Combined Flexure and Axial Force

48.6 Combined Flexure and Axial Force

When a member is subject to the combined action of bending and axial force, it must be designed to resist stresses and forces arising from both bending and axial actions. While a tensile axial force may induce a stiffening effect on the member, a compressive axial force tends to destabilize the member, and the instability effects due to member instability (P- d effect) and frame instability (P-D effect) must be properly accounted for. The P- d effect arises when the axial force acts through the lateral deflection of the member relative to its chord. The P-

D effect arises when the axial force acts through the relative displacements of the two ends of the member. Both effects tend to increase member deflection and moment; so they must be considered in the design. A number of approaches are available in the literature to handle these so-called P-delta effects (see, for example, Galambos [1998] and Chen and Lui [1991]). The design of members subject to combined bending and axial force is facilitated by the use of interaction equations. In these equations the effects of bending and axial actions are combined in a certain manner to reflect the capacity demand on the member.

Design for Combined Flexure and Axial Force Allowable Stress Design

The interaction equations are as follows: If the axial force is tensile,

f a f bx f by

F t F bx F by F t F bx F by

f a = the computed axial tensile stress

f bx and f by = the computed bending tensile stresses about the major and minor axes, respectively

F t = the allowable tensile stress (see Section 48.3)

F bx and F by = the allowable bending stresses about the major and minor axes, respectively (see Section 48.5).

If the axial force is compressive, Stability requirement:

Yield requirement:

However, if the axial force is small (when f a /F a £ 0.15), the following interaction equation can be used in lieu of the above equations.

The terms in the Eqs. (48.70) to (48.72) are defined as follows: f a ,f bx , and f by are the computed axial compressive stress, the computed bending stress about the major axis, and the computer bending stress about the minor axis, respectively. These stresses are to be computed based on a first-order analysis: F y is the minimum specified yield stress; F ex ¢ and F ey ¢ are the Euler stresses about the major and minor axes

( p 2 E/(Kl/r) x and p 2 E/(Kl/r) y ), respectively, divided by a factor of safety of 23/12; and C m is a coefficient to account for the effect of moment gradient on member and frame instabilities (C m is defined in the following section). The other terms are defined as in Eq. (48.69)

The terms in brackets in Eq. (48.70) are moment magnification factors. The computed bending stresses

f bx and f by are magnified by these magnification factors to account for the P-delta effects in the member.

Load and Resistance Factor Design

Doubly or singly symmetric members subject to combined flexure and axial force shall be designed in accordance with the following interaction equations:

For P u / fP n ≥ 0.2,

For P u / fP n < 0.2,

where, if P is tensile, P u is the factored tensile axial force, P n is the design tensile strength (see Section 48.3), M u is the factored moment (preferably obtained from a second-order analysis), M n is the design flexural

Original frame

Nonsway frame

Sway frame analysis

analysis for Mnt

FIGURE 48.11 Calculation of M nt and M lt .

strength (see Section 48.5), f=f t = resistance factor for tension = 0.90, and f b = resistance factor for flexure = 0.90. If P is compressive, P u is the factored compressive axial force, P n is the design compressive strength (see Section 48.4), M u is the required flexural strength (see discussion below), M n is the design

flexural strength (see Section 48.5), f=f c = resistance factor for compression = 0.85, and f b = resistance factor for flexure = 0.90. The required flexural strength M u shall be determined from a second-order elastic analysis. In lieu of such an analysis, the following equation may be used

(48.75) where M nt = the factored moment in a member, assuming the frame does not undergo lateral transla-

M u = 1 bt + B M 2 lt B M

tion (see Fig. 48.11 ) M lt = the factored moment in a member as a result of lateral translation (see Fig. 48.11)

B 1 =C m /(1 – P u /P e ) ≥ 1.0 and is the P-d moment magnification factor P e = p 2 EI/(KL) 2 , with K £ 1.0 in the plane of bending

C m = a coefficient to account for moment gradient (see discussion below)

B 2 = 1/[1 – ( SP u D oh / SHL)] or B 2 = 1/[1 – (SP u / SP e ) SP u = the sum of all factored loads acting on and above the story under consideration

D oh = the first-order interstory translation SH = the sum of all lateral loads acting on and above the story under consideration

L = the story height P e = p 2 EI/(KL) 2

For end-restrained members that do not undergo relative joint translation and are not subject to transverse loading between their supports in the plane of bending, C m is given by

C m = 0604 . - . 1 ËÁ M 2 ¯˜

where M 1 /M 2 is the ratio of the smaller to larger member end moments. The ratio is positive if the member bends in reverse curvature and negative if the member bends in single curvature. For end-restrained members that do not undergo relative joint translation and are subject to transverse loading between their supports in the plane of bending

C m = 0 85 .

For unrestrained members that do not undergo relative joint translation and are subject to transverse loading between their supports in the plane of bending

C m = 1 00 .

Although Eqs. (48.73) and (48.74) can be used directly for the design of beam-columns by using trial and error or by the use of design aids [Aminmansour, 2000], the selection of trial sections is facilitated by rewriting the interaction equations of Eqs. (48.73) and (48.74) into the so-called equivalent axial load form:

For P u / f c P n > 0.2,

(48.76) For P u / f c P n £ 0.2,

Numerical values for b, m, and n are provided in the AISC manual [AISC, 2001]. The advantage of using Eqs. (48.76) and (48.77) for preliminary design is that the terms on the left-hand side of the inequality can be regarded as an equivalent axial load, (P u ) eff , thus allowing the designer to take advantage of the column tables provided in the manual for selecting trial sections.