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?

To put it in a simple statement, "Stress-Strain relationship is not linear as in the case of linear analysis". Non-linearity may be classified as follows.

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

P-Delta effect falls under the Geometric Non-linearity category. This particular non-linearity is due to the excessive deformation or deflection of the material or structure, even though they are in the elastic limit.

### 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 is called a second-order analysis because it depends on the initial deflection of the structure. If the initial first-order deflection is more, then obviously the P-Delta effect will be more in the structure.

### P-Delta Analysis

Usually, we used to analyze a structure in a static position with loads being applied to it and determine their responses. P-Delta analysis is nothing but analyzing a structure by applying loads on the deflected form of a structure. A deflected structure may encounter significant moments because the ends of the members have changed their position.

Consider a column of length 'h', which is fixed at the bottom and free at the top. It is encountering an axial load P and lateral load H as shown in the following image.

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?

ReplyDeleteAsking 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..

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

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