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Building codes such as BS5950 provide clear guidelines on how to consider P-Δ effect due to sway of a vertically loaded portal frame or single storey building, causing additional moment in the columns.

Should this sway frame effect be extended to the horizontal members of the braced frame structure in an excavation if the walls are not so stiff (such as an interlocking pipe pile wall system in water) and can move longitudinally (in-plane)? If so, how would the effects of P-delta (I am only interested in the big delta) be applied to the waling-strut combination.

I would appreciate if answerers can point to relevant literature to support the answer.

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Generally, in geotechnical engineering you won't allow much movement, especially for excavation support. Nevertheless, you will always have small movements but this is usually too small to consider P-Δ effect and safety factors account for it. Most software automatically do this. If you really want to be conservative allow $0.02H$, where $H$ is your height of excavation as movement and calculate P-Δ.

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P-Δ obviously only need be considered in members with high axial forces. There are two elements of P-Δ to consider:

  • The effect of end restraints etc. inducing bending into a strut meaning that the axial force is no oriented along the centreline of the member. Many struts I see are designed with large physical pinned connections at the end to avoid this.

  • The effect of direct forces on a strut causing the strut to bend and moving the centre of thrust away from the centre of the member. The obvious examples of this could either bending due to differential heating of a strut (top vs. bottom) or the self weight of the strut causing the mid span to sag (I think this is covered by a French design standard for props).

In all cases the basic P-Δ approach can apply. Create your structural model of the system. Run the model and find the various deflections and axial forces. Then add in additional moments into the system based on the deflection multiplied by the axial force. Then check whether the new defections are sufficiently similar to the previous deflections. If not then repeat. The exact process for adding the additional moments will vary depending on the structural model but a lot of typical frame analysis software has this capability built in.

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