STRUCTURAL DESIGN FOR ARCHITECTURE

Structural Design for Architecture

254

(A2.8)

Fig. A2.4 If the conditions of restraint are the same in all

planes a strut will buckle about its weakest axis. Thus, the

element with the rectangular cross-section is less stable in

compression than that with the square cross-section of the

same total area.

/ = r^2 A

where: r = radius of gyration

A = area of cross-section

The effect of second moment of area on the

buckling characteristics of an element can be

seen by considering the behaviour of elements

with different shapes of cross-section. If the

conditions of lateral restraint are the same in

all directions and an increasing amount of axial

load is applied to an element, it will always

buckle about the axis through the centroid of

its cross-section about which the bending

strength is least. This is the axis which gives

the smallest second moment of area. In Fig.

A2.4, for example, the strut with the rectangular

cross-section will fail by compressive instability

at a lower load than that with the square cross-

section, even though their total cross-sectional

areas are the same, because the second

moment of area of the rectangular cross-

section, about one axis through its centroid, is

very small. The fact that the second moments

of area of its cross-section about other axes are

larger than those of the strut with the square

cross-section does not affect its compressive

strength since it buckles about its weakest axis.

Because the quantity second moment of

area must always be calculated with respect to

a particular axis through the cross-section of

an element, the critical load which is calcu-

lated by the Euler formula applies to buckling

in a particular plane. This is the plane which is

normal to the axis which is used to calculate /

(Fig. A2.4). For a given element cross-section, a

number of different critical loads can often be

calculated from the Euler formula depending

on the axis which is chosen for the / value.

Each relates to a particular plane of buckling. If

the conditions of lateral restraint are the same

for all possible planes of buckling, the true

critical load of the element is the value which

is calculated from the smallest value of / of the

cross-section. If the conditions of lateral

restraint are not the same for all buckling

planes the plane for which the ratio of L/l is

greatest determines the critical load.

The property of the cross-section of a

compressive element which is normally used

to gauge the critical load is not in fact the

second moment of area but a related quantity

called the radius of gyration. This is defined by

the equation,

Weak axis

Strong axis

Axis from which

/ calculated.

Plane to which assessment of

buckling based on / relates