P-Delta Analysis is a Non-linear second-order analysis which exhibits a considerable increase in the base moment when a structure is subjected to large lateral displacement due to wind or seismic forces.

Before moving further, there are two terms that need additional clarification to understand the behavior on the whole. They are “Non-linear” & “Second-order”.

##### What is Non-linear Analysis?

- Geometric Non-linearity
- Material Non-linearity
- Boundary condition Non-linearity

##### Why P-Delta analysis is called “Second-Order” Analysis?

For easy understanding, when a structure is loaded, it will deflect or deform to relieve the stress. This deflection is said to be the first-order effect. Without any additional loading, if any stresses or adverse effects induced in a structure due to the first-order deflection, it is called a second-order effect.

##### P-Delta Analysis

Notice that, in the case of linear static analysis, the lateral deflection depends on the lateral load, H. The deflection d1 due to the applied lateral load H, is the first-order deflection of the column. The moment at the base of the column, M0 will be “Hxh”, as shown in the figure.

Now, this is where it gets interesting. There will be this axial load P which is a result of gravity loads acting on the column. If the initial deflection d1 is large, then geometric non-linearity would occur and this axial load P combines with the initial deflection d1 results in an additional moment “Pxd1”. Now, the moment at the base would be “(Hxh) + (Pxd1)”.

Now, what happens next? wouldn’t the column deflect even more?

Yes, it will. And the new deflection will be d2 and the moment will be increased to “(Hxh) + (Pxd2)”. This goes up to several iterations until the increase in deflection becomes negligible. This iterative analysis process is termed as “P-Delta” Analysis.

It is obvious that a P-Delta effect is named after the secondary moment, Pxd.

There are two types of P-Delta effects. Namely,

- P-Large Delta effect
- P-Small Delta effect

What we discussed so far, comes under P-Large Delta effect. It deals with the entire structure. Whereas, P-Small Delta effect is elementary level. It is a second-order deflection in the particular member due to applied axial load and initial lateral deflection of the member. In most of the cases, P-Small delta won’t have a role, since it is elementary level, it requires huge deflection values to get effective.

##### When P-Delta Analysis is necessary?

- P-Delta effect usually becomes prevalent in a tall structure that is experiencing gravity loads and large lateral displacement due to wind or other forces.
- If the lateral displacement or the vertical axial loads are significant, P-Delta analysis should be performed. In many cases, a linear static analysis can severely underestimate displacement in comparison with the P-Delta effect.

##### Conclusion

P-Delta effect is a second-order effect experienced by any structure when subjected to lateral loads like earthquake or wind loads and is originated by an additional destabilizing moment generated due to the gravity load acting on the laterally deflected member further displacing it.

To be more precise, when a slender structure is subjected to lateral loads like wind or earthquake loads, it undergoes lateral displacement or sway. When this lateral displacement is reasonably large, gravity loads start to act with an eccentricity equal to the magnitude of elastic deflection causing an additional overturning moment. Due to which, the structure is pushed even further developing a second-order deflection. This second-order effect experienced is conveniently termed as P-Delta effect.

Is pdelta applicable to flat slab system?

Asking this because someone said me that.

It may come under P-small delta… In order for P-Large delta to occur, there should be lateral load in the system..

Thanks for posting.

how can we related this P delta effect to the structural system for example outrigger system?