The crushing behaviors of thin-walled composite structures subjected to quasi-static axial loading are comparatively evaluated using four different failure initiation criteria. Both available crushing tests of composite corrugated plate and square tube are used to validate the stiffness degradation-based damage model with the Maximum-stress criterion. Comparatively, Hashin, Maximum-stress, Stress-based Linde, and Modified criteria are respectively implemented in the damage model to predict crush behaviors of corrugated plate and square tube. To develop failure criteria, effects of shear coefficients and exponents in the Modified and Maximum-stress criteria on damage mechanisms of corrugated plate are discussed. Results show that numerical predictions successfully capture both of experimental failure modes and load–displacement responses. The Modified criterion and particularly Maximum-stress criterion are found to be more appropriate for present crush models of corrugated plate and square tube. When increasing the failure index, the crushing load is decreased, which also causes premature material failure. The shear coefficient and exponents have dramatic influence on the crushing load. Overall, an insight into the quantitative relation of failure initiation is obtained.