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	Applies To
	Product(s):	STAAD.Pro
	Version(s):	All
	Environment:	N/A
	Area:	Analysis Solutions
	Subarea:	PDelta Analysis
	Original Author:	Bentley Technical Support Group

What is PDelta analysis in STAAD.Pro?

The P-Delta effect is one of the primary parameter that every structural designer would not want to ignore accounting for while performing the stability analysis and design of slender member, tall structure or any structure that experiences significant gravity loads in addition to the lateral force.

This is a second order analysis induced by the geometric non linearity which accounts for both the P-Large delta ( P-Δ) and P-small delta (P-δ) effect. No practical column is ideally straight and vertical and hence the effect of the initial imperfections, out of plumbness must be accounted for in the stability analysis. So it is suggested to take the P-Δ and P-δ effect while performing the analysis.

 

In the conventional first order structural analysis, the equilibrium is expressed in terms of the geometry of the un-deformed structure. In case of the linearly elastic structure, relation between displacement and external force is proportional and hence the unknown deformations can be obtained in a simple and direct manner, whereas second-order analysis requires an iterative procedure to obtain the solutions. This is because the deformed geometry of the structure is not known during the formation of the equilibrium and kinematic relationship.

Thus, the analysis proceeds in a step-by-step incremental manner, using the deformed geometry of the structure obtained from an iterative cycles of the calculation (Iterative analysis).

 

The structure that are more sensitive to 2^nd order effects are the structures with low flexural stiffness and under high compressive force. If the axial force “P” acting vertically on the element is compressive in nature and that is closer to the elastic buckling load “Pcr”, the P-delta effect becomes more significant and prominent.

 

         

Fig-1

 

Also, in Staad, one can perform the PDelta Analysis directly by the consideration of the Geometric Stiffness Matrix [KG].

Here, the stress stiffening effect due to the axial stress is considered directly to modify the actual Stiffness Matrix [K].

Depending on the load “P” acting axially on the member, the lateral stiffness of the members starts decreasing. So more the compressive force experienced by the member, lesser will be its lateral load carrying capacity. This phenomena is known as Stress Softening effect which is the result of the change in GEOMETRIC STIFFNESS of that member. In Staad, the change in the GEOMETRIC STIFFNESS can be captured by invoking the K_G option intended to modify the Stiffness Matrix [K] to [K + K_G].

So, by this approach  the P-Delta stiffness equation is directly linearized by the [K + K_G] matrix and the solution can be  obtained directly and exactly, without iteration. 

Moreover, in addition to the above two methods of performing the PDelta Analysis, there is a simple and approximate way to determine the result of second order effect is to simply magnify the result reported from the first order analysis by the amplification factor 1/(1-P/Pcr), where Pcr is the elastic buckling load of the concerned member. This approach is also known as Amplification Method.

In case of the heavy gravity loading or the flexible structure, the accuracy of capturing the actual P-Delta effect falls and result thus obtained from the Amplification method becomes unreliable. In such situation, performing the iterative P-delta analysis or the PDelta KG analysis is must to predict the actual effect on the structure.

 

 

 

 

Following is a simple example of a cantilever column analyzed in Staad by the iterative analysis, where the “Small delta” (δ) is ignored and is verified and compared with that of the Hand- Calculation upto 4 iteration cycles.

 

 

 

 

 

Hand-calculation and Verification of Large P-delta effect in a Cantilever column with that of Staad result upto 4 iteration cycles.

 

 

Height, L = 10000 mm

E = 205000 N/mm²

I = 8.33 x10⁶ mm⁴

Vertical Load, P =50000 N

Horizontal Load, H = 2000N

 

First Iteration

 

Now, M= HxL = 20 kNm

So, the lateral displacement (Δ₁) at the column tip 

Δ₁ = (ML²)/(3EI)     = 0.3902 m

Also, the vertical load P acting on displaced Δ₁ column tip resulting in generation of the additional moment M₁ at the base.  

M₁ = P x Δ₁ = 19.51 KNm

Now, the total moment “M_t1” at the base = (M+M₁)   = 39.51 KNm

The modified horizontal displacement “Δ₂” undergone for the first modified moment M_t1

Δ₂ = (M_t1L²)/ (3EI) = 0.770 m

Second Iteration

Also, the vertical load “P” acting on the newly displaced column tip (Δ₂ - Δ₁) resulting in the additional moment M₂ at the base

M₂ = P x (Δ₂ - Δ₁) = 19.035 KNm

Now, the total moment at the base, M_t2 = (M+M₁+M₂) = (M_t1+M₂) = 58.54 KNm

The second modified horizontal displacement Δ₃ against the second modified Moment M_t2

Δ₃ = (M_t2L²)/3EI = 1.14 m

Third Iteration

Similarly, following the iteration steps of 1 and 2

M₃ = P x (Δ₃ – Δ₂) = 18.555 KNm

The total moment at the base, M_t3 = M+M₁+M₂+M₃=  77.1 KNm

Δ₄ = (M_t3L²)/3EI = 1.50 m

Fourth Iteration

M₄ = P x (Δ₄ – Δ₃) = 18.15 KNm

The total moment at the base, M_t4 = M+M₁+M₂+M₃= M_t3 + M₄ =

                                                        M_t4 = 95.25

Δ₅ = (M_t4L²)/3EI = 1.85 m

 

Fig-2

 

 

 

 communities.bentley.com/.../PDelta-Models.zip