section modulus - wikipedia, the free encyclopedia

Section modulusFrom Wikipedia, the free encyclopedia

Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members. Other geometric properties used in design

include area for tension, radius of gyration for compression, and moment of inertia for stiffness. Any relationship between these properties is highly dependent

on the shape in question. Equations for the section moduli of common shapes are given below. There are two types of section moduli, the elastic section modulus

(S) and the plastic section modulus (Z).

Contents

1 Notation

2 Elastic section modulus

3 Plastic section modulus

4 See also

5 References

6 External links

Notation

North American and British/Australian convention reverse the usage of S & Z. Elastic modulus is S in North America,[1]

but Z in Britain/Australia,[2]

and vice

versa for the plastic modulus. Eurocode 3 (EN 1993 - Steel Design) resolves this by using W for both, but distinguishes between them by the use of subscripts -

Wel and Wpl.

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 defined as S = 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.[3]

It is often reported using y = c, where c is the distance from the neutral axis to the most extreme fibre, as seen in the table below. It is also often

used to determine the yield moment (My) such that My = S y, where y is the yield strength of the material.[3]

Section modulus - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Section_modulus

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Section modulus equations[4]

Cross-sectional shape Figure Equation Comment

Rectangle Solid arrow represents neutral axis

doubly symmetric I-section (strong axis) NA indicates neutral axis

doubly symmetric I-section (weak axis) NA indicates neutral axis

Circle[4]

Solid arrow represents neutral axis


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Circular tube Solid arrow represents neutral axis

Rectangular tube NA indicates neutral axis

Diamond NA indicates neutral axis

C-channel NA indicates neutral axis

Plastic section modulus


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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 (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. So, for sections with constant yielding stress, the area above

and below the PNA will be equal, but for composite sections, this is not 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 may or may not be equal) multiplied by the distance

from the local centroids of the two areas to the PNA:

Description Figure Equation Comment

Rectangular section

Hollow rectangular

sectionwhere: b=width, h=height, t=wall thickness

For the two flanges of an

I-beam with the web

excluded

,[5]

where: =width, =thickness,

are the distances from the neutral axis to the

centroids of the flanges respectively.

For an I Beam including

the web[6]

For an I Beam (weak

axis)

Solid Circle


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Hollow Circle

The plastic section modulus is used to calculate the plastic moment, Mp, or full capacity of a cross-section. The two terms are related by the yield strength of the

material in question, Fy, by Mp=Fy*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

References

^ Specification for Structural Steel Buildings (http://www.aisc.org/2010spec). Chicago, Illinois: American Institute of Steel Construction, Inc. 2010. p. 16.1xxxiv.1.

^ AS4100 - Steel Structures (http://www.standards.org.au). Sydney, Australia: Standards Australia. 1998. p. 21.2.

^ a b Kulak, G.L. and Grondin, G.Y., 2006, Limit States Design in Structural Steel 8th Ed., Canadian Institute of Steel Construction.3.

^ a b Gere, J. M. and Timoshenko, S., 1997, Mechanics of Materials 4th Ed., PWS Publishing Co.4.

^ American Institute of Steel Construction: Load and Resistance Factor Design, 3rd Edition, pp. 17-34.5.

^ Megson, T H G (2005). Structural and stress analysis (http://books.google.co.uk/books?id=N2WyMxutXK4C&lpg=PP1&pg=PP1#v=onepage&q&f=false). elsever.

pp. 598 EQ (iv).

6.

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

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Categories: Structural analysis

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section modulus - wikipedia, the free encyclopedia

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