What is the P-Delta Effect?

P-Delta effect refers to the additional moments generated in a structure due to lateral displacement under vertical loads. When a structure is laterally displaced due to seismic or wind loads, the gravity (vertical) loads act through the displaced configuration, producing secondary moments — called the P-Delta moments.

It is a form of geometric nonlinearity — because the equilibrium equations are evaluated in the deformed shape.

In simple language, P-Delta analysis is a method used in structural engineering to account for the secondary effects of loads on structures. 

Now what is secondary Effect??

First let us know about primary and secondary effect in structural analysis. Which are also termed as first order and second order analysis.

If we apply the equilibrium equation on undeformed shape the bending moment at bottom of column would be, M = F * L. This is the default behavior that we get if we don’t consider geometric non-linearity This is known as linear analysis or elastic first order analysis.

But if we write the equilibrium equation on deformed shape of the column, we will get the bending moment at bottom, M = F * L + P * Δ. This P * Δ is the extra moment known as secondary moment that we get here, and this is what P-Δ effect is. This is also known as second order analysis.

We call this second order because we are considering the deformed shape while writing the equilibrium equation and so we are actually considering the geometric non-linearity

It is important to understand that in our example above, the axial load was compressive, so it has increasing effect on responses like BM and displacements. This is also known as softening effect. However, if the load was tensile, it would actually decrease the BM and displacements, which is known as stiffening effect.

P-𝛅 (small delta)

So far, we have only discussed the P-Δ effect, where Δ represents the lateral deflection at the top relative to its base. This phenomenon is also referred to the global or large delta.

In addition to Δ, there is also 𝛅 which is deflection of column locally with respect to reference line connecting end points of column as shown in figure.

This 𝛅 also causes secondary moments and forces in the frames this is often known as P-𝛅 (small delta) effect.

For design of frame structures, we have to consider both P-Δ and P-𝛅 effect.

P-Δ effect is defined in Analysis model during analysis.  But to capture P-𝛅 (small delta) effect during actual column design amplification factors is used as given in code

P-Delta Analysis in ETABS

There are two methods in ETABS to perform P-Delta Analysis

• Non-iterative Based on Mass
  In which load is automatically computed from the mass at each level. This is an approximate method which does not require an iterative solution, providing for faster computation. It is most effective with a single rigid diaphragm at each level. Local buckling is not captured as effectively. The benefit of this non-iterative method is that P-Delta may be considered in load cases which do not specify gravity load. When gravity load is specified, the Iterative Based on Load Cases method is generally recommended.

• Iterative Based on Load Cases
  In which load is computed from a specified combination of static load cases, then known as the P-Delta load combination. This is an iterative method which considers P-Delta on an element-by-element basis. Local buckling is captured more effectively. An example application may be when load includes the dead load case and a fraction of a live load case.