Robust identification of material parameters via vibration measurements is crucial for the structural health monitoring (SHM) of civil infrastructure like bridges. The accuracy of this identification critically depends on the spatial placement of a limited number of sensors, which is constrained by economic and practical factors. In this talk, we present a mathematical framework for optimal sensor placement (OSP) to maximize the information gained for parameter estimation. Departing from common discrete [2] or linearized approaches [1], our strategy operates primarily in the continuous domain. We formulate the OSP problem for a 1D Euler-Bernoulli beam as a bilevel optimization. The inner problem performs PDE-constrained parameter identification, simulating the future inversion of field data. The outer problem optimizes the sensor density function to minimize the trace of the posterior covariance (the A-optimality criterion), derived from the Fisher information matrix. The key contribution is the derivation and use of continuous adjoint-based gradients for both optimization levels, ensuring mathematical consistency and elegance before numerical discretization. We compare this bilevel strategy against a simpler linear-Gaussian model where the need for parameter estimation (the inner problem) is eliminated. While computationally more ntensive, our continuous bilevel approach provides a principled and general benchmark, free from linearization errors and discretization bias inherent in grid-based methods. Representative simulations demonstrate how the framework can guide sensor placement to increase parameter sensitivity while establishing a theoretically sound foundation for OSP in SHM applications.