C6.3 Virtual Work

If you have a laser-sharp eye, you would have noticed that in all the previous questions, we could only calculate our displacement Δ or slope θ at the point where the load is applied.

What if we wanted to find Δ or θ away from the point of load application? Or what if we have more than one load acting on the structure? How do we find our Δ or θ in such scenarios?

We can no longer use the method defined in Chapter 6.2. But fear not! We have another method, called virtual work.

Basically, this method works by applying a “virtual load” on the point where we want to find our Δ or θ. This “virtual load” will result in “virtual work” done on the structure, which we can then equate to our internal strain energy to find Δ or θ:

Virtual work formula

C6.3 Virtual Work

If you have a laser-sharp eye, you would have noticed that in all the previous questions, we could only calculate our displacement Δ or slope θ at the point where the load is applied.

What if we wanted to find Δ or θ away from the point of load application? Or what if we have more than one load acting on the structure? How do we find our Δ or θ in such scenarios?

We can no longer use the method defined in Chapter 6.2. But fear not! We have another method, called virtual work.

Basically, this method works by applying a “virtual load” on the point where we want to find our Δ or θ. This “virtual load” will result in “virtual work” done on the structure, which we can then equate to our internal strain energy to find Δ or θ:

Virtual work formula

Because the load applied is “virtual”, then surely our internal strain energy U_i is different as well! And so we have a slightly different method to calculate our strain energy:

Virtual work formula for external work done

Note:

Notice there are small and capital letters for our various loadings
N, V, M, T are still the real internal forces caused by the original external loadings.
n, v, m, t are the “virtual” internal forces caused by the external load.
Having two sets of internal forces means we have to perform our calculations twice, once for the real internal forces (N, V, M, T) based on the original external loadings, and another time for the virtual internal forces (n, v, m, t) based on the virtual load.
Section properties A, I, J and the moduli E, G are the same as Chapter 6.1

Let’s look at an example now to see how virtual work "works" (pun intended).

Because the load applied is “virtual”, then surely our internal strain energy U_i is different as well! And so we have a slightly different method to calculate our strain energy:

Virtual work formula for external work done

Note:

Notice there are small and capital letters for our various loadings
N, V, M, T are still the real internal forces caused by the original external loadings.
n, v, m, t are the “virtual” internal forces caused by the external load.
Having two sets of internal forces means we have to perform our calculations twice, once for the real internal forces (N, V, M, T) based on the original external loadings, and another time for the virtual internal forces (n, v, m, t) based on the virtual load.
Section properties A, I, J and the moduli E, G are the same as Chapter 6.1

Let’s look at an example now to see how virtual work "works" (pun intended).