A DIFFERENTIAL QUADRATURE PROCEDURE WITH REGULARIZATION OF THE DIRAC-DELTA FUNCTION FOR NUMERICAL SOLUTION OF MOVING LOAD PROBLEM

Abstract

THE DIFFERENTIAL QUADRATURE METHOD (DQM) IS ONE OF THE MOST ELEGANT AND EFFICIENT METHODS FOR THE NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS ARISING IN ENGINEERING AND APPLIED SCIENCES. IT IS SIMPLE TO USE AND ALSO STRAIGHTFORWARD TO IMPLEMENT. HOWEVER, THE DQM IS WELL-KNOWN TO HAVE SOME DIFFICULTY WHEN APPLIED TO PARTIAL DIFFERENTIAL EQUATIONS INVOLVING SINGULAR FUNCTIONS LIKE THE DIRAC-DELTA FUNCTION. THIS IS CAUSED BY THE FACT THAT THE DIRAC-DELTA FUNCTION CANNOT BE DIRECTLY DISCRETIZED BY THE DQM. TO OVERCOME THIS DIFFICULTY, THIS PAPER PRESENTS A SIMPLE DIFFERENTIAL QUADRATURE PROCEDURE IN WHICH THE DIRAC-DELTA FUNCTION IS REPLACED BY REGULARIZED SMOOTH FUNCTIONS. BY REGULARIZING THE DIRAC-DELTA FUNCTION, SUCH SINGULAR FUNCTION IS TREATED AS NON-SINGULAR FUNCTIONS AND CAN BE EASILY AND DIRECTLY DISCRETIZED USING THE DQM. TO DEMONSTRATE THE APPLICABILITY AND RELIABILITY OF THE PROPOSED METHOD, IT IS APPLIED HERE TO SOLVE SOME MOVING LOAD PROBLEMS OF BEAMS AND RECTANGULAR PLATES, WHERE THE LOCATION OF THE MOVING LOAD IS DESCRIBED BY A TIME-DEPENDENT DIRAC-DELTA FUNCTION. THE RESULTS GENERATED BY THE PROPOSED METHOD ARE COMPARED WITH ANALYTICAL AND NUMERICAL RESULTS AVAILABLE IN LITERATURE. NUMERICAL RESULTS REVEAL THAT THE PROPOSED METHOD CAN BE USED AS AN EFFICIENT TOOL FOR DYNAMIC ANALYSIS OF BEAM- AND PLATE-TYPE STRUCTURES TRAVERSED BY MOVING DYNAMIC LOADS.