Finite Elements for Modelling Beams Affected By A Distributed Load

Usually a finite element with cubic deflection approximation function is applied when evaluating the stress and strain field of bar structures. But such an element only approximately evaluates the actual strain field of the bar affected by a distributed load. The improved finite elements (Fig 1, 2) with fourth and fifth-order deflection approximation functions (1), (6) and (13) are presented in the actual manuscript. The fifth-order deflection approximation function is used for modelling the beams affected by a linearly distributed load (11). The plain bending of the finite element is modelled by 5 and 6 freedom degrees. The additional 5th and 6th freedom degrees are the deflection and deviation of the middle node of element (Fig 2). The element stiffness matrices (Table 1, 2) and node force vectors are presented. The created finite elements exactly modells the stress and strain field of bars, which are affected by distributed load, and also allow to compute directly the middle section displacements of bars. It creates conditions for diminishing the volume of problems and obtaining information, which is necessary to be analysed later. The reduced finite elements (Fig 4) are created by the elimination of the internal freedom degrees. Their number of freedom degrees is decreased up to the number of freedom degrees of a usually applied finite element. But the reduced finite elements have all afore-mentioned qualities. Formulas (20) and (21) are derived expressing the middle node displacements by the final node displacements. These formulas allow to compute the middle section displacements of the bar already after the solution of equation system. The proposed reduced elements can be introduced and applied in engineering practice very easily, because their stiffness matrix coincide with the stiffness matrix of a usual bar finite element. The created elements with internal freedom degrees are very important for the problems of structures optimization with displacement constraints, because the constraint of bar middle section displacement can form just in case, when this displacement is one of the problem's unknown. Also it is very important to decrease the number of unknowns of optimization problem.

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