This study investigates the application of the Quantum Approximate Optimization Algorithm (QAOA) [1] to deformation analysis in solid mechanics, with the aim of establishing a scalable quantum–classical framework suitable for near-term quantum devices. As a first step toward quantum-based structural optimization, we focus on linear truss structures with periodic boundary conditions and formulate the governing stiffness equations within the Variational Quantum Linear Solver (VQLS) paradigm. The deformation problem is mapped to a cost Hamiltonian whose expectation value corresponds to the residual of the linear system, enabling displacement fields to be obtained via variational optimization. To address the scalability limitations arising from problem-size-dependent Hamiltonian construction, we propose a structured representation of the stiffness operator based on discrete Laplacian operators under periodicity. By exploiting the translational symmetry of the system, the stiffness-related operators are expressed in a form that is diagonalizable through the Quantum Fourier Transform (QFT). This transformation allows the Hamiltonian components to be reused across different problem sizes, significantly reducing the measurement overhead required for expectation value estimation. Numerical simulations using noiseless quantum circuits demonstrate that the proposed formulation reproduces displacement fields consistent with classical finite element solutions, while maintaining stable accuracy as the system size increases. Moreover, the QFT-based representation enables the number of quantum measurements to remain constant under specific conditions, independent of the spatial resolution of the structural model. The results indicate that combining QAOA with symmetry-aware Hamiltonian design and QFT provides a promising pathway toward efficient quantum algorithms for large-scale structural analysis. This framework lays the groundwork for extending quantum variational methods to more complex solid mechanics problems and future topology optimization tasks.