平面应力三角元例子
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平面线性三角元实例
Problem 1: Using two CST elements, solve the simple shear problem depicted in the figure and determine whether the CST elements can represent the simple shear condition accurately. Material properties are given as E = 10 GPa, = 0.25, and thickness is h = 0.1 m. The distributed force f = 100 kN/m 2 is applied at the top edge.
Solution:
Given,
For element 1:
1 2
(1, 0) (1, 1)
x
y
The area of the element 1,
Thus the matrix can be written as,
The stress strain or elasticity matrix can be written as,
The element stiffness matrices for element 1 can be obtained as,
For element 2:
The area of the element 1,
Thus the matrix can be written as,
The stress strain or elasticity matrix can be written as,
The element stiffness matrices for element 2 can be obtained as,
The are assembled to form the global stiffness matrix,
The global matrix equation can be written as,
The total distributed load is acting at the top edge and will be equally divided into two nodes, node 4 and node 3.Thus the total load is,
Horizontal load at node 3 and node 4 is,
.
Now, we apply the boundary conditions by setting as, and the values of in the above global matrix equation and striking out the rows and columns of zero elements,
we finally get,
Solving these set of equations,
Strains for element 1:
Solving this equation we find,
Strains for element 2:
For element 1: For element 2:
Stress for element 1:
Stress for element 2:
Here in this problem, since only distributed shear stress of is applied on the top edge, the shear stress is exact. So, from the results of strain and stress for both elements, we can see that, there are no normal strains and stress and shear strains are same for both elements. That’s why as only shear stress exists; it satisfies the pure shear condition. Thus, the CST element can represent the pure shear condition accurately.
Problem 8: A 2m 2m 1mm square plate with E = 70 GPa and = 0.3 is subjected to a uniformly distributed load as shown in Figure (a). Due to symmetry, it is sufficient to model one quarter of the plate with artificial boundary conditions, as shown in Figure (b). Use two triangular elements to find the displacements, strains and stresses in the plate. Check the answers using simple calculations from mechanics of materials.
For element 1: For element 2:
(a) (b)
Solution:
Given,
For element 1:
The area of the element 1,
Thus the matrix can be written as,
The stress strain or elasticity matrix can be written as,
The element stiffness matrices for element 1 can be obtained as,
For element 2:
The area of the element 1,
Thus the matrix can be written as,
The stress strain or elasticity matrix can be written as,
The element stiffness matrices for element 2 can be obtained as,
The are assembled to form the global stiffness matrix,
The global matrix equation can be written as,
The total distributed stress is acting at the top edge and will be equally divided into two nodes,
node 4 and node 3.Thus the total load from figure (b) is,
Vertical Load at node 3 and node 4 is,
.
Now, we apply the boundary conditions by setting as, and the values of
in the above global matrix equation and striking out the rows and columns of zero elements,
we finally get,
which can be written as,
Solving these set of equations,
Strains for element 1:
Solving this equation we find,
Strains for element 2:
For element 1:
For element 2:
Stress for element 1:
Stress for element 2:
Solution by Mechanics of Materials:
If we consider the diagram in figure (a), it is given that,
The vertical displacement, :
For element 1: For element 2:
Vertical strain,
Horizontal strain,
The horizontal displacement,
So, the analytical solution completely matches with the finite element solution.。
