# 统计代写|主成分分析代写Principal Component Analysis代考|”Mathematical Analysis of Torsion in Non-Circular Cross-Sections”

The passage describes the mathematical treatment of torsion in non-circular cross-section members within the context of finite element analysis. When a structural member undergoes torsion due to an applied torque, the assumption that plane sections remain plane breaks down for non-circular sections, leading to additional complexities in the analysis. The key aspects are:

Displacements and Strains: The displacement components for an arbitrary point in the twisted member are given by v = -zθ and w = yθ, where θ is the angle of twist. The displacement in the x-direction (u) occurs due to warping and is a function of y and z. Normal strains in the x, y, and z directions are zero (εx = εy = εz = 0) while shear strains exist.

Stress Function (Prandtl’s Stress Function): Introducing the angle of twist per unit length \theta_x allows for expressing the displacement components. The shear stress components τxy and τxz are related to the derivatives of a scalar stress function (y,z) by:

τxy = ∂/∂z, τxz = -∂/∂y Equilibrium Equation: Using the stress-strain relationship and applying the equilibrium condition leads to a governing equation for the stress function, which resembles the Laplace equation:

∇² = -2Gθ_x Where G is the shear modulus and \theta_x plays a role analogous to internal heat generation in heat conduction problems.

Boundary Conditions: On the outer surface, the normal stress must be zero, which implies that the stress function is constant (d/ds = 0).

Torque Calculation: The applied torque T can be related to the stress function by integrating the shear stresses over the cross-sectional area, resulting in:

T = 2∫∫Ay∂/∂y – Az∂/∂z dA = -2∫∫A∂²/∂y² + ∂²/∂z² dA Finite Element Formulation: The stress function is approximated using interpolation functions Ni(y, z) and nodal values i. The stiffness matrix and nodal forces are derived by analogy with heat conduction problems, with the torsional term 2Gθ_x playing the role of heat source density.

Example Solution: For a specific example involving a shaft with a square cross-section, the author illustrates how to construct the stiffness matrices and nodal force vectors for triangular elements, and solves for the stress function at the interior node. The total torque is calculated by summing up contributions from each element.

Unknown Twist Angle: In the actual problem-solving process, the angle of twist per unit length \theta_x is initially unknown. After computing the torque, a scaling approach is suggested to match the known applied torque by adjusting \theta_x.

However, the text ends without providing the exact computation for the angle of twist per unit length \theta_x despite the computed torque. To finalize the solution, one would usually invert the system of equations to find the nodal values of the stress function, and then derive \theta_x from the constitutive relationships or equilibrium equations. In practice, iterative methods could be employed to solve for \theta_x until the calculated torque matches the externally applied torque.

### MATLAB代写

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