Compression Members

48.4 Compression Members

Members under compression can fail by yielding, inelastic buckling, or elastic buckling, depending on the slenderness ratio of the members. Members with low slenderness ratios tend to fail by yielding, while members with high slenderness ratios tend to fail by elastic buckling. Most compression members used in construction have intermediate slenderness ratios, so the predominant mode of failure is inelastic buckling. Overall member buckling can occur in one of three different modes: flexural, torsional, and flexural–tor- sional. Flexural buckling occurs in members with doubly symmetric or doubly antisymmetric cross sections (e.g., I or Z sections) and in members with singly symmetric sections (e.g., channel, tee, equal-legged angle, and double angle sections) when such sections are buckled about an axis that is perpendicular to the axis of symmetry. Torsional buckling occurs in members with doubly symmetric sections such as cruciform or built-up shapes with very thin walls. Flexural–torsional buckling occurs in members with singly symmetric cross sections (e.g., channel, tee, equal-legged angle, and double-angle sections) when such sections are buckled about the axis of symmetry and in members with unsymmetric cross sections (e.g., unequal-legged L). Normally, torsional buckling of symmetric shapes is not particularly important in the design of hot- rolled compression members. Either it does not govern or its buckling strength does not differ significantly from the corresponding weak-axis flexural buckling strengths. However, torsional buckling may become important for open sections with relatively thin component plates. It should be noted that for a given cross- sectional area, a closed section is much stiffer torsionally than an open section. Therefore, if torsional deformation is of concern, a closed section should be used. Regardless of the mode of buckling, the governing effective slenderness ratio (Kl/r) of the compression member preferably should not exceed 200.

In addition to the slenderness ratio and cross-sectional shape, the behavior of compression members is affected by the relative thickness of the component elements that constitute the cross section. The relative thickness of a component element is quantified by the width–thickness ratio (b/t) of the element. The width–thickness ratios of some selected steel shapes are shown in Fig. 48.6 . If the width–thickness ratio falls within a limiting value (denoted by the LRFD Specification [AISC, 1999] as l r ) as shown in Table 48.4 , the section will not experience local buckling prior to overall buckling of the member.

Web: Flange: bf/2tf h/tw

Web: Flange: bf/2tf

Flange: bf/tf

Flange: bf/tf

Both Legs: b/t

Web: Flange: (bf - 3t)/t (d - 3t)/t

Web: Flange: bf/tf h/tw

FIGURE 48.6 Definition of width–thickness ratio of selected cross sections.

TABLE 48.4

Limiting Width–Thickness Ratios for Compression Elements under Pure Compression

Width–Thickness Component Element

Ratio Limiting Value, l r

Flanges of I-shaped sections; plates projecting from

0.56 EF § y compression elements; outstanding legs of pairs of angles in continuous contact; flanges of channels

b/t

Flanges of square and rectangular box and hollow structural

1.40 EF § y sections of uniform thickness; flange cover plates and diaphragm plates between lines of fasteners or welds

b/t

Unsupported width of cover plates perforated with a

1.86 EF § y succession of access holes

b/t

Legs of single-angle struts; legs of double-angle struts with

0.45 EF § y separators; unstiffened elements (i.e., elements supported along one edge)

b/t

Flanges projecting from built-up members

0.64 E § ( F y § k c ) Stems of tees

b/t

0.75 EF § y All other uniformly compressed stiffened elements (i.e.,

d/t

1.49 EF § y elements supported along two edges)

b/t

h/t w a Circular hollow sections

D/t b 0.11E/F y a h = web depth, t w = web thickness.

b D = outside diameter, t = wall thickness. c E = modulus of elasticity, F y = specified minimum yield stress, k c = 4/ ÷(h/t w ); 0.35 £k c £ 0.763 for I-shaped

sections, and k c = 0.763 for other sections.

However, if the width–thickness ratio exceeds this limiting width–thickness value, consideration of local buckling in the design of the compression member is required.

