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. A plastic hinge forms once every fiber across the cross-section has reached the yield stress, σy; beyond that point, the section cannot resist any further increase in bending moment. This limiting condition defines the plastic moment, Mp, and the plastic section modulus through Mp = σyZp. 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.
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 = I c {\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
M y = S ⋅ σ y {\displaystyle M_{y}=S\cdot \sigma _{y}}
where σy is the yield strength of the material. The table below shows formulas for the elastic section modulus for various shapes.
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 ⋅ σ y {\displaystyle M_{p}=Z\cdot \sigma _{y}} where σy 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 {\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.
Z = A C y C + A T y T {\displaystyle 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 = M p M y = Z S {\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. The table below shows formulas for the plastic section modulus for various shapes.
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:
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