A square patch made of two triangular plate elements

Basic implementation test with applied loads.

Testing the tangent stiffness computation for a Triangle() (using linear shape functions).

Using

• elements.linear.Triangle (see Triangle class)

• materials.PlaneStress (see PlaneStress material class)

free   free
 ^     ^
 |     |
 3-----2 -> free
 |\  b | >
 | \   | >
 |  \  | > (w = 1.0)
 |   \ | >
 | a  \| >
 0-----1 -> free

width:  10.
height: 10.

Material parameters: St. Venant-Kirchhoff, plane stress
    E  = 10.0
    nu =  0.30
    t  =  1.0

Element loads:
    node 0: [ 0.0, 0.0]
    node 1: [ 5.0, 0.0]
    node 2: [ 5.0, 0.0]
    node 3: [ 0.0, 0.0]

2nd Piola-Kirchhoff stress:
    S_XX =  w                  =  1.000
    S_YY = S_XY = S_YX = S_ZZ  =  0.000

Green Lagrange strain:
    eps_XX = (1/E) ((1.000) - (0.30)(0.000)) =  0.100
    eps_YY = (1/E) ((0.000) - (0.30)(1.000)) = -0.030
    eps_XY = eps_YX                          =  0.000
    eps_ZZ = -nu * (eps_XX + eps_YY)         = -0.021

Stretches:
    lam_X = sqrt(1 + 2 eps_XX) = 1.095
    lam_Y = sqrt(1 + 2 eps_YY) = 0.9695

Displacements:
    ux = (lam_X - 1) * x, uy = (lam_Y - 1) * y
    node 0: [ 0.000,  0.000 ]
    node 1: [ 0.954,  0.000 ]
    node 2: [ 0.954, -0.305 ]
    node 3: [ 0.000, -0.305 ]

Author: Peter Mackenzie-Helnwein

from femedu.examples import Example

from femedu.domain import System, Node
from femedu.solver import NewtonRaphsonSolver
from femedu.elements.linear import Triangle
from femedu.materials import PlaneStress


class ExamplePlate02(Example):

    def problem(self):

        params = dict(
            E  = 10.,   # Young's modulus
            nu = 0.3,   # Poisson's ratio
            t  = 1.0,   # thickness of the plate
            fy = 1.e30  # yield stress
        )

        a = 10.     # length of the plate in the x-direction
        b = 10.     # length of the plate in the y-direction

        model = System()
        model.setSolver(NewtonRaphsonSolver())

        nd0 = Node( 0.0, 0.0)
        nd1 = Node(   a, 0.0)
        nd2 = Node(   a,   b)
        nd3 = Node( 0.0,   b)

        nd0.fixDOF('ux', 'uy')
        nd1.fixDOF('uy')
        nd3.fixDOF('ux')

        model.addNode(nd0, nd1, nd2, nd3)

        elemA = Triangle(nd0, nd1, nd3, PlaneStress(params))
        elemB = Triangle(nd2, nd3, nd1, PlaneStress(params))

        model.addElement(elemA, elemB)

        elemB.setSurfaceLoad(face=2, pn=1.0)

        model.setLoadFactor(0.0)
        model.solve()
        #model.report()  # activate this line for lots of debug info
        model.plot(factor=0.0, title="Undeformed system", filename="plate02_undeformed.png", show_bc=1)

        model.setLoadFactor(1.0)
        model.solve()
        model.plot(factor=1.0, filename="plate02_deformed.png")

        model.report()

Run the example by creating an instance of the problem and executing it by calling Example.run()

if __name__ == "__main__":
    ex = ExamplePlate02()
    ex.run()

• 
• 

+
/usr/local/lib/python3.11/site-packages/matplotlib/quiver.py:632: RuntimeWarning: Mean of empty slice.
  amean = a.mean()
/usr/local/lib/python3.11/site-packages/numpy/core/_methods.py:129: RuntimeWarning: invalid value encountered in scalar divide
  ret = ret.dtype.type(ret / rcount)
+

System Analysis Report
=======================

Nodes:
---------------------
  Node_633:
      x:    [0. 0.]
      fix:  ['ux', 'uy']
      u:    [0. 0.]
  Node_634:
      x:    [10.  0.]
      fix:  ['uy']
      u:    [1. 0.]
  Node_635:
      x:    [10. 10.]
      u:    [ 1.  -0.3]
  Node_636:
      x:    [ 0. 10.]
      fix:  ['ux']
      u:    [ 0.  -0.3]

Elements:
---------------------
  Triangle_917: nodes ( Node_633 Node_634 Node_636 )
      material: PlaneStress
      strain: xx=1.000e-01 yy=-3.000e-02 xy=0.000e+00 zz=-2.100e-02
      stress: xx=1.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
  Triangle_918: nodes ( Node_635 Node_636 Node_634 )
      material: PlaneStress
      strain: xx=1.000e-01 yy=-3.000e-02 xy=1.776e-16 zz=-2.100e-02
      stress: xx=1.000e+00 yy=2.442e-15 xy=6.832e-16 zz=0.000e+00