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Learning the Virtual Work Method in Statics:
What Is a Compatible Virtual Displacement?
Ing-Chang Jong
University of Arkansas
Session 3568: Innovative Teaching Techniques in Mechanics
2:15 4:00 p.m., Wednesday, June 21, 2006
Chicago, Illinois
Proceedings of the 2006 ASEE Annual Conference & Exposition
Mechanics Division 1
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Key concepts: displacement center, compatible virtual
displacement, & rigid-body virtual displacement
2( ) 02
LB B
A body undergoes a compatible
virtual displacement from position
AB to position B, where thedisplacement center is atD.
A body undergoes a rigid-body virtual
displacement fromAB to AB, and acompatible virtual displacement fromAB
toAB. The displacement centeris atA.
Mechanics Division 3
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Work of a force:||1 2U Fq = =F q
=1 2 ( )U MWork of a moment:Radian measure formula: s r =
Virtual work: It is the work done by a force or momenton a body during a virtual displacementof the body.
Principle of virtual work:
If a body is in equilibrium, the total virtual workUof
the external force system acting on its free body during
any compatible virtual displacementof its free body is
equal to zero; i.e.,
0 =U
Mechanics Division 4
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Example 1. Determine only the reaction moment MA at thefixed supportA of the frame loaded as shown.
U= 0:
( 2 ) 6(12 )
2( 4 ) 10 0
AM
+
+ + =
37AM = 37 kN mA = M
Mechanics Division 6
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Example 2. Determine only the reaction moment MA at thefixed supportA of the beam loaded as shown.
1400 lb ftA = M 1400
A
M =( ) 500( 4 ) 600( ) 0AM + + =U= 0:
Mechanics Division 7
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Example 3. Determine only the vertical reaction force Ay at thefixed supportA of the beam loaded as shown.
350 lby = AU= 0: (4 ) 500( 4 ) 600( ) 0yA
+ + = 350yA =
Mechanics Division 8
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Example 4. Determine only the vertical reaction force Ey at theroller supportEof the beam loaded as shown.
1050 lby
= E45
600( 2.5 ) (6 ) (1500)( 4 ) 0yE + + =U= 0:
1050y
E =
Mechanics Division 9
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Example 5. Determine only the vertical reaction force Hy at theroller supportHof the beam loaded as shown.
750 lby = HU= 0:
4
5600( ) (1500)( 2 ) 200( 6 ) (4 ) 0
yH + + + = 750yH =
Mechanics Division 10
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Example 6. Determine only the reaction moment MA at thefixed supportA of the frame loaded as shown.
Mechanics Division 11
U= 0:
( 4 ) 10(16 )
25(2 ) 15( 8 )
5(6 ) 16( ) 0
AM
+
+ + + + =
26 kNmA =M 26AM =
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Example 7. Determine only the horizontal reaction force Ax atthe fixed supportA of the frame loaded as shown.
Mechanics Division 12
U= 0:
( 16 ) 10(16 )
25(2 ) 15( 8 )
5(6 ) 16( ) 0
xA
+
+ + + + =
6.5 kNx = A6.5xA =
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Concluding Remarks
The key concepts in virtual work method include: work of aforce, work of a moment, displacement center, compatiblevirtual displacement, and radian measure formula.
The virtual work method in Statics consists ofthree majorsteps: (a) draw the FBD, (b) draw the VDD with a strategy,and (c) set U= 0 to solve for the unknown.
The strategy in drawing the VDD is to give the free body a
compatible virtual displacementin such a way that the onespecified unknown, but no other unknowns, will be involvedin the total virtual work done.
George Bernard Shaw once said, You see things; and you
say, Why? But I dream things that never were; and I say,Why not?
Mechanics Division 13
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QuestionsMechanics Division 14