近期，有研究學者提出 higher-order refined zigzag theory (HRZT)理論。此理論是
以 refined zigzag theory (RZT)為基礎並引入高階剪力變形項，以更準確描述面內的剪
力變形。HRZT 之解析解已被證實在三明治複合樑之線性靜態結果的表現上具有優
越性，而本研究目標是基於 HRZT 理論發展有限元素公式，用於分析三明治複合樑
之線性靜態彎矩分析。基於上述所提出之樑元素，另外發展出幾何非線性之有限元
素公式用以分析三明治複合樑的大位移與挫曲現象。
本研究所提出之 HRZT 樑元素包含 C0-2N11(具有 11 個自由度之C0連續兩節點元素)、C1-2N14(具有 14 個自由度之C1連續兩節點元素)、C0-3N16(具有 16 個自由度之C0連續三節點元素)、C1-3N21(具有 21 個自由度之C1連續三節點元素)。研究表明上述所提出之 HRZT 樑元素皆無發生剪力自鎖。此外，透過與二維解析解、HRZT 理論解以及二維 ANSYS 模型的結果進行比較，驗證了 HRZT 樑元素在三明治複合樑中包含軸向位移、撓度、正向應力及剪應力之結果的準確性。接著，將HRZT 樑元素與其它樑理論，包含一階剪力變形理論(FSDT)、高階剪力變形理論(HSDT)、RZT 和高階轉折理論(Higher-order zigzag theory)的樑元素進行比較，並討論不同長寬比、厚度比及剛度比之參數的影響。而為了準確描述幾何非線性的現象，本研究使用牛頓法(Newton-Raphson method)分析大位移，得出與二維 ANSYS 模型一致的結果。此外，也討論在不同邊界條件與厚度比下對臨界挫曲負載的影響。為了進一步分析挫曲前之非線性大位移現象，考慮各種初始幾何缺陷形狀與不同的最大初始撓度值，並探討其對於挫曲前之非線性負載-撓度曲線的影響。根據本研究中呈現的數值結果，證明了 HRZT 樑元素能夠準確描述三明治複合樑的線性與幾何非線性的現象，有限元素公式及其數值結果可應用於航空、軍事、土木工程和機械工程等領域中。

Recently, a novel higher-order refined zigzag theory (HRZT) was developed. This
theory is an extension of the refined zigzag theory (RZT) by incorporating a higher-order term in the axial displacement kinematics to account for higher-order shear deformation. The analytical solutions were proposed and were proven to have superior performance in solving the linear static response of a sandwich beam. This study aims at developing finite element formulations based on HRZT to analyze the linear static bending behavior of sandwich composite beams. Based on the beam elements developed in this study, the beam element formulations with geometric nonlinearity are also developed to solve sandwich composite beams with large deflections and solve the buckling behavior of the beam.
The HRZT beam element formulations developed in this study include C0-2N11 (2-node C0 continuity element with 11 DOFs), C1-2N14 (2-node C1 continuity element with 14 DOFs), C0-3N16 (3-node C0 continuity element with 16 DOFs) and C1-3N21 (3-node C1 continuity element with 21 DOFs). A comprehensive study on the numerical performance shows that all the developed beam elements are free of shear locking. By comparing with 2D analytical solutions, analytical solutions of HRZT, and 2D ANSYS models, the results by the HRZT beam elements including axial displacements, deflections, normal stresses, and shear stresses in sandwich composite beams are validated. Furthermore, the study compared different beam elements based on various beam theories such as First-order shear deformation theory (FSDT), Higher-order shear deformation theory (HSDT), RZT, and another type of higher-order zigzag theory. The influence of parameters such as aspect ratio, thickness ratio, and stiffness ratio are also discussed. To investigate geometry nonlinearity, the Newton-Raphson method was employed, yielding consistent results with 2D ANSYS models. Moreover, the study analyzed critical buckling loads under various thickness ratios and boundary conditions. For nonlinear buckling analysis, various shapes for initial imperfections and varying maximum initial deflection values are considered to investigate nonlinear load-deflection curves before buckling.
Based on the numerical results presented in this study, it is proven that the HRZT beam elements can accurately calculate the linear and geometric nonlinear responses of the sandwich composite beams. The formulations and results developed in this study can be applicable to engineering fields such as aviation, military, civil engineering and mechanical engineering.

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