What is the method to compute the effective bending stiffness "EI" of a composite structure, in this case a shaft (as seen below).

The shaft is hollow with outer radius $r_{o}$, inner radius $r_{i}$, and modulus of elasticity $E_{1}$. Four rods are inserted into the shaft's thickness. They are placed at an equal angular separation. They have a radius of $r_{m}$ and are at a distance $r_{c}$ from the shaft's center. They are all made of the same material with modulus of elasticity $E_2$.

Is the effective bending stiffness simply the sum of all bending stiffnesses $\sum_{}^{}E_{i}\int{}\int{r^2dA}$?

enter image description here


Regards, Omar

  • 1
    $\begingroup$ Are the rods a sliding fit, an interference fit or even ribbed ( like rebar)? $\endgroup$
    – Solar Mike
    Sep 21, 2020 at 12:12
  • 1
    $\begingroup$ We can assume a rigid connection between both materials, i.e. no slip. $\endgroup$
    – Omar
    Sep 21, 2020 at 13:08

1 Answer 1


The bending stiffness will be determined by the second moment of area ($I$). The formula you provide $\int\int r^2 da$ is for the Polar Moment of area ($J_p$), and is valid for torsional problems.

Apart from little issue you are on the right track. Assuming that:

  • x is the horizontal axis
  • y is the vertical axis

then you are after $I_{xx}$.

Additionall, I'm going to number 1 the right most hole and proceed CCW to number the holes (so 2 at the top, 3 to the left, and 4 bottom).

For the polymeric structure you need to take the $I_{xx}$ of the cylinder and subtract the $I_xx$ of the reinfocrement holes. i.e.

$$I_{xx,poly} = I_{xx,cyl} - \sum_{i=1}^4 I_{xx, hole\; i}$$


  • $I_{xx,cyl}= \frac{\pi (r_o^4-r_i^4)}{4}$
  • $I_{xx,hole\;1}=I_{xx,hole\;3}= \frac{\pi r_m^4}{4}$
  • $I_{xx,hole\;2}=I_{xx,hole\;3}= \frac{\pi r_m^4}{4} + \pi r_m^2 r_c^2$

So after substitution:

$$I_{xx,poly} = \frac{\pi (r_o^4-r_i^4)}{4} - 4 \frac{\pi r_m^4}{4} - 2\pi r_m^2 r_c^2$$

Additionally, the reinforcement moment of area is:

$$I_{xx,reinf} =\sum_{i=1}^4 I_{xx, hole\; i} =4 \frac{\pi r_m^4}{4} + 2\pi r_m^2 r_c^2 $$

Bottom line: Bending stiffness

The bending stiffness is calculated by :

$$EI_{total} = E_1\cdot I_{xx,poly} + E_2\cdot I_{xx,reinf} $$


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