Virtual Work Method: Flexural Strains – Beams

The virtual work method is another way of finding the deflection of structures. In this post, let’s use it on an example to demonstrate how to use it for beams or frames. If you’re interested in the derivation of the equation, we highly recommend reading this post first.

Virtual Work: Flexural Strains

To start let’s consider this beam example:

We are to find the deflection components at C: horizontal translation, vertical translation, and rotation using the virtual work method.

Virtual Work Flexural Strains Example 1

First, by inspection, we can say that the horizontal translation is zero because there are no horizontal loading components that would cause the beam to deflect along the x-axis; That leaves us only two components to work with.

Virtual Work

We begin the process by identifying the key equations to solve for the deflection at C:

  • The unit load or couple (1 kN or 1 kN•m) represents the forces being applied on the virtual beam.
  • Δ or θ represents real translation and rotation – the unknowns of the problem.
  • M represents the internal bending moment expressions of the real beam.
  • mv or mα are the internal bending moment expressions of the virtual beam.
  • EI is flexural rigidity.

Set-Up the Virtual Structures

Since the equations require the internal moment expression for the virtual beam (mv and mα), we need to set-up virtual structures for each deflection component and represent vertical translation and rotation as a unit load and unit couple at C:

Virtual Work Flexural Strains Example 2

We assume our unit load to act downward and the unit load to act clockwise; this means that we have also assumed that the translation and rotation to be downward and clockwise respectively. In case our answers are negative, it means we have the wrong assumption.

An important thing to take note is how to segment the beam. As you can see, there are five points (A, B, C, D, and E) which would result in four segments (AB, BC, CD, and DE). For each structure, virtual or real, we would need to indicate the position of these points.

Generally, these are the rules for segmenting the beam: (1) there is a change in loading condition, or (2) there is a change in cross-sectional properties.

Formulate the Moment Equations

The next thing we need to do is to set-up the M, mv, and mα for each segment of the beam. We do that by formulating moment equations by placing a cutting plane and considering either the left or right section of the structures. It is important to take note of our sign conventions:

Segment AB

We place a section between points A and B and analyse the left section:

  • Limits (x-coordinate): A to B (0 to 2)
  • Moment of Inertia: I

The moment equations of AB are the following:

Virtual Work Flexural Strains Example 3
Segment BC

Next, we place a section between points B and C and investigate the left section:

  • Limits (x-coordinate): B to C (2 to 6.5)
  • Moment of Inertia: I

The moment equations of BC are the following:

Virtual Work Flexural Strains Example 4
Segment CD

Next, we place a section between points C and D. This time, we analyse the right section because it is easier to formulate the moment equations. When creating these equations, remember our change in sign convention.

  • Limits (x-coordinate): C to D (6.5 to 11)
  • Moment of Inertia: I

The moment equations of CD are the following:

Virtual Work Flexural Strains Example 5
Segment DE

Next, we place a section between points D and E. Again, we investigate the right section because it is easier to formulate the equations:

  • Limits (x-coordinate): D to E (11 to 12.5)
  • Moment of Inertia: I

The moment equations of DE are the following:

Virtual Work Flexural Strains Example 6

Integrate the Equations

With the moment equations set-up, the only thing left to do is to apply the virtual work equations.

The deflection component is equal to the sum of definite integrals for every segment. For example, to solve for the vertical deflection at C:

Next, for the rotation at C:

At this point, you have now solve for the deflections at C. In terms of the directions, translation is positive which means our assumption that it is acting downward is correct. On the other hand, rotation is negative which means that the direction is counterclockwise (not clockwise which was our initial assumption).

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