Doubly symmetrical systems—basic critical loads

The basic critical loads are those which belong to the basic (uncoupled) critical modes: sway buckling in the principal planes and pure torsional buckling. This is the case with doubly symmetrical arrangements when the shear centre of the bracing system and the centre of the vertical load coincide and the three basic

modes develop independently of one another. The governing differential equations characterizing the doubly symmetrical case (and the basic modes) are obtained from equations (3.1), (3.2) and (3.3) which, in the uncoupled case when xc=yc=0 holds, become independent of each other and simplify to

The basic critical loads for buildings subjected to uniformly distributed load on each floor are given in the following. Based on the well-known classical formula for a cantilever subjected to uniformly distributed axial load [Timoshenko and Gere, 1961], the two sway critical loads in the principal directions are:

where parameter r_s is a reduction factor.

Formulae (3.11) only differ from Timoshenko’s formula in factor rs. It is a reduction factor which allows for the fact that the actual load of the structure consists of Fig. 3.1 Load diagrams, a) Load on the multistorey building, b) first part of the equivalent load: uniformly distributed load, c) second part: concentrated force on top.

concentrated forces on floor levels (Fig. 3.1/a) and is not uniformly distributed over the height (Fig. 3.1/b) as is assumed for the derivation of the original formula for buckling. The continuous load is obtained by distributing the concentrated forces downwards resulting in a more favourable load distribution. This unconservative manoeuvre leaves a concentrated force on top of the column (Fig.

3.1/c), which is not covered by the classical formula. The effect of this concentrated force can be accounted for by applying Dunkerley’s [1894] summation theorem.

(The summation theorems and their use in civil engineering are discussed in detail by Tarnai [1999].)

Dunkerley’s summation theorem applied to two load systems (uniformly distributed load and concentrated top load) leads to the formula

where F_cr is the critical load when the structure is subjected to a top load, N_cr is the critical load when the structure is subjected to a uniformly distributed load and is the total critical load when the two load systems act simultaneously. Figure 3.2 demonstrates that when the structure is under two load systems, the total critical load is always smaller than the critical load of either of the two loads.

It is advantageous to use the Dunkerley formula in a different form. When the magnitude of either of the applied loads is fixed, the magnitude of the other load can be calculated from

(3.12)

Fig. 3.2 Graphical interpretation of the Dunkerley formula.

(3.13)

In this case, the magnitude of the concentrated top load is F=N/2n, where n is the number of stories (Fig. 3.1/c). After substituting N/2n for F, formula (3.13) can be rearranged as

from where, in making use of the ratio N_cr/F_cr=3.176 of the classical column formulae, the value of the total vertical load can be calculated as

In this formula which now takes into account the real load on the equivalent column according to Fig. 3.1/a,

is the reduction factor which is introduced in formulae (3.11) and whose values are given in Table 3.1. The accuracy of the (conservative) Dunkerley formula is improved in Table 3.1 by making use of the exact solution for one and two storey structures under concentrated forces—the values of r_s in Table 3.1 for n=1 and n=2 reflect this modification.

The critical load for pure torsional buckling [solution of eigenvalue problem (3.10)] is obtained from

or, in the special case when the warping stiffness is zero, for example for thin-walled, closed cross-sections, from

N=r_sN_cr. ^(3.14)

(3.15)

Table 3.1 Reduction factor r_s

(3.16)

In formula (3.16) the values of critical load parameter α (eigenvalue of pure torsional buckling) are given in Fig. 3.3 as a function of

where

is the torsion parameter and r_s is the modifier whose values are given in Table 3.1.

The diagram in Fig. 3.3 covers the range 0 ≤ ks ≤ 2 where most practical cases fall. When the value of ks exceeds 2.0, or greater accuracy is needed, Table 3.2 can be used. When Table 3.2 was compiled, the special solution procedure (demonstrated in Appendix B) was used which makes it possible to obtain reliable solutions of good accuracy even for ill-conditioned eigenvalue problems.

(3.17)

(3.18)

(3.19)

Fig. 3.3 Critical load parameter α.

Table 3.2 Critical load parameter α

The evaluation of the formulae of the basic critical loads (3.11) and (3.16) shows that the most important characteristics that influence the values of the basic critical loads are:

• the height of the building,

• the bending stiffnesses of the bracing system,

• the warping stiffness of the bracing system,

• the radius of gyration.

The sway critical loads are in direct proportion to the bending stiffnesses of the bracing system and in inverse proportion to the square of the height of the building.

In a similar manner, the pure torsional critical load is in direct proportion to the warping stiffness of the bracing system and in inverse proportion to the square of the height. The Saint-Venant torsional stiffness affects the value of the critical load through the critical load parameter α(k_s) but its effect is normally small as in most practical cases k_s<1 holds. There is, however, a significant difference between the sway-and pure torsional critical loads. The value of the pure torsional critical load also depends on the radius of gyration. The effect of the radius of gyration is best shown by formula (2.11). According to the formula which assumes uniformly distributed floor load, the greater the size of the building (and the distance between the shear centre and the centre of vertical load), the greater the radius of gyration and consequently the smaller the pure torsional critical load. This is in sharp contrast to sway buckling where the geometrical characteristics of the layout of the building do not influence the critical load.

Formula (3.17) shows another interesting fact. When a structure is braced by a bracing element of zero warping stiffness (e.g. a core of thinwalled, closed cross-section), the value of the critical load for pure torsional buckling does not depend on the height of the building nor on the distribution of the load.