Simplifications of CQC method and CCQC method

The response-spectrum mode superposition method is widely used for seismic response analyses of linear systems. In using this method, the complete quadratic combination （CQC） is adopted for classically damped linear systems and the complex complete quadratic combination （CCQC） formula is adopted for non-classically damped linear systems. However, in both cases, the calculation of seismic response analyses is very time consuming. In this paper, the variation of the modal correlation coefficients of displacement, velocity and displacement-velocity with frequency and damping ratios of two modes of interest are studied, Moreover, the calculation errors generated by using CQC and square-root-of-the-sum-of-thesquares （SRSS） methods （or CCQC and CSRSS methods） for different damping combinations are compared. In these analyses, some boundary lines for classically and non-classically damped systems are plotted to distinguish the allowed minimum frequency ratio at given geometric mean of the damping ratios of both modes if their relativity is neglected. Furthermore, the simplified method, which is a special mode quadratic combination method considering only relativity of adjacent modes in CQC method and named simplified CQC or partial quadratic combination （PQC） method for classically damped linear system, is proposed to improve computational efficiency, and the criterion for determination of how many correlated modes should be adopted is proposed. Similarly, the simplified CCQC or complex partial quadratic combination （CPQC） method for the non-classically damped linear system and the corresponding criterion are also deduced. Finally, a numerical example is given to illustrate the applicability, computational accuracy and efficiency of the PQC and CPQC methods.

Complex complete quadratic combination method for damped system with repeated eigenvalues

A new response-spectrum mode superposition method, entirely in real value form, is developed to analyze the maximum structural response under earthquake ground motion for generally damped linear systems with repeated eigenvalues and defective eigenvectors. This algorithm has clear physical concepts and is similar to the complex complete quadratic combination （CCQC） method previously established. Since it can consider the effect of repeated eigenvalues, it is called the CCQC-R method, in which the correlation coefficients of high-order modal responses are enclosed in addition to the correlation coefficients in the normal CCQC method. As a result, the formulas for calculating the correlation coefficients of high-order modal responses are deduced in this study, including displacement, velocity and velocity-displacement correlation coefficients. Furthermore, the relationship between high-order displacement and velocity covariance is derived to make the CCQC-R algorithm only relevant to the high-order displacement response spectrum. Finally, a practical step-by-step integration procedure for calculating high-order displacement response spectrum is obtained by changing the earthquake ground motion input, which is evaluated by comparing it to the theory solution under the sine-wave input. The method derived here is suitable for generally linear systems with classical or non-classical damping.

Simplified Analysis of the Effect of Soil Liquefaction on the Earthquake Pile Response

The time-history response of a structure-pile system during soil liquefaction is highly complicated and several analytical methods have been proposed through the accuracy verification based on the comparison with the experimental works. However, the analytical methods with higher accuracy often require large computational loads and are not necessarily preferred in the actual design practice. On the other hand, while the response spectrum method is not accurate compared to the aforementioned methods, it can provide useful design guidelines in the preliminary stage for structure-pile systems under soil liquefaction with acceptable accuracy. In this paper, the previously proposed response spectrum method for a structure-pile-soil system is used where the effect of soil liquefaction is taken into account by introducing the so-called p-multiplier method. It is shown that, while in the case of inner partial liquefaction with a non-liquefied layer at the top, the demand on the pile moment is large due to the inertial effect of that non-liquefied layer at the top, in the case of overall liquefaction near the ground surface, the demand is smaller than the case of inner partial liquefaction.

Numerical Simulation of Atmosphere-Ocean-Sea Ice Interaction During Interannual Cycle in High Northern Latitudes

The interannual atmosphere-ocean-sea ice interaction （AOSI） in high northern latitudes is studied with a global atmosphere-ocean-sea ice coupled model system, in which the model components of atmosphere and land surface are from China National Climate Center and that of ocean and sea ice are from LASG, Institute of Atmospheric Physics, Chinese Academy of Sciences. A daily flux anomaly correction scheme is employed to couple the atmosphere model and the ocean model with the effect of inhomogenity of sea ice in high latitudes is considered. The coupled model system has been run for 50 yr and the results of the last 30 years are analyzed. After the sea level pressure （SLP）, surface air temperature （SAT）, sea surface temperature （SST）, sea ice concentration （SIC）, and sea surface sensible heat flux （SHF） are filtered with a digital filter firstly, their normalized anomalies are used to perform the decomposition of combined complex empirical orthogonal function （CCEOF） and then they are reconstructed with the leading mode. The atmosphere-ocean-sea ice interactions in high northern latitudes during a periodical cycle （approximately 4 yr） are analyzed. It is shown that： （1） When the North Atlantic Oscillation （NAO） is in its positive phase, the southerly anomaly appears in the Greenland Sea, SAT increases, the sea loses less SHF, SST increases and SIC decreases accordingly; when the NAO is in its negative phase, the northerly anomaly appears in the Greenland Sea, SAT decreases, the sea loses more SHF, SST decreases and SIC increases accordingly. There are similar features in the Barents Sea, but the phase of evolution in the Barents Sea is different from that in the Greenland Sea. （2） For an average of multi-years, there is a cold center in the inner part of the Arctic Ocean near the North Pole. When there is an anomaly of low pressure, which is closer to the Pacific Ocean, in the inner part of the Arctic Ocean, anomalies of warm advection appear in the region near the Pacific Ocean and anomalies of cold advection appear in the region near the Atlantic Ocean. Accompanying with these anomalies of warm and cold advection in these two regions~ warm and cold anomalies appear respectively. Accordingly, SHF sent to the atmosphere from the sea surface decreases and increases, and SST increases and decreases~ SIC decreases and increases in these two regions. When there is an anomaly of high pressure in the inner part of the Arctic Ocean, the former relationships reverse. From these results~ it can be deduced that, during the interannual cycle of the coupled atmosphere-ocean system, the variability of large-scale atmospheric circulation plays a dominant role and variations of SST and SIC are mainly responding to that of atmospheric circulation.