Consideration of P-Delta Effects (Second-Order Analysis) in the Response Spectrum Analysis According to ASCE 7-16

Technical Article on the Topic Structural Analysis Using Dlubal Software

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Technical Article

RFEM offers the option to perform a response spectrum analysis according to ASCE 7-16. This standard describes the determination of seismic loads for the American market. It might happen that the P-Delta effect has to be considered due to the stiffness of the entire structure in order to calculate the internal forces and carry out the design.

The P-Delta effect is also known colloquially as the calculation according to the second-order analysis with applying imperfection. According to ASCE 7-16 [1], a check must be made after calculating the equivalent seismic loads as to whether it is necessary to consider this P-Delta effect. The following formulas are given at Point 12.8-16 in the standard.

It is not necessary to consider P-Delta effects on floor shear and moments as well as floor deformations if the stability coefficient is smaller than 0.1. This coefficient can be determined with the following formula:

Stability coefficient

Θ = Px · Δ · IeVx · hsx · Cd

Θ Stability coefficient
Px Maximum design vertical load on and above story x
Δ Design deformation of the story defined in Chapter 12.8.6
le Importance factor according to 11.5.1
Vx Horizontal shear between floors x and x - 1
hsx Story height below the considered story x
Cd Deformation amplification factor according to Table 12.2-1

The stability coefficient must not exceed 0.25 to perform a second-order analysis. If the coefficient is greater than the maximum value, we recommend revising the structure, because it has no stability according to the second-order analysis. The maximum value is calculated as follows:

Maximum stability coefficient

Θmax = 0.5β · Cd  0.25

Θmax Maximum value of stability coefficient
Cd Deformation amplification factor according to Table 12.2-1

where β is the ratio between the shear capacity of the single levels x and x-1. The conservative approach is to apply 1.0 here.

If the stability coefficient is between 0.1 and 0.25, the calculation can be performed according to the P-Delta analysis. It is also possible to calculate the determined internal forces and the deformation according to the linear static analysis, then increase it with a factor. The following formula is used:

Formula 3

Value acc. to second-order analys. = Value acc. to linear static analys.  · 11 - Θ

The P-Delta effect according to ASCE 7-10 of 2016 does not have to be used consistently, as shown with the formulas. Therefore, it is worthwile to check the formulas according to 12.8.6 to avoid extra work.

Keywords

ASCE 7 ASCE 7-16 Seismic Earthquake P-Delta Stability

Reference

[1]   ASCE/SEI 7‑16, Minimum Design Loads and Associated Criteria for Buildings and Other Structures

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  • Updated 01/11/2023

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