11.3:
Virtual Work
Consider a body in static equilibrium undergoing an infinitesimal small virtual displacement or a rotation.
The virtual work done is the product of the virtual force and displacement. Similarly for a virtual rotation, the work done is the product of the moment and virtual angular displacement.
The principle of virtual work states that the algebraic sum of the virtual work done by all the forces and moments is zero for any virtual displacement or rotation.
When a ball undergoes a virtual downward displacement, its weight does positive virtual work, while the normal force does negative virtual work.
For equilibrium conditions, the sum of all the virtual work done must be zero.
Similarly, when a supported beam undergoes a virtual rotation, only two forces do the work.
The components of the reaction force at the support do not contribute to virtual work.
By considering the virtual displacements along the y-direction, and applying the principle of virtual work, the virtual work equation is derived.
The term in the parentheses indicates the state of rotational equilibrium.
11.3:
Virtual Work
The principle of virtual work states that if a body is in static and dynamic equilibrium, then the sum of all the virtual work done by all external forces and couple moments for any given virtual displacement must be zero.
In static equilibrium, a body can experience an imaginary or virtual movement, such as displacement or rotation. The virtual work done by a force is equal to the dot product of force and virtual displacement in the direction of the force. When it comes to virtually rotating a couple moment, the same principle applies. The virtual rotational work done is determined by multiplying the couple moment with its respective virtual rotation.
To illustrate this concept using an example, suppose there is a ball sitting on top of a flat surface. Drawing its full-body diagram will reveal that when there is a downward virtual displacement, the weight will do positive virtual work, while the normal force will do negative virtual work. To achieve equilibrium, the sum of all these forces must be zero, and so an equation representing this condition can be formulated accordingly.
The concept of virtual work is used to solve problems related to both particles and rigid bodies. When dealing with rigid bodies subjected to coplanar forces, three separate equations are required, since they relate to different types of displacements—translation in the x and y directions and rotations about an axis perpendicular to the x-y plane.
In conclusion, virtual work is a fundamental concept in mechanics that allows engineers and scientists to predict the behavior of structures and machines without physically testing them. It is a powerful tool that can be used to analyze the behavior of static and dynamic systems, and it has wide-ranging applications in engineering and science.
Suggested Reading
- Hibbeler, R.C. (2016). Engineering Mechanics: Statics. Fourteenth Edition, New Jersey: Pearson. Section 11.1, Pp. 583-584.
- Meriam, J.L., Kraige, L.G. and Bolton, J.N. (2016). Engineering Mechanics: Statics and Dynamics. Eighth Edition, Singapore: John Wiley & Sons. Section 7.2 Pp. 394-395.
- Beer, F.P., Johnston, E.R., Mazurek, D.F., Cornwell, P.J. and Self, B.P. (2016). Vector Mechanics For Engineers. Eleventh Edition, New York: McGraw-Hill Education. Pp. 577-578.