# P-Δ vs. P-𝛿

Overview: a look at the differences between P- and P- analysis, and how to investigate these effects in GSA.

# Overview

A P-delta analysis considers the effect of deflection on the loads carried by the structure: the difference between the two options being that the P- effects are global while the P- effects are local.

The features introduced to the design layer in GSA 10.1 make it much more straightforward to consider both P- and P- effects.

# What are P-Δ and P-𝛿 effects

To highlight the difference, consider a simple vertical column of height h subject to a transverse force F at the top and a vertical load P there will be a linear bending moment distribution from 0 at the top to at the base.

p-delta

# Global

In a linear analysis there is no interaction between the axial and transverse load. However, the effect of this transverse load is to create a deflection at the top of the column. If this deflection is then there is an additional moment at the base of the column of , so that the total base moment is:

This is the P- (or P-big-delta) effect that modifies the response of the structure.

Note: If the column is in tension the P- effect reduces the moment.

# Local

While this takes into account the global effect of the interaction of load and deflection, there is a local effect which this misses. The way the element deflects means that the moment at mid height can be more than that predicted by the P- effect.

At mid height there is an additional moment of . This is the P- (or P-small-delta) effect.

While this is not important for the response of the structure as a whole it can be significant to the design of individual members, especially if this effect coincides with the position of maximum bending in the member.

# When are P-Δ and P-𝛿 important to consider

# P-Δ

Typically P- effects are important when a structure is susceptible to buckling. This is usually the case if the buckling load factors are < 10. In these situations a P- analysis should be selected rather than a linear analysis.

Note: When the analysis is no longer linear, combination cases are not valid. Take care to define the design loads for the analysis cases rather than base load cases ***(e.g., dead, live, etc.)***.

# P-𝛿

P- is significant when displacements are large or columns are slender. The P- effect is important for local buckling and for design algorithms that expect member buckling to be accounted for by analysis.

This includes the AISC direct-analysis method, removing the need for a side sway buckling analysis to determine K.

See Static P Delta Analysis Options for more details on how GSA calculates these forces.

# How adjust your GSA model to account for P-Δ and P-𝛿 effects

P- effects can be captured using a Static P Delta Analysis. If the structure requires large deflection assumptions (if stiffness matrix must be generated from deflected geometry), a nonlinear static analysis is recommended.

P- effects can be captured by subdividing members into multiple elements. If a model is built in the GSA design layer column, members are generally defined from floor to floor. The simplest way to pick up the P- for analysis is to split the columns into two or more elements.

To do this:

Select the column members.
Set the element size to approximately half (or an appropriate fraction of ) the storey height.
When using the Create Elements from Members tool, GSA will generate multiple elements per column allowing the analysis to pick up both the P- and P- effects.

Remember: P- effects are global. P- effects are local.

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