# 统计代写|主成分分析代写Principal Component Analysis代考|”Converting Distributed Loads into Nodal Forces: Work Equivalence in Flexure Elements”

The passage discusses the treatment of distributed loads in a flexure element by converting them into equivalent nodal forces and moments. This conversion is based on the principle of work equivalence: the mechanical work done by the distributed load should be equal to the work done by the equivalent nodal loads. To achieve this, the nodal forces and moments (F1q, M1q, F2q, M2q) are determined by integrating the product of the distributed load (q(x)) and the interpolation functions (N1(x), N2(x), N3(x), N4(x)) over the element length.

An example illustrates this for a simply supported beam with a uniform load. The beam is modeled using two equal-length flexure elements, and the boundary conditions dictate that the displacements at nodes 1 and 3 are zero. The global stiffness matrix is assembled, and work-equivalent nodal loads are calculated according to Equation 4.58.

Global equilibrium equations are formulated considering both the stiffness matrix and the equivalent nodal loads. Upon solving these equations, the nodal displacements, rotations, and end-slope conditions are obtained. Despite the differences in the detailed deflection curve between the finite element solution and the exact solution from strength of materials, the nodal displacement results match perfectly due to the use of work-equivalent nodal loads.

However, it’s highlighted that the finite element solution approximates the actual deflection curve because the interpolation functions used are cubic while the true deflection equation is quartic. Nonetheless, the finite element method still captures the essential behavior of the beam accurately in this scenario.

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