This study addresses the analytical treatment of a closed-form buckling equation for lateral-torsional stability of thin web composite cantilever beams under mid-height tip force. The beam is composed of random ply fiber orientations. Classical lamination theory is embedded into the Vlasov plate formulation to make up the framework of the analytical treatment. A closed-form solution is realized when an innovative dimensional reduction is extended to the 3D constitutive stiffness matrix. This was made possible through a two-step process in which the shear strain, lateral curvature, and twisting curvature are retained first. By condensing the shear strain variable, effective lateral, torsional, and coupling stiffness terms were formulated. Applying the equilibrium conditions in the deformed configuration, two differential equations are obtained in terms of the lateral curvature and twisting angle. Eliminating the lateral curvature, the twisting angle differential equation with nonconstant coefficients is generated. This equation is solved using a hybrid numerical-analytical approach yielding an analytical buckling expression. Finite element results are generated to verify the accuracy of the buckling load predictions indicating very good correlation with the buckling equation results regardless of the random lamination applied.

中文翻译：

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无规叠层薄网复合悬臂梁的稳定性

本研究解决了在中等高度末端力作用下薄腹板复合悬臂梁横向扭转稳定性的闭式屈曲方程的解析处理。梁由随机的层纤维取向组成。经典的层压理论被嵌入到 Vlasov 板公式中，以构成分析处理的框架。当创新的降维扩展到 3D 本构刚度矩阵时，实现了封闭形式的解决方案。这是通过一个两步过程实现的，其中首先保留剪切应变、横向曲率和扭曲曲率。通过压缩剪切应变变量，制定了有效的横向、扭转和耦合刚度项。在变形配置中应用平衡条件，根据横向曲率和扭转角得到两个微分方程。消除侧向曲率，生成系数不定的扭转角微分方程。该方程使用混合数值解析方法求解，产生解析屈曲表达式。生成有限元结果以验证屈曲载荷预测的准确性，表明与屈曲方程结果具有非常好的相关性，而不管应用的随机层压如何。