## Structural.SteelBeamCompactSections History

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February 21, 2011
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!! Compact vs. Non-Compact vs. Slender Sections

To put your mind at ease, this section is not on weight loss (as a fellow engineer I understand how that could be worrisome to many). In addition to that, the vast majority of sections out there are compact, which is what you want. Discovering whether or not you have a compact, non-compact, or slender section all depends on the shape of the beam. Table B4.1 in the AISC 360 specs provides all the limits you need to determine which zone your section shape falls into (this will be explained in greater detail below).

When you have a '''compact section''' there is no possibility of local flange or web bucking to prevent attainment of the full sections yield strength. In common terms, the beam will not have a local failure (i.e. your web buckles) before the beam has global failure.

With a '''non-compact section''' one or more of the elements in the beam (e.g. in an I-beam which will be detailed below, those elements are the web and the flange) have the possibility of buckling before the beam is able to attain section plasticity.

A '''slender section''' is almost always avoided. But sometimes it is important to check this type of section. Understand a slender section is even more extreme than a non-compact section.

If:

* '''λ ≤ λ'_p_'''' (the section is compact)

* '''λ'_p_' < λ ≤ λ'_r_'''' (the section is non-compact)

* '''λ'_r_' < λ''' (the section is slender)

''where:''

->'''λ''' = Width to thickness ratio (dependent on section type and element)\\

'''λ'_p_'''' = Limiting ratio to see if a section is compact\\

'''λ'_r_'''' = Limiting ratio to see if a section is non-compact\\\

%lframe% '''Note:''' All current ASTM W, S, M, C, & MC shapes have compact flanges except W21x48, W14x99, W14x90, W12x65, W10x12, W8x31, W8x10, W6x15, W6x9, W6x8.5, & M4x6 (for F'_y_' = 50 ksi). All current ASTM W, S, M, HP, C, & MC shapes have compact webs for F'_y_' ≤ 65 ksi.

\\\\

!!! How to find if a rolled doubly symmetric I shaped sections is compact?

This example has been taken from Table B4.1 in the AISC 360 specs '^[1]^'.

(:table border=2 cellpadding=2 cellspacing=0 width=100%:)

(:cellnr bgcolor=#d4d7ba colspan=4 align=center:) '+'''Table 1: Is a Rolled I-shaped Section or Channels compact?'''+'

(:cellnr width=30%:)'''Description of Element'''

(:cell align=center width=20%:)'''Width Thickness Ratio (λ)'''

(:cell align=center width=25%:)'''λ'_p_''''

(:cell align=center width=25%:)'''λ'_r_''''

(:cellnr colspan=4:)

(:cellnr:)Flexure in '''flanges''' of rolled I-shaped sections & channels

(:cell align=center:)[++{$ \frac{b}{t} $}++]

(:cell align=center:)[++{$ 0.38\sqrt{E/F_y} $}++]

(:cell align=center:)[++{$ 1.0\sqrt{E/F_y} $}++]

(:cellnr:)Flexure in '''webs''' of rolled I-shaped sections & channels

(:cell align=center:)[++{$ \frac{h}{t_w} $}++]

(:cell align=center:)[++{$ 3.76\sqrt{E/F_y} $}++]

(:cell align=center:)[++{$ 5.70\sqrt{E/F_y} $}++]

(:tableend:)

''where:''

->'''b''' = the base of the I-shape or channel's flange\\

'''t''' = the thickness of the flange\\

'''h''' = the height of the web (does not include the flange thickness)\\

'''t'_w_'''' = the thickness of the web\\

'''E''' = the modulus of elasticity of steel (29,000 ksi)\\

'''F'_y_'''' = the yield strength of the steel (e.g. 36 ksi, 50 ksi, 65 ksi, etc.)\\

!! References

# American Institute of Steel Construction (AISC 360), "Steel Construction Manual 13th edition", 2005

** Table B4.1 in the Specifications (AISC 360) has been referenced to find λ'_p_' & λ'_r_'. Additional shapes can be found there (including t-sections, HSS rectangular and round sections, built up sections, etc.)

