# 统计代写|主成分分析代写Principal Component Analysis代考|Computing Strain Energy and Nodal Forces for Flexure Elements

The passage discusses the computation of strain energy and the derivation of nodal forces and moments for a flexure (beam) element using the discretized approximation of the displacement function. The total strain energy UeU_eU e ​ stored in the element due to bending is given by Equation 4.38, which involves the second derivative of the displacement with respect to xxx, integrated over the length of the beam and multiplied by the Young’s modulus EEE, the moment of inertia IzI_zI z ​ , and a scaling factor related to the beam length LLL.

Applying Castigliano’s first theorem, the partial derivatives of the strain energy with respect to nodal displacements v1v_1v 1 ​ and v2v_2v 2 ​ give the transverse forces at nodes 1 and 2 (Equations 4.40 and 4.42), respectively, while the partial derivatives with respect to rotational displacements θ1\theta_1θ 1 ​ and θ2\theta_2θ 2 ​ provide the bending moments at these nodes (Equations 4.41 and 4.43).

These relationships are summarized in matrix form (Equation 4.44), where the element stiffness matrix [Ke][K_e][K e ​ ] relates nodal displacements to applied forces and moments. The stiffness matrix is symmetric due to the linearly elastic behavior of the material.

To compute the stiffness coefficients kmnk_{mn}k mn ​ , the integrals are converted to a dimensionless length variable β=x/L\beta = x/Lβ=x/L, making the calculations more manageable. The stiffness coefficients are then derived explicitly (Equations following 4.48), resulting in a 4×4 stiffness matrix [Ke][K_e][K e ​ ] as shown in Equation 4.49.

It is important to note that the stiffness matrix is singular due to the possibility of rigid body motion if the element is not adequately constrained. Additionally, the matrix is valid for any consistent unit system, with rotational degrees of freedom expressed in radians. Once the stiffness matrix is known, it can be used to solve for the nodal displacements and rotations when external loads are applied to the beam element.

### MATLAB代写

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