## Bending Moments Beam Column Action

A5,23 Introduction

A beam-column is a member subjected to transverse loads or end moments plus axial loads. The transverse loading, or 9nd moments, produces bending moments which, in turn, produce lateral bending deflection of the member. The axial loads produce secondary bending moments due to the axial load times this lateral deflection. Compressive axial loads tend to increase the primary transverse bending moments, where as tensile axial loads tend to decrease them.

Beam-column members are quite common in airplane structures. For example, the beams of externally braced wing and tall surfaces are typical examples, the air loads producing transverse beam loads and the struts introducing axial beam loads. In landing gears, one member is usually subjected to large bending and axial loads. In tubular fuselage trusses, lateral loads due to installations supported on members between truss Joints produce beam-column action.

In general, beam column members in airplane structures are comparatively long and slender compared to those in buildings and bridges; thus, the secondary bending moments due to the axial loads are frequently of considerable proportion and need to be considered in the design of the members.

This chapter deals briefly on the theory of single span beam-column members. A summary of equations and design tables is included together with examples of their use. The information in this chapter is used frequently in other chapters where practical analysis and design of beam-column members is considered. For a completed and comprehensive treatment of beam-column theory and derivation of equations, see Nlles and Newell-"Airplane structures1*.

A5. 24 General Action of a Member Subjected to Combined Axial and Transverse Loads.

Sub-figure a of Fig. AS.65 shows a member subjected to transverse loads W end axial compressive loads P. The transverse loads W produce a primary bending distribution on the member as shown In Fig. b. This bending will produce a transverse deflection curve as illustrated in Fig. c. The end loads P now produce an additional secondary bending moment due to the end load p times the deflection 5 , or the bending moment diagram of Fig. d. This first secondary moment distribution produces the additional lateral deflection curve of Fig. e and the end load P will again produce further bending moments due to this deflection. If the axial load is not too large, these successive deflections will gradually converge and the member will reach a state of equilibrium. These secondary bending moments could be found by successive steps by the various deflection principles given in Chapter A7. However, for prismatic beams this convergency can be expressed as a mathematical series and thus save much time over the above successive step method. For members of variable moment of inertia, the secondary moments will usually have to be found by successive steps

If the end loads P are tension, they will tend to decrease the primary moments; thus, in general, the case of axial compression is more important in practical design, since buckling and instability enter into the problem.

A5. 25 Equations for a Compressive Axlally Loaded Strut with Uniformly Distributed Side Load.

Fig. AS.66 shows a prismatic beam of length L subjected to a concentric cempressive load P and a uniformly transverse distributed load w, with the beam supported laterally at each end, and with end restraining moments Mi and Mfl. It Is assumed that the general conditions for the beam theory hold, namely; that-plane sections remain plane after bending; that stress is proportional to strain in both tension' and compression.

At any point a distance x from the beam end, the moment expression is,

From applied mechanics, we know that therefore, differentiating equation (A5.1) twice with respect to x gives

DeffectjUdt* t«1»* Seesodanj Moment

DeffectjUdt* t«1»* Seesodanj Moment

For simplification, let J which, substituted in (A5.2), gives

daM dxa

+1 0