A DIFFERENTIAL QUADRATURE PROCEDURE WITH DIRECT PROJECTION OF THE HEAVISIDE FUNCTION FOR NUMERICAL SOLUTION OF MOVING LOAD PROBLEM

Abstract

OWING TO ITS PARTICULAR CHARACTERISTICS, THE DIRECT DISCRETIZATION OF THE DIRAC-DELTA FUNCTION IS NOT FEASIBLE WHEN POINT DISCRETIZATION METHODS LIKE THE DIFFERENTIAL QUADRATURE METHOD (DQM) ARE AP-PLIED. A WAY FOR OVERCOMING THIS DIFFICULTY IS TO APPROXIMATE (OR REGULARIZE) THE DIRAC-DELTA FUNCTION WITH SIMPLE MATHEMATICAL FUNC-TIONS. BY REGULARIZING THE DIRAC-DELTA FUNCTION, SUCH SINGULAR FUNC-TION IS TREATED AS NON-SINGULAR FUNCTIONS AND CAN BE EASILY AND DIRECTLY DISCRETIZED USING THE DQM. ON THE OTHER HAND, IT IS POSSI-BLE TO COMBINE THE DQM WITH THE INTEGRAL QUADRATURE METHOD (IQM) TO HANDLE THE DIRAC-DELTA FUNCTION. ALTERNATIVELY, ONE MAY USE ANOTHER DEϬNITION OF THE DIRAC-DELTA FUNCTION THAT THE DERIVA-TIVE OF THE HEAVISIDE FUNCTION, H(X), IS THE DIRAC-DELTA FUNCTION, δ(X), IN THE DISTRIBUTION SENSE, NAMELY, DH(X)/DX = δ(X). THIS APPROACH HAS BEEN REFERRED IN THE LITERATURE AS THE DIRECT PROJECTION APPROACH. IT HAS BEEN SHOWN THAT ALTHOUGH THIS APPROACH YIELDS HIGHLY OSCILLATORY APPROXIMATION OF THE DIRAC-DELTA FUNCTION, IT CAN YIELD A NON-OSCILLATORY APPROXIMATION OF THE SOLUTION. IN THIS PAPER, WE FIRST PRESENT A MODIFIED DIRECT PROJECTION APPROACH THAT ELIMI-NATES SUCH DIFFICULTY (OSCILLATORY APPROXIMATION OF THE DIRAC-DELTA FUNCTION). WE THEN DEMONSTRATE THE APPLICABILITY AND RELIABILITY OF THE PROPOSED METHOD BY APPLYING IT TO SOME MOVING LOAD PROBLEMS OF BEAMS AND RECTANGULAR PLATES.