To facilitate the design of compression members, column tables for W, tee, double-angle, square and rectangular tubular, and circular pipe sections are available in the AISC manuals for both allowable stress design [AISC, 1989] and load and resistance factor design [AISC, 2001].

Compression Member Design Allowable Stress Design

The computed compressive stress f a in a compression member shall not exceed its allowable value given by

Kl r 2 ˘

Ô Í 1 - () 2 ˙ y F

Ô 5 3 () Kl r () Kl r

Ô 23 () Kl r

if Kl r > C c

where Kl/r = the slenderness ratio

K = the effective length factor of the compression member in the plane of buckling

l = the unbraced member length in the plane of buckling r = the radius of gyration of the cross section about the axis of buckling

E = the modulus of elasticity

C c = ( 2 p 2 EF § y ) is the slenderness ratio that demarcates inelastic from elastic member buck- ling. Kl/r should be evaluated for both buckling axes, and the larger value should be used

in Eq. (48.16) to compute F a .

The first of Eq. (48.16) is the allowable stress for inelastic buckling; the second is the allowable stress for elastic buckling. In ASD, no distinction is made between flexural, torsional, and flexural–torsional buckling.

Load and Resistance Design

Compression members are to be designed so that the design compressive strength f c P n will exceed the required compressive strength P u . f c P n is to be calculated as follows for the different types of overall buckling modes:

Flexural buckling (with a width–thickness ratio of £l r ):

Ï . 0 85 È A

where l c = (KL/r p)÷(F y /E) is the slenderness parameter

A g = the gross cross-sectional area

F y = the specified minimum yield stress

E = the modulus of elasticity K = the effective length factor E = the modulus of elasticity K = the effective length factor

The first of Eq. (48.17) is the design strength for inelastic buckling; the second is the design strength for elastic buckling. The slenderness parameter l c = 1.5 demarcates inelastic from elastic behavior.

Torsional buckling (with a width–thickness ratio of £l r ):

f c P n is to be calculated from Eq. (48.17), but with l c replaced by l e given by

in which C w = the warping constant

G = the shear modulus, which equals 11,200 ksi (77,200 MPa)

I x and I y = the moments of inertia about the major and minor principal axes, respectively

J = the torsional constant K z = the effective length factor for torsional buckling

The warping constant C w and the torsional constant J are tabulated for various steel shapes in the AISC-LRFD manual [AISC, 2001]. Equations for calculating approximate values for these constants for some commonly used steel shapes are shown in Table 48.5 .

Flexural–torsional buckling (with a width–thickness ratio of £l r ):

Same as for torsional buckling, except F e is now given by:

For singly symmetric sections:

( es + ez ) ˚

where F es =F ex if the x axis is the axis of symmetry of the cross section, or = F ey if the y axis is the axis of symmetry of the cross section

F ex = p 2 E/(Kl/r) x 2 ;F ey =p 2 E/(Kl/r) x 2

H = 1 – (x o 2 + y o 2 )/r o 2 , in which K x and K y are the effective length factors for buckling about the x and y axes, respectively l = the unbraced member length in the plane of buckling r x and r y = the radii of gyration about the x and y axes, respectively x o and y o = the shear center coordinates with respect to the centroid ( Fig. 48.7 ), r o 2 =x o 2 +y o 2 +r x 2 +r y 2 .

Numerical values for r o and H are given for hot-rolled W, channel, tee, single-angle, and double-angle sections in the AISC-LRFD manual [AISC, 2001].

For unsymmetric sections:

F e is to be solved from the cubic equation

F - F F 2 2 - y F F - F - F F - F o - - F F F o ( ˆ e ex ) ( e ey ) ( e ez ) e ( e ey )

e ( e ex )

ËÁ r o ¯˜ The terms in the above equations are defined the same as in Eq. (48.20).

