1. Stiffness Matrices in Structural Elements

Each structural element has two types of stiffness matrices:

A. Mechanical Stiffness Matrix (Km):

Determined by the physical properties of the element.

Independent of applied forces.

Can be obtained from the first-order analysis step.

B. Geometric/Stress Stiffness Matrix (Kg):

Directly dependent on the element’s end joint forces and deformations.

Cannot be determined without performing the first-order analysis.

Formulated using the extracted forces and deformations from the first-order analysis.

The total stiffness of an element (Kt) is the sum of both matrices:

Kt=Km+KgKt = Km + KgKt=Km+Kg

If the forces from the first-order analysis are compressive, Kg becomes negative, reducing the total stiffness: Kt=Km−KgKt = Km - KgKt=Km−Kg

If the compressive force is large enough that Kg > Km, then Kt becomes negative, indicating structural instability.

2. P-Delta Effect

The P-Delta effect is a type of geometric nonlinearity that considers the equilibrium and compatibility relationships of a structure under its deformed configuration.

This effect is particularly critical for multi-story buildings subjected to gravity loads, as it magnifies story drift, alters mechanical behavior, and reduces deformation capacity.

Sources of P-Delta Effect:

A. P-Δ Effect (P-"Big Delta")

Associated with the displacement of member ends.

Significant for overall structural behavior under large axial loads.

To accurately account for P-Δ effects, gravity loads must be included in nonlinear analysis.

ETABS software captures the Δ deformation effects well but does not typically account for δ deformation effects.

B. P-δ Effect (P-"Small Delta")

Related to local deformations along the element’s length.

Critical for local buckling and design algorithms that consider member buckling.

In ETABS, columns can be subdivided along their length to evaluate slenderness effects, but this is not recommended for concrete elements.

Alternatively, per ACI Section R6.7.1.2, if the column is not subdivided, slenderness effects can be evaluated using the nonsway moment magnifier method (ACI 6.6.4.5), which aligns with the second-order elastic analysis used in ETABS.

Key Considerations for P-Delta Analysis

Column Stability Check:

If Pu > 0.75Pc, the column is unstable, and its cross-section must be increased.

Slenderness Effect Check:

If δ > 1.4, ACI Section 6.2.6 requires the column cross-section to be enlarged.

Sway vs. Nonsway Frames:

A frame is nonsway if second-order effects increase column-end moments by less than 5% of first-order moments (ACI 6.6.4.3(a)).

This condition is met when lateral deflections are restricted by walls or bracing elements.

P-Delta Analysis Parameters in ETABS:

Further details can be found here: ETABS P-Delta Analysis Parameters

References

ACI 318-14 – Building Code Requirements for Structural Concrete and Commentary.

Geometric Stiffness and P-Delta Effects – By E.D. Wilson.

ETABS Software Manual.

Reinforced Concrete Mechanics and Design – Structural Engineering Reference Book.