Response spectra for differential motion of columns, paper II: Out-of-plane response

It is shown that the common response spectrum method for synchronous ground motion can be extended to make it applicable for earthquake response analyses of extended structures experiencing differential out-of-plane ground motion. A relative displacementspectrum for design of first-story columns SDC (T, TT, z, zT, t, d) is defined. In addition to the natural period of the out-of-plane response, T, and the corresponding fraction of critical damping, z, this spectrum also depends on the fundamental period of torsional vibrations, TT, and the corresponding fraction of critical damping, zT, on the ‘‘travel time,’’ t (of the waves in the soil over a distance of about one-half the length of the structure), and on a dimensionless factor d, describing the relative response of the first floor. The new spectrum, SDC, can be estimated by using the empirical scaling equations for relative displacement spectra, SD, and for peak ground velocity, vmax. For recorded strong-motion acceleration, and for symmetric buildings, the new spectrum can be computed from Duhamel’s integrals of two uncoupled equations for dynamics equilibrium describing translation and rotation of a two-degree-offreedom system. This representation is accurate when the energy of the strong-motion is carried by waves in the ground the wavelengths of which are one order of magnitude or more longer than the characteristic length of the structure

Response spectra for differential motion of columns, paper II: Out-of-plane response

It is shown that the common response spectrum method for synchronous ground motion can be extended to make it applicable for earthquake response analyses of extended structures experiencing differential out-of-plane ground motion. A relative displacementspectrum for design of first-story columns SDC (T, TT, z, zT, t, d) is defined. In addition to the natural period of the out-of-plane response, T, and the corresponding fraction of critical damping, z, this spectrum also depends on the fundamental period of torsional vibrations, TT, and the corresponding fraction of critical damping, zT, on the ‘‘travel time,’’ t (of the waves in the soil over a distance of about one-half the length of the structure), and on a dimensionless factor d, describing the relative response of the first floor. The new spectrum, SDC, can be estimated by using the empirical scaling equations for relative displacement spectra, SD, and for peak ground velocity, vmax. For recorded strong-motion acceleration, and for symmetric buildings, the new spectrum can be computed from Duhamel’s integrals of two uncoupled equations for dynamics equilibrium describing translation and rotation of a two-degree-offreedom system. This representation is accurate when the energy of the strong-motion is carried by waves in the ground the wavelengths of which are one order of magnitude or more longer than the characteristic length of the structure

Soil Structure Interaction in Nonlinear Soil

A two-dimensional (2-D) model of a building supported by a semi-circular flexible foundation embedded in nonlinear soil is analyzed. The building, the foundation, and the soil have different physical properties. The model is excited by a half-sine SH wave pulse, which travels toward the foundation. The results show that the spatial distribution of permanent, nonlinear strain in the soil depends upon the incident angle, the amplitude, and the duration of the pulse. If the wave has a large amplitude and a short duration, a nonlinear zone in the soil appears immediately after the reflection from the half-space and is located close to the free surface. This results from interference of the reflected pulse from the free surface and the incoming part of the pulse that still has not reached the free surface. When the wave reaches the foundation, it is divided on two parts—the first part is reflected, and the second part enters the foundation. Further, there is separation of this second part at the foundation-building contact. One part is reflected back, and one part enters the building. After each contact of the part of the wave that enters the building with the foundation-building contact, one part of the wave energy is released back into the soil. This process continues until all of the energy in the building is released back into the soil. The work needed for the development of nonlinear strains spends part of the input wave energy, and thus a smaller amount of energy is available for exciting the building