Practical　engineering　issues　such　as　design　optimization,　design　space　exploration,　sensitivity　analyses,　and　reliability　analyses　need　many　simulations.　If　a　single　simulation　is　very　time-consuming,　engineers　cannot　perform　the　thousands　or　even　millions　of　simulations　needed　for　such　analyses.　The　polynomial　chaos　expansion　(PCE)　method　is　an　effective　method　that　allows　analyses　of　complex　problems.　This　paper　introduces　the　mathematical　theory　of　the　PCE　method　and　presents　a　structural　reliability　analysis　example.　The　performance　response　function　for　the　structural　reliability　analysis　is　expressed　as　a　PCE　using　Hermite　polynomials.　A　general　form　of　the　Hermite　polynomial,　which　is　suitable　for　use　in　a　computer　program,　is　used　to　generalize　the　PCE　analysis　program　and　the　adaptive　selection　of　the　polynomial　order.　Then,　the　accuracy　and　applicability　of　the　surrogate　model　are　verified　using　structural　reliability　analysis　examples　with　explicit　performance　functions.　The　results　show　that　the　model　has　an　excellent　convergence　rate　with　higher　order　PCE　giving　higher　accuracy.　The　examples　also　show　that　the　direct　use　of　explicit　performance　functions　is　the　easiest　way　to　investigate　PCE　surrogate　models.　©　2022,　Tsinghua　University　Press.　All　right　reserved.