P-Delta Effects and Seismic Design According to the ASCE 7-16 and NBC 2015

Technical Article

As gravity loads act on a structure, lateral displacement occurs. In turn, a secondary overturning moment is generated as the gravity load continues to act on the elements in the laterally displaced position. This effect is also known as "P-Delta (Δ)." Sect. 12.9.1.6 of the ASCE 7-16 Standard and the NBC 2015 Commentary specify when P-Delta effects should be considered during a modal response spectrum analysis.

ASCE 7-16 and P-Delta Effects

Sect. 12.9.1.6 of the ASCE 7-16 Standard [1] clarifies when P-Delta effects should be accounted for when calculating a modal response spectrum analysis for seismic design. This section further refers to Sect. 12.8.7 [1] which states that P-Delta does not need to be considered when the stability coefficient (θ) determined by the equation below is equal to or less than 0.10.

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

Where,
Px = total vertical design load at and above level x with all load factors equal to or less than 1.0
Δ = design story drift defined in Sect. 12.8.6 [1] occurring with Vx
Ie = Importance Factor from Sect. 11.5.1 [1]
Vx = seismic shear force between levels x and x-1
hsx = story height below level x
Cd = deflection amplification factor given in Table 12.2-1 [1]

The standard continues to state that θ should not exceed the lesser of θmax or 0.25 given by the equation below as the structure is potentially unsafe and should be redesigned.

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

When 0.10 ≤ θ ≤ θmax, all displacement and member forces should be multiplied by a factor of $\frac{1.0}{1\;-\;\mathrm\theta}$. Alternatively, P-Delta effects can be included in an automated analysis.

NBC 2015 and P-Delta Effects

In Part 4.1.8.3.8.c of the NBC 2015 Standard [2], only a short requirement is given that sway effects due to the interaction of gravity loads with the deformed structure should be considered. However, the NBC 2015 Commentary [3] gives further explanation similar to the ASCE 7 standard where the stability factor (θx) at level x should be calculated with the given equation below.

${\mathrm\theta}_{\mathrm x}\;=\;\frac{\sum_{\mathrm i=\mathrm x}^{\mathrm n}{\mathrm W}_{\mathrm i}}{{\mathrm R}_{\mathrm o}\sum_{\mathrm i=\mathrm x}^{\mathrm n}{\mathrm F}_{\mathrm i}}\;\cdot\;\frac{{\mathrm\Delta}_{\mathrm{mx}}}{{\mathrm h}_{\mathrm s}}$

Where,
ΣWi = portion of the factored dead plus live load at level x determined from Sent. 4.1.8.11.(7) [3]
ΣFi = sum of the design lateral seismic forces acting at or above level x
Ro = overstrength-related modification factor
Δmx = max inelastic interstory deflection defined in Sent. 4.1.8.13.(3) [3]
hs = interstory height

When θx is less than 0.10, then P-Delta effects can be ignored. When θx is greater than 0.40, the structure should be redesigned as it's considered unsafe during extreme earthquakes. For 0.10 ≤ θx ≤ 0.40, the seismic-induced forces and moments can be multiplied by an amplification factor of (1+θx) to account for P-Delta. This amplification factor does not need to be applied to displacements.

Approximate Consideration of P-Delta Effects with Amplification Factors

The stability factor value should be calculated in both orthogonal horizontal directions to determine if P-Delta is a concern. If one or both directions require second-order effects to be considered within the given ranges, the factor $\frac{1.0}{1\;-\;\mathrm\theta}$ from the ASCE 7-16 [1] or (1 + θx) from the NBC 2015 [3] can easily be accounted for in RF-/DYNAM Pro - Equivalent Loads. All resulting forces and/or deflections will be amplified by the set value.

Figure 01 - Approximate Consideration of P-Delta Effects with Amplification Factors in RF-DYNAM Pro - Equivalent Loads

More Exact Consideration of P-Delta Effects with the Geometric Stiffness Matrix

Although secondary effects can be estimated in the above manner, this is a more conservative approach. For scenarios where large story drifts occur or P-Delta effects need to be calculated with a more exact approach, the influence of axial forces can be activated in the RF-/DYNAM Pro modules.

When running a dynamic analysis, the typical nonlinear iterative calculations for second-order effects when considering a static analysis are no longer applicable. The problem must be linearlized which is carried out by activating the geometric stiffness matrix during the analysis. With this approach, it is assumed that vertical loads do not change due to horizontal effects and that deformations are small when compared to the overall dimensions of the structure.

The concept behind the geometric stiffness matrix is the stress stiffening effect. Tensile axial forces will lead to an increased bending stiffness of a member while compressive axial forces will lead to a reduced bending stiffness. This can be easily conveyed with the example of a cable or slender rod. When the member experiences a tension force, the bending stiffness is significantly greater than when the member is undergoing a compression force. In the case of compression, the member has very little if any bending stiffness to withstand an applied lateral load.

The geometric stiffness matrix Kg can be derived from the static equilibrium conditions. For the purpose of simplification, only the degrees of freedom of the horizontal displacement are displayed here. 

$\begin{bmatrix}{\mathrm F}_{\mathrm i}\\{\mathrm F}_{\mathrm i+1}\end{bmatrix}\;=\;\frac{{\mathrm N}_{\mathrm i}}{{\mathrm h}_{\mathrm i}}\;\cdot\;\begin{bmatrix}1.0&-1.0\\-1.0&1.0\end{bmatrix}\;\cdot\;\begin{bmatrix}{\mathrm u}_{\mathrm i}\\{\mathrm u}_{\mathrm i+1}\end{bmatrix}$

The derivation shown is based on the overturning moment approach due to the linear displacement application. This is a simplification for the bending element and an accurate assumption for the truss element. Notice how the matrix is only dependent on the element length and axial force.

More precise determination of the geometric stiffness matrix for bending beams can be obtained by using the cubic displacement approach or the analytical solution of the differential equation of the bending line. More information on theory and derivations are provided by Werkle [4].

The geometric stiffness matrix Kg is added to the system stiffness matrix K, and thus the modified stiffness matrix Kmod is obtained:

Kmod = K + Kg

In the case of compression normal forces, this consequently leads to the stiffness reduction.

Application of the Geometric Stiffness Matrix in RFEM and RF-DYNAM Pro

The application of stiffness reduction utilizing the geometric stiffness matrix to consider second-order effects (P-Delta) in a response spectrum analysis is performed partially in RFEM and partially in RF-DYNAM Pro.

A detailed example per the ASCE 7 can be found in the FAQ How can I consider the second-order analysis for seismic design? with access to the PDF download.

Additionally, the Dlubal Webinar: ASCE 7-16 Response Spectrum Analysis in RFEM specifically at time 52:25 will provide a detailed look at the workflow in RFEM and RF-DYNAM Pro for application of the geometric stiffness matrix to account for P-Delta effects per the ASCE 7.

Keywords

ASCE 7 ASCE 7-16 NBC NBCC NBC 2015 Seismic Earthquake P-Delta P-Δ Secondary effect Geometric stiffness matrix Stability factor Stability coefficient

Reference

[1]   ASCE/SEI 7‑16, Minimum Design Loads and Associated Criteria for Buildings and Other Structures
[2]   NBC 2015, National Building Code of Canada 2015
[3]   Structural commentaries (User's guide - NBC 2015: part 4 of division B)
[4]   Werkle, H. (2008). Finite Elemente in der Baustatik (3rd ed.). Wiesbaden: Vieweg & Sohn.

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