Graph Neural Networks, commonly known as GNNs, have emerged as powerful tools for solving physics-informed learning problems, particularly in modeling elasticity and structural mechanics. Traditional neural networks often struggle to enforce physical constraints and capture spatial correlations in complex domains. In this work, we develop a GNN-based framework for predicting the displacement field in a lattice-based elasticity problem under external forces. A key component of our approach is the normalization of the governing equations, which improves numerical stability and facilitates efficient learning. The model further integrates a physics-informed loss function that enforces equilibrium equations, boundary conditions, and smoothness constraints using graph-based message passing. Unlike conventional approaches, our method leverages graph connectivity and equation normalization to improve gradient approximations and enhance solution accuracy. We analyze the modifications introduced by the GNN in the loss function, demonstrating improved physical consistency and generalization capabilities. The results highlight the effectiveness of GNNs in learning physics-driven representations, offering a robust and scalable alternative to traditional deep learning methods for elasticity problems. Our findings indicate that GNN-based physics-informed learning, combined with normalization strategies, provides a promising direction for structural mechanics and continuum mechanics applications.