Conventional mechanical design paradigms rely on experts systematically refining concepts through experience-guided modification and FEA to meet specific requirements. However, this approach can be time-consuming and heavily dependent on prior knowledge and experience. While numerous machine learning models have been developed to streamline this intensive and expert-driven iterative process, these methods typically demand extensive training data and considerable computational resources. Furthermore, methods based on deep learning are usually restricted to the specific domains and tasks for which they were trained, limiting their applicability across different tasks. This creates a trade-off between the efficiency of automation and the demand for resources. In this study, we present a novel approach that integrates pre-trained LLMs with a FEM module. The FEM module evaluates each design and provides essential feedback, guiding the LLMs to continuously learn, plan, generate, and optimize designs without the need for domain-specific training. We demonstrate the effectiveness of our proposed framework in managing the iterative optimization of truss structures, showcasing its capability to reason about and refine designs according to structured feedback and criteria. Our results reveal that these LLM-based agents can successfully generate truss designs that comply with natural language specifications with a success rate of up to 90%, which varies according to the applied constraints. By employing prompt-based optimization techniques we show that LLM based agents exhibit optimization behavior when provided with solution-score pairs to iteratively refine designs to meet specifications. This ability of LLM agents to produce viable designs and optimize them based on their inherent reasoning capabilities highlights their potential to develop and implement effective design strategies autonomously.