# What is P-Delta Analysis?

P-Delta Analysis is a type of analysis that is particularly important for laterally displacing multi-story building structures experiencing a gravity load. In this article, SkyCiv Online Engineering Software will discuss the simple theory behind P-Delta effects and help you understand the importance of such an analysis when modeling your structures.

## What are P-Delta Effects?

Of course, any structural model will deflect when it is loaded. A deflected structure may encounter significant secondary moments because the ends of the members have changed position. To illustrate this, consider the simple cantilevered column example shown below:

In this example, a column of length L is encountering an axial load (P) and a lateral load (V). In a standard linear static analysis we would calculate the lateral deflection (Δ) as:

[math] \Delta = \dfrac{ML^2}{3EI} = \dfrac{VL^3}{3EI} \text{ since M=VL} [math]Notice that in the case of a linear static analysis the lateral deflection, Δ, depends on the lateral load (V). However, if the column is encountering an axial load (P), then wouldn't the column deflect even more? This is obvious because the axial load would induce a secondary moment with a value of P×Δ. To illustrate this, let's sum the moments about the base of the column:

[math] \sum{M}=(V \times L) + (P \times \Delta)= VL + P\Delta \\\\ M_{1} = VL \\\\ M_{2} = P\Delta [math]Here M_{1} is due to the lateral point load whereas, M_{2} is due to the axial load. Each of these moments contribute to lateral deflection differently (you can look up the cantilever formulae for the end deflection due to a point load and a moment, respectively for these formulae):

So really, the total lateral deflection would be closer to:

[math] \Delta_{new} = \Delta_{1} + \Delta_{2} = \dfrac{VL^3}{3EI} + \dfrac{P \Delta L^2}{2EI} [math]We can see that compared to the original deflection value, there is an extra term on the right in terms of P and Δ. If P or Δ are siginificant values, the standard linear static analysis would be underestimating the deflection of the column. It should be obvious now that a P-Delta Analysis is named after the secondary moment **PΔ**. Therefore, P-Delta effects are caused by geometric non-linearity. For this reason, a P-Delta Analysis is often called a **Non-Linear Analysis**. A proper P-Delta Analysis would continue to iterate the process above to update the value of Δ_{new}.

## When do I need to worry about conducting a P-Delta Analysis?

The good news is that SkyCiv Structural 3D can now perform a P-Delta Analysis for you. P-Delta effects usually become prevalent in tall structures that are experiencing gravity loads and lateral displacement due to wind or other forces. If the lateral displacement and/or the vertical axial loads through the structure are significant, a P-Delta Analysis should be performed to account for the non-linearities. In many cases, a linear static analysis can **severely underestimate** displacement (among other results) in comparison with a P-Delta (Non-Linear) Analysis.

The importance of a P-Delta Non-Linear Analysis will be illustrated in the example below. The multi-story frame of the building is 20m in height, with each story being 5m high. The columns are fully fixed at the base with distributed loads on each level. Additionally, there are vertical loads on the top floor and self weight is considered so gravity loads can be simulated. There is also a **relatively small** lateral load applied to the side of the structure.

Under these conditions let's compare the results between a Linear and P-Delta (Non-Linear) Analysis:

Linear | P-Delta (Non-Linear) | % Difference | |
---|---|---|---|

Max Total Displacement | 254 mm | 353 mm | + 39% |

Max Vertical Reaction | 629 kN | 668 kN | + 6% |

Max Moment Reaction | 42 kN-m | 60 kN-m | + 43% |

It is clear that P-Delta effects play an extremely significant role with this structure, increasing the max deflection and max moment results by around **40%**! Thus, a linear static analysis is inadequate in such a case.

In summary, P-Delta Analysis is preferable to Linear Static Analysis as it accounts for unforeseen non-linearities in your model. You can use SkyCiv Structural 3D to perform fast and effective P-Delta Analyses on your models; simply select "P-Delta Analysis" when clicking "Solve." Let the software do the work for you so all you have to worry about is the design!

**Paul Comino**

CTO and Co-Founder of SkyCiv

BEng Mechanical (Hons1), BCom

I think the Delta_new equation would be VL^3/3EI + PDeltaL^2/2EI (the denominator of the Pdelta term is 2EI instead of 3EI), Could you confirm 😀

BEST REGARDS!

Hi Sergio. The deflection of a cantilever subject to a point load on the end is: ML^2/3EI. This is where the 3EI comes from. Can you please explain why it would be 2EI in the denominator for the P-delta term?

Hi Paul, thanks for your answer. Maybe I’m wrong but this is my argument:

From the Free Body diagram, I want an expression of M(x) along the beam. Before deflection we know that the moment at the fixed end is Mf=VL and if we make a cut at a distance x, then M(x)=Vx – Mf = V(x-L).

Then, when deflected, I sum the PDelta to the fixed end moment Mf=VL+PD. Substituting this into the M(x) expression above, and integrating twice respect to x, and then evaluating x=L (max deflection) I got that Dnew = VL^3/3EI + PDL^2/2EI

I would appreciate If you let me know if the way im implementing it is wrong.. and why ;D

Hey Sergio. You are indeed correct. The end deflection due to a point load on a cantilever is ML^2/3EI, however the end deflection due to a couple moment on a cantilever is ML^2/2EI. The PD term is acting as a moment on the end of the member, not a point load so it contributes to deflection by PDL^2/2EI. I have updated the article. Thanks for pointing this out. I was using a general formula for deflection a substituting the moment but that formula only applies if the load is purely a point load, which in this case it wasn’t completely a point load, so the formula was not correct. It’s good to derive it from scratch as you have done rather than using general formula like I did in the original article. Here is the derivation from scratch if anyone is interested to verify the solution:

https://uploads.disquscdn.com/images/1c6363ef9d0f1545a0572d4d6ed0d31c59b5d47d9d591ece7d5baed5c2546296.jpg