C-B4B4 Member properties

B4.1 Classification of Sections for Local Buckling

Cross sections with a limiting width-to-thickness ratio, $\lambda$, greater than those provided in Table B4.1 are subject to local buckling limit states. Since the 2010 AISC Specification (AISC, 2010), Table B4.1 has been separated into two parts: B4.1a for compression members and B4.1b for flexural members. Separation of Table B4.1 into two parts reflects the fact that compression members are only categorized as either slender or nonslender, while flexural members may be slender, noncompact, or compact. In addition, separation of Table B4.1 into two parts clarifies ambiguities in $\lambda_{r}$. The width-to-thickness ratio, $\lambda_{r}$, may be different for columns and beams, even for the same element in a cross section, reflecting both the underlying stress state of the connected elements and the different design methodologies between columns,

Chapter E and Appendix 1, and beams, Chapter F and Appendix 1. A comprehensive review of the basis for the local buckling width-to-thickness limits of Table B4.1 is presented by Schafer et al. (2022).

Limiting Width-to-Thickness Ratios for Compression Elements in Members Subjected to Axial Compression. Compression members containing any elements with width-to-thickness ratios greater than $\lambda_{r}$ provided in Table B4.1a are designated as slender and are subject to the local buckling reductions detailed in Section E7. Nonslender compression members (all elements having width-to-thickness ratio $\leq \lambda_{r}$ ) are not subject to local buckling reductions.

Flanges of Built-Up I-Shaped Sections. In the 1993 Load and Resistance Factor Design Specification for Structural Steel Buildings (AISC, 1993), for built-up I-shaped sections under axial compression (Case 2 in Table B4.1a), modifications were made to the flange local buckling criterion to include web-flange interaction. The $k_{c}$ in the $\lambda_{r}$ limit is the same as that used for flexural members. Theory indicates that the web-flange interaction in axial compression is at least as severe as in flexure. Rolled shapes are excluded from this provision because there are no standard sections with proportions where the interaction would occur at commonly available yield stresses. In built-up sections where the interaction causes a reduction in the flange local buckling strength, it is likely that the web is also a thin stiffened element. The $k_{c}$ factor accounts for the interaction of flange and web local buckling demonstrated in experiments reported in Johnson (1985). The maximum limit of 0.76 corresponds to $F_{c r}=0.69 E / \lambda^{2}$, which was used as the local buckling strength in earlier editions of both the ASD and LRFD Specifications. An $h / t_{w}=27.5$ is required to reach $k_{c}=0.76$. Fully fixed restraint for an unstiffened compression element corresponds to $k_{c}=1.3$ while zero restraint gives $k_{c}=0.42$. Because of web-flange interactions, it is possible to get $k_{c}<0.42$ from the $k_{c}$ formula. If $h / t_{w}>5.70 \sqrt{E / F_{y}}$, use $h / t_{w}=5.70 \sqrt{E / F_{y}}$ in the $k_{c}$ equation, which corresponds to the 0.35 limit.

Rectangular HSS in Compression. The limits for rectangular HSS walls in uniform compression (Case 6 in Table B4.1a) have been used in AISC Specifications since 1969 (AISC, 1969). They are based on Winter (1968), where adjacent stiffened compression elements in box sections of uniform thickness were observed to provide negligible torsional restraint for one another along their corner edges.

Round HSS in Compression. The $\lambda_{r}$ limit for round HSS in compression (Case 9 in Table B4.1a) was first used in the 1978 Specification for the Design, Fabrication, and Erection of Structural Steel for Buildings (AISC, 1978). It was recommended in Schilling (1965) based upon research reported in Winter (1968). Excluding the use of round HSS with $D / t>0.45 E / F_{y}$ was also recommended in Schilling (1965). This is implied in Sections E7 and F8 where no criteria are given for round HSS with $D / t$ greater than this limit.

