Section modulus

In solid mechanics and structural engineering, section modulus is a geometric property of 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 second moment of area and polar second moment of area for stiffness. Any relationship between these properties is highly dependent on the shape in question. There are two types of section modulus, elastic and plastic:

• The elastic section modulus is used to calculate a cross-section's resistance to bending within the elastic range, where stress and strain are proportional.

• The plastic section modulus is used to calculate a cross-section's capacity to resist bending after yielding has occurred across the entire section. It is used for determining the plastic, or full moment, strength and is larger than the elastic section modulus, reflecting the section's strength beyond the elastic range.^[1]

Equations for the section moduli of common shapes are given below. The section moduli for various profiles are often available as numerical values in tables that list the properties of standard structural shapes.^[2]

Note: Both the elastic and plastic section moduli are different to the first moment of area. It is used to determine how shear forces are distributed.

Different codes use varying notations for the elastic and plastic section modulus, as illustrated in the table below.

The North American notation is used in this article.

The elastic section modulus is used for general design. It is applicable up to the yield point for most metals and other common materials. It is defined as^[1]

$${\displaystyle S={\frac {I}{c}}}$$

where:

I is the second moment of area (or area moment of inertia, not to be confused with moment of inertia), and

c is the distance from the neutral axis to the most extreme fibre.

It is used to determine the yield moment strength of a section^[1]

$${\displaystyle M_{y}=S\cdot \sigma _{y}}$$

where σ_y is the yield strength of the material.

The table below shows formulas for the elastic section modulus for various shapes.

Elastic Section Modulus Equations

Cross-sectional shape	Figure	Equation	Comment	Ref.
Rectangle		${\displaystyle S={\cfrac {bh^{2}}{6}}}$	Solid arrow represents neutral axis	[1]
doubly symmetric Ɪ-section (major axis)		${\displaystyle S_{x}={\cfrac {BH^{2}}{6}}-{\cfrac {bh^{3}}{6H}}}$ ${\displaystyle S_{x}={\tfrac {I_{x}}{y}}}$, with ${\displaystyle y={\cfrac {H}{2}}}$	NA indicates neutral axis	[12]
doubly symmetric Ɪ-section (minor axis)		${\displaystyle S_{y}={\cfrac {B^{2}(H-h)}{6}}+{\cfrac {(B-b)^{3}h}{6B}}}$	NA indicates neutral axis	[13]
Circle		${\displaystyle S={\cfrac {\pi d^{3}}{32}}}$	Solid arrow represents neutral axis	[12]
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	[12]
Rectangular hollow section		${\displaystyle S={\cfrac {BH^{2}}{6}}-{\cfrac {bh^{3}}{6H}}}$	NA indicates neutral axis	[12]
Diamond		${\displaystyle S={\cfrac {BH^{2}}{24}}}$	NA indicates neutral axis	[12]
C-channel		${\displaystyle S={\cfrac {BH^{2}}{6}}-{\cfrac {bh^{3}}{6H}}}$	NA indicates neutral axis	[12]
Equal and Unequal Angles	These sections require careful consideration because the axes for the maximum and minimum section modulus are not parallel with its flanges.[14] Tables of values for standard sections are available.[15]			[14] [15]

The plastic section modulus is used for materials and structures where limited plastic deformation is acceptable. It represents the section's capacity to resist bending once the material has yielded and entered the plastic range. It is used to determine the plastic, or full, moment strength of a section^[1]

$${\displaystyle M_{p}=Z\cdot \sigma _{y}}$$

where σ_y is the yield strength of the material.

Engineers often compare the plastic moment strength against factored applied moments to ensure that the structure can safely endure the required loads without significant or unacceptable permanent deformation. This is an integral part of the limit state design method.

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. For sections with constant, equal compressive and tensile yield strength, the area above and below the PNA will be equal^[16]

$${\displaystyle A_{C}=A_{T}}$$

These areas may differ in composite sections, which have differing material properties, resulting in unequal contributions to the plastic section modulus.

The plastic section modulus is calculated as the sum of the areas of the cross section on either side of the PNA, each multiplied by the distance from their respective local centroids to the PNA.^[16]

$${\displaystyle Z=A_{C}y_{C}+A_{T}y_{T}}$$

where:

A_C is the area in compression

A_T is the area in tension

y_C, y_T are the distances from the PNA to their centroids.

Plastic section modulus and elastic section modulus can be related by a shape factor k:

$${\displaystyle k={\frac {M_{p}}{M_{y}}}={\frac {Z}{S}}}$$

This is an indication of a section's capacity beyond the yield strength of material. The shape factor for a rectangular section is 1.5.^[1]

The table below shows formulas for the plastic section modulus for various shapes.

Plastic Section Modulus Equations

Description	Figure	Equation	Comment	Ref.
Rectangular section		${\displaystyle Z={\frac {bh^{2}}{4}}}$	${\displaystyle A_{C}=A_{T}={\frac {bh}{2}}}$ ${\displaystyle y_{C}=y_{T}={\frac {h}{4}}}$	[1] [17]
Rectangular hollow section		${\displaystyle Z={\cfrac {bh^{2}}{4}}-(b-2t)\left({\cfrac {h}{2}}-t\right)^{2}}$	b = width, h = height, t = wall thickness	[1]
For the two flanges of an Ɪ-beam with the web excluded		${\displaystyle Z=b_{1}t_{1}y_{1}+b_{2}t_{2}y_{2}\,}$	b1, b2 = width, t1, t2 = thickness, y1, y2 = distances from the neutral axis to the centroids of the flanges respectively.	[18]
For an I Beam including the web		${\displaystyle Z=bt_{f}(d-t_{f})+{\frac {t_{w}(d-2t_{f})^{2}}{4}}}$		[1] [19]
For an I Beam (weak axis)		${\displaystyle Z={\frac {b^{2}t_{f}}{2}}+{\frac {t_{w}^{2}(d-2t_{f})}{4}}}$	d = full height of the I beam	[1]
Solid Circle		${\displaystyle Z={\cfrac {d^{3}}{6}}}$		[1]
Circular hollow section		${\displaystyle Z={\cfrac {d_{2}^{3}-d_{1}^{3}}{6}}}$		[1]
Equal and Unequal Angles	These sections require careful consideration because the axes for the maximum and minimum section modulus are not parallel with its flanges.[14]			[14]

In structural engineering, the choice between utilizing the elastic or plastic (full moment) strength of a section is determined by the specific application. Engineers follow relevant codes that dictate whether an elastic or plastic design approach is appropriate, which in turn informs the use of either the elastic or plastic section modulus. While a detailed examination of all relevant codes is beyond the scope of this article, the following observations are noteworthy:

• When assessing the strength of long, slender beams, it is essential to evaluate their capacity to resist lateral torsional buckling in addition to determining their moment capacity based on the section modulus.^[20]
• Although T-sections may not be the most efficient choice for resisting bending, they are sometimes selected for their architectural appeal. In such cases, it is crucial to carefully assess their capacity to resist lateral torsional buckling.^[21]
• While standard uniform cross-section beams are often used, they may not be optimally utilized when subjected to load moments that vary along their length. For large beams with predictable loading conditions, strategically adjusting the section modulus along the length can significantly enhance efficiency and cost-effectiveness.^[22]
• In certain applications, such as cranes and aeronautical or space structures, relying solely on calculations is often deemed insufficient. In these cases, structural testing is conducted to validate the load capacity of the structure.

• Beam theory
• Buckling
• List of area moments of inertia
• Second moment of area
• Structural testing
• Yield strength