We propose Shuttling-based Distributed Quantum Computing (SDQC), a hybrid architecture that combines the strengths of physical qubit shuttling and distributed quantum computing to enable scalable trapped-ion quantum computing. SDQC performs non-local quantum operations by distributing entangled ion qubits via deterministic shuttling, combining the high-fidelity and deterministic operations of shuttling-based architectures with the parallelism and pipelining advantages of distributed quantum computing. We present (1) a practical architecture incorporating quantum error correction (QEC), (2) pipelining strategies to exploit parallelism in entanglement distribution and measurement, and (3) a performance evaluation in terms of logical error rate and clock speed. For a 256-bit elliptic-curve discrete logarithm problem (ECDLP) instance, which requires 2,871 logical qubits at code distance 13, SDQC achieves a logical error rate which is $1.20_{-0.45}^{+0.94}\times 10^{-8}$ of Photonic DQC error rate and $3.79_{-2.84}^{+5.09}\times 10^{-3}$ of Quantum Charge-Coupled Device (QCCD) error rate, while providing 2.82 times faster logical clock speed than QCCD.