task:$2\mathrm{.}26$###$!!!$Discuss key equations and concepts such as Hooke's Law, stress, and strain, elucidating their significance in civil engineering practice. Furthermore, investigate how elasticity theory informs the design of resilient structures capable of withstanding various external forces, ensuring safety and longevity. Additionally, consider practical examples and case studies to illustrate the implementation of elasticity theory in real$-$world civil engineering projects, highlighting its crucial role in ensuring structural integrity and stability.