# Section Moduli, Moment and Stress

In the previous section on Euler-Bernoulli beam bending theory the following equation was given to link beam properties, applied moment and resulting stress:

$$\sigma= \frac{My}{I}$$

Where, M is the applied moment, I is the second moment of area of the beam, \(\sigma\) is the resulting stress at a point on the beam cross section, y is the distance from the neutral axis to the location on the cross section where you would like to find your value of stress (typically called the fibre distance).

From this equation we can see that for a given moment and I value, the stress in the section varies linearly with the distance from the neutral axis. It’s worth highlighting that stress can be both positive and negative, depending on the sign convention used a positive stress could be a tensile stress, while a negative stress would be compressive. At the aptly named neutral axis there is zero stress, as y = 0, at the edges of the section the stress is the highest and lowest, at the most positive y value and most negative y value respectively. When a beam bends stresses in opposite directions are experienced by the tops and bottoms of the beams, this concept is typically expressed using “stress blocks”.

#### Elastic Stress Blocks

A beam cross section is shown below, along with the locations of the top and bottom fibres and the neutral axis. It is assumed that bending is imposed that gives rise to compression in the top face of the beam and tension in the bottom face (i.e. sagging bending).

The values of stress in the top and bottom values can be calculated using \( \sigma = \frac{My}{I}\) and by applying y values equal to the distance between the neutral axis and the bottom of the section and the distance between the neutral axis and the top of the section:

As the applied moment increases the stresses at the top and bottom of the section increase:

When the value of stress at either the top or bottom of the section reaches the materials yield stress then the beam is said to have reached its elastic limit and the moment associated with this will be the elastic moment resistance of the section.

By taking this example a step further, if a higher moment resistance was needed, then either a material with a higher yield stress could be used, or a section with a higher I or lower y. This is the basis of beam design.

#### Elastic Section Moduli

Typically engineers don’t need to know the stress at every point in a beam section, the governing elastic stress limit is either at the top or bottom of a section. Engineers will therefore calculate the I/y values for the top and bottom of the section and use this as a new variable called the section moduli:

$$Z_{top} = \frac{I}{y_{top}}$$ $$Z_{bottom} = \frac{I}{y_{bottom}}$$

Where \(Z_{top}\) and \(Z_{bottom}\) are the section moduli for the top and bottom of the section respectively, I is the second moment of area of the section and \(y_{top}\) and \(y_{bottom}\) are the distance from the neutral axis to the top and bottom of the section respectively (the fibre distances). Please click here for more information on the units for elastic section modulus.

Section moduli are useful as they are a quick way to find a beams capacity by working out whether the top or bottom of the beam yields first (i.e. has the lowest section moduli):

$$As$$ $$\qquad Z_{top}= \frac{I}{y_{top}}$$ $$\qquad Z_{bottom}= \frac{I}{y_{bottom}}, $$ $$and$$ $$\qquad \sigma= \frac{My}{I} \rightarrow M = \frac{\sigma I}{y} $$ $$\text{elastic moment resistance can be calculated}$$ $$\qquad M_{elastic,resistance}= min(Z_{top}, Z_{bottom})\times\sigma_{y}$$

#### Plastic Stress Blocks

Some materials are elastoplastic and can behave in a plastic manner under increasing load, this allows the material to deflect while still providing resistance. Steel is a good example of a elastoplastic material and can undergo significant plastic strain prior to failure.

Plastic behaviour allows a beam to withstand higher moments than an elastic section would alone. The stress blocks below show what happen when a section is loaded past the yield point of the outer fibres (i.e. past the elastic moment of resistance):

It is worth noting that the elastic neutral axis and the plastic neutral axis are often at different positions within the beam, as a beam is loaded past its elastic moment of resistance the neutral axis begins to shift towards the plastic neutral axis.

#### Plastic Section Modulus

To calculate the plastic moment of resistance of a beam the plastic section modulus is used. Unlike the elastic section moduli, which can be a different values for the top and bottom of a beam, the plastic section modulus is always a single value for a given axis.

More information on calculating the plastic section modulus is given in a separate article, however the following basic principles of the plastic behaviour are worth noting:

- At the plastic moment of resistance all points within the beam are stressed to the full yield point of the section, either in a compressive or tensile sense.
- The plastic neutral axis divides the section into two equal areas, such that the tensile force in one half of the section is balanced by the compressive force in the other half of the section (otherwise there would be an out of balance axial force on the section).
- The plastic section modulus can be calculated as the area in compression times by the lever arm from the centroid of the compression area to the plastic neutral axis added to the area in tension times by the lever arm from the centroid of the tensile area to the plastic neutral axis.

For information on the units of the plastic section modulus, please click here . The plastic section modulus can then be used in a similar manner to the elastic moduli to calculate a plastic moment of resistance:

$$\text{Plastic moment resistance can be calculated}$$ $$\qquad M_{plastic,resistance}= Z_{plastic}\times\sigma_{y}$$

#### Biaxial Bending

In the sections above we have focussed on bending about a single axis, however beams can be bent about both their major and minor axes at the same time. The stresses which result from bending about each axis can be added together to give a stress pattern that varies across the full section. This is shown on the figure below:

It is important when calculating biaxial bending to ensure that the fibre distances chosen relate to a physical point on the section and aren’t just the maximum and minimum fibre values. This is shown in the example below: