interbraced columns.
number of zero displacement points within a brace line. The determinant of eqns (4.28) and (4.29) is shown in Appendix 1 with a summary of other details.
The lower envelope to the relationships between the effective brace stiffness and axial load can be very closely approximated by a quadrant of a circle if the nondimensional ρ is used instead of t and if the spring stiffness is represented using the factor
(4.31) Figure 4.14 shows the relationship between true and approximate representations of this lower bound.
4.4 FLEXURAL-TORSIONAL BUCKLING OF COLUMNS
A uniform, doubly symmetric cross-section member under axial compression can buckle in three distinct forms. The lowest buckling load would normally involve bending about the minor axis in a simple Euler manner, at a load dictated by the section properties and end support conditions. If this mode is prevented, or considerably restrained by outside influence, both Euler buckling about the major axis and pure twist buckling about the longitudinal axis through the shear centre of the cross-section are possible. If the section has only one degree of
FIG. 4.15. Illustration of flexural-torsional buckling displacements.
symmetry (e.g. a channel section, an equal leg angle or an I-section with unequal flange widths) the shear centre will not coincide with the centroid and the buckling mode will be either pure displacement in the plane of the axis of symmetry or combined ‘lateral’
displacement and twist. If no symmetry exists, the buckling mode contains components of both lateral displacements as well as twist, as illustrated in Fig. 4.15. A doubly symmetric member on which the axial compression stress is not symmetrically disposed over the cross-section will behave in a similar manner. These latter cases are generally referred to as undergoing flexural-torsional buckling. Given simple support conditions at each end against each form of displacement, the buckling component forms will all vary sinusoidally within the length, each having its own amplitude dependent on its inherent stiffness in that mode.
In the following, such combined modes occur not as a result of the section properties (which are doubly symmetric) but because the elastic constraints (braces) are attached eccentrically to the compression member. As mentioned earlier, such eccentric attachments precipitate flexural torsional buckling but often provide some restraint to those movements in compensation.
FIG. 4.16. Torsional displacement: (a) free-to-warp; (b) warp restrained.
In members which are subjected to torsion, warping occurs. The torsional shears cause the basic ‘plane-sections-remain-plane’ assumption to be violated in all but circular cross-section members. If this distortion is resisted by a stiffening attachment or, more
subtly, by there being non-uniform torsion or torsional resistance within the member, additional local stresses are engendered. Resistance to warping considerably enhances the torsional stiffness of the member, particularly for I and channel shapes, but the resulting stresses can cause local problems. Figure 4.16(a) illustrates the distortions involved in the free twisting of an I-section member, while Fig. 4.16(b) shows the extra distortion and consequently torsional resistance and energy absorption which accompanies the twist where warping is resisted. The figure illustrates how the warping effect in a flanged member is dominated by the flanges moving out of plane with one another.
4.4.1 Interbraced Columns in Flexural-Torsional Buckling
In most interbraced structures of the type discussed in Section 4.3.4, the bracing members are not simply tension-compression members but, like girts and purlins, have considerable bending resistance themselves. The connection with the braced member is often eccentric to the shear centre of that member and is also capable of transferring rotational forces in the plane of the structure. In such cases the brace may provide torsional, rotational and warping support as well as the basic lateral, and at the same time causes the buckling mode of the braced member to involve those four components.
Figure 4.17 illustrates a typical connection within a grid and the associated support forms.
Since out-of-plane displacement is assumed to be zero in the buckling modes, the four further continuity equations of twist , rate of change of twist , torsional moment and warping moment must be satisfied. As an example, the torsional moment continuity across a
FIG. 4.17. Brace connection providing lateral, rotational and warping
constraint.
brace point at which the effective stiffness resisting torsion is KT (moment per unit twist), the linear in-plane spring stiffness is KL and the eccentricity of that brace from the shear centre is e, can be written
(4.32)
The coefficient of KT contains the direct and carryover twist effects, while the other two involve the contributions due to the net extension of the linear spring, attached eccentrically, by the amount e. The sign convention for displacements and forces is shown in Fig. 4.18. The form selected for isthe same as that for un,m in eqn (4.14). The with different coefficients and using α instead of λ. The full set of continuity equations across the general brace point can be written. These are presented in detail by Medland (1979). The factor α represents the axial load and is defined by
α2=(PI0/A−C)/C1
(4.32) Upon substitution of eqns (4.22) and (4.27), the set of four equations which are the four degrees of freedom equivalent of the pair of equations (4.28) and (4.29) may be written and their 4×4 determinant evaluated at increasing levels of axial force until it becomes zero at the