Abstract
Wind load is one of the important later loads for civil engineering,such as high-rise buildings[1]and long span bridges.Currently the methods to explore the wind-induced response of structure can be classified into computational fluid dynamics,wind tunnel test and field measurement[2].Recently many researchers focus on the field measurement as the reallife data provides valuable information to investigate the wind load characteristic and wind-induced response of structure.However,direct measurements of the wind load on large-scale buildings are unfeasible and difficult due to windstructure interaction and the limitation of measurement equipment.In comparison,measurement of acceleration and displacement responses is easier and more accurate than the measurement of force.It would be meaningful if the wind loads on structures can be estimated from limited measurements of responses.
This paper proposes an input-state estimation method that identifies the unknown time-varying wind load and unknown model parameters together from spatially-spare output response measurements.It employs the Unscented Kalman Filter(UKF)to estimate the mean vector and covariance matrix of the argument state vector by using a set of deterministic sampling points.As the argument state vector includes the unknown wind load and model parameters,the size of the state vector increases with the time increasing.To reduce the size of argument state vectors,the wind load is estimated with a finite time interval referred to as a time window and an overlapping time window technique is utilized to improve the accuracy of the estimation result.Figure 1 shows the procedure of the proposed method,where k is the time step.
Figure 1 The UKF method for jointly identifying the model parameters and wind load using the roll time window
are the posterior mean and covariance matrix of the state vector at time k;nx is the number of state vector;
is the sigma point,j=0,1,2,…,2(nx+1);
are the weight for mean and covariance of the state vector,respectively;Kk+1 is the Kalman gain;
is the predicted response.
To verify the proposed wind load estimation framework,a numerical model is created,which is a 40-story 4-bay steel frame structure(Figure 2).The wind loads are generated from a random wind model,which are applied to each floor as point loads.The structural responses are generated according to the 40-story building model,which are regarded as true values in the estimation stage after being artificially contaminated by Gaussian noise.
Figure 2 Basic information of the 40-story building
In the estimation process,the true displacement of floor 5,7,10,12,15,17,20,22,25,27,30,32,35,37 and 40 is regarded as the known displacement,which is then used to jointly identify the model parameters and wind load.Figure 3 shows the comparison between the estimated wind load and true values.Due to the limit length of paper,only the 40th floor estimated time-history wind load is presented in Figure 3(a),which has a good agreement with the true value.To quantify the accuracy of the estimated quantities,the relative root mean square error(RRMSE)between the true and estimated results is defined as follows:
where Nt denotes the total number of data sample;
denotes the true response;
denotes the response from estimation.The RRMSE of the wind load for floor 1～20 and floor 21～40 is shown in both Figure 3(b)and Figure 3(c),respectively.The values of RRMSE show large in the floor 1～7,whereas the RRMSE decreases as the number of floor increase.
The estimated displacement of the floor 40 is presented in Figure 4(a),which is consistent the true displacement.The RRMSE between the estimated and true results are also compared in Figure 4(b),with the value less than 1%.The estimated damping ratio is convergence to the true value after a few seconds.
In the above synthetic example,both the wind load and model parameters can be identified with relatively precision.This case study demonstrates that the proposed method can be used to estimate the unknown dynamic input excitation and unknown model parameters jointly from limited number measurement.This tool provides a promising route for wind loading identification.
Figure 3 Comparison of the true wind load and estimated results
Figure 4 The estimated and true results