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    Virtual Work

    A force does work when it produces a displacement along its line of action. Virtual work (\( dU \)) is the work produced by a force over an infinitesimally small displacement \( dr \).
    $$ dU = F\cdot dr\ $$
    Virtual work of a couple moment:
    $$ dU = \vec{M}\cdot d\theta \vec{k}\ $$
    Virtual work \( dU \) completed by a couple moment, \( M \) .
    We can use virtual work to solve for the forces in a system without needing to solve for all of the support reactions. For example, if a truss is completely pinned on one end, it doesn't move, which means there is no virtual work completed on that pin because there are no virtual displacements!

    Virtual displacements

    A virtual displacement is an infinitesimally small displacement (or rotation) that is possible in the system, denoted usually as \( dx \) or \( d\theta \). It's important to note that virtual displacements are assumed to be possible but don't actually exist.

    Principle of Virtual Work

    If a body is in equilibrium, the sum of the virtual work done by all the forces and couple moments acting on the body is zero.
    $$ \sum{dU} = \sum{\vec{F}d\vec{x}} + \sum{\vec{M}d\vec{\theta}} = 0 $$

    Virtual work analysis

    Steps for completing an analysis of a system using virtual work:

    1. Draw a free body diagram of the system, including a coordinate system
    2. Sketch the "deflected" position of the system
    3. Define the position coordinates (measured from a fixed point), select the parallel line of action component, and remove all the forces that do not complete any work
    4. Differentiate the position coordinates to obtain virtual displacement
    5. Write the virtual work equation, expressing the virtual work of every force and moment
    6. Factor out the common virtual displacement term and solve