Understanding Castigliano's First Theorem Proof

I'm trying to understand the derivation of Castigliano's first theorem in the book Fundamentals of Finite Element Analysis by David V. Hutton, which goes as follows.

The strain energy is defined as $$U=\sum_{j=1}^N\int_{0}^{\delta_j}F_jd\delta_j$$

Where each of the j loads has a corresponding displacement $$\delta_j$$

Consider some variation in the displacement $$\Delta\delta_j$$ caused by varying the corresponding force by $$\Delta F _j$$ and keeping all the other displacements constant. Then the change in strain energy is $$F_j\Delta\delta_j+\int_{0}^{\Delta\delta_j}\Delta Fd\delta_j$$

I understand where the first term comes from, but I don't understand why the integral has bounds from $$0$$ to $$\Delta\delta_j$$ rather than bounds from $$\delta_j$$ to $$\delta_j+\Delta\delta_j$$, like this

$$\Delta U=\int_{0}^{\Delta\delta_j+\delta_j}F_j+\Delta F_jd\delta-\int_{0}^{\delta_j}F_jd\delta\approx F_j\Delta\delta_j+\int_{0}^{\Delta\delta_j+\delta_j}\Delta F_jd\delta$$

The distinction doesn't really change the final result since the integral is negligible, but I still don't understand why it takes that form.

• Because the total strain energy equals to (F_j + delF_j)*del(sigma_j). or (force + force increment)*change in length.
– r13
Oct 10 at 21:17
• Shouldn't the total change in length still be from 0 to \delta+\Delta\delta because the original definition of the strain energy used \delta=0 as the point where there was 0 potential energy
– QED
Oct 11 at 0:46
• Is the displacement curve nonlinear?
– r13
Oct 11 at 2:03
• It's meant to be linear (within the elastic region)
– QED
Oct 11 at 3:54
• I am puzzled about the second term as Castigliano's first theorem is usually expressed as force = partial derivative of (strain energy/deflection). Let's see how others say.
– r13
Oct 11 at 4:01

If I am not mistaken Castigliano considers the elastic region of an element and therefore in that region displacement and force are proportional. So, a diagram like this describes the strain energy when $$F_j$$ is applied (a single element):

When an additional force $$\Delta F_j$$ is applied and the displacement $$\Delta \delta _j$$ is observed then:

Regarding the strain energies, the additional portions are the orange and the green denoted in the image below:

If my assumption above are correct (see note below) from the images, I think its obvious why:

• orange = $$F_j\cdot \Delta\delta_j$$ and,
• green = $$\int_0^{\Delta\delta_j}\Delta F_j\cdot d\delta_j$$

NOTE: I did this by memory and deduction, so please give some feedback (until I go back to my books and notes and find any flaws myself

• Wouldn't the bounds of the integral still be from the original displacement to the original displacement+the change in displacement because the area is between those values rather than 0 to the displacement
– QED
Oct 12 at 3:53
• The way I understand it, is that $\int_{\delta j}^{\delta j + \Delta \delta j} F_j d\delta j$ is the sum of the orange and green. Notice the subtle difference with $\int_{\delta j}^{\delta j + \Delta \delta j} \color{red}{\Delta} F_j d\delta j$ Oct 12 at 8:10