We develop a stochastic Galerkin finite element method for nonlinear elasticity and apply it to reinforced concrete members with random material properties. The strategy is based on the modified Newton-Raphson method, which consists of an incremental loading process and a linearization scheme applied at each load increment. We consider that the material properties are given by a stochastic expansion in the so-called generalized polynomial chaos (gPC) framework. We search the gPC expansion of the displacement, which is then used to update the gPC expansions of the stress, strain, and internal forces. The proposed method is applied to a reinforced concrete beam with uncertain initial concrete modulus of elasticity and a shear wall with uncertain maximum compressive stress of concrete, and the results are compared to those of stochastic collocation and Monte Carlo methods. Since the systems of equations obtained in the linearization scheme using the stochastic Galerkin method are very large, and there are typically many load increments, we also studied iterative solution using preconditioned conjugate gradients. The efficiency of the proposed method is illustrated by a set of numerical experiments.

中文翻译：

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非线性弹性的随机伽辽金有限元方法及其在钢筋混凝土构件上的应用

我们开发了一种用于非线性弹性的随机 Galerkin 有限元方法，并将其应用于具有随机材料特性的钢筋混凝土构件。该策略基于改进的 Newton-Raphson 方法，该方法由增量加载过程和在每个加载增量处应用的线性化方案组成。我们认为材料特性是由所谓的广义多项式混沌 (gPC) 框架中的随机展开给出的。我们搜索位移的 gPC 扩展，然后将其用于更新应力、应变和内力的 gPC 扩展。将该方法应用于初始混凝土弹性模量不确定的钢筋混凝土梁和混凝土最大压应力不确定的剪力墙，并将结果与​​随机搭配和蒙特卡罗方法的结果进行了比较。由于使用随机 Galerkin 方法在线性化方案中获得的方程组非常大，并且通常存在许多载荷增量，因此我们还研究了使用预条件共轭梯度的迭代求解。通过一组数值实验说明了所提出方法的效率。