## 1 Introduction  
In recent decades, multi-support structures, such as long-span bridges, in cities or on coasts have been constructed with the development of infrastructures around the world [1]. Meanwhile, destructive earthquakes have frequently occurred on earth, for instance, the Wenchuan earthquake of China in 2008 and the Fukushima earthquake of Japan in 2011, which seriously damaged many bridges. These disasters promoted the considerable researches on the anti-seismic capacity of structures. Furthermore, according to analytical and experimental achievements, the anti-seismic design codes of structures in the world were continuously improved [2; 3], leading to the survival of more structures in serious earthquakes [4; 5; 6; 7]. At present, a concept named as resilient city is established to make civil infrastructures enable to be quickly restored after an earthquake [8]. Actually, this resilience indicates that structures could work in a normal/stable working state or a limited elastic-plastic working state during a strong earthquake [9]. Thus, structural analysis, even for the structures in the elastic working state, is still necessary in order to pursue the goal of resilient structures. Meanwhile, the progress in engineering materials and structural designs has greatly improved the structural loading capacity, which has continuously provided higher performance products for structural engineering [10; 11; 12]. Hence, new structures could have a stronger capacity to work in the elastic state than the past structures under the same seismic magnitude [13; 14]. This tendency is promoting the development of analytical techniques for the structural elastic working behavior in both calculating efficiency and simulating accuracy [15]. Infrastructures, such as multi-support structures and long-span structures, are huge in scale and complex in configuration. These structures are important in social life and expensive in cost, so that they must satisfy of safety requirements under the severe earthquakes [16; 17]. So far, response spectrum methods have already become the basic approaches for analyzing multi-supported or long-span structures [18; 19; 20]. The conventional spectrum methods suppose that all the structural supports move synchronously according to the same rule. However, because the seismic response of multi-supported structures is in fact affected by wave passage, random disturbance, damping, and the local site, it is concluded that the motions at different supports of the structure with multi-supports are not completely consistent, particularly, for the long-span structures with multi-supports [21; 22; 23; 24]. [PERSON] and [PERSON] modified the spectral value of the response spectrum in consideration of the effects caused by different seismic wave inputs at various supports, and then used the modal combination method to calculate the structural response [25]. Actually, the multi-support excitations affect not only the spectral value, but also the correlation coefficient of the model combination. Hence, they modified the correlation coefficient in their following study and finally obtained the Berrah-Kausel (B-K) method [26]. Meanwhile, [PERSON] and [PERSON] derived out the multi-support excitation response spectrum (MSRS) method based on the stationary random vibration theory [18]. The MSRS method has a rigorous theoretical basis and reflects the wave passage effect, incoherence effect, and local site effect; also, the MSRS method reflects the ground motion correlation in different supports and the whole working relation in structural vibration modes. In the past twenty years, the MSRS method has made great contributions to the analysis of long-span and multi-support structures under incoherent ground motions. However, the MSRS method also exposes some shortcomings and insufficiencies with its applications:  
The displacements near to the supports of long-span and multi-support structures are not accurate in many cases, based on a conceptual and empirical adjustment.