In structural engineering, the P-Δ or P-delta effect refers to the abrupt changes in ground shear, overturning moment, and/or the axial force distribution at the base of a sufficiently tall structure or structural component when it is subject to a critical lateral displacement. A distinction can be made between P-delta effects on a multi-tiered building, written as P-Δ, and the effects on members deflecting within a tier, written as P-δ.:lii
P-delta effects are also known as second-order effects since their magnitude depends on the amount of initial displacement or deflection. The P-delta effect is a destabilizing moment equal to the force of gravity multiplied by the horizontal displacement a structure undergoes when loaded laterally.
To illustrate the effect, take the example of a typical statics case: in a perfectly rigid body subject only to small displacements, the effect of a gravitational or concentrated vertical load at the top of the structure is usually neglected in the computation of ground reactions. However, structures in real life are flexible and can exhibit large lateral displacements in unusual circumstances. The lateral displacements can be caused by wind or seismically induced inertial forces. Given the side displacement, the vertical loads present in the structure can adversely perturb the ground reactions. This is known as the P-Δ effect.
In some sense, the P-delta effect is similar to the buckling load of an elastic, small-scale solid column given the boundary conditions of a free end on top and a completely restrained end at the bottom, with the exception that there may exist an invariant vertical load at the top of the column. A rod planted firmly into the ground, given a constant cross-section, can only extend so far up before it buckles under its own weight; in this case the lateral displacement for the solid is an infinitesimal quantity governed by Euler buckling. If the lateral displacement and/or the vertical axial loads through the structure are significant then a P-delta analysis should be performed to account for the non-linearities.
- Lindeburg, M.R., Baradar, M. Seismic Design of Building Structures : A Professional's Introduction to Earthquake Forces and Design Details (8th ed.). Professional Publications, Inc. Belmont, CA (2001).
- Comino, P. What is P-Delta Analysis? SkyCiv Engineering. Sydney, Australia (2016).