacy
- Example: the pipe is also statically indeterminate because of
additional (or redundant) supports reactions
The pipe assembly has eight unknowns, for which only six
equilibrium equations can be written
Engineering Mechanics β Statics 5.59 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§7. Constrains and Statical Determinacy
Statically indeterminate
the number of unknown reactive forces > the number of the
derived static equilibrium equations
How to solve ?
- The additional equations needed to solve statically
indeterminate problems are generally obtained from the
deformation conditions at the points of supports
- This is done in courses dealing with βMechanics of Materialsβ
Engineering Mechanics β Statics 5.60 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/13/2013
11
Β§7. Constrains and Statical Determinacy
- Improper constraints: having the same number of unknown
reactive forces as available equations of equilibrium does not
always guarantee that a body will be stable when subjected to
a particular loading
- For example, the pin support at π΄ and the roller support at π΅
for the beam are placed in such away that the lines of action
the reactive forces are concurrent at point π΄
- Consequently, the applied loading π will cause the beam to
rotates lightly about π΄ , and so the beam is improperly
constrained
Engineering Mechanics β Statics 5.61 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§7. Constrains and Statical Determinacy
- In three dimensions, a body will be improperly constrained if
the lines of action of all the reactive forces intersect a common
axis
- For example, the reactive forces at the ball-and-socket
supports at π΄ and π΅ all intersect the axis passing through π΄
and π΅
- Note: Since the moments of these forces about π΄ and π΅ are all
zero, then the loading π will rotate the member about the π΄π΅
axis
Engineering Mechanics β Statics 5.62 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§7. Constrains and Statical Determinacy
- Another way in which improper constraining leads to instability
occurs when the reactive forces are all parallel
- Note: the summation of forces along the π₯ axis will not be
equal zero
Engineering Mechanics β Statics 5.63 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§7. Constrains and Statical Determinacy
- Example 5.15 The homogeneous plate has a mass of 100ππ
and is subjected to a force and
couple moment along its edges. If it
is supported in the horizontal plane
by a roller at π΄ , a ball-and-socket
joint at π΅, and a cord at πΆ, determine
the components of reaction at these
supports
Solution
Free-body Diagram
Engineering Mechanics β Statics 5.64 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§7. Constrains and Statical Determinacy
Equations of Equilibrium
βπΉπ₯ = 0: π΅π₯ = 0
βπΉπ¦ = 0: π΅π¦ =0
βπΉπ§ = 0: π΄π§ +π΅π§ +ππΆ β300β981= 0
βππ₯ = 0: ππΆ Γ2+981Γ1+π΅π§ Γ2= 0
βππ¦ = 0: 300Γ1.5+981Γ1.5
βπ΅π§ Γ3βπ΄π§ Γ3β200= 0
βΉ π΄π§ = 790π
π΅π₯ = 0
π΅π¦ = 0
π΅π§ = β217π
ππΆ = 707π
Engineering Mechanics β Statics 5.65 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§7. Constrains and Statical Determinacy
- Example 5.16 Determine the components of reaction that the
ball-and-socket joint at π΄, the smooth journal bearing at π΅, and
the roller support at πΆ exert on the rod assembly
Solution
Free-body Diagram
Equations of Equilibrium
βπΉπ₯ = 0: π΄π₯ + π΅π₯ = 0
βπΉπ¦ = 0: π΄π¦ = 0
βπΉπ§ = 0: π΄π§ β 900 + π΅π§ + πΉπΆ = 0
βππ₯ = 0: β900Γ0.4+π΅π§Γ0.8+πΉπΆΓ1.2=0
βππ¦ = 0: β900 Γ 0.4 + πΉπΆ Γ 0.6 = 0
βππ§ = 0: π΅π₯ Γ 0.8 = 0
βΉ π΄π¦ = 0, π΄π₯ = 0, π΄π§ = 750π, π΅π₯ = 0, π΅π§ = β450π, πΉπΆ = 600π
Engineering Mechanics β Statics 5.66 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/13/2013
12
Β§7. Constrains and Statical Determinacy
- Example 5.17 The boom is used to support the 75π flowerpot.