Limiting Width-to-Thickness Ratios for Compression Elements in Members Subjected to Flexure. Flexural members containing compression elements, all with width-to-thickness ratios less than or equal to $\lambda_{p}$ as provided in Table B4.1b, are designated as compact. Compact sections are capable of developing a fully plastic stress

distribution and they typically possess a rotation capacity, $R_{\text {cap }}$, of approximately 3 (see Figure C-A-1.2) before the onset of local buckling (Yura et al., 1978). Flexural members containing any compression element with width-to-thickness ratios greater than $\lambda_{p}$, but still with all compression elements having width-to-thickness ratios less than or equal to $\lambda_{r}$, are designated as noncompact. Noncompact sections can develop partial yielding in compression elements before local buckling occurs but will not resist inelastic local buckling at the strain levels required for a fully plastic stress distribution. Flexural members containing any compression elements with width-to-thickness ratios greater than $\lambda_{r}$ are designated as slender. Slender-element sections have one or more compression elements that will buckle elastically before the yield stress is achieved. Noncompact and slender-element sections are subject to flange local buckling or web local buckling reductions as provided in Chapter F, and summarized in Table User Note F1.1, and in Appendix 1.

The values of the limiting ratios, $\lambda_{p}$ and $\lambda_{r}$, specified in Table B4.1b are similar to those in the 1989 Specification for Structural Steel Buildings-Allowable Stress Design and Plastic Design (AISC, 1989) and Table 2.3.3.3 of Galambos (1978), except that $\lambda_{p}=0.38 \sqrt{E / F_{y}}$, limited in Galambos (1978) to determinate beams and to indeterminate beams when moments are determined by elastic analysis, was adopted for all conditions on the basis of Yura et al. (1978). For greater inelastic rotation capacities than provided by the limiting value of $\lambda_{p}$ given in Table B4.1b or for structures in areas of high seismicity, see AISC Seismic Provisions for Structural Steel Buildings, Chapter D and Table D1.1 (AISC, 2022c).

Webs in Flexure. In the 2010 Specification for Structural Steel Buildings (AISC, 2010), and continuing for this Specification, formulas for $\lambda_{p}$ were added as Case 16 in Table B4.1b for I -shaped beams with unequal flanges based on White (2008). In extreme cases where the plastic neutral axis is located in the compression flange, $h_{p}=0$ and the web is considered to be compact.

Rectangular HSS in Flexure. The $\lambda_{p}$ limit for compact sections is adopted from Limit States Design of Steel Structures (CSA, 2009). Lower values of $\lambda_{p}$ are specified for high-seismic design in the AISC Seismic Provisions for Structural Steel Buildings (AISC, 2022c), based upon tests (Lui and Goel, 1987) that have shown that rectangular HSS braces subjected to reversed axial load fracture catastrophically under relatively few cycles if a local buckle forms. This was confirmed in tests (Sherman, 1995a) where rectangular HSS braces sustained over 500 cycles when a local buckle did not form, even though general column buckling had occurred, but failed in less than 40 cycles when a local buckle developed. Since 2005, the $\lambda_{p}$ limit for webs in rectangular HSS flexural members (Case 19 in Table B4.1b) has been reduced from $\lambda_{p}=3.76 \sqrt{E / F_{y}}$ to $\lambda_{p}=2.42 \sqrt{E / F_{y}}$ based on the work of Wilkinson and Hancock (1998, 2002).

Box Sections in Flexure. Since the 2016 Specification (AISC, 2016), box sections have been defined separately from rectangular HSS. Thus, Case 21 in Table B4.1b addresses flanges of box sections and Case 19 addresses webs of box sections.

Round HSS in Flexure. The $\lambda_{p}$ values for round HSS in flexure (Case 20, Table B4.1b) are based on Sherman (1976), Sherman and Tanavde (1984), and Ziemian

(2010). Beyond this, the local buckling strength decreases rapidly, making it impractical to use these sections in building construction.

Limiting Width-to-Thickness Ratios Where the Web and Flange are not Continuously Attached. The Specification does not explicitly provide requirements for cases where the cross-section elements are not continuously attached; this is due to the fact that the buckling occurs over a short length and it is generally not practical to provide less than continuous connection and still limit local buckling. Of course, a stiffened element with one longitudinal edge intermittently connected may be conservatively considered as an unstiffened element for the purposes of determining limiting width-to-thickness ratios.

