Section modulus

Section modulus is a geometric property for a given cross-section often used in the design of beams or flexural members. There are two types of section moduli, the elastic section modulus (S) and the plastic section modulus (Z).

Notation

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

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 determined by 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 compression fibre, as seen in the table below. It is also often used to determine the yield moment (M_y) such that M_y = S × σ_y, where σ_y is the yield strength of the material.

Section modulus equations

Cross-sectional shape	Figure	Equation	Comment
Rectangle		$S = \cfrac{bh^2}{6}$	Solid arrow represents neutral axis
doubly symmetric I-section (strong axis)		$S = \cfrac{BH^2}{6} - \cfrac{bh^3}{6H}$	NA indicates neutral axis
doubly symmetric I-section (weak axis)		$S = \cfrac{B^2(H-h)}{6} + \cfrac{(B-b)^3h}{6B}$	NA indicates neutral axis
Circle		$S = \cfrac{\pi r^3}{4} = \cfrac{\pi d^3}{32}$	Solid arrow represents neutral axis
Circular tube		$S = \cfrac{\pi\left(r_2^4-r_1^4\right)}{4 r_2} = \cfrac{\pi (d_2^4 - d_1^4)}{32d_2}$	Solid arrow represents neutral axis
Rectangular tube		$S = \cfrac{BH^2}{6}-\cfrac{bh^3}{6H}$	NA indicates neutral axis
Diamond		$S = \cfrac{BH^2}{24}$	NA indicates neutral axis
C-channel		$S = \cfrac{BH^2}{6} - \cfrac{bh^3}{6H}$	NA indicates neutral axis

Plastic section modulus

The Plastic section modulus is used for materials where (irreversible) plastic behaviour is dominant. The majority of designs do not intentionally encounter this behaviour.

The plastic section modulus depends on the location of the plastic neutral axis, or PNA. The PNA is defined as the axis that splits the cross section into two equal areas so that the area of compression equals the area of tension. So, for a square cross section the plastic and elastic neutral axis coincide, but given a T-shape for example, this isn't necessarily the case.

The plastic section modulus is then the sum of the areas of the cross section on each side of the PNA (which are equal) multiplied by the distance from the local centroids of the two areas to the PNA:

Z = A_Cy_C + A_Ty_T

Rectangular section	$Z = \cfrac{bh^2}{4}$
For the two flanges of an I-beam with the web excluded	$Z = b_1t_1y_1+b_2t_2y_2\,$	where: b1,b2=width, t1,t2=thickness, y1,y2 are the distances from the neutral axis to the centroids of the flanges respectively.
Solid Circle	$Z = \cfrac{d^3}{6}$
Hollow Circle	$Z = \cfrac{d_2^3-d_1^3}{6}$

The plastic section modulus is used to calculate the plastic moment, M_p, or full capacity of a cross-section. The two terms are related by the yield strength of the material in question, F_y, by M_p=F_y*Z. Sometimes Z and S are related by defining a 'k' factor which is something of an indication of capacity beyond first yield. k=Z/S

Therefore for a rectangular section, k=1.5

See also

• Beam theory
• List of area moments of inertia
• Second moment of area

External links

• http://www.engineeringtoolbox.com/american-wide-flange-steel-beams-d_1318.html - List of section moduli for common beam shapes
• http://www.novanumeric.com/samples.php?CalcName=SectionModulus - Online Calculation for Section Modulus