section modulus

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

$$S\; =\; \backslash frac\{I\}\{c\}$$

where:

:{{mvar|I}} is the second moment of area (or area moment of inertia, not to be confused with moment of inertia), and

:{{mvar|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

$$M\_y\; =\; S\; \backslash cdot\; \backslash sigma\_y$$

where {{mvar|σ{{sub|y}}}} is the yield strength of the material.

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

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

$$M\_p\; =\; Z\; \backslash cdot\; \backslash sigma\_y$$

where {{mvar|σ{{sub|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{{Cite web |title=Plastic Modulus |url=https://www.rcet.org.in/uploads/academics/rohini_50184075199.pdf}}

$$A\_C\; =\; A\_T$$

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:

:{{mvar|A_C}} is the area in compression

:{{mvar|A_T}} is the area in tension

:{{math|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 {{mvar|k}}:

$$k\; =\; \backslash frac\{M\_p\}\{M\_y\}\; =\; \backslash frac\{Z\}\{S\}$$

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.

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.{{Cite book |title=Structural steel designer's handbook |date=1999 |publisher=McGraw-Hill |isbn=978-0-07-008782-8 |editor-last=Brockenbrough |editor-first=Roger L. |edition=3 |series=McGraw-Hill handbooks |location=New York |page=3.96 |editor-last2=Merritt |editor-first2=Frederick S.}}
• 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.{{Cite journal |last=Brown |first=David |date=2024-08-27 |title=The design of tee sections in bending |url=http://www.newsteelconstruction.com/wp/wp-content/uploads/TechPaper/TechNSCapril2016.pdf |journal=New Steel Construction}}
• 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.{{Cite journal |last=Vu |first=Huy Hoang |last2=Chu |first2=Thi Hoang Anh |date=2024 |title=Simply supported built-up I-beam optimization comparison |url=https://www.e3s-conferences.org/articles/e3sconf/abs/2024/63/e3sconf_form2024_02010/e3sconf_form2024_02010.html |journal=E3S Web of Conferences |language=en |volume=533 |pages=02010 |doi=10.1051/e3sconf/202453302010 |issn=2267-1242|doi-access=free }}
• 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

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Category:Structural analysis

Category:Mechanical quantities