# 12.8.7 P-Delta Effects

The stability coefficient θ and θ_{max} are determined by eq. 12.8-16 and 12.8-17 automatically.

Where the stability coefficient θ is greater than 0.10, but less than or equal to θ_{max}, ıt’s multiplied displacements and member forces by 1.0 / (1 − θ) automatically.

For both directions, P-Delta Effect is automatically calculated with **Equation 12.8-16** at each floor.

If the stability coefficient (θ) determined for all floors conforms to the limit value calculated by **Equation 12.8-17**, the stability coefficient are not taken into account in the calculation of the internal forces based on the design.

If the condition given in **Equation 12.8-17** is not fulfilled, all internal forces are automatically increased according to Section 12.8.7.

### Symbols

* Px =* Total vertical design load at and above level x [kip (kN)]

*Design story drift as defined in Section 12.8.6 occurring simultaneously with V*

**Δ =**_{x}[in. (mm)]

*Importance Factor*

**I**_{e }=*Seismic shear force acting between levels x and x − 1 [kip (kN)]*

**V**_{x}=*story height below level x [in. (mm)]*

**h**_{sx}=*Deflection amplification factor in Table 12.2-1.*

**C**_{d}=* β =* The ratio of shear demand to shear capacity for the story between levels x and x − 1.

Section 12.9.1.6 explains that P-delta effects is determined with Section 12.8.7. And, the base shear used to determine the story shears and the story drifts is determined with Section 12.8.6.

P-delta effects on story shears and moments, the resulting member forces and moments, and the story drifts induced by these effects are not required to be considered where the stability coefficient (θ) as determined by the following equation is equal to or less than 0.10:

The stability coefficient (θ) does not exceed θ_{max}, determined with 12.8-17:

Where the stability coefficient (θ) is greater than 0.10 but less than or equal to θ_{max}, the incremental factor related to P-delta effects on displacements and member forces is determined by rational analysis. Alternatively, it is permitted to multiply displacements and member forces by 1.0 / (1 − θ). Where θ is greater than θ_{max}, the structure is potentially unstable and should be redesigned.

Where the P-delta effect is included in an automated analysis, Eq. (12.8-17) is still satisfied; however, the value of θ computed from Eq. (12.8-16) using the results of the P-delta analysis is permitted to be divided by (1 + θ) before checking Eq. (12.8-17).

The load-bearing system calculation, which is made as a standard under the influence of various loads, is essentially made over an unchanged (not deformed) system. However, taking into account the deformed shape, additional behavior magnitudes (internal forces and displacements, etc.) can be calculated in the structural system. The effect of the deformed shape of the structural system can be taken into account in two ways. In the former, the deformed shape is taken into account in equilibrium conditions. This approach, called the "Large Displacement Theory", can only be applied in flexible systems with very large deformation. In practice, the approach used especially for building bearing systems is the "Second Order Theory", which opposes the deformed shape to be considered only in equilibrium equations.

It is necessary to calculate the "P-Delta effects" that occur as a result of horizontal loads occurring in the vertical elements of the bearing system under the effect of earthquake ground motion in general, and should be taken into account in the design. Because these effects cause both "stiffness loss" and "strength loss" especially due to nonlinear behavior in the conveyor system. If both effects take too large values, it may ultimately lead to loss of stability (buckling) in the conveyor system and eventually collapse.

However, it is not an easy task to obtain P-Delta effects in seismic calculation within the scope of Design Based on Strength, and it is contented with approximate methods. In the approximate methods considered in ASCE 7-16 and which are very similar to each other, rigid diaphragm is accepted and torsional effects are neglected.

Details of the calculations made in the P-Delta effects section of the earthquake regulation report are given.

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