A novel strain-displacement variational formulation for the flexural behaviour of laminated composite beams is presented, which accurately predicts three-dimensional stresses, yet is computationally more efficient than 3D finite element models. A global third-order and layer-wise zigzag profile is assumed for the axial deformation field to account for the effect of both stress-channelling and stress localisation. The axial and couple stresses are evaluated from the displacement field, while the transverse shear and transverse normal stresses are computed by the interlaminar-continuous equilibrium conditions within the framework of the modified couple stress theory. Then, axial and transverse force equilibrium conditions are imposed via two Lagrange multipliers, which correspond to the axial and transverse displacements. Using this mixed variational approach, both displacements and strains are treated as unknown quantities, resulting in more functional freedom to minimise the total strain energy. The differential quadrature method is used to solve the resulting governing and boundary equations for simply-supported, clamped and cantilever laminated beams. The deflections and stresses from this variational formulation for simply supported beams agree well with those from a Hellinger-Reissner stress-displacement mixed model found in the literature and the 3D elasticity solution given by Pagano. These strain-displacement models also accurately predict the localised stresses near clamped and free boundaries, which is confirmed by the high-fidelity Abaqus models.

提出了一种用于层合复合梁弯曲行为的新的应变位移变分公式，该公式可以精确预测三维应力，但在计算上比3D有限元模型更有效。假定轴向变形场具有全局三阶和逐层之字形轮廓，以说明应力通道效应和应力局部化的影响。轴向应力和耦合应力是从位移场评估的，而横向剪应力和横向法向应力是在改进的耦合应力理论的框架内通过层间连续平衡条件计算的。然后，通过两个拉格朗日乘数来施加轴向力和横向力的平衡条件，这对应于轴向位移和横向位移。使用这种混合变分方法，位移和应变都被视为未知量，从而获得了更大的功能自由度，可将总应变能降至最低。微分求积法用于求解简单支撑，夹紧和悬臂层合梁的控制方程和边界方程。简单支撑梁的这种变形公式的挠度和应力与文献中的Hellinger-Reissner应力-位移混合模型和Pagano给出的3D弹性解的挠度和应力非常吻合。夹紧的悬臂梁。简单支撑梁的这种变形公式的挠度和应力与文献中的Hellinger-Reissner应力-位移混合模型和Pagano给出的3D弹性解的挠度和应力非常吻合。夹紧的悬臂梁。简单支撑梁的这种变形公式的挠度和应力与文献中的Hellinger-Reissner应力-位移混合模型和Pagano给出的3D弹性解的挠度和应力非常吻合。这些应变位移模型还可以准确地预测夹紧和自由边界附近的局部应力，这已由高保真Abaqus模型证实。