# 统计代写|主成分分析代写Principal Component Analysis代考|Truss Structure Displacement and Reaction Forces Solution

The passage discusses the solution process for determining the displacements and reaction forces in a truss structure once the global stiffness matrix has been assembled. Due to the singularity of the stiffness matrix caused by the possibility of rigid body motion, the system displacement equations require consideration of boundary conditions to remove the indeterminacy.

In the example of the truss in Figure 3.2, certain displacement boundary conditions (3.43) are imposed at nodes 1 and 2, constraining them to have no displacement (U1=U2=U3=U4=0). These constraints reduce the system of equations to a smaller set (3.44) involving only the active degrees of freedom (U5 and U6).

In a more generalized scenario, the global stiffness matrix can be partitioned into submatrices corresponding to constrained (Uc) and unconstrained (Ua) displacements. The partitioned system equation (3.45) reflects this distinction. The constrained displacements are already known, while the unconstrained ones are solved for using Equation 3.46a or Equation 3.46b, which involves inverting the submatrix [Kaa] and considering both the applied external forces (Fa) and the influence of the constrained displacements (through [Kac]).

Once the active displacements (Ua) are calculated, the reaction forces (Fc) at the constrained nodes can be determined using Equation 3.47. Here, the symmetry of the stiffness matrix ensures that [Kca]=[Kac]^T.

It’s important to note that while Equation 3.46b suggests matrix inversion, in practice, more computationally efficient algorithms are often employed to solve the linear system without explicit matrix inversion, such as LU decomposition, QR decomposition, or iterative solvers.

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