## Design Of Members Subjected To Axial Actions

5.3.1 Members subjected to axial compression

These are members that are subjected to a compressive action acting parallel to the grain and along the centroidal x-x axis of the member, as shown in Figure 5.2. Such members function as columns, posts, stud members in walls or struts in pin jointed trusses.

When a member is subjected to axial compression its failure strength is dependent on several factors:

• strength/stiffness - compressive strength and modulus of elasticity of the timber;

• geometry of the member - cross-sectional sizes and length;

• support condition - the amount of lateral support and fixity at its ends;

• geometric imperfections - deviations from nominal sizes, initial curvature and inclination;

• material variations and imperfections - density, effect of knots, effect of compression wood and moisture content.

### The rules in EC5 take these factors into account.

When subjected to an axial load, because of imperfections in the geometry of the member or variations in its properties, or a combination of both, as the slenderness ratio, X, of the member increases there is a tendency for it to displace laterally and to eventually fail by buckling as shown in Figure 5.3.

The slenderness ratio is defined as the effective length of the member, Le, divided by its radius of gyration, i,

Axial compression.

where the radius of gyration about an axis i = VI/A, I is the second moment of area about the axis, and A is the cross-sectional area of the member.

For any member there will be a slenderness ratio, ky, about the y-y axis and, kz, about the z-z axis and when using a rectangular section, as shown in section A-A in Figure 5.3, the respective slenderness ratios are

. L e,y L e,y L e,z L e,z ky = —- =-^ and kz =-=-—

where Le y and Le z are the effective length about the y-y axis and the z-z axis respectively. Buckling will occur about the axis with the highest slenderness ratio.

The effective length Le (or buckling length) of a compression member is the distance along its length between adjacent points of contra-flexure. These are adjacent points at which the bending moment in the member is zero. Although EC5 gives no information on how to determine the effective length of a compression member, provided the end connections of the member ensure full positional and directional control where required, the guidance given in Table 5.2 can be used. The content of Table 5.2 covers the cases shown in Figure 5.4, where Le is the effective length and L is the actual column length. Where full positional and directional control cannot be assured and the stiffness properties of the end connections are known, approximate solutions can be determined using second-order analysis methods.

For an idealised perfectly straight column of length L having uniform properties and pin jointed at both ends, the theoretical axial load at which buckling will occur about the y-y or the z-z axes within the elastic limit of the column material will be the Euler buckling load for the respective axis. Expressing the Euler buckling loads in terms of the slenderness ratios of the member, they can be written as n 2 E0 05 A n 2 E0 05 A

Table 5.2 Effective length of compression members*

Support condition at the ends of the member

Held effectively in position and direction at both ends 0.7

Held effectively in position at both ends and in direction at one end 0.85

Held effectively in position at both ends but not in direction 1

Held effectively in position and direction at one end and in direction but not 1.5

position at the other end Held effectively in position and direction at one end and completely free at the 2 other end

where PE,y is the Euler buckling load about the y-y axis, PE,z is the Euler buckling load about the z-z axis, E005 is the characteristic modulus of elasticity of the member, A is the cross-sectional area of the member, ky is the slenderness ratio about the y-y axis = (1.0 x L)/iy, kz is the slenderness ratio about the z-z axis = (1.0 x L)/ iz.

Dividing the respective Euler buckling loads by the cross-sectional area of the member, A, the buckling strength of the member about the y-y axis, aE,y, and about the z-z axis, aE,z, is obtained as follows:

n2 E

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