Determine the tension developed in
wires π΄π΅ and π΄πΆ
Solution
Free-body Diagram
Equations of Equilibrium
πΉ π΄π΅ = πΉπ΄π΅π π΄π΅ππ΄π΅
= πΉπ΄π΅2π β 6π + 3π
22 + (β6)2+32
=2
7πΉπ΄π΅π β
6
7πΉπ΄π΅π +
3
7πΉπ΄π΅π
πΉ π΄πΆ = πΉπ΄πΆπ π΄πΆππ΄πΆ
= πΉπ΄πΆβ2π β 6π + 3π
(β2)2+(β6)2+32
= β2
7πΉπ΄πΆπ β
6
7πΉπ΄πΆπ +
3
7πΉπ΄πΆπ
Engineering Mechanics β Statics 5.67 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§7. Constrains and Statical Determinacy
πΉ π΄π΅ =2
7πΉπ΄π΅π β
6
7πΉπ΄π΅π +
3
7πΉπ΄π΅π
πΉ π΄πΆ = β2
7πΉπ΄πΆπ β
6
7πΉπ΄πΆπ +
3
7πΉπ΄πΆπ
π = β75π
βππ = 0: π π΄ Γ πΉ π΄π΅ + πΉ π΄πΆ +π = 0
βΉ6π Γ 2
7πΉπ΄π΅π β
6
7πΉπ΄π΅π +
3
7πΉπ΄π΅π +
β2
7πΉπ΄πΆπ β
6
7πΉπ΄πΆπ +
3
7πΉπ΄πΆπ β75π = 0
βΉ18
7πΉπ΄π΅ +
18
7πΉπ΄πΆ β450 π
+ β12
7πΉπ΄π΅+
12
7πΉπ΄πΆ π = 0
Engineering Mechanics β Statics 5.68 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§7. Constrains and Statical Determinacy
18
7πΉπ΄π΅ +
18
7πΉπ΄πΆ β 450 π
+ β12
7πΉπ΄π΅ +
12
7πΉπ΄πΆ π = 0
βΉ βππ₯ = 0: 18
7πΉπ΄π΅ +
18
7πΉπ΄πΆ β 450 = 0
βππ¦ = 0: 0 = 0
βππ§ = 0: β12
7πΉπ΄π΅ +
12
7πΉπ΄πΆ = 0
βΉ πΉπ΄π΅ = 87.5π
πΉπ΄πΆ = 87.5π
Engineering Mechanics β Statics 5.69 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§7. Constrains and Statical Determinacy
- Example 5.18 Rod π΄π΅ is subjected to the 200π force.
Determine the reactions at the ball-and-socket joint π΄ and the
tension in the cables π΅π· and π΅πΈ
Solution
Free-body Diagram
Equations of Equilibrium
πΉ π΄ = π΄π₯π + π΄π¦π + π΄π§π
ππΈ = ππΈπ
ππ· = ππ·π
πΉ = β200π
Engineering Mechanics β Statics 5.70 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§7. Constrains and Statical Determinacy
Applying the force equation of equilibrium
βπΉ = 0: πΉ π΄ + ππΈ + ππ· + πΉ = 0
βππ΄ = 0: π πΆ Γ πΉ + π π΅ Γ ππΈ + ππ· = 0
βΉ π΄π₯ + ππΈ π + π΄π¦ + ππ· π
+ π΄π§ β 200 π = 0
0.5π + π β π Γ β200π
+ π +2π β2π Γ ππΈπ +ππ·π = 0
βΉ π΄π₯ + ππΈ π + π΄π¦ + ππ· π
+ π΄π§ β 200 π = 0
2ππ· β 200 π + β2ππΈ + 100 π
+ ππ· β2ππΈ π = 0
βΉππ· = 100π, ππΈ = 50π, π΄π₯ =β50π, π΄π¦ =β100π, π΄π§ = 200π
Engineering Mechanics β Statics 5.71 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Β§7. Constrains and Statical Determinacy
- Example 5.19 The bent rod is supported at π΄ by a journal
bearing, at π· by a ball-and-socket joint, and
at π΅ by means of cable π΅πΆ. Using only one
equilibrium equation, obtain a direct solution
for the tension in cable π΅πΆ. The bearing at π΄
is capable of exerting force components
only in the π§ and π¦ directions since it is
properly aligned on the shaft
Solution
Free-body Diagram
Engineering Mechanics β Statics 5.72 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/13/2013
13
Β§7. Constrains and Statical Determinacy
Equations of Equilibrium
The cable tension may be obtained directly
by summing moments about an axis that
passes through points π· and π΄
π’ =π π·π΄ππ·π΄
= β1
2π β
1
2π = β0.7071(π + π )
The sum of the moments about this axis is zero
βππ·π΄ = π’β π Γ πΉ = 0
βΉ π’ π π΅ Γ ππ΅ + π πΈ Γπ = 0
β0.7071(π + π ) βπ Γππ΅π β0.5π Γ β981π = 0
β0.7071(π + π ) βππ΅ + 490.5 π = 0
β0.7071 βππ΅ + 490.5 + 0 + 0 = 0
βΉ ππ΅ = 490.5π
Engineering Mechanics β Statics 5.73 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Fundamental Problems
- F5.7 The uniform plate has a weight of 500π. Determine the
tension in each of the supporting cables
Engineering Mechanics β Statics 5.74 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Fundamental Problems
- F5.8 Determine the reactions at the roller support π΄, the ball-
and-socket joint π·, and the tension in cable π΅πΆ for the plate
Engineering Mechanics β Statics 5.75 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Fundamental Problems
- F5.9 The rod is supported by smooth journal bearings at π΄, π΅
and πΆ and is subjected to the two forces. Determine the
reactions at these supports
Engineering Mechanics β Statics 5.76 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Fundamental Problems
- F5.10 Determine the support reactions at the smooth journal
bearings π΄, π΅, and πΆ of the pipe assembly
Engineering Mechanics β Statics 5.77 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Fundamental Problems
- F5.11 Determine the force developed in cords π΅π·, πΆπΈ, and πΆπΉ
and the reactions of the ball-and-socket joint π΄ on the block
Engineering Mechanics β Statics 5.78 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
2/13/2013
14
Fundamental Problems
- F5.12 Determine the components of reaction that the thrust
bearing π΄ and cable π΅πΆ exert on the bar
Engineering Mechanics β Statics 5.79 Equilibrium of a Rigid Body
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien