Principal Moment Direction Research Articles

Uniformity Prediction of Bending Stiffness of Composite Space Mirrors With Ply Angle Misalignments by Deflection Angle

Based on the classical lamination theory, the principal moment direction of the laminate is derived. It is 45° apart from the generalized ply angle, which is the orientation of the ply where the ply angle misalignment occurs nominally. After adding a fixed one-ply angle or random multiple ply angle misalignments to six quasi-isotropic and symmetric stacking sequences, out-of-plane deformations of mirrors were simulated and compared. The result shows an angle difference between the principal moment and curvature directions. It is referred to as the deflection angle, approximately appearing as a trigonometric curve, which varies with laminates and is dependent on the generalized ply angle within each laminate. The deflection angle directly reflects the consistency of bending stiffness or laminate bending anisotropy; the smaller the deflection angle, the better will be the uniformity of bending stiffness. Consequently, it is an efficient approach to obtain a laminate with more uniformity of bending stiffness by comparing the deflection angle. These results enrich the research of out-of-plane deformation and could objectively promote the study of quasi-isotropic laminates for bending stiffness.

Rotation-free finite element ‘yield line’ analysis of non-isotropic slabs

A new, non-incremental advance in the lower bound finite element (FE) method for non-isotropic slab analysis is presented as a development of earlier work by the authors that was applicable only to isotropic slabs. The method provides a faster and simpler alternative to the full non-linear FE method in the non-linear analysis of slabs. Reinforced concrete was modelled as elastic perfectly plastic material, with rotation-free (RF) plate elements used to model slabs. They were modified to allow plastic yield lines to pass through elements satisfying non-isotropic yield conditions. Satisfying yield conditions and finding yield line directions differ significantly in their sophistication from their isotropic counterparts, as slab moment capacity is direction-dependent and yield line direction does not coincide with principal moment direction. Yield line effects were projected to element sides then used in curvature calculations. Analysis was performed incrementally, with the slab considered to behave linearly within each load increment. As a result, no iterations were required to reduce the unbalanced forces, as the behaviour within each load increment was in equilibrium. Accuracy was assessed against benchmark examples, and generally was within 2.5% of yield line collapse loads.

Finite element lower bound “yield line” analysis of isotropic slabs using rotation-free elements

A new lower bound finite element method for slab analysis is presented as a practical substitute to full, non-linear, finite element methods that require expert knowledge and long running times. The method provides a general, safe and efficient lower bound solution for the analysis of reinforced concrete slabs up to failure. As it is finite element based, the method is more general than the yield line and strip methods currently in use. Furthermore, its lower bound nature makes it safer than the yield line method. The method uses a rotation-free, plate finite element modified to allow plastic “yield lines” to pass through at any direction. Yield lines are generated at the principal moment directions when the plastic moment capacity is attained. The material is assumed to be elastic perfectly-plastic. Following the general spirit of yield line analysis, the effects of a yield line are projected to the sides of the triangular element and then used to calculate the bending curvatures. The method’s efficiency is achieved by using rotation-free plate elements with a single degree of freedom per node and by the incremental solution that does not require iterations. The method’s accuracy and convergence are assessed by comparing standard cases with known results. In all cases, results were close to the theoretical values with difference of less than 1%. It is also used to solve a practical sized flat slab problem in order to demonstrate the method’s efficiency, convergence, and speed.