Section modulus

Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members. Other geometric properties used in design include area for tension and shear, radius of gyration for compression, and moment of inertia and polar moment of inertia for stiffness. Any relationship between these properties is highly dependent on the shape in question. Equations for the section moduli of common shapes are given below. There are two types of section moduli, the elastic section modulus (S) and the plastic section modulus (Z). The section modulus of different profiles can also be found as numerical values for common profiles in tables listing properties of such.

Notation

North American and British/Australian convention reverse the usage of S & Z. Elastic modulus is S in North America,[1] but Z in Britain/Australia,[2] and vice versa for the plastic modulus. Eurocode 3 (EN 1993 - Steel Design) resolves this by using W for both, but distinguishes between them by the use of subscripts - Wel and Wpl.

Elastic section modulus

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 area (or moment of inertia) and y is the distance from the neutral axis to any given fibre. It is often reported using y = c, where c is the distance from the neutral axis to the most extreme fibre, as seen in the table below. It is also often used to determine the yield moment (My) such that My = S × σy, where σy is the yield strength of the material. Elastic Section Modulus can also be defined as the first moment of area.

Section modulus equations[3]
Cross-sectional shape Figure Equation Comment
Rectangle ${\displaystyle S={\cfrac {bh^{2}}{6}}}$ Solid arrow represents neutral axis
doubly symmetric I-section (major axis) ${\displaystyle Sx={\cfrac {BH^{2}}{6}}-{\cfrac {bh^{3}}{6H}}}$

${\displaystyle Sx={\tfrac {Ix}{y}}}$,

with ${\displaystyle y={\cfrac {H}{2}}}$

NA indicates neutral axis
doubly symmetric I-section (minor axis) ${\displaystyle Sy={\cfrac {B^{2}(H-h)}{6}}+{\cfrac {(B-b)^{3}h}{6B}}}$ NA indicates neutral axis
Circle ${\displaystyle S={\cfrac {\pi r^{3}}{4}}={\cfrac {\pi d^{3}}{32}}}$[3] Solid arrow represents neutral axis
Circular hollow section ${\displaystyle S={\cfrac {\pi \left(r_{2}^{4}-r_{1}^{4}\right)}{4r_{2}}}={\cfrac {\pi (d_{2}^{4}-d_{1}^{4})}{32d_{2}}}}$ Solid arrow represents neutral axis
Rectangular hollow section ${\displaystyle S={\cfrac {BH^{2}}{6}}-{\cfrac {bh^{3}}{6H}}}$ NA indicates neutral axis
Diamond ${\displaystyle S={\cfrac {BH^{2}}{24}}}$ NA indicates neutral axis
C-channel ${\displaystyle S={\cfrac {BH^{2}}{6}}-{\cfrac {bh^{3}}{6H}}}$ NA indicates neutral axis

Plastic section modulus

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:

${\displaystyle S_{P}=A_{C}y_{C}+A_{T}y_{T}}$

Description Figure Equation Comment
Rectangular section ${\displaystyle Z_{P}={\cfrac {bh^{2}}{4}}}$[4][5] ${\displaystyle A_{C}=A_{T}={\cfrac {bh}{2}}}$ , ${\displaystyle y_{C}=y_{T}={\cfrac {h}{4}}}$
Rectangular hollow section ${\displaystyle Z_{P}={\cfrac {bh^{2}}{4}}-(b-2t)({\cfrac {h}{2}}-t)^{2}}$ where: b=width, h=height, t=wall thickness
For the two flanges of an I-beam with the web excluded[6] ${\displaystyle Z_{P}=b_{1}t_{1}y_{1}+b_{2}t_{2}y_{2}\,}$ where:

${\displaystyle b_{1},b_{2}}$ =width, ${\displaystyle t_{1},t_{2}}$=thickness, ${\displaystyle y_{1},y_{2}}$ are the distances from the neutral axis to the centroids of the flanges respectively.

For an I Beam including the web ${\displaystyle Z_{P}=bt_{f}(d-t_{f})+0.25t_{w}(d-2t_{f})^{2}}$ [7]
For an I Beam (weak axis) ${\displaystyle Z_{P}=(b^{2}t_{f})/2+0.25t_{w}^{2}(d-2t_{f})}$
Solid Circle ${\displaystyle Z_{P}={\cfrac {d^{3}}{6}}}$
Circular hollow section ${\displaystyle Z_{P}={\cfrac {d_{2}^{3}-d_{1}^{3}}{6}}}$

The plastic section modulus is used to calculate the plastic moment, Mp, or full capacity of a cross-section. The two terms are related by the yield strength of the material in question, Fy, by Mp=Fy*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 :-

${\displaystyle k={\cfrac {Z}{S}}}$

Shape factor for a rectangular section is 1.5.

Use in structural engineering

Though generally section modulus is calculated for the extreme tensile or compressive fibres in a bending beam, often compression is the most critical case due to onset of flexural torsional (F/T) buckling. Generally (except for brittle materials like concrete) tensile extreme fibres have a higher allowable stress or capacity than compressive fibres.

In the case of T-sections if there are tensile fibres at the bottom of the T they may still be more critical than the compressive fibres at the top due to a generally much larger distance from the neutral axis so despite having a higher allowable the elastic section modulus is also lower. In this case F/T buckling must still be assessed as the beam length and restraints may result in reduced compressive member bending allowable stress or capacity.

There may also be a number of different critical cases that require consideration, such as there being different values for orthogonal and principal axes and in the case of unequal angle sections in the principal axes there is a section modulus for each corner.

For a conservative (safe) design, civil structural engineers are often concerned with the combination of the highest load (tensile or compressive) and lowest elastic section modulus for a given section station along a beam, although if the loading is well understood one can take advantage of different section modulus for tension and compression to get more out of the design. For aeronautical and space applications where designs must be much less conservative for weight saving, structural testing is often required to ensure safety as reliance on structural analysis alone is more difficult (and expensive) to justify.