Composite Beam Sections

In normal language, when we talk about “composites”, that usually refers to glass fiber composites (such as are used in the midsection of Darwin III). Such materials generally form a homogeneous material on the level of analysis.

In structural mechanics II, we’re talking about large-scale inhomogeneities in otherwise solid beams. Since material further from the neutral axis experiences greater stress in bending, you put your more expensive material further from the neutral axis.

Imagine a beam made of wood and steel:

The strain is evenly distributed through the material. The stress, however, is not. Recall that in the elastic region of stress, the amount of strain is proportional to the amount of stress:

\sigma = E \ \epsilon,

where E is the Young’s Modulus specific to that material.

Since E varies throughout the length of the section, the stress-height graph is not continuous.

You deal with this problem by refactoring the composite beam as a beam made entirely of one material (either the timber or the steel).

Take a thin slice of the material to be replaced of height dy and area dA:

dF = b_s \times dy \times E_s \times \epsilon,

where b_s is the breadth of the steel in that section, b_s \times dy is the area of the cross section, and E_s \times \epsilon is the stress in that section of the beam (Young’s Modulus for steel times the amount of strain in that section of the steel).

1 Flexural Rigidity

This quantity gives the stiffness of the material as a whole:

G = E\ I

It takes into account both the material stiffness E (the Young’s Modulus) and the geometric stiffness I (the second moment of area).