5.3. Bending Test

A bending test is a good way to experimentally determine the elastic modulus \(E\) of a material. Starting point is a material sample with a constant width \(w\) and height \(h\). It is important that the dimensions, especially the height, are fairly precise, since they have a large influence on the final result.

The useful length \(l\) of the sample should be much larger than the cross section dimensions such that it can bend enough for easy measuring while still staying in the elastic range of the material. On the other hand, the sample shouldn’t be so long that it bends significantly under its own weight either. A suggestion would be \(l \ge 100 \cdot h\).

To perform the test, the sample is either clamped on one end, as shown in Figure 27, or placed between two supports as shown in Figure 28. It is then subjected to a force \(F\) and the resulting deflection \(s\) is measured. A good way to apply an accurate force is to use a known weight \(m\), suspended for example on a thread, and calculate the force as \(F = m \cdot g\) with \(g=9.81\frac{m}{s^2}\). Alternatively, an accurate bow/hanging scale might also work.

Finally the elastic modulus can be calulated from the measured deflection depending on the chosen setup.

Clamped Setup

setup clamped
Figure 27. Bending test with a clamped setup

In this setup the sample is clamped on one end and the force is applied on the free end. The elastic modulus can be calculated as

\[s = \frac{Fl^3}{3EI} \quad \Rightarrow \quad E = \frac{Fl^3}{3Is} = \frac{4Fl^3}{wh^3 s}.\]

Three-point Setup

setup three point
Figure 28. Bending test with a three-point setup

In this setup the sample is placed between two supports with distance \(l\) and the force is applied in the middle. The elastic modulus can be calculated as

\[s = \frac{Fl^3}{48EI} \quad \Rightarrow \quad E = \frac{Fl^3}{48Is} = \frac{Fl^3}{4wh^3 s}.\]