Section Modulus calculation and optimization

Section Modulus

Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members. It is the direct measure of the strength of any section. Higher the section modulus of a structure, the more resistive it becomes to bending. The section modulus of the cross-sectional shape is of significant importance in designing.

A section that has a larger section modulus than another will be stronger and capable of supporting greater loads.

It includes the idea that most of the work in bending is being done by the extreme fibres of the section, i.e. the top and bottom fibres of the section. The distance of the fibres from top to bottom is therefore built into the calculation.

There are two types of section moduli, the elastic section modulus (S) and the plastic section modulus (Z). In the case for bending a steel section it is important to calculate ‘S’ by taking the moment of inertia of the area of the cross section of a structural member – divided by the distance from the neutral axis to the furthest point of the steel section. This is where the steel will bend first.

Elastic Section Modulus(S)

For general design, the elastic section modulus is used, applying up to the yield point for most metals and other common materials.

The elastic section modulus is defined as S = I / y, where I is the second moment of inertia (or area moment of inertia) and y is the distance from the neutral axis to any given fibre.

Plastic Section Modulus(Z)

The plastic section modulus is used for materials where elastic yielding is acceptable and plastic behavior is assumed to be an acceptable limit. Designs generally strive to ultimately remain below the plastic limit to avoid permanent deformations, often comparing the plastic capacity against amplified forces or stresses.

The plastic section modulus depends on the location of the plastic neutral axis (PNA). The PNA is defined as the axis that splits the cross section such that the compression force from the area in compression equals the tension force from the area in tension. So, for sections with constant yielding stress, the area above and below the PNA will be equal, but for composite sections, this is not necessarily the case.

The plastic section modulus is the sum of the areas of the cross section on each side of the PNA (which may or may not be equal) multiplied by the distance from the local centroids of the two areas to the PNA:

The plastic section modulus is used to calculate the plastic moment, M_p, or full capacity of a cross-section. The two terms are related by the yield strength of the material in question, F_y, by M_p = F_y ⋅ Z. Plastic section modulus and elastic section modulus are related by a shape factor which can be denoted by k, used for an indication of capacity beyond elastic limit of material. This could be shown mathematically with the formula :-

Shape factor for a rectangular section is 1.5.

Calculation of Elastic Section Modulus(S) for the designed Hood

Case 1:

On performing the Section Inertia Analysis, the above results were obtained and Section Modulus of the cross section was calculated. The minimum moment of inertia (MOI) of the cross section was considered for calculations since it is the threshold value.

Imin = Minimum moment of Inertia = 3.088e+05 [mm^4]

Y = Distance between the neutral axis and extreme end = 440.95mm

S = Section Modulus = I/Y = 700.30 mm^3

Case 2:

For the second case, the section was modified (the inner panel of hood was given an offset of 3mm. The same inner panel was only 2mm offset in the first case).

Imin = 3.156e+05 [mm^4]

Y = 440.95mm

S = I/Y = 715.72 mm^3

Conclusion: Thus, it is clearly evident that with increase in cross-section area, the section modulus of the particular section increases which in turn increases the strength of that section. An increase in offset of inner panel by 1mm is a direct increase in section modulus by 2.20%