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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 so far lacks a systematic method to find these extrema and compute the series expansion of the Hamiltonian in their vicinity for an arbitrary nonlinear superconducting circuit. We present such a method, which can aid the synthesis of new functionalities in quantum devices.