STRUCTURAL DESIGN FOR ARCHITECTURE

The radius of gyration is therefore given by,

(A2.9)

The substitution of I = r^2 A is normally made in
the Euler buckling formula which then
becomes,

Pcr = π^2 E[r/L]^2

This can be rearranged into the form,

Pcr/A = π^2 E[r/L]^2

fcr = π^2 E[r/L]^2 (A2.10)

The introduction of^2 A into the formula
instead of I therefore allows the critical load to
be expressed in terms of a critical average
stress. This is a more convenient form for use
in design.

Design of compression elements to resist buckling
The relationship between critical average stress
and slenderness ratio, which is expressed in
equation (A2.10), is shown in the form of a
sketch graph in Fig. A2.5. This can be used as
the basis of a design method for compression
elements.

Fig. A2.5 The relationship between critical load and

slenderness ratio. The more slender the element the lower

is the critical load. The graph shows the relationship as

predicted by the Euler formula (ideal strut) and gives an

indication of the type of relationship which occurs in

practice and which can be derived empirically. The discrep-

ancy is due to the invalidity in practice of one of the

assumptions on which the Euler formula is based, namely

that all of the material is stressed within the elastic range.

The design of a compression element is a

matter of determining a cross-section whose

size and shape are such that its buckling

strength is greater than the compressive load

which will be applied to it. The following

procedure can be used in conjunction with

versions of Fig. A2.5 for particular materials to

achieve this.

1 The compressive force to be carried is deter-

mined from the analysis of the structure and

the effective length of the strut judged from

its actual length and proposed end condi-

tions (see below for an explanation of effec-

tive length and the importance of

considering end conditions).

2 A trial size and shape of cross-section are

selected.

3 From the properties of the trial cross-

section and the effective length of the strut

the slenderness ratio is calculated and the

Euler formula (or graph) is used to calculate

the magnitude of the average compressive

stress at the critical load value. This is the

value of the compressive stress which must

not be exceeded if buckling failure is to be

avoided.

4 The magnitude of the average compressive

stress which will actually occur in the strut

is calculated from the applied compressive

force and the cross-sectional area of the

trial section.

5 If the relationship between the actual stress

and the critical stress is not satisfactory the

properties of the cross-section are amended

and the above sequence repeated. In the

interests of safety and economy it is desir-

able to achieve a cross-section which results

in the actual stress being slightly smaller

than the critical stress but not excessively

so.

The cyclic process is continued until a satisfac-

tory cross-section is achieved.

Slenderness ratio

The critical load of a compressive element is

affected by its length and the fact that the

length term L appears in the lower part of the 255

Appendix 2

Ideal strut

Slenderness ratio

Critical

load

Real strut