The research for mechanics stimulation method of nonlinear random vibration based on statistical linear theory

Earthquake disasters have brought great harm to people's life safety and economic property. Its effect on fabric mainly focus on random effects currently, the general pseudo excitation method could solve the inefficiency calculation problem of linear random earthquake. However it could not take the nonlinear problem factors into account for calculation. In this paper, we suggest that a nonlinear structural incentive method should be improved based on statistical linearity to calculate and solve absolute displacement value. Through the analysis and research for cases, we calculate the displacement, speed, random vibration spectrum of bridge’s accelerated speed, as well as the influencing situation of axial force. The results indicate that such perfect incentive method could not only perform nonlinear structure analysis, but also to be very accurate and high effective. Such method could reasonably avoid the displacement decomposition and solution of the pseudostatic model,thus it will be widely applied in common software.


INTRODUCTION
n recent years, earthquake disasters occurred frequently in our country, it has brought more and more loss to people.The prevention and control efforts that we have taken for earthquake disasters have been enlarged in China.However the seismic theory has been gradually advanced, and transformed from static force into dynamic force, and transferred from established pattern into random one.Now we should explore the transforming mechanism from linearity to nonlinearity.The effect of earthquake on structure mainly focuses on the random effects of seismic oscillation currently.The earthquake input process is generally assumed to be stationary stochastic process.When the actual earthquake occurs, its effect is small in early times, while it would be magnified in following period, afterwards it will be of attenuation gradually.Thus the earthquake input process should be considered as non-stationary random process .Accordingly, the effect of earthquake on structure could make construction be in linear elastic stage originally.As earthquake effect increases, the structure will enters into nonlinear phase gradually, and begins to have plastic accumulation.When plasticity accumulation reaches certain stage, the structure will be destroyed for it could not resist seismic action.In the seismic process, how architectural structure operates under the load of earthquake is the standard of measuring anti-seismic effect.In the original seismic experience, applying proper method into designing building structure is an approach suitable to actual circumstance.Nowadays, the widely applied method in China includes response spectrum method, response spectrum method and vibration method.But these methods are relatively fixed, and could not be flexibly applied into the seismic construction of actual building, thus a set of more scientific and reasonable method was born.As it could adopt to the approach of earthquake's random excitation, it is called random vibration method.It takes I power density as the core, and sufficiently considers the probability problem of earthquake occurring.It is very suitable for the building such as bridges, roads etc which have many structural support points with large span, and it could assist the more advanced analysis and be used as design tools.Random vibration method is taken as the design and calculation tool of structural seismic resistance by European structural seismic design code (Eurocode 8) in 1995.The highway bridge seismic design code in China also incorporates random vibration method into the calculating method of aseismic design currently.In recent years, Jia-Hao Lin [1] has proposed simple, convenient and accurate pseudo excitation method starting from computational mechanics; Jiwei Zhang etc [2]have proposed the simple calculation method of response value in structure peak under the non-stationary seismic excitation based on traditional incentive method; Yang Jiang etc [3]have proposed the pseudo-excitation algorithm with high precision which could only be obtained by a small amount of vibration mode, which is the modified absolute displacement method.So far, calculating working amount with pseudo excitation method could solve the response problem of structural linear stochastic earthquake.There is still no in-depth research for analyzing nonlinear seismic buildings.As architectural design has such response requirement to non-linear systematic random earthquake, such research method is urgently needed.Through referring to the solution of FAP numberical value by computer, this paper is to suggest simple and efficient approach of structural nonlinear pseudo excitation method, which could be applied into analyzing displacement, speed, random vibration spectrum of accelerated velocity as well as the influencing situation of axial force, thus we could realize the high accuracy and efficiency in mechanics stimulation method of nonlinear random vibration on the basis of statistical linear theory.

STATISTICAL LINEAR THEORY
tatistical linear theory originated from the 1960s, R. Isaacs etc in America and L.S. Pontryagin etc in former Soviet Union had performed the initial study to this [4].The numerical method applied in nonlinear differential game began taking shape in the 1980s, and it has developed into the method which could deal with random response problems in nonlinear dynamics system so far.For non-linear mechanics with n degree of freedom, it could be expressed with differential mode [5][6][7]: where [M] is the matrix of structural mass with n order; [ ({y},{y})] G  is the equivalent matrix of nonlinear restoring force and damping force.The equivalent linear equation of nonlinear system could be established according to this method: where e e [ ][ ] C K are the equivalent viscous drag coefficient matrix and stiffness matrix respectively, the following could be obtained through direct subtract between (1) type and (2) type: where E is the mathematical expectation, the solution of type (4) could be obtained through calculation, and the equivalent linear stiffness and viscous damp of nonlinear system is derived.

