# 统计代写|主成分分析代写Principal Component Analysis代考|”Flexure Element Method for Midspan Deflection of a Beam with Concentrated Load”

The passage describes how to apply the flexure element method to solve for the midspan deflection of a statically indeterminate beam subject to a transverse load at its midspan. In the finite element approach, the beam is divided into two flexure elements of length L/2L/2L/2 each, and the element stiffness matrices are calculated accordingly. The nodal forces and moments in the finite element formulation are equated to the respective shear forces and bending moments in beam theory through the sign convention in Equation 4.50.

The assembled global stiffness matrix reflects the connectivity and boundary conditions of the two-element model. Applying the boundary conditions, the system equations are reduced, and the nodal displacements v2v_2v 2 ​ , θ2\theta_2θ 2 ​ , and θ3\theta_3θ 3 ​ are solved for in terms of the applied load PPP.

The deformed shape of the beam is then plotted, and the reactions at supports are computed using the nodal displacement values and the constraint equations. The verification of the solution is done by checking the global equilibrium conditions, which are satisfied in this case.

This particular example demonstrates that the finite element method using flexure elements provides an exact solution when dealing with a simple case of a concentrated load applied at the midspan of a simply supported beam, where the deflection equation is cubic and matches the interpolation functions’ cubic nature. However, it is emphasized that the finite element method might not yield exact solutions in all cases, especially when distributed loads are present.

### MATLAB代写

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