ËÁ o ¯˜

TABLE 48.5

Approximate Equations for C w and J Structural Shape

Warping Constant, C w

Torsional Constant, J

E o =b ¢ 2 t f /(2b ¢t f +h ¢t w /3)

(b f 3 t f 3 /4 + h ≤ 3 t w 3 )/36

( ª0 for small t)

(l 1 3 t 1 3 +l 2 3 t 2 3 )/36

( ª0 for small t)

Correction F 0.25

0 2 4 6 8 10 Aspect Ratio, b/t

Note:

b ¢ = distance measured from toe of flange to centerline of web h ¢ = distance between centerlines of flanges h ≤ = distance from centerline of flange to tip of stem

l 1 ,l 2 = length of the legs of the angle t 1 ,t 2 = thickness of the legs of the angle b f = flange width t f = average thickness of flange

t w = thickness of web I c = moment of inertia of compression flange taken about the axis of

the web

I t = moment of inertia of tension flange taken about the axis of the web I x

= moment of inertia of the cross section taken about the major prin-

cipal axis a b i = width of component element i, t i = thickness of component element i,

C i = correction factor for component element i. Local buckling (with a width–thickness ratio of ≥l r ):

Local buckling in the component element of the cross section is accounted for in design by introducing

a reduction factor Q in Eq. (48.17) as follows:

Ï Q l 2 Ô ˘ . 0 85 È AQ

( ) F y ˚˙ l Q £

l ¯˜ ˚ where l=l c for flexural buckling and l=l e for flexural–torsional buckling.

The Q factor is given by Q = QQ s a (48.23)

where Q s = the reduction factor for unstiffened compression elements of the cross section (see Table 48.6 ) Q a = the reduction factor for stiffened compression elements of the cross section (see Table 48.7 ).

x0=y0=0

x0=y0=0

w x0

y0=0 x

y0

x0=0 y0=0

FIGURE 48.7 Location of shear center for selected cross sections.

TABLE 48.6

Formulas for Q s

Structural Element Range of b/t Q s Single angles 0.45 ÷(E/F y ) < b/t <0.91 ÷(E/F y )

1.340 – 0.76(b/t) ÷(F y /E)

0.53E/[F y (b/t) 2 ] Flanges, angles,

b/t ≥ 0.91÷(E/F y )

1.415 – 0.74(b/t) ÷(F y /E) and plates

0.56 ÷(E/F y ) < b/t < 1.03 ÷(E/F y )

0.69E/[F y (b/t) 2 ] projecting from columns or other compression members

b/t ≥ 1.03÷(E/F y )

Flanges, angles, 0.64 ÷(E/(F y /k c )] < b/t < 1.17 ÷[(E/(F y /k c )] 1.415 – 0.65(b/t) ÷(F y /k c E) and plates

0.90E k c /[F y (b/t) 2 ] projecting from built-up columns or other compression members

b/t ≥ 1.17÷[E/(F y /k c )]

Stems of tees 0.75 ÷ E/F y ) < d/t < 1.03 ÷(E/F y ) 1.908 – 1.22(d/t) ÷(F y /E)

0.69E/[F y (b/t) 2 ] Note: k c is defined in the footnote of Table 48.4 , E = modulus of elasticity, F y = specified

d/t ≥ 1.03÷(E/F y )

minimum yield stress, b = width of the component element, d = depth of tee sections, t = thickness of the component element.

TABLE 48.7

Formula for Q a

Q s = effective area actual area

The effective area is equal to the summation of the effective areas of the stiffened elements of the cross section. The effective area of a stiffened element is equal to the product of its thickness, t, and its effective width,

b e , given by For flanges of square and rectangular sections of uniform thickness, when b/t ≥ 1.40÷(E/f ) a :

0 38 E ˘

e = 1 91 t

f Î £ Í () bt f ˚ ˙

For other noncircular uniformly compressed elements, when b/t ≥ 1.49÷(E/f ) a :

˘ b e = . 1 91 t E 1 - . 0 34 E ˙ £ b

f Í Î Í () bt f ˚ ˙

For axially loaded circular sections with 0.11E/F y < D/t < 0.45E/F y :

+ FDt y () 3

. 0 038 E 2

Note: b = actual width of the stiffened element, t = wall thickness, E = modulus of elasticity, f = computed elastic compressive stress in the stiffened elements, D = outside diameter of circular sections. a b e = b otherwise.