THE SOLUTION OF PSEUDO-EXCITATION METHOD
or stimulation method, we generally establish the equation of motion through the variable generated on the basis of particle's relative displacement.The equation of motion is established according to d'Alembert principle, the inertia force is added on particle as load to avoid the matrix containing support quality in motion process.For the pseudo-excitation algorithm of multiple-support excitation, we could see from above solving idea that the dynamic S F relative displacement should be taken as the power balance equation of basic variable, then the dynamic relative displacement is derived, as well as the pseudo-static displacement of internal node caused by support movement will be solved, the sum of the two is the absolute displacement, and the solution procedure also avoids the participating computing of support quality matrix.But actually seismic action process is the quality mass vibration of support node generated by seismic excitation, the vibration of internal node is generated by the vibration of support node, earthquake force is directly added on support node, the quality of support node is not easy to determine, thus we usually solve the equation through applying this approach.Type (2) could be written into partitioned matrix model: where b y denotes the force displacement on ground of N supports, s y denotes the displacement of all unsupported node in construction system, b F denotes the force of ground effect on N supports, M, C, K denote mass matrix, damping matrix and stiffness matrix respectively, the small sign s, k correspond to the freedom degree of the internal structure node and support node respectively.We assume the damping force is in direct proportion to relative velocity in type ( 5), d u  , {0} is used to replace the freedom degree of internal nodes and support nodes in type (5), by which to derive the equation: Virtual acceleration excitation is constructed on Eq. ( 6): where superscript"*"represents taking complex conjugate, ,   denote the complex characteristics pair of power spectrum matrix of input ground motion.The product of virtual acceleration excitation and b M in Eq. ( 7) is taken as b F , then it will be substituted into Eq.(5): Type ( 8) could be expanded as: Both sides of type ( 9)is multiplied by 10) could obtain virtual acceleration excitation of support: In conclusion, as long as giving a larger mass on support and exerting stimulation, we could realize the load of virtual acceleration excitation.And support's virtual velocity and virtual displacement incentives are: Type ( 8) is expanded to be: Through substituting (12) into (13), virtual absolute displacement s y  is obtained.According to pseudo excitation method, the power spectrum matrix of absolute displacement is:

THE ANALYSIS AND RESEARCH FOR CASES
The Exposition of Main Information about Cases e take a prestressed concrete cable-stayed bridge with two span, single tower and double cable planes as an example whose length is 130 m, span arrangement is 75 + 55 m.Its tower pier beam is semi-consolidated structural system.The girder section is the section of double solid girder cantilever, the center height of girder is 1.9 m, roof width is 38 m, cantilever length is 4.5 m, lateral solid beam of girder is 3 m, the width of lateral solid beam across back is 4 m, the thickness of roof between solid beam is 0.28 m.The girder section across back is added with baseplate as construction due to needing balance weight, thus box cross-section forms.The girder adopts two-way prestressed system, the king-tower is reinforced concrete leaning tower, the included angle between the center line of the tower and horizon is 75°, the vertical height above bridge is 50.7 m, the king-tower adopts the filled rectangle of variable cross-section, cross section height along the bridge changes from 3 m (tower top) into 8 m (the root of tower above bridge): The width across bridge is 2.5 m.The model is constructed through general finite element software ANSYS, main girder and main tower adopt C50 concrete, stayed-cable adopts high tensile steel wire PSEM7 -241, cross section type of girder is simulated with finite element modeling, as it is shown in Fig. 1, calculation model performs dispersing to tower with space beam element -beam 4, the girder is simplified as fishbone shape through applying spatial beam-element beam 4 with rigid arm, stayed-cable adopts link 10 units, the left end of bridge is given vertical translational constraint of freedom degree, the right end of bridge is given transverse translational constraint of freedom degree, the bottom of cable support tower is completely constrained, cable element and beam element apply hinge constraint, as it is shown in Fig. 2.  Figure 3 The response power spectral density of main span, mid-span across back in cable-stayed bridge.
When comparing the pseudo excitation method directly obtained with absolute displacement and traditional random vibration method, the relative error of its calculation results is shown as Tab. 1.We could derive the following conclusion from the data in table that: (i) the calculation results of two algorithm are basically the same, it demonstrates that after linearization, the pseudo excitation method directly obtained with absolute displacement could replace traditional random vibration method in studying the seismic performance of structure.The excellence of this method has following points: the absolute displacement does not needs to be decomposed into quasi static displacement and dynamic relative displacement during calculating; quasi static modal matrix A does not needs calculation when constructing virtual stimuli for harmonic response analysis, thus it is simple and feasible; its essence does not changes the basic principle of traditional pseudo excitation method, its calculation efficiency is equivalent to traditional pseudo excitation method.
(ii) When comparing absolute displacement method with random vibration method, the error of acceleration response power spectral density is the smallest, the biggest error of main span is only 1.15%, the biggest error of back span is only 1.37%.The main reasons resulting into such phenomenon is that the inputted stimulus is the acceleration power spectrum density during the early analysis stage of model, thus after a series of calculation, its corresponding acceleration response power spectral density is closest to actual situation, while displacement and velocity response power spectral density have experienced more calculation steps than acceleration response power spectral density, then results into error accumulation, thus the maximum error of displacement power spectral density in main span reaches 11.1%, the maximum error of displacement power spectral density in back span reaches 9.14 %.

