Abstract A deployable M cross section thin-walled boom (M boom) can be flattened and coiled elastically around a hub; and can then be self-deployed by releasing the stored strain energy. The M boom has been proposed as the key member of membrane deployable structures. First, the covariant base vectors of geometrical relation of the single type I tape spring were analyzed by establishing three coordinate systems. Second, the constitutive relation between stress and strain was expressed according to the Kirchhoff-Love hypothesis. Third, the equilibrium and controlling equations of the single tape spring were modeled based on Calladine shell theory. Fourthly, the total strain energy model of the single type I tape spring was built by integration. Fifth, the strain energy of the M boom was modeled by the sum of the strain energies of the six tape springs. Then, the strain energies of the single type II and III tape springs were analyzed. The sum of the strain energies of the six tape springs equals the total strain energy of the M boom. The bending moment model was established based on the minimum potential energy principle. The experimental equipment and four M boom samples were processed. The bending force value of the M booms was tested 20 times. Then, the average peak bending moment was calculated. The relative error between the theoretical and experimental results of the peak bending moment does not exceed 6.5% verifying the accuracy of the theoretical model.

中文翻译：

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膜可展开结构M臂弯矩峰值解析解

摘要 一种可展开的 M 截面薄壁动臂（M 动臂）可以在轮毂周围展平和弹性盘绕；然后可以通过释放存储的应变能进行自我部署。M 臂已被提议作为膜可展开结构的关键成员。首先，通过建立三个坐标系，分析了单I型带簧几何关系的协变基向量。其次，应力和应变之间的本构关系是根据基尔霍夫-洛夫假设来表达的。第三，基于Calladine壳理论建立了单条带弹簧的平衡方程和控制方程。第四，通过集成建立了单一I型带簧的总应变能模型。第五，M 动臂的应变能由六个带式弹簧的应变能之和建模。然后，分析了单个 II 型和 III 型带簧的应变能。六个带式弹簧的应变能之和等于 M 动臂的总应变能。基于最小势能原理建立弯矩模型。处理了实验设备和四个 M 动臂样品。M臂的弯曲力值测试了20次。然后，计算平均峰值弯矩。峰值弯矩理论与实验结果的相对误差不超过6.5%，验证了理论模型的准确性。六个带式弹簧的应变能之和等于 M 动臂的总应变能。基于最小势能原理建立弯矩模型。处理了实验设备和四个 M 动臂样品。M臂的弯曲力值测试了20次。然后，计算平均峰值弯矩。峰值弯矩理论与实验结果的相对误差不超过6.5%，验证了理论模型的准确性。六个带式弹簧的应变能之和等于 M 动臂的总应变能。基于最小势能原理建立弯矩模型。处理了实验设备和四个 M 动臂样品。M臂的弯曲力值测试了20次。然后，计算平均峰值弯矩。峰值弯矩理论与实验结果的相对误差不超过6.5%，验证了理论模型的准确性。