关于矩形截面受均布载荷的问题

水胖娃儿

1，用有限元计算矩形截面直梁受均布载荷时，应力云图上显示的最大应力时最大弯应力吗？弹性力学上讲的描述一点应力状态需要六个应力分量，那应力云图上那一点是指什么应力？
2，用有限元软件求出的形截面直梁受均布载荷的最大应力，能够和弹性力学半逆解法算出来的形矩截面直梁纯弯曲的最大应力进行对比吗？就是说两者的假设是否相同。能否用解析解作为标准来判断有限元解的精度。
3，是否能列出矩形截面梁受均布载荷的平衡方程，几何方程，物理方程。和需要用有限元解的泛函。（给出参考书目最好）。以上三个问题归结于我想解决：此问题的有限元解通过怎么的处理收敛于弹性力学的精确解。

tonnyw

My understanding is the following way.

You have the mathematical model which can be expressed as equations and corresponding boundary conditions. In most cases, we don't have the analytical solution and thus we seek numerical one. In this case, we are using finite element solution to approximate the unknown exact solution. It is important to know that what kind of mathematical model you are trying to solve.

For instance, for a cantilever beam subject to uniform load f on its transverse surface, if you are using beam element, you are solving the following mathematical model
E*I*d^4y/d x^4 = f  0<x<L
Boundary conditions: x(0) = X'(0) = 0, E*I*d^2 y(L)/dx^2 = E*I*d^3 y(L)/dx^3 = 0
Therefore you should compare your finite element solution with the exact solution from this model.

However, if you are using brick element, you are solving the following mathematical model:
div(sigma) = f
sigma is the stress tensor.
Then you should compare your finite element solution with the exact solution from this model. This is an apple-to-apple comparison.

水胖娃儿

1.用beam单元和用brick单元，他们的数学模型不同，是否就是说用这两个单元，分别对问题进行了简化。如果简化了，我能用弹性力学的解析解来和这样的有限元解进行对比吗？
2.我想知道用弹性力学里导出的悬臂梁受均布载荷的数学模型（3D）及推导过程，在用有限元方法就是对这个数学模型进行求解，得到的近似解拿来和精确解对比。

综上所述，需要有相同的数学模型，我才能比较分别用数值解和解析解。我的问题在于弹性力学中数学模型的建立过程。