# Bending stiffness of composite shaft

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 $$\sum_{}^{}E_{i}\int{}\int{r^2dA}$$? Thanks!

Regards, Omar

• Are the rods a sliding fit, an interference fit or even ribbed ( like rebar)? – Solar Mike Sep 21 '20 at 12:12
• We can assume a rigid connection between both materials, i.e. no slip. – Omar Sep 21 '20 at 13:08

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.

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

where:

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

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 :

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