{'fraction_non_alphanumeric': 0.013567438148443736, 'fraction_numerical': 0.0, 'mean_word_length': 5.56544502617801, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Analytical Solution for Longitudinal Dynamic Responses of Long Tunnels under Arbitrary Excitations

In this paper, an analytical solution is proposed for longitudinal dynamic responses of long tunnels under arbitrary excitations. For the derivation, the tunnel is assumed as a Timoshenko beam resting on a visco-Pasternak foundation. The Timoshenko beam theory is employed to consider both effects of the shear distortion as well as the rotary inertia of the tunnel, which are neglected by the Euler–Bernoulli beam. The visco-Pasternak foundation is applied to represent the viscoelastic compressive and shear resistance of soil. The governing equations of motion are transformed from partial differential forms into algebraic forms through integral transformations, and thus the solutions are conveniently obtained. The analytical solutions of the tunnel under several specific dynamic loads, including impulsive loads, moving line loads as well as traveling loads, are presented in detail and verified by comparing to the known degraded solution in literature and finite element results. Several examples are also conducted to investigate the influence of the relative stiffness ratio between the soil and the tunnel structure on the tunnel responses.