Engineering reliability analysis often requires surrogate modeling of multiple correlated structural responses under limited simulation data. Conventional single-output Gaussian processes cannot directly exploit inter-response correlations, while shallow multi-task Gaussian processes may be insufficient for strongly nonlinear engineering systems. To address these issues, this paper develops a Multi-Task Deep Gaussian Process (MTDGP) framework for sparse multi-response surrogate modeling and reliability assessment. The proposed model combines the nonlinear latent representation of deep Gaussian processes with the task-correlation structure of multi-task Gaussian processes. To improve posterior inference, a Gibbs sampling framework with block hidden-layer updates in whitened coordinates is introduced, which enhances sampling stability and alleviates inefficient latent-state updates. The method is evaluated using two synthetic benchmarks and a portal crane reliability case. The numerical examples show that MTDGP can improve predictive accuracy when the target response is sparsely observed but correlated auxiliary responses are available, with particularly clear gains in the five-dimensional benchmark. Sampling diagnostics indicate that the block update strategy reduces non-moving latent updates compared with coordinate-wise sampling. In the portal crane case, MTDGP gives stable response-surface predictions and competitive reliability estimates under incomplete response observations. Nevertheless, its advantage over shallow multi-task Gaussian process models is not universal. Overall, the findings suggest that MTDGP is a promising surrogate framework for sparse multi-response engineering problems, while further improvements in posterior inference and reliability-oriented active learning remain necessary.
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