# Effective Moment of Inertia (If

In order to account for gradual yielding of different parts of the cross section under the applied loading, the concept of “effective moment of inertia” is incorporated in this analysis. Contribution of any steel

element of area As to the flexural rigidity (EI)s is the product of the tangent modulus and the element’s moment of inertia, as follows:

{EI I = EtAvi (7)

where Et is the tangent modulus based on the stress-strain curve of steel. If the idealized elastic-plastic stress-strain curve of steel with Young’s modulus Es is used, then: for ei + rs < £y, Et = Es and for ei + rs >Ey, Et = 0. This indicates that the flexural rigidity of the yielded parts becomes zero. Consequently, the effective bending stiffness {EI) of the entire section takes the following form:

CEI )seff = Es. £U. ty?) (8)

elastic steel

The effective moment of inertia I for the section can then be introduced in terms of the elastic parts

seff

only as follows:

Isef = Екк)

elastic steel

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(b) Axial displacements due to shortening Aaand P-6 effect

(c) Load vs. axial displacement of control specimen

Fig. 6 Variation of stiffness (EI) along member’s
length due to yielding

First yielding will typically occur at the extreme fibres of the member’s cross-section at mid­height. As the axial load and corresponding lateral deflection increase, yielding will spread within the cross-section and also in the longitudinal direction of the member, as shown in Figures 4(b) and 6(a). This indicates that the member will have a moment of inertia that varies with the applied load and also longitudinally within the yielded length as indicated by Eq. 9. As such, a more general expression for the Euler buckling load may be used in lieu of Eq. 6, which assumes a constant moment of inertia. The finite-difference method is used in this case (Ghali and Neville, 1989), where the member is divided into a number of segments of equal length X and the equivalent concentrated elastic loads at each of the m internal nodes can be obtained. A series of simultaneous equations are then solved. The solution of these equations is an eigenvalue problem. An iteration procedure is utilized until a stable eigenvector {8} is obtained. The buckling load Pcr can then be calculated from the largest eigenvalue у as in Eq. 10. Figure 6(b) shows the idealized compression member and the variation of stiffness using five segments.

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Pcr b

A conservative yet reliable simplification may be made by assuming that flexural stiffness of the critical section at mid-height (using Eq. 9) governs, and can then be assumed constant along the length of the member, as shown in Fig. 6(c). This will allow the use of the simple Euler buckling formula, as shown in Eq. 11.

For HSS sections with FRP layers, the transformed effective moment of inertia If should be

J Wf

used in lieu of I in Eq. 11. It is calculated using the following equation:

where I^ = (‘Af y2 ). A, Eft and Aft are the moment of inertia, Young’s modulus, and the area of intact

FRP element i, respectively.

The lateral displacement of the member can now be calculated at any point along the member’s length, at any load level, using Eq. 5.

Axial Displacement

The axial displacement A is the sum of two components, Aa and Ab, as shown in Fig. 5(b):

A = A + 4 (13)

where Aa and Ab are the displacements due to axial shortening and curvature arising from P-8 effect, respectively, and can be approximated as follows:

PL

Ea

where At is the transformed cross sectional area and is calculated as follows:

where, As is the cross sectional area of the HSS section

4 = L – S (16)

where S is the chord length of the deformed shape of the compressed member (Fig. 5(b)), and is calculated based on a sine curve of arc length L and amplitude £for the member’s deflected shape.

Figure 5(c) shows the predicted load-axial displacement response of the control specimen. The figure shows that the contribution of the ‘curvature’ component Ab is only significant near and after the peak load, where the overall buckling occurs, whereas the axial shortening component Aa is dominant within the linear elastic range, before buckling.