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Flux-biased loops including one or more Josephson junctions are ubiquitous elements in quantum information experiments based on superconducting circuits. These quantum circuits can be tuned to implement a variety of Hamiltonians, with applications ranging from decoherence-protected qubits to quantum limited converters and amplifiers. The extrema of the Hamiltonian of these circuits are of special interest because they govern their low-energy dynamics. However, the theory of superconducting quantum circuits lacks so far a systematic method to compute the series expansion of the Hamiltonian around these extrema for an arbitrary non-linear superconducting circuit. We present such method, which can aid the synthesis of new functionalities in quantum devices.