Section modulus

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title: "Section modulus" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["beam-theory", "structural-analysis", "mechanical-quantities"] description: "Geometric property of a structural member" topic_path: "general/beam-theory" source: "https://en.wikipedia.org/wiki/Section_modulus" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Geometric property of a structural member ::

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.

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.

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.

Notation

Different codes use varying notations for the elastic and plastic section modulus, as illustrated in the table below. ::data[format=table title="Section Modulus Notation"]

Region	Code	Section Modulus	Elastic	Plastic
North America	USA: ANSI/AISC 360-10	S	Z
Canada: CSA S16-14	S	Z
Europe	Europe (inc. Britain): Eurocode 3	Wel	Wpl
Britain (obsolete): BS 5950 a	Z	S
Asia	Japan: Standard Specifications for Steel and Composite Structures	W	Z
China: GB 50017	W	Wp
India: IS 800	Ze	Zp
Australia: AS 4100	Z	S
Notes:
::

The North American notation is used in this article.

Elastic section modulus

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

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

M_y = S \cdot \sigma_y

where σ is the yield strength of the material.

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

::data[format=table title="Elastic Section Modulus Equations"]

Cross-sectional shape	Figure	Equation	Comment	Ref.
Rectangle	[[File:Area moment of inertia of a rectangle.svg]]	S = \cfrac{bh^2}{6}	Solid arrow represents neutral axis
doubly symmetric -section (major axis)	[[File:Section modulus-I-beam-strong axis.svg	x150px]]	S_x = \cfrac{BH^2}{6} - \cfrac{bh^3}{6H}	NA indicates neutral axis
doubly symmetric -section (minor axis)	[[File:Section modulus-I-beam-weak axis.svg	x175px]]	S_y = \cfrac{B^2(H-h)}{6} + \cfrac{(B-b)^3 h}{6B}	NA indicates neutral axis
Circle	[[File:Area moment of inertia of a circle.svg]]	S = \cfrac{\pi d^3}{32}	Solid arrow represents neutral axis
Circular hollow section	[[File:Area moment of inertia of a circular area.svg]]	S = \cfrac{\pi\left(r_2^4-r_1^4\right)}{4 r_2} = \cfrac{\pi (d_2^4 - d_1^4)}{32d_2}	Solid arrow represents neutral axis
Rectangular hollow section	[[File:Section modulus-rectangular tube.svg	x150px]]	S = \cfrac{BH^2}{6}-\cfrac{bh^3}{6H}	NA indicates neutral axis
Diamond	[[File:Secion modulus-diamond.svg	x150px]]	S = \cfrac{BH^2}{24}	NA indicates neutral axis
C-channel	[[File:Section modulus-C-channel.svg	x150px]]	S = \cfrac{BH^2}{6} - \cfrac{bh^3}{6H}	NA indicates neutral axis
Equal and Unequal	These sections require careful consideration because the axes for the maximum and minimum
::

Plastic section modulus

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

M_p = Z \cdot \sigma_y

where σ 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

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.

Z = A_C y_C + A_T y_T

where: :AC is the area in compression :AT is the area in tension :yC, yT are the distances from the PNA to their centroids.

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

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.

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

::data[format=table title="Plastic Section Modulus Equations"] | Description || Figure || Equation || Comment | Ref. | |---|---| | Rectangular section | [[File:Area moment of inertia of a rectangle.svg]] | | Rectangular hollow section | | | For the two flanges of an -beam with the web excluded | | | For an I Beam including the web | | | For an I Beam (weak axis) | | | Solid Circle | | | Circular hollow section | | | Equal and Unequal Angles | These sections require careful consideration because the axes for the maximum and minimum | ::

Use in structural engineering

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.
• 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.
• 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.
• 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.

References

References

1. Young, Warren C.. (1989). "Roark's Formulas for Stress and Strain". McGraw Hill.
2. "'Blue Book' home - Blue Book - Steel for Life".
3. "Specification for Structural Steel Buildings (ANSI/AISC 360-10) - 2010 {{!}} American Institute of Steel Construction".
4. (2024-08-23). "S16-14 (R2019) Design of steel structures". Canadian Standards Association.
5. "Eurocode 3: Design of steel structures - Part 1-1: General rules and rules for buildings".
6. "BS 5950-1 Structural use of steelwork in building". BSI British Standards.
7. (2024-08-24). "Standard Specifications for Steel and Composite Structures". Japan Society of Civil Engineers.
8. (2003). "GB 50017 Code for Design of Steel Structures". Ministry of Construction of the People's Republic of China.
9. (2007). "IS800:2007 General Construction in Steel - Code of Practice". Bureau of Indian Standards.
10. (2020). "AS 4100- 2020 Steel Structures". Standards Australia Ltd.
11. (2024-08-23). "British Standards Institute".
12. Gere, J. M. and Timoshenko, S., 1997, Mechanics of Materials 4th Ed., PWS Publishing Co.
13. "Section Modulus Equations and Calculators Common Shapes".
14. Trahair, N. S.. (2002-11-01). "Moment Capacities of Steel Angle Sections". Journal of Structural Engineering.
15. "Section properties - Dimensions & properties - Blue Book - Steel for Life".
16. "Plastic Modulus".
17. "Calculating the section modulus".
18. American Institute of Steel Construction: Load and Resistance Factor Design, 3rd Edition, pp. 17-34.
19. Megson, T H G. (2005). "Structural and stress analysis". elsever.
20. (1999). "Structural steel designer's handbook". McGraw-Hill.
21. Brown, David. (2024-08-27). "The design of tee sections in bending". New Steel Construction.
22. Vu, Huy Hoang. (2024). "Simply supported built-up I-beam optimization comparison". E3S Web of Conferences.

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