C-E7E7 Members with slender elements

The structural engineer designing with hot-rolled shapes will seldom find an occasion to turn to Section E7. Among rolled shapes, the most frequently encountered cases requiring the application of this section are beam shapes used as columns, columns containing angles with thin legs, and tee-shaped columns having slender stems. Special attention to the determination of effective area must be given when columns are made by welding or bolting thin plates together or ultra-high strength steels are employed.

The provisions of Section E7 address the modifications to be made when one or more plate elements in the column cross section are slender. A plate element is considered to be slender if its width-to-thickness ratio exceeds the limiting value, $\lambda_{r}$, defined in Table B4.1a. If the plate element is not slender, it can support the full yield stress without local buckling. When the cross section contains slender elements, the potential reduction in capacity due to local-global buckling interaction must be accounted for.

Since the 2016 AISC Specification (AISC, 2016), local-global interaction is accounted for via the effective width approach. This method has been used for both stiffened and unstiffened cross-section elements in the North American Specification for the Design of Cold-Formed Steel Structural Members since 2001 (AISI, 2001) and was validated by Seif and Schafer (2014) for structural steel shapes. Prior to 2016, the Specification used the $Q$-factor approach for local-global interaction (Ziemian, 2010) and only stiffened elements included the potential for local postbuckling capacity, while unstiffened elements were assumed to be limited by their elastic local buckling stress.

E7.1 Slender Element Members Excluding Round HSS

The effective width method is employed for determining the reduction in capacity due to local buckling. The effective width method was developed by von Kármán et al. (1932), empirically modified by Winter (1947), and generalized for localglobal buckling interaction by Peköz (1987); see Ziemian (2010) for a complete summary. The point at which the slender element begins to influence column strength, $\lambda_{r} \sqrt{F_{y} / F_{n}}$, is a function of element slenderness from Table B4.1a and column slenderness as reflected through $F_{n}$. This reflects the unified effective width approach where the maximum stress in the effective width formulation is the column stress, $F_{n}$ (as opposed to $F_{y}$ ). This implies that columns designated as having slender elements by Table B4.1a may not necessarily see any reduction in strength due to local buckling, depending on the column stress, $F_{n}$.

Prior to the 2016 AISC Specification, the effective width, $b_{e}$, of a stiffened element was expressed as

(C-E7-1)

where

• $E=\operatorname{modulus}$ of elasticity, ksi (MPa)
• $b=$ width of stiffened compression element, in. (mm)
• $f$ = critical stress when slender element is not considered, ksi (MPa)
• $t=$ thickness of element, in. (mm)

This may be compared with the generalized effective width, Equation E7-3:

(C-E7-2)

where $F_{e l}$ is the local elastic buckling stress, and $c_{1}$ is the empirical correction factor typically associated with imperfection sensitivity. The two expressions are essentially equivalent if one recognizes that

(C-E7-3)

where

$\mathrm{v}=\mathrm{Poisson}^{\prime} \mathrm{~s}$ ratio

= 0.3

and utilizes $k=4.0$ for the stiffened element, $c_{1}=0.18$ for the imperfection sensitivity factor, and sets $f=F_{n}$.

Equation E7-3 provides an effective width expression applicable to both stiffened and unstiffened elements. Further, by making elastic local buckling explicit in the expression, the potential to use analysis to determine $F_{e l}$ is also allowed [see Seif and Schafer (2010)]. For ultra-high-strength steel sections or sections built-up from thin plates, this can be especially useful.

Equation E7-5 provides an explicit expression for elastic local buckling, $F_{e l}$. This expression is based on the assumptions implicit in Table B4.1a and was determined as follows. At the limiting width-to-thickness ratio, $\lambda=\lambda_{r}, b=b_{e}$, and $F_{e l}=F_{e l-r}$; therefore, at this limit, local elastic buckling implies

(C-E7-4)

and the effective width expression simplifies to

(C-E7-5)

which may be used to back-calculate the plate buckling coefficient, $k$, assumed in Table B4.1a:

(C-E7-6)

This relationship provides a prediction of the elastic local buckling stress consistent with the $k$ implicit in Table B4.1a, after substitution:

(C-E7-7)

Thus, $\lambda_{r}$ from Table B4.1a may be used to determine $k$, which may be used to determine the elastic local buckling stress. Further, $c_{2}$ is shown to be determined by $c_{1}$ alone, and is used only for convenience.

Equation E7-3 has long been used in the North American Specification for the Design of Cold-Formed Steel Structural Members (AISI, 2016) with $c_{1}=0.22$ for both stiffened and unstiffened elements. The same $c_{1}$ factor is adopted here for all elements, except those that prior to the 2016 AISC Specification had explicit (and calibrated) effective width expressions.

One disadvantage of Equation E7-3, and the explicit use of $F_{e l}$, is the loss of convenience when working with a particular slender element. If Equation E7-5 is utilized directly, then Equation E7-3 may be simplified to

(C-E7-8)

or, more specifically, for case (a), stiffened elements, except walls of square and rectangular sections of uniform thickness,

(C-E7-9)

for case (b), walls of square and rectangular sections of uniform thickness,

(C-E7-10)

or, case (c), all other elements,

(C-E7-11)

These equations may be further simplified if the constants associated with the slenderness limit, $\lambda_{r}$, are combined with the constants in Table E7.1. This results in

(C-E7-12)

where $c_{3}$ is the constant associated with slenderness limits given in Table B4.1a (Geschwindner and Troemner, 2016). Combining the constants in Equation C-E7-12 with $c_{4}=c_{2} c_{3}$ and $c_{5}=c_{1} c_{2} c_{3}$ yields

(C-E7-13)

Table C-E7.1 Constants for use in Equations C-E7-12 and C-E7-13

The constants c4 and c5 are given in Table C-E7.1 for each of the cases in Table B4.1a, excluding round HSS.

The impact of the changes to the effective width approach implemented in the 2016 AISC Specification (AISC, 2016) for treatment of slender-element compression members is greatest for unstiffened-element compression members and may be negligible for stiffened-element compression members as shown by Geschwindner and Troemner (2016).

E7.2 Round HSS

The classical theory of longitudinally compressed cylinders overestimates the actual buckling strength, often by 200% or more. Inevitable imperfections of shape and the eccentricity of the load are responsible for the reduction in actual strength below the theoretical strength. The limits in this section are based upon test evidence (Sherman, 1976), rather than theoretical calculations, that local buckling will not occur if $D / t \leq 0.11 E / F_{y}$. When $D / t$ exceeds this value but is less than $0.45 E / F_{y}$, Equation E7-7 provides a reduction in the local buckling effective area. This Specification does not recommend the use of round HSS or pipe columns with $D / t>0.45 E / F_{y}$.

Following the SSRC recommendations (Ziemian, 2010) and the approach used for other shapes with slender compression elements, an effective area is used in Section E7 for round sections to account for interaction between local and column buckling. The effective area is determined based on the ratio between the local buckling stress and the yield stress. The local buckling stress for the round section is taken from AISI provisions based on inelastic action (Winter, 1970) and is based on tests conducted on fabricated and manufactured cylinders. Subsequent tests on fabricated cylinders (Ziemian, 2010) confirm that this equation is conservative.