Motivated by the paradoxical results obtained from the differential nonlocal elasticity theory in some cases (e.g., bending and vibration problems of cantilevers), several attempts have been recently made to develop nonlocal beam models based on the integral (original) formulation of Eringen’s nonlocal theory. These models can be classified into two main groups including strain- and stress-driven ones which have the capability of capturing the softening and hardening behaviors of material caused by nanoscale (nonlocal) effects, respectively. In the present paper, a novel stress-driven nonlocal formulation is developed for the nonlinear analysis of Timoshenko beams made of functionally graded materials. To this end, the governing equations are first derived in the context of integral form of stress-driven nonlocal model. The proposed model can be used for arbitrary kernel functions, and the paradox related to cantilever is resolved by it. The governing equations of stress-driven model in differential form together with corresponding constitutive boundary conditions are also derived. The Timoshenko beam under various end conditions is considered as the problem under study whose nonlinear static bending is analyzed. Furthermore, the generalized differential quadrature method is employed in the solution procedure. The effects of nonlocal parameter, FG index, length-to-thickness ratio and nonlinearity on the deflection of fully clamped, fully simply supported, clamped–simply supported and clamped–free beams are investigated. The presented formulation and results may be helpful in understanding nonlocal phenomena in nano-electro-mechanical systems.

在某些情况下（例如，悬臂的弯曲和振动问题），从微分非局部弹性理论获得的悖论结果促使人们进行了一些尝试，以基于Eringen非局部理论的整体（原始）公式来开发非局部梁模型。这些模型可分为两大类，包括应变和应力驱动的模型，它们分别具有捕获由纳米级（非局部）效应引起的材料的软化和硬化行为的能力。在本文中，开发了一种新颖的应力驱动非局部公式，用于对功能梯度材料制成的Timoshenko梁进行非线性分析。为此，首先在应力驱动的非局部模型的积分形式的情况下导出控制方程。该模型可用于任意核函数，并解决了与悬臂梁有关的悖论。推导了微分形式应力驱动模型的控制方程以及相应的本构边界条件。研究中的Timoshenko梁在各种终端条件下均是研究的问题，并对其非线性静态弯曲进行了分析。此外，在求解过程中采用了广义差分正交方法。研究了非局部参数，FG指数，长厚比和非线性对完全夹紧，完全简单支撑，夹紧—简单支撑和自由束的挠度的影响。提出的公式和结果可能有助于理解纳米机电系统中的非局部现象。