# 统计代写|主成分分析代写Principal Component Analysis代考|Application of Galerkin Finite Element Method for Structural Analysis

This text discusses the application of the Galerkin finite element method to both a simple bar element and a beam element for structural analysis.

For the bar element, the governing equation is derived from the equilibrium and strain-displacement relations. Assuming constant strain and stress throughout the element, the equation is Ed2udx2=0E \frac{d^2u}{dx^2} = 0E dx 2

d 2 u ​ =0, where EEE is the Young’s modulus, and u(x)u(x)u(x) is the displacement. Discretizing the displacement field with a two-node linear element (Equation 5.32), the Galerkin residual equations (Equation 5.33) are derived, and after integrating by parts, they lead to Equation 5.35. This equation demonstrates that the gradient boundary conditions are equivalent to the nodal forces applied to the ends of the bar element. The final matrix form (Equation 5.37) is identical to the one derived using equilibrium and energy methods.

Moving to the beam element, the focus shifts to the deflection v(x)v(x)v(x) in the y-direction under distributed loading q(x)q(x)q(x). The governing equation for beam flexure is established as EI_z \frac{d^2^2v}{dx^2} = q(x), where EIzEI_zEI z ​ is the bending stiffness of the beam. The displacement solution for the beam is approximated using a four-node linear quadrilateral element (Equation 5.45) with interpolation functions. The Galerkin residual equations (Equation 5.46) are derived accordingly.

Upon integration by parts and rearrangement, the final element equations take the form of a symmetric stiffness matrix (Equation 5.50) and a nodal force vector (Equation 5.51), incorporating both shear force and bending moment conditions at the element nodes. When a constant distributed load is applied, the nodal forces and moments can be directly calculated (Equation 5.52).

When multiple beam elements share a node, the following conditions apply:

Without external forces or moments, the shear forces and moments from adjacent elements cancel each other out during assembly. With concentrated forces or moments applied at the node, the sum of boundary shear forces equals the applied force, and the sum of bending moments equals the applied moment. The software packages for finite element analysis typically allow users to input a “pressure” on the beam’s transverse face, which is translated into the equivalent nodal loads according to the distributed load distribution.

### MATLAB代写

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