To put your mind at ease, this section is not on weight loss (as a fellow engineer I understand how that could be worrisome to many). In addition to that, the vast majority of sections out there are compact, which is what you want. Discovering whether or not you have a compact, non-compact, or slender section all depends on the shape of the beam. Table B4.1 in the AISC 360 specs provides all the limits you need to determine which zone your section shape falls into (this will be explained in greater detail below).

When you have a '''compact section''' there is no possibility of local flange or web bucking to prevent attainment of the full sections yield strength. In common terms, the beam will not have a local failure (i.e. your web buckles) before the beam has global failure.

With a '''non-compact section''' one or more of the elements in the beam (e.g. in an I-beam which will be detailed below, those elements are the web and the flange) have the possibility of buckling before the beam is able to attain section plasticity.

A '''slender section''' is almost always avoided. But sometimes it is important to check this type of section. Understand a slender section is even more extreme than a non-compact section.

If:

* '''λ ≤ λ'_p_'''' (the section is compact)

* '''λ'_p_' < λ ≤ λ'_r_'''' (the section is non-compact)

* '''λ'_r_' < λ''' (the section is slender)

''where:''

->'''λ''' = Width to thickness ratio (dependent on section type and element)\\

'''λ'_p_'''' = Limiting ratio to see if a section is compact\\

'''λ'_r_'''' = Limiting ratio to see if a section is non-compact\\\

%lframe% '''Note:''' All current ASTM W, S, M, C, & MC shapes have compact flanges except W21x48, W14x99, W14x90, W12x65, W10x12, W8x31, W8x10, W6x15, W6x9, W6x8.5, & M4x6 (for F'_y_' = 50 ksi). All current ASTM W, S, M, HP, C, & MC shapes have compact webs for F'_y_' ≤ 65 ksi.

\\\\

!!! How to find if a rolled doubly symmetric I shaped sections is compact?

This example has been taken from Table B4.1 in the AISC 360 specs '^[1]^'.

(:table border=2 cellpadding=2 cellspacing=0 width=100%:)

(:cellnr bgcolor=#d4d7ba colspan=4 align=center:) '+'''Table 1: Is a Rolled I-shaped Section or Channels compact?'''+'

(:cellnr width=30%:)'''Description of Element'''

(:cell align=center width=20%:)'''Width Thickness Ratio (λ)'''

(:cell align=center width=25%:)'''λ'_p_''''

(:cell align=center width=25%:)'''λ'_r_''''

(:cellnr colspan=4:)

(:cellnr:)Flexure in '''flanges''' of rolled I-shaped sections & channels

(:cell align=center:)[++{$ \frac{b}{t} $}++]

(:cell align=center:)[++{$ 0.38\sqrt{E/F_y} $}++]

(:cell align=center:)[++{$ 1.0\sqrt{E/F_y} $}++]

(:cellnr:)Flexure in '''webs''' of rolled I-shaped sections & channels

(:cell align=center:)[++{$ \frac{h}{t_w} $}++]

(:cell align=center:)[++{$ 3.76\sqrt{E/F_y} $}++]

(:cell align=center:)[++{$ 5.70\sqrt{E/F_y} $}++]

(:tableend:)

''where:''

->'''b''' = the base of the I-shape or channel's flange\\

'''t''' = the thickness of the flange\\

'''h''' = the height of the web (does not include the flange thickness)\\

'''t'_w_'''' = the thickness of the web\\

'''E''' = the modulus of elasticity of steel (29,000 ksi)\\

'''F'_y_'''' = the yield strength of the steel (e.g. 36 ksi, 50 ksi, 65 ksi, etc.)\\

!! References

# American Institute of Steel Construction (AISC 360), "Steel Construction Manual 13th edition", 2005

** Table B4.1 in the Specifications (AISC 360) has been referenced to find λ'_p_' & λ'_r_'. Additional shapes can be found there (including t-sections, HSS rectangular and round sections, built up sections, etc.)