1. A beam AB is fixed at both ends A and B, with the middle point denoted as C, which is subjected to a uniformly distributed load q over the middle region, as shown by the figure. The bending stiffness of beam AB is EI.
(1) Use the Ritz’s method to determine the displacement and moment at point C, and discuss the accuracy as more base functions are included.
Hints: According to the shape of deflection curve, the base functions could be chosen in form of , where the coefficients and should be determined by boundary conditions at points A or B.
(2) Use 1-D finite element method to solve the problem, and discuss the accuracy as the mesh becomes finer.
Hints: Use 1-D Beam element.
1. 光束AB固定在A和B两端，中间点表示为C，在中间区域承受均匀分布的负载q，如图所示。光束 AB 的弯曲刚度为 EI。
• 使用 Ritz 的方法确定点 C 处的位移和位点，并讨论随着包含更多基本函数的准确性。提示：根据偏转曲线的形状，可以选择基函数的形式，其中系数，并且应该由点 A 或 B 处的边界条件确定。
（2）采用一维有限元法解决问题，并讨论网格越细时的准确性。提示：使用 1-D 光束元素。
2. Consider the wave scattering problem in an infinitely extended isotropic elastic solid media. A cylindrical cavity with infinite length in x3 direction is cut within the media, with radius r. Now an incident plane wave propagates in positive x1 direction, with the displacement field:
The incident wave is scattered by the cavity.
Problem: Please solve the scattering problem by BEM, and express the displacement of scattered and total wave fields (and ) along the cavity boundary.
Hints:
(1) The scattering wave is a typical SH wave field, satisfying governing equation:
(2) The traction is zero on the boundary for the total wave field. i.e. .
(3) Use constant element.
2. 考虑无限扩展各向异性弹性固体介质中的波散射问题。在 x3 方向上具有无限长度的圆柱形空腔在介质内切割，半径为 r。现在，入射平面波以正 x1 方向传播，置换场：入射波被腔散射。
事件波在腔中散射。
问题：请用BEM求解散射问题，并表达散射和总波场沿腔边界的位移。
提示：（1）散射波是典型的SH波场，满足控制方程：
（2）总波场边界上的牵引力为零。即.
（3） 使用常量元素。