section modulus

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 geometricproperties used in design include area for tension, radius of gyration for compression, and moment of inertia for stiffness. Anyrelationship between these properties is highly dependent on the shape in question. Equations for the section moduli of common shapesare given below. There are two types of section moduli, the elastic section modulus (S) and the plastic section modulus (Z).

Contents1 Notation2 Elastic section modulus3 Plastic section modulus4 Use in structural engineering5 See also6 References7 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 inBritain/Australia,[2] and vice versa for the plastic modulus. Eurocode 3 (EN 1993 - Steel Design) resolves this by using W for both, butdistinguishes between them by the use of subscripts - Wel and Wpl.

Elastic section modulusFor 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 distancefrom 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 mostextreme 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 theyield strength of the material.[3] Elastic Section Modulus can also be defined as the first moment of area.

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

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Section modulus equations[4]

Cross-sectional shape Figure Equation Comment

Rectangle Solid arrow represents neutralaxis

doubly symmetric I-section (strongaxis)

NA indicates neutral axis

doubly symmetric I-section (weakaxis)

NA indicates neutral axis

Circle [4] Solid arrow represents neutralaxis

Circular tube Solid arrow represents neutralaxis

Rectangular tube NA indicates neutral axis

Diamond NA indicates neutral axis

C-channel NA indicates neutral axis

Plastic section modulusThe Plastic section modulus is used for materials where elastic yielding is acceptable and plastic behavior is assumed to be anacceptable limit. Designs generally strive to ultimately remain below the plastic limit to avoid permanent deformations, often comparingthe plastic capacity against amplified forces or stresses.

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, forsections with constant yielding stress, the area above and below the PNA will be equal, but for composite sections, this is not necessarilythe 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 [5][6]

Hollow rectangularsection

where: b=width, h=height, t=wallthickness

For the two flanges ofan I-beam with theweb excluded[7]

where:

=width, =thickness, are the distances from the

neutral axis to the centroids of theflanges respectively.

For an I Beamincluding the web

[8]

For an I Beam (weakaxis)

Solid Circle

Circular tube

The plastic section modulus is used to calculate the plastic moment, Mp, or full capacity of a cross-section. The two terms are related bythe yield strength of the material in question, Fy, by Mp=Fy*Z. Plastic section modulus and elastic section modulus are related byshape factor which can be denoted by 'k', used for an indication of capacity beyond elastic limit of material. This could be shownmathematicaly with the formula :-

Shape factor for a rectangular section is 1.5.

Use in structural engineeringThough generally section modulus is calculated for the extreme tensile or compressive fibres in a bending beam, often compression isthe most critical case due to onset of flexural torsional buckling. Generally (except for brittle materials like concrete) tensile extremefibres have a higher allowable stress or capacity than compressive fibres.

In the case of T-sections if there are tensile fibres at the bottom of the T they may still be more critical than the compressive fibres at thetop due to a generally much larger distance from the neutral axis so despite having a higher allowable the elastic section modulus is alsolower. In this case F/T buckling must still be assessed as the beam length and restraints may result in reduced compressive memberbending allowable stress or capacity.

There may also be a number of different critical cases that require consideration, such as there being different values for orthogonal andprincipal axes and in the case of unequal angle sections in the principal axes there is a section modulus for each corner.

For a conservative (safe) design, civil structural engineers are often concerned with the combination of the highest load (tensile orcompressive) and lowest elastic section modulus for a given section station along a beam, although if the loading is well understood onecan take advantage of different section modulus for tension and compression to get more out of the design. For aeronautical and spaceapplications where designs must be much less conservative for weight saving, structural testing is often required to ensure safety asreliance on structural analysis alone is more difficult (and expensive) to justify.

See alsoBeam theoryList of area moments of inertiaSecond 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. Kulak, G.L. and Grondin, G.Y., 2006, Limit States Design in Structural Steel 8th Ed., Canadian Institute of Steel Construction.3. Gere, J. M. and Timoshenko, S., 1997, Mechanics of Materials 4th Ed., PWS Publishing Co.4. https://www.dlsweb.rmit.edu.au/toolbox/buildright/content/bcgbc4010a/03_properties/02_section_properties/page_008.htm5. Young, Warren C. (1989). Roark's Formulas for Stress and Strain. McGraw Hill. p. 217.6. American Institute of Steel Construction: Load and Resistance Factor Design, 3rd Edition, pp. 17-34.7. 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).

8.

External linkshttp://www.engineeringtoolbox.com/american-wide-flange-steel-beams-d_1318.html - List of section moduli for common beamshapeshttp://www.novanumeric.com/samples.php?CalcName=SectionModulus - Online Calculation for Section Modulushttp://academics.triton.edu/faculty/fheitzman/propertiesofsections.html - Properties of Sections From Triton collegehttp://www.amesweb.info/SectionalPropertiesTabs/SectionalPropertiesHollowCircle.aspx - SECTIONAL PROPERTIESCALCULATOR - HOLLOW CIRCLE

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Categories: Structural analysis

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http://www.aisc.org/2010spechttp://www.standards.org.auhttps://www.dlsweb.rmit.edu.au/toolbox/buildright/content/bcgbc4010a/03_properties/02_section_properties/page_008.htmhttp://books.google.co.uk/books?id=N2WyMxutXK4C&lpg=PP1&pg=PP1#v=onepage&http://www.engineeringtoolbox.com/american-wide-flange-steel-beams-d_1318.htmlhttp://www.novanumeric.com/samples.php?CalcName=SectionModulushttp://academics.triton.edu/faculty/fheitzman/propertiesofsections.htmlhttp://www.amesweb.info/SectionalPropertiesTabs/SectionalPropertiesHollowCircle.aspxhttp://en.wikipedia.org/w/index.php?title=Section_modulus&oldid=656673835https://en.wikipedia.org/wiki/Section_modulushttp://www.pdffactory.com

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