Size-dependent bending analysis of Timoshenko curved beams is performed with a modified nonlocal strain gradient integral model, in which the integral constitutive equation is transformed into an equivalent differential form equipped with two constitutive boundary equations. The governing equations and boundary conditions are derived via the minimum total potential energy principle and solved analytically using the Laplace transformation technique and its inverse version. In numerical examples, the inconsistency of the nonlocal strain gradient model is examined extendedly under different boundary and loading conditions, while consistent softening and stiffening responses can be observed via the modified nonlocal strain gradient integral model. In addition, within the modified nonlocal strain gradient model, numerical examples also show that the increase of the opening angle can affect the total size effects of the combination of the two scale parameters (i.e., nonlocal and gradient), and these effects are inconsistent for different beam boundaries. Finally, by comparing with the results of the Euler–Bernoulli theory, an interesting finding is that as the nonlocal (or gradient length-scale) parameter increases, the shear deformations of simply supported-simply supported beams (or clamped-clamped/simply supported beams) become more significant.

中文翻译：

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基于修正的非局部应变梯度模型的 Timoshenko 弯曲梁尺寸相关弯曲的精确解

Timoshenko 弯曲梁的尺寸相关弯曲分析使用修改后的非局部应变梯度积分模型进行，其中积分本构方程转换为配备两个本构边界方程的等效微分形式。控制方程和边界条件通过最小总势能原理导出，并使用拉普拉斯变换技术及其逆版本进行解析求解。在数值例子中，非局部应变梯度模型在不同边界和加载条件下的不一致性得到了扩展检查，同时通过修改后的非局部应变梯度积分模型可以观察到一致的软化和硬化响应。此外，在修正的非局部应变梯度模型中，数值例子还表明，张角的增加会影响两个尺度参数（即非局部和梯度）组合的总尺寸效应，并且这些效应对于不同的光束边界是不一致的。最后，通过与欧拉-伯努利理论的结果进行比较，一个有趣的发现是随着非局部（或梯度长度尺度）参数的增加，简支-简支梁（或夹-夹/简支）的剪切变形梁）变得更加重要。