Problem Statement
A cantilever beam is subjected to a uniform distributed load with value 30 N/m. The trapezoid beam has right vertical length 0.5m and left vertical length 1m. And top length is 2m.as shown in the graph.
B.C.: The vertical left side is fixed; the bottom and the vertical right side are free.
Material properties: isotropic material E=3×107Pa Poisson’s ratio ν=0.3
Regard this problem as a plane stress one. (beam width is 0.1m)
Math expression of the problem
PDE:
Boundary Conditions：
where
Weak form:
The integral over Ω in the weak form is computed as a sum of integrals over element domains Ωi
Li is gather matrix which will gather the nodal displacements of each element from the global matrix.
w’ is the portion of w corresponding to nodes that are not on an essential boundary, namely it is arbitrary except on an essential boundary condition.
Element Stiffness Matrix
Element external force matrix:
The weak form can then be written as
Using assembly operation, it can be written as:
Element type: 4 nodes bilinear quadrilateral element with two degrees of freedom in each node.
The element matrix will be integrated using 2×2 Gauss quadrature with the following coordinates in the parent element and weights:
The element stiffness matrix :
Continued……
Original Journey©2009
Original Journey 
Hi, I am doing a problem of solving deformed configuration and optimal shape design of a cantilever beam with concentrated load at free end. May I have a copy of your code of this analysis for references? Thanks.