# 统计代写|主成分分析代写Principal Component Analysis代考|”Formulation and Solution of Axisymmetric Stress Problems Using Finite Elements: A Mathematical Approach”

In summary, the provided text describes the process of formulating and solving axisymmetric stress problems using finite elements. The main takeaways are as follows:

Displacement Interpolation: The displacement field for an axisymmetric element can be interpolated using two-dimensional shape functions Ni(r, z) applied to the nodal displacements ui and wi.

Strain Calculation: The strain components are derived from the gradients of the displacement field, which leads to the strain-displacement matrix [B] where strains {ε} are expressed as {ε} = [B]{u,w} with {u} and {w} being vectors of nodal displacements.

Stiffness Matrix: The total strain energy is integrated over the volume of the element to derive the stiffness matrix [k] using the equation {k(e)} = 2 \int_{V(e)} [B]^T[D][B] dV. Due to the specific nature of axisymmetric problems, the integral is adjusted to 2 \int_{A(e)} [B]^T[D][B] rdrdz, accounting for the toroidal geometry.

Centroid Approximation: To simplify calculations, the stiffness matrix can be approximated by evaluating [B] at the centroid of the element in the rz-plane, resulting in {k(e)} ≈ 2 \bar{r} A [\bar{B}]^T [D] [\bar{B}].

Nodal Forces: External pressures and body forces are converted to nodal forces using the work equivalence principle. For a uniform radial pressure pr, nodal forces are calculated along the element edges, with the area modified by the factor 2r.

Example Calculations: For the axisymmetric element in Figure 9.11, nodal forces due to radial pressure are computed based on the pressure distribution and interpolation functions. Similarly, nodal forces due to body forces (radial and axial) resulting from rotation and gravity are calculated using the given density and gravitational acceleration.

Alternative Approach: An alternative method for calculating body forces is to approximate their influence at the centroid of the element cross-section, which distributes the forces evenly among the nodes. This method becomes more accurate as the element size decreases.

The solution to the problem includes computing nodal forces for both a uniform radial pressure load and for body forces due to rotation and gravity. In the case of radial pressure, only the end nodes experience a force (positive for node 1 and 3), and in the case of body forces, each node experiences both a radial and an axial force due to the combination of rotation and gravity. The detailed calculation steps and results are provided in the text.

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