## 46 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 PS 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, and so they must be considered in the design. A number of approaches are available in the literature to handle these so-called P-D effects (see, e.g., Chen and Lui 1991; Galambos 1998). The design of members subject to combined bending and axial force is facilitated by the use 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.

4.6.1 Design for Combined Flexure and Axial Force

4.6.1.1 Allowable Stress Design

The interaction equations are If the axial force is tensile:

Ft Pbx Fby

where fa is the computed axial tensile stress, fbx, fby are computed bending tensile stresses about the major and minor axes, respectively, Ft is the allowable tensile stress (see Section 4.3), and Fbx, Fby are allowable bending stresses about the major and minor axes, respectively (see Section 4.5).

If the axial force is compressive: Stability requirement

Yield requirement fa

0 66Fy Fbx Fby

However, if the axial force is small (when fa/Fa < 0.15), the following interaction equation can be used in lieu of the above equations.

Fa Fbx Fby

The terms in Equations 4.70 to 4.72 are defined as follows:

fa, _/bx, fby = Computed axial compressive stress, computed bending stresses about the major and minor axes, respectively. These stresses are to be computed based on a first-order analysis.

Fy = Minimum specified yield stress.

F0x, F0y = Euler stresses about the major and minor axes (n2E/(Kl/r)x, n2E/(Kl/r)y) divided by a factor of safety of 1|.

Cm = a coefficient to account for the effect of moment gradient on member and frame instabilities (Cm is defined in Section 4.6.1.2).

The other terms are defined as in (Equation 4.69).

The terms in brackets in (Equation 4.70) are moment magnification factors. The computed bending stresses fbx, fby are magnified by these magnification factors to account for the P-d effects in the member.

4.6.1.2 Load and Resistance Factor Design

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

where, if P is tensile,

2fPn^ VfbMnx + fbM

Mu ny

0 0