Effect of second-order forces on seismic response

In a building structure subjected to seismic forces, the gravity loads acting through the lateral displacements lead to additional shears and moments. This is generally referred to as the P-Δ effect; it tends to reduce the capacity of the structure to resist the seismic forces and may lead to instability. It has been suggested that an increase in structural strength, in stiffness, or in both would mitigate the P-Δ effect and ensure stability of the structure. It is shown here that instability results when the P-Δ effect causes the stiffness of the structure to become negative in the post-yield range, in which case increasing the strength, the stiffness, or both does not ensure stability. In a single-storey structure, stability can be ensured if there is sufficient strain hardening that the post-yield stiffness is positive even in the presence of the P-Δ effect. For a multistorey building the vulnerability of the structure to P-Δ instability can be judged by obtaining a pushover curve. It is shown that as long as the maximum displacement produced by the design earthquake lies in the region of positive slope of the pushover curve, the structure will remain stable.