The Comparison for Absolute Displacement Method, Time History Method and Spectra Analysis
In current vibration resistance analysis, in addition to above mentioned random vibration method and harmony response analysis, spectrum analysis and time history method are also very important research method, the essence of dynamic time history analysis method is transient analysis Full method, various nonlinear factors could be considered, thus for the algorithm cases studied in this paper, the linearization process could be skipped to perform dynamic time history analysis directly.Vibration isolation bearing is considered during calculation, the structure is non-orthogonal damping system, the inputted artificial wave is generated according to the above adopted condition, and seismic action adopts both horizontal and consistent earthquake excitation.According to seismic code, earthquake acceleration at the second-rate direction is multiplied by the coefficient of 0.85.In order to compare above analyzed nonlinear and linear results, the method in last section is equally applied.As it is shown in Fig. 4, the straight line in (a) and (b) represents spectra analysis, point and line interval represents time history method, dotted line represents absolute displacement method; (c),(d) square lines represent spectra analysis, dot line represents time history method, triangle line represents absolute displacement method.Through comparing the displacement time history diagram of nodes across main span and back span in Fig. 4(a) and (b), we could find out following rules: (i) as spectral analysis is to study the displacement response of cable-stayed bridge within linear elastic range, thus all the displacement response value of absolute displacement method are too small when compared with dynamic time history method and the absolute displacement method after linear processing, and the maximum deviation reaches more than 50% when compared with dynamic time history method which has the largest displacement response, it demonstrates that when studying the anti-seismic property of structure, the structural nonlinearity must be considered, otherwise more error will occur and the calculation results will be made deviate from true value.(ii)The calculation results of absolute displacement method and dynamic time history after taking account nonlinearity are very close, the maximum displacement error at main span X direction is only 7.8%, for back span Y direction ,its maximum error is 7.3%.And through research ,we find that there is similar regularity among the displacement response of main girder, main tower and stay-cables node, it demonstrates that the absolute displacement method could sufficiently consider structural nonlinearity after linearization, thus it could be taken as an effective method for aseismic design.Through observing axial force response mean contrast figure at Z direction of typical unit under the action of consistent earthquake for three calculation method in figure 3.4, we could see that the unit axial force response value obtained through absolute displacement method after linearization is the biggest, and it is basically the same with unit axial force value obtained through calculating with dynamic time history method, its maximum error is only 0.23% (the units near the middle section of main span), for spectral analysis, all its axial force response values at Z direction are too small, when compared with dynamic time history method, the maximum error reaches 8%, thus it demonstrates that nonlinearity has certain effect on structural anti-seismic property, which should be considered in anti-earthquake analysis of bridge structure.

CONCLUSIONS
ompared with traditional pseudo excitation method, the new pseudo excitation method directly obtained through the absolute displacement based on statistical linearization could sufficiently consider structural nonlinearity and make its seismic analysis results be more close to actual situation; secondly, absolute displacement needs not to be discomposed into quasi static displacement and dynamic relative displacement during calculation, and quasi static modal matrix needs not to be calculated when constructing virtual stimuli for harmonic response analysis, it could be directly applied in general finite element analysis software and convenient for the application of pseudo excitation method in actual engineering.After system linearization, the node response power spectral density obtained through calculating with absolute displacement method and random vibration method is basically the same, and it demonstrates the correctness of the absolute displacement method, and the accuracy of response power spectral density outputted in calculation depends on the type of input excitation.Compared with dynamic time history method and the absolute displacement after linearization method, spectrum analysis does not take into account the nonlinear structure, the value of node displacement response and the unit axial force response are too small, it demonstrates that nonlinearity has great influence on seismic performance of structure, and seismic analysis of bridge structure must be taken into account.The response value of node displacement and the unit axial force calculated through the absolute displacement method and dynamic time history method after linearization are basically the same.It demonstrates that the pseudo-excitation method directly solved through absolute displacement based on statistical linearization is equivalent to the dynamic time history method, as they can effectively analyze and study the seismic behavior of the nonlinear system.One point needs to distinguish is that performing harmonic response analysis with absolute displacement method is the required stimuli when taking frequency as interval input, while the dynamic time history method is the seismic wave data established by taking time as interval, but their essence is the same.C

Figure 2
Figure 2 The diagrammatic figure of concrete deck cable stayed bridge.
(a) The displacement course of main span at X direction.(b) The axial force response of back span unit at Z direction.(c)The axial force response of main span unit at Z direction.(d)The axial force response of main span unit at Z direction.

Figure 4 :
Figure 4: The comparison diagram of axial force response

Table 1 :
The error comparison for two methods.