2. Chapter Objectives Chapter Outline
Principle of virtual work and how to apply in statics problem for pin-connected members
Chapter Outline
Definition of Work
Principle of Virtual Work
Principle of Virtual Work for a System of Connected Rigid Bodies

3. 11.1 Definition of Work Work of a Force
The work occurs from the force F when there is a displacement in the direction of the force
Consider the force F and the displacement dr, and the angle between the angle and displacement is θ, hence the component force in the direction of the displacement is F cos θ
Work can be writtten as
dU = F dr cos θ
Using dot product
dU = F·dr
Work is scale with the unit Joule (J = N-m)

4. 11.1 Definition of Work Work of a Couple Moment
Consider RBD with the couple moment with the force F and -F and hence the couple is M = Fr
For the object moved from A by drA and from B by drB
drB = drA + dr’
The work from F and -F with drA is cancelled out
Hence only work dr’ (=r d θ )
dU = F dr’ = F r d θ = M d θ
The work from Moment couple
dU = M d θ

5. 11.1 Definition of Work
Virtual Work = Consider that there is a virtual displacement, i.e. what would happen if there is a displacement, but does not actually occur
Virtual is represented as
The virtual work from the force F with the virtual displacement with is the angle between F and
The virtual work from couple moment M

6. 11.2 Principle of Virtual Work
Principle of Virtual Work: If the object is in equilibrium, the sum of virtual work must be zero
Can be used in stead of equilibrium equations
Example, FBD of an object on the floor
If virtual displacement is acting downwards δy, there will be virtual work downward Wδy and the virtual work upward –Nδy
Total Virtual Work Done;
δU = Wδy –Nδy = (W-N)δy =0
Since δy ≠ 0, then N = W
= Equilibrium

7. 11.2 Principle of Virtual Work
Example, consider equilibrium of a beam
Apply a virtual work at point B with the virtual angle δθ
The virtual work is from P and Ay only
Since δy=L δθ and δy’=(L/2) δθ the virtual work is
δU = Ay(Lδθ) – P(L/2)δθ = (AyL– PL/2) δθ = 0
dure to δθ ≠ 0, Ay = P/2
L/2
L/2

8. 11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Virtual can be used to solve all equilibrium system with pin-connected member
Use when degrees of freedom is 1 or the motion in the system or can use only one unknown θ from the example below

9. 11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Procedure for Analysis
Free Body Diagram
Draw FBD of the system and then use q as one unknow displacement from degrees of freedom of the system
Sketch how the system can be moved with the virtual displacement δq

10. 11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Procedure for Analysis
Virtual Displacements
Locate the position coordinate s of the force in the system in the function of q from a static point
Write a reletionship s and q in Differentiation form and use δs in δq

11. 11.3 Principle of Virtual Work for a System of Connected Rigid Bodies
Procedure for Analysis
Virtual-Work Equation
Write Virtual-Work equation by assuming each point s moved by δs and use the write sign
No work from internal force, except deformable object such as spring
Convert δs in the form δq
Solve the equation using the virtual work principle

12. Example 11.1
Determine the angle θ for equilibrium of the two-member linkage. Each member has a mass of 10 kg.

13. Solution (by equilibrium)

14. Solution (by virtual work)
FBD
One degree of freedom since location of both links may be specified by a single independent coordinate. (q= θ)
θ undergoes a positive (CW) virtual rotation δθ, only the active forces, F and the 2 units of 9.81N weights do work.
Virtual Displacement
Position coordinates xB and yW

15. Solution Virtual Work Equation
If δ xB and δyw were both positive, forces W and F would do positive work.
For virtual work equation for displacement δθ,
Relating virtual displacements to common δθ,

16. Example 11.2
Determine the required force P needed to maintain equilibrium of the scissors linkage when θ=60°. The spring is unstretched when θ=30°. Neglect the mass of the links.

17. Solution
FBD
Virtual Displacement
Virtual Work Equation

18. Example 11.3
If the box has a mass of 10 kg, determine the couple moment M needed to maintain equilibrium when θ=60°. Neglect the mass of the members.

19. Solution
FBD
Virtual Displacement
Virtual Work Equation

20. Example 11.4
The mechanism supports the 500-N cylinder. Determine the angle θ for equilibrium if the spring has an unstretched length of 2 m when θ=0°. Neglect the mass of the members.

21. Solution
FBD
Virtual Displacement
Virtual Work Equation

22. Solution by equilibrium

23. Solution by virtual work