# 统计代写|主成分分析代写Principal Component Analysis代考|”Finite Element Analysis: Stiffness Method, Strain and Stress Calculations, and Failure Theories”

In summary, during the solution phase of a finite element analysis using the stiffness method, the primary goal is to compute the nodal displacements and reaction forces at constrained nodes. Once the displacements are known, strain and stress calculations follow as a post-processing step.

Strain Calculation: Strain components at each node are computed using Equation 9.119 which involves the strain-displacement matrix [B] and the nodal displacement vector {}. This process is repeated for every element in the model.

Stress Calculation: Stress components are then determined using Equation 9.120, which multiplies the strain components by the material property matrix [D]. The stresses are initially computed in the element’s local coordinate system, which typically aligns with the global coordinate system.

Principal Stresses and Failure Theories: Principal stresses (1, 2, 3) are calculated from the cubic equation (Equation 9.121). Two popular failure theories, Maximum Shear Stress Theory (MSST) and Distortion Energy Theory (DET), are used to assess the risk of yielding or failure. MSST uses the maximum shear stress (Equation 9.122), whereas DET evaluates the distortion energy (Equation 9.127) or the equivalent (Von Mises) stress (Equation 9.130).

Stress Data in Finite Element Analysis: Most FEA software packages provide stress and strain data in different formats, including nodal and elemental stresses. Nodal stresses are averages taken over all attached elements and may not accurately reflect actual stress distribution, especially near element boundaries due to their discontinuous nature. Elemental stresses, computed at the element centroid, are more precise and are recommended for evaluating failure criteria.

Illustration: Tables 9.1 and 9.2 from Example 9.4 demonstrate the difference between nodal and elemental stress computations. The nodal stresses show discontinuities across element boundaries, while the elemental stresses are continuous within each element and include principal stresses and the equivalent (Von Mises) stress. Engineers must understand these differences and rely on elemental stress data for making sound engineering judgments about structural integrity and potential failure points.

### MATLAB代写

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