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

### Technical Article

The P-Delta effect is colloquially also known as the calculation according to the second-order analysis with applying imperfection. According to ASCE 7-16 [1], it has to be checked after the calculation of the equivalent seismic loads if it is necessary to consider this P-Delta effect. The following formulae are indicated 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:

$\mathrm\Theta\;=\;\frac{{\mathrm P}_\mathrm x\;\cdot\;\mathrm\Delta\;\cdot\;{\mathrm I}_\mathrm e}{{\mathrm V}_\mathrm x\;\cdot\;{\mathrm h}_\mathrm{sx}\;\cdot\;{\mathrm C}_\mathrm d}$

with

P_{x} = total unfactored vertical design load at and above level x

Δ = design story drift as determined in Section 12.8.6

I_{e }= the importance factor in Section 11.5.1

V_{x} = horizontal shear between the levels x and x - 1

h_{sx} = the story height below level x

C_{d} = deflection amplification factor as given in Tables 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, it is recommended to revise the structure because it has no stability according to the second-order analysis. The maximum value is calculated as follows:

${\mathrm\Theta}_\max\;=\;\frac{0.5}{\mathrm\beta\;\cdot\;{\mathrm C}_\mathrm d}\;\leq\;0.25$

β is here the ratio between the shear capacity of the single levels x and x-1. The conservative approach is to 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 internal forces and deformation according to the linear static analysis and then increase it with a factor. The following formula is used here:

$\mathrm{internal}\;\mathrm{force}\;\mathrm{according}\;\mathrm{to}\;\mathrm{the}\;\mathrm{second}-\mathrm{order}\;\mathrm{analysis}\;=\;\mathrm{internal}\;\mathrm{force}\;\mathrm{according}\;\mathrm{to}\;\mathrm{the}\;\mathrm{linear}\;\mathrm{static}\;\mathrm{analysis}\;\;\cdot\;\left(\frac1{1\;-\;\mathrm\Theta}\right)$

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

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