Convolution as seen in a Virtual Work (Structural Analysis) Problem

Linear or nonlinear?

Recently, I read a post featuring an example virtual work problem in structural analysis.

On the first page, there was the usual integration equation in which the strain energy of a virtual moment was multiplied by the strain energy of a single point load. Nothing unusual at all about the problem, or the approach, or the equation.

My surprise was in seeing the shortcut calculation. The diagram is at the bottom of Page 1 (the diagram with two triangles). The integration equation and the shortcut are near the top of Page 2 (the third equation). Here’s the link to the PDF.

I had never seen the shortcut used, nor the table which explained it.

I decided to explore this on my own.

My exploration resulted in a few charts as follows. All of these are live interactive (built using Plotly).

Hover over chart elements (on a mobile device, touch) to see data values.

I derived another special case as shown below. This pattern has two linear elements and the integration is for the pattern squared.

# In this example, a <= b
x, a, b, c, d, L1, L2 = symbols('x, a, b, c, d, L1, L2')
Q = b + x * (c-b)/(L1)
R = c + (x-L1) * (d-c)/(L2-L1)
Q2 = integrate(Q**2, (x, 0, L1))
R2 = integrate(R**2, (x, L1, L2))
T = Q2 + R2
T = simplify(T)

I note that the most general convolution of two independent linear segments is:

# In this example, a <= b
x, a, b, c, d, L1, L2 = symbols('x, a, b, c, d, L1, L2')
R1 = a + (x-L1) * (b-a)/(L2-L1)
R2 = c + (x-L1) * (d-c)/(L2-L1)
T = integrate(R1*R2, (x, L1, L2))
T = simplify(T)