Tikhonov Regularization Method to Determine the Shear and Moment Functions of Euler-Bernoulli beams from noisy displacement

Abstract

Determination of shear and moment functions from displacements involves inverse problem solution compared to the conventional methods of directly obtaining them from the applied loads. Real displacements are obtained from field measurements that are inevitably noisy compared to the theoretically computed displacements due to simplifications of Euler- Bernoulli beam theory and instrument errors. Thus the inverse problem is ill-posed where a small error in the data might lead to large errors in the results, or the results might not depend on the data. The Tikhonov Regularization Method minimizes the errors by minimizing the least square distances between the theoretical displacements and the measured displacements and by simultaneously applying a smoothness criterion on the target function. This paper casts the Euler-Bernoulli beam theory into linear transformations that inversely maps displacements into bending moment and shear force. The real noisy displacements are
simulated by adding random numbers to the theoretical displacements. The machine generated random numbers have a certain statistical distribution within a certain width. The shear and moment functions obtained from noisy displacements data are compared with the respective functions from the direct problem results solved with the given applied load and
very good agreements were found.