If an intermittent connection is to provide restraint similar to a continuous connection, it must provide adequate stiffness and strength at a spacing within the local buckling half-wavelength of the cross-section element, in other words, at a spacing less than approximately one-half the width-to-thickness ratio from Table B4.1 multiplied times the thickness. In general, width-to-thickness ratios similar to those in Table B4.1 can be derived if the local buckling stress, $F_{c r}$, is accurately known. In classic form, the local buckling stress is expressed as

(C-B4-1)

and is, therefore, a function of the material properties, dimensions, loading, and boundary conditions as reflected through the plate buckling coefficient, $k$; see Ziemian (2010). As detailed in Seif and Schafer (2010), if $F_{c r}$ is known, then the limiting width-to-thickness ratio, $\lambda_{r}$, from Table B4.1 is the $b / t$ such that $\sqrt{F_{y} / F_{c r}}=0.7$ for elements in compression and $=1.0$ for elements in flexure. The limiting widthto-thickness ratio, $\lambda_{p}$, from Table B4.1b is the $b / t$ such that $\sqrt{F_{y} / F_{c r}}=0.46$ for unstiffened elements and $=0.58$ for stiffened elements. Thus, if the engineer can approximate $k$, or perform a local buckling analysis consistent with the intermittent connections for finding $F_{c r}$, then a limiting width-to-thickness ratio may be approximated. For an intermittently connected plate, AISI (2016) uses the approximation of a pin-ended column of length equal to the connector spacing for determining $F_{c r}$. For complex connection conditions, the provisions of Appendix 1 may provide an alternative path for Specification-compliant design.

B4.2 Design Wall Thickness for HSS

ASTM A500/A500M tolerances allow for a wall thickness that is not greater than ±10% of the nominal value. Because the plate and strip from which these HSS are made are produced to a much smaller thickness tolerance, manufacturers in the United States consistently produce these HSS with a wall thickness that is near the lower-bound wall thickness limit. Consequently, AISC and the Steel Tube Institute of North America (STI) recommend that 0.93 times the nominal wall thickness be used for calculations involving engineering design properties of these HSS. This results in a weight (mass) variation that is similar to that found in other structural shapes. The design wall thickness and section properties based upon this reduced thickness have been tabulated in AISC and STI publications since 1997.

Two new HSS material standards were added to the 2016 Specification (AISC, 2016). ASTM A1085/A1085M is a standard in which the wall thickness is permitted to be no more than 5% under the nominal thickness and the mass is permitted to be no more than 3.5% under the nominal mass. This is in addition to a Charpy V-notch toughness limit and a limit on the range of yield strength that makes ASTM A1085/ A1085M suitable for seismic applications. With these tolerances, the design wall thickness may be taken as the nominal thickness of the HSS. Other acceptable HSS products that do not have the same thickness and mass tolerances must still use the design thickness as 0.93 times the nominal thickness as discussed previously.

The other material standard added to the 2016 Specification was ASTM A1065/A1065M. These HSS are produced by cold-forming two C-shaped sections and joining them with two electric-fusion seam welds to form a square or rectangular HSS. These sections are available in larger sizes than those produced in a tube mill. Because the thickness meets plate tolerance limits, the design wall thickness may be taken as the nominal thickness. Prior to the 2016 AISC Specification, they were classified as box sections because they were not produced according to an ASTM standard. With the ASTM A1065/A1065M standard, they are included as acceptable HSS and the term box section is used for sections made by corner welding four plates to form a hollow box.

B4.3 Gross and Net Area Determination

B4.3a Gross Area

Gross area is the total area of the cross section without deductions for holes or ineffective portions of elements subjected to local buckling.

B4.3b Net Area

The net area is based on net width and load transfer at a particular chain. Because of possible damage around a bolt hole during drilling or punching operations, 1/16 in. (2 mm) is added to the nominal bolt hole diameter when computing the net area for tension or shear.

Unless indicated otherwise, the net area associated with net tensile checks in the Specification assumes the material that is removed represents bolt holes formed in accordance with Section M2.5. For statically loaded structures, open bolt holes and bolt holes filled with snug-tight or pretensioned bolts can be treated identically relative to the net area and net tensile checks. Holes for plug, slot, or fillet welds in holes or slots are treated in the same manner as bolt holes.