A square patch made of two triangular plate elements

Basic implementation test with all prescribed displacements.

Testing the internal force computation.

v=-5   v=-5
 |     |
 v     v
 3-----2 -> u=5
 |\  b | >
 | \   | >
 |  \  | > (w = 1.0)
 |   \ | >
 | a  \| >
 0-----1 -> u=5

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]

Green Lagrange strain:
    eps_XX = 0.5 * ((1.5)^2 - 1)      =  0.625
    eps_YY = 0.5 * ((0.5)^2 - 1)      = -0.375
    eps_XY = eps_YX                   =  0.000
    eps_ZZ = - nu * (eps_XX + eps_YY) = -0.075

2nd Piola-Kirchhoff stress:
    D = E t/(1 - nu^2) = 10.989
    S_XX = (10.989) * ((0.625) + (0.30)(-0.375)) =  5.632
    S_YY = (10.989) * ((-0.375) + (0.30)(0.625)) = -2.060
    S_XY = S_YX = S_ZZ                           =  0.000

Author: Peter Mackenzie-Helnwein

from femedu.examples import Example

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


class ExamplePlate01(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()

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

        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.plot(factor=0, title="Undeformed system", filename="plate01_undeformed.png", show_bc=1)

        model.setLoadFactor(1.0)

        nd0.setDisp( [0.0, 0.0] )
        nd1.setDisp( [5.0, 0.0] )
        nd2.setDisp( [5.0,-5.0] )
        nd3.setDisp( [0.0,-5.0] )

        elemA.updateState()
        elemB.updateState()

        model.report()

        model.plot(factor=1.0, filename="plate01_deformed.png")

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

if __name__ == "__main__":
    ex = ExamplePlate01()
    ex.run()

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

Nodes:
---------------------
  Node_629:
      x:    [0. 0.]
      u:    [0. 0.]
  Node_630:
      x:    [10.  0.]
      u:    [5. 0.]
  Node_631:
      x:    [10. 10.]
      u:    [ 5. -5.]
  Node_632:
      x:    [ 0. 10.]
      u:    [ 0. -5.]

Elements:
---------------------
  Triangle_915: nodes ( Node_629 Node_630 Node_632 )
      material: PlaneStress
      strain: xx=5.000e-01 yy=-5.000e-01 xy=0.000e+00 zz=-0.000e+00
      stress: xx=3.846e+00 yy=-3.846e+00 xy=0.000e+00 zz=0.000e+00
  Triangle_916: nodes ( Node_631 Node_632 Node_630 )
      material: PlaneStress
      strain: xx=5.000e-01 yy=-5.000e-01 xy=-0.000e+00 zz=-0.000e+00
      stress: xx=3.846e+00 yy=-3.846e+00 xy=0.000e+00 zz=0.000e+00